From aabf7eff51e9e54b238bdaea587b59870d2000b8 Mon Sep 17 00:00:00 2001 From: tonyfast Date: Sun, 7 Jan 2024 20:33:14 -0800 Subject: [PATCH 1/9] add the work to get the contents in bootstrapping --- src/nobook/bootstrap.ipynb | 321 +++++++++++++++++++++++++++++-------- 1 file changed, 251 insertions(+), 70 deletions(-) diff --git a/src/nobook/bootstrap.ipynb b/src/nobook/bootstrap.ipynb index a5baf5b..9a7384d 100644 --- a/src/nobook/bootstrap.ipynb +++ b/src/nobook/bootstrap.ipynb @@ -18,25 +18,30 @@ { "cell_type": "code", "execution_count": 1, - "id": "ba3b86d2-4953-4cff-89bb-c2160947c31a", + "id": "c4601326-6456-47e8-99c9-ae8c8f59e32b", "metadata": { "execution": { - "iopub.execute_input": "2024-01-07T07:16:44.823272Z", - "iopub.status.busy": "2024-01-07T07:16:44.823177Z", - "iopub.status.idle": "2024-01-07T07:16:45.075102Z", - "shell.execute_reply": "2024-01-07T07:16:45.074774Z", - "shell.execute_reply.started": "2024-01-07T07:16:44.823262Z" + "iopub.execute_input": "2024-01-08T04:32:10.613639Z", + "iopub.status.busy": "2024-01-08T04:32:10.613413Z", + "iopub.status.idle": "2024-01-08T04:32:11.095633Z", + "shell.execute_reply": "2024-01-08T04:32:11.095038Z", + "shell.execute_reply.started": "2024-01-08T04:32:10.613618Z" }, "tags": [] }, "outputs": [], "source": [ - " import docx, itertools, pandas, tempfile, shutil, typing, asyncio, subprocess, functools, unittest, operator\n", + " import docx, itertools, pandas, tempfile, shutil, typing, asyncio, subprocess, functools, unittest, operator, re\n", " from IPython.display import Markdown, display; from toolz.curried import *\n", " PANDOC, PANDOC_TO, PANDOC_FROM = shutil.which(\"pandoc\"), \"gfm\", \"docx\"\n", " MAIN, FILE = __name__ == \"__main__\", \"__file__\" in locals()\n", " INTERACTIVE = MAIN and not FILE\n", - " singleton = functools.lru_cache(1)" + " singleton = functools.lru_cache(1)\n", + " from pathlib import Path\n", + " HERE = Path(__file__).parent if \"__file__\" in globals() else Path(subprocess.check_output([\"pwd\"]).decode().strip())\n", + " PSU = HERE.parent.parent / \"psu410\" / \"src\" / \"psu410\"\n", + "\n", + " import nbformat.v4 as v4, anyio, json" ] }, { @@ -53,11 +58,11 @@ "id": "2f23bf91-3d5d-4bd5-8619-e05ac0df6c53", "metadata": { "execution": { - "iopub.execute_input": "2024-01-07T07:16:45.076028Z", - "iopub.status.busy": "2024-01-07T07:16:45.075852Z", - "iopub.status.idle": "2024-01-07T07:16:45.078060Z", - "shell.execute_reply": "2024-01-07T07:16:45.077781Z", - "shell.execute_reply.started": "2024-01-07T07:16:45.076018Z" + "iopub.execute_input": "2024-01-08T04:32:11.097354Z", + "iopub.status.busy": "2024-01-08T04:32:11.097071Z", + "iopub.status.idle": "2024-01-08T04:32:11.101711Z", + "shell.execute_reply": "2024-01-08T04:32:11.100541Z", + "shell.execute_reply.started": "2024-01-08T04:32:11.097336Z" }, "tags": [] }, @@ -91,11 +96,11 @@ "id": "bd674c8e-4a31-4544-b6b9-0aa2573385c8", "metadata": { "execution": { - "iopub.execute_input": "2024-01-07T07:16:45.078515Z", - "iopub.status.busy": "2024-01-07T07:16:45.078412Z", - "iopub.status.idle": "2024-01-07T07:16:45.090755Z", - "shell.execute_reply": "2024-01-07T07:16:45.090419Z", - "shell.execute_reply.started": "2024-01-07T07:16:45.078505Z" + "iopub.execute_input": "2024-01-08T04:32:11.103406Z", + "iopub.status.busy": "2024-01-08T04:32:11.102966Z", + "iopub.status.idle": "2024-01-08T04:32:11.113313Z", + "shell.execute_reply": "2024-01-08T04:32:11.112714Z", + "shell.execute_reply.started": "2024-01-08T04:32:11.103383Z" }, "tags": [] }, @@ -118,11 +123,11 @@ "id": "345b47f7-6fbf-40f5-a64f-5646db86dd2b", "metadata": { "execution": { - "iopub.execute_input": "2024-01-07T07:16:45.091391Z", - "iopub.status.busy": "2024-01-07T07:16:45.091272Z", - "iopub.status.idle": "2024-01-07T07:16:45.093825Z", - "shell.execute_reply": "2024-01-07T07:16:45.093529Z", - "shell.execute_reply.started": "2024-01-07T07:16:45.091380Z" + "iopub.execute_input": "2024-01-08T04:32:11.114386Z", + "iopub.status.busy": "2024-01-08T04:32:11.114084Z", + "iopub.status.idle": "2024-01-08T04:32:11.118742Z", + "shell.execute_reply": "2024-01-08T04:32:11.117965Z", + "shell.execute_reply.started": "2024-01-08T04:32:11.114364Z" }, "tags": [] }, @@ -152,11 +157,11 @@ "id": "ff516019-bebf-4610-9f16-5dc90d0b8da7", "metadata": { "execution": { - "iopub.execute_input": "2024-01-07T07:16:45.094389Z", - "iopub.status.busy": "2024-01-07T07:16:45.094237Z", - "iopub.status.idle": "2024-01-07T07:16:45.096534Z", - "shell.execute_reply": "2024-01-07T07:16:45.096185Z", - "shell.execute_reply.started": "2024-01-07T07:16:45.094378Z" + "iopub.execute_input": "2024-01-08T04:32:11.119935Z", + "iopub.status.busy": "2024-01-08T04:32:11.119620Z", + "iopub.status.idle": "2024-01-08T04:32:11.123408Z", + "shell.execute_reply": "2024-01-08T04:32:11.122958Z", + "shell.execute_reply.started": "2024-01-08T04:32:11.119915Z" }, "tags": [] }, @@ -181,11 +186,11 @@ "id": "d4e75784-0945-47e0-9260-4116ca2e0e5d", "metadata": { "execution": { - "iopub.execute_input": "2024-01-07T07:16:45.097110Z", - "iopub.status.busy": "2024-01-07T07:16:45.096941Z", - "iopub.status.idle": "2024-01-07T07:16:45.099624Z", - "shell.execute_reply": "2024-01-07T07:16:45.099235Z", - "shell.execute_reply.started": "2024-01-07T07:16:45.097100Z" + "iopub.execute_input": "2024-01-08T04:32:11.124996Z", + "iopub.status.busy": "2024-01-08T04:32:11.124539Z", + "iopub.status.idle": "2024-01-08T04:32:11.128171Z", + "shell.execute_reply": "2024-01-08T04:32:11.127491Z", + "shell.execute_reply.started": "2024-01-08T04:32:11.124980Z" }, "tags": [] }, @@ -211,11 +216,11 @@ "id": "dfc37237-2240-4585-949d-0dbc13d1e7cc", "metadata": { "execution": { - "iopub.execute_input": "2024-01-07T07:16:45.100743Z", - "iopub.status.busy": "2024-01-07T07:16:45.100523Z", - "iopub.status.idle": "2024-01-07T07:16:45.103120Z", - "shell.execute_reply": "2024-01-07T07:16:45.102825Z", - "shell.execute_reply.started": "2024-01-07T07:16:45.100732Z" + "iopub.execute_input": "2024-01-08T04:32:11.130135Z", + "iopub.status.busy": "2024-01-08T04:32:11.129877Z", + "iopub.status.idle": "2024-01-08T04:32:11.134681Z", + "shell.execute_reply": "2024-01-08T04:32:11.134109Z", + "shell.execute_reply.started": "2024-01-08T04:32:11.130115Z" }, "tags": [] }, @@ -225,9 +230,12 @@ " df = df.join(\n", " df[df.style_id.str.match(\"^Heading1\")].p.apply(operator.attrgetter(\"text\")).rename(\"chapter\")\n", " ).join(\n", - " df[df.style_id.str.match(\"^Heading\")].p.apply(operator.attrgetter(\"text\")).rename(\"notebook\")\n", - " ).ffill().fillna(\"empty\").set_index([\"chapter\", \"notebook\"])\n", - " return df.groupby([\"chapter\", \"notebook\"]).p.apply(get_new_docx).rename(\"docx\")" + " df[df.style_id.str.match(\"^Heading\")].p.apply(operator.attrgetter(\"text\")).rename(\"heading\")\n", + " ).ffill().fillna(\"empty\").set_index([\"chapter\", \"heading\"])\n", + " df = df.groupby([\"chapter\", \"heading\"]).p.apply(get_new_docx).rename(\"docx\")\n", + " df.pop((\"empty\", \"empty\"))\n", + " df = df.to_frame().set_index(pandas.RangeIndex(0, len(df), name=\"order\"), append=True)\n", + " return df.docx" ] }, { @@ -236,11 +244,11 @@ "id": "8f4bc119-c81b-40aa-9205-2f6b2f3bb63e", "metadata": { "execution": { - "iopub.execute_input": "2024-01-07T07:16:45.103586Z", - "iopub.status.busy": "2024-01-07T07:16:45.103490Z", - "iopub.status.idle": "2024-01-07T07:16:45.105690Z", - "shell.execute_reply": "2024-01-07T07:16:45.105386Z", - "shell.execute_reply.started": "2024-01-07T07:16:45.103576Z" + "iopub.execute_input": "2024-01-08T04:32:11.138093Z", + "iopub.status.busy": "2024-01-08T04:32:11.137666Z", + "iopub.status.idle": "2024-01-08T04:32:11.144323Z", + "shell.execute_reply": "2024-01-08T04:32:11.143572Z", + "shell.execute_reply.started": "2024-01-08T04:32:11.138067Z" }, "tags": [] }, @@ -258,11 +266,11 @@ "id": "225460f7-f090-4f8b-9ff5-b8adc1459ffa", "metadata": { "execution": { - "iopub.execute_input": "2024-01-07T07:16:45.106237Z", - "iopub.status.busy": "2024-01-07T07:16:45.106105Z", - "iopub.status.idle": "2024-01-07T07:16:45.108433Z", - "shell.execute_reply": "2024-01-07T07:16:45.108003Z", - "shell.execute_reply.started": "2024-01-07T07:16:45.106228Z" + "iopub.execute_input": "2024-01-08T04:32:11.145386Z", + "iopub.status.busy": "2024-01-08T04:32:11.145086Z", + "iopub.status.idle": "2024-01-08T04:32:11.153066Z", + "shell.execute_reply": "2024-01-08T04:32:11.152322Z", + "shell.execute_reply.started": "2024-01-08T04:32:11.145367Z" }, "tags": [] }, @@ -280,11 +288,11 @@ "id": "ec3b7821-e291-4bbb-b651-82f36a925d43", "metadata": { "execution": { - "iopub.execute_input": "2024-01-07T07:16:45.109243Z", - "iopub.status.busy": "2024-01-07T07:16:45.109004Z", - "iopub.status.idle": "2024-01-07T07:16:59.926888Z", - "shell.execute_reply": "2024-01-07T07:16:59.926145Z", - "shell.execute_reply.started": "2024-01-07T07:16:45.109229Z" + "iopub.execute_input": "2024-01-08T04:32:11.154163Z", + "iopub.status.busy": "2024-01-08T04:32:11.153916Z", + "iopub.status.idle": "2024-01-08T04:32:35.998123Z", + "shell.execute_reply": "2024-01-08T04:32:35.997791Z", + "shell.execute_reply.started": "2024-01-08T04:32:11.154142Z" }, "tags": [] }, @@ -294,18 +302,28 @@ "output_type": "stream", "text": [ "test_number_of_sections (__main__.Tests.test_number_of_sections)\n", - "the table of contents, 9 primary chapters, and the references are captured ... ok\n", + "the table of contents, 9 primary chapters, and the references are captured ... FAIL\n", + "\n", + "======================================================================\n", + "FAIL: test_number_of_sections (__main__.Tests.test_number_of_sections)\n", + "the table of contents, 9 primary chapters, and the references are captured\n", + "----------------------------------------------------------------------\n", + "Traceback (most recent call last):\n", + " File \"/tmp/ipykernel_431217/4110034353.py\", line 13, in test_number_of_sections\n", + " assert len(self.md.index.get_level_values(\"heading\").unique()) == 10\n", + " ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^\n", + "AssertionError\n", "\n", "----------------------------------------------------------------------\n", - "Ran 1 test in 14.809s\n", + "Ran 1 test in 24.823s\n", "\n", - "OK\n" + "FAILED (failures=1)\n" ] }, { "data": { "text/plain": [ - "" + "" ] }, "execution_count": 10, @@ -319,13 +337,14 @@ " self.md = arun(get_organized_markdown())\n", " self.toc, self.sections = get_frames().values()\n", " \n", + " # executing this test case in a notebook updates the global namespace\n", + " INTERACTIVE and globals().update(md=self.md, toc=self.toc, sections=self.sections)\n", + " \n", " def test_number_of_sections(self):\n", " \"\"\"the table of contents, 9 primary chapters, and the references are captured\"\"\"\n", " # all of the main headings are captured as Heading1s\n", " assert self.sections.style_id.eq(\"Heading1\").sum() == 10\n", - " # the organized has synthetic empty labels that add one more value\n", - " # still need to inspect these rows before remove them.\n", - " assert len(self.md.index.get_level_values(\"chapter\").unique()) == 11\n", + " assert len(self.md.index.get_level_values(\"heading\").unique()) == 10\n", " \n", " INTERACTIVE and unittest.main(argv=[\"discover\"], verbosity=2, exit=False)" ] @@ -333,14 +352,177 @@ { "cell_type": "code", "execution_count": 11, + "id": "94532190-795c-49c7-8540-5e01bbdb7126", + "metadata": { + "execution": { + "iopub.execute_input": "2024-01-08T04:32:35.998896Z", + "iopub.status.busy": "2024-01-08T04:32:35.998601Z", + "iopub.status.idle": "2024-01-08T04:32:36.001602Z", + "shell.execute_reply": "2024-01-08T04:32:36.001246Z", + "shell.execute_reply.started": "2024-01-08T04:32:35.998880Z" + }, + "tags": [] + }, + "outputs": [], + "source": [ + " def slugify_index(s): return s.str.lower().str.replace(\n", + " \"\\s\", \"_\", regex=True\n", + " ).str.replace(\"[^a-z\\_]\", \"\", regex=True)" + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "id": "03e3aca3-6205-49d3-b02c-a8acaee48457", + "metadata": { + "execution": { + "iopub.execute_input": "2024-01-08T04:32:36.002208Z", + "iopub.status.busy": "2024-01-08T04:32:36.002065Z", + "iopub.status.idle": "2024-01-08T04:32:36.014394Z", + "shell.execute_reply": "2024-01-08T04:32:36.014046Z", + "shell.execute_reply.started": "2024-01-08T04:32:36.002194Z" + }, + "tags": [] + }, + "outputs": [], + "source": [ + " def get_notebooks(md):\n", + " df = md.to_frame().assign(\n", + " level=md.str.extract(\"^(#{1,})\")[0].apply(len).rename(\"level\"),\n", + " chapter_slug=slugify_index(md.index.get_level_values(\"chapter\")),\n", + " heading_slug=slugify_index(md.index.get_level_values(\"heading\")),\n", + " )\n", + " df = df.join(\n", + " df.set_index(\"level\", append=True)[[]].assign(x=1).unstack().x[[1,2]].fillna(0).cumsum().apply(\n", + " tuple, axis=1\n", + " ).rename(\"group\")\n", + " ).groupby([\"chapter_slug\", \"group\"]).apply(\n", + " lambda x: pandas.Series({\n", + " PSU.parent / x.name[0] / ((\n", + " x.heading_slug.iloc[0] if x.level.iloc[0] > 1 else \"index\"\n", + " ) + \".ipynb\") : v4.new_notebook(cells=[\n", + " v4.new_markdown_cell(y) for y in x.markdown\n", + " ])\n", + " })\n", + " ).to_frame(\"nb\").reset_index([\"chapter_slug\", \"group\"], drop=True)\n", + " return df" + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "id": "636a096d-aad3-4af3-86ae-15ee357fc90c", + "metadata": { + "execution": { + "iopub.execute_input": "2024-01-08T04:32:36.015044Z", + "iopub.status.busy": "2024-01-08T04:32:36.014900Z", + "iopub.status.idle": "2024-01-08T04:32:36.017546Z", + "shell.execute_reply": "2024-01-08T04:32:36.017248Z", + "shell.execute_reply.started": "2024-01-08T04:32:36.015031Z" + }, + "tags": [] + }, + "outputs": [], + "source": [ + " async def write_nb(target, data):\n", + " target = anyio.Path(target)\n", + " await target.parent.mkdir(exist_ok=True, parents=True)\n", + " return await target.write_text(json.dumps(data))" + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "id": "1f7b0f6b-6b56-48a5-8562-baa57317fa09", + "metadata": { + "execution": { + "iopub.execute_input": "2024-01-08T04:32:36.018049Z", + "iopub.status.busy": "2024-01-08T04:32:36.017900Z", + "iopub.status.idle": "2024-01-08T04:32:36.157118Z", + "shell.execute_reply": "2024-01-08T04:32:36.156771Z", + "shell.execute_reply.started": "2024-01-08T04:32:36.018041Z" + }, + "tags": [] + }, + "outputs": [ + { + "data": { + "text/plain": [ + "[971,\n", + " 8496,\n", + " 5457,\n", + " 8652,\n", + " 3065,\n", + " 267,\n", + " 1367,\n", + " 12632,\n", + " 16503,\n", + " 1228,\n", + " 21406,\n", + " 2660,\n", + " 6401,\n", + " 3108,\n", + " 2920,\n", + " 9592,\n", + " 1650,\n", + " 7060,\n", + " 11650,\n", + " 3490,\n", + " 1195,\n", + " 5588,\n", + " 14873,\n", + " 1374,\n", + " 2724,\n", + " 2568,\n", + " 12801,\n", + " 2625,\n", + " 3057,\n", + " 14951,\n", + " 6998,\n", + " 10104,\n", + " 13692,\n", + " 6998,\n", + " 6763,\n", + " 11740,\n", + " 15037,\n", + " 7300,\n", + " 201,\n", + " 37183,\n", + " 10295,\n", + " 5905,\n", + " 3739,\n", + " 164,\n", + " 10191,\n", + " 6318,\n", + " 2718,\n", + " 18979,\n", + " 183,\n", + " 41750,\n", + " 11731]" + ] + }, + "execution_count": 14, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + " 10 and INTERACTIVE and await asyncio.gather(*(\n", + " write_nb(target, nb) for target, nb in get_notebooks(md).nb.items()))\n", + " " + ] + }, + { + "cell_type": "code", + "execution_count": 15, "id": "ae617c09-6440-48a7-a015-2c60896fb1a1", "metadata": { "execution": { - "iopub.execute_input": "2024-01-07T07:16:59.927836Z", - "iopub.status.busy": "2024-01-07T07:16:59.927607Z", - "iopub.status.idle": "2024-01-07T07:17:13.375055Z", - "shell.execute_reply": "2024-01-07T07:17:13.374743Z", - "shell.execute_reply.started": "2024-01-07T07:16:59.927813Z" + "iopub.execute_input": "2024-01-08T04:32:36.157946Z", + "iopub.status.busy": "2024-01-08T04:32:36.157626Z", + "iopub.status.idle": "2024-01-08T04:32:36.169388Z", + "shell.execute_reply": "2024-01-08T04:32:36.168922Z", + "shell.execute_reply.started": "2024-01-08T04:32:36.157927Z" }, "tags": [] }, @@ -1645,14 +1827,13 @@ ], "source": [ " if INTERACTIVE:\n", - " md = await get_organized_markdown()\n", " display(*md[md.str.contains(\"\\$\")].iloc[:10].apply(Markdown))" ] }, { "cell_type": "code", "execution_count": null, - "id": "28541e2b-2e80-4ae3-b148-784cafdaa641", + "id": "dc1a963d-8365-42c8-b50b-ff4e1918ef02", "metadata": {}, "outputs": [], "source": [] From 319f9e7644682be06b5503b2187fbc7d53fb5db6 Mon Sep 17 00:00:00 2001 From: tonyfast Date: Sun, 7 Jan 2024 20:33:40 -0800 Subject: [PATCH 2/9] add all the generated notebooks --- .../ellingham_diagram_and_buffered_systems.ipynb | 1 + psu410/src/applications_to_chemical_reactions/index.ipynb | 1 + ..._process_and_differential_and_integrated_driving_forces.ipynb | 1 + ...maximum_reaction_rate_and_chemical_transport_reactions_.ipynb | 1 + .../trends_of_entropies_of_reactions.ipynb | 1 + .../application_examples.ipynb | 1 + .../aqueous_solution_and_pourbaix_diagram.ipynb | 1 + ..._activities_and_reference_states_of_electrolyte_species.ipynb | 1 + .../electrochemical_cells_and_half_cell_potentials.ipynb | 1 + .../electrolyte_reactions_and_electrochemical_reactions.ipynb | 1 + psu410/src/applications_to_electrochemical_systems/index.ipynb | 1 + .../importance_of_lattice_stability.ipynb | 1 + psu410/src/calpahd_modeling_of_thermodynamics/index.ipynb | 1 + .../modeling_of_pure_elements.ipynb | 1 + .../modeling_of_random_solution_phases.ipynb | 1 + .../modeling_of_solution_phases_with_longrange_ordering.ipynb | 1 + .../modeling_of_stoichiometric_phases.ipynb | 1 + .../application_to_cerium.ipynb | 1 + .../application_to_fept.ipynb | 1 + .../concept_of_materials_genome.ipynb | 1 + .../index.ipynb | 1 + .../mms_model_applied_to_thermal_expansion.ipynb | 1 + .../additional_methods.ipynb | 1 + .../src/experimental_data_for_thermodynamic_modeling/index.ipynb | 1 + .../phase_equilibrium_data_from_experiments.ipynb | 1 + .../thermodynamic_data_from_experiments.ipynb | 1 + .../bornoppenheimer_approximation.ipynb | 1 + .../firstprinciples_approaches_to_disordered_alloys.ipynb | 1 + .../firstprinciples_formulation_of_thermodynamics.ipynb | 1 + psu410/src/firstprinciples_calculations_and_theory/index.ipynb | 1 + .../lattice_dynamics.ipynb | 1 + .../nickel_as_the_prototype.ipynb | 1 + .../quantum_theory_for_the_motion_of_electrons.ipynb | 1 + psu410/src/gibbs_energy_function/binary_random_solutions.ipynb | 1 + ...stic_magnetic_and_electric_contributions_to_free_energy.ipynb | 1 + psu410/src/gibbs_energy_function/index.ipynb | 1 + .../gibbs_energy_function/phases_with_fixed_compositions.ipynb | 1 + .../phases_with_variable_compositions_random_solutions.ipynb | 1 + ...ases_with_variable_compositions_solutions_with_ordering.ipynb | 1 + .../polymer_solutions_and_polymer_blends.ipynb | 1 + ...mbined_law_of_thermodynamics_and_equilibrium_conditions.ipynb | 1 + .../first_and_second_laws_of_thermodynamics.ipynb | 1 + psu410/src/laws_of_thermodynamics/gibbsduhem_equation.ipynb | 1 + psu410/src/laws_of_thermodynamics/index.ipynb | 1 + .../stability_at_equilibrium_and_property_anomaly.ipynb | 1 + .../general_condition_of_equilibrium.ipynb | 1 + .../gibbs_phase_rule.ipynb | 1 + psu410/src/phase_equilibria_in_heterogeneous_systems/index.ipynb | 1 + .../molar_phase_diagrams.ipynb | 1 + .../potential_phase_diagrams.ipynb | 1 + psu410/src/references/index.ipynb | 1 + 51 files changed, 51 insertions(+) create mode 100644 psu410/src/applications_to_chemical_reactions/ellingham_diagram_and_buffered_systems.ipynb create mode 100644 psu410/src/applications_to_chemical_reactions/index.ipynb create mode 100644 psu410/src/applications_to_chemical_reactions/internal_process_and_differential_and_integrated_driving_forces.ipynb create mode 100644 psu410/src/applications_to_chemical_reactions/maximum_reaction_rate_and_chemical_transport_reactions_.ipynb create mode 100644 psu410/src/applications_to_chemical_reactions/trends_of_entropies_of_reactions.ipynb create mode 100644 psu410/src/applications_to_electrochemical_systems/application_examples.ipynb create mode 100644 psu410/src/applications_to_electrochemical_systems/aqueous_solution_and_pourbaix_diagram.ipynb create mode 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+{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "8610ccff", "cell_type": "markdown", "source": "## Ellingham diagram and buffered systems\n\nOne type of chemical reactions is between a pure element in liquid or\nsolid states and its oxides involving one mole of oxygen gas, i.e.\n\n*Eq. 7\u20117* $\\frac{4}{v_{M}}M + O_{2}(gas) = M_{4/v_{M}}O_{2}$\n\nwith $v_{M}$ being the valence of the element $M$ in the oxide. Taking\nthe pure $M$ and the gaseous $O_{2}$ at the reaction temperature and one\natmospheric pressure as their respective reference states, both\nactivities of the pure $M$ solid or liquid phase and its oxide are\nunity, and the activity of $O_{2}$ equals to its partial pressure in an\nideal gas. becomes\n\n*Eq. 7\u20118*\n$RTlnP_{O_{2}} = \\mathrm{\\Delta}_{\\ }^{0}G = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S$\n\nBased on , one can plot $\\mathrm{\\Delta}_{\\ }^{0}G$ as a function of\ntemperature for various oxidation reactions, which is called Ellingham\ndiagram. The intercept at $T = 0K$ is given by\n$\\mathrm{\\Delta}_{\\ }^{0}H$, and the slope is represented by\n$- \\mathrm{\\Delta}_{\\ }^{0}S$, depicted by the following equation\n\n*Eq. 7\u20119*\n$\\mathrm{\\Delta}_{\\ }^{0}S = S_{M_{4/v_{M}}O_{2}} - S_{O_{2}} - \\frac{4}{v_{M}}S_{M}$\n\nSince the entropy of one mole of $O_{2}$ gas is significantly larger\nthan those of the pure element and its oxide when they are in solid or\nliquid states and the entropy difference between the pure element and\nits oxide, the entropy of reaction is thus dominated by the reduction of\nthe entropy by one mole of $O_{2}$ gas. Consequently, the entropies of\nreaction are approximately the same for most reactions when the pure\nelements and their oxides are solid as seen in where most lines have\nsimilar slopes.\n\nFigure \u2011: Ellingham Diagram for a number of metal-oxide systems.\n\nFor a chemical reaction on the Ellingham diagram at a given temperature,\n$\\mathrm{\\Delta}_{\\ }^{0}G$ can be read from the y-axis of the diagram,\nand the equilibrium partial pressure of $O_{2}$ gas can be calculated\nusing . Alternatively, one can plot the left part of for a given\n$P_{O_{2}}$ as a function of temperature on the Ellingham diagram, i.e.\n\n*Eq. 7\u201110* $\\mathrm{\\Delta}_{\\ }^{0}{G =}RTlnP_{O_{2}}$\n\nThis results in straight lines, representing iso-partial-pressure lines\nof $O_{2}$, with the intercept being zero at $T = 0K$ and slopes of\n$RlnP_{O_{2}}$, which are negative for $P_{O_{2}}$ lower than one\natmospheric pressure (1atm). The values of $P_{O_{2}}$ are marked on the\nsecondary axis on the right of the Ellingham diagram. The intersection\nof the isoactivitiy line and the equilibrium line of the chemical\nreaction thus gives the relation of equilibrium temperature and\nequilibrium partial pressure of $O_{2}$. This is demonstrated in .\n\nFigure \u2011: Intersection of iso-partial-pressure lines of $O_{2}$ and\nequilibrium lines in the Ellingham diagram\n\nFor each chemical reaction in the Ellingham diagram, the three phases\nare in equilibrium on the line represented by , i.e. metal, metal\noxides, and O2 gas. For conditions above the line, the value\nof $P_{O_{2}}$ is larger than its equilibrium value, and the metal will\nbe oxidized. For conditions below the line, the value of $P_{O_{2}}$ is\nlower than its equilibrium value, and the metal oxide will be reduced.\nTherefore, the metal oxides in the upper part of the Ellingham diagram\ncan be reduced by the metals in the lower part of the diagram, and vice\nversus, metals in the lower part of the diagram can be oxidized by the\nmetal oxides in the upper part of the diagram. For example, in , Ca can\nreduce all oxides, and Cu2O is the least stable oxide.\n\nFrom the above Ellingham diagram, it is noted that equilibrium partial\npressures of $O_{2}$ are very low for most chemical reactions with many\nof them lower than $10^{- 12}\\ atm$. One approach to obtain such a low\npressure is to use auxiliary reactions containing $O_{2}$ that are easy\nto control and are independent of the equilibrium system of interest\nexcept the sharing of the oxygen partial pressure. Two common auxiliary\nreactions are the $H_{2}$/$H_{2}O$ and $CO$/$CO_{2}$ mixtures. For the\n$H_{2}$/$H_{2}O$ mixture, the chemical reaction is\n\n*Eq. 7\u201111* $2H_{2}(gas) + O_{2}(gas) = 2H_{2}O(gas)$\n\nThe equilibrium oxygen partial pressure is obtained as\n\n*Eq. 7\u201112*\n$- RTln\\left\\{ {\\frac{1}{P_{O_{2}}}\\left( \\frac{P_{H_{2}O}}{P_{H_{2}}} \\right)}^{2} \\right\\} = \\mathrm{\\Delta}_{\\ }^{0}G = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S = - 498488\\ + 112.972T\\ (J)$\n\nwhere the $\\mathrm{\\Delta}_{\\ }^{0}H$ and $\\mathrm{\\Delta}_{\\ }^{0}S$\nare taken from the substance thermodynamic database (SSUB4) compiled by\nScientific Group Thermodata Europe (SGTE) \\[59\\], which are slightly\ndependent on temperature, and the values in are evaluated at 1273K using\nThermo-Calc \\[60\\]. At any given temperature, one has the following\nrelation\n\n*Eq. 7\u201113*\n$RTlnP_{O_{2}} = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S - 2RTln\\frac{P_{H_{2}}}{P_{H_{2}O}} = - 498488 + \\left( 112.972 - 2Rln\\frac{P_{H_{2}}}{P_{H_{2}O}} \\right)T$\n\nIts intercept at $T = 0K$ is given by\n$\\mathrm{\\Delta}_{\\ }^{0}H = - 498,488\\ (J)$, and the slope by\n$- \\mathrm{\\Delta}_{\\ }^{0}S - 2Rln\\left( \\frac{P_{H_{2}}}{P_{H_{2}O}} \\right) = 112.972 - 2Rln\\left( \\frac{P_{H_{2}}}{P_{H_{2}O}} \\right)$.\nThe values of the $\\frac{P_{H_{2}}}{P_{H_{2}O}}$ ratio are marked on\nanother secondary axis on the right of the Ellingham diagram. The\nintersection of the iso-partial-pressure-ratio line and the equilibrium\nline of the chemical reaction gives the relation of equilibrium\ntemperature and equilibrium partial pressure ratio\n$\\frac{P_{H_{2}}}{P_{H_{2}O}}$ for desired $P_{O_{2}}$ of the chemical\nequilibrium.\n\nFor the $CO$/$CO_{2}$mixture, the chemical reaction is shown below\n\n*Eq. 7\u201114* $2CO(gas) + O_{2}(gas) = 2CO_{2}(gas)$\n\nSimilar, from the SSUB database, the following equation is obtained\n\n*Eq. 7\u201115*\n$- RTln\\left\\{ {\\frac{1}{P_{O_{2}}}\\left( \\frac{P_{{CO}_{2}}}{P_{CO}} \\right)}^{2} \\right\\} = \\mathrm{\\Delta}_{\\ }^{0}G = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S = - 562,927 - 172.020T\\ (J)$\n\nwith $\\mathrm{\\Delta}_{\\ }^{0}H$ and $\\mathrm{\\Delta}_{\\ }^{0}S$\ncalculated at 1273K, which are also slightly temperature-dependent as in\nthe case of the $H_{2}$/$H_{2}O$ mixture. The iso-partial-pressure-ratio\nlines are written as\n\n*Eq. 7\u201116*\n$RTlnP_{O_{2}} = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S - 2RTln\\frac{P_{CO}}{P_{{CO}_{2}}} = - 562,927 + \\left( 172.020 - 2Rln\\frac{P_{CO}}{P_{{CO}_{2}}} \\right)T\\ $\n\nIts intercept at $T = 0K$ is given by\n$\\mathrm{\\Delta}_{\\ }^{0}H = - 562,927\\ (J)$, and the slope by\n$172.020 - - 2Rln\\left( \\frac{P_{CO}}{P_{{CO}_{2}}} \\right)$. The values\nof the $\\frac{P_{CO}}{P_{{CO}_{2}}}$ ratio are marked on the third\nsecondary axis on the right of the Ellingham diagram. The intersection\nof the iso-partial-pressure-ratio line and the equilibrium line of the\nchemical reaction gives the relation of equilibrium temperature and\nequilibrium partial pressure ratio $\\frac{P_{CO}}{P_{{CO}_{2}}}$ for\ndesired $P_{O_{2}}$ of the chemical equilibrium.\n\nTherefore, one can use a mixture of the $H_{2}$/$H_{2}O$ or\n$CO$/$CO_{2}$ to obtain the desired low $P_{O_{2}}$ values using and or\ncalculating from the SSUB database. For example,\n$P_{O_{2}} = 10^{- 15}\\ atm$ at 1273K, the calculated values from the\nSSUB database are $\\frac{P_{H_{2}}}{P_{H_{2}O}} \\approx 1.67$ and\n$\\frac{P_{CO}}{P_{{CO}_{2}}} \\approx 2.78$, respectively, in which the\ntemperature dependences of $\\mathrm{\\Delta}_{\\ }^{0}H$ and\n$\\mathrm{\\Delta}_{\\ }^{0}S$ and the many more gaseous species in the gas\nphase are taken into account. On the other hand, the reading from the\nEllingham diagram gives $\\frac{P_{H_{2}}}{P_{H_{2}O}} \\approx 2.0$ and\n$\\frac{P_{CO}}{P_{{CO}_{2}}} \\approx 2.4$, and the agreement with the\nmore accurate calculations above is remarkable keeping in mind the\nuncertainties in graphically drawing the lines and reading both values\nin the logarithmic scales from the diagram, indicating the robustness of\nthe Ellingham diagram.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/applications_to_chemical_reactions/index.ipynb b/psu410/src/applications_to_chemical_reactions/index.ipynb new file mode 100644 index 0000000..fb6fae2 --- /dev/null +++ b/psu410/src/applications_to_chemical_reactions/index.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "67f02270", "cell_type": "markdown", "source": "# Applications to chemical reactions\n\nA chemical reaction can be viewed as a framework dividing a system into\ntwo closed subsystems: reactants and products. The phases and species of\nreactants and products are selected from possible phases and species\nthat may form from the independent components of the system. A chemical\nreaction can thus be considered as an internal process to transfer heat\nand work between the two subsystems of reactants and products. It is\nevident that this subset of phases and species only represents partial\nequilibrium information of the system under given external conditions,\nand more stable equilibrium states may exist by including more phases\nand species with the global equilibrium depicted by phase diagrams\ndiscussed in previous chapters with all known phases and species\nincluded.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/applications_to_chemical_reactions/internal_process_and_differential_and_integrated_driving_forces.ipynb b/psu410/src/applications_to_chemical_reactions/internal_process_and_differential_and_integrated_driving_forces.ipynb new file mode 100644 index 0000000..b62cd0a --- /dev/null +++ b/psu410/src/applications_to_chemical_reactions/internal_process_and_differential_and_integrated_driving_forces.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "b3c62995", "cell_type": "markdown", "source": "## Internal process and differential and integrated driving forces\n\nThe driving force for an internal process can be defined as following\nusing $U$, $H$, $F$, or $G$ as discussed in Chapter and depending on\nwhat system variables are kept constant\n\n*Eq. 7\u20111*\n$- D = \\left( \\frac{\\partial U}{\\partial\\xi} \\right)_{S,\\ V,N_{i}\\ } = \\left( \\frac{\\partial H}{\\partial\\xi} \\right)_{S,\\ P,N_{i}\\ } = \\left( \\frac{\\partial F}{\\partial\\xi} \\right)_{T,\\ V,N_{i}\\ } = \\left( \\frac{\\partial G}{\\partial\\xi} \\right)_{T,P,N_{i}\\ }$\n\nThis can be termed as differential driving force as it relates the\nderivative of energy with respect to an internal process so the change\nis infinitesimally small and does not affect the properties of the\nsystem significantly. For a system under constant $T$ and $P$, let us\nconsider an internal process for forming a new phase $\\alpha$ with the\ncomposition $x_{i}^{\\alpha}$ and Gibbs energy\n$G_{m}^{\\alpha}\\left( x_{i}^{\\alpha} \\right)$. The differential driving\nforce for such an internal process can thus be defined as\n\n*Eq. 7\u20112*\n$- D = G_{m}^{\\alpha}\\left( x_{i}^{\\alpha} \\right) - \\sum_{i}^{}x_{i}^{\\alpha}\\mu_{i} = \\sum_{i}^{}x_{i}^{\\alpha}\\left( \\mu_{i}^{\\alpha} - \\mu_{i} \\right)$\n\nwhere $\\mu_{i}$ is the chemical potential of component *i* in the\nsystem. This may also be called as nucleation driving force as if the\n$\\alpha$ phase is nucleating in the system. As discussed in Chapter ,\nchemical potentials are the intercepts on the Gibbs energy axis by the\ntangent plane of Gibbs energy. thus represents the distance between the\ntangent planes of the original system and the $\\alpha$ phase at the\ncomposition of the $\\alpha$ phase. Evidently, this distance is at its\nmaximum when the two tangent planes are parallel to each other, i.e. the\ncommonly called parallel tangent construction when evaluating nucleation\ndriving force.\n\nThe situation is different for chemical reactions where the amount of\neach component in reactants is the same as that in products, i.e. there\nis a mass balance between reactants and products. The driving force for\na chemical reaction is defined by the Gibbs energy difference between\nthe reactants and products as if all the reactants are transferred to\nthe products. This driving force may thus be called integrated driving\nforce as it describes the energy difference of a system under two\ndifferent states, i.e.\n\n*Eq. 7\u20113*\n$- \\int_{}^{}{Dd\\xi} = \\mathrm{\\Delta}G = \\sum_{p}^{}{n_{p}G_{p}} - \\sum_{r}^{}{n_{r}G_{r}}$\n\nwhere superscripts $p$ and $r$ denote products and reactants, $n_{p}$\nand $n_{r}$ their corresponding moles, $G_{p}$and $G_{r}$ the Gibbs\nenergies in per mole of formula of their respective stoichiometries as\nwritten in the chemical reaction. Conventionally, the products and\nreactants are represented by species or stoichiometric compounds rather\nthan individual phases. This is particularly evident when various\ngaseous species are considered in a chemical reaction. Consequently,\n$G_{p}$and $G_{r}$ in represent the chemical potentials of product and\nreactant species. For species with a fixed composition, its chemical\npotential is the same as its Gibbs energy as shown by . For a species in\na solution phase, its chemical potential is related to its activity as\nshown in . can thus be further written as\n\n*Eq. 7\u20114*\n$\\mathrm{\\Delta}G = \\sum_{p}^{}{n_{p}_{\\ }^{0}G_{p}} - \\sum_{r}^{}{n_{r}_{\\ }^{0}G_{r}} + RTln\\frac{\\prod_{p}^{}\\left( a_{p} \\right)^{n_{p}}}{\\prod_{r}^{}\\left( a_{r} \\right)^{n_{r}}} = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{\\prod_{p}^{}\\left( a_{p} \\right)^{n_{p}}}{\\prod_{r}^{}\\left( a_{r} \\right)^{n_{r}}}$\n\nIt is evident from that for stoichiometric phases in a chemical\nreaction, their activities equal to one. At equilibrium, the integrated\ndriving force becomes zero, i.e. $\\mathrm{\\Delta}G = 0$, and is\nre-arranged to become\n\n*Eq. 7\u20115*\n$- RTln\\frac{\\prod_{p}^{}\\left( a_{p} \\right)^{n_{p}}}{\\prod_{r}^{}\\left( a_{r} \\right)^{n_{r}}} = - RTlnK_{e} = \\mathrm{\\Delta}_{\\ }^{0}G = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S$\n\nwhere $K_{e}$ is often called reaction constant relating the activities\nof products and reactants at equilibrium. In a system with\n$\\frac{\\prod_{p}^{}\\left( a_{p} \\right)^{n_{p}}}{\\prod_{r}^{}\\left( a_{r} \\right)^{n_{r}}} > K_{e}$,\nthe chemical reaction goes to left, and the products are decomposed,\nwhile\n$\\frac{\\prod_{p}^{}\\left( a_{p} \\right)^{n_{p}}}{\\prod_{r}^{}\\left( a_{r} \\right)^{n_{r}}} < K_{e}$,\nthe chemical reaction goes to right, and the products are formed. is\noften recast into the following form by dividing $\u2013RT$ on both sides of\nthe equation\n\n*Eq. 7\u20116*\n$\\ln K_{e} = - \\frac{\\mathrm{\\Delta}_{\\ }^{0}H}{RT} + \\frac{\\mathrm{\\Delta}_{\\ }^{0}S}{R}$\n\nWith $\\ln K_{e}$ plotted with respect to $1/T$, indicates that the slope\nis $- \\mathrm{\\Delta}_{\\ }^{0}H$/R and the intercept on the y axis is\n$\\mathrm{\\Delta}_{\\ }^{0}{S/R}$ as shown in .\n\nFigure \u2011: $\\ln K_{e}$ plotted with respect to $1/T$ for several M-O\nsystems at 1 bar and $K_{e}$ represented by the partial pressure ratio\nof CO2 and CO.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/applications_to_chemical_reactions/maximum_reaction_rate_and_chemical_transport_reactions_.ipynb b/psu410/src/applications_to_chemical_reactions/maximum_reaction_rate_and_chemical_transport_reactions_.ipynb new file mode 100644 index 0000000..004b1d3 --- /dev/null +++ b/psu410/src/applications_to_chemical_reactions/maximum_reaction_rate_and_chemical_transport_reactions_.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "7416df00", "cell_type": "markdown", "source": "## Maximum reaction rate and chemical transport reactions \n\nEquilibrium thermodynamics can be used to calculate the maximum rates of\nreaction that are possible in dynamically reacting systems, such as when\na corrosive gas passes over a heated sample. Other examples of such\nreactions include the reduction of a metal oxide in flowing hydrogen,\nthe evaporation of a material in a vacuum, and the deposition of a thin\nfilm by a chemical vapour deposition process. The basic assumption used\nin calculating these maximum reaction rates is that local equilibrium\nexists at the location of the considered reaction.\n\nThis section examines several examples in which maximum reaction rates\nare calculated. The system can typically be divided into three regions:\n(1) the input region, which is usually near room temperature, (2) the\nhigh temperature region in which the primary reaction of interest is\noccurring, and (3) the exit region. Such a system is almost always at\nconstant pressure throughout the system. Since the three regions have\ndifferent temperatures, the key is to use the mass conservation of the\ncarrier gas in all three regions. With the input gas at $T = 298K$ and\n$P = 1atm$, the volume of one mole of ideal gas is\n$0.0244\\left( m^{3} \\right) = 24.4(liter)$. The number of moles of input\ngas flowing through the system can be written as\n\n*Eq. 7\u201117*\n$n_{gas}^{0} = \\frac{Pf_{gas}}{RT} = \\frac{f_{gas}}{24.4} = 0.0409f_{gas}$\n\nwhere $f_{gas}$ is the input gas flow rate in liter per unit time at\n$T = 298K$ and $P = 1atm$, and $R$ the gas constant.\n\nThe first example is the evaporation of water in a flowing stream of dry\nhydrogen. A schematic diagram of the system being considered is given in\n. The goal of the calculation is to determine the maximum rate of\n$H_{2}O(liquid)$ loss, $n_{H_{2}O}^{\\ }$, through vaporization in a\nflowing stream of $H_{2}$, e.g. for generating a given $H_{2}O/H_{2}$\nratio for producing a given $O_{2}$ pressure in a high temperature\nsystem as related to the Ellingham diagram discussed in Chapter . The\nmaximum rate is determined by saturating the $H_{2}$ with the\nequilibrium vapour pressure of $H_{2}O(liquid)$ with the number of moles\nof $H_{2}$ being $n_{H_{2}}^{0} = 0.0409f_{H_{2}}$ and $f_{H_{2}}$ being\nthe input flow rate of $H_{2}$ in liter per unit time.\n\nFigure \u2011: Schematic diagram of vaporization of water in a flowing stream\nof dry hydrogen\n\nIn the first input region, the total number moles of gas is\n$N = n_{H_{2}}^{0}$. In the second high temperature region with the\ntemperature and pressure being $T_{sys}$ and $P_{sys}$, the total number\nmoles of gas is $N = n_{H_{2}}^{0} + n_{H_{2}O}^{\\ }$, which is the same\nfor the third exit region with $T_{exit}$. To avoid condensation, one\nneeds to maintain $T_{exit} > T_{sys}$. The vapour pressure of $H_{2}O$\nat $T_{sys}$, $P_{H_{2}O\\ }$, can be obtained from equilibrium\nthermodynamic calculations. Since the $H_{2}$ and $H_{2}O$ are occupying\nthe same volume at the same temperature, the maximum number of moles\n$H_{2}O$ can be calculated from the following relation based on the\nideal gas law\n\n*Eq. 7\u201118*\n$n_{H_{2}O}^{\\ } = \\frac{P_{H_{2}O}^{\\ }}{P_{H_{2}}^{\\ }}n_{H_{2}}^{0} = \\frac{P_{H_{2}O}^{\\ }}{P_{sys}^{\\ } - P_{H_{2}O}^{\\ }}n_{H_{2}}^{0} = \\frac{P_{H_{2}O}^{\\ }}{P_{sys}^{\\ } - P_{H_{2}O}^{\\ }}\\frac{f_{H_{2}}}{24.4}$\n\nFor $P_{sys}^{\\ } = 101325Pa$, one can calculate $P_{H_{2}O}^{\\ }$ and\nplot $\\frac{n_{H_{2}O}^{\\ }}{f_{H_{2}}}$ as a function of temperature\nfrom the SSUB database as shown in . The corresponding partial pressure\nratio, $P_{H_{2}O}/P_{H_{2}}$ , is plotted in , which can be used to\ncalculate $P_{O_{2}}^{\\ }$ at any given temperatures. An example is\nshown in with $T_{sys} = 348.15K$ and $P_{H_{2}O}/P_{H_{2}} = 0.607$.\n\nFigure \u2011: Ratio of maximum number of moles $H_{2}O$ with respect to\nhydrogen flow rate, $\\frac{n_{H_{2}O}^{\\ }}{f_{H_{2}}}$, plotted as a\nfunction of temperature.\n\nFigure \u2011: Partial pressure ratio, $P_{H_{2}O}/P_{H_{2}}$, corresponding\nto .\n\nFigure \u2011: Partial pressure of oxygen, $P_{O_{2}}^{\\ }$, as a function of\ntemperature with $T_{sys} = 348.15K$ and $P_{H_{2}O}/P_{H_{2}} = 0.607$.\n\nThe second example is the corrosion of $SiO_{2}$(s) by flowing $H_{2}$\ngas at high temperatures. Considering $T_{sys}^{\\ } = 1700K$,\n$P_{sys}^{\\ } = 101325Pa$, and $f_{H_{2}} = 1\\ liter/min$, the system is\nthus defined with 0.0409 moles of $H_{2}$ and an equilibrium between the\ngas phase and the tridymite $SiO_{2}$. The equilibrium calculation gives\n0.04092 moles of gas with its constitutions as\n$H_{2}:H_{2}O:SiO = 0.998887:4.833 \\bullet 10^{- 4}:4.832 \\bullet 10^{- 4}$.\nThe corrosion rate of SiO2(s) is thus\n$0.04092x4.832 \\bullet 10^{- 4} = 1.98 \\bullet 10^{- 5}mol/min = 1.19gram/min.$\n\nAnother example is to consider that the flowing $CO$ gas of 298K and\n1atm ($= 101325Pa$) at a rate of $1\\ liter/min$ passes through and\nequilibrates with single phase $C(s)$ at 1500K. The equilibrium system\nis defined by $T = 1500K$, $P = 1atm$, and 0.0409 moles of $CO$ with the\nequilibrium between the gas phase and C(s). The equilibrium calculation\ngives 0.040872 moles of gas phase with the mole fraction of $CO$ being\n0.999327, resulting in the loss of *CO* or the deposition of *C(s)* at a\nrate of\n$0.0409 - 0.040872 \\bullet 0.999327 = 5.55 \\bullet 10^{- 5}mole/min$.\n\nA chemical transport reaction is a reaction in which a condensed phase\nreacts with a gas phase to form vapour-phase products, which in turn\nundergo the reverse reformation of the condensed phase. Two well-known\nexamples of such reactions are\n\n*Eq. 7\u201119* $M(s) + \\frac{n}{2}I_{2}(g) = MI_{n}(g)$\n\n*Eq. 7\u201120* $Ni(s) + 4CO(g) = Ni{(CO)}_{4}(g)$\n\nwhich are used in the purification of metals by the iodide process and\nin the purification of nickel by the Mond-Langer process. In both\nprocesses the forward reaction is favoured by lower temperatures and the\nreverse reaction by higher temperatures, resulting in the deposition of\nthe metal. The most common technique for causing a chemical transport of\na condensed substance makes use of the temperature dependence of the\nequilibrium constant. As was discussed previously, the enthalpy of\nreaction, $\\mathrm{\\Delta}_{\\ }^{0}H$, determines the manner in which\n$K_{e}$ changes with temperature (see ). The value of $K_{e}$ increases\nwith increasing *T* for $\\mathrm{\\Delta}_{\\ }^{0}H > 0$, $K_{e}$\ndecreases with increasing *T* for $\\mathrm{\\Delta}_{\\ }^{0}H < 0$, and\n$K_{e}$ is independent of T for $\\mathrm{\\Delta}_{\\ }^{0}H = 0$. The\n$\\mathrm{\\Delta}_{\\ }^{0}H$ and $\\mathrm{\\Delta}_{\\ }^{0}S$ values for\nchemical transport reactions may be either positive or negative. For\nreactions by both are positive, and for reactions by both are negative.\n\nIn a typical experiment the starting solid is located at the point in a\ntemperature gradient that corresponds to the largest $K_{e}$ value for\nthe experimental condition. As the gaseous species migrate to other\nlocations in the system with temperatures corresponding to lower $K_{e}$\nvalues, the reverse reaction occurs to satisfy the new equilibrium\nrequirements, and the solid phase is deposited. The dependence of\n$K_{e}$ on $\\mathrm{\\Delta}_{\\ }^{0}H$ results in a material transport\nfrom hot to cold for $\\mathrm{\\Delta}_{\\ }^{0}H > 0$ (the same as for\nvaporization-condensation reactions), from cold to hot for\n$\\mathrm{\\Delta}_{\\ }^{0}H < 0$, and no transport for\n$\\mathrm{\\Delta}_{\\ }^{0}H = 0$.\n\nThe success of a particular reaction in causing an appreciable transport\nof a condensed phase depends mainly upon the partial pressure gradients\nor concentration gradients of the gaseous species in the system. A\nreaction whose equilibrium is extreme toward either the reactant side or\nthe product side will not give an appreciable transport of material. The\nconcentration gradients are too small in such a system. Reactions with\nequilibrium constants near unity at the experimental temperatures\nusually give the largest transport since small changes in $K_{e}$ cause\nlarge changes in concentrations. The general condition required for\nobtaining a $K_{e}$ value near unity at a reasonable temperature is that\n$\\mathrm{\\Delta}_{\\ }^{0}H$ and $\\mathrm{\\Delta}_{\\ }^{0}S$ both have\nthe same sign, resulting from the equalities of .\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/applications_to_chemical_reactions/trends_of_entropies_of_reactions.ipynb b/psu410/src/applications_to_chemical_reactions/trends_of_entropies_of_reactions.ipynb new file mode 100644 index 0000000..1840aa3 --- /dev/null +++ b/psu410/src/applications_to_chemical_reactions/trends_of_entropies_of_reactions.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "dbbb5455", "cell_type": "markdown", "source": "## Trends of entropies of reactions\n\nThe reaction entropy, $\\mathrm{\\Delta}_{\\ }^{0}S$ in , plays an\nimportant role in determining equilibria of high-temperature reactions.\nThe most important single factor that determines the entropy of a\nreaction is the net change in the number of moles of gas as briefly\nmentioned in the discussion of the Ellingham diagram above. The reason\nthis is true can be explained as follows.\n\nThe entropy of a substance can be thought of as being the sum of four\nparts: (i) translational, (ii) rotational, (iii) vibrational, and (iv)\nelectronic. The translational entropy of a gas is the largest entropy\nterm under most conditions. To the extent that the other contributions\ncancel between reactants and products, the entropy of reaction is\ndetermined by the change in the number of moles of gaseous molecules.\nBased on the literature data or calculations from the SSUB database, the\nnet change in the number of moles of gas in a reaction results\napproximately in an entropy of reaction of about 175\u00b145 J/K/mole-gas at\n298K for many halides and oxides. The chemical reactions of and\ndiscussed above both reduce the gas by one mole, and their entropies of\nreaction are -113 and -172 J/K at 1273K, and -89 and -173J/K at 298K,\nrespectively, indicating that the chemical reaction of is an exception\nof the empirical rule. For chemical reactions shown in the Ellingham\ndiagram, their entropies of reaction follow this empirical rule pretty\nwell with some of them shown in calculated from the SGTE database.\n\nTable \u2011: Entropies of reactions with gas at 298.15K, J/K\n\nReaction: Si+O2 =SiO2 -182\n\nReaction: Ti+O2=TiO2 -185\n\nReaction: 2Mg+O2=2MgO -217\n\nReaction: 2Ca+O2=2CaO -212\n\nReaction: 2Mn+O2=2MnO -150\n\nSince the entropy of a reaction is primarily determined by the net\nchange in the number of moles of gas, the entropies for reactions\ninvolving only condensed phases must be small. The entropies of fusion\nof monatomic solids are usually in the range 8-15 J/K/mole-atom as shown\nfor some elements in . Most metals and many ionic salts have values that\nlie in this range when given in terms of per mole of atom of material.\nThere are few exceptions such as silicon and boron shown in the table.\nFor solid-state reactions, the average values can be approximated as\n0\u00b18J/K/mole-atom as also shown in the table.\n\nTable \u2011: Entropies of reactions of condensed phases at 298.15K, J/K\n\nReaction: Si(s)=Si(l) 29.762\n\nReaction: Ti(s2)=Ti(l) 7.288\n\nReaction: Mg(s2)=Mg(l) 9.184\n\nReaction: Ca(s2)=Ca(l) 7.659\n\nReaction: Mn(s2)=Mn(l) 11.443\n\nReaction: W(s)=W(l) 14.158\n\nReaction: B(s)=B(l) 21.380\n\nReaction: 3Fe+C=CFe3 17.060\n\nReaction: S+Mn=MnS 13.909\n\nReaction: NiO+Fe2O3=Fe2NiO4\n0.464\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/applications_to_electrochemical_systems/application_examples.ipynb b/psu410/src/applications_to_electrochemical_systems/application_examples.ipynb new file mode 100644 index 0000000..e9bb59d --- /dev/null +++ b/psu410/src/applications_to_electrochemical_systems/application_examples.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "af90f499", "cell_type": "markdown", "source": "## Application examples\n\nAmong many applications of electrochemistry, several of them are briefly\ndiscussed in this section.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/applications_to_electrochemical_systems/aqueous_solution_and_pourbaix_diagram.ipynb b/psu410/src/applications_to_electrochemical_systems/aqueous_solution_and_pourbaix_diagram.ipynb new file mode 100644 index 0000000..34ed115 --- /dev/null +++ b/psu410/src/applications_to_electrochemical_systems/aqueous_solution_and_pourbaix_diagram.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "e064a684", "cell_type": "markdown", "source": "## Aqueous solution and Pourbaix diagram\n\nThe importance of aqueous solutions in all aspects of life is so well\nknown and needs not be discussed further. Since many electrochemical\nprocesses involve electrolyte solutions in an aqueous solvent,\nelectrochemical processes including water, hydrogen, and/or oxygen are\ndiscussed in more details. The hydrogen-oxygen cell can be described for\nboth acidic electrolytes and alkaline electrolytes. With acidic\nelectrolytes, H+ is in much higher concentrations than\nOH-, and thus half-cell reactions with H+ as an\nionic transport species are more important than those involving\nOH-. The reverse is true for alkaline electrolytes that\ncontain high OH-concentrations. Other than for nearly neutral\nacid-base systems, either H+ or OH- dominates the\nother by several orders of magnitude as can be seen from the value of\nthe 298 K dissociation constant for H2O:\n\n*Eq. 8\u201135* H2O(l) = H+(aq) + OH-(aq)\n\nwith the reaction constant being Ke =\n\\[H+\\]\\[OH-\\] = 10-14 and\n$\\mathrm{\\Delta}_{\\ }^{0}G$= -RT *ln* Ke *\u03b2*= +79,908 J. By\nconvention, one defines pH = - log \\[H+\\] and pOH = - log\n\\[OH-\\], and then pH + pOH = 14.\n\nUnder acidic electrolyte conditions of low pH (high \\[H+\\]\nconcentrations) the anode reaction in a hydrogen-oxygen cell is:\n\n*Eq. 8\u201136* \u00bd H2(g) = H+(aq) + e-\n\nwith \u03b51o = 0.0 V and\n$\\mathrm{\\Delta}_{\\ }^{0}G_{1}$= 0 J. The corresponding cathode\n(reduction) reaction is:\n\n*Eq. 8\u201137* 2 H+(aq) + \u00bd O2(g) + 2 e- =\nH2O(l)\n\nwith \u03b52o = 1.229 V and\n$\\mathrm{\\Delta}_{\\ }^{0}G_{2}$ = -2\\*1.229\\*96,485 J = -237,160 J. The\nnet cell reaction for acidic electrolytes is:\n\n*Eq. 8\u201138* H2(g) + \u00bd O2(g) = H2O(l)\n\nwith \u03b5ocell = 1.229 V and\n$\\mathrm{\\Delta}_{\\ }^{0}G_{cell}$ = -2\\*1.229\\*96,485 J = -237,160 J\n\nUnder alkaline electrolyte conditions of high pH (high\n\\[OH-\\] concentrations) the anode reaction in a\nhydrogen-oxygen cell is:\n\n*Eq. 8\u201139* 2 OH-(aq) + H2(g) = 2\nH2O(l) + 2 e-\n\nwith \u03b53o = 0.828 V and\n$\\mathrm{\\Delta}_{\\ }^{0}G_{3}$= -2\\*0.828\\*96,485 J = -159,779 J. The\ncorresponding cathode (reduction) reaction is:\n\n*Eq. 8\u201140* H2O(l) + \u00bd O2(g) + 2 e- = 2\nOH-(aq)\n\nwith \u03b54o = 0.401 V and\n$\\mathrm{\\Delta}_{\\ }^{0}G_{4}$ = -2\\*0.401\\*96,485 J = -77,381 J. The\nnet cell reaction for alkaline electrolytes is:\n\n*Eq. 8\u201141* H2(g) + \u00bd O2(g) = H2O(l)\n\nwith \u03b5ocell = 1.229 V and\n$\\mathrm{\\Delta}_{\\ }^{0}G_{cell}$ = -2\\*1.229\\*96,485 J = -237,160 J.\n\nPlots of \u03b5 versus pH for a given chemical system have been typically\nused to exhibit the stability relationships of ionic species and solid\nphases in aqueous-based electrochemical systems. These graphs are often\ncalled Pourbaix diagrams after the inventor and are at constant\ntemperature and constant pressure diagrams for a constant concentration,\nusually for one metallic element. By convention, the \u03b5 in a Pourbaix\ndiagram corresponds to the potential for the cathode reduction reactions\nin the electrochemical half-cell with electrons as reactants. Pourbaix\ndiagrams can be extended to multi-component materials when thermodynamic\nproperties of the components are available in both the materials and the\naqueous solution.\n\nAn example of an \u03b5 versus pH diagram is shown in for the\nNi-H2O system at a 298K, 1 bar, and $c_{{Ni}^{2 +}} = 0.001$\nmolality. Three stability regions for Ni species are shown: Ni(s),\nNiO(s), and \\[Ni2+\\]. The two dashed lines on this diagram\ncorrespond to hydrogen reduction () and oxygen reduction () reactions,\nrespectively.\n\nFigure \u2011: An \u03b5 versus pH, Pourbaix diagram for the Ni-H2O at\n298K, 1 bar, and $c_{{Ni}^{2 +}} = 0.001$ molality.\n\nFor the \u03b5 and pH conditions within the boundaries of the Ni(s) region,\nno solid phase other than Ni(s) is stable, no ionic species with a\nconcentration of 1 molarity is stable, and no gas species with a\npressure of 1 bar is stable. Similar statements could be made about the\nNiO(s) and \\[Ni2+\\] areas on the diagram. In the\n\\[Ni2+\\] area, introduction of Ni(s) or NiO(s) into the\nsystem would result in the dissolution of these solid phases since they\nare not stable with respect to the \\[Ni2+\\] aqueous solution.\nThe corresponding chemical reactions proceed spontaneously to the right\nas follows until the solid phases are consumed:\n\n*Eq. 8\u201142* Ni(s) \u2192 Ni2+(1 molarity) + 2 e-\n\n*Eq. 8\u201143* NiO(s) + 2 H+(aq) \u2192 Ni2+(1 molarity) +\nH2O(l)\n\nNo H+(aq) in involved in the first reaction, , so the\nboundary line separating Ni(s) and Ni2+ is independent of pH.\nNo oxidation or reduction occurs in the second reaction, , i.e. no\nelectrons are reactants or products in the reaction, the boundary line\nseparating NiO(s) and Ni2+ is independent of \u03b5.\n\nNote the convention that the \u03b5 is the potential for a cathode reduction\nreaction, and boundary lines between two stability regions depict\nconditions under which partial equilibrium occurs for the two species\nfor the \u03b5 and pH values at any point on these lines. For the boundary\nline separating Ni(s) and Ni2+ in an ideal aqueous solution,\ni.e. the reverse of , the following equation is obtained.\n\n*Eq. 8\u201144* \u03b5 = \u03b5o = -0.268 V\n\nFor the NiO(s)-Ni2+ boundary line of an ideal solution, the\nreaction, , is a complete equilibrium, and thus the relationship is\n\n*Eq. 8\u201145*\n$0 = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{1}{\\left( c_{H^{+}} \\right)^{2}} = \\mathrm{\\Delta}_{\\ }^{0}G + 2 \\cdot 2.303 \\cdot RT \\cdot pH$\n\n*Eq. 8\u201146*\n$pH = - \\frac{\\mathrm{\\Delta}_{\\ }^{0}G}{2 \\cdot 2.303 \\cdot RT}$\n\nwhere $\\mathrm{\\Delta}_{\\ }^{0}G$ is obtained as follows and can be\ncalculated from the SSUB database and the standard potential of Ni,\n\n*Eq. 8\u201147*\n\nAt a specified temperature, only one standard free energy and only one\nequilibrium constant exists for this chemical reaction, and thus only\none specific value of $pH = 6.631$ exists for the reaction represented\nby in this Pourbaix diagram.\n\nThe diagonal line in represents the equilibrium between Ni(s) and NiO(s)\nand is for a partial equilibrium reaction that is the sum of reactions\nof and\n\n*Eq. 8\u201148* NiO(s) + 2 H+(aq) + 2 e- \u2550 Ni(s) +\nH2O(l)\n\nThe reduction of Ni from a divalent state in NiO to metallic Ni(s)\noccurs, but the reaction also depends on the H+\nconcentration, the pH. The corresponding Gibbs energy and Nernst\nequations are,\n\n*Eq. 8\u201149*\n$\\mathrm{\\Delta}G = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{1}{\\left( c_{H^{+}} \\right)^{2}} = - 23,939 + 2 \\cdot 2.303 \\cdot RT \\cdot pH$\n\n*Eq. 8\u201150*\n$\\varepsilon\\ = \\ \\varepsilon^{0}\\ - \\ \\frac{RT}{2f}\\ln\\frac{1}{\\left( c_{H^{+}} \\right)^{2}}\\ = 0.124\\ \u2013\\ \\frac{2.303 \\cdot RT}{f}pH$\n\nwhere $\\mathrm{\\Delta}_{\\ }^{0}G$ can be calculated as follows\n\n*Eq. 8\u201151*\n\nThe two additional lines in correspond to the reduction reactions\nrelated to H2 and O2 gases, i.e. the stability of\nH2O. The lower one is for the reverse of under \u03b5o\n= 0 and $P_{H_{2}} = 1$ with the Nernst equation being\n\n*Eq. 8\u201152*\n$\\varepsilon\\ = \\ \\varepsilon^{0}\\ - \\ \\frac{RT}{f}\\ln\\frac{\\left( P_{H_{2}} \\right)^{1/2}}{c_{H^{+}}}\\ = - \\frac{2.303 \\cdot RT}{f}pH$\n\nAs the pH increases from 0, \u03b5 becomes more negative as is depicted. The\ntop dashed line corresponds to the oxygen reduction reaction represented\nby under \u03b5o = 1.225 calculated from the aqueous solution\ndatabase in Thermo-Calc \\[60\\] and $P_{O_{2}} = 1$ with the Nernst\nequation being\n\nEq. 8\u201153\n$\\varepsilon\\ = \\ \\varepsilon^{0}\\ - \\ \\frac{RT}{2f}\\ln\\frac{\\left( P_{O_{2}} \\right)^{1/2}}{\\left( c_{H^{+}} \\right)^{2}}\\ = 1.225\\ \u2013\\ \\frac{2.303 \\cdot RT}{f}pH$\n\nThe dependence of \u03b5 on pH is identical for both reduction reaction and ,\nand their intercepts at $pH = 0$ differ by their difference in their\n\u03b5o values.\n\nIn this simple Pourbaix diagram of Ni in an ideal aqueous solution, all\nboundary lines are straight because there is only one ionic species of\nNi in the aqueous solution, i.e. Ni2+. When there are more\nthan one ionic species in the aqueous solution, the boundary lines may\nno longer be straight due to the competition between species. One\nexample is Cu with two main ionic species of Cu+2 and\nCuOH+, and the reduction reaction between the metallic Cu and\nthe aqueous solution involves both two species, i.e.\n\n*Eq. 8\u201154*\n${xCu}^{2 + \\ }\\ + \\ (1 - x){CuOH}^{+} + (1 - x)H^{+} + 2\\ e - \\ \\ = \\ \\ Cu(s) + {(1 - x)H}_{2}O$\n\nwith\n\n*Eq. 8\u201155*\n$\\mathrm{\\Delta}G = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{1}{\\left( c_{{Cu}^{2 +}} \\right)^{x}\\left( c_{{CuOH}^{+}}c_{H^{+}} \\right)^{1 - x}} = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{1}{\\left( c_{{Cu}^{2 +}} \\right)^{x}\\left( c_{{CuOH}^{+}} \\right)^{1 - x}} + 2.303(1 - x) \\cdot RT \\cdot pH$\n\n*Eq. 8\u201156*\n$\\varepsilon = \\varepsilon^{0} - \\frac{RT}{2f}\\ln\\frac{1}{\\left( c_{{Cu}^{2 +}} \\right)^{x}\\left( c_{{CuOH}^{+}} \\right)^{1 - x}} - \\frac{2.303(1 - x) \\cdot RT}{2f}pH$.\n\nIt is evident that both the slope and the intercept at $pH = 0$ are a\nfunction of the concentration of ${CuOH}^{+}$, which is a function of\n$pH$. Consequently, the boundary between the metallic Cu and the aqueous\nsolution is no longer a straight line as shown in .\n\nFigure \u2011: An \u03b5 versus pH, Pourbaix diagram for the Cu-H2O\nsystem at 298K, 1 bar, and $c_{Cu} = 0.001$ molality.\n\nThe concentrations of various species in the aqueous solution, i.e.\ncommonly called speciation, are plotted in , showing the change of\ndominant species as a function of pH value.\n\nFigure \u2011: Concentrations of ionic species in the aqueous solution at\n$\\varepsilon = 0.3\\ V$ from .\n\nIn Pourbaix diagrams for alloys with two or more elements, activities of\nindividual elements are to be used in calculating the potentials of\nreduction reactions. Considering a Fe-Ni alloy with Fe2+ and\nNi2+ in the aqueous solution, the reduction reactions for Fe\nand Ni can be written separately as\n\n*Eq. 8\u201157* Ni2+(cNi) + 2 e- \u2192 Ni\n(aNi in alloy)\n\n*Eq. 8\u201158* Fe2+(cFe) + 2 e- \u2192 Fe\n(aFe in alloy)\n\nwith their potentials as\n\n*Eq. 8\u201159*\n$\\varepsilon_{Ni}\\ = \\ \\varepsilon_{Ni}^{0} - \\frac{2.303RT}{2f}\\ln\\frac{a_{Ni}}{c_{Ni}} = \\ - 0.268 - \\frac{2.303RT}{2f}\\ln\\frac{a_{Ni}}{c_{Ni}}$\n\n*Eq. 8\u201160*\n$\\varepsilon_{Fe}\\ = \\ \\varepsilon_{Fe}^{0} - \\frac{2.303RT}{2f}\\ln\\frac{a_{Fe}}{c_{Fe}} = \\ - 0.441 - \\frac{2.303RT}{2f}\\ln\\frac{a_{Fe}}{c_{Fe}}$\n\nIn principle, there are two scenarios for a given set of $a_{Ni}$ and\n$a_{Fe}$ of the alloy. The first scenario is at the limit of a dilute\naqueous solution, i.e. $c_{Ni} = c_{Fe} = 0.001\\ $molarity,\n$\\varepsilon_{Ni}$ and $\\varepsilon_{Fe}$ can be calculated, and the\nelement with the lower potential has the tendency to dissolve first,\nwhich can result in the so-called dialloying effect. The second scenario\nis for the equal potential, i.e. $\\varepsilon_{Ni} = \\varepsilon_{Fe}$\ndue to the externally imposed potential, and the equilibrium\nconcentrations of Fe+2 and Ni+2 can be calculated\nfrom and .\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/applications_to_electrochemical_systems/concentrations_activities_and_reference_states_of_electrolyte_species.ipynb b/psu410/src/applications_to_electrochemical_systems/concentrations_activities_and_reference_states_of_electrolyte_species.ipynb new file mode 100644 index 0000000..3351784 --- /dev/null +++ b/psu410/src/applications_to_electrochemical_systems/concentrations_activities_and_reference_states_of_electrolyte_species.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "ebc6e0c8", "cell_type": "markdown", "source": "## Concentrations, activities, and reference states of electrolyte species\n\nThermodynamic descriptions of ionic species in solutions are different\nfrom those of neutral species, which leads to a need for defining\nconcentration units, standard states, activities, and activity\ncoefficients of ionic solutions. In most studies of electrochemical\ncorrosion and electrodeposition, and in applied work of electrochemical\nengineers, ionic species concentrations are given in units of molarity,\nthe number of moles of a species in a liter of solution (mol/l)\nsymbolically represented in equations by either *ci* or\n\\[M+Z\\]. The other common concentration used for ionic\nspecies is molality, which is defined as the number of moles of a\nspecies in 1000g of solvent. For dilute aqueous solutions, molarity and\nmolality values are very similar.\n\nAs discussed in Chapter , a practical definition of the activity of a\nspecies *i* is the thermodynamic reactivity, or tendency to react, of\nspecies *i* in the system of interest as compared to *i* in its\nreference state form. The reference state of a species is typically\nchosen as a specific chemical/physical state of the species at 1 atm\nexternal pressure and the temperature of interest. Similarly, a typical\nreference state for ionic species in aqueous solutions is the 1 molar\nideal solution at 1 bar external pressure and the temperature of\ninterest. If an electrolyte solution behaves ideally, then the activity\nof species *i* in solution is\n\n*Eq. 8\u20116*\n$a_{i} = \\frac{c_{i}\\left( \\frac{mol}{l} \\right)}{c_{i}^{0}\\left( \\frac{mol}{l} \\right)} = \\frac{c_{i}\\left( \\frac{mol}{l} \\right)}{1\\left( \\frac{mol}{l} \\right)} = c_{i}(dimensionless)$\n\nwhere $c_{i}$ is the molar concentration of *i* in the solution divided\nby $c_{i}^{0}$, the 1 molar reference state ideal solution\nconcentration. Thus, in ideal solutions, the activity of an electrolyte\nspecies is numerically equal to its molar concentration. The above\ntreatment of ionic species is equivalent to the common practice of\ndepicting the activity of a gas by the value of its ideal gas partial\npressure in units of bar.\n\nThe activity coefficient corrects for the nonideality of the species in\nsolution as defined in . If the solution is ideal, $\\gamma_{i} = 1$ for\nall concentrations of a species in solution. For all solutions, one\nexpects $\\gamma_{i} \\rightarrow 1$ as $c_{i} \\rightarrow 1$. It is not\npossible to measure $\\gamma_{i^{+}}$ or $\\gamma_{i^{-}}$ for individual\ncharged ions, only a geometric mean of the positive and negative ion\nvalues. Consider the following ionic solution\n\nEq. 8\u20117\n\nIts chemical potential can be written as\n\n*Eq. 8\u20118*\n\nIts geometric average or mean activity and activity coefficient are\ndefined as\n\n*Eq. 8\u20119*\n\n*Eq. 8\u201110*\n\nFor example, one can define\n$\\gamma_{\\pm} = \\left( \\gamma_{{Na}^{+}}\\gamma_{{Cl}^{-}} \\right)^{1/2}$\nand\n$\\gamma_{\\pm} = \\left( \\gamma_{{Al}^{3 +}}^{2}\\gamma_{{{SO}_{4}}^{2 -}}^{3} \\right)^{1/5}$\nfor NaCl and Al2(SO4)3, respectively.\nFor idea, weak electrolytes, $\\gamma_{\\pm} = 1$, and for non-ideal,\nstrong electrolytes, $\\gamma_{\\pm} \\neq 1$.\n", "metadata": {}}, {"id": "5e28080f", "cell_type": "markdown", "source": "### Electrical batteries\n\nBatteries utilize electrochemical reactions to generate electricity for\nvarious devices. The theoretic voltage of a battery can be calculated\nfrom and , i.e.\n\n*Eq. 8\u201173*\n$\\varepsilon = - \\frac{\\mathrm{\\Delta}G}{zf} = \\varepsilon^{0} - \\frac{RTlnQ}{zf}$\n\nwith $\\mathrm{\\Delta}G$ being the driving force of the net cell reaction\nand $Q$ being the reaction activity quotient. The actual voltage of a\nbattery is lower than the theoretical one due to kinetic limitations of\ncell reactions and resistance to ion diffusion through the electrolyte.\nBased on whether the cell reactions are reversible or not, batteries\ntypically categorized as either primary disposable or secondary\nrechargeable batteries. The net cell reactions in primary disposable\nbatteries are not easily reversible, and electrode materials may not\nreturn to their original forms by applying a higher external potential\nof opposite sign. Consequently, primary batteries cannot be reliably\nrecharged. On the other hand, the net cell reactions in secondary\nbatteries are easily reversible. Furthermore, two half-cells in\nbatteries may use different electrolytes with each half-cell enclosed in\na container and a separator permeable to conducting ions but not the\nbulk of the electrolytes.\n\nOne common primary battery is zinc-carbon battery with a zinc anode\ncylinder and a carbon cathode central rod. The electrolytes are ammonium\nor zinc chloride next to the zinc anode and a mixture of ammonium\nchloride and manganese dioxide next to the carbon cathode. The half cell\nand net reactions with ammonium chloride are as follows\n\n*Eq. 8\u201174* Zn + 2NH3 \u2192\nZn(NH3)22+ + 2 e-\n\n*Eq. 8\u201175* 2NH4Cl + 2MnO2 + 2 e- \u2192\n2NH3 + Mn2O3 +\nH2O+2Cl\u2212\n\n*Eq. 8\u201176* Zn + 2MnO2 + 2NH4Cl \u2192\nMn2O3 +\nZn(NH3)2Cl2 + H2O.\n\nThe electric potential of the reaction is, treating all compounds as\nstoichiometric compounds\n\nEq. 8\u201177\n$\\varepsilon = - \\frac{\\mathrm{\\Delta}G}{2f} = - \\frac{\\mathrm{\\Delta}^{0}G}{2f} = \\frac{1}{2f}\\left(_{\\ }^{0}G^{Zn} + 2_{\\ }^{0}G^{{MnO}_{2}} + 2_{\\ }^{0}G^{{NH}_{4}Cl} -_{\\ }^{0}G^{H_{2}O} -_{\\ }^{0}G^{Zn\\left( {NH}_{3} \\right)_{2}{Cl}_{2}} -_{\\ }^{0}G^{{{Mn}_{2}O}_{3}} \\right)$.\n\nThe Gibbs energy of $Zn\\left( {NH}_{3} \\right)_{2}{Cl}_{2}$ is not\navailable in current databases and has been recently estimated to be\n\u2212505,375 J/mole-formula \\[62\\]. The value of at 298.15K is thus obtained\nas 1.67 V, which is pretty close to the actual operating voltage of the\nbattery around 1.5 V.\n\nWhile with zinc chloride, the cell reactions and electric potential may\nbe written as\n\nEq. 8\u201178 Zn + ZnCl2 + 2OH\u2212 \u2192 2ZnOHCl + 2\ne-\n\n*Eq. 8\u201179* MnO2 + H2O + e- \u2192 MnOOH +\nOH-\n\n*Eq. 8\u201180* Zn + 2 MnO2 + ZnCl2 + 2 H2O\n\u2192 2 MnOOH + 2 ZnOHCl\n\n*Eq. 8\u201181*\n$\\varepsilon = \\frac{1}{2f}\\left(_{\\ }^{0}G^{Zn} + 2_{\\ }^{0}G^{{MnO}_{2}} +_{\\ }^{0}G^{Zn{Cl}_{2}} + 2_{\\ }^{0}G^{H_{2}O} - 2_{\\ }^{0}G^{MnOOH} - 2_{\\ }^{0}G^{ZnOHCl} \\right)$\n\nSecondary batteries can be recharged by applying an external electrical\npotential, which reverses the net cell reaction that occur during\ndischarging. The oldest form of rechargeable battery is the lead-acid\nbatteries used in automotive, and the latest development is the\nlithium-ion (Li-ion) batteries. A lead-acid battery typically uses Pb\nand PbO2 as the cathode and anode electrodes and a 35% sulfuric acid and\n65% water solution as the electrolyte. Its anode and cathode reactions\ncan be simplified as follows\n\nEq. 8\u201182 $Pb + SO_{4}^{2 -} = PbSO_{4} + 2e^{-}$\n\nEq. 8\u201183 $PbO_{2} + 4H^{+} + SO_{4}^{2 -} + 2e^{-} = PbSO_{4} + 2H_{2}O$\n\nThe net cell reaction is\n\nEq. 8\u201184 $Pb + PbO_{2} + 2H_{2}SO_{4}^{\\ } = 2PbSO_{4} + 2H_{2}O$.\n\nIts electric potential is represented by the following equation\n\n*Eq. 8\u201185*\n$\\varepsilon = - \\frac{1}{2f}\\left( 2_{\\ }^{0}G^{H_{2}O} + 2_{\\ }^{0}G^{{PbSO}_{4}} -_{\\ }^{0}G^{Pb} -_{\\ }^{0}G^{{PbO}_{2}} - 2_{\\ }^{0}G^{H_{2}SO_{4}} \\right)$\n\nwith the value at 298.15K being 2.651 V calculated from Thermo-Calc\n\\[60\\]. During discharging, the reaction goes to right, and $PbSO_{4}$\nis formed on both anode and cathode, while during charging, the reaction\ngoes to the left, and $Pb$ and $PbO_{2}$ are restored. In practical\napplications, other ionic species such as ${H_{3}O}^{+}$ and\n$HSO_{4}^{-}$ may form in the electrolyte, complicating the reactions\nand affecting its potential.\n\nIn lithium ion batteries, lithium ions migrate in electrolytes between\nelectrodes made of intercalated lithium compounds during charging and\ndischarging. LiCoO2 and LiFePO4 are two of the\nseveral cathode materials used in lithium ion batteries, and the anode\nis typically made of carbon or metallic Li. The anode and cathode\nreactions for LiCoO2 batteries can be written in simple forms as follows\n\nEq. 8\u201186 ${Li}_{x}C_{6} = x{Li}^{+} + xe^{-} + 6C$\n\nEq. 8\u201187 $x{Li}^{+} + xe^{-} + {Li}_{1 - x}CoO_{2} = LiCoO_{2}$\n\nwith the net reaction and electric potential being\n\nEq. 8\u201188 ${Li}_{x}C_{6} + {Li}_{1 - x}CoO_{2} = LiCoO_{2} + 6C$\n\n*Eq. 8\u201189*\n$\\varepsilon = - \\frac{1}{xf}\\left\\{ 6_{\\ }^{0}G^{C} +_{\\ }^{0}G^{{LiCoO}_{2}} - G^{{{Li}_{x}C}_{6}} - G^{{{Li}_{1 - x}CoO}_{2}} \\right\\} = - \\frac{1}{f}\\left\\{ \\left( \\mu_{Li}^{{Li}_{1 - x}CoO_{2}} - \\mu_{Li}^{{Li}_{x}C} \\right) - \\frac{1}{x}\\left( \\mu_{LiCoO_{2}}^{{Li}_{1 - x}CoO_{2}} -_{\\ }^{0}G^{{LiCoO}_{2}} \\right) \\right\\}$\n\nThe electric potential is a function of $x$. The value in the first\nparenthesis in the above equation denotes the chemical potential\ndifference of Li between two electrodes, and the value in the second\nparenthesis represents the chemical potential difference of\n${LiCoO}_{2}$ between the states in the solution phase of\n${Li}_{1 - x}CoO_{2}$ and by itself. Gibbs energies of ${{Li}_{x}C}_{6}$\nand ${{Li}_{1 - x}CoO}_{2}$ need to be obtained as a function $x$ in\norder to calculate the electric potential of the battery.\n\nLiFePO4 uses metallic lithium as the anode with following\nhalf-cell and net cell reactions\n\nEq. 8\u201190 $xLi = x{Li}^{+} + xe^{-}$\n\nEq. 8\u201191 $x{Li}^{+} + xe^{-} + {Li}_{1 - x}FePO_{4} = LiFePO_{4}$\n\nEq. 8\u201192 $xLi + {Li}_{1 - x}FePO_{4} = LiFePO_{4}$.\n\nIts electric potential is also a function of $x$, i.e.\n\n*Eq. 8\u201193*\n$\\varepsilon = - \\frac{1}{xf}\\left\\{_{\\ }^{0}G^{LiFePO_{4}} - x\\ ^{0}G^{Li} - G^{{Li}_{1 - x}FePO_{4}} \\right\\} = - \\frac{1}{f}\\left\\{ \\left( \\mu_{Li}^{{Li}_{1 - x}FePO_{4}} - \\ ^{0}\\mu_{Li} \\right) - \\frac{1}{x}\\left( \\mu_{LiFePO_{4}}^{{Li}_{1 - x}FePO_{4}} -_{\\ }^{0}G^{LiFePO_{4}} \\right) \\right\\}$\n\nThe value in the first parenthesis in the above equation denotes the\nchemical potential difference of Li between two electrodes, and the\nvalue in the second parenthesis represents the chemical potential\ndifference of $LiFePO_{4}$ between the states in the solution phase of\n${Li}_{1 - x}FePO_{4}$ and by itself. Consequently, Gibbs energy of\n${Li}_{1 - x}FePO_{4}$ needs to be obtained as a function $x$ in order\nto calculate the electric potential of the battery. It is known that\nthere are several miscibility gaps in the $FePO_{4}$ and $LiFePO_{4}$\npsuedo-binary system, in which the chemical potentials are constants, so\nis the electric potential, resulting in stable battery output.\n", "metadata": {}}, {"id": "b49dd005", "cell_type": "markdown", "source": "### Electrochemical cells\n\nA potential difference, i.e. voltage difference, can be generated\nbetween the electrodes in a cell from differences in the potentials of\nthe half-cell reactions. This potential can originate from potential\ndifferences of two chemically different half-cells (a *galvanic cell*),\nor concentration differences in two otherwise identical half-cells (a\n*concentration cell*). Each type of cell is illustrated below.\n\nThe reaction between copper ions and zinc illustrated below represents\nthe *net cell reaction* of a *galvanic* cell in which the oxidation of\nZn(s) occurs at the *anode* electrode, and the reduction of\nCu+2 occurs at the *cathode* electrode\n\n*Eq. 8\u201111* Cu2+(aq) + Zn(s) = Cu(s) + Zn2+(aq).\n\nThe reaction at each electrode, the *half-cell reaction*, includes ions\nand electrons as reactant and/or product species. The anode, oxidation\nreaction is represented by\n\n*Eq. 8\u201112* Zn(s) = Zn2+(aq) + 2 e-.\n\nElectrons are products of anode reactions and flow externally from anode\nto cathode. By convention the activities of the electrons in an\nequilibrium cell are taken as unity. The cathode, reduction reaction is\nwritten as\n\n*Eq. 8\u201113* Cu2+(aq) + 2 e- = Cu(s)\n\nElectrons are reactants of cathode reactions and supplied by an external\nflow from the anode. In addition to consuming electrons at the cathode\nat the same rate as they are produced at the anode, charge balance is\nmaintained in the electrolyte by the generation of Zn2+ ions\nat the same rate that Cu2+ ions are consumed. A schematic\ndiagram in illustrates the simple physical relationships in such an\nelectrochemical cell.\n\nFigure \u2011: Schematic diagram of a galvanic electrochemical cell\nconsisting of a zinc electrode and a copper electrode.\n\nA concentration cell in which an electrochemical potential is developed\nbecause of concentration differences between otherwise equivalent anode\nand cathode reactions is illustrated below. Such a cell can be produced\nby the oxidation and reduction of copper at two separate electrodes as\nis depicted in the following reactions\n\n*Eq. 8\u201114* Cu(s) = Cu2+(aq, ca) + 2 e- (anode)\n\n*Eq. 8\u201115* Cu2+(aq, cb) + 2 e- = Cu(s) (cathode)\n\nwhere ca and cb are the respective concentrations\nof Cu2+ in the aqueous solutions at the anode and cathode,\nand ca \\< cb. The net cell reaction is\n\n*Eq. 8\u201116* Cu2+(aq, cb) = Cu2+(aq,\nca)\n\nwhere the reaction occurs spontaneously to decrease cb and to\nincrease ca until the two concentrations become the same,\ncb = ca. A schematic diagram of such a cell is\nshown in .\n\nFigure \u2011: Schematic diagram of a concentration electrochemical cell\nconsisting of two copper electrodes.\n\nA semi-impermeable membrane, or a salt bridge, must exist in such a cell\nto maintain charge balance. As Cu+2 ions are produced at the\nanode and consumed at the cathode, the negatively charged ions in the\nsolution, for example Cl-, must be transferred from the\ncathode region to the anode region of the cell to maintain electrically\nneutral solutions.\n\nThe above concentration cell provides a good example for illustrating a\nstandard notation for depicting an electrochemical cell. This cell can\nbe represented by .\n\nFigure \u2011: Standard notation of an electrochemical cell\n\nThe anode where oxidation occurs is always denoted on the left, and the\ncathode where reduction occurs is on the right. A *single line*\nseparating phases denotes an *interface* between two phases. The above\nanode electrode and reaction of are symbolically represented by\n\n*Eq. 8\u201117* \\| Cu(s) \\| Cu2+(ca).\n\nThe interface between the external conductor and Cu(s) is depicted by\nthe single line to the left of Cu(s), while the single line between the\nCu(s) and Cu2+(ca) depicts the interface between\nthe anode electrode and the electrolyte solution. Similarly, the cathode\nelectrode and reaction of are symbolically represented by\n\n*Eq. 8\u201118* Cu2+(cb) \\| Cu(s) \\|.\n\nA double line between the two copper ions in the notation denotes a\nphysical separation of two solution phases, the anode and cathode\nelectrolyte regions that exhibit different concentrations of copper ions\n\n*Eq. 8\u201119* Cu2+(ca) \\| \\|\nCu2+(cb).\n\nThese solution phases are physically connected by a semi-impermeable\nmembrane or salt bridge that allows a common negative ion, for example\nCl-, of the solution phases to diffuse from one region to the\nother in order to maintain charge balance as the cell reaction occurs.\nThe Cu2+ ions cannot be transported from one region to the\nother.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/applications_to_electrochemical_systems/electrochemical_cells_and_half_cell_potentials.ipynb b/psu410/src/applications_to_electrochemical_systems/electrochemical_cells_and_half_cell_potentials.ipynb new file mode 100644 index 0000000..d636f47 --- /dev/null +++ b/psu410/src/applications_to_electrochemical_systems/electrochemical_cells_and_half_cell_potentials.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "dfa135f5", "cell_type": "markdown", "source": "## Electrochemical cells and half cell potentials\n\nAn electrochemical system must fulfill certain requirements in order to\napply equilibrium thermodynamic descriptions of the system, including\n\n- The cell must be reversible when slight changes in conditions\n (potentials, concentrations, pressures, temperature) cause\n electrochemical reactions and an external flow of electrons to occur\n in the direction needed to re-establish equilibrium.\n\n- All non-electrochemical reactions in the system must be prevented as\n such reactions would cause a shift in equilibrium without causing a\n shift in cell potential and thus a driving force for external electron\n flow.\n\n- Chemical reactions must occur only when an external current flows.\n These finite distances for external electron transport can be a short\n as grain size-dimensions in many corrosion reactions, or this\n transport may be through an external electrical conductor connecting\n the anode and cathode half-cell as in batteries.\n\n- Charge balance as well as mass balance is required of all reactions.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/applications_to_electrochemical_systems/electrolyte_reactions_and_electrochemical_reactions.ipynb b/psu410/src/applications_to_electrochemical_systems/electrolyte_reactions_and_electrochemical_reactions.ipynb new file mode 100644 index 0000000..f2f0b01 --- /dev/null +++ b/psu410/src/applications_to_electrochemical_systems/electrolyte_reactions_and_electrochemical_reactions.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "a49607a0", "cell_type": "markdown", "source": "## Electrolyte reactions and electrochemical reactions\n\nElectrolytes that dissolve in a polar solvent such as water to produce\nionic species do not necessarily exhibit valence changes. A simple\nexample is the strong electrolyte NaCl(s) dissolving in water to produce\nsolvated ions\n\n*Eq. 8\u20111* $NaCl(s)\\ \\ = \\ \\ {Na}^{+}(aq)\\ \\ + \\ \\ {Cl}^{\u2013}(aq)$\n\nwhere the $(aq)$ indicates the ionic species in an aqueous solution. In\nthis system, the ion concentrations must become quite large before the\nsolution is saturated and can exist in equilibrium with $NaCl(s)$. Its\nreaction constant, defined by , is shown as\n$K_{e} = \\ \\ a_{{Na}^{+}}a_{{Cl}^{-}}$. If the product of the ion\nactivities is less than $K_{e}$, the solution is not saturated, and more\n$NaCl(s)\\ \\ $ can be dissolved.\n\nThe precipitation of $AgCl(s)$, a weak electrolyte, occurs quite readily\nwhen ${Cl}^{\u2013}$ ions are added to an aqueous solution containing\n${Ag}^{+}(aq)$:\n\n*Eq. 8\u20112* ${Ag}^{+}(aq)\\ + \\ \\ {Cl}^{\u2013}(aq)\\ = AgCl(s)$\n\nThe equilibrium constant for this reaction,\n$K_{e} = \\ \\frac{1}{\\left( a_{{Ag}^{+}}a_{{Cl}^{-}} \\right)}$ is quite\nlarge, so the equilibrium product of the ion activities, proportional to\ntheir concentrations, is quite small. In the laboratory, the above\nreaction could occur as a result of adding hydrochloric acid to a silver\nnitrate solution. The accompanying $H^{+}(aq)$ \\[or ${H_{3}O}^{+}(aq)$\\]\nand ${NO_{3}}^{-}(aq)$ ions in solution are not directly involved in the\nsilver chloride precipitation reaction so are not shown in reaction\nrepresented by .\n\nThe above ionic equilibria in the $AgCl(s) - H_{2}O$ system is not only\nimportant for understanding this electrolyte system, but also critical\nin electrochemical systems in which *Ag(s)* undergoes a valence change\nat one electrode and reacts with a ${Cl}^{\u2013}(aq)$ ion to produce\n*AgCl(s)*, and an electron that is externally transported finite\ndistances to another electrode. The *oxidation* reaction occurs at the\nAg/AgCl electrode (*anode* half-cell reaction where electrons are\n*added* into the system)\n\n*Eq. 8\u20113* $Ag(s)\\ \\ + \\ {Cl}^{-}(aq)\\ \\ = \\ \\ AgCl(s)\\ \\ + \\ \\ e^{-}$\n\nA *reduction* reaction occurs at the other electrode (*cathode*\nhalf-cell reaction where electrons are *consumed* by the reaction)\n\n*Eq. 8\u20114* $\\frac{1}{2}{Cl}_{2}(g) + \\ \\ e^{-} = \\ {Cl}^{-}(aq)\\ \\ $\n\nThe *net cell reaction* results in the formation AgCl(s) from its\nelements\n\n*Eq. 8\u20115* $Ag(s)\\ \\ + \\frac{1}{2}{Cl}_{2}\\ (g)\\ = \\ \\ AgCl(s)$\n\nWithout knowledge of the physical system under which the reaction is\noccurring, it would not be possible to know if reaction of was a result\nof chlorine gas reacting directly with Ag(s), or if the reaction was\npart of an electrochemical cell with a transport of electrons and ions\nover finite distances. The addition of the two half-cell reactions gives\nthe *net cell reaction*, which does not show electrons as either\nreactant or product species and may or may not include ionic species in\nthe reaction*.* A schematic of an electrochemical cell for the above\nsystem is shown in Figure 8\u20111.\n\nFigure \u2011: Schematic diagram of an electrochemical cell consisting of a\nchlorine electrode and a silver-silver chloride electrode.\n\nOxidation and reduction can occur in electrolyte reactions without\ncreating an electrochemical cell. This is the case when chlorine gas\nreacts directly with silver on a Ag(s) surface. Reaction of above is the\nnet reaction for this process, but the electrons produced from the\noxidation of Ag(s) are not transported over finite distances before\ncombining with Cl2(g) in its reduction to Cl\u2013(aq).\nNo anode or cathode half-cell reactions exist in this system. The\nelectrons and ions involved in the reaction move only over atomic-scale\ndistances.\n", "metadata": {}}, {"id": "9758eaea", "cell_type": "markdown", "source": "### Fuel cells\n\nFuel cells are devices to convert chemical energy to electricity and\nheat through electrochemical reactions with the fuel and oxygen supplied\nto the anode and cathode, respectively. Typical ions migrating through\nthe electrolyte are $H^{+}$, ${OH}^{-}$, ${CO}_{3}^{2 -}$, and\n$O^{2 -}$. In fuel cells with $H^{+}$ as migrating ions, H2\nmolecules are dissociated into $H^{+}$ on the anode, which are combined\nwith O2 on the cathode to form H2O and release\nheat with the half cell and the net cell reactions in their simplest\nform as shown by for the anode, for the cathode, and for the net cell,\nrespectively. Commonly used electrolytes are polymer and phosphoric\nacid, and both anode and cathode reactions are facilitated by catalyst,\ntypically platinum. The thermodynamic limit of power which can be\ngenerated by the fuel cell is represented by\n\n*Eq. 8\u201163*\n$w = - \\mathrm{\\Delta}G = - \\mathrm{\\Delta}_{\\ }^{0}G_{cell} + RTln\\left( P_{H_{2}}P_{O_{2}}^{1/2} \\right)$\n\nFor fuel cells with anions as migrating ions, the anions are generated\non the cathode with H2O formed and heat generated on the\nanode. Their representative cathode reactions are\n\n*Eq. 8\u201164* $\\frac{1}{2}O_{2} + H_{2}O + 2e^{-}{= 2OH}^{-}$\n\n*Eq. 8\u201165* ${\\frac{1}{2}O}_{2} + CO_{2} + 2e^{-} = {CO}_{3}^{2 -}$\n\n*Eq. 8\u201166* $\\frac{1}{2}O_{2} + 2e^{-} = O^{2 -}$.\n\nThe anode reaction for is the reaction represented by , operating at low\ntemperatures and using catalyst for both electrodes. The anode reactions\nfor and are\n\n*Eq. 8\u201167* ${CO}_{3}^{2 -} + H_{2} = H_{2}O + CO_{2} + 2e^{-}$\n\n*Eq. 8\u201168* $O^{2 -} + H_{2} = H_{2}O + 2e^{-}$\n\nrespectively. To enable the diffusion of ${CO}_{3}^{2 -}$ and $O^{2 -}$\nthrough the cathode and the electrolyte, both fuel cells are operated at\nrelative high temperatures, with the former typically in molten\ncarbonate solutions and the latter through solid oxides. Due to the high\noperating temperatures, fuels are converted to hydrogen within the fuel\ncell itself by a process called internal reforming, removing the need\nfor precious-metal catalyst and enabling the use of a variety of fuels.\nThe net cell reaction for all three fuel cells is the same as in the\ncase of $H^{+}$, represented by .\n", "metadata": {}}, {"id": "69d8c384", "cell_type": "markdown", "source": "### Galvanic protection\n\nA galvanic reaction takes place between two different materials at the\ntwo respective electrodes, each with different tendency to hold on to\nelectrons. Consider the following electrochemical cell used to protect\nCu tanks against oxidation by using a \u201csacrificial\u201d Fe electrode\n\nanode solution cathode\n\nFe(s) \\| Fe2+ \\| SO4= \\|\nCu2+\\| Cu(s) \\|\n\nCathode reduction: Cu2+ + 2 e- = Cu \u03b5o\n(volts) = 0.34\n\nAnode oxidation: Fe(s) = Fe2+ + 2 e- \u03b5o (volts) =\n0.44\n\nNet reaction: Cu2+ + Fe =\\> Cu + Fe2+\n\u03b5o (volts) = 0.78\n\nIf the cell has a direct connection between the electrodes, i.e. it has\na short circuit: \u2206G \u2192 0 and thus \u03b5cell \u2192 0. Since the value\nof \u03b5ocell \\> 0 for the net cell reaction, the equilibrium\nconstant *Ke* \\> 1, which means\n\\[Fe2+\\]/\\[Cu2+\\] \\> 1. By assuming an ideal\nelectrolyte solution, the activities in *Ke* can be\nrepresented by concentrations (in molar concentration units), assuming\nthat solid Fe and Cu are present at unit activities. If the electrodes\nof Cu and Fe are short circuited while in contact with the same\n\"electrolyte solution\", the final equilibrium concentrations can be\ncalculated by the standard equation,\n\n*Eq. 8\u201161* *Ke* = \\[Fe2+\\]/\\[Cu2+\\] =\nexp (-\u2206oG/RT)\n\nor, using the Nernst equations,\n\n*Eq. 8\u201162* \u03b5 = \u03b5o -(RT/z f) ln (\\[Fe2+\\]/\\[Cu2+\\])\n= 0\n\nand the above standard cell potential:\n\\[Fe2+\\]/\\[Cu2+\\] = 2.4\u20221026\n\nWith this large ratio it can be seen that the tendency to produce\nCu+2 ions, i.e., the tendency to corrode the Cu(s), is\nextremely small if a sacrificial Fe electrode is configured in an\nelectrochemical cell with the Cu tank.\n", "metadata": {}}, {"id": "1ee6e949", "cell_type": "markdown", "source": "### Half cell potentials\n\nWhen electron current flows between electrodes, reactions are occurring\nat the electrodes and concentration gradients causing polarization\ndevelop around the electrodes. These gradients result in extraneous\npotentials to occur at the electrodes. In such cases cell equilibrium is\nnot established and measured cell potentials are not those for true\npartial equilibrium. If a cell is short-circuited with the electrodes\nconnected by a conductor, current will flow until the external potential\nbecomes zero, i.e. \u03b5ext = 0, and equilibrium is established\nwith same conditions as non-electrochemical systems. If an external\npotential, \u03b5ext, is applied to the cell, chemical reactions\noccur until the cell potential balances to \u03b5ext, and no\ncurrent flows. This potential is called open-circuit voltage (OCV) in\nthe literature. It is important to realize that OCV includes all\nreactions that occur on the electrode surface when the electrode is in\ncontact with the electrolyte, such as passivation discussed in Chapter .\nPartial equilibrium in a cell is achieved when the cell potential is\nbalanced by an applied external potential. In such partial equilibrium\ncases, equilibrium thermodynamic analyses can be used even though the\ncell potential is not zero, i.e. \u03b5cell \u2260 0. This\ndifferentiates electrochemical systems from other equilibrium systems\ndiscussed previously.\n\nThe number of electrons involved in a net cell reaction is important in\nrelating cell potential and the Gibbs energy change for the cell\nreaction. As will be illustrated later in this section, this number\ndenotes the number of electrons involved in the half-cell reactions that\nwere added to yield the net cell reaction. The electrical work achieved\nby the transport of an electrical charge through a cell potential can be\nwritten as\n\n*Eq. 8\u201120* $w = z\\ f\\ \\varepsilon$\n\nwhere *z* represents the moles of electrons in cell reaction, *f* the\nFaraday constant equal to 96,485 J/V/mole-electron, and *\u03b5* the\npotential difference, often referred as electromotive force (emf) in the\nliterature. For a system at constant temperature, pressure, and\ncomposition, this work is the same as the Gibbs energy difference\nbetween the two electrodes, i.e.\n\n*Eq. 8\u201121* $- \\Delta G = w = zf\\varepsilon$\n\nwhere the negative sign is because the system does work to the\nsurrounding when the Gibbs energy of the system is decreased. When the\napplied external potential is larger than the cell potential, the\nsurrounding does work to the system, and a common example is the\ncharging of a battery. Thermodynamic relations discussed in previous\nchapters can thus be directly applied to electrochemical systems with\nsome common equations shown in .\n\nTable \u2011: Thermodynamic Equations for Electrochemical Cells\n\n*\u2206G = -z f \u03b5*\n\n*\u2206S = - (\u2202\u2206G/\u2202T)P = + z f (\u2202\u03b5/\u2202T)P*\n\n*\u2206H = \\[\u2202(\u2206G/T)/ \u2202(1/T)\\]P = - z f \\[\u2202(\u03b5/T)/\u2202(1/T)\\]P = z f \\[T(\u2202\u03b5/\u2202T)P\n\u2013 \u03b5\\]*\n\n*\u2206CP = (\u2202\u2206H/\u2202T)P = T z f (\u22022\u03b5/\u2202T2)P*\n\nA half-cell reaction potential cannot be measured directly, only its\npotential relative to another half-cell reaction. By convention, a\nstandard half-cell potential is measured relative to the standard\nhydrogen half-cell reduction reaction at 25oC (298K) and 1\nbar, which has a defined standard potential of zero volts,\n\n*Eq. 8\u201122* H+(aq, a=1) + e- = 1/2 H2(g,\n1 bar)\n\nwith \u03b5o (H+/H2,g) = 0.00 volts. The\nstandard half-cell reduction reactions of metals at 25oC are\nfor the general reaction\n\n*Eq. 8\u201123* Mz+(aq, a=1) + z e- = M(s)\n\nwith \u03b5o (Mz+/M) volts. Half-cell reactions with\nthe most positive standard electrode potentials have a tendency to\nspontaneously proceed toward reduction (cathode reactions). Half-cell\nreactions with the most negative standard electrode potentials have a\ntendency to spontaneously proceed toward oxidation (anode reactions).\n\nConsider, for example, a cell made up of a standard hydrogen electrode\nand a standard zinc electrode with \u03b5o\n(H+/H2,g) = 0.00 volts and \u03b5o\n(Zn2+/Zn) = -0.762 volts. Thus, the H+ would tend\nto be reduced, and the zinc metal would tend to be oxidized, and the\nspontaneous reaction if all species had unit activities would be\n\n*Eq. 8\u201124* 2 H+(aq, a=1)+ Zn = H2(1 bar) +\nZn2+(aq, a=1)\n\nwith \u03b5ocell = 0.762 volts and\n$\\mathrm{\\Delta}_{\\ }^{0}G = - 2*96485*\\varepsilon_{cell}^{0}\\ $. The\ncathode half-cell reaction would be the same as , while the anode\nhalf-cell reaction would be\n\n*Eq. 8\u201125* Zn = Zn2+(aq, a=1) + 2 e-\n\nWhen ion concentrations and H2 gas do not all have unit\nactivities, the Gibbs energy and cell potential of the cell reaction, ,\nbecomes\n\n*Eq. 8\u201126*\n$\\mathrm{\\Delta}G = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{a_{{Zn}^{2 +}}P_{H_{2}}}{\\left( a_{H^{+}} \\right)^{2}}$\n\n*Eq. 8\u201127*\n$\\varepsilon_{cell} = \\varepsilon_{cell}^{0} - \\frac{RT}{zf}\\ln\\frac{a_{{Zn}^{2 +}}P_{H_{2}}}{\\left( a_{H^{+}} \\right)^{2}}$\n\nThe standard reduction potentials of some common metals at\n25oC are given in .\n\nTable \u2011: Standard reduction potentials of some common metals\n\nA cell reaction can be established by different half-cell reactions. For\nexample, the following reaction can be derived from two different cells\n\n*Eq. 8\u201128* $3\\ {Fe}^{2 + \\ }\\ = \\ \\ 2\\ {Fe}^{3 +}\\ \\ + \\ \\ Fe(s)$\n\ncell A\n\n*Eq. 8\u201129* $3{Fe}^{2 +} + \\ 6\\ e - \\ = \\ \\ 3Fe(s)$ \u03b5o1 =\n-0.440 V\n\n*Eq. 8\u201130* $2\\ Fe(s)\\ \\ = \\ \\ 2\\ {Fe}^{3 +}\\ \\ + \\ 6\\ e -$\n\u03b5o2 = +0.036 V\n\ncell B\n\n*Eq. 8\u201131* $2{Fe}^{2 +}\\ = \\ 2{Fe}^{3 + \\ \\ } + 2\\ e^{-}$\n*\u03b5o4 = -0.772 V*\n\n*Eq. 8\u201132* ${Fe}^{2 + \\ }\\ + \\ 2\\ e - \\ \\ = \\ \\ Fe(s)$ *\u03b5o5\n= -0.440 V*\n\nBoth give the same net reaction shown by , but with 6 and 2 electrons\nand standard cell potentials being \u03b5ocell A = -0.404 V and\n\u03b5ocell B = -1.212 V, respectively. However, the standard\nGibbs energies of both cells are the same, i.e.\n\n*Eq. 8\u201133* $\\mathrm{\\Delta}_{\\ }^{0}G$*cell A = -6 f (-0.404)\n= + 2.424 f*\n\n*Eq. 8\u201134* $\\mathrm{\\Delta}_{\\ }^{0}G$*cell B = -2 f (-1.212)\n= + 2.424 f*\n\nIt is shown that $\\mathrm{\\Delta}_{\\ }^{0}G$ values are independent of\nhalf-cell reactions and depend only on the net reaction because the net\nreaction is neutral in electron and balanced in mass.\n", "metadata": {}}, {"id": "976f0090", "cell_type": "markdown", "source": "### Ion transport membranes\n\nIon transport membranes (ITMs) are ceramic membranes conducting both\nelectrons and oxygen ions, but no other species. The chemical potential\ndifference of oxygen between two sides of a membrane provides the\ndriving force for oxygen to diffuse through the membrane. Commonly used\nITM oxides include perovskite and fluorite, with the chemical formula of\nABO3 and AO2, respectively, typically with more\nthan one elements in the A-site and / or the B-site to tailor the\nelectron and ionic conductivities. Key thermodynamic properties of ITM\noxides are their stability in service environments, vacancy\nconcentrations in the oxygen and cation sites, valances of cations. On\nthe high oxygen partial pressure side, the reaction is as the following\n\n*Eq. 8\u201169* $\\frac{1}{2}O_{2} + 2e^{-} = O^{2 -}$\n\nAt the same time the number of oxygen vacancy is reduced, resulting in a\nlower concentration of oxygen vacancy and higher oxygen activity in the\noxide on the high oxygen partial pressure side. On the low oxygen\npartial pressure side, the reaction is reversed to produce oxygen\nmolecules, i.e.\n\n*Eq. 8\u201170* $O^{2 -} = \\frac{1}{2}O_{2} + 2e^{-}$\n\nThis reaction results in higher oxygen vacancy concentration and lower\noxygen activity. At both sides, the charge neutrality is compensated by\nthe valance changes of cations, resulting in the electron flow in the\nopposite direction of oxygen diffusion. The ionic conductivity is\ndictated by the oxygen transportation across the membrane with the\ndriving force of the following net reaction\n\n*Eq. 8\u201171*\n$\\frac{1}{2}O_{2}\\left( P_{high} \\right) = \\frac{1}{2}O_{2}\\left( P_{low} \\right)$\n\n*Eq. 8\u201172*\n$\\mathrm{\\Delta}G = 0.5RTln\\left( \\frac{P_{low}}{P_{high}} \\right)$.\n\nThe oxygen transportation is closely related to the concentration of\noxygen vacancy in the membrane, which is obtained by minimizing the\nGibbs energy of the phase under given temperature and oxygen partial\npressure conditions. High vacancy concentrations can be obtained by\ncation dopants with lower valances or small energy differences between\nvarious valance states. However, at the same time, high vacancy\nconcentrations reduce the thermodynamic stability of the membrane, which\nmay result in its decomposition into more stable phases.\n", "metadata": {}}, {"id": "90efadcf", "cell_type": "markdown", "source": "### Metastability and passivation\n\nOur modern industrial society is built on various metals such as Fe, Ni,\nAl, Ti, and Zr alloys which are reactive, but exhibit extraordinary\nkinetic stabilities in oxidizing environments due to the existence of a\nthin reaction product film on the surface. This film effectively\nisolates the metal from the corrosive environment, a phenomenon called\npassivation. One interesting experiment by Faraday in 1836 who reported\nthat iron corrodes freely in dilute nitric acid, while in concentrated\nnitric acid, no reaction apparently occurred. To understand this\nphenomenon, let us exam a simple, schematic Pourbaix diagram for the\niron-water system shown in .\n\nFigure \u2011: Schematic Pourbaix diagram for iron illustrating the\nresolution of the Faraday paradox in the corrosion of iron in nitric\nacid \\[61\\]. Lines (a) (b), and (c) correspond to the equilibria: (a)\n$H^{+} + e^{-} = 1/2H_{2}$; (b) $O_{2} + 4H^{+} + 4e^{-} = 2H_{2}O$; (c)\n$NO_{3}^{-} + 3H^{+} + 2e^{-} = HNO_{2} + H_{2}O$, respectively\n\nFor iron in deaerated acid solution, the partial anodic and cathodic\nreactions are Line 1 $\\left( Fe/{Fe}^{2 +} \\right)$ and Line (a),\nrespectively, resulting in a corrosion potential that lies between Lines\n$1$ and $a$. In oxygenated (aerated) solutions, the corrosion potential\nmay lie between Lines $1$ and $b$, because the reduction of oxygen is a\npossible (likely) cathodic reaction. Since dilute HNO3 is\nonly a weak oxidizing agent, the principal cathodic reaction was most\nlikely hydrogen evolution, and hence the corrosion potential is expected\nto lie between Lines $1$ and $a$ at relatively high pH, as shown. Since\nthe $Fe/{Fe}^{2 +}$ reaction is relatively fast compared with\n$H^{+}/H_{2}$ on iron, if the corrosion potential is situated below the\nextension of Line 2 (Fe/Fe3O4) into the\n${Fe}^{2 +}$ stability region, Fe3O4 cannot form\non the surface, even as a metastable phase.\n\nOn the other hand, concentrated HNO3 is a strong oxidizing\nagent due to the reaction:\n$NO_{3}^{-} + 3H^{+} + 2e^{-} = HNO_{2} + H_{2}O$, so that the corrosion\npotential can lie anywhere between Lines $1$ and $c$ at low pH. Since\nReaction (2) is likely to be fast, the corrosion potential will be high\nand certainly will be more positive than the extension of Line 2 into\nthe stability region for ${Fe}^{2 +}$ at low pH. Therefore,\nFe3O4 becomes metastable and can form between the\naqueous solution and iron. The thickness of this\nFe3O4 film depends on its dissolution rate into\nthe aqueous solution and its growth rate at the interface with iron,\nwhich depends on the diffusion of ionic species across the film. Its\nexistence results in passivity and the observed kinetic inactivity of\niron in this medium. When the potential becomes even more positive above\nthe extension of Line $3$, Fe2O3 may form on\nFe3O4 as an additional metastable phase, resulting\nin the commonly observed bilayer structure.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/applications_to_electrochemical_systems/index.ipynb b/psu410/src/applications_to_electrochemical_systems/index.ipynb new file mode 100644 index 0000000..080631b --- /dev/null +++ b/psu410/src/applications_to_electrochemical_systems/index.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "0488302e", "cell_type": "markdown", "source": "# Applications to electrochemical systems\n\nThe two basic types of chemical processes involving ions as reactant\nand/or product species are electrolyte reactions and electrochemical\nreactions. *Electrolyte reactions* are accompanied by the *atomic-scale\nmovement* of ionic species and possibly electrons. Chemical changes that\nproduce changes in valence and *electron and ion transport over finite\ndistances* constitute an area of science termed *electrochemistry*. The\nlatter chemical changes occur in an electrochemical cell comprised of\ntwo electrodes, an anode and a cathode, which are coupled by an\nelectrolyte and an external electron conductor. Most thermodynamic\nconcepts and analyses described in previous chapters remain unchanged\nwhen applied to electrochemistry, but the analysis of electrochemical\nsystems does require some new terminology, new definitions, and new\nconventions. The primary focus of this chapter is on applications of\nthermodynamics to electrochemical reactions that involve either aqueous\nelectrolyte solutions or solid state electrolytes. Since all\nelectrochemical systems include ionized species as reactant and/or\nproduct species, electrolyte reactions will also be discussed.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/calpahd_modeling_of_thermodynamics/importance_of_lattice_stability.ipynb b/psu410/src/calpahd_modeling_of_thermodynamics/importance_of_lattice_stability.ipynb new file mode 100644 index 0000000..0f113b4 --- /dev/null +++ b/psu410/src/calpahd_modeling_of_thermodynamics/importance_of_lattice_stability.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "ca973b9e", "cell_type": "markdown", "source": "## Importance of lattice stability\n\nFor modeling of Gibbs energy of individual phases, it is necessary to\ndefine the values of $_{\\ }^{0}G_{i}$ in . However, the independent\ncomponent *i* may not be stable in the crystal structure of the phase\nunder consideration, so its Gibbs energy could not be obtained directly\nfrom experiments and must be estimated with respect to the Gibbs energy\nof their stable crystal structure. In the pioneering work by Kaufman\n\\[47\\], this Gibbs energy difference was termed as lattice stability and\nobtained through extrapolations in either temperature-pressure or\ntemperature-composition phase diagrams. It is evident from that the\nvalues of $_{\\ }^{0}G_{i}$ and $_{\\ }^{M}G$ jointly contribute to the\nGibbs energy of the solution, and Kaufman had to simplify the treatment\nof $_{\\ }^{M}G$ in order to show the importance of the concept of\nlattice stability. Using ideal or regular solution models, Kaufman was\nable to define the lattice stability for pure elements and remarkably\nreproduce many features of binary phase diagrams by introducing the\ninteraction parameters afterwards.\n\nOver the years, there had been various revisions of lattice stability\nvalues for common crystal structures \\[48\\], and every revision\nnecessitates the re-evaluation of interaction parameters in the solution\nphase shown in . It was until the lattice stability values established\nby the Scientific Group Thermodata Europe (SGTE) \\[52\\] that the\ndevelopment of binary thermodynamic models using the same thermodynamic\nmodels of pure elements became possible, and those binary models thus\ndeveloped in different groups around the world can be combined to create\nthermodynamic models of ternary and multi-component systems. Clearly,\nany further modifications of the SGTE pure element database will require\nthe re-modeling of all binary and ternary systems in which the models of\npure elements are changed. This challenge is briefly addressed in the\nlater part of this chapter.\n\nA less addressed issue is the Gibbs energy of end-members in\nnon-stoichiometric compounds, i.e. , where each sublattice contains only\none element. In case all sublattices are occupied by the same element,\nit is the lattice stability of the elements in the structure of the\ncompound. Since the stable composition ranges of non-stoichiometric\ncompounds are typically small, the existing method cannot be used to\nreliably evaluate the Gibbs energy of end-members, and currently there\nis not a commonly accepted lattice stability database for compounds.\nMost values used in the existing databases have been either roughly\nestimated or computed from first-principles calculations. Such a\nstandard database is highly desirable in order to make various models of\ncompounds compatible.\n\nIn an effort to compare the lattice stability from the CALPHAD models\nand the first-principles calculations, Wang et al. \\[53\\] systematically\ncalculated the total energies of 78 pure elements at zero Kelvin in the\nface-centered-cubic (fcc), body-centered-cubic (bcc), and\nhexagonal-close-packed (hcp) crystal structures using the projector\naugmented-wave (PAW) method within the generalized gradient\napproximation (GGA). The calculated values are compared with the values\nin the SGTE database as shown in and . For non-transition metal\nelements, the differences between the SGTE data and the PAW-GGA data are\ntypically around 1\u223c2 kJ/mole-of-atoms or less, while for some transition\nmetal elements, the differences can be quite large, for example, as high\nas about 54 kJ/mole-of-atoms for and about 40 kJ/mole-of-atoms for . and\npresent the differences between the PAW-GGA data and the SGTE data, for\nelements from the Ti group to the Ni group, respectively.\n\nTable \u2011: Lattice stability Ebcc-fcc (kJ/mole-of-atoms).\n\nTable \u2011: Lattice stability Ehcp-fcc (kJ/mole-of-atoms).\n\nFigure \u2011: Lattice stability difference between bcc and fcc, , for\nselected elements between PAW-GGA and SGTE\n\nFigure \u2011: Lattice stability difference between hcp and fcc, , for\nselected elements between PAW-GGA and SGTE\n\nThe large differences between the first-principles calculations and the\nSGTE data could partly be attributed to the instability of the\nhigher-energy phases, the entropies of which at finite *T* become\nabnormal. The lattice instabilities along the tetragonal transformation\npath between fcc and bcc structures with the continuous change of the\nc/a ratio defined in a bcc-based tetragonal lattice are demonstrated for\nbcc Mo, Ta, W in and for fcc Al, Cu, Ni in . It is shown that the fcc\nstructure of bcc Mo, Ta, and W is a local maximum with respect to the\ntetragonal transformation, and the higher the maximum is, the larger the\ndiscrepancy between the SGTE data and the present PAW-GGA data, while\nfor fcc Al, Cu, Ni, the bcc structure is at a local maximum. Similarly,\nthe lattice instabilities along the tetragonal transformation path for\nthe hcp metals Ru and Os as shown in . The behavior of energy against\nc/a ratio of these two hcp metals is very similar to those of fcc\nelements.\n\nFigure \u2011: Total energy, , along the Bain deformation path between bcc\nand fcc for Mo, Ta, and W.\n\nFigure \u2011: Total energy, , along the tetragonal transformation path\nbetween bcc and fcc for Ni, Al, and Cu.\n\nFigure \u2011: Total energy, , along the tetragonal transformation path for\nRu and Os\n\nIt can be concluded that a fcc structure for elements with bcc being the\nground state or a bcc structure for elements with fcc being the ground\nstate, is unstable with respect to the tetragonal transformation. For an\nunstable structure, the harmonic description of its vibrational entropy\nis thermodynamically incorrect since the potential surface seen by the\nlattice ion can no longer be approximated by a parabola. If an unstable\nstructure of a pure element is stabilized at high temperatures, its\nentropy has to be abnormal. The instability issue has been recently\naddressed by ab initio molecular dynamics simulations at high\ntemperatures using W as an example \\[54\\], which is beyond the scope of\nthe book and thus not discussed here.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/calpahd_modeling_of_thermodynamics/index.ipynb b/psu410/src/calpahd_modeling_of_thermodynamics/index.ipynb new file mode 100644 index 0000000..0be6e16 --- /dev/null +++ b/psu410/src/calpahd_modeling_of_thermodynamics/index.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "ab0f51db", "cell_type": "markdown", "source": "# CALPAHD modeling of thermodynamics\n\nCALPHAD modeling of thermodynamics was pioneered by Kaufman \\[47\\] and\nhas been reviewed in details by Saunders and Miodownik \\[48\\] and Lukas,\nFries and Sundman \\[49\\]. Information on features of software tools for\nCALPHAD modeling can be found at two series of publications in the\nCALPHAD journal \\[50-51\\]. The key feature of the CALPHAD method is the\nmodeling of Gibbs energy of individual phases using both thermodynamic\nand phase equilibrium data. The main significances of the CALPHAD method\nare as follows\n\n1. It enabled the development of the concept of lattice stability, i.e.\n the energy difference of a pure element with stable and non-stable\n crystal structures;\n\n2. The Gibbs energy expression of each phase covers the full\n temperature, pressure, and composition spaces including both stable\n and non-stable regions of the phase. This enables the evaluation of\n the Gibbs energy of a system as a function of non-equilibrium state,\n i.e. with \u03be as an independent variable;\n\n3. Thermodynamic data are usually obtained by measurements of heat such\n as enthalpy of transition and heat capacity as discussed in Chapter\n , which bear large uncertainties typically in the range of\n kilojoules per mole-of-atom. On the other hand, phase equilibrium\n data as discussed in Chapter , though more accurate, only contain\n information on compositions of phases at equilibria, i.e., the\n relative Gibbs energy of phases at equilibrium. The combination of\n these two sets of data is foundational in CALPHAD modeling that\n allows for accurate modeling of thermodynamic properties of\n individual phases and reliable calculations of phase stability and\n driving forces;\n\n4. It provides a framework to model thermodynamic properties of\n multi-component systems of industrial importance, enabling the\n computational materials design. It has also been extended to model a\n range of properties of individual phases in multi-component systems\n such as diffusion coefficients, elastic coefficients, and thermal\n expansion, supplying input data for computational simulations of\n phase transformations during materials processing.\n\nIn this chapter, the basics of CALPHAD modeling of Gibbs energy of\nindividual phases are presented. For detailed implementations in various\nsoftware packages and modeling procedures, readers are referred to the\nreferences listed above.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_pure_elements.ipynb b/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_pure_elements.ipynb new file mode 100644 index 0000000..ede1115 --- /dev/null +++ b/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_pure_elements.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "981c197e", "cell_type": "markdown", "source": "## Modeling of pure elements\n\nIn modeling the Gibbs energy of pure elements in the SER structure in\nterms of , the coefficients in are evaluated using the heat capacity\ndata, $b^{'}$ in is evaluated by the value of $S_{298.15}$, and $a$ in\nis evaluated by $H_{298.15}^{SER} = 0$, respectively. For the high\ntemperature phase, the enthalpy of transformation from the low\ntemperature phase to the high temperature phase,\n${\\mathrm{\\Delta}H}_{trans}^{}$, can be measured by calorimetry methods\ndiscussed in Chapter , and the entropy of transformation,\n${\\mathrm{\\Delta}S}_{trans}^{}$, is then calculated using the\nequilibrium condition of equal Gibbs energy of the two phases, i.e.\n\n*Eq. 6\u20111*\n${\\mathrm{\\Delta}S}_{trans}^{} = \\frac{{\\mathrm{\\Delta}H}_{trans}^{}}{T_{trans}^{}}$\n\nwhere $T_{trans}^{}$ is the transition temperature.\n${\\mathrm{\\Delta}H}_{trans}^{}$ and ${\\mathrm{\\Delta}S}_{trans}^{}$ are\nthen used to evaluate the integration constants, $b^{'}$ and $a$, in the\nplace of $S_{298.15}$ and $H_{298.15}^{SER}$ for the structure in the\nSER state.\n\nThis works well for the stable temperature range of each phase. However,\nthere is an issue in extrapolation above and below the melting\ntemperature (Tm). It is known that the heat capacity of the\nsolid phase, , increases with temperature, while that of the liquid\nphase, , is typically constant. The extrapolation of the Gibbs energy of\na solid phase to above its melting temperature can result in the solid\nphase becoming more stable than the liquid phase at high temperatures.\nBy the same token, the extrapolation of the Gibbs energy of a liquid\nphase to below its melting temperature can result in the liquid phase\nbecoming more stable than the solid phase at low temperatures. To\naddress this problem, it is proposed by SGTE that the heat capacity of\nthe solid phase approaches that of the liquid at high temperatures, and\nthat of the liquid phase approaches that of the solid phase at low\ntemperatures using the following equations\n\n1. for solid at T\\>Tm\n\n*Eq. 6\u20112*\n\n*Eq. 6\u20113*\n\n2. for liquid at T\\m\n\n*Eq. 6\u20114*\n\n*Eq. 6\u20115*\n\nAs an example, the heat capacity of solid fcc Al and liquid Al in the\nSGTE pure element database is plotted in . It can be seen that the heat\ncapacity of fcc Al approaches that of liquid Al at high temperatures,\nwhile the heat capacity of liquid Al approaches that of fcc Al at low\ntemperatures. It ensures that the liquid Al is stable at high\ntemperatures, and fcc Al is stable at low temperatures. However, this\nsimple model for liquid is often not satisfactory, in comparison with\navailable experimental data in supercooled liquid, particularly those\nsystems with glass transitions. New models are thus needed and are being\ndeveloped in the CALPHAD community.\n\nFigure \u2011: Heat capacity of fcc Al solid and liquid as a function of\ntemperature\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_random_solution_phases.ipynb b/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_random_solution_phases.ipynb new file mode 100644 index 0000000..9b271cb --- /dev/null +++ b/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_random_solution_phases.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "a51f0d6e", "cell_type": "markdown", "source": "## Modeling of random solution phases\n\nDepending on the degree of short-range ordering in a solution phase,\nvarious Gibbs energy models are available as discussed in Chapter . When\nthe short-range ordering is weak, it can be accounted for by the\ncomposition-dependence of excess Gibbs energy in a binary system in\nterms of the Redlich-Kister polynomial as follows\n\n*Eq. 6\u20117*\n\nwhere the interaction parameters, , can be temperature dependent or even\nhave contributions from heat capacity in the form of when data is\navailable. shows that and are symmetrical with respect of composition,\nwhile is asymmetrical. Their individual contributions to the excess\nGibbs energy are shown in with all interaction parameters being -30,000\nJ/mole-of-atoms.\n\nFigure \u2011: Contributions of interactions parameters to the excess Gibbs\nenergy\n\nIt can be seen in that even though all interaction parameters are\nnegative, the asymmetrical feature of results in the curvature change in\nthe excess Gibbs energy as a function of composition. This indicates the\ntendency to form a miscibility gap at low temperatures. The interaction\nparameters are to be evaluated from the data of enthalpy, entropy, and\nheat capacity of mixing. The experimental data on enthalpy of mixing are\navailable for the liquid phase in some systems, but typically very\nlimited for solid solution phases. The first-principles calculations can\npredict the enthalpy, entropy, and heat capacity of mixing in solid\nsolution phases using the dilute solution approach with one solute atom\nin a supercell and the CPA/CE/SQS approach for concentrated solutions as\ndiscussed in Chapter . It demonstrates again that the interaction\nparameters and the lattice stability jointly determine the Gibbs energy\nof an individual phase. The change of lattice stability requires the\nre-evaluation of interaction parameters.\n\nFor individual phases with strong short-range ordering, quasichemical or\nassociated models can be used. As discussed in Chapter , with fixed\ncomposition in the system, the amounts of various chemical bonds or\nassociates are related through the mass conservation in the system and\nare calculated through the minimization of Gibbs energy of the phase\nwith given temperature, pressure, and the amount of each independent\ncomponents. The model parameters include the formation energy of bonds\nor associates and interaction between various bonds or associates,\nnoting that pure elements can be considered as the simplest associates.\nThe interactions between pure elements can be predicted from\nfirst-principles calculations as mentioned above, but currently there\nare no efficient approaches to predict the interactions between\nassociates from first-principles calculations.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_solution_phases_with_longrange_ordering.ipynb b/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_solution_phases_with_longrange_ordering.ipynb new file mode 100644 index 0000000..5b0e43d --- /dev/null +++ b/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_solution_phases_with_longrange_ordering.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "3bc1eecc", "cell_type": "markdown", "source": "## Modeling of solution phases with long-range ordering\n\nThe commonly used Gibbs energy model is shown in Chapter with the\ncrystal lattice divided into sublattices, often referred to as the\ncompound energy formalism \\[55\\]. The Gibbs energy of end-members\nrepresented by plays the same important role for solution phases with\nsublattice as the lattice stability for random solution phases. The\nend-members are modeled in the same way as the stoichiometric phases\ndiscussed in Chapter . The enthalpy and entropy of mixing in each\nsublattice can be predicted by first-principles calculations using the\ndilute solution and SQS approaches discussed in Chapter and modeled in\nthe same way as the random solution discussed in Chapter .\n\nIt is important to realize that with a simple two-sublattice model of\n(A,B)a(C,D)b, the miscibility gap can easily form\neven without any interaction parameters when the Gibbs energy of\nend-members differ from each other significantly. The contribution of\nend-members to the Gibbs energy of the phase, i.e. , is re-written as\nfollows and schematically shown in ,\n\n*Eq. 6\u20118*\n\nFigure \u2011: Schematic diagrams depicting (a) the concentration square with\nthe site fractions of B and D on the horizontal and vertical axes,\nrespectively, and (b) Gibbs energy reference plan for\n(A,B)a(C,D)b, as represented by .\n\nFrom b, it is evident that there is a strong tendency to form a\nmiscibility gap between the composition sets of\n(A)a(C)b and (B)a(D)b due to\ntheir lower Gibbs energies than the other two end-members. Since it\nwould be rare for all four end-members to have their Gibbs energy values\nequal, the miscibility gap in this type of phases is practically\ninevitable at low temperatures. An example is shown in for the complex\ntitanium niobium carbonitride (Ti,Nb)(C,N). The solid lines parallel to\nthe direction from NbC to TiN are tie-lines. The Gibbs energy values of\nTiC, TiN, NbC and NbN are -144495, -229236, -132324, and -179772\nJ/mole-of-atoms, respectively. The Gibbs energy value of TiN is\nsignificantly lower than the other values, resulting in the tie-lines\noriginating from the TiN corner.\n\nFigure \u2011: Miscibility gap in (Ti,Nb)(C,N) at 1673K with the straight\nlines in the middle of the plot being tie-lines.\n\nThe order-disorder transitions can be similarly described with the\nsimplest case being a two-sublattice model of\n(A,B)a(A,B)b. When the site fractions of A or B in\nboth sublattices are the same, it becomes a random solution model; when\nthey are different, the phase is partially ordered; and when there is\nonly one component in each sublattice, the phase is fully ordered as a\nstoichiometric compound. The Gibbs energy of this phase is obtained from\nas follows\n\n*Eq. 6\u20119*\n\n$G_{mf} = y_{A}^{I}y_{A}^{II}\\ _{\\ }^{0}G_{A:A} + y_{A}^{I}y_{B}^{II}\\ _{\\ }^{0}G_{A:B} + y_{B}^{I}y_{A}^{II}\\ _{\\ }^{0}G_{B:A} + y_{B}^{I}y_{B}^{II}\\ _{\\ }^{0}G_{B:B} + + aRT\\left( y_{A}^{I}\\ln y_{A}^{I} + y_{B}^{I}\\ln y_{B}^{I} \\right) + bRT\\left( y_{A}^{II}\\ln y_{A}^{II} + y_{B}^{II}\\ln y_{B}^{II} \\right) + y_{A}^{II}y_{A}^{I}y_{B}^{I}\\ L_{A,B:A} + y_{B}^{II}y_{A}^{I}y_{B}^{I}\\ L_{A,B:B} + y_{A}^{I}y_{A}^{II}y_{B}^{II}\\ L_{A:A,B} + y_{B}^{I}y_{A}^{II}y_{B}^{II}\\ L_{B:A,B} + y_{A}^{I}y_{B}^{I}\\ y_{A}^{II}y_{B}^{II}L_{A,B:A,B}$\n\nwhere the superscript denotes the sublattice, and column and comma\nseparate sublattices and interaction components, respectively. The\nrelationship between site fraction and over-all atomic fractions in such\na two-sublattice model can be represented by and schematically shown in\n. The two red dashed lines represent the phase with\n$x_{B} = \\frac{a}{a + b}$, but different a/b ratios. Along the red\ndashed lines, the phase can adjust the site fraction to minimize its\nGibbs energy, i.e. it has one internal degree of freedom to be either\ndisordered on the blue diagonal line between A:A and B:B, or ordered at\nanywhere else. The interplay of interaction parameters and site\nfractions is depicted where $L_{A,B:A}$, $L_{A,B:B}$, $L_{A:A,B}$, and\n$L_{B:A,B}$ affect the four sides, and $L_{A,B:A,B}$ influences the\ncenter part.\n\nFigure \u2011: Schematic composition square of\n(A,B)a(A,B)b.\n\nWhen fully disordered with $y_{A}^{I} = \\ y_{A}^{II} = x_{A}$ and\n$y_{B}^{I} = y_{B}^{II} = x_{B}$, becomes\n\n*Eq. 6\u201110*\n\n$$G_{mf} = x_{A}\\left( 1 - x_{B} \\right)_{\\ }^{0}G_{A:A} + x_{A}x_{B}\\ _{\\ }^{0}G_{A:B} + x_{A}x_{B}\\ _{\\ }^{0}G_{B:A} + x_{B}\\left( 1 - x_{A} \\right)\\ _{\\ }^{0}G_{B:B} + + (a + b)RT\\left( x_{A}\\ln x_{A} + x_{B}\\ln x_{B} \\right) + \\ x_{A}x_{B}x_{A}L_{A,B:A} + x_{A}x_{B}x_{B}\\ L_{A,B:B} + x_{A}x_{B}x_{A}\\ L_{A:A,B} + x_{A}x_{B}x_{B}\\ L_{B:A,B} + x_{A}x_{A}x_{B}x_{B}L_{A,B:A,B} = x_{A}_{\\ }^{0}G_{A:A} + x_{B}\\ _{\\ }^{0}G_{B:B} + (a + b)RT\\left( x_{A}\\ln x_{A} + x_{B}\\ln x_{B} \\right) + x_{A}x_{B}\\left\\lbrack \\left(_{\\ }^{0}G_{A:B} +_{\\ }^{0}G_{B:A} -_{\\ }^{0}G_{A:A} -_{\\ }^{0}G_{B:B} \\right) + x_{A}\\left( L_{A,B:A} + L_{A:A,B} \\right) + x_{B}\\ \\left( L_{A,B:B} + L_{B:A,B} \\right) + x_{A}x_{B}L_{A,B:A,B} \\right\\rbrack = (a + b)\\left\\lbrack x_{A}_{\\ }^{0}G_{A} + x_{B}\\ _{\\ }^{0}G_{B} + RT\\left( x_{A}\\ln x_{A} + x_{B}\\ln x_{B} \\right) + x_{A}x_{B}L_{A,B} \\right\\rbrack$$\n\nwith\n\n*Eq. 6\u201111* $_{\\ }^{0}G_{A:A} = (a + b)_{\\ }^{0}G_{A}$\n\n*Eq. 6\u201112* $_{\\ }^{0}G_{B:B} = (a + b)_{\\ }^{0}G_{B}$\n\n*Eq. 6\u201113*\n\n$$L_{A,B} = \\left\\lbrack \\left(_{\\ }^{0}G_{A:B} +_{\\ }^{0}G_{B:A} -_{\\ }^{0}G_{A:A} -_{\\ }^{0}G_{B:B} \\right) + x_{A}\\left( L_{A,B:A} + L_{A:A,B} \\right) + x_{B}\\ \\left( L_{A,B:B} + L_{B:A,B} \\right) + x_{A}x_{B}L_{A,B:A,B} \\right\\rbrack/(a + b)$$\n\nwhere $_{\\ }^{0}G_{A}$, $_{\\ }^{0}G_{B}$, and $L_{A,B}$ are the molar\nGibbs energy of pure A and B and the molar interaction parameter in the\ndisordered solid solution, respectively. It is evident that the\ninteratcation paramter $L_{A,B}$ is fully determined by the parameters\nin the ordered phase, but the parameters in the ordered phase are not\nuniquely determined by the interaction parameters in the disordered\nphase.\n\nDue to crystal symmetry, some of the parameters in are related. For\nexample, in the BCC A2/B2 ordering with a=b=0.5, the BCC corner or\ncenter lattice sites are favored by either one type of atoms, but the\ntwo sublattices are equivalent crystallographically, resulting in\nfollowing relations\n\n*Eq. 6\u201114* $\\ _{\\ }^{0}G_{A:B} = \\ _{\\ }^{0}G_{B:A}$\n\n*Eq. 6\u201115* $L_{A,B:A} = L_{A:A,B}$\n\n*Eq. 6\u201116* $L_{A,B:B} = \\ L_{B:A,B}$\n\nFor more complex ordering of BCC lattice such as B32, D03,\nand L21 shown in with ideal compositions being AB,\nA3B, and A2BC, respectively, more sublattices are\nneeded, noting that the L21 Heusler structure exists in\nternary systems only. To use one model to describe all ordering in the\nBCC lattice, the minimum cluster is an irregular tetrahedron with four\nsublattices as depicted in as discussed in the modeling of the Al-Fe\nsystem \\[56\\]. In such a four sublattice model of\n$(A,B)_{0.25}^{I}(A,B)_{0.25}^{II}(A,B)_{0.25}^{II}(A,B)_{0.25}^{IV}$,\nthe site fractions of A2, B2, B32, and D03, are represented\nby $y_{i}^{I} = \\ y_{i}^{II} = y_{i}^{III} = \\ y_{i}^{IV}$,\n$y_{i}^{I} = \\ y_{i}^{II} \\neq y_{i}^{III} = \\ y_{i}^{IV}$,\n$y_{i}^{I} = \\ y_{i}^{III} \\neq y_{i}^{II} = \\ y_{i}^{IV}$, and\n$y_{i}^{I} = \\ y_{i}^{II} \\neq y_{i}^{III} \\neq \\ y_{i}^{IV}$,\nrespectively. The site fractions of L21 are the same as those\nof D03 except with at least three components.\n\nFigure \u2011: Atomic structures and four sublattice tetrahedrons of BCC\ndisordered and ordered phases.\n\nAnother common ordering phenomenon is in the FCC lattice including the\ndisordered A1 structure and ordered L10 and L12\nstructures as shown in . In the L10 structure, the\nneighboring (001) planes are favored by different atoms, respectively,\nresulting in an ideal composition of AB. While in the L12\nstructure, the corners and faces are favored by different atoms,\nrespectively, resulting in an ideal composition of A3B. In a\nfour sublattice model of\n$(A,B)_{0.25}^{I}(A,B)_{0.25}^{II}(A,B)_{0.25}^{II}(A,B)_{0.25}^{IV}$,\nthe site fractions of A1, L10 and L12 are\nrepresented by $y_{i}^{I} = \\ y_{i}^{II} = y_{i}^{III} = \\ y_{i}^{IV}$,\n$y_{i}^{I} = \\ y_{i}^{II} \\neq y_{i}^{III} = \\ y_{i}^{IV}$, and\n$y_{i}^{I} = \\ y_{i}^{II} = y_{i}^{III} \\neq \\ y_{i}^{IV}$, respectively\n\\[57\\].\n\nFigure \u2011: Atomic structures and four sublattice tetrahedrons of FCC\ndisordered and ordered phases.\n\nAs mentioned above, the interaction parameters in each sublattice can be\npredicted by first-principles calculations using the dilute solution and\nSQS approaches when there is only one component in each of the remaining\nsublattices. For the interaction involving two components in two or more\nsublattices, i.e. $L_{A,B:A,B}$ in applicable to four sublattice models\n\\[58\\], the energetics from cluster expansion (CE) approach discussed in\nChapter can be used to evaluate the interaction parameters.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_stoichiometric_phases.ipynb b/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_stoichiometric_phases.ipynb new file mode 100644 index 0000000..b4a443b --- /dev/null +++ b/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_stoichiometric_phases.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "8ce82e48", "cell_type": "markdown", "source": "## Modeling of stoichiometric phases\n\nThe Gibbs energy of a stoichiometric phase can be modeled in the same\nway as that of pure elements discussed above using the data of heat\ncapacity, $S_{298.15}$, and enthalpy of formation at 298.15 K (). When\nthese data are not available from experiments, they can be predicted by\nfirst-principles calculations as discussed Chapter . It should be\npointed out that constraints placed on heat capacity of stoichiometric\ncompounds above melting temperatures, i.e. and , have not been\nrigorously implemented in the literature for such modeling, partially\nbecause the heat capacity of the corresponding liquid solution is not\nwell established, and the heat capacity of a compound is often not\navailable.\n\nWhen the data of heat capacity is not available, a simple approach,\ncommonly referred to as Neumann\u2013Kopp Rule assuming that the heat\ncapacity of formation of is zero, can be used. The Gibbs energy of the\ncompound is written as\n\n*Eq. 6\u20116*\n$G = \\sum_{}^{}N_{i}{_{\\ }^{0}G}_{i}^{ref} + \\mathrm{\\Delta}_{f}H - T\\mathrm{\\Delta}_{f}S$\n\nwith $\\mathrm{\\Delta}_{f}H$ and $\\mathrm{\\Delta}_{f}S$ modeled as\nconstants. An additional draw back of is that the melting temperature of\nthe compound can be higher than those of pure elements, and the heat\ncapacity of the compound may thus be questionable at temperatures higher\nthan the melting temperatures of pure elements due to the non-physical\ncontributions from pure elements based on .\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_cerium.ipynb b/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_cerium.ipynb new file mode 100644 index 0000000..99c2cbc --- /dev/null +++ b/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_cerium.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "5d3cd313", "cell_type": "markdown", "source": "## Application to cerium\n\nCerium (Ce) displays intriguing physical and chemical properties of\nwhich the most fascinating is its first-order isostructural phase\ntransition. This involves a magnetic, high temperature/high volume\n\u201c\u03b3-phase\u201d and a nonmagnetic, low temperature/low volume \u201c\u03b1-phase\u201d, both\nin the same face-centered-cubic (fcc) lattice structure. At 298 K and\n0.7 GPa, the \u03b3\u2192\u03b1 transition is accompanied by a 14-17% volume collapse.\nThe Ce phase transition has been studied extensively including our own\nworks with two or three microstates \\[55, 64\\].\n\nThe simplest model for the system is to consider two microstates:\nferromagnetic and nonmagnetic. The first-principles calculations of the\nfree energy of two Ce microstates are problematic in the absence of\nstrong correlation of the *f*-electrons in the DFT Hamiltonian. The\nrelative stability of the nonmagnetic (\u201cdelocalized\u201d) Ce 4*f* state to\nthat of the magnetic (\u201clocalized\u201d) Ce 4*f* state is greatly\noverestimated in the GGA \\[31-32\\] with spin polarization. The usual\napproach to surmount this is the Dudarev DFT + U method \\[65\\] with the\non-site Coulomb and exchange interactions as described with a\nHartree-Fock approximation added to the DFT Hamiltonian. This method\noffers the advantage that only the difference between the Hubbard U (due\nto the energy increase from an electron addition to a specific site) and\nthe J (due to the screened exchange energy) need to be specified *a\npriori*.\n\nEvaluation of numerous U \u2013 J values over a 1.0 \u2013 6.0 eV range revealed\nthat 1.6 eV gives the most consistent prediction of nonmagnetic Ce and\nmagnetic Ce energetics over the range of atomic volumes that includes\nboth microstates at 0 K. The energy-volume curves thus obtained is\nplotted in , showing that the nonmagnetic microstate is the ground\nstate, and the equilibrium between the two microstates at 0 K is at the\nnegative pressure of -0.87GPa. Since and , NTE does not exist in the\nsystem.\n\nFigure \u2011: Variation of cell energy (eV) with atomic volume\n(\u00c53) for Ce computed with strong correlation based upon\nDudarev\u2019s method with U \u2013 J = 1.6 eV.\n\nTo take into account the possible magnetic disordering in the\nferromagnetic microstate at finite temperatures, the following\ncontribution is added to the free energy of the ferromagnetic microstate\n\nEq. 9\u20119\n\nwhere *MS* is the spin moment, and *l* = 3 the orbital\nangular momentum of an *f*-electron. is a generalization of Hund\u2019s rule,\nwith total angular momentum . The Helmholtz energies thus obtained for\nboth microstates and the system are shown in at several temperatures\nwith the tie-lines included. In the figure, the blue, dot-dashed curves\nare for the nonmagnetic microstate, red, solid curves for ferromagnetic\nmicrostate, cyan shadows for entropy of mixing between two microstates,\nand the circle in (e) is the critical point. The numbers below the black\ndashed line, representing the common tangent curves, mark the transition\npressures. The 0-K static energies of the nonmagnetic microstate and the\nmagnetic microstate are also plotted in (a) using the solid circles and\ndotted lines.\n\nFigure \u2011: Helmholtz energies at (a) 0 K; (b) 100 K; (c) 165 K; (d) 300\nK; (e) 476 K; and (f) 600 K.\n\nThe temperature vs volume phase diagram is plotted and compared with\navailable experimental data as shown in \\[66\\]. In this figure, the\nvolume (V) is normalized to its equilibrium volume (VN) at\natmospheric pressure and room temperature. In the pressure range of 2.25\n\u2013 3.5 GPa, the system is within the single-phase region at all\ntemperatures considered as shown by the five continuous isobaric volumes\nas a function of temperature. In this pressure range, normal thermal\nexpansion is observed at both low and high temperatures on each isobaric\ncurve where the probability of each microstate does not change\nsignificantly with temperature. While in the middle temperature range on\neach isobaric curve, the colossal positive thermal expansion (CPTE),\nhighlighted by the pink open diamond symbols, exists due to the fast\nincrease of probability of the metastable ferromagnetic microstate with\nrespect to temperature, i.e. , , and . This CPTE is much higher than the\nindividual positive thermal expansions of the stable and metastable\nmicrostates, respectively.\n\nFigure \u2011: Calculated temperature-volume phase diagram of Ce.\n\nWith decreasing pressure, the system reaches a critical point (green\ncircle) where the homogeneous single phase becomes unstable, represented\nby and , and both entropy and volume change infinitely. At even lower\npressure, a miscibility gap forms, and the single phase separates into\ntwo phases with the same fcc crystal structures, but different magnetic\nspin structures. Inside the miscibility gap, the volume changes\ndiscontinuously with respect to temperature by the so-called first-order\ntransition as shown by the isobaric curve at zero pressure, compared\nwell with experimental volume data (solid squares) under ambient\npressure.\n\nThe fraction of the ferromagnetic microstate, *x*mag, in \u03b1-Ce\n(blue) and \u03b3-Ce (red) calculated using is plotted in as a function of\npressure along the miscibility gap phase boundary. It can be seen that\nthe fraction of the ferromagnetic microstate in \u03b1-Ce increases with\nincreasing pressure while the fraction of the ferromagnetic microstate\nin \u03b3-Ce decreases. At the critical point, the fraction of ferromagnetic\nmicrostate is calculated to be 0.58. This is in qualitative agreement\nwith the 0.67 value (filled circle) estimated experimentally at the\ncritical point.\n\nFigure \u2011: Fraction of ferromagnetic microstate in \u03b1-Ce (blue) and \u03b3-Ce\n(red) along the \u03b3-\u03b1 phase boundary.\n\nThe relative volume, *V/VN*, as a function of pressure, is\nplotted as the black solid lines in from 200 to 600 K at 50 K\nincrements. The blue and red solid lines correspond to \u03b1-Ce and \u03b3-Ce,\nrespectively. Symbols denote experimental data in the literature, except\nthe open green circle being the calculated critical point, in good\nagreement with the computed isotherms. In the two-phase miscibility gap\nregion, the \u03b3\u2192\u03b1 volume collapse is again noted, with the magnitude of\nthe collapse increasing with decreasing *T*. This is shown explicitly by\nthe dashed vertical lines at *T* = 200, 250, 300, 350, 400, and 450 K.\nFor *T* \\> 476 K, the calculated isotherms show an anomalous slope\nchange which closely matches the behavior near *V/VN* = ~0.85\nfrom experiment.\n\nFigure \u2011: Equation-of-states for Ce. The black solid lines represent the\ncalculated isotherms from 200 to 600 K at \u2206T = 50 K increments.\n\nA more complex model is to add the anti-ferromagnetic microstate. Thus\nobtained E-V and Helmholtz energy curves at 0 K are shown in . The\nequilibrium volume energies reveal that the energy of the\nanti-ferromagnetic microstate at the equilibrium volume is close to that\nof the nonmagnetic microstate but substantially lower than that of the\nferromagnetic microstate. It should be noted that the magnetic spin\ndisordering in the system is taken into account by the two magnetic\nmicrostates, and the contribution denoted by the mean-field theory, i.e.\n, should thus not be added to either magnetic microstate to avoid double\ncounting. The predicted critical point values are 546 K and 2.05 GPa,\ncloser to the experimental data than the previous predication with two\nmicrostates as shown in the temperature-pressure phase diagram in in\ncomparison with experimental data.\n\nFigure \u2011: (a) dot-dashed line with **\u25cb** (red), dashed line **\u25d1**\n(blue), and solid line with \u25cf (blue) represent the 0 K static total\nenergies for nonmagnetic (NM), anti-ferromagnetic (AFM), and\nferromagnetic (FM) microstates of Ce, respectively. (b) The solid lines\ndenote Helmholtz energy (per atom) from 0 to 600 K at \u2206T = 100 K; the\nheavy dot-dashed (\u03b1-Ce, red) and solid (\u03b3-Ce, blue) looping curves\nenclose the two-phase region with the light red dot-dashed lines\nconnecting the common tangents of each isotherm; the black dashed line\ndenotes zero pressure equilibrium state at given T; **\u25cb** (red) and \u25cf\n(blue) emphasize the phase boundary at 300 K while **\u25d1** (green) is the\ncritical point.\n\nFigure \u2011: Calculated temperature-pressure phase diagram along with\nexperimental data.\n\nThe calculated entropy changes are plotted in a in terms of lattice\nvibration only (black dashed line), lattice vibration plus thermal\nelectron (black dot-dashed line), and lattice vibration plus thermal\nelectron and plus configuration coupling (solid blue). The black square\nis the estimated vibrational entropy change at 0.7 GPa of \u03b3-Ce relative\nto \u03b1-Ce, and other open (solid) symbols are from experimental\nmeasurements of total entropy. Various contributions to the Helmholtz\nenergy along the \u03b3-\u03b1 phase boundary are plotted in b in terms of *T\u2206S*\n(blue diamonds), *\u2206E* (green circles), and *P\u2206V* (red squares), shown\nexcellent agreement with experimental data.\n\nFigure \u2011: Calculated (a) entropy; (b) various contributions to Helmholtz\nenergy of Ce along with experimental data.\n\nThe predicted fractions of three microstates as a function of\ntemperature and heat capacity at the critical pressure of 2.05 Pa are\nshown in a and b, respectively. Near the critical point, the theory\npredicts that the system is a mixture of the various microstates. a\ndepicts that for T \\< 300 K, the system consists mainly of the\nnonmagnetic Ce state which results in \u03b1-Ce. For *T* \\> 300 K, the\nthermal populations of the magnetic states increase with increasing\ntemperature. Finally, for *T* \\> 546 K (the critical point), 70% of the\nsystem is composed of the antiferromagnetic Ce state with the remaining\n30% consisting of the nonmagnetic and ferromagnetic Ce states. This is\nin agreement with the common belief that \u03b3-Ce is magnetic with a\npartially disordered local moment (paramagnetic) and that \u03b1-Ce is\nnonmagnetic.\n\nb shows the predicted temperature evolution of contributions to heat\ncapacity: vibrational and magnetic (*Cf/T*), electronic\n(*Cel*/*T*), and their sum (*Cf+el/T*) at 2.05\nGPa. The theory suggests the following: (a) below ~500 K,\n*Cf+el/T* shows an exponential temperature dependence due to\nthe statistic fluctuation among the non-magnetic, ferromagnetic, and\nantiferromagnetic states; (b) a peak appears at ~500 K in the\n*Cf+el/T* curve, which typically suggests the Schottky\nanomaly; (c) the electronic specific heat coefficient\n(*Cel*/*T*) is linear against *T*; (d) above *~*500 K the sum\nof *Cf/T* and *Cel*/*T* renders\n*Cf+el/T* temperature-independent.\n\nFigure \u2011: (a) Thermal populations of the nonmagnetic (red dot-dashed),\nanti-ferromagnetic (green dashed), and ferromagnetic (blue solid) as a\nfunction of temperature at the critical pressure of 2.05 GPa. (b)\nCel/T (black dashed line), Cf/T (black dot-dashed\nline), and their sum Cf+el/T (blue solid line) at 2.05 GPa.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_fept.ipynb b/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_fept.ipynb new file mode 100644 index 0000000..03a58d0 --- /dev/null +++ b/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_fept.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "d57f6e2d", "cell_type": "markdown", "source": "## Application to Fe3Pt\n\nInvar was first discovered in the intermetallic\nFe65Ni35 alloy and is characterized by \u201canomalies\u201d\nincluding thermal expansion, equation-of-state, elastic modulus, heat\ncapacity, magnetization, and Curie temperature. There are a number of\ntheoretical models for Invar such as the Weiss 2-\u03b3 model, the\nnon-collinear spin model, and the disordered local moment approach as\nreviewed in Ref. \\[67\\]. In this section, the application of the MMS\nmodel to the ordered L12 Fe3Pt is presented to\nstudy the Invar anomaly at finite temperatures.\n\nFor a supercell of 12 atoms with nine magnetic Fe atoms in\nFe3Pt, if only the up and down spins are considered, the\nsystem contains 29 = 512 spin configurations, which are, by\nsymmetry, reduced to 37 non-equivalent ones. They are the microstates in\nthe MMS model, and FMC and SFC are used to represent the ferromagnetic\nand spin flipping microstates, respectively. For the first-principles\ncalculations of each microstate, VASP package \\[14\\] within the\nprojector-augmented wave (PAW) method and the exchange-correlation part\nof the density functional treated within the GGA of\nPerdew-Burke-Ernzerhof (PBE) \\[32\\] are employed with details in Ref.\n\\[67\\]. For the lattice vibration, the Debye-Gr\u00fcneisen approach\ndescribed in Chapter 5.2.4 is used.\n\npresents the first-principles 0 K total energies of 36 non-equivalent\nSFCs as well as the FMC as a function of atomic volume. It can be seen\nthat there are a number of SFCs, whose energies are in the range of ~1\nmRy/atom to that of the FMC. It is noted that all the SFCs studied\nherein have the equilibrium averaged atomic volumes at least 1.8 %\nsmaller than that of the FMC, the 0 GPa ground state. In , the two\nlowest energy SFCs have been labeled as SFC55 and SFC41 with their spin\narrangements very similar to the double layer antiferromagnetic state.\nThe nonmagnetic configuration has a very small atomic volume of 11.66\n\u00c53/atom and much higher energy than both FMC and all SFCs,\nand is thus not shown here.\n\nFigure \u2011: 0 K total energies. The heavy black line represents the FMC.\nThe symbols **\u25cb**, \u25d4, \u25d1, and \u25d5 with dashed lines indicate the minima of\nthe energy-volume curves of the SFCs with spin polarization rates of\n1/9, 3/9, 5/9, and 7/9, respectively. The red **\u25cb** and \u25d4 with\ndot-dashed lines mark the two lowest SFCs in energy.\n\nThe normalized Helmholtz energies of all SFCs are plotted in with the\nHelmholtz energy of FMC as the reference state, shown that the FMC has\nthe lowest Helmholtz energy at all temperatures considered. If only the\nrelative Helmholtz energies of microstates are considered, FMC should be\nstable at all temperatures. However, the configurational entropy due to\nthe mixing among multiple microstates, i.e. , lowers the system free\nenergy by introducing the statistic probability of metastable SFCs.\n\nFigure \u2011: Normalized Helmholtz energy of all SFCs with respect to that\nof FMC.\n\nThrough the minimization of Helmholtz energy of , the\ntemperature-pressure and temperature-volume phase diagrams are obtained\nand shown in \\[66\\]. A critical point at 141 K and 5.81 GPa is predicted\nwith *V* = 12.61 \u00c53. Below the critical point, it is a\ntwo-phase miscibility gap (the shadow area) with the dominant\nmicrostates being FMC and SFCs, respectively, and the transition between\nthem is first-order. Above the critical point, the macroscopically\nhomogeneous single phase is stable, and the phase transitions between\nthe ferromagnetic-dominant phase with large volumes and the SFC-dominant\nparamagnetic phase with small volumes are of second-order. The\nsecond-order transition pressures and volumes are determined by the\ncondition that the weighted Helmholtz energy counting all SFCs equals to\nthe Helmholtz energy counting only FMC.\n\nFigure \u2011: Calculated (a) temperature-pressure and (b) temperature-volume\nphase diagrams of Fe3Pt.\n\nIn a, the data points are the measured pressure dependence of the Curie\ntemperature, and the agreement between the measurements and predictions\nis remarkable. It should be pointed out that the classical Weiss 2-\u03b3\nmodel predicts only first-order phase transitions while the\nnon-collinear spin model yields only second-order phase transitions at\nall temperatures. In b, four isobaric volume curves are also plotted\nwith the predicted NTE regions marked by the pink open diamonds and the\nexperimental volume data under ambient pressure superimposed, showing\nexcellent agreement. It also depicts the gigantic NTE around the\ncritical point on the isobaric curve at 7 GPa.\n\nindicates that the entropies of SFCs are larger than that of FMC so\ntheir Helmholtz energy differences decrease with temperature. This is in\nline with the origin of NTE in a single phase due to the statistic\nexistence of metastable microstates with lower volumes and higher\nentropies than the stable state in a temperature range where their\nprobabilities change dramatically. plots the calculated thermal\npopulations of the FMC (black solid line) and that of the sum over all\nSFCs (black dot-dashed line) under ambient pressure. The two major\ncontributions to the paramagnetic (PM) phase are from SFC55 and SFC41,\nwhich are also plotted using red dashed and long dashed lines,\nrespectively. The system is dominated by the FMC at temperatures below\nhalf of the transition temperature, and the populations of SFCs increase\nmonotonously at temperatures higher than half of the transition\ntemperature. As mentioned above, the transition temperature is defined\nwhen the population of all SFCs is the same as that of FMC due to their\nequal Helmholtz energy.\n\nFigure \u2011: Calculated thermal populations of FMC, SFC55, SFC41, and sum\nof all SFCS, respectively.\n\nThe predicted thermal volume expansion and the derived linear thermal\nexpansion coefficient (LTEC) under ambient pressure are plotted in . A\npositive thermal expansion is predicted from 100 K to 288 K, followed by\na negative thermal expansion in the range of 289 ~ 449 K, and then a\npositive thermal expansion again at \\>450 K, in excellent agreement with\nexperiments. The only disagreement between the predictions and\nexperiments occur at T \\< 100 K where the calculations did not reproduce\nthe negative thermal expansion for Fe3Pt. Large supercell or\nmore spin configurations may be needed.\n\nFigure \u2011: (a) Relative volume increase (V-\nV300)/V300 with V300 being the\nequilibrium volume at 300K and 0 GPa for the ordered Fe3Pt.\n(b) Linear thermal expansion coefficient (LTEC) along with experimental\ndata (symbols) with details in Ref. \\[67\\].\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/concept_of_materials_genome.ipynb b/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/concept_of_materials_genome.ipynb new file mode 100644 index 0000000..079d170 --- /dev/null +++ b/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/concept_of_materials_genome.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "e9a0ab3f", "cell_type": "markdown", "source": "## Concept of Materials Genome\u00ae\n\n\u201cA genome is a set of information encoded in the language of DNA that\nserves as a blueprint for an organism\u2019s growth and development. The word\ngenome, when applied in nonbiological contexts, connotes a fundamental\nbuilding block toward a larger purpose\u201d \\[68\\]. Materials Genome\u00ae (a\ntrademark of Materials Genome, Inc., Pennsylvania, USA) thus concerns\nthe building blocks of materials. Most part of this book focuses on the\nGibbs/Helmholtz energies of individual phases as a function of its\nnatural variables, and the same in the CALPHAD modeling of\nthermodynamics and other properties of individual phases.\nMulti-component materials systems and their properties are built on the\nindividual phases and their properties. Individual phases are thus\nnaturally considered as building blocks of materials. Consequently, the\nlanguage of CALPHAD thermodynamics and kinetics contains the genomics of\nmaterials by representing experimental and theoretical results in\ndatabases to make them applicable to a much wider context than the\noriginal experiments or calculations \\[69\\]. The change of individual\nphases in terms of their properties, amounts, and interactions with\nother phases with respect to external conditions thus determines the\nperformance of the materials.\n\nOn the other hand, at critical points and beyond, phases lose their\nindividuality and form one macroscopically homogeneous system, and the\nproperties of the system change dramatically with respect to external\nconditions. As shown in this chapter, these dramatically responses can\nbe predicted through the statistic competition of stable and metastable\nmicrostates. From the thermodynamic point of view, under any given\nconditions, one of the individual microstate has the lowest\nGibbs/Helmholtz energy and is stable, while all other microstates have\nhigher free energy and are metastable or unstable. These metastable or\neven unstable microstates are brought into statistic existence in the\nmatrix of the stable microstate due to the entropy of mixing of the\nstable and metastable configurations. Those microstates may thus be\nconsidered as the building blocks of individual phases \\[70\\].\n\nIt has demonstrated in this chapter that the properties of a\nmacroscopically homogeneous system with multiple microstates are not a\nlinear combination of properties of constituent microstates and depends\nsignificantly on the change rate of statistic probability of microstates\nwith respect to external fields. This change rate is determined by the\nfree energy difference between the stable and metastable microstates and\nits change with respect to external fields. As shown in and , this\nchange rate can be correlated to the distance of the system with respect\nto the critical point in the system. At the critical point, there is a\nmathematical singularity when the single phase becomes unstable. When\nthe macroscopically homogeneous single-phase system moves away from the\ncritical point, its properties become less and less dramatic, but always\nremain a certain degree of anomaly. The properties of a system can thus\nbe dramatically altered and designed by changing the position of the\ncritical point through adjustments of chemical compositions and strain\nenergy in thin films.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/index.ipynb b/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/index.ipynb new file mode 100644 index 0000000..08547d4 --- /dev/null +++ b/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/index.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "eb4fc5ff", "cell_type": "markdown", "source": "# Critical phenomena, thermal expansion, and Materials Genome\u00ae\n\nIn Chapter , it was shown that all molar quantities of a homogeneous\nsystem diverge at the critical point, i.e. the limit of stability,\nincluding the additional ones shown in . As illustrated by , even though\neach molar quantity changes in the same direction as its conjugate\npotential, i.e. with the same sign, its dependence with respect to a\nnonconjugate potentials can either be in the same sign or opposite\nsigns. It is often considered to be normal when they change in the same\ndirection, while abnormal when in different directions.\n\nIn this chapter, the thermal expansion defined by , is used as an\nexample for detailed discussion of those extraordinary phenomena in the\ncontext of a critical point based on the MMS model presented in Chapter\n. The MMS model is first discussed in terms of thermal expansion and\nthen applied to elemental cerium (Ce) with the colossal positive thermal\nexpansion (CPTE) and Fe3Pt with negative thermal expansion\n(NTE).\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/mms_model_applied_to_thermal_expansion.ipynb b/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/mms_model_applied_to_thermal_expansion.ipynb new file mode 100644 index 0000000..343d12d --- /dev/null +++ b/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/mms_model_applied_to_thermal_expansion.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "fad8b29b", "cell_type": "markdown", "source": "## MMS model applied to thermal expansion\n\nAs shown in , the thermal expansion of a system can be positive, zero\nand negative depending on the pressure dependence of entropy of the\nsystem. Let us carry out a virtual experiment by analyzing a system\nstarting with one microstate only, *\u03b1*, when the metastable *\u03b2*\nmicrostate has a higher entropy than the *\u03b1* microstate, i.e. and the\nrelative stability of the *\u03b2* microstate thus increases with\ntemperature. The cases with will be discussed after this starting with a\nmixture of the *\u03b1* and *\u03b2* microstates.\n\nWhen a metastable microstate, *\u03b2*, is introduced by changing pressure\nunder constant temperature, based on and , the entropy change of the\nsystem can be written as\n\nEq. 9\u20111\n\nwhere represents the statistic probability of the microstate in the\nsystem. With , this would results in a positive entropy change of , i.\ne. , since . If this entropy increase is due to the decrease of\npressure, i.e. because volume and its conjugate potential (negative\npressure) change in the same direction, the volume thermal expansion of\nthe system is positive due to the increase of the population of the *\u03b2*\nmicrostate with a larger volume. In this case,\n$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}} > 0$, and the\nvolume and entropy of the two microstates change in the same direction.\n\nOn the other hand, if this entropy increase is realized by increasing\npressure, i.e. , the volume thermal expansion of the system is negative\ndue to the increase of the population of the *\u03b2* microstate with a\nsmaller volume. In this case,\n$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}} < 0$, and the\nvolume and entropy of the two microstates change in the opposite\ndirections.\n\nTherefore, the sign of\n$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}}$ of two\nmicrostates can be used as a criterion to determine whether a system\npossesses NTE, i.e. positive\n$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}}$ for positive\nthermal expansion, and negative\n$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}}$ for NTE. At a\ncritical point, the entropy change with respect to temperature is\ninfinite, resulting in either infinite positive or infinite negative\nthermal expansion correspondingly. When the system moves away from the\ncritical point into the macroscopically homogeneous single-phase region,\nthe thermal expansion becomes less positive or negative. A number of\nsystems with\n$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}} < 0$, thus\npotentially NTE, are listed in the supplementary information of Ref.\n\\[63\\].\n\nNow let us consider the case that the metastable *\u03b2* microstate has\nlower entropy than the *\u03b1* microstate, i.e. , and the *\u03b2* microstate is\nthus more stable at low temperatures. The system at higher temperatures\ncontains thus only the *\u03b1* microstate and has positive thermal\nexpansion. When the metastable *\u03b2* microstate is introduced, the sign of\nthe entropy change in can be either positive or negative because the\nfirst term is negative and the second term is positive, and its sign\nthus depends on the value of the entropy difference between two\nmicrostates and the probability of the metastable *\u03b2* microstate. The\nvirtual experiment should thus be carried out in a system with the\nhighest MCE in , i.e. when the two microstates have the same free energy\nand are in equilibrium with each other. From and , the system entropy\ncan be written as\n\nEq. 9\u20112\n\nWith the change of pressure, will either increase or decrease, and the\nentropy of the system becomes\n\nEq. 9\u20113\n\nThe difference of and is obtained as\n\nEq. 9\u20114\n\nThe second term in is always negative, and the first term is also\nnegative if because . It is thus evident that if is increased by\ndecreasing pressure, the entropy of the system decreases, and the system\nwould possess negative thermal expansion because is negative, and the\nentropy is reduced by the decrease of pressure. At the same time, and\n$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}} < 0$, the latter\nbeing the same condition for a negative thermal expansion as in the\nfirst virtual experiment with . One can thus conclude that for a\ntwo-phase equilibrium line with\n$\\frac{dT}{dP} = \\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}} < 0$,\nboth phases can display negative thermal expansion. On the other hand,\nif is increased by increasing pressure, the system would possess\npositive thermal expansion because the entropy is reduced by the\nincrease of pressure.\n\nFurthermore, the thermal expansion of a system can be approximated as\nfollows using the rule of mixture of volumes\n\nEq. 9\u20115\n\nEq. 9\u20116\n\nwhere , , , , , and are the thermal expansion coefficients and volumes\nof the system and the \u03b1 and \u03b2 microstates, respectively. For\nsimplification, let us assume both microstates have similar positive\nthermal expansion, i.e. , and becomes\n\nEq. 9\u20117\n\nshows that it is the combination of volume difference and that\ndetermines the macroscopic thermal expansion. By setting , one obtains\n\nEq. 9\u20118\n\nFor , for $p^{\\beta} \\geq 0$, and for , at $p^{\\beta} \\rightarrow 0$ .\nThe readers are reminded that the sign of is the same as the sign of .\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/experimental_data_for_thermodynamic_modeling/additional_methods.ipynb b/psu410/src/experimental_data_for_thermodynamic_modeling/additional_methods.ipynb new file mode 100644 index 0000000..cd083fa --- /dev/null +++ b/psu410/src/experimental_data_for_thermodynamic_modeling/additional_methods.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "de75ddcb", "cell_type": "markdown", "source": "### Additional methods\n\nElectrical resistivities of different phases are usually different. A\nchange of slope of electric resistivity as a function of composition or\ntemperature or pressure reflects a phase transformation. This technique\nis simple and reliable.\n\nMagnetic transitions can be measured using a vibrating sample or\nsuperconducting quantum interference device (SQUID) magnetometer by\ndetermining the magnetic moment of a sample in presence of an applied\nmagnetic field. Magnetic field lines form closed loops, resulting in an\nexternal dipolar and demagnetizing field in a finite-sized sample. The\neffective field sensed by the sample is thus the difference between the\napplied field and the demagnetizing field. The magnetic transition\ntemperature is evaluated from the Arrott plots where the ratio of\nmagnetic field over magnetization with a proper exponent is plotted with\nrespect to the magnetization with another proper exponent for a series\nof temperatures. Those proper exponents result in parallel isotherm\nlines, and the isotherm line intersecting the origin corresponds to the\nmagnetic transition temperature.\n\nThin films with composition gradients, commonly referred to as\ncombinatorial libraries, can be used to study the phase relations\nsimilar to diffusion couples/multiples though the results may differ due\nto the effects of surface and potential interactions with the substrate.\n\nPhase relations at high pressures are measured in equipment using\ndiamond anvil cells (DAC) or multi-anvil devices. The high pressure is\nrealized by decreasing the area, i.e. the anvil culet size. Pressures up\nto 100 GPa can be created in DAC with a culet size of 0.3mm for small\nsamples in the order of 0.2 to 0.4mm. For large samples, the\nlarge-volume presses (LVP) technique is developed, typically using WC\nwith the pressure mostly limited to 30 GPa and the sample size ranging\nfrom 1mm3 to 1cm3. The pressure can be measured by\neither the ruby (Cr3+ doped Al2O3)\nfluorescence line shift or the molar volume of a pressure marker by\nX-ray diffraction. The samples in DAC apparatus are heated by laser or\nresistive wire or a small heater around the samples, while high\ntemperatures in LVPs are achieved by resistive heaters. Crystal\nstructures are detected by in-situ X-ray or synchrotron diffraction.\nAttentions need to be paid to temperature and pressure homogeneity and\nthe non-hydrostatic stresses, which are both better controlled in LVP\nequipment.\n", "metadata": {}}, {"id": "4eb26f12", "cell_type": "markdown", "source": "### Combustion, direct reaction, and heat capacity calorimetry\n\nIn combustion calorimetry, the sample is ignited and reacts with\nreactive gases like oxygen or fluorine. To accurately calculate the\nenthalpy of formation from the enthalpy of combustion, the reliable\ncharacterization of reactants and reaction products is critical, such as\nincomplete combustion, impurities in the reactants which are often\nill-defined, and more than one reaction gaseous species and condensed\nphases. Combustion calorimeters are usually of isoperibol type around\nroom temperature in a water bath.\n\nThe direct reaction calorimetry is similar to the combustion calorimetry\nthough at high temperatures in heat-flux or adiabatic environments. The\npartial enthalpy of reaction can also be measured if the partial\npressure of volatile species can be controlled and measured. The key\nfactor for accurate results is that both the reactants and reaction\nproducts are well characterized and the reaction goes to completion\nquickly like in the combustion calorimetry. For reactive reactants,\nspecial procedure is needed to avoid their loss before the reaction\ntakes place.\n\nHeat capacity is defined as the amount of heat needed to increase the\ntemperature by 1K as shown by , and its integration with respect to\ntemperature from 0K gives entropy as shown by . At low temperatures,\nadiabatic calorimetry gives more accurate data of heat capacity though\nit is time consuming and requires complex instruments. At high\ntemperatures, the efficient and less accurate DSC method is more widely\nused.\n", "metadata": {}}, {"id": "f0ee87a8", "cell_type": "markdown", "source": "### Diffusion couples/multiples\n\nThe major challenge in the equilibrated material approach is to ensure\nthat the whole sample reaches equilibrium. On the other hand the\ndiffusion couple/multiple technique does not require the whole sample to\nbe in equilibrium and is based on the assumption that any two phases in\ncontact are in equilibrium with each other at the phase interface, and\nthe phase compositions can be obtained by extrapolation of concentration\nprofiles in the two phases to the phase interface. Since the total\nsystem of a diffusion couple is not at equilibrium, many kinetic\nphenomena related to diffusion can be studied in a diffusion couple,\nsuch as interdiffusion coefficients, parabolic growth kinetics of\nproduct layer thickness, diffusion path (represented by the local\noverall compositions) in ternary and multi-component systems for\nvisualizing the microstructure of reaction zones, and other properties,\nall as a function of composition, which are beyond the scope of the\npresent book.\n\nTypical diffusion couples are in solid-state with two materials brought\ninto intimate contact to allow diffusion of elements between the two\nmaterials though solid-liquid diffusion couples are also used. The\ncontacting faces are commonly ground and polished flat, clamped together\nusing mechanical mechanisms, and annealed at high temperatures where\ndiffusion can take place to a significant degree in a matter of days,\nweeks or months. The samples are then quenched to retain the\nhigh-temperature equilibrium features. For metallic systems, diffusion\ncouples can also be prepared by eletrolytical and electroless plating\ntechniques. It is important to avoid the formation of liquid during\nannealing as it ruins the sample geometry. Furthermore, good adherence\nat the interfaces is critical for reliable data.\n\nSince diffusion couples are not in a fully equilibrium state, the\ntie-lines between two phases at the phase interfaces need to be obtained\nby extrapolations of concentration profiles in neighbouring phases. The\nelectron propagation in quantitative EPMA is typically in the range of\n1-2\u03bcm, yielding an excitation volume of approximately 2-4\u03bcm diameter and\na requirement of reasonable layer widths of phases on both sides of the\ninterface for accurate extrapolation. Therefore, the reliable\ncomposition of a single phase must be taken several micrometers away\nfrom the interface. When steep concentration gradients exist near the\ninterfaces, the extrapolation may lead to large errors, and analytical\nelectron microscopy is then needed. Furthermore, the fluorescence\neffects, where the primary excitation can be powerful enough to excite\nother elements in the sample, resulting in enhanced X-ray production and\nthe need of proper corrections. For a new phase to become observable\nexperimentally, it must nucleate and grow to reach the resolution limit\nof analytical tools. It is thus not uncommon that some known equilibrium\nphases are not found, and some non-equilibrium phases form. One way to\ncircumvent this issue is to use incremental diffusion couples with\nnarrow concentrations next to the phase of interest so that only this\nphase may be formed.\n\nFor ternary and higher-order systems, a more efficient approach can be\nused by placing a thin layer of the third alloy between two alloys. The\nthin central layer is eventually consumed, and the diffusion path is not\nfixed as in the semi-infinite diffusion couples. The phase compositions\nchange continuously with time as a result of the overlapping of two\nquasi-equilibrated diffusion zones.\n\nA diffusion multiple contains three or more pure elements or alloys of\ndifferent compositions and is a sample with multiple diffusion couples\nand diffusion triples in it. It is more efficient in terms of both\nmaterials and time in comparison with equilibrated alloys and diffusion\ncouples. All alloy blocks are prepared individually and sealed in vacuum\nin a cylinder which is also used as one alloy for the diffusion\nmultiple. The sealed cylinder with vacuum inside also serves as the can\nfor subsequent hot isostatic pressing to achieve intimate interfacial\ncontacts. The cylinder can then be cut into disks for further annealing\ntreatments. A broad range of design strategies is needed for complex\ndiffusion multiples along with automated plotting procedures due to the\nlarge amounts of EMPA data. The major source of error lies in the\nextraction of tie-lines from EPMA results due to very condensed\ninformation in a small area and the deviation of scanned lines from\nthose perpendicular to the interface.\n\nIn terms of local equilibrium characteristics of diffusion\ncouples/multiples, it is evident that in the equilibrated materials\napproach, it may not be necessary to reach the full equilibrium for the\nwhole sample if one is only interested in the local equilibrium\ncompositions between two neighboring phases, which can even provide\ninformation on metastable extensions of two phases if the two phases are\nin a metastable equilibrium at the annealing temperature.\n", "metadata": {}}, {"id": "1e4f1d1e", "cell_type": "markdown", "source": "### Equilibrated materials\n\nThe most common method to determine phase equilibria is to use\nequilibrated materials. This method typically involves material\npreparation through high temperature melting or powder metallurgy,\nhomogenization heat treatment, isothermal or cooling/heating procedure,\nand identifications of crystal structures and phase compositions. It is\nimportant to avoid macro-inhomogeneity as it can be difficult remove the\ninhomogeneity in subsequent treatments. It is also important to use\nstarting materials of highest purity and minimize the loss and\ncontamination of materials during the whole experiments using a\nprotective atmosphere of inert gas or vacuum. The typical melting\ntechniques include high temperature furnaces with crucibles, arc\nmelting, and induction melting. Attentions need to be paid for possible\nreactions between materials and crucibles/containers, which can be\navoided by levitating the materials through electromagnetic fields or\nother means. In addition to use pure elements as raw materials, master\nalloys with well-controlled compositions are often utilized because the\ncompositions and melting properties of master alloys are usually much\ncloser to those of final materials than the pure elements. For materials\nwith very high melting temperature or volatile components, the powder\nmetallurgy method can be used where compacts are made, capsulated, and\nsintered.\n\nHomogenization during subsequent heat treatment is achieved through\ndiffusion, in which time and temperature are two important parameters.\nTo accelerate the homogenization process, the heat treatment temperature\nshould be as close to the solidus temperature as possible taking into\naccount the composition inhomogeneity with variable solidus\ntemperatures. When there are phase transformations taking place during\nthe heat treatment, extra time is needed for the heat treatment.\n\nThe phase boundaries are then determined through measurements of either\ncompositions of individual phases that are in equilibrium under constant\ntemperature, pressure/stress/strain, and electric/magnetic fields or\ndiscontinuity in some physical properties of materials due to a phase\ntransition from the continuous change of temperature or\npressure/stress/strain or electric/magnetic fields. The measurement of\ncompositions is usually carried out at ambient conditions so it is\nnecessary that the phases are fully equilibrated at experimental\nconditions, which requires rigorous verifications, and can be \u201cquenched\u201d\nto ambient conditions to remain unaltered during \u201cquenching\u201d, at least\nin terms of compositions. The compositions are typically measured by\nscanning electron microscopy (SEM) equipped with energy dispersive\nspectrometer (EDS) or wavelength dispersive spectroscopy (WDS), with a\nmicro-level spatial resolution and a better compositional resolution of\nWDS than EDS. A dedicated SEM with WDS gives another important, widely\nused composition measurement technique called electron probe\nmicroanalysis (EMPA). For submicron-sized phases, analytical\ntransmission electron microscopy equipped with EDS can be used though\ncare must be taken to avoid interference from neighboring phases.\n\nTo accurately identify phases in equilibrium under experimental\nconditions, in-situ characterizations are necessary, which complicates\nthe experimentation. An alternative indirect method is to measure a\nphysical property that changes discontinuously or dramatically for a\nfirst- or second-order phase transition, such as heat, volume, electric\nconductivity, and magnetization. There are two widely used techniques in\nmeasuring heat: differential thermal analysis (DTA) and differential\nscanning calorimetry (DSC). Both attempt to measure the difference in\ntemperature with the same amount of power supplied between a sample and\nan inert standard during heating or cooling. A DSC may also measure the\ndifference in the amount of power supplied to keep their temperatures\nidentical. The deviation of this difference from a baseline indicates a\nphase transition in the sample and is plotted as a function of time or\ntemperature of either the sample or the inert standard. It is evident\nthat the major challenges in both DTA and DSC techniques are to reach\nthermal equilibrium between the sample/standard and the instrument and\nthe thermodynamic equilibrium within the sample due to the continuous\nheating or cooling. The thermal equilibrium can be improved or mitigated\nthrough various methods such as sample shape and size, crucible\nselection, mixture with the material used for the inert standard, and\nwith thermocouples in direct contact with the sample and the inert\nstandard. However, the thermodynamic equilibrium within the sample can\nonly be reached when the heating/cooling rate is comparable with the\nrate of the phase transition in the sample, which is almost impossible\nif the phase transition typically involves diffusion in solid phases.\nTherefore, extreme care is needed in interpreting the temperature\ndetermination and the amount of heat associated with the DTA/DSC curves\nas discussed in detail in the reference \\[11\\].\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/experimental_data_for_thermodynamic_modeling/index.ipynb b/psu410/src/experimental_data_for_thermodynamic_modeling/index.ipynb new file mode 100644 index 0000000..f62f6e8 --- /dev/null +++ b/psu410/src/experimental_data_for_thermodynamic_modeling/index.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "38eccd96", "cell_type": "markdown", "source": "# Experimental data for thermodynamic modeling\n\nThe most widely used thermodynamic modeling technique is the CALPHAD\n(CALculation of PHAse Diagram) method to be discussed in detail in\nChapter 6. The input data in the evaluations of thermodynamic model\nparameters have been primarily from experiments and estimations until\nfirst-principles calculations based on the density functional theory\n\\[8\\] became a user tool in the later 1990\u2019s. Experimental data include\nboth thermodynamic and phase equilibrium data, while the\nfirst-principles calculations, which only provide thermodynamic data of\nindividual phases, are discussed more extensively in Chapter .\n\nThe three recently published books summarize the commonly used methods\non experimental measurements of thermodynamic properties of single \\[9\\]\nand multiple phases \\[10\\] and phase diagrams \\[11\\]. They are briefly\ndiscussed here, and readers are referred to these books for details. The\nmain techniques for crystal structure analysis include X-ray\ndiffraction, electron backscatter diffraction (EBSD), electron\ndiffraction in transmission electron microscopy, neutral scattering, and\nsynchrotron scattering, which are not discussed in this book.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/experimental_data_for_thermodynamic_modeling/phase_equilibrium_data_from_experiments.ipynb b/psu410/src/experimental_data_for_thermodynamic_modeling/phase_equilibrium_data_from_experiments.ipynb new file mode 100644 index 0000000..ca07ada --- /dev/null +++ b/psu410/src/experimental_data_for_thermodynamic_modeling/phase_equilibrium_data_from_experiments.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "71eaef0b", "cell_type": "markdown", "source": "## Phase equilibrium data from experiments\n", "metadata": {}}, {"id": "c517027c", "cell_type": "markdown", "source": "### Solution calorimetry\n\nThe book edited by Marsh and O\u2019Hare \\[12\\] documented the detailed\nexperimental techniques used for solution calorimetry. In one\nexperiment, the enthalpy of solution of a single phase is measured in a\nparticular solvent. To convert this enthalpy of solution into enthalpy\nof formation of the phase, a thermodynamic cycle is setup for a chemical\nreaction to form this phase from either constitutive pure elements or\ncompounds. Therefore in another experiment, the enthalpy of solution of\nconstitutive pure elements or compounds is measured in the solvent as\nidentical as possible to that used in the first experiment. The\ndifference of the two enthalpies of solution thus gives the enthalpy of\nformation of the single phase from its constitutive elements or\ncompounds at the temperature of the samples before they are dropped into\nthe solvent, usually at room temperature.\n\nThe solvent can be aqueous solutions at ambient temperatures and\npressures or metallic/salt/oxide melts at high temperatures under either\nadiabatic or isoperibol conditions. The adiabatic calorimetry measures\nthe temperature change of the solvent and is usually more accurate than\nthe isoperibol calorimetry that measures the heat generated during the\ndissolution, though the adiabatic ones are with more complex\ninstruments. It is important that the choices of solvent and temperature\nensure the complete dissolution of all substances into the solvent to\nform a homogeneous solution. Furthermore, the effect of dilution and of\nchanges in solvent composition needs to be considered in the calculation\nof enthalpy of solution.\n\nA large number of solvents are used. For aqueous solvent, hydrofluoric\nacid or mixtures of HF and HCl are often used. For oxides, buffer-type\nsystems are typical such as lead and alkali borates and alkali\ntungstates/molybdates. For metallic phases, low melting metals such as\nSn, Bi, In, Pb, and Cd, sometimes Al and Cu, are used. Factors such as\nsolubility, dissolution kinetics, thermal history, stirring, heat flow,\nparticle size, and system size are to be optimized for accurate\nmeasurements. To enhance solution kinetics, the compound to be studied\ncan be mixed with another element or compound so that the mixture can\nform liquid in the solvent at higher reaction rate. In any case, it is\nimportant to calibrate the system with pure elements and compounds of\nknown enthalpy of formation.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/experimental_data_for_thermodynamic_modeling/thermodynamic_data_from_experiments.ipynb b/psu410/src/experimental_data_for_thermodynamic_modeling/thermodynamic_data_from_experiments.ipynb new file mode 100644 index 0000000..75e32ff --- /dev/null +++ b/psu410/src/experimental_data_for_thermodynamic_modeling/thermodynamic_data_from_experiments.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "3f372bef", "cell_type": "markdown", "source": "## Thermodynamic data from experiments\n\nBroadly speaking, thermodynamic data represents the values of Gibbs free\nenergy and all its first and second derivatives. For thermochemical\nproperties as the main topic of the book, calorimetric, electrochemical,\nand vapor pressure methods are the primary techniques used with the\nfirst for accurate measurement of enthalpy and entropy, and the latter\ntwo for direct determination of Gibbs energy and activity. The\nelectrochemical method is discussed in Chapter . The calorimetric method\nis divided into solution, combustion, direct reaction, and heat capacity\ncalorimetry, respectively, which all involve chemical reactions to be\ndiscussed in detail in Chapter . The vapor pressure method involves the\nequilibrium of volatile species between gas and samples and is divided\ninto Knudsen effusion and equilibration methods, respectively.\n", "metadata": {}}, {"id": "417cc01d", "cell_type": "markdown", "source": "### Vapor pressure method\n\nIn the Knudsen effusion method, a small amount of volatile species in\nthe gas phase effuses through a small orifice of 0.1 to 1 mm with\nnegligible influence of the equilibrium in the Knudsen cell. The vapor\nis ionized and analyzed with a mass spectrometer. The partial pressure\nof a species can be calculated from its ionization area and intensity\nthrough a calibration factor determined by a reference material with\nknown partial pressure. For high temperature measurements, care must be\ntaken to avoid the reactions between cell and sample and fragmentation\nof gas species on ionization. The typical vapor pressure range is\nbetween 10-7 and 10 Pa.\n\nIn various equilibration methods, the total vapor pressure is usually\nmeasured directly using pressure gauges in the range of 10-7\nand 100 kPa. Other direct or indirect methods include\n\nthermogravimetric method for measuring the vapor mass\n\natomic absorption spectroscopy for gas composition\n\nmeasurement of sample composition equilibrated with a gas of\nwell-defined activity of the volatile species\n\nthe dew point method in which the condensation temperature of the\nvolatile component is measured from the vapor equilibrated with the\nsample at a higher temperature,\n\nthe chemical transport method to be discussed in Chapter .\n\nThe main error in all these methods is often due to inadequate\nequilibration between vapor and sample.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/firstprinciples_calculations_and_theory/bornoppenheimer_approximation.ipynb b/psu410/src/firstprinciples_calculations_and_theory/bornoppenheimer_approximation.ipynb new file mode 100644 index 0000000..c9815d6 --- /dev/null +++ b/psu410/src/firstprinciples_calculations_and_theory/bornoppenheimer_approximation.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "5798c63c", "cell_type": "markdown", "source": "### Born-Oppenheimer Approximation\n\nFor a time independent atomic system, it is often accurate enough to\nwrite in or in terms of the electron coordinate **r** and nuclei\ncoordinate **R**,\n\n*Eq. 5\u201164* ,\n\nwhere *e* represents the electron charge and *i* and *j* label the\nelectrons, *I* and *J* the atomic nuclei, *ZI* the atomic\nnuclear charge number of atom *I*, *e* the electron charge,\n*me* the electron mass, *MI* the mass of atomic\nnuclei *I*, the Laplace operator for electron *i*, and the Laplace\noperator for atomic nuclei *I,* noting\n\n*Eq. 5\u201165* ,\n\nwith respect to the Cartesian axis *x*, *y*, and *z*.\n\nConsider the fact that the electron mass is thousand times smaller than\nthe mass of the atomic nuclei, implying that the motions of the\nelectrons are much faster than the atomic nuclei, Born and Oppenheimer\nproposed that the wave function of the whole system can be simply\napproximated as the product of the electron wave function and the atomic\nnuclei wave function as\n\n*Eq. 5\u201166* .\n\nWith the auxiliary approximation of neglecting the dynamic coupling\nbetween the motions of electrons and atomic nuclei, the Schr\u00f6dinger\nequation for the motion of electrons becomes\n\n*Eq. 5\u201167* ,\n\nwhere\n\n*Eq. 5\u201168* ,\n\nand the Schr\u00f6dinger equation for the motion of atomic nuclei becomes\n\n*Eq. 5\u201169* ,\n\nwhere\n\n*Eq. 5\u201170* ,\n\nwith\n\n*Eq. 5\u201171* ,\n\nwhere represents the complex conjugate of .\n", "metadata": {}}, {"id": "307fdf80", "cell_type": "markdown", "source": "### Cluster Expansions\n\nMany properties of a solution phase such as energy are dependent on the\n*configurations* - the arrangements of atoms on the lattice sites. In\ncluster expansion \\[35, 42\\], the configuration dependence of properties\nis formulated efficiently by a \u201clattice algebra\u201d which maps a\nsubstitutional configuration into an Ising-like lattice model. Taking a\nbinary A1-*x*B*x* solution phase for instance, A\natoms are represented by the \u201cdown\u201d spins (*Si* = -1) and B\natoms are represented by the \u201cup\u201d spins (*Si* = +1) as\nillustrated in . Using the cluster expansion technique, for a system\ncontaining *N* atoms, the total energy of any alloy configuration *\u03c3* =\n(*S*1, *S*2, \u2026, *SN*) can be\nconveniently evaluated using the following Ising-like Hamiltonian:\n\n*Eq. 5\u2011147*\n\nwhere *J*\u2019s are the effective cluster interactions (ECI\u2019s); is a number\nrepresenting the atomic occupation at the lattice *i* under the\nconfiguration *\u03c3*, which takes the values -1 and 1 for binary and -1, 0,\nand 1 for ternary etc. In , the 2-site, 3-site, and 4-site correlations\nare written as follows,\n\nEq. 5\u2011148\n\nEq. 5\u2011149\n\nEq. 5\u2011150 .\n\nThe expansion in would be exact as long as *all* the *n*-site\ninteractions are included. For a binary system, this can be observed by\nthe combination law that where is the number of *n*-site interactions.\nHowever, in actual calculations, one never does an expansion to the\norder *N* (containing 2*N* terms for binary) since it is too\nlong to be practical. In fact, since the interactions between widely\nseparated atoms are expected to be weaker than the interactions between\nnearer atoms for most of the important properties, the expansion in is\nusually truncated at certain distance to include only a few short-ranged\npair (2-site), triple (3-site), and up to the most, the quadruple\n(4-site) interactions.\n\nOnce a configuration is assigned, the *S*\u2019s are just the geometrical\nfactors. The common practice in the cluster expansion is that i) perform\nfirst-principles calculations of a selected set of configurations\n(around 20-100); ii) evaluate the values of *J*\u2019s using with the\nenergies from i); iii) use the fitted *J*\u2019s to predict the energy for a\ndesired set of configurations; and iv) make the ensemble average at a\ngiven temperature for the energetics of the random alloys through Monte\nCarlo simulations.\n", "metadata": {}}, {"id": "c8fd2663", "cell_type": "markdown", "source": "### Debye-Gr\u00fcneisen approximation to the vibrational contribution \n\nStrictly speaking, the Debye theory is only accurate at very low\ntemperatures. It assumes a parabolic type of dependence of PDOS\u00a0on the\nphonon frequency. This assumption is only correct at the scale of 10\u2019s K\nbecause at low temperatures,\u00a0only the low frequency acoustic phonons are\nactivated which play the major role for the parabolic type of PDOS for\nthe low frequency range as shown in . That is why\u00a0there are two kinds of\nDebye temperature: low-temperature Debye temperature and\nhigh-temperature\u00a0Debye temperature. The low-temperature Debye\ntemperature can be strictly derived by fitting low temperature heat\ncapacity data. The high-temperature\u00a0Debye temperature is mostly a\nphenomenological fitting parameter.\n\nThe Debye model approximates the PDOS in by\n\n*Eq. 5\u201138* ,\n\nwhere is the so-called Debye cutoff frequency related to the Debye\ntemperature as\n\n*Eq. 5\u201139* .\n\nAs the result, the vibrational contribution to Helmholtz energy under\nthe Debye approximation becomes,\n\n*Eq. 5\u201140*\n\nwhere *D*(*\u0398D/T*) is the Debye function given by .\n\nUnder the Debye approximation, the formulation to calculate the entropy\nbecomes\n\n*Eq. 5\u201141* ,\n\nthe formulation to calculate the internal energy becomes\n\n*Eq. 5\u201142* ,\n\nwhere\n\n*Eq. 5\u201143* ,\n\nand the formulation to calculate the heat capacity at constant volume\nbecomes\n\n*Eq. 5\u201144* .\n\nHere it is noted that is volume dependent which is often written in\nterms of Gr\u00fcneisen constant:\n\n*Eq. 5\u201145* .\n\nIt has been found that the dependence of on *V* usually is weak and\nhence is often approximated as a constant. With , the formulation to\ncalculate the pressure becomes\n\n*Eq. 5\u201146* .\n\nAn important result of the Debye approximation is that when , together\nwith , the heat capacity in is reduced to\n\n*Eq. 5\u201147* .\n\nThis gives the Debye T3 law when the thermal electron\ncontribution is neglected. In the analysis of superconductor and heavy\nFermion materials, is often rewritten as\n\n*Eq. 5\u201148* ,\n\nwhich is more convenient for examining the heat capacity measured at\nextremely low temperatures.\n", "metadata": {}}, {"id": "bb53a123", "cell_type": "markdown", "source": "### Density functional theory (DFT) and 0 K Kohn-Sham equations\n\nThe density functional theory advocates that the properties of a matter\nare solely dictated by its electronic density distribution (or equally\nsay charge density), , in the real space. This is to say, that for an\narbitrary the total energy of the system, *E*, is always larger or equal\nto a value, *E0*, called as the ground state energy:\n\n*Eq. 5\u201186* .\n\nIn terms of variational principle, is equivalent to\n\n*Eq. 5\u201187* .\n\nKohn and Sham \\[8\\] proposed to write the total energy as\n\n*Eq. 5\u201188* ,\n\nwhere represents the kinetic energy of the system, is the external\npotential acting on the system, is the Hartree energy, and is the\nso-called exchange-correlation energy with where and represent the\ncharge density of electrons with spin down and spin up, respectively .\nUsing\n\n*Eq. 5\u201189* ,\n\ntogether with the variational principle of , one can get the\none-electron Schr\u00f6dinger equation\n\n*Eq. 5\u201190*\n\nwhere *me* represents the mass of an electron and\n\n*Eq. 5\u201191*\n\nso that the total energy is obtained as\n\n*Eq. 5\u201192*\n\nThe major challenge within DFT is that the accurate formulation of the\nexchange-correlation energy is unknown. Except for the uniform electron\ngas, no exact analytical form for the exchange-correlation energy has\nyet been obtained. Therefore approximations must be made for the\nexchange-correlation energy in calculating a realistic system. Until\nnow, the two most popular approximations are the local density\napproximation (LDA) \\[30\\] and the generalized gradient approximation\n(GGA) \\[31-32\\].\n\nThe local density approximation (LDA) states that the\nexchange-correlation energy is the same as that for a locally uniform\nelectron gas. In this case one can write *Vxc* as\n\n*Eq. 5\u201193* .\n\nAlthough this approximation is extremely simple, it works reasonably\nwell for many systems. The only remaining problem is to find an\napproximate solution to . One of most employed parameterized expression\nfor is that by Perdew and Zunger \\[30\\].\n\nMany modern DFT codes use the more advanced generalized gradient\napproximation (GGA) \\[31-32\\] to the exchange-correlation energy to\nimprove accuracy for certain physical properties. As the LDA\napproximates the energy of the true density by the energy of a local\nconstant density, it fails in situations where the density undergoes\nrapid changes such as in molecules. An improvement to this can be made\nby considering the gradient of the electron density. Symbolically, this\ncan be written as\n\n*Eq. 5\u201194*\n\nThe commonly used GGA is those due to Perdew et al. \\[31-32\\].\n", "metadata": {}}, {"id": "0cf3bd9c", "cell_type": "markdown", "source": "### Dynamical matrix and phonon mode\n\nBecause of the periodicity of a crystal, one can make an initial guess\nthat the solutions of are elastic plane waves made of a collective\natomic vibrations \\[23, 33\\], from the harmonic approximation of ,\n\n*Eq. 5\u2011122* ,\n\nwhere *\u03c9* represents the frequency of the plane wave, and is a wave\nvector designating the wave number and direction along which the plane\nwave propagates. It should be pointed out that in is now independent of\nthe index *P.* That is, except for a phase factor, the atoms that are\nequivalent by the translational symmetry among different primitive unit\ncells will experience the same type of atomic motion, independent of the\npositions of the primitive unit cell in the system. This is equivalent\nto applying the periodic condition so that in obeys\n\n*Eq. 5\u2011123* .\n\nNote that is now independent of the index *P*.\n\nFurthermore, one wants to limit **q***t* in and to those\nknown as the exact wave vectors which represent a special set of points\nin the reciprocal space that satisfy the condition\n\n*Eq. 5\u2011124* ,\n\nwhere is the Kronecker delta function. In fact, the number of equals to\nthe number of primitive unit cells contained in the system.\n\nUtilizing the translational invariance by which in \\[or in \\] depends on\n*P* and *Q* only through the difference , the following Fourier\ntransformation can be employed to simplify\n\n*Eq. 5\u2011125* ,\n\nand one obtains\n\n*Eq. 5\u2011126* .\n\nThe counterpart of with respect to is\n\n*Eq. 5\u2011127*\n\nis now an equation with 3*Na* degrees of freedom. At each ,\none can always find 3*Na* eigenvalues of (*i* = 1, \u2026,\n3*Na*). The 3*Na* vibrations are often known as\nphonon modes, noting again that *Na* is the number of atoms\nin the primitive unit cell. Each of yields a set of which can be chosen\nsuch that\n\n*Eq. 5\u2011128* ,\n\nwhere represents the complex conjugate of , and\n\n*Eq. 5\u2011129*\n\nFinally, for a solid, the summation over *Q* in can be abbreviated out,\nresulting in, after transforming back to together with using\n\n*Eq. 5\u2011130* ,\n\nbecause of translational invariance by which depends on *P* and *Q* only\nthrough the difference .\n\nIn the case that a system is built by a parallelepiped multiplication of\nthe primitive unit cell with lattice vectors of **a***\u03b1*\n(\u03b1=*x*, *y*, and *z*) in the form , in and takes the form\n\n*Eq. 5\u2011131* ,\n\nwhere = 0, 1, \u2026, , and is the primitive lattice vector in the reciprocal\nspace as\n\n*Eq. 5\u2011132* ,\n\nwith *Va* representing the volume of the primitive unit cell\ngiven as\n\n*Eq. 5\u2011133* .\n\nCombine and , it is easy to demonstrate\n\n*Eq. 5\u2011134* ,\n\nwhere is the Kronecker delta symbol.\n\nIt can be shown that the **q** points defined in represent the exact\nwave points. First, can be written as\n\n*Eq. 5\u2011135* ,\n\nwhere = 0, 1, \u2026, . Then, the left hand side of for the definition of\nexact wave vector becomes\n\n*Eq. 5\u2011136* .\n\nKnowing the fact that\n\n*Eq. 5\u2011137* .\n\nHence, it is proved the **q** points defined in are the exact wave\npoints.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/firstprinciples_calculations_and_theory/firstprinciples_approaches_to_disordered_alloys.ipynb b/psu410/src/firstprinciples_calculations_and_theory/firstprinciples_approaches_to_disordered_alloys.ipynb new file mode 100644 index 0000000..91c43c9 --- /dev/null +++ b/psu410/src/firstprinciples_calculations_and_theory/firstprinciples_approaches_to_disordered_alloys.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "9965d400", "cell_type": "markdown", "source": "## First-principles approaches to disordered alloys\n\nFirst-principles calcuations discussed so far strictly rely on the exact\natomic positions in the unit cells. A brutal-force approach for random\nsolution phase would be to directly construct a large supercell and\nrandomly decorate the host lattice with different types of atoms. Such\nan approach would necessarily require very large supercells to\nadequately mimic the statistics of the random solutions. Since\nfirst-principles methods are computationally constrained by the number\nof atoms that one can treat, this brute-force approach is\ncomputationally prohibitive. Take a binary\nA1-*x*B*x* substitutional alloy as illustrated in\nas an example, for a system containing *N* atoms, there can be\n2*N* possible number of configurations, which is an\nastronomically large number when *N* is large. It is an impossible task\nto explore such a huge configuration space with available computing\nresources.\n\nFigure \u2011. Mapping of a substitutional A1-xBx alloy\ninto an Ising-like lattice model \\[39-40\\].\n\nAs a result, approximations must be made to the first-principles\ncalculations. At present, there are mainly three approaches to\ncalculating the disordered solution phases: the coherent potential\napproximation (CPA) \\[41\\], the cluster expansion (CE) \\[42\\], and the\nspecial quasirandom structures (SQS\u2019s) \\[43\\] approach.\n\nCPA \\[41\\] treats random alloys by considering the *average* occupations\nof lattice sites in solving the Kohn-Sham equation. Since a mean-field\napproach is employed, dependence of properties on the local environments\nsurrounding an atom is not treated explicitly in CPA. In a random\nsolution, there exists a distribution of local environments (e.g., in\nbcc alloys, A or B surrounded by the various\nA*X*B8-*X* coordination shells with *X* between 0\nand 8), resulting in local environmentally-dependent quantities such as\ncharge transfer and local displacements of atoms from their ideal\nlattice positions. Even in random A1-*x*B*x* solid\nsolutions, the average A-A, A-B and B-B bond lengths are generally\ndifferent. These effects can be considered by the CE and SQS approaches,\nwhich are the focus of the next two subsections. In following\nsubsections, unless specifically noted, the formulism for the binary\nsystem is discussed for the sake of simplicity.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/firstprinciples_calculations_and_theory/firstprinciples_formulation_of_thermodynamics.ipynb b/psu410/src/firstprinciples_calculations_and_theory/firstprinciples_formulation_of_thermodynamics.ipynb new file mode 100644 index 0000000..55da089 --- /dev/null +++ b/psu410/src/firstprinciples_calculations_and_theory/firstprinciples_formulation_of_thermodynamics.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "ff0a17fb", "cell_type": "markdown", "source": "## First-principles formulation of thermodynamics\n", "metadata": {}}, {"id": "f4ca09c6", "cell_type": "markdown", "source": "### Formation enthalpy of Ni3Al\n\nOne can do the similar calculations for the elemental metal Al and the\ncompound Ni3Al which has the L12 structure,\nfollowing the same steps in the calculations of Ni. The formation\nenthalpy in the unit of per mole atom can be calculated as\n\n*Eq. 5\u201115* ,\n\nwhere , , and represent the enthalpies of Ni3Al, Ni, and Al\nin the energy unit of per mole atom, respectively. Plotted in is the\ncalculated formation enthalpy of Ni3Al (curve) as a function\nof temperature from 0 to 1600 K at *P* = 0 compared with experimental\ndata (symbols) with details in Ref. \\[16\\].\n\n.\n\nFigure \u2011. Formation enthalpy of L12-Ni3Al with\nrespect to pure fcc Ni and Al.\n", "metadata": {}}, {"id": "66decf35", "cell_type": "markdown", "source": "### Hartree-Fock approximation to solve Schr\u00f6dinger equation\n\nIt was Hartree who first assumed that the electron wave function in can\nbe expressed as a product of a collection of *N* independent\none-electron wave functions, where *i* = 1, 2, \u2026, *N* with *N* being the\nnumber of electrons in a system, in terms of the its space coordinate\n**r** and spin state **s**. After that, Fock modified the Hartree\napproximation by considering the fact that the wave function of a\nmulti-fermionic system should satisfy anti-symmetry requirements and\nsubsequently the Pauli exclusion principle that the total wave function\nchanges sign upon the exchange of fermions. Accordingly, the wave\nfunction of *N* electrons system under the Hartree-Fock approximation is\nexpressed as the Slater determinant \\[29\\]\n\n*Eq. 5\u201172*\n\nFor brevity, one can use the atomic unit that treats in , so that\n\n*Eq. 5\u201173*\n\nwhere , and\n\n*Eq. 5\u201174*\n\nAccordingly, the total energy of the system is expressed as\n\n*Eq. 5\u201175*\n\nwhere *Jij* is called as Coulomb/Hartree term\n\n*Eq. 5\u201176*\n\nwhere *Kij* is called as exchange term\n\n*Eq. 5\u201177* ,\n\nwhere = 1 if spin **s***i* and **s***j* points to\nthe same direction and = 0 if spin **s***i* and\n**s***j* points to the opposite direction.\n\nBy utilizing the variational condition of , one gets\n\n*Eq. 5\u201178* ,\n\nwhere is called one-electron energy, and\n\n*Eq. 5\u201179* ,\n\nwith being the electronic charge density whose expression is\n\n*Eq. 5\u201180* ,\n\n*Eq. 5\u201181* .\n\nIt should be especially noted here that to solve the Hartree-Fock\nequation , the most time consuming part is due to the nonlocal exchange\nterm, knowing the fact that the being evaluated one-electron wave\nfunction is also contained in the expression in the left hand side of by\nmeans of .\n\nThe configurational interaction method is the generalization of the\nHartree-Fock approximation. In such a case, *Y*, the number of\none-electron wave functions can be larger than the number of electrons,\n*N*, in the system. Accordingly, from the number of one-electron wave\nfunctions, , *y* = 1, 2, \u2026, *Y*, one can build the number of Stater\ndeterminants, *M*, by the combinatorial mathematics that the maximum of\n*M* can be\n\n*Eq. 5\u201182* .\n\nAs a result, the wave function of a collection of *N* electron system\nbecomes the recombination of the *M* Stater determinants as\n\n*Eq. 5\u201183* ,\n\nwhere the coefficients is to be found from the multiple linear equation\n\n*Eq. 5\u201184* .\n\nThe matrix element in *Eq. 5\u201184* is determined by the integral\n\n*Eq. 5\u201185* .\n", "metadata": {}}, {"id": "57a76f3e", "cell_type": "markdown", "source": "### Helmholtz energy\n\nIn this chapter so far, all our discussions are limited to the case of a\nsystem which is made from a single microstate (microscopic state). Here,\nthe terminology microstate refers to the microscopic structure that is\ndistinguished by crystal structure, atom distributions in the lattice\nsites, and the arrangements of the local atomic spin and electronic\nangular momentum distributions among the lattice sites. From this\nsection on, the index \u03c3 is employed to label the microstate. For a solid\nmaterial at finite temperatures, a phase can be formed by a single\nmicrostate or a mixture of multiple microstates.\n\nLet us consider a canonical system made of *N* atoms with an average\natomic volume *V*. The study is limited to the motions of atomic nuclei\nand electrons. For such a system, one can use to index the energy\neigenvalues of the corresponding microscopic Hamiltonian associated with\nmicrostate *\u03c3*. The subscript **g** symbolically labels the different\nvibrational states for the motions of atomic nuclei and the subscript\n**n** symbolically labels the electronic states distinguished by the\ndifferent distributions of the electrons between the electronic valence\nand conduction bands. Neglecting electron-phonon coupling, one can\nassume that the contributions to between the vibrational and electronic\nstates are additive, so that\n\n*Eq. 5\u201116* ,\n\nwhere *Ec* is the static total energy of the microstate *\u03c3*.\nNote that in and represent the energetics of the vibrational state and\nthe electronic state respectively.\n\nOne then can formulate the canonical partition function of the\nmicrostate *\u03c3* at the given temperature *T* and volume *V* as\n\n*Eq. 5\u201117* ,\n\nwhere *\u03b2* = 1/*kBT* with *kB* being the\nBoltzmann\u2019s constant. As a result, with, the Helmholtz energy *F* per\natom for the microstate *\u03c3* is derived as follows:\n\n*Eq. 5\u201118* ,\n\nwhere the variable *N* has been abbreviated using\n\n*Eq. 5\u201119* ,\n\n*Eq. 5\u201120* ,\n\n*Eq. 5\u201121* .\n\nThe calculation of *Ec* is straightforward in most of the\nexisting first-principles codes as discussed earlier.\n", "metadata": {}}, {"id": "c9b00001", "cell_type": "markdown", "source": "### Helmholtz energy and quasiharmonic approximation\n\nAt present, the most rigorous method to predict the thermodynamic\nproperties of a material at finite temperatures is the phonon approach.\nIn such an approach, the microscopic Hamiltonian is expanded up to the\nsecond order. All the thermodynamic quantities are calculated using\nformulations derived from the statistical physics without further\napproximation. The great importance of the phonon theory is that all the\ninput parameters can be obtained by means of first-principles\ncalculations without using any phenomenological parameters.\n\nLet us consider a system with an averaged atomic volume *V*. Neglecting\nthe electron phonon coupling, it is a well demonstrated procedure \\[16\\]\nto decompose the Helmholtz energy *F*(*V*,*T*) of the system at\ntemperature *T* into three additive contributions as follows\n\n*Eq. 5\u20111* ,\n\nwhere *Ec* is the static total energy which is the total\nenergy of the system at 0 K without any atomic vibrations,\n*Fvib* is the vibrational contribution due to the lattice\nions, and *Fel* is the electronic contribution due to the\nthermal electronic excitation at finite temperature which can become\nimportant for metals at high temperature.\n\nThe terminology of \u201cquasiharmonic approximation\u201d arises from the fact\nthat for a given volume, *Fvib*(*V*,*T*) is calculated under\nthe harmonic approximation and the anharmonic effects are carried out\nsolely through volume dependence of the phonon frequency. The easiest\ncomputational implementation of is to first independently calculate the\nHelmholtz energy at several selected volumes near the equilibrium volume\nand then use the numerical interpolation to find the Helmholtz energy at\nan arbitrary volume. The volume interval is usually at the scale of 3~5%\nof the equilibrium volume. Too small volume interval can result in\nnumerical instability due to the numerical uncertainties in the static\ntotal energy calculation, in particular, when one numerically computes\nthe first- and especially the second-order derivatives of the Helmholtz\nenergy in deriving the thermodynamic quantities. It should be noted that\nwhenever available, analytic formulas should be used instead of the\nnumerical second-order derivative to avoid numerical errors. For\ninstance, when the phonon approach is employed, the constant volume heat\ncapacity has the analytic expression in terms of phonon density of\nstates.\n\nNickel metal adopts the fcc structure at ambient conditions and the\nprimitive unit cell contains one atom. Almost all the existing\nfirst-principles codes have the function to calculate the static total\nenergy. The static total energy *Ec* in should be calculated\nusing the primitive unit cell. As the Helmholtz energy is to be\ncalculated at several volumes, a good practice is to plot the calculated\nstatic total energy points together with the interpolated energy curve\nto examine the convergence of the static total energy calculation. Since\nthe first-principles method often employs the self-consistent technique,\nit could occur that calculations at certain volumes may not convergent,\nwhich should be fixed by trying the various algebraic schemes provided\nin most of the existing codes. Furthermore, since certain calculations\ninvolve the second order derivative of the Helmholtz energy, a minor\nuncertainty along the static total energy curve can result in large\ndeviation for the calculated properties such as thermal expansion\ncoefficient and bulk modulus. In that case, a reasonable solution is to\nsmoothen the static total energy using the modified Birch-Murnaghan\nequation of states (EOS) \\[17-18\\]\n\n*Eq. 5\u20112* .\n\nPlotted in is the calculated static total energy of the elemental metal\nNi with the circles representing the calculated values and the curve\nrepresenting that by EOS fitting.\n\nFigure \u2011. Static total energy of nickel.\n\nThe vibrational contribution to the Helmholtz energy by phonon theory\ncan be computed by \\[19\\]\n\n*Eq. 5\u20113* ,\n\nwhere is the Boltzmann\u2019s constant, *\u03c9* represents the phonon frequency,\nand is the phonon density of states. It is recommended that is\ncalculated at the same volume set at which the static total static\nenergies are calculated.\n\nFor the present prototype of Ni, the supercell method for the\ncalculation of has been employed. The procedure is follows:\n\n1. Make supercell by enlarging the primitive unit cell according to the\n > defined neighbor interaction distance; Employ the first-principles\n > code (VASP, \\[13-14\\] in this work) to calculate the interatomic\n > force constants.\n\n2. Assign the mesh in the wave vector (**q**) space; Make the dynamical\n > matrix at each **q** point; Diagonalize the dynamical matrix to\n > find out the phonon frequencies at each **q** point; And finally\n > collect all the phonon frequencies for all **q** points. The\n > detailed formulation for phonon calculations is given in Chapter .\n\nFor the phonon calculations, one can use the open source code YPHON\n\\[15\\] by the present authors. Other choices can be the free ATAT code\n\\[20\\] or the free PHON code \\[21\\]. For the calculation of the phonon\ndensity of states, we have made a supercell containing 64 atoms which is\na 4\u00d74\u00d74 supercell of the primitive unit cell. Plotted in is the\ncalculated phonon density of states using YPHON code at the calculated\nstatic equilibrium volume compared with the measured data at 10 K \\[22\\]\n(symbols).\n\nFigure \u2011. Phonon density of states of nickel.\n\nFor a first-principles thermodynamic calculation, an important step to\navoid possible calculation errors is to examine the phonon dispersions\nfirst. Phonon dispersion \\[23\\] depicts the evolution of phonon\nfrequencies along the designated direction for a crystal. Phonon\ndispersion can be measured rather accurately by inelastic neutron\nscattering \\[24-26\\] or inelastic x-ray scattering \\[27\\] experiment.\nPlotted in are the calculated phonon dispersions (curves) along the\n\\[00\u03b6\\], \\[0\u03b61\\], \\[0\u03b6\u03b6\\], and \\[\u03b6\u03b6\u03b6\\] directions of Ni using YPHON code\ncompared with the neutron scattering data at 296 K (symbols) with\ndetails in Ref. \\[16\\].\n\nFigure \u2011. Phonon dispersions of nickel. The solid lines represent\nresults calculated using a supercell containing 256 atoms which is 4\u00d74\u00d74\nsupercell of the conventional cubic unit cell. The dot-dashed lines\nrepresent results calculated using a supercell containing 64 atoms which\nis 4\u00d74\u00d74 supercell of the primitive unit cell.\n\nFor the calculation of *Fel* in , the most computationally\nconvenient approach is to use the Mermin statistics as follows\n\n*Eq. 5\u20114* ,\n\nwhere is the thermal electronic energy, and *Sel* is the bare\nelectronic entropy. Both the calculations of and *Sel* need\nthe electronic density of states (EDOS) as input. The electronic density\nof states can be obtained during the step of the static total energy\ncalculation. The detailed formulations for and *Sel* are\ngiven in Chapter . Since Ni is magnetic, the EDOS of Ni can be\npartitioned into those of spin up and spin down due to the spin freedom\nof electron. The calculated EDOS for Ni is shown in where the solid,\ndot-dashed, and dashed lines represent the total, spin up, and spin down\nEDOS with the Fermi energy set to zero.\n\nFigure \u2011. Electronic density of states of nickel. That due to spin up is\nplotted as positive value and that due to spin down is plotted as\nnegative value purely for the clarity of the figure. The \u201ctotal\u201d is the\nsum of the absolute values of those of spin up and spin down.\n\nThe calculated temperature evolution of Helmholtz energy as a function\nof volume for Ni are illustrated in . The circles represent the\ncalculated static total energies. The solid curves represent the\nHelmholtz energy curves from 0 to 1600 K at a temperature increment of\n100 K as displayed from top to bottom in . The dashed line marks the\nevolution of the equilibrium volume at *P*=0 with increasing\ntemperature. It is noted that Helmholtz energy always decreases with\nincreasing temperature due to the entropy term of \u2013*TS*. Note that the\nat 0 K the Helmholtz energy is higher than the static total energy due\nto the zero point vibrational energy as can be seen when *T* \u21920 which\nreduces to\n\n*Eq. 5\u20115* ,\n\nwhich is positive.\n\nFigure \u2011. Temperature evolution of the Helmholtz energy for nickel.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/firstprinciples_calculations_and_theory/index.ipynb b/psu410/src/firstprinciples_calculations_and_theory/index.ipynb new file mode 100644 index 0000000..e250c93 --- /dev/null +++ b/psu410/src/firstprinciples_calculations_and_theory/index.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "e58b1f1e", "cell_type": "markdown", "source": "# First-principles calculations and theory\n\nIn the previous chapter, the experimental techniques in obtaining the\nthermochemical and phase equilibrium data that are the inputs for the\nthermodynamic modeling of a system were summarized. However,\nexperimental data are not always available. This is due to the fact that\ni) the experiments are expensive, especially true in developing new\nmaterials; and ii) the experiments cannot reliably access the non-stable\nphases in most cases. The alternative approach is to predict the\nthermochemical data by first-principles calculations. The prediction of\nthe material properties, without using phenomenological parameters, is\nthe basic spirit of first-principles calculations. In particular, the\nsteady increase of both computer power and the efficiency of\ncomputational methods have made the first-principles predictions of most\nthermodynamic properties possible, including both enthalpy and entropy\nas a function of temperature, volume, and/or pressure.\n\nBy its definition, the term of \u201cfirst-principles\u201d represents a\nphilosophy that the prediction is to be based on a basic, fundamental\nproposition or assumption that cannot be deduced from any other\nproposition or assumption. This implies that the computational\nformulations are based on the most fundamental theory of quantum\nmechanics - Schr\u00f6dinger equation or density functional theory and the\ninputs to the calculations must be based on well-defined physical\nconstants \u2013 the nuclear and electronic charges. In another word, once\nthe atomic species of an assigned material are known, the theory should\npredict the energy of all possible crystalline structures, without\ninvoking any phenomenological fitting parameters.\n\nThis chapter organized in the sequence from thermodynamic calculations\nto fundamental theory to help those readers who are more interested in\nrealistic calculations using existing computer codes. The detailed\ntheoretical sections are presented to follow the section of\nthermodynamic calculations for those readers who are also interested in\nthe derivation of the formulations used in the thermodynamic\ncalculations. The following subsections are arranged accordingly in the\norder: (i) examples the commonly adopted calculation procedures for\nthermodynamic properties using the elemental metal nickel as the main\nprototype; (ii) derivation of the Helmholtz energy expression under the\nfirst-principles framework; (iii) introduction of the solution to the\nelectronic Schr\u00f6dinger equation within the two well developed frameworks\n\u2013 the quantum chemistry approach and the density functional theory; (iv)\ndetailed description of the procedure on how to solve the Schr\u00f6dinger\nequation for the motions of the atomic nuclei by means of lattice\ndynamics. The relation between the Helmholtz energy and Gibbs energy is\nshown by .\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/firstprinciples_calculations_and_theory/lattice_dynamics.ipynb b/psu410/src/firstprinciples_calculations_and_theory/lattice_dynamics.ipynb new file mode 100644 index 0000000..40cd460 --- /dev/null +++ b/psu410/src/firstprinciples_calculations_and_theory/lattice_dynamics.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "0a5cf3ba", "cell_type": "markdown", "source": "## Lattice Dynamics\n", "metadata": {}}, {"id": "fa5e392d", "cell_type": "markdown", "source": "### Linear-response method vs supercell method\n\nThe problem of lattice vibration for a solid is now transformed into\ncomputing the dynamical matrix in . The first-principles solution of the\nproblem is currently divided into two categories: the linear-response\nmethod \\[34\\] and the supercell method \\[35\\]. In the linear-response\nmethod, utilizing the electronic linear response upon the undistorted\ncrystals \\[36\\], the evaluations of the dynamical matrix can be\nperformed through the density-functional perturbation theory \\[34\\]\nwithout the approximation of the cutoff in neighboring interaction.\n\nCompared with the linear-response method, the supercell method is\nconceptually simple and is easy to implement computationally. The\nsupercell method adopts the frozen phonon approximation through which\nthe changes in total energy or forces are calculated in the direct space\nby displacing the atoms from their equilibrium positions. The advantage\nof the supercell method is that the phonon frequencies at the exact wave\nvectors, which are commensurable with the supercell, are calculated\nexactly with no further approximation \\[37\\]. The shortcoming of the\nsupercell method is that it is often limited by the size of the\nsupercell that can be handled with current computing resources.\n\nIn supercell approach, inaccuracies are thought to arise from the\ntruncation of the force constants \\[34-35\\]. This is only partially\ntrue. In the supercell method, the calculated represents the cumulative\ncontributions of the atom indexed by *k* and *P* in the supercell and\nall its images by translational transformation of the supercell in the\nwhole space. Let **L**\u03b1 represent the lattice vectors of the\nsupercell, then\n\n*Eq. 5\u2011138* .\n\nFor the exact wave vectors in , one has\n\n*Eq. 5\u2011139* **,**\n\nwhere is an integer. Replacing in with in , one obtains\n\nEq. 5\u2011140 .\n\nTherefore, the phonon frequencies calculated at the exact wave vectors\nby the cumulative force constants approach are exact, and the supercell\nsize will not lead to errors in the calculated phonon frequencies\n\\[37\\].\n\nGenerally speaking, if a supercell contains *Nc* primitive\nunit cells, one can always find *Nc* corresponding exact wave\nvectors in the **q** space. In most linear response calculations, the\ncommon choice of a 4\u00d74\u00d74 **q**-mesh is exactly equivalent to the 4\u00d74\u00d74\nsupercell in the real-space. Furthermore, since the supercell approach\ndoes not impose any approximation on the electronic response to the\ndistortion of the lattice geometry, the effects of electron-phonon\ninteractions can be accounted for by the supercell method.\n\nIn the supercell method, due to the impose of periodic condition, the\ncalculated force constant in the real-space cannot account for the\neffects of the vibration-induced electric field for the polar materials.\nIt has been demonstrated that such an effect adds an additional term to\ndynamical matrix in the reciprocal-space in the form,\n\n*Eq. 5\u2011141* ,\n\nwhere ***Z**\\* (j)* represents the Born effective charge\ntensor of the *j*th atom in the primitive unit cell and\n**\u03b5\u221e** the high frequency static dielectric tensor, i.e., the\ncontribution to the dielectric permittivity tensor from the electronic\npolarization. As a result for the polar materials, the matrix element at\nwave vector **q**\u21920 of the dynamical matrix in , by means of , should\nhave the form\n\n*Eq. 5\u2011142* .\n\nIt can be demonstrated \\[38\\] that this is equivalent to replacing the\nreal space force constant in by\n\n*Eq. 5\u2011143* .\n\nThe present implementation of the first-principles method in calculating\nthe phonon frequencies are mostly limited by the supercell size when\nusing or the number of exact wave vector points when using . A supercell\ncell built on the primitive unit cell or exact wave vector mesh is\nusually the common limit. If only the phonon frequencies derived from or\nare used, can be rather unsmooth which will result in inaccurate\nthermodynamic properties when it is used with . The mixed-space approach\n\\[38\\] can circumvent this bottleneck by using the Fourier interpolation\nto calculate the dynamical matrix for an arbitrary wave vector **q** as\n\n*Eq. 5\u2011144* ,\n\nwith the help of , , and for a polar materials. Note that the term is\nfor polar materials only.\n\nThe can be calculated as\n\n*Eq. 5\u2011145* ,\n\nwhere the function is usually taken as the Gaussian type\n\n*Eq. 5\u2011146* ,\n\nwhere is an adjustable damping (broadening) parameter whose role is to\nsmear the curve. in is the number of used **q** points. Empirically, a\n**q** mesh of is accurate enough for most purposes and doable with the\nYPHON code \\[15\\] efficiently, which is discussed in the Appendix A.\n", "metadata": {}}, {"id": "b4b6a83a", "cell_type": "markdown", "source": "### Mermin statistics to the thermal electronic contribution \n\nFor the calculation of *Fel* in , the most computationally\nflexible approach is to use the Mermin statistics \\[28\\] by which\n\n*Eq. 5\u201122*\n\nwhere the bare electronic entropy *Sel* takes the form, after\nreplacing the summation in over the electronic states with integration\n\n*Eq. 5\u201123* ,\n\nby means of utilizing , the electronic density of states *n*. *f* in is\nthe Fermi distribution that takes the form\n\n*Eq. 5\u201124* .\n\nNote that in is the electronic chemical potential that is strongly\ntemperature dependent. At each given *T*, should be carefully calculated\nkeeping the number of electrons unchanged in solving the following\nequation:\n\n*Eq. 5\u201125* ,\n\nnoting that is the Fermi energy calculated at 0 K. With respect to , the\nthermal electronic energy *Eel*, due to the thermal electron\nexcitations, can be calculated through\n\n*Eq. 5\u201126* .\n\nAt low temperature, , , and are reduced to\n\n*Eq. 5\u201127*\n\nwhere *\u03bb* is the so-called electronic specific heat coefficient\ncalculated as\n\n*Eq. 5\u201128* ,\n\nwhere is the electronic density of states at the Fermi level, and\n\n*Eq. 5\u201129*\n\n*Eq. 5\u201130*\n\nFrom , one can easily derive the electronic contribution to the specific\nheat at low temperature as\n\n*Eq. 5\u201131* .\n\nUsually, the dependence of on *V* is weak. Therefore for a normal\nconductor (except for the heavy Fermion metal or superconductor related\nmaterials), at low temperatures, the electronic contribution to the heat\ncapacity is linear against *T*. also indicates that for insulators, by\nmeans of , the electronic contribution to the heat capacity is zero\nsince for insulators .\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/firstprinciples_calculations_and_theory/nickel_as_the_prototype.ipynb b/psu410/src/firstprinciples_calculations_and_theory/nickel_as_the_prototype.ipynb new file mode 100644 index 0000000..dc17080 --- /dev/null +++ b/psu410/src/firstprinciples_calculations_and_theory/nickel_as_the_prototype.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "2f770a8c", "cell_type": "markdown", "source": "## Nickel as the prototype\n\nThis section exemplifies the step-by-step procedure in calculating the\nthermal properties within the framework of first-principles phonon\napproach, using the elemental metal Ni as the prototype. The calculation\nof the formation enthalpy of Ni3Al is given at the end. The\ncalculation in this section is limited to the case for the ferromagnetic\nphase, i.e. single microstate, implying that no configurational mixtures\nor magnetic phase transitions is considered which is discussed in\nChapter 5.2.5 and Chapter 6.\n\nThe Vienna Ab-initio Simulation Package (VASP) \\[13-14\\] has been\nemployed for electronic calculations, and the YPHON code \\[15\\] has been\nemployed for phonon calculations. VASP is a code based on the\npseudopotential approach to the density functional theory using plane\nwave function as the basis set, by which only the valence electrons are\nhandled explicitly and the core electrons are approximated by an\neffective pseudopotential. The same energy cutoff values, which\ndetermine the number of plane waves in the expansion of electronic wave\nfunction, have been used for Ni, Al, and Ni3Al. The rationale\nfor the derivations of the formulations used in this section is to be\ngiven in Chapter for readers who want to have an in-depth understanding\nof the physics behind the used formulations.\n", "metadata": {}}, {"id": "a104be41", "cell_type": "markdown", "source": "### Normal coordinates, eigenenergetics, and phonons\n\nOne way to simplify the solution to the Schr\u00f6dinger\u2019s equation for\nmotion of atomic nuclei is to follow the study of the vibrations of the\natoms in a crystal \u2013 lattice dynamics. The essential step in lattice\ndynamics is to transform the problem of the correlated motions of 3*N*\nparticle into the problem of 3*N* independent harmonics. For this\npurpose, one can introduce a set of new coordinates ( *i*=1, 2, \u2026,\n3*N*), known as normal coordinates, by the transformation\n\n*Eq. 5\u2011111* ,\n\nwhere is the transformation coefficient that can be determined by\nsolving 3*N* simultaneous equations\n\n*Eq. 5\u2011112* ,\n\nwhere is to be determined so that one can find 3*N* , which are, not all\nzero. The equations are linear and homogeneous. Follow the basic theorem\nin linear algebra that, to find the nonzero solutions of the equations,\nthe determinant formed by must equal zero\n\n*Eq. 5\u2011113* ,\n\nwhere is the Kronecker delta symbol. Since is an equation with 3*N*\ndegrees, one can always find 3*N* values of (*i* = 1, \u2026, 3*N*). Each of\nyields a set of which can be chosen such that\n\n*Eq. 5\u2011114* ,\n\nwhere represents the Kronecker delta symbol and\n\n*Eq. 5\u2011115*\n\nThen with the normal coordinates defined in and utilizing , defined in\nis obtained as\n\n*Eq. 5\u2011116* .\n\nWith this equation, is simplified by the following process\n\n*Eq. 5\u2011117* ,\n\nnoting that in the above process and have been utilized.\n\nFurthermore, using , the kinetic energy operator in can be transformed\nas follows\n\n*Eq. 5\u2011118* ,\n\nnoting that in the above process and are utilized again.\n\nAs a result, under the harmonic approximation, the Hamiltonian in is\nsimplified as\n\n*Eq. 5\u2011119*\n\nwhich represents a quantum system containing 3*N* independent harmonics.\nCorresponding to the each of the *\u03c9i*, the quantum theory\ntells that the eigenenergy of a harmonics has the form\n\n*Eq. 5\u2011120* ,\n\nwith *gi* = 0, 1, \u2026, \u221e. Such a type of energetics behaves\nlike a Boson particle with energy and forms the concept of phonon.\n\nFurthermore, a state of the whole system is to be specified by the set\nof 3*N* independent quantum numbers **g** = (*g1*,\n*g1*, \u2026, *g3N*,). Finally, the energetics of a\nstate of the system formed by 3*N* independent harmonics, introduced by\n, is obtained with the summation of the energies of 3*N* independent\nharmonics as\n\n*Eq. 5\u2011121* .\n\nThis concludes the rationale by which how is derived.\n", "metadata": {}}, {"id": "be96a8e8", "cell_type": "markdown", "source": "### Phonon calculations for SQS\n\nA somewhat more theoretically demanding application of the SQS approach\nis the calculation of the phonon dispersions of a random alloy.\nConsidering the fact that the size of an SQS cell in general is around\n8-32, phonon calculations based on SQS is doable, either using the SQS\ncell or its supercell, e.g., of it. However, one notes that while the\nphonon density of states can be calculated straightforwardly, the\ncalculations of the phonon dispersions run into a problem. That is,\nsince phonon calculation treats the SQS as primitive unit cell made of\nmore atoms than the primitive unit cell of the ideal lattice, the number\nof phonon dispersions derived from a regular phonon calculations is a\nlot greater than that measured for the random alloy. Say, one uses an\nSQS containing 16 atoms for an fcc solid solution, the regular phonon\ncalculations would produce phonon dispersions in comparison to that just\nthree phonon dispersions from measurement. By averaging over the force\nconstants of a SQS, the dynamical matrix can be calculated with respect\nto the wave vector space of the ideal lattice of the random alloys.\n\nOne consideration that must be taken into account is that the phonon\ndispersions measured from the inelastic neutron scattering experiments\nonly represent the averaged vibrations of the ideal lattice. For random\nalloys or phases with minor geometry distortion, it is suggested to\ncalculate the dynamical matrix by instead of *Eq. 5\u2011125* as (see \\[45\\])\n\n*Eq. 5\u2011155* ,\n\nwhere in the case of random alloy, represents the averaged atomic mass\nat the *j*th lattice site. The purpose of the summation over *Q* is to\naverage the effects of local distortions, making it possible of\ncomparing the calculated dispersions to the measured dispersions\nrepresenting the averaged vibrations of the ideal lattice. As a result,\nthe dimension of the SQS supercell dynamical matrix is thus reduced to\nmatch that of the primitive unit cell of the ideal lattice for the\ncalculation of the phonon frequencies. The calculational procedure is as\nfollows:\n\n1. Make an SQS supercell based on the primitive unit cell of ideal\n lattice to mimic the correlation functions of the random solution;\n\n2. Relax the SQS supercell with respect to the internal atomic\n positions while keeping the cell shape and volume fixed;\n\n3. Make the phonon supercell by further enlarging the SQS supercell and\n calculate the force constants; and\n\n4. Calculate the dynamical matrix , with the wave vector, **q**, being\n defined from the primitive unit cell of the ideal lattice, through\n the following Fourier transformation.\n\nThe calculated phonon dispersions, along the directions (00\u03be), (0\u03be\u03be),\nand (\u03be\u03be\u03be), are compared with the inelastic neutron scattering data in\nfor Cu3Au.\n\nFigure 5\u201113. Phonon dispersions for random Cu3Au. The solid\n(black) lines represent the present calculation and the open circles\nrepresent the inelastic neutron scattering data with details in Ref.\n\\[45\\]. The dashed (blue) lines represent the calculated results using\nthe ab initio transferable force-constant model by Dutta et al. \\[46\\].\n", "metadata": {}}, {"id": "ad6b512f", "cell_type": "markdown", "source": "### Quantum theory for motion of atomic nuclei\n\nFor the convenience of discussion, the following convention of notations\nare used: *\u03b1* and *\u03b2* to label the Cartesian axes which is either *x*,\n*y*, or *z*, *j* and *k* to label atoms in the primitive unit cell,\n*mj* the atomic mass of the *j*th atom in the\nprimitive unit cell, **r**(*j*) the position of the *j*th\natom in the primitive unit cell, *P* and *Q* the index of the primitive\nunit cell in the system, **R**(*P*) the position of the *P*th\nprimitive unit cell in the system, and *V* the averaged volume of the\nprimitive unit cell.\n\nThe quantum theory for motion of atomic nuclei replicates closely the\nquantum theory for motion of electrons. That is to solve the wave\nfunction for the motions of the atomic nuclei for a Schr\u00f6dinger\u2019s\nequation with the potential being the total electronic energy derived\nfrom . The symbol is replaced by to represent the static total\nelectronic energy with **R** representing the static equilibrium\npositions of the atomic nuclei, **u** the displacements of atomic nuclei\naway from their static equilibrium positions, and *\u03c3* the additional\ndegree of freedom such as electronic states. The Schr\u00f6dinger\u2019s equation\nfor the motion of atomic nuclei is then\n\n*Eq. 5\u2011103* ,\n\nwhere\n\n*Eq. 5\u2011104* ,\n\nwith representing the kinetic energy operator\n\n*Eq. 5\u2011105* ,\n\nwhere *Nc* is the numbers of primitive unit cells in the\nsystem, *Na* is the numbers of atoms in the primitive unit\ncell, is the Plack constant, represents the *\u03b1*th Cartesian\ncomponent of **u** for the atom at *j*th lattice site in the\nprimitive unit cell and the *P*th primitive unit cell in the\nsystem.\n\nThe harmonic approximation \\[23, 33\\] truncates the into the second\norder in its Taylor\u2019s series\n\n*Eq. 5\u2011106* ,\n\nwhere is the real-space interatomic force constant. With the\napproximation of , it can be demonstrated that finding the solution of\nis equivalent to finding the vibrational frequencies of a classical\nsystem with *NcNa* particles for small mechanical\noscillations.\n\nLet us rewrite as\n\n*Eq. 5\u2011107* ,\n\nwhere\n\n*Eq. 5\u2011108* , and\n\n*Eq. 5\u2011109* .\n\nAccordingly, the kinetic energy operator in becomes\n\n*Eq. 5\u2011110* .\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/firstprinciples_calculations_and_theory/quantum_theory_for_the_motion_of_electrons.ipynb b/psu410/src/firstprinciples_calculations_and_theory/quantum_theory_for_the_motion_of_electrons.ipynb new file mode 100644 index 0000000..18d0e5c --- /dev/null +++ b/psu410/src/firstprinciples_calculations_and_theory/quantum_theory_for_the_motion_of_electrons.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "f37e1df9", "cell_type": "markdown", "source": "## Quantum theory for the motion of electrons\n", "metadata": {}}, {"id": "285a35ab", "cell_type": "markdown", "source": "### Schr\u00f6dinger equation\n\nThe Schr\u00f6dinger equation is typically written as follows\n\n*Eq. 5\u201157* ,\n\nwhere is the reduced Planck constant, **X** an abbreviation of the space\ncoordinates and spin states of the multiple particle system, the energy\noperator called Hamiltonian. When is independent of time *t*, one can\nseparate the coordinate **X** from the time *t* in finding the solution\nof by writing\n\n*Eq. 5\u201158* ,\n\nby which the stationary solutions of can be expressed through letting\n\n*Eq. 5\u201159* ,\n\nresulting in\n\n*Eq. 5\u201160*\n\n*Eq. 5\u201161*\n\nNote that is the frequency of de Broglie matter wave.\n\nFor any trial function \u039b(**X**) (in the Hilbert space) for \u03a8(**X**), the\nvariational principle tells us that the energy of the system always has\na lower bound through a ground state with energy *E0*, as\n\n*Eq. 5\u201162* ,\n\nwhere represents the complex conjugate of \u039b(**X**), resulting in\n\n*Eq. 5\u201163*\n\nwhich is known as the Rayleigh-Ritz variational principle,\n", "metadata": {}}, {"id": "6d31abbc", "cell_type": "markdown", "source": "#### Solving the Kohn-Sham Equations for a Solid\n\nFor a solid, is still a mathematical challenge with infinite number of\none-electron wave functions to be solved and therefore cannot be solved\ndirectly in the real space. To reduce the dimension of the problem, one\ncan choose to solve the equation at a specific point in the reciprocal\nspace. According to Bloch\u2019s theorem, the wave function for a solid can\nbe written as the product of a wavelike part, , and a cell periodic\npart,\n\n*Eq. 5\u201195*\n\nwhere can be expressed as a sum of a finite number of plane waves whose\nwave vectors are reciprocal lattice vectors of the crystal\n\n*Eq. 5\u201196* .\n\nso that\n\n*Eq. 5\u201197*\n\nwhere the band index *j* is used to number the eigenenergy and the\neigenvector at a given . The number of plane waves is determined by the\nfollowing equation\n\n*Eq. 5\u201198*\n\nwhere is called energy cutoff.\n\nUtilize the obtained wave functions, the charge density can be\ncalculated with the Brillouin zone integration\n\n*Eq. 5\u201199*\n\nWhere\n\n*Eq. 5\u2011100*\n\nwhere the parameter is to make the integration over the charge density\nwithin the primitive unit cell equal to the number of electrons, *N,* in\nthe primitive unit cell. Numerically, the integration can be\napproximated by summation over a set of discrete *k*-mesh as\n\n*Eq. 5\u2011101*\n\nwhere *N*BZ represents the number of points in the first\nBrillouin zone in the *k*-mesh. When a solid possesses symmetry, the\nsummation in the above equation can be further reduced to the summation\nover the irreducible Brillouin zone (IBZ).\n\n*Eq. 5\u2011102*\n\nwhere is a weight factor that represents the number of points that are\nequivalent to by space group symmetry.\n", "metadata": {}}, {"id": "1f27a649", "cell_type": "markdown", "source": "### Special Quasirandom Structures\n\nSQS\u2019s \\[43-44\\] are specially designed *small-unit-cell* periodic\nstructures with minimal number of atoms per unit cell, which closely\nmimic the most relevant, near-neighbor pair and multisite correlation\nfunctions of random substitutional alloys. The correlation functions are\nclassified by their *n*-site component \u201cfigures\u201d *f* = (*n,m*), where\nthe index *n* is called vortex for pair, triple, and quadruple\ncorrelations (*n* = 2, 3, and 4); *m* measures the correlation distance.\n\nIn the SQS approach, a distribution of distinct local environments is\nmaintained and their average corresponds to the random alloy. Thus, a\nsingle DFT calculation of an SQS can give many important alloy\nproperties (e.g. equilibrium bond lengths, charge transfer, formation\nenthalpies, etc.) that depend on the existence of those distinct local\nenvironments. The SQS approach has been used extensively to study the\nformation enthalpies, bond length distributions, density of states, band\ngaps and optical properties in semiconductor alloys. It is noted that\nthe CE approach can treat short-range ordering efficiently, while it is\nnot clear how the SQS approach can be used to represent short-range\nordering.\n\nThe key quantities in the SQS approach are the *n*-site correlation\nfunctions. Specifically, the 2-site correlation function corresponding\nto the 2-site component \u201cfigures\u201d (2,*m*) is\n\nEq. 5\u2011151\n\nwhere represents the total number of possible pairs with correlation\ndistance (neighboring distance) *Rij* being equal to *m*. The\n3-site correlation function corresponding to the 3-site component\n\u201cfigures\u201d (3,*m*) is\n\nEq. 5\u2011152\n\nwhere represents the total number of all possible 3-site \u201cfigures\u201d with\nthe correlation distance (size and shape) *Rijk* being equal\nto *m*. The 4-site correlation function corresponding to the 4-site\ncomponent \u201cfigures\u201d (4,*m*) is\n\nEq. 5\u2011153 ,\n\nwhere represents the total number of all possible 4-site \u201cfigures\u201d with\ncorrelation distance (size and shape) *Rijkl* being equal to\n*m*.\n\nWith a given supercell size *N*, the essential task of the SQS approach\nis to search through all configurations that approach as close as\npossible to the correlation functions of the perfectly random (*R*)\nstructure and for the binary system it is\n\nEq. 5\u2011154 .\n\nDescribing random alloys by small unit-cell periodical structures surely\nintroduces erroneous correlations beyond a certain distance. However,\nsince interactions between nearest neighbors are generally more\nimportant than interactions between more distant neighbors, SQS\u2019s can be\nconstructed in such a way that they exactly reproduce the correlation\nfunctions of a random alloy between the first few nearest neighbors,\ndeferring errors due to periodicity to more distant neighbors. The\npractical procedure could be that to find the structures that match the\n2-site correlation functions up to a given neighboring distance, and\nthen add the conditions matching the high order correlation functions up\nto certain correlation distance.\n\nAppendix B is a collection of the SQS\u2019s with a variety of compositions\nfor binary fcc, bcc, hcp, and L12 structures, for ternary\nfcc, bcc, and B2 structures, and Perovskite in the cubic ABO3\nstructure. The used format is that of VASP.\n", "metadata": {}}, {"id": "1c9ba376", "cell_type": "markdown", "source": "### System with multiple microstates (MMS model)\n\nFor a system made of multiple microstates, the total partition function\nis the summation over the partition functions of all microstates, in ,\nas\n\n*Eq. 5\u201149* ,\n\nwhere is the multiplicity of the microstate *\u03c3.* It is immediately\napparent that is the thermal population of the microstate *\u03c3*.\nFurthermore, with , one obtains\n\n*Eq. 5\u201150* .\n\nrelates the total Helmholtz energy, *F(N,V,T)*, of a system with mixing\namong multiple microstates and the Helmholtz energy, , of individual\nmicrostates. An important result of is the configurational entropy due\nto the mixing among multiple microstates, named as microstate\nconfigurational entropy (MCE) in this book,\n\n*Eq. 5\u201151* ,\n\nwhich makes the entropy of a system with mixing among multiple\nmicrostates as\n\n*Eq. 5\u201152* .\n\nSimilarly, one can get the heat capacity at constant volume of a system\nwith mixing among multiple microstates as\n\n*Eq. 5\u201153* ,\n\nwhere the configurational contributions to the heat capacity due to the\nmixing among multiple microstates is\n\n*Eq. 5\u201154* .\n\nMoreover, the isothermal bulk modulus of a system with mixing among\nmultiple microstates can be also computed similarly\n\n*Eq. 5\u201155* ,\n\nWith\n\n*Eq. 5\u201156*\n\nwith being the partial pressure of the microstate *\u03c3*. This multiple\nmicrostate model (MMS model) is used in Chapter to quantitatively\npredict thermal expansion anomalies.\n", "metadata": {}}, {"id": "39adb4e2", "cell_type": "markdown", "source": "### Vibrational contribution by phonon theory\n\nUnder the harmonic/quasiharmonic approximation, the lattice dynamics or\nphonon theory is currently the most established method. It truncates the\ninteraction potential up to the second order. In such a case, in can be\nexpressed in terms of phonon frequency as follows\n\n*Eq. 5\u201132* ,\n\nwhere the label **g** has the meanings of (*g1*,\n*g2*, \u2026, *g3N*) that *gj* can take any\ninteger values from zero to infinite.\n\nAs a result, is reduced to\n\n*Eq. 5\u201133* ,\n\nor equivalently,\n\n*Eq. 5\u201134* ,\n\nwhere an integration has been used to replace the summation in by means\nof introducing a function, , named the phonon density of states (PDOS)\nwhose integration over *\u03c9* equals to three per atom.\n\nAccordingly, the formulation to calculate the entropy becomes\n\n*Eq. 5\u201135* ,\n\nthe formulation to calculate the internal energy becomes\n\n*Eq. 5\u201136* ,\n\nand the formulation to calculate the heat capacity at constant volume\nbecomes\n\n*Eq. 5\u201137* .\n", "metadata": {}}, {"id": "8355a666", "cell_type": "markdown", "source": "### Volume, entropy, enthalpy, thermal expansion, bulk modulus, and heat capacity \n\nThe equilibrium volume *Veq* (*P*,*T*) at given *T* and *P*\ncan be obtained by finding the root of the following equation\n\n*Eq. 5\u20116* .\n\nThe dashed line in illustrates *Veq* (*P*,*T*) as a function\nof *T* from 0 to 1600 K at *P* = 0 for Ni.\n\nThe entropy can be calculated through *F* by\n\n*Eq. 5\u20117* .\n\nPlotted in is the calculated entropy (curve) of Ni as a function of\ntemperature from 0 to 1600 K at *P* = 0 compared with the recommended\ndata (symbols) with details in Ref. \\[16\\].\n\nFigure \u2011. Entropy of nickel as a function of temperature.\n\nBased on *F* and *S*, the enthalpy at given *P* and *T* can be computed\nas\n\n*Eq. 5\u20118* .\n\nPlotted in is the calculated enthalpy (curve) of Ni as function of\ntemperature from 0 to 1600 K at *P* = 0 comparing with the recommended\ndata (open circles) with details in Ref. \\[16\\]\n\nFigure \u2011. Enthalpy of nickel as a function of temperature.\n\nWith the equilibrium volume *Veq* (*P*,*T*) calculated by ,\nthe volume thermal expansion coefficient defined by can be calculated by\n\n*Eq. 5\u20119* .\n\nPlotted in is the calculated thermal expansion coefficient (curve) of\nnickel as function of temperature from 0 to 1600 K at *P* = 0 comparing\nwith experimental data (symbols) with details in Ref. \\[16\\]\n\nFigure \u2011. Volume thermal expansion coefficient of nickel as a function\nof temperature.\n\nThe bulk modulus of a material represents the substance's resistance to\nuniform compression. Depending on how the temperature varies during\ncompression, a distinction should be made between the isothermal bulk\nmodulus (constant temperature) and adiabatic bulk modulus (constant\nentropy or no heat transfer). As a matter of fact, most of the\nexperimental data are adiabatic whereas most of the published\ntheoretical data are isothermal.\n\nThe isothermal bulk modulus, as defined in terms of Gibbs energy shown\nby , can be calculated by\n\n*Eq. 5\u201110*\n\nBased on the isothermal bulk modulus, the adiabatic bulk modulus can be\ncalculated by\n\n*Eq. 5\u201111* ,\n\nwhere *CP* and *CV* represent the constant\npressure heat capacity and constant volume heat capacity respectively.\nPlotted in is the calculated bulk moduli (curves) of Ni as function of\ntemperature from 0 to 1600 K at *P* = 0. The experimental data are from\nultrasonic measurements (symbols, see Ref. \\[28\\] for more details) that\nare therefore adiabatic bulk moduli calculated based on the measured\nadiabatic elastic constants using the relation\n\n*Eq. 5\u201112* .\n\nFigure \u2011. Bulk moduli of nickel as a function of temperature. Solid\nline: adiabatic; Dashed line: isothermal.\n\nThe heat capacity at constant volume, as defined in terms of Gibbs\nenergy shown by Eq. 2\u201128, can be calculated by\n\n*Eq. 5\u201113* ,\n\nwhere represents the internal energy. The heat capacity at constant\npressure (see ) can then be calculated as\n\n*Eq. 5\u201114* ,\n\nutilizing the calculated thermal expansion coefficient in and bulk\nmodulus in .\n\nIt can be advocated that thermal expansion makes the difference between\nthe heat capacity at constant volume and the heat capacity at constant\npressure. The calculated contributions to the heat capacity of Ni as\nfunction of temperature from 0 to 1600 K at *P* = 0 are illustrated in\nwhere the lattice vibration and the thermal electron contributions have\nbeen separated out. From , it is observed a large difference between the\ncalcualted *CP* (solid line) and the experimental data\n(symbols, see Ref. \\[18\\] for more details) at 600 K due to the magnetic\nphase transition which has not been considered in the calculation. It\nshould be pointed out that for Ni the thermal electronic contribution to\nthe heat capacity (dashed line in ) is substantial at high temperatures.\n\nFigure \u2011. Heat capacity of nickel as a function of temperature.\nrepresents the calculated lattice vibration contribution; represents the\ncalculated thermal electronic contribution.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/gibbs_energy_function/binary_random_solutions.ipynb b/psu410/src/gibbs_energy_function/binary_random_solutions.ipynb new file mode 100644 index 0000000..b60aeaf --- /dev/null +++ b/psu410/src/gibbs_energy_function/binary_random_solutions.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "d71e3c32", "cell_type": "markdown", "source": "### Binary random solutions\n\nFrom , the Gibbs-Duhem equation of a binary system consisting of\ncomponents $A$ and $B$ is written as\n\nEq. \u2011 $0 = - SdT - Vd( - P) - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B}$\n\nThis equation represents a four-dimensional surface. It is self-evident\nthat both and hold for stable binary solutions too, i.e. the directions\nand the curvature of the surface are all negative. To visualize the\nfour-dimensional surface in the three-dimensional space, one needs to\nfix one of the four potentials. As $T$ and $P$ are the natural variables\nof Gibbs energy, they are usually chosen to be kept constant. One can\ntypically investigate behaviors of systems consisting of condensed\nphases by varying the temperature at constant pressure. at constant\npressure thus becomes\n\nEq. \u2011 $0 = - SdT - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B}$\n\nSimilar to and , the property of a phase can be represented by a\ntwo-dimensional surface in the three-dimensional space composed of $T$,\n$\\mu_{A}$, and $\\mu_{B}$ under constant $P$, keeping in mind the\nfollowing\n\nEq. \u2011\n$G_{m} = x_{A}\\mu_{A} + x_{B}\\mu_{B} = x_{A}_{\\ }^{0}G_{A} + x_{B}_{\\ }^{0}G_{B} + RT\\left( x_{A}l{nx}_{A} + x_{B}l{nx}_{B} \\right) +_{\\ }^{E}G_{m}$\n\nSince $_{\\ }^{E}G_{m}$ must be zero for pure components $A$ and $B$, it\nneeds to be in the following form\n\nEq. \u2011 $_{\\ }^{E}G_{m} = x_{A}x_{B}L_{AB}$\n\nwith $L_{AB}$ being a parameter denoting the interaction between\ncomponents $A$ and $B$, called interaction parameter. When $L_{AB} = 0$,\nthe solution is an ideal solution. When $L_{AB}$ is a non-zero constant\nindependent of temperature and composition, the solution is called a\nregular solution. Its excess entropy and excess enthalpy of mixing are\nobtained as\n\nEq. \u2011 $_{\\ }^{E}S_{m} = \\frac{\\partial_{\\ }^{E}G_{m}}{\\partial T} = 0$\n\nEq. \u2011\n$_{\\ }^{E}H_{m} =_{\\ }^{E}G_{m} - T_{\\ }^{E}S_{m} = x_{A}x_{B}L_{AB}$\n\nThe chemical potential of component $A$ or $B$ in a binary regular\nsolution can be derived as\n\nEq. \u2011\n$\\mu_{i} =_{\\ }^{0}G_{i} + RTlnx_{i} + \\left( 1 - x_{i} \\right)^{2}L_{AB}$\n\nIn a dilute solution with $x_{i} \\rightarrow 0$, one can have\n\nEq. \u2011\n$RTln\\gamma_{i} = \\left( 1 - x_{i} \\right)^{2}L_{AB} \\approx L_{AB}$\n\nEq. \u2011 $\\gamma_{i} = e^{\\frac{L_{AB}}{RT}}$\n\nThe activity is thus proportional to its mole fraction, which is called\nHenry\u2019s law. By the same token, for the solvent, i.e.\n$x_{i} \\rightarrow 1$,\n\nEq. \u2011 $RTln\\gamma_{i} = \\left( 1 - x_{i} \\right)^{2}L_{AB} \\approx 0$\n\nwhich gives $\\gamma_{i} \\approx 1$, and its activity approaches its mole\nfraction. This is called Raoult\u2019s law.\n\nThe stability of a binary solution is derived from as\n\nEq. \u2011\n$\\left( \\frac{\\partial\\mu_{A}}{\\partial N_{A}} \\right)_{T,P,N_{B}} = \\left\\lbrack \\frac{RT}{x_{A}} - 2\\left( 1 - x_{A} \\right)L_{AB} \\right\\rbrack\\frac{1 - x_{A}}{N}$\n\nEq. \u2011\n$\\left( \\frac{\\partial\\mu_{B}}{\\partial N_{B}} \\right)_{T,P,N_{A}} = \\left\\lbrack \\frac{RT}{x_{B}} - 2\\left( 1 - x_{B} \\right)L_{AB} \\right\\rbrack\\frac{1 - x_{B}}{N}$\n\nIt should be noted that the two chemical potentials in a binary system\nat constant temperature and pressure are dependent on each other due to\nthe Gibbs-Duhem equation shown in , i.e.\n\nEq. \u2011 $0 = - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B}$\n\nand the two chemical potentials depend on each other by the following\nrelation\n\nEq. \u2011\n$\\left( \\frac{\\partial\\mu_{A}}{\\partial\\mu_{B}} \\right)_{T,P} = - \\frac{N_{B}}{N_{A}}$\n\nTherefore, at the limit of stability, both and go to zero at the same\ntime, which is obtained when\n\nEq. \u2011 $RT = 2{x_{A}x_{B}L}_{AB}$\n\nAs the absolute temperature cannot be negative, has no solution for a\nsolution phase with $L_{AB} < 0$, i.e. the solution phase is stable with\nrespect to the composition fluctuation. For a solution with\n$L_{AB} > 0$, its limit of stability is represented .\n\nA schematic molar Gibbs energy of a solution with $L_{AB} < 0$ at\nconstant temperature and pressure is shown in along with the ideal and\nexcess Gibbs energy of mixing. A tangent line on the molar Gibbs energy\nof the solution is drawn, and its two intercepts at $x_{B} = 0$ and\n$x_{B} = 1$ give the chemical potentials of components $A$ and $B$,\n$\\mu_{A}$ and $\\mu_{B}$ by , respectively. It is evident that $\\mu_{A}$\nand $\\mu_{B}$ are not independent on each other as they are two points\non the same straight line. This is a graphic representation of the\nGibbs-Duhem equation of . The chemical activity of component $B$ is also\ndepicted with the reference state being the pure B with the same\nstructure. As shown in , other structures of pure B can be selected as\nthe reference states of the chemical activity of component B, resulting\nin the different distances to its chemical potential in the solution,\nthus different values of its chemical activities. It is clear that this\nchange of reference state for chemical activity does not affect the\nchemical potential of the component in the solution.\n\nFigure \u2011: Schematic molar Gibbs energy diagram with $L_{AB} < 0$\n\nWhen $L_{AB} > 0$, represents a parabola in the $T - x_{i}$\ntwo-dimensional coordinate, symmetric with respect to $x_{A}$ and\n$x_{B}$, shown in , i.e. the spinodal of the solution. The consolute\npoint is obtained by applying to and letting equal to zero at the\nconsolute point\n\nEq. \u2011\n$\\left( \\frac{\\partial^{2}\\mu_{A}}{\\partial N_{A}^{2}} \\right)_{T,P,N_{B}} = \\left\\lbrack - \\frac{RT}{x_{A}^{2}} + 2L_{AB} \\right\\rbrack\\left( \\frac{1 - x_{A}}{N} \\right)^{2} = 0$\n\nwhich gives\n\nEq. \u2011 $T_{cons} = 2x_{A}^{2}L_{AB}$\n\nSolving and , one obtains $x_{A} = x_{B} = 0.5$ and\n\nEq. \u2011 $T_{cons} = \\frac{L_{AB}}{2R}$\n\nFigure \u2011: A Spinodal curve with $L_{AB} > 0$\n\nA schematic molar Gibbs energy diagram at temperatures below the\nconsolute point is shown in . It can be seen that part of the molar\nGibbs energy has negative curvature, and the solution becomes unstable.\nThe chemical potential thus does not change monotonically with respect\nto composition and its derivative changes sign at the inflexion point.\n\nFigure \u2011: Schematic molar Gibbs energy diagram with $L_{AB} > 0$\n\nFor more complex solutions, $L_{AB}$ can be a function of temperature,\npressure, and compositions. In principle, the temperature and pressure\ndependences can be treated by means of formula similar to . There are\nvarious approaches in the literature to consider the composition\ndependence of $L_{AB}$. The empirical Redlich-Kister polynomial stands\nout as the one most widely used because it can be extrapolated to\nternary and multi-component systems consistently, which will be\ndiscussed in Chapter .\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/gibbs_energy_function/elastic_magnetic_and_electric_contributions_to_free_energy.ipynb b/psu410/src/gibbs_energy_function/elastic_magnetic_and_electric_contributions_to_free_energy.ipynb new file mode 100644 index 0000000..59e0f56 --- /dev/null +++ b/psu410/src/gibbs_energy_function/elastic_magnetic_and_electric_contributions_to_free_energy.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "3e60ca75", "cell_type": "markdown", "source": "## Elastic, magnetic, and electric contributions to free energy\n\nChapters and focus on the thermal and hydrostatical pressure\ncontributions to Gibbs energy, which are the two prime variables\naffecting phase stability in typical experimental environments. However,\nthere are other internal and external contributions, which are\nparticularly important for crystalline phases. Two important internal\ncontributions are from magnetic and electric polarizations of materials\nwith the corresponding external contributions due to the magnetic and\nelectric fields. Furthermore, for non-hydrostatic pressing of solid\nphases, the $PV$ term in the combined law is to be replaced by elastic\nenergy calculated from elastic stress and elastic strain. The\ncorresponding works done to a system are as follows \\[3\\]\n\nEq. \u2011\n$dW_{elastic} = - V\\sum_{i,j,k,l}^{}{\\sigma_{ij}{d\\varepsilon}_{kl}}$\n\nEq. \u2011 $dW_{magnetic} = - V\\sum_{i}^{}{H_{i}{dB}_{i}}$\n\nEq. \u2011 $dW_{electric} = - V\\sum_{i}^{}{E_{i}{dD}_{i}}$\n\nwhere $i,j,\\ k,\\ l = 1,2,3$, $\\sigma_{ij}$ and $\\varepsilon_{kl}$ are\nthe components of stress and strain; $H_{i}$ and $B_{i}$ are the\ncomponents of magnetic field and magnetic induction; $E_{i}$ and $D_{i}$\nare the components of electric field and electric displacement; and $V$\nis the volume of the crystal. The negative sign in front of the equation\nis due to the fact that the system does work to the surroundings when it\nexpands its volume due to the strain, magnetic induction, and electric\ndisplacement.\n\nUsing the combined law of thermodynamics, can thus be re-written as\nfollows\n\nEq. \u2011\n$dU = TdS - V\\left( \\sum_{i,j,k,l}^{}{\\sigma_{ij}{d\\varepsilon}_{kl}} + \\sum_{i}^{}{H_{i}{dB}_{i}} + \\sum_{i}^{}{E_{i}{dD}_{i}} \\right) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$\n\nThe Legendre transformation similar to Helmholtz energy and Gibbs\nenergy, and , can be made to obtain the following characteristic free\nenergy functions\n\nEq. \u2011 $dF = d(U - TS)$\n\n$$= - SdT - V\\left( \\sum_{i,j,k,l}^{}{\\sigma_{ij}{d\\varepsilon}_{kl}} + \\sum_{i}^{}{H_{i}{dB}_{i}} + \\sum_{i}^{}{E_{i}{dD}_{i}} \\right) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$$\n\nEq. \u2011 $dF_{H} = d\\left( U - TS + \\sum_{i}^{}{H_{i}B_{i}} \\right)$\n\n$$= - SdT - V\\left( \\sum_{i,j,k,l}^{}{\\sigma_{ij}{d\\varepsilon}_{kl}} - \\sum_{i}^{}{B_{i}dH_{i}} + \\sum_{i}^{}{E_{i}{dD}_{i}} \\right) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$$\n\nEq. \u2011 $dF_{E} = d\\left( U - TS + \\sum_{i}^{}{E_{i}D_{i}} \\right)$\n\n$$= - SdT - V\\left( \\sum_{i,j,k,l}^{}{\\sigma_{ij}{d\\varepsilon}_{kl}} + \\sum_{i}^{}{H_{i}{dB}_{i}} - \\sum_{i}^{}{D_{i}{dE}_{i}} \\right) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$$\n\nEq. \u2011\n$dF_{EH} = d\\left( U - TS + \\sum_{i}^{}{H_{i}B_{i}} + \\sum_{i}^{}{E_{i}D_{i}} \\right)$\n\n$$= - SdT - V\\left( \\sum_{i,j,k,l}^{}{\\sigma_{ij}{d\\varepsilon}_{kl}} - \\sum_{i}^{}{B_{i}{dH}_{i}} - \\sum_{i}^{}{D_{i}{dE}_{i}} \\right) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$$\n\nEq. \u2011\n$dG = d\\left( U - TS + \\sum_{i,j,k,l}^{}{\\sigma_{ij}\\varepsilon_{kl}} + \\sum_{i}^{}{H_{i}B_{i}} + \\sum_{i}^{}{E_{i}D_{i}} \\right)$\n\n$$= - SdT + V\\left( \\sum_{i,j,k,l}^{}{\\varepsilon_{ij}d\\sigma}_{kl} + \\sum_{i}^{}{B_{i}dH_{i}} + \\sum_{i}^{}{D_{i}{dE}_{i}} \\right) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$$\n\nFrom above equations, it can be seen that the natural variables of\nvarious free energies are\n$F\\left( T,\\ \\varepsilon_{ij},\\ B_{i},\\ D_{i},\\ N_{i},\\ \\xi \\right)$,\n$F_{H}\\left( T,\\ \\varepsilon_{ij},\\ H_{i},\\ D_{i},\\ N_{i},\\ \\xi \\right)$,\n$F_{E}\\left( T,\\ \\varepsilon_{ij},\\ B_{i},\\ E_{i},\\ N_{i},\\ \\xi \\right)$,\n$F_{EH}\\left( T,\\ \\varepsilon_{ij},\\ H_{i},\\ E_{i},\\ N_{i},\\ \\xi \\right)$,\nand $G\\left( T,\\ \\sigma_{ij},\\ H_{i},\\ E_{i},\\ N_{i},\\ \\xi \\right)$.\nClearly, there can be more combinations when the components of\n$\\varepsilon_{ij}$, $D_{i}$, and $B_{i}$ are partially replaced by their\nconjugate potentials. The free energies listed above are useful\ndepending on how the system is constrained by the surroundings. For\npractical applications, the elastic, magnetic, and electric properties\nare usually considered for phases with fixed compositions, and at\nequilibrium can then be written as\n\nEq. \u2011\n$dG = - SdT + V\\left( \\sum_{i,j,k.l}^{}{\\varepsilon_{ij}d\\sigma}_{kl} + \\sum_{i}^{}{B_{i}dH_{i}} + \\sum_{i}^{}{D_{i}{dE}_{i}} \\right) + \\mu dN$\n\nThe corresponding Gibbs-Duhem equation follows as below\n\nEq. \u2011\n$0 = - SdT + V\\left( \\sum_{i,j,k,l}^{}{\\varepsilon_{ij}d\\sigma}_{kl} + \\sum_{i}^{}{B_{i}dH_{i}} + \\sum_{i}^{}{D_{i}{dE}_{i}} \\right) - Nd\\mu$\n\nThe general differential form of a molar quantity can be extended from\nas follows as a one-component system\n\nEq. \u2011\n\nEq. \u2011\n\nEq. \u2011\n\nEq. \u2011\n\nThe first derivatives in to are the second directives of Gibbs energy\nwith respect to its natural variables, i.e. potentials, and have their\nrespective nomenclatures as shown in . The limit of stability follows\nand can be re-written as\n\nEq. \u2011\n$\\left( \\frac{\\partial X_{i}}{\\partial Y_{i}} \\right)_{N,Y_{j}} = \\infty$\n\nThis means that the derivatives in to , i.e. the quantities in , diverge\nat the limit of stability.\n\nTable \u2011: Physical quantities related to the first derivatives in to .\nThe table is symmetric due to the Maxwell relations related to the\nsecond directives of Gibbs energy with respect to its natural variables.\n\n| | T | $$\\sigma_{kl}$$ | $$E_{k}$$ | $$H_{k}$$ |\n|----------------------|------------------------------------|------------------------------------|---------------------------------------|---------------------------------------|\n| S | C/T, heat capacity | $\\alpha_{kl}$, piezocaloric effect | $p_{k}$, electrocaloric effect | $m_{k}$, magnetocaloric effect |\n| $$\\varepsilon_{ij}$$ | $\\alpha_{ij}$, thermal expansion | $s_{ijkl}$, elastic compliance | $d_{ijk}$, converse piezoelectricity | $q_{ijk}$, piezomagnetic moduli |\n| $$D_{i}$$ | $p_{i}$, pyroelectric coefficients | $d_{ikl}$, piezoelectric moduli | $k_{ik}$, permittivities | $a_{ik}$, magnetoelectric coefficient |\n| $$B_{i}$$ | $m_{i}$, pyromagnetic coefficient | $q_{ikl}$, piezomagnetic moduli | $a_{ik}$, magnetoelectric coefficient | $\\mu_{ik}$, permeability |\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/gibbs_energy_function/index.ipynb b/psu410/src/gibbs_energy_function/index.ipynb new file mode 100644 index 0000000..d325a7a --- /dev/null +++ b/psu410/src/gibbs_energy_function/index.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "f31bed02", "cell_type": "markdown", "source": "# Gibbs energy function\n\nAs shown in through , all functions have $N_{i}$ and $\\xi$ as natural\nvariables while they differ in other two natural variables. In typical\nmaterials-related experiments, temperature and pressure are the two\nvariables controlled. They are also the natural variables of Gibbs\nenergy. Consequently, Gibbs energy is the most widely used function in\nthermodynamics of materials science. The rest of this book focuses on\nGibbs energy for this reason. In this chapter, the mathematical formulas\nfor Gibbs energy of phases with fixed and variable compositions are\ndiscussed which are needed for quantitative calculations of Gibbs energy\nunder given values of its natural variables.\n\nFrom , the molar Gibbs energy can be defined as\n\nEq. \u2011\n$G_{m}\\left( T,P,x_{i},\\xi \\right) = \\frac{G}{N} = \\sum_{}^{}\\mu_{i}x_{i}$\n\nThe molar entropy, molar volume, chemical potential, and the driving\nforce can be obtained from as\n\nEq. \u2011\n$S_{m} = \\frac{S}{N} = - \\frac{1}{N}\\left( \\frac{\\partial G}{\\partial T} \\right)_{P,\\ N_{i},\\ \\xi} = {- \\left( \\frac{\\partial G_{m}}{\\partial T} \\right)}_{P,\\ x_{i},\\ \\xi}$\n\nEq. \u2011\n$V_{m} = \\frac{V}{N} = \\frac{1}{N}\\left( \\frac{\\partial G}{\\partial P} \\right)_{T,\\ N_{i},\\ \\xi} = \\left( \\frac{\\partial G_{m}}{\\partial P} \\right)_{T,\\ x_{i},\\ \\xi}$\n\nEq. \u2011\n$\\mu_{i} = \\left( \\frac{\\partial G}{\\partial N_{i}} \\right)_{T,P,N_{j \\neq i},\\ \\xi}$\n\nEq. \u2011\n$- D = \\left( \\frac{\\partial G}{\\partial\\xi} \\right)_{T,P,N_{i}\\ }$\n\nBased on , the molar enthalpy is written as\n\nEq. \u2011 $H_{m} = G_{m} + TS_{m}$\n\nOther physical properties of the system can also be represented by the\npartial derivatives of Gibbs energy such as heat capacity, $C_{P}$,\nvolume thermal expansivity, $\\alpha_{V}$, isothermal compressibility,\n$\\kappa_{T}$, as follows under constant pressure or temperature\n\nEq. \u2011\n$C_{P} = \\left( \\frac{\\partial Q}{\\partial T} \\right)_{P} = \\left( \\frac{\\partial H}{\\partial T} \\right)_{P} = T\\left( \\frac{\\partial(G + TS)}{\\partial T} \\right)_{P} = T\\left( \\frac{\\partial S}{\\partial T} \\right)_{P} = - T\\left( \\frac{\\partial^{2}G}{\\partial T^{2}} \\right)_{P}$\n\nEq. \u2011\n$\\alpha_{V} = \\frac{\\left( \\frac{\\partial V}{\\partial T} \\right)_{P}}{V} = \\frac{\\left( \\frac{\\left( \\partial G/\\partial( - P) \\right)_{T}}{\\partial T} \\right)_{P}}{\\left( \\partial G/\\partial( - P) \\right)_{T}} = \\frac{\\frac{\\partial^{2}G}{\\partial T\\partial( - P)}}{\\left( \\partial G/\\partial( - P) \\right)_{T}}$\n\nEq. \u2011\n$\\kappa_{T} = \\frac{\\left( \\frac{\\partial V}{\\partial( - P)} \\right)_{T}}{V} = \\frac{\\left( \\frac{\\left( \\partial G/\\partial( - P) \\right)_{T}}{\\partial( - P)} \\right)_{T}}{\\left( \\partial G/\\partial( - P) \\right)_{T}} = \\frac{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}}{\\left( \\partial G/\\partial( - P) \\right)_{T}} = \\frac{1}{B}$\n\nwhere the $N_{i}$ and $\\xi$ are kept constant for all partial\nderivatives, and $B$ is the bulk modulus.\n\nIn , $G$ cannot be directly replaced by $G_{m}$ because *N* also depends\non *Ni*. The thermodynamic quantities under such conditions,\ni.e. varying the amount of a component at constant temperature and\npressure, are called partial quantities which are introduced in Eq. 1\u20118\nfor partial entropy and for partial enthalpy. This definition can be\nextended to all molar quantities such as partial volume and partial\nGibbs energy. Partial quantities of a molar quantity, $A$, can thus be\ndefined in general as\n\nEq. \u2011\n$A_{i} = \\left( \\frac{\\partial A}{\\partial N_{i}} \\right)_{T,P,N_{j \\neq i},\\ \\xi}$\n\nThe general differential form of a molar quantity for a system at\nequilibrium can be represented by its partial quantities as\n\nEq. \u2011\n$dA = \\left( \\frac{\\partial A}{\\partial T} \\right)dT + \\left( \\frac{\\partial A}{\\partial P} \\right)dP + \\sum_{}^{}\\left( \\frac{\\partial A}{\\partial N_{i}} \\right){dN}_{i}$\n\nwhere the subscripts representing variables kept constant, i.e. the\nremaining natural variables of Gibbs energy not in the denominator, are\nomitted for simplicity. This will be done throughout the book unless\nspecified otherwise.\n\nUsing the following relations: $A = NA_{m}$, $N = \\sum_{}^{}N_{j}$,\n$x_{i} = N_{i}/N$,\n$\\frac{{\\partial x}_{i}}{{\\partial N}_{i}} = \\left( 1 - x_{i} \\right)/N$,\nand $\\frac{{\\partial x}_{k}}{{\\partial N}_{i}} = {- x}_{k}/N$, can be\nderived as, under constant T and P,\n\nEq. \u2011\n$A_{i} = A_{m} + N\\sum_{j = 1}^{c}{\\frac{\\partial A_{m}}{\\partial x_{j}}\\frac{\\partial x_{j}}{\\partial N_{i}}} = A_{m} + \\frac{\\partial A_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial A_{m}}{\\partial x_{j}}$\n\nwhere the summation is for all *c* components and the partial\nderivatives are taken with other mole fractions kept constant. However,\nmole fractions are not independent, but follow the relation\n$\\sum_{}^{}x_{i} = 1$. Taking $x_{1} = 1 - \\sum_{j = 2}^{c}x_{j}$ as the\ndependent mole fraction, can be rewritten as\n\nEq. \u2011\n$A_{i} = A_{m} + \\left( \\frac{\\partial A_{m}}{\\partial x_{i}} - \\frac{\\partial A_{m}}{\\partial x_{1}} \\right) - \\sum_{j = 2}^{c}x_{j}\\left( \\frac{\\partial A_{m}}{\\partial x_{j}} - \\frac{\\partial A_{m}}{\\partial x_{1}} \\right)$\n\nThe difference of the partial derivatives in the parenthesis in\nrepresents the partial derivative of $A_{m}$ with respect to the mole\nfraction of one component when the first component is selected as the\ndependent component. Applying and to Gibbs energy, the partial Gibbs\nenergy or chemical potential of component $i$ is obtained as\n\nEq. \u2011\n$\\mu_{i} = G_{i} = G_{m} + \\frac{\\partial G_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial G_{m}}{\\partial x_{j}} = G_{m} + \\left( \\frac{\\partial G_{m}}{\\partial x_{i}} - \\frac{\\partial G_{m}}{\\partial x_{1}} \\right) - \\sum_{j = 2}^{c}x_{j}\\left( \\frac{\\partial G_{m}}{\\partial x_{j}} - \\frac{\\partial G_{m}}{\\partial x_{1}} \\right)$\n\nThe derivatives in the stability equation, , are defined with the molar\nquantities kept constant. On the other hand, Gibbs energy has two\npotentials, temperature and pressure, as natural variables instead. One\nwould thus need to compare the stability conditions when a variable kept\nfixed is changed from a molar quantity to its conjugate potential. This\ncan be carried out through the use of Jacobians to change the\nindependent variables\n\nEq. \u2011\n$\\frac{\\partial\\left( Y_{i},Y_{j} \\right)}{\\partial\\left( X_{i},X_{j} \\right)} = \\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{Y_{j}}\\left( \\frac{\\partial Y_{j}}{\\partial X_{j}} \\right)_{X_{i}} = \\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}}\\left( \\frac{\\partial Y_{j}}{\\partial X_{j}} \\right)_{X_{i}} - \\left( \\frac{\\partial Y_{i}}{\\partial X_{j}} \\right)_{X_{i}}\\left( \\frac{\\partial Y_{j}}{\\partial X_{i}} \\right)_{X_{j}}$\n\nFor a stable system, both\n$\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}}$ and\n$\\left( \\frac{\\partial Y_{j}}{\\partial X_{j}} \\right)_{X_{i}}$ are\npositive based on . Using the Maxwell relation shown by , one thus\nobtains\n\nEq. \u2011\n$\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}} - \\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{Y_{j}} = \\left( \\frac{\\partial Y_{i}}{\\partial X_{j}} \\right)_{X_{i}}\\left( \\frac{\\partial Y_{j}}{\\partial X_{i}} \\right)_{X_{j}}/\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}} \\geq 0$\n\nThis means that\n$\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{Y_{j}}$ will go\nto zero before\n$\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}}$ does. It\nindicates that the stability condition becomes more restrictive when\npotentials are kept constant in place of their conjugate molar\nquantities. Based on the Gibbs-Duhem equation of , the maximum number of\nindependent potentials is *c+1*, and the last potential is dependent,\ni.e.\n\nEq. \u2011\n$\\left( \\frac{\\partial Y_{c + 2}}{\\partial X_{c + 2}} \\right)_{Y_{j \\leq c + 1}} = 0$\n\nTherefore, the limit of stability is determined when the derivative\nbecomes zero with one molar quantity kept constant, e.g.\n\nEq. \u2011\n$\\left( \\frac{\\partial Y_{c + 1}}{\\partial X_{c + 1}} \\right)_{Y_{j < c + 1},X_{c + 2}} = 0$\n\nThis is because this derivative reaches zero faster than any other\nderivatives with more molar quantities kept constant. shows that all\nmolar quantities diverge at the limit of stability. The consolute point\nis obtained with $c$ additional conditions as follows based on\n\nEq. \u2011\n$\\left( \\frac{\\partial^{2}Y_{i}}{\\partial\\left( X_{i} \\right)^{2}} \\right)_{Y_{j \\leq c + 1, \\neq i},X_{c + 2}} = 0$\n\nTogether with , all $c + 1$ independent potentials at the consolute\npoint can be determined. It is evident that the consolute point is a\nzero-dimensional point in a two-dimensional space of independent\npotentials in a one-component system. With the addition of a second\ncomponent to form a binary system, this consolute point in the\none-component system extends into a one-dimensional line. This line\nrepresents the limit of stability of the binary system, and a consolute\npoint is located at the end of this line. It is thus evident that in a\nsystem with $c$ independent components, the limit of stability is a\n*c-1*-dimensional hypersurface in a space of $c + 1$ independent\npotentials, while the consolute point is a zero-dimensional point in all\nsystems, which may be called the invariant critical point.\n", "metadata": {}}, {"id": "71b23714", "cell_type": "markdown", "source": "### Multi-component random solutions\n\nSimilar to a ternary solution, the excess Gibbs energy of mixing of a\nmulti-component solution can be written as\n\nEq. \u2011\n$_{\\ }^{E}G_{m} = \\sum_{i}^{}{\\sum_{j}^{}{x_{i}x_{j}L_{ij}}} + \\sum_{i}^{}{\\sum_{j}^{}{\\sum_{k}^{}{x_{i}x_{j}x_{k}L_{ijk}}}}$\n\nIn principle, one can add interaction parameters for quaternary and\nhigher order systems, but their contributions to Gibbs energy are\nrelatively minor because the major contributions have already been taken\ninto account by the binary and ternary interactions. It is anticipated\nthat not only the interaction parameters of four or more components are\nsmall, but also the multiplication of mole fractions in front the\ninteraction parameters diminishes their contribution to the Gibbs energy\neven further.\n\nUnder the condition that all interaction parameters are constant, the\nchemical potential of a component in a multi-component system with\nbinary and ternary interaction parameters can be extended from as\n\nEq. \u2011\n$\\mu_{i} =_{\\ }^{0}G_{i} + RTlnx_{i} + \\sum_{j \\neq i}^{}{x_{j}^{2}L_{ij}} + \\sum_{k > j}^{}{\\sum_{j \\neq i}^{}{x_{j}x_{k}}}\\left\\lbrack L_{ij} + L_{ik} - L_{jk} + \\left( 1 - 2x_{i} \\right)L_{ijk} \\right\\rbrack$\n\nThe stability of the solution can also be extended from as\n\nEq. \u2011\n${N\\left( \\frac{\\partial\\mu_{i}}{\\partial N_{i}} \\right)}_{T,P,N_{j \\neq i}} = NG_{ii} = \\frac{RT\\left( 1 - x_{i} \\right)}{x_{i}} - 2\\sum_{j \\neq i}^{}{x_{j}^{2}L_{ij}} - 2\\sum_{k > j}^{}{\\sum_{j \\neq i}^{}{x_{j}x_{k}}}\\left\\lbrack L_{ij} + L_{ik} - L_{jk} + \\left( 2 - 3x_{i} \\right)L_{ijk} \\right\\rbrack$\n\nThe limit of stability of a multi-component random solution can be\nrepresented by or .\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/gibbs_energy_function/phases_with_fixed_compositions.ipynb b/psu410/src/gibbs_energy_function/phases_with_fixed_compositions.ipynb new file mode 100644 index 0000000..f2d79bd --- /dev/null +++ b/psu410/src/gibbs_energy_function/phases_with_fixed_compositions.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "b7f59679", "cell_type": "markdown", "source": "## Phases with fixed compositions\n\nThe homogeneous system discussed so far means that there is only one\nphase in the system, i.e. a single-phase system. A phase with a fixed\ncomposition can be a pure element or a stoichiometric compound. There is\nthus only one independent component in the system. A stoichiometric\ncompound contains more than one element, but the relative amounts of\neach element are fixed by the stoichiometry and cannot vary\nindependently, i.e., $dN_{i} = x_{i}dN$. The combined law of\nthermodynamics becomes\n\nEq. \u2011\n$dG = - SdT - Vd( - P) + \\left( \\sum_{}^{}{x_{i}\\mu_{i}} \\right)dN - Dd\\xi = - SdT - Vd( - P) + G_{m}dN - Dd\\xi$\n\n$G_{m}$ is the molar Gibbs energy of the stoichiometric compound and can\nbe regarded as the chemical potential of the stoichiometric\nphase,$\\ \\alpha$,\n\nEq. \u2011 $G_{m} = \\mu^{\\alpha} = \\sum_{}^{}{x_{i}\\mu_{i}}$\n\nThe chemical potential of individual components in the phase cannot be\ndefined because the amount of each component cannot be varied\nindependently. For a stoichiometric phase of $N$ moles of atoms at\nequilibrium with $dG$=$Nd\\mu^{\\alpha} + \\mu^{\\alpha}dN$, reduces to\n\nEq. \u2011 $\\ 0 = - SdT - Vd( - P) - Nd\\mu^{\\alpha}$\n\nwhich is the Gibbs-Duhem equation, , applied to a stoichiometric phase.\nIt can be represented graphically by a surface in a three-dimensional\nspace composed of $\\ \\mu^{\\alpha}$, *T* and *\u2013P*. The direction of the\nsurface is represented by the three partial directives between any two\nof $\\ \\mu^{\\alpha}$, *T* and *\u2013P* with the third one kept constant, i.e.\n\nEq. \u2011\n$\\ \\left( \\frac{\\partial\\mu^{\\alpha}}{\\partial T} \\right)_{P} = - \\frac{S}{N} = - S_{m}$\n\nEq. \u2011\n$\\ \\left( \\frac{\\partial\\mu^{\\alpha}}{\\partial( - P)} \\right)_{T} = - \\frac{V}{N} = - V_{m}$\n\nEq. \u2011\n$\\ \\left( \\frac{\\partial( - P)}{\\partial T} \\right)_{\\mu^{\\alpha}} = - \\frac{S}{V} = - \\frac{S_{m}}{V_{m}}$\n\nBased on Nernst\u2019s heat theorem, the entropy difference between two\ncrystals approaches zero when the temperature approaches absolute zero.\nIt is thus a common practice to put $S = 0$ for a crystal at 0 K. This\nis usually referred as the third law of thermodynamics. From the\ndefinition of entropy change by , $S$ or $S_{m}$ is always positive at\nfinite temperatures as the system or the crystal absorbs heat from the\nsurroundings to increase its temperature. $V$ or $V_{m}$ of a phase is a\nwell-defined physical quantity, and its absolute value can be given and\nis always positive. The above three equations can be written in a\ngeneral form as\n\nEq. \u2011\n$\\ \\left( \\frac{\\partial Y_{i}}{\\partial Y_{j}} \\right)_{Y_{k}} = - \\frac{X_{j}}{X_{i}} < 0$\n\nThe surface thus has negative slopes in all its directions. The\ncurvature of the surface can be derived from\n\nEq. \u2011\n$\\ \\left( \\frac{\\partial^{2}Y_{i}}{\\partial\\left( Y_{j} \\right)^{2}} \\right)_{Y_{k}} = - \\left( \\frac{\\partial\\left( \\frac{X_{j}}{X_{i}} \\right)}{\\partial Y_{j}} \\right)_{Y_{k}} = - \\frac{1}{X_{i}}\\left( \\frac{\\partial X_{j}}{\\partial Y_{j}} \\right)_{Y_{k}} + \\frac{X_{j}}{\\left( X_{i} \\right)^{2}}\\left( \\frac{\\partial X_{i}}{\\partial Y_{j}} \\right)_{Y_{k}} = - \\frac{1}{X_{i}}\\left\\lbrack \\left( \\frac{\\partial X_{j}}{\\partial Y_{j}} \\right)_{Y_{k}} - \\frac{X_{j}}{X_{i}}{\\left( \\frac{\\partial X_{i}}{\\partial Y_{i}} \\right)\\left( \\frac{\\partial Y_{i}}{\\partial Y_{j}} \\right)}_{Y_{k}} \\right\\rbrack = - \\frac{1}{X_{i}}\\left\\lbrack \\left( \\frac{\\partial X_{j}}{\\partial Y_{j}} \\right)_{Y_{k}} + \\left( \\frac{X_{j}}{X_{i}} \\right)^{2}\\left( \\frac{\\partial X_{i}}{\\partial Y_{i}} \\right)_{Y_{k}} \\right\\rbrack < 0$\n\nBoth terms inside the last bracket are positive for a system in a state\nof stable internal equilibrium, and the surface thus has a negative\ncurvature and is convex everywhere as shown in .\n\nFigure \u2011: Gibbs energy of a one-component phase as a function of\ntemperature and negative pressure, showing the convex shape\n\nFrom experimental observations, it is known that\n$S_{m}^{vapor} \\gg S_{m}^{liquid} > S_{m}^{solid}$. The curves of\n$G_{m}$ or $\\mu^{\\alpha}$ plotted with respect to $T$ at constant $P$\nwould thus have the most negative slope for a vapour phase followed by\nits liquid and solid phases. As an example, shows Gibbs energy of Zn in\nits solid, liquid, and vapour forms as a function of $T$ at constant\n$P = 1$ atmospheric pressure.\n\nFigure \u2011: Molar Gibbs energy of Zn as a function of T at constant P\n\nSimilarly it is common that\n$V_{m}^{vapor} \\gg V_{m}^{liquid} > V_{m}^{solid}$, and the curves of\n$G_{m}$ or $\\mu^{\\alpha}$ plotted with respect to $P$ at constant $T$\nwould thus have the most positive slope for a vapour phase followed by\nits liquid and solid phases, though there are cases that\n$V_{m}^{liquid} < V_{m}^{solid}$ such as those of water and ice. As an\nexample, shows Gibbs energy of Fe in its two solid (fcc and bcc),\nliquid, and vapour forms as a function of $P$ at constant $T = 1273K$.\n\nFigure \u2011: Molar Gibbs energy of Zn as a function of P at constant T\n\nThe quantities measurable by experiments typically include temperature,\npressure, volume, composition, and amount of heat flow in the combined\nlaw of thermodynamics discussed so far. By measuring the heat needed to\nincrease the temperature of a phase, the heat capacity of the phase is\nobtained as shown by Eq. 2\u20117. A typical heat capacity curve as a\nfunction of temperature is shown in for fcc-Al, hcp-Mg, and an\nintermetallic phase Al12Mg17.\n\nFigure \u2011: Heat capacity of fcc-Al, hcp-Mg, and\nAl12Mg17 as a function of temperature\n\nThere are various theoretical models for the heat capacity under\nconstant volume to be discussed in Chapter 5 of this book, which is\ndefined as\n\nEq. \u2011\n$C_{V} = \\left( \\frac{\\partial U}{\\partial T} \\right)_{V} = T\\left( \\frac{\\partial(F + TS)}{\\partial T} \\right)_{V} = T\\left( \\frac{\\partial S}{\\partial T} \\right)_{V} = - T\\left( \\frac{\\partial^{2}F}{\\partial T^{2}} \\right)_{V}$\n\nTo establish the relationship between $C_{P}$ defined by and $C_{V}$,\n$U$ needs to be represented as a function of $T$ and $V$ in terms of $G$\nand its derivatives with respect to Gibbs energy\u2019s natural variables of\n$T$ and $P$. It can be done as follows\n\nEq. \u2011\n$dV = \\frac{\\partial V}{\\partial T}dT + \\frac{\\partial V}{\\partial( - P)}d( - P) = - \\frac{\\partial^{2}G}{\\partial T( - P)}dT - \\frac{\\partial^{2}G}{\\partial( - P)^{2}}d( - P)$\n\nEq. \u2011\n$dU = \\frac{\\partial(G + TS - PV)}{\\partial T}dT + \\frac{\\partial(G + TS - PV)}{\\partial( - P)}d( - P) = - \\left( T\\frac{\\partial^{2}G}{\\partial T^{2}} - P\\frac{\\partial^{2}G}{\\partial T( - P)} \\right)dT\u2014\\left( T\\frac{\\partial^{2}G}{\\partial T( - P)} + P\\frac{\\partial^{2}G}{\\partial( - P)^{2}} \\right)\\left( - \\frac{1}{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}}dV + \\frac{\\frac{\\partial^{2}G}{\\partial T( - P)}}{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}}dT \\right) = - \\left\\lbrack T\\frac{\\partial^{2}G}{\\partial T^{2}} - T\\frac{\\left( \\frac{\\partial^{2}G}{\\partial T( - P)} \\right)^{2}}{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}} \\right\\rbrack dT + \\left( - T\\frac{\\frac{\\partial^{2}G}{\\partial T( - P)}}{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}} + P \\right)dV$\n\nEq. \u2011\n$C_{V} = C_{P} + T\\frac{\\left( \\frac{\\partial^{2}G}{\\partial T( - P)} \\right)^{2}}{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}} = C_{P} - \\frac{\\alpha_{V}^{2}VT}{\\kappa_{T}} = C_{P} - \\alpha_{V}^{2}BVT$\n\nwhere the thermal expansion, $\\alpha_{V}$, and the compressibility or\nbulk modulus, $\\kappa_{T}$ or $B$, are defined by and , respectively.\nFrom the heat capacity, the enthalpy and entropy can be obtained by\nintegration of at a constant pressure\n\nEq. \u2011\n$S = S_{0} + \\int_{0}^{T}\\frac{C_{P}}{T}dT = S_{0} + \\int_{0}^{298.15}\\frac{C_{P}}{T}dT + \\int_{298.15}^{T}\\frac{C_{P}}{T}dT = S_{298.15} + \\int_{298.15}^{T}\\frac{C_{P}}{T}dT$\n\nEq. \u2011\n$H = H_{0} + \\int_{0}^{T}C_{P}dT = H_{0} + \\int_{0}^{298.15}C_{P}dT + \\int_{298.15}^{T}C_{P}dT = H_{298.15} + \\int_{298.15}^{T}C_{P}dT$\n\nIn the above equations, two temperature ranges of integration are chosen\nfor practical applications as most processing procedures in the field of\nmaterials science and engineering take place at temperatures above the\nroom temperature. Based on the third-law of thermodynamics, $S_{0} = 0$,\n$S_{298.15}$ can be obtained by integration. On the other hand for\n$H_{0} = U_{0} + PV$, one does not know the absolute value of the\ninternal energy and thus have to select a reference state for $H$. In\nprinciple, the reference state can be arbitrarily chosen. A widely used\nreference state in the thermodynamic modeling practice is to set\n$H_{298.15}^{SER} = 0$ at ambient pressure for pure elements at their\nrespective stable structures at room temperature, called stable element\nreference (SER) state with\n\nEq. \u2011\n$G_{298.15}^{SER} = H_{298.15}^{SER} - TS_{298.15}^{SER} = - TS_{298.15}^{SER}$\n\nIt is further noted that after defining $S_{298.15}$ and $H_{298.15}$,\none only needs the heat capacity at higher temperatures. This makes the\nmathematical representation of heat capacity simpler due to a relatively\nsimple temperature dependence of heat capacity at higher temperatures in\ncomparison with the variation at lower temperatures. One common\nexpression for heat capacity at high temperatures and ambient pressure\nis as follows\n\nEq. \u2011 $C_{P} = c + dT + \\frac{e}{T^{2}} + fT^{2}$\n\nwhere c, d, e, and f are parameters fitted to experimental or theoretic\ndata and compiled in various handbooks.\n\nThe corresponding $S$, $H$, and $G$ are obtained as\n\nEq. \u2011 $S = b^{'} + clnT + dT - \\frac{e}{{2T}^{2}} + \\frac{f}{2}T^{2}$\n\nEq. \u2011 $H = a + cT + \\frac{d}{2}T^{2} - \\frac{e}{T} + \\frac{f}{3}T^{3}$\n\nEq. \u2011\n$G = H - TS = a - bT - cTlnT - \\frac{d}{2}T^{2} - \\frac{e}{2T} - \\frac{f}{6}T^{3}$\n\nwith $b = b^{'} - c$. The integration constants $b^{'}$ and $a$ are\nevaluated from $S_{298.15}$ and $H_{298.15}$. As an example, the\nenthalpy and entropy of Zn in solid (hcp), liquid, and gas forms are\nplotted in and , respectively. The distances between any two curves in\nand represent the enthalpy or entropy differences between the two\nphases. It can be seen that the gas has much higher enthalpy and entropy\nthan the solid and liquid.\n\nFigure \u2011: Enthalpy of Zn as a function of temperature at one atmospheric\npressure\n\nFigure \u2011: Entropy of Zn as a function of temperature at one atmospheric\npressure\n\nSimilarly, one can add the pressure dependence into the Gibbs energy\nfunction such as\n\nEq. \u2011\n$G = a - bT - cTlnT - \\frac{d}{2}T^{2} - \\frac{e}{2T} - \\frac{f}{6}T^{3} + gP + hTP + mP^{2}$\n\nwhere g, h, and m are parameters fitted to experimental or theoretic\ndata and compiled in various handbooks.\n\nThe expression for $V$ can be derived as\n\nEq. \u2011 $V = g + hT + 2mP$\n\nThe Helmholtz energy can be expressed as a function of its natural\nvariables by solving $P\\ $ from\n\nEq. \u2011\n$F = G - PV = a - bT - cTlnT - \\frac{d}{2}T^{2} - \\frac{e}{2T} - \\frac{f}{6}T^{3} - \\frac{(g + hT - V)^{2}}{4m}$\n\nIn the literature there are many models to represent the temperature and\npressure dependences of thermodynamic properties. The Gibbs energy\ndifference between a stoichiometric compound and the components at their\nreference states of which the compound is composed,\n${_{\\ }^{0}G}_{i}^{ref}$, is termed as Gibbs energy of formation, i.e.\n\nEq. \u2011 $\\mathrm{\\Delta}_{f}G = G - \\sum_{}^{}N_{i}{_{\\ }^{0}G}_{i}^{ref}$\n\nwith $N_{i}$ being the stoichiometry of the compound. Similarly,\nenthalpy of formation, entropy of formation, and heat capacity of\nformation with respect to components at their reference states,\n$_{\\ }^{0}H_{i}^{ref}$, $_{\\ }^{0}S_{i}^{ref}$, and\n$_{\\ }^{0}{C_{P}}_{i}^{ref}$, can be defined as\n\nEq. \u2011 $\\mathrm{\\Delta}_{f}H = H - \\sum_{}^{}{N_{i}_{\\ }^{0}H_{i}^{ref}}$\n\nEq. \u2011 $\\mathrm{\\Delta}_{f}S = S - \\sum_{}^{}{N_{i}_{\\ }^{0}S_{i}^{ref}}$\n\nEq. \u2011\n$\\mathrm{\\Delta}_{f}C_{P} = C_{P} - \\sum_{}^{}{N_{i}_{\\ }^{0}{C_{P}}_{i}^{ref}}$\n\nIt should be mentioned that one mole of a compound usually refers to one\nmole of formula of stoichiometry of the compound. With a formula like\n$A_{a}B_{b}C_{c}$, the compound is composed of total $(a + b + c)$ moles\nof components. One should thus be very careful when dealing with\nnumerical values to be sure whether the data is in terms of per mole of\nformula or per mole of components. At the same time the reference states\nmust be clearly defined. When the SER state defined in is selected as\nthe reference state, the above formation quantities are called standard\nformation quantities such as standard enthalpy of formation.\n\nSince there are only two independent potentials in a one-component\nsystem, its limit of stability can be evaluated with one potential kept\nconstant, i.e. either $T$ or $P$. Consequently, either Helmholtz energy\nor enthalpy is to be used in deriving the limit of stability of a\nhomogeneous system. For the practical usefulness, let us use Helmholtz\nenergy because its natural variables of $T$ and *V* are measurable\nquantities in typical experiments, while one of the natural variables of\nenthalpy, entropy, is not. From and , the limit of stability for a\none-component system at constant temperature can be written as\n\nEq. \u2011\n$\\left( \\frac{\\partial( - P)}{\\partial V} \\right)_{T,N} = F_{VV} = \\frac{1}{V\\kappa_{T}} = \\frac{B}{V} = 0$\n\nwhere the isothermal compressibility and bulk modulus, $\\kappa_{T}$ and\n$B$, are defined in . The limit of stability is thus determined when the\nisothermal compressibility diverges or the bulk modulus becomes zero\nbecause $V$ has finite values at any temperatures. It is evident that\nHelmholtz energy must have higher order dependence on volume than in for\na system with instability because $F_{VV}$ from is constant.\n\nFrom , the consolute point is defined by\n\nEq. \u2011\n$F_{VVV} = \\left( \\frac{\\partial^{2}( - P)}{\\partial V^{2}} \\right)_{T,N} = \\frac{\\partial\\left( \\frac{1}{V\\kappa_{T}} \\right)}{\\partial V} = - \\frac{1 + \\frac{V}{\\kappa_{T}}\\frac{\\partial\\kappa_{T}}{\\partial V}}{\\kappa_{T}V^{2}} = 0$\n\nSince $\\kappa_{T}$ becomes infinite at the limit of stability,\n$\\frac{\\partial\\kappa_{T}}{\\partial V}$ approaches negative infinite\nwhen the critical/consolute point is approached so that\n$\\frac{V}{\\kappa_{T}}\\frac{\\partial\\kappa_{T}}{\\partial V} = - 1$ and\n$F_{VVV} = 0$.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/gibbs_energy_function/phases_with_variable_compositions_random_solutions.ipynb b/psu410/src/gibbs_energy_function/phases_with_variable_compositions_random_solutions.ipynb new file mode 100644 index 0000000..812aaf7 --- /dev/null +++ b/psu410/src/gibbs_energy_function/phases_with_variable_compositions_random_solutions.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "8bfb087d", "cell_type": "markdown", "source": "## Phases with variable compositions: Random solutions\n\nThe combined law of thermodynamics and the Gibbs-Duhem equation of a\nsolution phase with variable compositions are shown by and ,\nrespectively. A phase can be represented by a *c+1*-dimensional surface\nin a *c+2*-dimensional space of potentials based on the Gibbs-Duhem\nequation. The directions and curvature of the surface are represented by\nthe partial derivatives shown by and secondary derivatives shown by ,\nboth being negative for a stable phase. To develop a mathematical\nformula for Gibbs energy of a phase with variable compositions, one can\nconsider a phase as a mixture of independent components that the phase\nis made of. Its Gibbs energy function can be postulated as the summation\nof Gibbs energy of the independent components of the same structure of\nthe solution, $_{\\ }^{0}G_{i}$, plus the contribution due to the mixing,\n$_{\\ }^{mixing}G$ or $_{\\ }^{M}G$\n\nEq. \u2011 $G = \\sum_{}^{}{N_{i}_{\\ }^{0}G_{i}} +_{\\ }^{M}G$\n\nSince the system size usually is not important in thermodynamics, its\nproperties are typically normalized to one mole with its composition\nrepresented by mole fractions of components. The molar Gibbs energy is\nobtained as shown below with the molar Gibbs energy of mixing separated\ninto two parts: ideal Gibbs energy of mixing assuming no chemical\ninteraction among components, $_{\\ }^{ideal}G_{m}$ or $_{\\ }^{I}G_{m}$,\nand excess Gibbs energy of mixing due to chemical reaction among\ncomponents, $_{\\ }^{excess}G_{m}$ or $_{\\ }^{E}G_{m}$\n\nEq. \u2011\n$G_{m} = \\sum_{}^{}{x_{i}_{\\ }^{0}G_{i}} +_{\\ }^{M}G_{m} = \\sum_{}^{}{x_{i}_{\\ }^{0}G_{i}} +_{\\ }^{I}G_{m} +_{\\ }^{E}G_{m}$\n\nFrom , the chemical potential of a component is thus\n\nEq. \u2011\n$\\mu_{i} =_{\\ }^{0}G_{i} +_{\\ }^{I}{G_{m} +}_{\\ }^{E}G_{m} + \\frac{\\partial\\left(_{\\ }^{I}{G_{m} +}_{\\ }^{E}G_{m} \\right)}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial\\left(_{\\ }^{I}{G_{m} +}_{\\ }^{E}G_{m} \\right)}{\\partial x_{j}}$\n\nOne may define the chemical activity of component *i*, $a_{i}^{\\ }$, as\nfollows\n\nEq. \u2011\n$RTlna_{i}^{\\ } = \\mu_{i} -_{\\ }^{0}G_{i}^{\\ } =_{\\ }^{I}{G_{m} +}\\frac{\\partial_{\\ }^{I}G_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial_{\\ }^{I}G_{m}}{\\partial x_{j}} +_{\\ }^{E}G_{m} + \\frac{\\partial_{\\ }^{E}G_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial_{\\ }^{E}G_{m}}{\\partial x_{j}}$\n\nIn this definition, the chemical activity or simply activity is\ncalculated with respect to the pure elements in the structure of the\nsolution for practical reasons as one would like to understand the\nchemical potential difference of components in the solution and by\nitself with the same structure. It should be noted that this reference\nstate for chemical activity is usually different from the SER reference\nstate defined in as the solution may have a different structure than\nthose of pure components in their SER states. On the other hand, the\nactivity under the SER reference state can be easily obtained by\nreplacing $_{\\ }^{0}G_{i}$ with ${_{\\ }^{0}G}_{i}^{SER}$ from . In\nprinciple, one may choose any structure as the reference state for\nactivity to be useful for practical applications, i.e.\n\nEq. \u2011 $RTlna_{i}^{ref} = \\mu_{i} -_{\\ }^{0}G_{i}^{ref}$\n\nFor example, the activity of a component in a liquid solution is defined\nwith respect to the pure component in its liquid form from , but can\nalso be referred to its SER state which is solid using . The following\nsections will discuss in more details how components mix when they are\nbrought together including concepts such as random mixing, short-range\nordering, and long-range ordering.\n\nThe limit of stability of a solution with respect to composition\nfluctuation under constant *T*, *P*, and *N1* can be derived\nas follows from and\n\nEq. \u2011\n$\\left( \\frac{\\partial\\mu_{i}}{\\partial N_{i}} \\right)_{T,P,N_{j \\neq i},i > 1} > \\left( \\frac{\\partial\\mu_{i}}{\\partial N_{i}} \\right)_{T,P,N_{1},{\\mu_{2},N}_{j \\neq i},i,j > 2}\\ldots... > \\left( \\frac{\\partial\\mu_{c}}{\\partial N_{c}} \\right)_{T,P,N_{1},\\mu_{2}..\\mu_{c - 1}} = 0$\n\nThe first term can be derived from as follows\n\nEq. \u2011\n$\\left( \\frac{\\partial\\mu_{i}}{\\partial N_{i}} \\right)_{T,P,N_{j \\neq i},i > 1} = \\sum_{j = 1}^{c}\\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{i}\\partial x_{j}}\\frac{\\partial x_{j}}{\\partial N_{i}} - \\sum_{j = 1}^{c}{x_{j}\\sum_{k = 1}^{c}{\\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{j}\\partial x_{k}}\\frac{\\partial x_{k}}{\\partial N_{i}}}} = \\frac{1}{N}\\left( \\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{i}^{2}} - \\sum_{j = 1}^{c}{x_{j}\\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{j}^{2}}} - \\sum_{j = 1}^{c}{x_{j}\\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{i}\\partial x_{j}}} + \\sum_{j = 1}^{c}{\\sum_{k = 1}^{c}{{x_{j}x}_{k}\\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{j}\\partial x_{k}}}} \\right)$\n\nDenoting\n$G_{ij} = \\left( \\frac{\\partial\\mu_{i}}{\\partial N_{j}} \\right)_{T,P,N_{k \\neq j}}$and\nusing to change the variables kept constant from molar quantities to\npotentials one-by-one (see \\[1\\]), the limit of stability can be further\nobtained as\n\nEq. \u2011\n$\\left( \\frac{\\partial\\mu_{c}}{\\partial N_{c}} \\right)_{T,P,N_{1},\\mu_{2}..\\mu_{c - 1}} = \\frac{\\det\\left( G_{ij}:2 \\leq i,j \\leq c \\right)}{\\det\\left( G_{ij}:2 \\leq i,j \\leq c - 1 \\right)} = 0$\n\nwhere $\\det$ stands for determinant. indicates that\n$\\det\\left( G_{ij}:2 \\leq i,j \\leq c \\right) = 0$ at the limit of\nstability. Considering $x_{1} = 1 - \\sum_{j \\neq 1}^{}x_{j}$, let us\nintroduce\n\nEq. \u2011\n$g_{i} = \\mu_{i} - \\mu_{1} = \\left( \\frac{\\partial G_{m}}{\\partial x_{i}} \\right)_{x_{k \\neq i}} - \\left( \\frac{\\partial G_{m}}{\\partial x_{1}} \\right)_{x_{k \\neq 1}}$\n\nand\n\nEq. \u2011\n$g_{ij} = \\frac{\\partial g_{i}}{\\partial x_{j}} = \\frac{\\partial\\left( \\mu_{i} - \\mu_{1} \\right)}{\\partial x_{j}} = \\frac{\\partial^{2}G_{m}}{\\partial x_{i}\\partial x_{j}} - \\frac{\\partial^{2}G_{m}}{\\partial x_{1}\\partial x_{j}} - \\frac{\\partial^{2}G_{m}}{\\partial x_{i}\\partial x_{1}} + \\frac{\\partial^{2}G_{m}}{\\partial\\left( x_{1} \\right)^{2}}$\n\nThe limit of stability can be re-written as\n\nEq. \u2011\n$\\left( \\frac{\\partial\\left( \\mu_{c} - \\mu_{1} \\right)}{\\partial x_{c}} \\right)_{T,P,N,\\mu_{2} - \\mu_{1},\\ ...\\mu_{c - 1} - \\mu_{1}} = \\frac{\\det\\left( g_{ij}:2 \\leq i,j \\leq c \\right)}{\\det\\left( g_{ij}:2 \\leq i,j \\leq c - 1 \\right)} = 0$\n\ni.e. $\\det\\left( g_{ij}:2 \\leq i,j \\leq c \\right) = 0$. The consolute\npoint can be defined following\n\nEq. \u2011\n$\\left( \\frac{\\partial^{2}\\mu_{c}}{\\partial\\left( N_{c} \\right)^{2}} \\right)_{T,P,N_{1},\\mu_{2}..\\mu_{c - 1}} = \\left( \\frac{\\partial^{2}\\left( \\mu_{c} - \\mu_{1} \\right)}{\\partial\\left( x_{c} \\right)^{2}} \\right)_{T,P,N,\\mu_{2} - \\mu_{1},\\ ...\\mu_{c - 1} - \\mu_{1}} = 0$\n\nNo closed mathematic form has been published in the literature.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/gibbs_energy_function/phases_with_variable_compositions_solutions_with_ordering.ipynb b/psu410/src/gibbs_energy_function/phases_with_variable_compositions_solutions_with_ordering.ipynb new file mode 100644 index 0000000..6d406a2 --- /dev/null +++ b/psu410/src/gibbs_energy_function/phases_with_variable_compositions_solutions_with_ordering.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "49e6341c", "cell_type": "markdown", "source": "## Phases with variable compositions: Solutions with ordering\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/gibbs_energy_function/polymer_solutions_and_polymer_blends.ipynb b/psu410/src/gibbs_energy_function/polymer_solutions_and_polymer_blends.ipynb new file mode 100644 index 0000000..e172c0a --- /dev/null +++ b/psu410/src/gibbs_energy_function/polymer_solutions_and_polymer_blends.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "efad1127", "cell_type": "markdown", "source": "## Polymer solutions and polymer blends\n\nA polymer solution is a mixture between polymer molecules and solvents,\nwhile a polymer blend is a mixture between different polymer molecules.\nA polymer molecule consists of the same repeating units of one or more\nmonomers, which can be an atom or a small molecule. The number of\nrepeating units is called the degree of polymerization and can be as\nlarge as 104\u2013105. It defines the molecular mass,\ni.e. the mass of one polymer molecule. There are three typical\narchitectures of polymerization: a linear chain, a branched chain, and a\ncross-linked polymer. Nearly all polymers are mixtures of molecules with\na different degree of polymerization with a molecular mass distribution,\ncomplicating the modeling of their thermodynamic properties because of\nthe dependence of properties on molecular mass.\n\nGibbs energy functions of polymers with a single molecular mass can be\ntreated similarly as in Chapter . For a polymer solution, the ideal\nentropy of mixing is quite different from that of atomically random\nsolutions discussed in Chapter because the monomers in a polymer\nmolecule are connected to each other and cannot move freely. One common\napproach to calculate the ideal entropy of mixing is to evoke a lattice\nmodel and assume that one monomer occupies a lattice site with a fixed\nvolume. The number of translational states of a single molecule is equal\nto the number of lattice sites available. In a homogeneous solution, the\ntotal number of lattice sites available is\n\nEq. \u2011 $n = \\sum_{i}^{}{m_{i}n_{i}}$\n\nwhere $n_{i}$ and $m_{i}$ are the number of molecule $i$ and the number\nof lattice sites per molecule $i$, respectively. While in its pure\nstate, i.e. before mixing, the number of states of molecule $i$ in terms\nof the number of lattice sites is\n\nEq. \u2011 $w_{i} = m_{i}n_{i} = n\\varphi_{i}$\n\nwhere $\\varphi_{i}$ is the volume fraction of molecule $i$ in the\nsolution. The entropy change per molecule $i$ is thus\n\nEq. \u2011\n$S_{i} = kln(n) - klnw_{i} = kln\\frac{1}{\\varphi_{i}} = - kln\\varphi_{i}$\n\nThe total entropy of mixing is the summation for all molecules,\nnormalized to per mole of lattice site\n\nEq. \u2011\n$_{}^{I}S_{m} = \\frac{N_{a}}{n}\\sum_{i}^{}{n_{i}S_{i}} = - R\\sum_{i}^{}{\\frac{\\varphi_{i}}{m_{i}}\\ln\\varphi_{i}}$\n\nwhere $N_{a}$ is the Avogadro number. When $m_{i} = 1$ for all\nmolecules, Eq. 2\u2011143 reduces to . Since $m_{i}$ values are typically\nvery large numbers for polymers, the entropy of mixing in polymer\nsolutions and blends is thus significantly lower than those in\nnon-polymer solutions as shown schematically in for binary systems with\nvarious $m_{i}$ values.\n\nFigure \u2011: Schematic entropy of mixing in solutions with the numbers of\nlattice sites per molecule shown.\n\nSimilar to , Gibbs energy of a multi-component random polymer\nsolution/blend can be written as\n\nEq. \u2011\n$G_{m} = \\sum_{}^{}\\frac{\\varphi_{i}}{m_{i}}_{\\ }^{0}G_{im} + RT\\left( \\sum_{}^{}{\\frac{\\varphi_{i}}{m_{i}}\\ln\\varphi_{i}} + \\sum_{}^{}{\\varphi_{i}\\varphi_{j}\\chi_{ij}} \\right)$\n\nwhere $_{\\ }^{0}G_{im}$ is the Gibbs energy of molecular $i$ per mole of\nlattice site, and $\\chi_{ij}$ the unitless interaction parameter between\nmolecule $i$ and $j$. Other equations shown in Chapter can be derived\nsimilarly too. It is to be noted that is very similar to the\nFlory\u2013Huggins solution equation widely used in the polymer community.\n", "metadata": {}}, {"id": "d26c203e", "cell_type": "markdown", "source": "### Random solutions\n\nThe ideal Gibbs energy of mixing represents an ideal solution in which\nall sites are equivalent and the distributions of components on the\nsites are completely random. The number of different configurations to\narrange all components is\n\nEq. \u2011 $w = \\frac{N!}{\\prod_{}^{}\\left( N_{i}! \\right)}$\n\nBased on Boltzmann\u2019s relation from statistic thermodynamics when all\nconfigurations have the same probability to be observed, the ideal\nconfigurational molar entropy of mixing due to the distribution is\n\nEq. \u2011\n$_{\\ }^{ideal}S_{m} =_{\\ }^{I}S_{m} = \\frac{Rlnw}{N} = R\\frac{lnN! - \\sum_{}^{}\\ln\\left( N_{i}! \\right)}{N} \\cong R\\frac{NlnN - \\sum_{}^{}{N_{i}l{nN}_{i}}}{N} = - R\\sum_{}^{}{x_{i}l{nx}_{i}}$\n\nwhere $R$ is the gas constant. represents the entropy difference between\nthe ideal solution and the states when individual components are by\nthemselves, i.e. the mechanical mixing of components. As $x_{i}$ is\nsmaller than unity, the entropy production to form an ideal solution\nfrom pure components is thus positive, indicating that it is a\nspontaneous process. In such an ideal solution, it is assumed that there\nare no interactions between components, and the enthalpy of mixing is\nthus zero as the internal energy and the volume of the system do not\nchange. The ideal Gibbs energy of mixing is written as\n\nEq. \u2011 $_{\\ }^{I}G = - T_{\\ }^{I}S_{m} = RT\\sum_{}^{}{x_{i}l{nx}_{i}}$\n\nThe Gibbs energy of real solutions, i.e. , becomes\n\nEq. \u2011\n$G_{m} = \\sum_{}^{}{x_{i}_{\\ }^{0}G_{i}} + RT\\sum_{}^{}{x_{i}l{nx}_{i}} +_{\\ }^{E}G_{m}$\n\nFrom , the chemical potential is obtained as\n\nEq. \u2011\n$\\mu_{i} = G_{i} =_{\\ }^{0}G_{i} + RTlnx_{i} +_{\\ }^{E}G_{m} + \\frac{\\partial_{\\ }^{E}G_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial_{\\ }^{E}G_{m}}{\\partial x_{j}}$\n\nFrom the chemical activity of , the activity coefficient, $\\gamma_{i}$,\ncan be defined as follows\n\nEq. \u2011\n$\\gamma_{i} = \\frac{a_{i}}{x_{i}} = \\frac{1}{x_{i}}\\exp\\frac{G_{i} -_{\\ }^{0}G_{i}\\ }{RT}$\n\nThe solution is an ideal solution with $\\gamma_{i} = 1$, and is said to\nbe positively or negatively deviating from an ideal solution with\n$\\gamma_{i} > 1$ or $\\gamma_{i} < 1$, respectively. The chemical\npotential is related to the activity and activity coefficient by the\nfollowing equation\n\nEq. \u2011\n$\\mu_{i} =_{\\ }^{0}G_{i} + RTlna_{i} =_{\\ }^{0}G_{i} + RTln\\gamma_{i}x_{i} =_{\\ }^{0}G_{i} + RTlnx_{i} + RTln\\gamma_{i}$\n\nLet us further exam in more details in order to better understand the\nrelation between $G_{m}$ and $\\mu_{i}$. The partial derivatives in\nrepresent the directions of molar Gibbs energy in the composition space,\ni.e. the tangents of molar Gibbs energy with respect to mole fractions\nof independent components. Collectively, they define the\nmulti-dimensional tangent plane of molar Gibbs energy at the given\ncomposition, $x_{i}^{0}$. The mathematical representation of this\ntangent plane, $z_{G_{m}}$, is defined by its directional derivatives\nand the distance from the point where the derivatives are taken,\n\nEq. \u2011\n$z_{G_{m}} = G_{m}\\left( x_{i}^{0} \\right) + \\sum_{i = 1}^{c}{\\left( \\frac{\\partial G_{m}}{\\partial x_{i}} \\right)_{x_{i}^{0}}\\left( x_{i} - x_{i}^{0} \\right)}$\n\nThe intercept of this tangent plane at each pure component axis, i.e.\n$x_{i} = 1$ and $x_{j \\neq i} = 0$, is obtained as\n\nEq. \u2011\n$z_{G_{m},x_{i} = 1} = G_{m}\\left( x_{i}^{0} \\right) + \\left( \\frac{\\partial G_{m}}{\\partial x_{i}} \\right)_{x_{i}^{\\ } = x_{i}^{0}} - \\sum_{j = 1}^{c}{x_{j}^{0}\\left( \\frac{\\partial G_{m}}{\\partial x_{j}} \\right)_{x_{i}^{\\ } = x_{i}^{0}}}$\n\nThis is identical to at the point $x_{i}^{0}$. It is thus shown that the\nchemical potential of a component in a solution is represented by the\nintercept of the tangent plane of Gibbs energy of the solution on the\n$G_{m}$ axis of the component. The distance between the intercept and\nthe Gibbs energy of the pure component in the same structure of the\nsolution is related to the chemical activity of the component as defined\nby . On the other hand, it is evident that one can choose any other\nstructure of the pure element to define the chemical activity to compare\nchemical potential of the component as shown by .\n\nThe stability of a solution is evaluated following , and the derivatives\nof chemical potential with respect to its moles, i.e. the elements in\nthe determinant, are obtained as follows from and ,\n\nEq. \u2011\n$\\frac{N}{RT}\\frac{\\partial\\mu_{i}}{\\partial N_{i}} = \\frac{N}{RT}G_{ii} = \\frac{1 - x_{i}}{x_{i}} + \\frac{1}{\\gamma_{i}}\\left( \\frac{\\partial\\gamma_{i}}{\\partial x_{i}} - \\sum_{j = 1}^{c}{x_{j}\\frac{\\partial\\gamma_{i}}{\\partial x_{j}}} \\right)$\n\nEq. \u2011\n$\\frac{N}{RT}\\frac{\\partial\\mu_{i}}{\\partial N_{k}} = \\frac{N}{RT}G_{ik} = - \\frac{x_{k}}{x_{i}} + \\frac{1}{\\gamma_{i}}\\left( 1 - \\sum_{j = 1}^{c}{x_{j}\\frac{\\partial\\gamma_{i}}{\\partial x_{j}}} \\right)$\n\nTo further study Gibbs energy of solution phases, let us discuss the\ndetails on the excess Gibbs energy of mixing. At this point, one can\nstart with lower-order systems with fewer components, i.e. two component\nand three-component systems, noting that the Gibbs energy of phases with\none component is already presented in Chapter .\n", "metadata": {}}, {"id": "ab833401", "cell_type": "markdown", "source": "### Solutions with both short-range and long-range ordering\n\nThe short-range ordering in a solution with long-rang ordering can take\nplace in each sublattice or between two sublattices. The short-range\nordering in one sublattice can be treated similarly as in Chapter 2.3.1\nwith mole fractions substituted by site fractions. In case that\nassociates form, the relation between mole fractions and site fractions\nbecomes more complicated as follows\n\nEq. \u2011\n$x_{i} = \\frac{\\sum_{}^{}{a^{t}\\sum_{k}^{}{i_{k}y}_{k}^{t}}}{\\sum_{}^{}{a^{t}\\left( 1 - y_{va}^{t} \\right)}}$\n\nwhere the summation for $k$ goes over all associates in sublattice $t$\ncontaining component $i$.\n\nThe short-range ordering between two-sublattices indicates that a\ncomponent in one sublattice has different interactions with different\ncomponents in another sublattice. This results in local ordering of one\ncomponent around another component in two neighboring sublattices. Such\nlocal ordering involves interactions between two sublattices shown as\nthe third term in .\n", "metadata": {}}, {"id": "09d5f96a", "cell_type": "markdown", "source": "### Solutions with charged species\n\nOne special type of solutions with both short-range and long-range\nordering is solutions with charged species, i.e. ionic solutions, plus\nelectrons and holes. There is an additional constraint on species\nconcentrations to maintain the charge neutrality of such solutions, i.e.\n\nEq. \u2011 $0 = \\sum_{t}^{}{\\sum_{i}^{}{a^{t}y_{i}^{t}v_{i}^{t}}}$\n\nwhere $v_{i}^{t}$ is the valance of species $i$ in sublattice $t$\nincluding its sign which is positive for cations, negative for anions,\nand zero for neutral species. The conventional defect chemistry theory\nis typically based on the ideal mass action laws and applicable to a\nsingle set of defects and at very low defect concentrations, i.e. in the\nlimit of ideal solutions. For interacting defects, their concentrations\nshould be replaced by their activities, which can be obtained from\nthermodynamic principles similarly as discussed in previous sections. It\nshould be emphasized that in addition to formation of many more charged\nspecies, one component may have different valences, particularly the\ntransition metals. Consequently, there can be many more species in an\nionic phase than the number of independent components in the system, and\ntheir concentrations can be found by equilibrium calculations as\ndiscussed in Chapter .\n", "metadata": {}}, {"id": "fe69a707", "cell_type": "markdown", "source": "### Solutions with long-range ordering\n\nSo far, solutions in which a component can occupy any site in a phase\nare discussed. In many phases, this is not the case. For example, in the\nfcc solid solution of Fe and C, Fe atoms take the fcc lattice sites, and\nC atoms occupy the interstitial sites between the fcc lattice sites.\nTherefore, Fe atoms do not mix with C atoms on the fcc lattice sites,\nand they rather develop long-rang ordering by occupying their own\ndistinct sites in the phase. Long-range ordering can also develop when\nshort-range ordering extends to the whole lattice. New formula for the\nGibbs energy of mixing is needed by considering the details how\ncomponents are distributed and mixed in various sites in a phase.\n\nOne way to group various sites in a phase is based on equivalent\ncrystallographic positions in a phase, i.e. Wyckoff positions. Various\nsets of equivalent positions divide the lattice into sub-sets of\nlattices. Each set of equivalent positions forms a sublattice. The\ndistributions of components on each sublattice can be represented by\nmole fractions of components in the sublattice, commonly referred to as\nsite fractions and defined as\n\nEq. \u2011 $y_{i}^{t} = \\frac{N_{i}^{t}}{\\sum_{j}^{}N_{j}^{t}}$\n\nEq. \u2011 $\\sum_{i}^{}y_{i}^{t} = 1$\n\nwhere the superscript $t$ denotes the sublattice in which the component\nresides, and the summation is for all species in sublattice $t$\nincluding vacancy. Site fractions and mole fractions are related through\nthe mass balance as follows\n\nEq. \u2011\n$x_{i} = \\frac{\\sum_{}^{}{a^{t}y_{i}^{t}}}{\\sum_{}^{}{a^{t}\\left( 1 - y_{va}^{t} \\right)}}$\n\nwhere $a^{t}$ and $y_{va}^{t}$ are the number of sites and the site\nfraction of vacancy in the sublattice *t*.\n\nRandom solutions form when each component enters all sublattices\nequally. Mole fractions and site fractions thus become identical.\nSolutions with both substitutional and interstitial component, like the\nfcc Fe-C solution mentioned above, can be represented by two\nsublattices. Stoichiometric compounds have its site fractions equal to\nunity in each sublattice. When site fractions in a compound deviate from\nunity, the compound is no longer stoichiometric and develops a\ncomposition range of homogeneity. When the composition range is small,\nthe deviations are often referred to as defects. Since many properties\nof a compound are determined by defects, a distinct field of defect\nchemistry exists, predominantly for charged species. As will be\ndemonstrated in Chapter and the rest of the book, defects can be treated\nas an integral part of the thermodynamics of a phase with more than one\nsublattices.\n\nLet us consider a case where there is only one component in each\nsublattice, which represents one possible stoichiometric composition of\nthe phase and is often called an end-member of the phase. The Gibbs\nenergy of an end-member is the same as that of a phase with a fixed\ncomposition shown by or or . By re-arranging , Gibbs energy of an\nend-member, $_{\\ }^{0}G_{em}$, is obtained as\n\nEq. \u2011\n$_{\\ }^{0}G_{em} = \\sum_{t}^{}{a^{t}{_{\\ }^{0}G}_{i}^{t,ref}} + \\mathrm{\\Delta}_{f}G_{em}$\n\nwhere ${_{\\ }^{0}G}_{i}^{t,ref}$ represents the Gibbs energy of\ncomponent $i$ in a given reference state which occupies sublattice $t$\nin the end-member. For vacancy, $_{\\ }^{0}G_{Va} = 0$ is defined. The\ncontribution of each end-member to the Gibbs energy of the phase is the\nproduct of site fraction of each component in their respective\nsublattices and the Gibbs energy of the end-member per mole of formula\nunit (*mf*), i.e.\n\nEq. \u2011\n$_{\\ }^{0}G_{mf} = \\sum_{em}^{}\\left( \\prod_{t}^{}y_{i}^{t}_{\\ }^{0}G_{em} \\right)$\n\nThe ideal mixing in each sublattice is similar to that in a random\nsolution with mole fractions substituted by site fractions. The excess\nGibbs energy of mixing consists of two contributions: (i) the mixing in\none sublattice with all other sublattices containing only one component\nin each sublattice and (ii) the mixing simultaneously in more than one\nsublattices. The Gibbs energy of a solution phase with multi-sublattices\ncan thus be written in terms of per mole of formula unit as\n\nEq. \u2011\n$G_{mf} =_{\\ }^{0}G_{mf} + RT\\sum_{t}^{}{a^{t}\\sum_{i}^{}{y_{i}^{t}\\ln y_{i}^{t}}} +_{\\ }^{E}G_{mf}$\n\nwith $_{\\ }^{E}G_{mf}$ given by\n\nEq. \u2011\n$_{\\ }^{E}G_{mf} = \\sum_{t}^{}{\\prod_{s \\neq t}^{}y_{l}^{s}\\sum_{i > j}^{}{\\sum_{j}^{}{y_{i}^{t}y_{j}^{t}L_{i,j:l}^{t}}}} + \\sum_{t}^{}{\\prod_{s \\neq t}^{}y_{l}^{s}\\sum_{i > j}^{}{\\sum_{j > k}^{}{\\sum_{k}^{}{y_{i}^{t}y_{j}^{t}y_{k}^{t}L_{i,j,k:l}^{t}}}}} + \\sum_{t}^{}{\\prod_{s \\neq t,u}^{}y_{l}^{s}\\sum_{i > j}^{}{\\sum_{j > k}^{}{\\sum_{k}^{}{y_{i}^{t}y_{j}^{t}y_{m}^{u}y_{n}^{u}L_{i,j:m,n:l}^{t}}}}}$\n\nThe first term in represents the binary interaction between component\n$i$ and $j$ in sublattice $t$ with sublattice $s$ occupied by component\n$l$ with comma separating interacting components in one sublattice and\ncolon separating sublattices. The product,\n$\\prod_{s \\neq t}^{}y_{l}^{s}$, runs through all other sublattices with\none component in each sublattice, except sublattice $t$ in which the\ninteraction is considered. The second term denotes the ternary\ninteraction among $i$, $j$, and $k$ in sublattice $t$ with sublattice\n$s$ occupied by component $l$. The third term depicts the interactions\nsimultaneously in both sublattice $t$ and $u$, and the product runs\nthrough all other sublattices with one component in each sublattice,\nexcept sublattice $t$ and $u$ in which the interactions are considered.\nThe third term thus partially reflects the short-range ordering among\ncomponents between two sublattices. In principles, high-order\ninteraction parameters such as quaternary, quinary, and multiple\nsublattice interaction parameters could be added, but their\ncontributions to $_{\\ }^{E}G_{mf}$ are small due to the physical\ninsignificance of co-location of four or five components indicated by\nthe product of their site fractions in front of the interaction\nparameters.\n\nIn , the chemical potential of a stoichiometric compound was defined in\nterms of a summation of chemical potential of individual components in\nthe compound because the relative amounts of components are constrained\nby the stoichiometry of the compound and chemical potentials of\nindividual components can not vary independently. By the same token, the\nchemical potential of an end-member in a solution can be written as\n\nEq. \u2011 $\\mu_{em} = G_{em} = \\sum_{t}^{}a^{t}\\mu_{i}^{t}$\n\nwhere $\\mu_{i}^{t}$ is the chemical potential of component $i$ that\noccupies the sublattice $t$ in the end-member, and can be derived using\n\nEq. \u2011\n$\\mu_{i}^{t} = a^{t}_{\\ }^{0}G_{i}^{t,ref} + a^{t}RTlny_{i}^{t} +_{\\ }^{E}G_{mf}^{\\ } + \\frac{\\partial_{\\ }^{E}G_{mf}^{\\ }}{\\partial y_{i}^{t}} - \\sum_{j}^{\\ }y_{j}^{t}\\frac{\\partial_{\\ }^{E}G_{mf}^{\\ }}{\\partial y_{j}^{t}}$\n\nFor constant interaction parameters in , for chemical potential reduces\nto the following expression from\n\nEq. \u2011\n$\\mu_{i}^{t} = a^{t}_{\\ }^{0}G_{i}^{t} + a^{t}RTlny_{i}^{t} + \\sum_{j \\neq i}^{}{\\left( y_{i}^{t} \\right)^{2}L}_{i,j}^{t} + \\sum_{k > j}^{}{\\sum_{j \\neq i}^{}{y_{j}^{t}y_{k}^{t}\\left\\lbrack L_{i,j}^{t} + L_{i,k}^{t} - L_{j,k}^{t} + \\left( 1 - 2y_{i}^{t} \\right)L_{i,j,k}^{t} \\right\\rbrack}}$\n\nThe stability of the solution is defined by\n$\\frac{\\partial\\mu_{em}}{\\partial N_{em}}$ with $N_{em}$ being the moles\nof the end-member in the solution given by\n$N_{em} = N\\prod_{u}^{}y_{j}^{u}$ and\n\nEq. \u2011\n$\\frac{\\partial N_{em}}{\\partial y_{j}^{u}} = \\frac{N\\prod_{t}^{}y_{i}^{t}}{y_{j}^{u}\\left( 1 - \\prod_{t}^{}y_{i}^{t} \\right)}$\n\nFollowing , one obtains\n\nEq. \u2011\n$\\frac{\\partial\\mu_{em}}{\\partial N_{em}} = \\sum_{u}^{}{\\frac{\\partial\\left( \\sum_{t}^{}a^{t}\\mu_{i}^{t} \\right)}{\\partial y_{j}^{u}}\\frac{\\partial y_{j}^{u}}{\\partial N_{em}}} = \\frac{\\left( 1 - \\prod_{t}^{}y_{j}^{t} \\right)}{N\\prod_{t}^{}y_{j}^{t}}\\sum_{u}^{}{a^{u}y_{i}^{u}\\frac{\\partial\\mu_{i}^{u}}{\\partial y_{i}^{u}}} = \\frac{\\left( 1 - \\prod_{t}^{}y_{j}^{t} \\right)}{N\\prod_{t}^{}y_{j}^{t}}\\sum_{u}^{}{a^{u}y_{i}^{u}\\left\\{ \\frac{RT\\left( 1 - y_{i}^{u} \\right)}{y_{i}^{u}} - 2\\sum_{j \\neq i}^{}\\left( y_{j}^{u} \\right)^{2}L_{i,j}^{u} - 2\\sum_{k > j}^{}{\\sum_{j \\neq i}^{}{y_{j}^{u}y_{k}^{u}\\left\\lbrack L_{i,j}^{u} + L_{i,k}^{u} - L_{j,k}^{u} + \\left( 2 - 3y_{i}^{u} \\right)L_{i,j,k}^{u} \\right\\rbrack}} \\right\\}}$\n\nIt is self-evident from that a site fraction is only uniquely defined\nfrom the mole fraction of the component when the component enters into\none sublattice only and does not form any associates. Therefore, in\ngeneral, the distribution of components on sublattices and different\nkinds of molecules can only be obtained by equilibrium calculations, and\nthe thermodynamic properties for such a phase can thus not be\nrepresented in a closed form using mole fractions of independent\ncomponents. This was demonstrated in Chapter when short-range ordering\nexists in solution phases, where the energy minimization procedure was\nused to obtain the distribution of components on different kinds of\nbonds and the amounts of individual associates.\n", "metadata": {}}, {"id": "82f8e314", "cell_type": "markdown", "source": "### Solutions with short-range ordering\n\nThe order in a system can be measured by correlation functions which\ndescribes how various components are correlated in space. For\nsimplicity, let us consider only the pairs between nearest neighbors\nwith the correlation function represented by the pair probability of\nnearest neighbor bonds between two components. In a random solution, the\nprobability to find nearest neighbor bonds between two components $i$\nand $j$ is\n\nEq. \u2011 $p_{ij} = x_{i}x_{j}$\n\nWhen $p_{ij} \\neq x_{i}x_{j}$, the nearest neighbors of component $i$\nare not occupied randomly by component $j$, rather certain components\nare favored, resulting in short-range ordering or local clustering in\nthe solution. When short-range ordering develops throughout the\nsolution, long-range ordering takes place, and each component has its\nown primary sites in the solution, to be discussed in Chapter . There\nare relations between bond probabilities and mole fractions of\ncomponents due to the mass balance as follows, with the assumption of\n$p_{ij} = p_{ji}$\n\nEq. \u2011 $\\sum_{i}^{}{\\sum_{j}^{}p_{ij}} = 1$\n\nEq. \u2011 $x_{i} = \\sum_{j}^{}p_{ij}$\n\nFor small deviations from a random solution, one can consider the\nformation of $i - j$ bonds from $i - i\\ $ and $j - j$ bonds and the\nideal mixing of three types of bonds, similar to a typical Ising model.\nThe bond reaction can be written as\n\nEq. \u2011 $(i - i\\ )bonds + (j - j)\\ bonds = 2\\ (i - j)\\ bonds$\n\nwith the Gibbs energy of reaction being\n\nEq. \u2011\n${\\mathrm{\\Delta}G}_{ij} = 2G_{ij} - \\left( G_{ii} + G_{jj} \\right)\\ $\n\nGibbs energy of the solution per mole of atom is thus represented by the\nbond energies and the ideal mixing of bonds plus non-ideal interactions\nbetween pairs,\n\nEq. \u2011\n$G_{m} = \\sum_{i}^{}{\\sum_{j}^{}{p_{ij}G_{ij}}} + \\frac{Z}{2}RT\\sum_{i}^{}{\\sum_{j}^{}{p_{ij}\\ln p_{ij}}} +_{\\ }^{E}G_{m}$\n\nwith $G_{ij}$ being the molar bond energy between components $i$ and\n$j$, $Z$ the number of bonds per atom which is divided by two in the\nequation due to two atoms needed to form one bond, and\n$_{\\ }^{E}G_{m} = \\sum_{}^{}{p_{ij}p_{kl}I_{ijkl}}$ the excess Gibbs\nenergy of mixing between bonds. This approach proposed by Guggenheim\n\\[2\\] is called quasi-chemical method as it is based on the chemical\nreaction shown by .\n\nHowever, the entropy of mixing in does not reduce to ideal entropy of\nmixing for a solution without short-range ordering as defined by . An\napproximated correction may be added for small degree of short-range\nordering as follows\n\nEq. \u2011\n$G_{m} = \\sum_{i}^{}{\\sum_{j}^{}{p_{ij}G_{ij}}} + \\frac{Z}{2}RT\\sum_{i}^{}{\\sum_{j}^{}{p_{ij}\\ln\\frac{p_{ij}}{x_{i}x_{j}}}} + RT\\sum_{}^{}{x_{i}l{nx}_{i}} +_{\\ }^{E}G_{m}$\n\nFor a random solution defined by , becomes\n\nEq. \u2011\n$G_{m} = \\sum_{}^{}{x_{i}_{\\ }^{0}G_{i}} + RT\\sum_{}^{}{x_{i}l{nx}_{i}} + \\sum_{}^{}{x_{i}x_{j}{\\mathrm{\\Delta}G}_{ij}} +_{\\ }^{E}G_{m}$\n\nwith $_{\\ }^{0}G_{i} = G_{ii}$, ${\\mathrm{\\Delta}G}_{ij}$ from Eq. 2\u2011107\nrepresenting the interaction parameter between components $i$ and $j$,\nand $_{\\ }^{E}G_{m} = \\sum_{}^{}{x_{i}{x_{j}x_{k}x}_{l}I_{ijkl}}$\ndenoting the higher order interactions, in comparison with .\n\nWhen short-range ordering exists in a solution, one typically uses the\nlaw of mass reaction for the chemical reaction represented by to define\nthe equilibrium among all bonds, i.e.\n\nEq. \u2011\n$\\frac{\\left( p_{ij} \\right)^{2}}{p_{ii}p_{jj}} = e^{- \\frac{{\\mathrm{\\Delta}G}_{ij}}{kT}}$\n\nHowever, this is under the assumption that the chemical activities of\nall bonds can be represented by their respective probabilities, which is\nonly true for an ideal solution even excluding dilute solutions due to\nthe Henry\u2019s law shown by . Preferrably, the bond probabilities can be\nobtained by calculating the driving force for the fluctuation of bond\nprobabilities under constant temperature, pressure, and amount of each\ncomponent along with the constraints defined by and and equating the\ndriving force to zero, i.e.\n\nEq. \u2011\n${\\frac{1}{N}\\left( \\frac{\\partial G}{\\partial\\xi} \\right)}_{T,P,N_{n}} = \\left( \\frac{\\partial G_{m}}{\\partial p_{ij}} \\right)_{T,P,N_{n}} = \\left( \\frac{\\partial G_{m}}{\\partial p_{ij}} \\right)_{T,P,x_{n},p_{kl \\neq ij}} - \\sum_{kl \\neq ij}^{}\\left( \\frac{\\partial G_{m}}{\\partial p_{kl}} \\right)_{T,P,x_{n},p_{op \\neq kl}} + \\left( \\frac{\\partial G_{m}}{\\partial x_{i}} \\right)_{T,P,x_{q \\neq i},p_{kl}} + \\left( \\frac{\\partial G_{m}}{\\partial x_{j}} \\right)_{T,P,x_{q \\neq j},p_{kl}} = 0$\n\nwhere $\\frac{\\partial p_{kl}}{\\partial p_{ij}} = - 1$ and\n$\\frac{\\partial x_{i}}{\\partial p_{ij}} = \\frac{\\partial x_{j}}{\\partial p_{ij}} = 1$\nare used from and . Numerical values of $p_{ij}$ can be obtained by\nminimization of Gibbs energy under constraints given by and .\n\nThe chemical potential of independent component $i$ is defined as in and\ncan be represented by the following equation\n\nEq. \u2011\n$\\mu_{i} = G_{m} + \\frac{\\partial G_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial G_{m}}{\\partial x_{j}} + 2\\sum_{j = 1}^{c}\\frac{\\partial G_{m}}{\\partial p_{ij}} - \\frac{\\partial G_{m}}{\\partial p_{ii}} - 2\\sum_{j = 1}^{c}x_{j}\\sum_{k = 1}^{c}\\frac{\\partial G_{m}}{\\partial p_{jk}} + \\sum_{j = 1}^{c}x_{j}\\frac{\\partial G_{m}}{\\partial p_{jj}}$\n\nThe stability of the solution can be derived similar to .\n\nWhen the bonding between components becomes very strong, distinctive new\ncomponents may form. They are not independent components and often\ncalled associates. Both the independent and dependent components are\ncollectively called species. The formation of an associate\n$i_{a_{i}}j_{b_{j}}$ consisting of $a_{i}$ mole of $i$ and $a_{j}$ mole\n$j$ can be written as\n\nEq. \u2011 $a_{i}i + a_{j}j = i_{a_{i}}j_{a_{j}}$\n\nGibbs energy of the associate follows the same format of that of a\nstoichiometric phase as shown by ,\n\nEq. \u2011\n${_{\\ }^{0}G}_{i_{a_{i}}j_{a_{j}}} = \\sum_{}^{}a_{i}{_{\\ }^{0}G}_{i}^{SER} + \\mathrm{\\Delta}_{f}G_{i_{a_{i}}j_{a_{j}}}$\n\nGibbs energy of the solution is obtained by extending to all species\n\nEq. \u2011\n$G_{m} = \\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}_{\\ }^{0}G_{i_{a_{i}}j_{a_{j}}}} + RT\\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}\\ln y_{i_{a_{i}}j_{a_{j}}}} +_{\\ }^{E}G_{m}$\n\nwhere $y_{i_{a_{i}}j_{a_{j}}}$ is the mole fraction of species\n$i_{a_{i}}j_{a_{j}}$ in the solution with $a_{i} = 1$ and $a_{j} = 0$\nfor component $i$ and $a_{i} = 0$ and $a_{j} = 1$ for component $j$. The\nequilibrium amount of each associate $i_{a_{i}}j_{a_{j}}$ is obtained in\ncombination of mass balance and the zero driving force for the variation\nof the amount of the associate similar to , i.e.\n\nEq. \u2011 $\\sum_{i}^{}{\\sum_{j}^{}y_{i_{a_{i}}j_{a_{j}}}} = 1$\n\nEq. \u2011 $x_{i} = \\sum_{}^{}{a_{i}y_{i_{a_{i}}j_{a_{j}}}}$\n\nEq. \u2011\n$\\left( \\frac{{\\partial G}_{m}}{\\partial y_{i_{a_{i}}j_{a_{j}}}} \\right)_{x_{i}} = 0$\n\nAssociates are particularly plentiful in a gas phase, and their amounts\nare significantly affected by pressure. For an ideal gas phase with\n$_{\\ }^{E}G_{m} = 0$ and $PV_{m} = RT$, the effect of pressure is added\nas follows\n\nEq. \u2011\n$G_{m} = \\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}_{\\ }^{0}G_{i_{a_{i}}j_{a_{j}}}} + RT\\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}\\ln y_{i_{a_{i}}j_{a_{j}}}} + \\int_{P_{0}}^{P}{V_{m}dP} = \\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}_{\\ }^{0}G_{i_{a_{i}}j_{a_{j}}}} + RT\\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}\\ln y_{i_{a_{i}}j_{a_{j}}}} + RT\\ln\\frac{P}{P_{0}}$\n\nwhere $P$ is the total pressure, and $P_{0}$ the reference pressure at\nwhich $_{\\ }^{0}G_{i_{a_{i}}j_{a_{j}}}$ is defined, usually chosen to be\none atmospheric pressure. thus becomes\n\nEq. \u2011\n$G_{m} = \\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}_{\\ }^{0}{G_{i_{a_{i}}j_{a_{j}}}(P = 1atm)}} + RT\\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}\\ln y_{i_{a_{i}}j_{a_{j}}}} + RTlnP$\n\nwith the unit of the total pressure $P$ being in atmospheric pressure\n(atm). The chemical potential of species $i_{a_{i}}j_{a_{j}}$ is equal\nto\n\nEq. \u2011\n$\\mu_{i_{a_{i}}j_{a_{j}}} =_{\\ }^{0}{G_{i_{a_{i}}j_{a_{j}}}(P = 1atm)} + RTlny_{i_{a_{i}}j_{a_{j}}}P =_{\\ }^{0}{G_{i_{a_{i}}j_{a_{j}}}(P = 1atm)} + RTlnP_{i_{a_{i}}j_{a_{j}}}$\n\nwhere $P_{i_{a_{i}}j_{a_{j}}}$ is the partial pressure of species\n$i_{a_{i}}j_{a_{j}}$ defined as\n\nEq. \u2011 $P_{i_{a_{i}}j_{a_{j}}} = y_{i_{a_{i}}j_{a_{j}}}P$\n\nCombining and with , the relation between chemical potentials of an\nassociate and its constituents is expressed as\n\nEq. \u2011\n$\\mu_{i_{a_{i}}j_{a_{j}}} = a_{i}\\mu_{i} + {a_{j}\\mu}_{j} = a_{i}_{\\ }^{0}G_{i} + a_{j}_{\\ }^{0}G_{j} + RTln\\left( {P_{i}}^{a_{i}}{P_{j}}^{a_{j}} \\right)$\n\nThe equilibrium condition for the chemical reaction of an associate\nforming from its constituents in an ideal gas phase is obtained as\n\nEq. \u2011\n$\\mathrm{\\Delta}_{f}G_{i_{a_{i}}j_{a_{j}}} + RTln\\frac{{P_{i}}^{a_{i}}{P_{j}}^{a_{j}}}{P_{i_{a_{i}}j_{a_{j}}}} = 0$\n\nFor non-ideal phases, the mole fractions of various associates can be\ncalculated numerically through the minimization of Gibbs energy under\nthe constraints of and .\n", "metadata": {}}, {"id": "d1d41fbf", "cell_type": "markdown", "source": "### Ternary random solutions\n\nFrom , the Gibbs-Duhem equation of a ternary system consisting of\ncomponents $A$, $B$ and $C$ is written as\n\nEq. \u2011\n$0 = - SdT - Vd( - P) - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B} - {N_{C}d\\mu}_{C}$\n\nThis equation represents a five-dimensional surface. It can be\nvisualized in a three-dimensional space with two of the five potentials\nfixed. Usually $T$ and $P$ are kept constant as they are the natural\nvariables of $G$, and reduces to\n\nEq. \u2011 $0 = - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B} - {N_{C}d\\mu}_{C}$\n\nA phase can thus be represented by a surface in the three-dimensional\nspace of $\\mu_{A}$, $\\mu_{B}$, and $\\mu_{C}$ at constant $T$ and $P$\nwith similar geometric appearance as .\n\nFrom , Gibbs energy of a ternary solution is written as\n\nEq. \u2011\n$G_{m} = x_{A}_{\\ }^{0}G_{A} + x_{B}_{\\ }^{0}G_{B} + x_{C}_{\\ }^{0}G_{C} + RT\\left( x_{A}l{nx}_{A} + x_{B}l{nx}_{B} + x_{C}l{nx}_{C} \\right) +_{\\ }^{E}G_{m}$\n\nWhen the mole fraction of one component approaches zero,\n$_{\\ }^{E}G_{m}$ reduces to the excess Gibbs energy of mixing of the\nbinary systems of the remaining two components, represented by .\nHowever, for a given composition of a ternary solution, there is no\nunique way to assign the contributions from $_{\\ }^{E}G_{m}$ of each\nbinary to $_{\\ }^{E}G_{m}$ of the ternary solution because\n$_{\\ }^{E}G_{m}$ of the ternary solution contains information of both\nbinary and ternary interactions. A variety of models is available in the\nliterature (see \\[1\\]). One intuitive approach would be to use the same\nformula as that in the binary system, i.e. , with the mole fractions\nsubstituted by the values in the ternary system, and $_{\\ }^{E}G_{m}$ of\na ternary solution may thus be defined as the following by including the\nternary interaction involving all three components,\n\nEq. \u2011\n$_{\\ }^{E}G_{m} = x_{A}x_{B}L_{AB} + x_{A}x_{C}L_{AC} + x_{B}x_{C}L_{BC} + x_{A}x_{B}x_{C}L_{ABC}$\n\nThe chemical potential of a component is represented by . When all\ninteraction parameters in are constant, i.e. a ternary regular solution,\nthe chemical potential of component $A$ can be derived as\n\nEq. \u2011\n$\\mu_{A} = G_{A} =_{\\ }^{0}G_{A} + RTlnx_{A} + x_{B}L_{AB} + x_{C}L_{AC} -_{\\ }^{E}G_{m} =_{\\ }^{0}G_{A} + RTlnx_{A} + x_{B}{\\left( 1 - x_{A} \\right)L}_{AB} + x_{C}\\left( 1 - x_{A} \\right)L_{AC} - x_{B}x_{C}L_{BC} + x_{B}x_{C}\\left( 1 - 2x_{A} \\right)L_{ABC} =_{\\ }^{0}G_{A} + RTlnx_{B} + x_{B}^{2}L_{AB} + x_{C}^{2}L_{AC} + x_{B}x_{C}\\left( L_{AB} + L_{AC} - L_{BC} \\right) + x_{B}x_{C}\\left( 1 - 2x_{A} \\right)L_{ABC}$\n\nSimilar equations can be derived for component $B$ and C with\n$L_{AB} = L_{BA}$, $L_{AC} = L_{CA}$, and $L_{BC} = L_{CB}$. A schematic\nmolar Gibbs energy diagram at constant temperature and pressure is shown\nin with all three binary systems having $L_{ij} < 0$ of similar values.\n\nFigure \u2011: Schematic ternary molar Gibbs energy diagram as a function of\ncompositions for given temperature and pressure\n\nTo evaluate the stability of a ternary solution, one needs to calculate\nthe elements in the determinant shown in . Using the mole of component\n$C$ as the independent molar quantity, the limit of stability is\nexpresses as\n\nEq. \u2011 $G_{AA}G_{BB} - G_{AB}G_{BA} = 0$\n\nAs an example, $G_{AA}$ is shown in the following equation, which must\nbe positive for the solution to be stable\n\nEq. \u2011\n${N\\left( \\frac{\\partial\\mu_{A}}{\\partial N_{A}} \\right)}_{T,P,N_{B},N_{C}} = NG_{AA} = \\frac{RT\\left( 1 - x_{A} \\right)}{x_{A}} - 2x_{B}^{2}L_{AB} - 2x_{C}^{2}L_{AC} - 2x_{B}x_{C}\\left( L_{AB} + L_{AC} - L_{BC} \\right) - 2x_{B}x_{C}\\left( 2 - 3x_{A} \\right)L_{ABC}$\n\nIt is evident that any instability in binary systems with positive\ninteraction parameters extends into the ternary system. It can also be\nseen that even if all binary interaction parameters are negative, i.e.\nno instability in the binary systems, it is possible that becomes\nnegative for some combinations of the binary interaction parameters such\nthat $\\mathrm{\\Delta}L = L_{AB} + L_{AC} - L_{BC}$ becomes very positive\nand overshadows the contributions due to $L_{AB}$ and $L_{AC}$, i.e.\n$L_{BC}$ is more negative than $L_{AB}$ and $L_{AC}$ combined. In an\nextreme case with $L_{AB} = L_{AC} = L_{ABC} = 0$ and $L_{BC} < 0$, i.e.\nideal solutions for the $A - B$ and $A - C$ binary systems, stable\nsolution in the $B - C$ binary system, and no additional ternary\ninteraction, reduces to\n\nEq. \u2011\n$N\\left( \\frac{\\partial\\mu_{A}}{\\partial N_{A}} \\right)_{T,P,N_{B},N_{C}} = \\frac{RT\\left( 1 - x_{A} \\right)}{x_{A}} + 2x_{B}x_{C}L_{BC}$\n\nSetting\n$\\left( \\frac{\\partial\\mu_{A}}{\\partial N_{A}} \\right)_{T,P,N_{B},N_{C}} = 0$,\none obtains\n\nEq. \u2011\n$- \\frac{RT}{2L_{BC}} = \\frac{x_{A}x_{B}x_{C}}{1 - x_{A}} = \\frac{{\\left( 1 - x_{B} - x_{C} \\right)x}_{B}x_{C}}{x_{B} + x_{C}}$\n\nWith $- \\frac{RT}{2L_{BC}}$ being positive due to $L_{BC} < 0$, there is\na parabola-shaped composition area in which the solution is unstable at\nconstant temperature and pressure. This is reasonable because the system\ntends to maximize the number of B-C bonds due to its lower energy, which\ncompetes with the entropy of mixing among the three elements and results\nin segregation of B-C bonds, thus miscibility gap at low temperatures.\n\nTo evaluate the ternary consolute point, the second derivatives for\ncomponent A and B are obtained as\n\nEq. \u2011\n${N\\left( \\frac{\\partial_{\\ }^{2}\\mu_{A}}{\\partial N_{A}^{2}} \\right)}_{T,P,N_{B},N_{C}} = \\frac{RT\\left( 1 - x_{A} \\right)}{x_{A}^{2}} + 4x_{B}^{2}L_{AB} + 4x_{C}^{2}L_{AC} + 4x_{B}x_{C}\\left( L_{AB} + L_{AC} - L_{BC} \\right) + 2x_{B}x_{C}\\left( 7 - 9x_{A} \\right)L_{ABC} = 0$\n\nEq. \u2011\n${N\\left( \\frac{\\partial_{\\ }^{2}\\mu_{B}}{\\partial N_{B}^{2}} \\right)}_{T,P,N_{A},N_{C}} = \\frac{RT\\left( 1 - x_{B} \\right)}{x_{B}^{2}} + 4x_{A}^{2}L_{AB} + 4x_{C}^{2}L_{BC} + 4x_{A}x_{C}\\left( L_{AB} + L_{BC} - L_{AC} \\right) + 2x_{A}x_{C}\\left( 7 - 9x_{B} \\right)L_{ABC} = 0$\n\nThe consolute point can then be obtained using , and .\n\nIt is observed in that $\\left( 1 - 2x_{A} \\right)L_{ABC} = 0$ at\n$x_{A} = 0.5$, i.e. the ternary interaction parameter does not\ncontribute to the chemical potential of $A$. It is also observed in that\nthe contribution from the ternary interaction parameter changes sign at\n$x_{i} = 2/3$ due to $\\left( 2 - 3x_{A} \\right)L_{ABC} = 0$.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/laws_of_thermodynamics/combined_law_of_thermodynamics_and_equilibrium_conditions.ipynb b/psu410/src/laws_of_thermodynamics/combined_law_of_thermodynamics_and_equilibrium_conditions.ipynb new file mode 100644 index 0000000..7e0d6ae --- /dev/null +++ b/psu410/src/laws_of_thermodynamics/combined_law_of_thermodynamics_and_equilibrium_conditions.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "e9a1e936", "cell_type": "markdown", "source": "## Combined law of thermodynamics and equilibrium conditions\n\nFor a system with an irreversible internal process taking place, the\nentropy change in the system thus consists of three parts: the heat\nexchange with the surrounding defined by , the entropy production due to\nthe internal process represented by , and the entropy of mass exchange\nwith the surrounding. The total entropy change of the system can thus be\nwritten as follows\n\nEq. \u2011 $dS = \\frac{dQ}{T} + d_{ip}S + \\sum_{}^{}{S_{i}dN_{i}}$\n\nwhere $S_{i}$ is the unit entropy of component *i* in the surroundings,\noften called partial entropy of component *i*, and will be further\ndiscussed in Chapter\n\nCombining and and re-arranging, one obtains\n\nEq. \u2011 $dQ = TdS - Dd\\xi - \\sum_{}^{}{TS_{i}dN_{i}}$\n\nInserting and into yields the combined law of thermodynamics from the\nfirst and second laws of thermodynamics,\n\nEq. \u2011\n$dU = TdS - PdV + \\sum_{}^{}\\left( H_{i} - TS_{i} \\right){dN}_{i} - Dd\\xi$\n\nThe internal energy of the system is thus a function of *S*, *V*,\n*Ni* and *\u03be* of the system, which are called natural\nvariables of the internal energy, i.e. *U*(*S*,*V*,*Ni*,*\u03be*).\nThe other variables are dependent variables and can be represented by\npartial derivatives of the internal energy with respect to their\nrespective natural variables with other natural variables kept constant\nas shown below\n\nEq. \u2011\n$T = \\left( \\frac{\\partial U}{\\partial S} \\right)_{V,\\ N_{i},\\ \\xi}$\n\nEq. \u2011\n$- P = \\left( \\frac{\\partial U}{\\partial V} \\right)_{S,\\ N_{i},\\ \\xi}$\n\nEq. \u2011\n$\\mu_{i} = H_{i} - TS_{i} = \\left( \\frac{\\partial U}{\\partial N_{i}} \\right)_{S,\\ V,N_{j \\neq i},\\ \\xi} = U_{i}$\n\nEq. \u2011\n$- D = \\left( \\frac{\\partial U}{\\partial\\xi} \\right)_{S,\\ V,N_{i}\\ }$\n\nIn , a new variable, $\\mu_{i}$, is introduced. It is called chemical\npotential and defined as the internal energy change with respect of the\naddition of the component *i* when the entropy, volume and the amount of\nother components of the system are kept constant. It may be worth\npointing out that for a system at equilibrium, i.e. $d_{ip}S = 0$, and\nwith constant entropy, $dS = 0$, if it exchanges mass with the\nsurroundings, $dN_{i} \\neq 0$, the system must also exchange heat with\nthe surroundings at the same time in order to keep the entropy invariant\nas demonstrated by .\n\nThe pairs of the natural variables and their corresponding partial\nderivatives are called conjugate variables, i.e. *S* and *T*, *V* and\n*\u2013P*, *Ni* and $\\mu_{i}$, and *\u03be* and *\u2013D*. There are minus\nsign in front of *P* and *D* as the increase of volume and the progress\nof the internal process decrease the internal energy of the system. The\nimportance of this conjugate relation will be evident when various forms\nof combined thermodynamic laws and various types of phase diagrams are\nintroduced in the book.\n\nThe last pair of conjugate variables, *\u03be* and *\u2013D*, is worthy of further\ndiscussion. Based on the second law of thermodynamics, i.e. , no\ninternal processes take place spontaneously if there is no entropy\nproductions, i.e. D\u22640 or *d\u03be*=0 and *D*\\>0. With D\u22640, there is no\ndriving for any internal processes, and the system is at a full\nequilibrium state. The last term in drops off, and *\u03be* becomes a\ndependent variable of the system and can be calculated from the\nequilibrium conditions. With *d\u03be*=0 and *D*\\>0, the system is under a\nconstrained equilibrium or freezing-in condition when the internal\nprocess is constrained not to take place, and *\u03be* remains to be an\nindependent variable of the system.\n\nThese two cases represent the two branches of thermodynamics:\nequilibrium, reversible thermodynamics and irreversible thermodynamics.\nIt is clear from the above discussions that these two branches are\nidentical if the internal energy is not only a function of *S*, *V*, and\n*Ni* , but also any internal process variable *\u03be*. This means\nthat one should be able to evaluate the internal energy of a system for\nany freezing-in equilibrium conditions in addition to the full\nequilibrium condition. In the rest of the book, the freezing-in\nequilibrium and full equilibrium are not differentiated unless\nspecified.\n\nAs the mechanical work under hydrostatic pressure is very important in\nexperiments, let us define a new quantity called enthalpy as follows\n\nEq. \u2011 $H = U + PV$\n\nIts differential form can be obtained from as\n\nEq. \u2011 $dH = dU + d(PV) = dQ + VdP + \\sum_{}^{}H_{i}{dN}_{i}$\n\nThere are two significant consequences of the above equation. First, for\na close system under constant pressure, i.e. ${dN}_{i} = dP = 0$, one\nhas $dH = dQ$. This implies that the enthalpy change in a system is\nequal to the heat exchange between the system and the surrounding of the\nsystem, which is why enthalpy and heat are often exchangeable in the\nliterature. Second, for an adiabatic system under constant pressure,\ni.e. $dQ = dP = 0$, can be re-arranged to the following equation\n\nEq. \u2011\n$H_{i} = \\left( \\frac{\\partial H}{\\partial N_{i}} \\right)_{N_{j \\neq i,\\ \\ dQ = dP = 0}}$\n\n$H_{i}$ is thus the partial enthalpy of component *i* and will be\nfurther discussed in Chapter . The chemical potential of component *i*\ndefined in is thus related to the partial enthalpy and partial entropy\nof the component.\n\nTo further define equilibrium conditions of a system, consider a\nhomogeneous system in a state of internal equilibrium, i.e. no\nspontaneous internal processes are possible with $Dd\\xi = 0$, and\nbecomes\n\nEq. \u2011\n$dU = TdS - PdV + \\sum_{}^{}\\mu_{i}{dN}_{i} = \\sum_{}^{}{Y_{i}dX_{i}}$\n\nwhere *X* represents *S*, *V*, *Ni*, and *Y* their conjugate\nvariables *T*, *-P*, $\\mu_{i}$. The state of the system with *c*\nindependent components is completely determined by the *c+2* variables,\ni.e. *S*, *V*, and *Ni* with *i* from 1 to *c*.\n\nTo simplify the situation, let us limit the discussion to an isolated\nequilibrium system, i.e. $dU = 0$, and conduct a virtual internal\nexperiment inside the system by moving an infinitesimal amount of\n$X_{i}$, ${dX}_{i}$, with other $X_{j}$ kept constant, from one region\nof the system to another region of the system as schematically shown in\n.\n\nFigure \u2011: Virtual experiment for a system at equilibrium\n\nAs the system is homogeneous and at equilibrium,\n$- dX_{i}^{'} = dX_{i}^{\"} = dX_{i}$. The total change of the internal\nenergy of this internal process is the combination of the changes in the\ntwo regions, i.e.\n\nEq. \u2011\n$dU = dU^{'} + dU^{\"} = Y_{i}^{'}dX_{i}^{'} + Y_{i}^{\"}dX_{i}^{\"} = \\left( - Y_{i}^{'} + Y_{i}^{\"} \\right)dX_{i} = 0$\n\nTherefore, $Y_{i}^{'} = Y_{i}^{\"}$ for *T*, *-P*, and $\\mu_{i}$,\nindicating that *T*, *-P*, and $\\mu_{i}$ are homogeneous in the system,\nrespectively, and are thus named as potentials of the system.\nFurthermore these potentials are independent of the size of the system\nand are often referred to as intensive variables in the literature. On\nthe other hand, all *X:s*, i.e. *S*, *V*, and *Ni*, are\nproportional to the size of the system and may be normalized with\nrespect to the size of the system, usually in terms of total moles,\n\nEq. \u2011 $N = \\sum_{}^{}N_{i}$\n\nThey are thus called molar quantities and often referred to as extensive\nvariables, and the respective normalized variables are molar entropy,\nmolar volume, and mole fractions, defined as follows\n\nEq. \u2011 $S_{m} = \\frac{S}{N}$\n\nEq. \u2011 $V_{m} = \\frac{V}{N}$\n\nEq. \u2011 $x_{i} = \\frac{N_{i}}{N}$\n\nConsider a small subsystem in this homogeneous system at equilibrium and\nlet the subsystem grow in size. The entropy, volume, and mass enclosed\nin the subsystem increase as follows\n\nEq. \u2011 ${dS = S}_{m}dN$\n\nEq. \u2011 $dV = V_{m}dN$\n\nEq. \u2011 $dN_{i} = x_{i}dN$\n\nThe corresponding change in the internal energy of the subsystem becomes\n\nEq. \u2011\n$dU = TdS - PdV + \\sum_{}^{}\\mu_{i}{dN}_{i} = \\left( TS_{m} - PV_{m} + \\sum_{}^{}\\mu_{i}x_{i} \\right)dN = U_{m}dN$\n\nBy integration one obtains the integral form of the internal energy as\n\nEq. \u2011\n$U = TS - PV + \\sum_{}^{}\\mu_{i}N_{i} = \\left( TS_{m} - PV_{m} + \\sum_{}^{}\\mu_{i}x_{i} \\right)N = U_{m}N$\n\nSimilarly, the molar enthalpy can be defined as follows\n\nEq. \u2011\n$H = U + PV = U_{m}N + PV_{m}N = \\left( U_{m} + PV_{m} \\right)N = H_{m}N$\n\nIn case a potential is not homogeneous in a system, the system will not\nbe in a state of equilibrium. Let us consider the same virtual\nexperiment as shown in for an isolated system that is not in\nequilibrium, i.e. by moving an infinitesimal amount of $X_{i}$,\n${dX}_{i}$, with other $X_{j}$ kept constant, from one region of the\nsystem to another region of the system with the two regions having\ndifferent potentials. The total internal energy change equals to zero as\nthe virtual experiment has $dU = 0$. Similarly, each region can be\nconsidered to be homogeneous by itself, and one has\n$- dX_{i}^{'} = dX_{i}^{\"} = dX_{i}$. The total internal energy change\nin the system is thus the sum of these two regions plus the entropy\nproduction due to the internal process with $d\\xi = dX_{i}$, i.e.\n\nEq. \u2011\n$dU = dU^{'} + dU^{\"} + Dd\\xi = Y_{i}^{'}dX_{i}^{'} + Y_{i}^{\"}dX_{i}^{\"} + Dd\\xi = \\left( - Y_{i}^{'} + Y_{i}^{\"} \\right)dX_{i} + Dd\\xi = 0$\n\nConsequently, one obtains the following\n\n*Eq. 1\u201131* $D = Y_{i}^{'} - Y_{i}^{\"}$\n\nThe driving force thus represents the difference of the potential at the\ntwo regions, and the internal process is to eliminate inhomogeneity of\nthe potential with the heat transfer from high temperature regions to\nlow temperature regions, volume shrink of low pressure regions (high\n$\u2013P$) and volume expansion of high pressure regions (low $\u2013P$), and\ntransport of components from high chemical potential regions to low\nchemical potential regions.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/laws_of_thermodynamics/first_and_second_laws_of_thermodynamics.ipynb b/psu410/src/laws_of_thermodynamics/first_and_second_laws_of_thermodynamics.ipynb new file mode 100644 index 0000000..6bc1f58 --- /dev/null +++ b/psu410/src/laws_of_thermodynamics/first_and_second_laws_of_thermodynamics.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "292875c5", "cell_type": "markdown", "source": "## First and second laws of thermodynamics\n\nA system typically consists of many chemical components. The first law\nof thermodynamics states that the exchanges of heat, work, and\nindividual components with its surroundings must obey the law of\nconversation of energy. In the domain of materials science and\nengineering, the energy of interest is at the atomic and molecular\nlevels. The energy at the higher and lower levels such as nuclear energy\nand kinetic and potential energies of a rigid body are usually excluded\nfrom the discussion of thermodynamics of materials.\n\nLet us consider a system receiving an amount of heat, *dQ*, and an\namount of work, *dW*, and an amount of each independent component *i*,\n*dNi*, from the surroundings. Such a system is called an open\nsystem in contrast to a closed system when *dNi*=0 for all\ncomponents, i.e. no exchange of mass between the system and the\nsurrounding. Other types of systems commonly defined in thermodynamics\ninclude adiabatic systems without exchange of heat, i.e. *dQ*=0, and\nisolated systems without exchange of any, i.e. *dQ*= *dW*=\n*dNi*=0.\n\nThe corresponding change of energy in the system, i.e. the internal\nenergy change, *dU*, is formulated in terms of the first law of\nthermodynamics as follows,\n\nEq. \u2011 $dU = dQ + dW + \\sum_{}^{}{H_{i}dN_{i}}$\n\nwhere $H_{i}$ is the unit energy of component *i* in the surroundings,\nand the summation is for all components in the system, which can be\ncontrolled independently from the surroundings, i.e. the independent\ncomponents of the system.\n\nIt is self-evident that the left-hand side of refers to the change\ninside the system, while its right-hand side is for the contributions\nfrom the surroundings to the system. In principle, no matter how the\nheat and mass are added, or how the work is done to the system, as far\nas their summation is the same, the change of the internal energy will\nbe the same based on the first law of thermodynamics, indicating that\nthe system reaches the same state for a closed system. The internal\nenergy is thus a state function in a close system as it does not depend\non how the state is reached.\n\nOn the other hand, for the purpose of easy mathematical treatment, a\nreversible process can be considered for a closed system, in which the\ninitial state of the system can be restored reversibly without any\nchanges left to the surroundings. Therefore, the heat transferred and\nthe work done to the system are identical to the heat and work lost by\nthe surroundings and vice versa. The classic example of reversible\nprocesses is the Carnot\u2019s cycle, which is shown in . It consists of four\nreversible processes for a closed system. The four reversible processes\nare compression at constant temperature *T1* (isothermal,),\ncompression without heat exchange (adiabatic) ending at *T2*,\nisothermal expansion at *T2*, and adiabatic expansion ending\nat *T1*.\n\nFigure \u2011: Schematics of the Carnot\u2019s cycle\n\nThe Carnot\u2019s cycle involves a simple type of mechanical work, either\nhydrostatic expansion or compression, with the work that the surrounding\ndoes to the system represented by\n\nEq. \u2011 $dW = - PdV$\n\nwith *P* being the external pressure that the surrounding exerts on the\nsystem and *V* the volume of the system. It is now necessary to\ndifferentiate the external and internal variables for further discussion\nwith the former representing variables in the surroundings and the\nlatter representing variables in the system. For the isothermal\nprocesses in the Carnot\u2019s cycle, the entropy change of the system, *dS*,\ncan be defined as the heat exchange divided by temperature\n\nEq. \u2011 $dS = \\frac{dQ}{T}$\n\nIn addition to processes involving heat, work, and mass exchanges\nbetween the system and the surroundings, there could be internal\nprocesses taking place inside the system. As the system cannot do work\nto itself, the criterion for whether an internal process can occur\nspontaneously must be related to the heat exchange, which is related to\nthe entropy change as shown by .\n\nIt is a known fact that heat can spontaneously transfer from a higher\ntemperature (*T2*) region to a lower temperature\n(*T1*) region inside a system if the heat conduction is\nallowed, and this process is irreversible because heat cannot be\nconducted from a low temperature region to a high temperature region by\nitself. indicates that for the same amount of heat change, the entropy\nchange at *T1* is higher than that at *T2*, and\nthe heat conduction thus results in a positive entropy change in the\nsystem, i.e.\n\nEq. \u2011\n$\\mathrm{\\Delta}S = - \\frac{dQ}{T_{2}} + \\frac{dQ}{T_{1}} = \\frac{dQ}{{T_{2}T}_{1}}\\left( {T_{2} - T}_{1} \\right) > 0$\n\nConsequently, the second law of thermodynamics is obtained, which states\nthat for an internal process to take place spontaneously or\nirreversibly, this internal process (*ip*) must have a positive entropy\nproduction, which can be written in a differential form as follows\n\nEq. \u2011 $d_{ip}S > 0$\n\nFrom the definition of entropy change shown by , the amount of heat\nproduced by this irreversible internal process can be calculated as\nfollows\n\nEq. \u2011 $d_{ip}Q = Td_{ip}S$\n\nLet us represent this internal process by *d\u03be* and define the driving\nforce for this internal process by *D*. The work done by this internal\nprocess is thus *Dd\u03be*, which is released as heat, i.e.\n\nEq. \u2011 $Dd\\xi = d_{ip}Q = Td_{ip}S$\n\nAn irreversible process thus must have a positive driving force in order\nfor it to take place spontaneously.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/laws_of_thermodynamics/gibbsduhem_equation.ipynb b/psu410/src/laws_of_thermodynamics/gibbsduhem_equation.ipynb new file mode 100644 index 0000000..16be524 --- /dev/null +++ b/psu410/src/laws_of_thermodynamics/gibbsduhem_equation.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "37f679d4", "cell_type": "markdown", "source": "## Gibbs-Duhem equation\n\nIn experiments, it is difficult to control *S* and *V* of a system in\ncomparison with their conjugate variables *T* and *-P*. It is thus\ndesirable to construct new functions to represent the system with *T*\nand *-P* as natural variables of the functions. One of them is enthalpy\ndefined in , and other two can be defined as follows\n\nEq. \u2011 $F = U - TS$\n\nEq. \u2011 $G = U - TS + PV = \\sum_{}^{}\\mu_{i}N_{i} = H - TS = F + PV$\n\nwith *F* and *G* called Helmholtz energy and Gibbs energy, respectively.\nThe middle part of is obtained using $U$ from . The corresponding\ncombined law of thermodynamics in terms of *H*, *F*, and *G* can be\nobtained through the Legendre transformation of as\n\nEq. \u2011 $dH = TdS - Vd( - P) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$\n\nEq. \u2011 $dF = - SdT - PdV + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$\n\nEq. \u2011 $dG = - SdT - Vd( - P) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$\n\nThe independent variables in each of the above form are regarded as the\nnatural variables to the corresponding function. The integral forms of\nall the functions can thus be written as the following with their\nnatural variables listed in the parenthesis\n\nEq. \u2011 $U = U\\left( S,V,N_{i},\\xi \\right)$\n\nEq. \u2011 $H = H\\left( S, - P,N_{i},\\xi \\right)$\n\nEq. \u2011 $F = F\\left( T,V,N_{i},\\xi \\right)$\n\nEq. \u2011 $G = G\\left( T, - P,N_{i},\\xi \\right)$\n\nBy differentiating , one obtains\n\nEq. \u2011\n$dG = \\sum_{}^{}\\mu_{i}{dN}_{i} + \\sum_{}^{}{N_{i}d\\mu}_{i} = - SdT - Vd( - P) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$\n\nFor a system at equilibrium, $Dd\\xi = 0$, re-arranging gives the\nGibbs-Duhem equation\n\nEq. \u2011 $0 = - SdT - Vd( - P) - \\sum_{}^{}{N_{i}d\\mu}_{i}$\n\nThis equation indicates that for a homogeneous system with *c*\nindependent components at equilibrium, there is a direct relation among\nall the *c+2* potentials, and they are $c$ chemical potentials\n($\\mu_{i}$), temperature, and pressure. Consequently, only *c+1*\npotentials can change independently, and the remaining potential is\ndependent on the other potentials. As discussed in connection with ,\nthere are $c + 2$ independent variables for an equilibrium system with\n*c* independent components, where all of them are molar quantities.\n\nWith the relationships between potentials and molar quantities defined\nby to , one can switch between potentials and molar quantities as\nnatural variables of the system. For example, one can define a new free\nenergy function when the chemical potential of one component is\ncontrolled from the surroundings instead of its content and obtain the\nfollowing combined first and second law of thermodynamics\n\nEq. \u2011 $\\Phi = G - \\mu_{1}N_{1} = \\sum_{i = 2}^{c}{\\mu_{i}N_{i}}$\n\nEq. \u2011\n$d\\Phi = - SdT - Vd( - P) - N_{1}{d\\mu}_{1} + \\sum_{i = 2}^{c}{\\mu_{i}{dN}_{i}} - Dd\\xi$\n\nHowever, even though the $c + 2$ molar quantities are independent of\neach other, indicates that not all the $c + 2$ potentials are\nindependent, i.e., if chemical potentials of all components are changed\nto natural variables, one would obtain . Therefore, among the *c+2*\nindependent variables used to define the system, the maximum number of\nindependent potential is *c+1*, and at least one of the *c+2*\nindependent variables must be a molar quantity. This variable is usually\nchosen to be the size of the system or the major element in the system.\nThe Gibbs-Duhem equation is used to derive Gibbs phase rule in\nheterogeneous systems, which is discussed in Chapter of the book.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/laws_of_thermodynamics/index.ipynb b/psu410/src/laws_of_thermodynamics/index.ipynb new file mode 100644 index 0000000..3900ce3 --- /dev/null +++ b/psu410/src/laws_of_thermodynamics/index.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "df14ed3f", "cell_type": "markdown", "source": "# Laws of thermodynamics\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/laws_of_thermodynamics/stability_at_equilibrium_and_property_anomaly.ipynb b/psu410/src/laws_of_thermodynamics/stability_at_equilibrium_and_property_anomaly.ipynb new file mode 100644 index 0000000..76c6fde --- /dev/null +++ b/psu410/src/laws_of_thermodynamics/stability_at_equilibrium_and_property_anomaly.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "0e50c529", "cell_type": "markdown", "source": "## Stability at equilibrium and property anomaly\n\nAs shown by , potentials are homogenous for a homogeneous system in a\nstate of internal equilibrium. To study the stability of the equilibrium\nstate, one considers the entropy production due to a fluctuation of a\nmolar quantity as an internal process. Based on the second law of\nthermodynamics, the driving force, as the first derivative of the\nentropy production with respect to the internal process, is zero for\nsuch a fluctuation at equilibrium, i.e. *D*=0, and the entropy of\nproduction thus depends on the second derivative. It can be written as\nfollows\n\nEq. \u2011\n$Td_{ip}S = \\frac{\\partial_{ip}S}{\\partial\\xi}d\\xi + \\frac{1}{2}{\\frac{\\partial_{ip}^{2}S}{\\partial\\xi^{2}}(d\\xi)}^{2} = Dd\\xi - \\frac{1}{2}D_{2}(d\\xi)^{2}$\n\nwith $D_{2} = - \\frac{\\partial_{ip}^{2}S}{\\partial\\xi^{2}}$. When\n$\\frac{\\partial_{ip}^{2}S}{\\partial\\xi^{2}} < 0$ or $D_{2} > 0$ along\nwith $D = 0$, the fluctuation does not produce positive entropy of\nproduction and can thus not develop further. The equilibrium state of\nthe system is therefore stable against the fluctuation. On the other\nhand, when $\\frac{\\partial_{ip}^{2}S}{\\partial\\xi^{2}} > 0$ or\n$D_{2} < 0$ along with $D = 0$, the fluctuation creates positive entropy\nof production and can continue to grow. The equilibrium state of the\nsystem is therefore unstable against the fluctuation. In connection with\n, one can realize that for a system at stable equilibrium without heat\nand mass exchanges with the surroundings, its entropy is at its maximum,\nand there are no other internal processes, which could produce any more\nentropy. This is schematically shown in .\n\nFigure \u2011: Schematic diagram showing maximum entropy\n\nUsing , , and , the combined law of thermodynamics can be written as\n\n*Eq. 1\u201133*\n$dU = \\sum_{}^{}{Y_{i}dX_{i}} - Dd\\xi + \\frac{1}{2}D_{2}(d\\xi)^{2}$\n\nLet us carry out the same virtual internal experiment shown in Chapter ,\ni.e. moving an infinitesimal amount of *Xi* in a homogenous\nsystem with other $X_{j}$ kept constant in an isolated system, i.e.\n$dU = 0$ and $D = 0$. The internal energy change due to this internal\nprocess is\n\nEq. \u2011\n$dU = \\frac{1}{2}D_{2}\\left\\{ \\left( dX_{i}^{'} \\right)^{2} + \\left( dX_{i}^{\"} \\right)^{2} \\right\\}$\n\nFor a homogeneous system in a state of stable equilibrium with\n$\\left( dX_{i}^{'} \\right)^{2} = \\left( dX_{i}^{\"} \\right)^{2} = \\left( dX_{i} \\right)^{2}$,\nthis internal process must result in an increase of internal energy,\n$dU > 0$, and thus gives\n\nEq. \u2011\n$D_{2} = 2\\left( \\frac{\\partial^{2}U}{\\partial\\left( X_{i} \\right)^{2}} \\right)_{X_{j}} = 2\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}} > 0$\n\nshows that for a system to be stable, any pair of conjugate variables\nmust change in the same direction when other independent molar\nquantities are kept constant. For the conjugate variables discussed so\nfar, it means that for a stable system, the addition of entropy\nincreases temperature with $\\frac{\\partial T}{\\partial S}$\\>0, the\nvolume decreases with pressure or increases with the negative of\npressure with $\\frac{\\partial( - P)}{\\partial V} > 0$, and the chemical\npotential of a component increases with its amount, i.e.\n$\\frac{\\partial\\mu_{i}}{\\partial N_{i}} > 0$, where the derivatives are\ntaken with all other molar quantities kept constant. The limit of\nstability is reached when becomes zero, i.e.\n\nEq. \u2011\n$D_{2} = 2\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}} = 0$\n\nshows schematically the energy as a function of configurations including\nthree states: unstable, stable, and metastable. Both the stable and\nmetastable states have positive curvatures due to $D_{2} > 0$, while the\nunstable state has a negative curvature due to $D_{2} < 0$. There is an\ninflection point of $D_{2} = 0$ for a state between a stable or\nmetastable state with $D_{2} > 0$ and an unstable state with\n$D_{2} < 0$. These two inflection points, called spinodal, represent the\nlimit of stability. The states between the two inflection points are\nunstable, and other states are either stable or metastable. The two\ninflection points can move apart from or close to each other depending\non the change of external conditions, i.e. the natural variables. One\nextreme situation is when these two inflection points merge into one\npoint, and the instability occurs only at this particular point. It is\nevident that all three states, stable, metastable, and unstable, also\nmerge into one point. This point is called critical or consolute point,\nbeyond which the instability no longer exists.\n\nFigure \u2011: Schematic diagram showing the stable and unstable equilibrium\nstates\n\nTo mathematically define the consolute point, the third derivative needs\nto be added to because both $D$ and $D_{2}$ vanish at this point, i.e.\n\nEq. \u2011\n$Td_{ip}S = \\frac{\\partial_{ip}S}{\\partial\\xi}d\\xi + \\frac{1}{2}{\\frac{\\partial_{ip}^{2}S}{\\partial\\xi^{2}}(d\\xi)}^{2} + \\frac{1}{6}{\\frac{\\partial_{ip}^{3}S}{\\partial\\xi^{3}}(d\\xi)}^{3} = Dd\\xi - \\frac{1}{2}D_{2}(d\\xi)^{2} + \\frac{1}{6}{D_{3}(d\\xi)}^{3}$\n\nEq. \u2011\n$dU = \\sum_{}^{}{Y_{i}dX_{i}} - Dd\\xi + \\frac{1}{2}D_{2}(d\\xi)^{2} - \\frac{1}{6}{D_{3}(d\\xi)}^{3}$\n\nAt the consolute point, the third derivative also becomes zero, i.e.\n\nEq. \u2011 $D_{3} = \\frac{\\partial_{ip}^{3}S}{\\partial\\xi^{3}}^{3} = 0$\n\nLet us further discuss the properties of the system in relation to the\ncritical point. By taking the inverse of the equation of the limit of\nstability, , one obtains\n\nEq. \u2011\n$\\left( \\frac{\\partial X_{i}}{\\partial Y_{i}} \\right)_{X_{j}} = + \\ \\infty$\n\ni.e. all $X_{i}$ quantities diverge at the critical point. Therefore,\nwhen a system approaches the critical point from its stable region, the\nchange of a molar quantity with respect to its conjugate potential\nvaries dramatically and becomes infinite at the critical point,\nresulting in property anomalies in the system. In the unstable region,\nthe system would thus separate into stable subsystems and becomes\nheterogeneous, and $X_{i}$:s change discontinuously between subsystems.\nWhile in the stable region, the change of a molar quantity with respect\nto its conjugate potential decreases as the system moves away from the\ncritical point and remains positive due to the stability criteria\ndenoted by .\n\nHowever, it is not clear how a molar quantity changes with respect to a\nnon-conjugate potential at the critical point. From the Maxwell\nrelation, one has\n\nEq. \u2011\n$\\left( \\frac{\\partial Y_{i}}{\\partial X_{j}} \\right)_{X_{k \\neq j}} = \\frac{\\partial^{2}U}{\\partial X_{i}\\partial X_{j}} = \\left( \\frac{\\partial Y_{j}}{\\partial X_{i}} \\right)_{X_{k \\neq i}}$\n\nEq. \u2011\n$\\left( \\frac{\\partial X_{j}}{\\partial Y_{i}} \\right)_{X_{k \\neq j}} = \\left( \\frac{\\partial X_{i}}{\\partial Y_{j}} \\right)_{X_{k \\neq i}}$\n\nSince all $X_{i}$:s diverge at the critical point, both derivatives in\nshould also go to infinite at the critical point. To investigate their\nsigns, let us carry out a virtual experiment similar to the one in\nderiving the stability condition ( and ). In this case, two internal\nprocesses are needed for moving two molar quantities simultaneously in\nan isolated system, i.e.\n\n*Eq. 1\u201143*\n$dU = - D_{\\xi_{1}}d\\xi_{1} - D_{\\xi_{2}}d\\xi_{2} + D_{\\xi_{1}\\xi_{2}}d\\xi_{1}d\\xi_{2} + \\frac{1}{2}D_{2\\xi_{1}}\\left( d\\xi_{1} \\right)^{2} + \\frac{1}{2}D_{2\\xi_{2}}\\left( d\\xi_{2} \\right)^{2}$\n\nBased on the above discussions, in a stable system at equilibrium with\n$D_{\\xi_{1}} = D_{\\xi_{2}} = 0$, $D_{2\\xi_{1}} > 0$ and\n$D_{2\\xi_{2}} > 0$, the sign of $D_{\\xi_{1}\\xi_{2}}$ cannot be\nunambiguously determined in keeping the change of internal energy\npositive, i.e. $dU > 0$. This indicates that the quantities in can be\neither positive or negative in the stable region and become zero at the\ncritical point. By the same token, the quantities in can be either\npositive or negative and become positive or negative infinite at the\ncritical point.\n\nA profound conclusion from this analysis is that in a stable system even\nthough a molar quantity always changes in the same direction as its\nconjugate potential, the same molar quantity may change in the opposite\ndirection of a non-conjugate potential, resulting in additional\nanomalies represented by Eq. 1\u201140. One example of is the thermal\nexpansion in a closed system, i.e. $dN_{i} = 0$, as follows\n\nEq. \u2011\n$\\left( \\frac{\\partial V}{\\partial T} \\right)_{S} = \\left( \\frac{\\partial S}{\\partial( - P)} \\right)_{V}$\n\nThe left-hand side of can be understood as follows: with the increase of\ntemperature, the system regulates its pressure in order to keep the\nentropy from increasing, which results in the volume change of the\nsystem. The behavior of the system depends on whether the pressure\ndecreases or increases in order to maintain the entropy of the system\nconstant. If the pressure decreases to maintain the entropy of the\nsystem constant, the volume would increase with the increase of\ntemperature, i.e. the left-hand side of the equation has a positive\nsign, which is also shown by the right-hand side of the equation as the\nchanges of $S$ and $\u2013P$ have the same sign. That the volume increases\nwith temperature is the normal scenario. On the other hand, if the\npressure increases to maintain the entropy of the system constant, the\nvolume would decrease with the increase of temperature, resulting in a\nnegative sign for the left-hand side of the equation. This decrease of\nvolume with the increase of temperature is usually considered to be\nanomalous, originated from the increase of entropy by the decrease of\n$\u2013P$ or the increase of pressure. More discussions on entropy will\nfollow in Chapter 5.2.5 and Chapter 9.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/phase_equilibria_in_heterogeneous_systems/general_condition_of_equilibrium.ipynb b/psu410/src/phase_equilibria_in_heterogeneous_systems/general_condition_of_equilibrium.ipynb new file mode 100644 index 0000000..0fdb4a4 --- /dev/null +++ b/psu410/src/phase_equilibria_in_heterogeneous_systems/general_condition_of_equilibrium.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "d796a0d0", "cell_type": "markdown", "source": "## General condition of equilibrium\n\nA system is heterogeneous when some properties have different values at\ndifferent portions of the system when the system is at equilibrium. Two\nscenarios may exist where the variations of properties can be either\ncontinuous or discontinuous. In the scenario of continuous variations,\nthe gradients of those variations must be coupled so that the system\nremains at equilibrium. The number of independent variables is thus\nreduced. These gradients must also be constrained along the boundaries\nbetween the system and the surroundings. This type of constrained\nequilibrium is not discussed in the book as it involves heterogeneous\nboundary conditions between the system and the surroundings and depends\non the morphology of the system.\n\nIn the second scenario with discontinuous variations, those properties\nhave different values in different portions of the system, but remain\nhomogenous within each portion. The system is in equilibrium as each\nportion is in equilibrium with all other portions of the system. Those\nhomogeneous portions represent different phases in the system with the\nproperties in each phase being homogeneous at equilibrium. In the\nprevious chapter, it has been shown that all potentials are homogeneous\nin a homogeneous system.\n\nFor a heterogeneous system, the same conclusion can be obtained. If the\ntemperature is inhomogeneous, heat can be conducted from high\ntemperature locations to low temperature locations, and this process is\nirreversible based on the second law of thermodynamics because it\nincreases the internal entropy of the system. If the pressure is\ninhomogeneous, the amounts of lower molar volume phases will increase to\nreduce the internal energy of the system. If the chemical potential of a\ncomponent is inhomogeneous, the chemical potential difference of the\ncomponent will drive that component to the locations with a lower\nchemical potential so the internal energy of the system can be\ndecreased. Therefore, it can be concluded that all potentials are\nhomogeneous in a heterogeneous system at equilibrium, and the variables\nthat are not homogeneous are thus their conjugate molar quantities.\nUnder certain special circumstances to be discussed in later part of\nthis book, some molar quantities may also have the same values in\ndifference phases.\n\nIn a system at equilibrium with $c$ independent components, there are\n$c + 2$ pairs of conjugate variables based on or though more can be\nadded as shown by depending on experimental conditions. For simplicity,\nmost discussions in this book are limited to systems with $c + 2$ pairs\nof conjugate variables unless otherwise specified, with number \u201c2\u201d\nrepresenting conjugate variables of $T - S$ and $( - P) - V$.\n\nFor a system under constant temperature, pressure, and moles of each\nindependent component, the equilibrium condition derives from as\n\nEq. \u2011\n$dG = - SdT - Vd( - P) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi = - Dd\\xi = 0$\n\nConsequently, the equilibrium state is defined by the minimization of\nGibbs energy of the system at constant $T$, $P$ and $N_{i}$ because the\nsecond derivatives need to be positive for the equilibrium system to be\nstable as stipulated by . For heterogeneous systems with two or more\nphases, Gibbs energy of the system is the weighted summation of Gibbs\nenergies of individual phases, i.e.\n\nEq. \u2011 $\\frac{G}{N} = G_{m} = \\sum_{\\beta}^{}{f^{\\beta}G_{m}^{\\beta}}$\n\nwhere $f^{\\beta}$ and $G_{m}^{\\beta}$ are the mole fraction and molar\nGibbs energy of the phase $\\beta$, respectively, and the summation goes\nover all phases in the system. $f^{\\beta}$ is equal to zero for phases\nnot present in the equilibrium state.\n\nThe minimization of Gibbs energy of the system is carried out under the\nfollowing mass balance conditions\n\nEq. \u2011\n$x_{i} = \\sum_{\\beta}^{}{f^{\\beta}x_{i}^{\\beta}} = \\sum_{\\beta}^{}{f^{\\beta}\\frac{\\sum_{\\beta - t}^{}{a^{\\beta - t}\\sum_{k}^{}{i_{k}^{\\beta - t}y_{k}^{\\beta - t}}}}{\\sum_{\\beta - t}^{}{a^{\\beta - t}\\left( 1 - y_{va}^{\\beta - t} \\right)}}}$\n\nEq. \u2011 $\\sum_{i}^{}x_{i} = 1$\n\nEq. \u2011 $\\sum_{k}^{}y_{k}^{\\beta - t} = 1$\n\nwhere $a^{\\beta - t}$ and $y_{k}^{\\beta - t}$ are the number of site in\nsublattice $t$ in the $\\beta$ phase and the corresponding site fraction\nof species $k$ in the sublattice, respectively, and $i_{k}^{\\beta - t}$\nis the stoichiometry of the component $i$ in the species $k$, as used in\n. The summation in runs over species for each sublattice. For phases\ncontaining ionic species, electroneutrality also needs to be maintained,\ni.e. is applied to each phase. This minimization problem of Gibbs energy\nunder the constraint of mass conservation can be solved by means of a\nrange of algorithms. It should be noted that the mole fractions of\nphases and site fractions of species are bounded between 0 and 1.\n\nThis minimization procedure must result in that potentials are\nhomogeneous in the system as discussed above. Since the present book\ndeals with thermodynamics of materials, the chemical potential of each\ncomponent is of particular interest and must be homogenous in all phases\nof the system at equilibrium, i.e.\n\nEq. \u2011\n$\\mu_{i}^{\\alpha} = \\mu_{i}^{\\beta} = \\mu_{i}^{\\gamma}\\ldots\\ldots$\n\nFor phases in which the chemical potentials of individual components\ncannot be evaluated due to stoichiometry, the combined chemical\npotentials can be used to relate individual potentials as shown by and .\nAs proved in Chapter shown by , the chemical potential of a component in\na solution is represented by the intercept on the Gibbs energy axis by\nthe multi-dimensional tangent surface of Gibbs energy of the solution\nplotted with respect to mole fractions of independent components. The\nGibbs energy functions of all phases in equilibrium must thus share the\nsame tangent surface. This is usually referred to as the common tangent\nconstruction for phases at equilibrium. Any phase with its Gibbs energy\ncurve above the tangent surface is not stable under given compositions\nof the system.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/phase_equilibria_in_heterogeneous_systems/gibbs_phase_rule.ipynb b/psu410/src/phase_equilibria_in_heterogeneous_systems/gibbs_phase_rule.ipynb new file mode 100644 index 0000000..f9d21a5 --- /dev/null +++ b/psu410/src/phase_equilibria_in_heterogeneous_systems/gibbs_phase_rule.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "60a57083", "cell_type": "markdown", "source": "## Gibbs phase rule\n\nThe Gibbs-Duhem equation, i.e. , states that only $c + 1$ potentials are\nindependent in a homogeneous system with $c$ independent components and\nthe additional two variables of temperature and pressure. In a\nheterogeneous system at equilibrium, this equation can be applied to\nindividual phases as each phase is homogeneous. Noting that each\npotential has the same value in all phases at equilibrium, can be\nwritten as follows for each individual phase, $\\ \\beta$, in the system\nat equilibrium\n\nEq. \u2011\n$0 = - S^{\\beta}dT - V^{\\beta}d( - P) - \\sum_{}^{}{N_{i}^{\\beta}d\\mu}_{i}$\n\nFor a system with $p$ phases at equilibrium, there are $p$ such\nequations relating the potentials in the system. The number of\nindependent potentials thus becomes\n\nEq. \u2011 $\\upsilon = c + 2 - p$\n\nis called Gibbs phase rule. It dictates the number of potentials that\ncan change independently for a given number of phases co-existing at\nequilibrium, commonly called degree of freedom of the system at\nequilibrium. It stipulates that the maximum number of phases can\nco-exist in a system at equilibrium is obtained by setting\n$\\upsilon = 0$, called an invariant equilibrium due to the zero degree\nof freedom,\n\nEq. \u2011 $p_{\\max} = c + 2$\n\nThere are thus maximum three phases in a one-component system, four\nphases in a binary system, five phases in a ternary system, and so on,\nthat can co-exist simultaneously at equilibrium with all potentials in\nthe system at fixed values. This should not be confused with the total\nnumber of phases that could exist, but not co-exist in a system, which\nof course are not limited by Gibbs phase rule.\n\nIt should be emphasized that the degree of freedom, $\\upsilon$, is\nreferred to the number of potentials only, not to molar quantities of\nthe system because molar quantities are generally not homogeneous in a\nheterogeneous system. For example, in a system at equilibrium with\n$\\upsilon = 0$, the amount of each component can be varied, while\nkeeping the number of phases at $p_{\\max} = c + 2$. This can be done\nthrough changing the amount of each phase in the system through the mass\nbalance equation, , without altering the composition of each phase and\nthus the chemical potentials in the system. As mentioned at the\nbeginning of Chapter , the number of independent variables in a system\nat equilibrium, i.e. the sum of independent potentials and independent\nmolar quantities, is $c + 2$ with the maximum number of independent\npotentials determined by Gibbs phase rule, .\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/phase_equilibria_in_heterogeneous_systems/index.ipynb b/psu410/src/phase_equilibria_in_heterogeneous_systems/index.ipynb new file mode 100644 index 0000000..21c3fc8 --- /dev/null +++ b/psu410/src/phase_equilibria_in_heterogeneous_systems/index.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "b01733f8", "cell_type": "markdown", "source": "# Phase equilibria in heterogeneous systems\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/phase_equilibria_in_heterogeneous_systems/molar_phase_diagrams.ipynb b/psu410/src/phase_equilibria_in_heterogeneous_systems/molar_phase_diagrams.ipynb new file mode 100644 index 0000000..be0f2ed --- /dev/null +++ b/psu410/src/phase_equilibria_in_heterogeneous_systems/molar_phase_diagrams.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "1ae66619", "cell_type": "markdown", "source": "## Molar phase diagrams\n\nThe potential phase diagrams discussed in Chapter present information on\nwhich phases are in equilibrium under given values of potentials, but do\nnot have any information on the properties of phases in equilibrium. On\nthe other hand, there are direct relations between potentials and their\nconjugate molar quantities for each phase at equilibrium depicted by to\n. One may thus substitute the potentials by their conjugate molar\nquantities in the potential phase diagrams as molar quantities provide\nmore information on the properties of phases and the system. This is\nparticularly true when chemical potentials are replaced by compositions\nas the compositions of the system are often the variables controlled and\nmeasured in experiments instead of chemical potentials.\n", "metadata": {}}, {"id": "3179cb21", "cell_type": "markdown", "source": "### Phase diagrams with both potentials and molar quantities\n\nBased on Gibbs phase rule discussed in Chapters and , the\ndimensionalities of phase regions in a potential phase diagram are given\nby or . With every potential substituted by its conjugate molar\nquantity, the dimensionalities of phase regions increase by one until\nthe phase region reaches the dimensionality of the phase diagram. The\naxes of this phase diagram now consist of both potentials and molar\nquantities. The dimensionality of a phase region can thus be represented\nby the following equation based on\n\nEq. \u2011\n$\\upsilon_{m} = \\left( c - n_{s} \\right) + 2 - p + n_{m} \\leq \\left( c - n_{s} \\right) + 1$\n\nwhere $n_{m}$ is the number of molar axes. This equation is applicable\nto phase regions with more than $n_{m} + 1$ phases. For phase regions\nwith $n_{m} + 1$ phases or fewer, the dimensionalities are the same as\nthe phase diagram, i.e. $\\upsilon_{m} = \\left( c - n_{s} \\right) + 1$,\nand no longer vary with the number of molar quantity axes. When all\n$\\left( c - n_{s} \\right) + 1$ potentials are substituted by their\nconjugate molar quantities, one obtains a complete molar phase diagrams\nto be discussed in Chapter , and all phase regions have the\nsame-dimensionality of $\\left( c - n_{s} \\right) + 1$.\n\nFor the sake of graphic visualization, let us exam a two-dimensional\nphase diagram of a one-component system. Topologically, it is equivalent\nto a multi-component system with $n_{s} = c - 1$. In , three\ntwo-dimensional phase diagrams are shown for pure Fe. In principle, one\ncan use any one of them to illustrate mixed potential and molar phase\ndiagrams. For the purpose of practical usefulness, one selects the\n$T - ( - P)$ potential diagram as temperature and pressure are the two\ntypical variables controlled in experiments of one-component systems.\nThe conjugate molar quantities of $- P$ and $T$ are molar volume and\nmolar entropy, respectively. For stable phases, any pair of conjugate\nvariables changes in the same direction as illustrated by and , i.e. the\nphase stable at higher $T$ has higher molar entropy, and the phase\nstable at higher $\u2013P$, i.e. lower pressure, has higher molar volume. Let\nus first substitute $\u2013P$ by $V_{m}$ as shown in a. The dimensionality of\na sing-phase region remains unchanged because of\n$p = 1 < 2 = n_{m} + 1$. The dimensionality of two-phase regions is\nchanged from 1 to 2 due to $\\upsilon_{m} = 3 - 2 + 1 = 2$ from .\n\nFigure \u2011: T-Vm and Sm-(-P) phase diagrams of Fe\n\nAs both phases in a two-phase equilibrium have the same temperature, all\ntie-lines, depicted by dotted lines, are perpendicular to the\ntemperature axis in a. When the molar volume of the system changes from\none end of a tie-line to another end at a constant temperature, the mole\nfraction of one phase increases from 0 to 1, and the mole fraction of\nanother phase decreases from 1 to 0. The tine-lines at various\ntemperatures combine together to form a two-dimensional two-phase\nregion. The two curves at the two ends of tie-lines represent the\nboundaries between the single-phase and two-phase regions and are no\nlonger phase regions themselves. They are thus called phase boundaries.\n\nBy the same token, by changing the temperature at a constant molar\nvolume of the system, the system will locate on different tie-lines with\nthe amounts of the two phases determined by the lever rule. It is thus\nclear that the system maintains the two-phase equilibrium state with\nboth $T$ and $V_{m}$ changing independently inside the two-dimensional\ntwo-phase region. This seems in contradiction to Gibbs phase rule being\n$\\upsilon = 3 - p = 1$ from , but it is not because Gibbs phase rule\napplies strictly to potential phase diagrams only, while the $T - V_{m}$\nphase diagram has one of its axis being a molar quantity. As an\nalternative, one may consider as a modified Gibbs phase rule in\ndescribing the dimensionality of a phase region in a mixed potential and\nmolar phase diagram with $p \\geq n_{m} + 1$.\n\nThe two three-phase equilibria in pure Fe are also represented by\ntie-lines connecting all three phases. The dimensionality of three-phase\nregions is $\\upsilon_{m} = 3 - 3 + 1 = 1$ from , and the three two-phase\ntie-lines for a three-phase equilibrium thus overlap each other with\ntheir three molar volumes on the same tie-line.\n\nLet us examine the two three-phase equilibria in more detail. In the\n\u03b3/\u03b1/\u03b5 three-phase equilibrium at $T_{E} = 756.6\\ K$ and\n$\\left( - P_{E} \\right) = - 1.046 \\bullet 10^{10}\\ Pa$, the molar\nvolumes of \u03b1, \u03b3, and \u03b5 are 6.837, 6.677, 6.582 $10^{- 6}\\ m^{3}/mol$,\nrespectively. There are two two-phase regions at higher temperatures and\none two-phase region at lower temperatures. This is also shown in the\npotential phase diagram of a with two two-phase curves entering into and\none two-phase curve leaving from the three-phase equilibrium point with\ndecreasing temperature. Considering the system with the fixed molar\nvolume equal to that of the $\\gamma$ phase, i.e.\n$V_{m}^{\\ }\\left( T_{E} \\right) = 6.677 \\bullet 10^{- 6}\\ m^{3}/mol$. At\n$T > T_{E}$, the system is in the single \u03b3 phase region. With decrease\nin temperature across $T_{E}$, it enters into the $\\alpha + \\varepsilon$\ntwo-phase region. This transformation can be written as follows and is\ncalled a eutectoid reaction\n\nEq. \u2011 $\\gamma \\rightarrow \\alpha + \\varepsilon$\n\nThis type of transformations is named as eutectic reaction if the high\ntemperature phase is a liquid phase. If the system molar volume is\nbetween $V_{m}^{\\gamma}$ and $V_{m}^{\\alpha}$ at $756.6K$, with decrease\nof temperature, the system first moves from the single \u03b3 phase region to\nthe $\\alpha + \\gamma$ two-phase region when the\n$\\gamma/(\\alpha + \\gamma)$ phase boundary is crossed. When the\ntemperature reaches $T_{E}$, the eutectoid transformation takes place in\nthe remaining \u03b3 phase. The $\\alpha$ formed prior to the eutectoid\ntransformation is called proeutectoid $\\alpha$. By the same token, when\nthe system molar volume is between $V_{m}^{\\gamma}$ and\n$V_{m}^{\\varepsilon}$ at $756.6K$, proeutectoid $\\varepsilon$ would form\nfollowed by the eutectoid transformation with decrease in temperature.\n\nOn the other hand, the $L/\\delta/\\gamma$ three-phase equilibrium at\n$T_{P} = 1977.9\\ K$ and\n$\\left( - P_{P} \\right) = - 5.111 \\bullet \\ 10^{9}\\ Pa$ has different\ncharacteristics, where the subscript P will be defined shortly. There\nare one two-phase equilibrium above and two two-phase equilibria below\nthe invariant temperature, shown in the potential phase diagram of a\nwith one two-phase curve entering into and two two-phase curves leaving\nfrom the three-phase equilibrium point with decreasing temperature. The\nmolar volumes of $L$, $\\delta$, and $\\gamma$ at $T_{P}$ are 7.735,\n7.542, and 7.498 $10^{- 6}\\ m^{3}/mol$, respectively. For $T > T_{P}$,\nthe two-phase region is $L + \\gamma$. If the system molar volume is\nbetween 7.735 and 7.498 $10^{- 6}\\ m^{3}/mol$, i.e.\n$V_{m}^{L}\\left( T_{P} \\right)$ and\n$V_{m}^{\\gamma}\\left( T_{P} \\right)$, when the temperature reaches\n$T_{P}$, $L$ and $\\gamma$ are combined to form $\\delta$ with the\ntransformation written as\n\nEq. \u2011 $L + \\gamma \\rightarrow \\delta$\n\nThis type of reactions is called as peritectic reaction or peritectoid\nreaction when the liquid phase is replaced by a solid phase, denoted by\nthe subscript P. At $T < T_{P}$, one or both high temperature phases may\nno longer be present in equilibrium depending on the value of the system\nmolar volume. For\n$V_{m}^{\\ } = V_{m}^{\\delta}\\left( T_{P} \\right) = 7.542 \\bullet 10^{- 6}\\ m^{3}/mol$,\nthe peritectic reaction, , can come to completion with no $L$ and\n$\\gamma$ left upon decreasing temperature. For\n$V_{m}^{\\gamma}\\left( T_{P} \\right) = 7.498 < V_{m}^{\\ } < V_{m}^{\\delta}\\left( T_{P} \\right) = 7.542\\ \\left( 10^{- 6}\\ m^{3}/mol \\right)$,\nthe liquid phase is consumed, and the system enters the\n$\\gamma + \\delta$ two-phase region. On the other hand, for\n$V_{m}^{\\delta}\\left( T_{P} \\right) = 7.542 < V_{m}^{\\ } < V_{m}^{\\ L}\\left( T_{P} \\right) = 7.735\\ \\ \\left( 10^{- 6}\\ m^{3}/mol \\right)$,\nthe $\\gamma$ phase is consumed instead, and the system enters into the\n$L + \\delta$ two-phase equilibrium region upon cooling.\n\nLet us now replace $T$ by $S_{m}$ to obtain the $( - P) - S_{m}$ phase\ndiagram shown in b. The morphology of this phase diagram is identical to\nthe $T - V_{m}$ phase diagram just discussed with all tie-lines\nperpendicular to the pressure axis. The transformations at the two\nthree-phase equilibria with $( - P)$ decreasing or $P$ increasing are as\nfollows\n\nEq. \u2011 $\\gamma + \\alpha \\rightarrow \\varepsilon$\n\nEq. \u2011 $\\delta \\rightarrow L + \\gamma$\n\nTo visualize two-dimensional phase diagrams of binary systems, one\nusually keeps the pressure constant. One type of commonly used binary\nphase diagram is the temperature-composition ($T - x$) phase diagram. As\nan example, let us re-plot the $T - \\mu_{C}$ potential diagram shown in\ninto a $T - x_{C}$ mixed potential and molar phase diagram by replacing\nthe chemical potential of $C$ by its mole fraction. The $T - x_{C}$\nphase diagram thus obtained is shown in . In this phase diagram there\nare one peritectic reaction and two eutectic reactions as follows\n\nEq. \u2011 $L + \\delta \\rightarrow \\gamma$\n\nEq. \u2011 $L \\rightarrow \\gamma + C$\n\nEq. 3\u201161 $\\gamma \\rightarrow \\alpha + C$\n\nFigure \u2011: $T - x_{C}\\ $ phase diagram of the Fe-C binary system\n\nIn in Chapter , it was discussed the formation of miscibility gaps due\nto repulsive interactions between components. One example is shown in\nfor the Al-Zn binary system in terms of both $T - \\mu_{Zn}$ potential\nphase diagram and $\\ T - x_{Zn}\\ $ mixed potential and molar phase\ndiagram.\n\nFigure \u2011: $T - \\mu_{Zn}$ potential phase diagram (a) and\n$\\ T - x_{Zn}\\ $ mixed potential and molar phase diagram (b) of the\nAl-Zn binary system\n\nIn ,there are one eutectic reaction and one eutectoid reaction as\nfollows\n\nEq. \u2011 $L \\rightarrow fcc + hcp$\n\nEq. \u2011 $fcc\\# 1 \\rightarrow fcc\\# 2 + hcp$\n\nThe eutectoid reaction, , is also termed as monotectoid reaction because\nthe fcc phase appears on both sides of the reaction with different\ncompositions, $fcc\\# 1$ and $fcc\\# 2$, due to the miscibility gap. The\nhighest temperature of the miscibility gap is called the consolute point\nas discussed in Chapter , which can be clearly seen in the\n$T - \\mu_{Zn}$ potential phase diagram shown in . This is a critical\npoint, marking the limit of instability as shown in .\n\nWhen there is only one phase on either side of the reaction, i.e. both\nphases have the same composition, the reaction is called a congruent\nreaction. One example is shown in for the $T - x_{SiO_{2}}\\ $ mixed\npotential and molar phase diagram of the CaO-SiO2\npseudo-binary system with two congruent reactions as follows\n\nEq. 3\u201164 $L \\rightarrow CaSiO_{3}$\n\nEq. 3\u201165 $L \\rightarrow {Ca}_{2}SiO_{4}$\n\nThey are not invariant reactions based on the Gibbs phase rule. In , it\nis noted that there are one miscibility gap in the liquid phase close to\nthe $SiO_{2}$ side, four eutectic reactions with one being monotectic\ninvolving two liquid phases due to the miscibility gap, and three\nperitectic reactions.\n\nFigure \u2011: $T - x_{SiO_{2}}\\ $ mixed potential and molar phase diagram of\nthe CaO-SiO2 pseudo-binary system.\n\nLet us generalize the above discussion to phase diagrams with\n$\\left( c - n_{s} \\right) + 1$ axes. In such a phase diagram, the\nmaximum number of phases is given by as\n$p_{\\max} = \\left( c - n_{s} \\right) + 2$. The number of phases on the\neither side of an invariant reaction can vary from one phase to\n$p_{\\max} - 1 = \\left( c - n_{s} \\right) + 1$ phases with the remaining\nphases on the other side of the reaction, typically with the potential\ndecreasing from left to right. The invariant reaction with one phase on\nthe left of the reaction is named as eutectic, or eutectoid reaction\ndepending on if the phase on the left of the reaction is liquid or\nsolid. The rest invariant reactions are named as peritectic or\nperitectoid reactions with or without a liquid phase.\n", "metadata": {}}, {"id": "7fa35f9c", "cell_type": "markdown", "source": "### Phase diagrams with only molar quantities\n\nWhen all $\\left( c - n_{s} \\right) + 1$ potentials in a potential phase\ndiagrams are replaced by their conjugate molar quantities, one obtains a\nmolar phase diagram with molar quantities on all axes of the phase\ndiagram. For regions with the number of phases\n$p \\leq n_{m} + 1 = \\left( c - n_{s} \\right) + 2 = p_{\\max}$ (see ), the\nphase regions have the same-dimensionality as that of the phase diagram,\ni.e. all phase regions have the same-dimensionality of\n$\\left( c - n_{s} \\right) + 1$, and any geometric feature with\nlower-dimensionalities, i.e. from 0 to $\\ \\left( c - n_{s} \\right)$, is\nnot phase regions, but phase boundaries between neighbouring phase\nregions. For the sake of graphic visualization, the molar phase diagram\nof pure Fe is shown in obtained by combining the two mixed phase\ndiagrams in .\n\nFigure \u2011: Molar phase diagram of Fe\n\nIn this molar phase diagram, all one-, two-, and three-phase regions are\ntwo-dimensional, the same as the dimensionality of the phase diagram. A\ntwo-phase region is made up by tie-lines connecting the two phases in\nequilibrium, while a three phase-region is surrounded by three two-phase\ntie-lines, i.e. a tie-triangle. The amount of each phase in the\ntie-triangle can be obtained using the lever rule represented by and .\nAs can be seen, phase boundaries between a one-phase region and a\ntwo-phase region are one-dimensional. When the system crosses such a\nphase boundary, the number of phases changes by one from two to one or\nvice versa. Phase boundaries between two- and three-phase regions are\nrepresented by two-phase tie-lines. When the system crosses such a phase\nboundary, the number of phases also changes by one. The\nlowest-dimensional phase boundaries are the points between one- and\nthree-phase regions that are zero-dimensional and the intercept of four\none-dimensional phase boundaries. When the system crosses such a phase\nboundary, the number of phases changes by two.\n\nFor multi-component systems, the phase relations cannot be directly\nvisualized. By representing the system equations in terms of equilibrium\nconditions and level rules on a phase boundary using phases separately\nfrom the two adjacent phase regions, Palatnik and Landau \\[4\\]\npostulated that the difference between the number of unknowns and\nequations gives the dimensionality of the phase boundary and derived the\nfollowing relationships\n\nEq. \u2011 $D^{+} + D^{-} = r - b = c - n_{s} - b$\n\nwhere $D^{+}$ and $D^{-}$ are the numbers of phases added and removed\nwhen the phase boundary is crossed, and $r$ and $b$ are the\ndimensionalities of the phase diagram and the phase boundary,\nrespectively. They termed the as the contact rule, which is named as the\nMPL boundary rule by Hillert \\[1\\].\n\nBy the same token, is applicable to any phase boundary where the two\nadjacent phase regions have the same-dimensionality of the phase\ndiagram, even in phase diagrams with a mixture of potentials and molar\nquantities as the diagram axes. This can be understood because the\npotentials are homogeneous in all phases in equilibrium on the phase\nboundary. The phase boundary is thus equivalent to those in a complete\nmolar phase diagram with its number of components equal to the number of\nmolar axes in a mixed potential and molar phase diagrams minus one, i.e.\n\nEq. \u2011 $c^{'} = n_{m} - 1 = c - n_{s}$\n\nThe last part of stems from the discussion related to when all\n$c^{'} + 1$ potentials are replaced by their conjugate molar quantities,\nwhich is analogous to a molar phase diagram with $\\ c$ independent\ncomponents and $n_{s}$ potentials fixed.\n\nFor a two-dimensional phase diagram with $r = 2$, the phase boundary can\nbe either zero- or one--dimensional. As shown in , the basic element of\na molar phase diagram is a joint of four one-dimensional phase boundary\nlines. When a phase boundary line is crossed, the number of phases is\neither increased or decreased by one. The joint of four one-dimensional\nphase boundary lines is zero-dimensional. The number of phases differs\nby two between the phase regions across the zero-dimensional joint. Two\nscenarios are possible\n\nTwo phases are added or removed, i.e. $D^{+} = 2$ and $D^{-} = 0$ or\n$D^{+} = 0$ and $D^{-} = 2$, and the number of phases in the two phase\nregions differs by two;\n\nOne phase is added, one phase is removed, i.e. $D^{+} = D^{-} = 1$, and\nthe phase regions have the same number of phases.\n\nBy combining the contact rules for both zero- and one-dimensional phase\nboundaries, it is evident that the above two scenarios co-exist in a\njoint of four-phase regions with two-phase regions following the first\nscenario and other two-phase regions following the second scenario.\nBased on the Schreinemakers\u2019 rule generalized by Hillert \\[5\\], each of\nthe two-phase regions with the same number of phases contains one\nextrapolation of the phase boundaries, while the other two-phase regions\ncontain either zero or two extrapolations of the phase boundaries. This\ncan be observed for all the zero-dimensional phase boundaries in and is\nfurther schematically illustrated in for general cases.\n\nFigure \u2011: Schematic molar phase diagram, demonstrating the\nSchreinemakers\u2019 rule\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/phase_equilibria_in_heterogeneous_systems/potential_phase_diagrams.ipynb b/psu410/src/phase_equilibria_in_heterogeneous_systems/potential_phase_diagrams.ipynb new file mode 100644 index 0000000..e2e0f84 --- /dev/null +++ b/psu410/src/phase_equilibria_in_heterogeneous_systems/potential_phase_diagrams.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "5cff8694", "cell_type": "markdown", "source": "## Potential phase diagrams\n\nGibbs phase rule can be further understood through the\n$c + 2$-dimensional space of potentials consisting of $T$, $- P$, and\n$\\mu_{i}$ with $i$ from 1 to $c$. Each phase is a $c + 1$-dimensional\nfeature in this $c + 2$-dimensional space characterized by . The\ndirections of this $c + 1$-dimensional feature are represented by their\nmolar quantities as shown by following equations,\n\nEq. \u2011\n$\\ \\left( \\frac{\\partial\\mu_{i}}{\\partial T} \\right)_{P,\\mu_{j \\neq i}} = - \\frac{S^{\\beta}}{N_{i}^{\\beta}} = - \\frac{S_{m}^{\\beta}}{x_{i}^{\\beta}}$\n\nEq. \u2011\n$\\ \\left( \\frac{\\partial\\mu_{i}}{\\partial( - P)} \\right)_{T,\\mu_{j \\neq i}} = - \\frac{V^{\\beta}}{N_{i}^{\\beta}} = - \\frac{V_{m}^{\\beta}}{x_{i}^{\\beta}}$\n\nEq. \u2011\n$\\ \\left( \\frac{\\partial( - P)}{\\partial T} \\right)_{\\mu_{i}} = - \\frac{S^{\\beta}}{V^{\\beta}} = - \\frac{S_{m}^{\\beta}}{V_{m}^{\\beta}}$\n\nAs can be seen, all the direction derivatives are negative, indicating\nthat the $c + 1$-dimensional feature is convex. The intercept of any two\n$c + 1$-dimensional features is thus a $c$-dimensional feature. On this\n$c$-dimensional feature, these two phases are in equilibrium with each\nother because each potential has the same value in both phases. This\nfeature thus represents a two-phase equilibrium. By the same token, the\nintercept of any three $c + 1$-dimensional features is a\n$c - 1$-dimensional feature in the $c + 2$-dimensional space of\npotentials and represents a three-phase equilibrium. This continues\nuntil the number of phases reaches $c + 2$ with all $c + 2$ potentials\ncompletely determined, and the dimension of their intercepts becomes\nzero.\n\nThose $c + 1$ to zero-dimensional geometrical features in the\n$c + 2$-dimensional space of potentials thus denote one-phase,\ntwo-phase, three-phase to $(c + 2)$-phase equilibria of the system with\nthe dimensionality of the feature and the number of phases in\nequilibrium related by , i.e. Gibbs phase rule. Their arrangements in\nthe $c + 2$-dimensional space of potentials thus depict the phase\nrelations in the system and are commonly called phase diagrams. Since\nall the diagram axes in the phase diagram discussed above are\npotentials, the diagram is called potential phase diagram in order to\ndifferentiate it from phase diagrams with some or all diagram axes being\ntheir conjugate molar quantities. Both potential and molar phase\ndiagrams are discussed in this chapter.\n", "metadata": {}}, {"id": "1706759a", "cell_type": "markdown", "source": "### Potential phase diagrams of one-component systems\n\nAs physical vision of human being is limited to three-dimensions, only\none-component system can be completely visualized as shown in for one\nphase where any two of the three potentials can change independently.\nFrom Gibbs phase rule, when two phases are in equilibrium, only one\npotential can vary freely if the two-phase equilibrium is to be\nmaintained. While three phases are in equilibrium, the degree of freedom\nis zero, and all three potentials are fixed.\n\nFor a two-phase equilibrium, two surfaces intersect each other as\ndepicted by the dashed line in . This two-phase equilibrium line is\nobtained by applying to both phases in a one-component system. Since one\nof the potentials is dependent on the other two, one can eliminate it by\ndividing the equation by its conjugate molar quantity and subtracting\nthe two equations, resulting in the following three equations\n\nEq. \u2011\n$0 = \\left( \\frac{S^{\\alpha}}{N_{A}^{\\alpha}} - \\frac{S^{\\beta}}{N_{A}^{\\beta}} \\right)dT + \\left( \\frac{V^{\\alpha}}{N_{A}^{\\alpha}} - \\frac{V^{\\beta}}{N_{A}^{\\beta}} \\right)d( - P) = \\mathrm{\\Delta}S_{m}^{\\alpha\\beta}dT + \\mathrm{\\Delta}V_{m}^{\\alpha\\beta}d( - P)$\n\nEq. \u2011\n$0 = \\left( \\frac{S^{\\alpha}}{V^{\\alpha}} - \\frac{S^{\\beta}}{V^{\\beta}} \\right)dT + \\left( \\frac{N_{A}^{\\alpha}}{V^{\\alpha}} - \\frac{N_{A}^{\\beta}}{V^{\\beta}} \\right){d\\mu}_{A} = \\mathrm{\\Delta}\\left( \\frac{S_{m}}{V_{m}} \\right)^{\\alpha\\beta}dT + \\mathrm{\\Delta}\\left( \\frac{1}{V_{m}} \\right)^{\\alpha\\beta}{d\\mu}_{A}$\n\nEq. \u2011\n$0 = \\left( \\frac{V^{\\alpha}}{S^{\\alpha}} - \\frac{V^{\\beta}}{S^{\\beta}} \\right)d( - P) + \\left( \\frac{N_{A}^{\\alpha}}{S^{\\alpha}} - \\frac{N_{A}^{\\beta}}{S^{\\beta}} \\right){d\\mu}_{A} = \\mathrm{\\Delta}\\left( \\frac{V_{m}}{S_{m}} \\right)^{\\alpha\\beta}d( - P) + \\mathrm{\\Delta}\\left( \\frac{1}{S_{m}} \\right)^{\\alpha\\beta}{d\\mu}_{A}$\n\nFigure \u2011: Gibbs energy surfaces of two phases and their intersection,\nrepresenting the two-phase equilibrium.\n\nThe directions of the two-phase equilibrium line can thus be obtained as\n\nEq. \u2011\n$\\frac{d( - P)}{dT} = - \\frac{\\mathrm{\\Delta}S_{m}^{\\alpha\\beta}}{\\mathrm{\\Delta}V_{m}^{\\alpha\\beta}}$\n\nEq. \u2011\n$\\frac{dT}{{d\\mu}_{A}} = - \\frac{\\mathrm{\\Delta}\\left( \\frac{1}{V_{m}} \\right)^{\\alpha\\beta}}{\\mathrm{\\Delta}\\left( \\frac{S_{m}}{V_{m}} \\right)^{\\alpha\\beta}}$\n\nEq. \u2011\n$\\frac{d( - P)}{{d\\mu}_{A}} = - \\frac{\\mathrm{\\Delta}\\left( \\frac{1}{S_{m}} \\right)^{\\alpha\\beta}}{\\mathrm{\\Delta}\\left( \\frac{V_{m}}{S_{m}} \\right)^{\\alpha\\beta}}$\n\nThese three equations define the mathematical forms of the two-phase\nequilibrium line in the two-dimensional spaces of $T - ( - P)$,\n$\\mu_{A} - T$ and $\\mu_{A} - ( - P)$, respectively, and can thus be\nplotted as two-dimensional diagrams. is commonly called\nClausius-Clapeyron equation in the literature. One may thus call all\nthree equations above as generalized Clausius-Clapeyron equations. At\nequilibrium, the chemical potentials of the component in both phases are\nequal to each other, so are their Gibbs energies. One thus has\n\nEq. \u2011\n$G_{m}^{\\alpha} - G_{m}^{\\beta} = 0 = \\mathrm{\\Delta}G_{m} = \\mathrm{\\Delta}H_{m}^{\\alpha\\beta} - T\\mathrm{\\Delta}S_{m}^{\\alpha\\beta}$\n\nThe Clausius-Clapeyron equation, , can be re-written as\n\nEq. \u2011\n$\\frac{d( - P)}{dT} = - \\frac{\\mathrm{\\Delta}H_{m}^{\\alpha\\beta}}{T\\mathrm{\\Delta}V_{m}^{\\alpha\\beta}}$\n\nAs an example, three potential phase diagrams of pure Fe are shown in .\nThere are four phases in the system, bcc, fcc, hcp, and liquid. In the\nliterature, the high temperature and low temperature bcc phases are\nusually denoted by \u03b4 (high temperature) and \u03b1 (low temperature), the fcc\nand hcp phases by \u03b3 and \u03b5, and the liquid phase by L, respectively. In\nthese figures, the two-dimensional areas are single-phase regions where\ntwo potentials can change independently with the system remaining as\nsingle-phase. The lines denote two-phase equilibrium regions where only\none potential can vary independently if the two-phase equilibrium is to\nbe maintained. The points where three two-phase equilibrium lines meet\nrepresent the invariant three-phase equilibria with three potentials\nfixed.\n\nFigure \u2011 $T - ( - P)$, $\\mu_{A} - T$ and $\\mu_{A} - ( - P)$ phase\ndiagrams of pure Fe\n\nBased on the discussions in Chapter , enthalpy and entropy of a phase\nincrease monotonically with temperature, and phases stable at higher\ntemperatures have higher enthalpy and entropy than phases stable at\nlower temperatures. Consequently, the two-phase equilibrium lines in a\n$T - ( - P)$ potential phase diagram have negative slopes if the phase\nstable at higher temperatures also has larger molar volume than the\nphase stable at lower temperatures (note that if $P$ is plotted instead\nof $- P$, the slope is positive). This is the case for the two-phase\nequilibrium lines of \u03b4/L, \u03b3/L, and \u03b3/\u03b4 at high temperatures, and \u03b5/\u03b3\nshown in a. On the other hand, the two-phase equilibrium lines of \u03b1/\u03b5\nand \u03b1/\u03b3 at low temperatures have positive slopes, indicating that \u03b5 and\n\u03b3 have smaller molar volume than \u03b1 as \u03b5 and \u03b3 are more stable at higher\npressures than \u03b1 at constant temperatures. It is thus evident that the\nphase stable at higher pressure can have either higher or lower entropy\nthan the phase stable at lower pressure, and the phase stable at higher\ntemperature can have either higher or lower volume than the phase stable\nat lower temperature. This is the property anomaly discussed in Chapter\n.\n\nAnother useful example of potential phase diagram is the\npressure-temperature phase diagram of $H_{2}O$ shown in with three\nphases: ice, water, and vapour. It is known that the solid ice has many\npolymorphic structures at high pressures, which are not included in this\ndiagram. As in the pure Fe potential phase diagram discussed above, the\nsingle-phase regions of ice, water, and vapour are represented by the\ntwo-dimensional areas with two-degree of freedom based on the Gibbs\nphase rule, the lines are for the two-phase regions of ice-water,\nice-vapour, and water-vapour, and the three-phase equilibrium has\nzero-degree of freedom represented by a point at $273.16\\ K$ and\n$611.73\\ Pa$.\n\nThere are two features in which are different from those of Fe shown in\na. The first feature is that the slope of the liquid-solid two-phase\nequilibrium line in has the opposite sign of that in a. This is because\nsolid ice has larger molar volume than liquid water, while the molar\nvolume of liquid Fe is larger than those of fcc-Fe and bcc-Fe. The\nsecond feature is that the two-phase equilibrium line of water-vapour\nends at $647\\ K$ and $22.064 \\cdot 10^{6}\\ Pa$. Beyond this point, the\ndifference between vapour and water disappears when the pressure and\ntemperature are changed, i.e. it behaves like one phase. This point is a\ncritical point, as discussed in Chapter . However, it should be pointed\nout that it does not represent an invariant reaction as the degree of\nfreedom based on the Gibbs phase rule is equal to one and not zero. On\nthe other hand, both the temperature and pressure of the critical point\nare invariant due to the two constraints introduced by the limit of\nstability of a single phase, i.e. the second and third derivatives of\ntemperature to entropy or pressure to volume are zero.\n\nFigure \u2011: P-T phase diagram of $H_{2}O$\n", "metadata": {}}, {"id": "8b8697d3", "cell_type": "markdown", "source": "### Potential phase diagrams of two-component systems\n\nFrom Gibbs-Duhem equation (see ), a single phase in a two-component\nsystem has three independent potentials, out of the four potentials of\n$T$, $- P$, $\\mu_{A}$, and $\\mu_{B}$, and is a three-dimensional\ngeometric feature in a four-dimensional space. Alternatively, it can be\nrepresented by a three-dimensional space constructed by the three\nindependent potentials. A two-phase equilibrium is thus a\ntwo-dimensional surface in this three-dimensional space created by the\nintercept of two three-dimensional spaces, and a three-phase equilibrium\nis a one-dimensional line, and a four-phase equilibrium is a\nzero-dimensional point. This is shown in for the Fe-C binary system\ninvolving four phases: fcc, bcc, Fe3C and graphite. Since any\none of the four potentials can be chosen as the dependent one, four\nthree-dimensional potential phase diagrams are depicted in .\n\nFigure \u2011: Projected potential phase diagram of Fe-C system with fcc,\nbcc, Fe3C, and graphite\n\nThe two-phase equilibrium surfaces are obtained by choosing any one of\nthe four potentials as the dependent one and solving the Gibbs-Duhem\nequations for both phases, resulting in following four equations\n\nEq. \u2011\n$0 = \\left( \\frac{S^{\\alpha}}{N_{A}^{\\alpha}} - \\frac{S^{\\beta}}{N_{A}^{\\beta}} \\right)dT + \\left( \\frac{V^{\\alpha}}{N_{A}^{\\alpha}} - \\frac{V^{\\beta}}{N_{A}^{\\beta}} \\right)d( - P) + \\left( \\frac{N_{B}^{\\alpha}}{N_{A}^{\\alpha}} - \\frac{N_{B}^{\\beta}}{N_{A}^{\\beta}} \\right){d\\mu}_{B} = \\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}dT + \\mathrm{\\Delta}V_{mA}^{\\alpha\\beta}d( - P) + \\mathrm{\\Delta}z_{B}^{\\alpha\\beta}{d\\mu}_{B}$\n\nEq. \u2011\n$0 = \\left( \\frac{S^{\\alpha}}{N_{B}^{\\alpha}} - \\frac{S^{\\beta}}{N_{B}^{\\beta}} \\right)dT + \\left( \\frac{V^{\\alpha}}{N_{B}^{\\alpha}} - \\frac{V^{\\beta}}{N_{B}^{\\beta}} \\right)d( - P) + \\left( \\frac{N_{B}^{\\alpha}}{N_{B}^{\\alpha}} - \\frac{N_{B}^{\\beta}}{N_{B}^{\\beta}} \\right){d\\mu}_{B} = \\mathrm{\\Delta}S_{mB}^{\\alpha\\beta}dT + \\mathrm{\\Delta}V_{mB}^{\\alpha\\beta}d( - P) + \\mathrm{\\Delta}z_{A}^{\\alpha\\beta}{d\\mu}_{A}$\n\nEq. \u2011\n$0 = \\left( \\frac{S^{\\alpha}}{V^{\\alpha}} - \\frac{S^{\\beta}}{V^{\\beta}} \\right)dT + \\left( \\frac{N_{A}^{\\alpha}}{V^{\\alpha}} - \\frac{N_{A}^{\\beta}}{V^{\\beta}} \\right){d\\mu}_{A} + \\left( \\frac{N_{B}^{\\alpha}}{V^{\\alpha}} - \\frac{N_{B}^{\\beta}}{V^{\\beta}} \\right){d\\mu}_{B} = \\mathrm{\\Delta}\\left( \\frac{S_{m}}{V_{m}} \\right)^{\\alpha\\beta}dT + \\mathrm{\\Delta}\\left( \\frac{1}{V_{mA}} \\right)^{\\alpha\\beta}{d\\mu}_{A} + \\mathrm{\\Delta}\\left( \\frac{1}{V_{mB}} \\right)^{\\alpha\\beta}{d\\mu}_{B}$\n\nEq. \u2011\n$0 = \\left( \\frac{V^{\\alpha}}{S^{\\alpha}} - \\frac{V^{\\beta}}{S^{\\beta}} \\right)d( - P) + \\left( \\frac{N_{A}^{\\alpha}}{S^{\\alpha}} - \\frac{N_{A}^{\\beta}}{S^{\\beta}} \\right){d\\mu}_{A} + \\left( \\frac{N_{B}^{\\alpha}}{S^{\\alpha}} - \\frac{N_{B}^{\\beta}}{S^{\\beta}} \\right){d\\mu}_{B} = \\mathrm{\\Delta}\\left( \\frac{V_{m}}{S_{m}} \\right)^{\\alpha\\beta}d( - P) + \\mathrm{\\Delta}\\left( \\frac{1}{S_{mA}} \\right)^{\\alpha\\beta}{d\\mu}_{A} + \\mathrm{\\Delta}\\left( \\frac{1}{S_{mB}} \\right)^{\\alpha\\beta}{d\\mu}_{B}$\n\nA three-phase equilibrium line is represented by the intercept of two\ntwo-phase surfaces by applying any one of the above four equations to\ntwo two-phase equilibria. Let us use as an example\n\nEq. \u2011\n$0 = \\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}dT + \\mathrm{\\Delta}V_{mA}^{\\alpha\\beta}d( - P) + {\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}{d\\mu}_{B}$\n\nEq. \u2011\n$0 = \\mathrm{\\Delta}S_{mA}^{\\alpha\\gamma}dT + \\mathrm{\\Delta}V_{mA}^{\\alpha\\gamma}d( - P) + \\mathrm{\\Delta}{z_{B}^{\\alpha\\gamma}d\\mu}_{B}$\n\nIt is self-evident that the two-phase equilibrium surface between\n$\\beta$ and $\\gamma$ is not independent and can be obtained by\nsubtraction of and\n\nEq. \u2011\n$0 = \\left( \\mathrm{\\Delta}S_{mA}^{\\alpha\\beta} - \\mathrm{\\Delta}S_{mA}^{\\alpha\\gamma} \\right)dT + \\left( \\mathrm{\\Delta}V_{mA}^{\\alpha\\beta} - \\mathrm{\\Delta}V_{mA}^{\\alpha\\gamma} \\right)d( - P) + \\left( {\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta} - \\mathrm{\\Delta}z_{B}^{\\alpha\\gamma} \\right){d\\mu}_{B} = \\mathrm{\\Delta}S_{mA}^{\\gamma\\beta}dT + \\mathrm{\\Delta}V_{mA}^{\\gamma\\beta}d( - P) + {\\mathrm{\\Delta}z}_{B}^{\\gamma\\beta}{d\\mu}_{B}$\n\nEliminating one of three potentials in and , one can obtain three\nequations for the three-phase equilibrium line\n\nEq. \u2011\n$\\frac{d( - P)}{dT} = - \\frac{\\frac{\\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}}{{\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}S_{mA}^{\\alpha\\gamma}}{\\mathrm{\\Delta}z_{B}^{\\alpha\\gamma}}}{\\frac{\\mathrm{\\Delta}V_{mA}^{\\alpha\\beta}}{{\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}V_{mA}^{\\alpha\\gamma}}{\\mathrm{\\Delta}z_{B}^{\\alpha\\gamma}}}$\n\nEq. \u2011\n$\\frac{dT}{{d\\mu}_{B}} = - \\frac{\\frac{{\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}}{\\mathrm{\\Delta}V_{mA}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}z_{B}^{\\alpha\\gamma}}{\\mathrm{\\Delta}V_{mA}^{\\alpha\\gamma}}}{\\frac{\\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}}{\\mathrm{\\Delta}V_{mA}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}S_{mA}^{\\alpha\\gamma}}{\\mathrm{\\Delta}V_{mA}^{\\alpha\\gamma}}}$\n\nEq. \u2011\n$\\frac{d( - P)}{{d\\mu}_{B}} = - \\frac{\\frac{{\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}}{\\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}z_{B}^{\\alpha\\gamma}}{\\mathrm{\\Delta}S_{mA}^{\\alpha\\gamma}}}{\\frac{\\mathrm{\\Delta}V_{mA}^{\\alpha\\beta}}{\\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}V_{mA}^{\\alpha\\gamma}}{\\mathrm{\\Delta}S_{mA}^{\\alpha\\gamma}}}$\n\nto can be referred as generalized Clausius-Clapeyron equations for\nbinary systems. Similar equations can be derived for\n$T - ( - P) - \\mu_{A}$, $T - \\mu_{A} - \\mu_{B}$, and\n$( - P) - \\mu_{A} - \\mu_{B}$ potential phase diagrams from to , and are\nlisted below\n\n- Generalized Clausius-Clapeyron equations for a three-phase equilibrium\n in $T - ( - P) - \\mu_{A}$ potential phase diagrams\n\nEq. \u2011\n$\\frac{d( - P)}{dT} = - \\frac{\\frac{\\mathrm{\\Delta}S_{mB}^{\\alpha\\beta}}{{\\mathrm{\\Delta}z}_{A}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}S_{mB}^{\\alpha\\gamma}}{\\mathrm{\\Delta}z_{A}^{\\alpha\\gamma}}}{\\frac{\\mathrm{\\Delta}V_{mB}^{\\alpha\\beta}}{{\\mathrm{\\Delta}z}_{A}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}V_{mB}^{\\alpha\\gamma}}{\\mathrm{\\Delta}z_{A}^{\\alpha\\gamma}}}$\n\nEq. \u2011\n$\\frac{dT}{{d\\mu}_{A}} = - \\frac{\\frac{{\\mathrm{\\Delta}z}_{A}^{\\alpha\\beta}}{\\mathrm{\\Delta}V_{mB}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}z_{A}^{\\alpha\\gamma}}{\\mathrm{\\Delta}V_{mB}^{\\alpha\\gamma}}}{\\frac{\\mathrm{\\Delta}S_{mB}^{\\alpha\\beta}}{\\mathrm{\\Delta}V_{mB}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}S_{mB}^{\\alpha\\gamma}}{\\mathrm{\\Delta}V_{mB}^{\\alpha\\gamma}}}$\n\nEq. \u2011\n$\\frac{d( - P)}{{d\\mu}_{A}} = - \\frac{\\frac{{\\mathrm{\\Delta}z}_{A}^{\\alpha\\beta}}{\\mathrm{\\Delta}S_{mB}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}z_{A}^{\\alpha\\gamma}}{\\mathrm{\\Delta}S_{mB}^{\\alpha\\gamma}}}{\\frac{\\mathrm{\\Delta}V_{mB}^{\\alpha\\beta}}{\\mathrm{\\Delta}S_{mB}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}V_{mB}^{\\alpha\\gamma}}{\\mathrm{\\Delta}S_{mB}^{\\alpha\\gamma}}}$\n\n- Generalized Clausius-Clapeyron equations for a three-phase equilibrium\n in $\\ T - \\mu_{A} - \\mu_{B}$ potential phase diagrams\n\nEq. \u2011\n$\\frac{dT}{{d\\mu}_{A}} = - \\frac{\\frac{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}}{\\frac{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}}$\n\nEq. \u2011\n$\\frac{dT}{{d\\mu}_{B}} = - \\frac{\\frac{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}}{\\frac{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}}$\n\nEq. \u2011\n$\\frac{{d\\mu}_{A}}{{d\\mu}_{B}} = - \\frac{\\frac{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\gamma}}}{\\frac{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\gamma}}}$\n\n- Generalized Clausius-Clapeyron equations for a three-phase equilibrium\n in $( - P) - \\mu_{A} - \\mu_{B}$ potential phase diagrams\n\nEq. \u2011\n$\\frac{d( - P)}{{d\\mu}_{A}} = - \\frac{\\frac{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}}{\\frac{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}}$\n\nEq. \u2011\n$\\frac{d( - P)}{{d\\mu}_{B}} = - \\frac{\\frac{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}}{\\frac{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}}$\n\nEq. \u2011\n$\\frac{{d\\mu}_{A}}{{d\\mu}_{B}} = - \\frac{\\frac{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\gamma}}}{\\frac{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\gamma}}}$\n", "metadata": {}}, {"id": "9e783eec", "cell_type": "markdown", "source": "### Projection and section of phase diagrams with potential and molar quantities\n\nAs discussed in Chapter , projections of high-dimensional phase diagrams\nusually cannot keep all information. However, there is one type of\nwidely used projection in the literature, i.e. the liquidus surface in\nternary systems under constant pressure with temperature and mole\nfractions of two components as its axes. The projection along the\ntemperature axis reveals the composition regions for primary phases that\nsolidify from liquid upon cooling. These regions are separated by\nunivariant lines of three-phase equilibria. The projections along one of\nthe two mole fractions show the temperature as a function of composition\non the univariant three-phase equilibrium lines and also depict whether\na four-phase equilibrium is peritectic or eutectic. There are four\nscenarios for the three univariant three-phase equilibrium lines to meet\nat the four-phase equilibrium as depicted in and discussed individually\nbelow.\n\nFigure \u2011: Schematic four options for three univariant three-phase\nequilibrium lines to meet at the invariant four-phase equilibrium\n\nThe first scenario is that with decreasing temperature, all three\nunivariant lines merge into the four-phase equilibrium. It indicates\nthat the liquid phase does not exist at temperatures below the\nfour-phase invariant reaction. This invariant reaction is thus a ternary\neutectic reaction with liquid completely transformed to three solid\nphases upon cooling, i.e.\n\nEq. \u2011 $L \\rightarrow \\alpha + \\beta + \\gamma$\n\nIn the second scenario, two univariant lines merge into and one leaves\nfrom the four-phase equilibrium with decreasing temperature. This means\nthat one solid phase at higher temperature is no longer stable at lower\ntemperature, and it must react with the liquid phase to form the\nremaining two solid phases. The four-phase invariant reaction is thus\nperitectic. The solid phase common to both univariant lines at high\ntemperatures reacts with the liquid phase. Assuming that this phase is\n$\\alpha$, the four-phase invariant reaction becomes\n\nEq. \u2011 $L + \\alpha \\rightarrow \\beta + \\gamma$\n\nIn the third scenario, one univariant line points to and two leave from\nthe four-phase equilibrium with decreasing temperature. A new phase\nforms at low temperatures from the three high temperature phases, e.g.\nliquid, $\\alpha$, and $\\beta$, with the four-phase invariant reaction as\n\nEq. \u2011 $L + \\alpha + \\beta \\rightarrow \\gamma$\n\nThe fourth scenario is the inverse of the first scenario, indicating the\nformation of liquid from solid phases upon cooling, i.e.\n\nEq. \u2011 $\\alpha + \\beta + \\gamma \\rightarrow L$\n\nThis case has not been observed in reality.\n\nAs an example, the liquidus projections of the Al-Fe-Si ternary system\nare shown in in two formats\\[6\\], i.e. (a) three-dimensional liquidus\nsurface with the isotherms showing the liquidus contours; (b)\nconventional projection to the composition axis with the temperature\ndecrease shown by arrows; (c) projection to the temperature and weight\nfraction of Si. The first to third scenarios of invariant reactions\ndiscussed above can clearly be identified and listed in . It is evident\nthat c provides the easiest route to visualize the type of invariant\nreactions as show by .\n\nFigure \u2011: Liquidus of the Al-Fe-Si ternary system\\[6\\], (a)\nthree-dimensional presentation of the liquidus; (b) projection to the\ncomposition triangle with isotherms (dotted lines) superimposed and\ntheir temperatures indicated close to the horizontal axis.\n\nTable \u2011: Invariant liquidus reactions of the Al-Fe-Si ternary system\nwith the composition of the liquid phase \\[6\\]\n\n| Reaction | T, \u00b0C | wFe, % | wSi, % |\n|----------|-------|-------------------|-------------------|\n| | 1178 | 77.9 | 21.0 |\n| | 1155 | 49.0 | 0.16 |\n| | 1127 | 52.7 | 2.81 |\n| -H-L | 1076 | 41.6 | 44.0 |\n| | 1073 | 51.0 | 7.19 |\n| | 1050 | 53.8 | 18.4 |\n| -H-L | 1019 | 34.8 | 44.4 |\n| | 1004 | 49.0 | 12.6 |\n| | 1000 | 46.7 | 13.2 |\n| -L | 940 | 37.9 | 32.0 |\n| | 921 | 33.7 | 20.1 |\n| | 899 | 33.8 | 32.2 |\n| | 884 | 30.8 | 26.0 |\n| -L | 877 | 29.5 | 35.2 |\n| | 851 | 23.3 | 21.6 |\n| | 834 | 22.2 | 31.7 |\n| | 825 | 22.1 | 25.7 |\n| | 823 | 21.8 | 25.4 |\n| | 715 | 6.64 | 10.8 |\n| | 694 | 6.11 | 17.1 |\n| | 680 | 4.68 | 11.6 |\n| | 630 | 2.11 | 4.10 |\n| | 616 | 1.76 | 6.56 |\n| | 598 | 1.22 | 14.3 |\n| | 575 | 0.73 | 12.7 |\n\nIn contrast to projections, sectioning is used more often to understand\nphase relations in multi-component systems. Sectioning of a potential\nphase diagrams is relatively simple as the resulted phase diagram\nbehaves like a system with one component less. The same is true if\npotentials are sectioned in phase diagrams with both potential and molar\nquantities as the section is along the tie-lines of the fixed\npotentials. As an example, shows the ternary Al-Fe-Si potential and\nmolar phase diagrams sectioned at T=1273K and P=1atm, commonly referred\nto as isothermal section. It is evident that the geometric features of\nboth phase diagrams are identical to those of pure Fe shown in and ,\nrespectively, with one-, two-, and three-phase regions and corresponding\nphase boundaries.\n\nFigure \u2011: Ternary isothermal section of the Al-Fe-Si ternary system at\nT=1273K and P=1atm\n\nOn the other hand, when the phase diagram is sectioned along a molar\nquantity, it would usually not follow a tie-line because phases in\nequilibrium usually have different values for the same molar quantity.\nConsequently, there are no tie-lines inside such phase diagrams in\ngeneral, and any phase regions only show which phases are in equilibrium\nwith each other without any information on the values of molar\nquantities of individual phases.\n\nThis type of sectioning reduces both the dimensionalities of the phase\ndiagram and phase boundary by the same number, but does not alter the\nnumber of phases in the adjacent phase regions. The contact rule, i.e. ,\nthus remains valid and is applicable to phase regions with the\nsame-dimensionality as that of the sectioned phase diagram. Similarly,\nthe Schreinemakers\u2019 rule shown in is valid under the same conditions.\n\nFor example, the two-dimensional phase diagram of the Mg-Al-Zn ternary\nsystem sectioned with one atmospheric pressure and the weight fraction\nof Zn fixed at 0.01 is shown in plotted with temperature and mole\nfraction of Al \\[7\\]. This phase diagram is commonly called isopleth and\nis generated by fixing one potential, the pressure, $P$, changing the\nchemical potentials of Al and Zn to their conjugate molar quantities\nrepresented by weight fractions of Al and Zn, and sectioning at\n$w_{Zn} = 0.01$. From the discussions in Chapter , the phase regions\nwith the number of phases equal to three or fewer, i.e.\n$p \\leq n_{m} + 1 = 3$, have the same-dimensionality as the phase\ndiagram, i.e. two-dimensional in the present case, and the phase\nboundary rule is applicable. The maximum number of phases co-existing at\nequilibrium is given by as following for the present case\n\nEq. \u2011 $p_{\\max} = \\left( c - n_{s} \\right) + 2 = 3 - 1 + 2 = 4$\n\nThis is because introducing molar quantities only increases the\ndimensionality of phase regions and does not change the maximum number\nof co-existing phases.\n\nThe dimensionality of a four-phase region is calculated from as\n\nEq. \u2011\n$\\upsilon_{m} = \\left( c - n_{s} \\right) + 2 - p + n_{m} - n_{ms} = 3 - 1 + 2 - 4 + 2 - 1 = 1$\n\nwhere $n_{ms}$ is the number of sectioned molar quantities. Since the\ndimensionality of a four-phase region is lower than that of the phase\ndiagram, the phase boundary rule cannot be applied directly. Such a\nfour-phase region, liquid+Mg+\u03b3+\u03c6, is shown in between three three-phase\nregions of liquid+Mg+\u03b3, liquid+Mg+\u03c6, and Mg+\u03b3+\u03c6.\n\nFigure \u2011: Isopleth with the weight fraction of Zn fixed at 0.01 of the\nMg-Al-Zn ternary system.\n\nalso displays information on what phases are in equilibrium for a given\nalloy at various temperatures. One example is shown by the dotted\nvertical line marking the weight fraction of Al being 0.09, a widely\nused Mg alloy called AZ91. Various phases are present at different\ntemperature ranges, but the equilibrium phase fractions and phase\ncompositions are not shown in the figure as the tie-lines are not in the\nplane of the phase diagram and have to be calculated at each temperature\nindividually. shows the amount of each phase of the AZ91 alloy as a\nfunction of temperature with the dotted lines depicting the values under\nthe equilibrium condition and the solid lines depicting the values under\nthe so-called Scheil condition assuming no diffusion in solid phases and\ninfinitelyfast diffusion in liquid. Similarly, the composition of each\nphase can also be plotted as shown in .\n\nFigure \u2011: Mole fraction of individual phases under equilibrium (dotted\ncurves) and Scheil (solid curves) conditions in the AZ91 alloy\n\nFigure \u2011: Mass fraction of Al and Zn in the Mg solid solution phase\nunder equilibrium (dotted curves) and Scheil (solid curves) conditions\nin AZ91\n", "metadata": {}}, {"id": "f06dd6a3", "cell_type": "markdown", "source": "### Section of potential phase diagrams\n\nBased on the Gibbs-Duhem equation (see ), a single-phase equilibrium in\na system with more than two independent components has more than three\nindependent potentials. There is no problem in representing them using\nthe mathematical formulas discussed so far, but it is not possible for\nus to visualize graphically the full potential phase diagrams in\nmulti-component systems with more than two independent components. In\nprinciples, there are two options. One option is to project the\nmulti-dimensional potential phase diagram into a two- or three-dimension\ndiagram, and another option is to section the multi-dimensional\npotential phase diagram by fixing the values of some potentials.\n\nThe projection approach is used for one-component systems in Chapter .\nSince a two-phase equilibrium in a one-component potential phase diagram\nis one-dimensional, the projection does not lose any information, and\nthe same is true for a three-phase equilibrium in a one-component\npotential phase diagram. In a binary system, the projections of three-\nand four-phase equilibria do not lose any information, while the\nprojections of two-phase equilibria become two-dimensional and cannot\nretain all the information as the original-dimensionality of these\ntwo-phase equilibria is three. Consequently, sectioning at fixed values\nof some potentials is necessary in order to visualize the phase\nrelations in systems with two or more components. Gibbs phase rule shown\nin and are thus modified to\n\nEq. \u2011 $\\upsilon = c + 2 - p - n_{s} = \\left( c - n_{s} \\right) + 2 - p$\n\nEq. \u2011 $p_{\\max} = \\left( c - n_{s} \\right) + 2$\n\nwhere $n_{s}$ is the number of potentials fixed in sectioning. As can be\nseen in the last part of , the number of sectioning is equivalent to the\nreduction of the effective number of independent components. Therefore,\nany multi-component systems with $n_{s} = c - i$ behave like an\n$i$-component system. The equations presented in Chapter and are thus\ndirectly applicable to multi-component systems with $n_{s} = c - 1$ and\n$n_{s} = c - 2$, respectively.\n\nA common practice in experiments is to fix pressure, temperature or\nchemical potentials of volatile components as they are usually the\nvariables controlled experimentally. In a binary system, the potential\nphase diagram at constant pressure can be represented by any two of the\nthree potentials, i.e. two chemical potentials and temperature, with the\nremaining potential being dependent, and has the identical morphology as\na one-component system. The Gibbs-Duhem equation under such conditions\nbecomes\n\nEq. \u2011 $0 = - SdT - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B}$\n\nThe corresponding two-phase Clausius-Clapeyron equations are written as\n\nEq. \u2011\n$\\frac{dT}{{d\\mu}_{A}} = - \\frac{{\\mathrm{\\Delta}z}_{A}^{\\alpha\\beta}}{\\mathrm{\\Delta}S_{mB}^{\\alpha\\beta}}$\n\nEq. \u2011\n$\\frac{dT}{{d\\mu}_{B}} = - \\frac{{\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}}{\\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}}$\n\nEq. \u2011\n$\\frac{{d\\mu}_{A}}{{d\\mu}_{B}} = - \\frac{\\mathrm{\\Delta}\\left( \\frac{1}{S_{mB}^{}} \\right)^{\\alpha\\beta}}{\\mathrm{\\Delta}\\left( \\frac{1}{S_{mA}^{}} \\right)^{\\alpha\\beta}}$\n\nAs an example, the $T$-$\\mu_{C}$ potential phase diagram for the Fe-C\nbinary system at one atmospheric pressure is shown in .\n\nFigure \u2011: $T$-$\\mu_{C}$ potential phase diagram for the Fe-C binary\nsystem at $P = 1atm$\n\nIn a ternary system, two potentials need to be fixed in order to obtain\ntwo-dimensional potential phase diagrams. When the pressure and the\nchemical potential of one species are fixed, the system behaves like a\nbinary system discussed above. When the system temperature and pressure\nare fixed, the Gibbs-Duhem equation is written as\n\nEq. \u2011 $0 = - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B} - {N_{C}d\\mu}_{C}$\n\nTaking the component A as the dependent element, the two-phase\nClausius-Clapeyron equation is simplified as\n\nEq. \u2011\n$\\frac{{d\\mu}_{B}}{{d\\mu}_{C}} = \\frac{d\\left( \\ln a_{B} \\right)}{d\\left( \\ln a_{C} \\right)} = - \\frac{\\mathrm{\\Delta}z_{C}^{\\alpha\\beta}}{{\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}}$\n\nWhen the two phases in equilibrium are stoichiometric phases, the\ntwo-phase equilibrium is thus a straight line. For example, the Ti-O-Cl\npotential phase diagram at 600\u00b0C and one atmospheric pressure is shown\nin . Since both O and Cl are volatile components, their activities are\nusually represented by their partial pressures with the pure\nO2 and Cl2 gas as their respective reference\nstates at the given temperature and pressure.\n\nFigure \u2011: Ti-O-Cl potential phase diagram at 600\u00b0C and one atmospheric\npressure\n\nFor systems with more than three components, the chemical potentials of\none or more components must be fixed in order to obtain a\ntwo-dimensional potential phase diagram similar to the potential phase\ndiagrams discussed above.\n", "metadata": {}}, {"id": "bd5e4695", "cell_type": "markdown", "source": "### Tie-lines and lever rule\n\nIt is self-evident from to Eq. 2\u20114 that while potentials are homogeneous\nin all phases in a heterogeneous system at equilibrium, the molar\nquantities usually have different values in individual phases. This is\nalso stipulated in various Clausius-Clapeyron equations such as to and\nto . The difference in molar quantities thus increases the\ndimensionality of the phase region by the number of potentials replaced\nby their conjugate molar quantities. The maximum dimensionality of a\nphase region is the dimensionality of the phase diagram under\nconsideration. This thus creates a finite space between phases in\nequilibrium in the phase diagram with some axes being molar quantities.\n\nFor an equilibrium system under constant $T$, $P$ and $N_{i}$, the\npotentials in the system and their conjugate molar quantities in each\nphase are all uniquely defined. In a phase diagram with one or more\npotentials replaced by their conjugate molar quantities, two phases in\nequilibrium in a system with $c$ independent components are connected by\na $c$ dimensional line in a *c*+2 dimensional space or\nits$\\ $*c+1*-dimensional projection as discussed in Chapter . These\nlines are called tie-lines and collectively represent a two-phase\nequilibrium region. For a $k$-phase equilibrium, there are total\n$C_{k}^{2} = \\frac{k(k - 1)}{2}$ tie-lines connecting every two phases\nwith $k - 1\\ $of them independent because the number of independent\ntie-lines increases by one with each new phase added. For the invariant\nequilibrium with zero degree of freedom, the number of phases in\nequilibrium is $c + 2$ shown by containing\n$C_{c + 2}^{2} = \\frac{(c + 2)(c + 1)}{2}$ tie-lines with $c + 1$ of\nthem being independent.\n\nInside the space encapsulated by the tie-lines, the axis variables of\nthe phase diagram (a mixture of potentials and molar quantities) can be\nchanged independently without changing the phases in equilibrium and\ntheir properties. Only the relative amounts of individual phases are\nadjusted accordingly to maintain the conservation of the molar\nquantities in the system specified by the molar quantity axes of the\nphase diagram. The geometric feature circumscribing the space\nencapsulated by the tie-lines no longer represents any phase regions,\nbut a boundary between the neighbouring phase regions. Its\ncharacteristics will be discussed in more details in the next few\nsections. As properties in each phase are homogeneous, the values of\nmolar quantities of a system are simply the sum of individual phases and\ncan be represented by the following equation\n\nEq. \u2011 $A_{m} = \\sum_{\\alpha}^{}{f^{\\alpha}A_{m}^{\\alpha}}$\n\nwhere $A_{m}$ and $A_{m}^{\\alpha}$ represent the values of a molar\nquantity of the system and the $\\alpha$ phase, respectively,\n$f^{\\alpha}$ the mole fraction of the $\\alpha$ phase, and the summation\ngoes over all phases in equilibrium with each other. With\n$\\sum_{\\alpha}^{}f^{\\alpha} = 1$, can be re-arranged into the following\nequation\n\nEq. \u2011\n$\\sum_{\\alpha}^{}{f^{\\alpha}\\left( {A_{m} - A}_{m}^{\\alpha} \\right)} = 0$\n\nis commonly referred as lever rule. For a two-phase equilibrium of\n$\\alpha$ and $\\beta$, it becomes\n\nEq. \u2011\n$f^{\\alpha} = \\frac{A_{m}^{\\beta} - A_{m}}{A_{m}^{\\beta} - A_{m}^{\\alpha}}$\n\nEq. \u2011\n$f^{\\beta} = \\frac{A_{m}^{\\alpha} - A_{m}}{A_{m}^{\\alpha} - A_{m}^{\\beta}}$\n\nFor a phase diagram with the number of axes being\n$n = \\left( c - n_{s} \\right) + 1$, the number of possible axes being\nmolar quantities is thus $k \\leq n$. There are thus $k$ equations\nsimilar to with one for each molar quantity, $A_{mi}$, resulting in the\nfollowing $k + 1$ equations\n\nEq. \u2011\n$\\sum_{\\alpha}^{}{f^{\\alpha}\\left( {A_{mi} - A}_{mi}^{\\alpha} \\right)} = 0$\n\nEq. \u2011 $1 - \\sum_{\\alpha}^{}f^{\\alpha} = 0$\n\nThe summations in and go over the phases in equilibrium, and the amount\nof each phase is obtained by solving these $k + 1$ equations\nsimultaneously along with the equilibrium conditions.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/references/index.ipynb b/psu410/src/references/index.ipynb new file mode 100644 index 0000000..7b7d2c0 --- /dev/null +++ b/psu410/src/references/index.ipynb @@ -0,0 +1 @@ +{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "fc2eb200", "cell_type": "markdown", "source": "# References\n\n1\\. M. Hillert, *Phase Equilibria, Phase Diagrams and Phase\nTransformations,* Cambridge University Press, Cambridge, ed. 2nd, 2007.\n\n2\\. E. 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Bull.*\n**59** (2014) 1619-1623.\n", "metadata": {}}]} \ No newline at end of file From fd17246d0fb709a0026dfacc47abb498bc42ccd0 Mon Sep 17 00:00:00 2001 From: tonyfast Date: Mon, 8 Jan 2024 10:25:14 -0800 Subject: [PATCH 3/9] fix test --- src/nobook/bootstrap.ipynb | 178 +++++++++++++++++-------------------- 1 file changed, 80 insertions(+), 98 deletions(-) diff --git a/src/nobook/bootstrap.ipynb b/src/nobook/bootstrap.ipynb index 9a7384d..591232c 100644 --- a/src/nobook/bootstrap.ipynb +++ b/src/nobook/bootstrap.ipynb @@ -21,11 +21,11 @@ "id": "c4601326-6456-47e8-99c9-ae8c8f59e32b", "metadata": { "execution": { - "iopub.execute_input": "2024-01-08T04:32:10.613639Z", - "iopub.status.busy": "2024-01-08T04:32:10.613413Z", - "iopub.status.idle": "2024-01-08T04:32:11.095633Z", - "shell.execute_reply": "2024-01-08T04:32:11.095038Z", - "shell.execute_reply.started": "2024-01-08T04:32:10.613618Z" + "iopub.execute_input": "2024-01-08T18:23:39.490266Z", + "iopub.status.busy": "2024-01-08T18:23:39.490022Z", + "iopub.status.idle": "2024-01-08T18:23:39.779513Z", + "shell.execute_reply": "2024-01-08T18:23:39.779085Z", + "shell.execute_reply.started": "2024-01-08T18:23:39.490247Z" }, "tags": [] }, @@ -58,11 +58,11 @@ "id": "2f23bf91-3d5d-4bd5-8619-e05ac0df6c53", "metadata": { "execution": { - "iopub.execute_input": "2024-01-08T04:32:11.097354Z", - "iopub.status.busy": "2024-01-08T04:32:11.097071Z", - "iopub.status.idle": "2024-01-08T04:32:11.101711Z", - "shell.execute_reply": "2024-01-08T04:32:11.100541Z", - "shell.execute_reply.started": "2024-01-08T04:32:11.097336Z" + "iopub.execute_input": "2024-01-08T18:23:40.341269Z", + "iopub.status.busy": "2024-01-08T18:23:40.340600Z", + "iopub.status.idle": "2024-01-08T18:23:40.345868Z", + "shell.execute_reply": "2024-01-08T18:23:40.345198Z", + "shell.execute_reply.started": "2024-01-08T18:23:40.341229Z" }, "tags": [] }, @@ -96,11 +96,11 @@ "id": "bd674c8e-4a31-4544-b6b9-0aa2573385c8", "metadata": { "execution": { - "iopub.execute_input": "2024-01-08T04:32:11.103406Z", - "iopub.status.busy": "2024-01-08T04:32:11.102966Z", - "iopub.status.idle": "2024-01-08T04:32:11.113313Z", - "shell.execute_reply": "2024-01-08T04:32:11.112714Z", - "shell.execute_reply.started": "2024-01-08T04:32:11.103383Z" + "iopub.execute_input": "2024-01-08T18:23:41.060965Z", + "iopub.status.busy": "2024-01-08T18:23:41.060528Z", + "iopub.status.idle": "2024-01-08T18:23:41.066993Z", + "shell.execute_reply": "2024-01-08T18:23:41.066040Z", + "shell.execute_reply.started": "2024-01-08T18:23:41.060920Z" }, "tags": [] }, @@ -123,11 +123,11 @@ "id": "345b47f7-6fbf-40f5-a64f-5646db86dd2b", "metadata": { "execution": { - "iopub.execute_input": "2024-01-08T04:32:11.114386Z", - "iopub.status.busy": "2024-01-08T04:32:11.114084Z", - "iopub.status.idle": "2024-01-08T04:32:11.118742Z", - "shell.execute_reply": "2024-01-08T04:32:11.117965Z", - "shell.execute_reply.started": "2024-01-08T04:32:11.114364Z" + "iopub.execute_input": "2024-01-08T18:23:41.380582Z", + "iopub.status.busy": "2024-01-08T18:23:41.380109Z", + "iopub.status.idle": "2024-01-08T18:23:41.387025Z", + "shell.execute_reply": "2024-01-08T18:23:41.386298Z", + "shell.execute_reply.started": "2024-01-08T18:23:41.380541Z" }, "tags": [] }, @@ -157,11 +157,11 @@ "id": "ff516019-bebf-4610-9f16-5dc90d0b8da7", "metadata": { "execution": { - "iopub.execute_input": "2024-01-08T04:32:11.119935Z", - "iopub.status.busy": "2024-01-08T04:32:11.119620Z", - "iopub.status.idle": "2024-01-08T04:32:11.123408Z", - "shell.execute_reply": "2024-01-08T04:32:11.122958Z", - "shell.execute_reply.started": "2024-01-08T04:32:11.119915Z" + "iopub.execute_input": "2024-01-08T18:23:42.136240Z", + "iopub.status.busy": "2024-01-08T18:23:42.135848Z", + "iopub.status.idle": "2024-01-08T18:23:42.140652Z", + "shell.execute_reply": "2024-01-08T18:23:42.140104Z", + "shell.execute_reply.started": "2024-01-08T18:23:42.136205Z" }, "tags": [] }, @@ -186,11 +186,11 @@ "id": "d4e75784-0945-47e0-9260-4116ca2e0e5d", "metadata": { "execution": { - "iopub.execute_input": "2024-01-08T04:32:11.124996Z", - "iopub.status.busy": "2024-01-08T04:32:11.124539Z", - "iopub.status.idle": "2024-01-08T04:32:11.128171Z", - "shell.execute_reply": "2024-01-08T04:32:11.127491Z", - "shell.execute_reply.started": "2024-01-08T04:32:11.124980Z" + "iopub.execute_input": "2024-01-08T18:23:42.799746Z", + "iopub.status.busy": "2024-01-08T18:23:42.799588Z", + "iopub.status.idle": "2024-01-08T18:23:42.802160Z", + "shell.execute_reply": "2024-01-08T18:23:42.801870Z", + "shell.execute_reply.started": "2024-01-08T18:23:42.799733Z" }, "tags": [] }, @@ -216,11 +216,11 @@ "id": "dfc37237-2240-4585-949d-0dbc13d1e7cc", "metadata": { "execution": { - "iopub.execute_input": "2024-01-08T04:32:11.130135Z", - "iopub.status.busy": "2024-01-08T04:32:11.129877Z", - "iopub.status.idle": "2024-01-08T04:32:11.134681Z", - "shell.execute_reply": "2024-01-08T04:32:11.134109Z", - "shell.execute_reply.started": "2024-01-08T04:32:11.130115Z" + "iopub.execute_input": "2024-01-08T18:23:43.582571Z", + "iopub.status.busy": "2024-01-08T18:23:43.582417Z", + "iopub.status.idle": "2024-01-08T18:23:43.585729Z", + "shell.execute_reply": "2024-01-08T18:23:43.585405Z", + "shell.execute_reply.started": "2024-01-08T18:23:43.582559Z" }, "tags": [] }, @@ -244,11 +244,11 @@ "id": "8f4bc119-c81b-40aa-9205-2f6b2f3bb63e", "metadata": { "execution": { - "iopub.execute_input": "2024-01-08T04:32:11.138093Z", - "iopub.status.busy": "2024-01-08T04:32:11.137666Z", - "iopub.status.idle": "2024-01-08T04:32:11.144323Z", - "shell.execute_reply": "2024-01-08T04:32:11.143572Z", - "shell.execute_reply.started": "2024-01-08T04:32:11.138067Z" + "iopub.execute_input": "2024-01-08T18:23:44.109666Z", + "iopub.status.busy": "2024-01-08T18:23:44.109230Z", + "iopub.status.idle": "2024-01-08T18:23:44.115130Z", + "shell.execute_reply": "2024-01-08T18:23:44.114394Z", + "shell.execute_reply.started": "2024-01-08T18:23:44.109629Z" }, "tags": [] }, @@ -266,11 +266,11 @@ "id": "225460f7-f090-4f8b-9ff5-b8adc1459ffa", "metadata": { "execution": { - "iopub.execute_input": "2024-01-08T04:32:11.145386Z", - "iopub.status.busy": "2024-01-08T04:32:11.145086Z", - "iopub.status.idle": "2024-01-08T04:32:11.153066Z", - "shell.execute_reply": "2024-01-08T04:32:11.152322Z", - "shell.execute_reply.started": "2024-01-08T04:32:11.145367Z" + "iopub.execute_input": "2024-01-08T18:23:44.460637Z", + "iopub.status.busy": "2024-01-08T18:23:44.460242Z", + "iopub.status.idle": "2024-01-08T18:23:44.466824Z", + "shell.execute_reply": "2024-01-08T18:23:44.465680Z", + "shell.execute_reply.started": "2024-01-08T18:23:44.460603Z" }, "tags": [] }, @@ -288,11 +288,11 @@ "id": "ec3b7821-e291-4bbb-b651-82f36a925d43", "metadata": { "execution": { - "iopub.execute_input": "2024-01-08T04:32:11.154163Z", - "iopub.status.busy": "2024-01-08T04:32:11.153916Z", - "iopub.status.idle": "2024-01-08T04:32:35.998123Z", - "shell.execute_reply": "2024-01-08T04:32:35.997791Z", - "shell.execute_reply.started": "2024-01-08T04:32:11.154142Z" + "iopub.execute_input": "2024-01-08T18:23:44.864971Z", + "iopub.status.busy": "2024-01-08T18:23:44.864690Z", + "iopub.status.idle": "2024-01-08T18:23:58.876212Z", + "shell.execute_reply": "2024-01-08T18:23:58.875832Z", + "shell.execute_reply.started": "2024-01-08T18:23:44.864946Z" }, "tags": [] }, @@ -302,28 +302,18 @@ "output_type": "stream", "text": [ "test_number_of_sections (__main__.Tests.test_number_of_sections)\n", - "the table of contents, 9 primary chapters, and the references are captured ... FAIL\n", + "the table of contents, 9 primary chapters, and the references are captured ... ok\n", "\n", - "======================================================================\n", - "FAIL: test_number_of_sections (__main__.Tests.test_number_of_sections)\n", - "the table of contents, 9 primary chapters, and the references are captured\n", "----------------------------------------------------------------------\n", - "Traceback (most recent call last):\n", - " File \"/tmp/ipykernel_431217/4110034353.py\", line 13, in test_number_of_sections\n", - " assert len(self.md.index.get_level_values(\"heading\").unique()) == 10\n", - " ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^\n", - "AssertionError\n", + "Ran 1 test in 14.002s\n", "\n", - "----------------------------------------------------------------------\n", - "Ran 1 test in 24.823s\n", - "\n", - "FAILED (failures=1)\n" + "OK\n" ] }, { "data": { "text/plain": [ - "" + "" ] }, "execution_count": 10, @@ -344,7 +334,7 @@ " \"\"\"the table of contents, 9 primary chapters, and the references are captured\"\"\"\n", " # all of the main headings are captured as Heading1s\n", " assert self.sections.style_id.eq(\"Heading1\").sum() == 10\n", - " assert len(self.md.index.get_level_values(\"heading\").unique()) == 10\n", + " assert len(self.md.index.get_level_values(\"chapter\").unique()) == 10\n", " \n", " INTERACTIVE and unittest.main(argv=[\"discover\"], verbosity=2, exit=False)" ] @@ -355,11 +345,11 @@ "id": "94532190-795c-49c7-8540-5e01bbdb7126", "metadata": { "execution": { - "iopub.execute_input": "2024-01-08T04:32:35.998896Z", - "iopub.status.busy": "2024-01-08T04:32:35.998601Z", - "iopub.status.idle": "2024-01-08T04:32:36.001602Z", - "shell.execute_reply": "2024-01-08T04:32:36.001246Z", - "shell.execute_reply.started": "2024-01-08T04:32:35.998880Z" + "iopub.execute_input": "2024-01-08T18:23:58.876959Z", + "iopub.status.busy": "2024-01-08T18:23:58.876758Z", + "iopub.status.idle": "2024-01-08T18:23:58.879559Z", + "shell.execute_reply": "2024-01-08T18:23:58.879227Z", + "shell.execute_reply.started": "2024-01-08T18:23:58.876929Z" }, "tags": [] }, @@ -376,11 +366,11 @@ "id": "03e3aca3-6205-49d3-b02c-a8acaee48457", "metadata": { "execution": { - "iopub.execute_input": "2024-01-08T04:32:36.002208Z", - "iopub.status.busy": "2024-01-08T04:32:36.002065Z", - "iopub.status.idle": "2024-01-08T04:32:36.014394Z", - "shell.execute_reply": "2024-01-08T04:32:36.014046Z", - "shell.execute_reply.started": "2024-01-08T04:32:36.002194Z" + "iopub.execute_input": "2024-01-08T18:23:58.880268Z", + "iopub.status.busy": "2024-01-08T18:23:58.880060Z", + "iopub.status.idle": "2024-01-08T18:23:58.892271Z", + "shell.execute_reply": "2024-01-08T18:23:58.891848Z", + "shell.execute_reply.started": "2024-01-08T18:23:58.880253Z" }, "tags": [] }, @@ -414,11 +404,11 @@ "id": "636a096d-aad3-4af3-86ae-15ee357fc90c", "metadata": { "execution": { - "iopub.execute_input": "2024-01-08T04:32:36.015044Z", - "iopub.status.busy": "2024-01-08T04:32:36.014900Z", - "iopub.status.idle": "2024-01-08T04:32:36.017546Z", - "shell.execute_reply": "2024-01-08T04:32:36.017248Z", - "shell.execute_reply.started": "2024-01-08T04:32:36.015031Z" + "iopub.execute_input": "2024-01-08T18:23:58.893274Z", + "iopub.status.busy": "2024-01-08T18:23:58.892953Z", + "iopub.status.idle": "2024-01-08T18:23:58.895739Z", + "shell.execute_reply": "2024-01-08T18:23:58.895391Z", + "shell.execute_reply.started": "2024-01-08T18:23:58.893256Z" }, "tags": [] }, @@ -436,11 +426,11 @@ "id": "1f7b0f6b-6b56-48a5-8562-baa57317fa09", "metadata": { "execution": { - "iopub.execute_input": "2024-01-08T04:32:36.018049Z", - "iopub.status.busy": "2024-01-08T04:32:36.017900Z", - "iopub.status.idle": "2024-01-08T04:32:36.157118Z", - "shell.execute_reply": "2024-01-08T04:32:36.156771Z", - "shell.execute_reply.started": "2024-01-08T04:32:36.018041Z" + "iopub.execute_input": "2024-01-08T18:24:10.529538Z", + "iopub.status.busy": "2024-01-08T18:24:10.528957Z", + "iopub.status.idle": "2024-01-08T18:24:10.672463Z", + "shell.execute_reply": "2024-01-08T18:24:10.672025Z", + "shell.execute_reply.started": "2024-01-08T18:24:10.529483Z" }, "tags": [] }, @@ -518,11 +508,11 @@ "id": "ae617c09-6440-48a7-a015-2c60896fb1a1", "metadata": { "execution": { - "iopub.execute_input": "2024-01-08T04:32:36.157946Z", - "iopub.status.busy": "2024-01-08T04:32:36.157626Z", - "iopub.status.idle": "2024-01-08T04:32:36.169388Z", - "shell.execute_reply": "2024-01-08T04:32:36.168922Z", - "shell.execute_reply.started": "2024-01-08T04:32:36.157927Z" + "iopub.execute_input": "2024-01-08T18:24:11.246794Z", + "iopub.status.busy": "2024-01-08T18:24:11.246598Z", + "iopub.status.idle": "2024-01-08T18:24:11.254289Z", + "shell.execute_reply": "2024-01-08T18:24:11.254032Z", + "shell.execute_reply.started": "2024-01-08T18:24:11.246780Z" }, "tags": [] }, @@ -1829,14 +1819,6 @@ " if INTERACTIVE:\n", " display(*md[md.str.contains(\"\\$\")].iloc[:10].apply(Markdown))" ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "dc1a963d-8365-42c8-b50b-ff4e1918ef02", - "metadata": {}, - "outputs": [], - "source": [] } ], "metadata": { From 5ac069ba90c4bad2f08072be018d95ed641d525c Mon Sep 17 00:00:00 2001 From: tonyfast Date: Mon, 8 Jan 2024 19:59:14 -0800 Subject: [PATCH 4/9] move the location of the generated contents --- ...lingham_diagram_and_buffered_systems.ipynb | 1 - .../index.ipynb | 1 - ...ential_and_integrated_driving_forces.ipynb | 1 - ...te_and_chemical_transport_reactions_.ipynb | 1 - .../trends_of_entropies_of_reactions.ipynb | 1 - .../application_examples.ipynb | 1 - ...queous_solution_and_pourbaix_diagram.ipynb | 1 - ...erence_states_of_electrolyte_species.ipynb | 1 - ...mical_cells_and_half_cell_potentials.ipynb | 1 - ...ctions_and_electrochemical_reactions.ipynb | 1 - .../index.ipynb | 1 - .../importance_of_lattice_stability.ipynb | 1 - .../index.ipynb | 1 - .../modeling_of_pure_elements.ipynb | 1 - .../modeling_of_random_solution_phases.ipynb | 1 - ...ution_phases_with_longrange_ordering.ipynb | 1 - .../modeling_of_stoichiometric_phases.ipynb | 1 - .../application_to_cerium.ipynb | 1 - .../application_to_fept.ipynb | 1 - .../concept_of_materials_genome.ipynb | 1 - .../index.ipynb | 1 - ...s_model_applied_to_thermal_expansion.ipynb | 1 - .../additional_methods.ipynb | 1 - .../index.ipynb | 1 - ...se_equilibrium_data_from_experiments.ipynb | 1 - .../thermodynamic_data_from_experiments.ipynb | 1 - .../bornoppenheimer_approximation.ipynb | 1 - ...ples_approaches_to_disordered_alloys.ipynb | 1 - ...ciples_formulation_of_thermodynamics.ipynb | 1 - .../index.ipynb | 1 - .../lattice_dynamics.ipynb | 1 - .../nickel_as_the_prototype.ipynb | 1 - ...m_theory_for_the_motion_of_electrons.ipynb | 1 - .../binary_random_solutions.ipynb | 1 - ...lectric_contributions_to_free_energy.ipynb | 1 - psu410/src/gibbs_energy_function/index.ipynb | 1 - .../phases_with_fixed_compositions.ipynb | 1 - ...riable_compositions_random_solutions.ipynb | 1 - ...compositions_solutions_with_ordering.ipynb | 1 - ...polymer_solutions_and_polymer_blends.ipynb | 1 - ...odynamics_and_equilibrium_conditions.ipynb | 1 - ...st_and_second_laws_of_thermodynamics.ipynb | 1 - .../gibbsduhem_equation.ipynb | 1 - psu410/src/laws_of_thermodynamics/index.ipynb | 1 - ..._at_equilibrium_and_property_anomaly.ipynb | 1 - .../general_condition_of_equilibrium.ipynb | 1 - .../gibbs_phase_rule.ipynb | 1 - .../index.ipynb | 1 - .../molar_phase_diagrams.ipynb | 1 - .../potential_phase_diagrams.ipynb | 1 - psu410/src/references/index.ipynb | 1 - src/.gitignore | 2 + src/nobook/bootstrap.ipynb | 422 +++++++---- ...lingham_diagram_and_buffered_systems.ipynb | 162 ++++ .../index.ipynb | 28 + ...ential_and_integrated_driving_forces.ipynb | 102 +++ ...te_and_chemical_transport_reactions_.ipynb | 159 ++++ .../trends_of_entropies_of_reactions.ipynb | 84 +++ .../application_examples.ipynb | 18 + ...queous_solution_and_pourbaix_diagram.ipynb | 263 +++++++ ...erence_states_of_electrolyte_species.ipynb | 349 +++++++++ ...mical_cells_and_half_cell_potentials.ipynb | 36 + ...ctions_and_electrochemical_reactions.ipynb | 465 ++++++++++++ .../index.ipynb | 33 + .../importance_of_lattice_stability.ipynb | 121 +++ .../index.ipynb | 59 ++ .../modeling_of_pure_elements.ipynb | 76 ++ .../modeling_of_random_solution_phases.ipynb | 63 ++ ...ution_phases_with_longrange_ordering.ipynb | 170 +++++ .../modeling_of_stoichiometric_phases.ipynb | 42 ++ .../application_to_cerium.ipynb | 209 +++++ .../application_to_fept.ipynb | 131 ++++ .../concept_of_materials_genome.ipynb | 66 ++ .../index.ipynb | 32 + ...s_model_applied_to_thermal_expansion.ipynb | 127 ++++ .../additional_methods.ipynb | 268 +++++++ .../index.ipynb | 34 + ...se_equilibrium_data_from_experiments.ipynb | 62 ++ .../thermodynamic_data_from_experiments.ipynb | 67 ++ .../bornoppenheimer_approximation.ipynb | 388 ++++++++++ ...ples_approaches_to_disordered_alloys.ipynb | 56 ++ ...ciples_formulation_of_thermodynamics.ipynb | 356 +++++++++ .../index.ipynb | 60 ++ .../lattice_dynamics.ipynb | 189 +++++ .../nickel_as_the_prototype.ipynb | 257 +++++++ ...m_theory_for_the_motion_of_electrons.ipynb | 405 ++++++++++ .../binary_random_solutions.ipynb | 164 ++++ ...lectric_contributions_to_free_energy.ipynb | 131 ++++ .../src/gibbs_energy_function/index.ipynb | 215 ++++++ .../phases_with_fixed_compositions.ipynb | 279 +++++++ ...riable_compositions_random_solutions.ipynb | 123 +++ ...compositions_solutions_with_ordering.ipynb | 15 + ...polymer_solutions_and_polymer_blends.ipynb | 713 ++++++++++++++++++ ...odynamics_and_equilibrium_conditions.ipynb | 229 ++++++ ...st_and_second_laws_of_thermodynamics.ipynb | 128 ++++ .../gibbsduhem_equation.ipynb | 91 +++ .../src/laws_of_thermodynamics/index.ipynb | 15 + ..._at_equilibrium_and_property_anomaly.ipynb | 192 +++++ .../general_condition_of_equilibrium.ipynb | 123 +++ .../gibbs_phase_rule.ipynb | 64 ++ .../index.ipynb | 15 + .../molar_phase_diagrams.ipynb | 360 +++++++++ .../potential_phase_diagrams.ipynb | 678 +++++++++++++++++ src/psu410/src/references/index.ipynb | 274 +++++++ 104 files changed, 9010 insertions(+), 211 deletions(-) delete mode 100644 psu410/src/applications_to_chemical_reactions/ellingham_diagram_and_buffered_systems.ipynb delete mode 100644 psu410/src/applications_to_chemical_reactions/index.ipynb delete mode 100644 psu410/src/applications_to_chemical_reactions/internal_process_and_differential_and_integrated_driving_forces.ipynb delete mode 100644 psu410/src/applications_to_chemical_reactions/maximum_reaction_rate_and_chemical_transport_reactions_.ipynb delete mode 100644 psu410/src/applications_to_chemical_reactions/trends_of_entropies_of_reactions.ipynb delete mode 100644 psu410/src/applications_to_electrochemical_systems/application_examples.ipynb delete mode 100644 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"8610ccff", "cell_type": "markdown", "source": "## Ellingham diagram and buffered systems\n\nOne type of chemical reactions is between a pure element in liquid or\nsolid states and its oxides involving one mole of oxygen gas, i.e.\n\n*Eq. 7\u20117* $\\frac{4}{v_{M}}M + O_{2}(gas) = M_{4/v_{M}}O_{2}$\n\nwith $v_{M}$ being the valence of the element $M$ in the oxide. Taking\nthe pure $M$ and the gaseous $O_{2}$ at the reaction temperature and one\natmospheric pressure as their respective reference states, both\nactivities of the pure $M$ solid or liquid phase and its oxide are\nunity, and the activity of $O_{2}$ equals to its partial pressure in an\nideal gas. becomes\n\n*Eq. 7\u20118*\n$RTlnP_{O_{2}} = \\mathrm{\\Delta}_{\\ }^{0}G = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S$\n\nBased on , one can plot $\\mathrm{\\Delta}_{\\ }^{0}G$ as a function of\ntemperature for various oxidation reactions, which is called Ellingham\ndiagram. The intercept at $T = 0K$ is given by\n$\\mathrm{\\Delta}_{\\ }^{0}H$, and the slope is represented by\n$- \\mathrm{\\Delta}_{\\ }^{0}S$, depicted by the following equation\n\n*Eq. 7\u20119*\n$\\mathrm{\\Delta}_{\\ }^{0}S = S_{M_{4/v_{M}}O_{2}} - S_{O_{2}} - \\frac{4}{v_{M}}S_{M}$\n\nSince the entropy of one mole of $O_{2}$ gas is significantly larger\nthan those of the pure element and its oxide when they are in solid or\nliquid states and the entropy difference between the pure element and\nits oxide, the entropy of reaction is thus dominated by the reduction of\nthe entropy by one mole of $O_{2}$ gas. Consequently, the entropies of\nreaction are approximately the same for most reactions when the pure\nelements and their oxides are solid as seen in where most lines have\nsimilar slopes.\n\nFigure \u2011: Ellingham Diagram for a number of metal-oxide systems.\n\nFor a chemical reaction on the Ellingham diagram at a given temperature,\n$\\mathrm{\\Delta}_{\\ }^{0}G$ can be read from the y-axis of the diagram,\nand the equilibrium partial pressure of $O_{2}$ gas can be calculated\nusing . Alternatively, one can plot the left part of for a given\n$P_{O_{2}}$ as a function of temperature on the Ellingham diagram, i.e.\n\n*Eq. 7\u201110* $\\mathrm{\\Delta}_{\\ }^{0}{G =}RTlnP_{O_{2}}$\n\nThis results in straight lines, representing iso-partial-pressure lines\nof $O_{2}$, with the intercept being zero at $T = 0K$ and slopes of\n$RlnP_{O_{2}}$, which are negative for $P_{O_{2}}$ lower than one\natmospheric pressure (1atm). The values of $P_{O_{2}}$ are marked on the\nsecondary axis on the right of the Ellingham diagram. The intersection\nof the isoactivitiy line and the equilibrium line of the chemical\nreaction thus gives the relation of equilibrium temperature and\nequilibrium partial pressure of $O_{2}$. This is demonstrated in .\n\nFigure \u2011: Intersection of iso-partial-pressure lines of $O_{2}$ and\nequilibrium lines in the Ellingham diagram\n\nFor each chemical reaction in the Ellingham diagram, the three phases\nare in equilibrium on the line represented by , i.e. metal, metal\noxides, and O2 gas. For conditions above the line, the value\nof $P_{O_{2}}$ is larger than its equilibrium value, and the metal will\nbe oxidized. For conditions below the line, the value of $P_{O_{2}}$ is\nlower than its equilibrium value, and the metal oxide will be reduced.\nTherefore, the metal oxides in the upper part of the Ellingham diagram\ncan be reduced by the metals in the lower part of the diagram, and vice\nversus, metals in the lower part of the diagram can be oxidized by the\nmetal oxides in the upper part of the diagram. For example, in , Ca can\nreduce all oxides, and Cu2O is the least stable oxide.\n\nFrom the above Ellingham diagram, it is noted that equilibrium partial\npressures of $O_{2}$ are very low for most chemical reactions with many\nof them lower than $10^{- 12}\\ atm$. One approach to obtain such a low\npressure is to use auxiliary reactions containing $O_{2}$ that are easy\nto control and are independent of the equilibrium system of interest\nexcept the sharing of the oxygen partial pressure. Two common auxiliary\nreactions are the $H_{2}$/$H_{2}O$ and $CO$/$CO_{2}$ mixtures. For the\n$H_{2}$/$H_{2}O$ mixture, the chemical reaction is\n\n*Eq. 7\u201111* $2H_{2}(gas) + O_{2}(gas) = 2H_{2}O(gas)$\n\nThe equilibrium oxygen partial pressure is obtained as\n\n*Eq. 7\u201112*\n$- RTln\\left\\{ {\\frac{1}{P_{O_{2}}}\\left( \\frac{P_{H_{2}O}}{P_{H_{2}}} \\right)}^{2} \\right\\} = \\mathrm{\\Delta}_{\\ }^{0}G = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S = - 498488\\ + 112.972T\\ (J)$\n\nwhere the $\\mathrm{\\Delta}_{\\ }^{0}H$ and $\\mathrm{\\Delta}_{\\ }^{0}S$\nare taken from the substance thermodynamic database (SSUB4) compiled by\nScientific Group Thermodata Europe (SGTE) \\[59\\], which are slightly\ndependent on temperature, and the values in are evaluated at 1273K using\nThermo-Calc \\[60\\]. At any given temperature, one has the following\nrelation\n\n*Eq. 7\u201113*\n$RTlnP_{O_{2}} = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S - 2RTln\\frac{P_{H_{2}}}{P_{H_{2}O}} = - 498488 + \\left( 112.972 - 2Rln\\frac{P_{H_{2}}}{P_{H_{2}O}} \\right)T$\n\nIts intercept at $T = 0K$ is given by\n$\\mathrm{\\Delta}_{\\ }^{0}H = - 498,488\\ (J)$, and the slope by\n$- \\mathrm{\\Delta}_{\\ }^{0}S - 2Rln\\left( \\frac{P_{H_{2}}}{P_{H_{2}O}} \\right) = 112.972 - 2Rln\\left( \\frac{P_{H_{2}}}{P_{H_{2}O}} \\right)$.\nThe values of the $\\frac{P_{H_{2}}}{P_{H_{2}O}}$ ratio are marked on\nanother secondary axis on the right of the Ellingham diagram. The\nintersection of the iso-partial-pressure-ratio line and the equilibrium\nline of the chemical reaction gives the relation of equilibrium\ntemperature and equilibrium partial pressure ratio\n$\\frac{P_{H_{2}}}{P_{H_{2}O}}$ for desired $P_{O_{2}}$ of the chemical\nequilibrium.\n\nFor the $CO$/$CO_{2}$mixture, the chemical reaction is shown below\n\n*Eq. 7\u201114* $2CO(gas) + O_{2}(gas) = 2CO_{2}(gas)$\n\nSimilar, from the SSUB database, the following equation is obtained\n\n*Eq. 7\u201115*\n$- RTln\\left\\{ {\\frac{1}{P_{O_{2}}}\\left( \\frac{P_{{CO}_{2}}}{P_{CO}} \\right)}^{2} \\right\\} = \\mathrm{\\Delta}_{\\ }^{0}G = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S = - 562,927 - 172.020T\\ (J)$\n\nwith $\\mathrm{\\Delta}_{\\ }^{0}H$ and $\\mathrm{\\Delta}_{\\ }^{0}S$\ncalculated at 1273K, which are also slightly temperature-dependent as in\nthe case of the $H_{2}$/$H_{2}O$ mixture. The iso-partial-pressure-ratio\nlines are written as\n\n*Eq. 7\u201116*\n$RTlnP_{O_{2}} = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S - 2RTln\\frac{P_{CO}}{P_{{CO}_{2}}} = - 562,927 + \\left( 172.020 - 2Rln\\frac{P_{CO}}{P_{{CO}_{2}}} \\right)T\\ $\n\nIts intercept at $T = 0K$ is given by\n$\\mathrm{\\Delta}_{\\ }^{0}H = - 562,927\\ (J)$, and the slope by\n$172.020 - - 2Rln\\left( \\frac{P_{CO}}{P_{{CO}_{2}}} \\right)$. The values\nof the $\\frac{P_{CO}}{P_{{CO}_{2}}}$ ratio are marked on the third\nsecondary axis on the right of the Ellingham diagram. The intersection\nof the iso-partial-pressure-ratio line and the equilibrium line of the\nchemical reaction gives the relation of equilibrium temperature and\nequilibrium partial pressure ratio $\\frac{P_{CO}}{P_{{CO}_{2}}}$ for\ndesired $P_{O_{2}}$ of the chemical equilibrium.\n\nTherefore, one can use a mixture of the $H_{2}$/$H_{2}O$ or\n$CO$/$CO_{2}$ to obtain the desired low $P_{O_{2}}$ values using and or\ncalculating from the SSUB database. For example,\n$P_{O_{2}} = 10^{- 15}\\ atm$ at 1273K, the calculated values from the\nSSUB database are $\\frac{P_{H_{2}}}{P_{H_{2}O}} \\approx 1.67$ and\n$\\frac{P_{CO}}{P_{{CO}_{2}}} \\approx 2.78$, respectively, in which the\ntemperature dependences of $\\mathrm{\\Delta}_{\\ }^{0}H$ and\n$\\mathrm{\\Delta}_{\\ }^{0}S$ and the many more gaseous species in the gas\nphase are taken into account. On the other hand, the reading from the\nEllingham diagram gives $\\frac{P_{H_{2}}}{P_{H_{2}O}} \\approx 2.0$ and\n$\\frac{P_{CO}}{P_{{CO}_{2}}} \\approx 2.4$, and the agreement with the\nmore accurate calculations above is remarkable keeping in mind the\nuncertainties in graphically drawing the lines and reading both values\nin the logarithmic scales from the diagram, indicating the robustness of\nthe Ellingham diagram.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/applications_to_chemical_reactions/index.ipynb b/psu410/src/applications_to_chemical_reactions/index.ipynb deleted file mode 100644 index fb6fae2..0000000 --- a/psu410/src/applications_to_chemical_reactions/index.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "67f02270", "cell_type": "markdown", "source": "# Applications to chemical reactions\n\nA chemical reaction can be viewed as a framework dividing a system into\ntwo closed subsystems: reactants and products. The phases and species of\nreactants and products are selected from possible phases and species\nthat may form from the independent components of the system. A chemical\nreaction can thus be considered as an internal process to transfer heat\nand work between the two subsystems of reactants and products. It is\nevident that this subset of phases and species only represents partial\nequilibrium information of the system under given external conditions,\nand more stable equilibrium states may exist by including more phases\nand species with the global equilibrium depicted by phase diagrams\ndiscussed in previous chapters with all known phases and species\nincluded.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/applications_to_chemical_reactions/internal_process_and_differential_and_integrated_driving_forces.ipynb b/psu410/src/applications_to_chemical_reactions/internal_process_and_differential_and_integrated_driving_forces.ipynb deleted file mode 100644 index b62cd0a..0000000 --- a/psu410/src/applications_to_chemical_reactions/internal_process_and_differential_and_integrated_driving_forces.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "b3c62995", "cell_type": "markdown", "source": "## Internal process and differential and integrated driving forces\n\nThe driving force for an internal process can be defined as following\nusing $U$, $H$, $F$, or $G$ as discussed in Chapter and depending on\nwhat system variables are kept constant\n\n*Eq. 7\u20111*\n$- D = \\left( \\frac{\\partial U}{\\partial\\xi} \\right)_{S,\\ V,N_{i}\\ } = \\left( \\frac{\\partial H}{\\partial\\xi} \\right)_{S,\\ P,N_{i}\\ } = \\left( \\frac{\\partial F}{\\partial\\xi} \\right)_{T,\\ V,N_{i}\\ } = \\left( \\frac{\\partial G}{\\partial\\xi} \\right)_{T,P,N_{i}\\ }$\n\nThis can be termed as differential driving force as it relates the\nderivative of energy with respect to an internal process so the change\nis infinitesimally small and does not affect the properties of the\nsystem significantly. For a system under constant $T$ and $P$, let us\nconsider an internal process for forming a new phase $\\alpha$ with the\ncomposition $x_{i}^{\\alpha}$ and Gibbs energy\n$G_{m}^{\\alpha}\\left( x_{i}^{\\alpha} \\right)$. The differential driving\nforce for such an internal process can thus be defined as\n\n*Eq. 7\u20112*\n$- D = G_{m}^{\\alpha}\\left( x_{i}^{\\alpha} \\right) - \\sum_{i}^{}x_{i}^{\\alpha}\\mu_{i} = \\sum_{i}^{}x_{i}^{\\alpha}\\left( \\mu_{i}^{\\alpha} - \\mu_{i} \\right)$\n\nwhere $\\mu_{i}$ is the chemical potential of component *i* in the\nsystem. This may also be called as nucleation driving force as if the\n$\\alpha$ phase is nucleating in the system. As discussed in Chapter ,\nchemical potentials are the intercepts on the Gibbs energy axis by the\ntangent plane of Gibbs energy. thus represents the distance between the\ntangent planes of the original system and the $\\alpha$ phase at the\ncomposition of the $\\alpha$ phase. Evidently, this distance is at its\nmaximum when the two tangent planes are parallel to each other, i.e. the\ncommonly called parallel tangent construction when evaluating nucleation\ndriving force.\n\nThe situation is different for chemical reactions where the amount of\neach component in reactants is the same as that in products, i.e. there\nis a mass balance between reactants and products. The driving force for\na chemical reaction is defined by the Gibbs energy difference between\nthe reactants and products as if all the reactants are transferred to\nthe products. This driving force may thus be called integrated driving\nforce as it describes the energy difference of a system under two\ndifferent states, i.e.\n\n*Eq. 7\u20113*\n$- \\int_{}^{}{Dd\\xi} = \\mathrm{\\Delta}G = \\sum_{p}^{}{n_{p}G_{p}} - \\sum_{r}^{}{n_{r}G_{r}}$\n\nwhere superscripts $p$ and $r$ denote products and reactants, $n_{p}$\nand $n_{r}$ their corresponding moles, $G_{p}$and $G_{r}$ the Gibbs\nenergies in per mole of formula of their respective stoichiometries as\nwritten in the chemical reaction. Conventionally, the products and\nreactants are represented by species or stoichiometric compounds rather\nthan individual phases. This is particularly evident when various\ngaseous species are considered in a chemical reaction. Consequently,\n$G_{p}$and $G_{r}$ in represent the chemical potentials of product and\nreactant species. For species with a fixed composition, its chemical\npotential is the same as its Gibbs energy as shown by . For a species in\na solution phase, its chemical potential is related to its activity as\nshown in . can thus be further written as\n\n*Eq. 7\u20114*\n$\\mathrm{\\Delta}G = \\sum_{p}^{}{n_{p}_{\\ }^{0}G_{p}} - \\sum_{r}^{}{n_{r}_{\\ }^{0}G_{r}} + RTln\\frac{\\prod_{p}^{}\\left( a_{p} \\right)^{n_{p}}}{\\prod_{r}^{}\\left( a_{r} \\right)^{n_{r}}} = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{\\prod_{p}^{}\\left( a_{p} \\right)^{n_{p}}}{\\prod_{r}^{}\\left( a_{r} \\right)^{n_{r}}}$\n\nIt is evident from that for stoichiometric phases in a chemical\nreaction, their activities equal to one. At equilibrium, the integrated\ndriving force becomes zero, i.e. $\\mathrm{\\Delta}G = 0$, and is\nre-arranged to become\n\n*Eq. 7\u20115*\n$- RTln\\frac{\\prod_{p}^{}\\left( a_{p} \\right)^{n_{p}}}{\\prod_{r}^{}\\left( a_{r} \\right)^{n_{r}}} = - RTlnK_{e} = \\mathrm{\\Delta}_{\\ }^{0}G = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S$\n\nwhere $K_{e}$ is often called reaction constant relating the activities\nof products and reactants at equilibrium. In a system with\n$\\frac{\\prod_{p}^{}\\left( a_{p} \\right)^{n_{p}}}{\\prod_{r}^{}\\left( a_{r} \\right)^{n_{r}}} > K_{e}$,\nthe chemical reaction goes to left, and the products are decomposed,\nwhile\n$\\frac{\\prod_{p}^{}\\left( a_{p} \\right)^{n_{p}}}{\\prod_{r}^{}\\left( a_{r} \\right)^{n_{r}}} < K_{e}$,\nthe chemical reaction goes to right, and the products are formed. is\noften recast into the following form by dividing $\u2013RT$ on both sides of\nthe equation\n\n*Eq. 7\u20116*\n$\\ln K_{e} = - \\frac{\\mathrm{\\Delta}_{\\ }^{0}H}{RT} + \\frac{\\mathrm{\\Delta}_{\\ }^{0}S}{R}$\n\nWith $\\ln K_{e}$ plotted with respect to $1/T$, indicates that the slope\nis $- \\mathrm{\\Delta}_{\\ }^{0}H$/R and the intercept on the y axis is\n$\\mathrm{\\Delta}_{\\ }^{0}{S/R}$ as shown in .\n\nFigure \u2011: $\\ln K_{e}$ plotted with respect to $1/T$ for several M-O\nsystems at 1 bar and $K_{e}$ represented by the partial pressure ratio\nof CO2 and CO.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/applications_to_chemical_reactions/maximum_reaction_rate_and_chemical_transport_reactions_.ipynb b/psu410/src/applications_to_chemical_reactions/maximum_reaction_rate_and_chemical_transport_reactions_.ipynb deleted file mode 100644 index 004b1d3..0000000 --- a/psu410/src/applications_to_chemical_reactions/maximum_reaction_rate_and_chemical_transport_reactions_.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "7416df00", "cell_type": "markdown", "source": "## Maximum reaction rate and chemical transport reactions \n\nEquilibrium thermodynamics can be used to calculate the maximum rates of\nreaction that are possible in dynamically reacting systems, such as when\na corrosive gas passes over a heated sample. Other examples of such\nreactions include the reduction of a metal oxide in flowing hydrogen,\nthe evaporation of a material in a vacuum, and the deposition of a thin\nfilm by a chemical vapour deposition process. The basic assumption used\nin calculating these maximum reaction rates is that local equilibrium\nexists at the location of the considered reaction.\n\nThis section examines several examples in which maximum reaction rates\nare calculated. The system can typically be divided into three regions:\n(1) the input region, which is usually near room temperature, (2) the\nhigh temperature region in which the primary reaction of interest is\noccurring, and (3) the exit region. Such a system is almost always at\nconstant pressure throughout the system. Since the three regions have\ndifferent temperatures, the key is to use the mass conservation of the\ncarrier gas in all three regions. With the input gas at $T = 298K$ and\n$P = 1atm$, the volume of one mole of ideal gas is\n$0.0244\\left( m^{3} \\right) = 24.4(liter)$. The number of moles of input\ngas flowing through the system can be written as\n\n*Eq. 7\u201117*\n$n_{gas}^{0} = \\frac{Pf_{gas}}{RT} = \\frac{f_{gas}}{24.4} = 0.0409f_{gas}$\n\nwhere $f_{gas}$ is the input gas flow rate in liter per unit time at\n$T = 298K$ and $P = 1atm$, and $R$ the gas constant.\n\nThe first example is the evaporation of water in a flowing stream of dry\nhydrogen. A schematic diagram of the system being considered is given in\n. The goal of the calculation is to determine the maximum rate of\n$H_{2}O(liquid)$ loss, $n_{H_{2}O}^{\\ }$, through vaporization in a\nflowing stream of $H_{2}$, e.g. for generating a given $H_{2}O/H_{2}$\nratio for producing a given $O_{2}$ pressure in a high temperature\nsystem as related to the Ellingham diagram discussed in Chapter . The\nmaximum rate is determined by saturating the $H_{2}$ with the\nequilibrium vapour pressure of $H_{2}O(liquid)$ with the number of moles\nof $H_{2}$ being $n_{H_{2}}^{0} = 0.0409f_{H_{2}}$ and $f_{H_{2}}$ being\nthe input flow rate of $H_{2}$ in liter per unit time.\n\nFigure \u2011: Schematic diagram of vaporization of water in a flowing stream\nof dry hydrogen\n\nIn the first input region, the total number moles of gas is\n$N = n_{H_{2}}^{0}$. In the second high temperature region with the\ntemperature and pressure being $T_{sys}$ and $P_{sys}$, the total number\nmoles of gas is $N = n_{H_{2}}^{0} + n_{H_{2}O}^{\\ }$, which is the same\nfor the third exit region with $T_{exit}$. To avoid condensation, one\nneeds to maintain $T_{exit} > T_{sys}$. The vapour pressure of $H_{2}O$\nat $T_{sys}$, $P_{H_{2}O\\ }$, can be obtained from equilibrium\nthermodynamic calculations. Since the $H_{2}$ and $H_{2}O$ are occupying\nthe same volume at the same temperature, the maximum number of moles\n$H_{2}O$ can be calculated from the following relation based on the\nideal gas law\n\n*Eq. 7\u201118*\n$n_{H_{2}O}^{\\ } = \\frac{P_{H_{2}O}^{\\ }}{P_{H_{2}}^{\\ }}n_{H_{2}}^{0} = \\frac{P_{H_{2}O}^{\\ }}{P_{sys}^{\\ } - P_{H_{2}O}^{\\ }}n_{H_{2}}^{0} = \\frac{P_{H_{2}O}^{\\ }}{P_{sys}^{\\ } - P_{H_{2}O}^{\\ }}\\frac{f_{H_{2}}}{24.4}$\n\nFor $P_{sys}^{\\ } = 101325Pa$, one can calculate $P_{H_{2}O}^{\\ }$ and\nplot $\\frac{n_{H_{2}O}^{\\ }}{f_{H_{2}}}$ as a function of temperature\nfrom the SSUB database as shown in . The corresponding partial pressure\nratio, $P_{H_{2}O}/P_{H_{2}}$ , is plotted in , which can be used to\ncalculate $P_{O_{2}}^{\\ }$ at any given temperatures. An example is\nshown in with $T_{sys} = 348.15K$ and $P_{H_{2}O}/P_{H_{2}} = 0.607$.\n\nFigure \u2011: Ratio of maximum number of moles $H_{2}O$ with respect to\nhydrogen flow rate, $\\frac{n_{H_{2}O}^{\\ }}{f_{H_{2}}}$, plotted as a\nfunction of temperature.\n\nFigure \u2011: Partial pressure ratio, $P_{H_{2}O}/P_{H_{2}}$, corresponding\nto .\n\nFigure \u2011: Partial pressure of oxygen, $P_{O_{2}}^{\\ }$, as a function of\ntemperature with $T_{sys} = 348.15K$ and $P_{H_{2}O}/P_{H_{2}} = 0.607$.\n\nThe second example is the corrosion of $SiO_{2}$(s) by flowing $H_{2}$\ngas at high temperatures. Considering $T_{sys}^{\\ } = 1700K$,\n$P_{sys}^{\\ } = 101325Pa$, and $f_{H_{2}} = 1\\ liter/min$, the system is\nthus defined with 0.0409 moles of $H_{2}$ and an equilibrium between the\ngas phase and the tridymite $SiO_{2}$. The equilibrium calculation gives\n0.04092 moles of gas with its constitutions as\n$H_{2}:H_{2}O:SiO = 0.998887:4.833 \\bullet 10^{- 4}:4.832 \\bullet 10^{- 4}$.\nThe corrosion rate of SiO2(s) is thus\n$0.04092x4.832 \\bullet 10^{- 4} = 1.98 \\bullet 10^{- 5}mol/min = 1.19gram/min.$\n\nAnother example is to consider that the flowing $CO$ gas of 298K and\n1atm ($= 101325Pa$) at a rate of $1\\ liter/min$ passes through and\nequilibrates with single phase $C(s)$ at 1500K. The equilibrium system\nis defined by $T = 1500K$, $P = 1atm$, and 0.0409 moles of $CO$ with the\nequilibrium between the gas phase and C(s). The equilibrium calculation\ngives 0.040872 moles of gas phase with the mole fraction of $CO$ being\n0.999327, resulting in the loss of *CO* or the deposition of *C(s)* at a\nrate of\n$0.0409 - 0.040872 \\bullet 0.999327 = 5.55 \\bullet 10^{- 5}mole/min$.\n\nA chemical transport reaction is a reaction in which a condensed phase\nreacts with a gas phase to form vapour-phase products, which in turn\nundergo the reverse reformation of the condensed phase. Two well-known\nexamples of such reactions are\n\n*Eq. 7\u201119* $M(s) + \\frac{n}{2}I_{2}(g) = MI_{n}(g)$\n\n*Eq. 7\u201120* $Ni(s) + 4CO(g) = Ni{(CO)}_{4}(g)$\n\nwhich are used in the purification of metals by the iodide process and\nin the purification of nickel by the Mond-Langer process. In both\nprocesses the forward reaction is favoured by lower temperatures and the\nreverse reaction by higher temperatures, resulting in the deposition of\nthe metal. The most common technique for causing a chemical transport of\na condensed substance makes use of the temperature dependence of the\nequilibrium constant. As was discussed previously, the enthalpy of\nreaction, $\\mathrm{\\Delta}_{\\ }^{0}H$, determines the manner in which\n$K_{e}$ changes with temperature (see ). The value of $K_{e}$ increases\nwith increasing *T* for $\\mathrm{\\Delta}_{\\ }^{0}H > 0$, $K_{e}$\ndecreases with increasing *T* for $\\mathrm{\\Delta}_{\\ }^{0}H < 0$, and\n$K_{e}$ is independent of T for $\\mathrm{\\Delta}_{\\ }^{0}H = 0$. The\n$\\mathrm{\\Delta}_{\\ }^{0}H$ and $\\mathrm{\\Delta}_{\\ }^{0}S$ values for\nchemical transport reactions may be either positive or negative. For\nreactions by both are positive, and for reactions by both are negative.\n\nIn a typical experiment the starting solid is located at the point in a\ntemperature gradient that corresponds to the largest $K_{e}$ value for\nthe experimental condition. As the gaseous species migrate to other\nlocations in the system with temperatures corresponding to lower $K_{e}$\nvalues, the reverse reaction occurs to satisfy the new equilibrium\nrequirements, and the solid phase is deposited. The dependence of\n$K_{e}$ on $\\mathrm{\\Delta}_{\\ }^{0}H$ results in a material transport\nfrom hot to cold for $\\mathrm{\\Delta}_{\\ }^{0}H > 0$ (the same as for\nvaporization-condensation reactions), from cold to hot for\n$\\mathrm{\\Delta}_{\\ }^{0}H < 0$, and no transport for\n$\\mathrm{\\Delta}_{\\ }^{0}H = 0$.\n\nThe success of a particular reaction in causing an appreciable transport\nof a condensed phase depends mainly upon the partial pressure gradients\nor concentration gradients of the gaseous species in the system. A\nreaction whose equilibrium is extreme toward either the reactant side or\nthe product side will not give an appreciable transport of material. The\nconcentration gradients are too small in such a system. Reactions with\nequilibrium constants near unity at the experimental temperatures\nusually give the largest transport since small changes in $K_{e}$ cause\nlarge changes in concentrations. The general condition required for\nobtaining a $K_{e}$ value near unity at a reasonable temperature is that\n$\\mathrm{\\Delta}_{\\ }^{0}H$ and $\\mathrm{\\Delta}_{\\ }^{0}S$ both have\nthe same sign, resulting from the equalities of .\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/applications_to_chemical_reactions/trends_of_entropies_of_reactions.ipynb b/psu410/src/applications_to_chemical_reactions/trends_of_entropies_of_reactions.ipynb deleted file mode 100644 index 1840aa3..0000000 --- a/psu410/src/applications_to_chemical_reactions/trends_of_entropies_of_reactions.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "dbbb5455", "cell_type": "markdown", "source": "## Trends of entropies of reactions\n\nThe reaction entropy, $\\mathrm{\\Delta}_{\\ }^{0}S$ in , plays an\nimportant role in determining equilibria of high-temperature reactions.\nThe most important single factor that determines the entropy of a\nreaction is the net change in the number of moles of gas as briefly\nmentioned in the discussion of the Ellingham diagram above. The reason\nthis is true can be explained as follows.\n\nThe entropy of a substance can be thought of as being the sum of four\nparts: (i) translational, (ii) rotational, (iii) vibrational, and (iv)\nelectronic. The translational entropy of a gas is the largest entropy\nterm under most conditions. To the extent that the other contributions\ncancel between reactants and products, the entropy of reaction is\ndetermined by the change in the number of moles of gaseous molecules.\nBased on the literature data or calculations from the SSUB database, the\nnet change in the number of moles of gas in a reaction results\napproximately in an entropy of reaction of about 175\u00b145 J/K/mole-gas at\n298K for many halides and oxides. The chemical reactions of and\ndiscussed above both reduce the gas by one mole, and their entropies of\nreaction are -113 and -172 J/K at 1273K, and -89 and -173J/K at 298K,\nrespectively, indicating that the chemical reaction of is an exception\nof the empirical rule. For chemical reactions shown in the Ellingham\ndiagram, their entropies of reaction follow this empirical rule pretty\nwell with some of them shown in calculated from the SGTE database.\n\nTable \u2011: Entropies of reactions with gas at 298.15K, J/K\n\nReaction: Si+O2 =SiO2 -182\n\nReaction: Ti+O2=TiO2 -185\n\nReaction: 2Mg+O2=2MgO -217\n\nReaction: 2Ca+O2=2CaO -212\n\nReaction: 2Mn+O2=2MnO -150\n\nSince the entropy of a reaction is primarily determined by the net\nchange in the number of moles of gas, the entropies for reactions\ninvolving only condensed phases must be small. The entropies of fusion\nof monatomic solids are usually in the range 8-15 J/K/mole-atom as shown\nfor some elements in . Most metals and many ionic salts have values that\nlie in this range when given in terms of per mole of atom of material.\nThere are few exceptions such as silicon and boron shown in the table.\nFor solid-state reactions, the average values can be approximated as\n0\u00b18J/K/mole-atom as also shown in the table.\n\nTable \u2011: Entropies of reactions of condensed phases at 298.15K, J/K\n\nReaction: Si(s)=Si(l) 29.762\n\nReaction: Ti(s2)=Ti(l) 7.288\n\nReaction: Mg(s2)=Mg(l) 9.184\n\nReaction: Ca(s2)=Ca(l) 7.659\n\nReaction: Mn(s2)=Mn(l) 11.443\n\nReaction: W(s)=W(l) 14.158\n\nReaction: B(s)=B(l) 21.380\n\nReaction: 3Fe+C=CFe3 17.060\n\nReaction: S+Mn=MnS 13.909\n\nReaction: NiO+Fe2O3=Fe2NiO4\n0.464\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/applications_to_electrochemical_systems/application_examples.ipynb b/psu410/src/applications_to_electrochemical_systems/application_examples.ipynb deleted file mode 100644 index e9bb59d..0000000 --- a/psu410/src/applications_to_electrochemical_systems/application_examples.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "af90f499", "cell_type": "markdown", "source": "## Application examples\n\nAmong many applications of electrochemistry, several of them are briefly\ndiscussed in this section.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/applications_to_electrochemical_systems/aqueous_solution_and_pourbaix_diagram.ipynb b/psu410/src/applications_to_electrochemical_systems/aqueous_solution_and_pourbaix_diagram.ipynb deleted file mode 100644 index 34ed115..0000000 --- a/psu410/src/applications_to_electrochemical_systems/aqueous_solution_and_pourbaix_diagram.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "e064a684", "cell_type": "markdown", "source": "## Aqueous solution and Pourbaix diagram\n\nThe importance of aqueous solutions in all aspects of life is so well\nknown and needs not be discussed further. Since many electrochemical\nprocesses involve electrolyte solutions in an aqueous solvent,\nelectrochemical processes including water, hydrogen, and/or oxygen are\ndiscussed in more details. The hydrogen-oxygen cell can be described for\nboth acidic electrolytes and alkaline electrolytes. With acidic\nelectrolytes, H+ is in much higher concentrations than\nOH-, and thus half-cell reactions with H+ as an\nionic transport species are more important than those involving\nOH-. The reverse is true for alkaline electrolytes that\ncontain high OH-concentrations. Other than for nearly neutral\nacid-base systems, either H+ or OH- dominates the\nother by several orders of magnitude as can be seen from the value of\nthe 298 K dissociation constant for H2O:\n\n*Eq. 8\u201135* H2O(l) = H+(aq) + OH-(aq)\n\nwith the reaction constant being Ke =\n\\[H+\\]\\[OH-\\] = 10-14 and\n$\\mathrm{\\Delta}_{\\ }^{0}G$= -RT *ln* Ke *\u03b2*= +79,908 J. By\nconvention, one defines pH = - log \\[H+\\] and pOH = - log\n\\[OH-\\], and then pH + pOH = 14.\n\nUnder acidic electrolyte conditions of low pH (high \\[H+\\]\nconcentrations) the anode reaction in a hydrogen-oxygen cell is:\n\n*Eq. 8\u201136* \u00bd H2(g) = H+(aq) + e-\n\nwith \u03b51o = 0.0 V and\n$\\mathrm{\\Delta}_{\\ }^{0}G_{1}$= 0 J. The corresponding cathode\n(reduction) reaction is:\n\n*Eq. 8\u201137* 2 H+(aq) + \u00bd O2(g) + 2 e- =\nH2O(l)\n\nwith \u03b52o = 1.229 V and\n$\\mathrm{\\Delta}_{\\ }^{0}G_{2}$ = -2\\*1.229\\*96,485 J = -237,160 J. The\nnet cell reaction for acidic electrolytes is:\n\n*Eq. 8\u201138* H2(g) + \u00bd O2(g) = H2O(l)\n\nwith \u03b5ocell = 1.229 V and\n$\\mathrm{\\Delta}_{\\ }^{0}G_{cell}$ = -2\\*1.229\\*96,485 J = -237,160 J\n\nUnder alkaline electrolyte conditions of high pH (high\n\\[OH-\\] concentrations) the anode reaction in a\nhydrogen-oxygen cell is:\n\n*Eq. 8\u201139* 2 OH-(aq) + H2(g) = 2\nH2O(l) + 2 e-\n\nwith \u03b53o = 0.828 V and\n$\\mathrm{\\Delta}_{\\ }^{0}G_{3}$= -2\\*0.828\\*96,485 J = -159,779 J. The\ncorresponding cathode (reduction) reaction is:\n\n*Eq. 8\u201140* H2O(l) + \u00bd O2(g) + 2 e- = 2\nOH-(aq)\n\nwith \u03b54o = 0.401 V and\n$\\mathrm{\\Delta}_{\\ }^{0}G_{4}$ = -2\\*0.401\\*96,485 J = -77,381 J. The\nnet cell reaction for alkaline electrolytes is:\n\n*Eq. 8\u201141* H2(g) + \u00bd O2(g) = H2O(l)\n\nwith \u03b5ocell = 1.229 V and\n$\\mathrm{\\Delta}_{\\ }^{0}G_{cell}$ = -2\\*1.229\\*96,485 J = -237,160 J.\n\nPlots of \u03b5 versus pH for a given chemical system have been typically\nused to exhibit the stability relationships of ionic species and solid\nphases in aqueous-based electrochemical systems. These graphs are often\ncalled Pourbaix diagrams after the inventor and are at constant\ntemperature and constant pressure diagrams for a constant concentration,\nusually for one metallic element. By convention, the \u03b5 in a Pourbaix\ndiagram corresponds to the potential for the cathode reduction reactions\nin the electrochemical half-cell with electrons as reactants. Pourbaix\ndiagrams can be extended to multi-component materials when thermodynamic\nproperties of the components are available in both the materials and the\naqueous solution.\n\nAn example of an \u03b5 versus pH diagram is shown in for the\nNi-H2O system at a 298K, 1 bar, and $c_{{Ni}^{2 +}} = 0.001$\nmolality. Three stability regions for Ni species are shown: Ni(s),\nNiO(s), and \\[Ni2+\\]. The two dashed lines on this diagram\ncorrespond to hydrogen reduction () and oxygen reduction () reactions,\nrespectively.\n\nFigure \u2011: An \u03b5 versus pH, Pourbaix diagram for the Ni-H2O at\n298K, 1 bar, and $c_{{Ni}^{2 +}} = 0.001$ molality.\n\nFor the \u03b5 and pH conditions within the boundaries of the Ni(s) region,\nno solid phase other than Ni(s) is stable, no ionic species with a\nconcentration of 1 molarity is stable, and no gas species with a\npressure of 1 bar is stable. Similar statements could be made about the\nNiO(s) and \\[Ni2+\\] areas on the diagram. In the\n\\[Ni2+\\] area, introduction of Ni(s) or NiO(s) into the\nsystem would result in the dissolution of these solid phases since they\nare not stable with respect to the \\[Ni2+\\] aqueous solution.\nThe corresponding chemical reactions proceed spontaneously to the right\nas follows until the solid phases are consumed:\n\n*Eq. 8\u201142* Ni(s) \u2192 Ni2+(1 molarity) + 2 e-\n\n*Eq. 8\u201143* NiO(s) + 2 H+(aq) \u2192 Ni2+(1 molarity) +\nH2O(l)\n\nNo H+(aq) in involved in the first reaction, , so the\nboundary line separating Ni(s) and Ni2+ is independent of pH.\nNo oxidation or reduction occurs in the second reaction, , i.e. no\nelectrons are reactants or products in the reaction, the boundary line\nseparating NiO(s) and Ni2+ is independent of \u03b5.\n\nNote the convention that the \u03b5 is the potential for a cathode reduction\nreaction, and boundary lines between two stability regions depict\nconditions under which partial equilibrium occurs for the two species\nfor the \u03b5 and pH values at any point on these lines. For the boundary\nline separating Ni(s) and Ni2+ in an ideal aqueous solution,\ni.e. the reverse of , the following equation is obtained.\n\n*Eq. 8\u201144* \u03b5 = \u03b5o = -0.268 V\n\nFor the NiO(s)-Ni2+ boundary line of an ideal solution, the\nreaction, , is a complete equilibrium, and thus the relationship is\n\n*Eq. 8\u201145*\n$0 = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{1}{\\left( c_{H^{+}} \\right)^{2}} = \\mathrm{\\Delta}_{\\ }^{0}G + 2 \\cdot 2.303 \\cdot RT \\cdot pH$\n\n*Eq. 8\u201146*\n$pH = - \\frac{\\mathrm{\\Delta}_{\\ }^{0}G}{2 \\cdot 2.303 \\cdot RT}$\n\nwhere $\\mathrm{\\Delta}_{\\ }^{0}G$ is obtained as follows and can be\ncalculated from the SSUB database and the standard potential of Ni,\n\n*Eq. 8\u201147*\n\nAt a specified temperature, only one standard free energy and only one\nequilibrium constant exists for this chemical reaction, and thus only\none specific value of $pH = 6.631$ exists for the reaction represented\nby in this Pourbaix diagram.\n\nThe diagonal line in represents the equilibrium between Ni(s) and NiO(s)\nand is for a partial equilibrium reaction that is the sum of reactions\nof and\n\n*Eq. 8\u201148* NiO(s) + 2 H+(aq) + 2 e- \u2550 Ni(s) +\nH2O(l)\n\nThe reduction of Ni from a divalent state in NiO to metallic Ni(s)\noccurs, but the reaction also depends on the H+\nconcentration, the pH. The corresponding Gibbs energy and Nernst\nequations are,\n\n*Eq. 8\u201149*\n$\\mathrm{\\Delta}G = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{1}{\\left( c_{H^{+}} \\right)^{2}} = - 23,939 + 2 \\cdot 2.303 \\cdot RT \\cdot pH$\n\n*Eq. 8\u201150*\n$\\varepsilon\\ = \\ \\varepsilon^{0}\\ - \\ \\frac{RT}{2f}\\ln\\frac{1}{\\left( c_{H^{+}} \\right)^{2}}\\ = 0.124\\ \u2013\\ \\frac{2.303 \\cdot RT}{f}pH$\n\nwhere $\\mathrm{\\Delta}_{\\ }^{0}G$ can be calculated as follows\n\n*Eq. 8\u201151*\n\nThe two additional lines in correspond to the reduction reactions\nrelated to H2 and O2 gases, i.e. the stability of\nH2O. The lower one is for the reverse of under \u03b5o\n= 0 and $P_{H_{2}} = 1$ with the Nernst equation being\n\n*Eq. 8\u201152*\n$\\varepsilon\\ = \\ \\varepsilon^{0}\\ - \\ \\frac{RT}{f}\\ln\\frac{\\left( P_{H_{2}} \\right)^{1/2}}{c_{H^{+}}}\\ = - \\frac{2.303 \\cdot RT}{f}pH$\n\nAs the pH increases from 0, \u03b5 becomes more negative as is depicted. The\ntop dashed line corresponds to the oxygen reduction reaction represented\nby under \u03b5o = 1.225 calculated from the aqueous solution\ndatabase in Thermo-Calc \\[60\\] and $P_{O_{2}} = 1$ with the Nernst\nequation being\n\nEq. 8\u201153\n$\\varepsilon\\ = \\ \\varepsilon^{0}\\ - \\ \\frac{RT}{2f}\\ln\\frac{\\left( P_{O_{2}} \\right)^{1/2}}{\\left( c_{H^{+}} \\right)^{2}}\\ = 1.225\\ \u2013\\ \\frac{2.303 \\cdot RT}{f}pH$\n\nThe dependence of \u03b5 on pH is identical for both reduction reaction and ,\nand their intercepts at $pH = 0$ differ by their difference in their\n\u03b5o values.\n\nIn this simple Pourbaix diagram of Ni in an ideal aqueous solution, all\nboundary lines are straight because there is only one ionic species of\nNi in the aqueous solution, i.e. Ni2+. When there are more\nthan one ionic species in the aqueous solution, the boundary lines may\nno longer be straight due to the competition between species. One\nexample is Cu with two main ionic species of Cu+2 and\nCuOH+, and the reduction reaction between the metallic Cu and\nthe aqueous solution involves both two species, i.e.\n\n*Eq. 8\u201154*\n${xCu}^{2 + \\ }\\ + \\ (1 - x){CuOH}^{+} + (1 - x)H^{+} + 2\\ e - \\ \\ = \\ \\ Cu(s) + {(1 - x)H}_{2}O$\n\nwith\n\n*Eq. 8\u201155*\n$\\mathrm{\\Delta}G = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{1}{\\left( c_{{Cu}^{2 +}} \\right)^{x}\\left( c_{{CuOH}^{+}}c_{H^{+}} \\right)^{1 - x}} = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{1}{\\left( c_{{Cu}^{2 +}} \\right)^{x}\\left( c_{{CuOH}^{+}} \\right)^{1 - x}} + 2.303(1 - x) \\cdot RT \\cdot pH$\n\n*Eq. 8\u201156*\n$\\varepsilon = \\varepsilon^{0} - \\frac{RT}{2f}\\ln\\frac{1}{\\left( c_{{Cu}^{2 +}} \\right)^{x}\\left( c_{{CuOH}^{+}} \\right)^{1 - x}} - \\frac{2.303(1 - x) \\cdot RT}{2f}pH$.\n\nIt is evident that both the slope and the intercept at $pH = 0$ are a\nfunction of the concentration of ${CuOH}^{+}$, which is a function of\n$pH$. Consequently, the boundary between the metallic Cu and the aqueous\nsolution is no longer a straight line as shown in .\n\nFigure \u2011: An \u03b5 versus pH, Pourbaix diagram for the Cu-H2O\nsystem at 298K, 1 bar, and $c_{Cu} = 0.001$ molality.\n\nThe concentrations of various species in the aqueous solution, i.e.\ncommonly called speciation, are plotted in , showing the change of\ndominant species as a function of pH value.\n\nFigure \u2011: Concentrations of ionic species in the aqueous solution at\n$\\varepsilon = 0.3\\ V$ from .\n\nIn Pourbaix diagrams for alloys with two or more elements, activities of\nindividual elements are to be used in calculating the potentials of\nreduction reactions. Considering a Fe-Ni alloy with Fe2+ and\nNi2+ in the aqueous solution, the reduction reactions for Fe\nand Ni can be written separately as\n\n*Eq. 8\u201157* Ni2+(cNi) + 2 e- \u2192 Ni\n(aNi in alloy)\n\n*Eq. 8\u201158* Fe2+(cFe) + 2 e- \u2192 Fe\n(aFe in alloy)\n\nwith their potentials as\n\n*Eq. 8\u201159*\n$\\varepsilon_{Ni}\\ = \\ \\varepsilon_{Ni}^{0} - \\frac{2.303RT}{2f}\\ln\\frac{a_{Ni}}{c_{Ni}} = \\ - 0.268 - \\frac{2.303RT}{2f}\\ln\\frac{a_{Ni}}{c_{Ni}}$\n\n*Eq. 8\u201160*\n$\\varepsilon_{Fe}\\ = \\ \\varepsilon_{Fe}^{0} - \\frac{2.303RT}{2f}\\ln\\frac{a_{Fe}}{c_{Fe}} = \\ - 0.441 - \\frac{2.303RT}{2f}\\ln\\frac{a_{Fe}}{c_{Fe}}$\n\nIn principle, there are two scenarios for a given set of $a_{Ni}$ and\n$a_{Fe}$ of the alloy. The first scenario is at the limit of a dilute\naqueous solution, i.e. $c_{Ni} = c_{Fe} = 0.001\\ $molarity,\n$\\varepsilon_{Ni}$ and $\\varepsilon_{Fe}$ can be calculated, and the\nelement with the lower potential has the tendency to dissolve first,\nwhich can result in the so-called dialloying effect. The second scenario\nis for the equal potential, i.e. $\\varepsilon_{Ni} = \\varepsilon_{Fe}$\ndue to the externally imposed potential, and the equilibrium\nconcentrations of Fe+2 and Ni+2 can be calculated\nfrom and .\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/applications_to_electrochemical_systems/concentrations_activities_and_reference_states_of_electrolyte_species.ipynb b/psu410/src/applications_to_electrochemical_systems/concentrations_activities_and_reference_states_of_electrolyte_species.ipynb deleted file mode 100644 index 3351784..0000000 --- a/psu410/src/applications_to_electrochemical_systems/concentrations_activities_and_reference_states_of_electrolyte_species.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "ebc6e0c8", "cell_type": "markdown", "source": "## Concentrations, activities, and reference states of electrolyte species\n\nThermodynamic descriptions of ionic species in solutions are different\nfrom those of neutral species, which leads to a need for defining\nconcentration units, standard states, activities, and activity\ncoefficients of ionic solutions. In most studies of electrochemical\ncorrosion and electrodeposition, and in applied work of electrochemical\nengineers, ionic species concentrations are given in units of molarity,\nthe number of moles of a species in a liter of solution (mol/l)\nsymbolically represented in equations by either *ci* or\n\\[M+Z\\]. The other common concentration used for ionic\nspecies is molality, which is defined as the number of moles of a\nspecies in 1000g of solvent. For dilute aqueous solutions, molarity and\nmolality values are very similar.\n\nAs discussed in Chapter , a practical definition of the activity of a\nspecies *i* is the thermodynamic reactivity, or tendency to react, of\nspecies *i* in the system of interest as compared to *i* in its\nreference state form. The reference state of a species is typically\nchosen as a specific chemical/physical state of the species at 1 atm\nexternal pressure and the temperature of interest. Similarly, a typical\nreference state for ionic species in aqueous solutions is the 1 molar\nideal solution at 1 bar external pressure and the temperature of\ninterest. If an electrolyte solution behaves ideally, then the activity\nof species *i* in solution is\n\n*Eq. 8\u20116*\n$a_{i} = \\frac{c_{i}\\left( \\frac{mol}{l} \\right)}{c_{i}^{0}\\left( \\frac{mol}{l} \\right)} = \\frac{c_{i}\\left( \\frac{mol}{l} \\right)}{1\\left( \\frac{mol}{l} \\right)} = c_{i}(dimensionless)$\n\nwhere $c_{i}$ is the molar concentration of *i* in the solution divided\nby $c_{i}^{0}$, the 1 molar reference state ideal solution\nconcentration. Thus, in ideal solutions, the activity of an electrolyte\nspecies is numerically equal to its molar concentration. The above\ntreatment of ionic species is equivalent to the common practice of\ndepicting the activity of a gas by the value of its ideal gas partial\npressure in units of bar.\n\nThe activity coefficient corrects for the nonideality of the species in\nsolution as defined in . If the solution is ideal, $\\gamma_{i} = 1$ for\nall concentrations of a species in solution. For all solutions, one\nexpects $\\gamma_{i} \\rightarrow 1$ as $c_{i} \\rightarrow 1$. It is not\npossible to measure $\\gamma_{i^{+}}$ or $\\gamma_{i^{-}}$ for individual\ncharged ions, only a geometric mean of the positive and negative ion\nvalues. Consider the following ionic solution\n\nEq. 8\u20117\n\nIts chemical potential can be written as\n\n*Eq. 8\u20118*\n\nIts geometric average or mean activity and activity coefficient are\ndefined as\n\n*Eq. 8\u20119*\n\n*Eq. 8\u201110*\n\nFor example, one can define\n$\\gamma_{\\pm} = \\left( \\gamma_{{Na}^{+}}\\gamma_{{Cl}^{-}} \\right)^{1/2}$\nand\n$\\gamma_{\\pm} = \\left( \\gamma_{{Al}^{3 +}}^{2}\\gamma_{{{SO}_{4}}^{2 -}}^{3} \\right)^{1/5}$\nfor NaCl and Al2(SO4)3, respectively.\nFor idea, weak electrolytes, $\\gamma_{\\pm} = 1$, and for non-ideal,\nstrong electrolytes, $\\gamma_{\\pm} \\neq 1$.\n", "metadata": {}}, {"id": "5e28080f", "cell_type": "markdown", "source": "### Electrical batteries\n\nBatteries utilize electrochemical reactions to generate electricity for\nvarious devices. The theoretic voltage of a battery can be calculated\nfrom and , i.e.\n\n*Eq. 8\u201173*\n$\\varepsilon = - \\frac{\\mathrm{\\Delta}G}{zf} = \\varepsilon^{0} - \\frac{RTlnQ}{zf}$\n\nwith $\\mathrm{\\Delta}G$ being the driving force of the net cell reaction\nand $Q$ being the reaction activity quotient. The actual voltage of a\nbattery is lower than the theoretical one due to kinetic limitations of\ncell reactions and resistance to ion diffusion through the electrolyte.\nBased on whether the cell reactions are reversible or not, batteries\ntypically categorized as either primary disposable or secondary\nrechargeable batteries. The net cell reactions in primary disposable\nbatteries are not easily reversible, and electrode materials may not\nreturn to their original forms by applying a higher external potential\nof opposite sign. Consequently, primary batteries cannot be reliably\nrecharged. On the other hand, the net cell reactions in secondary\nbatteries are easily reversible. Furthermore, two half-cells in\nbatteries may use different electrolytes with each half-cell enclosed in\na container and a separator permeable to conducting ions but not the\nbulk of the electrolytes.\n\nOne common primary battery is zinc-carbon battery with a zinc anode\ncylinder and a carbon cathode central rod. The electrolytes are ammonium\nor zinc chloride next to the zinc anode and a mixture of ammonium\nchloride and manganese dioxide next to the carbon cathode. The half cell\nand net reactions with ammonium chloride are as follows\n\n*Eq. 8\u201174* Zn + 2NH3 \u2192\nZn(NH3)22+ + 2 e-\n\n*Eq. 8\u201175* 2NH4Cl + 2MnO2 + 2 e- \u2192\n2NH3 + Mn2O3 +\nH2O+2Cl\u2212\n\n*Eq. 8\u201176* Zn + 2MnO2 + 2NH4Cl \u2192\nMn2O3 +\nZn(NH3)2Cl2 + H2O.\n\nThe electric potential of the reaction is, treating all compounds as\nstoichiometric compounds\n\nEq. 8\u201177\n$\\varepsilon = - \\frac{\\mathrm{\\Delta}G}{2f} = - \\frac{\\mathrm{\\Delta}^{0}G}{2f} = \\frac{1}{2f}\\left(_{\\ }^{0}G^{Zn} + 2_{\\ }^{0}G^{{MnO}_{2}} + 2_{\\ }^{0}G^{{NH}_{4}Cl} -_{\\ }^{0}G^{H_{2}O} -_{\\ }^{0}G^{Zn\\left( {NH}_{3} \\right)_{2}{Cl}_{2}} -_{\\ }^{0}G^{{{Mn}_{2}O}_{3}} \\right)$.\n\nThe Gibbs energy of $Zn\\left( {NH}_{3} \\right)_{2}{Cl}_{2}$ is not\navailable in current databases and has been recently estimated to be\n\u2212505,375 J/mole-formula \\[62\\]. The value of at 298.15K is thus obtained\nas 1.67 V, which is pretty close to the actual operating voltage of the\nbattery around 1.5 V.\n\nWhile with zinc chloride, the cell reactions and electric potential may\nbe written as\n\nEq. 8\u201178 Zn + ZnCl2 + 2OH\u2212 \u2192 2ZnOHCl + 2\ne-\n\n*Eq. 8\u201179* MnO2 + H2O + e- \u2192 MnOOH +\nOH-\n\n*Eq. 8\u201180* Zn + 2 MnO2 + ZnCl2 + 2 H2O\n\u2192 2 MnOOH + 2 ZnOHCl\n\n*Eq. 8\u201181*\n$\\varepsilon = \\frac{1}{2f}\\left(_{\\ }^{0}G^{Zn} + 2_{\\ }^{0}G^{{MnO}_{2}} +_{\\ }^{0}G^{Zn{Cl}_{2}} + 2_{\\ }^{0}G^{H_{2}O} - 2_{\\ }^{0}G^{MnOOH} - 2_{\\ }^{0}G^{ZnOHCl} \\right)$\n\nSecondary batteries can be recharged by applying an external electrical\npotential, which reverses the net cell reaction that occur during\ndischarging. The oldest form of rechargeable battery is the lead-acid\nbatteries used in automotive, and the latest development is the\nlithium-ion (Li-ion) batteries. A lead-acid battery typically uses Pb\nand PbO2 as the cathode and anode electrodes and a 35% sulfuric acid and\n65% water solution as the electrolyte. Its anode and cathode reactions\ncan be simplified as follows\n\nEq. 8\u201182 $Pb + SO_{4}^{2 -} = PbSO_{4} + 2e^{-}$\n\nEq. 8\u201183 $PbO_{2} + 4H^{+} + SO_{4}^{2 -} + 2e^{-} = PbSO_{4} + 2H_{2}O$\n\nThe net cell reaction is\n\nEq. 8\u201184 $Pb + PbO_{2} + 2H_{2}SO_{4}^{\\ } = 2PbSO_{4} + 2H_{2}O$.\n\nIts electric potential is represented by the following equation\n\n*Eq. 8\u201185*\n$\\varepsilon = - \\frac{1}{2f}\\left( 2_{\\ }^{0}G^{H_{2}O} + 2_{\\ }^{0}G^{{PbSO}_{4}} -_{\\ }^{0}G^{Pb} -_{\\ }^{0}G^{{PbO}_{2}} - 2_{\\ }^{0}G^{H_{2}SO_{4}} \\right)$\n\nwith the value at 298.15K being 2.651 V calculated from Thermo-Calc\n\\[60\\]. During discharging, the reaction goes to right, and $PbSO_{4}$\nis formed on both anode and cathode, while during charging, the reaction\ngoes to the left, and $Pb$ and $PbO_{2}$ are restored. In practical\napplications, other ionic species such as ${H_{3}O}^{+}$ and\n$HSO_{4}^{-}$ may form in the electrolyte, complicating the reactions\nand affecting its potential.\n\nIn lithium ion batteries, lithium ions migrate in electrolytes between\nelectrodes made of intercalated lithium compounds during charging and\ndischarging. LiCoO2 and LiFePO4 are two of the\nseveral cathode materials used in lithium ion batteries, and the anode\nis typically made of carbon or metallic Li. The anode and cathode\nreactions for LiCoO2 batteries can be written in simple forms as follows\n\nEq. 8\u201186 ${Li}_{x}C_{6} = x{Li}^{+} + xe^{-} + 6C$\n\nEq. 8\u201187 $x{Li}^{+} + xe^{-} + {Li}_{1 - x}CoO_{2} = LiCoO_{2}$\n\nwith the net reaction and electric potential being\n\nEq. 8\u201188 ${Li}_{x}C_{6} + {Li}_{1 - x}CoO_{2} = LiCoO_{2} + 6C$\n\n*Eq. 8\u201189*\n$\\varepsilon = - \\frac{1}{xf}\\left\\{ 6_{\\ }^{0}G^{C} +_{\\ }^{0}G^{{LiCoO}_{2}} - G^{{{Li}_{x}C}_{6}} - G^{{{Li}_{1 - x}CoO}_{2}} \\right\\} = - \\frac{1}{f}\\left\\{ \\left( \\mu_{Li}^{{Li}_{1 - x}CoO_{2}} - \\mu_{Li}^{{Li}_{x}C} \\right) - \\frac{1}{x}\\left( \\mu_{LiCoO_{2}}^{{Li}_{1 - x}CoO_{2}} -_{\\ }^{0}G^{{LiCoO}_{2}} \\right) \\right\\}$\n\nThe electric potential is a function of $x$. The value in the first\nparenthesis in the above equation denotes the chemical potential\ndifference of Li between two electrodes, and the value in the second\nparenthesis represents the chemical potential difference of\n${LiCoO}_{2}$ between the states in the solution phase of\n${Li}_{1 - x}CoO_{2}$ and by itself. Gibbs energies of ${{Li}_{x}C}_{6}$\nand ${{Li}_{1 - x}CoO}_{2}$ need to be obtained as a function $x$ in\norder to calculate the electric potential of the battery.\n\nLiFePO4 uses metallic lithium as the anode with following\nhalf-cell and net cell reactions\n\nEq. 8\u201190 $xLi = x{Li}^{+} + xe^{-}$\n\nEq. 8\u201191 $x{Li}^{+} + xe^{-} + {Li}_{1 - x}FePO_{4} = LiFePO_{4}$\n\nEq. 8\u201192 $xLi + {Li}_{1 - x}FePO_{4} = LiFePO_{4}$.\n\nIts electric potential is also a function of $x$, i.e.\n\n*Eq. 8\u201193*\n$\\varepsilon = - \\frac{1}{xf}\\left\\{_{\\ }^{0}G^{LiFePO_{4}} - x\\ ^{0}G^{Li} - G^{{Li}_{1 - x}FePO_{4}} \\right\\} = - \\frac{1}{f}\\left\\{ \\left( \\mu_{Li}^{{Li}_{1 - x}FePO_{4}} - \\ ^{0}\\mu_{Li} \\right) - \\frac{1}{x}\\left( \\mu_{LiFePO_{4}}^{{Li}_{1 - x}FePO_{4}} -_{\\ }^{0}G^{LiFePO_{4}} \\right) \\right\\}$\n\nThe value in the first parenthesis in the above equation denotes the\nchemical potential difference of Li between two electrodes, and the\nvalue in the second parenthesis represents the chemical potential\ndifference of $LiFePO_{4}$ between the states in the solution phase of\n${Li}_{1 - x}FePO_{4}$ and by itself. Consequently, Gibbs energy of\n${Li}_{1 - x}FePO_{4}$ needs to be obtained as a function $x$ in order\nto calculate the electric potential of the battery. It is known that\nthere are several miscibility gaps in the $FePO_{4}$ and $LiFePO_{4}$\npsuedo-binary system, in which the chemical potentials are constants, so\nis the electric potential, resulting in stable battery output.\n", "metadata": {}}, {"id": "b49dd005", "cell_type": "markdown", "source": "### Electrochemical cells\n\nA potential difference, i.e. voltage difference, can be generated\nbetween the electrodes in a cell from differences in the potentials of\nthe half-cell reactions. This potential can originate from potential\ndifferences of two chemically different half-cells (a *galvanic cell*),\nor concentration differences in two otherwise identical half-cells (a\n*concentration cell*). Each type of cell is illustrated below.\n\nThe reaction between copper ions and zinc illustrated below represents\nthe *net cell reaction* of a *galvanic* cell in which the oxidation of\nZn(s) occurs at the *anode* electrode, and the reduction of\nCu+2 occurs at the *cathode* electrode\n\n*Eq. 8\u201111* Cu2+(aq) + Zn(s) = Cu(s) + Zn2+(aq).\n\nThe reaction at each electrode, the *half-cell reaction*, includes ions\nand electrons as reactant and/or product species. The anode, oxidation\nreaction is represented by\n\n*Eq. 8\u201112* Zn(s) = Zn2+(aq) + 2 e-.\n\nElectrons are products of anode reactions and flow externally from anode\nto cathode. By convention the activities of the electrons in an\nequilibrium cell are taken as unity. The cathode, reduction reaction is\nwritten as\n\n*Eq. 8\u201113* Cu2+(aq) + 2 e- = Cu(s)\n\nElectrons are reactants of cathode reactions and supplied by an external\nflow from the anode. In addition to consuming electrons at the cathode\nat the same rate as they are produced at the anode, charge balance is\nmaintained in the electrolyte by the generation of Zn2+ ions\nat the same rate that Cu2+ ions are consumed. A schematic\ndiagram in illustrates the simple physical relationships in such an\nelectrochemical cell.\n\nFigure \u2011: Schematic diagram of a galvanic electrochemical cell\nconsisting of a zinc electrode and a copper electrode.\n\nA concentration cell in which an electrochemical potential is developed\nbecause of concentration differences between otherwise equivalent anode\nand cathode reactions is illustrated below. Such a cell can be produced\nby the oxidation and reduction of copper at two separate electrodes as\nis depicted in the following reactions\n\n*Eq. 8\u201114* Cu(s) = Cu2+(aq, ca) + 2 e- (anode)\n\n*Eq. 8\u201115* Cu2+(aq, cb) + 2 e- = Cu(s) (cathode)\n\nwhere ca and cb are the respective concentrations\nof Cu2+ in the aqueous solutions at the anode and cathode,\nand ca \\< cb. The net cell reaction is\n\n*Eq. 8\u201116* Cu2+(aq, cb) = Cu2+(aq,\nca)\n\nwhere the reaction occurs spontaneously to decrease cb and to\nincrease ca until the two concentrations become the same,\ncb = ca. A schematic diagram of such a cell is\nshown in .\n\nFigure \u2011: Schematic diagram of a concentration electrochemical cell\nconsisting of two copper electrodes.\n\nA semi-impermeable membrane, or a salt bridge, must exist in such a cell\nto maintain charge balance. As Cu+2 ions are produced at the\nanode and consumed at the cathode, the negatively charged ions in the\nsolution, for example Cl-, must be transferred from the\ncathode region to the anode region of the cell to maintain electrically\nneutral solutions.\n\nThe above concentration cell provides a good example for illustrating a\nstandard notation for depicting an electrochemical cell. This cell can\nbe represented by .\n\nFigure \u2011: Standard notation of an electrochemical cell\n\nThe anode where oxidation occurs is always denoted on the left, and the\ncathode where reduction occurs is on the right. A *single line*\nseparating phases denotes an *interface* between two phases. The above\nanode electrode and reaction of are symbolically represented by\n\n*Eq. 8\u201117* \\| Cu(s) \\| Cu2+(ca).\n\nThe interface between the external conductor and Cu(s) is depicted by\nthe single line to the left of Cu(s), while the single line between the\nCu(s) and Cu2+(ca) depicts the interface between\nthe anode electrode and the electrolyte solution. Similarly, the cathode\nelectrode and reaction of are symbolically represented by\n\n*Eq. 8\u201118* Cu2+(cb) \\| Cu(s) \\|.\n\nA double line between the two copper ions in the notation denotes a\nphysical separation of two solution phases, the anode and cathode\nelectrolyte regions that exhibit different concentrations of copper ions\n\n*Eq. 8\u201119* Cu2+(ca) \\| \\|\nCu2+(cb).\n\nThese solution phases are physically connected by a semi-impermeable\nmembrane or salt bridge that allows a common negative ion, for example\nCl-, of the solution phases to diffuse from one region to the\nother in order to maintain charge balance as the cell reaction occurs.\nThe Cu2+ ions cannot be transported from one region to the\nother.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/applications_to_electrochemical_systems/electrochemical_cells_and_half_cell_potentials.ipynb b/psu410/src/applications_to_electrochemical_systems/electrochemical_cells_and_half_cell_potentials.ipynb deleted file mode 100644 index d636f47..0000000 --- a/psu410/src/applications_to_electrochemical_systems/electrochemical_cells_and_half_cell_potentials.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "dfa135f5", "cell_type": "markdown", "source": "## Electrochemical cells and half cell potentials\n\nAn electrochemical system must fulfill certain requirements in order to\napply equilibrium thermodynamic descriptions of the system, including\n\n- The cell must be reversible when slight changes in conditions\n (potentials, concentrations, pressures, temperature) cause\n electrochemical reactions and an external flow of electrons to occur\n in the direction needed to re-establish equilibrium.\n\n- All non-electrochemical reactions in the system must be prevented as\n such reactions would cause a shift in equilibrium without causing a\n shift in cell potential and thus a driving force for external electron\n flow.\n\n- Chemical reactions must occur only when an external current flows.\n These finite distances for external electron transport can be a short\n as grain size-dimensions in many corrosion reactions, or this\n transport may be through an external electrical conductor connecting\n the anode and cathode half-cell as in batteries.\n\n- Charge balance as well as mass balance is required of all reactions.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/applications_to_electrochemical_systems/electrolyte_reactions_and_electrochemical_reactions.ipynb b/psu410/src/applications_to_electrochemical_systems/electrolyte_reactions_and_electrochemical_reactions.ipynb deleted file mode 100644 index f2f0b01..0000000 --- a/psu410/src/applications_to_electrochemical_systems/electrolyte_reactions_and_electrochemical_reactions.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "a49607a0", "cell_type": "markdown", "source": "## Electrolyte reactions and electrochemical reactions\n\nElectrolytes that dissolve in a polar solvent such as water to produce\nionic species do not necessarily exhibit valence changes. A simple\nexample is the strong electrolyte NaCl(s) dissolving in water to produce\nsolvated ions\n\n*Eq. 8\u20111* $NaCl(s)\\ \\ = \\ \\ {Na}^{+}(aq)\\ \\ + \\ \\ {Cl}^{\u2013}(aq)$\n\nwhere the $(aq)$ indicates the ionic species in an aqueous solution. In\nthis system, the ion concentrations must become quite large before the\nsolution is saturated and can exist in equilibrium with $NaCl(s)$. Its\nreaction constant, defined by , is shown as\n$K_{e} = \\ \\ a_{{Na}^{+}}a_{{Cl}^{-}}$. If the product of the ion\nactivities is less than $K_{e}$, the solution is not saturated, and more\n$NaCl(s)\\ \\ $ can be dissolved.\n\nThe precipitation of $AgCl(s)$, a weak electrolyte, occurs quite readily\nwhen ${Cl}^{\u2013}$ ions are added to an aqueous solution containing\n${Ag}^{+}(aq)$:\n\n*Eq. 8\u20112* ${Ag}^{+}(aq)\\ + \\ \\ {Cl}^{\u2013}(aq)\\ = AgCl(s)$\n\nThe equilibrium constant for this reaction,\n$K_{e} = \\ \\frac{1}{\\left( a_{{Ag}^{+}}a_{{Cl}^{-}} \\right)}$ is quite\nlarge, so the equilibrium product of the ion activities, proportional to\ntheir concentrations, is quite small. In the laboratory, the above\nreaction could occur as a result of adding hydrochloric acid to a silver\nnitrate solution. The accompanying $H^{+}(aq)$ \\[or ${H_{3}O}^{+}(aq)$\\]\nand ${NO_{3}}^{-}(aq)$ ions in solution are not directly involved in the\nsilver chloride precipitation reaction so are not shown in reaction\nrepresented by .\n\nThe above ionic equilibria in the $AgCl(s) - H_{2}O$ system is not only\nimportant for understanding this electrolyte system, but also critical\nin electrochemical systems in which *Ag(s)* undergoes a valence change\nat one electrode and reacts with a ${Cl}^{\u2013}(aq)$ ion to produce\n*AgCl(s)*, and an electron that is externally transported finite\ndistances to another electrode. The *oxidation* reaction occurs at the\nAg/AgCl electrode (*anode* half-cell reaction where electrons are\n*added* into the system)\n\n*Eq. 8\u20113* $Ag(s)\\ \\ + \\ {Cl}^{-}(aq)\\ \\ = \\ \\ AgCl(s)\\ \\ + \\ \\ e^{-}$\n\nA *reduction* reaction occurs at the other electrode (*cathode*\nhalf-cell reaction where electrons are *consumed* by the reaction)\n\n*Eq. 8\u20114* $\\frac{1}{2}{Cl}_{2}(g) + \\ \\ e^{-} = \\ {Cl}^{-}(aq)\\ \\ $\n\nThe *net cell reaction* results in the formation AgCl(s) from its\nelements\n\n*Eq. 8\u20115* $Ag(s)\\ \\ + \\frac{1}{2}{Cl}_{2}\\ (g)\\ = \\ \\ AgCl(s)$\n\nWithout knowledge of the physical system under which the reaction is\noccurring, it would not be possible to know if reaction of was a result\nof chlorine gas reacting directly with Ag(s), or if the reaction was\npart of an electrochemical cell with a transport of electrons and ions\nover finite distances. The addition of the two half-cell reactions gives\nthe *net cell reaction*, which does not show electrons as either\nreactant or product species and may or may not include ionic species in\nthe reaction*.* A schematic of an electrochemical cell for the above\nsystem is shown in Figure 8\u20111.\n\nFigure \u2011: Schematic diagram of an electrochemical cell consisting of a\nchlorine electrode and a silver-silver chloride electrode.\n\nOxidation and reduction can occur in electrolyte reactions without\ncreating an electrochemical cell. This is the case when chlorine gas\nreacts directly with silver on a Ag(s) surface. Reaction of above is the\nnet reaction for this process, but the electrons produced from the\noxidation of Ag(s) are not transported over finite distances before\ncombining with Cl2(g) in its reduction to Cl\u2013(aq).\nNo anode or cathode half-cell reactions exist in this system. The\nelectrons and ions involved in the reaction move only over atomic-scale\ndistances.\n", "metadata": {}}, {"id": "9758eaea", "cell_type": "markdown", "source": "### Fuel cells\n\nFuel cells are devices to convert chemical energy to electricity and\nheat through electrochemical reactions with the fuel and oxygen supplied\nto the anode and cathode, respectively. Typical ions migrating through\nthe electrolyte are $H^{+}$, ${OH}^{-}$, ${CO}_{3}^{2 -}$, and\n$O^{2 -}$. In fuel cells with $H^{+}$ as migrating ions, H2\nmolecules are dissociated into $H^{+}$ on the anode, which are combined\nwith O2 on the cathode to form H2O and release\nheat with the half cell and the net cell reactions in their simplest\nform as shown by for the anode, for the cathode, and for the net cell,\nrespectively. Commonly used electrolytes are polymer and phosphoric\nacid, and both anode and cathode reactions are facilitated by catalyst,\ntypically platinum. The thermodynamic limit of power which can be\ngenerated by the fuel cell is represented by\n\n*Eq. 8\u201163*\n$w = - \\mathrm{\\Delta}G = - \\mathrm{\\Delta}_{\\ }^{0}G_{cell} + RTln\\left( P_{H_{2}}P_{O_{2}}^{1/2} \\right)$\n\nFor fuel cells with anions as migrating ions, the anions are generated\non the cathode with H2O formed and heat generated on the\nanode. Their representative cathode reactions are\n\n*Eq. 8\u201164* $\\frac{1}{2}O_{2} + H_{2}O + 2e^{-}{= 2OH}^{-}$\n\n*Eq. 8\u201165* ${\\frac{1}{2}O}_{2} + CO_{2} + 2e^{-} = {CO}_{3}^{2 -}$\n\n*Eq. 8\u201166* $\\frac{1}{2}O_{2} + 2e^{-} = O^{2 -}$.\n\nThe anode reaction for is the reaction represented by , operating at low\ntemperatures and using catalyst for both electrodes. The anode reactions\nfor and are\n\n*Eq. 8\u201167* ${CO}_{3}^{2 -} + H_{2} = H_{2}O + CO_{2} + 2e^{-}$\n\n*Eq. 8\u201168* $O^{2 -} + H_{2} = H_{2}O + 2e^{-}$\n\nrespectively. To enable the diffusion of ${CO}_{3}^{2 -}$ and $O^{2 -}$\nthrough the cathode and the electrolyte, both fuel cells are operated at\nrelative high temperatures, with the former typically in molten\ncarbonate solutions and the latter through solid oxides. Due to the high\noperating temperatures, fuels are converted to hydrogen within the fuel\ncell itself by a process called internal reforming, removing the need\nfor precious-metal catalyst and enabling the use of a variety of fuels.\nThe net cell reaction for all three fuel cells is the same as in the\ncase of $H^{+}$, represented by .\n", "metadata": {}}, {"id": "69d8c384", "cell_type": "markdown", "source": "### Galvanic protection\n\nA galvanic reaction takes place between two different materials at the\ntwo respective electrodes, each with different tendency to hold on to\nelectrons. Consider the following electrochemical cell used to protect\nCu tanks against oxidation by using a \u201csacrificial\u201d Fe electrode\n\nanode solution cathode\n\nFe(s) \\| Fe2+ \\| SO4= \\|\nCu2+\\| Cu(s) \\|\n\nCathode reduction: Cu2+ + 2 e- = Cu \u03b5o\n(volts) = 0.34\n\nAnode oxidation: Fe(s) = Fe2+ + 2 e- \u03b5o (volts) =\n0.44\n\nNet reaction: Cu2+ + Fe =\\> Cu + Fe2+\n\u03b5o (volts) = 0.78\n\nIf the cell has a direct connection between the electrodes, i.e. it has\na short circuit: \u2206G \u2192 0 and thus \u03b5cell \u2192 0. Since the value\nof \u03b5ocell \\> 0 for the net cell reaction, the equilibrium\nconstant *Ke* \\> 1, which means\n\\[Fe2+\\]/\\[Cu2+\\] \\> 1. By assuming an ideal\nelectrolyte solution, the activities in *Ke* can be\nrepresented by concentrations (in molar concentration units), assuming\nthat solid Fe and Cu are present at unit activities. If the electrodes\nof Cu and Fe are short circuited while in contact with the same\n\"electrolyte solution\", the final equilibrium concentrations can be\ncalculated by the standard equation,\n\n*Eq. 8\u201161* *Ke* = \\[Fe2+\\]/\\[Cu2+\\] =\nexp (-\u2206oG/RT)\n\nor, using the Nernst equations,\n\n*Eq. 8\u201162* \u03b5 = \u03b5o -(RT/z f) ln (\\[Fe2+\\]/\\[Cu2+\\])\n= 0\n\nand the above standard cell potential:\n\\[Fe2+\\]/\\[Cu2+\\] = 2.4\u20221026\n\nWith this large ratio it can be seen that the tendency to produce\nCu+2 ions, i.e., the tendency to corrode the Cu(s), is\nextremely small if a sacrificial Fe electrode is configured in an\nelectrochemical cell with the Cu tank.\n", "metadata": {}}, {"id": "1ee6e949", "cell_type": "markdown", "source": "### Half cell potentials\n\nWhen electron current flows between electrodes, reactions are occurring\nat the electrodes and concentration gradients causing polarization\ndevelop around the electrodes. These gradients result in extraneous\npotentials to occur at the electrodes. In such cases cell equilibrium is\nnot established and measured cell potentials are not those for true\npartial equilibrium. If a cell is short-circuited with the electrodes\nconnected by a conductor, current will flow until the external potential\nbecomes zero, i.e. \u03b5ext = 0, and equilibrium is established\nwith same conditions as non-electrochemical systems. If an external\npotential, \u03b5ext, is applied to the cell, chemical reactions\noccur until the cell potential balances to \u03b5ext, and no\ncurrent flows. This potential is called open-circuit voltage (OCV) in\nthe literature. It is important to realize that OCV includes all\nreactions that occur on the electrode surface when the electrode is in\ncontact with the electrolyte, such as passivation discussed in Chapter .\nPartial equilibrium in a cell is achieved when the cell potential is\nbalanced by an applied external potential. In such partial equilibrium\ncases, equilibrium thermodynamic analyses can be used even though the\ncell potential is not zero, i.e. \u03b5cell \u2260 0. This\ndifferentiates electrochemical systems from other equilibrium systems\ndiscussed previously.\n\nThe number of electrons involved in a net cell reaction is important in\nrelating cell potential and the Gibbs energy change for the cell\nreaction. As will be illustrated later in this section, this number\ndenotes the number of electrons involved in the half-cell reactions that\nwere added to yield the net cell reaction. The electrical work achieved\nby the transport of an electrical charge through a cell potential can be\nwritten as\n\n*Eq. 8\u201120* $w = z\\ f\\ \\varepsilon$\n\nwhere *z* represents the moles of electrons in cell reaction, *f* the\nFaraday constant equal to 96,485 J/V/mole-electron, and *\u03b5* the\npotential difference, often referred as electromotive force (emf) in the\nliterature. For a system at constant temperature, pressure, and\ncomposition, this work is the same as the Gibbs energy difference\nbetween the two electrodes, i.e.\n\n*Eq. 8\u201121* $- \\Delta G = w = zf\\varepsilon$\n\nwhere the negative sign is because the system does work to the\nsurrounding when the Gibbs energy of the system is decreased. When the\napplied external potential is larger than the cell potential, the\nsurrounding does work to the system, and a common example is the\ncharging of a battery. Thermodynamic relations discussed in previous\nchapters can thus be directly applied to electrochemical systems with\nsome common equations shown in .\n\nTable \u2011: Thermodynamic Equations for Electrochemical Cells\n\n*\u2206G = -z f \u03b5*\n\n*\u2206S = - (\u2202\u2206G/\u2202T)P = + z f (\u2202\u03b5/\u2202T)P*\n\n*\u2206H = \\[\u2202(\u2206G/T)/ \u2202(1/T)\\]P = - z f \\[\u2202(\u03b5/T)/\u2202(1/T)\\]P = z f \\[T(\u2202\u03b5/\u2202T)P\n\u2013 \u03b5\\]*\n\n*\u2206CP = (\u2202\u2206H/\u2202T)P = T z f (\u22022\u03b5/\u2202T2)P*\n\nA half-cell reaction potential cannot be measured directly, only its\npotential relative to another half-cell reaction. By convention, a\nstandard half-cell potential is measured relative to the standard\nhydrogen half-cell reduction reaction at 25oC (298K) and 1\nbar, which has a defined standard potential of zero volts,\n\n*Eq. 8\u201122* H+(aq, a=1) + e- = 1/2 H2(g,\n1 bar)\n\nwith \u03b5o (H+/H2,g) = 0.00 volts. The\nstandard half-cell reduction reactions of metals at 25oC are\nfor the general reaction\n\n*Eq. 8\u201123* Mz+(aq, a=1) + z e- = M(s)\n\nwith \u03b5o (Mz+/M) volts. Half-cell reactions with\nthe most positive standard electrode potentials have a tendency to\nspontaneously proceed toward reduction (cathode reactions). Half-cell\nreactions with the most negative standard electrode potentials have a\ntendency to spontaneously proceed toward oxidation (anode reactions).\n\nConsider, for example, a cell made up of a standard hydrogen electrode\nand a standard zinc electrode with \u03b5o\n(H+/H2,g) = 0.00 volts and \u03b5o\n(Zn2+/Zn) = -0.762 volts. Thus, the H+ would tend\nto be reduced, and the zinc metal would tend to be oxidized, and the\nspontaneous reaction if all species had unit activities would be\n\n*Eq. 8\u201124* 2 H+(aq, a=1)+ Zn = H2(1 bar) +\nZn2+(aq, a=1)\n\nwith \u03b5ocell = 0.762 volts and\n$\\mathrm{\\Delta}_{\\ }^{0}G = - 2*96485*\\varepsilon_{cell}^{0}\\ $. The\ncathode half-cell reaction would be the same as , while the anode\nhalf-cell reaction would be\n\n*Eq. 8\u201125* Zn = Zn2+(aq, a=1) + 2 e-\n\nWhen ion concentrations and H2 gas do not all have unit\nactivities, the Gibbs energy and cell potential of the cell reaction, ,\nbecomes\n\n*Eq. 8\u201126*\n$\\mathrm{\\Delta}G = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{a_{{Zn}^{2 +}}P_{H_{2}}}{\\left( a_{H^{+}} \\right)^{2}}$\n\n*Eq. 8\u201127*\n$\\varepsilon_{cell} = \\varepsilon_{cell}^{0} - \\frac{RT}{zf}\\ln\\frac{a_{{Zn}^{2 +}}P_{H_{2}}}{\\left( a_{H^{+}} \\right)^{2}}$\n\nThe standard reduction potentials of some common metals at\n25oC are given in .\n\nTable \u2011: Standard reduction potentials of some common metals\n\nA cell reaction can be established by different half-cell reactions. For\nexample, the following reaction can be derived from two different cells\n\n*Eq. 8\u201128* $3\\ {Fe}^{2 + \\ }\\ = \\ \\ 2\\ {Fe}^{3 +}\\ \\ + \\ \\ Fe(s)$\n\ncell A\n\n*Eq. 8\u201129* $3{Fe}^{2 +} + \\ 6\\ e - \\ = \\ \\ 3Fe(s)$ \u03b5o1 =\n-0.440 V\n\n*Eq. 8\u201130* $2\\ Fe(s)\\ \\ = \\ \\ 2\\ {Fe}^{3 +}\\ \\ + \\ 6\\ e -$\n\u03b5o2 = +0.036 V\n\ncell B\n\n*Eq. 8\u201131* $2{Fe}^{2 +}\\ = \\ 2{Fe}^{3 + \\ \\ } + 2\\ e^{-}$\n*\u03b5o4 = -0.772 V*\n\n*Eq. 8\u201132* ${Fe}^{2 + \\ }\\ + \\ 2\\ e - \\ \\ = \\ \\ Fe(s)$ *\u03b5o5\n= -0.440 V*\n\nBoth give the same net reaction shown by , but with 6 and 2 electrons\nand standard cell potentials being \u03b5ocell A = -0.404 V and\n\u03b5ocell B = -1.212 V, respectively. However, the standard\nGibbs energies of both cells are the same, i.e.\n\n*Eq. 8\u201133* $\\mathrm{\\Delta}_{\\ }^{0}G$*cell A = -6 f (-0.404)\n= + 2.424 f*\n\n*Eq. 8\u201134* $\\mathrm{\\Delta}_{\\ }^{0}G$*cell B = -2 f (-1.212)\n= + 2.424 f*\n\nIt is shown that $\\mathrm{\\Delta}_{\\ }^{0}G$ values are independent of\nhalf-cell reactions and depend only on the net reaction because the net\nreaction is neutral in electron and balanced in mass.\n", "metadata": {}}, {"id": "976f0090", "cell_type": "markdown", "source": "### Ion transport membranes\n\nIon transport membranes (ITMs) are ceramic membranes conducting both\nelectrons and oxygen ions, but no other species. The chemical potential\ndifference of oxygen between two sides of a membrane provides the\ndriving force for oxygen to diffuse through the membrane. Commonly used\nITM oxides include perovskite and fluorite, with the chemical formula of\nABO3 and AO2, respectively, typically with more\nthan one elements in the A-site and / or the B-site to tailor the\nelectron and ionic conductivities. Key thermodynamic properties of ITM\noxides are their stability in service environments, vacancy\nconcentrations in the oxygen and cation sites, valances of cations. On\nthe high oxygen partial pressure side, the reaction is as the following\n\n*Eq. 8\u201169* $\\frac{1}{2}O_{2} + 2e^{-} = O^{2 -}$\n\nAt the same time the number of oxygen vacancy is reduced, resulting in a\nlower concentration of oxygen vacancy and higher oxygen activity in the\noxide on the high oxygen partial pressure side. On the low oxygen\npartial pressure side, the reaction is reversed to produce oxygen\nmolecules, i.e.\n\n*Eq. 8\u201170* $O^{2 -} = \\frac{1}{2}O_{2} + 2e^{-}$\n\nThis reaction results in higher oxygen vacancy concentration and lower\noxygen activity. At both sides, the charge neutrality is compensated by\nthe valance changes of cations, resulting in the electron flow in the\nopposite direction of oxygen diffusion. The ionic conductivity is\ndictated by the oxygen transportation across the membrane with the\ndriving force of the following net reaction\n\n*Eq. 8\u201171*\n$\\frac{1}{2}O_{2}\\left( P_{high} \\right) = \\frac{1}{2}O_{2}\\left( P_{low} \\right)$\n\n*Eq. 8\u201172*\n$\\mathrm{\\Delta}G = 0.5RTln\\left( \\frac{P_{low}}{P_{high}} \\right)$.\n\nThe oxygen transportation is closely related to the concentration of\noxygen vacancy in the membrane, which is obtained by minimizing the\nGibbs energy of the phase under given temperature and oxygen partial\npressure conditions. High vacancy concentrations can be obtained by\ncation dopants with lower valances or small energy differences between\nvarious valance states. However, at the same time, high vacancy\nconcentrations reduce the thermodynamic stability of the membrane, which\nmay result in its decomposition into more stable phases.\n", "metadata": {}}, {"id": "90efadcf", "cell_type": "markdown", "source": "### Metastability and passivation\n\nOur modern industrial society is built on various metals such as Fe, Ni,\nAl, Ti, and Zr alloys which are reactive, but exhibit extraordinary\nkinetic stabilities in oxidizing environments due to the existence of a\nthin reaction product film on the surface. This film effectively\nisolates the metal from the corrosive environment, a phenomenon called\npassivation. One interesting experiment by Faraday in 1836 who reported\nthat iron corrodes freely in dilute nitric acid, while in concentrated\nnitric acid, no reaction apparently occurred. To understand this\nphenomenon, let us exam a simple, schematic Pourbaix diagram for the\niron-water system shown in .\n\nFigure \u2011: Schematic Pourbaix diagram for iron illustrating the\nresolution of the Faraday paradox in the corrosion of iron in nitric\nacid \\[61\\]. Lines (a) (b), and (c) correspond to the equilibria: (a)\n$H^{+} + e^{-} = 1/2H_{2}$; (b) $O_{2} + 4H^{+} + 4e^{-} = 2H_{2}O$; (c)\n$NO_{3}^{-} + 3H^{+} + 2e^{-} = HNO_{2} + H_{2}O$, respectively\n\nFor iron in deaerated acid solution, the partial anodic and cathodic\nreactions are Line 1 $\\left( Fe/{Fe}^{2 +} \\right)$ and Line (a),\nrespectively, resulting in a corrosion potential that lies between Lines\n$1$ and $a$. In oxygenated (aerated) solutions, the corrosion potential\nmay lie between Lines $1$ and $b$, because the reduction of oxygen is a\npossible (likely) cathodic reaction. Since dilute HNO3 is\nonly a weak oxidizing agent, the principal cathodic reaction was most\nlikely hydrogen evolution, and hence the corrosion potential is expected\nto lie between Lines $1$ and $a$ at relatively high pH, as shown. Since\nthe $Fe/{Fe}^{2 +}$ reaction is relatively fast compared with\n$H^{+}/H_{2}$ on iron, if the corrosion potential is situated below the\nextension of Line 2 (Fe/Fe3O4) into the\n${Fe}^{2 +}$ stability region, Fe3O4 cannot form\non the surface, even as a metastable phase.\n\nOn the other hand, concentrated HNO3 is a strong oxidizing\nagent due to the reaction:\n$NO_{3}^{-} + 3H^{+} + 2e^{-} = HNO_{2} + H_{2}O$, so that the corrosion\npotential can lie anywhere between Lines $1$ and $c$ at low pH. Since\nReaction (2) is likely to be fast, the corrosion potential will be high\nand certainly will be more positive than the extension of Line 2 into\nthe stability region for ${Fe}^{2 +}$ at low pH. Therefore,\nFe3O4 becomes metastable and can form between the\naqueous solution and iron. The thickness of this\nFe3O4 film depends on its dissolution rate into\nthe aqueous solution and its growth rate at the interface with iron,\nwhich depends on the diffusion of ionic species across the film. Its\nexistence results in passivity and the observed kinetic inactivity of\niron in this medium. When the potential becomes even more positive above\nthe extension of Line $3$, Fe2O3 may form on\nFe3O4 as an additional metastable phase, resulting\nin the commonly observed bilayer structure.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/applications_to_electrochemical_systems/index.ipynb b/psu410/src/applications_to_electrochemical_systems/index.ipynb deleted file mode 100644 index 080631b..0000000 --- a/psu410/src/applications_to_electrochemical_systems/index.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "0488302e", "cell_type": "markdown", "source": "# Applications to electrochemical systems\n\nThe two basic types of chemical processes involving ions as reactant\nand/or product species are electrolyte reactions and electrochemical\nreactions. *Electrolyte reactions* are accompanied by the *atomic-scale\nmovement* of ionic species and possibly electrons. Chemical changes that\nproduce changes in valence and *electron and ion transport over finite\ndistances* constitute an area of science termed *electrochemistry*. The\nlatter chemical changes occur in an electrochemical cell comprised of\ntwo electrodes, an anode and a cathode, which are coupled by an\nelectrolyte and an external electron conductor. Most thermodynamic\nconcepts and analyses described in previous chapters remain unchanged\nwhen applied to electrochemistry, but the analysis of electrochemical\nsystems does require some new terminology, new definitions, and new\nconventions. The primary focus of this chapter is on applications of\nthermodynamics to electrochemical reactions that involve either aqueous\nelectrolyte solutions or solid state electrolytes. Since all\nelectrochemical systems include ionized species as reactant and/or\nproduct species, electrolyte reactions will also be discussed.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/calpahd_modeling_of_thermodynamics/importance_of_lattice_stability.ipynb b/psu410/src/calpahd_modeling_of_thermodynamics/importance_of_lattice_stability.ipynb deleted file mode 100644 index 0f113b4..0000000 --- a/psu410/src/calpahd_modeling_of_thermodynamics/importance_of_lattice_stability.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "ca973b9e", "cell_type": "markdown", "source": "## Importance of lattice stability\n\nFor modeling of Gibbs energy of individual phases, it is necessary to\ndefine the values of $_{\\ }^{0}G_{i}$ in . However, the independent\ncomponent *i* may not be stable in the crystal structure of the phase\nunder consideration, so its Gibbs energy could not be obtained directly\nfrom experiments and must be estimated with respect to the Gibbs energy\nof their stable crystal structure. In the pioneering work by Kaufman\n\\[47\\], this Gibbs energy difference was termed as lattice stability and\nobtained through extrapolations in either temperature-pressure or\ntemperature-composition phase diagrams. It is evident from that the\nvalues of $_{\\ }^{0}G_{i}$ and $_{\\ }^{M}G$ jointly contribute to the\nGibbs energy of the solution, and Kaufman had to simplify the treatment\nof $_{\\ }^{M}G$ in order to show the importance of the concept of\nlattice stability. Using ideal or regular solution models, Kaufman was\nable to define the lattice stability for pure elements and remarkably\nreproduce many features of binary phase diagrams by introducing the\ninteraction parameters afterwards.\n\nOver the years, there had been various revisions of lattice stability\nvalues for common crystal structures \\[48\\], and every revision\nnecessitates the re-evaluation of interaction parameters in the solution\nphase shown in . It was until the lattice stability values established\nby the Scientific Group Thermodata Europe (SGTE) \\[52\\] that the\ndevelopment of binary thermodynamic models using the same thermodynamic\nmodels of pure elements became possible, and those binary models thus\ndeveloped in different groups around the world can be combined to create\nthermodynamic models of ternary and multi-component systems. Clearly,\nany further modifications of the SGTE pure element database will require\nthe re-modeling of all binary and ternary systems in which the models of\npure elements are changed. This challenge is briefly addressed in the\nlater part of this chapter.\n\nA less addressed issue is the Gibbs energy of end-members in\nnon-stoichiometric compounds, i.e. , where each sublattice contains only\none element. In case all sublattices are occupied by the same element,\nit is the lattice stability of the elements in the structure of the\ncompound. Since the stable composition ranges of non-stoichiometric\ncompounds are typically small, the existing method cannot be used to\nreliably evaluate the Gibbs energy of end-members, and currently there\nis not a commonly accepted lattice stability database for compounds.\nMost values used in the existing databases have been either roughly\nestimated or computed from first-principles calculations. Such a\nstandard database is highly desirable in order to make various models of\ncompounds compatible.\n\nIn an effort to compare the lattice stability from the CALPHAD models\nand the first-principles calculations, Wang et al. \\[53\\] systematically\ncalculated the total energies of 78 pure elements at zero Kelvin in the\nface-centered-cubic (fcc), body-centered-cubic (bcc), and\nhexagonal-close-packed (hcp) crystal structures using the projector\naugmented-wave (PAW) method within the generalized gradient\napproximation (GGA). The calculated values are compared with the values\nin the SGTE database as shown in and . For non-transition metal\nelements, the differences between the SGTE data and the PAW-GGA data are\ntypically around 1\u223c2 kJ/mole-of-atoms or less, while for some transition\nmetal elements, the differences can be quite large, for example, as high\nas about 54 kJ/mole-of-atoms for and about 40 kJ/mole-of-atoms for . and\npresent the differences between the PAW-GGA data and the SGTE data, for\nelements from the Ti group to the Ni group, respectively.\n\nTable \u2011: Lattice stability Ebcc-fcc (kJ/mole-of-atoms).\n\nTable \u2011: Lattice stability Ehcp-fcc (kJ/mole-of-atoms).\n\nFigure \u2011: Lattice stability difference between bcc and fcc, , for\nselected elements between PAW-GGA and SGTE\n\nFigure \u2011: Lattice stability difference between hcp and fcc, , for\nselected elements between PAW-GGA and SGTE\n\nThe large differences between the first-principles calculations and the\nSGTE data could partly be attributed to the instability of the\nhigher-energy phases, the entropies of which at finite *T* become\nabnormal. The lattice instabilities along the tetragonal transformation\npath between fcc and bcc structures with the continuous change of the\nc/a ratio defined in a bcc-based tetragonal lattice are demonstrated for\nbcc Mo, Ta, W in and for fcc Al, Cu, Ni in . It is shown that the fcc\nstructure of bcc Mo, Ta, and W is a local maximum with respect to the\ntetragonal transformation, and the higher the maximum is, the larger the\ndiscrepancy between the SGTE data and the present PAW-GGA data, while\nfor fcc Al, Cu, Ni, the bcc structure is at a local maximum. Similarly,\nthe lattice instabilities along the tetragonal transformation path for\nthe hcp metals Ru and Os as shown in . The behavior of energy against\nc/a ratio of these two hcp metals is very similar to those of fcc\nelements.\n\nFigure \u2011: Total energy, , along the Bain deformation path between bcc\nand fcc for Mo, Ta, and W.\n\nFigure \u2011: Total energy, , along the tetragonal transformation path\nbetween bcc and fcc for Ni, Al, and Cu.\n\nFigure \u2011: Total energy, , along the tetragonal transformation path for\nRu and Os\n\nIt can be concluded that a fcc structure for elements with bcc being the\nground state or a bcc structure for elements with fcc being the ground\nstate, is unstable with respect to the tetragonal transformation. For an\nunstable structure, the harmonic description of its vibrational entropy\nis thermodynamically incorrect since the potential surface seen by the\nlattice ion can no longer be approximated by a parabola. If an unstable\nstructure of a pure element is stabilized at high temperatures, its\nentropy has to be abnormal. The instability issue has been recently\naddressed by ab initio molecular dynamics simulations at high\ntemperatures using W as an example \\[54\\], which is beyond the scope of\nthe book and thus not discussed here.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/calpahd_modeling_of_thermodynamics/index.ipynb b/psu410/src/calpahd_modeling_of_thermodynamics/index.ipynb deleted file mode 100644 index 0be6e16..0000000 --- a/psu410/src/calpahd_modeling_of_thermodynamics/index.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "ab0f51db", "cell_type": "markdown", "source": "# CALPAHD modeling of thermodynamics\n\nCALPHAD modeling of thermodynamics was pioneered by Kaufman \\[47\\] and\nhas been reviewed in details by Saunders and Miodownik \\[48\\] and Lukas,\nFries and Sundman \\[49\\]. Information on features of software tools for\nCALPHAD modeling can be found at two series of publications in the\nCALPHAD journal \\[50-51\\]. The key feature of the CALPHAD method is the\nmodeling of Gibbs energy of individual phases using both thermodynamic\nand phase equilibrium data. The main significances of the CALPHAD method\nare as follows\n\n1. It enabled the development of the concept of lattice stability, i.e.\n the energy difference of a pure element with stable and non-stable\n crystal structures;\n\n2. The Gibbs energy expression of each phase covers the full\n temperature, pressure, and composition spaces including both stable\n and non-stable regions of the phase. This enables the evaluation of\n the Gibbs energy of a system as a function of non-equilibrium state,\n i.e. with \u03be as an independent variable;\n\n3. Thermodynamic data are usually obtained by measurements of heat such\n as enthalpy of transition and heat capacity as discussed in Chapter\n , which bear large uncertainties typically in the range of\n kilojoules per mole-of-atom. On the other hand, phase equilibrium\n data as discussed in Chapter , though more accurate, only contain\n information on compositions of phases at equilibria, i.e., the\n relative Gibbs energy of phases at equilibrium. The combination of\n these two sets of data is foundational in CALPHAD modeling that\n allows for accurate modeling of thermodynamic properties of\n individual phases and reliable calculations of phase stability and\n driving forces;\n\n4. It provides a framework to model thermodynamic properties of\n multi-component systems of industrial importance, enabling the\n computational materials design. It has also been extended to model a\n range of properties of individual phases in multi-component systems\n such as diffusion coefficients, elastic coefficients, and thermal\n expansion, supplying input data for computational simulations of\n phase transformations during materials processing.\n\nIn this chapter, the basics of CALPHAD modeling of Gibbs energy of\nindividual phases are presented. For detailed implementations in various\nsoftware packages and modeling procedures, readers are referred to the\nreferences listed above.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_pure_elements.ipynb b/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_pure_elements.ipynb deleted file mode 100644 index ede1115..0000000 --- a/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_pure_elements.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "981c197e", "cell_type": "markdown", "source": "## Modeling of pure elements\n\nIn modeling the Gibbs energy of pure elements in the SER structure in\nterms of , the coefficients in are evaluated using the heat capacity\ndata, $b^{'}$ in is evaluated by the value of $S_{298.15}$, and $a$ in\nis evaluated by $H_{298.15}^{SER} = 0$, respectively. For the high\ntemperature phase, the enthalpy of transformation from the low\ntemperature phase to the high temperature phase,\n${\\mathrm{\\Delta}H}_{trans}^{}$, can be measured by calorimetry methods\ndiscussed in Chapter , and the entropy of transformation,\n${\\mathrm{\\Delta}S}_{trans}^{}$, is then calculated using the\nequilibrium condition of equal Gibbs energy of the two phases, i.e.\n\n*Eq. 6\u20111*\n${\\mathrm{\\Delta}S}_{trans}^{} = \\frac{{\\mathrm{\\Delta}H}_{trans}^{}}{T_{trans}^{}}$\n\nwhere $T_{trans}^{}$ is the transition temperature.\n${\\mathrm{\\Delta}H}_{trans}^{}$ and ${\\mathrm{\\Delta}S}_{trans}^{}$ are\nthen used to evaluate the integration constants, $b^{'}$ and $a$, in the\nplace of $S_{298.15}$ and $H_{298.15}^{SER}$ for the structure in the\nSER state.\n\nThis works well for the stable temperature range of each phase. However,\nthere is an issue in extrapolation above and below the melting\ntemperature (Tm). It is known that the heat capacity of the\nsolid phase, , increases with temperature, while that of the liquid\nphase, , is typically constant. The extrapolation of the Gibbs energy of\na solid phase to above its melting temperature can result in the solid\nphase becoming more stable than the liquid phase at high temperatures.\nBy the same token, the extrapolation of the Gibbs energy of a liquid\nphase to below its melting temperature can result in the liquid phase\nbecoming more stable than the solid phase at low temperatures. To\naddress this problem, it is proposed by SGTE that the heat capacity of\nthe solid phase approaches that of the liquid at high temperatures, and\nthat of the liquid phase approaches that of the solid phase at low\ntemperatures using the following equations\n\n1. for solid at T\\>Tm\n\n*Eq. 6\u20112*\n\n*Eq. 6\u20113*\n\n2. for liquid at T\\m\n\n*Eq. 6\u20114*\n\n*Eq. 6\u20115*\n\nAs an example, the heat capacity of solid fcc Al and liquid Al in the\nSGTE pure element database is plotted in . It can be seen that the heat\ncapacity of fcc Al approaches that of liquid Al at high temperatures,\nwhile the heat capacity of liquid Al approaches that of fcc Al at low\ntemperatures. It ensures that the liquid Al is stable at high\ntemperatures, and fcc Al is stable at low temperatures. However, this\nsimple model for liquid is often not satisfactory, in comparison with\navailable experimental data in supercooled liquid, particularly those\nsystems with glass transitions. New models are thus needed and are being\ndeveloped in the CALPHAD community.\n\nFigure \u2011: Heat capacity of fcc Al solid and liquid as a function of\ntemperature\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_random_solution_phases.ipynb b/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_random_solution_phases.ipynb deleted file mode 100644 index 9b271cb..0000000 --- a/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_random_solution_phases.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "a51f0d6e", "cell_type": "markdown", "source": "## Modeling of random solution phases\n\nDepending on the degree of short-range ordering in a solution phase,\nvarious Gibbs energy models are available as discussed in Chapter . When\nthe short-range ordering is weak, it can be accounted for by the\ncomposition-dependence of excess Gibbs energy in a binary system in\nterms of the Redlich-Kister polynomial as follows\n\n*Eq. 6\u20117*\n\nwhere the interaction parameters, , can be temperature dependent or even\nhave contributions from heat capacity in the form of when data is\navailable. shows that and are symmetrical with respect of composition,\nwhile is asymmetrical. Their individual contributions to the excess\nGibbs energy are shown in with all interaction parameters being -30,000\nJ/mole-of-atoms.\n\nFigure \u2011: Contributions of interactions parameters to the excess Gibbs\nenergy\n\nIt can be seen in that even though all interaction parameters are\nnegative, the asymmetrical feature of results in the curvature change in\nthe excess Gibbs energy as a function of composition. This indicates the\ntendency to form a miscibility gap at low temperatures. The interaction\nparameters are to be evaluated from the data of enthalpy, entropy, and\nheat capacity of mixing. The experimental data on enthalpy of mixing are\navailable for the liquid phase in some systems, but typically very\nlimited for solid solution phases. The first-principles calculations can\npredict the enthalpy, entropy, and heat capacity of mixing in solid\nsolution phases using the dilute solution approach with one solute atom\nin a supercell and the CPA/CE/SQS approach for concentrated solutions as\ndiscussed in Chapter . It demonstrates again that the interaction\nparameters and the lattice stability jointly determine the Gibbs energy\nof an individual phase. The change of lattice stability requires the\nre-evaluation of interaction parameters.\n\nFor individual phases with strong short-range ordering, quasichemical or\nassociated models can be used. As discussed in Chapter , with fixed\ncomposition in the system, the amounts of various chemical bonds or\nassociates are related through the mass conservation in the system and\nare calculated through the minimization of Gibbs energy of the phase\nwith given temperature, pressure, and the amount of each independent\ncomponents. The model parameters include the formation energy of bonds\nor associates and interaction between various bonds or associates,\nnoting that pure elements can be considered as the simplest associates.\nThe interactions between pure elements can be predicted from\nfirst-principles calculations as mentioned above, but currently there\nare no efficient approaches to predict the interactions between\nassociates from first-principles calculations.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_solution_phases_with_longrange_ordering.ipynb b/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_solution_phases_with_longrange_ordering.ipynb deleted file mode 100644 index 5b0e43d..0000000 --- a/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_solution_phases_with_longrange_ordering.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "3bc1eecc", "cell_type": "markdown", "source": "## Modeling of solution phases with long-range ordering\n\nThe commonly used Gibbs energy model is shown in Chapter with the\ncrystal lattice divided into sublattices, often referred to as the\ncompound energy formalism \\[55\\]. The Gibbs energy of end-members\nrepresented by plays the same important role for solution phases with\nsublattice as the lattice stability for random solution phases. The\nend-members are modeled in the same way as the stoichiometric phases\ndiscussed in Chapter . The enthalpy and entropy of mixing in each\nsublattice can be predicted by first-principles calculations using the\ndilute solution and SQS approaches discussed in Chapter and modeled in\nthe same way as the random solution discussed in Chapter .\n\nIt is important to realize that with a simple two-sublattice model of\n(A,B)a(C,D)b, the miscibility gap can easily form\neven without any interaction parameters when the Gibbs energy of\nend-members differ from each other significantly. The contribution of\nend-members to the Gibbs energy of the phase, i.e. , is re-written as\nfollows and schematically shown in ,\n\n*Eq. 6\u20118*\n\nFigure \u2011: Schematic diagrams depicting (a) the concentration square with\nthe site fractions of B and D on the horizontal and vertical axes,\nrespectively, and (b) Gibbs energy reference plan for\n(A,B)a(C,D)b, as represented by .\n\nFrom b, it is evident that there is a strong tendency to form a\nmiscibility gap between the composition sets of\n(A)a(C)b and (B)a(D)b due to\ntheir lower Gibbs energies than the other two end-members. Since it\nwould be rare for all four end-members to have their Gibbs energy values\nequal, the miscibility gap in this type of phases is practically\ninevitable at low temperatures. An example is shown in for the complex\ntitanium niobium carbonitride (Ti,Nb)(C,N). The solid lines parallel to\nthe direction from NbC to TiN are tie-lines. The Gibbs energy values of\nTiC, TiN, NbC and NbN are -144495, -229236, -132324, and -179772\nJ/mole-of-atoms, respectively. The Gibbs energy value of TiN is\nsignificantly lower than the other values, resulting in the tie-lines\noriginating from the TiN corner.\n\nFigure \u2011: Miscibility gap in (Ti,Nb)(C,N) at 1673K with the straight\nlines in the middle of the plot being tie-lines.\n\nThe order-disorder transitions can be similarly described with the\nsimplest case being a two-sublattice model of\n(A,B)a(A,B)b. When the site fractions of A or B in\nboth sublattices are the same, it becomes a random solution model; when\nthey are different, the phase is partially ordered; and when there is\nonly one component in each sublattice, the phase is fully ordered as a\nstoichiometric compound. The Gibbs energy of this phase is obtained from\nas follows\n\n*Eq. 6\u20119*\n\n$G_{mf} = y_{A}^{I}y_{A}^{II}\\ _{\\ }^{0}G_{A:A} + y_{A}^{I}y_{B}^{II}\\ _{\\ }^{0}G_{A:B} + y_{B}^{I}y_{A}^{II}\\ _{\\ }^{0}G_{B:A} + y_{B}^{I}y_{B}^{II}\\ _{\\ }^{0}G_{B:B} + + aRT\\left( y_{A}^{I}\\ln y_{A}^{I} + y_{B}^{I}\\ln y_{B}^{I} \\right) + bRT\\left( y_{A}^{II}\\ln y_{A}^{II} + y_{B}^{II}\\ln y_{B}^{II} \\right) + y_{A}^{II}y_{A}^{I}y_{B}^{I}\\ L_{A,B:A} + y_{B}^{II}y_{A}^{I}y_{B}^{I}\\ L_{A,B:B} + y_{A}^{I}y_{A}^{II}y_{B}^{II}\\ L_{A:A,B} + y_{B}^{I}y_{A}^{II}y_{B}^{II}\\ L_{B:A,B} + y_{A}^{I}y_{B}^{I}\\ y_{A}^{II}y_{B}^{II}L_{A,B:A,B}$\n\nwhere the superscript denotes the sublattice, and column and comma\nseparate sublattices and interaction components, respectively. The\nrelationship between site fraction and over-all atomic fractions in such\na two-sublattice model can be represented by and schematically shown in\n. The two red dashed lines represent the phase with\n$x_{B} = \\frac{a}{a + b}$, but different a/b ratios. Along the red\ndashed lines, the phase can adjust the site fraction to minimize its\nGibbs energy, i.e. it has one internal degree of freedom to be either\ndisordered on the blue diagonal line between A:A and B:B, or ordered at\nanywhere else. The interplay of interaction parameters and site\nfractions is depicted where $L_{A,B:A}$, $L_{A,B:B}$, $L_{A:A,B}$, and\n$L_{B:A,B}$ affect the four sides, and $L_{A,B:A,B}$ influences the\ncenter part.\n\nFigure \u2011: Schematic composition square of\n(A,B)a(A,B)b.\n\nWhen fully disordered with $y_{A}^{I} = \\ y_{A}^{II} = x_{A}$ and\n$y_{B}^{I} = y_{B}^{II} = x_{B}$, becomes\n\n*Eq. 6\u201110*\n\n$$G_{mf} = x_{A}\\left( 1 - x_{B} \\right)_{\\ }^{0}G_{A:A} + x_{A}x_{B}\\ _{\\ }^{0}G_{A:B} + x_{A}x_{B}\\ _{\\ }^{0}G_{B:A} + x_{B}\\left( 1 - x_{A} \\right)\\ _{\\ }^{0}G_{B:B} + + (a + b)RT\\left( x_{A}\\ln x_{A} + x_{B}\\ln x_{B} \\right) + \\ x_{A}x_{B}x_{A}L_{A,B:A} + x_{A}x_{B}x_{B}\\ L_{A,B:B} + x_{A}x_{B}x_{A}\\ L_{A:A,B} + x_{A}x_{B}x_{B}\\ L_{B:A,B} + x_{A}x_{A}x_{B}x_{B}L_{A,B:A,B} = x_{A}_{\\ }^{0}G_{A:A} + x_{B}\\ _{\\ }^{0}G_{B:B} + (a + b)RT\\left( x_{A}\\ln x_{A} + x_{B}\\ln x_{B} \\right) + x_{A}x_{B}\\left\\lbrack \\left(_{\\ }^{0}G_{A:B} +_{\\ }^{0}G_{B:A} -_{\\ }^{0}G_{A:A} -_{\\ }^{0}G_{B:B} \\right) + x_{A}\\left( L_{A,B:A} + L_{A:A,B} \\right) + x_{B}\\ \\left( L_{A,B:B} + L_{B:A,B} \\right) + x_{A}x_{B}L_{A,B:A,B} \\right\\rbrack = (a + b)\\left\\lbrack x_{A}_{\\ }^{0}G_{A} + x_{B}\\ _{\\ }^{0}G_{B} + RT\\left( x_{A}\\ln x_{A} + x_{B}\\ln x_{B} \\right) + x_{A}x_{B}L_{A,B} \\right\\rbrack$$\n\nwith\n\n*Eq. 6\u201111* $_{\\ }^{0}G_{A:A} = (a + b)_{\\ }^{0}G_{A}$\n\n*Eq. 6\u201112* $_{\\ }^{0}G_{B:B} = (a + b)_{\\ }^{0}G_{B}$\n\n*Eq. 6\u201113*\n\n$$L_{A,B} = \\left\\lbrack \\left(_{\\ }^{0}G_{A:B} +_{\\ }^{0}G_{B:A} -_{\\ }^{0}G_{A:A} -_{\\ }^{0}G_{B:B} \\right) + x_{A}\\left( L_{A,B:A} + L_{A:A,B} \\right) + x_{B}\\ \\left( L_{A,B:B} + L_{B:A,B} \\right) + x_{A}x_{B}L_{A,B:A,B} \\right\\rbrack/(a + b)$$\n\nwhere $_{\\ }^{0}G_{A}$, $_{\\ }^{0}G_{B}$, and $L_{A,B}$ are the molar\nGibbs energy of pure A and B and the molar interaction parameter in the\ndisordered solid solution, respectively. It is evident that the\ninteratcation paramter $L_{A,B}$ is fully determined by the parameters\nin the ordered phase, but the parameters in the ordered phase are not\nuniquely determined by the interaction parameters in the disordered\nphase.\n\nDue to crystal symmetry, some of the parameters in are related. For\nexample, in the BCC A2/B2 ordering with a=b=0.5, the BCC corner or\ncenter lattice sites are favored by either one type of atoms, but the\ntwo sublattices are equivalent crystallographically, resulting in\nfollowing relations\n\n*Eq. 6\u201114* $\\ _{\\ }^{0}G_{A:B} = \\ _{\\ }^{0}G_{B:A}$\n\n*Eq. 6\u201115* $L_{A,B:A} = L_{A:A,B}$\n\n*Eq. 6\u201116* $L_{A,B:B} = \\ L_{B:A,B}$\n\nFor more complex ordering of BCC lattice such as B32, D03,\nand L21 shown in with ideal compositions being AB,\nA3B, and A2BC, respectively, more sublattices are\nneeded, noting that the L21 Heusler structure exists in\nternary systems only. To use one model to describe all ordering in the\nBCC lattice, the minimum cluster is an irregular tetrahedron with four\nsublattices as depicted in as discussed in the modeling of the Al-Fe\nsystem \\[56\\]. In such a four sublattice model of\n$(A,B)_{0.25}^{I}(A,B)_{0.25}^{II}(A,B)_{0.25}^{II}(A,B)_{0.25}^{IV}$,\nthe site fractions of A2, B2, B32, and D03, are represented\nby $y_{i}^{I} = \\ y_{i}^{II} = y_{i}^{III} = \\ y_{i}^{IV}$,\n$y_{i}^{I} = \\ y_{i}^{II} \\neq y_{i}^{III} = \\ y_{i}^{IV}$,\n$y_{i}^{I} = \\ y_{i}^{III} \\neq y_{i}^{II} = \\ y_{i}^{IV}$, and\n$y_{i}^{I} = \\ y_{i}^{II} \\neq y_{i}^{III} \\neq \\ y_{i}^{IV}$,\nrespectively. The site fractions of L21 are the same as those\nof D03 except with at least three components.\n\nFigure \u2011: Atomic structures and four sublattice tetrahedrons of BCC\ndisordered and ordered phases.\n\nAnother common ordering phenomenon is in the FCC lattice including the\ndisordered A1 structure and ordered L10 and L12\nstructures as shown in . In the L10 structure, the\nneighboring (001) planes are favored by different atoms, respectively,\nresulting in an ideal composition of AB. While in the L12\nstructure, the corners and faces are favored by different atoms,\nrespectively, resulting in an ideal composition of A3B. In a\nfour sublattice model of\n$(A,B)_{0.25}^{I}(A,B)_{0.25}^{II}(A,B)_{0.25}^{II}(A,B)_{0.25}^{IV}$,\nthe site fractions of A1, L10 and L12 are\nrepresented by $y_{i}^{I} = \\ y_{i}^{II} = y_{i}^{III} = \\ y_{i}^{IV}$,\n$y_{i}^{I} = \\ y_{i}^{II} \\neq y_{i}^{III} = \\ y_{i}^{IV}$, and\n$y_{i}^{I} = \\ y_{i}^{II} = y_{i}^{III} \\neq \\ y_{i}^{IV}$, respectively\n\\[57\\].\n\nFigure \u2011: Atomic structures and four sublattice tetrahedrons of FCC\ndisordered and ordered phases.\n\nAs mentioned above, the interaction parameters in each sublattice can be\npredicted by first-principles calculations using the dilute solution and\nSQS approaches when there is only one component in each of the remaining\nsublattices. For the interaction involving two components in two or more\nsublattices, i.e. $L_{A,B:A,B}$ in applicable to four sublattice models\n\\[58\\], the energetics from cluster expansion (CE) approach discussed in\nChapter can be used to evaluate the interaction parameters.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_stoichiometric_phases.ipynb b/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_stoichiometric_phases.ipynb deleted file mode 100644 index b4a443b..0000000 --- a/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_stoichiometric_phases.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "8ce82e48", "cell_type": "markdown", "source": "## Modeling of stoichiometric phases\n\nThe Gibbs energy of a stoichiometric phase can be modeled in the same\nway as that of pure elements discussed above using the data of heat\ncapacity, $S_{298.15}$, and enthalpy of formation at 298.15 K (). When\nthese data are not available from experiments, they can be predicted by\nfirst-principles calculations as discussed Chapter . It should be\npointed out that constraints placed on heat capacity of stoichiometric\ncompounds above melting temperatures, i.e. and , have not been\nrigorously implemented in the literature for such modeling, partially\nbecause the heat capacity of the corresponding liquid solution is not\nwell established, and the heat capacity of a compound is often not\navailable.\n\nWhen the data of heat capacity is not available, a simple approach,\ncommonly referred to as Neumann\u2013Kopp Rule assuming that the heat\ncapacity of formation of is zero, can be used. The Gibbs energy of the\ncompound is written as\n\n*Eq. 6\u20116*\n$G = \\sum_{}^{}N_{i}{_{\\ }^{0}G}_{i}^{ref} + \\mathrm{\\Delta}_{f}H - T\\mathrm{\\Delta}_{f}S$\n\nwith $\\mathrm{\\Delta}_{f}H$ and $\\mathrm{\\Delta}_{f}S$ modeled as\nconstants. An additional draw back of is that the melting temperature of\nthe compound can be higher than those of pure elements, and the heat\ncapacity of the compound may thus be questionable at temperatures higher\nthan the melting temperatures of pure elements due to the non-physical\ncontributions from pure elements based on .\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_cerium.ipynb b/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_cerium.ipynb deleted file mode 100644 index 99c2cbc..0000000 --- a/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_cerium.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "5d3cd313", "cell_type": "markdown", "source": "## Application to cerium\n\nCerium (Ce) displays intriguing physical and chemical properties of\nwhich the most fascinating is its first-order isostructural phase\ntransition. This involves a magnetic, high temperature/high volume\n\u201c\u03b3-phase\u201d and a nonmagnetic, low temperature/low volume \u201c\u03b1-phase\u201d, both\nin the same face-centered-cubic (fcc) lattice structure. At 298 K and\n0.7 GPa, the \u03b3\u2192\u03b1 transition is accompanied by a 14-17% volume collapse.\nThe Ce phase transition has been studied extensively including our own\nworks with two or three microstates \\[55, 64\\].\n\nThe simplest model for the system is to consider two microstates:\nferromagnetic and nonmagnetic. The first-principles calculations of the\nfree energy of two Ce microstates are problematic in the absence of\nstrong correlation of the *f*-electrons in the DFT Hamiltonian. The\nrelative stability of the nonmagnetic (\u201cdelocalized\u201d) Ce 4*f* state to\nthat of the magnetic (\u201clocalized\u201d) Ce 4*f* state is greatly\noverestimated in the GGA \\[31-32\\] with spin polarization. The usual\napproach to surmount this is the Dudarev DFT + U method \\[65\\] with the\non-site Coulomb and exchange interactions as described with a\nHartree-Fock approximation added to the DFT Hamiltonian. This method\noffers the advantage that only the difference between the Hubbard U (due\nto the energy increase from an electron addition to a specific site) and\nthe J (due to the screened exchange energy) need to be specified *a\npriori*.\n\nEvaluation of numerous U \u2013 J values over a 1.0 \u2013 6.0 eV range revealed\nthat 1.6 eV gives the most consistent prediction of nonmagnetic Ce and\nmagnetic Ce energetics over the range of atomic volumes that includes\nboth microstates at 0 K. The energy-volume curves thus obtained is\nplotted in , showing that the nonmagnetic microstate is the ground\nstate, and the equilibrium between the two microstates at 0 K is at the\nnegative pressure of -0.87GPa. Since and , NTE does not exist in the\nsystem.\n\nFigure \u2011: Variation of cell energy (eV) with atomic volume\n(\u00c53) for Ce computed with strong correlation based upon\nDudarev\u2019s method with U \u2013 J = 1.6 eV.\n\nTo take into account the possible magnetic disordering in the\nferromagnetic microstate at finite temperatures, the following\ncontribution is added to the free energy of the ferromagnetic microstate\n\nEq. 9\u20119\n\nwhere *MS* is the spin moment, and *l* = 3 the orbital\nangular momentum of an *f*-electron. is a generalization of Hund\u2019s rule,\nwith total angular momentum . The Helmholtz energies thus obtained for\nboth microstates and the system are shown in at several temperatures\nwith the tie-lines included. In the figure, the blue, dot-dashed curves\nare for the nonmagnetic microstate, red, solid curves for ferromagnetic\nmicrostate, cyan shadows for entropy of mixing between two microstates,\nand the circle in (e) is the critical point. The numbers below the black\ndashed line, representing the common tangent curves, mark the transition\npressures. The 0-K static energies of the nonmagnetic microstate and the\nmagnetic microstate are also plotted in (a) using the solid circles and\ndotted lines.\n\nFigure \u2011: Helmholtz energies at (a) 0 K; (b) 100 K; (c) 165 K; (d) 300\nK; (e) 476 K; and (f) 600 K.\n\nThe temperature vs volume phase diagram is plotted and compared with\navailable experimental data as shown in \\[66\\]. In this figure, the\nvolume (V) is normalized to its equilibrium volume (VN) at\natmospheric pressure and room temperature. In the pressure range of 2.25\n\u2013 3.5 GPa, the system is within the single-phase region at all\ntemperatures considered as shown by the five continuous isobaric volumes\nas a function of temperature. In this pressure range, normal thermal\nexpansion is observed at both low and high temperatures on each isobaric\ncurve where the probability of each microstate does not change\nsignificantly with temperature. While in the middle temperature range on\neach isobaric curve, the colossal positive thermal expansion (CPTE),\nhighlighted by the pink open diamond symbols, exists due to the fast\nincrease of probability of the metastable ferromagnetic microstate with\nrespect to temperature, i.e. , , and . This CPTE is much higher than the\nindividual positive thermal expansions of the stable and metastable\nmicrostates, respectively.\n\nFigure \u2011: Calculated temperature-volume phase diagram of Ce.\n\nWith decreasing pressure, the system reaches a critical point (green\ncircle) where the homogeneous single phase becomes unstable, represented\nby and , and both entropy and volume change infinitely. At even lower\npressure, a miscibility gap forms, and the single phase separates into\ntwo phases with the same fcc crystal structures, but different magnetic\nspin structures. Inside the miscibility gap, the volume changes\ndiscontinuously with respect to temperature by the so-called first-order\ntransition as shown by the isobaric curve at zero pressure, compared\nwell with experimental volume data (solid squares) under ambient\npressure.\n\nThe fraction of the ferromagnetic microstate, *x*mag, in \u03b1-Ce\n(blue) and \u03b3-Ce (red) calculated using is plotted in as a function of\npressure along the miscibility gap phase boundary. It can be seen that\nthe fraction of the ferromagnetic microstate in \u03b1-Ce increases with\nincreasing pressure while the fraction of the ferromagnetic microstate\nin \u03b3-Ce decreases. At the critical point, the fraction of ferromagnetic\nmicrostate is calculated to be 0.58. This is in qualitative agreement\nwith the 0.67 value (filled circle) estimated experimentally at the\ncritical point.\n\nFigure \u2011: Fraction of ferromagnetic microstate in \u03b1-Ce (blue) and \u03b3-Ce\n(red) along the \u03b3-\u03b1 phase boundary.\n\nThe relative volume, *V/VN*, as a function of pressure, is\nplotted as the black solid lines in from 200 to 600 K at 50 K\nincrements. The blue and red solid lines correspond to \u03b1-Ce and \u03b3-Ce,\nrespectively. Symbols denote experimental data in the literature, except\nthe open green circle being the calculated critical point, in good\nagreement with the computed isotherms. In the two-phase miscibility gap\nregion, the \u03b3\u2192\u03b1 volume collapse is again noted, with the magnitude of\nthe collapse increasing with decreasing *T*. This is shown explicitly by\nthe dashed vertical lines at *T* = 200, 250, 300, 350, 400, and 450 K.\nFor *T* \\> 476 K, the calculated isotherms show an anomalous slope\nchange which closely matches the behavior near *V/VN* = ~0.85\nfrom experiment.\n\nFigure \u2011: Equation-of-states for Ce. The black solid lines represent the\ncalculated isotherms from 200 to 600 K at \u2206T = 50 K increments.\n\nA more complex model is to add the anti-ferromagnetic microstate. Thus\nobtained E-V and Helmholtz energy curves at 0 K are shown in . The\nequilibrium volume energies reveal that the energy of the\nanti-ferromagnetic microstate at the equilibrium volume is close to that\nof the nonmagnetic microstate but substantially lower than that of the\nferromagnetic microstate. It should be noted that the magnetic spin\ndisordering in the system is taken into account by the two magnetic\nmicrostates, and the contribution denoted by the mean-field theory, i.e.\n, should thus not be added to either magnetic microstate to avoid double\ncounting. The predicted critical point values are 546 K and 2.05 GPa,\ncloser to the experimental data than the previous predication with two\nmicrostates as shown in the temperature-pressure phase diagram in in\ncomparison with experimental data.\n\nFigure \u2011: (a) dot-dashed line with **\u25cb** (red), dashed line **\u25d1**\n(blue), and solid line with \u25cf (blue) represent the 0 K static total\nenergies for nonmagnetic (NM), anti-ferromagnetic (AFM), and\nferromagnetic (FM) microstates of Ce, respectively. (b) The solid lines\ndenote Helmholtz energy (per atom) from 0 to 600 K at \u2206T = 100 K; the\nheavy dot-dashed (\u03b1-Ce, red) and solid (\u03b3-Ce, blue) looping curves\nenclose the two-phase region with the light red dot-dashed lines\nconnecting the common tangents of each isotherm; the black dashed line\ndenotes zero pressure equilibrium state at given T; **\u25cb** (red) and \u25cf\n(blue) emphasize the phase boundary at 300 K while **\u25d1** (green) is the\ncritical point.\n\nFigure \u2011: Calculated temperature-pressure phase diagram along with\nexperimental data.\n\nThe calculated entropy changes are plotted in a in terms of lattice\nvibration only (black dashed line), lattice vibration plus thermal\nelectron (black dot-dashed line), and lattice vibration plus thermal\nelectron and plus configuration coupling (solid blue). The black square\nis the estimated vibrational entropy change at 0.7 GPa of \u03b3-Ce relative\nto \u03b1-Ce, and other open (solid) symbols are from experimental\nmeasurements of total entropy. Various contributions to the Helmholtz\nenergy along the \u03b3-\u03b1 phase boundary are plotted in b in terms of *T\u2206S*\n(blue diamonds), *\u2206E* (green circles), and *P\u2206V* (red squares), shown\nexcellent agreement with experimental data.\n\nFigure \u2011: Calculated (a) entropy; (b) various contributions to Helmholtz\nenergy of Ce along with experimental data.\n\nThe predicted fractions of three microstates as a function of\ntemperature and heat capacity at the critical pressure of 2.05 Pa are\nshown in a and b, respectively. Near the critical point, the theory\npredicts that the system is a mixture of the various microstates. a\ndepicts that for T \\< 300 K, the system consists mainly of the\nnonmagnetic Ce state which results in \u03b1-Ce. For *T* \\> 300 K, the\nthermal populations of the magnetic states increase with increasing\ntemperature. Finally, for *T* \\> 546 K (the critical point), 70% of the\nsystem is composed of the antiferromagnetic Ce state with the remaining\n30% consisting of the nonmagnetic and ferromagnetic Ce states. This is\nin agreement with the common belief that \u03b3-Ce is magnetic with a\npartially disordered local moment (paramagnetic) and that \u03b1-Ce is\nnonmagnetic.\n\nb shows the predicted temperature evolution of contributions to heat\ncapacity: vibrational and magnetic (*Cf/T*), electronic\n(*Cel*/*T*), and their sum (*Cf+el/T*) at 2.05\nGPa. The theory suggests the following: (a) below ~500 K,\n*Cf+el/T* shows an exponential temperature dependence due to\nthe statistic fluctuation among the non-magnetic, ferromagnetic, and\nantiferromagnetic states; (b) a peak appears at ~500 K in the\n*Cf+el/T* curve, which typically suggests the Schottky\nanomaly; (c) the electronic specific heat coefficient\n(*Cel*/*T*) is linear against *T*; (d) above *~*500 K the sum\nof *Cf/T* and *Cel*/*T* renders\n*Cf+el/T* temperature-independent.\n\nFigure \u2011: (a) Thermal populations of the nonmagnetic (red dot-dashed),\nanti-ferromagnetic (green dashed), and ferromagnetic (blue solid) as a\nfunction of temperature at the critical pressure of 2.05 GPa. (b)\nCel/T (black dashed line), Cf/T (black dot-dashed\nline), and their sum Cf+el/T (blue solid line) at 2.05 GPa.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_fept.ipynb b/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_fept.ipynb deleted file mode 100644 index 03a58d0..0000000 --- a/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_fept.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "d57f6e2d", "cell_type": "markdown", "source": "## Application to Fe3Pt\n\nInvar was first discovered in the intermetallic\nFe65Ni35 alloy and is characterized by \u201canomalies\u201d\nincluding thermal expansion, equation-of-state, elastic modulus, heat\ncapacity, magnetization, and Curie temperature. There are a number of\ntheoretical models for Invar such as the Weiss 2-\u03b3 model, the\nnon-collinear spin model, and the disordered local moment approach as\nreviewed in Ref. \\[67\\]. In this section, the application of the MMS\nmodel to the ordered L12 Fe3Pt is presented to\nstudy the Invar anomaly at finite temperatures.\n\nFor a supercell of 12 atoms with nine magnetic Fe atoms in\nFe3Pt, if only the up and down spins are considered, the\nsystem contains 29 = 512 spin configurations, which are, by\nsymmetry, reduced to 37 non-equivalent ones. They are the microstates in\nthe MMS model, and FMC and SFC are used to represent the ferromagnetic\nand spin flipping microstates, respectively. For the first-principles\ncalculations of each microstate, VASP package \\[14\\] within the\nprojector-augmented wave (PAW) method and the exchange-correlation part\nof the density functional treated within the GGA of\nPerdew-Burke-Ernzerhof (PBE) \\[32\\] are employed with details in Ref.\n\\[67\\]. For the lattice vibration, the Debye-Gr\u00fcneisen approach\ndescribed in Chapter 5.2.4 is used.\n\npresents the first-principles 0 K total energies of 36 non-equivalent\nSFCs as well as the FMC as a function of atomic volume. It can be seen\nthat there are a number of SFCs, whose energies are in the range of ~1\nmRy/atom to that of the FMC. It is noted that all the SFCs studied\nherein have the equilibrium averaged atomic volumes at least 1.8 %\nsmaller than that of the FMC, the 0 GPa ground state. In , the two\nlowest energy SFCs have been labeled as SFC55 and SFC41 with their spin\narrangements very similar to the double layer antiferromagnetic state.\nThe nonmagnetic configuration has a very small atomic volume of 11.66\n\u00c53/atom and much higher energy than both FMC and all SFCs,\nand is thus not shown here.\n\nFigure \u2011: 0 K total energies. The heavy black line represents the FMC.\nThe symbols **\u25cb**, \u25d4, \u25d1, and \u25d5 with dashed lines indicate the minima of\nthe energy-volume curves of the SFCs with spin polarization rates of\n1/9, 3/9, 5/9, and 7/9, respectively. The red **\u25cb** and \u25d4 with\ndot-dashed lines mark the two lowest SFCs in energy.\n\nThe normalized Helmholtz energies of all SFCs are plotted in with the\nHelmholtz energy of FMC as the reference state, shown that the FMC has\nthe lowest Helmholtz energy at all temperatures considered. If only the\nrelative Helmholtz energies of microstates are considered, FMC should be\nstable at all temperatures. However, the configurational entropy due to\nthe mixing among multiple microstates, i.e. , lowers the system free\nenergy by introducing the statistic probability of metastable SFCs.\n\nFigure \u2011: Normalized Helmholtz energy of all SFCs with respect to that\nof FMC.\n\nThrough the minimization of Helmholtz energy of , the\ntemperature-pressure and temperature-volume phase diagrams are obtained\nand shown in \\[66\\]. A critical point at 141 K and 5.81 GPa is predicted\nwith *V* = 12.61 \u00c53. Below the critical point, it is a\ntwo-phase miscibility gap (the shadow area) with the dominant\nmicrostates being FMC and SFCs, respectively, and the transition between\nthem is first-order. Above the critical point, the macroscopically\nhomogeneous single phase is stable, and the phase transitions between\nthe ferromagnetic-dominant phase with large volumes and the SFC-dominant\nparamagnetic phase with small volumes are of second-order. The\nsecond-order transition pressures and volumes are determined by the\ncondition that the weighted Helmholtz energy counting all SFCs equals to\nthe Helmholtz energy counting only FMC.\n\nFigure \u2011: Calculated (a) temperature-pressure and (b) temperature-volume\nphase diagrams of Fe3Pt.\n\nIn a, the data points are the measured pressure dependence of the Curie\ntemperature, and the agreement between the measurements and predictions\nis remarkable. It should be pointed out that the classical Weiss 2-\u03b3\nmodel predicts only first-order phase transitions while the\nnon-collinear spin model yields only second-order phase transitions at\nall temperatures. In b, four isobaric volume curves are also plotted\nwith the predicted NTE regions marked by the pink open diamonds and the\nexperimental volume data under ambient pressure superimposed, showing\nexcellent agreement. It also depicts the gigantic NTE around the\ncritical point on the isobaric curve at 7 GPa.\n\nindicates that the entropies of SFCs are larger than that of FMC so\ntheir Helmholtz energy differences decrease with temperature. This is in\nline with the origin of NTE in a single phase due to the statistic\nexistence of metastable microstates with lower volumes and higher\nentropies than the stable state in a temperature range where their\nprobabilities change dramatically. plots the calculated thermal\npopulations of the FMC (black solid line) and that of the sum over all\nSFCs (black dot-dashed line) under ambient pressure. The two major\ncontributions to the paramagnetic (PM) phase are from SFC55 and SFC41,\nwhich are also plotted using red dashed and long dashed lines,\nrespectively. The system is dominated by the FMC at temperatures below\nhalf of the transition temperature, and the populations of SFCs increase\nmonotonously at temperatures higher than half of the transition\ntemperature. As mentioned above, the transition temperature is defined\nwhen the population of all SFCs is the same as that of FMC due to their\nequal Helmholtz energy.\n\nFigure \u2011: Calculated thermal populations of FMC, SFC55, SFC41, and sum\nof all SFCS, respectively.\n\nThe predicted thermal volume expansion and the derived linear thermal\nexpansion coefficient (LTEC) under ambient pressure are plotted in . A\npositive thermal expansion is predicted from 100 K to 288 K, followed by\na negative thermal expansion in the range of 289 ~ 449 K, and then a\npositive thermal expansion again at \\>450 K, in excellent agreement with\nexperiments. The only disagreement between the predictions and\nexperiments occur at T \\< 100 K where the calculations did not reproduce\nthe negative thermal expansion for Fe3Pt. Large supercell or\nmore spin configurations may be needed.\n\nFigure \u2011: (a) Relative volume increase (V-\nV300)/V300 with V300 being the\nequilibrium volume at 300K and 0 GPa for the ordered Fe3Pt.\n(b) Linear thermal expansion coefficient (LTEC) along with experimental\ndata (symbols) with details in Ref. \\[67\\].\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/concept_of_materials_genome.ipynb b/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/concept_of_materials_genome.ipynb deleted file mode 100644 index 079d170..0000000 --- a/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/concept_of_materials_genome.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "e9a0ab3f", "cell_type": "markdown", "source": "## Concept of Materials Genome\u00ae\n\n\u201cA genome is a set of information encoded in the language of DNA that\nserves as a blueprint for an organism\u2019s growth and development. The word\ngenome, when applied in nonbiological contexts, connotes a fundamental\nbuilding block toward a larger purpose\u201d \\[68\\]. Materials Genome\u00ae (a\ntrademark of Materials Genome, Inc., Pennsylvania, USA) thus concerns\nthe building blocks of materials. Most part of this book focuses on the\nGibbs/Helmholtz energies of individual phases as a function of its\nnatural variables, and the same in the CALPHAD modeling of\nthermodynamics and other properties of individual phases.\nMulti-component materials systems and their properties are built on the\nindividual phases and their properties. Individual phases are thus\nnaturally considered as building blocks of materials. Consequently, the\nlanguage of CALPHAD thermodynamics and kinetics contains the genomics of\nmaterials by representing experimental and theoretical results in\ndatabases to make them applicable to a much wider context than the\noriginal experiments or calculations \\[69\\]. The change of individual\nphases in terms of their properties, amounts, and interactions with\nother phases with respect to external conditions thus determines the\nperformance of the materials.\n\nOn the other hand, at critical points and beyond, phases lose their\nindividuality and form one macroscopically homogeneous system, and the\nproperties of the system change dramatically with respect to external\nconditions. As shown in this chapter, these dramatically responses can\nbe predicted through the statistic competition of stable and metastable\nmicrostates. From the thermodynamic point of view, under any given\nconditions, one of the individual microstate has the lowest\nGibbs/Helmholtz energy and is stable, while all other microstates have\nhigher free energy and are metastable or unstable. These metastable or\neven unstable microstates are brought into statistic existence in the\nmatrix of the stable microstate due to the entropy of mixing of the\nstable and metastable configurations. Those microstates may thus be\nconsidered as the building blocks of individual phases \\[70\\].\n\nIt has demonstrated in this chapter that the properties of a\nmacroscopically homogeneous system with multiple microstates are not a\nlinear combination of properties of constituent microstates and depends\nsignificantly on the change rate of statistic probability of microstates\nwith respect to external fields. This change rate is determined by the\nfree energy difference between the stable and metastable microstates and\nits change with respect to external fields. As shown in and , this\nchange rate can be correlated to the distance of the system with respect\nto the critical point in the system. At the critical point, there is a\nmathematical singularity when the single phase becomes unstable. When\nthe macroscopically homogeneous single-phase system moves away from the\ncritical point, its properties become less and less dramatic, but always\nremain a certain degree of anomaly. The properties of a system can thus\nbe dramatically altered and designed by changing the position of the\ncritical point through adjustments of chemical compositions and strain\nenergy in thin films.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/index.ipynb b/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/index.ipynb deleted file mode 100644 index 08547d4..0000000 --- a/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/index.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "eb4fc5ff", "cell_type": "markdown", "source": "# Critical phenomena, thermal expansion, and Materials Genome\u00ae\n\nIn Chapter , it was shown that all molar quantities of a homogeneous\nsystem diverge at the critical point, i.e. the limit of stability,\nincluding the additional ones shown in . As illustrated by , even though\neach molar quantity changes in the same direction as its conjugate\npotential, i.e. with the same sign, its dependence with respect to a\nnonconjugate potentials can either be in the same sign or opposite\nsigns. It is often considered to be normal when they change in the same\ndirection, while abnormal when in different directions.\n\nIn this chapter, the thermal expansion defined by , is used as an\nexample for detailed discussion of those extraordinary phenomena in the\ncontext of a critical point based on the MMS model presented in Chapter\n. The MMS model is first discussed in terms of thermal expansion and\nthen applied to elemental cerium (Ce) with the colossal positive thermal\nexpansion (CPTE) and Fe3Pt with negative thermal expansion\n(NTE).\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/mms_model_applied_to_thermal_expansion.ipynb b/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/mms_model_applied_to_thermal_expansion.ipynb deleted file mode 100644 index 343d12d..0000000 --- a/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/mms_model_applied_to_thermal_expansion.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "fad8b29b", "cell_type": "markdown", "source": "## MMS model applied to thermal expansion\n\nAs shown in , the thermal expansion of a system can be positive, zero\nand negative depending on the pressure dependence of entropy of the\nsystem. Let us carry out a virtual experiment by analyzing a system\nstarting with one microstate only, *\u03b1*, when the metastable *\u03b2*\nmicrostate has a higher entropy than the *\u03b1* microstate, i.e. and the\nrelative stability of the *\u03b2* microstate thus increases with\ntemperature. The cases with will be discussed after this starting with a\nmixture of the *\u03b1* and *\u03b2* microstates.\n\nWhen a metastable microstate, *\u03b2*, is introduced by changing pressure\nunder constant temperature, based on and , the entropy change of the\nsystem can be written as\n\nEq. 9\u20111\n\nwhere represents the statistic probability of the microstate in the\nsystem. With , this would results in a positive entropy change of , i.\ne. , since . If this entropy increase is due to the decrease of\npressure, i.e. because volume and its conjugate potential (negative\npressure) change in the same direction, the volume thermal expansion of\nthe system is positive due to the increase of the population of the *\u03b2*\nmicrostate with a larger volume. In this case,\n$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}} > 0$, and the\nvolume and entropy of the two microstates change in the same direction.\n\nOn the other hand, if this entropy increase is realized by increasing\npressure, i.e. , the volume thermal expansion of the system is negative\ndue to the increase of the population of the *\u03b2* microstate with a\nsmaller volume. In this case,\n$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}} < 0$, and the\nvolume and entropy of the two microstates change in the opposite\ndirections.\n\nTherefore, the sign of\n$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}}$ of two\nmicrostates can be used as a criterion to determine whether a system\npossesses NTE, i.e. positive\n$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}}$ for positive\nthermal expansion, and negative\n$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}}$ for NTE. At a\ncritical point, the entropy change with respect to temperature is\ninfinite, resulting in either infinite positive or infinite negative\nthermal expansion correspondingly. When the system moves away from the\ncritical point into the macroscopically homogeneous single-phase region,\nthe thermal expansion becomes less positive or negative. A number of\nsystems with\n$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}} < 0$, thus\npotentially NTE, are listed in the supplementary information of Ref.\n\\[63\\].\n\nNow let us consider the case that the metastable *\u03b2* microstate has\nlower entropy than the *\u03b1* microstate, i.e. , and the *\u03b2* microstate is\nthus more stable at low temperatures. The system at higher temperatures\ncontains thus only the *\u03b1* microstate and has positive thermal\nexpansion. When the metastable *\u03b2* microstate is introduced, the sign of\nthe entropy change in can be either positive or negative because the\nfirst term is negative and the second term is positive, and its sign\nthus depends on the value of the entropy difference between two\nmicrostates and the probability of the metastable *\u03b2* microstate. The\nvirtual experiment should thus be carried out in a system with the\nhighest MCE in , i.e. when the two microstates have the same free energy\nand are in equilibrium with each other. From and , the system entropy\ncan be written as\n\nEq. 9\u20112\n\nWith the change of pressure, will either increase or decrease, and the\nentropy of the system becomes\n\nEq. 9\u20113\n\nThe difference of and is obtained as\n\nEq. 9\u20114\n\nThe second term in is always negative, and the first term is also\nnegative if because . It is thus evident that if is increased by\ndecreasing pressure, the entropy of the system decreases, and the system\nwould possess negative thermal expansion because is negative, and the\nentropy is reduced by the decrease of pressure. At the same time, and\n$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}} < 0$, the latter\nbeing the same condition for a negative thermal expansion as in the\nfirst virtual experiment with . One can thus conclude that for a\ntwo-phase equilibrium line with\n$\\frac{dT}{dP} = \\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}} < 0$,\nboth phases can display negative thermal expansion. On the other hand,\nif is increased by increasing pressure, the system would possess\npositive thermal expansion because the entropy is reduced by the\nincrease of pressure.\n\nFurthermore, the thermal expansion of a system can be approximated as\nfollows using the rule of mixture of volumes\n\nEq. 9\u20115\n\nEq. 9\u20116\n\nwhere , , , , , and are the thermal expansion coefficients and volumes\nof the system and the \u03b1 and \u03b2 microstates, respectively. For\nsimplification, let us assume both microstates have similar positive\nthermal expansion, i.e. , and becomes\n\nEq. 9\u20117\n\nshows that it is the combination of volume difference and that\ndetermines the macroscopic thermal expansion. By setting , one obtains\n\nEq. 9\u20118\n\nFor , for $p^{\\beta} \\geq 0$, and for , at $p^{\\beta} \\rightarrow 0$ .\nThe readers are reminded that the sign of is the same as the sign of .\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/experimental_data_for_thermodynamic_modeling/additional_methods.ipynb b/psu410/src/experimental_data_for_thermodynamic_modeling/additional_methods.ipynb deleted file mode 100644 index cd083fa..0000000 --- a/psu410/src/experimental_data_for_thermodynamic_modeling/additional_methods.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "de75ddcb", "cell_type": "markdown", "source": "### Additional methods\n\nElectrical resistivities of different phases are usually different. A\nchange of slope of electric resistivity as a function of composition or\ntemperature or pressure reflects a phase transformation. This technique\nis simple and reliable.\n\nMagnetic transitions can be measured using a vibrating sample or\nsuperconducting quantum interference device (SQUID) magnetometer by\ndetermining the magnetic moment of a sample in presence of an applied\nmagnetic field. Magnetic field lines form closed loops, resulting in an\nexternal dipolar and demagnetizing field in a finite-sized sample. The\neffective field sensed by the sample is thus the difference between the\napplied field and the demagnetizing field. The magnetic transition\ntemperature is evaluated from the Arrott plots where the ratio of\nmagnetic field over magnetization with a proper exponent is plotted with\nrespect to the magnetization with another proper exponent for a series\nof temperatures. Those proper exponents result in parallel isotherm\nlines, and the isotherm line intersecting the origin corresponds to the\nmagnetic transition temperature.\n\nThin films with composition gradients, commonly referred to as\ncombinatorial libraries, can be used to study the phase relations\nsimilar to diffusion couples/multiples though the results may differ due\nto the effects of surface and potential interactions with the substrate.\n\nPhase relations at high pressures are measured in equipment using\ndiamond anvil cells (DAC) or multi-anvil devices. The high pressure is\nrealized by decreasing the area, i.e. the anvil culet size. Pressures up\nto 100 GPa can be created in DAC with a culet size of 0.3mm for small\nsamples in the order of 0.2 to 0.4mm. For large samples, the\nlarge-volume presses (LVP) technique is developed, typically using WC\nwith the pressure mostly limited to 30 GPa and the sample size ranging\nfrom 1mm3 to 1cm3. The pressure can be measured by\neither the ruby (Cr3+ doped Al2O3)\nfluorescence line shift or the molar volume of a pressure marker by\nX-ray diffraction. The samples in DAC apparatus are heated by laser or\nresistive wire or a small heater around the samples, while high\ntemperatures in LVPs are achieved by resistive heaters. Crystal\nstructures are detected by in-situ X-ray or synchrotron diffraction.\nAttentions need to be paid to temperature and pressure homogeneity and\nthe non-hydrostatic stresses, which are both better controlled in LVP\nequipment.\n", "metadata": {}}, {"id": "4eb26f12", "cell_type": "markdown", "source": "### Combustion, direct reaction, and heat capacity calorimetry\n\nIn combustion calorimetry, the sample is ignited and reacts with\nreactive gases like oxygen or fluorine. To accurately calculate the\nenthalpy of formation from the enthalpy of combustion, the reliable\ncharacterization of reactants and reaction products is critical, such as\nincomplete combustion, impurities in the reactants which are often\nill-defined, and more than one reaction gaseous species and condensed\nphases. Combustion calorimeters are usually of isoperibol type around\nroom temperature in a water bath.\n\nThe direct reaction calorimetry is similar to the combustion calorimetry\nthough at high temperatures in heat-flux or adiabatic environments. The\npartial enthalpy of reaction can also be measured if the partial\npressure of volatile species can be controlled and measured. The key\nfactor for accurate results is that both the reactants and reaction\nproducts are well characterized and the reaction goes to completion\nquickly like in the combustion calorimetry. For reactive reactants,\nspecial procedure is needed to avoid their loss before the reaction\ntakes place.\n\nHeat capacity is defined as the amount of heat needed to increase the\ntemperature by 1K as shown by , and its integration with respect to\ntemperature from 0K gives entropy as shown by . At low temperatures,\nadiabatic calorimetry gives more accurate data of heat capacity though\nit is time consuming and requires complex instruments. At high\ntemperatures, the efficient and less accurate DSC method is more widely\nused.\n", "metadata": {}}, {"id": "f0ee87a8", "cell_type": "markdown", "source": "### Diffusion couples/multiples\n\nThe major challenge in the equilibrated material approach is to ensure\nthat the whole sample reaches equilibrium. On the other hand the\ndiffusion couple/multiple technique does not require the whole sample to\nbe in equilibrium and is based on the assumption that any two phases in\ncontact are in equilibrium with each other at the phase interface, and\nthe phase compositions can be obtained by extrapolation of concentration\nprofiles in the two phases to the phase interface. Since the total\nsystem of a diffusion couple is not at equilibrium, many kinetic\nphenomena related to diffusion can be studied in a diffusion couple,\nsuch as interdiffusion coefficients, parabolic growth kinetics of\nproduct layer thickness, diffusion path (represented by the local\noverall compositions) in ternary and multi-component systems for\nvisualizing the microstructure of reaction zones, and other properties,\nall as a function of composition, which are beyond the scope of the\npresent book.\n\nTypical diffusion couples are in solid-state with two materials brought\ninto intimate contact to allow diffusion of elements between the two\nmaterials though solid-liquid diffusion couples are also used. The\ncontacting faces are commonly ground and polished flat, clamped together\nusing mechanical mechanisms, and annealed at high temperatures where\ndiffusion can take place to a significant degree in a matter of days,\nweeks or months. The samples are then quenched to retain the\nhigh-temperature equilibrium features. For metallic systems, diffusion\ncouples can also be prepared by eletrolytical and electroless plating\ntechniques. It is important to avoid the formation of liquid during\nannealing as it ruins the sample geometry. Furthermore, good adherence\nat the interfaces is critical for reliable data.\n\nSince diffusion couples are not in a fully equilibrium state, the\ntie-lines between two phases at the phase interfaces need to be obtained\nby extrapolations of concentration profiles in neighbouring phases. The\nelectron propagation in quantitative EPMA is typically in the range of\n1-2\u03bcm, yielding an excitation volume of approximately 2-4\u03bcm diameter and\na requirement of reasonable layer widths of phases on both sides of the\ninterface for accurate extrapolation. Therefore, the reliable\ncomposition of a single phase must be taken several micrometers away\nfrom the interface. When steep concentration gradients exist near the\ninterfaces, the extrapolation may lead to large errors, and analytical\nelectron microscopy is then needed. Furthermore, the fluorescence\neffects, where the primary excitation can be powerful enough to excite\nother elements in the sample, resulting in enhanced X-ray production and\nthe need of proper corrections. For a new phase to become observable\nexperimentally, it must nucleate and grow to reach the resolution limit\nof analytical tools. It is thus not uncommon that some known equilibrium\nphases are not found, and some non-equilibrium phases form. One way to\ncircumvent this issue is to use incremental diffusion couples with\nnarrow concentrations next to the phase of interest so that only this\nphase may be formed.\n\nFor ternary and higher-order systems, a more efficient approach can be\nused by placing a thin layer of the third alloy between two alloys. The\nthin central layer is eventually consumed, and the diffusion path is not\nfixed as in the semi-infinite diffusion couples. The phase compositions\nchange continuously with time as a result of the overlapping of two\nquasi-equilibrated diffusion zones.\n\nA diffusion multiple contains three or more pure elements or alloys of\ndifferent compositions and is a sample with multiple diffusion couples\nand diffusion triples in it. It is more efficient in terms of both\nmaterials and time in comparison with equilibrated alloys and diffusion\ncouples. All alloy blocks are prepared individually and sealed in vacuum\nin a cylinder which is also used as one alloy for the diffusion\nmultiple. The sealed cylinder with vacuum inside also serves as the can\nfor subsequent hot isostatic pressing to achieve intimate interfacial\ncontacts. The cylinder can then be cut into disks for further annealing\ntreatments. A broad range of design strategies is needed for complex\ndiffusion multiples along with automated plotting procedures due to the\nlarge amounts of EMPA data. The major source of error lies in the\nextraction of tie-lines from EPMA results due to very condensed\ninformation in a small area and the deviation of scanned lines from\nthose perpendicular to the interface.\n\nIn terms of local equilibrium characteristics of diffusion\ncouples/multiples, it is evident that in the equilibrated materials\napproach, it may not be necessary to reach the full equilibrium for the\nwhole sample if one is only interested in the local equilibrium\ncompositions between two neighboring phases, which can even provide\ninformation on metastable extensions of two phases if the two phases are\nin a metastable equilibrium at the annealing temperature.\n", "metadata": {}}, {"id": "1e4f1d1e", "cell_type": "markdown", "source": "### Equilibrated materials\n\nThe most common method to determine phase equilibria is to use\nequilibrated materials. This method typically involves material\npreparation through high temperature melting or powder metallurgy,\nhomogenization heat treatment, isothermal or cooling/heating procedure,\nand identifications of crystal structures and phase compositions. It is\nimportant to avoid macro-inhomogeneity as it can be difficult remove the\ninhomogeneity in subsequent treatments. It is also important to use\nstarting materials of highest purity and minimize the loss and\ncontamination of materials during the whole experiments using a\nprotective atmosphere of inert gas or vacuum. The typical melting\ntechniques include high temperature furnaces with crucibles, arc\nmelting, and induction melting. Attentions need to be paid for possible\nreactions between materials and crucibles/containers, which can be\navoided by levitating the materials through electromagnetic fields or\nother means. In addition to use pure elements as raw materials, master\nalloys with well-controlled compositions are often utilized because the\ncompositions and melting properties of master alloys are usually much\ncloser to those of final materials than the pure elements. For materials\nwith very high melting temperature or volatile components, the powder\nmetallurgy method can be used where compacts are made, capsulated, and\nsintered.\n\nHomogenization during subsequent heat treatment is achieved through\ndiffusion, in which time and temperature are two important parameters.\nTo accelerate the homogenization process, the heat treatment temperature\nshould be as close to the solidus temperature as possible taking into\naccount the composition inhomogeneity with variable solidus\ntemperatures. When there are phase transformations taking place during\nthe heat treatment, extra time is needed for the heat treatment.\n\nThe phase boundaries are then determined through measurements of either\ncompositions of individual phases that are in equilibrium under constant\ntemperature, pressure/stress/strain, and electric/magnetic fields or\ndiscontinuity in some physical properties of materials due to a phase\ntransition from the continuous change of temperature or\npressure/stress/strain or electric/magnetic fields. The measurement of\ncompositions is usually carried out at ambient conditions so it is\nnecessary that the phases are fully equilibrated at experimental\nconditions, which requires rigorous verifications, and can be \u201cquenched\u201d\nto ambient conditions to remain unaltered during \u201cquenching\u201d, at least\nin terms of compositions. The compositions are typically measured by\nscanning electron microscopy (SEM) equipped with energy dispersive\nspectrometer (EDS) or wavelength dispersive spectroscopy (WDS), with a\nmicro-level spatial resolution and a better compositional resolution of\nWDS than EDS. A dedicated SEM with WDS gives another important, widely\nused composition measurement technique called electron probe\nmicroanalysis (EMPA). For submicron-sized phases, analytical\ntransmission electron microscopy equipped with EDS can be used though\ncare must be taken to avoid interference from neighboring phases.\n\nTo accurately identify phases in equilibrium under experimental\nconditions, in-situ characterizations are necessary, which complicates\nthe experimentation. An alternative indirect method is to measure a\nphysical property that changes discontinuously or dramatically for a\nfirst- or second-order phase transition, such as heat, volume, electric\nconductivity, and magnetization. There are two widely used techniques in\nmeasuring heat: differential thermal analysis (DTA) and differential\nscanning calorimetry (DSC). Both attempt to measure the difference in\ntemperature with the same amount of power supplied between a sample and\nan inert standard during heating or cooling. A DSC may also measure the\ndifference in the amount of power supplied to keep their temperatures\nidentical. The deviation of this difference from a baseline indicates a\nphase transition in the sample and is plotted as a function of time or\ntemperature of either the sample or the inert standard. It is evident\nthat the major challenges in both DTA and DSC techniques are to reach\nthermal equilibrium between the sample/standard and the instrument and\nthe thermodynamic equilibrium within the sample due to the continuous\nheating or cooling. The thermal equilibrium can be improved or mitigated\nthrough various methods such as sample shape and size, crucible\nselection, mixture with the material used for the inert standard, and\nwith thermocouples in direct contact with the sample and the inert\nstandard. However, the thermodynamic equilibrium within the sample can\nonly be reached when the heating/cooling rate is comparable with the\nrate of the phase transition in the sample, which is almost impossible\nif the phase transition typically involves diffusion in solid phases.\nTherefore, extreme care is needed in interpreting the temperature\ndetermination and the amount of heat associated with the DTA/DSC curves\nas discussed in detail in the reference \\[11\\].\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/experimental_data_for_thermodynamic_modeling/index.ipynb b/psu410/src/experimental_data_for_thermodynamic_modeling/index.ipynb deleted file mode 100644 index f62f6e8..0000000 --- a/psu410/src/experimental_data_for_thermodynamic_modeling/index.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "38eccd96", "cell_type": "markdown", "source": "# Experimental data for thermodynamic modeling\n\nThe most widely used thermodynamic modeling technique is the CALPHAD\n(CALculation of PHAse Diagram) method to be discussed in detail in\nChapter 6. The input data in the evaluations of thermodynamic model\nparameters have been primarily from experiments and estimations until\nfirst-principles calculations based on the density functional theory\n\\[8\\] became a user tool in the later 1990\u2019s. Experimental data include\nboth thermodynamic and phase equilibrium data, while the\nfirst-principles calculations, which only provide thermodynamic data of\nindividual phases, are discussed more extensively in Chapter .\n\nThe three recently published books summarize the commonly used methods\non experimental measurements of thermodynamic properties of single \\[9\\]\nand multiple phases \\[10\\] and phase diagrams \\[11\\]. They are briefly\ndiscussed here, and readers are referred to these books for details. The\nmain techniques for crystal structure analysis include X-ray\ndiffraction, electron backscatter diffraction (EBSD), electron\ndiffraction in transmission electron microscopy, neutral scattering, and\nsynchrotron scattering, which are not discussed in this book.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/experimental_data_for_thermodynamic_modeling/phase_equilibrium_data_from_experiments.ipynb b/psu410/src/experimental_data_for_thermodynamic_modeling/phase_equilibrium_data_from_experiments.ipynb deleted file mode 100644 index ca07ada..0000000 --- a/psu410/src/experimental_data_for_thermodynamic_modeling/phase_equilibrium_data_from_experiments.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "71eaef0b", "cell_type": "markdown", "source": "## Phase equilibrium data from experiments\n", "metadata": {}}, {"id": "c517027c", "cell_type": "markdown", "source": "### Solution calorimetry\n\nThe book edited by Marsh and O\u2019Hare \\[12\\] documented the detailed\nexperimental techniques used for solution calorimetry. In one\nexperiment, the enthalpy of solution of a single phase is measured in a\nparticular solvent. To convert this enthalpy of solution into enthalpy\nof formation of the phase, a thermodynamic cycle is setup for a chemical\nreaction to form this phase from either constitutive pure elements or\ncompounds. Therefore in another experiment, the enthalpy of solution of\nconstitutive pure elements or compounds is measured in the solvent as\nidentical as possible to that used in the first experiment. The\ndifference of the two enthalpies of solution thus gives the enthalpy of\nformation of the single phase from its constitutive elements or\ncompounds at the temperature of the samples before they are dropped into\nthe solvent, usually at room temperature.\n\nThe solvent can be aqueous solutions at ambient temperatures and\npressures or metallic/salt/oxide melts at high temperatures under either\nadiabatic or isoperibol conditions. The adiabatic calorimetry measures\nthe temperature change of the solvent and is usually more accurate than\nthe isoperibol calorimetry that measures the heat generated during the\ndissolution, though the adiabatic ones are with more complex\ninstruments. It is important that the choices of solvent and temperature\nensure the complete dissolution of all substances into the solvent to\nform a homogeneous solution. Furthermore, the effect of dilution and of\nchanges in solvent composition needs to be considered in the calculation\nof enthalpy of solution.\n\nA large number of solvents are used. For aqueous solvent, hydrofluoric\nacid or mixtures of HF and HCl are often used. For oxides, buffer-type\nsystems are typical such as lead and alkali borates and alkali\ntungstates/molybdates. For metallic phases, low melting metals such as\nSn, Bi, In, Pb, and Cd, sometimes Al and Cu, are used. Factors such as\nsolubility, dissolution kinetics, thermal history, stirring, heat flow,\nparticle size, and system size are to be optimized for accurate\nmeasurements. To enhance solution kinetics, the compound to be studied\ncan be mixed with another element or compound so that the mixture can\nform liquid in the solvent at higher reaction rate. In any case, it is\nimportant to calibrate the system with pure elements and compounds of\nknown enthalpy of formation.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/experimental_data_for_thermodynamic_modeling/thermodynamic_data_from_experiments.ipynb b/psu410/src/experimental_data_for_thermodynamic_modeling/thermodynamic_data_from_experiments.ipynb deleted file mode 100644 index 75e32ff..0000000 --- a/psu410/src/experimental_data_for_thermodynamic_modeling/thermodynamic_data_from_experiments.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "3f372bef", "cell_type": "markdown", "source": "## Thermodynamic data from experiments\n\nBroadly speaking, thermodynamic data represents the values of Gibbs free\nenergy and all its first and second derivatives. For thermochemical\nproperties as the main topic of the book, calorimetric, electrochemical,\nand vapor pressure methods are the primary techniques used with the\nfirst for accurate measurement of enthalpy and entropy, and the latter\ntwo for direct determination of Gibbs energy and activity. The\nelectrochemical method is discussed in Chapter . The calorimetric method\nis divided into solution, combustion, direct reaction, and heat capacity\ncalorimetry, respectively, which all involve chemical reactions to be\ndiscussed in detail in Chapter . The vapor pressure method involves the\nequilibrium of volatile species between gas and samples and is divided\ninto Knudsen effusion and equilibration methods, respectively.\n", "metadata": {}}, {"id": "417cc01d", "cell_type": "markdown", "source": "### Vapor pressure method\n\nIn the Knudsen effusion method, a small amount of volatile species in\nthe gas phase effuses through a small orifice of 0.1 to 1 mm with\nnegligible influence of the equilibrium in the Knudsen cell. The vapor\nis ionized and analyzed with a mass spectrometer. The partial pressure\nof a species can be calculated from its ionization area and intensity\nthrough a calibration factor determined by a reference material with\nknown partial pressure. For high temperature measurements, care must be\ntaken to avoid the reactions between cell and sample and fragmentation\nof gas species on ionization. The typical vapor pressure range is\nbetween 10-7 and 10 Pa.\n\nIn various equilibration methods, the total vapor pressure is usually\nmeasured directly using pressure gauges in the range of 10-7\nand 100 kPa. Other direct or indirect methods include\n\nthermogravimetric method for measuring the vapor mass\n\natomic absorption spectroscopy for gas composition\n\nmeasurement of sample composition equilibrated with a gas of\nwell-defined activity of the volatile species\n\nthe dew point method in which the condensation temperature of the\nvolatile component is measured from the vapor equilibrated with the\nsample at a higher temperature,\n\nthe chemical transport method to be discussed in Chapter .\n\nThe main error in all these methods is often due to inadequate\nequilibration between vapor and sample.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/firstprinciples_calculations_and_theory/bornoppenheimer_approximation.ipynb b/psu410/src/firstprinciples_calculations_and_theory/bornoppenheimer_approximation.ipynb deleted file mode 100644 index c9815d6..0000000 --- a/psu410/src/firstprinciples_calculations_and_theory/bornoppenheimer_approximation.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "5798c63c", "cell_type": "markdown", "source": "### Born-Oppenheimer Approximation\n\nFor a time independent atomic system, it is often accurate enough to\nwrite in or in terms of the electron coordinate **r** and nuclei\ncoordinate **R**,\n\n*Eq. 5\u201164* ,\n\nwhere *e* represents the electron charge and *i* and *j* label the\nelectrons, *I* and *J* the atomic nuclei, *ZI* the atomic\nnuclear charge number of atom *I*, *e* the electron charge,\n*me* the electron mass, *MI* the mass of atomic\nnuclei *I*, the Laplace operator for electron *i*, and the Laplace\noperator for atomic nuclei *I,* noting\n\n*Eq. 5\u201165* ,\n\nwith respect to the Cartesian axis *x*, *y*, and *z*.\n\nConsider the fact that the electron mass is thousand times smaller than\nthe mass of the atomic nuclei, implying that the motions of the\nelectrons are much faster than the atomic nuclei, Born and Oppenheimer\nproposed that the wave function of the whole system can be simply\napproximated as the product of the electron wave function and the atomic\nnuclei wave function as\n\n*Eq. 5\u201166* .\n\nWith the auxiliary approximation of neglecting the dynamic coupling\nbetween the motions of electrons and atomic nuclei, the Schr\u00f6dinger\nequation for the motion of electrons becomes\n\n*Eq. 5\u201167* ,\n\nwhere\n\n*Eq. 5\u201168* ,\n\nand the Schr\u00f6dinger equation for the motion of atomic nuclei becomes\n\n*Eq. 5\u201169* ,\n\nwhere\n\n*Eq. 5\u201170* ,\n\nwith\n\n*Eq. 5\u201171* ,\n\nwhere represents the complex conjugate of .\n", "metadata": {}}, {"id": "307fdf80", "cell_type": "markdown", "source": "### Cluster Expansions\n\nMany properties of a solution phase such as energy are dependent on the\n*configurations* - the arrangements of atoms on the lattice sites. In\ncluster expansion \\[35, 42\\], the configuration dependence of properties\nis formulated efficiently by a \u201clattice algebra\u201d which maps a\nsubstitutional configuration into an Ising-like lattice model. Taking a\nbinary A1-*x*B*x* solution phase for instance, A\natoms are represented by the \u201cdown\u201d spins (*Si* = -1) and B\natoms are represented by the \u201cup\u201d spins (*Si* = +1) as\nillustrated in . Using the cluster expansion technique, for a system\ncontaining *N* atoms, the total energy of any alloy configuration *\u03c3* =\n(*S*1, *S*2, \u2026, *SN*) can be\nconveniently evaluated using the following Ising-like Hamiltonian:\n\n*Eq. 5\u2011147*\n\nwhere *J*\u2019s are the effective cluster interactions (ECI\u2019s); is a number\nrepresenting the atomic occupation at the lattice *i* under the\nconfiguration *\u03c3*, which takes the values -1 and 1 for binary and -1, 0,\nand 1 for ternary etc. In , the 2-site, 3-site, and 4-site correlations\nare written as follows,\n\nEq. 5\u2011148\n\nEq. 5\u2011149\n\nEq. 5\u2011150 .\n\nThe expansion in would be exact as long as *all* the *n*-site\ninteractions are included. For a binary system, this can be observed by\nthe combination law that where is the number of *n*-site interactions.\nHowever, in actual calculations, one never does an expansion to the\norder *N* (containing 2*N* terms for binary) since it is too\nlong to be practical. In fact, since the interactions between widely\nseparated atoms are expected to be weaker than the interactions between\nnearer atoms for most of the important properties, the expansion in is\nusually truncated at certain distance to include only a few short-ranged\npair (2-site), triple (3-site), and up to the most, the quadruple\n(4-site) interactions.\n\nOnce a configuration is assigned, the *S*\u2019s are just the geometrical\nfactors. The common practice in the cluster expansion is that i) perform\nfirst-principles calculations of a selected set of configurations\n(around 20-100); ii) evaluate the values of *J*\u2019s using with the\nenergies from i); iii) use the fitted *J*\u2019s to predict the energy for a\ndesired set of configurations; and iv) make the ensemble average at a\ngiven temperature for the energetics of the random alloys through Monte\nCarlo simulations.\n", "metadata": {}}, {"id": "c8fd2663", "cell_type": "markdown", "source": "### Debye-Gr\u00fcneisen approximation to the vibrational contribution \n\nStrictly speaking, the Debye theory is only accurate at very low\ntemperatures. It assumes a parabolic type of dependence of PDOS\u00a0on the\nphonon frequency. This assumption is only correct at the scale of 10\u2019s K\nbecause at low temperatures,\u00a0only the low frequency acoustic phonons are\nactivated which play the major role for the parabolic type of PDOS for\nthe low frequency range as shown in . That is why\u00a0there are two kinds of\nDebye temperature: low-temperature Debye temperature and\nhigh-temperature\u00a0Debye temperature. The low-temperature Debye\ntemperature can be strictly derived by fitting low temperature heat\ncapacity data. The high-temperature\u00a0Debye temperature is mostly a\nphenomenological fitting parameter.\n\nThe Debye model approximates the PDOS in by\n\n*Eq. 5\u201138* ,\n\nwhere is the so-called Debye cutoff frequency related to the Debye\ntemperature as\n\n*Eq. 5\u201139* .\n\nAs the result, the vibrational contribution to Helmholtz energy under\nthe Debye approximation becomes,\n\n*Eq. 5\u201140*\n\nwhere *D*(*\u0398D/T*) is the Debye function given by .\n\nUnder the Debye approximation, the formulation to calculate the entropy\nbecomes\n\n*Eq. 5\u201141* ,\n\nthe formulation to calculate the internal energy becomes\n\n*Eq. 5\u201142* ,\n\nwhere\n\n*Eq. 5\u201143* ,\n\nand the formulation to calculate the heat capacity at constant volume\nbecomes\n\n*Eq. 5\u201144* .\n\nHere it is noted that is volume dependent which is often written in\nterms of Gr\u00fcneisen constant:\n\n*Eq. 5\u201145* .\n\nIt has been found that the dependence of on *V* usually is weak and\nhence is often approximated as a constant. With , the formulation to\ncalculate the pressure becomes\n\n*Eq. 5\u201146* .\n\nAn important result of the Debye approximation is that when , together\nwith , the heat capacity in is reduced to\n\n*Eq. 5\u201147* .\n\nThis gives the Debye T3 law when the thermal electron\ncontribution is neglected. In the analysis of superconductor and heavy\nFermion materials, is often rewritten as\n\n*Eq. 5\u201148* ,\n\nwhich is more convenient for examining the heat capacity measured at\nextremely low temperatures.\n", "metadata": {}}, {"id": "bb53a123", "cell_type": "markdown", "source": "### Density functional theory (DFT) and 0 K Kohn-Sham equations\n\nThe density functional theory advocates that the properties of a matter\nare solely dictated by its electronic density distribution (or equally\nsay charge density), , in the real space. This is to say, that for an\narbitrary the total energy of the system, *E*, is always larger or equal\nto a value, *E0*, called as the ground state energy:\n\n*Eq. 5\u201186* .\n\nIn terms of variational principle, is equivalent to\n\n*Eq. 5\u201187* .\n\nKohn and Sham \\[8\\] proposed to write the total energy as\n\n*Eq. 5\u201188* ,\n\nwhere represents the kinetic energy of the system, is the external\npotential acting on the system, is the Hartree energy, and is the\nso-called exchange-correlation energy with where and represent the\ncharge density of electrons with spin down and spin up, respectively .\nUsing\n\n*Eq. 5\u201189* ,\n\ntogether with the variational principle of , one can get the\none-electron Schr\u00f6dinger equation\n\n*Eq. 5\u201190*\n\nwhere *me* represents the mass of an electron and\n\n*Eq. 5\u201191*\n\nso that the total energy is obtained as\n\n*Eq. 5\u201192*\n\nThe major challenge within DFT is that the accurate formulation of the\nexchange-correlation energy is unknown. Except for the uniform electron\ngas, no exact analytical form for the exchange-correlation energy has\nyet been obtained. Therefore approximations must be made for the\nexchange-correlation energy in calculating a realistic system. Until\nnow, the two most popular approximations are the local density\napproximation (LDA) \\[30\\] and the generalized gradient approximation\n(GGA) \\[31-32\\].\n\nThe local density approximation (LDA) states that the\nexchange-correlation energy is the same as that for a locally uniform\nelectron gas. In this case one can write *Vxc* as\n\n*Eq. 5\u201193* .\n\nAlthough this approximation is extremely simple, it works reasonably\nwell for many systems. The only remaining problem is to find an\napproximate solution to . One of most employed parameterized expression\nfor is that by Perdew and Zunger \\[30\\].\n\nMany modern DFT codes use the more advanced generalized gradient\napproximation (GGA) \\[31-32\\] to the exchange-correlation energy to\nimprove accuracy for certain physical properties. As the LDA\napproximates the energy of the true density by the energy of a local\nconstant density, it fails in situations where the density undergoes\nrapid changes such as in molecules. An improvement to this can be made\nby considering the gradient of the electron density. Symbolically, this\ncan be written as\n\n*Eq. 5\u201194*\n\nThe commonly used GGA is those due to Perdew et al. \\[31-32\\].\n", "metadata": {}}, {"id": "0cf3bd9c", "cell_type": "markdown", "source": "### Dynamical matrix and phonon mode\n\nBecause of the periodicity of a crystal, one can make an initial guess\nthat the solutions of are elastic plane waves made of a collective\natomic vibrations \\[23, 33\\], from the harmonic approximation of ,\n\n*Eq. 5\u2011122* ,\n\nwhere *\u03c9* represents the frequency of the plane wave, and is a wave\nvector designating the wave number and direction along which the plane\nwave propagates. It should be pointed out that in is now independent of\nthe index *P.* That is, except for a phase factor, the atoms that are\nequivalent by the translational symmetry among different primitive unit\ncells will experience the same type of atomic motion, independent of the\npositions of the primitive unit cell in the system. This is equivalent\nto applying the periodic condition so that in obeys\n\n*Eq. 5\u2011123* .\n\nNote that is now independent of the index *P*.\n\nFurthermore, one wants to limit **q***t* in and to those\nknown as the exact wave vectors which represent a special set of points\nin the reciprocal space that satisfy the condition\n\n*Eq. 5\u2011124* ,\n\nwhere is the Kronecker delta function. In fact, the number of equals to\nthe number of primitive unit cells contained in the system.\n\nUtilizing the translational invariance by which in \\[or in \\] depends on\n*P* and *Q* only through the difference , the following Fourier\ntransformation can be employed to simplify\n\n*Eq. 5\u2011125* ,\n\nand one obtains\n\n*Eq. 5\u2011126* .\n\nThe counterpart of with respect to is\n\n*Eq. 5\u2011127*\n\nis now an equation with 3*Na* degrees of freedom. At each ,\none can always find 3*Na* eigenvalues of (*i* = 1, \u2026,\n3*Na*). The 3*Na* vibrations are often known as\nphonon modes, noting again that *Na* is the number of atoms\nin the primitive unit cell. Each of yields a set of which can be chosen\nsuch that\n\n*Eq. 5\u2011128* ,\n\nwhere represents the complex conjugate of , and\n\n*Eq. 5\u2011129*\n\nFinally, for a solid, the summation over *Q* in can be abbreviated out,\nresulting in, after transforming back to together with using\n\n*Eq. 5\u2011130* ,\n\nbecause of translational invariance by which depends on *P* and *Q* only\nthrough the difference .\n\nIn the case that a system is built by a parallelepiped multiplication of\nthe primitive unit cell with lattice vectors of **a***\u03b1*\n(\u03b1=*x*, *y*, and *z*) in the form , in and takes the form\n\n*Eq. 5\u2011131* ,\n\nwhere = 0, 1, \u2026, , and is the primitive lattice vector in the reciprocal\nspace as\n\n*Eq. 5\u2011132* ,\n\nwith *Va* representing the volume of the primitive unit cell\ngiven as\n\n*Eq. 5\u2011133* .\n\nCombine and , it is easy to demonstrate\n\n*Eq. 5\u2011134* ,\n\nwhere is the Kronecker delta symbol.\n\nIt can be shown that the **q** points defined in represent the exact\nwave points. First, can be written as\n\n*Eq. 5\u2011135* ,\n\nwhere = 0, 1, \u2026, . Then, the left hand side of for the definition of\nexact wave vector becomes\n\n*Eq. 5\u2011136* .\n\nKnowing the fact that\n\n*Eq. 5\u2011137* .\n\nHence, it is proved the **q** points defined in are the exact wave\npoints.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/firstprinciples_calculations_and_theory/firstprinciples_approaches_to_disordered_alloys.ipynb b/psu410/src/firstprinciples_calculations_and_theory/firstprinciples_approaches_to_disordered_alloys.ipynb deleted file mode 100644 index 91c43c9..0000000 --- a/psu410/src/firstprinciples_calculations_and_theory/firstprinciples_approaches_to_disordered_alloys.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "9965d400", "cell_type": "markdown", "source": "## First-principles approaches to disordered alloys\n\nFirst-principles calcuations discussed so far strictly rely on the exact\natomic positions in the unit cells. A brutal-force approach for random\nsolution phase would be to directly construct a large supercell and\nrandomly decorate the host lattice with different types of atoms. Such\nan approach would necessarily require very large supercells to\nadequately mimic the statistics of the random solutions. Since\nfirst-principles methods are computationally constrained by the number\nof atoms that one can treat, this brute-force approach is\ncomputationally prohibitive. Take a binary\nA1-*x*B*x* substitutional alloy as illustrated in\nas an example, for a system containing *N* atoms, there can be\n2*N* possible number of configurations, which is an\nastronomically large number when *N* is large. It is an impossible task\nto explore such a huge configuration space with available computing\nresources.\n\nFigure \u2011. Mapping of a substitutional A1-xBx alloy\ninto an Ising-like lattice model \\[39-40\\].\n\nAs a result, approximations must be made to the first-principles\ncalculations. At present, there are mainly three approaches to\ncalculating the disordered solution phases: the coherent potential\napproximation (CPA) \\[41\\], the cluster expansion (CE) \\[42\\], and the\nspecial quasirandom structures (SQS\u2019s) \\[43\\] approach.\n\nCPA \\[41\\] treats random alloys by considering the *average* occupations\nof lattice sites in solving the Kohn-Sham equation. Since a mean-field\napproach is employed, dependence of properties on the local environments\nsurrounding an atom is not treated explicitly in CPA. In a random\nsolution, there exists a distribution of local environments (e.g., in\nbcc alloys, A or B surrounded by the various\nA*X*B8-*X* coordination shells with *X* between 0\nand 8), resulting in local environmentally-dependent quantities such as\ncharge transfer and local displacements of atoms from their ideal\nlattice positions. Even in random A1-*x*B*x* solid\nsolutions, the average A-A, A-B and B-B bond lengths are generally\ndifferent. These effects can be considered by the CE and SQS approaches,\nwhich are the focus of the next two subsections. In following\nsubsections, unless specifically noted, the formulism for the binary\nsystem is discussed for the sake of simplicity.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/firstprinciples_calculations_and_theory/firstprinciples_formulation_of_thermodynamics.ipynb b/psu410/src/firstprinciples_calculations_and_theory/firstprinciples_formulation_of_thermodynamics.ipynb deleted file mode 100644 index 55da089..0000000 --- a/psu410/src/firstprinciples_calculations_and_theory/firstprinciples_formulation_of_thermodynamics.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "ff0a17fb", "cell_type": "markdown", "source": "## First-principles formulation of thermodynamics\n", "metadata": {}}, {"id": "f4ca09c6", "cell_type": "markdown", "source": "### Formation enthalpy of Ni3Al\n\nOne can do the similar calculations for the elemental metal Al and the\ncompound Ni3Al which has the L12 structure,\nfollowing the same steps in the calculations of Ni. The formation\nenthalpy in the unit of per mole atom can be calculated as\n\n*Eq. 5\u201115* ,\n\nwhere , , and represent the enthalpies of Ni3Al, Ni, and Al\nin the energy unit of per mole atom, respectively. Plotted in is the\ncalculated formation enthalpy of Ni3Al (curve) as a function\nof temperature from 0 to 1600 K at *P* = 0 compared with experimental\ndata (symbols) with details in Ref. \\[16\\].\n\n.\n\nFigure \u2011. Formation enthalpy of L12-Ni3Al with\nrespect to pure fcc Ni and Al.\n", "metadata": {}}, {"id": "66decf35", "cell_type": "markdown", "source": "### Hartree-Fock approximation to solve Schr\u00f6dinger equation\n\nIt was Hartree who first assumed that the electron wave function in can\nbe expressed as a product of a collection of *N* independent\none-electron wave functions, where *i* = 1, 2, \u2026, *N* with *N* being the\nnumber of electrons in a system, in terms of the its space coordinate\n**r** and spin state **s**. After that, Fock modified the Hartree\napproximation by considering the fact that the wave function of a\nmulti-fermionic system should satisfy anti-symmetry requirements and\nsubsequently the Pauli exclusion principle that the total wave function\nchanges sign upon the exchange of fermions. Accordingly, the wave\nfunction of *N* electrons system under the Hartree-Fock approximation is\nexpressed as the Slater determinant \\[29\\]\n\n*Eq. 5\u201172*\n\nFor brevity, one can use the atomic unit that treats in , so that\n\n*Eq. 5\u201173*\n\nwhere , and\n\n*Eq. 5\u201174*\n\nAccordingly, the total energy of the system is expressed as\n\n*Eq. 5\u201175*\n\nwhere *Jij* is called as Coulomb/Hartree term\n\n*Eq. 5\u201176*\n\nwhere *Kij* is called as exchange term\n\n*Eq. 5\u201177* ,\n\nwhere = 1 if spin **s***i* and **s***j* points to\nthe same direction and = 0 if spin **s***i* and\n**s***j* points to the opposite direction.\n\nBy utilizing the variational condition of , one gets\n\n*Eq. 5\u201178* ,\n\nwhere is called one-electron energy, and\n\n*Eq. 5\u201179* ,\n\nwith being the electronic charge density whose expression is\n\n*Eq. 5\u201180* ,\n\n*Eq. 5\u201181* .\n\nIt should be especially noted here that to solve the Hartree-Fock\nequation , the most time consuming part is due to the nonlocal exchange\nterm, knowing the fact that the being evaluated one-electron wave\nfunction is also contained in the expression in the left hand side of by\nmeans of .\n\nThe configurational interaction method is the generalization of the\nHartree-Fock approximation. In such a case, *Y*, the number of\none-electron wave functions can be larger than the number of electrons,\n*N*, in the system. Accordingly, from the number of one-electron wave\nfunctions, , *y* = 1, 2, \u2026, *Y*, one can build the number of Stater\ndeterminants, *M*, by the combinatorial mathematics that the maximum of\n*M* can be\n\n*Eq. 5\u201182* .\n\nAs a result, the wave function of a collection of *N* electron system\nbecomes the recombination of the *M* Stater determinants as\n\n*Eq. 5\u201183* ,\n\nwhere the coefficients is to be found from the multiple linear equation\n\n*Eq. 5\u201184* .\n\nThe matrix element in *Eq. 5\u201184* is determined by the integral\n\n*Eq. 5\u201185* .\n", "metadata": {}}, {"id": "57a76f3e", "cell_type": "markdown", "source": "### Helmholtz energy\n\nIn this chapter so far, all our discussions are limited to the case of a\nsystem which is made from a single microstate (microscopic state). Here,\nthe terminology microstate refers to the microscopic structure that is\ndistinguished by crystal structure, atom distributions in the lattice\nsites, and the arrangements of the local atomic spin and electronic\nangular momentum distributions among the lattice sites. From this\nsection on, the index \u03c3 is employed to label the microstate. For a solid\nmaterial at finite temperatures, a phase can be formed by a single\nmicrostate or a mixture of multiple microstates.\n\nLet us consider a canonical system made of *N* atoms with an average\natomic volume *V*. The study is limited to the motions of atomic nuclei\nand electrons. For such a system, one can use to index the energy\neigenvalues of the corresponding microscopic Hamiltonian associated with\nmicrostate *\u03c3*. The subscript **g** symbolically labels the different\nvibrational states for the motions of atomic nuclei and the subscript\n**n** symbolically labels the electronic states distinguished by the\ndifferent distributions of the electrons between the electronic valence\nand conduction bands. Neglecting electron-phonon coupling, one can\nassume that the contributions to between the vibrational and electronic\nstates are additive, so that\n\n*Eq. 5\u201116* ,\n\nwhere *Ec* is the static total energy of the microstate *\u03c3*.\nNote that in and represent the energetics of the vibrational state and\nthe electronic state respectively.\n\nOne then can formulate the canonical partition function of the\nmicrostate *\u03c3* at the given temperature *T* and volume *V* as\n\n*Eq. 5\u201117* ,\n\nwhere *\u03b2* = 1/*kBT* with *kB* being the\nBoltzmann\u2019s constant. As a result, with, the Helmholtz energy *F* per\natom for the microstate *\u03c3* is derived as follows:\n\n*Eq. 5\u201118* ,\n\nwhere the variable *N* has been abbreviated using\n\n*Eq. 5\u201119* ,\n\n*Eq. 5\u201120* ,\n\n*Eq. 5\u201121* .\n\nThe calculation of *Ec* is straightforward in most of the\nexisting first-principles codes as discussed earlier.\n", "metadata": {}}, {"id": "c9b00001", "cell_type": "markdown", "source": "### Helmholtz energy and quasiharmonic approximation\n\nAt present, the most rigorous method to predict the thermodynamic\nproperties of a material at finite temperatures is the phonon approach.\nIn such an approach, the microscopic Hamiltonian is expanded up to the\nsecond order. All the thermodynamic quantities are calculated using\nformulations derived from the statistical physics without further\napproximation. The great importance of the phonon theory is that all the\ninput parameters can be obtained by means of first-principles\ncalculations without using any phenomenological parameters.\n\nLet us consider a system with an averaged atomic volume *V*. Neglecting\nthe electron phonon coupling, it is a well demonstrated procedure \\[16\\]\nto decompose the Helmholtz energy *F*(*V*,*T*) of the system at\ntemperature *T* into three additive contributions as follows\n\n*Eq. 5\u20111* ,\n\nwhere *Ec* is the static total energy which is the total\nenergy of the system at 0 K without any atomic vibrations,\n*Fvib* is the vibrational contribution due to the lattice\nions, and *Fel* is the electronic contribution due to the\nthermal electronic excitation at finite temperature which can become\nimportant for metals at high temperature.\n\nThe terminology of \u201cquasiharmonic approximation\u201d arises from the fact\nthat for a given volume, *Fvib*(*V*,*T*) is calculated under\nthe harmonic approximation and the anharmonic effects are carried out\nsolely through volume dependence of the phonon frequency. The easiest\ncomputational implementation of is to first independently calculate the\nHelmholtz energy at several selected volumes near the equilibrium volume\nand then use the numerical interpolation to find the Helmholtz energy at\nan arbitrary volume. The volume interval is usually at the scale of 3~5%\nof the equilibrium volume. Too small volume interval can result in\nnumerical instability due to the numerical uncertainties in the static\ntotal energy calculation, in particular, when one numerically computes\nthe first- and especially the second-order derivatives of the Helmholtz\nenergy in deriving the thermodynamic quantities. It should be noted that\nwhenever available, analytic formulas should be used instead of the\nnumerical second-order derivative to avoid numerical errors. For\ninstance, when the phonon approach is employed, the constant volume heat\ncapacity has the analytic expression in terms of phonon density of\nstates.\n\nNickel metal adopts the fcc structure at ambient conditions and the\nprimitive unit cell contains one atom. Almost all the existing\nfirst-principles codes have the function to calculate the static total\nenergy. The static total energy *Ec* in should be calculated\nusing the primitive unit cell. As the Helmholtz energy is to be\ncalculated at several volumes, a good practice is to plot the calculated\nstatic total energy points together with the interpolated energy curve\nto examine the convergence of the static total energy calculation. Since\nthe first-principles method often employs the self-consistent technique,\nit could occur that calculations at certain volumes may not convergent,\nwhich should be fixed by trying the various algebraic schemes provided\nin most of the existing codes. Furthermore, since certain calculations\ninvolve the second order derivative of the Helmholtz energy, a minor\nuncertainty along the static total energy curve can result in large\ndeviation for the calculated properties such as thermal expansion\ncoefficient and bulk modulus. In that case, a reasonable solution is to\nsmoothen the static total energy using the modified Birch-Murnaghan\nequation of states (EOS) \\[17-18\\]\n\n*Eq. 5\u20112* .\n\nPlotted in is the calculated static total energy of the elemental metal\nNi with the circles representing the calculated values and the curve\nrepresenting that by EOS fitting.\n\nFigure \u2011. Static total energy of nickel.\n\nThe vibrational contribution to the Helmholtz energy by phonon theory\ncan be computed by \\[19\\]\n\n*Eq. 5\u20113* ,\n\nwhere is the Boltzmann\u2019s constant, *\u03c9* represents the phonon frequency,\nand is the phonon density of states. It is recommended that is\ncalculated at the same volume set at which the static total static\nenergies are calculated.\n\nFor the present prototype of Ni, the supercell method for the\ncalculation of has been employed. The procedure is follows:\n\n1. Make supercell by enlarging the primitive unit cell according to the\n > defined neighbor interaction distance; Employ the first-principles\n > code (VASP, \\[13-14\\] in this work) to calculate the interatomic\n > force constants.\n\n2. Assign the mesh in the wave vector (**q**) space; Make the dynamical\n > matrix at each **q** point; Diagonalize the dynamical matrix to\n > find out the phonon frequencies at each **q** point; And finally\n > collect all the phonon frequencies for all **q** points. The\n > detailed formulation for phonon calculations is given in Chapter .\n\nFor the phonon calculations, one can use the open source code YPHON\n\\[15\\] by the present authors. Other choices can be the free ATAT code\n\\[20\\] or the free PHON code \\[21\\]. For the calculation of the phonon\ndensity of states, we have made a supercell containing 64 atoms which is\na 4\u00d74\u00d74 supercell of the primitive unit cell. Plotted in is the\ncalculated phonon density of states using YPHON code at the calculated\nstatic equilibrium volume compared with the measured data at 10 K \\[22\\]\n(symbols).\n\nFigure \u2011. Phonon density of states of nickel.\n\nFor a first-principles thermodynamic calculation, an important step to\navoid possible calculation errors is to examine the phonon dispersions\nfirst. Phonon dispersion \\[23\\] depicts the evolution of phonon\nfrequencies along the designated direction for a crystal. Phonon\ndispersion can be measured rather accurately by inelastic neutron\nscattering \\[24-26\\] or inelastic x-ray scattering \\[27\\] experiment.\nPlotted in are the calculated phonon dispersions (curves) along the\n\\[00\u03b6\\], \\[0\u03b61\\], \\[0\u03b6\u03b6\\], and \\[\u03b6\u03b6\u03b6\\] directions of Ni using YPHON code\ncompared with the neutron scattering data at 296 K (symbols) with\ndetails in Ref. \\[16\\].\n\nFigure \u2011. Phonon dispersions of nickel. The solid lines represent\nresults calculated using a supercell containing 256 atoms which is 4\u00d74\u00d74\nsupercell of the conventional cubic unit cell. The dot-dashed lines\nrepresent results calculated using a supercell containing 64 atoms which\nis 4\u00d74\u00d74 supercell of the primitive unit cell.\n\nFor the calculation of *Fel* in , the most computationally\nconvenient approach is to use the Mermin statistics as follows\n\n*Eq. 5\u20114* ,\n\nwhere is the thermal electronic energy, and *Sel* is the bare\nelectronic entropy. Both the calculations of and *Sel* need\nthe electronic density of states (EDOS) as input. The electronic density\nof states can be obtained during the step of the static total energy\ncalculation. The detailed formulations for and *Sel* are\ngiven in Chapter . Since Ni is magnetic, the EDOS of Ni can be\npartitioned into those of spin up and spin down due to the spin freedom\nof electron. The calculated EDOS for Ni is shown in where the solid,\ndot-dashed, and dashed lines represent the total, spin up, and spin down\nEDOS with the Fermi energy set to zero.\n\nFigure \u2011. Electronic density of states of nickel. That due to spin up is\nplotted as positive value and that due to spin down is plotted as\nnegative value purely for the clarity of the figure. The \u201ctotal\u201d is the\nsum of the absolute values of those of spin up and spin down.\n\nThe calculated temperature evolution of Helmholtz energy as a function\nof volume for Ni are illustrated in . The circles represent the\ncalculated static total energies. The solid curves represent the\nHelmholtz energy curves from 0 to 1600 K at a temperature increment of\n100 K as displayed from top to bottom in . The dashed line marks the\nevolution of the equilibrium volume at *P*=0 with increasing\ntemperature. It is noted that Helmholtz energy always decreases with\nincreasing temperature due to the entropy term of \u2013*TS*. Note that the\nat 0 K the Helmholtz energy is higher than the static total energy due\nto the zero point vibrational energy as can be seen when *T* \u21920 which\nreduces to\n\n*Eq. 5\u20115* ,\n\nwhich is positive.\n\nFigure \u2011. Temperature evolution of the Helmholtz energy for nickel.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/firstprinciples_calculations_and_theory/index.ipynb b/psu410/src/firstprinciples_calculations_and_theory/index.ipynb deleted file mode 100644 index e250c93..0000000 --- a/psu410/src/firstprinciples_calculations_and_theory/index.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "e58b1f1e", "cell_type": "markdown", "source": "# First-principles calculations and theory\n\nIn the previous chapter, the experimental techniques in obtaining the\nthermochemical and phase equilibrium data that are the inputs for the\nthermodynamic modeling of a system were summarized. However,\nexperimental data are not always available. This is due to the fact that\ni) the experiments are expensive, especially true in developing new\nmaterials; and ii) the experiments cannot reliably access the non-stable\nphases in most cases. The alternative approach is to predict the\nthermochemical data by first-principles calculations. The prediction of\nthe material properties, without using phenomenological parameters, is\nthe basic spirit of first-principles calculations. In particular, the\nsteady increase of both computer power and the efficiency of\ncomputational methods have made the first-principles predictions of most\nthermodynamic properties possible, including both enthalpy and entropy\nas a function of temperature, volume, and/or pressure.\n\nBy its definition, the term of \u201cfirst-principles\u201d represents a\nphilosophy that the prediction is to be based on a basic, fundamental\nproposition or assumption that cannot be deduced from any other\nproposition or assumption. This implies that the computational\nformulations are based on the most fundamental theory of quantum\nmechanics - Schr\u00f6dinger equation or density functional theory and the\ninputs to the calculations must be based on well-defined physical\nconstants \u2013 the nuclear and electronic charges. In another word, once\nthe atomic species of an assigned material are known, the theory should\npredict the energy of all possible crystalline structures, without\ninvoking any phenomenological fitting parameters.\n\nThis chapter organized in the sequence from thermodynamic calculations\nto fundamental theory to help those readers who are more interested in\nrealistic calculations using existing computer codes. The detailed\ntheoretical sections are presented to follow the section of\nthermodynamic calculations for those readers who are also interested in\nthe derivation of the formulations used in the thermodynamic\ncalculations. The following subsections are arranged accordingly in the\norder: (i) examples the commonly adopted calculation procedures for\nthermodynamic properties using the elemental metal nickel as the main\nprototype; (ii) derivation of the Helmholtz energy expression under the\nfirst-principles framework; (iii) introduction of the solution to the\nelectronic Schr\u00f6dinger equation within the two well developed frameworks\n\u2013 the quantum chemistry approach and the density functional theory; (iv)\ndetailed description of the procedure on how to solve the Schr\u00f6dinger\nequation for the motions of the atomic nuclei by means of lattice\ndynamics. The relation between the Helmholtz energy and Gibbs energy is\nshown by .\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/firstprinciples_calculations_and_theory/lattice_dynamics.ipynb b/psu410/src/firstprinciples_calculations_and_theory/lattice_dynamics.ipynb deleted file mode 100644 index 40cd460..0000000 --- a/psu410/src/firstprinciples_calculations_and_theory/lattice_dynamics.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "0a5cf3ba", "cell_type": "markdown", "source": "## Lattice Dynamics\n", "metadata": {}}, {"id": "fa5e392d", "cell_type": "markdown", "source": "### Linear-response method vs supercell method\n\nThe problem of lattice vibration for a solid is now transformed into\ncomputing the dynamical matrix in . The first-principles solution of the\nproblem is currently divided into two categories: the linear-response\nmethod \\[34\\] and the supercell method \\[35\\]. In the linear-response\nmethod, utilizing the electronic linear response upon the undistorted\ncrystals \\[36\\], the evaluations of the dynamical matrix can be\nperformed through the density-functional perturbation theory \\[34\\]\nwithout the approximation of the cutoff in neighboring interaction.\n\nCompared with the linear-response method, the supercell method is\nconceptually simple and is easy to implement computationally. The\nsupercell method adopts the frozen phonon approximation through which\nthe changes in total energy or forces are calculated in the direct space\nby displacing the atoms from their equilibrium positions. The advantage\nof the supercell method is that the phonon frequencies at the exact wave\nvectors, which are commensurable with the supercell, are calculated\nexactly with no further approximation \\[37\\]. The shortcoming of the\nsupercell method is that it is often limited by the size of the\nsupercell that can be handled with current computing resources.\n\nIn supercell approach, inaccuracies are thought to arise from the\ntruncation of the force constants \\[34-35\\]. This is only partially\ntrue. In the supercell method, the calculated represents the cumulative\ncontributions of the atom indexed by *k* and *P* in the supercell and\nall its images by translational transformation of the supercell in the\nwhole space. Let **L**\u03b1 represent the lattice vectors of the\nsupercell, then\n\n*Eq. 5\u2011138* .\n\nFor the exact wave vectors in , one has\n\n*Eq. 5\u2011139* **,**\n\nwhere is an integer. Replacing in with in , one obtains\n\nEq. 5\u2011140 .\n\nTherefore, the phonon frequencies calculated at the exact wave vectors\nby the cumulative force constants approach are exact, and the supercell\nsize will not lead to errors in the calculated phonon frequencies\n\\[37\\].\n\nGenerally speaking, if a supercell contains *Nc* primitive\nunit cells, one can always find *Nc* corresponding exact wave\nvectors in the **q** space. In most linear response calculations, the\ncommon choice of a 4\u00d74\u00d74 **q**-mesh is exactly equivalent to the 4\u00d74\u00d74\nsupercell in the real-space. Furthermore, since the supercell approach\ndoes not impose any approximation on the electronic response to the\ndistortion of the lattice geometry, the effects of electron-phonon\ninteractions can be accounted for by the supercell method.\n\nIn the supercell method, due to the impose of periodic condition, the\ncalculated force constant in the real-space cannot account for the\neffects of the vibration-induced electric field for the polar materials.\nIt has been demonstrated that such an effect adds an additional term to\ndynamical matrix in the reciprocal-space in the form,\n\n*Eq. 5\u2011141* ,\n\nwhere ***Z**\\* (j)* represents the Born effective charge\ntensor of the *j*th atom in the primitive unit cell and\n**\u03b5\u221e** the high frequency static dielectric tensor, i.e., the\ncontribution to the dielectric permittivity tensor from the electronic\npolarization. As a result for the polar materials, the matrix element at\nwave vector **q**\u21920 of the dynamical matrix in , by means of , should\nhave the form\n\n*Eq. 5\u2011142* .\n\nIt can be demonstrated \\[38\\] that this is equivalent to replacing the\nreal space force constant in by\n\n*Eq. 5\u2011143* .\n\nThe present implementation of the first-principles method in calculating\nthe phonon frequencies are mostly limited by the supercell size when\nusing or the number of exact wave vector points when using . A supercell\ncell built on the primitive unit cell or exact wave vector mesh is\nusually the common limit. If only the phonon frequencies derived from or\nare used, can be rather unsmooth which will result in inaccurate\nthermodynamic properties when it is used with . The mixed-space approach\n\\[38\\] can circumvent this bottleneck by using the Fourier interpolation\nto calculate the dynamical matrix for an arbitrary wave vector **q** as\n\n*Eq. 5\u2011144* ,\n\nwith the help of , , and for a polar materials. Note that the term is\nfor polar materials only.\n\nThe can be calculated as\n\n*Eq. 5\u2011145* ,\n\nwhere the function is usually taken as the Gaussian type\n\n*Eq. 5\u2011146* ,\n\nwhere is an adjustable damping (broadening) parameter whose role is to\nsmear the curve. in is the number of used **q** points. Empirically, a\n**q** mesh of is accurate enough for most purposes and doable with the\nYPHON code \\[15\\] efficiently, which is discussed in the Appendix A.\n", "metadata": {}}, {"id": "b4b6a83a", "cell_type": "markdown", "source": "### Mermin statistics to the thermal electronic contribution \n\nFor the calculation of *Fel* in , the most computationally\nflexible approach is to use the Mermin statistics \\[28\\] by which\n\n*Eq. 5\u201122*\n\nwhere the bare electronic entropy *Sel* takes the form, after\nreplacing the summation in over the electronic states with integration\n\n*Eq. 5\u201123* ,\n\nby means of utilizing , the electronic density of states *n*. *f* in is\nthe Fermi distribution that takes the form\n\n*Eq. 5\u201124* .\n\nNote that in is the electronic chemical potential that is strongly\ntemperature dependent. At each given *T*, should be carefully calculated\nkeeping the number of electrons unchanged in solving the following\nequation:\n\n*Eq. 5\u201125* ,\n\nnoting that is the Fermi energy calculated at 0 K. With respect to , the\nthermal electronic energy *Eel*, due to the thermal electron\nexcitations, can be calculated through\n\n*Eq. 5\u201126* .\n\nAt low temperature, , , and are reduced to\n\n*Eq. 5\u201127*\n\nwhere *\u03bb* is the so-called electronic specific heat coefficient\ncalculated as\n\n*Eq. 5\u201128* ,\n\nwhere is the electronic density of states at the Fermi level, and\n\n*Eq. 5\u201129*\n\n*Eq. 5\u201130*\n\nFrom , one can easily derive the electronic contribution to the specific\nheat at low temperature as\n\n*Eq. 5\u201131* .\n\nUsually, the dependence of on *V* is weak. Therefore for a normal\nconductor (except for the heavy Fermion metal or superconductor related\nmaterials), at low temperatures, the electronic contribution to the heat\ncapacity is linear against *T*. also indicates that for insulators, by\nmeans of , the electronic contribution to the heat capacity is zero\nsince for insulators .\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/firstprinciples_calculations_and_theory/nickel_as_the_prototype.ipynb b/psu410/src/firstprinciples_calculations_and_theory/nickel_as_the_prototype.ipynb deleted file mode 100644 index dc17080..0000000 --- a/psu410/src/firstprinciples_calculations_and_theory/nickel_as_the_prototype.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "2f770a8c", "cell_type": "markdown", "source": "## Nickel as the prototype\n\nThis section exemplifies the step-by-step procedure in calculating the\nthermal properties within the framework of first-principles phonon\napproach, using the elemental metal Ni as the prototype. The calculation\nof the formation enthalpy of Ni3Al is given at the end. The\ncalculation in this section is limited to the case for the ferromagnetic\nphase, i.e. single microstate, implying that no configurational mixtures\nor magnetic phase transitions is considered which is discussed in\nChapter 5.2.5 and Chapter 6.\n\nThe Vienna Ab-initio Simulation Package (VASP) \\[13-14\\] has been\nemployed for electronic calculations, and the YPHON code \\[15\\] has been\nemployed for phonon calculations. VASP is a code based on the\npseudopotential approach to the density functional theory using plane\nwave function as the basis set, by which only the valence electrons are\nhandled explicitly and the core electrons are approximated by an\neffective pseudopotential. The same energy cutoff values, which\ndetermine the number of plane waves in the expansion of electronic wave\nfunction, have been used for Ni, Al, and Ni3Al. The rationale\nfor the derivations of the formulations used in this section is to be\ngiven in Chapter for readers who want to have an in-depth understanding\nof the physics behind the used formulations.\n", "metadata": {}}, {"id": "a104be41", "cell_type": "markdown", "source": "### Normal coordinates, eigenenergetics, and phonons\n\nOne way to simplify the solution to the Schr\u00f6dinger\u2019s equation for\nmotion of atomic nuclei is to follow the study of the vibrations of the\natoms in a crystal \u2013 lattice dynamics. The essential step in lattice\ndynamics is to transform the problem of the correlated motions of 3*N*\nparticle into the problem of 3*N* independent harmonics. For this\npurpose, one can introduce a set of new coordinates ( *i*=1, 2, \u2026,\n3*N*), known as normal coordinates, by the transformation\n\n*Eq. 5\u2011111* ,\n\nwhere is the transformation coefficient that can be determined by\nsolving 3*N* simultaneous equations\n\n*Eq. 5\u2011112* ,\n\nwhere is to be determined so that one can find 3*N* , which are, not all\nzero. The equations are linear and homogeneous. Follow the basic theorem\nin linear algebra that, to find the nonzero solutions of the equations,\nthe determinant formed by must equal zero\n\n*Eq. 5\u2011113* ,\n\nwhere is the Kronecker delta symbol. Since is an equation with 3*N*\ndegrees, one can always find 3*N* values of (*i* = 1, \u2026, 3*N*). Each of\nyields a set of which can be chosen such that\n\n*Eq. 5\u2011114* ,\n\nwhere represents the Kronecker delta symbol and\n\n*Eq. 5\u2011115*\n\nThen with the normal coordinates defined in and utilizing , defined in\nis obtained as\n\n*Eq. 5\u2011116* .\n\nWith this equation, is simplified by the following process\n\n*Eq. 5\u2011117* ,\n\nnoting that in the above process and have been utilized.\n\nFurthermore, using , the kinetic energy operator in can be transformed\nas follows\n\n*Eq. 5\u2011118* ,\n\nnoting that in the above process and are utilized again.\n\nAs a result, under the harmonic approximation, the Hamiltonian in is\nsimplified as\n\n*Eq. 5\u2011119*\n\nwhich represents a quantum system containing 3*N* independent harmonics.\nCorresponding to the each of the *\u03c9i*, the quantum theory\ntells that the eigenenergy of a harmonics has the form\n\n*Eq. 5\u2011120* ,\n\nwith *gi* = 0, 1, \u2026, \u221e. Such a type of energetics behaves\nlike a Boson particle with energy and forms the concept of phonon.\n\nFurthermore, a state of the whole system is to be specified by the set\nof 3*N* independent quantum numbers **g** = (*g1*,\n*g1*, \u2026, *g3N*,). Finally, the energetics of a\nstate of the system formed by 3*N* independent harmonics, introduced by\n, is obtained with the summation of the energies of 3*N* independent\nharmonics as\n\n*Eq. 5\u2011121* .\n\nThis concludes the rationale by which how is derived.\n", "metadata": {}}, {"id": "be96a8e8", "cell_type": "markdown", "source": "### Phonon calculations for SQS\n\nA somewhat more theoretically demanding application of the SQS approach\nis the calculation of the phonon dispersions of a random alloy.\nConsidering the fact that the size of an SQS cell in general is around\n8-32, phonon calculations based on SQS is doable, either using the SQS\ncell or its supercell, e.g., of it. However, one notes that while the\nphonon density of states can be calculated straightforwardly, the\ncalculations of the phonon dispersions run into a problem. That is,\nsince phonon calculation treats the SQS as primitive unit cell made of\nmore atoms than the primitive unit cell of the ideal lattice, the number\nof phonon dispersions derived from a regular phonon calculations is a\nlot greater than that measured for the random alloy. Say, one uses an\nSQS containing 16 atoms for an fcc solid solution, the regular phonon\ncalculations would produce phonon dispersions in comparison to that just\nthree phonon dispersions from measurement. By averaging over the force\nconstants of a SQS, the dynamical matrix can be calculated with respect\nto the wave vector space of the ideal lattice of the random alloys.\n\nOne consideration that must be taken into account is that the phonon\ndispersions measured from the inelastic neutron scattering experiments\nonly represent the averaged vibrations of the ideal lattice. For random\nalloys or phases with minor geometry distortion, it is suggested to\ncalculate the dynamical matrix by instead of *Eq. 5\u2011125* as (see \\[45\\])\n\n*Eq. 5\u2011155* ,\n\nwhere in the case of random alloy, represents the averaged atomic mass\nat the *j*th lattice site. The purpose of the summation over *Q* is to\naverage the effects of local distortions, making it possible of\ncomparing the calculated dispersions to the measured dispersions\nrepresenting the averaged vibrations of the ideal lattice. As a result,\nthe dimension of the SQS supercell dynamical matrix is thus reduced to\nmatch that of the primitive unit cell of the ideal lattice for the\ncalculation of the phonon frequencies. The calculational procedure is as\nfollows:\n\n1. Make an SQS supercell based on the primitive unit cell of ideal\n lattice to mimic the correlation functions of the random solution;\n\n2. Relax the SQS supercell with respect to the internal atomic\n positions while keeping the cell shape and volume fixed;\n\n3. Make the phonon supercell by further enlarging the SQS supercell and\n calculate the force constants; and\n\n4. Calculate the dynamical matrix , with the wave vector, **q**, being\n defined from the primitive unit cell of the ideal lattice, through\n the following Fourier transformation.\n\nThe calculated phonon dispersions, along the directions (00\u03be), (0\u03be\u03be),\nand (\u03be\u03be\u03be), are compared with the inelastic neutron scattering data in\nfor Cu3Au.\n\nFigure 5\u201113. Phonon dispersions for random Cu3Au. The solid\n(black) lines represent the present calculation and the open circles\nrepresent the inelastic neutron scattering data with details in Ref.\n\\[45\\]. The dashed (blue) lines represent the calculated results using\nthe ab initio transferable force-constant model by Dutta et al. \\[46\\].\n", "metadata": {}}, {"id": "ad6b512f", "cell_type": "markdown", "source": "### Quantum theory for motion of atomic nuclei\n\nFor the convenience of discussion, the following convention of notations\nare used: *\u03b1* and *\u03b2* to label the Cartesian axes which is either *x*,\n*y*, or *z*, *j* and *k* to label atoms in the primitive unit cell,\n*mj* the atomic mass of the *j*th atom in the\nprimitive unit cell, **r**(*j*) the position of the *j*th\natom in the primitive unit cell, *P* and *Q* the index of the primitive\nunit cell in the system, **R**(*P*) the position of the *P*th\nprimitive unit cell in the system, and *V* the averaged volume of the\nprimitive unit cell.\n\nThe quantum theory for motion of atomic nuclei replicates closely the\nquantum theory for motion of electrons. That is to solve the wave\nfunction for the motions of the atomic nuclei for a Schr\u00f6dinger\u2019s\nequation with the potential being the total electronic energy derived\nfrom . The symbol is replaced by to represent the static total\nelectronic energy with **R** representing the static equilibrium\npositions of the atomic nuclei, **u** the displacements of atomic nuclei\naway from their static equilibrium positions, and *\u03c3* the additional\ndegree of freedom such as electronic states. The Schr\u00f6dinger\u2019s equation\nfor the motion of atomic nuclei is then\n\n*Eq. 5\u2011103* ,\n\nwhere\n\n*Eq. 5\u2011104* ,\n\nwith representing the kinetic energy operator\n\n*Eq. 5\u2011105* ,\n\nwhere *Nc* is the numbers of primitive unit cells in the\nsystem, *Na* is the numbers of atoms in the primitive unit\ncell, is the Plack constant, represents the *\u03b1*th Cartesian\ncomponent of **u** for the atom at *j*th lattice site in the\nprimitive unit cell and the *P*th primitive unit cell in the\nsystem.\n\nThe harmonic approximation \\[23, 33\\] truncates the into the second\norder in its Taylor\u2019s series\n\n*Eq. 5\u2011106* ,\n\nwhere is the real-space interatomic force constant. With the\napproximation of , it can be demonstrated that finding the solution of\nis equivalent to finding the vibrational frequencies of a classical\nsystem with *NcNa* particles for small mechanical\noscillations.\n\nLet us rewrite as\n\n*Eq. 5\u2011107* ,\n\nwhere\n\n*Eq. 5\u2011108* , and\n\n*Eq. 5\u2011109* .\n\nAccordingly, the kinetic energy operator in becomes\n\n*Eq. 5\u2011110* .\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/firstprinciples_calculations_and_theory/quantum_theory_for_the_motion_of_electrons.ipynb b/psu410/src/firstprinciples_calculations_and_theory/quantum_theory_for_the_motion_of_electrons.ipynb deleted file mode 100644 index 18d0e5c..0000000 --- a/psu410/src/firstprinciples_calculations_and_theory/quantum_theory_for_the_motion_of_electrons.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "f37e1df9", "cell_type": "markdown", "source": "## Quantum theory for the motion of electrons\n", "metadata": {}}, {"id": "285a35ab", "cell_type": "markdown", "source": "### Schr\u00f6dinger equation\n\nThe Schr\u00f6dinger equation is typically written as follows\n\n*Eq. 5\u201157* ,\n\nwhere is the reduced Planck constant, **X** an abbreviation of the space\ncoordinates and spin states of the multiple particle system, the energy\noperator called Hamiltonian. When is independent of time *t*, one can\nseparate the coordinate **X** from the time *t* in finding the solution\nof by writing\n\n*Eq. 5\u201158* ,\n\nby which the stationary solutions of can be expressed through letting\n\n*Eq. 5\u201159* ,\n\nresulting in\n\n*Eq. 5\u201160*\n\n*Eq. 5\u201161*\n\nNote that is the frequency of de Broglie matter wave.\n\nFor any trial function \u039b(**X**) (in the Hilbert space) for \u03a8(**X**), the\nvariational principle tells us that the energy of the system always has\na lower bound through a ground state with energy *E0*, as\n\n*Eq. 5\u201162* ,\n\nwhere represents the complex conjugate of \u039b(**X**), resulting in\n\n*Eq. 5\u201163*\n\nwhich is known as the Rayleigh-Ritz variational principle,\n", "metadata": {}}, {"id": "6d31abbc", "cell_type": "markdown", "source": "#### Solving the Kohn-Sham Equations for a Solid\n\nFor a solid, is still a mathematical challenge with infinite number of\none-electron wave functions to be solved and therefore cannot be solved\ndirectly in the real space. To reduce the dimension of the problem, one\ncan choose to solve the equation at a specific point in the reciprocal\nspace. According to Bloch\u2019s theorem, the wave function for a solid can\nbe written as the product of a wavelike part, , and a cell periodic\npart,\n\n*Eq. 5\u201195*\n\nwhere can be expressed as a sum of a finite number of plane waves whose\nwave vectors are reciprocal lattice vectors of the crystal\n\n*Eq. 5\u201196* .\n\nso that\n\n*Eq. 5\u201197*\n\nwhere the band index *j* is used to number the eigenenergy and the\neigenvector at a given . The number of plane waves is determined by the\nfollowing equation\n\n*Eq. 5\u201198*\n\nwhere is called energy cutoff.\n\nUtilize the obtained wave functions, the charge density can be\ncalculated with the Brillouin zone integration\n\n*Eq. 5\u201199*\n\nWhere\n\n*Eq. 5\u2011100*\n\nwhere the parameter is to make the integration over the charge density\nwithin the primitive unit cell equal to the number of electrons, *N,* in\nthe primitive unit cell. Numerically, the integration can be\napproximated by summation over a set of discrete *k*-mesh as\n\n*Eq. 5\u2011101*\n\nwhere *N*BZ represents the number of points in the first\nBrillouin zone in the *k*-mesh. When a solid possesses symmetry, the\nsummation in the above equation can be further reduced to the summation\nover the irreducible Brillouin zone (IBZ).\n\n*Eq. 5\u2011102*\n\nwhere is a weight factor that represents the number of points that are\nequivalent to by space group symmetry.\n", "metadata": {}}, {"id": "1f27a649", "cell_type": "markdown", "source": "### Special Quasirandom Structures\n\nSQS\u2019s \\[43-44\\] are specially designed *small-unit-cell* periodic\nstructures with minimal number of atoms per unit cell, which closely\nmimic the most relevant, near-neighbor pair and multisite correlation\nfunctions of random substitutional alloys. The correlation functions are\nclassified by their *n*-site component \u201cfigures\u201d *f* = (*n,m*), where\nthe index *n* is called vortex for pair, triple, and quadruple\ncorrelations (*n* = 2, 3, and 4); *m* measures the correlation distance.\n\nIn the SQS approach, a distribution of distinct local environments is\nmaintained and their average corresponds to the random alloy. Thus, a\nsingle DFT calculation of an SQS can give many important alloy\nproperties (e.g. equilibrium bond lengths, charge transfer, formation\nenthalpies, etc.) that depend on the existence of those distinct local\nenvironments. The SQS approach has been used extensively to study the\nformation enthalpies, bond length distributions, density of states, band\ngaps and optical properties in semiconductor alloys. It is noted that\nthe CE approach can treat short-range ordering efficiently, while it is\nnot clear how the SQS approach can be used to represent short-range\nordering.\n\nThe key quantities in the SQS approach are the *n*-site correlation\nfunctions. Specifically, the 2-site correlation function corresponding\nto the 2-site component \u201cfigures\u201d (2,*m*) is\n\nEq. 5\u2011151\n\nwhere represents the total number of possible pairs with correlation\ndistance (neighboring distance) *Rij* being equal to *m*. The\n3-site correlation function corresponding to the 3-site component\n\u201cfigures\u201d (3,*m*) is\n\nEq. 5\u2011152\n\nwhere represents the total number of all possible 3-site \u201cfigures\u201d with\nthe correlation distance (size and shape) *Rijk* being equal\nto *m*. The 4-site correlation function corresponding to the 4-site\ncomponent \u201cfigures\u201d (4,*m*) is\n\nEq. 5\u2011153 ,\n\nwhere represents the total number of all possible 4-site \u201cfigures\u201d with\ncorrelation distance (size and shape) *Rijkl* being equal to\n*m*.\n\nWith a given supercell size *N*, the essential task of the SQS approach\nis to search through all configurations that approach as close as\npossible to the correlation functions of the perfectly random (*R*)\nstructure and for the binary system it is\n\nEq. 5\u2011154 .\n\nDescribing random alloys by small unit-cell periodical structures surely\nintroduces erroneous correlations beyond a certain distance. However,\nsince interactions between nearest neighbors are generally more\nimportant than interactions between more distant neighbors, SQS\u2019s can be\nconstructed in such a way that they exactly reproduce the correlation\nfunctions of a random alloy between the first few nearest neighbors,\ndeferring errors due to periodicity to more distant neighbors. The\npractical procedure could be that to find the structures that match the\n2-site correlation functions up to a given neighboring distance, and\nthen add the conditions matching the high order correlation functions up\nto certain correlation distance.\n\nAppendix B is a collection of the SQS\u2019s with a variety of compositions\nfor binary fcc, bcc, hcp, and L12 structures, for ternary\nfcc, bcc, and B2 structures, and Perovskite in the cubic ABO3\nstructure. The used format is that of VASP.\n", "metadata": {}}, {"id": "1c9ba376", "cell_type": "markdown", "source": "### System with multiple microstates (MMS model)\n\nFor a system made of multiple microstates, the total partition function\nis the summation over the partition functions of all microstates, in ,\nas\n\n*Eq. 5\u201149* ,\n\nwhere is the multiplicity of the microstate *\u03c3.* It is immediately\napparent that is the thermal population of the microstate *\u03c3*.\nFurthermore, with , one obtains\n\n*Eq. 5\u201150* .\n\nrelates the total Helmholtz energy, *F(N,V,T)*, of a system with mixing\namong multiple microstates and the Helmholtz energy, , of individual\nmicrostates. An important result of is the configurational entropy due\nto the mixing among multiple microstates, named as microstate\nconfigurational entropy (MCE) in this book,\n\n*Eq. 5\u201151* ,\n\nwhich makes the entropy of a system with mixing among multiple\nmicrostates as\n\n*Eq. 5\u201152* .\n\nSimilarly, one can get the heat capacity at constant volume of a system\nwith mixing among multiple microstates as\n\n*Eq. 5\u201153* ,\n\nwhere the configurational contributions to the heat capacity due to the\nmixing among multiple microstates is\n\n*Eq. 5\u201154* .\n\nMoreover, the isothermal bulk modulus of a system with mixing among\nmultiple microstates can be also computed similarly\n\n*Eq. 5\u201155* ,\n\nWith\n\n*Eq. 5\u201156*\n\nwith being the partial pressure of the microstate *\u03c3*. This multiple\nmicrostate model (MMS model) is used in Chapter to quantitatively\npredict thermal expansion anomalies.\n", "metadata": {}}, {"id": "39adb4e2", "cell_type": "markdown", "source": "### Vibrational contribution by phonon theory\n\nUnder the harmonic/quasiharmonic approximation, the lattice dynamics or\nphonon theory is currently the most established method. It truncates the\ninteraction potential up to the second order. In such a case, in can be\nexpressed in terms of phonon frequency as follows\n\n*Eq. 5\u201132* ,\n\nwhere the label **g** has the meanings of (*g1*,\n*g2*, \u2026, *g3N*) that *gj* can take any\ninteger values from zero to infinite.\n\nAs a result, is reduced to\n\n*Eq. 5\u201133* ,\n\nor equivalently,\n\n*Eq. 5\u201134* ,\n\nwhere an integration has been used to replace the summation in by means\nof introducing a function, , named the phonon density of states (PDOS)\nwhose integration over *\u03c9* equals to three per atom.\n\nAccordingly, the formulation to calculate the entropy becomes\n\n*Eq. 5\u201135* ,\n\nthe formulation to calculate the internal energy becomes\n\n*Eq. 5\u201136* ,\n\nand the formulation to calculate the heat capacity at constant volume\nbecomes\n\n*Eq. 5\u201137* .\n", "metadata": {}}, {"id": "8355a666", "cell_type": "markdown", "source": "### Volume, entropy, enthalpy, thermal expansion, bulk modulus, and heat capacity \n\nThe equilibrium volume *Veq* (*P*,*T*) at given *T* and *P*\ncan be obtained by finding the root of the following equation\n\n*Eq. 5\u20116* .\n\nThe dashed line in illustrates *Veq* (*P*,*T*) as a function\nof *T* from 0 to 1600 K at *P* = 0 for Ni.\n\nThe entropy can be calculated through *F* by\n\n*Eq. 5\u20117* .\n\nPlotted in is the calculated entropy (curve) of Ni as a function of\ntemperature from 0 to 1600 K at *P* = 0 compared with the recommended\ndata (symbols) with details in Ref. \\[16\\].\n\nFigure \u2011. Entropy of nickel as a function of temperature.\n\nBased on *F* and *S*, the enthalpy at given *P* and *T* can be computed\nas\n\n*Eq. 5\u20118* .\n\nPlotted in is the calculated enthalpy (curve) of Ni as function of\ntemperature from 0 to 1600 K at *P* = 0 comparing with the recommended\ndata (open circles) with details in Ref. \\[16\\]\n\nFigure \u2011. Enthalpy of nickel as a function of temperature.\n\nWith the equilibrium volume *Veq* (*P*,*T*) calculated by ,\nthe volume thermal expansion coefficient defined by can be calculated by\n\n*Eq. 5\u20119* .\n\nPlotted in is the calculated thermal expansion coefficient (curve) of\nnickel as function of temperature from 0 to 1600 K at *P* = 0 comparing\nwith experimental data (symbols) with details in Ref. \\[16\\]\n\nFigure \u2011. Volume thermal expansion coefficient of nickel as a function\nof temperature.\n\nThe bulk modulus of a material represents the substance's resistance to\nuniform compression. Depending on how the temperature varies during\ncompression, a distinction should be made between the isothermal bulk\nmodulus (constant temperature) and adiabatic bulk modulus (constant\nentropy or no heat transfer). As a matter of fact, most of the\nexperimental data are adiabatic whereas most of the published\ntheoretical data are isothermal.\n\nThe isothermal bulk modulus, as defined in terms of Gibbs energy shown\nby , can be calculated by\n\n*Eq. 5\u201110*\n\nBased on the isothermal bulk modulus, the adiabatic bulk modulus can be\ncalculated by\n\n*Eq. 5\u201111* ,\n\nwhere *CP* and *CV* represent the constant\npressure heat capacity and constant volume heat capacity respectively.\nPlotted in is the calculated bulk moduli (curves) of Ni as function of\ntemperature from 0 to 1600 K at *P* = 0. The experimental data are from\nultrasonic measurements (symbols, see Ref. \\[28\\] for more details) that\nare therefore adiabatic bulk moduli calculated based on the measured\nadiabatic elastic constants using the relation\n\n*Eq. 5\u201112* .\n\nFigure \u2011. Bulk moduli of nickel as a function of temperature. Solid\nline: adiabatic; Dashed line: isothermal.\n\nThe heat capacity at constant volume, as defined in terms of Gibbs\nenergy shown by Eq. 2\u201128, can be calculated by\n\n*Eq. 5\u201113* ,\n\nwhere represents the internal energy. The heat capacity at constant\npressure (see ) can then be calculated as\n\n*Eq. 5\u201114* ,\n\nutilizing the calculated thermal expansion coefficient in and bulk\nmodulus in .\n\nIt can be advocated that thermal expansion makes the difference between\nthe heat capacity at constant volume and the heat capacity at constant\npressure. The calculated contributions to the heat capacity of Ni as\nfunction of temperature from 0 to 1600 K at *P* = 0 are illustrated in\nwhere the lattice vibration and the thermal electron contributions have\nbeen separated out. From , it is observed a large difference between the\ncalcualted *CP* (solid line) and the experimental data\n(symbols, see Ref. \\[18\\] for more details) at 600 K due to the magnetic\nphase transition which has not been considered in the calculation. It\nshould be pointed out that for Ni the thermal electronic contribution to\nthe heat capacity (dashed line in ) is substantial at high temperatures.\n\nFigure \u2011. Heat capacity of nickel as a function of temperature.\nrepresents the calculated lattice vibration contribution; represents the\ncalculated thermal electronic contribution.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/gibbs_energy_function/binary_random_solutions.ipynb b/psu410/src/gibbs_energy_function/binary_random_solutions.ipynb deleted file mode 100644 index b60aeaf..0000000 --- a/psu410/src/gibbs_energy_function/binary_random_solutions.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "d71e3c32", "cell_type": "markdown", "source": "### Binary random solutions\n\nFrom , the Gibbs-Duhem equation of a binary system consisting of\ncomponents $A$ and $B$ is written as\n\nEq. \u2011 $0 = - SdT - Vd( - P) - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B}$\n\nThis equation represents a four-dimensional surface. It is self-evident\nthat both and hold for stable binary solutions too, i.e. the directions\nand the curvature of the surface are all negative. To visualize the\nfour-dimensional surface in the three-dimensional space, one needs to\nfix one of the four potentials. As $T$ and $P$ are the natural variables\nof Gibbs energy, they are usually chosen to be kept constant. One can\ntypically investigate behaviors of systems consisting of condensed\nphases by varying the temperature at constant pressure. at constant\npressure thus becomes\n\nEq. \u2011 $0 = - SdT - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B}$\n\nSimilar to and , the property of a phase can be represented by a\ntwo-dimensional surface in the three-dimensional space composed of $T$,\n$\\mu_{A}$, and $\\mu_{B}$ under constant $P$, keeping in mind the\nfollowing\n\nEq. \u2011\n$G_{m} = x_{A}\\mu_{A} + x_{B}\\mu_{B} = x_{A}_{\\ }^{0}G_{A} + x_{B}_{\\ }^{0}G_{B} + RT\\left( x_{A}l{nx}_{A} + x_{B}l{nx}_{B} \\right) +_{\\ }^{E}G_{m}$\n\nSince $_{\\ }^{E}G_{m}$ must be zero for pure components $A$ and $B$, it\nneeds to be in the following form\n\nEq. \u2011 $_{\\ }^{E}G_{m} = x_{A}x_{B}L_{AB}$\n\nwith $L_{AB}$ being a parameter denoting the interaction between\ncomponents $A$ and $B$, called interaction parameter. When $L_{AB} = 0$,\nthe solution is an ideal solution. When $L_{AB}$ is a non-zero constant\nindependent of temperature and composition, the solution is called a\nregular solution. Its excess entropy and excess enthalpy of mixing are\nobtained as\n\nEq. \u2011 $_{\\ }^{E}S_{m} = \\frac{\\partial_{\\ }^{E}G_{m}}{\\partial T} = 0$\n\nEq. \u2011\n$_{\\ }^{E}H_{m} =_{\\ }^{E}G_{m} - T_{\\ }^{E}S_{m} = x_{A}x_{B}L_{AB}$\n\nThe chemical potential of component $A$ or $B$ in a binary regular\nsolution can be derived as\n\nEq. \u2011\n$\\mu_{i} =_{\\ }^{0}G_{i} + RTlnx_{i} + \\left( 1 - x_{i} \\right)^{2}L_{AB}$\n\nIn a dilute solution with $x_{i} \\rightarrow 0$, one can have\n\nEq. \u2011\n$RTln\\gamma_{i} = \\left( 1 - x_{i} \\right)^{2}L_{AB} \\approx L_{AB}$\n\nEq. \u2011 $\\gamma_{i} = e^{\\frac{L_{AB}}{RT}}$\n\nThe activity is thus proportional to its mole fraction, which is called\nHenry\u2019s law. By the same token, for the solvent, i.e.\n$x_{i} \\rightarrow 1$,\n\nEq. \u2011 $RTln\\gamma_{i} = \\left( 1 - x_{i} \\right)^{2}L_{AB} \\approx 0$\n\nwhich gives $\\gamma_{i} \\approx 1$, and its activity approaches its mole\nfraction. This is called Raoult\u2019s law.\n\nThe stability of a binary solution is derived from as\n\nEq. \u2011\n$\\left( \\frac{\\partial\\mu_{A}}{\\partial N_{A}} \\right)_{T,P,N_{B}} = \\left\\lbrack \\frac{RT}{x_{A}} - 2\\left( 1 - x_{A} \\right)L_{AB} \\right\\rbrack\\frac{1 - x_{A}}{N}$\n\nEq. \u2011\n$\\left( \\frac{\\partial\\mu_{B}}{\\partial N_{B}} \\right)_{T,P,N_{A}} = \\left\\lbrack \\frac{RT}{x_{B}} - 2\\left( 1 - x_{B} \\right)L_{AB} \\right\\rbrack\\frac{1 - x_{B}}{N}$\n\nIt should be noted that the two chemical potentials in a binary system\nat constant temperature and pressure are dependent on each other due to\nthe Gibbs-Duhem equation shown in , i.e.\n\nEq. \u2011 $0 = - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B}$\n\nand the two chemical potentials depend on each other by the following\nrelation\n\nEq. \u2011\n$\\left( \\frac{\\partial\\mu_{A}}{\\partial\\mu_{B}} \\right)_{T,P} = - \\frac{N_{B}}{N_{A}}$\n\nTherefore, at the limit of stability, both and go to zero at the same\ntime, which is obtained when\n\nEq. \u2011 $RT = 2{x_{A}x_{B}L}_{AB}$\n\nAs the absolute temperature cannot be negative, has no solution for a\nsolution phase with $L_{AB} < 0$, i.e. the solution phase is stable with\nrespect to the composition fluctuation. For a solution with\n$L_{AB} > 0$, its limit of stability is represented .\n\nA schematic molar Gibbs energy of a solution with $L_{AB} < 0$ at\nconstant temperature and pressure is shown in along with the ideal and\nexcess Gibbs energy of mixing. A tangent line on the molar Gibbs energy\nof the solution is drawn, and its two intercepts at $x_{B} = 0$ and\n$x_{B} = 1$ give the chemical potentials of components $A$ and $B$,\n$\\mu_{A}$ and $\\mu_{B}$ by , respectively. It is evident that $\\mu_{A}$\nand $\\mu_{B}$ are not independent on each other as they are two points\non the same straight line. This is a graphic representation of the\nGibbs-Duhem equation of . The chemical activity of component $B$ is also\ndepicted with the reference state being the pure B with the same\nstructure. As shown in , other structures of pure B can be selected as\nthe reference states of the chemical activity of component B, resulting\nin the different distances to its chemical potential in the solution,\nthus different values of its chemical activities. It is clear that this\nchange of reference state for chemical activity does not affect the\nchemical potential of the component in the solution.\n\nFigure \u2011: Schematic molar Gibbs energy diagram with $L_{AB} < 0$\n\nWhen $L_{AB} > 0$, represents a parabola in the $T - x_{i}$\ntwo-dimensional coordinate, symmetric with respect to $x_{A}$ and\n$x_{B}$, shown in , i.e. the spinodal of the solution. The consolute\npoint is obtained by applying to and letting equal to zero at the\nconsolute point\n\nEq. \u2011\n$\\left( \\frac{\\partial^{2}\\mu_{A}}{\\partial N_{A}^{2}} \\right)_{T,P,N_{B}} = \\left\\lbrack - \\frac{RT}{x_{A}^{2}} + 2L_{AB} \\right\\rbrack\\left( \\frac{1 - x_{A}}{N} \\right)^{2} = 0$\n\nwhich gives\n\nEq. \u2011 $T_{cons} = 2x_{A}^{2}L_{AB}$\n\nSolving and , one obtains $x_{A} = x_{B} = 0.5$ and\n\nEq. \u2011 $T_{cons} = \\frac{L_{AB}}{2R}$\n\nFigure \u2011: A Spinodal curve with $L_{AB} > 0$\n\nA schematic molar Gibbs energy diagram at temperatures below the\nconsolute point is shown in . It can be seen that part of the molar\nGibbs energy has negative curvature, and the solution becomes unstable.\nThe chemical potential thus does not change monotonically with respect\nto composition and its derivative changes sign at the inflexion point.\n\nFigure \u2011: Schematic molar Gibbs energy diagram with $L_{AB} > 0$\n\nFor more complex solutions, $L_{AB}$ can be a function of temperature,\npressure, and compositions. In principle, the temperature and pressure\ndependences can be treated by means of formula similar to . There are\nvarious approaches in the literature to consider the composition\ndependence of $L_{AB}$. The empirical Redlich-Kister polynomial stands\nout as the one most widely used because it can be extrapolated to\nternary and multi-component systems consistently, which will be\ndiscussed in Chapter .\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/gibbs_energy_function/elastic_magnetic_and_electric_contributions_to_free_energy.ipynb b/psu410/src/gibbs_energy_function/elastic_magnetic_and_electric_contributions_to_free_energy.ipynb deleted file mode 100644 index 59e0f56..0000000 --- a/psu410/src/gibbs_energy_function/elastic_magnetic_and_electric_contributions_to_free_energy.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "3e60ca75", "cell_type": "markdown", "source": "## Elastic, magnetic, and electric contributions to free energy\n\nChapters and focus on the thermal and hydrostatical pressure\ncontributions to Gibbs energy, which are the two prime variables\naffecting phase stability in typical experimental environments. However,\nthere are other internal and external contributions, which are\nparticularly important for crystalline phases. Two important internal\ncontributions are from magnetic and electric polarizations of materials\nwith the corresponding external contributions due to the magnetic and\nelectric fields. Furthermore, for non-hydrostatic pressing of solid\nphases, the $PV$ term in the combined law is to be replaced by elastic\nenergy calculated from elastic stress and elastic strain. The\ncorresponding works done to a system are as follows \\[3\\]\n\nEq. \u2011\n$dW_{elastic} = - V\\sum_{i,j,k,l}^{}{\\sigma_{ij}{d\\varepsilon}_{kl}}$\n\nEq. \u2011 $dW_{magnetic} = - V\\sum_{i}^{}{H_{i}{dB}_{i}}$\n\nEq. \u2011 $dW_{electric} = - V\\sum_{i}^{}{E_{i}{dD}_{i}}$\n\nwhere $i,j,\\ k,\\ l = 1,2,3$, $\\sigma_{ij}$ and $\\varepsilon_{kl}$ are\nthe components of stress and strain; $H_{i}$ and $B_{i}$ are the\ncomponents of magnetic field and magnetic induction; $E_{i}$ and $D_{i}$\nare the components of electric field and electric displacement; and $V$\nis the volume of the crystal. The negative sign in front of the equation\nis due to the fact that the system does work to the surroundings when it\nexpands its volume due to the strain, magnetic induction, and electric\ndisplacement.\n\nUsing the combined law of thermodynamics, can thus be re-written as\nfollows\n\nEq. \u2011\n$dU = TdS - V\\left( \\sum_{i,j,k,l}^{}{\\sigma_{ij}{d\\varepsilon}_{kl}} + \\sum_{i}^{}{H_{i}{dB}_{i}} + \\sum_{i}^{}{E_{i}{dD}_{i}} \\right) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$\n\nThe Legendre transformation similar to Helmholtz energy and Gibbs\nenergy, and , can be made to obtain the following characteristic free\nenergy functions\n\nEq. \u2011 $dF = d(U - TS)$\n\n$$= - SdT - V\\left( \\sum_{i,j,k,l}^{}{\\sigma_{ij}{d\\varepsilon}_{kl}} + \\sum_{i}^{}{H_{i}{dB}_{i}} + \\sum_{i}^{}{E_{i}{dD}_{i}} \\right) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$$\n\nEq. \u2011 $dF_{H} = d\\left( U - TS + \\sum_{i}^{}{H_{i}B_{i}} \\right)$\n\n$$= - SdT - V\\left( \\sum_{i,j,k,l}^{}{\\sigma_{ij}{d\\varepsilon}_{kl}} - \\sum_{i}^{}{B_{i}dH_{i}} + \\sum_{i}^{}{E_{i}{dD}_{i}} \\right) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$$\n\nEq. \u2011 $dF_{E} = d\\left( U - TS + \\sum_{i}^{}{E_{i}D_{i}} \\right)$\n\n$$= - SdT - V\\left( \\sum_{i,j,k,l}^{}{\\sigma_{ij}{d\\varepsilon}_{kl}} + \\sum_{i}^{}{H_{i}{dB}_{i}} - \\sum_{i}^{}{D_{i}{dE}_{i}} \\right) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$$\n\nEq. \u2011\n$dF_{EH} = d\\left( U - TS + \\sum_{i}^{}{H_{i}B_{i}} + \\sum_{i}^{}{E_{i}D_{i}} \\right)$\n\n$$= - SdT - V\\left( \\sum_{i,j,k,l}^{}{\\sigma_{ij}{d\\varepsilon}_{kl}} - \\sum_{i}^{}{B_{i}{dH}_{i}} - \\sum_{i}^{}{D_{i}{dE}_{i}} \\right) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$$\n\nEq. \u2011\n$dG = d\\left( U - TS + \\sum_{i,j,k,l}^{}{\\sigma_{ij}\\varepsilon_{kl}} + \\sum_{i}^{}{H_{i}B_{i}} + \\sum_{i}^{}{E_{i}D_{i}} \\right)$\n\n$$= - SdT + V\\left( \\sum_{i,j,k,l}^{}{\\varepsilon_{ij}d\\sigma}_{kl} + \\sum_{i}^{}{B_{i}dH_{i}} + \\sum_{i}^{}{D_{i}{dE}_{i}} \\right) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$$\n\nFrom above equations, it can be seen that the natural variables of\nvarious free energies are\n$F\\left( T,\\ \\varepsilon_{ij},\\ B_{i},\\ D_{i},\\ N_{i},\\ \\xi \\right)$,\n$F_{H}\\left( T,\\ \\varepsilon_{ij},\\ H_{i},\\ D_{i},\\ N_{i},\\ \\xi \\right)$,\n$F_{E}\\left( T,\\ \\varepsilon_{ij},\\ B_{i},\\ E_{i},\\ N_{i},\\ \\xi \\right)$,\n$F_{EH}\\left( T,\\ \\varepsilon_{ij},\\ H_{i},\\ E_{i},\\ N_{i},\\ \\xi \\right)$,\nand $G\\left( T,\\ \\sigma_{ij},\\ H_{i},\\ E_{i},\\ N_{i},\\ \\xi \\right)$.\nClearly, there can be more combinations when the components of\n$\\varepsilon_{ij}$, $D_{i}$, and $B_{i}$ are partially replaced by their\nconjugate potentials. The free energies listed above are useful\ndepending on how the system is constrained by the surroundings. For\npractical applications, the elastic, magnetic, and electric properties\nare usually considered for phases with fixed compositions, and at\nequilibrium can then be written as\n\nEq. \u2011\n$dG = - SdT + V\\left( \\sum_{i,j,k.l}^{}{\\varepsilon_{ij}d\\sigma}_{kl} + \\sum_{i}^{}{B_{i}dH_{i}} + \\sum_{i}^{}{D_{i}{dE}_{i}} \\right) + \\mu dN$\n\nThe corresponding Gibbs-Duhem equation follows as below\n\nEq. \u2011\n$0 = - SdT + V\\left( \\sum_{i,j,k,l}^{}{\\varepsilon_{ij}d\\sigma}_{kl} + \\sum_{i}^{}{B_{i}dH_{i}} + \\sum_{i}^{}{D_{i}{dE}_{i}} \\right) - Nd\\mu$\n\nThe general differential form of a molar quantity can be extended from\nas follows as a one-component system\n\nEq. \u2011\n\nEq. \u2011\n\nEq. \u2011\n\nEq. \u2011\n\nThe first derivatives in to are the second directives of Gibbs energy\nwith respect to its natural variables, i.e. potentials, and have their\nrespective nomenclatures as shown in . The limit of stability follows\nand can be re-written as\n\nEq. \u2011\n$\\left( \\frac{\\partial X_{i}}{\\partial Y_{i}} \\right)_{N,Y_{j}} = \\infty$\n\nThis means that the derivatives in to , i.e. the quantities in , diverge\nat the limit of stability.\n\nTable \u2011: Physical quantities related to the first derivatives in to .\nThe table is symmetric due to the Maxwell relations related to the\nsecond directives of Gibbs energy with respect to its natural variables.\n\n| | T | $$\\sigma_{kl}$$ | $$E_{k}$$ | $$H_{k}$$ |\n|----------------------|------------------------------------|------------------------------------|---------------------------------------|---------------------------------------|\n| S | C/T, heat capacity | $\\alpha_{kl}$, piezocaloric effect | $p_{k}$, electrocaloric effect | $m_{k}$, magnetocaloric effect |\n| $$\\varepsilon_{ij}$$ | $\\alpha_{ij}$, thermal expansion | $s_{ijkl}$, elastic compliance | $d_{ijk}$, converse piezoelectricity | $q_{ijk}$, piezomagnetic moduli |\n| $$D_{i}$$ | $p_{i}$, pyroelectric coefficients | $d_{ikl}$, piezoelectric moduli | $k_{ik}$, permittivities | $a_{ik}$, magnetoelectric coefficient |\n| $$B_{i}$$ | $m_{i}$, pyromagnetic coefficient | $q_{ikl}$, piezomagnetic moduli | $a_{ik}$, magnetoelectric coefficient | $\\mu_{ik}$, permeability |\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/gibbs_energy_function/index.ipynb b/psu410/src/gibbs_energy_function/index.ipynb deleted file mode 100644 index d325a7a..0000000 --- a/psu410/src/gibbs_energy_function/index.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "f31bed02", "cell_type": "markdown", "source": "# Gibbs energy function\n\nAs shown in through , all functions have $N_{i}$ and $\\xi$ as natural\nvariables while they differ in other two natural variables. In typical\nmaterials-related experiments, temperature and pressure are the two\nvariables controlled. They are also the natural variables of Gibbs\nenergy. Consequently, Gibbs energy is the most widely used function in\nthermodynamics of materials science. The rest of this book focuses on\nGibbs energy for this reason. In this chapter, the mathematical formulas\nfor Gibbs energy of phases with fixed and variable compositions are\ndiscussed which are needed for quantitative calculations of Gibbs energy\nunder given values of its natural variables.\n\nFrom , the molar Gibbs energy can be defined as\n\nEq. \u2011\n$G_{m}\\left( T,P,x_{i},\\xi \\right) = \\frac{G}{N} = \\sum_{}^{}\\mu_{i}x_{i}$\n\nThe molar entropy, molar volume, chemical potential, and the driving\nforce can be obtained from as\n\nEq. \u2011\n$S_{m} = \\frac{S}{N} = - \\frac{1}{N}\\left( \\frac{\\partial G}{\\partial T} \\right)_{P,\\ N_{i},\\ \\xi} = {- \\left( \\frac{\\partial G_{m}}{\\partial T} \\right)}_{P,\\ x_{i},\\ \\xi}$\n\nEq. \u2011\n$V_{m} = \\frac{V}{N} = \\frac{1}{N}\\left( \\frac{\\partial G}{\\partial P} \\right)_{T,\\ N_{i},\\ \\xi} = \\left( \\frac{\\partial G_{m}}{\\partial P} \\right)_{T,\\ x_{i},\\ \\xi}$\n\nEq. \u2011\n$\\mu_{i} = \\left( \\frac{\\partial G}{\\partial N_{i}} \\right)_{T,P,N_{j \\neq i},\\ \\xi}$\n\nEq. \u2011\n$- D = \\left( \\frac{\\partial G}{\\partial\\xi} \\right)_{T,P,N_{i}\\ }$\n\nBased on , the molar enthalpy is written as\n\nEq. \u2011 $H_{m} = G_{m} + TS_{m}$\n\nOther physical properties of the system can also be represented by the\npartial derivatives of Gibbs energy such as heat capacity, $C_{P}$,\nvolume thermal expansivity, $\\alpha_{V}$, isothermal compressibility,\n$\\kappa_{T}$, as follows under constant pressure or temperature\n\nEq. \u2011\n$C_{P} = \\left( \\frac{\\partial Q}{\\partial T} \\right)_{P} = \\left( \\frac{\\partial H}{\\partial T} \\right)_{P} = T\\left( \\frac{\\partial(G + TS)}{\\partial T} \\right)_{P} = T\\left( \\frac{\\partial S}{\\partial T} \\right)_{P} = - T\\left( \\frac{\\partial^{2}G}{\\partial T^{2}} \\right)_{P}$\n\nEq. \u2011\n$\\alpha_{V} = \\frac{\\left( \\frac{\\partial V}{\\partial T} \\right)_{P}}{V} = \\frac{\\left( \\frac{\\left( \\partial G/\\partial( - P) \\right)_{T}}{\\partial T} \\right)_{P}}{\\left( \\partial G/\\partial( - P) \\right)_{T}} = \\frac{\\frac{\\partial^{2}G}{\\partial T\\partial( - P)}}{\\left( \\partial G/\\partial( - P) \\right)_{T}}$\n\nEq. \u2011\n$\\kappa_{T} = \\frac{\\left( \\frac{\\partial V}{\\partial( - P)} \\right)_{T}}{V} = \\frac{\\left( \\frac{\\left( \\partial G/\\partial( - P) \\right)_{T}}{\\partial( - P)} \\right)_{T}}{\\left( \\partial G/\\partial( - P) \\right)_{T}} = \\frac{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}}{\\left( \\partial G/\\partial( - P) \\right)_{T}} = \\frac{1}{B}$\n\nwhere the $N_{i}$ and $\\xi$ are kept constant for all partial\nderivatives, and $B$ is the bulk modulus.\n\nIn , $G$ cannot be directly replaced by $G_{m}$ because *N* also depends\non *Ni*. The thermodynamic quantities under such conditions,\ni.e. varying the amount of a component at constant temperature and\npressure, are called partial quantities which are introduced in Eq. 1\u20118\nfor partial entropy and for partial enthalpy. This definition can be\nextended to all molar quantities such as partial volume and partial\nGibbs energy. Partial quantities of a molar quantity, $A$, can thus be\ndefined in general as\n\nEq. \u2011\n$A_{i} = \\left( \\frac{\\partial A}{\\partial N_{i}} \\right)_{T,P,N_{j \\neq i},\\ \\xi}$\n\nThe general differential form of a molar quantity for a system at\nequilibrium can be represented by its partial quantities as\n\nEq. \u2011\n$dA = \\left( \\frac{\\partial A}{\\partial T} \\right)dT + \\left( \\frac{\\partial A}{\\partial P} \\right)dP + \\sum_{}^{}\\left( \\frac{\\partial A}{\\partial N_{i}} \\right){dN}_{i}$\n\nwhere the subscripts representing variables kept constant, i.e. the\nremaining natural variables of Gibbs energy not in the denominator, are\nomitted for simplicity. This will be done throughout the book unless\nspecified otherwise.\n\nUsing the following relations: $A = NA_{m}$, $N = \\sum_{}^{}N_{j}$,\n$x_{i} = N_{i}/N$,\n$\\frac{{\\partial x}_{i}}{{\\partial N}_{i}} = \\left( 1 - x_{i} \\right)/N$,\nand $\\frac{{\\partial x}_{k}}{{\\partial N}_{i}} = {- x}_{k}/N$, can be\nderived as, under constant T and P,\n\nEq. \u2011\n$A_{i} = A_{m} + N\\sum_{j = 1}^{c}{\\frac{\\partial A_{m}}{\\partial x_{j}}\\frac{\\partial x_{j}}{\\partial N_{i}}} = A_{m} + \\frac{\\partial A_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial A_{m}}{\\partial x_{j}}$\n\nwhere the summation is for all *c* components and the partial\nderivatives are taken with other mole fractions kept constant. However,\nmole fractions are not independent, but follow the relation\n$\\sum_{}^{}x_{i} = 1$. Taking $x_{1} = 1 - \\sum_{j = 2}^{c}x_{j}$ as the\ndependent mole fraction, can be rewritten as\n\nEq. \u2011\n$A_{i} = A_{m} + \\left( \\frac{\\partial A_{m}}{\\partial x_{i}} - \\frac{\\partial A_{m}}{\\partial x_{1}} \\right) - \\sum_{j = 2}^{c}x_{j}\\left( \\frac{\\partial A_{m}}{\\partial x_{j}} - \\frac{\\partial A_{m}}{\\partial x_{1}} \\right)$\n\nThe difference of the partial derivatives in the parenthesis in\nrepresents the partial derivative of $A_{m}$ with respect to the mole\nfraction of one component when the first component is selected as the\ndependent component. Applying and to Gibbs energy, the partial Gibbs\nenergy or chemical potential of component $i$ is obtained as\n\nEq. \u2011\n$\\mu_{i} = G_{i} = G_{m} + \\frac{\\partial G_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial G_{m}}{\\partial x_{j}} = G_{m} + \\left( \\frac{\\partial G_{m}}{\\partial x_{i}} - \\frac{\\partial G_{m}}{\\partial x_{1}} \\right) - \\sum_{j = 2}^{c}x_{j}\\left( \\frac{\\partial G_{m}}{\\partial x_{j}} - \\frac{\\partial G_{m}}{\\partial x_{1}} \\right)$\n\nThe derivatives in the stability equation, , are defined with the molar\nquantities kept constant. On the other hand, Gibbs energy has two\npotentials, temperature and pressure, as natural variables instead. One\nwould thus need to compare the stability conditions when a variable kept\nfixed is changed from a molar quantity to its conjugate potential. This\ncan be carried out through the use of Jacobians to change the\nindependent variables\n\nEq. \u2011\n$\\frac{\\partial\\left( Y_{i},Y_{j} \\right)}{\\partial\\left( X_{i},X_{j} \\right)} = \\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{Y_{j}}\\left( \\frac{\\partial Y_{j}}{\\partial X_{j}} \\right)_{X_{i}} = \\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}}\\left( \\frac{\\partial Y_{j}}{\\partial X_{j}} \\right)_{X_{i}} - \\left( \\frac{\\partial Y_{i}}{\\partial X_{j}} \\right)_{X_{i}}\\left( \\frac{\\partial Y_{j}}{\\partial X_{i}} \\right)_{X_{j}}$\n\nFor a stable system, both\n$\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}}$ and\n$\\left( \\frac{\\partial Y_{j}}{\\partial X_{j}} \\right)_{X_{i}}$ are\npositive based on . Using the Maxwell relation shown by , one thus\nobtains\n\nEq. \u2011\n$\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}} - \\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{Y_{j}} = \\left( \\frac{\\partial Y_{i}}{\\partial X_{j}} \\right)_{X_{i}}\\left( \\frac{\\partial Y_{j}}{\\partial X_{i}} \\right)_{X_{j}}/\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}} \\geq 0$\n\nThis means that\n$\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{Y_{j}}$ will go\nto zero before\n$\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}}$ does. It\nindicates that the stability condition becomes more restrictive when\npotentials are kept constant in place of their conjugate molar\nquantities. Based on the Gibbs-Duhem equation of , the maximum number of\nindependent potentials is *c+1*, and the last potential is dependent,\ni.e.\n\nEq. \u2011\n$\\left( \\frac{\\partial Y_{c + 2}}{\\partial X_{c + 2}} \\right)_{Y_{j \\leq c + 1}} = 0$\n\nTherefore, the limit of stability is determined when the derivative\nbecomes zero with one molar quantity kept constant, e.g.\n\nEq. \u2011\n$\\left( \\frac{\\partial Y_{c + 1}}{\\partial X_{c + 1}} \\right)_{Y_{j < c + 1},X_{c + 2}} = 0$\n\nThis is because this derivative reaches zero faster than any other\nderivatives with more molar quantities kept constant. shows that all\nmolar quantities diverge at the limit of stability. The consolute point\nis obtained with $c$ additional conditions as follows based on\n\nEq. \u2011\n$\\left( \\frac{\\partial^{2}Y_{i}}{\\partial\\left( X_{i} \\right)^{2}} \\right)_{Y_{j \\leq c + 1, \\neq i},X_{c + 2}} = 0$\n\nTogether with , all $c + 1$ independent potentials at the consolute\npoint can be determined. It is evident that the consolute point is a\nzero-dimensional point in a two-dimensional space of independent\npotentials in a one-component system. With the addition of a second\ncomponent to form a binary system, this consolute point in the\none-component system extends into a one-dimensional line. This line\nrepresents the limit of stability of the binary system, and a consolute\npoint is located at the end of this line. It is thus evident that in a\nsystem with $c$ independent components, the limit of stability is a\n*c-1*-dimensional hypersurface in a space of $c + 1$ independent\npotentials, while the consolute point is a zero-dimensional point in all\nsystems, which may be called the invariant critical point.\n", "metadata": {}}, {"id": "71b23714", "cell_type": "markdown", "source": "### Multi-component random solutions\n\nSimilar to a ternary solution, the excess Gibbs energy of mixing of a\nmulti-component solution can be written as\n\nEq. \u2011\n$_{\\ }^{E}G_{m} = \\sum_{i}^{}{\\sum_{j}^{}{x_{i}x_{j}L_{ij}}} + \\sum_{i}^{}{\\sum_{j}^{}{\\sum_{k}^{}{x_{i}x_{j}x_{k}L_{ijk}}}}$\n\nIn principle, one can add interaction parameters for quaternary and\nhigher order systems, but their contributions to Gibbs energy are\nrelatively minor because the major contributions have already been taken\ninto account by the binary and ternary interactions. It is anticipated\nthat not only the interaction parameters of four or more components are\nsmall, but also the multiplication of mole fractions in front the\ninteraction parameters diminishes their contribution to the Gibbs energy\neven further.\n\nUnder the condition that all interaction parameters are constant, the\nchemical potential of a component in a multi-component system with\nbinary and ternary interaction parameters can be extended from as\n\nEq. \u2011\n$\\mu_{i} =_{\\ }^{0}G_{i} + RTlnx_{i} + \\sum_{j \\neq i}^{}{x_{j}^{2}L_{ij}} + \\sum_{k > j}^{}{\\sum_{j \\neq i}^{}{x_{j}x_{k}}}\\left\\lbrack L_{ij} + L_{ik} - L_{jk} + \\left( 1 - 2x_{i} \\right)L_{ijk} \\right\\rbrack$\n\nThe stability of the solution can also be extended from as\n\nEq. \u2011\n${N\\left( \\frac{\\partial\\mu_{i}}{\\partial N_{i}} \\right)}_{T,P,N_{j \\neq i}} = NG_{ii} = \\frac{RT\\left( 1 - x_{i} \\right)}{x_{i}} - 2\\sum_{j \\neq i}^{}{x_{j}^{2}L_{ij}} - 2\\sum_{k > j}^{}{\\sum_{j \\neq i}^{}{x_{j}x_{k}}}\\left\\lbrack L_{ij} + L_{ik} - L_{jk} + \\left( 2 - 3x_{i} \\right)L_{ijk} \\right\\rbrack$\n\nThe limit of stability of a multi-component random solution can be\nrepresented by or .\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/gibbs_energy_function/phases_with_fixed_compositions.ipynb b/psu410/src/gibbs_energy_function/phases_with_fixed_compositions.ipynb deleted file mode 100644 index f2d79bd..0000000 --- a/psu410/src/gibbs_energy_function/phases_with_fixed_compositions.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "b7f59679", "cell_type": "markdown", "source": "## Phases with fixed compositions\n\nThe homogeneous system discussed so far means that there is only one\nphase in the system, i.e. a single-phase system. A phase with a fixed\ncomposition can be a pure element or a stoichiometric compound. There is\nthus only one independent component in the system. A stoichiometric\ncompound contains more than one element, but the relative amounts of\neach element are fixed by the stoichiometry and cannot vary\nindependently, i.e., $dN_{i} = x_{i}dN$. The combined law of\nthermodynamics becomes\n\nEq. \u2011\n$dG = - SdT - Vd( - P) + \\left( \\sum_{}^{}{x_{i}\\mu_{i}} \\right)dN - Dd\\xi = - SdT - Vd( - P) + G_{m}dN - Dd\\xi$\n\n$G_{m}$ is the molar Gibbs energy of the stoichiometric compound and can\nbe regarded as the chemical potential of the stoichiometric\nphase,$\\ \\alpha$,\n\nEq. \u2011 $G_{m} = \\mu^{\\alpha} = \\sum_{}^{}{x_{i}\\mu_{i}}$\n\nThe chemical potential of individual components in the phase cannot be\ndefined because the amount of each component cannot be varied\nindependently. For a stoichiometric phase of $N$ moles of atoms at\nequilibrium with $dG$=$Nd\\mu^{\\alpha} + \\mu^{\\alpha}dN$, reduces to\n\nEq. \u2011 $\\ 0 = - SdT - Vd( - P) - Nd\\mu^{\\alpha}$\n\nwhich is the Gibbs-Duhem equation, , applied to a stoichiometric phase.\nIt can be represented graphically by a surface in a three-dimensional\nspace composed of $\\ \\mu^{\\alpha}$, *T* and *\u2013P*. The direction of the\nsurface is represented by the three partial directives between any two\nof $\\ \\mu^{\\alpha}$, *T* and *\u2013P* with the third one kept constant, i.e.\n\nEq. \u2011\n$\\ \\left( \\frac{\\partial\\mu^{\\alpha}}{\\partial T} \\right)_{P} = - \\frac{S}{N} = - S_{m}$\n\nEq. \u2011\n$\\ \\left( \\frac{\\partial\\mu^{\\alpha}}{\\partial( - P)} \\right)_{T} = - \\frac{V}{N} = - V_{m}$\n\nEq. \u2011\n$\\ \\left( \\frac{\\partial( - P)}{\\partial T} \\right)_{\\mu^{\\alpha}} = - \\frac{S}{V} = - \\frac{S_{m}}{V_{m}}$\n\nBased on Nernst\u2019s heat theorem, the entropy difference between two\ncrystals approaches zero when the temperature approaches absolute zero.\nIt is thus a common practice to put $S = 0$ for a crystal at 0 K. This\nis usually referred as the third law of thermodynamics. From the\ndefinition of entropy change by , $S$ or $S_{m}$ is always positive at\nfinite temperatures as the system or the crystal absorbs heat from the\nsurroundings to increase its temperature. $V$ or $V_{m}$ of a phase is a\nwell-defined physical quantity, and its absolute value can be given and\nis always positive. The above three equations can be written in a\ngeneral form as\n\nEq. \u2011\n$\\ \\left( \\frac{\\partial Y_{i}}{\\partial Y_{j}} \\right)_{Y_{k}} = - \\frac{X_{j}}{X_{i}} < 0$\n\nThe surface thus has negative slopes in all its directions. The\ncurvature of the surface can be derived from\n\nEq. \u2011\n$\\ \\left( \\frac{\\partial^{2}Y_{i}}{\\partial\\left( Y_{j} \\right)^{2}} \\right)_{Y_{k}} = - \\left( \\frac{\\partial\\left( \\frac{X_{j}}{X_{i}} \\right)}{\\partial Y_{j}} \\right)_{Y_{k}} = - \\frac{1}{X_{i}}\\left( \\frac{\\partial X_{j}}{\\partial Y_{j}} \\right)_{Y_{k}} + \\frac{X_{j}}{\\left( X_{i} \\right)^{2}}\\left( \\frac{\\partial X_{i}}{\\partial Y_{j}} \\right)_{Y_{k}} = - \\frac{1}{X_{i}}\\left\\lbrack \\left( \\frac{\\partial X_{j}}{\\partial Y_{j}} \\right)_{Y_{k}} - \\frac{X_{j}}{X_{i}}{\\left( \\frac{\\partial X_{i}}{\\partial Y_{i}} \\right)\\left( \\frac{\\partial Y_{i}}{\\partial Y_{j}} \\right)}_{Y_{k}} \\right\\rbrack = - \\frac{1}{X_{i}}\\left\\lbrack \\left( \\frac{\\partial X_{j}}{\\partial Y_{j}} \\right)_{Y_{k}} + \\left( \\frac{X_{j}}{X_{i}} \\right)^{2}\\left( \\frac{\\partial X_{i}}{\\partial Y_{i}} \\right)_{Y_{k}} \\right\\rbrack < 0$\n\nBoth terms inside the last bracket are positive for a system in a state\nof stable internal equilibrium, and the surface thus has a negative\ncurvature and is convex everywhere as shown in .\n\nFigure \u2011: Gibbs energy of a one-component phase as a function of\ntemperature and negative pressure, showing the convex shape\n\nFrom experimental observations, it is known that\n$S_{m}^{vapor} \\gg S_{m}^{liquid} > S_{m}^{solid}$. The curves of\n$G_{m}$ or $\\mu^{\\alpha}$ plotted with respect to $T$ at constant $P$\nwould thus have the most negative slope for a vapour phase followed by\nits liquid and solid phases. As an example, shows Gibbs energy of Zn in\nits solid, liquid, and vapour forms as a function of $T$ at constant\n$P = 1$ atmospheric pressure.\n\nFigure \u2011: Molar Gibbs energy of Zn as a function of T at constant P\n\nSimilarly it is common that\n$V_{m}^{vapor} \\gg V_{m}^{liquid} > V_{m}^{solid}$, and the curves of\n$G_{m}$ or $\\mu^{\\alpha}$ plotted with respect to $P$ at constant $T$\nwould thus have the most positive slope for a vapour phase followed by\nits liquid and solid phases, though there are cases that\n$V_{m}^{liquid} < V_{m}^{solid}$ such as those of water and ice. As an\nexample, shows Gibbs energy of Fe in its two solid (fcc and bcc),\nliquid, and vapour forms as a function of $P$ at constant $T = 1273K$.\n\nFigure \u2011: Molar Gibbs energy of Zn as a function of P at constant T\n\nThe quantities measurable by experiments typically include temperature,\npressure, volume, composition, and amount of heat flow in the combined\nlaw of thermodynamics discussed so far. By measuring the heat needed to\nincrease the temperature of a phase, the heat capacity of the phase is\nobtained as shown by Eq. 2\u20117. A typical heat capacity curve as a\nfunction of temperature is shown in for fcc-Al, hcp-Mg, and an\nintermetallic phase Al12Mg17.\n\nFigure \u2011: Heat capacity of fcc-Al, hcp-Mg, and\nAl12Mg17 as a function of temperature\n\nThere are various theoretical models for the heat capacity under\nconstant volume to be discussed in Chapter 5 of this book, which is\ndefined as\n\nEq. \u2011\n$C_{V} = \\left( \\frac{\\partial U}{\\partial T} \\right)_{V} = T\\left( \\frac{\\partial(F + TS)}{\\partial T} \\right)_{V} = T\\left( \\frac{\\partial S}{\\partial T} \\right)_{V} = - T\\left( \\frac{\\partial^{2}F}{\\partial T^{2}} \\right)_{V}$\n\nTo establish the relationship between $C_{P}$ defined by and $C_{V}$,\n$U$ needs to be represented as a function of $T$ and $V$ in terms of $G$\nand its derivatives with respect to Gibbs energy\u2019s natural variables of\n$T$ and $P$. It can be done as follows\n\nEq. \u2011\n$dV = \\frac{\\partial V}{\\partial T}dT + \\frac{\\partial V}{\\partial( - P)}d( - P) = - \\frac{\\partial^{2}G}{\\partial T( - P)}dT - \\frac{\\partial^{2}G}{\\partial( - P)^{2}}d( - P)$\n\nEq. \u2011\n$dU = \\frac{\\partial(G + TS - PV)}{\\partial T}dT + \\frac{\\partial(G + TS - PV)}{\\partial( - P)}d( - P) = - \\left( T\\frac{\\partial^{2}G}{\\partial T^{2}} - P\\frac{\\partial^{2}G}{\\partial T( - P)} \\right)dT\u2014\\left( T\\frac{\\partial^{2}G}{\\partial T( - P)} + P\\frac{\\partial^{2}G}{\\partial( - P)^{2}} \\right)\\left( - \\frac{1}{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}}dV + \\frac{\\frac{\\partial^{2}G}{\\partial T( - P)}}{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}}dT \\right) = - \\left\\lbrack T\\frac{\\partial^{2}G}{\\partial T^{2}} - T\\frac{\\left( \\frac{\\partial^{2}G}{\\partial T( - P)} \\right)^{2}}{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}} \\right\\rbrack dT + \\left( - T\\frac{\\frac{\\partial^{2}G}{\\partial T( - P)}}{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}} + P \\right)dV$\n\nEq. \u2011\n$C_{V} = C_{P} + T\\frac{\\left( \\frac{\\partial^{2}G}{\\partial T( - P)} \\right)^{2}}{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}} = C_{P} - \\frac{\\alpha_{V}^{2}VT}{\\kappa_{T}} = C_{P} - \\alpha_{V}^{2}BVT$\n\nwhere the thermal expansion, $\\alpha_{V}$, and the compressibility or\nbulk modulus, $\\kappa_{T}$ or $B$, are defined by and , respectively.\nFrom the heat capacity, the enthalpy and entropy can be obtained by\nintegration of at a constant pressure\n\nEq. \u2011\n$S = S_{0} + \\int_{0}^{T}\\frac{C_{P}}{T}dT = S_{0} + \\int_{0}^{298.15}\\frac{C_{P}}{T}dT + \\int_{298.15}^{T}\\frac{C_{P}}{T}dT = S_{298.15} + \\int_{298.15}^{T}\\frac{C_{P}}{T}dT$\n\nEq. \u2011\n$H = H_{0} + \\int_{0}^{T}C_{P}dT = H_{0} + \\int_{0}^{298.15}C_{P}dT + \\int_{298.15}^{T}C_{P}dT = H_{298.15} + \\int_{298.15}^{T}C_{P}dT$\n\nIn the above equations, two temperature ranges of integration are chosen\nfor practical applications as most processing procedures in the field of\nmaterials science and engineering take place at temperatures above the\nroom temperature. Based on the third-law of thermodynamics, $S_{0} = 0$,\n$S_{298.15}$ can be obtained by integration. On the other hand for\n$H_{0} = U_{0} + PV$, one does not know the absolute value of the\ninternal energy and thus have to select a reference state for $H$. In\nprinciple, the reference state can be arbitrarily chosen. A widely used\nreference state in the thermodynamic modeling practice is to set\n$H_{298.15}^{SER} = 0$ at ambient pressure for pure elements at their\nrespective stable structures at room temperature, called stable element\nreference (SER) state with\n\nEq. \u2011\n$G_{298.15}^{SER} = H_{298.15}^{SER} - TS_{298.15}^{SER} = - TS_{298.15}^{SER}$\n\nIt is further noted that after defining $S_{298.15}$ and $H_{298.15}$,\none only needs the heat capacity at higher temperatures. This makes the\nmathematical representation of heat capacity simpler due to a relatively\nsimple temperature dependence of heat capacity at higher temperatures in\ncomparison with the variation at lower temperatures. One common\nexpression for heat capacity at high temperatures and ambient pressure\nis as follows\n\nEq. \u2011 $C_{P} = c + dT + \\frac{e}{T^{2}} + fT^{2}$\n\nwhere c, d, e, and f are parameters fitted to experimental or theoretic\ndata and compiled in various handbooks.\n\nThe corresponding $S$, $H$, and $G$ are obtained as\n\nEq. \u2011 $S = b^{'} + clnT + dT - \\frac{e}{{2T}^{2}} + \\frac{f}{2}T^{2}$\n\nEq. \u2011 $H = a + cT + \\frac{d}{2}T^{2} - \\frac{e}{T} + \\frac{f}{3}T^{3}$\n\nEq. \u2011\n$G = H - TS = a - bT - cTlnT - \\frac{d}{2}T^{2} - \\frac{e}{2T} - \\frac{f}{6}T^{3}$\n\nwith $b = b^{'} - c$. The integration constants $b^{'}$ and $a$ are\nevaluated from $S_{298.15}$ and $H_{298.15}$. As an example, the\nenthalpy and entropy of Zn in solid (hcp), liquid, and gas forms are\nplotted in and , respectively. The distances between any two curves in\nand represent the enthalpy or entropy differences between the two\nphases. It can be seen that the gas has much higher enthalpy and entropy\nthan the solid and liquid.\n\nFigure \u2011: Enthalpy of Zn as a function of temperature at one atmospheric\npressure\n\nFigure \u2011: Entropy of Zn as a function of temperature at one atmospheric\npressure\n\nSimilarly, one can add the pressure dependence into the Gibbs energy\nfunction such as\n\nEq. \u2011\n$G = a - bT - cTlnT - \\frac{d}{2}T^{2} - \\frac{e}{2T} - \\frac{f}{6}T^{3} + gP + hTP + mP^{2}$\n\nwhere g, h, and m are parameters fitted to experimental or theoretic\ndata and compiled in various handbooks.\n\nThe expression for $V$ can be derived as\n\nEq. \u2011 $V = g + hT + 2mP$\n\nThe Helmholtz energy can be expressed as a function of its natural\nvariables by solving $P\\ $ from\n\nEq. \u2011\n$F = G - PV = a - bT - cTlnT - \\frac{d}{2}T^{2} - \\frac{e}{2T} - \\frac{f}{6}T^{3} - \\frac{(g + hT - V)^{2}}{4m}$\n\nIn the literature there are many models to represent the temperature and\npressure dependences of thermodynamic properties. The Gibbs energy\ndifference between a stoichiometric compound and the components at their\nreference states of which the compound is composed,\n${_{\\ }^{0}G}_{i}^{ref}$, is termed as Gibbs energy of formation, i.e.\n\nEq. \u2011 $\\mathrm{\\Delta}_{f}G = G - \\sum_{}^{}N_{i}{_{\\ }^{0}G}_{i}^{ref}$\n\nwith $N_{i}$ being the stoichiometry of the compound. Similarly,\nenthalpy of formation, entropy of formation, and heat capacity of\nformation with respect to components at their reference states,\n$_{\\ }^{0}H_{i}^{ref}$, $_{\\ }^{0}S_{i}^{ref}$, and\n$_{\\ }^{0}{C_{P}}_{i}^{ref}$, can be defined as\n\nEq. \u2011 $\\mathrm{\\Delta}_{f}H = H - \\sum_{}^{}{N_{i}_{\\ }^{0}H_{i}^{ref}}$\n\nEq. \u2011 $\\mathrm{\\Delta}_{f}S = S - \\sum_{}^{}{N_{i}_{\\ }^{0}S_{i}^{ref}}$\n\nEq. \u2011\n$\\mathrm{\\Delta}_{f}C_{P} = C_{P} - \\sum_{}^{}{N_{i}_{\\ }^{0}{C_{P}}_{i}^{ref}}$\n\nIt should be mentioned that one mole of a compound usually refers to one\nmole of formula of stoichiometry of the compound. With a formula like\n$A_{a}B_{b}C_{c}$, the compound is composed of total $(a + b + c)$ moles\nof components. One should thus be very careful when dealing with\nnumerical values to be sure whether the data is in terms of per mole of\nformula or per mole of components. At the same time the reference states\nmust be clearly defined. When the SER state defined in is selected as\nthe reference state, the above formation quantities are called standard\nformation quantities such as standard enthalpy of formation.\n\nSince there are only two independent potentials in a one-component\nsystem, its limit of stability can be evaluated with one potential kept\nconstant, i.e. either $T$ or $P$. Consequently, either Helmholtz energy\nor enthalpy is to be used in deriving the limit of stability of a\nhomogeneous system. For the practical usefulness, let us use Helmholtz\nenergy because its natural variables of $T$ and *V* are measurable\nquantities in typical experiments, while one of the natural variables of\nenthalpy, entropy, is not. From and , the limit of stability for a\none-component system at constant temperature can be written as\n\nEq. \u2011\n$\\left( \\frac{\\partial( - P)}{\\partial V} \\right)_{T,N} = F_{VV} = \\frac{1}{V\\kappa_{T}} = \\frac{B}{V} = 0$\n\nwhere the isothermal compressibility and bulk modulus, $\\kappa_{T}$ and\n$B$, are defined in . The limit of stability is thus determined when the\nisothermal compressibility diverges or the bulk modulus becomes zero\nbecause $V$ has finite values at any temperatures. It is evident that\nHelmholtz energy must have higher order dependence on volume than in for\na system with instability because $F_{VV}$ from is constant.\n\nFrom , the consolute point is defined by\n\nEq. \u2011\n$F_{VVV} = \\left( \\frac{\\partial^{2}( - P)}{\\partial V^{2}} \\right)_{T,N} = \\frac{\\partial\\left( \\frac{1}{V\\kappa_{T}} \\right)}{\\partial V} = - \\frac{1 + \\frac{V}{\\kappa_{T}}\\frac{\\partial\\kappa_{T}}{\\partial V}}{\\kappa_{T}V^{2}} = 0$\n\nSince $\\kappa_{T}$ becomes infinite at the limit of stability,\n$\\frac{\\partial\\kappa_{T}}{\\partial V}$ approaches negative infinite\nwhen the critical/consolute point is approached so that\n$\\frac{V}{\\kappa_{T}}\\frac{\\partial\\kappa_{T}}{\\partial V} = - 1$ and\n$F_{VVV} = 0$.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/gibbs_energy_function/phases_with_variable_compositions_random_solutions.ipynb b/psu410/src/gibbs_energy_function/phases_with_variable_compositions_random_solutions.ipynb deleted file mode 100644 index 812aaf7..0000000 --- a/psu410/src/gibbs_energy_function/phases_with_variable_compositions_random_solutions.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "8bfb087d", "cell_type": "markdown", "source": "## Phases with variable compositions: Random solutions\n\nThe combined law of thermodynamics and the Gibbs-Duhem equation of a\nsolution phase with variable compositions are shown by and ,\nrespectively. A phase can be represented by a *c+1*-dimensional surface\nin a *c+2*-dimensional space of potentials based on the Gibbs-Duhem\nequation. The directions and curvature of the surface are represented by\nthe partial derivatives shown by and secondary derivatives shown by ,\nboth being negative for a stable phase. To develop a mathematical\nformula for Gibbs energy of a phase with variable compositions, one can\nconsider a phase as a mixture of independent components that the phase\nis made of. Its Gibbs energy function can be postulated as the summation\nof Gibbs energy of the independent components of the same structure of\nthe solution, $_{\\ }^{0}G_{i}$, plus the contribution due to the mixing,\n$_{\\ }^{mixing}G$ or $_{\\ }^{M}G$\n\nEq. \u2011 $G = \\sum_{}^{}{N_{i}_{\\ }^{0}G_{i}} +_{\\ }^{M}G$\n\nSince the system size usually is not important in thermodynamics, its\nproperties are typically normalized to one mole with its composition\nrepresented by mole fractions of components. The molar Gibbs energy is\nobtained as shown below with the molar Gibbs energy of mixing separated\ninto two parts: ideal Gibbs energy of mixing assuming no chemical\ninteraction among components, $_{\\ }^{ideal}G_{m}$ or $_{\\ }^{I}G_{m}$,\nand excess Gibbs energy of mixing due to chemical reaction among\ncomponents, $_{\\ }^{excess}G_{m}$ or $_{\\ }^{E}G_{m}$\n\nEq. \u2011\n$G_{m} = \\sum_{}^{}{x_{i}_{\\ }^{0}G_{i}} +_{\\ }^{M}G_{m} = \\sum_{}^{}{x_{i}_{\\ }^{0}G_{i}} +_{\\ }^{I}G_{m} +_{\\ }^{E}G_{m}$\n\nFrom , the chemical potential of a component is thus\n\nEq. \u2011\n$\\mu_{i} =_{\\ }^{0}G_{i} +_{\\ }^{I}{G_{m} +}_{\\ }^{E}G_{m} + \\frac{\\partial\\left(_{\\ }^{I}{G_{m} +}_{\\ }^{E}G_{m} \\right)}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial\\left(_{\\ }^{I}{G_{m} +}_{\\ }^{E}G_{m} \\right)}{\\partial x_{j}}$\n\nOne may define the chemical activity of component *i*, $a_{i}^{\\ }$, as\nfollows\n\nEq. \u2011\n$RTlna_{i}^{\\ } = \\mu_{i} -_{\\ }^{0}G_{i}^{\\ } =_{\\ }^{I}{G_{m} +}\\frac{\\partial_{\\ }^{I}G_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial_{\\ }^{I}G_{m}}{\\partial x_{j}} +_{\\ }^{E}G_{m} + \\frac{\\partial_{\\ }^{E}G_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial_{\\ }^{E}G_{m}}{\\partial x_{j}}$\n\nIn this definition, the chemical activity or simply activity is\ncalculated with respect to the pure elements in the structure of the\nsolution for practical reasons as one would like to understand the\nchemical potential difference of components in the solution and by\nitself with the same structure. It should be noted that this reference\nstate for chemical activity is usually different from the SER reference\nstate defined in as the solution may have a different structure than\nthose of pure components in their SER states. On the other hand, the\nactivity under the SER reference state can be easily obtained by\nreplacing $_{\\ }^{0}G_{i}$ with ${_{\\ }^{0}G}_{i}^{SER}$ from . In\nprinciple, one may choose any structure as the reference state for\nactivity to be useful for practical applications, i.e.\n\nEq. \u2011 $RTlna_{i}^{ref} = \\mu_{i} -_{\\ }^{0}G_{i}^{ref}$\n\nFor example, the activity of a component in a liquid solution is defined\nwith respect to the pure component in its liquid form from , but can\nalso be referred to its SER state which is solid using . The following\nsections will discuss in more details how components mix when they are\nbrought together including concepts such as random mixing, short-range\nordering, and long-range ordering.\n\nThe limit of stability of a solution with respect to composition\nfluctuation under constant *T*, *P*, and *N1* can be derived\nas follows from and\n\nEq. \u2011\n$\\left( \\frac{\\partial\\mu_{i}}{\\partial N_{i}} \\right)_{T,P,N_{j \\neq i},i > 1} > \\left( \\frac{\\partial\\mu_{i}}{\\partial N_{i}} \\right)_{T,P,N_{1},{\\mu_{2},N}_{j \\neq i},i,j > 2}\\ldots... > \\left( \\frac{\\partial\\mu_{c}}{\\partial N_{c}} \\right)_{T,P,N_{1},\\mu_{2}..\\mu_{c - 1}} = 0$\n\nThe first term can be derived from as follows\n\nEq. \u2011\n$\\left( \\frac{\\partial\\mu_{i}}{\\partial N_{i}} \\right)_{T,P,N_{j \\neq i},i > 1} = \\sum_{j = 1}^{c}\\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{i}\\partial x_{j}}\\frac{\\partial x_{j}}{\\partial N_{i}} - \\sum_{j = 1}^{c}{x_{j}\\sum_{k = 1}^{c}{\\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{j}\\partial x_{k}}\\frac{\\partial x_{k}}{\\partial N_{i}}}} = \\frac{1}{N}\\left( \\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{i}^{2}} - \\sum_{j = 1}^{c}{x_{j}\\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{j}^{2}}} - \\sum_{j = 1}^{c}{x_{j}\\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{i}\\partial x_{j}}} + \\sum_{j = 1}^{c}{\\sum_{k = 1}^{c}{{x_{j}x}_{k}\\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{j}\\partial x_{k}}}} \\right)$\n\nDenoting\n$G_{ij} = \\left( \\frac{\\partial\\mu_{i}}{\\partial N_{j}} \\right)_{T,P,N_{k \\neq j}}$and\nusing to change the variables kept constant from molar quantities to\npotentials one-by-one (see \\[1\\]), the limit of stability can be further\nobtained as\n\nEq. \u2011\n$\\left( \\frac{\\partial\\mu_{c}}{\\partial N_{c}} \\right)_{T,P,N_{1},\\mu_{2}..\\mu_{c - 1}} = \\frac{\\det\\left( G_{ij}:2 \\leq i,j \\leq c \\right)}{\\det\\left( G_{ij}:2 \\leq i,j \\leq c - 1 \\right)} = 0$\n\nwhere $\\det$ stands for determinant. indicates that\n$\\det\\left( G_{ij}:2 \\leq i,j \\leq c \\right) = 0$ at the limit of\nstability. Considering $x_{1} = 1 - \\sum_{j \\neq 1}^{}x_{j}$, let us\nintroduce\n\nEq. \u2011\n$g_{i} = \\mu_{i} - \\mu_{1} = \\left( \\frac{\\partial G_{m}}{\\partial x_{i}} \\right)_{x_{k \\neq i}} - \\left( \\frac{\\partial G_{m}}{\\partial x_{1}} \\right)_{x_{k \\neq 1}}$\n\nand\n\nEq. \u2011\n$g_{ij} = \\frac{\\partial g_{i}}{\\partial x_{j}} = \\frac{\\partial\\left( \\mu_{i} - \\mu_{1} \\right)}{\\partial x_{j}} = \\frac{\\partial^{2}G_{m}}{\\partial x_{i}\\partial x_{j}} - \\frac{\\partial^{2}G_{m}}{\\partial x_{1}\\partial x_{j}} - \\frac{\\partial^{2}G_{m}}{\\partial x_{i}\\partial x_{1}} + \\frac{\\partial^{2}G_{m}}{\\partial\\left( x_{1} \\right)^{2}}$\n\nThe limit of stability can be re-written as\n\nEq. \u2011\n$\\left( \\frac{\\partial\\left( \\mu_{c} - \\mu_{1} \\right)}{\\partial x_{c}} \\right)_{T,P,N,\\mu_{2} - \\mu_{1},\\ ...\\mu_{c - 1} - \\mu_{1}} = \\frac{\\det\\left( g_{ij}:2 \\leq i,j \\leq c \\right)}{\\det\\left( g_{ij}:2 \\leq i,j \\leq c - 1 \\right)} = 0$\n\ni.e. $\\det\\left( g_{ij}:2 \\leq i,j \\leq c \\right) = 0$. The consolute\npoint can be defined following\n\nEq. \u2011\n$\\left( \\frac{\\partial^{2}\\mu_{c}}{\\partial\\left( N_{c} \\right)^{2}} \\right)_{T,P,N_{1},\\mu_{2}..\\mu_{c - 1}} = \\left( \\frac{\\partial^{2}\\left( \\mu_{c} - \\mu_{1} \\right)}{\\partial\\left( x_{c} \\right)^{2}} \\right)_{T,P,N,\\mu_{2} - \\mu_{1},\\ ...\\mu_{c - 1} - \\mu_{1}} = 0$\n\nNo closed mathematic form has been published in the literature.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/gibbs_energy_function/phases_with_variable_compositions_solutions_with_ordering.ipynb b/psu410/src/gibbs_energy_function/phases_with_variable_compositions_solutions_with_ordering.ipynb deleted file mode 100644 index 6d406a2..0000000 --- a/psu410/src/gibbs_energy_function/phases_with_variable_compositions_solutions_with_ordering.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "49e6341c", "cell_type": "markdown", "source": "## Phases with variable compositions: Solutions with ordering\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/gibbs_energy_function/polymer_solutions_and_polymer_blends.ipynb b/psu410/src/gibbs_energy_function/polymer_solutions_and_polymer_blends.ipynb deleted file mode 100644 index e172c0a..0000000 --- a/psu410/src/gibbs_energy_function/polymer_solutions_and_polymer_blends.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "efad1127", "cell_type": "markdown", "source": "## Polymer solutions and polymer blends\n\nA polymer solution is a mixture between polymer molecules and solvents,\nwhile a polymer blend is a mixture between different polymer molecules.\nA polymer molecule consists of the same repeating units of one or more\nmonomers, which can be an atom or a small molecule. The number of\nrepeating units is called the degree of polymerization and can be as\nlarge as 104\u2013105. It defines the molecular mass,\ni.e. the mass of one polymer molecule. There are three typical\narchitectures of polymerization: a linear chain, a branched chain, and a\ncross-linked polymer. Nearly all polymers are mixtures of molecules with\na different degree of polymerization with a molecular mass distribution,\ncomplicating the modeling of their thermodynamic properties because of\nthe dependence of properties on molecular mass.\n\nGibbs energy functions of polymers with a single molecular mass can be\ntreated similarly as in Chapter . For a polymer solution, the ideal\nentropy of mixing is quite different from that of atomically random\nsolutions discussed in Chapter because the monomers in a polymer\nmolecule are connected to each other and cannot move freely. One common\napproach to calculate the ideal entropy of mixing is to evoke a lattice\nmodel and assume that one monomer occupies a lattice site with a fixed\nvolume. The number of translational states of a single molecule is equal\nto the number of lattice sites available. In a homogeneous solution, the\ntotal number of lattice sites available is\n\nEq. \u2011 $n = \\sum_{i}^{}{m_{i}n_{i}}$\n\nwhere $n_{i}$ and $m_{i}$ are the number of molecule $i$ and the number\nof lattice sites per molecule $i$, respectively. While in its pure\nstate, i.e. before mixing, the number of states of molecule $i$ in terms\nof the number of lattice sites is\n\nEq. \u2011 $w_{i} = m_{i}n_{i} = n\\varphi_{i}$\n\nwhere $\\varphi_{i}$ is the volume fraction of molecule $i$ in the\nsolution. The entropy change per molecule $i$ is thus\n\nEq. \u2011\n$S_{i} = kln(n) - klnw_{i} = kln\\frac{1}{\\varphi_{i}} = - kln\\varphi_{i}$\n\nThe total entropy of mixing is the summation for all molecules,\nnormalized to per mole of lattice site\n\nEq. \u2011\n$_{}^{I}S_{m} = \\frac{N_{a}}{n}\\sum_{i}^{}{n_{i}S_{i}} = - R\\sum_{i}^{}{\\frac{\\varphi_{i}}{m_{i}}\\ln\\varphi_{i}}$\n\nwhere $N_{a}$ is the Avogadro number. When $m_{i} = 1$ for all\nmolecules, Eq. 2\u2011143 reduces to . Since $m_{i}$ values are typically\nvery large numbers for polymers, the entropy of mixing in polymer\nsolutions and blends is thus significantly lower than those in\nnon-polymer solutions as shown schematically in for binary systems with\nvarious $m_{i}$ values.\n\nFigure \u2011: Schematic entropy of mixing in solutions with the numbers of\nlattice sites per molecule shown.\n\nSimilar to , Gibbs energy of a multi-component random polymer\nsolution/blend can be written as\n\nEq. \u2011\n$G_{m} = \\sum_{}^{}\\frac{\\varphi_{i}}{m_{i}}_{\\ }^{0}G_{im} + RT\\left( \\sum_{}^{}{\\frac{\\varphi_{i}}{m_{i}}\\ln\\varphi_{i}} + \\sum_{}^{}{\\varphi_{i}\\varphi_{j}\\chi_{ij}} \\right)$\n\nwhere $_{\\ }^{0}G_{im}$ is the Gibbs energy of molecular $i$ per mole of\nlattice site, and $\\chi_{ij}$ the unitless interaction parameter between\nmolecule $i$ and $j$. Other equations shown in Chapter can be derived\nsimilarly too. It is to be noted that is very similar to the\nFlory\u2013Huggins solution equation widely used in the polymer community.\n", "metadata": {}}, {"id": "d26c203e", "cell_type": "markdown", "source": "### Random solutions\n\nThe ideal Gibbs energy of mixing represents an ideal solution in which\nall sites are equivalent and the distributions of components on the\nsites are completely random. The number of different configurations to\narrange all components is\n\nEq. \u2011 $w = \\frac{N!}{\\prod_{}^{}\\left( N_{i}! \\right)}$\n\nBased on Boltzmann\u2019s relation from statistic thermodynamics when all\nconfigurations have the same probability to be observed, the ideal\nconfigurational molar entropy of mixing due to the distribution is\n\nEq. \u2011\n$_{\\ }^{ideal}S_{m} =_{\\ }^{I}S_{m} = \\frac{Rlnw}{N} = R\\frac{lnN! - \\sum_{}^{}\\ln\\left( N_{i}! \\right)}{N} \\cong R\\frac{NlnN - \\sum_{}^{}{N_{i}l{nN}_{i}}}{N} = - R\\sum_{}^{}{x_{i}l{nx}_{i}}$\n\nwhere $R$ is the gas constant. represents the entropy difference between\nthe ideal solution and the states when individual components are by\nthemselves, i.e. the mechanical mixing of components. As $x_{i}$ is\nsmaller than unity, the entropy production to form an ideal solution\nfrom pure components is thus positive, indicating that it is a\nspontaneous process. In such an ideal solution, it is assumed that there\nare no interactions between components, and the enthalpy of mixing is\nthus zero as the internal energy and the volume of the system do not\nchange. The ideal Gibbs energy of mixing is written as\n\nEq. \u2011 $_{\\ }^{I}G = - T_{\\ }^{I}S_{m} = RT\\sum_{}^{}{x_{i}l{nx}_{i}}$\n\nThe Gibbs energy of real solutions, i.e. , becomes\n\nEq. \u2011\n$G_{m} = \\sum_{}^{}{x_{i}_{\\ }^{0}G_{i}} + RT\\sum_{}^{}{x_{i}l{nx}_{i}} +_{\\ }^{E}G_{m}$\n\nFrom , the chemical potential is obtained as\n\nEq. \u2011\n$\\mu_{i} = G_{i} =_{\\ }^{0}G_{i} + RTlnx_{i} +_{\\ }^{E}G_{m} + \\frac{\\partial_{\\ }^{E}G_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial_{\\ }^{E}G_{m}}{\\partial x_{j}}$\n\nFrom the chemical activity of , the activity coefficient, $\\gamma_{i}$,\ncan be defined as follows\n\nEq. \u2011\n$\\gamma_{i} = \\frac{a_{i}}{x_{i}} = \\frac{1}{x_{i}}\\exp\\frac{G_{i} -_{\\ }^{0}G_{i}\\ }{RT}$\n\nThe solution is an ideal solution with $\\gamma_{i} = 1$, and is said to\nbe positively or negatively deviating from an ideal solution with\n$\\gamma_{i} > 1$ or $\\gamma_{i} < 1$, respectively. The chemical\npotential is related to the activity and activity coefficient by the\nfollowing equation\n\nEq. \u2011\n$\\mu_{i} =_{\\ }^{0}G_{i} + RTlna_{i} =_{\\ }^{0}G_{i} + RTln\\gamma_{i}x_{i} =_{\\ }^{0}G_{i} + RTlnx_{i} + RTln\\gamma_{i}$\n\nLet us further exam in more details in order to better understand the\nrelation between $G_{m}$ and $\\mu_{i}$. The partial derivatives in\nrepresent the directions of molar Gibbs energy in the composition space,\ni.e. the tangents of molar Gibbs energy with respect to mole fractions\nof independent components. Collectively, they define the\nmulti-dimensional tangent plane of molar Gibbs energy at the given\ncomposition, $x_{i}^{0}$. The mathematical representation of this\ntangent plane, $z_{G_{m}}$, is defined by its directional derivatives\nand the distance from the point where the derivatives are taken,\n\nEq. \u2011\n$z_{G_{m}} = G_{m}\\left( x_{i}^{0} \\right) + \\sum_{i = 1}^{c}{\\left( \\frac{\\partial G_{m}}{\\partial x_{i}} \\right)_{x_{i}^{0}}\\left( x_{i} - x_{i}^{0} \\right)}$\n\nThe intercept of this tangent plane at each pure component axis, i.e.\n$x_{i} = 1$ and $x_{j \\neq i} = 0$, is obtained as\n\nEq. \u2011\n$z_{G_{m},x_{i} = 1} = G_{m}\\left( x_{i}^{0} \\right) + \\left( \\frac{\\partial G_{m}}{\\partial x_{i}} \\right)_{x_{i}^{\\ } = x_{i}^{0}} - \\sum_{j = 1}^{c}{x_{j}^{0}\\left( \\frac{\\partial G_{m}}{\\partial x_{j}} \\right)_{x_{i}^{\\ } = x_{i}^{0}}}$\n\nThis is identical to at the point $x_{i}^{0}$. It is thus shown that the\nchemical potential of a component in a solution is represented by the\nintercept of the tangent plane of Gibbs energy of the solution on the\n$G_{m}$ axis of the component. The distance between the intercept and\nthe Gibbs energy of the pure component in the same structure of the\nsolution is related to the chemical activity of the component as defined\nby . On the other hand, it is evident that one can choose any other\nstructure of the pure element to define the chemical activity to compare\nchemical potential of the component as shown by .\n\nThe stability of a solution is evaluated following , and the derivatives\nof chemical potential with respect to its moles, i.e. the elements in\nthe determinant, are obtained as follows from and ,\n\nEq. \u2011\n$\\frac{N}{RT}\\frac{\\partial\\mu_{i}}{\\partial N_{i}} = \\frac{N}{RT}G_{ii} = \\frac{1 - x_{i}}{x_{i}} + \\frac{1}{\\gamma_{i}}\\left( \\frac{\\partial\\gamma_{i}}{\\partial x_{i}} - \\sum_{j = 1}^{c}{x_{j}\\frac{\\partial\\gamma_{i}}{\\partial x_{j}}} \\right)$\n\nEq. \u2011\n$\\frac{N}{RT}\\frac{\\partial\\mu_{i}}{\\partial N_{k}} = \\frac{N}{RT}G_{ik} = - \\frac{x_{k}}{x_{i}} + \\frac{1}{\\gamma_{i}}\\left( 1 - \\sum_{j = 1}^{c}{x_{j}\\frac{\\partial\\gamma_{i}}{\\partial x_{j}}} \\right)$\n\nTo further study Gibbs energy of solution phases, let us discuss the\ndetails on the excess Gibbs energy of mixing. At this point, one can\nstart with lower-order systems with fewer components, i.e. two component\nand three-component systems, noting that the Gibbs energy of phases with\none component is already presented in Chapter .\n", "metadata": {}}, {"id": "ab833401", "cell_type": "markdown", "source": "### Solutions with both short-range and long-range ordering\n\nThe short-range ordering in a solution with long-rang ordering can take\nplace in each sublattice or between two sublattices. The short-range\nordering in one sublattice can be treated similarly as in Chapter 2.3.1\nwith mole fractions substituted by site fractions. In case that\nassociates form, the relation between mole fractions and site fractions\nbecomes more complicated as follows\n\nEq. \u2011\n$x_{i} = \\frac{\\sum_{}^{}{a^{t}\\sum_{k}^{}{i_{k}y}_{k}^{t}}}{\\sum_{}^{}{a^{t}\\left( 1 - y_{va}^{t} \\right)}}$\n\nwhere the summation for $k$ goes over all associates in sublattice $t$\ncontaining component $i$.\n\nThe short-range ordering between two-sublattices indicates that a\ncomponent in one sublattice has different interactions with different\ncomponents in another sublattice. This results in local ordering of one\ncomponent around another component in two neighboring sublattices. Such\nlocal ordering involves interactions between two sublattices shown as\nthe third term in .\n", "metadata": {}}, {"id": "09d5f96a", "cell_type": "markdown", "source": "### Solutions with charged species\n\nOne special type of solutions with both short-range and long-range\nordering is solutions with charged species, i.e. ionic solutions, plus\nelectrons and holes. There is an additional constraint on species\nconcentrations to maintain the charge neutrality of such solutions, i.e.\n\nEq. \u2011 $0 = \\sum_{t}^{}{\\sum_{i}^{}{a^{t}y_{i}^{t}v_{i}^{t}}}$\n\nwhere $v_{i}^{t}$ is the valance of species $i$ in sublattice $t$\nincluding its sign which is positive for cations, negative for anions,\nand zero for neutral species. The conventional defect chemistry theory\nis typically based on the ideal mass action laws and applicable to a\nsingle set of defects and at very low defect concentrations, i.e. in the\nlimit of ideal solutions. For interacting defects, their concentrations\nshould be replaced by their activities, which can be obtained from\nthermodynamic principles similarly as discussed in previous sections. It\nshould be emphasized that in addition to formation of many more charged\nspecies, one component may have different valences, particularly the\ntransition metals. Consequently, there can be many more species in an\nionic phase than the number of independent components in the system, and\ntheir concentrations can be found by equilibrium calculations as\ndiscussed in Chapter .\n", "metadata": {}}, {"id": "fe69a707", "cell_type": "markdown", "source": "### Solutions with long-range ordering\n\nSo far, solutions in which a component can occupy any site in a phase\nare discussed. In many phases, this is not the case. For example, in the\nfcc solid solution of Fe and C, Fe atoms take the fcc lattice sites, and\nC atoms occupy the interstitial sites between the fcc lattice sites.\nTherefore, Fe atoms do not mix with C atoms on the fcc lattice sites,\nand they rather develop long-rang ordering by occupying their own\ndistinct sites in the phase. Long-range ordering can also develop when\nshort-range ordering extends to the whole lattice. New formula for the\nGibbs energy of mixing is needed by considering the details how\ncomponents are distributed and mixed in various sites in a phase.\n\nOne way to group various sites in a phase is based on equivalent\ncrystallographic positions in a phase, i.e. Wyckoff positions. Various\nsets of equivalent positions divide the lattice into sub-sets of\nlattices. Each set of equivalent positions forms a sublattice. The\ndistributions of components on each sublattice can be represented by\nmole fractions of components in the sublattice, commonly referred to as\nsite fractions and defined as\n\nEq. \u2011 $y_{i}^{t} = \\frac{N_{i}^{t}}{\\sum_{j}^{}N_{j}^{t}}$\n\nEq. \u2011 $\\sum_{i}^{}y_{i}^{t} = 1$\n\nwhere the superscript $t$ denotes the sublattice in which the component\nresides, and the summation is for all species in sublattice $t$\nincluding vacancy. Site fractions and mole fractions are related through\nthe mass balance as follows\n\nEq. \u2011\n$x_{i} = \\frac{\\sum_{}^{}{a^{t}y_{i}^{t}}}{\\sum_{}^{}{a^{t}\\left( 1 - y_{va}^{t} \\right)}}$\n\nwhere $a^{t}$ and $y_{va}^{t}$ are the number of sites and the site\nfraction of vacancy in the sublattice *t*.\n\nRandom solutions form when each component enters all sublattices\nequally. Mole fractions and site fractions thus become identical.\nSolutions with both substitutional and interstitial component, like the\nfcc Fe-C solution mentioned above, can be represented by two\nsublattices. Stoichiometric compounds have its site fractions equal to\nunity in each sublattice. When site fractions in a compound deviate from\nunity, the compound is no longer stoichiometric and develops a\ncomposition range of homogeneity. When the composition range is small,\nthe deviations are often referred to as defects. Since many properties\nof a compound are determined by defects, a distinct field of defect\nchemistry exists, predominantly for charged species. As will be\ndemonstrated in Chapter and the rest of the book, defects can be treated\nas an integral part of the thermodynamics of a phase with more than one\nsublattices.\n\nLet us consider a case where there is only one component in each\nsublattice, which represents one possible stoichiometric composition of\nthe phase and is often called an end-member of the phase. The Gibbs\nenergy of an end-member is the same as that of a phase with a fixed\ncomposition shown by or or . By re-arranging , Gibbs energy of an\nend-member, $_{\\ }^{0}G_{em}$, is obtained as\n\nEq. \u2011\n$_{\\ }^{0}G_{em} = \\sum_{t}^{}{a^{t}{_{\\ }^{0}G}_{i}^{t,ref}} + \\mathrm{\\Delta}_{f}G_{em}$\n\nwhere ${_{\\ }^{0}G}_{i}^{t,ref}$ represents the Gibbs energy of\ncomponent $i$ in a given reference state which occupies sublattice $t$\nin the end-member. For vacancy, $_{\\ }^{0}G_{Va} = 0$ is defined. The\ncontribution of each end-member to the Gibbs energy of the phase is the\nproduct of site fraction of each component in their respective\nsublattices and the Gibbs energy of the end-member per mole of formula\nunit (*mf*), i.e.\n\nEq. \u2011\n$_{\\ }^{0}G_{mf} = \\sum_{em}^{}\\left( \\prod_{t}^{}y_{i}^{t}_{\\ }^{0}G_{em} \\right)$\n\nThe ideal mixing in each sublattice is similar to that in a random\nsolution with mole fractions substituted by site fractions. The excess\nGibbs energy of mixing consists of two contributions: (i) the mixing in\none sublattice with all other sublattices containing only one component\nin each sublattice and (ii) the mixing simultaneously in more than one\nsublattices. The Gibbs energy of a solution phase with multi-sublattices\ncan thus be written in terms of per mole of formula unit as\n\nEq. \u2011\n$G_{mf} =_{\\ }^{0}G_{mf} + RT\\sum_{t}^{}{a^{t}\\sum_{i}^{}{y_{i}^{t}\\ln y_{i}^{t}}} +_{\\ }^{E}G_{mf}$\n\nwith $_{\\ }^{E}G_{mf}$ given by\n\nEq. \u2011\n$_{\\ }^{E}G_{mf} = \\sum_{t}^{}{\\prod_{s \\neq t}^{}y_{l}^{s}\\sum_{i > j}^{}{\\sum_{j}^{}{y_{i}^{t}y_{j}^{t}L_{i,j:l}^{t}}}} + \\sum_{t}^{}{\\prod_{s \\neq t}^{}y_{l}^{s}\\sum_{i > j}^{}{\\sum_{j > k}^{}{\\sum_{k}^{}{y_{i}^{t}y_{j}^{t}y_{k}^{t}L_{i,j,k:l}^{t}}}}} + \\sum_{t}^{}{\\prod_{s \\neq t,u}^{}y_{l}^{s}\\sum_{i > j}^{}{\\sum_{j > k}^{}{\\sum_{k}^{}{y_{i}^{t}y_{j}^{t}y_{m}^{u}y_{n}^{u}L_{i,j:m,n:l}^{t}}}}}$\n\nThe first term in represents the binary interaction between component\n$i$ and $j$ in sublattice $t$ with sublattice $s$ occupied by component\n$l$ with comma separating interacting components in one sublattice and\ncolon separating sublattices. The product,\n$\\prod_{s \\neq t}^{}y_{l}^{s}$, runs through all other sublattices with\none component in each sublattice, except sublattice $t$ in which the\ninteraction is considered. The second term denotes the ternary\ninteraction among $i$, $j$, and $k$ in sublattice $t$ with sublattice\n$s$ occupied by component $l$. The third term depicts the interactions\nsimultaneously in both sublattice $t$ and $u$, and the product runs\nthrough all other sublattices with one component in each sublattice,\nexcept sublattice $t$ and $u$ in which the interactions are considered.\nThe third term thus partially reflects the short-range ordering among\ncomponents between two sublattices. In principles, high-order\ninteraction parameters such as quaternary, quinary, and multiple\nsublattice interaction parameters could be added, but their\ncontributions to $_{\\ }^{E}G_{mf}$ are small due to the physical\ninsignificance of co-location of four or five components indicated by\nthe product of their site fractions in front of the interaction\nparameters.\n\nIn , the chemical potential of a stoichiometric compound was defined in\nterms of a summation of chemical potential of individual components in\nthe compound because the relative amounts of components are constrained\nby the stoichiometry of the compound and chemical potentials of\nindividual components can not vary independently. By the same token, the\nchemical potential of an end-member in a solution can be written as\n\nEq. \u2011 $\\mu_{em} = G_{em} = \\sum_{t}^{}a^{t}\\mu_{i}^{t}$\n\nwhere $\\mu_{i}^{t}$ is the chemical potential of component $i$ that\noccupies the sublattice $t$ in the end-member, and can be derived using\n\nEq. \u2011\n$\\mu_{i}^{t} = a^{t}_{\\ }^{0}G_{i}^{t,ref} + a^{t}RTlny_{i}^{t} +_{\\ }^{E}G_{mf}^{\\ } + \\frac{\\partial_{\\ }^{E}G_{mf}^{\\ }}{\\partial y_{i}^{t}} - \\sum_{j}^{\\ }y_{j}^{t}\\frac{\\partial_{\\ }^{E}G_{mf}^{\\ }}{\\partial y_{j}^{t}}$\n\nFor constant interaction parameters in , for chemical potential reduces\nto the following expression from\n\nEq. \u2011\n$\\mu_{i}^{t} = a^{t}_{\\ }^{0}G_{i}^{t} + a^{t}RTlny_{i}^{t} + \\sum_{j \\neq i}^{}{\\left( y_{i}^{t} \\right)^{2}L}_{i,j}^{t} + \\sum_{k > j}^{}{\\sum_{j \\neq i}^{}{y_{j}^{t}y_{k}^{t}\\left\\lbrack L_{i,j}^{t} + L_{i,k}^{t} - L_{j,k}^{t} + \\left( 1 - 2y_{i}^{t} \\right)L_{i,j,k}^{t} \\right\\rbrack}}$\n\nThe stability of the solution is defined by\n$\\frac{\\partial\\mu_{em}}{\\partial N_{em}}$ with $N_{em}$ being the moles\nof the end-member in the solution given by\n$N_{em} = N\\prod_{u}^{}y_{j}^{u}$ and\n\nEq. \u2011\n$\\frac{\\partial N_{em}}{\\partial y_{j}^{u}} = \\frac{N\\prod_{t}^{}y_{i}^{t}}{y_{j}^{u}\\left( 1 - \\prod_{t}^{}y_{i}^{t} \\right)}$\n\nFollowing , one obtains\n\nEq. \u2011\n$\\frac{\\partial\\mu_{em}}{\\partial N_{em}} = \\sum_{u}^{}{\\frac{\\partial\\left( \\sum_{t}^{}a^{t}\\mu_{i}^{t} \\right)}{\\partial y_{j}^{u}}\\frac{\\partial y_{j}^{u}}{\\partial N_{em}}} = \\frac{\\left( 1 - \\prod_{t}^{}y_{j}^{t} \\right)}{N\\prod_{t}^{}y_{j}^{t}}\\sum_{u}^{}{a^{u}y_{i}^{u}\\frac{\\partial\\mu_{i}^{u}}{\\partial y_{i}^{u}}} = \\frac{\\left( 1 - \\prod_{t}^{}y_{j}^{t} \\right)}{N\\prod_{t}^{}y_{j}^{t}}\\sum_{u}^{}{a^{u}y_{i}^{u}\\left\\{ \\frac{RT\\left( 1 - y_{i}^{u} \\right)}{y_{i}^{u}} - 2\\sum_{j \\neq i}^{}\\left( y_{j}^{u} \\right)^{2}L_{i,j}^{u} - 2\\sum_{k > j}^{}{\\sum_{j \\neq i}^{}{y_{j}^{u}y_{k}^{u}\\left\\lbrack L_{i,j}^{u} + L_{i,k}^{u} - L_{j,k}^{u} + \\left( 2 - 3y_{i}^{u} \\right)L_{i,j,k}^{u} \\right\\rbrack}} \\right\\}}$\n\nIt is self-evident from that a site fraction is only uniquely defined\nfrom the mole fraction of the component when the component enters into\none sublattice only and does not form any associates. Therefore, in\ngeneral, the distribution of components on sublattices and different\nkinds of molecules can only be obtained by equilibrium calculations, and\nthe thermodynamic properties for such a phase can thus not be\nrepresented in a closed form using mole fractions of independent\ncomponents. This was demonstrated in Chapter when short-range ordering\nexists in solution phases, where the energy minimization procedure was\nused to obtain the distribution of components on different kinds of\nbonds and the amounts of individual associates.\n", "metadata": {}}, {"id": "82f8e314", "cell_type": "markdown", "source": "### Solutions with short-range ordering\n\nThe order in a system can be measured by correlation functions which\ndescribes how various components are correlated in space. For\nsimplicity, let us consider only the pairs between nearest neighbors\nwith the correlation function represented by the pair probability of\nnearest neighbor bonds between two components. In a random solution, the\nprobability to find nearest neighbor bonds between two components $i$\nand $j$ is\n\nEq. \u2011 $p_{ij} = x_{i}x_{j}$\n\nWhen $p_{ij} \\neq x_{i}x_{j}$, the nearest neighbors of component $i$\nare not occupied randomly by component $j$, rather certain components\nare favored, resulting in short-range ordering or local clustering in\nthe solution. When short-range ordering develops throughout the\nsolution, long-range ordering takes place, and each component has its\nown primary sites in the solution, to be discussed in Chapter . There\nare relations between bond probabilities and mole fractions of\ncomponents due to the mass balance as follows, with the assumption of\n$p_{ij} = p_{ji}$\n\nEq. \u2011 $\\sum_{i}^{}{\\sum_{j}^{}p_{ij}} = 1$\n\nEq. \u2011 $x_{i} = \\sum_{j}^{}p_{ij}$\n\nFor small deviations from a random solution, one can consider the\nformation of $i - j$ bonds from $i - i\\ $ and $j - j$ bonds and the\nideal mixing of three types of bonds, similar to a typical Ising model.\nThe bond reaction can be written as\n\nEq. \u2011 $(i - i\\ )bonds + (j - j)\\ bonds = 2\\ (i - j)\\ bonds$\n\nwith the Gibbs energy of reaction being\n\nEq. \u2011\n${\\mathrm{\\Delta}G}_{ij} = 2G_{ij} - \\left( G_{ii} + G_{jj} \\right)\\ $\n\nGibbs energy of the solution per mole of atom is thus represented by the\nbond energies and the ideal mixing of bonds plus non-ideal interactions\nbetween pairs,\n\nEq. \u2011\n$G_{m} = \\sum_{i}^{}{\\sum_{j}^{}{p_{ij}G_{ij}}} + \\frac{Z}{2}RT\\sum_{i}^{}{\\sum_{j}^{}{p_{ij}\\ln p_{ij}}} +_{\\ }^{E}G_{m}$\n\nwith $G_{ij}$ being the molar bond energy between components $i$ and\n$j$, $Z$ the number of bonds per atom which is divided by two in the\nequation due to two atoms needed to form one bond, and\n$_{\\ }^{E}G_{m} = \\sum_{}^{}{p_{ij}p_{kl}I_{ijkl}}$ the excess Gibbs\nenergy of mixing between bonds. This approach proposed by Guggenheim\n\\[2\\] is called quasi-chemical method as it is based on the chemical\nreaction shown by .\n\nHowever, the entropy of mixing in does not reduce to ideal entropy of\nmixing for a solution without short-range ordering as defined by . An\napproximated correction may be added for small degree of short-range\nordering as follows\n\nEq. \u2011\n$G_{m} = \\sum_{i}^{}{\\sum_{j}^{}{p_{ij}G_{ij}}} + \\frac{Z}{2}RT\\sum_{i}^{}{\\sum_{j}^{}{p_{ij}\\ln\\frac{p_{ij}}{x_{i}x_{j}}}} + RT\\sum_{}^{}{x_{i}l{nx}_{i}} +_{\\ }^{E}G_{m}$\n\nFor a random solution defined by , becomes\n\nEq. \u2011\n$G_{m} = \\sum_{}^{}{x_{i}_{\\ }^{0}G_{i}} + RT\\sum_{}^{}{x_{i}l{nx}_{i}} + \\sum_{}^{}{x_{i}x_{j}{\\mathrm{\\Delta}G}_{ij}} +_{\\ }^{E}G_{m}$\n\nwith $_{\\ }^{0}G_{i} = G_{ii}$, ${\\mathrm{\\Delta}G}_{ij}$ from Eq. 2\u2011107\nrepresenting the interaction parameter between components $i$ and $j$,\nand $_{\\ }^{E}G_{m} = \\sum_{}^{}{x_{i}{x_{j}x_{k}x}_{l}I_{ijkl}}$\ndenoting the higher order interactions, in comparison with .\n\nWhen short-range ordering exists in a solution, one typically uses the\nlaw of mass reaction for the chemical reaction represented by to define\nthe equilibrium among all bonds, i.e.\n\nEq. \u2011\n$\\frac{\\left( p_{ij} \\right)^{2}}{p_{ii}p_{jj}} = e^{- \\frac{{\\mathrm{\\Delta}G}_{ij}}{kT}}$\n\nHowever, this is under the assumption that the chemical activities of\nall bonds can be represented by their respective probabilities, which is\nonly true for an ideal solution even excluding dilute solutions due to\nthe Henry\u2019s law shown by . Preferrably, the bond probabilities can be\nobtained by calculating the driving force for the fluctuation of bond\nprobabilities under constant temperature, pressure, and amount of each\ncomponent along with the constraints defined by and and equating the\ndriving force to zero, i.e.\n\nEq. \u2011\n${\\frac{1}{N}\\left( \\frac{\\partial G}{\\partial\\xi} \\right)}_{T,P,N_{n}} = \\left( \\frac{\\partial G_{m}}{\\partial p_{ij}} \\right)_{T,P,N_{n}} = \\left( \\frac{\\partial G_{m}}{\\partial p_{ij}} \\right)_{T,P,x_{n},p_{kl \\neq ij}} - \\sum_{kl \\neq ij}^{}\\left( \\frac{\\partial G_{m}}{\\partial p_{kl}} \\right)_{T,P,x_{n},p_{op \\neq kl}} + \\left( \\frac{\\partial G_{m}}{\\partial x_{i}} \\right)_{T,P,x_{q \\neq i},p_{kl}} + \\left( \\frac{\\partial G_{m}}{\\partial x_{j}} \\right)_{T,P,x_{q \\neq j},p_{kl}} = 0$\n\nwhere $\\frac{\\partial p_{kl}}{\\partial p_{ij}} = - 1$ and\n$\\frac{\\partial x_{i}}{\\partial p_{ij}} = \\frac{\\partial x_{j}}{\\partial p_{ij}} = 1$\nare used from and . Numerical values of $p_{ij}$ can be obtained by\nminimization of Gibbs energy under constraints given by and .\n\nThe chemical potential of independent component $i$ is defined as in and\ncan be represented by the following equation\n\nEq. \u2011\n$\\mu_{i} = G_{m} + \\frac{\\partial G_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial G_{m}}{\\partial x_{j}} + 2\\sum_{j = 1}^{c}\\frac{\\partial G_{m}}{\\partial p_{ij}} - \\frac{\\partial G_{m}}{\\partial p_{ii}} - 2\\sum_{j = 1}^{c}x_{j}\\sum_{k = 1}^{c}\\frac{\\partial G_{m}}{\\partial p_{jk}} + \\sum_{j = 1}^{c}x_{j}\\frac{\\partial G_{m}}{\\partial p_{jj}}$\n\nThe stability of the solution can be derived similar to .\n\nWhen the bonding between components becomes very strong, distinctive new\ncomponents may form. They are not independent components and often\ncalled associates. Both the independent and dependent components are\ncollectively called species. The formation of an associate\n$i_{a_{i}}j_{b_{j}}$ consisting of $a_{i}$ mole of $i$ and $a_{j}$ mole\n$j$ can be written as\n\nEq. \u2011 $a_{i}i + a_{j}j = i_{a_{i}}j_{a_{j}}$\n\nGibbs energy of the associate follows the same format of that of a\nstoichiometric phase as shown by ,\n\nEq. \u2011\n${_{\\ }^{0}G}_{i_{a_{i}}j_{a_{j}}} = \\sum_{}^{}a_{i}{_{\\ }^{0}G}_{i}^{SER} + \\mathrm{\\Delta}_{f}G_{i_{a_{i}}j_{a_{j}}}$\n\nGibbs energy of the solution is obtained by extending to all species\n\nEq. \u2011\n$G_{m} = \\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}_{\\ }^{0}G_{i_{a_{i}}j_{a_{j}}}} + RT\\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}\\ln y_{i_{a_{i}}j_{a_{j}}}} +_{\\ }^{E}G_{m}$\n\nwhere $y_{i_{a_{i}}j_{a_{j}}}$ is the mole fraction of species\n$i_{a_{i}}j_{a_{j}}$ in the solution with $a_{i} = 1$ and $a_{j} = 0$\nfor component $i$ and $a_{i} = 0$ and $a_{j} = 1$ for component $j$. The\nequilibrium amount of each associate $i_{a_{i}}j_{a_{j}}$ is obtained in\ncombination of mass balance and the zero driving force for the variation\nof the amount of the associate similar to , i.e.\n\nEq. \u2011 $\\sum_{i}^{}{\\sum_{j}^{}y_{i_{a_{i}}j_{a_{j}}}} = 1$\n\nEq. \u2011 $x_{i} = \\sum_{}^{}{a_{i}y_{i_{a_{i}}j_{a_{j}}}}$\n\nEq. \u2011\n$\\left( \\frac{{\\partial G}_{m}}{\\partial y_{i_{a_{i}}j_{a_{j}}}} \\right)_{x_{i}} = 0$\n\nAssociates are particularly plentiful in a gas phase, and their amounts\nare significantly affected by pressure. For an ideal gas phase with\n$_{\\ }^{E}G_{m} = 0$ and $PV_{m} = RT$, the effect of pressure is added\nas follows\n\nEq. \u2011\n$G_{m} = \\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}_{\\ }^{0}G_{i_{a_{i}}j_{a_{j}}}} + RT\\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}\\ln y_{i_{a_{i}}j_{a_{j}}}} + \\int_{P_{0}}^{P}{V_{m}dP} = \\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}_{\\ }^{0}G_{i_{a_{i}}j_{a_{j}}}} + RT\\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}\\ln y_{i_{a_{i}}j_{a_{j}}}} + RT\\ln\\frac{P}{P_{0}}$\n\nwhere $P$ is the total pressure, and $P_{0}$ the reference pressure at\nwhich $_{\\ }^{0}G_{i_{a_{i}}j_{a_{j}}}$ is defined, usually chosen to be\none atmospheric pressure. thus becomes\n\nEq. \u2011\n$G_{m} = \\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}_{\\ }^{0}{G_{i_{a_{i}}j_{a_{j}}}(P = 1atm)}} + RT\\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}\\ln y_{i_{a_{i}}j_{a_{j}}}} + RTlnP$\n\nwith the unit of the total pressure $P$ being in atmospheric pressure\n(atm). The chemical potential of species $i_{a_{i}}j_{a_{j}}$ is equal\nto\n\nEq. \u2011\n$\\mu_{i_{a_{i}}j_{a_{j}}} =_{\\ }^{0}{G_{i_{a_{i}}j_{a_{j}}}(P = 1atm)} + RTlny_{i_{a_{i}}j_{a_{j}}}P =_{\\ }^{0}{G_{i_{a_{i}}j_{a_{j}}}(P = 1atm)} + RTlnP_{i_{a_{i}}j_{a_{j}}}$\n\nwhere $P_{i_{a_{i}}j_{a_{j}}}$ is the partial pressure of species\n$i_{a_{i}}j_{a_{j}}$ defined as\n\nEq. \u2011 $P_{i_{a_{i}}j_{a_{j}}} = y_{i_{a_{i}}j_{a_{j}}}P$\n\nCombining and with , the relation between chemical potentials of an\nassociate and its constituents is expressed as\n\nEq. \u2011\n$\\mu_{i_{a_{i}}j_{a_{j}}} = a_{i}\\mu_{i} + {a_{j}\\mu}_{j} = a_{i}_{\\ }^{0}G_{i} + a_{j}_{\\ }^{0}G_{j} + RTln\\left( {P_{i}}^{a_{i}}{P_{j}}^{a_{j}} \\right)$\n\nThe equilibrium condition for the chemical reaction of an associate\nforming from its constituents in an ideal gas phase is obtained as\n\nEq. \u2011\n$\\mathrm{\\Delta}_{f}G_{i_{a_{i}}j_{a_{j}}} + RTln\\frac{{P_{i}}^{a_{i}}{P_{j}}^{a_{j}}}{P_{i_{a_{i}}j_{a_{j}}}} = 0$\n\nFor non-ideal phases, the mole fractions of various associates can be\ncalculated numerically through the minimization of Gibbs energy under\nthe constraints of and .\n", "metadata": {}}, {"id": "d1d41fbf", "cell_type": "markdown", "source": "### Ternary random solutions\n\nFrom , the Gibbs-Duhem equation of a ternary system consisting of\ncomponents $A$, $B$ and $C$ is written as\n\nEq. \u2011\n$0 = - SdT - Vd( - P) - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B} - {N_{C}d\\mu}_{C}$\n\nThis equation represents a five-dimensional surface. It can be\nvisualized in a three-dimensional space with two of the five potentials\nfixed. Usually $T$ and $P$ are kept constant as they are the natural\nvariables of $G$, and reduces to\n\nEq. \u2011 $0 = - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B} - {N_{C}d\\mu}_{C}$\n\nA phase can thus be represented by a surface in the three-dimensional\nspace of $\\mu_{A}$, $\\mu_{B}$, and $\\mu_{C}$ at constant $T$ and $P$\nwith similar geometric appearance as .\n\nFrom , Gibbs energy of a ternary solution is written as\n\nEq. \u2011\n$G_{m} = x_{A}_{\\ }^{0}G_{A} + x_{B}_{\\ }^{0}G_{B} + x_{C}_{\\ }^{0}G_{C} + RT\\left( x_{A}l{nx}_{A} + x_{B}l{nx}_{B} + x_{C}l{nx}_{C} \\right) +_{\\ }^{E}G_{m}$\n\nWhen the mole fraction of one component approaches zero,\n$_{\\ }^{E}G_{m}$ reduces to the excess Gibbs energy of mixing of the\nbinary systems of the remaining two components, represented by .\nHowever, for a given composition of a ternary solution, there is no\nunique way to assign the contributions from $_{\\ }^{E}G_{m}$ of each\nbinary to $_{\\ }^{E}G_{m}$ of the ternary solution because\n$_{\\ }^{E}G_{m}$ of the ternary solution contains information of both\nbinary and ternary interactions. A variety of models is available in the\nliterature (see \\[1\\]). One intuitive approach would be to use the same\nformula as that in the binary system, i.e. , with the mole fractions\nsubstituted by the values in the ternary system, and $_{\\ }^{E}G_{m}$ of\na ternary solution may thus be defined as the following by including the\nternary interaction involving all three components,\n\nEq. \u2011\n$_{\\ }^{E}G_{m} = x_{A}x_{B}L_{AB} + x_{A}x_{C}L_{AC} + x_{B}x_{C}L_{BC} + x_{A}x_{B}x_{C}L_{ABC}$\n\nThe chemical potential of a component is represented by . When all\ninteraction parameters in are constant, i.e. a ternary regular solution,\nthe chemical potential of component $A$ can be derived as\n\nEq. \u2011\n$\\mu_{A} = G_{A} =_{\\ }^{0}G_{A} + RTlnx_{A} + x_{B}L_{AB} + x_{C}L_{AC} -_{\\ }^{E}G_{m} =_{\\ }^{0}G_{A} + RTlnx_{A} + x_{B}{\\left( 1 - x_{A} \\right)L}_{AB} + x_{C}\\left( 1 - x_{A} \\right)L_{AC} - x_{B}x_{C}L_{BC} + x_{B}x_{C}\\left( 1 - 2x_{A} \\right)L_{ABC} =_{\\ }^{0}G_{A} + RTlnx_{B} + x_{B}^{2}L_{AB} + x_{C}^{2}L_{AC} + x_{B}x_{C}\\left( L_{AB} + L_{AC} - L_{BC} \\right) + x_{B}x_{C}\\left( 1 - 2x_{A} \\right)L_{ABC}$\n\nSimilar equations can be derived for component $B$ and C with\n$L_{AB} = L_{BA}$, $L_{AC} = L_{CA}$, and $L_{BC} = L_{CB}$. A schematic\nmolar Gibbs energy diagram at constant temperature and pressure is shown\nin with all three binary systems having $L_{ij} < 0$ of similar values.\n\nFigure \u2011: Schematic ternary molar Gibbs energy diagram as a function of\ncompositions for given temperature and pressure\n\nTo evaluate the stability of a ternary solution, one needs to calculate\nthe elements in the determinant shown in . Using the mole of component\n$C$ as the independent molar quantity, the limit of stability is\nexpresses as\n\nEq. \u2011 $G_{AA}G_{BB} - G_{AB}G_{BA} = 0$\n\nAs an example, $G_{AA}$ is shown in the following equation, which must\nbe positive for the solution to be stable\n\nEq. \u2011\n${N\\left( \\frac{\\partial\\mu_{A}}{\\partial N_{A}} \\right)}_{T,P,N_{B},N_{C}} = NG_{AA} = \\frac{RT\\left( 1 - x_{A} \\right)}{x_{A}} - 2x_{B}^{2}L_{AB} - 2x_{C}^{2}L_{AC} - 2x_{B}x_{C}\\left( L_{AB} + L_{AC} - L_{BC} \\right) - 2x_{B}x_{C}\\left( 2 - 3x_{A} \\right)L_{ABC}$\n\nIt is evident that any instability in binary systems with positive\ninteraction parameters extends into the ternary system. It can also be\nseen that even if all binary interaction parameters are negative, i.e.\nno instability in the binary systems, it is possible that becomes\nnegative for some combinations of the binary interaction parameters such\nthat $\\mathrm{\\Delta}L = L_{AB} + L_{AC} - L_{BC}$ becomes very positive\nand overshadows the contributions due to $L_{AB}$ and $L_{AC}$, i.e.\n$L_{BC}$ is more negative than $L_{AB}$ and $L_{AC}$ combined. In an\nextreme case with $L_{AB} = L_{AC} = L_{ABC} = 0$ and $L_{BC} < 0$, i.e.\nideal solutions for the $A - B$ and $A - C$ binary systems, stable\nsolution in the $B - C$ binary system, and no additional ternary\ninteraction, reduces to\n\nEq. \u2011\n$N\\left( \\frac{\\partial\\mu_{A}}{\\partial N_{A}} \\right)_{T,P,N_{B},N_{C}} = \\frac{RT\\left( 1 - x_{A} \\right)}{x_{A}} + 2x_{B}x_{C}L_{BC}$\n\nSetting\n$\\left( \\frac{\\partial\\mu_{A}}{\\partial N_{A}} \\right)_{T,P,N_{B},N_{C}} = 0$,\none obtains\n\nEq. \u2011\n$- \\frac{RT}{2L_{BC}} = \\frac{x_{A}x_{B}x_{C}}{1 - x_{A}} = \\frac{{\\left( 1 - x_{B} - x_{C} \\right)x}_{B}x_{C}}{x_{B} + x_{C}}$\n\nWith $- \\frac{RT}{2L_{BC}}$ being positive due to $L_{BC} < 0$, there is\na parabola-shaped composition area in which the solution is unstable at\nconstant temperature and pressure. This is reasonable because the system\ntends to maximize the number of B-C bonds due to its lower energy, which\ncompetes with the entropy of mixing among the three elements and results\nin segregation of B-C bonds, thus miscibility gap at low temperatures.\n\nTo evaluate the ternary consolute point, the second derivatives for\ncomponent A and B are obtained as\n\nEq. \u2011\n${N\\left( \\frac{\\partial_{\\ }^{2}\\mu_{A}}{\\partial N_{A}^{2}} \\right)}_{T,P,N_{B},N_{C}} = \\frac{RT\\left( 1 - x_{A} \\right)}{x_{A}^{2}} + 4x_{B}^{2}L_{AB} + 4x_{C}^{2}L_{AC} + 4x_{B}x_{C}\\left( L_{AB} + L_{AC} - L_{BC} \\right) + 2x_{B}x_{C}\\left( 7 - 9x_{A} \\right)L_{ABC} = 0$\n\nEq. \u2011\n${N\\left( \\frac{\\partial_{\\ }^{2}\\mu_{B}}{\\partial N_{B}^{2}} \\right)}_{T,P,N_{A},N_{C}} = \\frac{RT\\left( 1 - x_{B} \\right)}{x_{B}^{2}} + 4x_{A}^{2}L_{AB} + 4x_{C}^{2}L_{BC} + 4x_{A}x_{C}\\left( L_{AB} + L_{BC} - L_{AC} \\right) + 2x_{A}x_{C}\\left( 7 - 9x_{B} \\right)L_{ABC} = 0$\n\nThe consolute point can then be obtained using , and .\n\nIt is observed in that $\\left( 1 - 2x_{A} \\right)L_{ABC} = 0$ at\n$x_{A} = 0.5$, i.e. the ternary interaction parameter does not\ncontribute to the chemical potential of $A$. It is also observed in that\nthe contribution from the ternary interaction parameter changes sign at\n$x_{i} = 2/3$ due to $\\left( 2 - 3x_{A} \\right)L_{ABC} = 0$.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/laws_of_thermodynamics/combined_law_of_thermodynamics_and_equilibrium_conditions.ipynb b/psu410/src/laws_of_thermodynamics/combined_law_of_thermodynamics_and_equilibrium_conditions.ipynb deleted file mode 100644 index 7e0d6ae..0000000 --- a/psu410/src/laws_of_thermodynamics/combined_law_of_thermodynamics_and_equilibrium_conditions.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "e9a1e936", "cell_type": "markdown", "source": "## Combined law of thermodynamics and equilibrium conditions\n\nFor a system with an irreversible internal process taking place, the\nentropy change in the system thus consists of three parts: the heat\nexchange with the surrounding defined by , the entropy production due to\nthe internal process represented by , and the entropy of mass exchange\nwith the surrounding. The total entropy change of the system can thus be\nwritten as follows\n\nEq. \u2011 $dS = \\frac{dQ}{T} + d_{ip}S + \\sum_{}^{}{S_{i}dN_{i}}$\n\nwhere $S_{i}$ is the unit entropy of component *i* in the surroundings,\noften called partial entropy of component *i*, and will be further\ndiscussed in Chapter\n\nCombining and and re-arranging, one obtains\n\nEq. \u2011 $dQ = TdS - Dd\\xi - \\sum_{}^{}{TS_{i}dN_{i}}$\n\nInserting and into yields the combined law of thermodynamics from the\nfirst and second laws of thermodynamics,\n\nEq. \u2011\n$dU = TdS - PdV + \\sum_{}^{}\\left( H_{i} - TS_{i} \\right){dN}_{i} - Dd\\xi$\n\nThe internal energy of the system is thus a function of *S*, *V*,\n*Ni* and *\u03be* of the system, which are called natural\nvariables of the internal energy, i.e. *U*(*S*,*V*,*Ni*,*\u03be*).\nThe other variables are dependent variables and can be represented by\npartial derivatives of the internal energy with respect to their\nrespective natural variables with other natural variables kept constant\nas shown below\n\nEq. \u2011\n$T = \\left( \\frac{\\partial U}{\\partial S} \\right)_{V,\\ N_{i},\\ \\xi}$\n\nEq. \u2011\n$- P = \\left( \\frac{\\partial U}{\\partial V} \\right)_{S,\\ N_{i},\\ \\xi}$\n\nEq. \u2011\n$\\mu_{i} = H_{i} - TS_{i} = \\left( \\frac{\\partial U}{\\partial N_{i}} \\right)_{S,\\ V,N_{j \\neq i},\\ \\xi} = U_{i}$\n\nEq. \u2011\n$- D = \\left( \\frac{\\partial U}{\\partial\\xi} \\right)_{S,\\ V,N_{i}\\ }$\n\nIn , a new variable, $\\mu_{i}$, is introduced. It is called chemical\npotential and defined as the internal energy change with respect of the\naddition of the component *i* when the entropy, volume and the amount of\nother components of the system are kept constant. It may be worth\npointing out that for a system at equilibrium, i.e. $d_{ip}S = 0$, and\nwith constant entropy, $dS = 0$, if it exchanges mass with the\nsurroundings, $dN_{i} \\neq 0$, the system must also exchange heat with\nthe surroundings at the same time in order to keep the entropy invariant\nas demonstrated by .\n\nThe pairs of the natural variables and their corresponding partial\nderivatives are called conjugate variables, i.e. *S* and *T*, *V* and\n*\u2013P*, *Ni* and $\\mu_{i}$, and *\u03be* and *\u2013D*. There are minus\nsign in front of *P* and *D* as the increase of volume and the progress\nof the internal process decrease the internal energy of the system. The\nimportance of this conjugate relation will be evident when various forms\nof combined thermodynamic laws and various types of phase diagrams are\nintroduced in the book.\n\nThe last pair of conjugate variables, *\u03be* and *\u2013D*, is worthy of further\ndiscussion. Based on the second law of thermodynamics, i.e. , no\ninternal processes take place spontaneously if there is no entropy\nproductions, i.e. D\u22640 or *d\u03be*=0 and *D*\\>0. With D\u22640, there is no\ndriving for any internal processes, and the system is at a full\nequilibrium state. The last term in drops off, and *\u03be* becomes a\ndependent variable of the system and can be calculated from the\nequilibrium conditions. With *d\u03be*=0 and *D*\\>0, the system is under a\nconstrained equilibrium or freezing-in condition when the internal\nprocess is constrained not to take place, and *\u03be* remains to be an\nindependent variable of the system.\n\nThese two cases represent the two branches of thermodynamics:\nequilibrium, reversible thermodynamics and irreversible thermodynamics.\nIt is clear from the above discussions that these two branches are\nidentical if the internal energy is not only a function of *S*, *V*, and\n*Ni* , but also any internal process variable *\u03be*. This means\nthat one should be able to evaluate the internal energy of a system for\nany freezing-in equilibrium conditions in addition to the full\nequilibrium condition. In the rest of the book, the freezing-in\nequilibrium and full equilibrium are not differentiated unless\nspecified.\n\nAs the mechanical work under hydrostatic pressure is very important in\nexperiments, let us define a new quantity called enthalpy as follows\n\nEq. \u2011 $H = U + PV$\n\nIts differential form can be obtained from as\n\nEq. \u2011 $dH = dU + d(PV) = dQ + VdP + \\sum_{}^{}H_{i}{dN}_{i}$\n\nThere are two significant consequences of the above equation. First, for\na close system under constant pressure, i.e. ${dN}_{i} = dP = 0$, one\nhas $dH = dQ$. This implies that the enthalpy change in a system is\nequal to the heat exchange between the system and the surrounding of the\nsystem, which is why enthalpy and heat are often exchangeable in the\nliterature. Second, for an adiabatic system under constant pressure,\ni.e. $dQ = dP = 0$, can be re-arranged to the following equation\n\nEq. \u2011\n$H_{i} = \\left( \\frac{\\partial H}{\\partial N_{i}} \\right)_{N_{j \\neq i,\\ \\ dQ = dP = 0}}$\n\n$H_{i}$ is thus the partial enthalpy of component *i* and will be\nfurther discussed in Chapter . The chemical potential of component *i*\ndefined in is thus related to the partial enthalpy and partial entropy\nof the component.\n\nTo further define equilibrium conditions of a system, consider a\nhomogeneous system in a state of internal equilibrium, i.e. no\nspontaneous internal processes are possible with $Dd\\xi = 0$, and\nbecomes\n\nEq. \u2011\n$dU = TdS - PdV + \\sum_{}^{}\\mu_{i}{dN}_{i} = \\sum_{}^{}{Y_{i}dX_{i}}$\n\nwhere *X* represents *S*, *V*, *Ni*, and *Y* their conjugate\nvariables *T*, *-P*, $\\mu_{i}$. The state of the system with *c*\nindependent components is completely determined by the *c+2* variables,\ni.e. *S*, *V*, and *Ni* with *i* from 1 to *c*.\n\nTo simplify the situation, let us limit the discussion to an isolated\nequilibrium system, i.e. $dU = 0$, and conduct a virtual internal\nexperiment inside the system by moving an infinitesimal amount of\n$X_{i}$, ${dX}_{i}$, with other $X_{j}$ kept constant, from one region\nof the system to another region of the system as schematically shown in\n.\n\nFigure \u2011: Virtual experiment for a system at equilibrium\n\nAs the system is homogeneous and at equilibrium,\n$- dX_{i}^{'} = dX_{i}^{\"} = dX_{i}$. The total change of the internal\nenergy of this internal process is the combination of the changes in the\ntwo regions, i.e.\n\nEq. \u2011\n$dU = dU^{'} + dU^{\"} = Y_{i}^{'}dX_{i}^{'} + Y_{i}^{\"}dX_{i}^{\"} = \\left( - Y_{i}^{'} + Y_{i}^{\"} \\right)dX_{i} = 0$\n\nTherefore, $Y_{i}^{'} = Y_{i}^{\"}$ for *T*, *-P*, and $\\mu_{i}$,\nindicating that *T*, *-P*, and $\\mu_{i}$ are homogeneous in the system,\nrespectively, and are thus named as potentials of the system.\nFurthermore these potentials are independent of the size of the system\nand are often referred to as intensive variables in the literature. On\nthe other hand, all *X:s*, i.e. *S*, *V*, and *Ni*, are\nproportional to the size of the system and may be normalized with\nrespect to the size of the system, usually in terms of total moles,\n\nEq. \u2011 $N = \\sum_{}^{}N_{i}$\n\nThey are thus called molar quantities and often referred to as extensive\nvariables, and the respective normalized variables are molar entropy,\nmolar volume, and mole fractions, defined as follows\n\nEq. \u2011 $S_{m} = \\frac{S}{N}$\n\nEq. \u2011 $V_{m} = \\frac{V}{N}$\n\nEq. \u2011 $x_{i} = \\frac{N_{i}}{N}$\n\nConsider a small subsystem in this homogeneous system at equilibrium and\nlet the subsystem grow in size. The entropy, volume, and mass enclosed\nin the subsystem increase as follows\n\nEq. \u2011 ${dS = S}_{m}dN$\n\nEq. \u2011 $dV = V_{m}dN$\n\nEq. \u2011 $dN_{i} = x_{i}dN$\n\nThe corresponding change in the internal energy of the subsystem becomes\n\nEq. \u2011\n$dU = TdS - PdV + \\sum_{}^{}\\mu_{i}{dN}_{i} = \\left( TS_{m} - PV_{m} + \\sum_{}^{}\\mu_{i}x_{i} \\right)dN = U_{m}dN$\n\nBy integration one obtains the integral form of the internal energy as\n\nEq. \u2011\n$U = TS - PV + \\sum_{}^{}\\mu_{i}N_{i} = \\left( TS_{m} - PV_{m} + \\sum_{}^{}\\mu_{i}x_{i} \\right)N = U_{m}N$\n\nSimilarly, the molar enthalpy can be defined as follows\n\nEq. \u2011\n$H = U + PV = U_{m}N + PV_{m}N = \\left( U_{m} + PV_{m} \\right)N = H_{m}N$\n\nIn case a potential is not homogeneous in a system, the system will not\nbe in a state of equilibrium. Let us consider the same virtual\nexperiment as shown in for an isolated system that is not in\nequilibrium, i.e. by moving an infinitesimal amount of $X_{i}$,\n${dX}_{i}$, with other $X_{j}$ kept constant, from one region of the\nsystem to another region of the system with the two regions having\ndifferent potentials. The total internal energy change equals to zero as\nthe virtual experiment has $dU = 0$. Similarly, each region can be\nconsidered to be homogeneous by itself, and one has\n$- dX_{i}^{'} = dX_{i}^{\"} = dX_{i}$. The total internal energy change\nin the system is thus the sum of these two regions plus the entropy\nproduction due to the internal process with $d\\xi = dX_{i}$, i.e.\n\nEq. \u2011\n$dU = dU^{'} + dU^{\"} + Dd\\xi = Y_{i}^{'}dX_{i}^{'} + Y_{i}^{\"}dX_{i}^{\"} + Dd\\xi = \\left( - Y_{i}^{'} + Y_{i}^{\"} \\right)dX_{i} + Dd\\xi = 0$\n\nConsequently, one obtains the following\n\n*Eq. 1\u201131* $D = Y_{i}^{'} - Y_{i}^{\"}$\n\nThe driving force thus represents the difference of the potential at the\ntwo regions, and the internal process is to eliminate inhomogeneity of\nthe potential with the heat transfer from high temperature regions to\nlow temperature regions, volume shrink of low pressure regions (high\n$\u2013P$) and volume expansion of high pressure regions (low $\u2013P$), and\ntransport of components from high chemical potential regions to low\nchemical potential regions.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/laws_of_thermodynamics/first_and_second_laws_of_thermodynamics.ipynb b/psu410/src/laws_of_thermodynamics/first_and_second_laws_of_thermodynamics.ipynb deleted file mode 100644 index 6bc1f58..0000000 --- a/psu410/src/laws_of_thermodynamics/first_and_second_laws_of_thermodynamics.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "292875c5", "cell_type": "markdown", "source": "## First and second laws of thermodynamics\n\nA system typically consists of many chemical components. The first law\nof thermodynamics states that the exchanges of heat, work, and\nindividual components with its surroundings must obey the law of\nconversation of energy. In the domain of materials science and\nengineering, the energy of interest is at the atomic and molecular\nlevels. The energy at the higher and lower levels such as nuclear energy\nand kinetic and potential energies of a rigid body are usually excluded\nfrom the discussion of thermodynamics of materials.\n\nLet us consider a system receiving an amount of heat, *dQ*, and an\namount of work, *dW*, and an amount of each independent component *i*,\n*dNi*, from the surroundings. Such a system is called an open\nsystem in contrast to a closed system when *dNi*=0 for all\ncomponents, i.e. no exchange of mass between the system and the\nsurrounding. Other types of systems commonly defined in thermodynamics\ninclude adiabatic systems without exchange of heat, i.e. *dQ*=0, and\nisolated systems without exchange of any, i.e. *dQ*= *dW*=\n*dNi*=0.\n\nThe corresponding change of energy in the system, i.e. the internal\nenergy change, *dU*, is formulated in terms of the first law of\nthermodynamics as follows,\n\nEq. \u2011 $dU = dQ + dW + \\sum_{}^{}{H_{i}dN_{i}}$\n\nwhere $H_{i}$ is the unit energy of component *i* in the surroundings,\nand the summation is for all components in the system, which can be\ncontrolled independently from the surroundings, i.e. the independent\ncomponents of the system.\n\nIt is self-evident that the left-hand side of refers to the change\ninside the system, while its right-hand side is for the contributions\nfrom the surroundings to the system. In principle, no matter how the\nheat and mass are added, or how the work is done to the system, as far\nas their summation is the same, the change of the internal energy will\nbe the same based on the first law of thermodynamics, indicating that\nthe system reaches the same state for a closed system. The internal\nenergy is thus a state function in a close system as it does not depend\non how the state is reached.\n\nOn the other hand, for the purpose of easy mathematical treatment, a\nreversible process can be considered for a closed system, in which the\ninitial state of the system can be restored reversibly without any\nchanges left to the surroundings. Therefore, the heat transferred and\nthe work done to the system are identical to the heat and work lost by\nthe surroundings and vice versa. The classic example of reversible\nprocesses is the Carnot\u2019s cycle, which is shown in . It consists of four\nreversible processes for a closed system. The four reversible processes\nare compression at constant temperature *T1* (isothermal,),\ncompression without heat exchange (adiabatic) ending at *T2*,\nisothermal expansion at *T2*, and adiabatic expansion ending\nat *T1*.\n\nFigure \u2011: Schematics of the Carnot\u2019s cycle\n\nThe Carnot\u2019s cycle involves a simple type of mechanical work, either\nhydrostatic expansion or compression, with the work that the surrounding\ndoes to the system represented by\n\nEq. \u2011 $dW = - PdV$\n\nwith *P* being the external pressure that the surrounding exerts on the\nsystem and *V* the volume of the system. It is now necessary to\ndifferentiate the external and internal variables for further discussion\nwith the former representing variables in the surroundings and the\nlatter representing variables in the system. For the isothermal\nprocesses in the Carnot\u2019s cycle, the entropy change of the system, *dS*,\ncan be defined as the heat exchange divided by temperature\n\nEq. \u2011 $dS = \\frac{dQ}{T}$\n\nIn addition to processes involving heat, work, and mass exchanges\nbetween the system and the surroundings, there could be internal\nprocesses taking place inside the system. As the system cannot do work\nto itself, the criterion for whether an internal process can occur\nspontaneously must be related to the heat exchange, which is related to\nthe entropy change as shown by .\n\nIt is a known fact that heat can spontaneously transfer from a higher\ntemperature (*T2*) region to a lower temperature\n(*T1*) region inside a system if the heat conduction is\nallowed, and this process is irreversible because heat cannot be\nconducted from a low temperature region to a high temperature region by\nitself. indicates that for the same amount of heat change, the entropy\nchange at *T1* is higher than that at *T2*, and\nthe heat conduction thus results in a positive entropy change in the\nsystem, i.e.\n\nEq. \u2011\n$\\mathrm{\\Delta}S = - \\frac{dQ}{T_{2}} + \\frac{dQ}{T_{1}} = \\frac{dQ}{{T_{2}T}_{1}}\\left( {T_{2} - T}_{1} \\right) > 0$\n\nConsequently, the second law of thermodynamics is obtained, which states\nthat for an internal process to take place spontaneously or\nirreversibly, this internal process (*ip*) must have a positive entropy\nproduction, which can be written in a differential form as follows\n\nEq. \u2011 $d_{ip}S > 0$\n\nFrom the definition of entropy change shown by , the amount of heat\nproduced by this irreversible internal process can be calculated as\nfollows\n\nEq. \u2011 $d_{ip}Q = Td_{ip}S$\n\nLet us represent this internal process by *d\u03be* and define the driving\nforce for this internal process by *D*. The work done by this internal\nprocess is thus *Dd\u03be*, which is released as heat, i.e.\n\nEq. \u2011 $Dd\\xi = d_{ip}Q = Td_{ip}S$\n\nAn irreversible process thus must have a positive driving force in order\nfor it to take place spontaneously.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/laws_of_thermodynamics/gibbsduhem_equation.ipynb b/psu410/src/laws_of_thermodynamics/gibbsduhem_equation.ipynb deleted file mode 100644 index 16be524..0000000 --- a/psu410/src/laws_of_thermodynamics/gibbsduhem_equation.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "37f679d4", "cell_type": "markdown", "source": "## Gibbs-Duhem equation\n\nIn experiments, it is difficult to control *S* and *V* of a system in\ncomparison with their conjugate variables *T* and *-P*. It is thus\ndesirable to construct new functions to represent the system with *T*\nand *-P* as natural variables of the functions. One of them is enthalpy\ndefined in , and other two can be defined as follows\n\nEq. \u2011 $F = U - TS$\n\nEq. \u2011 $G = U - TS + PV = \\sum_{}^{}\\mu_{i}N_{i} = H - TS = F + PV$\n\nwith *F* and *G* called Helmholtz energy and Gibbs energy, respectively.\nThe middle part of is obtained using $U$ from . The corresponding\ncombined law of thermodynamics in terms of *H*, *F*, and *G* can be\nobtained through the Legendre transformation of as\n\nEq. \u2011 $dH = TdS - Vd( - P) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$\n\nEq. \u2011 $dF = - SdT - PdV + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$\n\nEq. \u2011 $dG = - SdT - Vd( - P) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$\n\nThe independent variables in each of the above form are regarded as the\nnatural variables to the corresponding function. The integral forms of\nall the functions can thus be written as the following with their\nnatural variables listed in the parenthesis\n\nEq. \u2011 $U = U\\left( S,V,N_{i},\\xi \\right)$\n\nEq. \u2011 $H = H\\left( S, - P,N_{i},\\xi \\right)$\n\nEq. \u2011 $F = F\\left( T,V,N_{i},\\xi \\right)$\n\nEq. \u2011 $G = G\\left( T, - P,N_{i},\\xi \\right)$\n\nBy differentiating , one obtains\n\nEq. \u2011\n$dG = \\sum_{}^{}\\mu_{i}{dN}_{i} + \\sum_{}^{}{N_{i}d\\mu}_{i} = - SdT - Vd( - P) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$\n\nFor a system at equilibrium, $Dd\\xi = 0$, re-arranging gives the\nGibbs-Duhem equation\n\nEq. \u2011 $0 = - SdT - Vd( - P) - \\sum_{}^{}{N_{i}d\\mu}_{i}$\n\nThis equation indicates that for a homogeneous system with *c*\nindependent components at equilibrium, there is a direct relation among\nall the *c+2* potentials, and they are $c$ chemical potentials\n($\\mu_{i}$), temperature, and pressure. Consequently, only *c+1*\npotentials can change independently, and the remaining potential is\ndependent on the other potentials. As discussed in connection with ,\nthere are $c + 2$ independent variables for an equilibrium system with\n*c* independent components, where all of them are molar quantities.\n\nWith the relationships between potentials and molar quantities defined\nby to , one can switch between potentials and molar quantities as\nnatural variables of the system. For example, one can define a new free\nenergy function when the chemical potential of one component is\ncontrolled from the surroundings instead of its content and obtain the\nfollowing combined first and second law of thermodynamics\n\nEq. \u2011 $\\Phi = G - \\mu_{1}N_{1} = \\sum_{i = 2}^{c}{\\mu_{i}N_{i}}$\n\nEq. \u2011\n$d\\Phi = - SdT - Vd( - P) - N_{1}{d\\mu}_{1} + \\sum_{i = 2}^{c}{\\mu_{i}{dN}_{i}} - Dd\\xi$\n\nHowever, even though the $c + 2$ molar quantities are independent of\neach other, indicates that not all the $c + 2$ potentials are\nindependent, i.e., if chemical potentials of all components are changed\nto natural variables, one would obtain . Therefore, among the *c+2*\nindependent variables used to define the system, the maximum number of\nindependent potential is *c+1*, and at least one of the *c+2*\nindependent variables must be a molar quantity. This variable is usually\nchosen to be the size of the system or the major element in the system.\nThe Gibbs-Duhem equation is used to derive Gibbs phase rule in\nheterogeneous systems, which is discussed in Chapter of the book.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/laws_of_thermodynamics/index.ipynb b/psu410/src/laws_of_thermodynamics/index.ipynb deleted file mode 100644 index 3900ce3..0000000 --- a/psu410/src/laws_of_thermodynamics/index.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "df14ed3f", "cell_type": "markdown", "source": "# Laws of thermodynamics\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/laws_of_thermodynamics/stability_at_equilibrium_and_property_anomaly.ipynb b/psu410/src/laws_of_thermodynamics/stability_at_equilibrium_and_property_anomaly.ipynb deleted file mode 100644 index 76c6fde..0000000 --- a/psu410/src/laws_of_thermodynamics/stability_at_equilibrium_and_property_anomaly.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "0e50c529", "cell_type": "markdown", "source": "## Stability at equilibrium and property anomaly\n\nAs shown by , potentials are homogenous for a homogeneous system in a\nstate of internal equilibrium. To study the stability of the equilibrium\nstate, one considers the entropy production due to a fluctuation of a\nmolar quantity as an internal process. Based on the second law of\nthermodynamics, the driving force, as the first derivative of the\nentropy production with respect to the internal process, is zero for\nsuch a fluctuation at equilibrium, i.e. *D*=0, and the entropy of\nproduction thus depends on the second derivative. It can be written as\nfollows\n\nEq. \u2011\n$Td_{ip}S = \\frac{\\partial_{ip}S}{\\partial\\xi}d\\xi + \\frac{1}{2}{\\frac{\\partial_{ip}^{2}S}{\\partial\\xi^{2}}(d\\xi)}^{2} = Dd\\xi - \\frac{1}{2}D_{2}(d\\xi)^{2}$\n\nwith $D_{2} = - \\frac{\\partial_{ip}^{2}S}{\\partial\\xi^{2}}$. When\n$\\frac{\\partial_{ip}^{2}S}{\\partial\\xi^{2}} < 0$ or $D_{2} > 0$ along\nwith $D = 0$, the fluctuation does not produce positive entropy of\nproduction and can thus not develop further. The equilibrium state of\nthe system is therefore stable against the fluctuation. On the other\nhand, when $\\frac{\\partial_{ip}^{2}S}{\\partial\\xi^{2}} > 0$ or\n$D_{2} < 0$ along with $D = 0$, the fluctuation creates positive entropy\nof production and can continue to grow. The equilibrium state of the\nsystem is therefore unstable against the fluctuation. In connection with\n, one can realize that for a system at stable equilibrium without heat\nand mass exchanges with the surroundings, its entropy is at its maximum,\nand there are no other internal processes, which could produce any more\nentropy. This is schematically shown in .\n\nFigure \u2011: Schematic diagram showing maximum entropy\n\nUsing , , and , the combined law of thermodynamics can be written as\n\n*Eq. 1\u201133*\n$dU = \\sum_{}^{}{Y_{i}dX_{i}} - Dd\\xi + \\frac{1}{2}D_{2}(d\\xi)^{2}$\n\nLet us carry out the same virtual internal experiment shown in Chapter ,\ni.e. moving an infinitesimal amount of *Xi* in a homogenous\nsystem with other $X_{j}$ kept constant in an isolated system, i.e.\n$dU = 0$ and $D = 0$. The internal energy change due to this internal\nprocess is\n\nEq. \u2011\n$dU = \\frac{1}{2}D_{2}\\left\\{ \\left( dX_{i}^{'} \\right)^{2} + \\left( dX_{i}^{\"} \\right)^{2} \\right\\}$\n\nFor a homogeneous system in a state of stable equilibrium with\n$\\left( dX_{i}^{'} \\right)^{2} = \\left( dX_{i}^{\"} \\right)^{2} = \\left( dX_{i} \\right)^{2}$,\nthis internal process must result in an increase of internal energy,\n$dU > 0$, and thus gives\n\nEq. \u2011\n$D_{2} = 2\\left( \\frac{\\partial^{2}U}{\\partial\\left( X_{i} \\right)^{2}} \\right)_{X_{j}} = 2\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}} > 0$\n\nshows that for a system to be stable, any pair of conjugate variables\nmust change in the same direction when other independent molar\nquantities are kept constant. For the conjugate variables discussed so\nfar, it means that for a stable system, the addition of entropy\nincreases temperature with $\\frac{\\partial T}{\\partial S}$\\>0, the\nvolume decreases with pressure or increases with the negative of\npressure with $\\frac{\\partial( - P)}{\\partial V} > 0$, and the chemical\npotential of a component increases with its amount, i.e.\n$\\frac{\\partial\\mu_{i}}{\\partial N_{i}} > 0$, where the derivatives are\ntaken with all other molar quantities kept constant. The limit of\nstability is reached when becomes zero, i.e.\n\nEq. \u2011\n$D_{2} = 2\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}} = 0$\n\nshows schematically the energy as a function of configurations including\nthree states: unstable, stable, and metastable. Both the stable and\nmetastable states have positive curvatures due to $D_{2} > 0$, while the\nunstable state has a negative curvature due to $D_{2} < 0$. There is an\ninflection point of $D_{2} = 0$ for a state between a stable or\nmetastable state with $D_{2} > 0$ and an unstable state with\n$D_{2} < 0$. These two inflection points, called spinodal, represent the\nlimit of stability. The states between the two inflection points are\nunstable, and other states are either stable or metastable. The two\ninflection points can move apart from or close to each other depending\non the change of external conditions, i.e. the natural variables. One\nextreme situation is when these two inflection points merge into one\npoint, and the instability occurs only at this particular point. It is\nevident that all three states, stable, metastable, and unstable, also\nmerge into one point. This point is called critical or consolute point,\nbeyond which the instability no longer exists.\n\nFigure \u2011: Schematic diagram showing the stable and unstable equilibrium\nstates\n\nTo mathematically define the consolute point, the third derivative needs\nto be added to because both $D$ and $D_{2}$ vanish at this point, i.e.\n\nEq. \u2011\n$Td_{ip}S = \\frac{\\partial_{ip}S}{\\partial\\xi}d\\xi + \\frac{1}{2}{\\frac{\\partial_{ip}^{2}S}{\\partial\\xi^{2}}(d\\xi)}^{2} + \\frac{1}{6}{\\frac{\\partial_{ip}^{3}S}{\\partial\\xi^{3}}(d\\xi)}^{3} = Dd\\xi - \\frac{1}{2}D_{2}(d\\xi)^{2} + \\frac{1}{6}{D_{3}(d\\xi)}^{3}$\n\nEq. \u2011\n$dU = \\sum_{}^{}{Y_{i}dX_{i}} - Dd\\xi + \\frac{1}{2}D_{2}(d\\xi)^{2} - \\frac{1}{6}{D_{3}(d\\xi)}^{3}$\n\nAt the consolute point, the third derivative also becomes zero, i.e.\n\nEq. \u2011 $D_{3} = \\frac{\\partial_{ip}^{3}S}{\\partial\\xi^{3}}^{3} = 0$\n\nLet us further discuss the properties of the system in relation to the\ncritical point. By taking the inverse of the equation of the limit of\nstability, , one obtains\n\nEq. \u2011\n$\\left( \\frac{\\partial X_{i}}{\\partial Y_{i}} \\right)_{X_{j}} = + \\ \\infty$\n\ni.e. all $X_{i}$ quantities diverge at the critical point. Therefore,\nwhen a system approaches the critical point from its stable region, the\nchange of a molar quantity with respect to its conjugate potential\nvaries dramatically and becomes infinite at the critical point,\nresulting in property anomalies in the system. In the unstable region,\nthe system would thus separate into stable subsystems and becomes\nheterogeneous, and $X_{i}$:s change discontinuously between subsystems.\nWhile in the stable region, the change of a molar quantity with respect\nto its conjugate potential decreases as the system moves away from the\ncritical point and remains positive due to the stability criteria\ndenoted by .\n\nHowever, it is not clear how a molar quantity changes with respect to a\nnon-conjugate potential at the critical point. From the Maxwell\nrelation, one has\n\nEq. \u2011\n$\\left( \\frac{\\partial Y_{i}}{\\partial X_{j}} \\right)_{X_{k \\neq j}} = \\frac{\\partial^{2}U}{\\partial X_{i}\\partial X_{j}} = \\left( \\frac{\\partial Y_{j}}{\\partial X_{i}} \\right)_{X_{k \\neq i}}$\n\nEq. \u2011\n$\\left( \\frac{\\partial X_{j}}{\\partial Y_{i}} \\right)_{X_{k \\neq j}} = \\left( \\frac{\\partial X_{i}}{\\partial Y_{j}} \\right)_{X_{k \\neq i}}$\n\nSince all $X_{i}$:s diverge at the critical point, both derivatives in\nshould also go to infinite at the critical point. To investigate their\nsigns, let us carry out a virtual experiment similar to the one in\nderiving the stability condition ( and ). In this case, two internal\nprocesses are needed for moving two molar quantities simultaneously in\nan isolated system, i.e.\n\n*Eq. 1\u201143*\n$dU = - D_{\\xi_{1}}d\\xi_{1} - D_{\\xi_{2}}d\\xi_{2} + D_{\\xi_{1}\\xi_{2}}d\\xi_{1}d\\xi_{2} + \\frac{1}{2}D_{2\\xi_{1}}\\left( d\\xi_{1} \\right)^{2} + \\frac{1}{2}D_{2\\xi_{2}}\\left( d\\xi_{2} \\right)^{2}$\n\nBased on the above discussions, in a stable system at equilibrium with\n$D_{\\xi_{1}} = D_{\\xi_{2}} = 0$, $D_{2\\xi_{1}} > 0$ and\n$D_{2\\xi_{2}} > 0$, the sign of $D_{\\xi_{1}\\xi_{2}}$ cannot be\nunambiguously determined in keeping the change of internal energy\npositive, i.e. $dU > 0$. This indicates that the quantities in can be\neither positive or negative in the stable region and become zero at the\ncritical point. By the same token, the quantities in can be either\npositive or negative and become positive or negative infinite at the\ncritical point.\n\nA profound conclusion from this analysis is that in a stable system even\nthough a molar quantity always changes in the same direction as its\nconjugate potential, the same molar quantity may change in the opposite\ndirection of a non-conjugate potential, resulting in additional\nanomalies represented by Eq. 1\u201140. One example of is the thermal\nexpansion in a closed system, i.e. $dN_{i} = 0$, as follows\n\nEq. \u2011\n$\\left( \\frac{\\partial V}{\\partial T} \\right)_{S} = \\left( \\frac{\\partial S}{\\partial( - P)} \\right)_{V}$\n\nThe left-hand side of can be understood as follows: with the increase of\ntemperature, the system regulates its pressure in order to keep the\nentropy from increasing, which results in the volume change of the\nsystem. The behavior of the system depends on whether the pressure\ndecreases or increases in order to maintain the entropy of the system\nconstant. If the pressure decreases to maintain the entropy of the\nsystem constant, the volume would increase with the increase of\ntemperature, i.e. the left-hand side of the equation has a positive\nsign, which is also shown by the right-hand side of the equation as the\nchanges of $S$ and $\u2013P$ have the same sign. That the volume increases\nwith temperature is the normal scenario. On the other hand, if the\npressure increases to maintain the entropy of the system constant, the\nvolume would decrease with the increase of temperature, resulting in a\nnegative sign for the left-hand side of the equation. This decrease of\nvolume with the increase of temperature is usually considered to be\nanomalous, originated from the increase of entropy by the decrease of\n$\u2013P$ or the increase of pressure. More discussions on entropy will\nfollow in Chapter 5.2.5 and Chapter 9.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/phase_equilibria_in_heterogeneous_systems/general_condition_of_equilibrium.ipynb b/psu410/src/phase_equilibria_in_heterogeneous_systems/general_condition_of_equilibrium.ipynb deleted file mode 100644 index 0fdb4a4..0000000 --- a/psu410/src/phase_equilibria_in_heterogeneous_systems/general_condition_of_equilibrium.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "d796a0d0", "cell_type": "markdown", "source": "## General condition of equilibrium\n\nA system is heterogeneous when some properties have different values at\ndifferent portions of the system when the system is at equilibrium. Two\nscenarios may exist where the variations of properties can be either\ncontinuous or discontinuous. In the scenario of continuous variations,\nthe gradients of those variations must be coupled so that the system\nremains at equilibrium. The number of independent variables is thus\nreduced. These gradients must also be constrained along the boundaries\nbetween the system and the surroundings. This type of constrained\nequilibrium is not discussed in the book as it involves heterogeneous\nboundary conditions between the system and the surroundings and depends\non the morphology of the system.\n\nIn the second scenario with discontinuous variations, those properties\nhave different values in different portions of the system, but remain\nhomogenous within each portion. The system is in equilibrium as each\nportion is in equilibrium with all other portions of the system. Those\nhomogeneous portions represent different phases in the system with the\nproperties in each phase being homogeneous at equilibrium. In the\nprevious chapter, it has been shown that all potentials are homogeneous\nin a homogeneous system.\n\nFor a heterogeneous system, the same conclusion can be obtained. If the\ntemperature is inhomogeneous, heat can be conducted from high\ntemperature locations to low temperature locations, and this process is\nirreversible based on the second law of thermodynamics because it\nincreases the internal entropy of the system. If the pressure is\ninhomogeneous, the amounts of lower molar volume phases will increase to\nreduce the internal energy of the system. If the chemical potential of a\ncomponent is inhomogeneous, the chemical potential difference of the\ncomponent will drive that component to the locations with a lower\nchemical potential so the internal energy of the system can be\ndecreased. Therefore, it can be concluded that all potentials are\nhomogeneous in a heterogeneous system at equilibrium, and the variables\nthat are not homogeneous are thus their conjugate molar quantities.\nUnder certain special circumstances to be discussed in later part of\nthis book, some molar quantities may also have the same values in\ndifference phases.\n\nIn a system at equilibrium with $c$ independent components, there are\n$c + 2$ pairs of conjugate variables based on or though more can be\nadded as shown by depending on experimental conditions. For simplicity,\nmost discussions in this book are limited to systems with $c + 2$ pairs\nof conjugate variables unless otherwise specified, with number \u201c2\u201d\nrepresenting conjugate variables of $T - S$ and $( - P) - V$.\n\nFor a system under constant temperature, pressure, and moles of each\nindependent component, the equilibrium condition derives from as\n\nEq. \u2011\n$dG = - SdT - Vd( - P) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi = - Dd\\xi = 0$\n\nConsequently, the equilibrium state is defined by the minimization of\nGibbs energy of the system at constant $T$, $P$ and $N_{i}$ because the\nsecond derivatives need to be positive for the equilibrium system to be\nstable as stipulated by . For heterogeneous systems with two or more\nphases, Gibbs energy of the system is the weighted summation of Gibbs\nenergies of individual phases, i.e.\n\nEq. \u2011 $\\frac{G}{N} = G_{m} = \\sum_{\\beta}^{}{f^{\\beta}G_{m}^{\\beta}}$\n\nwhere $f^{\\beta}$ and $G_{m}^{\\beta}$ are the mole fraction and molar\nGibbs energy of the phase $\\beta$, respectively, and the summation goes\nover all phases in the system. $f^{\\beta}$ is equal to zero for phases\nnot present in the equilibrium state.\n\nThe minimization of Gibbs energy of the system is carried out under the\nfollowing mass balance conditions\n\nEq. \u2011\n$x_{i} = \\sum_{\\beta}^{}{f^{\\beta}x_{i}^{\\beta}} = \\sum_{\\beta}^{}{f^{\\beta}\\frac{\\sum_{\\beta - t}^{}{a^{\\beta - t}\\sum_{k}^{}{i_{k}^{\\beta - t}y_{k}^{\\beta - t}}}}{\\sum_{\\beta - t}^{}{a^{\\beta - t}\\left( 1 - y_{va}^{\\beta - t} \\right)}}}$\n\nEq. \u2011 $\\sum_{i}^{}x_{i} = 1$\n\nEq. \u2011 $\\sum_{k}^{}y_{k}^{\\beta - t} = 1$\n\nwhere $a^{\\beta - t}$ and $y_{k}^{\\beta - t}$ are the number of site in\nsublattice $t$ in the $\\beta$ phase and the corresponding site fraction\nof species $k$ in the sublattice, respectively, and $i_{k}^{\\beta - t}$\nis the stoichiometry of the component $i$ in the species $k$, as used in\n. The summation in runs over species for each sublattice. For phases\ncontaining ionic species, electroneutrality also needs to be maintained,\ni.e. is applied to each phase. This minimization problem of Gibbs energy\nunder the constraint of mass conservation can be solved by means of a\nrange of algorithms. It should be noted that the mole fractions of\nphases and site fractions of species are bounded between 0 and 1.\n\nThis minimization procedure must result in that potentials are\nhomogeneous in the system as discussed above. Since the present book\ndeals with thermodynamics of materials, the chemical potential of each\ncomponent is of particular interest and must be homogenous in all phases\nof the system at equilibrium, i.e.\n\nEq. \u2011\n$\\mu_{i}^{\\alpha} = \\mu_{i}^{\\beta} = \\mu_{i}^{\\gamma}\\ldots\\ldots$\n\nFor phases in which the chemical potentials of individual components\ncannot be evaluated due to stoichiometry, the combined chemical\npotentials can be used to relate individual potentials as shown by and .\nAs proved in Chapter shown by , the chemical potential of a component in\na solution is represented by the intercept on the Gibbs energy axis by\nthe multi-dimensional tangent surface of Gibbs energy of the solution\nplotted with respect to mole fractions of independent components. The\nGibbs energy functions of all phases in equilibrium must thus share the\nsame tangent surface. This is usually referred to as the common tangent\nconstruction for phases at equilibrium. Any phase with its Gibbs energy\ncurve above the tangent surface is not stable under given compositions\nof the system.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/phase_equilibria_in_heterogeneous_systems/gibbs_phase_rule.ipynb b/psu410/src/phase_equilibria_in_heterogeneous_systems/gibbs_phase_rule.ipynb deleted file mode 100644 index f9d21a5..0000000 --- a/psu410/src/phase_equilibria_in_heterogeneous_systems/gibbs_phase_rule.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "60a57083", "cell_type": "markdown", "source": "## Gibbs phase rule\n\nThe Gibbs-Duhem equation, i.e. , states that only $c + 1$ potentials are\nindependent in a homogeneous system with $c$ independent components and\nthe additional two variables of temperature and pressure. In a\nheterogeneous system at equilibrium, this equation can be applied to\nindividual phases as each phase is homogeneous. Noting that each\npotential has the same value in all phases at equilibrium, can be\nwritten as follows for each individual phase, $\\ \\beta$, in the system\nat equilibrium\n\nEq. \u2011\n$0 = - S^{\\beta}dT - V^{\\beta}d( - P) - \\sum_{}^{}{N_{i}^{\\beta}d\\mu}_{i}$\n\nFor a system with $p$ phases at equilibrium, there are $p$ such\nequations relating the potentials in the system. The number of\nindependent potentials thus becomes\n\nEq. \u2011 $\\upsilon = c + 2 - p$\n\nis called Gibbs phase rule. It dictates the number of potentials that\ncan change independently for a given number of phases co-existing at\nequilibrium, commonly called degree of freedom of the system at\nequilibrium. It stipulates that the maximum number of phases can\nco-exist in a system at equilibrium is obtained by setting\n$\\upsilon = 0$, called an invariant equilibrium due to the zero degree\nof freedom,\n\nEq. \u2011 $p_{\\max} = c + 2$\n\nThere are thus maximum three phases in a one-component system, four\nphases in a binary system, five phases in a ternary system, and so on,\nthat can co-exist simultaneously at equilibrium with all potentials in\nthe system at fixed values. This should not be confused with the total\nnumber of phases that could exist, but not co-exist in a system, which\nof course are not limited by Gibbs phase rule.\n\nIt should be emphasized that the degree of freedom, $\\upsilon$, is\nreferred to the number of potentials only, not to molar quantities of\nthe system because molar quantities are generally not homogeneous in a\nheterogeneous system. For example, in a system at equilibrium with\n$\\upsilon = 0$, the amount of each component can be varied, while\nkeeping the number of phases at $p_{\\max} = c + 2$. This can be done\nthrough changing the amount of each phase in the system through the mass\nbalance equation, , without altering the composition of each phase and\nthus the chemical potentials in the system. As mentioned at the\nbeginning of Chapter , the number of independent variables in a system\nat equilibrium, i.e. the sum of independent potentials and independent\nmolar quantities, is $c + 2$ with the maximum number of independent\npotentials determined by Gibbs phase rule, .\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/phase_equilibria_in_heterogeneous_systems/index.ipynb b/psu410/src/phase_equilibria_in_heterogeneous_systems/index.ipynb deleted file mode 100644 index 21c3fc8..0000000 --- a/psu410/src/phase_equilibria_in_heterogeneous_systems/index.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "b01733f8", "cell_type": "markdown", "source": "# Phase equilibria in heterogeneous systems\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/phase_equilibria_in_heterogeneous_systems/molar_phase_diagrams.ipynb b/psu410/src/phase_equilibria_in_heterogeneous_systems/molar_phase_diagrams.ipynb deleted file mode 100644 index be0f2ed..0000000 --- a/psu410/src/phase_equilibria_in_heterogeneous_systems/molar_phase_diagrams.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "1ae66619", "cell_type": "markdown", "source": "## Molar phase diagrams\n\nThe potential phase diagrams discussed in Chapter present information on\nwhich phases are in equilibrium under given values of potentials, but do\nnot have any information on the properties of phases in equilibrium. On\nthe other hand, there are direct relations between potentials and their\nconjugate molar quantities for each phase at equilibrium depicted by to\n. One may thus substitute the potentials by their conjugate molar\nquantities in the potential phase diagrams as molar quantities provide\nmore information on the properties of phases and the system. This is\nparticularly true when chemical potentials are replaced by compositions\nas the compositions of the system are often the variables controlled and\nmeasured in experiments instead of chemical potentials.\n", "metadata": {}}, {"id": "3179cb21", "cell_type": "markdown", "source": "### Phase diagrams with both potentials and molar quantities\n\nBased on Gibbs phase rule discussed in Chapters and , the\ndimensionalities of phase regions in a potential phase diagram are given\nby or . With every potential substituted by its conjugate molar\nquantity, the dimensionalities of phase regions increase by one until\nthe phase region reaches the dimensionality of the phase diagram. The\naxes of this phase diagram now consist of both potentials and molar\nquantities. The dimensionality of a phase region can thus be represented\nby the following equation based on\n\nEq. \u2011\n$\\upsilon_{m} = \\left( c - n_{s} \\right) + 2 - p + n_{m} \\leq \\left( c - n_{s} \\right) + 1$\n\nwhere $n_{m}$ is the number of molar axes. This equation is applicable\nto phase regions with more than $n_{m} + 1$ phases. For phase regions\nwith $n_{m} + 1$ phases or fewer, the dimensionalities are the same as\nthe phase diagram, i.e. $\\upsilon_{m} = \\left( c - n_{s} \\right) + 1$,\nand no longer vary with the number of molar quantity axes. When all\n$\\left( c - n_{s} \\right) + 1$ potentials are substituted by their\nconjugate molar quantities, one obtains a complete molar phase diagrams\nto be discussed in Chapter , and all phase regions have the\nsame-dimensionality of $\\left( c - n_{s} \\right) + 1$.\n\nFor the sake of graphic visualization, let us exam a two-dimensional\nphase diagram of a one-component system. Topologically, it is equivalent\nto a multi-component system with $n_{s} = c - 1$. In , three\ntwo-dimensional phase diagrams are shown for pure Fe. In principle, one\ncan use any one of them to illustrate mixed potential and molar phase\ndiagrams. For the purpose of practical usefulness, one selects the\n$T - ( - P)$ potential diagram as temperature and pressure are the two\ntypical variables controlled in experiments of one-component systems.\nThe conjugate molar quantities of $- P$ and $T$ are molar volume and\nmolar entropy, respectively. For stable phases, any pair of conjugate\nvariables changes in the same direction as illustrated by and , i.e. the\nphase stable at higher $T$ has higher molar entropy, and the phase\nstable at higher $\u2013P$, i.e. lower pressure, has higher molar volume. Let\nus first substitute $\u2013P$ by $V_{m}$ as shown in a. The dimensionality of\na sing-phase region remains unchanged because of\n$p = 1 < 2 = n_{m} + 1$. The dimensionality of two-phase regions is\nchanged from 1 to 2 due to $\\upsilon_{m} = 3 - 2 + 1 = 2$ from .\n\nFigure \u2011: T-Vm and Sm-(-P) phase diagrams of Fe\n\nAs both phases in a two-phase equilibrium have the same temperature, all\ntie-lines, depicted by dotted lines, are perpendicular to the\ntemperature axis in a. When the molar volume of the system changes from\none end of a tie-line to another end at a constant temperature, the mole\nfraction of one phase increases from 0 to 1, and the mole fraction of\nanother phase decreases from 1 to 0. The tine-lines at various\ntemperatures combine together to form a two-dimensional two-phase\nregion. The two curves at the two ends of tie-lines represent the\nboundaries between the single-phase and two-phase regions and are no\nlonger phase regions themselves. They are thus called phase boundaries.\n\nBy the same token, by changing the temperature at a constant molar\nvolume of the system, the system will locate on different tie-lines with\nthe amounts of the two phases determined by the lever rule. It is thus\nclear that the system maintains the two-phase equilibrium state with\nboth $T$ and $V_{m}$ changing independently inside the two-dimensional\ntwo-phase region. This seems in contradiction to Gibbs phase rule being\n$\\upsilon = 3 - p = 1$ from , but it is not because Gibbs phase rule\napplies strictly to potential phase diagrams only, while the $T - V_{m}$\nphase diagram has one of its axis being a molar quantity. As an\nalternative, one may consider as a modified Gibbs phase rule in\ndescribing the dimensionality of a phase region in a mixed potential and\nmolar phase diagram with $p \\geq n_{m} + 1$.\n\nThe two three-phase equilibria in pure Fe are also represented by\ntie-lines connecting all three phases. The dimensionality of three-phase\nregions is $\\upsilon_{m} = 3 - 3 + 1 = 1$ from , and the three two-phase\ntie-lines for a three-phase equilibrium thus overlap each other with\ntheir three molar volumes on the same tie-line.\n\nLet us examine the two three-phase equilibria in more detail. In the\n\u03b3/\u03b1/\u03b5 three-phase equilibrium at $T_{E} = 756.6\\ K$ and\n$\\left( - P_{E} \\right) = - 1.046 \\bullet 10^{10}\\ Pa$, the molar\nvolumes of \u03b1, \u03b3, and \u03b5 are 6.837, 6.677, 6.582 $10^{- 6}\\ m^{3}/mol$,\nrespectively. There are two two-phase regions at higher temperatures and\none two-phase region at lower temperatures. This is also shown in the\npotential phase diagram of a with two two-phase curves entering into and\none two-phase curve leaving from the three-phase equilibrium point with\ndecreasing temperature. Considering the system with the fixed molar\nvolume equal to that of the $\\gamma$ phase, i.e.\n$V_{m}^{\\ }\\left( T_{E} \\right) = 6.677 \\bullet 10^{- 6}\\ m^{3}/mol$. At\n$T > T_{E}$, the system is in the single \u03b3 phase region. With decrease\nin temperature across $T_{E}$, it enters into the $\\alpha + \\varepsilon$\ntwo-phase region. This transformation can be written as follows and is\ncalled a eutectoid reaction\n\nEq. \u2011 $\\gamma \\rightarrow \\alpha + \\varepsilon$\n\nThis type of transformations is named as eutectic reaction if the high\ntemperature phase is a liquid phase. If the system molar volume is\nbetween $V_{m}^{\\gamma}$ and $V_{m}^{\\alpha}$ at $756.6K$, with decrease\nof temperature, the system first moves from the single \u03b3 phase region to\nthe $\\alpha + \\gamma$ two-phase region when the\n$\\gamma/(\\alpha + \\gamma)$ phase boundary is crossed. When the\ntemperature reaches $T_{E}$, the eutectoid transformation takes place in\nthe remaining \u03b3 phase. The $\\alpha$ formed prior to the eutectoid\ntransformation is called proeutectoid $\\alpha$. By the same token, when\nthe system molar volume is between $V_{m}^{\\gamma}$ and\n$V_{m}^{\\varepsilon}$ at $756.6K$, proeutectoid $\\varepsilon$ would form\nfollowed by the eutectoid transformation with decrease in temperature.\n\nOn the other hand, the $L/\\delta/\\gamma$ three-phase equilibrium at\n$T_{P} = 1977.9\\ K$ and\n$\\left( - P_{P} \\right) = - 5.111 \\bullet \\ 10^{9}\\ Pa$ has different\ncharacteristics, where the subscript P will be defined shortly. There\nare one two-phase equilibrium above and two two-phase equilibria below\nthe invariant temperature, shown in the potential phase diagram of a\nwith one two-phase curve entering into and two two-phase curves leaving\nfrom the three-phase equilibrium point with decreasing temperature. The\nmolar volumes of $L$, $\\delta$, and $\\gamma$ at $T_{P}$ are 7.735,\n7.542, and 7.498 $10^{- 6}\\ m^{3}/mol$, respectively. For $T > T_{P}$,\nthe two-phase region is $L + \\gamma$. If the system molar volume is\nbetween 7.735 and 7.498 $10^{- 6}\\ m^{3}/mol$, i.e.\n$V_{m}^{L}\\left( T_{P} \\right)$ and\n$V_{m}^{\\gamma}\\left( T_{P} \\right)$, when the temperature reaches\n$T_{P}$, $L$ and $\\gamma$ are combined to form $\\delta$ with the\ntransformation written as\n\nEq. \u2011 $L + \\gamma \\rightarrow \\delta$\n\nThis type of reactions is called as peritectic reaction or peritectoid\nreaction when the liquid phase is replaced by a solid phase, denoted by\nthe subscript P. At $T < T_{P}$, one or both high temperature phases may\nno longer be present in equilibrium depending on the value of the system\nmolar volume. For\n$V_{m}^{\\ } = V_{m}^{\\delta}\\left( T_{P} \\right) = 7.542 \\bullet 10^{- 6}\\ m^{3}/mol$,\nthe peritectic reaction, , can come to completion with no $L$ and\n$\\gamma$ left upon decreasing temperature. For\n$V_{m}^{\\gamma}\\left( T_{P} \\right) = 7.498 < V_{m}^{\\ } < V_{m}^{\\delta}\\left( T_{P} \\right) = 7.542\\ \\left( 10^{- 6}\\ m^{3}/mol \\right)$,\nthe liquid phase is consumed, and the system enters the\n$\\gamma + \\delta$ two-phase region. On the other hand, for\n$V_{m}^{\\delta}\\left( T_{P} \\right) = 7.542 < V_{m}^{\\ } < V_{m}^{\\ L}\\left( T_{P} \\right) = 7.735\\ \\ \\left( 10^{- 6}\\ m^{3}/mol \\right)$,\nthe $\\gamma$ phase is consumed instead, and the system enters into the\n$L + \\delta$ two-phase equilibrium region upon cooling.\n\nLet us now replace $T$ by $S_{m}$ to obtain the $( - P) - S_{m}$ phase\ndiagram shown in b. The morphology of this phase diagram is identical to\nthe $T - V_{m}$ phase diagram just discussed with all tie-lines\nperpendicular to the pressure axis. The transformations at the two\nthree-phase equilibria with $( - P)$ decreasing or $P$ increasing are as\nfollows\n\nEq. \u2011 $\\gamma + \\alpha \\rightarrow \\varepsilon$\n\nEq. \u2011 $\\delta \\rightarrow L + \\gamma$\n\nTo visualize two-dimensional phase diagrams of binary systems, one\nusually keeps the pressure constant. One type of commonly used binary\nphase diagram is the temperature-composition ($T - x$) phase diagram. As\nan example, let us re-plot the $T - \\mu_{C}$ potential diagram shown in\ninto a $T - x_{C}$ mixed potential and molar phase diagram by replacing\nthe chemical potential of $C$ by its mole fraction. The $T - x_{C}$\nphase diagram thus obtained is shown in . In this phase diagram there\nare one peritectic reaction and two eutectic reactions as follows\n\nEq. \u2011 $L + \\delta \\rightarrow \\gamma$\n\nEq. \u2011 $L \\rightarrow \\gamma + C$\n\nEq. 3\u201161 $\\gamma \\rightarrow \\alpha + C$\n\nFigure \u2011: $T - x_{C}\\ $ phase diagram of the Fe-C binary system\n\nIn in Chapter , it was discussed the formation of miscibility gaps due\nto repulsive interactions between components. One example is shown in\nfor the Al-Zn binary system in terms of both $T - \\mu_{Zn}$ potential\nphase diagram and $\\ T - x_{Zn}\\ $ mixed potential and molar phase\ndiagram.\n\nFigure \u2011: $T - \\mu_{Zn}$ potential phase diagram (a) and\n$\\ T - x_{Zn}\\ $ mixed potential and molar phase diagram (b) of the\nAl-Zn binary system\n\nIn ,there are one eutectic reaction and one eutectoid reaction as\nfollows\n\nEq. \u2011 $L \\rightarrow fcc + hcp$\n\nEq. \u2011 $fcc\\# 1 \\rightarrow fcc\\# 2 + hcp$\n\nThe eutectoid reaction, , is also termed as monotectoid reaction because\nthe fcc phase appears on both sides of the reaction with different\ncompositions, $fcc\\# 1$ and $fcc\\# 2$, due to the miscibility gap. The\nhighest temperature of the miscibility gap is called the consolute point\nas discussed in Chapter , which can be clearly seen in the\n$T - \\mu_{Zn}$ potential phase diagram shown in . This is a critical\npoint, marking the limit of instability as shown in .\n\nWhen there is only one phase on either side of the reaction, i.e. both\nphases have the same composition, the reaction is called a congruent\nreaction. One example is shown in for the $T - x_{SiO_{2}}\\ $ mixed\npotential and molar phase diagram of the CaO-SiO2\npseudo-binary system with two congruent reactions as follows\n\nEq. 3\u201164 $L \\rightarrow CaSiO_{3}$\n\nEq. 3\u201165 $L \\rightarrow {Ca}_{2}SiO_{4}$\n\nThey are not invariant reactions based on the Gibbs phase rule. In , it\nis noted that there are one miscibility gap in the liquid phase close to\nthe $SiO_{2}$ side, four eutectic reactions with one being monotectic\ninvolving two liquid phases due to the miscibility gap, and three\nperitectic reactions.\n\nFigure \u2011: $T - x_{SiO_{2}}\\ $ mixed potential and molar phase diagram of\nthe CaO-SiO2 pseudo-binary system.\n\nLet us generalize the above discussion to phase diagrams with\n$\\left( c - n_{s} \\right) + 1$ axes. In such a phase diagram, the\nmaximum number of phases is given by as\n$p_{\\max} = \\left( c - n_{s} \\right) + 2$. The number of phases on the\neither side of an invariant reaction can vary from one phase to\n$p_{\\max} - 1 = \\left( c - n_{s} \\right) + 1$ phases with the remaining\nphases on the other side of the reaction, typically with the potential\ndecreasing from left to right. The invariant reaction with one phase on\nthe left of the reaction is named as eutectic, or eutectoid reaction\ndepending on if the phase on the left of the reaction is liquid or\nsolid. The rest invariant reactions are named as peritectic or\nperitectoid reactions with or without a liquid phase.\n", "metadata": {}}, {"id": "7fa35f9c", "cell_type": "markdown", "source": "### Phase diagrams with only molar quantities\n\nWhen all $\\left( c - n_{s} \\right) + 1$ potentials in a potential phase\ndiagrams are replaced by their conjugate molar quantities, one obtains a\nmolar phase diagram with molar quantities on all axes of the phase\ndiagram. For regions with the number of phases\n$p \\leq n_{m} + 1 = \\left( c - n_{s} \\right) + 2 = p_{\\max}$ (see ), the\nphase regions have the same-dimensionality as that of the phase diagram,\ni.e. all phase regions have the same-dimensionality of\n$\\left( c - n_{s} \\right) + 1$, and any geometric feature with\nlower-dimensionalities, i.e. from 0 to $\\ \\left( c - n_{s} \\right)$, is\nnot phase regions, but phase boundaries between neighbouring phase\nregions. For the sake of graphic visualization, the molar phase diagram\nof pure Fe is shown in obtained by combining the two mixed phase\ndiagrams in .\n\nFigure \u2011: Molar phase diagram of Fe\n\nIn this molar phase diagram, all one-, two-, and three-phase regions are\ntwo-dimensional, the same as the dimensionality of the phase diagram. A\ntwo-phase region is made up by tie-lines connecting the two phases in\nequilibrium, while a three phase-region is surrounded by three two-phase\ntie-lines, i.e. a tie-triangle. The amount of each phase in the\ntie-triangle can be obtained using the lever rule represented by and .\nAs can be seen, phase boundaries between a one-phase region and a\ntwo-phase region are one-dimensional. When the system crosses such a\nphase boundary, the number of phases changes by one from two to one or\nvice versa. Phase boundaries between two- and three-phase regions are\nrepresented by two-phase tie-lines. When the system crosses such a phase\nboundary, the number of phases also changes by one. The\nlowest-dimensional phase boundaries are the points between one- and\nthree-phase regions that are zero-dimensional and the intercept of four\none-dimensional phase boundaries. When the system crosses such a phase\nboundary, the number of phases changes by two.\n\nFor multi-component systems, the phase relations cannot be directly\nvisualized. By representing the system equations in terms of equilibrium\nconditions and level rules on a phase boundary using phases separately\nfrom the two adjacent phase regions, Palatnik and Landau \\[4\\]\npostulated that the difference between the number of unknowns and\nequations gives the dimensionality of the phase boundary and derived the\nfollowing relationships\n\nEq. \u2011 $D^{+} + D^{-} = r - b = c - n_{s} - b$\n\nwhere $D^{+}$ and $D^{-}$ are the numbers of phases added and removed\nwhen the phase boundary is crossed, and $r$ and $b$ are the\ndimensionalities of the phase diagram and the phase boundary,\nrespectively. They termed the as the contact rule, which is named as the\nMPL boundary rule by Hillert \\[1\\].\n\nBy the same token, is applicable to any phase boundary where the two\nadjacent phase regions have the same-dimensionality of the phase\ndiagram, even in phase diagrams with a mixture of potentials and molar\nquantities as the diagram axes. This can be understood because the\npotentials are homogeneous in all phases in equilibrium on the phase\nboundary. The phase boundary is thus equivalent to those in a complete\nmolar phase diagram with its number of components equal to the number of\nmolar axes in a mixed potential and molar phase diagrams minus one, i.e.\n\nEq. \u2011 $c^{'} = n_{m} - 1 = c - n_{s}$\n\nThe last part of stems from the discussion related to when all\n$c^{'} + 1$ potentials are replaced by their conjugate molar quantities,\nwhich is analogous to a molar phase diagram with $\\ c$ independent\ncomponents and $n_{s}$ potentials fixed.\n\nFor a two-dimensional phase diagram with $r = 2$, the phase boundary can\nbe either zero- or one--dimensional. As shown in , the basic element of\na molar phase diagram is a joint of four one-dimensional phase boundary\nlines. When a phase boundary line is crossed, the number of phases is\neither increased or decreased by one. The joint of four one-dimensional\nphase boundary lines is zero-dimensional. The number of phases differs\nby two between the phase regions across the zero-dimensional joint. Two\nscenarios are possible\n\nTwo phases are added or removed, i.e. $D^{+} = 2$ and $D^{-} = 0$ or\n$D^{+} = 0$ and $D^{-} = 2$, and the number of phases in the two phase\nregions differs by two;\n\nOne phase is added, one phase is removed, i.e. $D^{+} = D^{-} = 1$, and\nthe phase regions have the same number of phases.\n\nBy combining the contact rules for both zero- and one-dimensional phase\nboundaries, it is evident that the above two scenarios co-exist in a\njoint of four-phase regions with two-phase regions following the first\nscenario and other two-phase regions following the second scenario.\nBased on the Schreinemakers\u2019 rule generalized by Hillert \\[5\\], each of\nthe two-phase regions with the same number of phases contains one\nextrapolation of the phase boundaries, while the other two-phase regions\ncontain either zero or two extrapolations of the phase boundaries. This\ncan be observed for all the zero-dimensional phase boundaries in and is\nfurther schematically illustrated in for general cases.\n\nFigure \u2011: Schematic molar phase diagram, demonstrating the\nSchreinemakers\u2019 rule\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/phase_equilibria_in_heterogeneous_systems/potential_phase_diagrams.ipynb b/psu410/src/phase_equilibria_in_heterogeneous_systems/potential_phase_diagrams.ipynb deleted file mode 100644 index e2e0f84..0000000 --- a/psu410/src/phase_equilibria_in_heterogeneous_systems/potential_phase_diagrams.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "5cff8694", "cell_type": "markdown", "source": "## Potential phase diagrams\n\nGibbs phase rule can be further understood through the\n$c + 2$-dimensional space of potentials consisting of $T$, $- P$, and\n$\\mu_{i}$ with $i$ from 1 to $c$. Each phase is a $c + 1$-dimensional\nfeature in this $c + 2$-dimensional space characterized by . The\ndirections of this $c + 1$-dimensional feature are represented by their\nmolar quantities as shown by following equations,\n\nEq. \u2011\n$\\ \\left( \\frac{\\partial\\mu_{i}}{\\partial T} \\right)_{P,\\mu_{j \\neq i}} = - \\frac{S^{\\beta}}{N_{i}^{\\beta}} = - \\frac{S_{m}^{\\beta}}{x_{i}^{\\beta}}$\n\nEq. \u2011\n$\\ \\left( \\frac{\\partial\\mu_{i}}{\\partial( - P)} \\right)_{T,\\mu_{j \\neq i}} = - \\frac{V^{\\beta}}{N_{i}^{\\beta}} = - \\frac{V_{m}^{\\beta}}{x_{i}^{\\beta}}$\n\nEq. \u2011\n$\\ \\left( \\frac{\\partial( - P)}{\\partial T} \\right)_{\\mu_{i}} = - \\frac{S^{\\beta}}{V^{\\beta}} = - \\frac{S_{m}^{\\beta}}{V_{m}^{\\beta}}$\n\nAs can be seen, all the direction derivatives are negative, indicating\nthat the $c + 1$-dimensional feature is convex. The intercept of any two\n$c + 1$-dimensional features is thus a $c$-dimensional feature. On this\n$c$-dimensional feature, these two phases are in equilibrium with each\nother because each potential has the same value in both phases. This\nfeature thus represents a two-phase equilibrium. By the same token, the\nintercept of any three $c + 1$-dimensional features is a\n$c - 1$-dimensional feature in the $c + 2$-dimensional space of\npotentials and represents a three-phase equilibrium. This continues\nuntil the number of phases reaches $c + 2$ with all $c + 2$ potentials\ncompletely determined, and the dimension of their intercepts becomes\nzero.\n\nThose $c + 1$ to zero-dimensional geometrical features in the\n$c + 2$-dimensional space of potentials thus denote one-phase,\ntwo-phase, three-phase to $(c + 2)$-phase equilibria of the system with\nthe dimensionality of the feature and the number of phases in\nequilibrium related by , i.e. Gibbs phase rule. Their arrangements in\nthe $c + 2$-dimensional space of potentials thus depict the phase\nrelations in the system and are commonly called phase diagrams. Since\nall the diagram axes in the phase diagram discussed above are\npotentials, the diagram is called potential phase diagram in order to\ndifferentiate it from phase diagrams with some or all diagram axes being\ntheir conjugate molar quantities. Both potential and molar phase\ndiagrams are discussed in this chapter.\n", "metadata": {}}, {"id": "1706759a", "cell_type": "markdown", "source": "### Potential phase diagrams of one-component systems\n\nAs physical vision of human being is limited to three-dimensions, only\none-component system can be completely visualized as shown in for one\nphase where any two of the three potentials can change independently.\nFrom Gibbs phase rule, when two phases are in equilibrium, only one\npotential can vary freely if the two-phase equilibrium is to be\nmaintained. While three phases are in equilibrium, the degree of freedom\nis zero, and all three potentials are fixed.\n\nFor a two-phase equilibrium, two surfaces intersect each other as\ndepicted by the dashed line in . This two-phase equilibrium line is\nobtained by applying to both phases in a one-component system. Since one\nof the potentials is dependent on the other two, one can eliminate it by\ndividing the equation by its conjugate molar quantity and subtracting\nthe two equations, resulting in the following three equations\n\nEq. \u2011\n$0 = \\left( \\frac{S^{\\alpha}}{N_{A}^{\\alpha}} - \\frac{S^{\\beta}}{N_{A}^{\\beta}} \\right)dT + \\left( \\frac{V^{\\alpha}}{N_{A}^{\\alpha}} - \\frac{V^{\\beta}}{N_{A}^{\\beta}} \\right)d( - P) = \\mathrm{\\Delta}S_{m}^{\\alpha\\beta}dT + \\mathrm{\\Delta}V_{m}^{\\alpha\\beta}d( - P)$\n\nEq. \u2011\n$0 = \\left( \\frac{S^{\\alpha}}{V^{\\alpha}} - \\frac{S^{\\beta}}{V^{\\beta}} \\right)dT + \\left( \\frac{N_{A}^{\\alpha}}{V^{\\alpha}} - \\frac{N_{A}^{\\beta}}{V^{\\beta}} \\right){d\\mu}_{A} = \\mathrm{\\Delta}\\left( \\frac{S_{m}}{V_{m}} \\right)^{\\alpha\\beta}dT + \\mathrm{\\Delta}\\left( \\frac{1}{V_{m}} \\right)^{\\alpha\\beta}{d\\mu}_{A}$\n\nEq. \u2011\n$0 = \\left( \\frac{V^{\\alpha}}{S^{\\alpha}} - \\frac{V^{\\beta}}{S^{\\beta}} \\right)d( - P) + \\left( \\frac{N_{A}^{\\alpha}}{S^{\\alpha}} - \\frac{N_{A}^{\\beta}}{S^{\\beta}} \\right){d\\mu}_{A} = \\mathrm{\\Delta}\\left( \\frac{V_{m}}{S_{m}} \\right)^{\\alpha\\beta}d( - P) + \\mathrm{\\Delta}\\left( \\frac{1}{S_{m}} \\right)^{\\alpha\\beta}{d\\mu}_{A}$\n\nFigure \u2011: Gibbs energy surfaces of two phases and their intersection,\nrepresenting the two-phase equilibrium.\n\nThe directions of the two-phase equilibrium line can thus be obtained as\n\nEq. \u2011\n$\\frac{d( - P)}{dT} = - \\frac{\\mathrm{\\Delta}S_{m}^{\\alpha\\beta}}{\\mathrm{\\Delta}V_{m}^{\\alpha\\beta}}$\n\nEq. \u2011\n$\\frac{dT}{{d\\mu}_{A}} = - \\frac{\\mathrm{\\Delta}\\left( \\frac{1}{V_{m}} \\right)^{\\alpha\\beta}}{\\mathrm{\\Delta}\\left( \\frac{S_{m}}{V_{m}} \\right)^{\\alpha\\beta}}$\n\nEq. \u2011\n$\\frac{d( - P)}{{d\\mu}_{A}} = - \\frac{\\mathrm{\\Delta}\\left( \\frac{1}{S_{m}} \\right)^{\\alpha\\beta}}{\\mathrm{\\Delta}\\left( \\frac{V_{m}}{S_{m}} \\right)^{\\alpha\\beta}}$\n\nThese three equations define the mathematical forms of the two-phase\nequilibrium line in the two-dimensional spaces of $T - ( - P)$,\n$\\mu_{A} - T$ and $\\mu_{A} - ( - P)$, respectively, and can thus be\nplotted as two-dimensional diagrams. is commonly called\nClausius-Clapeyron equation in the literature. One may thus call all\nthree equations above as generalized Clausius-Clapeyron equations. At\nequilibrium, the chemical potentials of the component in both phases are\nequal to each other, so are their Gibbs energies. One thus has\n\nEq. \u2011\n$G_{m}^{\\alpha} - G_{m}^{\\beta} = 0 = \\mathrm{\\Delta}G_{m} = \\mathrm{\\Delta}H_{m}^{\\alpha\\beta} - T\\mathrm{\\Delta}S_{m}^{\\alpha\\beta}$\n\nThe Clausius-Clapeyron equation, , can be re-written as\n\nEq. \u2011\n$\\frac{d( - P)}{dT} = - \\frac{\\mathrm{\\Delta}H_{m}^{\\alpha\\beta}}{T\\mathrm{\\Delta}V_{m}^{\\alpha\\beta}}$\n\nAs an example, three potential phase diagrams of pure Fe are shown in .\nThere are four phases in the system, bcc, fcc, hcp, and liquid. In the\nliterature, the high temperature and low temperature bcc phases are\nusually denoted by \u03b4 (high temperature) and \u03b1 (low temperature), the fcc\nand hcp phases by \u03b3 and \u03b5, and the liquid phase by L, respectively. In\nthese figures, the two-dimensional areas are single-phase regions where\ntwo potentials can change independently with the system remaining as\nsingle-phase. The lines denote two-phase equilibrium regions where only\none potential can vary independently if the two-phase equilibrium is to\nbe maintained. The points where three two-phase equilibrium lines meet\nrepresent the invariant three-phase equilibria with three potentials\nfixed.\n\nFigure \u2011 $T - ( - P)$, $\\mu_{A} - T$ and $\\mu_{A} - ( - P)$ phase\ndiagrams of pure Fe\n\nBased on the discussions in Chapter , enthalpy and entropy of a phase\nincrease monotonically with temperature, and phases stable at higher\ntemperatures have higher enthalpy and entropy than phases stable at\nlower temperatures. Consequently, the two-phase equilibrium lines in a\n$T - ( - P)$ potential phase diagram have negative slopes if the phase\nstable at higher temperatures also has larger molar volume than the\nphase stable at lower temperatures (note that if $P$ is plotted instead\nof $- P$, the slope is positive). This is the case for the two-phase\nequilibrium lines of \u03b4/L, \u03b3/L, and \u03b3/\u03b4 at high temperatures, and \u03b5/\u03b3\nshown in a. On the other hand, the two-phase equilibrium lines of \u03b1/\u03b5\nand \u03b1/\u03b3 at low temperatures have positive slopes, indicating that \u03b5 and\n\u03b3 have smaller molar volume than \u03b1 as \u03b5 and \u03b3 are more stable at higher\npressures than \u03b1 at constant temperatures. It is thus evident that the\nphase stable at higher pressure can have either higher or lower entropy\nthan the phase stable at lower pressure, and the phase stable at higher\ntemperature can have either higher or lower volume than the phase stable\nat lower temperature. This is the property anomaly discussed in Chapter\n.\n\nAnother useful example of potential phase diagram is the\npressure-temperature phase diagram of $H_{2}O$ shown in with three\nphases: ice, water, and vapour. It is known that the solid ice has many\npolymorphic structures at high pressures, which are not included in this\ndiagram. As in the pure Fe potential phase diagram discussed above, the\nsingle-phase regions of ice, water, and vapour are represented by the\ntwo-dimensional areas with two-degree of freedom based on the Gibbs\nphase rule, the lines are for the two-phase regions of ice-water,\nice-vapour, and water-vapour, and the three-phase equilibrium has\nzero-degree of freedom represented by a point at $273.16\\ K$ and\n$611.73\\ Pa$.\n\nThere are two features in which are different from those of Fe shown in\na. The first feature is that the slope of the liquid-solid two-phase\nequilibrium line in has the opposite sign of that in a. This is because\nsolid ice has larger molar volume than liquid water, while the molar\nvolume of liquid Fe is larger than those of fcc-Fe and bcc-Fe. The\nsecond feature is that the two-phase equilibrium line of water-vapour\nends at $647\\ K$ and $22.064 \\cdot 10^{6}\\ Pa$. Beyond this point, the\ndifference between vapour and water disappears when the pressure and\ntemperature are changed, i.e. it behaves like one phase. This point is a\ncritical point, as discussed in Chapter . However, it should be pointed\nout that it does not represent an invariant reaction as the degree of\nfreedom based on the Gibbs phase rule is equal to one and not zero. On\nthe other hand, both the temperature and pressure of the critical point\nare invariant due to the two constraints introduced by the limit of\nstability of a single phase, i.e. the second and third derivatives of\ntemperature to entropy or pressure to volume are zero.\n\nFigure \u2011: P-T phase diagram of $H_{2}O$\n", "metadata": {}}, {"id": "8b8697d3", "cell_type": "markdown", "source": "### Potential phase diagrams of two-component systems\n\nFrom Gibbs-Duhem equation (see ), a single phase in a two-component\nsystem has three independent potentials, out of the four potentials of\n$T$, $- P$, $\\mu_{A}$, and $\\mu_{B}$, and is a three-dimensional\ngeometric feature in a four-dimensional space. Alternatively, it can be\nrepresented by a three-dimensional space constructed by the three\nindependent potentials. A two-phase equilibrium is thus a\ntwo-dimensional surface in this three-dimensional space created by the\nintercept of two three-dimensional spaces, and a three-phase equilibrium\nis a one-dimensional line, and a four-phase equilibrium is a\nzero-dimensional point. This is shown in for the Fe-C binary system\ninvolving four phases: fcc, bcc, Fe3C and graphite. Since any\none of the four potentials can be chosen as the dependent one, four\nthree-dimensional potential phase diagrams are depicted in .\n\nFigure \u2011: Projected potential phase diagram of Fe-C system with fcc,\nbcc, Fe3C, and graphite\n\nThe two-phase equilibrium surfaces are obtained by choosing any one of\nthe four potentials as the dependent one and solving the Gibbs-Duhem\nequations for both phases, resulting in following four equations\n\nEq. \u2011\n$0 = \\left( \\frac{S^{\\alpha}}{N_{A}^{\\alpha}} - \\frac{S^{\\beta}}{N_{A}^{\\beta}} \\right)dT + \\left( \\frac{V^{\\alpha}}{N_{A}^{\\alpha}} - \\frac{V^{\\beta}}{N_{A}^{\\beta}} \\right)d( - P) + \\left( \\frac{N_{B}^{\\alpha}}{N_{A}^{\\alpha}} - \\frac{N_{B}^{\\beta}}{N_{A}^{\\beta}} \\right){d\\mu}_{B} = \\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}dT + \\mathrm{\\Delta}V_{mA}^{\\alpha\\beta}d( - P) + \\mathrm{\\Delta}z_{B}^{\\alpha\\beta}{d\\mu}_{B}$\n\nEq. \u2011\n$0 = \\left( \\frac{S^{\\alpha}}{N_{B}^{\\alpha}} - \\frac{S^{\\beta}}{N_{B}^{\\beta}} \\right)dT + \\left( \\frac{V^{\\alpha}}{N_{B}^{\\alpha}} - \\frac{V^{\\beta}}{N_{B}^{\\beta}} \\right)d( - P) + \\left( \\frac{N_{B}^{\\alpha}}{N_{B}^{\\alpha}} - \\frac{N_{B}^{\\beta}}{N_{B}^{\\beta}} \\right){d\\mu}_{B} = \\mathrm{\\Delta}S_{mB}^{\\alpha\\beta}dT + \\mathrm{\\Delta}V_{mB}^{\\alpha\\beta}d( - P) + \\mathrm{\\Delta}z_{A}^{\\alpha\\beta}{d\\mu}_{A}$\n\nEq. \u2011\n$0 = \\left( \\frac{S^{\\alpha}}{V^{\\alpha}} - \\frac{S^{\\beta}}{V^{\\beta}} \\right)dT + \\left( \\frac{N_{A}^{\\alpha}}{V^{\\alpha}} - \\frac{N_{A}^{\\beta}}{V^{\\beta}} \\right){d\\mu}_{A} + \\left( \\frac{N_{B}^{\\alpha}}{V^{\\alpha}} - \\frac{N_{B}^{\\beta}}{V^{\\beta}} \\right){d\\mu}_{B} = \\mathrm{\\Delta}\\left( \\frac{S_{m}}{V_{m}} \\right)^{\\alpha\\beta}dT + \\mathrm{\\Delta}\\left( \\frac{1}{V_{mA}} \\right)^{\\alpha\\beta}{d\\mu}_{A} + \\mathrm{\\Delta}\\left( \\frac{1}{V_{mB}} \\right)^{\\alpha\\beta}{d\\mu}_{B}$\n\nEq. \u2011\n$0 = \\left( \\frac{V^{\\alpha}}{S^{\\alpha}} - \\frac{V^{\\beta}}{S^{\\beta}} \\right)d( - P) + \\left( \\frac{N_{A}^{\\alpha}}{S^{\\alpha}} - \\frac{N_{A}^{\\beta}}{S^{\\beta}} \\right){d\\mu}_{A} + \\left( \\frac{N_{B}^{\\alpha}}{S^{\\alpha}} - \\frac{N_{B}^{\\beta}}{S^{\\beta}} \\right){d\\mu}_{B} = \\mathrm{\\Delta}\\left( \\frac{V_{m}}{S_{m}} \\right)^{\\alpha\\beta}d( - P) + \\mathrm{\\Delta}\\left( \\frac{1}{S_{mA}} \\right)^{\\alpha\\beta}{d\\mu}_{A} + \\mathrm{\\Delta}\\left( \\frac{1}{S_{mB}} \\right)^{\\alpha\\beta}{d\\mu}_{B}$\n\nA three-phase equilibrium line is represented by the intercept of two\ntwo-phase surfaces by applying any one of the above four equations to\ntwo two-phase equilibria. Let us use as an example\n\nEq. \u2011\n$0 = \\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}dT + \\mathrm{\\Delta}V_{mA}^{\\alpha\\beta}d( - P) + {\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}{d\\mu}_{B}$\n\nEq. \u2011\n$0 = \\mathrm{\\Delta}S_{mA}^{\\alpha\\gamma}dT + \\mathrm{\\Delta}V_{mA}^{\\alpha\\gamma}d( - P) + \\mathrm{\\Delta}{z_{B}^{\\alpha\\gamma}d\\mu}_{B}$\n\nIt is self-evident that the two-phase equilibrium surface between\n$\\beta$ and $\\gamma$ is not independent and can be obtained by\nsubtraction of and\n\nEq. \u2011\n$0 = \\left( \\mathrm{\\Delta}S_{mA}^{\\alpha\\beta} - \\mathrm{\\Delta}S_{mA}^{\\alpha\\gamma} \\right)dT + \\left( \\mathrm{\\Delta}V_{mA}^{\\alpha\\beta} - \\mathrm{\\Delta}V_{mA}^{\\alpha\\gamma} \\right)d( - P) + \\left( {\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta} - \\mathrm{\\Delta}z_{B}^{\\alpha\\gamma} \\right){d\\mu}_{B} = \\mathrm{\\Delta}S_{mA}^{\\gamma\\beta}dT + \\mathrm{\\Delta}V_{mA}^{\\gamma\\beta}d( - P) + {\\mathrm{\\Delta}z}_{B}^{\\gamma\\beta}{d\\mu}_{B}$\n\nEliminating one of three potentials in and , one can obtain three\nequations for the three-phase equilibrium line\n\nEq. \u2011\n$\\frac{d( - P)}{dT} = - \\frac{\\frac{\\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}}{{\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}S_{mA}^{\\alpha\\gamma}}{\\mathrm{\\Delta}z_{B}^{\\alpha\\gamma}}}{\\frac{\\mathrm{\\Delta}V_{mA}^{\\alpha\\beta}}{{\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}V_{mA}^{\\alpha\\gamma}}{\\mathrm{\\Delta}z_{B}^{\\alpha\\gamma}}}$\n\nEq. \u2011\n$\\frac{dT}{{d\\mu}_{B}} = - \\frac{\\frac{{\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}}{\\mathrm{\\Delta}V_{mA}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}z_{B}^{\\alpha\\gamma}}{\\mathrm{\\Delta}V_{mA}^{\\alpha\\gamma}}}{\\frac{\\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}}{\\mathrm{\\Delta}V_{mA}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}S_{mA}^{\\alpha\\gamma}}{\\mathrm{\\Delta}V_{mA}^{\\alpha\\gamma}}}$\n\nEq. \u2011\n$\\frac{d( - P)}{{d\\mu}_{B}} = - \\frac{\\frac{{\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}}{\\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}z_{B}^{\\alpha\\gamma}}{\\mathrm{\\Delta}S_{mA}^{\\alpha\\gamma}}}{\\frac{\\mathrm{\\Delta}V_{mA}^{\\alpha\\beta}}{\\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}V_{mA}^{\\alpha\\gamma}}{\\mathrm{\\Delta}S_{mA}^{\\alpha\\gamma}}}$\n\nto can be referred as generalized Clausius-Clapeyron equations for\nbinary systems. Similar equations can be derived for\n$T - ( - P) - \\mu_{A}$, $T - \\mu_{A} - \\mu_{B}$, and\n$( - P) - \\mu_{A} - \\mu_{B}$ potential phase diagrams from to , and are\nlisted below\n\n- Generalized Clausius-Clapeyron equations for a three-phase equilibrium\n in $T - ( - P) - \\mu_{A}$ potential phase diagrams\n\nEq. \u2011\n$\\frac{d( - P)}{dT} = - \\frac{\\frac{\\mathrm{\\Delta}S_{mB}^{\\alpha\\beta}}{{\\mathrm{\\Delta}z}_{A}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}S_{mB}^{\\alpha\\gamma}}{\\mathrm{\\Delta}z_{A}^{\\alpha\\gamma}}}{\\frac{\\mathrm{\\Delta}V_{mB}^{\\alpha\\beta}}{{\\mathrm{\\Delta}z}_{A}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}V_{mB}^{\\alpha\\gamma}}{\\mathrm{\\Delta}z_{A}^{\\alpha\\gamma}}}$\n\nEq. \u2011\n$\\frac{dT}{{d\\mu}_{A}} = - \\frac{\\frac{{\\mathrm{\\Delta}z}_{A}^{\\alpha\\beta}}{\\mathrm{\\Delta}V_{mB}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}z_{A}^{\\alpha\\gamma}}{\\mathrm{\\Delta}V_{mB}^{\\alpha\\gamma}}}{\\frac{\\mathrm{\\Delta}S_{mB}^{\\alpha\\beta}}{\\mathrm{\\Delta}V_{mB}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}S_{mB}^{\\alpha\\gamma}}{\\mathrm{\\Delta}V_{mB}^{\\alpha\\gamma}}}$\n\nEq. \u2011\n$\\frac{d( - P)}{{d\\mu}_{A}} = - \\frac{\\frac{{\\mathrm{\\Delta}z}_{A}^{\\alpha\\beta}}{\\mathrm{\\Delta}S_{mB}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}z_{A}^{\\alpha\\gamma}}{\\mathrm{\\Delta}S_{mB}^{\\alpha\\gamma}}}{\\frac{\\mathrm{\\Delta}V_{mB}^{\\alpha\\beta}}{\\mathrm{\\Delta}S_{mB}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}V_{mB}^{\\alpha\\gamma}}{\\mathrm{\\Delta}S_{mB}^{\\alpha\\gamma}}}$\n\n- Generalized Clausius-Clapeyron equations for a three-phase equilibrium\n in $\\ T - \\mu_{A} - \\mu_{B}$ potential phase diagrams\n\nEq. \u2011\n$\\frac{dT}{{d\\mu}_{A}} = - \\frac{\\frac{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}}{\\frac{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}}$\n\nEq. \u2011\n$\\frac{dT}{{d\\mu}_{B}} = - \\frac{\\frac{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}}{\\frac{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}}$\n\nEq. \u2011\n$\\frac{{d\\mu}_{A}}{{d\\mu}_{B}} = - \\frac{\\frac{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\gamma}}}{\\frac{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\gamma}}}$\n\n- Generalized Clausius-Clapeyron equations for a three-phase equilibrium\n in $( - P) - \\mu_{A} - \\mu_{B}$ potential phase diagrams\n\nEq. \u2011\n$\\frac{d( - P)}{{d\\mu}_{A}} = - \\frac{\\frac{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}}{\\frac{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}}$\n\nEq. \u2011\n$\\frac{d( - P)}{{d\\mu}_{B}} = - \\frac{\\frac{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}}{\\frac{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}}$\n\nEq. \u2011\n$\\frac{{d\\mu}_{A}}{{d\\mu}_{B}} = - \\frac{\\frac{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\gamma}}}{\\frac{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\gamma}}}$\n", "metadata": {}}, {"id": "9e783eec", "cell_type": "markdown", "source": "### Projection and section of phase diagrams with potential and molar quantities\n\nAs discussed in Chapter , projections of high-dimensional phase diagrams\nusually cannot keep all information. However, there is one type of\nwidely used projection in the literature, i.e. the liquidus surface in\nternary systems under constant pressure with temperature and mole\nfractions of two components as its axes. The projection along the\ntemperature axis reveals the composition regions for primary phases that\nsolidify from liquid upon cooling. These regions are separated by\nunivariant lines of three-phase equilibria. The projections along one of\nthe two mole fractions show the temperature as a function of composition\non the univariant three-phase equilibrium lines and also depict whether\na four-phase equilibrium is peritectic or eutectic. There are four\nscenarios for the three univariant three-phase equilibrium lines to meet\nat the four-phase equilibrium as depicted in and discussed individually\nbelow.\n\nFigure \u2011: Schematic four options for three univariant three-phase\nequilibrium lines to meet at the invariant four-phase equilibrium\n\nThe first scenario is that with decreasing temperature, all three\nunivariant lines merge into the four-phase equilibrium. It indicates\nthat the liquid phase does not exist at temperatures below the\nfour-phase invariant reaction. This invariant reaction is thus a ternary\neutectic reaction with liquid completely transformed to three solid\nphases upon cooling, i.e.\n\nEq. \u2011 $L \\rightarrow \\alpha + \\beta + \\gamma$\n\nIn the second scenario, two univariant lines merge into and one leaves\nfrom the four-phase equilibrium with decreasing temperature. This means\nthat one solid phase at higher temperature is no longer stable at lower\ntemperature, and it must react with the liquid phase to form the\nremaining two solid phases. The four-phase invariant reaction is thus\nperitectic. The solid phase common to both univariant lines at high\ntemperatures reacts with the liquid phase. Assuming that this phase is\n$\\alpha$, the four-phase invariant reaction becomes\n\nEq. \u2011 $L + \\alpha \\rightarrow \\beta + \\gamma$\n\nIn the third scenario, one univariant line points to and two leave from\nthe four-phase equilibrium with decreasing temperature. A new phase\nforms at low temperatures from the three high temperature phases, e.g.\nliquid, $\\alpha$, and $\\beta$, with the four-phase invariant reaction as\n\nEq. \u2011 $L + \\alpha + \\beta \\rightarrow \\gamma$\n\nThe fourth scenario is the inverse of the first scenario, indicating the\nformation of liquid from solid phases upon cooling, i.e.\n\nEq. \u2011 $\\alpha + \\beta + \\gamma \\rightarrow L$\n\nThis case has not been observed in reality.\n\nAs an example, the liquidus projections of the Al-Fe-Si ternary system\nare shown in in two formats\\[6\\], i.e. (a) three-dimensional liquidus\nsurface with the isotherms showing the liquidus contours; (b)\nconventional projection to the composition axis with the temperature\ndecrease shown by arrows; (c) projection to the temperature and weight\nfraction of Si. The first to third scenarios of invariant reactions\ndiscussed above can clearly be identified and listed in . It is evident\nthat c provides the easiest route to visualize the type of invariant\nreactions as show by .\n\nFigure \u2011: Liquidus of the Al-Fe-Si ternary system\\[6\\], (a)\nthree-dimensional presentation of the liquidus; (b) projection to the\ncomposition triangle with isotherms (dotted lines) superimposed and\ntheir temperatures indicated close to the horizontal axis.\n\nTable \u2011: Invariant liquidus reactions of the Al-Fe-Si ternary system\nwith the composition of the liquid phase \\[6\\]\n\n| Reaction | T, \u00b0C | wFe, % | wSi, % |\n|----------|-------|-------------------|-------------------|\n| | 1178 | 77.9 | 21.0 |\n| | 1155 | 49.0 | 0.16 |\n| | 1127 | 52.7 | 2.81 |\n| -H-L | 1076 | 41.6 | 44.0 |\n| | 1073 | 51.0 | 7.19 |\n| | 1050 | 53.8 | 18.4 |\n| -H-L | 1019 | 34.8 | 44.4 |\n| | 1004 | 49.0 | 12.6 |\n| | 1000 | 46.7 | 13.2 |\n| -L | 940 | 37.9 | 32.0 |\n| | 921 | 33.7 | 20.1 |\n| | 899 | 33.8 | 32.2 |\n| | 884 | 30.8 | 26.0 |\n| -L | 877 | 29.5 | 35.2 |\n| | 851 | 23.3 | 21.6 |\n| | 834 | 22.2 | 31.7 |\n| | 825 | 22.1 | 25.7 |\n| | 823 | 21.8 | 25.4 |\n| | 715 | 6.64 | 10.8 |\n| | 694 | 6.11 | 17.1 |\n| | 680 | 4.68 | 11.6 |\n| | 630 | 2.11 | 4.10 |\n| | 616 | 1.76 | 6.56 |\n| | 598 | 1.22 | 14.3 |\n| | 575 | 0.73 | 12.7 |\n\nIn contrast to projections, sectioning is used more often to understand\nphase relations in multi-component systems. Sectioning of a potential\nphase diagrams is relatively simple as the resulted phase diagram\nbehaves like a system with one component less. The same is true if\npotentials are sectioned in phase diagrams with both potential and molar\nquantities as the section is along the tie-lines of the fixed\npotentials. As an example, shows the ternary Al-Fe-Si potential and\nmolar phase diagrams sectioned at T=1273K and P=1atm, commonly referred\nto as isothermal section. It is evident that the geometric features of\nboth phase diagrams are identical to those of pure Fe shown in and ,\nrespectively, with one-, two-, and three-phase regions and corresponding\nphase boundaries.\n\nFigure \u2011: Ternary isothermal section of the Al-Fe-Si ternary system at\nT=1273K and P=1atm\n\nOn the other hand, when the phase diagram is sectioned along a molar\nquantity, it would usually not follow a tie-line because phases in\nequilibrium usually have different values for the same molar quantity.\nConsequently, there are no tie-lines inside such phase diagrams in\ngeneral, and any phase regions only show which phases are in equilibrium\nwith each other without any information on the values of molar\nquantities of individual phases.\n\nThis type of sectioning reduces both the dimensionalities of the phase\ndiagram and phase boundary by the same number, but does not alter the\nnumber of phases in the adjacent phase regions. The contact rule, i.e. ,\nthus remains valid and is applicable to phase regions with the\nsame-dimensionality as that of the sectioned phase diagram. Similarly,\nthe Schreinemakers\u2019 rule shown in is valid under the same conditions.\n\nFor example, the two-dimensional phase diagram of the Mg-Al-Zn ternary\nsystem sectioned with one atmospheric pressure and the weight fraction\nof Zn fixed at 0.01 is shown in plotted with temperature and mole\nfraction of Al \\[7\\]. This phase diagram is commonly called isopleth and\nis generated by fixing one potential, the pressure, $P$, changing the\nchemical potentials of Al and Zn to their conjugate molar quantities\nrepresented by weight fractions of Al and Zn, and sectioning at\n$w_{Zn} = 0.01$. From the discussions in Chapter , the phase regions\nwith the number of phases equal to three or fewer, i.e.\n$p \\leq n_{m} + 1 = 3$, have the same-dimensionality as the phase\ndiagram, i.e. two-dimensional in the present case, and the phase\nboundary rule is applicable. The maximum number of phases co-existing at\nequilibrium is given by as following for the present case\n\nEq. \u2011 $p_{\\max} = \\left( c - n_{s} \\right) + 2 = 3 - 1 + 2 = 4$\n\nThis is because introducing molar quantities only increases the\ndimensionality of phase regions and does not change the maximum number\nof co-existing phases.\n\nThe dimensionality of a four-phase region is calculated from as\n\nEq. \u2011\n$\\upsilon_{m} = \\left( c - n_{s} \\right) + 2 - p + n_{m} - n_{ms} = 3 - 1 + 2 - 4 + 2 - 1 = 1$\n\nwhere $n_{ms}$ is the number of sectioned molar quantities. Since the\ndimensionality of a four-phase region is lower than that of the phase\ndiagram, the phase boundary rule cannot be applied directly. Such a\nfour-phase region, liquid+Mg+\u03b3+\u03c6, is shown in between three three-phase\nregions of liquid+Mg+\u03b3, liquid+Mg+\u03c6, and Mg+\u03b3+\u03c6.\n\nFigure \u2011: Isopleth with the weight fraction of Zn fixed at 0.01 of the\nMg-Al-Zn ternary system.\n\nalso displays information on what phases are in equilibrium for a given\nalloy at various temperatures. One example is shown by the dotted\nvertical line marking the weight fraction of Al being 0.09, a widely\nused Mg alloy called AZ91. Various phases are present at different\ntemperature ranges, but the equilibrium phase fractions and phase\ncompositions are not shown in the figure as the tie-lines are not in the\nplane of the phase diagram and have to be calculated at each temperature\nindividually. shows the amount of each phase of the AZ91 alloy as a\nfunction of temperature with the dotted lines depicting the values under\nthe equilibrium condition and the solid lines depicting the values under\nthe so-called Scheil condition assuming no diffusion in solid phases and\ninfinitelyfast diffusion in liquid. Similarly, the composition of each\nphase can also be plotted as shown in .\n\nFigure \u2011: Mole fraction of individual phases under equilibrium (dotted\ncurves) and Scheil (solid curves) conditions in the AZ91 alloy\n\nFigure \u2011: Mass fraction of Al and Zn in the Mg solid solution phase\nunder equilibrium (dotted curves) and Scheil (solid curves) conditions\nin AZ91\n", "metadata": {}}, {"id": "f06dd6a3", "cell_type": "markdown", "source": "### Section of potential phase diagrams\n\nBased on the Gibbs-Duhem equation (see ), a single-phase equilibrium in\na system with more than two independent components has more than three\nindependent potentials. There is no problem in representing them using\nthe mathematical formulas discussed so far, but it is not possible for\nus to visualize graphically the full potential phase diagrams in\nmulti-component systems with more than two independent components. In\nprinciples, there are two options. One option is to project the\nmulti-dimensional potential phase diagram into a two- or three-dimension\ndiagram, and another option is to section the multi-dimensional\npotential phase diagram by fixing the values of some potentials.\n\nThe projection approach is used for one-component systems in Chapter .\nSince a two-phase equilibrium in a one-component potential phase diagram\nis one-dimensional, the projection does not lose any information, and\nthe same is true for a three-phase equilibrium in a one-component\npotential phase diagram. In a binary system, the projections of three-\nand four-phase equilibria do not lose any information, while the\nprojections of two-phase equilibria become two-dimensional and cannot\nretain all the information as the original-dimensionality of these\ntwo-phase equilibria is three. Consequently, sectioning at fixed values\nof some potentials is necessary in order to visualize the phase\nrelations in systems with two or more components. Gibbs phase rule shown\nin and are thus modified to\n\nEq. \u2011 $\\upsilon = c + 2 - p - n_{s} = \\left( c - n_{s} \\right) + 2 - p$\n\nEq. \u2011 $p_{\\max} = \\left( c - n_{s} \\right) + 2$\n\nwhere $n_{s}$ is the number of potentials fixed in sectioning. As can be\nseen in the last part of , the number of sectioning is equivalent to the\nreduction of the effective number of independent components. Therefore,\nany multi-component systems with $n_{s} = c - i$ behave like an\n$i$-component system. The equations presented in Chapter and are thus\ndirectly applicable to multi-component systems with $n_{s} = c - 1$ and\n$n_{s} = c - 2$, respectively.\n\nA common practice in experiments is to fix pressure, temperature or\nchemical potentials of volatile components as they are usually the\nvariables controlled experimentally. In a binary system, the potential\nphase diagram at constant pressure can be represented by any two of the\nthree potentials, i.e. two chemical potentials and temperature, with the\nremaining potential being dependent, and has the identical morphology as\na one-component system. The Gibbs-Duhem equation under such conditions\nbecomes\n\nEq. \u2011 $0 = - SdT - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B}$\n\nThe corresponding two-phase Clausius-Clapeyron equations are written as\n\nEq. \u2011\n$\\frac{dT}{{d\\mu}_{A}} = - \\frac{{\\mathrm{\\Delta}z}_{A}^{\\alpha\\beta}}{\\mathrm{\\Delta}S_{mB}^{\\alpha\\beta}}$\n\nEq. \u2011\n$\\frac{dT}{{d\\mu}_{B}} = - \\frac{{\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}}{\\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}}$\n\nEq. \u2011\n$\\frac{{d\\mu}_{A}}{{d\\mu}_{B}} = - \\frac{\\mathrm{\\Delta}\\left( \\frac{1}{S_{mB}^{}} \\right)^{\\alpha\\beta}}{\\mathrm{\\Delta}\\left( \\frac{1}{S_{mA}^{}} \\right)^{\\alpha\\beta}}$\n\nAs an example, the $T$-$\\mu_{C}$ potential phase diagram for the Fe-C\nbinary system at one atmospheric pressure is shown in .\n\nFigure \u2011: $T$-$\\mu_{C}$ potential phase diagram for the Fe-C binary\nsystem at $P = 1atm$\n\nIn a ternary system, two potentials need to be fixed in order to obtain\ntwo-dimensional potential phase diagrams. When the pressure and the\nchemical potential of one species are fixed, the system behaves like a\nbinary system discussed above. When the system temperature and pressure\nare fixed, the Gibbs-Duhem equation is written as\n\nEq. \u2011 $0 = - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B} - {N_{C}d\\mu}_{C}$\n\nTaking the component A as the dependent element, the two-phase\nClausius-Clapeyron equation is simplified as\n\nEq. \u2011\n$\\frac{{d\\mu}_{B}}{{d\\mu}_{C}} = \\frac{d\\left( \\ln a_{B} \\right)}{d\\left( \\ln a_{C} \\right)} = - \\frac{\\mathrm{\\Delta}z_{C}^{\\alpha\\beta}}{{\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}}$\n\nWhen the two phases in equilibrium are stoichiometric phases, the\ntwo-phase equilibrium is thus a straight line. For example, the Ti-O-Cl\npotential phase diagram at 600\u00b0C and one atmospheric pressure is shown\nin . Since both O and Cl are volatile components, their activities are\nusually represented by their partial pressures with the pure\nO2 and Cl2 gas as their respective reference\nstates at the given temperature and pressure.\n\nFigure \u2011: Ti-O-Cl potential phase diagram at 600\u00b0C and one atmospheric\npressure\n\nFor systems with more than three components, the chemical potentials of\none or more components must be fixed in order to obtain a\ntwo-dimensional potential phase diagram similar to the potential phase\ndiagrams discussed above.\n", "metadata": {}}, {"id": "bd5e4695", "cell_type": "markdown", "source": "### Tie-lines and lever rule\n\nIt is self-evident from to Eq. 2\u20114 that while potentials are homogeneous\nin all phases in a heterogeneous system at equilibrium, the molar\nquantities usually have different values in individual phases. This is\nalso stipulated in various Clausius-Clapeyron equations such as to and\nto . The difference in molar quantities thus increases the\ndimensionality of the phase region by the number of potentials replaced\nby their conjugate molar quantities. The maximum dimensionality of a\nphase region is the dimensionality of the phase diagram under\nconsideration. This thus creates a finite space between phases in\nequilibrium in the phase diagram with some axes being molar quantities.\n\nFor an equilibrium system under constant $T$, $P$ and $N_{i}$, the\npotentials in the system and their conjugate molar quantities in each\nphase are all uniquely defined. In a phase diagram with one or more\npotentials replaced by their conjugate molar quantities, two phases in\nequilibrium in a system with $c$ independent components are connected by\na $c$ dimensional line in a *c*+2 dimensional space or\nits$\\ $*c+1*-dimensional projection as discussed in Chapter . These\nlines are called tie-lines and collectively represent a two-phase\nequilibrium region. For a $k$-phase equilibrium, there are total\n$C_{k}^{2} = \\frac{k(k - 1)}{2}$ tie-lines connecting every two phases\nwith $k - 1\\ $of them independent because the number of independent\ntie-lines increases by one with each new phase added. For the invariant\nequilibrium with zero degree of freedom, the number of phases in\nequilibrium is $c + 2$ shown by containing\n$C_{c + 2}^{2} = \\frac{(c + 2)(c + 1)}{2}$ tie-lines with $c + 1$ of\nthem being independent.\n\nInside the space encapsulated by the tie-lines, the axis variables of\nthe phase diagram (a mixture of potentials and molar quantities) can be\nchanged independently without changing the phases in equilibrium and\ntheir properties. Only the relative amounts of individual phases are\nadjusted accordingly to maintain the conservation of the molar\nquantities in the system specified by the molar quantity axes of the\nphase diagram. The geometric feature circumscribing the space\nencapsulated by the tie-lines no longer represents any phase regions,\nbut a boundary between the neighbouring phase regions. Its\ncharacteristics will be discussed in more details in the next few\nsections. As properties in each phase are homogeneous, the values of\nmolar quantities of a system are simply the sum of individual phases and\ncan be represented by the following equation\n\nEq. \u2011 $A_{m} = \\sum_{\\alpha}^{}{f^{\\alpha}A_{m}^{\\alpha}}$\n\nwhere $A_{m}$ and $A_{m}^{\\alpha}$ represent the values of a molar\nquantity of the system and the $\\alpha$ phase, respectively,\n$f^{\\alpha}$ the mole fraction of the $\\alpha$ phase, and the summation\ngoes over all phases in equilibrium with each other. With\n$\\sum_{\\alpha}^{}f^{\\alpha} = 1$, can be re-arranged into the following\nequation\n\nEq. \u2011\n$\\sum_{\\alpha}^{}{f^{\\alpha}\\left( {A_{m} - A}_{m}^{\\alpha} \\right)} = 0$\n\nis commonly referred as lever rule. For a two-phase equilibrium of\n$\\alpha$ and $\\beta$, it becomes\n\nEq. \u2011\n$f^{\\alpha} = \\frac{A_{m}^{\\beta} - A_{m}}{A_{m}^{\\beta} - A_{m}^{\\alpha}}$\n\nEq. \u2011\n$f^{\\beta} = \\frac{A_{m}^{\\alpha} - A_{m}}{A_{m}^{\\alpha} - A_{m}^{\\beta}}$\n\nFor a phase diagram with the number of axes being\n$n = \\left( c - n_{s} \\right) + 1$, the number of possible axes being\nmolar quantities is thus $k \\leq n$. There are thus $k$ equations\nsimilar to with one for each molar quantity, $A_{mi}$, resulting in the\nfollowing $k + 1$ equations\n\nEq. \u2011\n$\\sum_{\\alpha}^{}{f^{\\alpha}\\left( {A_{mi} - A}_{mi}^{\\alpha} \\right)} = 0$\n\nEq. \u2011 $1 - \\sum_{\\alpha}^{}f^{\\alpha} = 0$\n\nThe summations in and go over the phases in equilibrium, and the amount\nof each phase is obtained by solving these $k + 1$ equations\nsimultaneously along with the equilibrium conditions.\n", "metadata": {}}]} \ No newline at end of file diff --git a/psu410/src/references/index.ipynb b/psu410/src/references/index.ipynb deleted file mode 100644 index 7b7d2c0..0000000 --- a/psu410/src/references/index.ipynb +++ /dev/null @@ -1 +0,0 @@ -{"nbformat": 4, "nbformat_minor": 5, "metadata": {}, "cells": [{"id": "fc2eb200", "cell_type": "markdown", "source": "# References\n\n1\\. M. Hillert, *Phase Equilibria, Phase Diagrams and Phase\nTransformations,* Cambridge University Press, Cambridge, ed. 2nd, 2007.\n\n2\\. 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Shang, \"Thermal Expansion Anomaly\nRegulated by Entropy\", **4** (2014).\n\n67\\. Y. Wang, S. L. Shang, H. Zhang, L. Q. Chen and Z. K. Liu,\n\"Thermodynamic fluctuations in magnetic states: Fe3Pt as a prototype\",\n*Phil. Mag. Lett.* **90** (2010) 851-859.\n\n68\\. National Science and Technology Council, \u201cMaterials Genome\nInitiative for Global Competitiveness\u201d,\n*[http://www.whitehouse.gov/sites/default/files/microsites/ostp/materials_genome_initiative-final.pdf]()\nOffice of Science and Technology Policy, Washington DC*, June, 2011.\n\n69\\. L. Kaufman and J. Agren, \"CALPHAD, first and second generation -\nBirth of the materials genome\", **70** (2014) 3-6.\n\n70\\. Z. K. Liu, \"Perspective on Materials Genome\u00ae\", *Chin. Sci. Bull.*\n**59** (2014) 1619-1623.\n", "metadata": {}}]} \ No newline at end of file diff --git a/src/.gitignore b/src/.gitignore new file mode 100644 index 0000000..eeb514c --- /dev/null +++ b/src/.gitignore @@ -0,0 +1,2 @@ +__pycache__ +.ipynb_checkpoints diff --git a/src/nobook/bootstrap.ipynb b/src/nobook/bootstrap.ipynb index 591232c..68b0a92 100644 --- a/src/nobook/bootstrap.ipynb +++ b/src/nobook/bootstrap.ipynb @@ -17,15 +17,15 @@ }, { "cell_type": "code", - "execution_count": 1, + "execution_count": 17, "id": "c4601326-6456-47e8-99c9-ae8c8f59e32b", "metadata": { "execution": { - "iopub.execute_input": "2024-01-08T18:23:39.490266Z", - "iopub.status.busy": "2024-01-08T18:23:39.490022Z", - "iopub.status.idle": "2024-01-08T18:23:39.779513Z", - "shell.execute_reply": "2024-01-08T18:23:39.779085Z", - "shell.execute_reply.started": "2024-01-08T18:23:39.490247Z" + "iopub.execute_input": "2024-01-09T03:51:36.901615Z", + "iopub.status.busy": "2024-01-09T03:51:36.901463Z", + "iopub.status.idle": "2024-01-09T03:51:36.906101Z", + "shell.execute_reply": "2024-01-09T03:51:36.905665Z", + "shell.execute_reply.started": "2024-01-09T03:51:36.901603Z" }, "tags": [] }, @@ -39,7 +39,7 @@ " singleton = functools.lru_cache(1)\n", " from pathlib import Path\n", " HERE = Path(__file__).parent if \"__file__\" in globals() else Path(subprocess.check_output([\"pwd\"]).decode().strip())\n", - " PSU = HERE.parent.parent / \"psu410\" / \"src\" / \"psu410\"\n", + " PSU = HERE.parent / \"psu410\" / \"src\" / \"psu410\"\n", "\n", " import nbformat.v4 as v4, anyio, json" ] @@ -58,11 +58,11 @@ "id": "2f23bf91-3d5d-4bd5-8619-e05ac0df6c53", "metadata": { "execution": { - "iopub.execute_input": "2024-01-08T18:23:40.341269Z", - "iopub.status.busy": "2024-01-08T18:23:40.340600Z", - "iopub.status.idle": "2024-01-08T18:23:40.345868Z", - "shell.execute_reply": "2024-01-08T18:23:40.345198Z", - "shell.execute_reply.started": "2024-01-08T18:23:40.341229Z" + "iopub.execute_input": "2024-01-09T03:49:55.457320Z", + "iopub.status.busy": "2024-01-09T03:49:55.457169Z", + "iopub.status.idle": "2024-01-09T03:49:55.459361Z", + "shell.execute_reply": "2024-01-09T03:49:55.459058Z", + "shell.execute_reply.started": "2024-01-09T03:49:55.457309Z" }, "tags": [] }, @@ -96,11 +96,11 @@ "id": "bd674c8e-4a31-4544-b6b9-0aa2573385c8", "metadata": { "execution": { - "iopub.execute_input": "2024-01-08T18:23:41.060965Z", - "iopub.status.busy": "2024-01-08T18:23:41.060528Z", - "iopub.status.idle": "2024-01-08T18:23:41.066993Z", - "shell.execute_reply": "2024-01-08T18:23:41.066040Z", - "shell.execute_reply.started": "2024-01-08T18:23:41.060920Z" + "iopub.execute_input": "2024-01-09T03:49:55.460025Z", + "iopub.status.busy": "2024-01-09T03:49:55.459752Z", + "iopub.status.idle": "2024-01-09T03:49:55.462463Z", + "shell.execute_reply": "2024-01-09T03:49:55.462096Z", + "shell.execute_reply.started": "2024-01-09T03:49:55.460015Z" }, "tags": [] }, @@ -111,10 +111,7 @@ " \"\"\"expand the docx into a pandas dataframe and extract features for organization\"\"\"\n", " return (\n", " paragraphs := pandas.Series(get_docx().iter_inner_content())\n", - " ).to_frame(\"p\").join(\n", - " paragraphs.apply(operator.attrgetter(\"style.style_id\")).rename(\"style_id\")\n", - " )\n", - " " + " ).to_frame(\"p\").join(paragraphs.apply(operator.attrgetter(\"style.style_id\")).rename(\"style_id\"))" ] }, { @@ -123,24 +120,21 @@ "id": "345b47f7-6fbf-40f5-a64f-5646db86dd2b", "metadata": { "execution": { - "iopub.execute_input": "2024-01-08T18:23:41.380582Z", - "iopub.status.busy": "2024-01-08T18:23:41.380109Z", - "iopub.status.idle": "2024-01-08T18:23:41.387025Z", - "shell.execute_reply": "2024-01-08T18:23:41.386298Z", - "shell.execute_reply.started": "2024-01-08T18:23:41.380541Z" + "iopub.execute_input": "2024-01-09T03:49:55.463034Z", + "iopub.status.busy": "2024-01-09T03:49:55.462877Z", + "iopub.status.idle": "2024-01-09T03:49:55.465145Z", + "shell.execute_reply": "2024-01-09T03:49:55.464904Z", + "shell.execute_reply.started": "2024-01-09T03:49:55.463023Z" }, "tags": [] }, "outputs": [], "source": [ " @singleton\n", - " def get_frames() -> tuple[\"pandas.DataFrame\", \"pandas.DataFrame\", \"pandas.DataFrame\"]:\n", + " def get_frames() -> dict[str, pandas.DataFrame]:\n", " df = get_paragraphs()\n", " end_of_toc = df.style_id.str.match(\"^TOC\").pipe(lambda x: x[x]).index[-1] + 1\n", - " return dict(zip(\n", - " (\"toc\", \"sections\"), \n", - " (df.iloc[:end_of_toc], df.iloc[end_of_toc:])\n", - " ))" + " return dict(zip((\"toc\", \"sections\"), (df.iloc[:end_of_toc], df.iloc[end_of_toc:])))" ] }, { @@ -157,19 +151,19 @@ "id": "ff516019-bebf-4610-9f16-5dc90d0b8da7", "metadata": { "execution": { - "iopub.execute_input": "2024-01-08T18:23:42.136240Z", - "iopub.status.busy": "2024-01-08T18:23:42.135848Z", - "iopub.status.idle": "2024-01-08T18:23:42.140652Z", - "shell.execute_reply": "2024-01-08T18:23:42.140104Z", - "shell.execute_reply.started": "2024-01-08T18:23:42.136205Z" + "iopub.execute_input": "2024-01-09T03:49:55.465569Z", + "iopub.status.busy": "2024-01-09T03:49:55.465475Z", + "iopub.status.idle": "2024-01-09T03:49:55.467586Z", + "shell.execute_reply": "2024-01-09T03:49:55.467332Z", + "shell.execute_reply.started": "2024-01-09T03:49:55.465559Z" }, "tags": [] }, "outputs": [], "source": [ " def get_new_docx(elements: list[\"docx.Type\"]) -> \"docx.document.Document\":\n", - " (new := docx.Document())._body._element.extend(x._element for x in elements)\n", - " return new" + " try: return (new := docx.Document())\n", + " finally: new._body._element.extend(x._element for x in elements)" ] }, { @@ -186,19 +180,18 @@ "id": "d4e75784-0945-47e0-9260-4116ca2e0e5d", "metadata": { "execution": { - "iopub.execute_input": "2024-01-08T18:23:42.799746Z", - "iopub.status.busy": "2024-01-08T18:23:42.799588Z", - "iopub.status.idle": "2024-01-08T18:23:42.802160Z", - "shell.execute_reply": "2024-01-08T18:23:42.801870Z", - "shell.execute_reply.started": "2024-01-08T18:23:42.799733Z" + "iopub.execute_input": "2024-01-09T03:49:55.468061Z", + "iopub.status.busy": "2024-01-09T03:49:55.467947Z", + "iopub.status.idle": "2024-01-09T03:49:55.470072Z", + "shell.execute_reply": "2024-01-09T03:49:55.469738Z", + "shell.execute_reply.started": "2024-01-09T03:49:55.468051Z" }, "tags": [] }, "outputs": [], "source": [ " async def get_docx_markdown(document) -> str:\n", - " tmp = tempfile.NamedTemporaryFile(suffix=\".docx\")\n", - " document.save(tmp.name)\n", + " document.save((tmp := tempfile.NamedTemporaryFile(suffix=\".docx\")).name)\n", " return subprocess.check_output([PANDOC, \"--to\", PANDOC_TO, \"--from\", PANDOC_FROM, tmp.name]).decode()" ] }, @@ -216,11 +209,11 @@ "id": "dfc37237-2240-4585-949d-0dbc13d1e7cc", "metadata": { "execution": { - "iopub.execute_input": "2024-01-08T18:23:43.582571Z", - "iopub.status.busy": "2024-01-08T18:23:43.582417Z", - "iopub.status.idle": "2024-01-08T18:23:43.585729Z", - "shell.execute_reply": "2024-01-08T18:23:43.585405Z", - "shell.execute_reply.started": "2024-01-08T18:23:43.582559Z" + "iopub.execute_input": "2024-01-09T03:49:55.471139Z", + "iopub.status.busy": "2024-01-09T03:49:55.470974Z", + "iopub.status.idle": "2024-01-09T03:49:55.473776Z", + "shell.execute_reply": "2024-01-09T03:49:55.473495Z", + "shell.execute_reply.started": "2024-01-09T03:49:55.471129Z" }, "tags": [] }, @@ -244,11 +237,11 @@ "id": "8f4bc119-c81b-40aa-9205-2f6b2f3bb63e", "metadata": { "execution": { - "iopub.execute_input": "2024-01-08T18:23:44.109666Z", - "iopub.status.busy": "2024-01-08T18:23:44.109230Z", - "iopub.status.idle": "2024-01-08T18:23:44.115130Z", - "shell.execute_reply": "2024-01-08T18:23:44.114394Z", - "shell.execute_reply.started": "2024-01-08T18:23:44.109629Z" + "iopub.execute_input": "2024-01-09T03:49:55.474390Z", + "iopub.status.busy": "2024-01-09T03:49:55.474225Z", + "iopub.status.idle": "2024-01-09T03:49:55.476236Z", + "shell.execute_reply": "2024-01-09T03:49:55.475972Z", + "shell.execute_reply.started": "2024-01-09T03:49:55.474378Z" }, "tags": [] }, @@ -266,11 +259,11 @@ "id": "225460f7-f090-4f8b-9ff5-b8adc1459ffa", "metadata": { "execution": { - "iopub.execute_input": "2024-01-08T18:23:44.460637Z", - "iopub.status.busy": "2024-01-08T18:23:44.460242Z", - "iopub.status.idle": "2024-01-08T18:23:44.466824Z", - "shell.execute_reply": "2024-01-08T18:23:44.465680Z", - "shell.execute_reply.started": "2024-01-08T18:23:44.460603Z" + "iopub.execute_input": "2024-01-09T03:49:55.476636Z", + "iopub.status.busy": "2024-01-09T03:49:55.476547Z", + "iopub.status.idle": "2024-01-09T03:49:55.478473Z", + "shell.execute_reply": "2024-01-09T03:49:55.478221Z", + "shell.execute_reply.started": "2024-01-09T03:49:55.476627Z" }, "tags": [] }, @@ -288,11 +281,11 @@ "id": "ec3b7821-e291-4bbb-b651-82f36a925d43", "metadata": { "execution": { - "iopub.execute_input": "2024-01-08T18:23:44.864971Z", - "iopub.status.busy": "2024-01-08T18:23:44.864690Z", - "iopub.status.idle": "2024-01-08T18:23:58.876212Z", - "shell.execute_reply": "2024-01-08T18:23:58.875832Z", - "shell.execute_reply.started": "2024-01-08T18:23:44.864946Z" + "iopub.execute_input": "2024-01-09T03:49:55.478861Z", + "iopub.status.busy": "2024-01-09T03:49:55.478774Z", + "iopub.status.idle": "2024-01-09T03:50:08.876085Z", + "shell.execute_reply": "2024-01-09T03:50:08.875809Z", + "shell.execute_reply.started": "2024-01-09T03:49:55.478853Z" }, "tags": [] }, @@ -305,7 +298,7 @@ "the table of contents, 9 primary chapters, and the references are captured ... ok\n", "\n", "----------------------------------------------------------------------\n", - "Ran 1 test in 14.002s\n", + "Ran 1 test in 13.391s\n", "\n", "OK\n" ] @@ -313,7 +306,7 @@ { "data": { "text/plain": [ - "" + "" ] }, "execution_count": 10, @@ -342,22 +335,37 @@ { "cell_type": "code", "execution_count": 11, - "id": "94532190-795c-49c7-8540-5e01bbdb7126", + "id": "30ca19bc-ea48-47de-add9-59856cfc7529", "metadata": { "execution": { - "iopub.execute_input": "2024-01-08T18:23:58.876959Z", - "iopub.status.busy": "2024-01-08T18:23:58.876758Z", - "iopub.status.idle": "2024-01-08T18:23:58.879559Z", - "shell.execute_reply": "2024-01-08T18:23:58.879227Z", - "shell.execute_reply.started": "2024-01-08T18:23:58.876929Z" + "iopub.execute_input": "2024-01-09T03:50:08.876601Z", + "iopub.status.busy": "2024-01-09T03:50:08.876498Z", + "iopub.status.idle": "2024-01-09T03:50:08.878913Z", + "shell.execute_reply": "2024-01-09T03:50:08.878586Z", + "shell.execute_reply.started": "2024-01-09T03:50:08.876592Z" }, "tags": [] }, "outputs": [], "source": [ - " def slugify_index(s): return s.str.lower().str.replace(\n", - " \"\\s\", \"_\", regex=True\n", - " ).str.replace(\"[^a-z\\_]\", \"\", regex=True)" + " def slugify_index(s): return pipe(s, str.lower, partial(re.sub, \"\\s\", \"_\"), partial(re.sub, \"[^\\.a-z\\_]\", \"\"))" + ] + }, + { + "cell_type": "markdown", + "id": "3666bff3-621a-44ba-a62f-2e9d5aa14a4c", + "metadata": { + "execution": { + "iopub.execute_input": "2024-01-09T02:31:44.903477Z", + "iopub.status.busy": "2024-01-09T02:31:44.903224Z", + "iopub.status.idle": "2024-01-09T02:31:44.906381Z", + "shell.execute_reply": "2024-01-09T02:31:44.905839Z", + "shell.execute_reply.started": "2024-01-09T02:31:44.903455Z" + }, + "tags": [] + }, + "source": [ + "`get_notebooks` processes a dataframe of markdown into a dataframe of notebooks." ] }, { @@ -366,49 +374,53 @@ "id": "03e3aca3-6205-49d3-b02c-a8acaee48457", "metadata": { "execution": { - "iopub.execute_input": "2024-01-08T18:23:58.880268Z", - "iopub.status.busy": "2024-01-08T18:23:58.880060Z", - "iopub.status.idle": "2024-01-08T18:23:58.892271Z", - "shell.execute_reply": "2024-01-08T18:23:58.891848Z", - "shell.execute_reply.started": "2024-01-08T18:23:58.880253Z" + "iopub.execute_input": "2024-01-09T03:50:08.879354Z", + "iopub.status.busy": "2024-01-09T03:50:08.879258Z", + "iopub.status.idle": "2024-01-09T03:50:08.891467Z", + "shell.execute_reply": "2024-01-09T03:50:08.891166Z", + "shell.execute_reply.started": "2024-01-09T03:50:08.879342Z" }, "tags": [] }, "outputs": [], "source": [ " def get_notebooks(md):\n", - " df = md.to_frame().assign(\n", - " level=md.str.extract(\"^(#{1,})\")[0].apply(len).rename(\"level\"),\n", - " chapter_slug=slugify_index(md.index.get_level_values(\"chapter\")),\n", - " heading_slug=slugify_index(md.index.get_level_values(\"heading\")),\n", - " )\n", + " df = md.to_frame().assign(level=md.str.extract(\"^(#{1,})\")[0].apply(len).rename(\"level\"),)\n", " df = df.join(\n", " df.set_index(\"level\", append=True)[[]].assign(x=1).unstack().x[[1,2]].fillna(0).cumsum().apply(\n", " tuple, axis=1\n", " ).rename(\"group\")\n", - " ).groupby([\"chapter_slug\", \"group\"]).apply(\n", + " ).groupby([\"chapter\", \"group\"]).apply(\n", " lambda x: pandas.Series({\n", - " PSU.parent / x.name[0] / ((\n", - " x.heading_slug.iloc[0] if x.level.iloc[0] > 1 else \"index\"\n", + " PSU.parent / slugify_index(x.name[0]) / slugify_index((\n", + " x.index[0][1] if x.level.iloc[0] > 1 else \"index\"\n", " ) + \".ipynb\") : v4.new_notebook(cells=[\n", - " v4.new_markdown_cell(y) for y in x.markdown\n", + " v4.new_markdown_cell(y.splitlines(True)) for y in x.markdown\n", " ])\n", " })\n", - " ).to_frame(\"nb\").reset_index([\"chapter_slug\", \"group\"], drop=True)\n", + " ).to_frame(\"nb\").reset_index([\"chapter\", \"group\"], drop=True)\n", " return df" ] }, + { + "cell_type": "markdown", + "id": "6d11170a-85d2-4da9-84db-aa7617070287", + "metadata": {}, + "source": [ + "an asynchronous method to write notebooks `json` to disc." + ] + }, { "cell_type": "code", "execution_count": 13, "id": "636a096d-aad3-4af3-86ae-15ee357fc90c", "metadata": { "execution": { - "iopub.execute_input": "2024-01-08T18:23:58.893274Z", - "iopub.status.busy": "2024-01-08T18:23:58.892953Z", - "iopub.status.idle": "2024-01-08T18:23:58.895739Z", - "shell.execute_reply": "2024-01-08T18:23:58.895391Z", - "shell.execute_reply.started": "2024-01-08T18:23:58.893256Z" + "iopub.execute_input": "2024-01-09T03:50:08.892008Z", + "iopub.status.busy": "2024-01-09T03:50:08.891885Z", + "iopub.status.idle": "2024-01-09T03:50:08.894194Z", + "shell.execute_reply": "2024-01-09T03:50:08.893808Z", + "shell.execute_reply.started": "2024-01-09T03:50:08.891999Z" }, "tags": [] }, @@ -417,20 +429,101 @@ " async def write_nb(target, data):\n", " target = anyio.Path(target)\n", " await target.parent.mkdir(exist_ok=True, parents=True)\n", - " return await target.write_text(json.dumps(data))" + " return await target.write_text(json.dumps(data, indent=2))" ] }, { "cell_type": "code", "execution_count": 14, + "id": "d5876162-6884-4d21-98dd-cb5ab3197e74", + "metadata": { + "execution": { + "iopub.execute_input": "2024-01-09T03:50:08.894927Z", + "iopub.status.busy": "2024-01-09T03:50:08.894696Z", + "iopub.status.idle": "2024-01-09T03:50:09.035807Z", + "shell.execute_reply": "2024-01-09T03:50:09.035420Z", + "shell.execute_reply.started": "2024-01-09T03:50:08.894917Z" + }, + "tags": [] + }, + "outputs": [ + { + "data": { + "text/plain": [ + "Index([ /home/tbone/Documents/nobook/psu410/src/applications_to_chemical_reactions/index.ipynb,\n", + " /home/tbone/Documents/nobook/psu410/src/applications_to_chemical_reactions/ellingham_diagram_and_buffered_systems.ipynb,\n", + " /home/tbone/Documents/nobook/psu410/src/applications_to_chemical_reactions/internal_process_and_differential_and_integrated_driving_forces.ipynb,\n", + " /home/tbone/Documents/nobook/psu410/src/applications_to_chemical_reactions/maximum_reaction_rate_and_chemical_transport_reactions_.ipynb,\n", + " /home/tbone/Documents/nobook/psu410/src/applications_to_chemical_reactions/trends_of_entropies_of_reactions.ipynb,\n", + " /home/tbone/Documents/nobook/psu410/src/applications_to_electrochemical_systems/application_examples.ipynb,\n", + " /home/tbone/Documents/nobook/psu410/src/applications_to_electrochemical_systems/index.ipynb,\n", + " /home/tbone/Documents/nobook/psu410/src/applications_to_electrochemical_systems/aqueous_solution_and_pourbaix_diagram.ipynb,\n", + " /home/tbone/Documents/nobook/psu410/src/applications_to_electrochemical_systems/concentrations_activities_and_reference_states_of_electrolyte_species.ipynb,\n", + " /home/tbone/Documents/nobook/psu410/src/applications_to_electrochemical_systems/electrochemical_cells_and_half_cell_potentials.ipynb,\n", + " /home/tbone/Documents/nobook/psu410/src/applications_to_electrochemical_systems/electrolyte_reactions_and_electrochemical_reactions.ipynb,\n", + " /home/tbone/Documents/nobook/psu410/src/calpahd_modeling_of_thermodynamics/index.ipynb,\n", + " /home/tbone/Documents/nobook/psu410/src/calpahd_modeling_of_thermodynamics/importance_of_lattice_stability.ipynb,\n", + " /home/tbone/Documents/nobook/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_pure_elements.ipynb,\n", + " /home/tbone/Documents/nobook/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_random_solution_phases.ipynb,\n", + " /home/tbone/Documents/nobook/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_solution_phases_with_longrange_ordering.ipynb,\n", + " /home/tbone/Documents/nobook/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_stoichiometric_phases.ipynb,\n", + " /home/tbone/Documents/nobook/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_fept.ipynb,\n", + " /home/tbone/Documents/nobook/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_cerium.ipynb,\n", + " /home/tbone/Documents/nobook/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/concept_of_materials_genome.ipynb,\n", + " /home/tbone/Documents/nobook/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/index.ipynb,\n", + " /home/tbone/Documents/nobook/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/mms_model_applied_to_thermal_expansion.ipynb,\n", + " /home/tbone/Documents/nobook/psu410/src/experimental_data_for_thermodynamic_modeling/additional_methods.ipynb,\n", + " /home/tbone/Documents/nobook/psu410/src/experimental_data_for_thermodynamic_modeling/index.ipynb,\n", + " /home/tbone/Documents/nobook/psu410/src/experimental_data_for_thermodynamic_modeling/phase_equilibrium_data_from_experiments.ipynb,\n", + " /home/tbone/Documents/nobook/psu410/src/experimental_data_for_thermodynamic_modeling/thermodynamic_data_from_experiments.ipynb,\n", + " /home/tbone/Documents/nobook/psu410/src/firstprinciples_calculations_and_theory/bornoppenheimer_approximation.ipynb,\n", + " /home/tbone/Documents/nobook/psu410/src/firstprinciples_calculations_and_theory/firstprinciples_approaches_to_disordered_alloys.ipynb,\n", + " /home/tbone/Documents/nobook/psu410/src/firstprinciples_calculations_and_theory/index.ipynb,\n", + " /home/tbone/Documents/nobook/psu410/src/firstprinciples_calculations_and_theory/firstprinciples_formulation_of_thermodynamics.ipynb,\n", + " /home/tbone/Documents/nobook/psu410/src/firstprinciples_calculations_and_theory/lattice_dynamics.ipynb,\n", + " /home/tbone/Documents/nobook/psu410/src/firstprinciples_calculations_and_theory/nickel_as_the_prototype.ipynb,\n", + " /home/tbone/Documents/nobook/psu410/src/firstprinciples_calculations_and_theory/quantum_theory_for_the_motion_of_electrons.ipynb,\n", + " /home/tbone/Documents/nobook/psu410/src/gibbs_energy_function/binary_random_solutions.ipynb,\n", + " /home/tbone/Documents/nobook/psu410/src/gibbs_energy_function/elastic_magnetic_and_electric_contributions_to_free_energy.ipynb,\n", + " 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+ " /home/tbone/Documents/nobook/psu410/src/laws_of_thermodynamics/stability_at_equilibrium_and_property_anomaly.ipynb,\n", + " /home/tbone/Documents/nobook/psu410/src/phase_equilibria_in_heterogeneous_systems/general_condition_of_equilibrium.ipynb,\n", + " /home/tbone/Documents/nobook/psu410/src/phase_equilibria_in_heterogeneous_systems/gibbs_phase_rule.ipynb,\n", + " /home/tbone/Documents/nobook/psu410/src/phase_equilibria_in_heterogeneous_systems/molar_phase_diagrams.ipynb,\n", + " /home/tbone/Documents/nobook/psu410/src/phase_equilibria_in_heterogeneous_systems/index.ipynb,\n", + " /home/tbone/Documents/nobook/psu410/src/phase_equilibria_in_heterogeneous_systems/potential_phase_diagrams.ipynb,\n", + " /home/tbone/Documents/nobook/psu410/src/references/index.ipynb],\n", + " dtype='object')" + ] + }, + "execution_count": 14, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + " get_notebooks(md).index" + ] + }, + { + "cell_type": "code", + "execution_count": 15, "id": "1f7b0f6b-6b56-48a5-8562-baa57317fa09", "metadata": { "execution": { - "iopub.execute_input": "2024-01-08T18:24:10.529538Z", - "iopub.status.busy": "2024-01-08T18:24:10.528957Z", - "iopub.status.idle": "2024-01-08T18:24:10.672463Z", - "shell.execute_reply": "2024-01-08T18:24:10.672025Z", - "shell.execute_reply.started": "2024-01-08T18:24:10.529483Z" + "iopub.execute_input": "2024-01-09T03:50:09.036339Z", + "iopub.status.busy": "2024-01-09T03:50:09.036212Z", + "iopub.status.idle": "2024-01-09T03:50:09.119950Z", + "shell.execute_reply": "2024-01-09T03:50:09.119540Z", + "shell.execute_reply.started": "2024-01-09T03:50:09.036330Z" }, "tags": [] }, @@ -438,81 +531,82 @@ { "data": { "text/plain": [ - "[971,\n", - " 8496,\n", - " 5457,\n", - " 8652,\n", - " 3065,\n", - " 267,\n", - " 1367,\n", - " 12632,\n", - " 16503,\n", - " 1228,\n", - " 21406,\n", - " 2660,\n", - " 6401,\n", - " 3108,\n", - " 2920,\n", - " 9592,\n", - " 1650,\n", - " 7060,\n", - " 11650,\n", - " 3490,\n", - " 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"iopub.status.busy": "2024-01-09T03:50:09.120557Z", + "iopub.status.idle": "2024-01-09T03:50:09.130521Z", + "shell.execute_reply": "2024-01-09T03:50:09.130155Z", + "shell.execute_reply.started": "2024-01-09T03:50:09.120799Z" }, "tags": [] }, @@ -1819,6 +1913,14 @@ " if INTERACTIVE:\n", " display(*md[md.str.contains(\"\\$\")].iloc[:10].apply(Markdown))" ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "d3046373-4dc9-4214-ab7d-169ce2317917", + "metadata": {}, + "outputs": [], + "source": [] } ], "metadata": { diff --git a/src/psu410/src/applications_to_chemical_reactions/ellingham_diagram_and_buffered_systems.ipynb b/src/psu410/src/applications_to_chemical_reactions/ellingham_diagram_and_buffered_systems.ipynb new file mode 100644 index 0000000..e53f112 --- /dev/null +++ b/src/psu410/src/applications_to_chemical_reactions/ellingham_diagram_and_buffered_systems.ipynb @@ -0,0 +1,162 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "899cf1bf", + "cell_type": "markdown", + "source": [ + "## Ellingham diagram and buffered systems\n", + "\n", + "One type of chemical reactions is between a pure element in liquid or\n", + "solid states and its oxides involving one mole of oxygen gas, i.e.\n", + "\n", + "*Eq. 7\u20117* $\\frac{4}{v_{M}}M + O_{2}(gas) = M_{4/v_{M}}O_{2}$\n", + "\n", + "with $v_{M}$ being the valence of the element $M$ in the oxide. Taking\n", + "the pure $M$ and the gaseous $O_{2}$ at the reaction temperature and one\n", + "atmospheric pressure as their respective reference states, both\n", + "activities of the pure $M$ solid or liquid phase and its oxide are\n", + "unity, and the activity of $O_{2}$ equals to its partial pressure in an\n", + "ideal gas. becomes\n", + "\n", + "*Eq. 7\u20118*\n", + "$RTlnP_{O_{2}} = \\mathrm{\\Delta}_{\\ }^{0}G = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S$\n", + "\n", + "Based on , one can plot $\\mathrm{\\Delta}_{\\ }^{0}G$ as a function of\n", + "temperature for various oxidation reactions, which is called Ellingham\n", + "diagram. The intercept at $T = 0K$ is given by\n", + "$\\mathrm{\\Delta}_{\\ }^{0}H$, and the slope is represented by\n", + "$- \\mathrm{\\Delta}_{\\ }^{0}S$, depicted by the following equation\n", + "\n", + "*Eq. 7\u20119*\n", + "$\\mathrm{\\Delta}_{\\ }^{0}S = S_{M_{4/v_{M}}O_{2}} - S_{O_{2}} - \\frac{4}{v_{M}}S_{M}$\n", + "\n", + "Since the entropy of one mole of $O_{2}$ gas is significantly larger\n", + "than those of the pure element and its oxide when they are in solid or\n", + "liquid states and the entropy difference between the pure element and\n", + "its oxide, the entropy of reaction is thus dominated by the reduction of\n", + "the entropy by one mole of $O_{2}$ gas. Consequently, the entropies of\n", + "reaction are approximately the same for most reactions when the pure\n", + "elements and their oxides are solid as seen in where most lines have\n", + "similar slopes.\n", + "\n", + "Figure \u2011: Ellingham Diagram for a number of metal-oxide systems.\n", + "\n", + "For a chemical reaction on the Ellingham diagram at a given temperature,\n", + "$\\mathrm{\\Delta}_{\\ }^{0}G$ can be read from the y-axis of the diagram,\n", + "and the equilibrium partial pressure of $O_{2}$ gas can be calculated\n", + "using . Alternatively, one can plot the left part of for a given\n", + "$P_{O_{2}}$ as a function of temperature on the Ellingham diagram, i.e.\n", + "\n", + "*Eq. 7\u201110* $\\mathrm{\\Delta}_{\\ }^{0}{G =}RTlnP_{O_{2}}$\n", + "\n", + "This results in straight lines, representing iso-partial-pressure lines\n", + "of $O_{2}$, with the intercept being zero at $T = 0K$ and slopes of\n", + "$RlnP_{O_{2}}$, which are negative for $P_{O_{2}}$ lower than one\n", + "atmospheric pressure (1atm). The values of $P_{O_{2}}$ are marked on the\n", + "secondary axis on the right of the Ellingham diagram. The intersection\n", + "of the isoactivitiy line and the equilibrium line of the chemical\n", + "reaction thus gives the relation of equilibrium temperature and\n", + "equilibrium partial pressure of $O_{2}$. This is demonstrated in .\n", + "\n", + "Figure \u2011: Intersection of iso-partial-pressure lines of $O_{2}$ and\n", + "equilibrium lines in the Ellingham diagram\n", + "\n", + "For each chemical reaction in the Ellingham diagram, the three phases\n", + "are in equilibrium on the line represented by , i.e. metal, metal\n", + "oxides, and O2 gas. For conditions above the line, the value\n", + "of $P_{O_{2}}$ is larger than its equilibrium value, and the metal will\n", + "be oxidized. For conditions below the line, the value of $P_{O_{2}}$ is\n", + "lower than its equilibrium value, and the metal oxide will be reduced.\n", + "Therefore, the metal oxides in the upper part of the Ellingham diagram\n", + "can be reduced by the metals in the lower part of the diagram, and vice\n", + "versus, metals in the lower part of the diagram can be oxidized by the\n", + "metal oxides in the upper part of the diagram. For example, in , Ca can\n", + "reduce all oxides, and Cu2O is the least stable oxide.\n", + "\n", + "From the above Ellingham diagram, it is noted that equilibrium partial\n", + "pressures of $O_{2}$ are very low for most chemical reactions with many\n", + "of them lower than $10^{- 12}\\ atm$. One approach to obtain such a low\n", + "pressure is to use auxiliary reactions containing $O_{2}$ that are easy\n", + "to control and are independent of the equilibrium system of interest\n", + "except the sharing of the oxygen partial pressure. Two common auxiliary\n", + "reactions are the $H_{2}$/$H_{2}O$ and $CO$/$CO_{2}$ mixtures. For the\n", + "$H_{2}$/$H_{2}O$ mixture, the chemical reaction is\n", + "\n", + "*Eq. 7\u201111* $2H_{2}(gas) + O_{2}(gas) = 2H_{2}O(gas)$\n", + "\n", + "The equilibrium oxygen partial pressure is obtained as\n", + "\n", + "*Eq. 7\u201112*\n", + "$- RTln\\left\\{ {\\frac{1}{P_{O_{2}}}\\left( \\frac{P_{H_{2}O}}{P_{H_{2}}} \\right)}^{2} \\right\\} = \\mathrm{\\Delta}_{\\ }^{0}G = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S = - 498488\\ + 112.972T\\ (J)$\n", + "\n", + "where the $\\mathrm{\\Delta}_{\\ }^{0}H$ and $\\mathrm{\\Delta}_{\\ }^{0}S$\n", + "are taken from the substance thermodynamic database (SSUB4) compiled by\n", + "Scientific Group Thermodata Europe (SGTE) \\[59\\], which are slightly\n", + "dependent on temperature, and the values in are evaluated at 1273K using\n", + "Thermo-Calc \\[60\\]. At any given temperature, one has the following\n", + "relation\n", + "\n", + "*Eq. 7\u201113*\n", + "$RTlnP_{O_{2}} = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S - 2RTln\\frac{P_{H_{2}}}{P_{H_{2}O}} = - 498488 + \\left( 112.972 - 2Rln\\frac{P_{H_{2}}}{P_{H_{2}O}} \\right)T$\n", + "\n", + "Its intercept at $T = 0K$ is given by\n", + "$\\mathrm{\\Delta}_{\\ }^{0}H = - 498,488\\ (J)$, and the slope by\n", + "$- \\mathrm{\\Delta}_{\\ }^{0}S - 2Rln\\left( \\frac{P_{H_{2}}}{P_{H_{2}O}} \\right) = 112.972 - 2Rln\\left( \\frac{P_{H_{2}}}{P_{H_{2}O}} \\right)$.\n", + "The values of the $\\frac{P_{H_{2}}}{P_{H_{2}O}}$ ratio are marked on\n", + "another secondary axis on the right of the Ellingham diagram. The\n", + "intersection of the iso-partial-pressure-ratio line and the equilibrium\n", + "line of the chemical reaction gives the relation of equilibrium\n", + "temperature and equilibrium partial pressure ratio\n", + "$\\frac{P_{H_{2}}}{P_{H_{2}O}}$ for desired $P_{O_{2}}$ of the chemical\n", + "equilibrium.\n", + "\n", + "For the $CO$/$CO_{2}$mixture, the chemical reaction is shown below\n", + "\n", + "*Eq. 7\u201114* $2CO(gas) + O_{2}(gas) = 2CO_{2}(gas)$\n", + "\n", + "Similar, from the SSUB database, the following equation is obtained\n", + "\n", + "*Eq. 7\u201115*\n", + "$- RTln\\left\\{ {\\frac{1}{P_{O_{2}}}\\left( \\frac{P_{{CO}_{2}}}{P_{CO}} \\right)}^{2} \\right\\} = \\mathrm{\\Delta}_{\\ }^{0}G = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S = - 562,927 - 172.020T\\ (J)$\n", + "\n", + "with $\\mathrm{\\Delta}_{\\ }^{0}H$ and $\\mathrm{\\Delta}_{\\ }^{0}S$\n", + "calculated at 1273K, which are also slightly temperature-dependent as in\n", + "the case of the $H_{2}$/$H_{2}O$ mixture. The iso-partial-pressure-ratio\n", + "lines are written as\n", + "\n", + "*Eq. 7\u201116*\n", + "$RTlnP_{O_{2}} = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S - 2RTln\\frac{P_{CO}}{P_{{CO}_{2}}} = - 562,927 + \\left( 172.020 - 2Rln\\frac{P_{CO}}{P_{{CO}_{2}}} \\right)T\\ $\n", + "\n", + "Its intercept at $T = 0K$ is given by\n", + "$\\mathrm{\\Delta}_{\\ }^{0}H = - 562,927\\ (J)$, and the slope by\n", + "$172.020 - - 2Rln\\left( \\frac{P_{CO}}{P_{{CO}_{2}}} \\right)$. The values\n", + "of the $\\frac{P_{CO}}{P_{{CO}_{2}}}$ ratio are marked on the third\n", + "secondary axis on the right of the Ellingham diagram. The intersection\n", + "of the iso-partial-pressure-ratio line and the equilibrium line of the\n", + "chemical reaction gives the relation of equilibrium temperature and\n", + "equilibrium partial pressure ratio $\\frac{P_{CO}}{P_{{CO}_{2}}}$ for\n", + "desired $P_{O_{2}}$ of the chemical equilibrium.\n", + "\n", + "Therefore, one can use a mixture of the $H_{2}$/$H_{2}O$ or\n", + "$CO$/$CO_{2}$ to obtain the desired low $P_{O_{2}}$ values using and or\n", + "calculating from the SSUB database. For example,\n", + "$P_{O_{2}} = 10^{- 15}\\ atm$ at 1273K, the calculated values from the\n", + "SSUB database are $\\frac{P_{H_{2}}}{P_{H_{2}O}} \\approx 1.67$ and\n", + "$\\frac{P_{CO}}{P_{{CO}_{2}}} \\approx 2.78$, respectively, in which the\n", + "temperature dependences of $\\mathrm{\\Delta}_{\\ }^{0}H$ and\n", + "$\\mathrm{\\Delta}_{\\ }^{0}S$ and the many more gaseous species in the gas\n", + "phase are taken into account. On the other hand, the reading from the\n", + "Ellingham diagram gives $\\frac{P_{H_{2}}}{P_{H_{2}O}} \\approx 2.0$ and\n", + "$\\frac{P_{CO}}{P_{{CO}_{2}}} \\approx 2.4$, and the agreement with the\n", + "more accurate calculations above is remarkable keeping in mind the\n", + "uncertainties in graphically drawing the lines and reading both values\n", + "in the logarithmic scales from the diagram, indicating the robustness of\n", + "the Ellingham diagram.\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/applications_to_chemical_reactions/index.ipynb b/src/psu410/src/applications_to_chemical_reactions/index.ipynb new file mode 100644 index 0000000..bc79398 --- /dev/null +++ b/src/psu410/src/applications_to_chemical_reactions/index.ipynb @@ -0,0 +1,28 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "ce48ca09", + "cell_type": "markdown", + "source": [ + "# Applications to chemical reactions\n", + "\n", + "A chemical reaction can be viewed as a framework dividing a system into\n", + "two closed subsystems: reactants and products. The phases and species of\n", + "reactants and products are selected from possible phases and species\n", + "that may form from the independent components of the system. A chemical\n", + "reaction can thus be considered as an internal process to transfer heat\n", + "and work between the two subsystems of reactants and products. It is\n", + "evident that this subset of phases and species only represents partial\n", + "equilibrium information of the system under given external conditions,\n", + "and more stable equilibrium states may exist by including more phases\n", + "and species with the global equilibrium depicted by phase diagrams\n", + "discussed in previous chapters with all known phases and species\n", + "included.\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/applications_to_chemical_reactions/internal_process_and_differential_and_integrated_driving_forces.ipynb b/src/psu410/src/applications_to_chemical_reactions/internal_process_and_differential_and_integrated_driving_forces.ipynb new file mode 100644 index 0000000..91f9412 --- /dev/null +++ b/src/psu410/src/applications_to_chemical_reactions/internal_process_and_differential_and_integrated_driving_forces.ipynb @@ -0,0 +1,102 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "38ff033b", + "cell_type": "markdown", + "source": [ + "## Internal process and differential and integrated driving forces\n", + "\n", + "The driving force for an internal process can be defined as following\n", + "using $U$, $H$, $F$, or $G$ as discussed in Chapter and depending on\n", + "what system variables are kept constant\n", + "\n", + "*Eq. 7\u20111*\n", + "$- D = \\left( \\frac{\\partial U}{\\partial\\xi} \\right)_{S,\\ V,N_{i}\\ } = \\left( \\frac{\\partial H}{\\partial\\xi} \\right)_{S,\\ P,N_{i}\\ } = \\left( \\frac{\\partial F}{\\partial\\xi} \\right)_{T,\\ V,N_{i}\\ } = \\left( \\frac{\\partial G}{\\partial\\xi} \\right)_{T,P,N_{i}\\ }$\n", + "\n", + "This can be termed as differential driving force as it relates the\n", + "derivative of energy with respect to an internal process so the change\n", + "is infinitesimally small and does not affect the properties of the\n", + "system significantly. For a system under constant $T$ and $P$, let us\n", + "consider an internal process for forming a new phase $\\alpha$ with the\n", + "composition $x_{i}^{\\alpha}$ and Gibbs energy\n", + "$G_{m}^{\\alpha}\\left( x_{i}^{\\alpha} \\right)$. The differential driving\n", + "force for such an internal process can thus be defined as\n", + "\n", + "*Eq. 7\u20112*\n", + "$- D = G_{m}^{\\alpha}\\left( x_{i}^{\\alpha} \\right) - \\sum_{i}^{}x_{i}^{\\alpha}\\mu_{i} = \\sum_{i}^{}x_{i}^{\\alpha}\\left( \\mu_{i}^{\\alpha} - \\mu_{i} \\right)$\n", + "\n", + "where $\\mu_{i}$ is the chemical potential of component *i* in the\n", + "system. This may also be called as nucleation driving force as if the\n", + "$\\alpha$ phase is nucleating in the system. As discussed in Chapter ,\n", + "chemical potentials are the intercepts on the Gibbs energy axis by the\n", + "tangent plane of Gibbs energy. thus represents the distance between the\n", + "tangent planes of the original system and the $\\alpha$ phase at the\n", + "composition of the $\\alpha$ phase. Evidently, this distance is at its\n", + "maximum when the two tangent planes are parallel to each other, i.e. the\n", + "commonly called parallel tangent construction when evaluating nucleation\n", + "driving force.\n", + "\n", + "The situation is different for chemical reactions where the amount of\n", + "each component in reactants is the same as that in products, i.e. there\n", + "is a mass balance between reactants and products. The driving force for\n", + "a chemical reaction is defined by the Gibbs energy difference between\n", + "the reactants and products as if all the reactants are transferred to\n", + "the products. This driving force may thus be called integrated driving\n", + "force as it describes the energy difference of a system under two\n", + "different states, i.e.\n", + "\n", + "*Eq. 7\u20113*\n", + "$- \\int_{}^{}{Dd\\xi} = \\mathrm{\\Delta}G = \\sum_{p}^{}{n_{p}G_{p}} - \\sum_{r}^{}{n_{r}G_{r}}$\n", + "\n", + "where superscripts $p$ and $r$ denote products and reactants, $n_{p}$\n", + "and $n_{r}$ their corresponding moles, $G_{p}$and $G_{r}$ the Gibbs\n", + "energies in per mole of formula of their respective stoichiometries as\n", + "written in the chemical reaction. Conventionally, the products and\n", + "reactants are represented by species or stoichiometric compounds rather\n", + "than individual phases. This is particularly evident when various\n", + "gaseous species are considered in a chemical reaction. Consequently,\n", + "$G_{p}$and $G_{r}$ in represent the chemical potentials of product and\n", + "reactant species. For species with a fixed composition, its chemical\n", + "potential is the same as its Gibbs energy as shown by . For a species in\n", + "a solution phase, its chemical potential is related to its activity as\n", + "shown in . can thus be further written as\n", + "\n", + "*Eq. 7\u20114*\n", + "$\\mathrm{\\Delta}G = \\sum_{p}^{}{n_{p}_{\\ }^{0}G_{p}} - \\sum_{r}^{}{n_{r}_{\\ }^{0}G_{r}} + RTln\\frac{\\prod_{p}^{}\\left( a_{p} \\right)^{n_{p}}}{\\prod_{r}^{}\\left( a_{r} \\right)^{n_{r}}} = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{\\prod_{p}^{}\\left( a_{p} \\right)^{n_{p}}}{\\prod_{r}^{}\\left( a_{r} \\right)^{n_{r}}}$\n", + "\n", + "It is evident from that for stoichiometric phases in a chemical\n", + "reaction, their activities equal to one. At equilibrium, the integrated\n", + "driving force becomes zero, i.e. $\\mathrm{\\Delta}G = 0$, and is\n", + "re-arranged to become\n", + "\n", + "*Eq. 7\u20115*\n", + "$- RTln\\frac{\\prod_{p}^{}\\left( a_{p} \\right)^{n_{p}}}{\\prod_{r}^{}\\left( a_{r} \\right)^{n_{r}}} = - RTlnK_{e} = \\mathrm{\\Delta}_{\\ }^{0}G = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S$\n", + "\n", + "where $K_{e}$ is often called reaction constant relating the activities\n", + "of products and reactants at equilibrium. In a system with\n", + "$\\frac{\\prod_{p}^{}\\left( a_{p} \\right)^{n_{p}}}{\\prod_{r}^{}\\left( a_{r} \\right)^{n_{r}}} > K_{e}$,\n", + "the chemical reaction goes to left, and the products are decomposed,\n", + "while\n", + "$\\frac{\\prod_{p}^{}\\left( a_{p} \\right)^{n_{p}}}{\\prod_{r}^{}\\left( a_{r} \\right)^{n_{r}}} < K_{e}$,\n", + "the chemical reaction goes to right, and the products are formed. is\n", + "often recast into the following form by dividing $\u2013RT$ on both sides of\n", + "the equation\n", + "\n", + "*Eq. 7\u20116*\n", + "$\\ln K_{e} = - \\frac{\\mathrm{\\Delta}_{\\ }^{0}H}{RT} + \\frac{\\mathrm{\\Delta}_{\\ }^{0}S}{R}$\n", + "\n", + "With $\\ln K_{e}$ plotted with respect to $1/T$, indicates that the slope\n", + "is $- \\mathrm{\\Delta}_{\\ }^{0}H$/R and the intercept on the y axis is\n", + "$\\mathrm{\\Delta}_{\\ }^{0}{S/R}$ as shown in .\n", + "\n", + "Figure \u2011: $\\ln K_{e}$ plotted with respect to $1/T$ for several M-O\n", + "systems at 1 bar and $K_{e}$ represented by the partial pressure ratio\n", + "of CO2 and CO.\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/applications_to_chemical_reactions/maximum_reaction_rate_and_chemical_transport_reactions_.ipynb b/src/psu410/src/applications_to_chemical_reactions/maximum_reaction_rate_and_chemical_transport_reactions_.ipynb new file mode 100644 index 0000000..e93d0d8 --- /dev/null +++ b/src/psu410/src/applications_to_chemical_reactions/maximum_reaction_rate_and_chemical_transport_reactions_.ipynb @@ -0,0 +1,159 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "2e089526", + "cell_type": "markdown", + "source": [ + "## Maximum reaction rate and chemical transport reactions \n", + "\n", + "Equilibrium thermodynamics can be used to calculate the maximum rates of\n", + "reaction that are possible in dynamically reacting systems, such as when\n", + "a corrosive gas passes over a heated sample. Other examples of such\n", + "reactions include the reduction of a metal oxide in flowing hydrogen,\n", + "the evaporation of a material in a vacuum, and the deposition of a thin\n", + "film by a chemical vapour deposition process. The basic assumption used\n", + "in calculating these maximum reaction rates is that local equilibrium\n", + "exists at the location of the considered reaction.\n", + "\n", + "This section examines several examples in which maximum reaction rates\n", + "are calculated. The system can typically be divided into three regions:\n", + "(1) the input region, which is usually near room temperature, (2) the\n", + "high temperature region in which the primary reaction of interest is\n", + "occurring, and (3) the exit region. Such a system is almost always at\n", + "constant pressure throughout the system. Since the three regions have\n", + "different temperatures, the key is to use the mass conservation of the\n", + "carrier gas in all three regions. With the input gas at $T = 298K$ and\n", + "$P = 1atm$, the volume of one mole of ideal gas is\n", + "$0.0244\\left( m^{3} \\right) = 24.4(liter)$. The number of moles of input\n", + "gas flowing through the system can be written as\n", + "\n", + "*Eq. 7\u201117*\n", + "$n_{gas}^{0} = \\frac{Pf_{gas}}{RT} = \\frac{f_{gas}}{24.4} = 0.0409f_{gas}$\n", + "\n", + "where $f_{gas}$ is the input gas flow rate in liter per unit time at\n", + "$T = 298K$ and $P = 1atm$, and $R$ the gas constant.\n", + "\n", + "The first example is the evaporation of water in a flowing stream of dry\n", + "hydrogen. A schematic diagram of the system being considered is given in\n", + ". The goal of the calculation is to determine the maximum rate of\n", + "$H_{2}O(liquid)$ loss, $n_{H_{2}O}^{\\ }$, through vaporization in a\n", + "flowing stream of $H_{2}$, e.g. for generating a given $H_{2}O/H_{2}$\n", + "ratio for producing a given $O_{2}$ pressure in a high temperature\n", + "system as related to the Ellingham diagram discussed in Chapter . The\n", + "maximum rate is determined by saturating the $H_{2}$ with the\n", + "equilibrium vapour pressure of $H_{2}O(liquid)$ with the number of moles\n", + "of $H_{2}$ being $n_{H_{2}}^{0} = 0.0409f_{H_{2}}$ and $f_{H_{2}}$ being\n", + "the input flow rate of $H_{2}$ in liter per unit time.\n", + "\n", + "Figure \u2011: Schematic diagram of vaporization of water in a flowing stream\n", + "of dry hydrogen\n", + "\n", + "In the first input region, the total number moles of gas is\n", + "$N = n_{H_{2}}^{0}$. In the second high temperature region with the\n", + "temperature and pressure being $T_{sys}$ and $P_{sys}$, the total number\n", + "moles of gas is $N = n_{H_{2}}^{0} + n_{H_{2}O}^{\\ }$, which is the same\n", + "for the third exit region with $T_{exit}$. To avoid condensation, one\n", + "needs to maintain $T_{exit} > T_{sys}$. The vapour pressure of $H_{2}O$\n", + "at $T_{sys}$, $P_{H_{2}O\\ }$, can be obtained from equilibrium\n", + "thermodynamic calculations. Since the $H_{2}$ and $H_{2}O$ are occupying\n", + "the same volume at the same temperature, the maximum number of moles\n", + "$H_{2}O$ can be calculated from the following relation based on the\n", + "ideal gas law\n", + "\n", + "*Eq. 7\u201118*\n", + "$n_{H_{2}O}^{\\ } = \\frac{P_{H_{2}O}^{\\ }}{P_{H_{2}}^{\\ }}n_{H_{2}}^{0} = \\frac{P_{H_{2}O}^{\\ }}{P_{sys}^{\\ } - P_{H_{2}O}^{\\ }}n_{H_{2}}^{0} = \\frac{P_{H_{2}O}^{\\ }}{P_{sys}^{\\ } - P_{H_{2}O}^{\\ }}\\frac{f_{H_{2}}}{24.4}$\n", + "\n", + "For $P_{sys}^{\\ } = 101325Pa$, one can calculate $P_{H_{2}O}^{\\ }$ and\n", + "plot $\\frac{n_{H_{2}O}^{\\ }}{f_{H_{2}}}$ as a function of temperature\n", + "from the SSUB database as shown in . The corresponding partial pressure\n", + "ratio, $P_{H_{2}O}/P_{H_{2}}$ , is plotted in , which can be used to\n", + "calculate $P_{O_{2}}^{\\ }$ at any given temperatures. An example is\n", + "shown in with $T_{sys} = 348.15K$ and $P_{H_{2}O}/P_{H_{2}} = 0.607$.\n", + "\n", + "Figure \u2011: Ratio of maximum number of moles $H_{2}O$ with respect to\n", + "hydrogen flow rate, $\\frac{n_{H_{2}O}^{\\ }}{f_{H_{2}}}$, plotted as a\n", + "function of temperature.\n", + "\n", + "Figure \u2011: Partial pressure ratio, $P_{H_{2}O}/P_{H_{2}}$, corresponding\n", + "to .\n", + "\n", + "Figure \u2011: Partial pressure of oxygen, $P_{O_{2}}^{\\ }$, as a function of\n", + "temperature with $T_{sys} = 348.15K$ and $P_{H_{2}O}/P_{H_{2}} = 0.607$.\n", + "\n", + "The second example is the corrosion of $SiO_{2}$(s) by flowing $H_{2}$\n", + "gas at high temperatures. Considering $T_{sys}^{\\ } = 1700K$,\n", + "$P_{sys}^{\\ } = 101325Pa$, and $f_{H_{2}} = 1\\ liter/min$, the system is\n", + "thus defined with 0.0409 moles of $H_{2}$ and an equilibrium between the\n", + "gas phase and the tridymite $SiO_{2}$. The equilibrium calculation gives\n", + "0.04092 moles of gas with its constitutions as\n", + "$H_{2}:H_{2}O:SiO = 0.998887:4.833 \\bullet 10^{- 4}:4.832 \\bullet 10^{- 4}$.\n", + "The corrosion rate of SiO2(s) is thus\n", + "$0.04092x4.832 \\bullet 10^{- 4} = 1.98 \\bullet 10^{- 5}mol/min = 1.19gram/min.$\n", + "\n", + "Another example is to consider that the flowing $CO$ gas of 298K and\n", + "1atm ($= 101325Pa$) at a rate of $1\\ liter/min$ passes through and\n", + "equilibrates with single phase $C(s)$ at 1500K. The equilibrium system\n", + "is defined by $T = 1500K$, $P = 1atm$, and 0.0409 moles of $CO$ with the\n", + "equilibrium between the gas phase and C(s). The equilibrium calculation\n", + "gives 0.040872 moles of gas phase with the mole fraction of $CO$ being\n", + "0.999327, resulting in the loss of *CO* or the deposition of *C(s)* at a\n", + "rate of\n", + "$0.0409 - 0.040872 \\bullet 0.999327 = 5.55 \\bullet 10^{- 5}mole/min$.\n", + "\n", + "A chemical transport reaction is a reaction in which a condensed phase\n", + "reacts with a gas phase to form vapour-phase products, which in turn\n", + "undergo the reverse reformation of the condensed phase. Two well-known\n", + "examples of such reactions are\n", + "\n", + "*Eq. 7\u201119* $M(s) + \\frac{n}{2}I_{2}(g) = MI_{n}(g)$\n", + "\n", + "*Eq. 7\u201120* $Ni(s) + 4CO(g) = Ni{(CO)}_{4}(g)$\n", + "\n", + "which are used in the purification of metals by the iodide process and\n", + "in the purification of nickel by the Mond-Langer process. In both\n", + "processes the forward reaction is favoured by lower temperatures and the\n", + "reverse reaction by higher temperatures, resulting in the deposition of\n", + "the metal. The most common technique for causing a chemical transport of\n", + "a condensed substance makes use of the temperature dependence of the\n", + "equilibrium constant. As was discussed previously, the enthalpy of\n", + "reaction, $\\mathrm{\\Delta}_{\\ }^{0}H$, determines the manner in which\n", + "$K_{e}$ changes with temperature (see ). The value of $K_{e}$ increases\n", + "with increasing *T* for $\\mathrm{\\Delta}_{\\ }^{0}H > 0$, $K_{e}$\n", + "decreases with increasing *T* for $\\mathrm{\\Delta}_{\\ }^{0}H < 0$, and\n", + "$K_{e}$ is independent of T for $\\mathrm{\\Delta}_{\\ }^{0}H = 0$. The\n", + "$\\mathrm{\\Delta}_{\\ }^{0}H$ and $\\mathrm{\\Delta}_{\\ }^{0}S$ values for\n", + "chemical transport reactions may be either positive or negative. For\n", + "reactions by both are positive, and for reactions by both are negative.\n", + "\n", + "In a typical experiment the starting solid is located at the point in a\n", + "temperature gradient that corresponds to the largest $K_{e}$ value for\n", + "the experimental condition. As the gaseous species migrate to other\n", + "locations in the system with temperatures corresponding to lower $K_{e}$\n", + "values, the reverse reaction occurs to satisfy the new equilibrium\n", + "requirements, and the solid phase is deposited. The dependence of\n", + "$K_{e}$ on $\\mathrm{\\Delta}_{\\ }^{0}H$ results in a material transport\n", + "from hot to cold for $\\mathrm{\\Delta}_{\\ }^{0}H > 0$ (the same as for\n", + "vaporization-condensation reactions), from cold to hot for\n", + "$\\mathrm{\\Delta}_{\\ }^{0}H < 0$, and no transport for\n", + "$\\mathrm{\\Delta}_{\\ }^{0}H = 0$.\n", + "\n", + "The success of a particular reaction in causing an appreciable transport\n", + "of a condensed phase depends mainly upon the partial pressure gradients\n", + "or concentration gradients of the gaseous species in the system. A\n", + "reaction whose equilibrium is extreme toward either the reactant side or\n", + "the product side will not give an appreciable transport of material. The\n", + "concentration gradients are too small in such a system. Reactions with\n", + "equilibrium constants near unity at the experimental temperatures\n", + "usually give the largest transport since small changes in $K_{e}$ cause\n", + "large changes in concentrations. The general condition required for\n", + "obtaining a $K_{e}$ value near unity at a reasonable temperature is that\n", + "$\\mathrm{\\Delta}_{\\ }^{0}H$ and $\\mathrm{\\Delta}_{\\ }^{0}S$ both have\n", + "the same sign, resulting from the equalities of .\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/applications_to_chemical_reactions/trends_of_entropies_of_reactions.ipynb b/src/psu410/src/applications_to_chemical_reactions/trends_of_entropies_of_reactions.ipynb new file mode 100644 index 0000000..198ee50 --- /dev/null +++ b/src/psu410/src/applications_to_chemical_reactions/trends_of_entropies_of_reactions.ipynb @@ -0,0 +1,84 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "f0c53519", + "cell_type": "markdown", + "source": [ + "## Trends of entropies of reactions\n", + "\n", + "The reaction entropy, $\\mathrm{\\Delta}_{\\ }^{0}S$ in , plays an\n", + "important role in determining equilibria of high-temperature reactions.\n", + "The most important single factor that determines the entropy of a\n", + "reaction is the net change in the number of moles of gas as briefly\n", + "mentioned in the discussion of the Ellingham diagram above. The reason\n", + "this is true can be explained as follows.\n", + "\n", + "The entropy of a substance can be thought of as being the sum of four\n", + "parts: (i) translational, (ii) rotational, (iii) vibrational, and (iv)\n", + "electronic. The translational entropy of a gas is the largest entropy\n", + "term under most conditions. To the extent that the other contributions\n", + "cancel between reactants and products, the entropy of reaction is\n", + "determined by the change in the number of moles of gaseous molecules.\n", + "Based on the literature data or calculations from the SSUB database, the\n", + "net change in the number of moles of gas in a reaction results\n", + "approximately in an entropy of reaction of about 175\u00b145 J/K/mole-gas at\n", + "298K for many halides and oxides. The chemical reactions of and\n", + "discussed above both reduce the gas by one mole, and their entropies of\n", + "reaction are -113 and -172 J/K at 1273K, and -89 and -173J/K at 298K,\n", + "respectively, indicating that the chemical reaction of is an exception\n", + "of the empirical rule. For chemical reactions shown in the Ellingham\n", + "diagram, their entropies of reaction follow this empirical rule pretty\n", + "well with some of them shown in calculated from the SGTE database.\n", + "\n", + "Table \u2011: Entropies of reactions with gas at 298.15K, J/K\n", + "\n", + "Reaction: Si+O2 =SiO2 -182\n", + "\n", + "Reaction: Ti+O2=TiO2 -185\n", + "\n", + "Reaction: 2Mg+O2=2MgO -217\n", + "\n", + "Reaction: 2Ca+O2=2CaO -212\n", + "\n", + "Reaction: 2Mn+O2=2MnO -150\n", + "\n", + "Since the entropy of a reaction is primarily determined by the net\n", + "change in the number of moles of gas, the entropies for reactions\n", + "involving only condensed phases must be small. The entropies of fusion\n", + "of monatomic solids are usually in the range 8-15 J/K/mole-atom as shown\n", + "for some elements in . Most metals and many ionic salts have values that\n", + "lie in this range when given in terms of per mole of atom of material.\n", + "There are few exceptions such as silicon and boron shown in the table.\n", + "For solid-state reactions, the average values can be approximated as\n", + "0\u00b18J/K/mole-atom as also shown in the table.\n", + "\n", + "Table \u2011: Entropies of reactions of condensed phases at 298.15K, J/K\n", + "\n", + "Reaction: Si(s)=Si(l) 29.762\n", + "\n", + "Reaction: Ti(s2)=Ti(l) 7.288\n", + "\n", + "Reaction: Mg(s2)=Mg(l) 9.184\n", + "\n", + "Reaction: Ca(s2)=Ca(l) 7.659\n", + "\n", + "Reaction: Mn(s2)=Mn(l) 11.443\n", + "\n", + "Reaction: W(s)=W(l) 14.158\n", + "\n", + "Reaction: B(s)=B(l) 21.380\n", + "\n", + "Reaction: 3Fe+C=CFe3 17.060\n", + "\n", + "Reaction: S+Mn=MnS 13.909\n", + "\n", + "Reaction: NiO+Fe2O3=Fe2NiO4\n", + "0.464\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/applications_to_electrochemical_systems/application_examples.ipynb b/src/psu410/src/applications_to_electrochemical_systems/application_examples.ipynb new file mode 100644 index 0000000..3fbf77a --- /dev/null +++ b/src/psu410/src/applications_to_electrochemical_systems/application_examples.ipynb @@ -0,0 +1,18 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "cde26072", + "cell_type": "markdown", + "source": [ + "## Application examples\n", + "\n", + "Among many applications of electrochemistry, several of them are briefly\n", + "discussed in this section.\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/applications_to_electrochemical_systems/aqueous_solution_and_pourbaix_diagram.ipynb b/src/psu410/src/applications_to_electrochemical_systems/aqueous_solution_and_pourbaix_diagram.ipynb new file mode 100644 index 0000000..ba52c57 --- /dev/null +++ b/src/psu410/src/applications_to_electrochemical_systems/aqueous_solution_and_pourbaix_diagram.ipynb @@ -0,0 +1,263 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "91f80ded", + "cell_type": "markdown", + "source": [ + "## Aqueous solution and Pourbaix diagram\n", + "\n", + "The importance of aqueous solutions in all aspects of life is so well\n", + "known and needs not be discussed further. Since many electrochemical\n", + "processes involve electrolyte solutions in an aqueous solvent,\n", + "electrochemical processes including water, hydrogen, and/or oxygen are\n", + "discussed in more details. The hydrogen-oxygen cell can be described for\n", + "both acidic electrolytes and alkaline electrolytes. With acidic\n", + "electrolytes, H+ is in much higher concentrations than\n", + "OH-, and thus half-cell reactions with H+ as an\n", + "ionic transport species are more important than those involving\n", + "OH-. The reverse is true for alkaline electrolytes that\n", + "contain high OH-concentrations. Other than for nearly neutral\n", + "acid-base systems, either H+ or OH- dominates the\n", + "other by several orders of magnitude as can be seen from the value of\n", + "the 298 K dissociation constant for H2O:\n", + "\n", + "*Eq. 8\u201135* H2O(l) = H+(aq) + OH-(aq)\n", + "\n", + "with the reaction constant being Ke =\n", + "\\[H+\\]\\[OH-\\] = 10-14 and\n", + "$\\mathrm{\\Delta}_{\\ }^{0}G$= -RT *ln* Ke *\u03b2*= +79,908 J. By\n", + "convention, one defines pH = - log \\[H+\\] and pOH = - log\n", + "\\[OH-\\], and then pH + pOH = 14.\n", + "\n", + "Under acidic electrolyte conditions of low pH (high \\[H+\\]\n", + "concentrations) the anode reaction in a hydrogen-oxygen cell is:\n", + "\n", + "*Eq. 8\u201136* \u00bd H2(g) = H+(aq) + e-\n", + "\n", + "with \u03b51o = 0.0 V and\n", + "$\\mathrm{\\Delta}_{\\ }^{0}G_{1}$= 0 J. The corresponding cathode\n", + "(reduction) reaction is:\n", + "\n", + "*Eq. 8\u201137* 2 H+(aq) + \u00bd O2(g) + 2 e- =\n", + "H2O(l)\n", + "\n", + "with \u03b52o = 1.229 V and\n", + "$\\mathrm{\\Delta}_{\\ }^{0}G_{2}$ = -2\\*1.229\\*96,485 J = -237,160 J. The\n", + "net cell reaction for acidic electrolytes is:\n", + "\n", + "*Eq. 8\u201138* H2(g) + \u00bd O2(g) = H2O(l)\n", + "\n", + "with \u03b5ocell = 1.229 V and\n", + "$\\mathrm{\\Delta}_{\\ }^{0}G_{cell}$ = -2\\*1.229\\*96,485 J = -237,160 J\n", + "\n", + "Under alkaline electrolyte conditions of high pH (high\n", + "\\[OH-\\] concentrations) the anode reaction in a\n", + "hydrogen-oxygen cell is:\n", + "\n", + "*Eq. 8\u201139* 2 OH-(aq) + H2(g) = 2\n", + "H2O(l) + 2 e-\n", + "\n", + "with \u03b53o = 0.828 V and\n", + "$\\mathrm{\\Delta}_{\\ }^{0}G_{3}$= -2\\*0.828\\*96,485 J = -159,779 J. The\n", + "corresponding cathode (reduction) reaction is:\n", + "\n", + "*Eq. 8\u201140* H2O(l) + \u00bd O2(g) + 2 e- = 2\n", + "OH-(aq)\n", + "\n", + "with \u03b54o = 0.401 V and\n", + "$\\mathrm{\\Delta}_{\\ }^{0}G_{4}$ = -2\\*0.401\\*96,485 J = -77,381 J. The\n", + "net cell reaction for alkaline electrolytes is:\n", + "\n", + "*Eq. 8\u201141* H2(g) + \u00bd O2(g) = H2O(l)\n", + "\n", + "with \u03b5ocell = 1.229 V and\n", + "$\\mathrm{\\Delta}_{\\ }^{0}G_{cell}$ = -2\\*1.229\\*96,485 J = -237,160 J.\n", + "\n", + "Plots of \u03b5 versus pH for a given chemical system have been typically\n", + "used to exhibit the stability relationships of ionic species and solid\n", + "phases in aqueous-based electrochemical systems. These graphs are often\n", + "called Pourbaix diagrams after the inventor and are at constant\n", + "temperature and constant pressure diagrams for a constant concentration,\n", + "usually for one metallic element. By convention, the \u03b5 in a Pourbaix\n", + "diagram corresponds to the potential for the cathode reduction reactions\n", + "in the electrochemical half-cell with electrons as reactants. Pourbaix\n", + "diagrams can be extended to multi-component materials when thermodynamic\n", + "properties of the components are available in both the materials and the\n", + "aqueous solution.\n", + "\n", + "An example of an \u03b5 versus pH diagram is shown in for the\n", + "Ni-H2O system at a 298K, 1 bar, and $c_{{Ni}^{2 +}} = 0.001$\n", + "molality. Three stability regions for Ni species are shown: Ni(s),\n", + "NiO(s), and \\[Ni2+\\]. The two dashed lines on this diagram\n", + "correspond to hydrogen reduction () and oxygen reduction () reactions,\n", + "respectively.\n", + "\n", + "Figure \u2011: An \u03b5 versus pH, Pourbaix diagram for the Ni-H2O at\n", + "298K, 1 bar, and $c_{{Ni}^{2 +}} = 0.001$ molality.\n", + "\n", + "For the \u03b5 and pH conditions within the boundaries of the Ni(s) region,\n", + "no solid phase other than Ni(s) is stable, no ionic species with a\n", + "concentration of 1 molarity is stable, and no gas species with a\n", + "pressure of 1 bar is stable. Similar statements could be made about the\n", + "NiO(s) and \\[Ni2+\\] areas on the diagram. In the\n", + "\\[Ni2+\\] area, introduction of Ni(s) or NiO(s) into the\n", + "system would result in the dissolution of these solid phases since they\n", + "are not stable with respect to the \\[Ni2+\\] aqueous solution.\n", + "The corresponding chemical reactions proceed spontaneously to the right\n", + "as follows until the solid phases are consumed:\n", + "\n", + "*Eq. 8\u201142* Ni(s) \u2192 Ni2+(1 molarity) + 2 e-\n", + "\n", + "*Eq. 8\u201143* NiO(s) + 2 H+(aq) \u2192 Ni2+(1 molarity) +\n", + "H2O(l)\n", + "\n", + "No H+(aq) in involved in the first reaction, , so the\n", + "boundary line separating Ni(s) and Ni2+ is independent of pH.\n", + "No oxidation or reduction occurs in the second reaction, , i.e. no\n", + "electrons are reactants or products in the reaction, the boundary line\n", + "separating NiO(s) and Ni2+ is independent of \u03b5.\n", + "\n", + "Note the convention that the \u03b5 is the potential for a cathode reduction\n", + "reaction, and boundary lines between two stability regions depict\n", + "conditions under which partial equilibrium occurs for the two species\n", + "for the \u03b5 and pH values at any point on these lines. For the boundary\n", + "line separating Ni(s) and Ni2+ in an ideal aqueous solution,\n", + "i.e. the reverse of , the following equation is obtained.\n", + "\n", + "*Eq. 8\u201144* \u03b5 = \u03b5o = -0.268 V\n", + "\n", + "For the NiO(s)-Ni2+ boundary line of an ideal solution, the\n", + "reaction, , is a complete equilibrium, and thus the relationship is\n", + "\n", + "*Eq. 8\u201145*\n", + "$0 = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{1}{\\left( c_{H^{+}} \\right)^{2}} = \\mathrm{\\Delta}_{\\ }^{0}G + 2 \\cdot 2.303 \\cdot RT \\cdot pH$\n", + "\n", + "*Eq. 8\u201146*\n", + "$pH = - \\frac{\\mathrm{\\Delta}_{\\ }^{0}G}{2 \\cdot 2.303 \\cdot RT}$\n", + "\n", + "where $\\mathrm{\\Delta}_{\\ }^{0}G$ is obtained as follows and can be\n", + "calculated from the SSUB database and the standard potential of Ni,\n", + "\n", + "*Eq. 8\u201147*\n", + "\n", + "At a specified temperature, only one standard free energy and only one\n", + "equilibrium constant exists for this chemical reaction, and thus only\n", + "one specific value of $pH = 6.631$ exists for the reaction represented\n", + "by in this Pourbaix diagram.\n", + "\n", + "The diagonal line in represents the equilibrium between Ni(s) and NiO(s)\n", + "and is for a partial equilibrium reaction that is the sum of reactions\n", + "of and\n", + "\n", + "*Eq. 8\u201148* NiO(s) + 2 H+(aq) + 2 e- \u2550 Ni(s) +\n", + "H2O(l)\n", + "\n", + "The reduction of Ni from a divalent state in NiO to metallic Ni(s)\n", + "occurs, but the reaction also depends on the H+\n", + "concentration, the pH. The corresponding Gibbs energy and Nernst\n", + "equations are,\n", + "\n", + "*Eq. 8\u201149*\n", + "$\\mathrm{\\Delta}G = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{1}{\\left( c_{H^{+}} \\right)^{2}} = - 23,939 + 2 \\cdot 2.303 \\cdot RT \\cdot pH$\n", + "\n", + "*Eq. 8\u201150*\n", + "$\\varepsilon\\ = \\ \\varepsilon^{0}\\ - \\ \\frac{RT}{2f}\\ln\\frac{1}{\\left( c_{H^{+}} \\right)^{2}}\\ = 0.124\\ \u2013\\ \\frac{2.303 \\cdot RT}{f}pH$\n", + "\n", + "where $\\mathrm{\\Delta}_{\\ }^{0}G$ can be calculated as follows\n", + "\n", + "*Eq. 8\u201151*\n", + "\n", + "The two additional lines in correspond to the reduction reactions\n", + "related to H2 and O2 gases, i.e. the stability of\n", + "H2O. The lower one is for the reverse of under \u03b5o\n", + "= 0 and $P_{H_{2}} = 1$ with the Nernst equation being\n", + "\n", + "*Eq. 8\u201152*\n", + "$\\varepsilon\\ = \\ \\varepsilon^{0}\\ - \\ \\frac{RT}{f}\\ln\\frac{\\left( P_{H_{2}} \\right)^{1/2}}{c_{H^{+}}}\\ = - \\frac{2.303 \\cdot RT}{f}pH$\n", + "\n", + "As the pH increases from 0, \u03b5 becomes more negative as is depicted. The\n", + "top dashed line corresponds to the oxygen reduction reaction represented\n", + "by under \u03b5o = 1.225 calculated from the aqueous solution\n", + "database in Thermo-Calc \\[60\\] and $P_{O_{2}} = 1$ with the Nernst\n", + "equation being\n", + "\n", + "Eq. 8\u201153\n", + "$\\varepsilon\\ = \\ \\varepsilon^{0}\\ - \\ \\frac{RT}{2f}\\ln\\frac{\\left( P_{O_{2}} \\right)^{1/2}}{\\left( c_{H^{+}} \\right)^{2}}\\ = 1.225\\ \u2013\\ \\frac{2.303 \\cdot RT}{f}pH$\n", + "\n", + "The dependence of \u03b5 on pH is identical for both reduction reaction and ,\n", + "and their intercepts at $pH = 0$ differ by their difference in their\n", + "\u03b5o values.\n", + "\n", + "In this simple Pourbaix diagram of Ni in an ideal aqueous solution, all\n", + "boundary lines are straight because there is only one ionic species of\n", + "Ni in the aqueous solution, i.e. Ni2+. When there are more\n", + "than one ionic species in the aqueous solution, the boundary lines may\n", + "no longer be straight due to the competition between species. One\n", + "example is Cu with two main ionic species of Cu+2 and\n", + "CuOH+, and the reduction reaction between the metallic Cu and\n", + "the aqueous solution involves both two species, i.e.\n", + "\n", + "*Eq. 8\u201154*\n", + "${xCu}^{2 + \\ }\\ + \\ (1 - x){CuOH}^{+} + (1 - x)H^{+} + 2\\ e - \\ \\ = \\ \\ Cu(s) + {(1 - x)H}_{2}O$\n", + "\n", + "with\n", + "\n", + "*Eq. 8\u201155*\n", + "$\\mathrm{\\Delta}G = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{1}{\\left( c_{{Cu}^{2 +}} \\right)^{x}\\left( c_{{CuOH}^{+}}c_{H^{+}} \\right)^{1 - x}} = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{1}{\\left( c_{{Cu}^{2 +}} \\right)^{x}\\left( c_{{CuOH}^{+}} \\right)^{1 - x}} + 2.303(1 - x) \\cdot RT \\cdot pH$\n", + "\n", + "*Eq. 8\u201156*\n", + "$\\varepsilon = \\varepsilon^{0} - \\frac{RT}{2f}\\ln\\frac{1}{\\left( c_{{Cu}^{2 +}} \\right)^{x}\\left( c_{{CuOH}^{+}} \\right)^{1 - x}} - \\frac{2.303(1 - x) \\cdot RT}{2f}pH$.\n", + "\n", + "It is evident that both the slope and the intercept at $pH = 0$ are a\n", + "function of the concentration of ${CuOH}^{+}$, which is a function of\n", + "$pH$. Consequently, the boundary between the metallic Cu and the aqueous\n", + "solution is no longer a straight line as shown in .\n", + "\n", + "Figure \u2011: An \u03b5 versus pH, Pourbaix diagram for the Cu-H2O\n", + "system at 298K, 1 bar, and $c_{Cu} = 0.001$ molality.\n", + "\n", + "The concentrations of various species in the aqueous solution, i.e.\n", + "commonly called speciation, are plotted in , showing the change of\n", + "dominant species as a function of pH value.\n", + "\n", + "Figure \u2011: Concentrations of ionic species in the aqueous solution at\n", + "$\\varepsilon = 0.3\\ V$ from .\n", + "\n", + "In Pourbaix diagrams for alloys with two or more elements, activities of\n", + "individual elements are to be used in calculating the potentials of\n", + "reduction reactions. Considering a Fe-Ni alloy with Fe2+ and\n", + "Ni2+ in the aqueous solution, the reduction reactions for Fe\n", + "and Ni can be written separately as\n", + "\n", + "*Eq. 8\u201157* Ni2+(cNi) + 2 e- \u2192 Ni\n", + "(aNi in alloy)\n", + "\n", + "*Eq. 8\u201158* Fe2+(cFe) + 2 e- \u2192 Fe\n", + "(aFe in alloy)\n", + "\n", + "with their potentials as\n", + "\n", + "*Eq. 8\u201159*\n", + "$\\varepsilon_{Ni}\\ = \\ \\varepsilon_{Ni}^{0} - \\frac{2.303RT}{2f}\\ln\\frac{a_{Ni}}{c_{Ni}} = \\ - 0.268 - \\frac{2.303RT}{2f}\\ln\\frac{a_{Ni}}{c_{Ni}}$\n", + "\n", + "*Eq. 8\u201160*\n", + "$\\varepsilon_{Fe}\\ = \\ \\varepsilon_{Fe}^{0} - \\frac{2.303RT}{2f}\\ln\\frac{a_{Fe}}{c_{Fe}} = \\ - 0.441 - \\frac{2.303RT}{2f}\\ln\\frac{a_{Fe}}{c_{Fe}}$\n", + "\n", + "In principle, there are two scenarios for a given set of $a_{Ni}$ and\n", + "$a_{Fe}$ of the alloy. The first scenario is at the limit of a dilute\n", + "aqueous solution, i.e. $c_{Ni} = c_{Fe} = 0.001\\ $molarity,\n", + "$\\varepsilon_{Ni}$ and $\\varepsilon_{Fe}$ can be calculated, and the\n", + "element with the lower potential has the tendency to dissolve first,\n", + "which can result in the so-called dialloying effect. The second scenario\n", + "is for the equal potential, i.e. $\\varepsilon_{Ni} = \\varepsilon_{Fe}$\n", + "due to the externally imposed potential, and the equilibrium\n", + "concentrations of Fe+2 and Ni+2 can be calculated\n", + "from and .\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/applications_to_electrochemical_systems/concentrations_activities_and_reference_states_of_electrolyte_species.ipynb b/src/psu410/src/applications_to_electrochemical_systems/concentrations_activities_and_reference_states_of_electrolyte_species.ipynb new file mode 100644 index 0000000..ef38ce0 --- /dev/null +++ b/src/psu410/src/applications_to_electrochemical_systems/concentrations_activities_and_reference_states_of_electrolyte_species.ipynb @@ -0,0 +1,349 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "6bfb8286", + "cell_type": "markdown", + "source": [ + "## Concentrations, activities, and reference states of electrolyte species\n", + "\n", + "Thermodynamic descriptions of ionic species in solutions are different\n", + "from those of neutral species, which leads to a need for defining\n", + "concentration units, standard states, activities, and activity\n", + "coefficients of ionic solutions. In most studies of electrochemical\n", + "corrosion and electrodeposition, and in applied work of electrochemical\n", + "engineers, ionic species concentrations are given in units of molarity,\n", + "the number of moles of a species in a liter of solution (mol/l)\n", + "symbolically represented in equations by either *ci* or\n", + "\\[M+Z\\]. The other common concentration used for ionic\n", + "species is molality, which is defined as the number of moles of a\n", + "species in 1000g of solvent. For dilute aqueous solutions, molarity and\n", + "molality values are very similar.\n", + "\n", + "As discussed in Chapter , a practical definition of the activity of a\n", + "species *i* is the thermodynamic reactivity, or tendency to react, of\n", + "species *i* in the system of interest as compared to *i* in its\n", + "reference state form. The reference state of a species is typically\n", + "chosen as a specific chemical/physical state of the species at 1 atm\n", + "external pressure and the temperature of interest. Similarly, a typical\n", + "reference state for ionic species in aqueous solutions is the 1 molar\n", + "ideal solution at 1 bar external pressure and the temperature of\n", + "interest. If an electrolyte solution behaves ideally, then the activity\n", + "of species *i* in solution is\n", + "\n", + "*Eq. 8\u20116*\n", + "$a_{i} = \\frac{c_{i}\\left( \\frac{mol}{l} \\right)}{c_{i}^{0}\\left( \\frac{mol}{l} \\right)} = \\frac{c_{i}\\left( \\frac{mol}{l} \\right)}{1\\left( \\frac{mol}{l} \\right)} = c_{i}(dimensionless)$\n", + "\n", + "where $c_{i}$ is the molar concentration of *i* in the solution divided\n", + "by $c_{i}^{0}$, the 1 molar reference state ideal solution\n", + "concentration. Thus, in ideal solutions, the activity of an electrolyte\n", + "species is numerically equal to its molar concentration. The above\n", + "treatment of ionic species is equivalent to the common practice of\n", + "depicting the activity of a gas by the value of its ideal gas partial\n", + "pressure in units of bar.\n", + "\n", + "The activity coefficient corrects for the nonideality of the species in\n", + "solution as defined in . If the solution is ideal, $\\gamma_{i} = 1$ for\n", + "all concentrations of a species in solution. For all solutions, one\n", + "expects $\\gamma_{i} \\rightarrow 1$ as $c_{i} \\rightarrow 1$. It is not\n", + "possible to measure $\\gamma_{i^{+}}$ or $\\gamma_{i^{-}}$ for individual\n", + "charged ions, only a geometric mean of the positive and negative ion\n", + "values. Consider the following ionic solution\n", + "\n", + "Eq. 8\u20117\n", + "\n", + "Its chemical potential can be written as\n", + "\n", + "*Eq. 8\u20118*\n", + "\n", + "Its geometric average or mean activity and activity coefficient are\n", + "defined as\n", + "\n", + "*Eq. 8\u20119*\n", + "\n", + "*Eq. 8\u201110*\n", + "\n", + "For example, one can define\n", + "$\\gamma_{\\pm} = \\left( \\gamma_{{Na}^{+}}\\gamma_{{Cl}^{-}} \\right)^{1/2}$\n", + "and\n", + "$\\gamma_{\\pm} = \\left( \\gamma_{{Al}^{3 +}}^{2}\\gamma_{{{SO}_{4}}^{2 -}}^{3} \\right)^{1/5}$\n", + "for NaCl and Al2(SO4)3, respectively.\n", + "For idea, weak electrolytes, $\\gamma_{\\pm} = 1$, and for non-ideal,\n", + "strong electrolytes, $\\gamma_{\\pm} \\neq 1$.\n" + ], + "metadata": {} + }, + { + "id": "ce234b05", + "cell_type": "markdown", + "source": [ + "### Electrical batteries\n", + "\n", + "Batteries utilize electrochemical reactions to generate electricity for\n", + "various devices. The theoretic voltage of a battery can be calculated\n", + "from and , i.e.\n", + "\n", + "*Eq. 8\u201173*\n", + "$\\varepsilon = - \\frac{\\mathrm{\\Delta}G}{zf} = \\varepsilon^{0} - \\frac{RTlnQ}{zf}$\n", + "\n", + "with $\\mathrm{\\Delta}G$ being the driving force of the net cell reaction\n", + "and $Q$ being the reaction activity quotient. The actual voltage of a\n", + "battery is lower than the theoretical one due to kinetic limitations of\n", + "cell reactions and resistance to ion diffusion through the electrolyte.\n", + "Based on whether the cell reactions are reversible or not, batteries\n", + "typically categorized as either primary disposable or secondary\n", + "rechargeable batteries. The net cell reactions in primary disposable\n", + "batteries are not easily reversible, and electrode materials may not\n", + "return to their original forms by applying a higher external potential\n", + "of opposite sign. Consequently, primary batteries cannot be reliably\n", + "recharged. On the other hand, the net cell reactions in secondary\n", + "batteries are easily reversible. Furthermore, two half-cells in\n", + "batteries may use different electrolytes with each half-cell enclosed in\n", + "a container and a separator permeable to conducting ions but not the\n", + "bulk of the electrolytes.\n", + "\n", + "One common primary battery is zinc-carbon battery with a zinc anode\n", + "cylinder and a carbon cathode central rod. The electrolytes are ammonium\n", + "or zinc chloride next to the zinc anode and a mixture of ammonium\n", + "chloride and manganese dioxide next to the carbon cathode. The half cell\n", + "and net reactions with ammonium chloride are as follows\n", + "\n", + "*Eq. 8\u201174* Zn + 2NH3 \u2192\n", + "Zn(NH3)22+ + 2 e-\n", + "\n", + "*Eq. 8\u201175* 2NH4Cl + 2MnO2 + 2 e- \u2192\n", + "2NH3 + Mn2O3 +\n", + "H2O+2Cl\u2212\n", + "\n", + "*Eq. 8\u201176* Zn + 2MnO2 + 2NH4Cl \u2192\n", + "Mn2O3 +\n", + "Zn(NH3)2Cl2 + H2O.\n", + "\n", + "The electric potential of the reaction is, treating all compounds as\n", + "stoichiometric compounds\n", + "\n", + "Eq. 8\u201177\n", + "$\\varepsilon = - \\frac{\\mathrm{\\Delta}G}{2f} = - \\frac{\\mathrm{\\Delta}^{0}G}{2f} = \\frac{1}{2f}\\left(_{\\ }^{0}G^{Zn} + 2_{\\ }^{0}G^{{MnO}_{2}} + 2_{\\ }^{0}G^{{NH}_{4}Cl} -_{\\ }^{0}G^{H_{2}O} -_{\\ }^{0}G^{Zn\\left( {NH}_{3} \\right)_{2}{Cl}_{2}} -_{\\ }^{0}G^{{{Mn}_{2}O}_{3}} \\right)$.\n", + "\n", + "The Gibbs energy of $Zn\\left( {NH}_{3} \\right)_{2}{Cl}_{2}$ is not\n", + "available in current databases and has been recently estimated to be\n", + "\u2212505,375 J/mole-formula \\[62\\]. The value of at 298.15K is thus obtained\n", + "as 1.67 V, which is pretty close to the actual operating voltage of the\n", + "battery around 1.5 V.\n", + "\n", + "While with zinc chloride, the cell reactions and electric potential may\n", + "be written as\n", + "\n", + "Eq. 8\u201178 Zn + ZnCl2 + 2OH\u2212 \u2192 2ZnOHCl + 2\n", + "e-\n", + "\n", + "*Eq. 8\u201179* MnO2 + H2O + e- \u2192 MnOOH +\n", + "OH-\n", + "\n", + "*Eq. 8\u201180* Zn + 2 MnO2 + ZnCl2 + 2 H2O\n", + "\u2192 2 MnOOH + 2 ZnOHCl\n", + "\n", + "*Eq. 8\u201181*\n", + "$\\varepsilon = \\frac{1}{2f}\\left(_{\\ }^{0}G^{Zn} + 2_{\\ }^{0}G^{{MnO}_{2}} +_{\\ }^{0}G^{Zn{Cl}_{2}} + 2_{\\ }^{0}G^{H_{2}O} - 2_{\\ }^{0}G^{MnOOH} - 2_{\\ }^{0}G^{ZnOHCl} \\right)$\n", + "\n", + "Secondary batteries can be recharged by applying an external electrical\n", + "potential, which reverses the net cell reaction that occur during\n", + "discharging. The oldest form of rechargeable battery is the lead-acid\n", + "batteries used in automotive, and the latest development is the\n", + "lithium-ion (Li-ion) batteries. A lead-acid battery typically uses Pb\n", + "and PbO2 as the cathode and anode electrodes and a 35% sulfuric acid and\n", + "65% water solution as the electrolyte. Its anode and cathode reactions\n", + "can be simplified as follows\n", + "\n", + "Eq. 8\u201182 $Pb + SO_{4}^{2 -} = PbSO_{4} + 2e^{-}$\n", + "\n", + "Eq. 8\u201183 $PbO_{2} + 4H^{+} + SO_{4}^{2 -} + 2e^{-} = PbSO_{4} + 2H_{2}O$\n", + "\n", + "The net cell reaction is\n", + "\n", + "Eq. 8\u201184 $Pb + PbO_{2} + 2H_{2}SO_{4}^{\\ } = 2PbSO_{4} + 2H_{2}O$.\n", + "\n", + "Its electric potential is represented by the following equation\n", + "\n", + "*Eq. 8\u201185*\n", + "$\\varepsilon = - \\frac{1}{2f}\\left( 2_{\\ }^{0}G^{H_{2}O} + 2_{\\ }^{0}G^{{PbSO}_{4}} -_{\\ }^{0}G^{Pb} -_{\\ }^{0}G^{{PbO}_{2}} - 2_{\\ }^{0}G^{H_{2}SO_{4}} \\right)$\n", + "\n", + "with the value at 298.15K being 2.651 V calculated from Thermo-Calc\n", + "\\[60\\]. During discharging, the reaction goes to right, and $PbSO_{4}$\n", + "is formed on both anode and cathode, while during charging, the reaction\n", + "goes to the left, and $Pb$ and $PbO_{2}$ are restored. In practical\n", + "applications, other ionic species such as ${H_{3}O}^{+}$ and\n", + "$HSO_{4}^{-}$ may form in the electrolyte, complicating the reactions\n", + "and affecting its potential.\n", + "\n", + "In lithium ion batteries, lithium ions migrate in electrolytes between\n", + "electrodes made of intercalated lithium compounds during charging and\n", + "discharging. LiCoO2 and LiFePO4 are two of the\n", + "several cathode materials used in lithium ion batteries, and the anode\n", + "is typically made of carbon or metallic Li. The anode and cathode\n", + "reactions for LiCoO2 batteries can be written in simple forms as follows\n", + "\n", + "Eq. 8\u201186 ${Li}_{x}C_{6} = x{Li}^{+} + xe^{-} + 6C$\n", + "\n", + "Eq. 8\u201187 $x{Li}^{+} + xe^{-} + {Li}_{1 - x}CoO_{2} = LiCoO_{2}$\n", + "\n", + "with the net reaction and electric potential being\n", + "\n", + "Eq. 8\u201188 ${Li}_{x}C_{6} + {Li}_{1 - x}CoO_{2} = LiCoO_{2} + 6C$\n", + "\n", + "*Eq. 8\u201189*\n", + "$\\varepsilon = - \\frac{1}{xf}\\left\\{ 6_{\\ }^{0}G^{C} +_{\\ }^{0}G^{{LiCoO}_{2}} - G^{{{Li}_{x}C}_{6}} - G^{{{Li}_{1 - x}CoO}_{2}} \\right\\} = - \\frac{1}{f}\\left\\{ \\left( \\mu_{Li}^{{Li}_{1 - x}CoO_{2}} - \\mu_{Li}^{{Li}_{x}C} \\right) - \\frac{1}{x}\\left( \\mu_{LiCoO_{2}}^{{Li}_{1 - x}CoO_{2}} -_{\\ }^{0}G^{{LiCoO}_{2}} \\right) \\right\\}$\n", + "\n", + "The electric potential is a function of $x$. The value in the first\n", + "parenthesis in the above equation denotes the chemical potential\n", + "difference of Li between two electrodes, and the value in the second\n", + "parenthesis represents the chemical potential difference of\n", + "${LiCoO}_{2}$ between the states in the solution phase of\n", + "${Li}_{1 - x}CoO_{2}$ and by itself. Gibbs energies of ${{Li}_{x}C}_{6}$\n", + "and ${{Li}_{1 - x}CoO}_{2}$ need to be obtained as a function $x$ in\n", + "order to calculate the electric potential of the battery.\n", + "\n", + "LiFePO4 uses metallic lithium as the anode with following\n", + "half-cell and net cell reactions\n", + "\n", + "Eq. 8\u201190 $xLi = x{Li}^{+} + xe^{-}$\n", + "\n", + "Eq. 8\u201191 $x{Li}^{+} + xe^{-} + {Li}_{1 - x}FePO_{4} = LiFePO_{4}$\n", + "\n", + "Eq. 8\u201192 $xLi + {Li}_{1 - x}FePO_{4} = LiFePO_{4}$.\n", + "\n", + "Its electric potential is also a function of $x$, i.e.\n", + "\n", + "*Eq. 8\u201193*\n", + "$\\varepsilon = - \\frac{1}{xf}\\left\\{_{\\ }^{0}G^{LiFePO_{4}} - x\\ ^{0}G^{Li} - G^{{Li}_{1 - x}FePO_{4}} \\right\\} = - \\frac{1}{f}\\left\\{ \\left( \\mu_{Li}^{{Li}_{1 - x}FePO_{4}} - \\ ^{0}\\mu_{Li} \\right) - \\frac{1}{x}\\left( \\mu_{LiFePO_{4}}^{{Li}_{1 - x}FePO_{4}} -_{\\ }^{0}G^{LiFePO_{4}} \\right) \\right\\}$\n", + "\n", + "The value in the first parenthesis in the above equation denotes the\n", + "chemical potential difference of Li between two electrodes, and the\n", + "value in the second parenthesis represents the chemical potential\n", + "difference of $LiFePO_{4}$ between the states in the solution phase of\n", + "${Li}_{1 - x}FePO_{4}$ and by itself. Consequently, Gibbs energy of\n", + "${Li}_{1 - x}FePO_{4}$ needs to be obtained as a function $x$ in order\n", + "to calculate the electric potential of the battery. It is known that\n", + "there are several miscibility gaps in the $FePO_{4}$ and $LiFePO_{4}$\n", + "psuedo-binary system, in which the chemical potentials are constants, so\n", + "is the electric potential, resulting in stable battery output.\n" + ], + "metadata": {} + }, + { + "id": "16f36a6d", + "cell_type": "markdown", + "source": [ + "### Electrochemical cells\n", + "\n", + "A potential difference, i.e. voltage difference, can be generated\n", + "between the electrodes in a cell from differences in the potentials of\n", + "the half-cell reactions. This potential can originate from potential\n", + "differences of two chemically different half-cells (a *galvanic cell*),\n", + "or concentration differences in two otherwise identical half-cells (a\n", + "*concentration cell*). Each type of cell is illustrated below.\n", + "\n", + "The reaction between copper ions and zinc illustrated below represents\n", + "the *net cell reaction* of a *galvanic* cell in which the oxidation of\n", + "Zn(s) occurs at the *anode* electrode, and the reduction of\n", + "Cu+2 occurs at the *cathode* electrode\n", + "\n", + "*Eq. 8\u201111* Cu2+(aq) + Zn(s) = Cu(s) + Zn2+(aq).\n", + "\n", + "The reaction at each electrode, the *half-cell reaction*, includes ions\n", + "and electrons as reactant and/or product species. The anode, oxidation\n", + "reaction is represented by\n", + "\n", + "*Eq. 8\u201112* Zn(s) = Zn2+(aq) + 2 e-.\n", + "\n", + "Electrons are products of anode reactions and flow externally from anode\n", + "to cathode. By convention the activities of the electrons in an\n", + "equilibrium cell are taken as unity. The cathode, reduction reaction is\n", + "written as\n", + "\n", + "*Eq. 8\u201113* Cu2+(aq) + 2 e- = Cu(s)\n", + "\n", + "Electrons are reactants of cathode reactions and supplied by an external\n", + "flow from the anode. In addition to consuming electrons at the cathode\n", + "at the same rate as they are produced at the anode, charge balance is\n", + "maintained in the electrolyte by the generation of Zn2+ ions\n", + "at the same rate that Cu2+ ions are consumed. A schematic\n", + "diagram in illustrates the simple physical relationships in such an\n", + "electrochemical cell.\n", + "\n", + "Figure \u2011: Schematic diagram of a galvanic electrochemical cell\n", + "consisting of a zinc electrode and a copper electrode.\n", + "\n", + "A concentration cell in which an electrochemical potential is developed\n", + "because of concentration differences between otherwise equivalent anode\n", + "and cathode reactions is illustrated below. Such a cell can be produced\n", + "by the oxidation and reduction of copper at two separate electrodes as\n", + "is depicted in the following reactions\n", + "\n", + "*Eq. 8\u201114* Cu(s) = Cu2+(aq, ca) + 2 e- (anode)\n", + "\n", + "*Eq. 8\u201115* Cu2+(aq, cb) + 2 e- = Cu(s) (cathode)\n", + "\n", + "where ca and cb are the respective concentrations\n", + "of Cu2+ in the aqueous solutions at the anode and cathode,\n", + "and ca \\< cb. The net cell reaction is\n", + "\n", + "*Eq. 8\u201116* Cu2+(aq, cb) = Cu2+(aq,\n", + "ca)\n", + "\n", + "where the reaction occurs spontaneously to decrease cb and to\n", + "increase ca until the two concentrations become the same,\n", + "cb = ca. A schematic diagram of such a cell is\n", + "shown in .\n", + "\n", + "Figure \u2011: Schematic diagram of a concentration electrochemical cell\n", + "consisting of two copper electrodes.\n", + "\n", + "A semi-impermeable membrane, or a salt bridge, must exist in such a cell\n", + "to maintain charge balance. As Cu+2 ions are produced at the\n", + "anode and consumed at the cathode, the negatively charged ions in the\n", + "solution, for example Cl-, must be transferred from the\n", + "cathode region to the anode region of the cell to maintain electrically\n", + "neutral solutions.\n", + "\n", + "The above concentration cell provides a good example for illustrating a\n", + "standard notation for depicting an electrochemical cell. This cell can\n", + "be represented by .\n", + "\n", + "Figure \u2011: Standard notation of an electrochemical cell\n", + "\n", + "The anode where oxidation occurs is always denoted on the left, and the\n", + "cathode where reduction occurs is on the right. A *single line*\n", + "separating phases denotes an *interface* between two phases. The above\n", + "anode electrode and reaction of are symbolically represented by\n", + "\n", + "*Eq. 8\u201117* \\| Cu(s) \\| Cu2+(ca).\n", + "\n", + "The interface between the external conductor and Cu(s) is depicted by\n", + "the single line to the left of Cu(s), while the single line between the\n", + "Cu(s) and Cu2+(ca) depicts the interface between\n", + "the anode electrode and the electrolyte solution. Similarly, the cathode\n", + "electrode and reaction of are symbolically represented by\n", + "\n", + "*Eq. 8\u201118* Cu2+(cb) \\| Cu(s) \\|.\n", + "\n", + "A double line between the two copper ions in the notation denotes a\n", + "physical separation of two solution phases, the anode and cathode\n", + "electrolyte regions that exhibit different concentrations of copper ions\n", + "\n", + "*Eq. 8\u201119* Cu2+(ca) \\| \\|\n", + "Cu2+(cb).\n", + "\n", + "These solution phases are physically connected by a semi-impermeable\n", + "membrane or salt bridge that allows a common negative ion, for example\n", + "Cl-, of the solution phases to diffuse from one region to the\n", + "other in order to maintain charge balance as the cell reaction occurs.\n", + "The Cu2+ ions cannot be transported from one region to the\n", + "other.\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/applications_to_electrochemical_systems/electrochemical_cells_and_half_cell_potentials.ipynb b/src/psu410/src/applications_to_electrochemical_systems/electrochemical_cells_and_half_cell_potentials.ipynb new file mode 100644 index 0000000..5782fb8 --- /dev/null +++ b/src/psu410/src/applications_to_electrochemical_systems/electrochemical_cells_and_half_cell_potentials.ipynb @@ -0,0 +1,36 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "7ac97a5b", + "cell_type": "markdown", + "source": [ + "## Electrochemical cells and half cell potentials\n", + "\n", + "An electrochemical system must fulfill certain requirements in order to\n", + "apply equilibrium thermodynamic descriptions of the system, including\n", + "\n", + "- The cell must be reversible when slight changes in conditions\n", + " (potentials, concentrations, pressures, temperature) cause\n", + " electrochemical reactions and an external flow of electrons to occur\n", + " in the direction needed to re-establish equilibrium.\n", + "\n", + "- All non-electrochemical reactions in the system must be prevented as\n", + " such reactions would cause a shift in equilibrium without causing a\n", + " shift in cell potential and thus a driving force for external electron\n", + " flow.\n", + "\n", + "- Chemical reactions must occur only when an external current flows.\n", + " These finite distances for external electron transport can be a short\n", + " as grain size-dimensions in many corrosion reactions, or this\n", + " transport may be through an external electrical conductor connecting\n", + " the anode and cathode half-cell as in batteries.\n", + "\n", + "- Charge balance as well as mass balance is required of all reactions.\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/applications_to_electrochemical_systems/electrolyte_reactions_and_electrochemical_reactions.ipynb b/src/psu410/src/applications_to_electrochemical_systems/electrolyte_reactions_and_electrochemical_reactions.ipynb new file mode 100644 index 0000000..4c3982d --- /dev/null +++ b/src/psu410/src/applications_to_electrochemical_systems/electrolyte_reactions_and_electrochemical_reactions.ipynb @@ -0,0 +1,465 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "7a91931d", + "cell_type": "markdown", + "source": [ + "## Electrolyte reactions and electrochemical reactions\n", + "\n", + "Electrolytes that dissolve in a polar solvent such as water to produce\n", + "ionic species do not necessarily exhibit valence changes. A simple\n", + "example is the strong electrolyte NaCl(s) dissolving in water to produce\n", + "solvated ions\n", + "\n", + "*Eq. 8\u20111* $NaCl(s)\\ \\ = \\ \\ {Na}^{+}(aq)\\ \\ + \\ \\ {Cl}^{\u2013}(aq)$\n", + "\n", + "where the $(aq)$ indicates the ionic species in an aqueous solution. In\n", + "this system, the ion concentrations must become quite large before the\n", + "solution is saturated and can exist in equilibrium with $NaCl(s)$. Its\n", + "reaction constant, defined by , is shown as\n", + "$K_{e} = \\ \\ a_{{Na}^{+}}a_{{Cl}^{-}}$. If the product of the ion\n", + "activities is less than $K_{e}$, the solution is not saturated, and more\n", + "$NaCl(s)\\ \\ $ can be dissolved.\n", + "\n", + "The precipitation of $AgCl(s)$, a weak electrolyte, occurs quite readily\n", + "when ${Cl}^{\u2013}$ ions are added to an aqueous solution containing\n", + "${Ag}^{+}(aq)$:\n", + "\n", + "*Eq. 8\u20112* ${Ag}^{+}(aq)\\ + \\ \\ {Cl}^{\u2013}(aq)\\ = AgCl(s)$\n", + "\n", + "The equilibrium constant for this reaction,\n", + "$K_{e} = \\ \\frac{1}{\\left( a_{{Ag}^{+}}a_{{Cl}^{-}} \\right)}$ is quite\n", + "large, so the equilibrium product of the ion activities, proportional to\n", + "their concentrations, is quite small. In the laboratory, the above\n", + "reaction could occur as a result of adding hydrochloric acid to a silver\n", + "nitrate solution. The accompanying $H^{+}(aq)$ \\[or ${H_{3}O}^{+}(aq)$\\]\n", + "and ${NO_{3}}^{-}(aq)$ ions in solution are not directly involved in the\n", + "silver chloride precipitation reaction so are not shown in reaction\n", + "represented by .\n", + "\n", + "The above ionic equilibria in the $AgCl(s) - H_{2}O$ system is not only\n", + "important for understanding this electrolyte system, but also critical\n", + "in electrochemical systems in which *Ag(s)* undergoes a valence change\n", + "at one electrode and reacts with a ${Cl}^{\u2013}(aq)$ ion to produce\n", + "*AgCl(s)*, and an electron that is externally transported finite\n", + "distances to another electrode. The *oxidation* reaction occurs at the\n", + "Ag/AgCl electrode (*anode* half-cell reaction where electrons are\n", + "*added* into the system)\n", + "\n", + "*Eq. 8\u20113* $Ag(s)\\ \\ + \\ {Cl}^{-}(aq)\\ \\ = \\ \\ AgCl(s)\\ \\ + \\ \\ e^{-}$\n", + "\n", + "A *reduction* reaction occurs at the other electrode (*cathode*\n", + "half-cell reaction where electrons are *consumed* by the reaction)\n", + "\n", + "*Eq. 8\u20114* $\\frac{1}{2}{Cl}_{2}(g) + \\ \\ e^{-} = \\ {Cl}^{-}(aq)\\ \\ $\n", + "\n", + "The *net cell reaction* results in the formation AgCl(s) from its\n", + "elements\n", + "\n", + "*Eq. 8\u20115* $Ag(s)\\ \\ + \\frac{1}{2}{Cl}_{2}\\ (g)\\ = \\ \\ AgCl(s)$\n", + "\n", + "Without knowledge of the physical system under which the reaction is\n", + "occurring, it would not be possible to know if reaction of was a result\n", + "of chlorine gas reacting directly with Ag(s), or if the reaction was\n", + "part of an electrochemical cell with a transport of electrons and ions\n", + "over finite distances. The addition of the two half-cell reactions gives\n", + "the *net cell reaction*, which does not show electrons as either\n", + "reactant or product species and may or may not include ionic species in\n", + "the reaction*.* A schematic of an electrochemical cell for the above\n", + "system is shown in Figure 8\u20111.\n", + "\n", + "Figure \u2011: Schematic diagram of an electrochemical cell consisting of a\n", + "chlorine electrode and a silver-silver chloride electrode.\n", + "\n", + "Oxidation and reduction can occur in electrolyte reactions without\n", + "creating an electrochemical cell. This is the case when chlorine gas\n", + "reacts directly with silver on a Ag(s) surface. Reaction of above is the\n", + "net reaction for this process, but the electrons produced from the\n", + "oxidation of Ag(s) are not transported over finite distances before\n", + "combining with Cl2(g) in its reduction to Cl\u2013(aq).\n", + "No anode or cathode half-cell reactions exist in this system. The\n", + "electrons and ions involved in the reaction move only over atomic-scale\n", + "distances.\n" + ], + "metadata": {} + }, + { + "id": "0b1b59f6", + "cell_type": "markdown", + "source": [ + "### Fuel cells\n", + "\n", + "Fuel cells are devices to convert chemical energy to electricity and\n", + "heat through electrochemical reactions with the fuel and oxygen supplied\n", + "to the anode and cathode, respectively. Typical ions migrating through\n", + "the electrolyte are $H^{+}$, ${OH}^{-}$, ${CO}_{3}^{2 -}$, and\n", + "$O^{2 -}$. In fuel cells with $H^{+}$ as migrating ions, H2\n", + "molecules are dissociated into $H^{+}$ on the anode, which are combined\n", + "with O2 on the cathode to form H2O and release\n", + "heat with the half cell and the net cell reactions in their simplest\n", + "form as shown by for the anode, for the cathode, and for the net cell,\n", + "respectively. Commonly used electrolytes are polymer and phosphoric\n", + "acid, and both anode and cathode reactions are facilitated by catalyst,\n", + "typically platinum. The thermodynamic limit of power which can be\n", + "generated by the fuel cell is represented by\n", + "\n", + "*Eq. 8\u201163*\n", + "$w = - \\mathrm{\\Delta}G = - \\mathrm{\\Delta}_{\\ }^{0}G_{cell} + RTln\\left( P_{H_{2}}P_{O_{2}}^{1/2} \\right)$\n", + "\n", + "For fuel cells with anions as migrating ions, the anions are generated\n", + "on the cathode with H2O formed and heat generated on the\n", + "anode. Their representative cathode reactions are\n", + "\n", + "*Eq. 8\u201164* $\\frac{1}{2}O_{2} + H_{2}O + 2e^{-}{= 2OH}^{-}$\n", + "\n", + "*Eq. 8\u201165* ${\\frac{1}{2}O}_{2} + CO_{2} + 2e^{-} = {CO}_{3}^{2 -}$\n", + "\n", + "*Eq. 8\u201166* $\\frac{1}{2}O_{2} + 2e^{-} = O^{2 -}$.\n", + "\n", + "The anode reaction for is the reaction represented by , operating at low\n", + "temperatures and using catalyst for both electrodes. The anode reactions\n", + "for and are\n", + "\n", + "*Eq. 8\u201167* ${CO}_{3}^{2 -} + H_{2} = H_{2}O + CO_{2} + 2e^{-}$\n", + "\n", + "*Eq. 8\u201168* $O^{2 -} + H_{2} = H_{2}O + 2e^{-}$\n", + "\n", + "respectively. To enable the diffusion of ${CO}_{3}^{2 -}$ and $O^{2 -}$\n", + "through the cathode and the electrolyte, both fuel cells are operated at\n", + "relative high temperatures, with the former typically in molten\n", + "carbonate solutions and the latter through solid oxides. Due to the high\n", + "operating temperatures, fuels are converted to hydrogen within the fuel\n", + "cell itself by a process called internal reforming, removing the need\n", + "for precious-metal catalyst and enabling the use of a variety of fuels.\n", + "The net cell reaction for all three fuel cells is the same as in the\n", + "case of $H^{+}$, represented by .\n" + ], + "metadata": {} + }, + { + "id": "560fe529", + "cell_type": "markdown", + "source": [ + "### Galvanic protection\n", + "\n", + "A galvanic reaction takes place between two different materials at the\n", + "two respective electrodes, each with different tendency to hold on to\n", + "electrons. Consider the following electrochemical cell used to protect\n", + "Cu tanks against oxidation by using a \u201csacrificial\u201d Fe electrode\n", + "\n", + "anode solution cathode\n", + "\n", + "Fe(s) \\| Fe2+ \\| SO4= \\|\n", + "Cu2+\\| Cu(s) \\|\n", + "\n", + "Cathode reduction: Cu2+ + 2 e- = Cu \u03b5o\n", + "(volts) = 0.34\n", + "\n", + "Anode oxidation: Fe(s) = Fe2+ + 2 e- \u03b5o (volts) =\n", + "0.44\n", + "\n", + "Net reaction: Cu2+ + Fe =\\> Cu + Fe2+\n", + "\u03b5o (volts) = 0.78\n", + "\n", + "If the cell has a direct connection between the electrodes, i.e. it has\n", + "a short circuit: \u2206G \u2192 0 and thus \u03b5cell \u2192 0. Since the value\n", + "of \u03b5ocell \\> 0 for the net cell reaction, the equilibrium\n", + "constant *Ke* \\> 1, which means\n", + "\\[Fe2+\\]/\\[Cu2+\\] \\> 1. By assuming an ideal\n", + "electrolyte solution, the activities in *Ke* can be\n", + "represented by concentrations (in molar concentration units), assuming\n", + "that solid Fe and Cu are present at unit activities. If the electrodes\n", + "of Cu and Fe are short circuited while in contact with the same\n", + "\"electrolyte solution\", the final equilibrium concentrations can be\n", + "calculated by the standard equation,\n", + "\n", + "*Eq. 8\u201161* *Ke* = \\[Fe2+\\]/\\[Cu2+\\] =\n", + "exp (-\u2206oG/RT)\n", + "\n", + "or, using the Nernst equations,\n", + "\n", + "*Eq. 8\u201162* \u03b5 = \u03b5o -(RT/z f) ln (\\[Fe2+\\]/\\[Cu2+\\])\n", + "= 0\n", + "\n", + "and the above standard cell potential:\n", + "\\[Fe2+\\]/\\[Cu2+\\] = 2.4\u20221026\n", + "\n", + "With this large ratio it can be seen that the tendency to produce\n", + "Cu+2 ions, i.e., the tendency to corrode the Cu(s), is\n", + "extremely small if a sacrificial Fe electrode is configured in an\n", + "electrochemical cell with the Cu tank.\n" + ], + "metadata": {} + }, + { + "id": "6dd6c6c3", + "cell_type": "markdown", + "source": [ + "### Half cell potentials\n", + "\n", + "When electron current flows between electrodes, reactions are occurring\n", + "at the electrodes and concentration gradients causing polarization\n", + "develop around the electrodes. These gradients result in extraneous\n", + "potentials to occur at the electrodes. In such cases cell equilibrium is\n", + "not established and measured cell potentials are not those for true\n", + "partial equilibrium. If a cell is short-circuited with the electrodes\n", + "connected by a conductor, current will flow until the external potential\n", + "becomes zero, i.e. \u03b5ext = 0, and equilibrium is established\n", + "with same conditions as non-electrochemical systems. If an external\n", + "potential, \u03b5ext, is applied to the cell, chemical reactions\n", + "occur until the cell potential balances to \u03b5ext, and no\n", + "current flows. This potential is called open-circuit voltage (OCV) in\n", + "the literature. It is important to realize that OCV includes all\n", + "reactions that occur on the electrode surface when the electrode is in\n", + "contact with the electrolyte, such as passivation discussed in Chapter .\n", + "Partial equilibrium in a cell is achieved when the cell potential is\n", + "balanced by an applied external potential. In such partial equilibrium\n", + "cases, equilibrium thermodynamic analyses can be used even though the\n", + "cell potential is not zero, i.e. \u03b5cell \u2260 0. This\n", + "differentiates electrochemical systems from other equilibrium systems\n", + "discussed previously.\n", + "\n", + "The number of electrons involved in a net cell reaction is important in\n", + "relating cell potential and the Gibbs energy change for the cell\n", + "reaction. As will be illustrated later in this section, this number\n", + "denotes the number of electrons involved in the half-cell reactions that\n", + "were added to yield the net cell reaction. The electrical work achieved\n", + "by the transport of an electrical charge through a cell potential can be\n", + "written as\n", + "\n", + "*Eq. 8\u201120* $w = z\\ f\\ \\varepsilon$\n", + "\n", + "where *z* represents the moles of electrons in cell reaction, *f* the\n", + "Faraday constant equal to 96,485 J/V/mole-electron, and *\u03b5* the\n", + "potential difference, often referred as electromotive force (emf) in the\n", + "literature. For a system at constant temperature, pressure, and\n", + "composition, this work is the same as the Gibbs energy difference\n", + "between the two electrodes, i.e.\n", + "\n", + "*Eq. 8\u201121* $- \\Delta G = w = zf\\varepsilon$\n", + "\n", + "where the negative sign is because the system does work to the\n", + "surrounding when the Gibbs energy of the system is decreased. When the\n", + "applied external potential is larger than the cell potential, the\n", + "surrounding does work to the system, and a common example is the\n", + "charging of a battery. Thermodynamic relations discussed in previous\n", + "chapters can thus be directly applied to electrochemical systems with\n", + "some common equations shown in .\n", + "\n", + "Table \u2011: Thermodynamic Equations for Electrochemical Cells\n", + "\n", + "*\u2206G = -z f \u03b5*\n", + "\n", + "*\u2206S = - (\u2202\u2206G/\u2202T)P = + z f (\u2202\u03b5/\u2202T)P*\n", + "\n", + "*\u2206H = \\[\u2202(\u2206G/T)/ \u2202(1/T)\\]P = - z f \\[\u2202(\u03b5/T)/\u2202(1/T)\\]P = z f \\[T(\u2202\u03b5/\u2202T)P\n", + "\u2013 \u03b5\\]*\n", + "\n", + "*\u2206CP = (\u2202\u2206H/\u2202T)P = T z f (\u22022\u03b5/\u2202T2)P*\n", + "\n", + "A half-cell reaction potential cannot be measured directly, only its\n", + "potential relative to another half-cell reaction. By convention, a\n", + "standard half-cell potential is measured relative to the standard\n", + "hydrogen half-cell reduction reaction at 25oC (298K) and 1\n", + "bar, which has a defined standard potential of zero volts,\n", + "\n", + "*Eq. 8\u201122* H+(aq, a=1) + e- = 1/2 H2(g,\n", + "1 bar)\n", + "\n", + "with \u03b5o (H+/H2,g) = 0.00 volts. The\n", + "standard half-cell reduction reactions of metals at 25oC are\n", + "for the general reaction\n", + "\n", + "*Eq. 8\u201123* Mz+(aq, a=1) + z e- = M(s)\n", + "\n", + "with \u03b5o (Mz+/M) volts. Half-cell reactions with\n", + "the most positive standard electrode potentials have a tendency to\n", + "spontaneously proceed toward reduction (cathode reactions). Half-cell\n", + "reactions with the most negative standard electrode potentials have a\n", + "tendency to spontaneously proceed toward oxidation (anode reactions).\n", + "\n", + "Consider, for example, a cell made up of a standard hydrogen electrode\n", + "and a standard zinc electrode with \u03b5o\n", + "(H+/H2,g) = 0.00 volts and \u03b5o\n", + "(Zn2+/Zn) = -0.762 volts. Thus, the H+ would tend\n", + "to be reduced, and the zinc metal would tend to be oxidized, and the\n", + "spontaneous reaction if all species had unit activities would be\n", + "\n", + "*Eq. 8\u201124* 2 H+(aq, a=1)+ Zn = H2(1 bar) +\n", + "Zn2+(aq, a=1)\n", + "\n", + "with \u03b5ocell = 0.762 volts and\n", + "$\\mathrm{\\Delta}_{\\ }^{0}G = - 2*96485*\\varepsilon_{cell}^{0}\\ $. The\n", + "cathode half-cell reaction would be the same as , while the anode\n", + "half-cell reaction would be\n", + "\n", + "*Eq. 8\u201125* Zn = Zn2+(aq, a=1) + 2 e-\n", + "\n", + "When ion concentrations and H2 gas do not all have unit\n", + "activities, the Gibbs energy and cell potential of the cell reaction, ,\n", + "becomes\n", + "\n", + "*Eq. 8\u201126*\n", + "$\\mathrm{\\Delta}G = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{a_{{Zn}^{2 +}}P_{H_{2}}}{\\left( a_{H^{+}} \\right)^{2}}$\n", + "\n", + "*Eq. 8\u201127*\n", + "$\\varepsilon_{cell} = \\varepsilon_{cell}^{0} - \\frac{RT}{zf}\\ln\\frac{a_{{Zn}^{2 +}}P_{H_{2}}}{\\left( a_{H^{+}} \\right)^{2}}$\n", + "\n", + "The standard reduction potentials of some common metals at\n", + "25oC are given in .\n", + "\n", + "Table \u2011: Standard reduction potentials of some common metals\n", + "\n", + "A cell reaction can be established by different half-cell reactions. For\n", + "example, the following reaction can be derived from two different cells\n", + "\n", + "*Eq. 8\u201128* $3\\ {Fe}^{2 + \\ }\\ = \\ \\ 2\\ {Fe}^{3 +}\\ \\ + \\ \\ Fe(s)$\n", + "\n", + "cell A\n", + "\n", + "*Eq. 8\u201129* $3{Fe}^{2 +} + \\ 6\\ e - \\ = \\ \\ 3Fe(s)$ \u03b5o1 =\n", + "-0.440 V\n", + "\n", + "*Eq. 8\u201130* $2\\ Fe(s)\\ \\ = \\ \\ 2\\ {Fe}^{3 +}\\ \\ + \\ 6\\ e -$\n", + "\u03b5o2 = +0.036 V\n", + "\n", + "cell B\n", + "\n", + "*Eq. 8\u201131* $2{Fe}^{2 +}\\ = \\ 2{Fe}^{3 + \\ \\ } + 2\\ e^{-}$\n", + "*\u03b5o4 = -0.772 V*\n", + "\n", + "*Eq. 8\u201132* ${Fe}^{2 + \\ }\\ + \\ 2\\ e - \\ \\ = \\ \\ Fe(s)$ *\u03b5o5\n", + "= -0.440 V*\n", + "\n", + "Both give the same net reaction shown by , but with 6 and 2 electrons\n", + "and standard cell potentials being \u03b5ocell A = -0.404 V and\n", + "\u03b5ocell B = -1.212 V, respectively. However, the standard\n", + "Gibbs energies of both cells are the same, i.e.\n", + "\n", + "*Eq. 8\u201133* $\\mathrm{\\Delta}_{\\ }^{0}G$*cell A = -6 f (-0.404)\n", + "= + 2.424 f*\n", + "\n", + "*Eq. 8\u201134* $\\mathrm{\\Delta}_{\\ }^{0}G$*cell B = -2 f (-1.212)\n", + "= + 2.424 f*\n", + "\n", + "It is shown that $\\mathrm{\\Delta}_{\\ }^{0}G$ values are independent of\n", + "half-cell reactions and depend only on the net reaction because the net\n", + "reaction is neutral in electron and balanced in mass.\n" + ], + "metadata": {} + }, + { + "id": "d098364d", + "cell_type": "markdown", + "source": [ + "### Ion transport membranes\n", + "\n", + "Ion transport membranes (ITMs) are ceramic membranes conducting both\n", + "electrons and oxygen ions, but no other species. The chemical potential\n", + "difference of oxygen between two sides of a membrane provides the\n", + "driving force for oxygen to diffuse through the membrane. Commonly used\n", + "ITM oxides include perovskite and fluorite, with the chemical formula of\n", + "ABO3 and AO2, respectively, typically with more\n", + "than one elements in the A-site and / or the B-site to tailor the\n", + "electron and ionic conductivities. Key thermodynamic properties of ITM\n", + "oxides are their stability in service environments, vacancy\n", + "concentrations in the oxygen and cation sites, valances of cations. On\n", + "the high oxygen partial pressure side, the reaction is as the following\n", + "\n", + "*Eq. 8\u201169* $\\frac{1}{2}O_{2} + 2e^{-} = O^{2 -}$\n", + "\n", + "At the same time the number of oxygen vacancy is reduced, resulting in a\n", + "lower concentration of oxygen vacancy and higher oxygen activity in the\n", + "oxide on the high oxygen partial pressure side. On the low oxygen\n", + "partial pressure side, the reaction is reversed to produce oxygen\n", + "molecules, i.e.\n", + "\n", + "*Eq. 8\u201170* $O^{2 -} = \\frac{1}{2}O_{2} + 2e^{-}$\n", + "\n", + "This reaction results in higher oxygen vacancy concentration and lower\n", + "oxygen activity. At both sides, the charge neutrality is compensated by\n", + "the valance changes of cations, resulting in the electron flow in the\n", + "opposite direction of oxygen diffusion. The ionic conductivity is\n", + "dictated by the oxygen transportation across the membrane with the\n", + "driving force of the following net reaction\n", + "\n", + "*Eq. 8\u201171*\n", + "$\\frac{1}{2}O_{2}\\left( P_{high} \\right) = \\frac{1}{2}O_{2}\\left( P_{low} \\right)$\n", + "\n", + "*Eq. 8\u201172*\n", + "$\\mathrm{\\Delta}G = 0.5RTln\\left( \\frac{P_{low}}{P_{high}} \\right)$.\n", + "\n", + "The oxygen transportation is closely related to the concentration of\n", + "oxygen vacancy in the membrane, which is obtained by minimizing the\n", + "Gibbs energy of the phase under given temperature and oxygen partial\n", + "pressure conditions. High vacancy concentrations can be obtained by\n", + "cation dopants with lower valances or small energy differences between\n", + "various valance states. However, at the same time, high vacancy\n", + "concentrations reduce the thermodynamic stability of the membrane, which\n", + "may result in its decomposition into more stable phases.\n" + ], + "metadata": {} + }, + { + "id": "728bc14f", + "cell_type": "markdown", + "source": [ + "### Metastability and passivation\n", + "\n", + "Our modern industrial society is built on various metals such as Fe, Ni,\n", + "Al, Ti, and Zr alloys which are reactive, but exhibit extraordinary\n", + "kinetic stabilities in oxidizing environments due to the existence of a\n", + "thin reaction product film on the surface. This film effectively\n", + "isolates the metal from the corrosive environment, a phenomenon called\n", + "passivation. One interesting experiment by Faraday in 1836 who reported\n", + "that iron corrodes freely in dilute nitric acid, while in concentrated\n", + "nitric acid, no reaction apparently occurred. To understand this\n", + "phenomenon, let us exam a simple, schematic Pourbaix diagram for the\n", + "iron-water system shown in .\n", + "\n", + "Figure \u2011: Schematic Pourbaix diagram for iron illustrating the\n", + "resolution of the Faraday paradox in the corrosion of iron in nitric\n", + "acid \\[61\\]. Lines (a) (b), and (c) correspond to the equilibria: (a)\n", + "$H^{+} + e^{-} = 1/2H_{2}$; (b) $O_{2} + 4H^{+} + 4e^{-} = 2H_{2}O$; (c)\n", + "$NO_{3}^{-} + 3H^{+} + 2e^{-} = HNO_{2} + H_{2}O$, respectively\n", + "\n", + "For iron in deaerated acid solution, the partial anodic and cathodic\n", + "reactions are Line 1 $\\left( Fe/{Fe}^{2 +} \\right)$ and Line (a),\n", + "respectively, resulting in a corrosion potential that lies between Lines\n", + "$1$ and $a$. In oxygenated (aerated) solutions, the corrosion potential\n", + "may lie between Lines $1$ and $b$, because the reduction of oxygen is a\n", + "possible (likely) cathodic reaction. Since dilute HNO3 is\n", + "only a weak oxidizing agent, the principal cathodic reaction was most\n", + "likely hydrogen evolution, and hence the corrosion potential is expected\n", + "to lie between Lines $1$ and $a$ at relatively high pH, as shown. Since\n", + "the $Fe/{Fe}^{2 +}$ reaction is relatively fast compared with\n", + "$H^{+}/H_{2}$ on iron, if the corrosion potential is situated below the\n", + "extension of Line 2 (Fe/Fe3O4) into the\n", + "${Fe}^{2 +}$ stability region, Fe3O4 cannot form\n", + "on the surface, even as a metastable phase.\n", + "\n", + "On the other hand, concentrated HNO3 is a strong oxidizing\n", + "agent due to the reaction:\n", + "$NO_{3}^{-} + 3H^{+} + 2e^{-} = HNO_{2} + H_{2}O$, so that the corrosion\n", + "potential can lie anywhere between Lines $1$ and $c$ at low pH. Since\n", + "Reaction (2) is likely to be fast, the corrosion potential will be high\n", + "and certainly will be more positive than the extension of Line 2 into\n", + "the stability region for ${Fe}^{2 +}$ at low pH. Therefore,\n", + "Fe3O4 becomes metastable and can form between the\n", + "aqueous solution and iron. The thickness of this\n", + "Fe3O4 film depends on its dissolution rate into\n", + "the aqueous solution and its growth rate at the interface with iron,\n", + "which depends on the diffusion of ionic species across the film. Its\n", + "existence results in passivity and the observed kinetic inactivity of\n", + "iron in this medium. When the potential becomes even more positive above\n", + "the extension of Line $3$, Fe2O3 may form on\n", + "Fe3O4 as an additional metastable phase, resulting\n", + "in the commonly observed bilayer structure.\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/applications_to_electrochemical_systems/index.ipynb b/src/psu410/src/applications_to_electrochemical_systems/index.ipynb new file mode 100644 index 0000000..7f8042b --- /dev/null +++ b/src/psu410/src/applications_to_electrochemical_systems/index.ipynb @@ -0,0 +1,33 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "b425decf", + "cell_type": "markdown", + "source": [ + "# Applications to electrochemical systems\n", + "\n", + "The two basic types of chemical processes involving ions as reactant\n", + "and/or product species are electrolyte reactions and electrochemical\n", + "reactions. *Electrolyte reactions* are accompanied by the *atomic-scale\n", + "movement* of ionic species and possibly electrons. Chemical changes that\n", + "produce changes in valence and *electron and ion transport over finite\n", + "distances* constitute an area of science termed *electrochemistry*. The\n", + "latter chemical changes occur in an electrochemical cell comprised of\n", + "two electrodes, an anode and a cathode, which are coupled by an\n", + "electrolyte and an external electron conductor. Most thermodynamic\n", + "concepts and analyses described in previous chapters remain unchanged\n", + "when applied to electrochemistry, but the analysis of electrochemical\n", + "systems does require some new terminology, new definitions, and new\n", + "conventions. The primary focus of this chapter is on applications of\n", + "thermodynamics to electrochemical reactions that involve either aqueous\n", + "electrolyte solutions or solid state electrolytes. Since all\n", + "electrochemical systems include ionized species as reactant and/or\n", + "product species, electrolyte reactions will also be discussed.\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/calpahd_modeling_of_thermodynamics/importance_of_lattice_stability.ipynb b/src/psu410/src/calpahd_modeling_of_thermodynamics/importance_of_lattice_stability.ipynb new file mode 100644 index 0000000..d25171d --- /dev/null +++ b/src/psu410/src/calpahd_modeling_of_thermodynamics/importance_of_lattice_stability.ipynb @@ -0,0 +1,121 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "ba6b61d7", + "cell_type": "markdown", + "source": [ + "## Importance of lattice stability\n", + "\n", + "For modeling of Gibbs energy of individual phases, it is necessary to\n", + "define the values of $_{\\ }^{0}G_{i}$ in . However, the independent\n", + "component *i* may not be stable in the crystal structure of the phase\n", + "under consideration, so its Gibbs energy could not be obtained directly\n", + "from experiments and must be estimated with respect to the Gibbs energy\n", + "of their stable crystal structure. In the pioneering work by Kaufman\n", + "\\[47\\], this Gibbs energy difference was termed as lattice stability and\n", + "obtained through extrapolations in either temperature-pressure or\n", + "temperature-composition phase diagrams. It is evident from that the\n", + "values of $_{\\ }^{0}G_{i}$ and $_{\\ }^{M}G$ jointly contribute to the\n", + "Gibbs energy of the solution, and Kaufman had to simplify the treatment\n", + "of $_{\\ }^{M}G$ in order to show the importance of the concept of\n", + "lattice stability. Using ideal or regular solution models, Kaufman was\n", + "able to define the lattice stability for pure elements and remarkably\n", + "reproduce many features of binary phase diagrams by introducing the\n", + "interaction parameters afterwards.\n", + "\n", + "Over the years, there had been various revisions of lattice stability\n", + "values for common crystal structures \\[48\\], and every revision\n", + "necessitates the re-evaluation of interaction parameters in the solution\n", + "phase shown in . It was until the lattice stability values established\n", + "by the Scientific Group Thermodata Europe (SGTE) \\[52\\] that the\n", + "development of binary thermodynamic models using the same thermodynamic\n", + "models of pure elements became possible, and those binary models thus\n", + "developed in different groups around the world can be combined to create\n", + "thermodynamic models of ternary and multi-component systems. Clearly,\n", + "any further modifications of the SGTE pure element database will require\n", + "the re-modeling of all binary and ternary systems in which the models of\n", + "pure elements are changed. This challenge is briefly addressed in the\n", + "later part of this chapter.\n", + "\n", + "A less addressed issue is the Gibbs energy of end-members in\n", + "non-stoichiometric compounds, i.e. , where each sublattice contains only\n", + "one element. In case all sublattices are occupied by the same element,\n", + "it is the lattice stability of the elements in the structure of the\n", + "compound. Since the stable composition ranges of non-stoichiometric\n", + "compounds are typically small, the existing method cannot be used to\n", + "reliably evaluate the Gibbs energy of end-members, and currently there\n", + "is not a commonly accepted lattice stability database for compounds.\n", + "Most values used in the existing databases have been either roughly\n", + "estimated or computed from first-principles calculations. Such a\n", + "standard database is highly desirable in order to make various models of\n", + "compounds compatible.\n", + "\n", + "In an effort to compare the lattice stability from the CALPHAD models\n", + "and the first-principles calculations, Wang et al. \\[53\\] systematically\n", + "calculated the total energies of 78 pure elements at zero Kelvin in the\n", + "face-centered-cubic (fcc), body-centered-cubic (bcc), and\n", + "hexagonal-close-packed (hcp) crystal structures using the projector\n", + "augmented-wave (PAW) method within the generalized gradient\n", + "approximation (GGA). The calculated values are compared with the values\n", + "in the SGTE database as shown in and . For non-transition metal\n", + "elements, the differences between the SGTE data and the PAW-GGA data are\n", + "typically around 1\u223c2 kJ/mole-of-atoms or less, while for some transition\n", + "metal elements, the differences can be quite large, for example, as high\n", + "as about 54 kJ/mole-of-atoms for and about 40 kJ/mole-of-atoms for . and\n", + "present the differences between the PAW-GGA data and the SGTE data, for\n", + "elements from the Ti group to the Ni group, respectively.\n", + "\n", + "Table \u2011: Lattice stability Ebcc-fcc (kJ/mole-of-atoms).\n", + "\n", + "Table \u2011: Lattice stability Ehcp-fcc (kJ/mole-of-atoms).\n", + "\n", + "Figure \u2011: Lattice stability difference between bcc and fcc, , for\n", + "selected elements between PAW-GGA and SGTE\n", + "\n", + "Figure \u2011: Lattice stability difference between hcp and fcc, , for\n", + "selected elements between PAW-GGA and SGTE\n", + "\n", + "The large differences between the first-principles calculations and the\n", + "SGTE data could partly be attributed to the instability of the\n", + "higher-energy phases, the entropies of which at finite *T* become\n", + "abnormal. The lattice instabilities along the tetragonal transformation\n", + "path between fcc and bcc structures with the continuous change of the\n", + "c/a ratio defined in a bcc-based tetragonal lattice are demonstrated for\n", + "bcc Mo, Ta, W in and for fcc Al, Cu, Ni in . It is shown that the fcc\n", + "structure of bcc Mo, Ta, and W is a local maximum with respect to the\n", + "tetragonal transformation, and the higher the maximum is, the larger the\n", + "discrepancy between the SGTE data and the present PAW-GGA data, while\n", + "for fcc Al, Cu, Ni, the bcc structure is at a local maximum. Similarly,\n", + "the lattice instabilities along the tetragonal transformation path for\n", + "the hcp metals Ru and Os as shown in . The behavior of energy against\n", + "c/a ratio of these two hcp metals is very similar to those of fcc\n", + "elements.\n", + "\n", + "Figure \u2011: Total energy, , along the Bain deformation path between bcc\n", + "and fcc for Mo, Ta, and W.\n", + "\n", + "Figure \u2011: Total energy, , along the tetragonal transformation path\n", + "between bcc and fcc for Ni, Al, and Cu.\n", + "\n", + "Figure \u2011: Total energy, , along the tetragonal transformation path for\n", + "Ru and Os\n", + "\n", + "It can be concluded that a fcc structure for elements with bcc being the\n", + "ground state or a bcc structure for elements with fcc being the ground\n", + "state, is unstable with respect to the tetragonal transformation. For an\n", + "unstable structure, the harmonic description of its vibrational entropy\n", + "is thermodynamically incorrect since the potential surface seen by the\n", + "lattice ion can no longer be approximated by a parabola. If an unstable\n", + "structure of a pure element is stabilized at high temperatures, its\n", + "entropy has to be abnormal. The instability issue has been recently\n", + "addressed by ab initio molecular dynamics simulations at high\n", + "temperatures using W as an example \\[54\\], which is beyond the scope of\n", + "the book and thus not discussed here.\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/calpahd_modeling_of_thermodynamics/index.ipynb b/src/psu410/src/calpahd_modeling_of_thermodynamics/index.ipynb new file mode 100644 index 0000000..4037370 --- /dev/null +++ b/src/psu410/src/calpahd_modeling_of_thermodynamics/index.ipynb @@ -0,0 +1,59 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "58cfeeff", + "cell_type": "markdown", + "source": [ + "# CALPAHD modeling of thermodynamics\n", + "\n", + "CALPHAD modeling of thermodynamics was pioneered by Kaufman \\[47\\] and\n", + "has been reviewed in details by Saunders and Miodownik \\[48\\] and Lukas,\n", + "Fries and Sundman \\[49\\]. Information on features of software tools for\n", + "CALPHAD modeling can be found at two series of publications in the\n", + "CALPHAD journal \\[50-51\\]. The key feature of the CALPHAD method is the\n", + "modeling of Gibbs energy of individual phases using both thermodynamic\n", + "and phase equilibrium data. The main significances of the CALPHAD method\n", + "are as follows\n", + "\n", + "1. It enabled the development of the concept of lattice stability, i.e.\n", + " the energy difference of a pure element with stable and non-stable\n", + " crystal structures;\n", + "\n", + "2. The Gibbs energy expression of each phase covers the full\n", + " temperature, pressure, and composition spaces including both stable\n", + " and non-stable regions of the phase. This enables the evaluation of\n", + " the Gibbs energy of a system as a function of non-equilibrium state,\n", + " i.e. with \u03be as an independent variable;\n", + "\n", + "3. Thermodynamic data are usually obtained by measurements of heat such\n", + " as enthalpy of transition and heat capacity as discussed in Chapter\n", + " , which bear large uncertainties typically in the range of\n", + " kilojoules per mole-of-atom. On the other hand, phase equilibrium\n", + " data as discussed in Chapter , though more accurate, only contain\n", + " information on compositions of phases at equilibria, i.e., the\n", + " relative Gibbs energy of phases at equilibrium. The combination of\n", + " these two sets of data is foundational in CALPHAD modeling that\n", + " allows for accurate modeling of thermodynamic properties of\n", + " individual phases and reliable calculations of phase stability and\n", + " driving forces;\n", + "\n", + "4. It provides a framework to model thermodynamic properties of\n", + " multi-component systems of industrial importance, enabling the\n", + " computational materials design. It has also been extended to model a\n", + " range of properties of individual phases in multi-component systems\n", + " such as diffusion coefficients, elastic coefficients, and thermal\n", + " expansion, supplying input data for computational simulations of\n", + " phase transformations during materials processing.\n", + "\n", + "In this chapter, the basics of CALPHAD modeling of Gibbs energy of\n", + "individual phases are presented. For detailed implementations in various\n", + "software packages and modeling procedures, readers are referred to the\n", + "references listed above.\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_pure_elements.ipynb b/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_pure_elements.ipynb new file mode 100644 index 0000000..36b8aa5 --- /dev/null +++ b/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_pure_elements.ipynb @@ -0,0 +1,76 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "04834bb0", + "cell_type": "markdown", + "source": [ + "## Modeling of pure elements\n", + "\n", + "In modeling the Gibbs energy of pure elements in the SER structure in\n", + "terms of , the coefficients in are evaluated using the heat capacity\n", + "data, $b^{'}$ in is evaluated by the value of $S_{298.15}$, and $a$ in\n", + "is evaluated by $H_{298.15}^{SER} = 0$, respectively. For the high\n", + "temperature phase, the enthalpy of transformation from the low\n", + "temperature phase to the high temperature phase,\n", + "${\\mathrm{\\Delta}H}_{trans}^{}$, can be measured by calorimetry methods\n", + "discussed in Chapter , and the entropy of transformation,\n", + "${\\mathrm{\\Delta}S}_{trans}^{}$, is then calculated using the\n", + "equilibrium condition of equal Gibbs energy of the two phases, i.e.\n", + "\n", + "*Eq. 6\u20111*\n", + "${\\mathrm{\\Delta}S}_{trans}^{} = \\frac{{\\mathrm{\\Delta}H}_{trans}^{}}{T_{trans}^{}}$\n", + "\n", + "where $T_{trans}^{}$ is the transition temperature.\n", + "${\\mathrm{\\Delta}H}_{trans}^{}$ and ${\\mathrm{\\Delta}S}_{trans}^{}$ are\n", + "then used to evaluate the integration constants, $b^{'}$ and $a$, in the\n", + "place of $S_{298.15}$ and $H_{298.15}^{SER}$ for the structure in the\n", + "SER state.\n", + "\n", + "This works well for the stable temperature range of each phase. However,\n", + "there is an issue in extrapolation above and below the melting\n", + "temperature (Tm). It is known that the heat capacity of the\n", + "solid phase, , increases with temperature, while that of the liquid\n", + "phase, , is typically constant. The extrapolation of the Gibbs energy of\n", + "a solid phase to above its melting temperature can result in the solid\n", + "phase becoming more stable than the liquid phase at high temperatures.\n", + "By the same token, the extrapolation of the Gibbs energy of a liquid\n", + "phase to below its melting temperature can result in the liquid phase\n", + "becoming more stable than the solid phase at low temperatures. To\n", + "address this problem, it is proposed by SGTE that the heat capacity of\n", + "the solid phase approaches that of the liquid at high temperatures, and\n", + "that of the liquid phase approaches that of the solid phase at low\n", + "temperatures using the following equations\n", + "\n", + "1. for solid at T\\>Tm\n", + "\n", + "*Eq. 6\u20112*\n", + "\n", + "*Eq. 6\u20113*\n", + "\n", + "2. for liquid at T\\m\n", + "\n", + "*Eq. 6\u20114*\n", + "\n", + "*Eq. 6\u20115*\n", + "\n", + "As an example, the heat capacity of solid fcc Al and liquid Al in the\n", + "SGTE pure element database is plotted in . It can be seen that the heat\n", + "capacity of fcc Al approaches that of liquid Al at high temperatures,\n", + "while the heat capacity of liquid Al approaches that of fcc Al at low\n", + "temperatures. It ensures that the liquid Al is stable at high\n", + "temperatures, and fcc Al is stable at low temperatures. However, this\n", + "simple model for liquid is often not satisfactory, in comparison with\n", + "available experimental data in supercooled liquid, particularly those\n", + "systems with glass transitions. New models are thus needed and are being\n", + "developed in the CALPHAD community.\n", + "\n", + "Figure \u2011: Heat capacity of fcc Al solid and liquid as a function of\n", + "temperature\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_random_solution_phases.ipynb b/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_random_solution_phases.ipynb new file mode 100644 index 0000000..7789f29 --- /dev/null +++ b/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_random_solution_phases.ipynb @@ -0,0 +1,63 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "420083e9", + "cell_type": "markdown", + "source": [ + "## Modeling of random solution phases\n", + "\n", + "Depending on the degree of short-range ordering in a solution phase,\n", + "various Gibbs energy models are available as discussed in Chapter . When\n", + "the short-range ordering is weak, it can be accounted for by the\n", + "composition-dependence of excess Gibbs energy in a binary system in\n", + "terms of the Redlich-Kister polynomial as follows\n", + "\n", + "*Eq. 6\u20117*\n", + "\n", + "where the interaction parameters, , can be temperature dependent or even\n", + "have contributions from heat capacity in the form of when data is\n", + "available. shows that and are symmetrical with respect of composition,\n", + "while is asymmetrical. Their individual contributions to the excess\n", + "Gibbs energy are shown in with all interaction parameters being -30,000\n", + "J/mole-of-atoms.\n", + "\n", + "Figure \u2011: Contributions of interactions parameters to the excess Gibbs\n", + "energy\n", + "\n", + "It can be seen in that even though all interaction parameters are\n", + "negative, the asymmetrical feature of results in the curvature change in\n", + "the excess Gibbs energy as a function of composition. This indicates the\n", + "tendency to form a miscibility gap at low temperatures. The interaction\n", + "parameters are to be evaluated from the data of enthalpy, entropy, and\n", + "heat capacity of mixing. The experimental data on enthalpy of mixing are\n", + "available for the liquid phase in some systems, but typically very\n", + "limited for solid solution phases. The first-principles calculations can\n", + "predict the enthalpy, entropy, and heat capacity of mixing in solid\n", + "solution phases using the dilute solution approach with one solute atom\n", + "in a supercell and the CPA/CE/SQS approach for concentrated solutions as\n", + "discussed in Chapter . It demonstrates again that the interaction\n", + "parameters and the lattice stability jointly determine the Gibbs energy\n", + "of an individual phase. The change of lattice stability requires the\n", + "re-evaluation of interaction parameters.\n", + "\n", + "For individual phases with strong short-range ordering, quasichemical or\n", + "associated models can be used. As discussed in Chapter , with fixed\n", + "composition in the system, the amounts of various chemical bonds or\n", + "associates are related through the mass conservation in the system and\n", + "are calculated through the minimization of Gibbs energy of the phase\n", + "with given temperature, pressure, and the amount of each independent\n", + "components. The model parameters include the formation energy of bonds\n", + "or associates and interaction between various bonds or associates,\n", + "noting that pure elements can be considered as the simplest associates.\n", + "The interactions between pure elements can be predicted from\n", + "first-principles calculations as mentioned above, but currently there\n", + "are no efficient approaches to predict the interactions between\n", + "associates from first-principles calculations.\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_solution_phases_with_longrange_ordering.ipynb b/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_solution_phases_with_longrange_ordering.ipynb new file mode 100644 index 0000000..eeaf83f --- /dev/null +++ b/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_solution_phases_with_longrange_ordering.ipynb @@ -0,0 +1,170 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "73e38aa7", + "cell_type": "markdown", + "source": [ + "## Modeling of solution phases with long-range ordering\n", + "\n", + "The commonly used Gibbs energy model is shown in Chapter with the\n", + "crystal lattice divided into sublattices, often referred to as the\n", + "compound energy formalism \\[55\\]. The Gibbs energy of end-members\n", + "represented by plays the same important role for solution phases with\n", + "sublattice as the lattice stability for random solution phases. The\n", + "end-members are modeled in the same way as the stoichiometric phases\n", + "discussed in Chapter . The enthalpy and entropy of mixing in each\n", + "sublattice can be predicted by first-principles calculations using the\n", + "dilute solution and SQS approaches discussed in Chapter and modeled in\n", + "the same way as the random solution discussed in Chapter .\n", + "\n", + "It is important to realize that with a simple two-sublattice model of\n", + "(A,B)a(C,D)b, the miscibility gap can easily form\n", + "even without any interaction parameters when the Gibbs energy of\n", + "end-members differ from each other significantly. The contribution of\n", + "end-members to the Gibbs energy of the phase, i.e. , is re-written as\n", + "follows and schematically shown in ,\n", + "\n", + "*Eq. 6\u20118*\n", + "\n", + "Figure \u2011: Schematic diagrams depicting (a) the concentration square with\n", + "the site fractions of B and D on the horizontal and vertical axes,\n", + "respectively, and (b) Gibbs energy reference plan for\n", + "(A,B)a(C,D)b, as represented by .\n", + "\n", + "From b, it is evident that there is a strong tendency to form a\n", + "miscibility gap between the composition sets of\n", + "(A)a(C)b and (B)a(D)b due to\n", + "their lower Gibbs energies than the other two end-members. Since it\n", + "would be rare for all four end-members to have their Gibbs energy values\n", + "equal, the miscibility gap in this type of phases is practically\n", + "inevitable at low temperatures. An example is shown in for the complex\n", + "titanium niobium carbonitride (Ti,Nb)(C,N). The solid lines parallel to\n", + "the direction from NbC to TiN are tie-lines. The Gibbs energy values of\n", + "TiC, TiN, NbC and NbN are -144495, -229236, -132324, and -179772\n", + "J/mole-of-atoms, respectively. The Gibbs energy value of TiN is\n", + "significantly lower than the other values, resulting in the tie-lines\n", + "originating from the TiN corner.\n", + "\n", + "Figure \u2011: Miscibility gap in (Ti,Nb)(C,N) at 1673K with the straight\n", + "lines in the middle of the plot being tie-lines.\n", + "\n", + "The order-disorder transitions can be similarly described with the\n", + "simplest case being a two-sublattice model of\n", + "(A,B)a(A,B)b. When the site fractions of A or B in\n", + "both sublattices are the same, it becomes a random solution model; when\n", + "they are different, the phase is partially ordered; and when there is\n", + "only one component in each sublattice, the phase is fully ordered as a\n", + "stoichiometric compound. The Gibbs energy of this phase is obtained from\n", + "as follows\n", + "\n", + "*Eq. 6\u20119*\n", + "\n", + "$G_{mf} = y_{A}^{I}y_{A}^{II}\\ _{\\ }^{0}G_{A:A} + y_{A}^{I}y_{B}^{II}\\ _{\\ }^{0}G_{A:B} + y_{B}^{I}y_{A}^{II}\\ _{\\ }^{0}G_{B:A} + y_{B}^{I}y_{B}^{II}\\ _{\\ }^{0}G_{B:B} + + aRT\\left( y_{A}^{I}\\ln y_{A}^{I} + y_{B}^{I}\\ln y_{B}^{I} \\right) + bRT\\left( y_{A}^{II}\\ln y_{A}^{II} + y_{B}^{II}\\ln y_{B}^{II} \\right) + y_{A}^{II}y_{A}^{I}y_{B}^{I}\\ L_{A,B:A} + y_{B}^{II}y_{A}^{I}y_{B}^{I}\\ L_{A,B:B} + y_{A}^{I}y_{A}^{II}y_{B}^{II}\\ L_{A:A,B} + y_{B}^{I}y_{A}^{II}y_{B}^{II}\\ L_{B:A,B} + y_{A}^{I}y_{B}^{I}\\ y_{A}^{II}y_{B}^{II}L_{A,B:A,B}$\n", + "\n", + "where the superscript denotes the sublattice, and column and comma\n", + "separate sublattices and interaction components, respectively. The\n", + "relationship between site fraction and over-all atomic fractions in such\n", + "a two-sublattice model can be represented by and schematically shown in\n", + ". The two red dashed lines represent the phase with\n", + "$x_{B} = \\frac{a}{a + b}$, but different a/b ratios. Along the red\n", + "dashed lines, the phase can adjust the site fraction to minimize its\n", + "Gibbs energy, i.e. it has one internal degree of freedom to be either\n", + "disordered on the blue diagonal line between A:A and B:B, or ordered at\n", + "anywhere else. The interplay of interaction parameters and site\n", + "fractions is depicted where $L_{A,B:A}$, $L_{A,B:B}$, $L_{A:A,B}$, and\n", + "$L_{B:A,B}$ affect the four sides, and $L_{A,B:A,B}$ influences the\n", + "center part.\n", + "\n", + "Figure \u2011: Schematic composition square of\n", + "(A,B)a(A,B)b.\n", + "\n", + "When fully disordered with $y_{A}^{I} = \\ y_{A}^{II} = x_{A}$ and\n", + "$y_{B}^{I} = y_{B}^{II} = x_{B}$, becomes\n", + "\n", + "*Eq. 6\u201110*\n", + "\n", + "$$G_{mf} = x_{A}\\left( 1 - x_{B} \\right)_{\\ }^{0}G_{A:A} + x_{A}x_{B}\\ _{\\ }^{0}G_{A:B} + x_{A}x_{B}\\ _{\\ }^{0}G_{B:A} + x_{B}\\left( 1 - x_{A} \\right)\\ _{\\ }^{0}G_{B:B} + + (a + b)RT\\left( x_{A}\\ln x_{A} + x_{B}\\ln x_{B} \\right) + \\ x_{A}x_{B}x_{A}L_{A,B:A} + x_{A}x_{B}x_{B}\\ L_{A,B:B} + x_{A}x_{B}x_{A}\\ L_{A:A,B} + x_{A}x_{B}x_{B}\\ L_{B:A,B} + x_{A}x_{A}x_{B}x_{B}L_{A,B:A,B} = x_{A}_{\\ }^{0}G_{A:A} + x_{B}\\ _{\\ }^{0}G_{B:B} + (a + b)RT\\left( x_{A}\\ln x_{A} + x_{B}\\ln x_{B} \\right) + x_{A}x_{B}\\left\\lbrack \\left(_{\\ }^{0}G_{A:B} +_{\\ }^{0}G_{B:A} -_{\\ }^{0}G_{A:A} -_{\\ }^{0}G_{B:B} \\right) + x_{A}\\left( L_{A,B:A} + L_{A:A,B} \\right) + x_{B}\\ \\left( L_{A,B:B} + L_{B:A,B} \\right) + x_{A}x_{B}L_{A,B:A,B} \\right\\rbrack = (a + b)\\left\\lbrack x_{A}_{\\ }^{0}G_{A} + x_{B}\\ _{\\ }^{0}G_{B} + RT\\left( x_{A}\\ln x_{A} + x_{B}\\ln x_{B} \\right) + x_{A}x_{B}L_{A,B} \\right\\rbrack$$\n", + "\n", + "with\n", + "\n", + "*Eq. 6\u201111* $_{\\ }^{0}G_{A:A} = (a + b)_{\\ }^{0}G_{A}$\n", + "\n", + "*Eq. 6\u201112* $_{\\ }^{0}G_{B:B} = (a + b)_{\\ }^{0}G_{B}$\n", + "\n", + "*Eq. 6\u201113*\n", + "\n", + "$$L_{A,B} = \\left\\lbrack \\left(_{\\ }^{0}G_{A:B} +_{\\ }^{0}G_{B:A} -_{\\ }^{0}G_{A:A} -_{\\ }^{0}G_{B:B} \\right) + x_{A}\\left( L_{A,B:A} + L_{A:A,B} \\right) + x_{B}\\ \\left( L_{A,B:B} + L_{B:A,B} \\right) + x_{A}x_{B}L_{A,B:A,B} \\right\\rbrack/(a + b)$$\n", + "\n", + "where $_{\\ }^{0}G_{A}$, $_{\\ }^{0}G_{B}$, and $L_{A,B}$ are the molar\n", + "Gibbs energy of pure A and B and the molar interaction parameter in the\n", + "disordered solid solution, respectively. It is evident that the\n", + "interatcation paramter $L_{A,B}$ is fully determined by the parameters\n", + "in the ordered phase, but the parameters in the ordered phase are not\n", + "uniquely determined by the interaction parameters in the disordered\n", + "phase.\n", + "\n", + "Due to crystal symmetry, some of the parameters in are related. For\n", + "example, in the BCC A2/B2 ordering with a=b=0.5, the BCC corner or\n", + "center lattice sites are favored by either one type of atoms, but the\n", + "two sublattices are equivalent crystallographically, resulting in\n", + "following relations\n", + "\n", + "*Eq. 6\u201114* $\\ _{\\ }^{0}G_{A:B} = \\ _{\\ }^{0}G_{B:A}$\n", + "\n", + "*Eq. 6\u201115* $L_{A,B:A} = L_{A:A,B}$\n", + "\n", + "*Eq. 6\u201116* $L_{A,B:B} = \\ L_{B:A,B}$\n", + "\n", + "For more complex ordering of BCC lattice such as B32, D03,\n", + "and L21 shown in with ideal compositions being AB,\n", + "A3B, and A2BC, respectively, more sublattices are\n", + "needed, noting that the L21 Heusler structure exists in\n", + "ternary systems only. To use one model to describe all ordering in the\n", + "BCC lattice, the minimum cluster is an irregular tetrahedron with four\n", + "sublattices as depicted in as discussed in the modeling of the Al-Fe\n", + "system \\[56\\]. In such a four sublattice model of\n", + "$(A,B)_{0.25}^{I}(A,B)_{0.25}^{II}(A,B)_{0.25}^{II}(A,B)_{0.25}^{IV}$,\n", + "the site fractions of A2, B2, B32, and D03, are represented\n", + "by $y_{i}^{I} = \\ y_{i}^{II} = y_{i}^{III} = \\ y_{i}^{IV}$,\n", + "$y_{i}^{I} = \\ y_{i}^{II} \\neq y_{i}^{III} = \\ y_{i}^{IV}$,\n", + "$y_{i}^{I} = \\ y_{i}^{III} \\neq y_{i}^{II} = \\ y_{i}^{IV}$, and\n", + "$y_{i}^{I} = \\ y_{i}^{II} \\neq y_{i}^{III} \\neq \\ y_{i}^{IV}$,\n", + "respectively. The site fractions of L21 are the same as those\n", + "of D03 except with at least three components.\n", + "\n", + "Figure \u2011: Atomic structures and four sublattice tetrahedrons of BCC\n", + "disordered and ordered phases.\n", + "\n", + "Another common ordering phenomenon is in the FCC lattice including the\n", + "disordered A1 structure and ordered L10 and L12\n", + "structures as shown in . In the L10 structure, the\n", + "neighboring (001) planes are favored by different atoms, respectively,\n", + "resulting in an ideal composition of AB. While in the L12\n", + "structure, the corners and faces are favored by different atoms,\n", + "respectively, resulting in an ideal composition of A3B. In a\n", + "four sublattice model of\n", + "$(A,B)_{0.25}^{I}(A,B)_{0.25}^{II}(A,B)_{0.25}^{II}(A,B)_{0.25}^{IV}$,\n", + "the site fractions of A1, L10 and L12 are\n", + "represented by $y_{i}^{I} = \\ y_{i}^{II} = y_{i}^{III} = \\ y_{i}^{IV}$,\n", + "$y_{i}^{I} = \\ y_{i}^{II} \\neq y_{i}^{III} = \\ y_{i}^{IV}$, and\n", + "$y_{i}^{I} = \\ y_{i}^{II} = y_{i}^{III} \\neq \\ y_{i}^{IV}$, respectively\n", + "\\[57\\].\n", + "\n", + "Figure \u2011: Atomic structures and four sublattice tetrahedrons of FCC\n", + "disordered and ordered phases.\n", + "\n", + "As mentioned above, the interaction parameters in each sublattice can be\n", + "predicted by first-principles calculations using the dilute solution and\n", + "SQS approaches when there is only one component in each of the remaining\n", + "sublattices. For the interaction involving two components in two or more\n", + "sublattices, i.e. $L_{A,B:A,B}$ in applicable to four sublattice models\n", + "\\[58\\], the energetics from cluster expansion (CE) approach discussed in\n", + "Chapter can be used to evaluate the interaction parameters.\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_stoichiometric_phases.ipynb b/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_stoichiometric_phases.ipynb new file mode 100644 index 0000000..cf55c4c --- /dev/null +++ b/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_stoichiometric_phases.ipynb @@ -0,0 +1,42 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "289b5213", + "cell_type": "markdown", + "source": [ + "## Modeling of stoichiometric phases\n", + "\n", + "The Gibbs energy of a stoichiometric phase can be modeled in the same\n", + "way as that of pure elements discussed above using the data of heat\n", + "capacity, $S_{298.15}$, and enthalpy of formation at 298.15 K (). When\n", + "these data are not available from experiments, they can be predicted by\n", + "first-principles calculations as discussed Chapter . It should be\n", + "pointed out that constraints placed on heat capacity of stoichiometric\n", + "compounds above melting temperatures, i.e. and , have not been\n", + "rigorously implemented in the literature for such modeling, partially\n", + "because the heat capacity of the corresponding liquid solution is not\n", + "well established, and the heat capacity of a compound is often not\n", + "available.\n", + "\n", + "When the data of heat capacity is not available, a simple approach,\n", + "commonly referred to as Neumann\u2013Kopp Rule assuming that the heat\n", + "capacity of formation of is zero, can be used. The Gibbs energy of the\n", + "compound is written as\n", + "\n", + "*Eq. 6\u20116*\n", + "$G = \\sum_{}^{}N_{i}{_{\\ }^{0}G}_{i}^{ref} + \\mathrm{\\Delta}_{f}H - T\\mathrm{\\Delta}_{f}S$\n", + "\n", + "with $\\mathrm{\\Delta}_{f}H$ and $\\mathrm{\\Delta}_{f}S$ modeled as\n", + "constants. An additional draw back of is that the melting temperature of\n", + "the compound can be higher than those of pure elements, and the heat\n", + "capacity of the compound may thus be questionable at temperatures higher\n", + "than the melting temperatures of pure elements due to the non-physical\n", + "contributions from pure elements based on .\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_cerium.ipynb b/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_cerium.ipynb new file mode 100644 index 0000000..7aab4c6 --- /dev/null +++ b/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_cerium.ipynb @@ -0,0 +1,209 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "a5a4b22a", + "cell_type": "markdown", + "source": [ + "## Application to cerium\n", + "\n", + "Cerium (Ce) displays intriguing physical and chemical properties of\n", + "which the most fascinating is its first-order isostructural phase\n", + "transition. This involves a magnetic, high temperature/high volume\n", + "\u201c\u03b3-phase\u201d and a nonmagnetic, low temperature/low volume \u201c\u03b1-phase\u201d, both\n", + "in the same face-centered-cubic (fcc) lattice structure. At 298 K and\n", + "0.7 GPa, the \u03b3\u2192\u03b1 transition is accompanied by a 14-17% volume collapse.\n", + "The Ce phase transition has been studied extensively including our own\n", + "works with two or three microstates \\[55, 64\\].\n", + "\n", + "The simplest model for the system is to consider two microstates:\n", + "ferromagnetic and nonmagnetic. The first-principles calculations of the\n", + "free energy of two Ce microstates are problematic in the absence of\n", + "strong correlation of the *f*-electrons in the DFT Hamiltonian. The\n", + "relative stability of the nonmagnetic (\u201cdelocalized\u201d) Ce 4*f* state to\n", + "that of the magnetic (\u201clocalized\u201d) Ce 4*f* state is greatly\n", + "overestimated in the GGA \\[31-32\\] with spin polarization. The usual\n", + "approach to surmount this is the Dudarev DFT + U method \\[65\\] with the\n", + "on-site Coulomb and exchange interactions as described with a\n", + "Hartree-Fock approximation added to the DFT Hamiltonian. This method\n", + "offers the advantage that only the difference between the Hubbard U (due\n", + "to the energy increase from an electron addition to a specific site) and\n", + "the J (due to the screened exchange energy) need to be specified *a\n", + "priori*.\n", + "\n", + "Evaluation of numerous U \u2013 J values over a 1.0 \u2013 6.0 eV range revealed\n", + "that 1.6 eV gives the most consistent prediction of nonmagnetic Ce and\n", + "magnetic Ce energetics over the range of atomic volumes that includes\n", + "both microstates at 0 K. The energy-volume curves thus obtained is\n", + "plotted in , showing that the nonmagnetic microstate is the ground\n", + "state, and the equilibrium between the two microstates at 0 K is at the\n", + "negative pressure of -0.87GPa. Since and , NTE does not exist in the\n", + "system.\n", + "\n", + "Figure \u2011: Variation of cell energy (eV) with atomic volume\n", + "(\u00c53) for Ce computed with strong correlation based upon\n", + "Dudarev\u2019s method with U \u2013 J = 1.6 eV.\n", + "\n", + "To take into account the possible magnetic disordering in the\n", + "ferromagnetic microstate at finite temperatures, the following\n", + "contribution is added to the free energy of the ferromagnetic microstate\n", + "\n", + "Eq. 9\u20119\n", + "\n", + "where *MS* is the spin moment, and *l* = 3 the orbital\n", + "angular momentum of an *f*-electron. is a generalization of Hund\u2019s rule,\n", + "with total angular momentum . The Helmholtz energies thus obtained for\n", + "both microstates and the system are shown in at several temperatures\n", + "with the tie-lines included. In the figure, the blue, dot-dashed curves\n", + "are for the nonmagnetic microstate, red, solid curves for ferromagnetic\n", + "microstate, cyan shadows for entropy of mixing between two microstates,\n", + "and the circle in (e) is the critical point. The numbers below the black\n", + "dashed line, representing the common tangent curves, mark the transition\n", + "pressures. The 0-K static energies of the nonmagnetic microstate and the\n", + "magnetic microstate are also plotted in (a) using the solid circles and\n", + "dotted lines.\n", + "\n", + "Figure \u2011: Helmholtz energies at (a) 0 K; (b) 100 K; (c) 165 K; (d) 300\n", + "K; (e) 476 K; and (f) 600 K.\n", + "\n", + "The temperature vs volume phase diagram is plotted and compared with\n", + "available experimental data as shown in \\[66\\]. In this figure, the\n", + "volume (V) is normalized to its equilibrium volume (VN) at\n", + "atmospheric pressure and room temperature. In the pressure range of 2.25\n", + "\u2013 3.5 GPa, the system is within the single-phase region at all\n", + "temperatures considered as shown by the five continuous isobaric volumes\n", + "as a function of temperature. In this pressure range, normal thermal\n", + "expansion is observed at both low and high temperatures on each isobaric\n", + "curve where the probability of each microstate does not change\n", + "significantly with temperature. While in the middle temperature range on\n", + "each isobaric curve, the colossal positive thermal expansion (CPTE),\n", + "highlighted by the pink open diamond symbols, exists due to the fast\n", + "increase of probability of the metastable ferromagnetic microstate with\n", + "respect to temperature, i.e. , , and . This CPTE is much higher than the\n", + "individual positive thermal expansions of the stable and metastable\n", + "microstates, respectively.\n", + "\n", + "Figure \u2011: Calculated temperature-volume phase diagram of Ce.\n", + "\n", + "With decreasing pressure, the system reaches a critical point (green\n", + "circle) where the homogeneous single phase becomes unstable, represented\n", + "by and , and both entropy and volume change infinitely. At even lower\n", + "pressure, a miscibility gap forms, and the single phase separates into\n", + "two phases with the same fcc crystal structures, but different magnetic\n", + "spin structures. Inside the miscibility gap, the volume changes\n", + "discontinuously with respect to temperature by the so-called first-order\n", + "transition as shown by the isobaric curve at zero pressure, compared\n", + "well with experimental volume data (solid squares) under ambient\n", + "pressure.\n", + "\n", + "The fraction of the ferromagnetic microstate, *x*mag, in \u03b1-Ce\n", + "(blue) and \u03b3-Ce (red) calculated using is plotted in as a function of\n", + "pressure along the miscibility gap phase boundary. It can be seen that\n", + "the fraction of the ferromagnetic microstate in \u03b1-Ce increases with\n", + "increasing pressure while the fraction of the ferromagnetic microstate\n", + "in \u03b3-Ce decreases. At the critical point, the fraction of ferromagnetic\n", + "microstate is calculated to be 0.58. This is in qualitative agreement\n", + "with the 0.67 value (filled circle) estimated experimentally at the\n", + "critical point.\n", + "\n", + "Figure \u2011: Fraction of ferromagnetic microstate in \u03b1-Ce (blue) and \u03b3-Ce\n", + "(red) along the \u03b3-\u03b1 phase boundary.\n", + "\n", + "The relative volume, *V/VN*, as a function of pressure, is\n", + "plotted as the black solid lines in from 200 to 600 K at 50 K\n", + "increments. The blue and red solid lines correspond to \u03b1-Ce and \u03b3-Ce,\n", + "respectively. Symbols denote experimental data in the literature, except\n", + "the open green circle being the calculated critical point, in good\n", + "agreement with the computed isotherms. In the two-phase miscibility gap\n", + "region, the \u03b3\u2192\u03b1 volume collapse is again noted, with the magnitude of\n", + "the collapse increasing with decreasing *T*. This is shown explicitly by\n", + "the dashed vertical lines at *T* = 200, 250, 300, 350, 400, and 450 K.\n", + "For *T* \\> 476 K, the calculated isotherms show an anomalous slope\n", + "change which closely matches the behavior near *V/VN* = ~0.85\n", + "from experiment.\n", + "\n", + "Figure \u2011: Equation-of-states for Ce. The black solid lines represent the\n", + "calculated isotherms from 200 to 600 K at \u2206T = 50 K increments.\n", + "\n", + "A more complex model is to add the anti-ferromagnetic microstate. Thus\n", + "obtained E-V and Helmholtz energy curves at 0 K are shown in . The\n", + "equilibrium volume energies reveal that the energy of the\n", + "anti-ferromagnetic microstate at the equilibrium volume is close to that\n", + "of the nonmagnetic microstate but substantially lower than that of the\n", + "ferromagnetic microstate. It should be noted that the magnetic spin\n", + "disordering in the system is taken into account by the two magnetic\n", + "microstates, and the contribution denoted by the mean-field theory, i.e.\n", + ", should thus not be added to either magnetic microstate to avoid double\n", + "counting. The predicted critical point values are 546 K and 2.05 GPa,\n", + "closer to the experimental data than the previous predication with two\n", + "microstates as shown in the temperature-pressure phase diagram in in\n", + "comparison with experimental data.\n", + "\n", + "Figure \u2011: (a) dot-dashed line with **\u25cb** (red), dashed line **\u25d1**\n", + "(blue), and solid line with \u25cf (blue) represent the 0 K static total\n", + "energies for nonmagnetic (NM), anti-ferromagnetic (AFM), and\n", + "ferromagnetic (FM) microstates of Ce, respectively. (b) The solid lines\n", + "denote Helmholtz energy (per atom) from 0 to 600 K at \u2206T = 100 K; the\n", + "heavy dot-dashed (\u03b1-Ce, red) and solid (\u03b3-Ce, blue) looping curves\n", + "enclose the two-phase region with the light red dot-dashed lines\n", + "connecting the common tangents of each isotherm; the black dashed line\n", + "denotes zero pressure equilibrium state at given T; **\u25cb** (red) and \u25cf\n", + "(blue) emphasize the phase boundary at 300 K while **\u25d1** (green) is the\n", + "critical point.\n", + "\n", + "Figure \u2011: Calculated temperature-pressure phase diagram along with\n", + "experimental data.\n", + "\n", + "The calculated entropy changes are plotted in a in terms of lattice\n", + "vibration only (black dashed line), lattice vibration plus thermal\n", + "electron (black dot-dashed line), and lattice vibration plus thermal\n", + "electron and plus configuration coupling (solid blue). The black square\n", + "is the estimated vibrational entropy change at 0.7 GPa of \u03b3-Ce relative\n", + "to \u03b1-Ce, and other open (solid) symbols are from experimental\n", + "measurements of total entropy. Various contributions to the Helmholtz\n", + "energy along the \u03b3-\u03b1 phase boundary are plotted in b in terms of *T\u2206S*\n", + "(blue diamonds), *\u2206E* (green circles), and *P\u2206V* (red squares), shown\n", + "excellent agreement with experimental data.\n", + "\n", + "Figure \u2011: Calculated (a) entropy; (b) various contributions to Helmholtz\n", + "energy of Ce along with experimental data.\n", + "\n", + "The predicted fractions of three microstates as a function of\n", + "temperature and heat capacity at the critical pressure of 2.05 Pa are\n", + "shown in a and b, respectively. Near the critical point, the theory\n", + "predicts that the system is a mixture of the various microstates. a\n", + "depicts that for T \\< 300 K, the system consists mainly of the\n", + "nonmagnetic Ce state which results in \u03b1-Ce. For *T* \\> 300 K, the\n", + "thermal populations of the magnetic states increase with increasing\n", + "temperature. Finally, for *T* \\> 546 K (the critical point), 70% of the\n", + "system is composed of the antiferromagnetic Ce state with the remaining\n", + "30% consisting of the nonmagnetic and ferromagnetic Ce states. This is\n", + "in agreement with the common belief that \u03b3-Ce is magnetic with a\n", + "partially disordered local moment (paramagnetic) and that \u03b1-Ce is\n", + "nonmagnetic.\n", + "\n", + "b shows the predicted temperature evolution of contributions to heat\n", + "capacity: vibrational and magnetic (*Cf/T*), electronic\n", + "(*Cel*/*T*), and their sum (*Cf+el/T*) at 2.05\n", + "GPa. The theory suggests the following: (a) below ~500 K,\n", + "*Cf+el/T* shows an exponential temperature dependence due to\n", + "the statistic fluctuation among the non-magnetic, ferromagnetic, and\n", + "antiferromagnetic states; (b) a peak appears at ~500 K in the\n", + "*Cf+el/T* curve, which typically suggests the Schottky\n", + "anomaly; (c) the electronic specific heat coefficient\n", + "(*Cel*/*T*) is linear against *T*; (d) above *~*500 K the sum\n", + "of *Cf/T* and *Cel*/*T* renders\n", + "*Cf+el/T* temperature-independent.\n", + "\n", + "Figure \u2011: (a) Thermal populations of the nonmagnetic (red dot-dashed),\n", + "anti-ferromagnetic (green dashed), and ferromagnetic (blue solid) as a\n", + "function of temperature at the critical pressure of 2.05 GPa. (b)\n", + "Cel/T (black dashed line), Cf/T (black dot-dashed\n", + "line), and their sum Cf+el/T (blue solid line) at 2.05 GPa.\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_fept.ipynb b/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_fept.ipynb new file mode 100644 index 0000000..4a5e384 --- /dev/null +++ b/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_fept.ipynb @@ -0,0 +1,131 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "844edfab", + "cell_type": "markdown", + "source": [ + "## Application to Fe3Pt\n", + "\n", + "Invar was first discovered in the intermetallic\n", + "Fe65Ni35 alloy and is characterized by \u201canomalies\u201d\n", + "including thermal expansion, equation-of-state, elastic modulus, heat\n", + "capacity, magnetization, and Curie temperature. There are a number of\n", + "theoretical models for Invar such as the Weiss 2-\u03b3 model, the\n", + "non-collinear spin model, and the disordered local moment approach as\n", + "reviewed in Ref. \\[67\\]. In this section, the application of the MMS\n", + "model to the ordered L12 Fe3Pt is presented to\n", + "study the Invar anomaly at finite temperatures.\n", + "\n", + "For a supercell of 12 atoms with nine magnetic Fe atoms in\n", + "Fe3Pt, if only the up and down spins are considered, the\n", + "system contains 29 = 512 spin configurations, which are, by\n", + "symmetry, reduced to 37 non-equivalent ones. They are the microstates in\n", + "the MMS model, and FMC and SFC are used to represent the ferromagnetic\n", + "and spin flipping microstates, respectively. For the first-principles\n", + "calculations of each microstate, VASP package \\[14\\] within the\n", + "projector-augmented wave (PAW) method and the exchange-correlation part\n", + "of the density functional treated within the GGA of\n", + "Perdew-Burke-Ernzerhof (PBE) \\[32\\] are employed with details in Ref.\n", + "\\[67\\]. For the lattice vibration, the Debye-Gr\u00fcneisen approach\n", + "described in Chapter 5.2.4 is used.\n", + "\n", + "presents the first-principles 0 K total energies of 36 non-equivalent\n", + "SFCs as well as the FMC as a function of atomic volume. It can be seen\n", + "that there are a number of SFCs, whose energies are in the range of ~1\n", + "mRy/atom to that of the FMC. It is noted that all the SFCs studied\n", + "herein have the equilibrium averaged atomic volumes at least 1.8 %\n", + "smaller than that of the FMC, the 0 GPa ground state. In , the two\n", + "lowest energy SFCs have been labeled as SFC55 and SFC41 with their spin\n", + "arrangements very similar to the double layer antiferromagnetic state.\n", + "The nonmagnetic configuration has a very small atomic volume of 11.66\n", + "\u00c53/atom and much higher energy than both FMC and all SFCs,\n", + "and is thus not shown here.\n", + "\n", + "Figure \u2011: 0 K total energies. The heavy black line represents the FMC.\n", + "The symbols **\u25cb**, \u25d4, \u25d1, and \u25d5 with dashed lines indicate the minima of\n", + "the energy-volume curves of the SFCs with spin polarization rates of\n", + "1/9, 3/9, 5/9, and 7/9, respectively. The red **\u25cb** and \u25d4 with\n", + "dot-dashed lines mark the two lowest SFCs in energy.\n", + "\n", + "The normalized Helmholtz energies of all SFCs are plotted in with the\n", + "Helmholtz energy of FMC as the reference state, shown that the FMC has\n", + "the lowest Helmholtz energy at all temperatures considered. If only the\n", + "relative Helmholtz energies of microstates are considered, FMC should be\n", + "stable at all temperatures. However, the configurational entropy due to\n", + "the mixing among multiple microstates, i.e. , lowers the system free\n", + "energy by introducing the statistic probability of metastable SFCs.\n", + "\n", + "Figure \u2011: Normalized Helmholtz energy of all SFCs with respect to that\n", + "of FMC.\n", + "\n", + "Through the minimization of Helmholtz energy of , the\n", + "temperature-pressure and temperature-volume phase diagrams are obtained\n", + "and shown in \\[66\\]. A critical point at 141 K and 5.81 GPa is predicted\n", + "with *V* = 12.61 \u00c53. Below the critical point, it is a\n", + "two-phase miscibility gap (the shadow area) with the dominant\n", + "microstates being FMC and SFCs, respectively, and the transition between\n", + "them is first-order. Above the critical point, the macroscopically\n", + "homogeneous single phase is stable, and the phase transitions between\n", + "the ferromagnetic-dominant phase with large volumes and the SFC-dominant\n", + "paramagnetic phase with small volumes are of second-order. The\n", + "second-order transition pressures and volumes are determined by the\n", + "condition that the weighted Helmholtz energy counting all SFCs equals to\n", + "the Helmholtz energy counting only FMC.\n", + "\n", + "Figure \u2011: Calculated (a) temperature-pressure and (b) temperature-volume\n", + "phase diagrams of Fe3Pt.\n", + "\n", + "In a, the data points are the measured pressure dependence of the Curie\n", + "temperature, and the agreement between the measurements and predictions\n", + "is remarkable. It should be pointed out that the classical Weiss 2-\u03b3\n", + "model predicts only first-order phase transitions while the\n", + "non-collinear spin model yields only second-order phase transitions at\n", + "all temperatures. In b, four isobaric volume curves are also plotted\n", + "with the predicted NTE regions marked by the pink open diamonds and the\n", + "experimental volume data under ambient pressure superimposed, showing\n", + "excellent agreement. It also depicts the gigantic NTE around the\n", + "critical point on the isobaric curve at 7 GPa.\n", + "\n", + "indicates that the entropies of SFCs are larger than that of FMC so\n", + "their Helmholtz energy differences decrease with temperature. This is in\n", + "line with the origin of NTE in a single phase due to the statistic\n", + "existence of metastable microstates with lower volumes and higher\n", + "entropies than the stable state in a temperature range where their\n", + "probabilities change dramatically. plots the calculated thermal\n", + "populations of the FMC (black solid line) and that of the sum over all\n", + "SFCs (black dot-dashed line) under ambient pressure. The two major\n", + "contributions to the paramagnetic (PM) phase are from SFC55 and SFC41,\n", + "which are also plotted using red dashed and long dashed lines,\n", + "respectively. The system is dominated by the FMC at temperatures below\n", + "half of the transition temperature, and the populations of SFCs increase\n", + "monotonously at temperatures higher than half of the transition\n", + "temperature. As mentioned above, the transition temperature is defined\n", + "when the population of all SFCs is the same as that of FMC due to their\n", + "equal Helmholtz energy.\n", + "\n", + "Figure \u2011: Calculated thermal populations of FMC, SFC55, SFC41, and sum\n", + "of all SFCS, respectively.\n", + "\n", + "The predicted thermal volume expansion and the derived linear thermal\n", + "expansion coefficient (LTEC) under ambient pressure are plotted in . A\n", + "positive thermal expansion is predicted from 100 K to 288 K, followed by\n", + "a negative thermal expansion in the range of 289 ~ 449 K, and then a\n", + "positive thermal expansion again at \\>450 K, in excellent agreement with\n", + "experiments. The only disagreement between the predictions and\n", + "experiments occur at T \\< 100 K where the calculations did not reproduce\n", + "the negative thermal expansion for Fe3Pt. Large supercell or\n", + "more spin configurations may be needed.\n", + "\n", + "Figure \u2011: (a) Relative volume increase (V-\n", + "V300)/V300 with V300 being the\n", + "equilibrium volume at 300K and 0 GPa for the ordered Fe3Pt.\n", + "(b) Linear thermal expansion coefficient (LTEC) along with experimental\n", + "data (symbols) with details in Ref. \\[67\\].\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/concept_of_materials_genome.ipynb b/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/concept_of_materials_genome.ipynb new file mode 100644 index 0000000..e7e53a2 --- /dev/null +++ b/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/concept_of_materials_genome.ipynb @@ -0,0 +1,66 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "93eed2fd", + "cell_type": "markdown", + "source": [ + "## Concept of Materials Genome\u00ae\n", + "\n", + "\u201cA genome is a set of information encoded in the language of DNA that\n", + "serves as a blueprint for an organism\u2019s growth and development. The word\n", + "genome, when applied in nonbiological contexts, connotes a fundamental\n", + "building block toward a larger purpose\u201d \\[68\\]. Materials Genome\u00ae (a\n", + "trademark of Materials Genome, Inc., Pennsylvania, USA) thus concerns\n", + "the building blocks of materials. Most part of this book focuses on the\n", + "Gibbs/Helmholtz energies of individual phases as a function of its\n", + "natural variables, and the same in the CALPHAD modeling of\n", + "thermodynamics and other properties of individual phases.\n", + "Multi-component materials systems and their properties are built on the\n", + "individual phases and their properties. Individual phases are thus\n", + "naturally considered as building blocks of materials. Consequently, the\n", + "language of CALPHAD thermodynamics and kinetics contains the genomics of\n", + "materials by representing experimental and theoretical results in\n", + "databases to make them applicable to a much wider context than the\n", + "original experiments or calculations \\[69\\]. The change of individual\n", + "phases in terms of their properties, amounts, and interactions with\n", + "other phases with respect to external conditions thus determines the\n", + "performance of the materials.\n", + "\n", + "On the other hand, at critical points and beyond, phases lose their\n", + "individuality and form one macroscopically homogeneous system, and the\n", + "properties of the system change dramatically with respect to external\n", + "conditions. As shown in this chapter, these dramatically responses can\n", + "be predicted through the statistic competition of stable and metastable\n", + "microstates. From the thermodynamic point of view, under any given\n", + "conditions, one of the individual microstate has the lowest\n", + "Gibbs/Helmholtz energy and is stable, while all other microstates have\n", + "higher free energy and are metastable or unstable. These metastable or\n", + "even unstable microstates are brought into statistic existence in the\n", + "matrix of the stable microstate due to the entropy of mixing of the\n", + "stable and metastable configurations. Those microstates may thus be\n", + "considered as the building blocks of individual phases \\[70\\].\n", + "\n", + "It has demonstrated in this chapter that the properties of a\n", + "macroscopically homogeneous system with multiple microstates are not a\n", + "linear combination of properties of constituent microstates and depends\n", + "significantly on the change rate of statistic probability of microstates\n", + "with respect to external fields. This change rate is determined by the\n", + "free energy difference between the stable and metastable microstates and\n", + "its change with respect to external fields. As shown in and , this\n", + "change rate can be correlated to the distance of the system with respect\n", + "to the critical point in the system. At the critical point, there is a\n", + "mathematical singularity when the single phase becomes unstable. When\n", + "the macroscopically homogeneous single-phase system moves away from the\n", + "critical point, its properties become less and less dramatic, but always\n", + "remain a certain degree of anomaly. The properties of a system can thus\n", + "be dramatically altered and designed by changing the position of the\n", + "critical point through adjustments of chemical compositions and strain\n", + "energy in thin films.\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/index.ipynb b/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/index.ipynb new file mode 100644 index 0000000..54cceed --- /dev/null +++ b/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/index.ipynb @@ -0,0 +1,32 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "7ac53aa2", + "cell_type": "markdown", + "source": [ + "# Critical phenomena, thermal expansion, and Materials Genome\u00ae\n", + "\n", + "In Chapter , it was shown that all molar quantities of a homogeneous\n", + "system diverge at the critical point, i.e. the limit of stability,\n", + "including the additional ones shown in . As illustrated by , even though\n", + "each molar quantity changes in the same direction as its conjugate\n", + "potential, i.e. with the same sign, its dependence with respect to a\n", + "nonconjugate potentials can either be in the same sign or opposite\n", + "signs. It is often considered to be normal when they change in the same\n", + "direction, while abnormal when in different directions.\n", + "\n", + "In this chapter, the thermal expansion defined by , is used as an\n", + "example for detailed discussion of those extraordinary phenomena in the\n", + "context of a critical point based on the MMS model presented in Chapter\n", + ". The MMS model is first discussed in terms of thermal expansion and\n", + "then applied to elemental cerium (Ce) with the colossal positive thermal\n", + "expansion (CPTE) and Fe3Pt with negative thermal expansion\n", + "(NTE).\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/mms_model_applied_to_thermal_expansion.ipynb b/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/mms_model_applied_to_thermal_expansion.ipynb new file mode 100644 index 0000000..faf4f8c --- /dev/null +++ b/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/mms_model_applied_to_thermal_expansion.ipynb @@ -0,0 +1,127 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "c636c3f0", + "cell_type": "markdown", + "source": [ + "## MMS model applied to thermal expansion\n", + "\n", + "As shown in , the thermal expansion of a system can be positive, zero\n", + "and negative depending on the pressure dependence of entropy of the\n", + "system. Let us carry out a virtual experiment by analyzing a system\n", + "starting with one microstate only, *\u03b1*, when the metastable *\u03b2*\n", + "microstate has a higher entropy than the *\u03b1* microstate, i.e. and the\n", + "relative stability of the *\u03b2* microstate thus increases with\n", + "temperature. The cases with will be discussed after this starting with a\n", + "mixture of the *\u03b1* and *\u03b2* microstates.\n", + "\n", + "When a metastable microstate, *\u03b2*, is introduced by changing pressure\n", + "under constant temperature, based on and , the entropy change of the\n", + "system can be written as\n", + "\n", + "Eq. 9\u20111\n", + "\n", + "where represents the statistic probability of the microstate in the\n", + "system. With , this would results in a positive entropy change of , i.\n", + "e. , since . If this entropy increase is due to the decrease of\n", + "pressure, i.e. because volume and its conjugate potential (negative\n", + "pressure) change in the same direction, the volume thermal expansion of\n", + "the system is positive due to the increase of the population of the *\u03b2*\n", + "microstate with a larger volume. In this case,\n", + "$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}} > 0$, and the\n", + "volume and entropy of the two microstates change in the same direction.\n", + "\n", + "On the other hand, if this entropy increase is realized by increasing\n", + "pressure, i.e. , the volume thermal expansion of the system is negative\n", + "due to the increase of the population of the *\u03b2* microstate with a\n", + "smaller volume. In this case,\n", + "$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}} < 0$, and the\n", + "volume and entropy of the two microstates change in the opposite\n", + "directions.\n", + "\n", + "Therefore, the sign of\n", + "$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}}$ of two\n", + "microstates can be used as a criterion to determine whether a system\n", + "possesses NTE, i.e. positive\n", + "$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}}$ for positive\n", + "thermal expansion, and negative\n", + "$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}}$ for NTE. At a\n", + "critical point, the entropy change with respect to temperature is\n", + "infinite, resulting in either infinite positive or infinite negative\n", + "thermal expansion correspondingly. When the system moves away from the\n", + "critical point into the macroscopically homogeneous single-phase region,\n", + "the thermal expansion becomes less positive or negative. A number of\n", + "systems with\n", + "$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}} < 0$, thus\n", + "potentially NTE, are listed in the supplementary information of Ref.\n", + "\\[63\\].\n", + "\n", + "Now let us consider the case that the metastable *\u03b2* microstate has\n", + "lower entropy than the *\u03b1* microstate, i.e. , and the *\u03b2* microstate is\n", + "thus more stable at low temperatures. The system at higher temperatures\n", + "contains thus only the *\u03b1* microstate and has positive thermal\n", + "expansion. When the metastable *\u03b2* microstate is introduced, the sign of\n", + "the entropy change in can be either positive or negative because the\n", + "first term is negative and the second term is positive, and its sign\n", + "thus depends on the value of the entropy difference between two\n", + "microstates and the probability of the metastable *\u03b2* microstate. The\n", + "virtual experiment should thus be carried out in a system with the\n", + "highest MCE in , i.e. when the two microstates have the same free energy\n", + "and are in equilibrium with each other. From and , the system entropy\n", + "can be written as\n", + "\n", + "Eq. 9\u20112\n", + "\n", + "With the change of pressure, will either increase or decrease, and the\n", + "entropy of the system becomes\n", + "\n", + "Eq. 9\u20113\n", + "\n", + "The difference of and is obtained as\n", + "\n", + "Eq. 9\u20114\n", + "\n", + "The second term in is always negative, and the first term is also\n", + "negative if because . It is thus evident that if is increased by\n", + "decreasing pressure, the entropy of the system decreases, and the system\n", + "would possess negative thermal expansion because is negative, and the\n", + "entropy is reduced by the decrease of pressure. At the same time, and\n", + "$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}} < 0$, the latter\n", + "being the same condition for a negative thermal expansion as in the\n", + "first virtual experiment with . One can thus conclude that for a\n", + "two-phase equilibrium line with\n", + "$\\frac{dT}{dP} = \\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}} < 0$,\n", + "both phases can display negative thermal expansion. On the other hand,\n", + "if is increased by increasing pressure, the system would possess\n", + "positive thermal expansion because the entropy is reduced by the\n", + "increase of pressure.\n", + "\n", + "Furthermore, the thermal expansion of a system can be approximated as\n", + "follows using the rule of mixture of volumes\n", + "\n", + "Eq. 9\u20115\n", + "\n", + "Eq. 9\u20116\n", + "\n", + "where , , , , , and are the thermal expansion coefficients and volumes\n", + "of the system and the \u03b1 and \u03b2 microstates, respectively. For\n", + "simplification, let us assume both microstates have similar positive\n", + "thermal expansion, i.e. , and becomes\n", + "\n", + "Eq. 9\u20117\n", + "\n", + "shows that it is the combination of volume difference and that\n", + "determines the macroscopic thermal expansion. By setting , one obtains\n", + "\n", + "Eq. 9\u20118\n", + "\n", + "For , for $p^{\\beta} \\geq 0$, and for , at $p^{\\beta} \\rightarrow 0$ .\n", + "The readers are reminded that the sign of is the same as the sign of .\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/experimental_data_for_thermodynamic_modeling/additional_methods.ipynb b/src/psu410/src/experimental_data_for_thermodynamic_modeling/additional_methods.ipynb new file mode 100644 index 0000000..ffb33f4 --- /dev/null +++ b/src/psu410/src/experimental_data_for_thermodynamic_modeling/additional_methods.ipynb @@ -0,0 +1,268 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "442c290a", + "cell_type": "markdown", + "source": [ + "### Additional methods\n", + "\n", + "Electrical resistivities of different phases are usually different. A\n", + "change of slope of electric resistivity as a function of composition or\n", + "temperature or pressure reflects a phase transformation. This technique\n", + "is simple and reliable.\n", + "\n", + "Magnetic transitions can be measured using a vibrating sample or\n", + "superconducting quantum interference device (SQUID) magnetometer by\n", + "determining the magnetic moment of a sample in presence of an applied\n", + "magnetic field. Magnetic field lines form closed loops, resulting in an\n", + "external dipolar and demagnetizing field in a finite-sized sample. The\n", + "effective field sensed by the sample is thus the difference between the\n", + "applied field and the demagnetizing field. The magnetic transition\n", + "temperature is evaluated from the Arrott plots where the ratio of\n", + "magnetic field over magnetization with a proper exponent is plotted with\n", + "respect to the magnetization with another proper exponent for a series\n", + "of temperatures. Those proper exponents result in parallel isotherm\n", + "lines, and the isotherm line intersecting the origin corresponds to the\n", + "magnetic transition temperature.\n", + "\n", + "Thin films with composition gradients, commonly referred to as\n", + "combinatorial libraries, can be used to study the phase relations\n", + "similar to diffusion couples/multiples though the results may differ due\n", + "to the effects of surface and potential interactions with the substrate.\n", + "\n", + "Phase relations at high pressures are measured in equipment using\n", + "diamond anvil cells (DAC) or multi-anvil devices. The high pressure is\n", + "realized by decreasing the area, i.e. the anvil culet size. Pressures up\n", + "to 100 GPa can be created in DAC with a culet size of 0.3mm for small\n", + "samples in the order of 0.2 to 0.4mm. For large samples, the\n", + "large-volume presses (LVP) technique is developed, typically using WC\n", + "with the pressure mostly limited to 30 GPa and the sample size ranging\n", + "from 1mm3 to 1cm3. The pressure can be measured by\n", + "either the ruby (Cr3+ doped Al2O3)\n", + "fluorescence line shift or the molar volume of a pressure marker by\n", + "X-ray diffraction. The samples in DAC apparatus are heated by laser or\n", + "resistive wire or a small heater around the samples, while high\n", + "temperatures in LVPs are achieved by resistive heaters. Crystal\n", + "structures are detected by in-situ X-ray or synchrotron diffraction.\n", + "Attentions need to be paid to temperature and pressure homogeneity and\n", + "the non-hydrostatic stresses, which are both better controlled in LVP\n", + "equipment.\n" + ], + "metadata": {} + }, + { + "id": "9d5856c5", + "cell_type": "markdown", + "source": [ + "### Combustion, direct reaction, and heat capacity calorimetry\n", + "\n", + "In combustion calorimetry, the sample is ignited and reacts with\n", + "reactive gases like oxygen or fluorine. To accurately calculate the\n", + "enthalpy of formation from the enthalpy of combustion, the reliable\n", + "characterization of reactants and reaction products is critical, such as\n", + "incomplete combustion, impurities in the reactants which are often\n", + "ill-defined, and more than one reaction gaseous species and condensed\n", + "phases. Combustion calorimeters are usually of isoperibol type around\n", + "room temperature in a water bath.\n", + "\n", + "The direct reaction calorimetry is similar to the combustion calorimetry\n", + "though at high temperatures in heat-flux or adiabatic environments. The\n", + "partial enthalpy of reaction can also be measured if the partial\n", + "pressure of volatile species can be controlled and measured. The key\n", + "factor for accurate results is that both the reactants and reaction\n", + "products are well characterized and the reaction goes to completion\n", + "quickly like in the combustion calorimetry. For reactive reactants,\n", + "special procedure is needed to avoid their loss before the reaction\n", + "takes place.\n", + "\n", + "Heat capacity is defined as the amount of heat needed to increase the\n", + "temperature by 1K as shown by , and its integration with respect to\n", + "temperature from 0K gives entropy as shown by . At low temperatures,\n", + "adiabatic calorimetry gives more accurate data of heat capacity though\n", + "it is time consuming and requires complex instruments. At high\n", + "temperatures, the efficient and less accurate DSC method is more widely\n", + "used.\n" + ], + "metadata": {} + }, + { + "id": "4c2375c8", + "cell_type": "markdown", + "source": [ + "### Diffusion couples/multiples\n", + "\n", + "The major challenge in the equilibrated material approach is to ensure\n", + "that the whole sample reaches equilibrium. On the other hand the\n", + "diffusion couple/multiple technique does not require the whole sample to\n", + "be in equilibrium and is based on the assumption that any two phases in\n", + "contact are in equilibrium with each other at the phase interface, and\n", + "the phase compositions can be obtained by extrapolation of concentration\n", + "profiles in the two phases to the phase interface. Since the total\n", + "system of a diffusion couple is not at equilibrium, many kinetic\n", + "phenomena related to diffusion can be studied in a diffusion couple,\n", + "such as interdiffusion coefficients, parabolic growth kinetics of\n", + "product layer thickness, diffusion path (represented by the local\n", + "overall compositions) in ternary and multi-component systems for\n", + "visualizing the microstructure of reaction zones, and other properties,\n", + "all as a function of composition, which are beyond the scope of the\n", + "present book.\n", + "\n", + "Typical diffusion couples are in solid-state with two materials brought\n", + "into intimate contact to allow diffusion of elements between the two\n", + "materials though solid-liquid diffusion couples are also used. The\n", + "contacting faces are commonly ground and polished flat, clamped together\n", + "using mechanical mechanisms, and annealed at high temperatures where\n", + "diffusion can take place to a significant degree in a matter of days,\n", + "weeks or months. The samples are then quenched to retain the\n", + "high-temperature equilibrium features. For metallic systems, diffusion\n", + "couples can also be prepared by eletrolytical and electroless plating\n", + "techniques. It is important to avoid the formation of liquid during\n", + "annealing as it ruins the sample geometry. Furthermore, good adherence\n", + "at the interfaces is critical for reliable data.\n", + "\n", + "Since diffusion couples are not in a fully equilibrium state, the\n", + "tie-lines between two phases at the phase interfaces need to be obtained\n", + "by extrapolations of concentration profiles in neighbouring phases. The\n", + "electron propagation in quantitative EPMA is typically in the range of\n", + "1-2\u03bcm, yielding an excitation volume of approximately 2-4\u03bcm diameter and\n", + "a requirement of reasonable layer widths of phases on both sides of the\n", + "interface for accurate extrapolation. Therefore, the reliable\n", + "composition of a single phase must be taken several micrometers away\n", + "from the interface. When steep concentration gradients exist near the\n", + "interfaces, the extrapolation may lead to large errors, and analytical\n", + "electron microscopy is then needed. Furthermore, the fluorescence\n", + "effects, where the primary excitation can be powerful enough to excite\n", + "other elements in the sample, resulting in enhanced X-ray production and\n", + "the need of proper corrections. For a new phase to become observable\n", + "experimentally, it must nucleate and grow to reach the resolution limit\n", + "of analytical tools. It is thus not uncommon that some known equilibrium\n", + "phases are not found, and some non-equilibrium phases form. One way to\n", + "circumvent this issue is to use incremental diffusion couples with\n", + "narrow concentrations next to the phase of interest so that only this\n", + "phase may be formed.\n", + "\n", + "For ternary and higher-order systems, a more efficient approach can be\n", + "used by placing a thin layer of the third alloy between two alloys. The\n", + "thin central layer is eventually consumed, and the diffusion path is not\n", + "fixed as in the semi-infinite diffusion couples. The phase compositions\n", + "change continuously with time as a result of the overlapping of two\n", + "quasi-equilibrated diffusion zones.\n", + "\n", + "A diffusion multiple contains three or more pure elements or alloys of\n", + "different compositions and is a sample with multiple diffusion couples\n", + "and diffusion triples in it. It is more efficient in terms of both\n", + "materials and time in comparison with equilibrated alloys and diffusion\n", + "couples. All alloy blocks are prepared individually and sealed in vacuum\n", + "in a cylinder which is also used as one alloy for the diffusion\n", + "multiple. The sealed cylinder with vacuum inside also serves as the can\n", + "for subsequent hot isostatic pressing to achieve intimate interfacial\n", + "contacts. The cylinder can then be cut into disks for further annealing\n", + "treatments. A broad range of design strategies is needed for complex\n", + "diffusion multiples along with automated plotting procedures due to the\n", + "large amounts of EMPA data. The major source of error lies in the\n", + "extraction of tie-lines from EPMA results due to very condensed\n", + "information in a small area and the deviation of scanned lines from\n", + "those perpendicular to the interface.\n", + "\n", + "In terms of local equilibrium characteristics of diffusion\n", + "couples/multiples, it is evident that in the equilibrated materials\n", + "approach, it may not be necessary to reach the full equilibrium for the\n", + "whole sample if one is only interested in the local equilibrium\n", + "compositions between two neighboring phases, which can even provide\n", + "information on metastable extensions of two phases if the two phases are\n", + "in a metastable equilibrium at the annealing temperature.\n" + ], + "metadata": {} + }, + { + "id": "1b1bfe27", + "cell_type": "markdown", + "source": [ + "### Equilibrated materials\n", + "\n", + "The most common method to determine phase equilibria is to use\n", + "equilibrated materials. This method typically involves material\n", + "preparation through high temperature melting or powder metallurgy,\n", + "homogenization heat treatment, isothermal or cooling/heating procedure,\n", + "and identifications of crystal structures and phase compositions. It is\n", + "important to avoid macro-inhomogeneity as it can be difficult remove the\n", + "inhomogeneity in subsequent treatments. It is also important to use\n", + "starting materials of highest purity and minimize the loss and\n", + "contamination of materials during the whole experiments using a\n", + "protective atmosphere of inert gas or vacuum. The typical melting\n", + "techniques include high temperature furnaces with crucibles, arc\n", + "melting, and induction melting. Attentions need to be paid for possible\n", + "reactions between materials and crucibles/containers, which can be\n", + "avoided by levitating the materials through electromagnetic fields or\n", + "other means. In addition to use pure elements as raw materials, master\n", + "alloys with well-controlled compositions are often utilized because the\n", + "compositions and melting properties of master alloys are usually much\n", + "closer to those of final materials than the pure elements. For materials\n", + "with very high melting temperature or volatile components, the powder\n", + "metallurgy method can be used where compacts are made, capsulated, and\n", + "sintered.\n", + "\n", + "Homogenization during subsequent heat treatment is achieved through\n", + "diffusion, in which time and temperature are two important parameters.\n", + "To accelerate the homogenization process, the heat treatment temperature\n", + "should be as close to the solidus temperature as possible taking into\n", + "account the composition inhomogeneity with variable solidus\n", + "temperatures. When there are phase transformations taking place during\n", + "the heat treatment, extra time is needed for the heat treatment.\n", + "\n", + "The phase boundaries are then determined through measurements of either\n", + "compositions of individual phases that are in equilibrium under constant\n", + "temperature, pressure/stress/strain, and electric/magnetic fields or\n", + "discontinuity in some physical properties of materials due to a phase\n", + "transition from the continuous change of temperature or\n", + "pressure/stress/strain or electric/magnetic fields. The measurement of\n", + "compositions is usually carried out at ambient conditions so it is\n", + "necessary that the phases are fully equilibrated at experimental\n", + "conditions, which requires rigorous verifications, and can be \u201cquenched\u201d\n", + "to ambient conditions to remain unaltered during \u201cquenching\u201d, at least\n", + "in terms of compositions. The compositions are typically measured by\n", + "scanning electron microscopy (SEM) equipped with energy dispersive\n", + "spectrometer (EDS) or wavelength dispersive spectroscopy (WDS), with a\n", + "micro-level spatial resolution and a better compositional resolution of\n", + "WDS than EDS. A dedicated SEM with WDS gives another important, widely\n", + "used composition measurement technique called electron probe\n", + "microanalysis (EMPA). For submicron-sized phases, analytical\n", + "transmission electron microscopy equipped with EDS can be used though\n", + "care must be taken to avoid interference from neighboring phases.\n", + "\n", + "To accurately identify phases in equilibrium under experimental\n", + "conditions, in-situ characterizations are necessary, which complicates\n", + "the experimentation. An alternative indirect method is to measure a\n", + "physical property that changes discontinuously or dramatically for a\n", + "first- or second-order phase transition, such as heat, volume, electric\n", + "conductivity, and magnetization. There are two widely used techniques in\n", + "measuring heat: differential thermal analysis (DTA) and differential\n", + "scanning calorimetry (DSC). Both attempt to measure the difference in\n", + "temperature with the same amount of power supplied between a sample and\n", + "an inert standard during heating or cooling. A DSC may also measure the\n", + "difference in the amount of power supplied to keep their temperatures\n", + "identical. The deviation of this difference from a baseline indicates a\n", + "phase transition in the sample and is plotted as a function of time or\n", + "temperature of either the sample or the inert standard. It is evident\n", + "that the major challenges in both DTA and DSC techniques are to reach\n", + "thermal equilibrium between the sample/standard and the instrument and\n", + "the thermodynamic equilibrium within the sample due to the continuous\n", + "heating or cooling. The thermal equilibrium can be improved or mitigated\n", + "through various methods such as sample shape and size, crucible\n", + "selection, mixture with the material used for the inert standard, and\n", + "with thermocouples in direct contact with the sample and the inert\n", + "standard. However, the thermodynamic equilibrium within the sample can\n", + "only be reached when the heating/cooling rate is comparable with the\n", + "rate of the phase transition in the sample, which is almost impossible\n", + "if the phase transition typically involves diffusion in solid phases.\n", + "Therefore, extreme care is needed in interpreting the temperature\n", + "determination and the amount of heat associated with the DTA/DSC curves\n", + "as discussed in detail in the reference \\[11\\].\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/experimental_data_for_thermodynamic_modeling/index.ipynb b/src/psu410/src/experimental_data_for_thermodynamic_modeling/index.ipynb new file mode 100644 index 0000000..84a2a3f --- /dev/null +++ b/src/psu410/src/experimental_data_for_thermodynamic_modeling/index.ipynb @@ -0,0 +1,34 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "8fcdf199", + "cell_type": "markdown", + "source": [ + "# Experimental data for thermodynamic modeling\n", + "\n", + "The most widely used thermodynamic modeling technique is the CALPHAD\n", + "(CALculation of PHAse Diagram) method to be discussed in detail in\n", + "Chapter 6. The input data in the evaluations of thermodynamic model\n", + "parameters have been primarily from experiments and estimations until\n", + "first-principles calculations based on the density functional theory\n", + "\\[8\\] became a user tool in the later 1990\u2019s. Experimental data include\n", + "both thermodynamic and phase equilibrium data, while the\n", + "first-principles calculations, which only provide thermodynamic data of\n", + "individual phases, are discussed more extensively in Chapter .\n", + "\n", + "The three recently published books summarize the commonly used methods\n", + "on experimental measurements of thermodynamic properties of single \\[9\\]\n", + "and multiple phases \\[10\\] and phase diagrams \\[11\\]. They are briefly\n", + "discussed here, and readers are referred to these books for details. The\n", + "main techniques for crystal structure analysis include X-ray\n", + "diffraction, electron backscatter diffraction (EBSD), electron\n", + "diffraction in transmission electron microscopy, neutral scattering, and\n", + "synchrotron scattering, which are not discussed in this book.\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/experimental_data_for_thermodynamic_modeling/phase_equilibrium_data_from_experiments.ipynb b/src/psu410/src/experimental_data_for_thermodynamic_modeling/phase_equilibrium_data_from_experiments.ipynb new file mode 100644 index 0000000..ef58950 --- /dev/null +++ b/src/psu410/src/experimental_data_for_thermodynamic_modeling/phase_equilibrium_data_from_experiments.ipynb @@ -0,0 +1,62 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "18183ae3", + "cell_type": "markdown", + "source": [ + "## Phase equilibrium data from experiments\n" + ], + "metadata": {} + }, + { + "id": "18e4172d", + "cell_type": "markdown", + "source": [ + "### Solution calorimetry\n", + "\n", + "The book edited by Marsh and O\u2019Hare \\[12\\] documented the detailed\n", + "experimental techniques used for solution calorimetry. In one\n", + "experiment, the enthalpy of solution of a single phase is measured in a\n", + "particular solvent. To convert this enthalpy of solution into enthalpy\n", + "of formation of the phase, a thermodynamic cycle is setup for a chemical\n", + "reaction to form this phase from either constitutive pure elements or\n", + "compounds. Therefore in another experiment, the enthalpy of solution of\n", + "constitutive pure elements or compounds is measured in the solvent as\n", + "identical as possible to that used in the first experiment. The\n", + "difference of the two enthalpies of solution thus gives the enthalpy of\n", + "formation of the single phase from its constitutive elements or\n", + "compounds at the temperature of the samples before they are dropped into\n", + "the solvent, usually at room temperature.\n", + "\n", + "The solvent can be aqueous solutions at ambient temperatures and\n", + "pressures or metallic/salt/oxide melts at high temperatures under either\n", + "adiabatic or isoperibol conditions. The adiabatic calorimetry measures\n", + "the temperature change of the solvent and is usually more accurate than\n", + "the isoperibol calorimetry that measures the heat generated during the\n", + "dissolution, though the adiabatic ones are with more complex\n", + "instruments. It is important that the choices of solvent and temperature\n", + "ensure the complete dissolution of all substances into the solvent to\n", + "form a homogeneous solution. Furthermore, the effect of dilution and of\n", + "changes in solvent composition needs to be considered in the calculation\n", + "of enthalpy of solution.\n", + "\n", + "A large number of solvents are used. For aqueous solvent, hydrofluoric\n", + "acid or mixtures of HF and HCl are often used. For oxides, buffer-type\n", + "systems are typical such as lead and alkali borates and alkali\n", + "tungstates/molybdates. For metallic phases, low melting metals such as\n", + "Sn, Bi, In, Pb, and Cd, sometimes Al and Cu, are used. Factors such as\n", + "solubility, dissolution kinetics, thermal history, stirring, heat flow,\n", + "particle size, and system size are to be optimized for accurate\n", + "measurements. To enhance solution kinetics, the compound to be studied\n", + "can be mixed with another element or compound so that the mixture can\n", + "form liquid in the solvent at higher reaction rate. In any case, it is\n", + "important to calibrate the system with pure elements and compounds of\n", + "known enthalpy of formation.\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/experimental_data_for_thermodynamic_modeling/thermodynamic_data_from_experiments.ipynb b/src/psu410/src/experimental_data_for_thermodynamic_modeling/thermodynamic_data_from_experiments.ipynb new file mode 100644 index 0000000..d79ab67 --- /dev/null +++ b/src/psu410/src/experimental_data_for_thermodynamic_modeling/thermodynamic_data_from_experiments.ipynb @@ -0,0 +1,67 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "2b11bcc5", + "cell_type": "markdown", + "source": [ + "## Thermodynamic data from experiments\n", + "\n", + "Broadly speaking, thermodynamic data represents the values of Gibbs free\n", + "energy and all its first and second derivatives. For thermochemical\n", + "properties as the main topic of the book, calorimetric, electrochemical,\n", + "and vapor pressure methods are the primary techniques used with the\n", + "first for accurate measurement of enthalpy and entropy, and the latter\n", + "two for direct determination of Gibbs energy and activity. The\n", + "electrochemical method is discussed in Chapter . The calorimetric method\n", + "is divided into solution, combustion, direct reaction, and heat capacity\n", + "calorimetry, respectively, which all involve chemical reactions to be\n", + "discussed in detail in Chapter . The vapor pressure method involves the\n", + "equilibrium of volatile species between gas and samples and is divided\n", + "into Knudsen effusion and equilibration methods, respectively.\n" + ], + "metadata": {} + }, + { + "id": "ee7088c1", + "cell_type": "markdown", + "source": [ + "### Vapor pressure method\n", + "\n", + "In the Knudsen effusion method, a small amount of volatile species in\n", + "the gas phase effuses through a small orifice of 0.1 to 1 mm with\n", + "negligible influence of the equilibrium in the Knudsen cell. The vapor\n", + "is ionized and analyzed with a mass spectrometer. The partial pressure\n", + "of a species can be calculated from its ionization area and intensity\n", + "through a calibration factor determined by a reference material with\n", + "known partial pressure. For high temperature measurements, care must be\n", + "taken to avoid the reactions between cell and sample and fragmentation\n", + "of gas species on ionization. The typical vapor pressure range is\n", + "between 10-7 and 10 Pa.\n", + "\n", + "In various equilibration methods, the total vapor pressure is usually\n", + "measured directly using pressure gauges in the range of 10-7\n", + "and 100 kPa. Other direct or indirect methods include\n", + "\n", + "thermogravimetric method for measuring the vapor mass\n", + "\n", + "atomic absorption spectroscopy for gas composition\n", + "\n", + "measurement of sample composition equilibrated with a gas of\n", + "well-defined activity of the volatile species\n", + "\n", + "the dew point method in which the condensation temperature of the\n", + "volatile component is measured from the vapor equilibrated with the\n", + "sample at a higher temperature,\n", + "\n", + "the chemical transport method to be discussed in Chapter .\n", + "\n", + "The main error in all these methods is often due to inadequate\n", + "equilibration between vapor and sample.\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/firstprinciples_calculations_and_theory/bornoppenheimer_approximation.ipynb b/src/psu410/src/firstprinciples_calculations_and_theory/bornoppenheimer_approximation.ipynb new file mode 100644 index 0000000..37d7cdf --- /dev/null +++ b/src/psu410/src/firstprinciples_calculations_and_theory/bornoppenheimer_approximation.ipynb @@ -0,0 +1,388 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "1ad8e47b", + "cell_type": "markdown", + "source": [ + "### Born-Oppenheimer Approximation\n", + "\n", + "For a time independent atomic system, it is often accurate enough to\n", + "write in or in terms of the electron coordinate **r** and nuclei\n", + "coordinate **R**,\n", + "\n", + "*Eq. 5\u201164* ,\n", + "\n", + "where *e* represents the electron charge and *i* and *j* label the\n", + "electrons, *I* and *J* the atomic nuclei, *ZI* the atomic\n", + "nuclear charge number of atom *I*, *e* the electron charge,\n", + "*me* the electron mass, *MI* the mass of atomic\n", + "nuclei *I*, the Laplace operator for electron *i*, and the Laplace\n", + "operator for atomic nuclei *I,* noting\n", + "\n", + "*Eq. 5\u201165* ,\n", + "\n", + "with respect to the Cartesian axis *x*, *y*, and *z*.\n", + "\n", + "Consider the fact that the electron mass is thousand times smaller than\n", + "the mass of the atomic nuclei, implying that the motions of the\n", + "electrons are much faster than the atomic nuclei, Born and Oppenheimer\n", + "proposed that the wave function of the whole system can be simply\n", + "approximated as the product of the electron wave function and the atomic\n", + "nuclei wave function as\n", + "\n", + "*Eq. 5\u201166* .\n", + "\n", + "With the auxiliary approximation of neglecting the dynamic coupling\n", + "between the motions of electrons and atomic nuclei, the Schr\u00f6dinger\n", + "equation for the motion of electrons becomes\n", + "\n", + "*Eq. 5\u201167* ,\n", + "\n", + "where\n", + "\n", + "*Eq. 5\u201168* ,\n", + "\n", + "and the Schr\u00f6dinger equation for the motion of atomic nuclei becomes\n", + "\n", + "*Eq. 5\u201169* ,\n", + "\n", + "where\n", + "\n", + "*Eq. 5\u201170* ,\n", + "\n", + "with\n", + "\n", + "*Eq. 5\u201171* ,\n", + "\n", + "where represents the complex conjugate of .\n" + ], + "metadata": {} + }, + { + "id": "63c06677", + "cell_type": "markdown", + "source": [ + "### Cluster Expansions\n", + "\n", + "Many properties of a solution phase such as energy are dependent on the\n", + "*configurations* - the arrangements of atoms on the lattice sites. In\n", + "cluster expansion \\[35, 42\\], the configuration dependence of properties\n", + "is formulated efficiently by a \u201clattice algebra\u201d which maps a\n", + "substitutional configuration into an Ising-like lattice model. Taking a\n", + "binary A1-*x*B*x* solution phase for instance, A\n", + "atoms are represented by the \u201cdown\u201d spins (*Si* = -1) and B\n", + "atoms are represented by the \u201cup\u201d spins (*Si* = +1) as\n", + "illustrated in . Using the cluster expansion technique, for a system\n", + "containing *N* atoms, the total energy of any alloy configuration *\u03c3* =\n", + "(*S*1, *S*2, \u2026, *SN*) can be\n", + "conveniently evaluated using the following Ising-like Hamiltonian:\n", + "\n", + "*Eq. 5\u2011147*\n", + "\n", + "where *J*\u2019s are the effective cluster interactions (ECI\u2019s); is a number\n", + "representing the atomic occupation at the lattice *i* under the\n", + "configuration *\u03c3*, which takes the values -1 and 1 for binary and -1, 0,\n", + "and 1 for ternary etc. In , the 2-site, 3-site, and 4-site correlations\n", + "are written as follows,\n", + "\n", + "Eq. 5\u2011148\n", + "\n", + "Eq. 5\u2011149\n", + "\n", + "Eq. 5\u2011150 .\n", + "\n", + "The expansion in would be exact as long as *all* the *n*-site\n", + "interactions are included. For a binary system, this can be observed by\n", + "the combination law that where is the number of *n*-site interactions.\n", + "However, in actual calculations, one never does an expansion to the\n", + "order *N* (containing 2*N* terms for binary) since it is too\n", + "long to be practical. In fact, since the interactions between widely\n", + "separated atoms are expected to be weaker than the interactions between\n", + "nearer atoms for most of the important properties, the expansion in is\n", + "usually truncated at certain distance to include only a few short-ranged\n", + "pair (2-site), triple (3-site), and up to the most, the quadruple\n", + "(4-site) interactions.\n", + "\n", + "Once a configuration is assigned, the *S*\u2019s are just the geometrical\n", + "factors. The common practice in the cluster expansion is that i) perform\n", + "first-principles calculations of a selected set of configurations\n", + "(around 20-100); ii) evaluate the values of *J*\u2019s using with the\n", + "energies from i); iii) use the fitted *J*\u2019s to predict the energy for a\n", + "desired set of configurations; and iv) make the ensemble average at a\n", + "given temperature for the energetics of the random alloys through Monte\n", + "Carlo simulations.\n" + ], + "metadata": {} + }, + { + "id": "69ae036f", + "cell_type": "markdown", + "source": [ + "### Debye-Gr\u00fcneisen approximation to the vibrational contribution \n", + "\n", + "Strictly speaking, the Debye theory is only accurate at very low\n", + "temperatures. It assumes a parabolic type of dependence of PDOS\u00a0on the\n", + "phonon frequency. This assumption is only correct at the scale of 10\u2019s K\n", + "because at low temperatures,\u00a0only the low frequency acoustic phonons are\n", + "activated which play the major role for the parabolic type of PDOS for\n", + "the low frequency range as shown in . That is why\u00a0there are two kinds of\n", + "Debye temperature: low-temperature Debye temperature and\n", + "high-temperature\u00a0Debye temperature. The low-temperature Debye\n", + "temperature can be strictly derived by fitting low temperature heat\n", + "capacity data. The high-temperature\u00a0Debye temperature is mostly a\n", + "phenomenological fitting parameter.\n", + "\n", + "The Debye model approximates the PDOS in by\n", + "\n", + "*Eq. 5\u201138* ,\n", + "\n", + "where is the so-called Debye cutoff frequency related to the Debye\n", + "temperature as\n", + "\n", + "*Eq. 5\u201139* .\n", + "\n", + "As the result, the vibrational contribution to Helmholtz energy under\n", + "the Debye approximation becomes,\n", + "\n", + "*Eq. 5\u201140*\n", + "\n", + "where *D*(*\u0398D/T*) is the Debye function given by .\n", + "\n", + "Under the Debye approximation, the formulation to calculate the entropy\n", + "becomes\n", + "\n", + "*Eq. 5\u201141* ,\n", + "\n", + "the formulation to calculate the internal energy becomes\n", + "\n", + "*Eq. 5\u201142* ,\n", + "\n", + "where\n", + "\n", + "*Eq. 5\u201143* ,\n", + "\n", + "and the formulation to calculate the heat capacity at constant volume\n", + "becomes\n", + "\n", + "*Eq. 5\u201144* .\n", + "\n", + "Here it is noted that is volume dependent which is often written in\n", + "terms of Gr\u00fcneisen constant:\n", + "\n", + "*Eq. 5\u201145* .\n", + "\n", + "It has been found that the dependence of on *V* usually is weak and\n", + "hence is often approximated as a constant. With , the formulation to\n", + "calculate the pressure becomes\n", + "\n", + "*Eq. 5\u201146* .\n", + "\n", + "An important result of the Debye approximation is that when , together\n", + "with , the heat capacity in is reduced to\n", + "\n", + "*Eq. 5\u201147* .\n", + "\n", + "This gives the Debye T3 law when the thermal electron\n", + "contribution is neglected. In the analysis of superconductor and heavy\n", + "Fermion materials, is often rewritten as\n", + "\n", + "*Eq. 5\u201148* ,\n", + "\n", + "which is more convenient for examining the heat capacity measured at\n", + "extremely low temperatures.\n" + ], + "metadata": {} + }, + { + "id": "72592df0", + "cell_type": "markdown", + "source": [ + "### Density functional theory (DFT) and 0 K Kohn-Sham equations\n", + "\n", + "The density functional theory advocates that the properties of a matter\n", + "are solely dictated by its electronic density distribution (or equally\n", + "say charge density), , in the real space. This is to say, that for an\n", + "arbitrary the total energy of the system, *E*, is always larger or equal\n", + "to a value, *E0*, called as the ground state energy:\n", + "\n", + "*Eq. 5\u201186* .\n", + "\n", + "In terms of variational principle, is equivalent to\n", + "\n", + "*Eq. 5\u201187* .\n", + "\n", + "Kohn and Sham \\[8\\] proposed to write the total energy as\n", + "\n", + "*Eq. 5\u201188* ,\n", + "\n", + "where represents the kinetic energy of the system, is the external\n", + "potential acting on the system, is the Hartree energy, and is the\n", + "so-called exchange-correlation energy with where and represent the\n", + "charge density of electrons with spin down and spin up, respectively .\n", + "Using\n", + "\n", + "*Eq. 5\u201189* ,\n", + "\n", + "together with the variational principle of , one can get the\n", + "one-electron Schr\u00f6dinger equation\n", + "\n", + "*Eq. 5\u201190*\n", + "\n", + "where *me* represents the mass of an electron and\n", + "\n", + "*Eq. 5\u201191*\n", + "\n", + "so that the total energy is obtained as\n", + "\n", + "*Eq. 5\u201192*\n", + "\n", + "The major challenge within DFT is that the accurate formulation of the\n", + "exchange-correlation energy is unknown. Except for the uniform electron\n", + "gas, no exact analytical form for the exchange-correlation energy has\n", + "yet been obtained. Therefore approximations must be made for the\n", + "exchange-correlation energy in calculating a realistic system. Until\n", + "now, the two most popular approximations are the local density\n", + "approximation (LDA) \\[30\\] and the generalized gradient approximation\n", + "(GGA) \\[31-32\\].\n", + "\n", + "The local density approximation (LDA) states that the\n", + "exchange-correlation energy is the same as that for a locally uniform\n", + "electron gas. In this case one can write *Vxc* as\n", + "\n", + "*Eq. 5\u201193* .\n", + "\n", + "Although this approximation is extremely simple, it works reasonably\n", + "well for many systems. The only remaining problem is to find an\n", + "approximate solution to . One of most employed parameterized expression\n", + "for is that by Perdew and Zunger \\[30\\].\n", + "\n", + "Many modern DFT codes use the more advanced generalized gradient\n", + "approximation (GGA) \\[31-32\\] to the exchange-correlation energy to\n", + "improve accuracy for certain physical properties. As the LDA\n", + "approximates the energy of the true density by the energy of a local\n", + "constant density, it fails in situations where the density undergoes\n", + "rapid changes such as in molecules. An improvement to this can be made\n", + "by considering the gradient of the electron density. Symbolically, this\n", + "can be written as\n", + "\n", + "*Eq. 5\u201194*\n", + "\n", + "The commonly used GGA is those due to Perdew et al. \\[31-32\\].\n" + ], + "metadata": {} + }, + { + "id": "68a79395", + "cell_type": "markdown", + "source": [ + "### Dynamical matrix and phonon mode\n", + "\n", + "Because of the periodicity of a crystal, one can make an initial guess\n", + "that the solutions of are elastic plane waves made of a collective\n", + "atomic vibrations \\[23, 33\\], from the harmonic approximation of ,\n", + "\n", + "*Eq. 5\u2011122* ,\n", + "\n", + "where *\u03c9* represents the frequency of the plane wave, and is a wave\n", + "vector designating the wave number and direction along which the plane\n", + "wave propagates. It should be pointed out that in is now independent of\n", + "the index *P.* That is, except for a phase factor, the atoms that are\n", + "equivalent by the translational symmetry among different primitive unit\n", + "cells will experience the same type of atomic motion, independent of the\n", + "positions of the primitive unit cell in the system. This is equivalent\n", + "to applying the periodic condition so that in obeys\n", + "\n", + "*Eq. 5\u2011123* .\n", + "\n", + "Note that is now independent of the index *P*.\n", + "\n", + "Furthermore, one wants to limit **q***t* in and to those\n", + "known as the exact wave vectors which represent a special set of points\n", + "in the reciprocal space that satisfy the condition\n", + "\n", + "*Eq. 5\u2011124* ,\n", + "\n", + "where is the Kronecker delta function. In fact, the number of equals to\n", + "the number of primitive unit cells contained in the system.\n", + "\n", + "Utilizing the translational invariance by which in \\[or in \\] depends on\n", + "*P* and *Q* only through the difference , the following Fourier\n", + "transformation can be employed to simplify\n", + "\n", + "*Eq. 5\u2011125* ,\n", + "\n", + "and one obtains\n", + "\n", + "*Eq. 5\u2011126* .\n", + "\n", + "The counterpart of with respect to is\n", + "\n", + "*Eq. 5\u2011127*\n", + "\n", + "is now an equation with 3*Na* degrees of freedom. At each ,\n", + "one can always find 3*Na* eigenvalues of (*i* = 1, \u2026,\n", + "3*Na*). The 3*Na* vibrations are often known as\n", + "phonon modes, noting again that *Na* is the number of atoms\n", + "in the primitive unit cell. Each of yields a set of which can be chosen\n", + "such that\n", + "\n", + "*Eq. 5\u2011128* ,\n", + "\n", + "where represents the complex conjugate of , and\n", + "\n", + "*Eq. 5\u2011129*\n", + "\n", + "Finally, for a solid, the summation over *Q* in can be abbreviated out,\n", + "resulting in, after transforming back to together with using\n", + "\n", + "*Eq. 5\u2011130* ,\n", + "\n", + "because of translational invariance by which depends on *P* and *Q* only\n", + "through the difference .\n", + "\n", + "In the case that a system is built by a parallelepiped multiplication of\n", + "the primitive unit cell with lattice vectors of **a***\u03b1*\n", + "(\u03b1=*x*, *y*, and *z*) in the form , in and takes the form\n", + "\n", + "*Eq. 5\u2011131* ,\n", + "\n", + "where = 0, 1, \u2026, , and is the primitive lattice vector in the reciprocal\n", + "space as\n", + "\n", + "*Eq. 5\u2011132* ,\n", + "\n", + "with *Va* representing the volume of the primitive unit cell\n", + "given as\n", + "\n", + "*Eq. 5\u2011133* .\n", + "\n", + "Combine and , it is easy to demonstrate\n", + "\n", + "*Eq. 5\u2011134* ,\n", + "\n", + "where is the Kronecker delta symbol.\n", + "\n", + "It can be shown that the **q** points defined in represent the exact\n", + "wave points. First, can be written as\n", + "\n", + "*Eq. 5\u2011135* ,\n", + "\n", + "where = 0, 1, \u2026, . Then, the left hand side of for the definition of\n", + "exact wave vector becomes\n", + "\n", + "*Eq. 5\u2011136* .\n", + "\n", + "Knowing the fact that\n", + "\n", + "*Eq. 5\u2011137* .\n", + "\n", + "Hence, it is proved the **q** points defined in are the exact wave\n", + "points.\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/firstprinciples_calculations_and_theory/firstprinciples_approaches_to_disordered_alloys.ipynb b/src/psu410/src/firstprinciples_calculations_and_theory/firstprinciples_approaches_to_disordered_alloys.ipynb new file mode 100644 index 0000000..44553a1 --- /dev/null +++ b/src/psu410/src/firstprinciples_calculations_and_theory/firstprinciples_approaches_to_disordered_alloys.ipynb @@ -0,0 +1,56 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "92653dd4", + "cell_type": "markdown", + "source": [ + "## First-principles approaches to disordered alloys\n", + "\n", + "First-principles calcuations discussed so far strictly rely on the exact\n", + "atomic positions in the unit cells. A brutal-force approach for random\n", + "solution phase would be to directly construct a large supercell and\n", + "randomly decorate the host lattice with different types of atoms. Such\n", + "an approach would necessarily require very large supercells to\n", + "adequately mimic the statistics of the random solutions. Since\n", + "first-principles methods are computationally constrained by the number\n", + "of atoms that one can treat, this brute-force approach is\n", + "computationally prohibitive. Take a binary\n", + "A1-*x*B*x* substitutional alloy as illustrated in\n", + "as an example, for a system containing *N* atoms, there can be\n", + "2*N* possible number of configurations, which is an\n", + "astronomically large number when *N* is large. It is an impossible task\n", + "to explore such a huge configuration space with available computing\n", + "resources.\n", + "\n", + "Figure \u2011. Mapping of a substitutional A1-xBx alloy\n", + "into an Ising-like lattice model \\[39-40\\].\n", + "\n", + "As a result, approximations must be made to the first-principles\n", + "calculations. At present, there are mainly three approaches to\n", + "calculating the disordered solution phases: the coherent potential\n", + "approximation (CPA) \\[41\\], the cluster expansion (CE) \\[42\\], and the\n", + "special quasirandom structures (SQS\u2019s) \\[43\\] approach.\n", + "\n", + "CPA \\[41\\] treats random alloys by considering the *average* occupations\n", + "of lattice sites in solving the Kohn-Sham equation. Since a mean-field\n", + "approach is employed, dependence of properties on the local environments\n", + "surrounding an atom is not treated explicitly in CPA. In a random\n", + "solution, there exists a distribution of local environments (e.g., in\n", + "bcc alloys, A or B surrounded by the various\n", + "A*X*B8-*X* coordination shells with *X* between 0\n", + "and 8), resulting in local environmentally-dependent quantities such as\n", + "charge transfer and local displacements of atoms from their ideal\n", + "lattice positions. Even in random A1-*x*B*x* solid\n", + "solutions, the average A-A, A-B and B-B bond lengths are generally\n", + "different. These effects can be considered by the CE and SQS approaches,\n", + "which are the focus of the next two subsections. In following\n", + "subsections, unless specifically noted, the formulism for the binary\n", + "system is discussed for the sake of simplicity.\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/firstprinciples_calculations_and_theory/firstprinciples_formulation_of_thermodynamics.ipynb b/src/psu410/src/firstprinciples_calculations_and_theory/firstprinciples_formulation_of_thermodynamics.ipynb new file mode 100644 index 0000000..4daeccb --- /dev/null +++ b/src/psu410/src/firstprinciples_calculations_and_theory/firstprinciples_formulation_of_thermodynamics.ipynb @@ -0,0 +1,356 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "771a08a9", + "cell_type": "markdown", + "source": [ + "## First-principles formulation of thermodynamics\n" + ], + "metadata": {} + }, + { + "id": "9f6cde7e", + "cell_type": "markdown", + "source": [ + "### Formation enthalpy of Ni3Al\n", + "\n", + "One can do the similar calculations for the elemental metal Al and the\n", + "compound Ni3Al which has the L12 structure,\n", + "following the same steps in the calculations of Ni. The formation\n", + "enthalpy in the unit of per mole atom can be calculated as\n", + "\n", + "*Eq. 5\u201115* ,\n", + "\n", + "where , , and represent the enthalpies of Ni3Al, Ni, and Al\n", + "in the energy unit of per mole atom, respectively. Plotted in is the\n", + "calculated formation enthalpy of Ni3Al (curve) as a function\n", + "of temperature from 0 to 1600 K at *P* = 0 compared with experimental\n", + "data (symbols) with details in Ref. \\[16\\].\n", + "\n", + ".\n", + "\n", + "Figure \u2011. Formation enthalpy of L12-Ni3Al with\n", + "respect to pure fcc Ni and Al.\n" + ], + "metadata": {} + }, + { + "id": "6e3ebf45", + "cell_type": "markdown", + "source": [ + "### Hartree-Fock approximation to solve Schr\u00f6dinger equation\n", + "\n", + "It was Hartree who first assumed that the electron wave function in can\n", + "be expressed as a product of a collection of *N* independent\n", + "one-electron wave functions, where *i* = 1, 2, \u2026, *N* with *N* being the\n", + "number of electrons in a system, in terms of the its space coordinate\n", + "**r** and spin state **s**. After that, Fock modified the Hartree\n", + "approximation by considering the fact that the wave function of a\n", + "multi-fermionic system should satisfy anti-symmetry requirements and\n", + "subsequently the Pauli exclusion principle that the total wave function\n", + "changes sign upon the exchange of fermions. Accordingly, the wave\n", + "function of *N* electrons system under the Hartree-Fock approximation is\n", + "expressed as the Slater determinant \\[29\\]\n", + "\n", + "*Eq. 5\u201172*\n", + "\n", + "For brevity, one can use the atomic unit that treats in , so that\n", + "\n", + "*Eq. 5\u201173*\n", + "\n", + "where , and\n", + "\n", + "*Eq. 5\u201174*\n", + "\n", + "Accordingly, the total energy of the system is expressed as\n", + "\n", + "*Eq. 5\u201175*\n", + "\n", + "where *Jij* is called as Coulomb/Hartree term\n", + "\n", + "*Eq. 5\u201176*\n", + "\n", + "where *Kij* is called as exchange term\n", + "\n", + "*Eq. 5\u201177* ,\n", + "\n", + "where = 1 if spin **s***i* and **s***j* points to\n", + "the same direction and = 0 if spin **s***i* and\n", + "**s***j* points to the opposite direction.\n", + "\n", + "By utilizing the variational condition of , one gets\n", + "\n", + "*Eq. 5\u201178* ,\n", + "\n", + "where is called one-electron energy, and\n", + "\n", + "*Eq. 5\u201179* ,\n", + "\n", + "with being the electronic charge density whose expression is\n", + "\n", + "*Eq. 5\u201180* ,\n", + "\n", + "*Eq. 5\u201181* .\n", + "\n", + "It should be especially noted here that to solve the Hartree-Fock\n", + "equation , the most time consuming part is due to the nonlocal exchange\n", + "term, knowing the fact that the being evaluated one-electron wave\n", + "function is also contained in the expression in the left hand side of by\n", + "means of .\n", + "\n", + "The configurational interaction method is the generalization of the\n", + "Hartree-Fock approximation. In such a case, *Y*, the number of\n", + "one-electron wave functions can be larger than the number of electrons,\n", + "*N*, in the system. Accordingly, from the number of one-electron wave\n", + "functions, , *y* = 1, 2, \u2026, *Y*, one can build the number of Stater\n", + "determinants, *M*, by the combinatorial mathematics that the maximum of\n", + "*M* can be\n", + "\n", + "*Eq. 5\u201182* .\n", + "\n", + "As a result, the wave function of a collection of *N* electron system\n", + "becomes the recombination of the *M* Stater determinants as\n", + "\n", + "*Eq. 5\u201183* ,\n", + "\n", + "where the coefficients is to be found from the multiple linear equation\n", + "\n", + "*Eq. 5\u201184* .\n", + "\n", + "The matrix element in *Eq. 5\u201184* is determined by the integral\n", + "\n", + "*Eq. 5\u201185* .\n" + ], + "metadata": {} + }, + { + "id": "1249586e", + "cell_type": "markdown", + "source": [ + "### Helmholtz energy\n", + "\n", + "In this chapter so far, all our discussions are limited to the case of a\n", + "system which is made from a single microstate (microscopic state). Here,\n", + "the terminology microstate refers to the microscopic structure that is\n", + "distinguished by crystal structure, atom distributions in the lattice\n", + "sites, and the arrangements of the local atomic spin and electronic\n", + "angular momentum distributions among the lattice sites. From this\n", + "section on, the index \u03c3 is employed to label the microstate. For a solid\n", + "material at finite temperatures, a phase can be formed by a single\n", + "microstate or a mixture of multiple microstates.\n", + "\n", + "Let us consider a canonical system made of *N* atoms with an average\n", + "atomic volume *V*. The study is limited to the motions of atomic nuclei\n", + "and electrons. For such a system, one can use to index the energy\n", + "eigenvalues of the corresponding microscopic Hamiltonian associated with\n", + "microstate *\u03c3*. The subscript **g** symbolically labels the different\n", + "vibrational states for the motions of atomic nuclei and the subscript\n", + "**n** symbolically labels the electronic states distinguished by the\n", + "different distributions of the electrons between the electronic valence\n", + "and conduction bands. Neglecting electron-phonon coupling, one can\n", + "assume that the contributions to between the vibrational and electronic\n", + "states are additive, so that\n", + "\n", + "*Eq. 5\u201116* ,\n", + "\n", + "where *Ec* is the static total energy of the microstate *\u03c3*.\n", + "Note that in and represent the energetics of the vibrational state and\n", + "the electronic state respectively.\n", + "\n", + "One then can formulate the canonical partition function of the\n", + "microstate *\u03c3* at the given temperature *T* and volume *V* as\n", + "\n", + "*Eq. 5\u201117* ,\n", + "\n", + "where *\u03b2* = 1/*kBT* with *kB* being the\n", + "Boltzmann\u2019s constant. As a result, with, the Helmholtz energy *F* per\n", + "atom for the microstate *\u03c3* is derived as follows:\n", + "\n", + "*Eq. 5\u201118* ,\n", + "\n", + "where the variable *N* has been abbreviated using\n", + "\n", + "*Eq. 5\u201119* ,\n", + "\n", + "*Eq. 5\u201120* ,\n", + "\n", + "*Eq. 5\u201121* .\n", + "\n", + "The calculation of *Ec* is straightforward in most of the\n", + "existing first-principles codes as discussed earlier.\n" + ], + "metadata": {} + }, + { + "id": "0887f571", + "cell_type": "markdown", + "source": [ + "### Helmholtz energy and quasiharmonic approximation\n", + "\n", + "At present, the most rigorous method to predict the thermodynamic\n", + "properties of a material at finite temperatures is the phonon approach.\n", + "In such an approach, the microscopic Hamiltonian is expanded up to the\n", + "second order. All the thermodynamic quantities are calculated using\n", + "formulations derived from the statistical physics without further\n", + "approximation. The great importance of the phonon theory is that all the\n", + "input parameters can be obtained by means of first-principles\n", + "calculations without using any phenomenological parameters.\n", + "\n", + "Let us consider a system with an averaged atomic volume *V*. Neglecting\n", + "the electron phonon coupling, it is a well demonstrated procedure \\[16\\]\n", + "to decompose the Helmholtz energy *F*(*V*,*T*) of the system at\n", + "temperature *T* into three additive contributions as follows\n", + "\n", + "*Eq. 5\u20111* ,\n", + "\n", + "where *Ec* is the static total energy which is the total\n", + "energy of the system at 0 K without any atomic vibrations,\n", + "*Fvib* is the vibrational contribution due to the lattice\n", + "ions, and *Fel* is the electronic contribution due to the\n", + "thermal electronic excitation at finite temperature which can become\n", + "important for metals at high temperature.\n", + "\n", + "The terminology of \u201cquasiharmonic approximation\u201d arises from the fact\n", + "that for a given volume, *Fvib*(*V*,*T*) is calculated under\n", + "the harmonic approximation and the anharmonic effects are carried out\n", + "solely through volume dependence of the phonon frequency. The easiest\n", + "computational implementation of is to first independently calculate the\n", + "Helmholtz energy at several selected volumes near the equilibrium volume\n", + "and then use the numerical interpolation to find the Helmholtz energy at\n", + "an arbitrary volume. The volume interval is usually at the scale of 3~5%\n", + "of the equilibrium volume. Too small volume interval can result in\n", + "numerical instability due to the numerical uncertainties in the static\n", + "total energy calculation, in particular, when one numerically computes\n", + "the first- and especially the second-order derivatives of the Helmholtz\n", + "energy in deriving the thermodynamic quantities. It should be noted that\n", + "whenever available, analytic formulas should be used instead of the\n", + "numerical second-order derivative to avoid numerical errors. For\n", + "instance, when the phonon approach is employed, the constant volume heat\n", + "capacity has the analytic expression in terms of phonon density of\n", + "states.\n", + "\n", + "Nickel metal adopts the fcc structure at ambient conditions and the\n", + "primitive unit cell contains one atom. Almost all the existing\n", + "first-principles codes have the function to calculate the static total\n", + "energy. The static total energy *Ec* in should be calculated\n", + "using the primitive unit cell. As the Helmholtz energy is to be\n", + "calculated at several volumes, a good practice is to plot the calculated\n", + "static total energy points together with the interpolated energy curve\n", + "to examine the convergence of the static total energy calculation. Since\n", + "the first-principles method often employs the self-consistent technique,\n", + "it could occur that calculations at certain volumes may not convergent,\n", + "which should be fixed by trying the various algebraic schemes provided\n", + "in most of the existing codes. Furthermore, since certain calculations\n", + "involve the second order derivative of the Helmholtz energy, a minor\n", + "uncertainty along the static total energy curve can result in large\n", + "deviation for the calculated properties such as thermal expansion\n", + "coefficient and bulk modulus. In that case, a reasonable solution is to\n", + "smoothen the static total energy using the modified Birch-Murnaghan\n", + "equation of states (EOS) \\[17-18\\]\n", + "\n", + "*Eq. 5\u20112* .\n", + "\n", + "Plotted in is the calculated static total energy of the elemental metal\n", + "Ni with the circles representing the calculated values and the curve\n", + "representing that by EOS fitting.\n", + "\n", + "Figure \u2011. Static total energy of nickel.\n", + "\n", + "The vibrational contribution to the Helmholtz energy by phonon theory\n", + "can be computed by \\[19\\]\n", + "\n", + "*Eq. 5\u20113* ,\n", + "\n", + "where is the Boltzmann\u2019s constant, *\u03c9* represents the phonon frequency,\n", + "and is the phonon density of states. It is recommended that is\n", + "calculated at the same volume set at which the static total static\n", + "energies are calculated.\n", + "\n", + "For the present prototype of Ni, the supercell method for the\n", + "calculation of has been employed. The procedure is follows:\n", + "\n", + "1. Make supercell by enlarging the primitive unit cell according to the\n", + " > defined neighbor interaction distance; Employ the first-principles\n", + " > code (VASP, \\[13-14\\] in this work) to calculate the interatomic\n", + " > force constants.\n", + "\n", + "2. Assign the mesh in the wave vector (**q**) space; Make the dynamical\n", + " > matrix at each **q** point; Diagonalize the dynamical matrix to\n", + " > find out the phonon frequencies at each **q** point; And finally\n", + " > collect all the phonon frequencies for all **q** points. The\n", + " > detailed formulation for phonon calculations is given in Chapter .\n", + "\n", + "For the phonon calculations, one can use the open source code YPHON\n", + "\\[15\\] by the present authors. Other choices can be the free ATAT code\n", + "\\[20\\] or the free PHON code \\[21\\]. For the calculation of the phonon\n", + "density of states, we have made a supercell containing 64 atoms which is\n", + "a 4\u00d74\u00d74 supercell of the primitive unit cell. Plotted in is the\n", + "calculated phonon density of states using YPHON code at the calculated\n", + "static equilibrium volume compared with the measured data at 10 K \\[22\\]\n", + "(symbols).\n", + "\n", + "Figure \u2011. Phonon density of states of nickel.\n", + "\n", + "For a first-principles thermodynamic calculation, an important step to\n", + "avoid possible calculation errors is to examine the phonon dispersions\n", + "first. Phonon dispersion \\[23\\] depicts the evolution of phonon\n", + "frequencies along the designated direction for a crystal. Phonon\n", + "dispersion can be measured rather accurately by inelastic neutron\n", + "scattering \\[24-26\\] or inelastic x-ray scattering \\[27\\] experiment.\n", + "Plotted in are the calculated phonon dispersions (curves) along the\n", + "\\[00\u03b6\\], \\[0\u03b61\\], \\[0\u03b6\u03b6\\], and \\[\u03b6\u03b6\u03b6\\] directions of Ni using YPHON code\n", + "compared with the neutron scattering data at 296 K (symbols) with\n", + "details in Ref. \\[16\\].\n", + "\n", + "Figure \u2011. Phonon dispersions of nickel. The solid lines represent\n", + "results calculated using a supercell containing 256 atoms which is 4\u00d74\u00d74\n", + "supercell of the conventional cubic unit cell. The dot-dashed lines\n", + "represent results calculated using a supercell containing 64 atoms which\n", + "is 4\u00d74\u00d74 supercell of the primitive unit cell.\n", + "\n", + "For the calculation of *Fel* in , the most computationally\n", + "convenient approach is to use the Mermin statistics as follows\n", + "\n", + "*Eq. 5\u20114* ,\n", + "\n", + "where is the thermal electronic energy, and *Sel* is the bare\n", + "electronic entropy. Both the calculations of and *Sel* need\n", + "the electronic density of states (EDOS) as input. The electronic density\n", + "of states can be obtained during the step of the static total energy\n", + "calculation. The detailed formulations for and *Sel* are\n", + "given in Chapter . Since Ni is magnetic, the EDOS of Ni can be\n", + "partitioned into those of spin up and spin down due to the spin freedom\n", + "of electron. The calculated EDOS for Ni is shown in where the solid,\n", + "dot-dashed, and dashed lines represent the total, spin up, and spin down\n", + "EDOS with the Fermi energy set to zero.\n", + "\n", + "Figure \u2011. Electronic density of states of nickel. That due to spin up is\n", + "plotted as positive value and that due to spin down is plotted as\n", + "negative value purely for the clarity of the figure. The \u201ctotal\u201d is the\n", + "sum of the absolute values of those of spin up and spin down.\n", + "\n", + "The calculated temperature evolution of Helmholtz energy as a function\n", + "of volume for Ni are illustrated in . The circles represent the\n", + "calculated static total energies. The solid curves represent the\n", + "Helmholtz energy curves from 0 to 1600 K at a temperature increment of\n", + "100 K as displayed from top to bottom in . The dashed line marks the\n", + "evolution of the equilibrium volume at *P*=0 with increasing\n", + "temperature. It is noted that Helmholtz energy always decreases with\n", + "increasing temperature due to the entropy term of \u2013*TS*. Note that the\n", + "at 0 K the Helmholtz energy is higher than the static total energy due\n", + "to the zero point vibrational energy as can be seen when *T* \u21920 which\n", + "reduces to\n", + "\n", + "*Eq. 5\u20115* ,\n", + "\n", + "which is positive.\n", + "\n", + "Figure \u2011. Temperature evolution of the Helmholtz energy for nickel.\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/firstprinciples_calculations_and_theory/index.ipynb b/src/psu410/src/firstprinciples_calculations_and_theory/index.ipynb new file mode 100644 index 0000000..dbc3fa2 --- /dev/null +++ b/src/psu410/src/firstprinciples_calculations_and_theory/index.ipynb @@ -0,0 +1,60 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "121ad3f6", + "cell_type": "markdown", + "source": [ + "# First-principles calculations and theory\n", + "\n", + "In the previous chapter, the experimental techniques in obtaining the\n", + "thermochemical and phase equilibrium data that are the inputs for the\n", + "thermodynamic modeling of a system were summarized. However,\n", + "experimental data are not always available. This is due to the fact that\n", + "i) the experiments are expensive, especially true in developing new\n", + "materials; and ii) the experiments cannot reliably access the non-stable\n", + "phases in most cases. The alternative approach is to predict the\n", + "thermochemical data by first-principles calculations. The prediction of\n", + "the material properties, without using phenomenological parameters, is\n", + "the basic spirit of first-principles calculations. In particular, the\n", + "steady increase of both computer power and the efficiency of\n", + "computational methods have made the first-principles predictions of most\n", + "thermodynamic properties possible, including both enthalpy and entropy\n", + "as a function of temperature, volume, and/or pressure.\n", + "\n", + "By its definition, the term of \u201cfirst-principles\u201d represents a\n", + "philosophy that the prediction is to be based on a basic, fundamental\n", + "proposition or assumption that cannot be deduced from any other\n", + "proposition or assumption. This implies that the computational\n", + "formulations are based on the most fundamental theory of quantum\n", + "mechanics - Schr\u00f6dinger equation or density functional theory and the\n", + "inputs to the calculations must be based on well-defined physical\n", + "constants \u2013 the nuclear and electronic charges. In another word, once\n", + "the atomic species of an assigned material are known, the theory should\n", + "predict the energy of all possible crystalline structures, without\n", + "invoking any phenomenological fitting parameters.\n", + "\n", + "This chapter organized in the sequence from thermodynamic calculations\n", + "to fundamental theory to help those readers who are more interested in\n", + "realistic calculations using existing computer codes. The detailed\n", + "theoretical sections are presented to follow the section of\n", + "thermodynamic calculations for those readers who are also interested in\n", + "the derivation of the formulations used in the thermodynamic\n", + "calculations. The following subsections are arranged accordingly in the\n", + "order: (i) examples the commonly adopted calculation procedures for\n", + "thermodynamic properties using the elemental metal nickel as the main\n", + "prototype; (ii) derivation of the Helmholtz energy expression under the\n", + "first-principles framework; (iii) introduction of the solution to the\n", + "electronic Schr\u00f6dinger equation within the two well developed frameworks\n", + "\u2013 the quantum chemistry approach and the density functional theory; (iv)\n", + "detailed description of the procedure on how to solve the Schr\u00f6dinger\n", + "equation for the motions of the atomic nuclei by means of lattice\n", + "dynamics. The relation between the Helmholtz energy and Gibbs energy is\n", + "shown by .\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/firstprinciples_calculations_and_theory/lattice_dynamics.ipynb b/src/psu410/src/firstprinciples_calculations_and_theory/lattice_dynamics.ipynb new file mode 100644 index 0000000..4893825 --- /dev/null +++ b/src/psu410/src/firstprinciples_calculations_and_theory/lattice_dynamics.ipynb @@ -0,0 +1,189 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "18f2b40f", + "cell_type": "markdown", + "source": [ + "## Lattice Dynamics\n" + ], + "metadata": {} + }, + { + "id": "33b80c96", + "cell_type": "markdown", + "source": [ + "### Linear-response method vs supercell method\n", + "\n", + "The problem of lattice vibration for a solid is now transformed into\n", + "computing the dynamical matrix in . The first-principles solution of the\n", + "problem is currently divided into two categories: the linear-response\n", + "method \\[34\\] and the supercell method \\[35\\]. In the linear-response\n", + "method, utilizing the electronic linear response upon the undistorted\n", + "crystals \\[36\\], the evaluations of the dynamical matrix can be\n", + "performed through the density-functional perturbation theory \\[34\\]\n", + "without the approximation of the cutoff in neighboring interaction.\n", + "\n", + "Compared with the linear-response method, the supercell method is\n", + "conceptually simple and is easy to implement computationally. The\n", + "supercell method adopts the frozen phonon approximation through which\n", + "the changes in total energy or forces are calculated in the direct space\n", + "by displacing the atoms from their equilibrium positions. The advantage\n", + "of the supercell method is that the phonon frequencies at the exact wave\n", + "vectors, which are commensurable with the supercell, are calculated\n", + "exactly with no further approximation \\[37\\]. The shortcoming of the\n", + "supercell method is that it is often limited by the size of the\n", + "supercell that can be handled with current computing resources.\n", + "\n", + "In supercell approach, inaccuracies are thought to arise from the\n", + "truncation of the force constants \\[34-35\\]. This is only partially\n", + "true. In the supercell method, the calculated represents the cumulative\n", + "contributions of the atom indexed by *k* and *P* in the supercell and\n", + "all its images by translational transformation of the supercell in the\n", + "whole space. Let **L**\u03b1 represent the lattice vectors of the\n", + "supercell, then\n", + "\n", + "*Eq. 5\u2011138* .\n", + "\n", + "For the exact wave vectors in , one has\n", + "\n", + "*Eq. 5\u2011139* **,**\n", + "\n", + "where is an integer. Replacing in with in , one obtains\n", + "\n", + "Eq. 5\u2011140 .\n", + "\n", + "Therefore, the phonon frequencies calculated at the exact wave vectors\n", + "by the cumulative force constants approach are exact, and the supercell\n", + "size will not lead to errors in the calculated phonon frequencies\n", + "\\[37\\].\n", + "\n", + "Generally speaking, if a supercell contains *Nc* primitive\n", + "unit cells, one can always find *Nc* corresponding exact wave\n", + "vectors in the **q** space. In most linear response calculations, the\n", + "common choice of a 4\u00d74\u00d74 **q**-mesh is exactly equivalent to the 4\u00d74\u00d74\n", + "supercell in the real-space. Furthermore, since the supercell approach\n", + "does not impose any approximation on the electronic response to the\n", + "distortion of the lattice geometry, the effects of electron-phonon\n", + "interactions can be accounted for by the supercell method.\n", + "\n", + "In the supercell method, due to the impose of periodic condition, the\n", + "calculated force constant in the real-space cannot account for the\n", + "effects of the vibration-induced electric field for the polar materials.\n", + "It has been demonstrated that such an effect adds an additional term to\n", + "dynamical matrix in the reciprocal-space in the form,\n", + "\n", + "*Eq. 5\u2011141* ,\n", + "\n", + "where ***Z**\\* (j)* represents the Born effective charge\n", + "tensor of the *j*th atom in the primitive unit cell and\n", + "**\u03b5\u221e** the high frequency static dielectric tensor, i.e., the\n", + "contribution to the dielectric permittivity tensor from the electronic\n", + "polarization. As a result for the polar materials, the matrix element at\n", + "wave vector **q**\u21920 of the dynamical matrix in , by means of , should\n", + "have the form\n", + "\n", + "*Eq. 5\u2011142* .\n", + "\n", + "It can be demonstrated \\[38\\] that this is equivalent to replacing the\n", + "real space force constant in by\n", + "\n", + "*Eq. 5\u2011143* .\n", + "\n", + "The present implementation of the first-principles method in calculating\n", + "the phonon frequencies are mostly limited by the supercell size when\n", + "using or the number of exact wave vector points when using . A supercell\n", + "cell built on the primitive unit cell or exact wave vector mesh is\n", + "usually the common limit. If only the phonon frequencies derived from or\n", + "are used, can be rather unsmooth which will result in inaccurate\n", + "thermodynamic properties when it is used with . The mixed-space approach\n", + "\\[38\\] can circumvent this bottleneck by using the Fourier interpolation\n", + "to calculate the dynamical matrix for an arbitrary wave vector **q** as\n", + "\n", + "*Eq. 5\u2011144* ,\n", + "\n", + "with the help of , , and for a polar materials. Note that the term is\n", + "for polar materials only.\n", + "\n", + "The can be calculated as\n", + "\n", + "*Eq. 5\u2011145* ,\n", + "\n", + "where the function is usually taken as the Gaussian type\n", + "\n", + "*Eq. 5\u2011146* ,\n", + "\n", + "where is an adjustable damping (broadening) parameter whose role is to\n", + "smear the curve. in is the number of used **q** points. Empirically, a\n", + "**q** mesh of is accurate enough for most purposes and doable with the\n", + "YPHON code \\[15\\] efficiently, which is discussed in the Appendix A.\n" + ], + "metadata": {} + }, + { + "id": "156976ea", + "cell_type": "markdown", + "source": [ + "### Mermin statistics to the thermal electronic contribution \n", + "\n", + "For the calculation of *Fel* in , the most computationally\n", + "flexible approach is to use the Mermin statistics \\[28\\] by which\n", + "\n", + "*Eq. 5\u201122*\n", + "\n", + "where the bare electronic entropy *Sel* takes the form, after\n", + "replacing the summation in over the electronic states with integration\n", + "\n", + "*Eq. 5\u201123* ,\n", + "\n", + "by means of utilizing , the electronic density of states *n*. *f* in is\n", + "the Fermi distribution that takes the form\n", + "\n", + "*Eq. 5\u201124* .\n", + "\n", + "Note that in is the electronic chemical potential that is strongly\n", + "temperature dependent. At each given *T*, should be carefully calculated\n", + "keeping the number of electrons unchanged in solving the following\n", + "equation:\n", + "\n", + "*Eq. 5\u201125* ,\n", + "\n", + "noting that is the Fermi energy calculated at 0 K. With respect to , the\n", + "thermal electronic energy *Eel*, due to the thermal electron\n", + "excitations, can be calculated through\n", + "\n", + "*Eq. 5\u201126* .\n", + "\n", + "At low temperature, , , and are reduced to\n", + "\n", + "*Eq. 5\u201127*\n", + "\n", + "where *\u03bb* is the so-called electronic specific heat coefficient\n", + "calculated as\n", + "\n", + "*Eq. 5\u201128* ,\n", + "\n", + "where is the electronic density of states at the Fermi level, and\n", + "\n", + "*Eq. 5\u201129*\n", + "\n", + "*Eq. 5\u201130*\n", + "\n", + "From , one can easily derive the electronic contribution to the specific\n", + "heat at low temperature as\n", + "\n", + "*Eq. 5\u201131* .\n", + "\n", + "Usually, the dependence of on *V* is weak. Therefore for a normal\n", + "conductor (except for the heavy Fermion metal or superconductor related\n", + "materials), at low temperatures, the electronic contribution to the heat\n", + "capacity is linear against *T*. also indicates that for insulators, by\n", + "means of , the electronic contribution to the heat capacity is zero\n", + "since for insulators .\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/firstprinciples_calculations_and_theory/nickel_as_the_prototype.ipynb b/src/psu410/src/firstprinciples_calculations_and_theory/nickel_as_the_prototype.ipynb new file mode 100644 index 0000000..e25d418 --- /dev/null +++ b/src/psu410/src/firstprinciples_calculations_and_theory/nickel_as_the_prototype.ipynb @@ -0,0 +1,257 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "55dfbde7", + "cell_type": "markdown", + "source": [ + "## Nickel as the prototype\n", + "\n", + "This section exemplifies the step-by-step procedure in calculating the\n", + "thermal properties within the framework of first-principles phonon\n", + "approach, using the elemental metal Ni as the prototype. The calculation\n", + "of the formation enthalpy of Ni3Al is given at the end. The\n", + "calculation in this section is limited to the case for the ferromagnetic\n", + "phase, i.e. single microstate, implying that no configurational mixtures\n", + "or magnetic phase transitions is considered which is discussed in\n", + "Chapter 5.2.5 and Chapter 6.\n", + "\n", + "The Vienna Ab-initio Simulation Package (VASP) \\[13-14\\] has been\n", + "employed for electronic calculations, and the YPHON code \\[15\\] has been\n", + "employed for phonon calculations. VASP is a code based on the\n", + "pseudopotential approach to the density functional theory using plane\n", + "wave function as the basis set, by which only the valence electrons are\n", + "handled explicitly and the core electrons are approximated by an\n", + "effective pseudopotential. The same energy cutoff values, which\n", + "determine the number of plane waves in the expansion of electronic wave\n", + "function, have been used for Ni, Al, and Ni3Al. The rationale\n", + "for the derivations of the formulations used in this section is to be\n", + "given in Chapter for readers who want to have an in-depth understanding\n", + "of the physics behind the used formulations.\n" + ], + "metadata": {} + }, + { + "id": "5c3e2e16", + "cell_type": "markdown", + "source": [ + "### Normal coordinates, eigenenergetics, and phonons\n", + "\n", + "One way to simplify the solution to the Schr\u00f6dinger\u2019s equation for\n", + "motion of atomic nuclei is to follow the study of the vibrations of the\n", + "atoms in a crystal \u2013 lattice dynamics. The essential step in lattice\n", + "dynamics is to transform the problem of the correlated motions of 3*N*\n", + "particle into the problem of 3*N* independent harmonics. For this\n", + "purpose, one can introduce a set of new coordinates ( *i*=1, 2, \u2026,\n", + "3*N*), known as normal coordinates, by the transformation\n", + "\n", + "*Eq. 5\u2011111* ,\n", + "\n", + "where is the transformation coefficient that can be determined by\n", + "solving 3*N* simultaneous equations\n", + "\n", + "*Eq. 5\u2011112* ,\n", + "\n", + "where is to be determined so that one can find 3*N* , which are, not all\n", + "zero. The equations are linear and homogeneous. Follow the basic theorem\n", + "in linear algebra that, to find the nonzero solutions of the equations,\n", + "the determinant formed by must equal zero\n", + "\n", + "*Eq. 5\u2011113* ,\n", + "\n", + "where is the Kronecker delta symbol. Since is an equation with 3*N*\n", + "degrees, one can always find 3*N* values of (*i* = 1, \u2026, 3*N*). Each of\n", + "yields a set of which can be chosen such that\n", + "\n", + "*Eq. 5\u2011114* ,\n", + "\n", + "where represents the Kronecker delta symbol and\n", + "\n", + "*Eq. 5\u2011115*\n", + "\n", + "Then with the normal coordinates defined in and utilizing , defined in\n", + "is obtained as\n", + "\n", + "*Eq. 5\u2011116* .\n", + "\n", + "With this equation, is simplified by the following process\n", + "\n", + "*Eq. 5\u2011117* ,\n", + "\n", + "noting that in the above process and have been utilized.\n", + "\n", + "Furthermore, using , the kinetic energy operator in can be transformed\n", + "as follows\n", + "\n", + "*Eq. 5\u2011118* ,\n", + "\n", + "noting that in the above process and are utilized again.\n", + "\n", + "As a result, under the harmonic approximation, the Hamiltonian in is\n", + "simplified as\n", + "\n", + "*Eq. 5\u2011119*\n", + "\n", + "which represents a quantum system containing 3*N* independent harmonics.\n", + "Corresponding to the each of the *\u03c9i*, the quantum theory\n", + "tells that the eigenenergy of a harmonics has the form\n", + "\n", + "*Eq. 5\u2011120* ,\n", + "\n", + "with *gi* = 0, 1, \u2026, \u221e. Such a type of energetics behaves\n", + "like a Boson particle with energy and forms the concept of phonon.\n", + "\n", + "Furthermore, a state of the whole system is to be specified by the set\n", + "of 3*N* independent quantum numbers **g** = (*g1*,\n", + "*g1*, \u2026, *g3N*,). Finally, the energetics of a\n", + "state of the system formed by 3*N* independent harmonics, introduced by\n", + ", is obtained with the summation of the energies of 3*N* independent\n", + "harmonics as\n", + "\n", + "*Eq. 5\u2011121* .\n", + "\n", + "This concludes the rationale by which how is derived.\n" + ], + "metadata": {} + }, + { + "id": "5cba0f25", + "cell_type": "markdown", + "source": [ + "### Phonon calculations for SQS\n", + "\n", + "A somewhat more theoretically demanding application of the SQS approach\n", + "is the calculation of the phonon dispersions of a random alloy.\n", + "Considering the fact that the size of an SQS cell in general is around\n", + "8-32, phonon calculations based on SQS is doable, either using the SQS\n", + "cell or its supercell, e.g., of it. However, one notes that while the\n", + "phonon density of states can be calculated straightforwardly, the\n", + "calculations of the phonon dispersions run into a problem. That is,\n", + "since phonon calculation treats the SQS as primitive unit cell made of\n", + "more atoms than the primitive unit cell of the ideal lattice, the number\n", + "of phonon dispersions derived from a regular phonon calculations is a\n", + "lot greater than that measured for the random alloy. Say, one uses an\n", + "SQS containing 16 atoms for an fcc solid solution, the regular phonon\n", + "calculations would produce phonon dispersions in comparison to that just\n", + "three phonon dispersions from measurement. By averaging over the force\n", + "constants of a SQS, the dynamical matrix can be calculated with respect\n", + "to the wave vector space of the ideal lattice of the random alloys.\n", + "\n", + "One consideration that must be taken into account is that the phonon\n", + "dispersions measured from the inelastic neutron scattering experiments\n", + "only represent the averaged vibrations of the ideal lattice. For random\n", + "alloys or phases with minor geometry distortion, it is suggested to\n", + "calculate the dynamical matrix by instead of *Eq. 5\u2011125* as (see \\[45\\])\n", + "\n", + "*Eq. 5\u2011155* ,\n", + "\n", + "where in the case of random alloy, represents the averaged atomic mass\n", + "at the *j*th lattice site. The purpose of the summation over *Q* is to\n", + "average the effects of local distortions, making it possible of\n", + "comparing the calculated dispersions to the measured dispersions\n", + "representing the averaged vibrations of the ideal lattice. As a result,\n", + "the dimension of the SQS supercell dynamical matrix is thus reduced to\n", + "match that of the primitive unit cell of the ideal lattice for the\n", + "calculation of the phonon frequencies. The calculational procedure is as\n", + "follows:\n", + "\n", + "1. Make an SQS supercell based on the primitive unit cell of ideal\n", + " lattice to mimic the correlation functions of the random solution;\n", + "\n", + "2. Relax the SQS supercell with respect to the internal atomic\n", + " positions while keeping the cell shape and volume fixed;\n", + "\n", + "3. Make the phonon supercell by further enlarging the SQS supercell and\n", + " calculate the force constants; and\n", + "\n", + "4. Calculate the dynamical matrix , with the wave vector, **q**, being\n", + " defined from the primitive unit cell of the ideal lattice, through\n", + " the following Fourier transformation.\n", + "\n", + "The calculated phonon dispersions, along the directions (00\u03be), (0\u03be\u03be),\n", + "and (\u03be\u03be\u03be), are compared with the inelastic neutron scattering data in\n", + "for Cu3Au.\n", + "\n", + "Figure 5\u201113. Phonon dispersions for random Cu3Au. The solid\n", + "(black) lines represent the present calculation and the open circles\n", + "represent the inelastic neutron scattering data with details in Ref.\n", + "\\[45\\]. The dashed (blue) lines represent the calculated results using\n", + "the ab initio transferable force-constant model by Dutta et al. \\[46\\].\n" + ], + "metadata": {} + }, + { + "id": "687b1264", + "cell_type": "markdown", + "source": [ + "### Quantum theory for motion of atomic nuclei\n", + "\n", + "For the convenience of discussion, the following convention of notations\n", + "are used: *\u03b1* and *\u03b2* to label the Cartesian axes which is either *x*,\n", + "*y*, or *z*, *j* and *k* to label atoms in the primitive unit cell,\n", + "*mj* the atomic mass of the *j*th atom in the\n", + "primitive unit cell, **r**(*j*) the position of the *j*th\n", + "atom in the primitive unit cell, *P* and *Q* the index of the primitive\n", + "unit cell in the system, **R**(*P*) the position of the *P*th\n", + "primitive unit cell in the system, and *V* the averaged volume of the\n", + "primitive unit cell.\n", + "\n", + "The quantum theory for motion of atomic nuclei replicates closely the\n", + "quantum theory for motion of electrons. That is to solve the wave\n", + "function for the motions of the atomic nuclei for a Schr\u00f6dinger\u2019s\n", + "equation with the potential being the total electronic energy derived\n", + "from . The symbol is replaced by to represent the static total\n", + "electronic energy with **R** representing the static equilibrium\n", + "positions of the atomic nuclei, **u** the displacements of atomic nuclei\n", + "away from their static equilibrium positions, and *\u03c3* the additional\n", + "degree of freedom such as electronic states. The Schr\u00f6dinger\u2019s equation\n", + "for the motion of atomic nuclei is then\n", + "\n", + "*Eq. 5\u2011103* ,\n", + "\n", + "where\n", + "\n", + "*Eq. 5\u2011104* ,\n", + "\n", + "with representing the kinetic energy operator\n", + "\n", + "*Eq. 5\u2011105* ,\n", + "\n", + "where *Nc* is the numbers of primitive unit cells in the\n", + "system, *Na* is the numbers of atoms in the primitive unit\n", + "cell, is the Plack constant, represents the *\u03b1*th Cartesian\n", + "component of **u** for the atom at *j*th lattice site in the\n", + "primitive unit cell and the *P*th primitive unit cell in the\n", + "system.\n", + "\n", + "The harmonic approximation \\[23, 33\\] truncates the into the second\n", + "order in its Taylor\u2019s series\n", + "\n", + "*Eq. 5\u2011106* ,\n", + "\n", + "where is the real-space interatomic force constant. With the\n", + "approximation of , it can be demonstrated that finding the solution of\n", + "is equivalent to finding the vibrational frequencies of a classical\n", + "system with *NcNa* particles for small mechanical\n", + "oscillations.\n", + "\n", + "Let us rewrite as\n", + "\n", + "*Eq. 5\u2011107* ,\n", + "\n", + "where\n", + "\n", + "*Eq. 5\u2011108* , and\n", + "\n", + "*Eq. 5\u2011109* .\n", + "\n", + "Accordingly, the kinetic energy operator in becomes\n", + "\n", + "*Eq. 5\u2011110* .\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/firstprinciples_calculations_and_theory/quantum_theory_for_the_motion_of_electrons.ipynb b/src/psu410/src/firstprinciples_calculations_and_theory/quantum_theory_for_the_motion_of_electrons.ipynb new file mode 100644 index 0000000..2ae4fe6 --- /dev/null +++ b/src/psu410/src/firstprinciples_calculations_and_theory/quantum_theory_for_the_motion_of_electrons.ipynb @@ -0,0 +1,405 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "6f45af32", + "cell_type": "markdown", + "source": [ + "## Quantum theory for the motion of electrons\n" + ], + "metadata": {} + }, + { + "id": "9128fb36", + "cell_type": "markdown", + "source": [ + "### Schr\u00f6dinger equation\n", + "\n", + "The Schr\u00f6dinger equation is typically written as follows\n", + "\n", + "*Eq. 5\u201157* ,\n", + "\n", + "where is the reduced Planck constant, **X** an abbreviation of the space\n", + "coordinates and spin states of the multiple particle system, the energy\n", + "operator called Hamiltonian. When is independent of time *t*, one can\n", + "separate the coordinate **X** from the time *t* in finding the solution\n", + "of by writing\n", + "\n", + "*Eq. 5\u201158* ,\n", + "\n", + "by which the stationary solutions of can be expressed through letting\n", + "\n", + "*Eq. 5\u201159* ,\n", + "\n", + "resulting in\n", + "\n", + "*Eq. 5\u201160*\n", + "\n", + "*Eq. 5\u201161*\n", + "\n", + "Note that is the frequency of de Broglie matter wave.\n", + "\n", + "For any trial function \u039b(**X**) (in the Hilbert space) for \u03a8(**X**), the\n", + "variational principle tells us that the energy of the system always has\n", + "a lower bound through a ground state with energy *E0*, as\n", + "\n", + "*Eq. 5\u201162* ,\n", + "\n", + "where represents the complex conjugate of \u039b(**X**), resulting in\n", + "\n", + "*Eq. 5\u201163*\n", + "\n", + "which is known as the Rayleigh-Ritz variational principle,\n" + ], + "metadata": {} + }, + { + "id": "c21545db", + "cell_type": "markdown", + "source": [ + "#### Solving the Kohn-Sham Equations for a Solid\n", + "\n", + "For a solid, is still a mathematical challenge with infinite number of\n", + "one-electron wave functions to be solved and therefore cannot be solved\n", + "directly in the real space. To reduce the dimension of the problem, one\n", + "can choose to solve the equation at a specific point in the reciprocal\n", + "space. According to Bloch\u2019s theorem, the wave function for a solid can\n", + "be written as the product of a wavelike part, , and a cell periodic\n", + "part,\n", + "\n", + "*Eq. 5\u201195*\n", + "\n", + "where can be expressed as a sum of a finite number of plane waves whose\n", + "wave vectors are reciprocal lattice vectors of the crystal\n", + "\n", + "*Eq. 5\u201196* .\n", + "\n", + "so that\n", + "\n", + "*Eq. 5\u201197*\n", + "\n", + "where the band index *j* is used to number the eigenenergy and the\n", + "eigenvector at a given . The number of plane waves is determined by the\n", + "following equation\n", + "\n", + "*Eq. 5\u201198*\n", + "\n", + "where is called energy cutoff.\n", + "\n", + "Utilize the obtained wave functions, the charge density can be\n", + "calculated with the Brillouin zone integration\n", + "\n", + "*Eq. 5\u201199*\n", + "\n", + "Where\n", + "\n", + "*Eq. 5\u2011100*\n", + "\n", + "where the parameter is to make the integration over the charge density\n", + "within the primitive unit cell equal to the number of electrons, *N,* in\n", + "the primitive unit cell. Numerically, the integration can be\n", + "approximated by summation over a set of discrete *k*-mesh as\n", + "\n", + "*Eq. 5\u2011101*\n", + "\n", + "where *N*BZ represents the number of points in the first\n", + "Brillouin zone in the *k*-mesh. When a solid possesses symmetry, the\n", + "summation in the above equation can be further reduced to the summation\n", + "over the irreducible Brillouin zone (IBZ).\n", + "\n", + "*Eq. 5\u2011102*\n", + "\n", + "where is a weight factor that represents the number of points that are\n", + "equivalent to by space group symmetry.\n" + ], + "metadata": {} + }, + { + "id": "028b08b6", + "cell_type": "markdown", + "source": [ + "### Special Quasirandom Structures\n", + "\n", + "SQS\u2019s \\[43-44\\] are specially designed *small-unit-cell* periodic\n", + "structures with minimal number of atoms per unit cell, which closely\n", + "mimic the most relevant, near-neighbor pair and multisite correlation\n", + "functions of random substitutional alloys. The correlation functions are\n", + "classified by their *n*-site component \u201cfigures\u201d *f* = (*n,m*), where\n", + "the index *n* is called vortex for pair, triple, and quadruple\n", + "correlations (*n* = 2, 3, and 4); *m* measures the correlation distance.\n", + "\n", + "In the SQS approach, a distribution of distinct local environments is\n", + "maintained and their average corresponds to the random alloy. Thus, a\n", + "single DFT calculation of an SQS can give many important alloy\n", + "properties (e.g. equilibrium bond lengths, charge transfer, formation\n", + "enthalpies, etc.) that depend on the existence of those distinct local\n", + "environments. The SQS approach has been used extensively to study the\n", + "formation enthalpies, bond length distributions, density of states, band\n", + "gaps and optical properties in semiconductor alloys. It is noted that\n", + "the CE approach can treat short-range ordering efficiently, while it is\n", + "not clear how the SQS approach can be used to represent short-range\n", + "ordering.\n", + "\n", + "The key quantities in the SQS approach are the *n*-site correlation\n", + "functions. Specifically, the 2-site correlation function corresponding\n", + "to the 2-site component \u201cfigures\u201d (2,*m*) is\n", + "\n", + "Eq. 5\u2011151\n", + "\n", + "where represents the total number of possible pairs with correlation\n", + "distance (neighboring distance) *Rij* being equal to *m*. The\n", + "3-site correlation function corresponding to the 3-site component\n", + "\u201cfigures\u201d (3,*m*) is\n", + "\n", + "Eq. 5\u2011152\n", + "\n", + "where represents the total number of all possible 3-site \u201cfigures\u201d with\n", + "the correlation distance (size and shape) *Rijk* being equal\n", + "to *m*. The 4-site correlation function corresponding to the 4-site\n", + "component \u201cfigures\u201d (4,*m*) is\n", + "\n", + "Eq. 5\u2011153 ,\n", + "\n", + "where represents the total number of all possible 4-site \u201cfigures\u201d with\n", + "correlation distance (size and shape) *Rijkl* being equal to\n", + "*m*.\n", + "\n", + "With a given supercell size *N*, the essential task of the SQS approach\n", + "is to search through all configurations that approach as close as\n", + "possible to the correlation functions of the perfectly random (*R*)\n", + "structure and for the binary system it is\n", + "\n", + "Eq. 5\u2011154 .\n", + "\n", + "Describing random alloys by small unit-cell periodical structures surely\n", + "introduces erroneous correlations beyond a certain distance. However,\n", + "since interactions between nearest neighbors are generally more\n", + "important than interactions between more distant neighbors, SQS\u2019s can be\n", + "constructed in such a way that they exactly reproduce the correlation\n", + "functions of a random alloy between the first few nearest neighbors,\n", + "deferring errors due to periodicity to more distant neighbors. The\n", + "practical procedure could be that to find the structures that match the\n", + "2-site correlation functions up to a given neighboring distance, and\n", + "then add the conditions matching the high order correlation functions up\n", + "to certain correlation distance.\n", + "\n", + "Appendix B is a collection of the SQS\u2019s with a variety of compositions\n", + "for binary fcc, bcc, hcp, and L12 structures, for ternary\n", + "fcc, bcc, and B2 structures, and Perovskite in the cubic ABO3\n", + "structure. The used format is that of VASP.\n" + ], + "metadata": {} + }, + { + "id": "3e725123", + "cell_type": "markdown", + "source": [ + "### System with multiple microstates (MMS model)\n", + "\n", + "For a system made of multiple microstates, the total partition function\n", + "is the summation over the partition functions of all microstates, in ,\n", + "as\n", + "\n", + "*Eq. 5\u201149* ,\n", + "\n", + "where is the multiplicity of the microstate *\u03c3.* It is immediately\n", + "apparent that is the thermal population of the microstate *\u03c3*.\n", + "Furthermore, with , one obtains\n", + "\n", + "*Eq. 5\u201150* .\n", + "\n", + "relates the total Helmholtz energy, *F(N,V,T)*, of a system with mixing\n", + "among multiple microstates and the Helmholtz energy, , of individual\n", + "microstates. An important result of is the configurational entropy due\n", + "to the mixing among multiple microstates, named as microstate\n", + "configurational entropy (MCE) in this book,\n", + "\n", + "*Eq. 5\u201151* ,\n", + "\n", + "which makes the entropy of a system with mixing among multiple\n", + "microstates as\n", + "\n", + "*Eq. 5\u201152* .\n", + "\n", + "Similarly, one can get the heat capacity at constant volume of a system\n", + "with mixing among multiple microstates as\n", + "\n", + "*Eq. 5\u201153* ,\n", + "\n", + "where the configurational contributions to the heat capacity due to the\n", + "mixing among multiple microstates is\n", + "\n", + "*Eq. 5\u201154* .\n", + "\n", + "Moreover, the isothermal bulk modulus of a system with mixing among\n", + "multiple microstates can be also computed similarly\n", + "\n", + "*Eq. 5\u201155* ,\n", + "\n", + "With\n", + "\n", + "*Eq. 5\u201156*\n", + "\n", + "with being the partial pressure of the microstate *\u03c3*. This multiple\n", + "microstate model (MMS model) is used in Chapter to quantitatively\n", + "predict thermal expansion anomalies.\n" + ], + "metadata": {} + }, + { + "id": "f2f7173c", + "cell_type": "markdown", + "source": [ + "### Vibrational contribution by phonon theory\n", + "\n", + "Under the harmonic/quasiharmonic approximation, the lattice dynamics or\n", + "phonon theory is currently the most established method. It truncates the\n", + "interaction potential up to the second order. In such a case, in can be\n", + "expressed in terms of phonon frequency as follows\n", + "\n", + "*Eq. 5\u201132* ,\n", + "\n", + "where the label **g** has the meanings of (*g1*,\n", + "*g2*, \u2026, *g3N*) that *gj* can take any\n", + "integer values from zero to infinite.\n", + "\n", + "As a result, is reduced to\n", + "\n", + "*Eq. 5\u201133* ,\n", + "\n", + "or equivalently,\n", + "\n", + "*Eq. 5\u201134* ,\n", + "\n", + "where an integration has been used to replace the summation in by means\n", + "of introducing a function, , named the phonon density of states (PDOS)\n", + "whose integration over *\u03c9* equals to three per atom.\n", + "\n", + "Accordingly, the formulation to calculate the entropy becomes\n", + "\n", + "*Eq. 5\u201135* ,\n", + "\n", + "the formulation to calculate the internal energy becomes\n", + "\n", + "*Eq. 5\u201136* ,\n", + "\n", + "and the formulation to calculate the heat capacity at constant volume\n", + "becomes\n", + "\n", + "*Eq. 5\u201137* .\n" + ], + "metadata": {} + }, + { + "id": "533ec4a5", + "cell_type": "markdown", + "source": [ + "### Volume, entropy, enthalpy, thermal expansion, bulk modulus, and heat capacity \n", + "\n", + "The equilibrium volume *Veq* (*P*,*T*) at given *T* and *P*\n", + "can be obtained by finding the root of the following equation\n", + "\n", + "*Eq. 5\u20116* .\n", + "\n", + "The dashed line in illustrates *Veq* (*P*,*T*) as a function\n", + "of *T* from 0 to 1600 K at *P* = 0 for Ni.\n", + "\n", + "The entropy can be calculated through *F* by\n", + "\n", + "*Eq. 5\u20117* .\n", + "\n", + "Plotted in is the calculated entropy (curve) of Ni as a function of\n", + "temperature from 0 to 1600 K at *P* = 0 compared with the recommended\n", + "data (symbols) with details in Ref. \\[16\\].\n", + "\n", + "Figure \u2011. Entropy of nickel as a function of temperature.\n", + "\n", + "Based on *F* and *S*, the enthalpy at given *P* and *T* can be computed\n", + "as\n", + "\n", + "*Eq. 5\u20118* .\n", + "\n", + "Plotted in is the calculated enthalpy (curve) of Ni as function of\n", + "temperature from 0 to 1600 K at *P* = 0 comparing with the recommended\n", + "data (open circles) with details in Ref. \\[16\\]\n", + "\n", + "Figure \u2011. Enthalpy of nickel as a function of temperature.\n", + "\n", + "With the equilibrium volume *Veq* (*P*,*T*) calculated by ,\n", + "the volume thermal expansion coefficient defined by can be calculated by\n", + "\n", + "*Eq. 5\u20119* .\n", + "\n", + "Plotted in is the calculated thermal expansion coefficient (curve) of\n", + "nickel as function of temperature from 0 to 1600 K at *P* = 0 comparing\n", + "with experimental data (symbols) with details in Ref. \\[16\\]\n", + "\n", + "Figure \u2011. Volume thermal expansion coefficient of nickel as a function\n", + "of temperature.\n", + "\n", + "The bulk modulus of a material represents the substance's resistance to\n", + "uniform compression. Depending on how the temperature varies during\n", + "compression, a distinction should be made between the isothermal bulk\n", + "modulus (constant temperature) and adiabatic bulk modulus (constant\n", + "entropy or no heat transfer). As a matter of fact, most of the\n", + "experimental data are adiabatic whereas most of the published\n", + "theoretical data are isothermal.\n", + "\n", + "The isothermal bulk modulus, as defined in terms of Gibbs energy shown\n", + "by , can be calculated by\n", + "\n", + "*Eq. 5\u201110*\n", + "\n", + "Based on the isothermal bulk modulus, the adiabatic bulk modulus can be\n", + "calculated by\n", + "\n", + "*Eq. 5\u201111* ,\n", + "\n", + "where *CP* and *CV* represent the constant\n", + "pressure heat capacity and constant volume heat capacity respectively.\n", + "Plotted in is the calculated bulk moduli (curves) of Ni as function of\n", + "temperature from 0 to 1600 K at *P* = 0. The experimental data are from\n", + "ultrasonic measurements (symbols, see Ref. \\[28\\] for more details) that\n", + "are therefore adiabatic bulk moduli calculated based on the measured\n", + "adiabatic elastic constants using the relation\n", + "\n", + "*Eq. 5\u201112* .\n", + "\n", + "Figure \u2011. Bulk moduli of nickel as a function of temperature. Solid\n", + "line: adiabatic; Dashed line: isothermal.\n", + "\n", + "The heat capacity at constant volume, as defined in terms of Gibbs\n", + "energy shown by Eq. 2\u201128, can be calculated by\n", + "\n", + "*Eq. 5\u201113* ,\n", + "\n", + "where represents the internal energy. The heat capacity at constant\n", + "pressure (see ) can then be calculated as\n", + "\n", + "*Eq. 5\u201114* ,\n", + "\n", + "utilizing the calculated thermal expansion coefficient in and bulk\n", + "modulus in .\n", + "\n", + "It can be advocated that thermal expansion makes the difference between\n", + "the heat capacity at constant volume and the heat capacity at constant\n", + "pressure. The calculated contributions to the heat capacity of Ni as\n", + "function of temperature from 0 to 1600 K at *P* = 0 are illustrated in\n", + "where the lattice vibration and the thermal electron contributions have\n", + "been separated out. From , it is observed a large difference between the\n", + "calcualted *CP* (solid line) and the experimental data\n", + "(symbols, see Ref. \\[18\\] for more details) at 600 K due to the magnetic\n", + "phase transition which has not been considered in the calculation. It\n", + "should be pointed out that for Ni the thermal electronic contribution to\n", + "the heat capacity (dashed line in ) is substantial at high temperatures.\n", + "\n", + "Figure \u2011. Heat capacity of nickel as a function of temperature.\n", + "represents the calculated lattice vibration contribution; represents the\n", + "calculated thermal electronic contribution.\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/gibbs_energy_function/binary_random_solutions.ipynb b/src/psu410/src/gibbs_energy_function/binary_random_solutions.ipynb new file mode 100644 index 0000000..b8cb041 --- /dev/null +++ b/src/psu410/src/gibbs_energy_function/binary_random_solutions.ipynb @@ -0,0 +1,164 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "ecc765e4", + "cell_type": "markdown", + "source": [ + "### Binary random solutions\n", + "\n", + "From , the Gibbs-Duhem equation of a binary system consisting of\n", + "components $A$ and $B$ is written as\n", + "\n", + "Eq. \u2011 $0 = - SdT - Vd( - P) - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B}$\n", + "\n", + "This equation represents a four-dimensional surface. It is self-evident\n", + "that both and hold for stable binary solutions too, i.e. the directions\n", + "and the curvature of the surface are all negative. To visualize the\n", + "four-dimensional surface in the three-dimensional space, one needs to\n", + "fix one of the four potentials. As $T$ and $P$ are the natural variables\n", + "of Gibbs energy, they are usually chosen to be kept constant. One can\n", + "typically investigate behaviors of systems consisting of condensed\n", + "phases by varying the temperature at constant pressure. at constant\n", + "pressure thus becomes\n", + "\n", + "Eq. \u2011 $0 = - SdT - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B}$\n", + "\n", + "Similar to and , the property of a phase can be represented by a\n", + "two-dimensional surface in the three-dimensional space composed of $T$,\n", + "$\\mu_{A}$, and $\\mu_{B}$ under constant $P$, keeping in mind the\n", + "following\n", + "\n", + "Eq. \u2011\n", + "$G_{m} = x_{A}\\mu_{A} + x_{B}\\mu_{B} = x_{A}_{\\ }^{0}G_{A} + x_{B}_{\\ }^{0}G_{B} + RT\\left( x_{A}l{nx}_{A} + x_{B}l{nx}_{B} \\right) +_{\\ }^{E}G_{m}$\n", + "\n", + "Since $_{\\ }^{E}G_{m}$ must be zero for pure components $A$ and $B$, it\n", + "needs to be in the following form\n", + "\n", + "Eq. \u2011 $_{\\ }^{E}G_{m} = x_{A}x_{B}L_{AB}$\n", + "\n", + "with $L_{AB}$ being a parameter denoting the interaction between\n", + "components $A$ and $B$, called interaction parameter. When $L_{AB} = 0$,\n", + "the solution is an ideal solution. When $L_{AB}$ is a non-zero constant\n", + "independent of temperature and composition, the solution is called a\n", + "regular solution. Its excess entropy and excess enthalpy of mixing are\n", + "obtained as\n", + "\n", + "Eq. \u2011 $_{\\ }^{E}S_{m} = \\frac{\\partial_{\\ }^{E}G_{m}}{\\partial T} = 0$\n", + "\n", + "Eq. \u2011\n", + "$_{\\ }^{E}H_{m} =_{\\ }^{E}G_{m} - T_{\\ }^{E}S_{m} = x_{A}x_{B}L_{AB}$\n", + "\n", + "The chemical potential of component $A$ or $B$ in a binary regular\n", + "solution can be derived as\n", + "\n", + "Eq. \u2011\n", + "$\\mu_{i} =_{\\ }^{0}G_{i} + RTlnx_{i} + \\left( 1 - x_{i} \\right)^{2}L_{AB}$\n", + "\n", + "In a dilute solution with $x_{i} \\rightarrow 0$, one can have\n", + "\n", + "Eq. \u2011\n", + "$RTln\\gamma_{i} = \\left( 1 - x_{i} \\right)^{2}L_{AB} \\approx L_{AB}$\n", + "\n", + "Eq. \u2011 $\\gamma_{i} = e^{\\frac{L_{AB}}{RT}}$\n", + "\n", + "The activity is thus proportional to its mole fraction, which is called\n", + "Henry\u2019s law. By the same token, for the solvent, i.e.\n", + "$x_{i} \\rightarrow 1$,\n", + "\n", + "Eq. \u2011 $RTln\\gamma_{i} = \\left( 1 - x_{i} \\right)^{2}L_{AB} \\approx 0$\n", + "\n", + "which gives $\\gamma_{i} \\approx 1$, and its activity approaches its mole\n", + "fraction. This is called Raoult\u2019s law.\n", + "\n", + "The stability of a binary solution is derived from as\n", + "\n", + "Eq. \u2011\n", + "$\\left( \\frac{\\partial\\mu_{A}}{\\partial N_{A}} \\right)_{T,P,N_{B}} = \\left\\lbrack \\frac{RT}{x_{A}} - 2\\left( 1 - x_{A} \\right)L_{AB} \\right\\rbrack\\frac{1 - x_{A}}{N}$\n", + "\n", + "Eq. \u2011\n", + "$\\left( \\frac{\\partial\\mu_{B}}{\\partial N_{B}} \\right)_{T,P,N_{A}} = \\left\\lbrack \\frac{RT}{x_{B}} - 2\\left( 1 - x_{B} \\right)L_{AB} \\right\\rbrack\\frac{1 - x_{B}}{N}$\n", + "\n", + "It should be noted that the two chemical potentials in a binary system\n", + "at constant temperature and pressure are dependent on each other due to\n", + "the Gibbs-Duhem equation shown in , i.e.\n", + "\n", + "Eq. \u2011 $0 = - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B}$\n", + "\n", + "and the two chemical potentials depend on each other by the following\n", + "relation\n", + "\n", + "Eq. \u2011\n", + "$\\left( \\frac{\\partial\\mu_{A}}{\\partial\\mu_{B}} \\right)_{T,P} = - \\frac{N_{B}}{N_{A}}$\n", + "\n", + "Therefore, at the limit of stability, both and go to zero at the same\n", + "time, which is obtained when\n", + "\n", + "Eq. \u2011 $RT = 2{x_{A}x_{B}L}_{AB}$\n", + "\n", + "As the absolute temperature cannot be negative, has no solution for a\n", + "solution phase with $L_{AB} < 0$, i.e. the solution phase is stable with\n", + "respect to the composition fluctuation. For a solution with\n", + "$L_{AB} > 0$, its limit of stability is represented .\n", + "\n", + "A schematic molar Gibbs energy of a solution with $L_{AB} < 0$ at\n", + "constant temperature and pressure is shown in along with the ideal and\n", + "excess Gibbs energy of mixing. A tangent line on the molar Gibbs energy\n", + "of the solution is drawn, and its two intercepts at $x_{B} = 0$ and\n", + "$x_{B} = 1$ give the chemical potentials of components $A$ and $B$,\n", + "$\\mu_{A}$ and $\\mu_{B}$ by , respectively. It is evident that $\\mu_{A}$\n", + "and $\\mu_{B}$ are not independent on each other as they are two points\n", + "on the same straight line. This is a graphic representation of the\n", + "Gibbs-Duhem equation of . The chemical activity of component $B$ is also\n", + "depicted with the reference state being the pure B with the same\n", + "structure. As shown in , other structures of pure B can be selected as\n", + "the reference states of the chemical activity of component B, resulting\n", + "in the different distances to its chemical potential in the solution,\n", + "thus different values of its chemical activities. It is clear that this\n", + "change of reference state for chemical activity does not affect the\n", + "chemical potential of the component in the solution.\n", + "\n", + "Figure \u2011: Schematic molar Gibbs energy diagram with $L_{AB} < 0$\n", + "\n", + "When $L_{AB} > 0$, represents a parabola in the $T - x_{i}$\n", + "two-dimensional coordinate, symmetric with respect to $x_{A}$ and\n", + "$x_{B}$, shown in , i.e. the spinodal of the solution. The consolute\n", + "point is obtained by applying to and letting equal to zero at the\n", + "consolute point\n", + "\n", + "Eq. \u2011\n", + "$\\left( \\frac{\\partial^{2}\\mu_{A}}{\\partial N_{A}^{2}} \\right)_{T,P,N_{B}} = \\left\\lbrack - \\frac{RT}{x_{A}^{2}} + 2L_{AB} \\right\\rbrack\\left( \\frac{1 - x_{A}}{N} \\right)^{2} = 0$\n", + "\n", + "which gives\n", + "\n", + "Eq. \u2011 $T_{cons} = 2x_{A}^{2}L_{AB}$\n", + "\n", + "Solving and , one obtains $x_{A} = x_{B} = 0.5$ and\n", + "\n", + "Eq. \u2011 $T_{cons} = \\frac{L_{AB}}{2R}$\n", + "\n", + "Figure \u2011: A Spinodal curve with $L_{AB} > 0$\n", + "\n", + "A schematic molar Gibbs energy diagram at temperatures below the\n", + "consolute point is shown in . It can be seen that part of the molar\n", + "Gibbs energy has negative curvature, and the solution becomes unstable.\n", + "The chemical potential thus does not change monotonically with respect\n", + "to composition and its derivative changes sign at the inflexion point.\n", + "\n", + "Figure \u2011: Schematic molar Gibbs energy diagram with $L_{AB} > 0$\n", + "\n", + "For more complex solutions, $L_{AB}$ can be a function of temperature,\n", + "pressure, and compositions. In principle, the temperature and pressure\n", + "dependences can be treated by means of formula similar to . There are\n", + "various approaches in the literature to consider the composition\n", + "dependence of $L_{AB}$. The empirical Redlich-Kister polynomial stands\n", + "out as the one most widely used because it can be extrapolated to\n", + "ternary and multi-component systems consistently, which will be\n", + "discussed in Chapter .\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/gibbs_energy_function/elastic_magnetic_and_electric_contributions_to_free_energy.ipynb b/src/psu410/src/gibbs_energy_function/elastic_magnetic_and_electric_contributions_to_free_energy.ipynb new file mode 100644 index 0000000..1ccae10 --- /dev/null +++ b/src/psu410/src/gibbs_energy_function/elastic_magnetic_and_electric_contributions_to_free_energy.ipynb @@ -0,0 +1,131 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "93e921a2", + "cell_type": "markdown", + "source": [ + "## Elastic, magnetic, and electric contributions to free energy\n", + "\n", + "Chapters and focus on the thermal and hydrostatical pressure\n", + "contributions to Gibbs energy, which are the two prime variables\n", + "affecting phase stability in typical experimental environments. However,\n", + "there are other internal and external contributions, which are\n", + "particularly important for crystalline phases. Two important internal\n", + "contributions are from magnetic and electric polarizations of materials\n", + "with the corresponding external contributions due to the magnetic and\n", + "electric fields. Furthermore, for non-hydrostatic pressing of solid\n", + "phases, the $PV$ term in the combined law is to be replaced by elastic\n", + "energy calculated from elastic stress and elastic strain. The\n", + "corresponding works done to a system are as follows \\[3\\]\n", + "\n", + "Eq. \u2011\n", + "$dW_{elastic} = - V\\sum_{i,j,k,l}^{}{\\sigma_{ij}{d\\varepsilon}_{kl}}$\n", + "\n", + "Eq. \u2011 $dW_{magnetic} = - V\\sum_{i}^{}{H_{i}{dB}_{i}}$\n", + "\n", + "Eq. \u2011 $dW_{electric} = - V\\sum_{i}^{}{E_{i}{dD}_{i}}$\n", + "\n", + "where $i,j,\\ k,\\ l = 1,2,3$, $\\sigma_{ij}$ and $\\varepsilon_{kl}$ are\n", + "the components of stress and strain; $H_{i}$ and $B_{i}$ are the\n", + "components of magnetic field and magnetic induction; $E_{i}$ and $D_{i}$\n", + "are the components of electric field and electric displacement; and $V$\n", + "is the volume of the crystal. The negative sign in front of the equation\n", + "is due to the fact that the system does work to the surroundings when it\n", + "expands its volume due to the strain, magnetic induction, and electric\n", + "displacement.\n", + "\n", + "Using the combined law of thermodynamics, can thus be re-written as\n", + "follows\n", + "\n", + "Eq. \u2011\n", + "$dU = TdS - V\\left( \\sum_{i,j,k,l}^{}{\\sigma_{ij}{d\\varepsilon}_{kl}} + \\sum_{i}^{}{H_{i}{dB}_{i}} + \\sum_{i}^{}{E_{i}{dD}_{i}} \\right) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$\n", + "\n", + "The Legendre transformation similar to Helmholtz energy and Gibbs\n", + "energy, and , can be made to obtain the following characteristic free\n", + "energy functions\n", + "\n", + "Eq. \u2011 $dF = d(U - TS)$\n", + "\n", + "$$= - SdT - V\\left( \\sum_{i,j,k,l}^{}{\\sigma_{ij}{d\\varepsilon}_{kl}} + \\sum_{i}^{}{H_{i}{dB}_{i}} + \\sum_{i}^{}{E_{i}{dD}_{i}} \\right) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$$\n", + "\n", + "Eq. \u2011 $dF_{H} = d\\left( U - TS + \\sum_{i}^{}{H_{i}B_{i}} \\right)$\n", + "\n", + "$$= - SdT - V\\left( \\sum_{i,j,k,l}^{}{\\sigma_{ij}{d\\varepsilon}_{kl}} - \\sum_{i}^{}{B_{i}dH_{i}} + \\sum_{i}^{}{E_{i}{dD}_{i}} \\right) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$$\n", + "\n", + "Eq. \u2011 $dF_{E} = d\\left( U - TS + \\sum_{i}^{}{E_{i}D_{i}} \\right)$\n", + "\n", + "$$= - SdT - V\\left( \\sum_{i,j,k,l}^{}{\\sigma_{ij}{d\\varepsilon}_{kl}} + \\sum_{i}^{}{H_{i}{dB}_{i}} - \\sum_{i}^{}{D_{i}{dE}_{i}} \\right) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$$\n", + "\n", + "Eq. \u2011\n", + "$dF_{EH} = d\\left( U - TS + \\sum_{i}^{}{H_{i}B_{i}} + \\sum_{i}^{}{E_{i}D_{i}} \\right)$\n", + "\n", + "$$= - SdT - V\\left( \\sum_{i,j,k,l}^{}{\\sigma_{ij}{d\\varepsilon}_{kl}} - \\sum_{i}^{}{B_{i}{dH}_{i}} - \\sum_{i}^{}{D_{i}{dE}_{i}} \\right) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$$\n", + "\n", + "Eq. \u2011\n", + "$dG = d\\left( U - TS + \\sum_{i,j,k,l}^{}{\\sigma_{ij}\\varepsilon_{kl}} + \\sum_{i}^{}{H_{i}B_{i}} + \\sum_{i}^{}{E_{i}D_{i}} \\right)$\n", + "\n", + "$$= - SdT + V\\left( \\sum_{i,j,k,l}^{}{\\varepsilon_{ij}d\\sigma}_{kl} + \\sum_{i}^{}{B_{i}dH_{i}} + \\sum_{i}^{}{D_{i}{dE}_{i}} \\right) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$$\n", + "\n", + "From above equations, it can be seen that the natural variables of\n", + "various free energies are\n", + "$F\\left( T,\\ \\varepsilon_{ij},\\ B_{i},\\ D_{i},\\ N_{i},\\ \\xi \\right)$,\n", + "$F_{H}\\left( T,\\ \\varepsilon_{ij},\\ H_{i},\\ D_{i},\\ N_{i},\\ \\xi \\right)$,\n", + "$F_{E}\\left( T,\\ \\varepsilon_{ij},\\ B_{i},\\ E_{i},\\ N_{i},\\ \\xi \\right)$,\n", + "$F_{EH}\\left( T,\\ \\varepsilon_{ij},\\ H_{i},\\ E_{i},\\ N_{i},\\ \\xi \\right)$,\n", + "and $G\\left( T,\\ \\sigma_{ij},\\ H_{i},\\ E_{i},\\ N_{i},\\ \\xi \\right)$.\n", + "Clearly, there can be more combinations when the components of\n", + "$\\varepsilon_{ij}$, $D_{i}$, and $B_{i}$ are partially replaced by their\n", + "conjugate potentials. The free energies listed above are useful\n", + "depending on how the system is constrained by the surroundings. For\n", + "practical applications, the elastic, magnetic, and electric properties\n", + "are usually considered for phases with fixed compositions, and at\n", + "equilibrium can then be written as\n", + "\n", + "Eq. \u2011\n", + "$dG = - SdT + V\\left( \\sum_{i,j,k.l}^{}{\\varepsilon_{ij}d\\sigma}_{kl} + \\sum_{i}^{}{B_{i}dH_{i}} + \\sum_{i}^{}{D_{i}{dE}_{i}} \\right) + \\mu dN$\n", + "\n", + "The corresponding Gibbs-Duhem equation follows as below\n", + "\n", + "Eq. \u2011\n", + "$0 = - SdT + V\\left( \\sum_{i,j,k,l}^{}{\\varepsilon_{ij}d\\sigma}_{kl} + \\sum_{i}^{}{B_{i}dH_{i}} + \\sum_{i}^{}{D_{i}{dE}_{i}} \\right) - Nd\\mu$\n", + "\n", + "The general differential form of a molar quantity can be extended from\n", + "as follows as a one-component system\n", + "\n", + "Eq. \u2011\n", + "\n", + "Eq. \u2011\n", + "\n", + "Eq. \u2011\n", + "\n", + "Eq. \u2011\n", + "\n", + "The first derivatives in to are the second directives of Gibbs energy\n", + "with respect to its natural variables, i.e. potentials, and have their\n", + "respective nomenclatures as shown in . The limit of stability follows\n", + "and can be re-written as\n", + "\n", + "Eq. \u2011\n", + "$\\left( \\frac{\\partial X_{i}}{\\partial Y_{i}} \\right)_{N,Y_{j}} = \\infty$\n", + "\n", + "This means that the derivatives in to , i.e. the quantities in , diverge\n", + "at the limit of stability.\n", + "\n", + "Table \u2011: Physical quantities related to the first derivatives in to .\n", + "The table is symmetric due to the Maxwell relations related to the\n", + "second directives of Gibbs energy with respect to its natural variables.\n", + "\n", + "| | T | $$\\sigma_{kl}$$ | $$E_{k}$$ | $$H_{k}$$ |\n", + "|----------------------|------------------------------------|------------------------------------|---------------------------------------|---------------------------------------|\n", + "| S | C/T, heat capacity | $\\alpha_{kl}$, piezocaloric effect | $p_{k}$, electrocaloric effect | $m_{k}$, magnetocaloric effect |\n", + "| $$\\varepsilon_{ij}$$ | $\\alpha_{ij}$, thermal expansion | $s_{ijkl}$, elastic compliance | $d_{ijk}$, converse piezoelectricity | $q_{ijk}$, piezomagnetic moduli |\n", + "| $$D_{i}$$ | $p_{i}$, pyroelectric coefficients | $d_{ikl}$, piezoelectric moduli | $k_{ik}$, permittivities | $a_{ik}$, magnetoelectric coefficient |\n", + "| $$B_{i}$$ | $m_{i}$, pyromagnetic coefficient | $q_{ikl}$, piezomagnetic moduli | $a_{ik}$, magnetoelectric coefficient | $\\mu_{ik}$, permeability |\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/gibbs_energy_function/index.ipynb b/src/psu410/src/gibbs_energy_function/index.ipynb new file mode 100644 index 0000000..92a6983 --- /dev/null +++ b/src/psu410/src/gibbs_energy_function/index.ipynb @@ -0,0 +1,215 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "94948523", + "cell_type": "markdown", + "source": [ + "# Gibbs energy function\n", + "\n", + "As shown in through , all functions have $N_{i}$ and $\\xi$ as natural\n", + "variables while they differ in other two natural variables. In typical\n", + "materials-related experiments, temperature and pressure are the two\n", + "variables controlled. They are also the natural variables of Gibbs\n", + "energy. Consequently, Gibbs energy is the most widely used function in\n", + "thermodynamics of materials science. The rest of this book focuses on\n", + "Gibbs energy for this reason. In this chapter, the mathematical formulas\n", + "for Gibbs energy of phases with fixed and variable compositions are\n", + "discussed which are needed for quantitative calculations of Gibbs energy\n", + "under given values of its natural variables.\n", + "\n", + "From , the molar Gibbs energy can be defined as\n", + "\n", + "Eq. \u2011\n", + "$G_{m}\\left( T,P,x_{i},\\xi \\right) = \\frac{G}{N} = \\sum_{}^{}\\mu_{i}x_{i}$\n", + "\n", + "The molar entropy, molar volume, chemical potential, and the driving\n", + "force can be obtained from as\n", + "\n", + "Eq. \u2011\n", + "$S_{m} = \\frac{S}{N} = - \\frac{1}{N}\\left( \\frac{\\partial G}{\\partial T} \\right)_{P,\\ N_{i},\\ \\xi} = {- \\left( \\frac{\\partial G_{m}}{\\partial T} \\right)}_{P,\\ x_{i},\\ \\xi}$\n", + "\n", + "Eq. \u2011\n", + "$V_{m} = \\frac{V}{N} = \\frac{1}{N}\\left( \\frac{\\partial G}{\\partial P} \\right)_{T,\\ N_{i},\\ \\xi} = \\left( \\frac{\\partial G_{m}}{\\partial P} \\right)_{T,\\ x_{i},\\ \\xi}$\n", + "\n", + "Eq. \u2011\n", + "$\\mu_{i} = \\left( \\frac{\\partial G}{\\partial N_{i}} \\right)_{T,P,N_{j \\neq i},\\ \\xi}$\n", + "\n", + "Eq. \u2011\n", + "$- D = \\left( \\frac{\\partial G}{\\partial\\xi} \\right)_{T,P,N_{i}\\ }$\n", + "\n", + "Based on , the molar enthalpy is written as\n", + "\n", + "Eq. \u2011 $H_{m} = G_{m} + TS_{m}$\n", + "\n", + "Other physical properties of the system can also be represented by the\n", + "partial derivatives of Gibbs energy such as heat capacity, $C_{P}$,\n", + "volume thermal expansivity, $\\alpha_{V}$, isothermal compressibility,\n", + "$\\kappa_{T}$, as follows under constant pressure or temperature\n", + "\n", + "Eq. \u2011\n", + "$C_{P} = \\left( \\frac{\\partial Q}{\\partial T} \\right)_{P} = \\left( \\frac{\\partial H}{\\partial T} \\right)_{P} = T\\left( \\frac{\\partial(G + TS)}{\\partial T} \\right)_{P} = T\\left( \\frac{\\partial S}{\\partial T} \\right)_{P} = - T\\left( \\frac{\\partial^{2}G}{\\partial T^{2}} \\right)_{P}$\n", + "\n", + "Eq. \u2011\n", + "$\\alpha_{V} = \\frac{\\left( \\frac{\\partial V}{\\partial T} \\right)_{P}}{V} = \\frac{\\left( \\frac{\\left( \\partial G/\\partial( - P) \\right)_{T}}{\\partial T} \\right)_{P}}{\\left( \\partial G/\\partial( - P) \\right)_{T}} = \\frac{\\frac{\\partial^{2}G}{\\partial T\\partial( - P)}}{\\left( \\partial G/\\partial( - P) \\right)_{T}}$\n", + "\n", + "Eq. \u2011\n", + "$\\kappa_{T} = \\frac{\\left( \\frac{\\partial V}{\\partial( - P)} \\right)_{T}}{V} = \\frac{\\left( \\frac{\\left( \\partial G/\\partial( - P) \\right)_{T}}{\\partial( - P)} \\right)_{T}}{\\left( \\partial G/\\partial( - P) \\right)_{T}} = \\frac{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}}{\\left( \\partial G/\\partial( - P) \\right)_{T}} = \\frac{1}{B}$\n", + "\n", + "where the $N_{i}$ and $\\xi$ are kept constant for all partial\n", + "derivatives, and $B$ is the bulk modulus.\n", + "\n", + "In , $G$ cannot be directly replaced by $G_{m}$ because *N* also depends\n", + "on *Ni*. The thermodynamic quantities under such conditions,\n", + "i.e. varying the amount of a component at constant temperature and\n", + "pressure, are called partial quantities which are introduced in Eq. 1\u20118\n", + "for partial entropy and for partial enthalpy. This definition can be\n", + "extended to all molar quantities such as partial volume and partial\n", + "Gibbs energy. Partial quantities of a molar quantity, $A$, can thus be\n", + "defined in general as\n", + "\n", + "Eq. \u2011\n", + "$A_{i} = \\left( \\frac{\\partial A}{\\partial N_{i}} \\right)_{T,P,N_{j \\neq i},\\ \\xi}$\n", + "\n", + "The general differential form of a molar quantity for a system at\n", + "equilibrium can be represented by its partial quantities as\n", + "\n", + "Eq. \u2011\n", + "$dA = \\left( \\frac{\\partial A}{\\partial T} \\right)dT + \\left( \\frac{\\partial A}{\\partial P} \\right)dP + \\sum_{}^{}\\left( \\frac{\\partial A}{\\partial N_{i}} \\right){dN}_{i}$\n", + "\n", + "where the subscripts representing variables kept constant, i.e. the\n", + "remaining natural variables of Gibbs energy not in the denominator, are\n", + "omitted for simplicity. This will be done throughout the book unless\n", + "specified otherwise.\n", + "\n", + "Using the following relations: $A = NA_{m}$, $N = \\sum_{}^{}N_{j}$,\n", + "$x_{i} = N_{i}/N$,\n", + "$\\frac{{\\partial x}_{i}}{{\\partial N}_{i}} = \\left( 1 - x_{i} \\right)/N$,\n", + "and $\\frac{{\\partial x}_{k}}{{\\partial N}_{i}} = {- x}_{k}/N$, can be\n", + "derived as, under constant T and P,\n", + "\n", + "Eq. \u2011\n", + "$A_{i} = A_{m} + N\\sum_{j = 1}^{c}{\\frac{\\partial A_{m}}{\\partial x_{j}}\\frac{\\partial x_{j}}{\\partial N_{i}}} = A_{m} + \\frac{\\partial A_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial A_{m}}{\\partial x_{j}}$\n", + "\n", + "where the summation is for all *c* components and the partial\n", + "derivatives are taken with other mole fractions kept constant. However,\n", + "mole fractions are not independent, but follow the relation\n", + "$\\sum_{}^{}x_{i} = 1$. Taking $x_{1} = 1 - \\sum_{j = 2}^{c}x_{j}$ as the\n", + "dependent mole fraction, can be rewritten as\n", + "\n", + "Eq. \u2011\n", + "$A_{i} = A_{m} + \\left( \\frac{\\partial A_{m}}{\\partial x_{i}} - \\frac{\\partial A_{m}}{\\partial x_{1}} \\right) - \\sum_{j = 2}^{c}x_{j}\\left( \\frac{\\partial A_{m}}{\\partial x_{j}} - \\frac{\\partial A_{m}}{\\partial x_{1}} \\right)$\n", + "\n", + "The difference of the partial derivatives in the parenthesis in\n", + "represents the partial derivative of $A_{m}$ with respect to the mole\n", + "fraction of one component when the first component is selected as the\n", + "dependent component. Applying and to Gibbs energy, the partial Gibbs\n", + "energy or chemical potential of component $i$ is obtained as\n", + "\n", + "Eq. \u2011\n", + "$\\mu_{i} = G_{i} = G_{m} + \\frac{\\partial G_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial G_{m}}{\\partial x_{j}} = G_{m} + \\left( \\frac{\\partial G_{m}}{\\partial x_{i}} - \\frac{\\partial G_{m}}{\\partial x_{1}} \\right) - \\sum_{j = 2}^{c}x_{j}\\left( \\frac{\\partial G_{m}}{\\partial x_{j}} - \\frac{\\partial G_{m}}{\\partial x_{1}} \\right)$\n", + "\n", + "The derivatives in the stability equation, , are defined with the molar\n", + "quantities kept constant. On the other hand, Gibbs energy has two\n", + "potentials, temperature and pressure, as natural variables instead. One\n", + "would thus need to compare the stability conditions when a variable kept\n", + "fixed is changed from a molar quantity to its conjugate potential. This\n", + "can be carried out through the use of Jacobians to change the\n", + "independent variables\n", + "\n", + "Eq. \u2011\n", + "$\\frac{\\partial\\left( Y_{i},Y_{j} \\right)}{\\partial\\left( X_{i},X_{j} \\right)} = \\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{Y_{j}}\\left( \\frac{\\partial Y_{j}}{\\partial X_{j}} \\right)_{X_{i}} = \\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}}\\left( \\frac{\\partial Y_{j}}{\\partial X_{j}} \\right)_{X_{i}} - \\left( \\frac{\\partial Y_{i}}{\\partial X_{j}} \\right)_{X_{i}}\\left( \\frac{\\partial Y_{j}}{\\partial X_{i}} \\right)_{X_{j}}$\n", + "\n", + "For a stable system, both\n", + "$\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}}$ and\n", + "$\\left( \\frac{\\partial Y_{j}}{\\partial X_{j}} \\right)_{X_{i}}$ are\n", + "positive based on . Using the Maxwell relation shown by , one thus\n", + "obtains\n", + "\n", + "Eq. \u2011\n", + "$\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}} - \\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{Y_{j}} = \\left( \\frac{\\partial Y_{i}}{\\partial X_{j}} \\right)_{X_{i}}\\left( \\frac{\\partial Y_{j}}{\\partial X_{i}} \\right)_{X_{j}}/\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}} \\geq 0$\n", + "\n", + "This means that\n", + "$\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{Y_{j}}$ will go\n", + "to zero before\n", + "$\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}}$ does. It\n", + "indicates that the stability condition becomes more restrictive when\n", + "potentials are kept constant in place of their conjugate molar\n", + "quantities. Based on the Gibbs-Duhem equation of , the maximum number of\n", + "independent potentials is *c+1*, and the last potential is dependent,\n", + "i.e.\n", + "\n", + "Eq. \u2011\n", + "$\\left( \\frac{\\partial Y_{c + 2}}{\\partial X_{c + 2}} \\right)_{Y_{j \\leq c + 1}} = 0$\n", + "\n", + "Therefore, the limit of stability is determined when the derivative\n", + "becomes zero with one molar quantity kept constant, e.g.\n", + "\n", + "Eq. \u2011\n", + "$\\left( \\frac{\\partial Y_{c + 1}}{\\partial X_{c + 1}} \\right)_{Y_{j < c + 1},X_{c + 2}} = 0$\n", + "\n", + "This is because this derivative reaches zero faster than any other\n", + "derivatives with more molar quantities kept constant. shows that all\n", + "molar quantities diverge at the limit of stability. The consolute point\n", + "is obtained with $c$ additional conditions as follows based on\n", + "\n", + "Eq. \u2011\n", + "$\\left( \\frac{\\partial^{2}Y_{i}}{\\partial\\left( X_{i} \\right)^{2}} \\right)_{Y_{j \\leq c + 1, \\neq i},X_{c + 2}} = 0$\n", + "\n", + "Together with , all $c + 1$ independent potentials at the consolute\n", + "point can be determined. It is evident that the consolute point is a\n", + "zero-dimensional point in a two-dimensional space of independent\n", + "potentials in a one-component system. With the addition of a second\n", + "component to form a binary system, this consolute point in the\n", + "one-component system extends into a one-dimensional line. This line\n", + "represents the limit of stability of the binary system, and a consolute\n", + "point is located at the end of this line. It is thus evident that in a\n", + "system with $c$ independent components, the limit of stability is a\n", + "*c-1*-dimensional hypersurface in a space of $c + 1$ independent\n", + "potentials, while the consolute point is a zero-dimensional point in all\n", + "systems, which may be called the invariant critical point.\n" + ], + "metadata": {} + }, + { + "id": "2252bace", + "cell_type": "markdown", + "source": [ + "### Multi-component random solutions\n", + "\n", + "Similar to a ternary solution, the excess Gibbs energy of mixing of a\n", + "multi-component solution can be written as\n", + "\n", + "Eq. \u2011\n", + "$_{\\ }^{E}G_{m} = \\sum_{i}^{}{\\sum_{j}^{}{x_{i}x_{j}L_{ij}}} + \\sum_{i}^{}{\\sum_{j}^{}{\\sum_{k}^{}{x_{i}x_{j}x_{k}L_{ijk}}}}$\n", + "\n", + "In principle, one can add interaction parameters for quaternary and\n", + "higher order systems, but their contributions to Gibbs energy are\n", + "relatively minor because the major contributions have already been taken\n", + "into account by the binary and ternary interactions. It is anticipated\n", + "that not only the interaction parameters of four or more components are\n", + "small, but also the multiplication of mole fractions in front the\n", + "interaction parameters diminishes their contribution to the Gibbs energy\n", + "even further.\n", + "\n", + "Under the condition that all interaction parameters are constant, the\n", + "chemical potential of a component in a multi-component system with\n", + "binary and ternary interaction parameters can be extended from as\n", + "\n", + "Eq. \u2011\n", + "$\\mu_{i} =_{\\ }^{0}G_{i} + RTlnx_{i} + \\sum_{j \\neq i}^{}{x_{j}^{2}L_{ij}} + \\sum_{k > j}^{}{\\sum_{j \\neq i}^{}{x_{j}x_{k}}}\\left\\lbrack L_{ij} + L_{ik} - L_{jk} + \\left( 1 - 2x_{i} \\right)L_{ijk} \\right\\rbrack$\n", + "\n", + "The stability of the solution can also be extended from as\n", + "\n", + "Eq. \u2011\n", + "${N\\left( \\frac{\\partial\\mu_{i}}{\\partial N_{i}} \\right)}_{T,P,N_{j \\neq i}} = NG_{ii} = \\frac{RT\\left( 1 - x_{i} \\right)}{x_{i}} - 2\\sum_{j \\neq i}^{}{x_{j}^{2}L_{ij}} - 2\\sum_{k > j}^{}{\\sum_{j \\neq i}^{}{x_{j}x_{k}}}\\left\\lbrack L_{ij} + L_{ik} - L_{jk} + \\left( 2 - 3x_{i} \\right)L_{ijk} \\right\\rbrack$\n", + "\n", + "The limit of stability of a multi-component random solution can be\n", + "represented by or .\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/gibbs_energy_function/phases_with_fixed_compositions.ipynb b/src/psu410/src/gibbs_energy_function/phases_with_fixed_compositions.ipynb new file mode 100644 index 0000000..030a892 --- /dev/null +++ b/src/psu410/src/gibbs_energy_function/phases_with_fixed_compositions.ipynb @@ -0,0 +1,279 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "286da7ba", + "cell_type": "markdown", + "source": [ + "## Phases with fixed compositions\n", + "\n", + "The homogeneous system discussed so far means that there is only one\n", + "phase in the system, i.e. a single-phase system. A phase with a fixed\n", + "composition can be a pure element or a stoichiometric compound. There is\n", + "thus only one independent component in the system. A stoichiometric\n", + "compound contains more than one element, but the relative amounts of\n", + "each element are fixed by the stoichiometry and cannot vary\n", + "independently, i.e., $dN_{i} = x_{i}dN$. The combined law of\n", + "thermodynamics becomes\n", + "\n", + "Eq. \u2011\n", + "$dG = - SdT - Vd( - P) + \\left( \\sum_{}^{}{x_{i}\\mu_{i}} \\right)dN - Dd\\xi = - SdT - Vd( - P) + G_{m}dN - Dd\\xi$\n", + "\n", + "$G_{m}$ is the molar Gibbs energy of the stoichiometric compound and can\n", + "be regarded as the chemical potential of the stoichiometric\n", + "phase,$\\ \\alpha$,\n", + "\n", + "Eq. \u2011 $G_{m} = \\mu^{\\alpha} = \\sum_{}^{}{x_{i}\\mu_{i}}$\n", + "\n", + "The chemical potential of individual components in the phase cannot be\n", + "defined because the amount of each component cannot be varied\n", + "independently. For a stoichiometric phase of $N$ moles of atoms at\n", + "equilibrium with $dG$=$Nd\\mu^{\\alpha} + \\mu^{\\alpha}dN$, reduces to\n", + "\n", + "Eq. \u2011 $\\ 0 = - SdT - Vd( - P) - Nd\\mu^{\\alpha}$\n", + "\n", + "which is the Gibbs-Duhem equation, , applied to a stoichiometric phase.\n", + "It can be represented graphically by a surface in a three-dimensional\n", + "space composed of $\\ \\mu^{\\alpha}$, *T* and *\u2013P*. The direction of the\n", + "surface is represented by the three partial directives between any two\n", + "of $\\ \\mu^{\\alpha}$, *T* and *\u2013P* with the third one kept constant, i.e.\n", + "\n", + "Eq. \u2011\n", + "$\\ \\left( \\frac{\\partial\\mu^{\\alpha}}{\\partial T} \\right)_{P} = - \\frac{S}{N} = - S_{m}$\n", + "\n", + "Eq. \u2011\n", + "$\\ \\left( \\frac{\\partial\\mu^{\\alpha}}{\\partial( - P)} \\right)_{T} = - \\frac{V}{N} = - V_{m}$\n", + "\n", + "Eq. \u2011\n", + "$\\ \\left( \\frac{\\partial( - P)}{\\partial T} \\right)_{\\mu^{\\alpha}} = - \\frac{S}{V} = - \\frac{S_{m}}{V_{m}}$\n", + "\n", + "Based on Nernst\u2019s heat theorem, the entropy difference between two\n", + "crystals approaches zero when the temperature approaches absolute zero.\n", + "It is thus a common practice to put $S = 0$ for a crystal at 0 K. This\n", + "is usually referred as the third law of thermodynamics. From the\n", + "definition of entropy change by , $S$ or $S_{m}$ is always positive at\n", + "finite temperatures as the system or the crystal absorbs heat from the\n", + "surroundings to increase its temperature. $V$ or $V_{m}$ of a phase is a\n", + "well-defined physical quantity, and its absolute value can be given and\n", + "is always positive. The above three equations can be written in a\n", + "general form as\n", + "\n", + "Eq. \u2011\n", + "$\\ \\left( \\frac{\\partial Y_{i}}{\\partial Y_{j}} \\right)_{Y_{k}} = - \\frac{X_{j}}{X_{i}} < 0$\n", + "\n", + "The surface thus has negative slopes in all its directions. The\n", + "curvature of the surface can be derived from\n", + "\n", + "Eq. \u2011\n", + "$\\ \\left( \\frac{\\partial^{2}Y_{i}}{\\partial\\left( Y_{j} \\right)^{2}} \\right)_{Y_{k}} = - \\left( \\frac{\\partial\\left( \\frac{X_{j}}{X_{i}} \\right)}{\\partial Y_{j}} \\right)_{Y_{k}} = - \\frac{1}{X_{i}}\\left( \\frac{\\partial X_{j}}{\\partial Y_{j}} \\right)_{Y_{k}} + \\frac{X_{j}}{\\left( X_{i} \\right)^{2}}\\left( \\frac{\\partial X_{i}}{\\partial Y_{j}} \\right)_{Y_{k}} = - \\frac{1}{X_{i}}\\left\\lbrack \\left( \\frac{\\partial X_{j}}{\\partial Y_{j}} \\right)_{Y_{k}} - \\frac{X_{j}}{X_{i}}{\\left( \\frac{\\partial X_{i}}{\\partial Y_{i}} \\right)\\left( \\frac{\\partial Y_{i}}{\\partial Y_{j}} \\right)}_{Y_{k}} \\right\\rbrack = - \\frac{1}{X_{i}}\\left\\lbrack \\left( \\frac{\\partial X_{j}}{\\partial Y_{j}} \\right)_{Y_{k}} + \\left( \\frac{X_{j}}{X_{i}} \\right)^{2}\\left( \\frac{\\partial X_{i}}{\\partial Y_{i}} \\right)_{Y_{k}} \\right\\rbrack < 0$\n", + "\n", + "Both terms inside the last bracket are positive for a system in a state\n", + "of stable internal equilibrium, and the surface thus has a negative\n", + "curvature and is convex everywhere as shown in .\n", + "\n", + "Figure \u2011: Gibbs energy of a one-component phase as a function of\n", + "temperature and negative pressure, showing the convex shape\n", + "\n", + "From experimental observations, it is known that\n", + "$S_{m}^{vapor} \\gg S_{m}^{liquid} > S_{m}^{solid}$. The curves of\n", + "$G_{m}$ or $\\mu^{\\alpha}$ plotted with respect to $T$ at constant $P$\n", + "would thus have the most negative slope for a vapour phase followed by\n", + "its liquid and solid phases. As an example, shows Gibbs energy of Zn in\n", + "its solid, liquid, and vapour forms as a function of $T$ at constant\n", + "$P = 1$ atmospheric pressure.\n", + "\n", + "Figure \u2011: Molar Gibbs energy of Zn as a function of T at constant P\n", + "\n", + "Similarly it is common that\n", + "$V_{m}^{vapor} \\gg V_{m}^{liquid} > V_{m}^{solid}$, and the curves of\n", + "$G_{m}$ or $\\mu^{\\alpha}$ plotted with respect to $P$ at constant $T$\n", + "would thus have the most positive slope for a vapour phase followed by\n", + "its liquid and solid phases, though there are cases that\n", + "$V_{m}^{liquid} < V_{m}^{solid}$ such as those of water and ice. As an\n", + "example, shows Gibbs energy of Fe in its two solid (fcc and bcc),\n", + "liquid, and vapour forms as a function of $P$ at constant $T = 1273K$.\n", + "\n", + "Figure \u2011: Molar Gibbs energy of Zn as a function of P at constant T\n", + "\n", + "The quantities measurable by experiments typically include temperature,\n", + "pressure, volume, composition, and amount of heat flow in the combined\n", + "law of thermodynamics discussed so far. By measuring the heat needed to\n", + "increase the temperature of a phase, the heat capacity of the phase is\n", + "obtained as shown by Eq. 2\u20117. A typical heat capacity curve as a\n", + "function of temperature is shown in for fcc-Al, hcp-Mg, and an\n", + "intermetallic phase Al12Mg17.\n", + "\n", + "Figure \u2011: Heat capacity of fcc-Al, hcp-Mg, and\n", + "Al12Mg17 as a function of temperature\n", + "\n", + "There are various theoretical models for the heat capacity under\n", + "constant volume to be discussed in Chapter 5 of this book, which is\n", + "defined as\n", + "\n", + "Eq. \u2011\n", + "$C_{V} = \\left( \\frac{\\partial U}{\\partial T} \\right)_{V} = T\\left( \\frac{\\partial(F + TS)}{\\partial T} \\right)_{V} = T\\left( \\frac{\\partial S}{\\partial T} \\right)_{V} = - T\\left( \\frac{\\partial^{2}F}{\\partial T^{2}} \\right)_{V}$\n", + "\n", + "To establish the relationship between $C_{P}$ defined by and $C_{V}$,\n", + "$U$ needs to be represented as a function of $T$ and $V$ in terms of $G$\n", + "and its derivatives with respect to Gibbs energy\u2019s natural variables of\n", + "$T$ and $P$. It can be done as follows\n", + "\n", + "Eq. \u2011\n", + "$dV = \\frac{\\partial V}{\\partial T}dT + \\frac{\\partial V}{\\partial( - P)}d( - P) = - \\frac{\\partial^{2}G}{\\partial T( - P)}dT - \\frac{\\partial^{2}G}{\\partial( - P)^{2}}d( - P)$\n", + "\n", + "Eq. \u2011\n", + "$dU = \\frac{\\partial(G + TS - PV)}{\\partial T}dT + \\frac{\\partial(G + TS - PV)}{\\partial( - P)}d( - P) = - \\left( T\\frac{\\partial^{2}G}{\\partial T^{2}} - P\\frac{\\partial^{2}G}{\\partial T( - P)} \\right)dT\u2014\\left( T\\frac{\\partial^{2}G}{\\partial T( - P)} + P\\frac{\\partial^{2}G}{\\partial( - P)^{2}} \\right)\\left( - \\frac{1}{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}}dV + \\frac{\\frac{\\partial^{2}G}{\\partial T( - P)}}{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}}dT \\right) = - \\left\\lbrack T\\frac{\\partial^{2}G}{\\partial T^{2}} - T\\frac{\\left( \\frac{\\partial^{2}G}{\\partial T( - P)} \\right)^{2}}{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}} \\right\\rbrack dT + \\left( - T\\frac{\\frac{\\partial^{2}G}{\\partial T( - P)}}{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}} + P \\right)dV$\n", + "\n", + "Eq. \u2011\n", + "$C_{V} = C_{P} + T\\frac{\\left( \\frac{\\partial^{2}G}{\\partial T( - P)} \\right)^{2}}{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}} = C_{P} - \\frac{\\alpha_{V}^{2}VT}{\\kappa_{T}} = C_{P} - \\alpha_{V}^{2}BVT$\n", + "\n", + "where the thermal expansion, $\\alpha_{V}$, and the compressibility or\n", + "bulk modulus, $\\kappa_{T}$ or $B$, are defined by and , respectively.\n", + "From the heat capacity, the enthalpy and entropy can be obtained by\n", + "integration of at a constant pressure\n", + "\n", + "Eq. \u2011\n", + "$S = S_{0} + \\int_{0}^{T}\\frac{C_{P}}{T}dT = S_{0} + \\int_{0}^{298.15}\\frac{C_{P}}{T}dT + \\int_{298.15}^{T}\\frac{C_{P}}{T}dT = S_{298.15} + \\int_{298.15}^{T}\\frac{C_{P}}{T}dT$\n", + "\n", + "Eq. \u2011\n", + "$H = H_{0} + \\int_{0}^{T}C_{P}dT = H_{0} + \\int_{0}^{298.15}C_{P}dT + \\int_{298.15}^{T}C_{P}dT = H_{298.15} + \\int_{298.15}^{T}C_{P}dT$\n", + "\n", + "In the above equations, two temperature ranges of integration are chosen\n", + "for practical applications as most processing procedures in the field of\n", + "materials science and engineering take place at temperatures above the\n", + "room temperature. Based on the third-law of thermodynamics, $S_{0} = 0$,\n", + "$S_{298.15}$ can be obtained by integration. On the other hand for\n", + "$H_{0} = U_{0} + PV$, one does not know the absolute value of the\n", + "internal energy and thus have to select a reference state for $H$. In\n", + "principle, the reference state can be arbitrarily chosen. A widely used\n", + "reference state in the thermodynamic modeling practice is to set\n", + "$H_{298.15}^{SER} = 0$ at ambient pressure for pure elements at their\n", + "respective stable structures at room temperature, called stable element\n", + "reference (SER) state with\n", + "\n", + "Eq. \u2011\n", + "$G_{298.15}^{SER} = H_{298.15}^{SER} - TS_{298.15}^{SER} = - TS_{298.15}^{SER}$\n", + "\n", + "It is further noted that after defining $S_{298.15}$ and $H_{298.15}$,\n", + "one only needs the heat capacity at higher temperatures. This makes the\n", + "mathematical representation of heat capacity simpler due to a relatively\n", + "simple temperature dependence of heat capacity at higher temperatures in\n", + "comparison with the variation at lower temperatures. One common\n", + "expression for heat capacity at high temperatures and ambient pressure\n", + "is as follows\n", + "\n", + "Eq. \u2011 $C_{P} = c + dT + \\frac{e}{T^{2}} + fT^{2}$\n", + "\n", + "where c, d, e, and f are parameters fitted to experimental or theoretic\n", + "data and compiled in various handbooks.\n", + "\n", + "The corresponding $S$, $H$, and $G$ are obtained as\n", + "\n", + "Eq. \u2011 $S = b^{'} + clnT + dT - \\frac{e}{{2T}^{2}} + \\frac{f}{2}T^{2}$\n", + "\n", + "Eq. \u2011 $H = a + cT + \\frac{d}{2}T^{2} - \\frac{e}{T} + \\frac{f}{3}T^{3}$\n", + "\n", + "Eq. \u2011\n", + "$G = H - TS = a - bT - cTlnT - \\frac{d}{2}T^{2} - \\frac{e}{2T} - \\frac{f}{6}T^{3}$\n", + "\n", + "with $b = b^{'} - c$. The integration constants $b^{'}$ and $a$ are\n", + "evaluated from $S_{298.15}$ and $H_{298.15}$. As an example, the\n", + "enthalpy and entropy of Zn in solid (hcp), liquid, and gas forms are\n", + "plotted in and , respectively. The distances between any two curves in\n", + "and represent the enthalpy or entropy differences between the two\n", + "phases. It can be seen that the gas has much higher enthalpy and entropy\n", + "than the solid and liquid.\n", + "\n", + "Figure \u2011: Enthalpy of Zn as a function of temperature at one atmospheric\n", + "pressure\n", + "\n", + "Figure \u2011: Entropy of Zn as a function of temperature at one atmospheric\n", + "pressure\n", + "\n", + "Similarly, one can add the pressure dependence into the Gibbs energy\n", + "function such as\n", + "\n", + "Eq. \u2011\n", + "$G = a - bT - cTlnT - \\frac{d}{2}T^{2} - \\frac{e}{2T} - \\frac{f}{6}T^{3} + gP + hTP + mP^{2}$\n", + "\n", + "where g, h, and m are parameters fitted to experimental or theoretic\n", + "data and compiled in various handbooks.\n", + "\n", + "The expression for $V$ can be derived as\n", + "\n", + "Eq. \u2011 $V = g + hT + 2mP$\n", + "\n", + "The Helmholtz energy can be expressed as a function of its natural\n", + "variables by solving $P\\ $ from\n", + "\n", + "Eq. \u2011\n", + "$F = G - PV = a - bT - cTlnT - \\frac{d}{2}T^{2} - \\frac{e}{2T} - \\frac{f}{6}T^{3} - \\frac{(g + hT - V)^{2}}{4m}$\n", + "\n", + "In the literature there are many models to represent the temperature and\n", + "pressure dependences of thermodynamic properties. The Gibbs energy\n", + "difference between a stoichiometric compound and the components at their\n", + "reference states of which the compound is composed,\n", + "${_{\\ }^{0}G}_{i}^{ref}$, is termed as Gibbs energy of formation, i.e.\n", + "\n", + "Eq. \u2011 $\\mathrm{\\Delta}_{f}G = G - \\sum_{}^{}N_{i}{_{\\ }^{0}G}_{i}^{ref}$\n", + "\n", + "with $N_{i}$ being the stoichiometry of the compound. Similarly,\n", + "enthalpy of formation, entropy of formation, and heat capacity of\n", + "formation with respect to components at their reference states,\n", + "$_{\\ }^{0}H_{i}^{ref}$, $_{\\ }^{0}S_{i}^{ref}$, and\n", + "$_{\\ }^{0}{C_{P}}_{i}^{ref}$, can be defined as\n", + "\n", + "Eq. \u2011 $\\mathrm{\\Delta}_{f}H = H - \\sum_{}^{}{N_{i}_{\\ }^{0}H_{i}^{ref}}$\n", + "\n", + "Eq. \u2011 $\\mathrm{\\Delta}_{f}S = S - \\sum_{}^{}{N_{i}_{\\ }^{0}S_{i}^{ref}}$\n", + "\n", + "Eq. \u2011\n", + "$\\mathrm{\\Delta}_{f}C_{P} = C_{P} - \\sum_{}^{}{N_{i}_{\\ }^{0}{C_{P}}_{i}^{ref}}$\n", + "\n", + "It should be mentioned that one mole of a compound usually refers to one\n", + "mole of formula of stoichiometry of the compound. With a formula like\n", + "$A_{a}B_{b}C_{c}$, the compound is composed of total $(a + b + c)$ moles\n", + "of components. One should thus be very careful when dealing with\n", + "numerical values to be sure whether the data is in terms of per mole of\n", + "formula or per mole of components. At the same time the reference states\n", + "must be clearly defined. When the SER state defined in is selected as\n", + "the reference state, the above formation quantities are called standard\n", + "formation quantities such as standard enthalpy of formation.\n", + "\n", + "Since there are only two independent potentials in a one-component\n", + "system, its limit of stability can be evaluated with one potential kept\n", + "constant, i.e. either $T$ or $P$. Consequently, either Helmholtz energy\n", + "or enthalpy is to be used in deriving the limit of stability of a\n", + "homogeneous system. For the practical usefulness, let us use Helmholtz\n", + "energy because its natural variables of $T$ and *V* are measurable\n", + "quantities in typical experiments, while one of the natural variables of\n", + "enthalpy, entropy, is not. From and , the limit of stability for a\n", + "one-component system at constant temperature can be written as\n", + "\n", + "Eq. \u2011\n", + "$\\left( \\frac{\\partial( - P)}{\\partial V} \\right)_{T,N} = F_{VV} = \\frac{1}{V\\kappa_{T}} = \\frac{B}{V} = 0$\n", + "\n", + "where the isothermal compressibility and bulk modulus, $\\kappa_{T}$ and\n", + "$B$, are defined in . The limit of stability is thus determined when the\n", + "isothermal compressibility diverges or the bulk modulus becomes zero\n", + "because $V$ has finite values at any temperatures. It is evident that\n", + "Helmholtz energy must have higher order dependence on volume than in for\n", + "a system with instability because $F_{VV}$ from is constant.\n", + "\n", + "From , the consolute point is defined by\n", + "\n", + "Eq. \u2011\n", + "$F_{VVV} = \\left( \\frac{\\partial^{2}( - P)}{\\partial V^{2}} \\right)_{T,N} = \\frac{\\partial\\left( \\frac{1}{V\\kappa_{T}} \\right)}{\\partial V} = - \\frac{1 + \\frac{V}{\\kappa_{T}}\\frac{\\partial\\kappa_{T}}{\\partial V}}{\\kappa_{T}V^{2}} = 0$\n", + "\n", + "Since $\\kappa_{T}$ becomes infinite at the limit of stability,\n", + "$\\frac{\\partial\\kappa_{T}}{\\partial V}$ approaches negative infinite\n", + "when the critical/consolute point is approached so that\n", + "$\\frac{V}{\\kappa_{T}}\\frac{\\partial\\kappa_{T}}{\\partial V} = - 1$ and\n", + "$F_{VVV} = 0$.\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/gibbs_energy_function/phases_with_variable_compositions_random_solutions.ipynb b/src/psu410/src/gibbs_energy_function/phases_with_variable_compositions_random_solutions.ipynb new file mode 100644 index 0000000..36dd8fe --- /dev/null +++ b/src/psu410/src/gibbs_energy_function/phases_with_variable_compositions_random_solutions.ipynb @@ -0,0 +1,123 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "2236d2f8", + "cell_type": "markdown", + "source": [ + "## Phases with variable compositions: Random solutions\n", + "\n", + "The combined law of thermodynamics and the Gibbs-Duhem equation of a\n", + "solution phase with variable compositions are shown by and ,\n", + "respectively. A phase can be represented by a *c+1*-dimensional surface\n", + "in a *c+2*-dimensional space of potentials based on the Gibbs-Duhem\n", + "equation. The directions and curvature of the surface are represented by\n", + "the partial derivatives shown by and secondary derivatives shown by ,\n", + "both being negative for a stable phase. To develop a mathematical\n", + "formula for Gibbs energy of a phase with variable compositions, one can\n", + "consider a phase as a mixture of independent components that the phase\n", + "is made of. Its Gibbs energy function can be postulated as the summation\n", + "of Gibbs energy of the independent components of the same structure of\n", + "the solution, $_{\\ }^{0}G_{i}$, plus the contribution due to the mixing,\n", + "$_{\\ }^{mixing}G$ or $_{\\ }^{M}G$\n", + "\n", + "Eq. \u2011 $G = \\sum_{}^{}{N_{i}_{\\ }^{0}G_{i}} +_{\\ }^{M}G$\n", + "\n", + "Since the system size usually is not important in thermodynamics, its\n", + "properties are typically normalized to one mole with its composition\n", + "represented by mole fractions of components. The molar Gibbs energy is\n", + "obtained as shown below with the molar Gibbs energy of mixing separated\n", + "into two parts: ideal Gibbs energy of mixing assuming no chemical\n", + "interaction among components, $_{\\ }^{ideal}G_{m}$ or $_{\\ }^{I}G_{m}$,\n", + "and excess Gibbs energy of mixing due to chemical reaction among\n", + "components, $_{\\ }^{excess}G_{m}$ or $_{\\ }^{E}G_{m}$\n", + "\n", + "Eq. \u2011\n", + "$G_{m} = \\sum_{}^{}{x_{i}_{\\ }^{0}G_{i}} +_{\\ }^{M}G_{m} = \\sum_{}^{}{x_{i}_{\\ }^{0}G_{i}} +_{\\ }^{I}G_{m} +_{\\ }^{E}G_{m}$\n", + "\n", + "From , the chemical potential of a component is thus\n", + "\n", + "Eq. \u2011\n", + "$\\mu_{i} =_{\\ }^{0}G_{i} +_{\\ }^{I}{G_{m} +}_{\\ }^{E}G_{m} + \\frac{\\partial\\left(_{\\ }^{I}{G_{m} +}_{\\ }^{E}G_{m} \\right)}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial\\left(_{\\ }^{I}{G_{m} +}_{\\ }^{E}G_{m} \\right)}{\\partial x_{j}}$\n", + "\n", + "One may define the chemical activity of component *i*, $a_{i}^{\\ }$, as\n", + "follows\n", + "\n", + "Eq. \u2011\n", + "$RTlna_{i}^{\\ } = \\mu_{i} -_{\\ }^{0}G_{i}^{\\ } =_{\\ }^{I}{G_{m} +}\\frac{\\partial_{\\ }^{I}G_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial_{\\ }^{I}G_{m}}{\\partial x_{j}} +_{\\ }^{E}G_{m} + \\frac{\\partial_{\\ }^{E}G_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial_{\\ }^{E}G_{m}}{\\partial x_{j}}$\n", + "\n", + "In this definition, the chemical activity or simply activity is\n", + "calculated with respect to the pure elements in the structure of the\n", + "solution for practical reasons as one would like to understand the\n", + "chemical potential difference of components in the solution and by\n", + "itself with the same structure. It should be noted that this reference\n", + "state for chemical activity is usually different from the SER reference\n", + "state defined in as the solution may have a different structure than\n", + "those of pure components in their SER states. On the other hand, the\n", + "activity under the SER reference state can be easily obtained by\n", + "replacing $_{\\ }^{0}G_{i}$ with ${_{\\ }^{0}G}_{i}^{SER}$ from . In\n", + "principle, one may choose any structure as the reference state for\n", + "activity to be useful for practical applications, i.e.\n", + "\n", + "Eq. \u2011 $RTlna_{i}^{ref} = \\mu_{i} -_{\\ }^{0}G_{i}^{ref}$\n", + "\n", + "For example, the activity of a component in a liquid solution is defined\n", + "with respect to the pure component in its liquid form from , but can\n", + "also be referred to its SER state which is solid using . The following\n", + "sections will discuss in more details how components mix when they are\n", + "brought together including concepts such as random mixing, short-range\n", + "ordering, and long-range ordering.\n", + "\n", + "The limit of stability of a solution with respect to composition\n", + "fluctuation under constant *T*, *P*, and *N1* can be derived\n", + "as follows from and\n", + "\n", + "Eq. \u2011\n", + "$\\left( \\frac{\\partial\\mu_{i}}{\\partial N_{i}} \\right)_{T,P,N_{j \\neq i},i > 1} > \\left( \\frac{\\partial\\mu_{i}}{\\partial N_{i}} \\right)_{T,P,N_{1},{\\mu_{2},N}_{j \\neq i},i,j > 2}\\ldots... > \\left( \\frac{\\partial\\mu_{c}}{\\partial N_{c}} \\right)_{T,P,N_{1},\\mu_{2}..\\mu_{c - 1}} = 0$\n", + "\n", + "The first term can be derived from as follows\n", + "\n", + "Eq. \u2011\n", + "$\\left( \\frac{\\partial\\mu_{i}}{\\partial N_{i}} \\right)_{T,P,N_{j \\neq i},i > 1} = \\sum_{j = 1}^{c}\\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{i}\\partial x_{j}}\\frac{\\partial x_{j}}{\\partial N_{i}} - \\sum_{j = 1}^{c}{x_{j}\\sum_{k = 1}^{c}{\\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{j}\\partial x_{k}}\\frac{\\partial x_{k}}{\\partial N_{i}}}} = \\frac{1}{N}\\left( \\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{i}^{2}} - \\sum_{j = 1}^{c}{x_{j}\\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{j}^{2}}} - \\sum_{j = 1}^{c}{x_{j}\\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{i}\\partial x_{j}}} + \\sum_{j = 1}^{c}{\\sum_{k = 1}^{c}{{x_{j}x}_{k}\\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{j}\\partial x_{k}}}} \\right)$\n", + "\n", + "Denoting\n", + "$G_{ij} = \\left( \\frac{\\partial\\mu_{i}}{\\partial N_{j}} \\right)_{T,P,N_{k \\neq j}}$and\n", + "using to change the variables kept constant from molar quantities to\n", + "potentials one-by-one (see \\[1\\]), the limit of stability can be further\n", + "obtained as\n", + "\n", + "Eq. \u2011\n", + "$\\left( \\frac{\\partial\\mu_{c}}{\\partial N_{c}} \\right)_{T,P,N_{1},\\mu_{2}..\\mu_{c - 1}} = \\frac{\\det\\left( G_{ij}:2 \\leq i,j \\leq c \\right)}{\\det\\left( G_{ij}:2 \\leq i,j \\leq c - 1 \\right)} = 0$\n", + "\n", + "where $\\det$ stands for determinant. indicates that\n", + "$\\det\\left( G_{ij}:2 \\leq i,j \\leq c \\right) = 0$ at the limit of\n", + "stability. Considering $x_{1} = 1 - \\sum_{j \\neq 1}^{}x_{j}$, let us\n", + "introduce\n", + "\n", + "Eq. \u2011\n", + "$g_{i} = \\mu_{i} - \\mu_{1} = \\left( \\frac{\\partial G_{m}}{\\partial x_{i}} \\right)_{x_{k \\neq i}} - \\left( \\frac{\\partial G_{m}}{\\partial x_{1}} \\right)_{x_{k \\neq 1}}$\n", + "\n", + "and\n", + "\n", + "Eq. \u2011\n", + "$g_{ij} = \\frac{\\partial g_{i}}{\\partial x_{j}} = \\frac{\\partial\\left( \\mu_{i} - \\mu_{1} \\right)}{\\partial x_{j}} = \\frac{\\partial^{2}G_{m}}{\\partial x_{i}\\partial x_{j}} - \\frac{\\partial^{2}G_{m}}{\\partial x_{1}\\partial x_{j}} - \\frac{\\partial^{2}G_{m}}{\\partial x_{i}\\partial x_{1}} + \\frac{\\partial^{2}G_{m}}{\\partial\\left( x_{1} \\right)^{2}}$\n", + "\n", + "The limit of stability can be re-written as\n", + "\n", + "Eq. \u2011\n", + "$\\left( \\frac{\\partial\\left( \\mu_{c} - \\mu_{1} \\right)}{\\partial x_{c}} \\right)_{T,P,N,\\mu_{2} - \\mu_{1},\\ ...\\mu_{c - 1} - \\mu_{1}} = \\frac{\\det\\left( g_{ij}:2 \\leq i,j \\leq c \\right)}{\\det\\left( g_{ij}:2 \\leq i,j \\leq c - 1 \\right)} = 0$\n", + "\n", + "i.e. $\\det\\left( g_{ij}:2 \\leq i,j \\leq c \\right) = 0$. The consolute\n", + "point can be defined following\n", + "\n", + "Eq. \u2011\n", + "$\\left( \\frac{\\partial^{2}\\mu_{c}}{\\partial\\left( N_{c} \\right)^{2}} \\right)_{T,P,N_{1},\\mu_{2}..\\mu_{c - 1}} = \\left( \\frac{\\partial^{2}\\left( \\mu_{c} - \\mu_{1} \\right)}{\\partial\\left( x_{c} \\right)^{2}} \\right)_{T,P,N,\\mu_{2} - \\mu_{1},\\ ...\\mu_{c - 1} - \\mu_{1}} = 0$\n", + "\n", + "No closed mathematic form has been published in the literature.\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/gibbs_energy_function/phases_with_variable_compositions_solutions_with_ordering.ipynb b/src/psu410/src/gibbs_energy_function/phases_with_variable_compositions_solutions_with_ordering.ipynb new file mode 100644 index 0000000..a0c94f1 --- /dev/null +++ b/src/psu410/src/gibbs_energy_function/phases_with_variable_compositions_solutions_with_ordering.ipynb @@ -0,0 +1,15 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "04ebd1df", + "cell_type": "markdown", + "source": [ + "## Phases with variable compositions: Solutions with ordering\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/gibbs_energy_function/polymer_solutions_and_polymer_blends.ipynb b/src/psu410/src/gibbs_energy_function/polymer_solutions_and_polymer_blends.ipynb new file mode 100644 index 0000000..327d655 --- /dev/null +++ b/src/psu410/src/gibbs_energy_function/polymer_solutions_and_polymer_blends.ipynb @@ -0,0 +1,713 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "184c02c4", + "cell_type": "markdown", + "source": [ + "## Polymer solutions and polymer blends\n", + "\n", + "A polymer solution is a mixture between polymer molecules and solvents,\n", + "while a polymer blend is a mixture between different polymer molecules.\n", + "A polymer molecule consists of the same repeating units of one or more\n", + "monomers, which can be an atom or a small molecule. The number of\n", + "repeating units is called the degree of polymerization and can be as\n", + "large as 104\u2013105. It defines the molecular mass,\n", + "i.e. the mass of one polymer molecule. There are three typical\n", + "architectures of polymerization: a linear chain, a branched chain, and a\n", + "cross-linked polymer. Nearly all polymers are mixtures of molecules with\n", + "a different degree of polymerization with a molecular mass distribution,\n", + "complicating the modeling of their thermodynamic properties because of\n", + "the dependence of properties on molecular mass.\n", + "\n", + "Gibbs energy functions of polymers with a single molecular mass can be\n", + "treated similarly as in Chapter . For a polymer solution, the ideal\n", + "entropy of mixing is quite different from that of atomically random\n", + "solutions discussed in Chapter because the monomers in a polymer\n", + "molecule are connected to each other and cannot move freely. One common\n", + "approach to calculate the ideal entropy of mixing is to evoke a lattice\n", + "model and assume that one monomer occupies a lattice site with a fixed\n", + "volume. The number of translational states of a single molecule is equal\n", + "to the number of lattice sites available. In a homogeneous solution, the\n", + "total number of lattice sites available is\n", + "\n", + "Eq. \u2011 $n = \\sum_{i}^{}{m_{i}n_{i}}$\n", + "\n", + "where $n_{i}$ and $m_{i}$ are the number of molecule $i$ and the number\n", + "of lattice sites per molecule $i$, respectively. While in its pure\n", + "state, i.e. before mixing, the number of states of molecule $i$ in terms\n", + "of the number of lattice sites is\n", + "\n", + "Eq. \u2011 $w_{i} = m_{i}n_{i} = n\\varphi_{i}$\n", + "\n", + "where $\\varphi_{i}$ is the volume fraction of molecule $i$ in the\n", + "solution. The entropy change per molecule $i$ is thus\n", + "\n", + "Eq. \u2011\n", + "$S_{i} = kln(n) - klnw_{i} = kln\\frac{1}{\\varphi_{i}} = - kln\\varphi_{i}$\n", + "\n", + "The total entropy of mixing is the summation for all molecules,\n", + "normalized to per mole of lattice site\n", + "\n", + "Eq. \u2011\n", + "$_{}^{I}S_{m} = \\frac{N_{a}}{n}\\sum_{i}^{}{n_{i}S_{i}} = - R\\sum_{i}^{}{\\frac{\\varphi_{i}}{m_{i}}\\ln\\varphi_{i}}$\n", + "\n", + "where $N_{a}$ is the Avogadro number. When $m_{i} = 1$ for all\n", + "molecules, Eq. 2\u2011143 reduces to . Since $m_{i}$ values are typically\n", + "very large numbers for polymers, the entropy of mixing in polymer\n", + "solutions and blends is thus significantly lower than those in\n", + "non-polymer solutions as shown schematically in for binary systems with\n", + "various $m_{i}$ values.\n", + "\n", + "Figure \u2011: Schematic entropy of mixing in solutions with the numbers of\n", + "lattice sites per molecule shown.\n", + "\n", + "Similar to , Gibbs energy of a multi-component random polymer\n", + "solution/blend can be written as\n", + "\n", + "Eq. \u2011\n", + "$G_{m} = \\sum_{}^{}\\frac{\\varphi_{i}}{m_{i}}_{\\ }^{0}G_{im} + RT\\left( \\sum_{}^{}{\\frac{\\varphi_{i}}{m_{i}}\\ln\\varphi_{i}} + \\sum_{}^{}{\\varphi_{i}\\varphi_{j}\\chi_{ij}} \\right)$\n", + "\n", + "where $_{\\ }^{0}G_{im}$ is the Gibbs energy of molecular $i$ per mole of\n", + "lattice site, and $\\chi_{ij}$ the unitless interaction parameter between\n", + "molecule $i$ and $j$. Other equations shown in Chapter can be derived\n", + "similarly too. It is to be noted that is very similar to the\n", + "Flory\u2013Huggins solution equation widely used in the polymer community.\n" + ], + "metadata": {} + }, + { + "id": "8736a1bd", + "cell_type": "markdown", + "source": [ + "### Random solutions\n", + "\n", + "The ideal Gibbs energy of mixing represents an ideal solution in which\n", + "all sites are equivalent and the distributions of components on the\n", + "sites are completely random. The number of different configurations to\n", + "arrange all components is\n", + "\n", + "Eq. \u2011 $w = \\frac{N!}{\\prod_{}^{}\\left( N_{i}! \\right)}$\n", + "\n", + "Based on Boltzmann\u2019s relation from statistic thermodynamics when all\n", + "configurations have the same probability to be observed, the ideal\n", + "configurational molar entropy of mixing due to the distribution is\n", + "\n", + "Eq. \u2011\n", + "$_{\\ }^{ideal}S_{m} =_{\\ }^{I}S_{m} = \\frac{Rlnw}{N} = R\\frac{lnN! - \\sum_{}^{}\\ln\\left( N_{i}! \\right)}{N} \\cong R\\frac{NlnN - \\sum_{}^{}{N_{i}l{nN}_{i}}}{N} = - R\\sum_{}^{}{x_{i}l{nx}_{i}}$\n", + "\n", + "where $R$ is the gas constant. represents the entropy difference between\n", + "the ideal solution and the states when individual components are by\n", + "themselves, i.e. the mechanical mixing of components. As $x_{i}$ is\n", + "smaller than unity, the entropy production to form an ideal solution\n", + "from pure components is thus positive, indicating that it is a\n", + "spontaneous process. In such an ideal solution, it is assumed that there\n", + "are no interactions between components, and the enthalpy of mixing is\n", + "thus zero as the internal energy and the volume of the system do not\n", + "change. The ideal Gibbs energy of mixing is written as\n", + "\n", + "Eq. \u2011 $_{\\ }^{I}G = - T_{\\ }^{I}S_{m} = RT\\sum_{}^{}{x_{i}l{nx}_{i}}$\n", + "\n", + "The Gibbs energy of real solutions, i.e. , becomes\n", + "\n", + "Eq. \u2011\n", + "$G_{m} = \\sum_{}^{}{x_{i}_{\\ }^{0}G_{i}} + RT\\sum_{}^{}{x_{i}l{nx}_{i}} +_{\\ }^{E}G_{m}$\n", + "\n", + "From , the chemical potential is obtained as\n", + "\n", + "Eq. \u2011\n", + "$\\mu_{i} = G_{i} =_{\\ }^{0}G_{i} + RTlnx_{i} +_{\\ }^{E}G_{m} + \\frac{\\partial_{\\ }^{E}G_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial_{\\ }^{E}G_{m}}{\\partial x_{j}}$\n", + "\n", + "From the chemical activity of , the activity coefficient, $\\gamma_{i}$,\n", + "can be defined as follows\n", + "\n", + "Eq. \u2011\n", + "$\\gamma_{i} = \\frac{a_{i}}{x_{i}} = \\frac{1}{x_{i}}\\exp\\frac{G_{i} -_{\\ }^{0}G_{i}\\ }{RT}$\n", + "\n", + "The solution is an ideal solution with $\\gamma_{i} = 1$, and is said to\n", + "be positively or negatively deviating from an ideal solution with\n", + "$\\gamma_{i} > 1$ or $\\gamma_{i} < 1$, respectively. The chemical\n", + "potential is related to the activity and activity coefficient by the\n", + "following equation\n", + "\n", + "Eq. \u2011\n", + "$\\mu_{i} =_{\\ }^{0}G_{i} + RTlna_{i} =_{\\ }^{0}G_{i} + RTln\\gamma_{i}x_{i} =_{\\ }^{0}G_{i} + RTlnx_{i} + RTln\\gamma_{i}$\n", + "\n", + "Let us further exam in more details in order to better understand the\n", + "relation between $G_{m}$ and $\\mu_{i}$. The partial derivatives in\n", + "represent the directions of molar Gibbs energy in the composition space,\n", + "i.e. the tangents of molar Gibbs energy with respect to mole fractions\n", + "of independent components. Collectively, they define the\n", + "multi-dimensional tangent plane of molar Gibbs energy at the given\n", + "composition, $x_{i}^{0}$. The mathematical representation of this\n", + "tangent plane, $z_{G_{m}}$, is defined by its directional derivatives\n", + "and the distance from the point where the derivatives are taken,\n", + "\n", + "Eq. \u2011\n", + "$z_{G_{m}} = G_{m}\\left( x_{i}^{0} \\right) + \\sum_{i = 1}^{c}{\\left( \\frac{\\partial G_{m}}{\\partial x_{i}} \\right)_{x_{i}^{0}}\\left( x_{i} - x_{i}^{0} \\right)}$\n", + "\n", + "The intercept of this tangent plane at each pure component axis, i.e.\n", + "$x_{i} = 1$ and $x_{j \\neq i} = 0$, is obtained as\n", + "\n", + "Eq. \u2011\n", + "$z_{G_{m},x_{i} = 1} = G_{m}\\left( x_{i}^{0} \\right) + \\left( \\frac{\\partial G_{m}}{\\partial x_{i}} \\right)_{x_{i}^{\\ } = x_{i}^{0}} - \\sum_{j = 1}^{c}{x_{j}^{0}\\left( \\frac{\\partial G_{m}}{\\partial x_{j}} \\right)_{x_{i}^{\\ } = x_{i}^{0}}}$\n", + "\n", + "This is identical to at the point $x_{i}^{0}$. It is thus shown that the\n", + "chemical potential of a component in a solution is represented by the\n", + "intercept of the tangent plane of Gibbs energy of the solution on the\n", + "$G_{m}$ axis of the component. The distance between the intercept and\n", + "the Gibbs energy of the pure component in the same structure of the\n", + "solution is related to the chemical activity of the component as defined\n", + "by . On the other hand, it is evident that one can choose any other\n", + "structure of the pure element to define the chemical activity to compare\n", + "chemical potential of the component as shown by .\n", + "\n", + "The stability of a solution is evaluated following , and the derivatives\n", + "of chemical potential with respect to its moles, i.e. the elements in\n", + "the determinant, are obtained as follows from and ,\n", + "\n", + "Eq. \u2011\n", + "$\\frac{N}{RT}\\frac{\\partial\\mu_{i}}{\\partial N_{i}} = \\frac{N}{RT}G_{ii} = \\frac{1 - x_{i}}{x_{i}} + \\frac{1}{\\gamma_{i}}\\left( \\frac{\\partial\\gamma_{i}}{\\partial x_{i}} - \\sum_{j = 1}^{c}{x_{j}\\frac{\\partial\\gamma_{i}}{\\partial x_{j}}} \\right)$\n", + "\n", + "Eq. \u2011\n", + "$\\frac{N}{RT}\\frac{\\partial\\mu_{i}}{\\partial N_{k}} = \\frac{N}{RT}G_{ik} = - \\frac{x_{k}}{x_{i}} + \\frac{1}{\\gamma_{i}}\\left( 1 - \\sum_{j = 1}^{c}{x_{j}\\frac{\\partial\\gamma_{i}}{\\partial x_{j}}} \\right)$\n", + "\n", + "To further study Gibbs energy of solution phases, let us discuss the\n", + "details on the excess Gibbs energy of mixing. At this point, one can\n", + "start with lower-order systems with fewer components, i.e. two component\n", + "and three-component systems, noting that the Gibbs energy of phases with\n", + "one component is already presented in Chapter .\n" + ], + "metadata": {} + }, + { + "id": "084be49b", + "cell_type": "markdown", + "source": [ + "### Solutions with both short-range and long-range ordering\n", + "\n", + "The short-range ordering in a solution with long-rang ordering can take\n", + "place in each sublattice or between two sublattices. The short-range\n", + "ordering in one sublattice can be treated similarly as in Chapter 2.3.1\n", + "with mole fractions substituted by site fractions. In case that\n", + "associates form, the relation between mole fractions and site fractions\n", + "becomes more complicated as follows\n", + "\n", + "Eq. \u2011\n", + "$x_{i} = \\frac{\\sum_{}^{}{a^{t}\\sum_{k}^{}{i_{k}y}_{k}^{t}}}{\\sum_{}^{}{a^{t}\\left( 1 - y_{va}^{t} \\right)}}$\n", + "\n", + "where the summation for $k$ goes over all associates in sublattice $t$\n", + "containing component $i$.\n", + "\n", + "The short-range ordering between two-sublattices indicates that a\n", + "component in one sublattice has different interactions with different\n", + "components in another sublattice. This results in local ordering of one\n", + "component around another component in two neighboring sublattices. Such\n", + "local ordering involves interactions between two sublattices shown as\n", + "the third term in .\n" + ], + "metadata": {} + }, + { + "id": "8c40d224", + "cell_type": "markdown", + "source": [ + "### Solutions with charged species\n", + "\n", + "One special type of solutions with both short-range and long-range\n", + "ordering is solutions with charged species, i.e. ionic solutions, plus\n", + "electrons and holes. There is an additional constraint on species\n", + "concentrations to maintain the charge neutrality of such solutions, i.e.\n", + "\n", + "Eq. \u2011 $0 = \\sum_{t}^{}{\\sum_{i}^{}{a^{t}y_{i}^{t}v_{i}^{t}}}$\n", + "\n", + "where $v_{i}^{t}$ is the valance of species $i$ in sublattice $t$\n", + "including its sign which is positive for cations, negative for anions,\n", + "and zero for neutral species. The conventional defect chemistry theory\n", + "is typically based on the ideal mass action laws and applicable to a\n", + "single set of defects and at very low defect concentrations, i.e. in the\n", + "limit of ideal solutions. For interacting defects, their concentrations\n", + "should be replaced by their activities, which can be obtained from\n", + "thermodynamic principles similarly as discussed in previous sections. It\n", + "should be emphasized that in addition to formation of many more charged\n", + "species, one component may have different valences, particularly the\n", + "transition metals. Consequently, there can be many more species in an\n", + "ionic phase than the number of independent components in the system, and\n", + "their concentrations can be found by equilibrium calculations as\n", + "discussed in Chapter .\n" + ], + "metadata": {} + }, + { + "id": "be4c0142", + "cell_type": "markdown", + "source": [ + "### Solutions with long-range ordering\n", + "\n", + "So far, solutions in which a component can occupy any site in a phase\n", + "are discussed. In many phases, this is not the case. For example, in the\n", + "fcc solid solution of Fe and C, Fe atoms take the fcc lattice sites, and\n", + "C atoms occupy the interstitial sites between the fcc lattice sites.\n", + "Therefore, Fe atoms do not mix with C atoms on the fcc lattice sites,\n", + "and they rather develop long-rang ordering by occupying their own\n", + "distinct sites in the phase. Long-range ordering can also develop when\n", + "short-range ordering extends to the whole lattice. New formula for the\n", + "Gibbs energy of mixing is needed by considering the details how\n", + "components are distributed and mixed in various sites in a phase.\n", + "\n", + "One way to group various sites in a phase is based on equivalent\n", + "crystallographic positions in a phase, i.e. Wyckoff positions. Various\n", + "sets of equivalent positions divide the lattice into sub-sets of\n", + "lattices. Each set of equivalent positions forms a sublattice. The\n", + "distributions of components on each sublattice can be represented by\n", + "mole fractions of components in the sublattice, commonly referred to as\n", + "site fractions and defined as\n", + "\n", + "Eq. \u2011 $y_{i}^{t} = \\frac{N_{i}^{t}}{\\sum_{j}^{}N_{j}^{t}}$\n", + "\n", + "Eq. \u2011 $\\sum_{i}^{}y_{i}^{t} = 1$\n", + "\n", + "where the superscript $t$ denotes the sublattice in which the component\n", + "resides, and the summation is for all species in sublattice $t$\n", + "including vacancy. Site fractions and mole fractions are related through\n", + "the mass balance as follows\n", + "\n", + "Eq. \u2011\n", + "$x_{i} = \\frac{\\sum_{}^{}{a^{t}y_{i}^{t}}}{\\sum_{}^{}{a^{t}\\left( 1 - y_{va}^{t} \\right)}}$\n", + "\n", + "where $a^{t}$ and $y_{va}^{t}$ are the number of sites and the site\n", + "fraction of vacancy in the sublattice *t*.\n", + "\n", + "Random solutions form when each component enters all sublattices\n", + "equally. Mole fractions and site fractions thus become identical.\n", + "Solutions with both substitutional and interstitial component, like the\n", + "fcc Fe-C solution mentioned above, can be represented by two\n", + "sublattices. Stoichiometric compounds have its site fractions equal to\n", + "unity in each sublattice. When site fractions in a compound deviate from\n", + "unity, the compound is no longer stoichiometric and develops a\n", + "composition range of homogeneity. When the composition range is small,\n", + "the deviations are often referred to as defects. Since many properties\n", + "of a compound are determined by defects, a distinct field of defect\n", + "chemistry exists, predominantly for charged species. As will be\n", + "demonstrated in Chapter and the rest of the book, defects can be treated\n", + "as an integral part of the thermodynamics of a phase with more than one\n", + "sublattices.\n", + "\n", + "Let us consider a case where there is only one component in each\n", + "sublattice, which represents one possible stoichiometric composition of\n", + "the phase and is often called an end-member of the phase. The Gibbs\n", + "energy of an end-member is the same as that of a phase with a fixed\n", + "composition shown by or or . By re-arranging , Gibbs energy of an\n", + "end-member, $_{\\ }^{0}G_{em}$, is obtained as\n", + "\n", + "Eq. \u2011\n", + "$_{\\ }^{0}G_{em} = \\sum_{t}^{}{a^{t}{_{\\ }^{0}G}_{i}^{t,ref}} + \\mathrm{\\Delta}_{f}G_{em}$\n", + "\n", + "where ${_{\\ }^{0}G}_{i}^{t,ref}$ represents the Gibbs energy of\n", + "component $i$ in a given reference state which occupies sublattice $t$\n", + "in the end-member. For vacancy, $_{\\ }^{0}G_{Va} = 0$ is defined. The\n", + "contribution of each end-member to the Gibbs energy of the phase is the\n", + "product of site fraction of each component in their respective\n", + "sublattices and the Gibbs energy of the end-member per mole of formula\n", + "unit (*mf*), i.e.\n", + "\n", + "Eq. \u2011\n", + "$_{\\ }^{0}G_{mf} = \\sum_{em}^{}\\left( \\prod_{t}^{}y_{i}^{t}_{\\ }^{0}G_{em} \\right)$\n", + "\n", + "The ideal mixing in each sublattice is similar to that in a random\n", + "solution with mole fractions substituted by site fractions. The excess\n", + "Gibbs energy of mixing consists of two contributions: (i) the mixing in\n", + "one sublattice with all other sublattices containing only one component\n", + "in each sublattice and (ii) the mixing simultaneously in more than one\n", + "sublattices. The Gibbs energy of a solution phase with multi-sublattices\n", + "can thus be written in terms of per mole of formula unit as\n", + "\n", + "Eq. \u2011\n", + "$G_{mf} =_{\\ }^{0}G_{mf} + RT\\sum_{t}^{}{a^{t}\\sum_{i}^{}{y_{i}^{t}\\ln y_{i}^{t}}} +_{\\ }^{E}G_{mf}$\n", + "\n", + "with $_{\\ }^{E}G_{mf}$ given by\n", + "\n", + "Eq. \u2011\n", + "$_{\\ }^{E}G_{mf} = \\sum_{t}^{}{\\prod_{s \\neq t}^{}y_{l}^{s}\\sum_{i > j}^{}{\\sum_{j}^{}{y_{i}^{t}y_{j}^{t}L_{i,j:l}^{t}}}} + \\sum_{t}^{}{\\prod_{s \\neq t}^{}y_{l}^{s}\\sum_{i > j}^{}{\\sum_{j > k}^{}{\\sum_{k}^{}{y_{i}^{t}y_{j}^{t}y_{k}^{t}L_{i,j,k:l}^{t}}}}} + \\sum_{t}^{}{\\prod_{s \\neq t,u}^{}y_{l}^{s}\\sum_{i > j}^{}{\\sum_{j > k}^{}{\\sum_{k}^{}{y_{i}^{t}y_{j}^{t}y_{m}^{u}y_{n}^{u}L_{i,j:m,n:l}^{t}}}}}$\n", + "\n", + "The first term in represents the binary interaction between component\n", + "$i$ and $j$ in sublattice $t$ with sublattice $s$ occupied by component\n", + "$l$ with comma separating interacting components in one sublattice and\n", + "colon separating sublattices. The product,\n", + "$\\prod_{s \\neq t}^{}y_{l}^{s}$, runs through all other sublattices with\n", + "one component in each sublattice, except sublattice $t$ in which the\n", + "interaction is considered. The second term denotes the ternary\n", + "interaction among $i$, $j$, and $k$ in sublattice $t$ with sublattice\n", + "$s$ occupied by component $l$. The third term depicts the interactions\n", + "simultaneously in both sublattice $t$ and $u$, and the product runs\n", + "through all other sublattices with one component in each sublattice,\n", + "except sublattice $t$ and $u$ in which the interactions are considered.\n", + "The third term thus partially reflects the short-range ordering among\n", + "components between two sublattices. In principles, high-order\n", + "interaction parameters such as quaternary, quinary, and multiple\n", + "sublattice interaction parameters could be added, but their\n", + "contributions to $_{\\ }^{E}G_{mf}$ are small due to the physical\n", + "insignificance of co-location of four or five components indicated by\n", + "the product of their site fractions in front of the interaction\n", + "parameters.\n", + "\n", + "In , the chemical potential of a stoichiometric compound was defined in\n", + "terms of a summation of chemical potential of individual components in\n", + "the compound because the relative amounts of components are constrained\n", + "by the stoichiometry of the compound and chemical potentials of\n", + "individual components can not vary independently. By the same token, the\n", + "chemical potential of an end-member in a solution can be written as\n", + "\n", + "Eq. \u2011 $\\mu_{em} = G_{em} = \\sum_{t}^{}a^{t}\\mu_{i}^{t}$\n", + "\n", + "where $\\mu_{i}^{t}$ is the chemical potential of component $i$ that\n", + "occupies the sublattice $t$ in the end-member, and can be derived using\n", + "\n", + "Eq. \u2011\n", + "$\\mu_{i}^{t} = a^{t}_{\\ }^{0}G_{i}^{t,ref} + a^{t}RTlny_{i}^{t} +_{\\ }^{E}G_{mf}^{\\ } + \\frac{\\partial_{\\ }^{E}G_{mf}^{\\ }}{\\partial y_{i}^{t}} - \\sum_{j}^{\\ }y_{j}^{t}\\frac{\\partial_{\\ }^{E}G_{mf}^{\\ }}{\\partial y_{j}^{t}}$\n", + "\n", + "For constant interaction parameters in , for chemical potential reduces\n", + "to the following expression from\n", + "\n", + "Eq. \u2011\n", + "$\\mu_{i}^{t} = a^{t}_{\\ }^{0}G_{i}^{t} + a^{t}RTlny_{i}^{t} + \\sum_{j \\neq i}^{}{\\left( y_{i}^{t} \\right)^{2}L}_{i,j}^{t} + \\sum_{k > j}^{}{\\sum_{j \\neq i}^{}{y_{j}^{t}y_{k}^{t}\\left\\lbrack L_{i,j}^{t} + L_{i,k}^{t} - L_{j,k}^{t} + \\left( 1 - 2y_{i}^{t} \\right)L_{i,j,k}^{t} \\right\\rbrack}}$\n", + "\n", + "The stability of the solution is defined by\n", + "$\\frac{\\partial\\mu_{em}}{\\partial N_{em}}$ with $N_{em}$ being the moles\n", + "of the end-member in the solution given by\n", + "$N_{em} = N\\prod_{u}^{}y_{j}^{u}$ and\n", + "\n", + "Eq. \u2011\n", + "$\\frac{\\partial N_{em}}{\\partial y_{j}^{u}} = \\frac{N\\prod_{t}^{}y_{i}^{t}}{y_{j}^{u}\\left( 1 - \\prod_{t}^{}y_{i}^{t} \\right)}$\n", + "\n", + "Following , one obtains\n", + "\n", + "Eq. \u2011\n", + "$\\frac{\\partial\\mu_{em}}{\\partial N_{em}} = \\sum_{u}^{}{\\frac{\\partial\\left( \\sum_{t}^{}a^{t}\\mu_{i}^{t} \\right)}{\\partial y_{j}^{u}}\\frac{\\partial y_{j}^{u}}{\\partial N_{em}}} = \\frac{\\left( 1 - \\prod_{t}^{}y_{j}^{t} \\right)}{N\\prod_{t}^{}y_{j}^{t}}\\sum_{u}^{}{a^{u}y_{i}^{u}\\frac{\\partial\\mu_{i}^{u}}{\\partial y_{i}^{u}}} = \\frac{\\left( 1 - \\prod_{t}^{}y_{j}^{t} \\right)}{N\\prod_{t}^{}y_{j}^{t}}\\sum_{u}^{}{a^{u}y_{i}^{u}\\left\\{ \\frac{RT\\left( 1 - y_{i}^{u} \\right)}{y_{i}^{u}} - 2\\sum_{j \\neq i}^{}\\left( y_{j}^{u} \\right)^{2}L_{i,j}^{u} - 2\\sum_{k > j}^{}{\\sum_{j \\neq i}^{}{y_{j}^{u}y_{k}^{u}\\left\\lbrack L_{i,j}^{u} + L_{i,k}^{u} - L_{j,k}^{u} + \\left( 2 - 3y_{i}^{u} \\right)L_{i,j,k}^{u} \\right\\rbrack}} \\right\\}}$\n", + "\n", + "It is self-evident from that a site fraction is only uniquely defined\n", + "from the mole fraction of the component when the component enters into\n", + "one sublattice only and does not form any associates. Therefore, in\n", + "general, the distribution of components on sublattices and different\n", + "kinds of molecules can only be obtained by equilibrium calculations, and\n", + "the thermodynamic properties for such a phase can thus not be\n", + "represented in a closed form using mole fractions of independent\n", + "components. This was demonstrated in Chapter when short-range ordering\n", + "exists in solution phases, where the energy minimization procedure was\n", + "used to obtain the distribution of components on different kinds of\n", + "bonds and the amounts of individual associates.\n" + ], + "metadata": {} + }, + { + "id": "2b814171", + "cell_type": "markdown", + "source": [ + "### Solutions with short-range ordering\n", + "\n", + "The order in a system can be measured by correlation functions which\n", + "describes how various components are correlated in space. For\n", + "simplicity, let us consider only the pairs between nearest neighbors\n", + "with the correlation function represented by the pair probability of\n", + "nearest neighbor bonds between two components. In a random solution, the\n", + "probability to find nearest neighbor bonds between two components $i$\n", + "and $j$ is\n", + "\n", + "Eq. \u2011 $p_{ij} = x_{i}x_{j}$\n", + "\n", + "When $p_{ij} \\neq x_{i}x_{j}$, the nearest neighbors of component $i$\n", + "are not occupied randomly by component $j$, rather certain components\n", + "are favored, resulting in short-range ordering or local clustering in\n", + "the solution. When short-range ordering develops throughout the\n", + "solution, long-range ordering takes place, and each component has its\n", + "own primary sites in the solution, to be discussed in Chapter . There\n", + "are relations between bond probabilities and mole fractions of\n", + "components due to the mass balance as follows, with the assumption of\n", + "$p_{ij} = p_{ji}$\n", + "\n", + "Eq. \u2011 $\\sum_{i}^{}{\\sum_{j}^{}p_{ij}} = 1$\n", + "\n", + "Eq. \u2011 $x_{i} = \\sum_{j}^{}p_{ij}$\n", + "\n", + "For small deviations from a random solution, one can consider the\n", + "formation of $i - j$ bonds from $i - i\\ $ and $j - j$ bonds and the\n", + "ideal mixing of three types of bonds, similar to a typical Ising model.\n", + "The bond reaction can be written as\n", + "\n", + "Eq. \u2011 $(i - i\\ )bonds + (j - j)\\ bonds = 2\\ (i - j)\\ bonds$\n", + "\n", + "with the Gibbs energy of reaction being\n", + "\n", + "Eq. \u2011\n", + "${\\mathrm{\\Delta}G}_{ij} = 2G_{ij} - \\left( G_{ii} + G_{jj} \\right)\\ $\n", + "\n", + "Gibbs energy of the solution per mole of atom is thus represented by the\n", + "bond energies and the ideal mixing of bonds plus non-ideal interactions\n", + "between pairs,\n", + "\n", + "Eq. \u2011\n", + "$G_{m} = \\sum_{i}^{}{\\sum_{j}^{}{p_{ij}G_{ij}}} + \\frac{Z}{2}RT\\sum_{i}^{}{\\sum_{j}^{}{p_{ij}\\ln p_{ij}}} +_{\\ }^{E}G_{m}$\n", + "\n", + "with $G_{ij}$ being the molar bond energy between components $i$ and\n", + "$j$, $Z$ the number of bonds per atom which is divided by two in the\n", + "equation due to two atoms needed to form one bond, and\n", + "$_{\\ }^{E}G_{m} = \\sum_{}^{}{p_{ij}p_{kl}I_{ijkl}}$ the excess Gibbs\n", + "energy of mixing between bonds. This approach proposed by Guggenheim\n", + "\\[2\\] is called quasi-chemical method as it is based on the chemical\n", + "reaction shown by .\n", + "\n", + "However, the entropy of mixing in does not reduce to ideal entropy of\n", + "mixing for a solution without short-range ordering as defined by . An\n", + "approximated correction may be added for small degree of short-range\n", + "ordering as follows\n", + "\n", + "Eq. \u2011\n", + "$G_{m} = \\sum_{i}^{}{\\sum_{j}^{}{p_{ij}G_{ij}}} + \\frac{Z}{2}RT\\sum_{i}^{}{\\sum_{j}^{}{p_{ij}\\ln\\frac{p_{ij}}{x_{i}x_{j}}}} + RT\\sum_{}^{}{x_{i}l{nx}_{i}} +_{\\ }^{E}G_{m}$\n", + "\n", + "For a random solution defined by , becomes\n", + "\n", + "Eq. \u2011\n", + "$G_{m} = \\sum_{}^{}{x_{i}_{\\ }^{0}G_{i}} + RT\\sum_{}^{}{x_{i}l{nx}_{i}} + \\sum_{}^{}{x_{i}x_{j}{\\mathrm{\\Delta}G}_{ij}} +_{\\ }^{E}G_{m}$\n", + "\n", + "with $_{\\ }^{0}G_{i} = G_{ii}$, ${\\mathrm{\\Delta}G}_{ij}$ from Eq. 2\u2011107\n", + "representing the interaction parameter between components $i$ and $j$,\n", + "and $_{\\ }^{E}G_{m} = \\sum_{}^{}{x_{i}{x_{j}x_{k}x}_{l}I_{ijkl}}$\n", + "denoting the higher order interactions, in comparison with .\n", + "\n", + "When short-range ordering exists in a solution, one typically uses the\n", + "law of mass reaction for the chemical reaction represented by to define\n", + "the equilibrium among all bonds, i.e.\n", + "\n", + "Eq. \u2011\n", + "$\\frac{\\left( p_{ij} \\right)^{2}}{p_{ii}p_{jj}} = e^{- \\frac{{\\mathrm{\\Delta}G}_{ij}}{kT}}$\n", + "\n", + "However, this is under the assumption that the chemical activities of\n", + "all bonds can be represented by their respective probabilities, which is\n", + "only true for an ideal solution even excluding dilute solutions due to\n", + "the Henry\u2019s law shown by . Preferrably, the bond probabilities can be\n", + "obtained by calculating the driving force for the fluctuation of bond\n", + "probabilities under constant temperature, pressure, and amount of each\n", + "component along with the constraints defined by and and equating the\n", + "driving force to zero, i.e.\n", + "\n", + "Eq. \u2011\n", + "${\\frac{1}{N}\\left( \\frac{\\partial G}{\\partial\\xi} \\right)}_{T,P,N_{n}} = \\left( \\frac{\\partial G_{m}}{\\partial p_{ij}} \\right)_{T,P,N_{n}} = \\left( \\frac{\\partial G_{m}}{\\partial p_{ij}} \\right)_{T,P,x_{n},p_{kl \\neq ij}} - \\sum_{kl \\neq ij}^{}\\left( \\frac{\\partial G_{m}}{\\partial p_{kl}} \\right)_{T,P,x_{n},p_{op \\neq kl}} + \\left( \\frac{\\partial G_{m}}{\\partial x_{i}} \\right)_{T,P,x_{q \\neq i},p_{kl}} + \\left( \\frac{\\partial G_{m}}{\\partial x_{j}} \\right)_{T,P,x_{q \\neq j},p_{kl}} = 0$\n", + "\n", + "where $\\frac{\\partial p_{kl}}{\\partial p_{ij}} = - 1$ and\n", + "$\\frac{\\partial x_{i}}{\\partial p_{ij}} = \\frac{\\partial x_{j}}{\\partial p_{ij}} = 1$\n", + "are used from and . Numerical values of $p_{ij}$ can be obtained by\n", + "minimization of Gibbs energy under constraints given by and .\n", + "\n", + "The chemical potential of independent component $i$ is defined as in and\n", + "can be represented by the following equation\n", + "\n", + "Eq. \u2011\n", + "$\\mu_{i} = G_{m} + \\frac{\\partial G_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial G_{m}}{\\partial x_{j}} + 2\\sum_{j = 1}^{c}\\frac{\\partial G_{m}}{\\partial p_{ij}} - \\frac{\\partial G_{m}}{\\partial p_{ii}} - 2\\sum_{j = 1}^{c}x_{j}\\sum_{k = 1}^{c}\\frac{\\partial G_{m}}{\\partial p_{jk}} + \\sum_{j = 1}^{c}x_{j}\\frac{\\partial G_{m}}{\\partial p_{jj}}$\n", + "\n", + "The stability of the solution can be derived similar to .\n", + "\n", + "When the bonding between components becomes very strong, distinctive new\n", + "components may form. They are not independent components and often\n", + "called associates. Both the independent and dependent components are\n", + "collectively called species. The formation of an associate\n", + "$i_{a_{i}}j_{b_{j}}$ consisting of $a_{i}$ mole of $i$ and $a_{j}$ mole\n", + "$j$ can be written as\n", + "\n", + "Eq. \u2011 $a_{i}i + a_{j}j = i_{a_{i}}j_{a_{j}}$\n", + "\n", + "Gibbs energy of the associate follows the same format of that of a\n", + "stoichiometric phase as shown by ,\n", + "\n", + "Eq. \u2011\n", + "${_{\\ }^{0}G}_{i_{a_{i}}j_{a_{j}}} = \\sum_{}^{}a_{i}{_{\\ }^{0}G}_{i}^{SER} + \\mathrm{\\Delta}_{f}G_{i_{a_{i}}j_{a_{j}}}$\n", + "\n", + "Gibbs energy of the solution is obtained by extending to all species\n", + "\n", + "Eq. \u2011\n", + "$G_{m} = \\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}_{\\ }^{0}G_{i_{a_{i}}j_{a_{j}}}} + RT\\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}\\ln y_{i_{a_{i}}j_{a_{j}}}} +_{\\ }^{E}G_{m}$\n", + "\n", + "where $y_{i_{a_{i}}j_{a_{j}}}$ is the mole fraction of species\n", + "$i_{a_{i}}j_{a_{j}}$ in the solution with $a_{i} = 1$ and $a_{j} = 0$\n", + "for component $i$ and $a_{i} = 0$ and $a_{j} = 1$ for component $j$. The\n", + "equilibrium amount of each associate $i_{a_{i}}j_{a_{j}}$ is obtained in\n", + "combination of mass balance and the zero driving force for the variation\n", + "of the amount of the associate similar to , i.e.\n", + "\n", + "Eq. \u2011 $\\sum_{i}^{}{\\sum_{j}^{}y_{i_{a_{i}}j_{a_{j}}}} = 1$\n", + "\n", + "Eq. \u2011 $x_{i} = \\sum_{}^{}{a_{i}y_{i_{a_{i}}j_{a_{j}}}}$\n", + "\n", + "Eq. \u2011\n", + "$\\left( \\frac{{\\partial G}_{m}}{\\partial y_{i_{a_{i}}j_{a_{j}}}} \\right)_{x_{i}} = 0$\n", + "\n", + "Associates are particularly plentiful in a gas phase, and their amounts\n", + "are significantly affected by pressure. For an ideal gas phase with\n", + "$_{\\ }^{E}G_{m} = 0$ and $PV_{m} = RT$, the effect of pressure is added\n", + "as follows\n", + "\n", + "Eq. \u2011\n", + "$G_{m} = \\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}_{\\ }^{0}G_{i_{a_{i}}j_{a_{j}}}} + RT\\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}\\ln y_{i_{a_{i}}j_{a_{j}}}} + \\int_{P_{0}}^{P}{V_{m}dP} = \\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}_{\\ }^{0}G_{i_{a_{i}}j_{a_{j}}}} + RT\\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}\\ln y_{i_{a_{i}}j_{a_{j}}}} + RT\\ln\\frac{P}{P_{0}}$\n", + "\n", + "where $P$ is the total pressure, and $P_{0}$ the reference pressure at\n", + "which $_{\\ }^{0}G_{i_{a_{i}}j_{a_{j}}}$ is defined, usually chosen to be\n", + "one atmospheric pressure. thus becomes\n", + "\n", + "Eq. \u2011\n", + "$G_{m} = \\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}_{\\ }^{0}{G_{i_{a_{i}}j_{a_{j}}}(P = 1atm)}} + RT\\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}\\ln y_{i_{a_{i}}j_{a_{j}}}} + RTlnP$\n", + "\n", + "with the unit of the total pressure $P$ being in atmospheric pressure\n", + "(atm). The chemical potential of species $i_{a_{i}}j_{a_{j}}$ is equal\n", + "to\n", + "\n", + "Eq. \u2011\n", + "$\\mu_{i_{a_{i}}j_{a_{j}}} =_{\\ }^{0}{G_{i_{a_{i}}j_{a_{j}}}(P = 1atm)} + RTlny_{i_{a_{i}}j_{a_{j}}}P =_{\\ }^{0}{G_{i_{a_{i}}j_{a_{j}}}(P = 1atm)} + RTlnP_{i_{a_{i}}j_{a_{j}}}$\n", + "\n", + "where $P_{i_{a_{i}}j_{a_{j}}}$ is the partial pressure of species\n", + "$i_{a_{i}}j_{a_{j}}$ defined as\n", + "\n", + "Eq. \u2011 $P_{i_{a_{i}}j_{a_{j}}} = y_{i_{a_{i}}j_{a_{j}}}P$\n", + "\n", + "Combining and with , the relation between chemical potentials of an\n", + "associate and its constituents is expressed as\n", + "\n", + "Eq. \u2011\n", + "$\\mu_{i_{a_{i}}j_{a_{j}}} = a_{i}\\mu_{i} + {a_{j}\\mu}_{j} = a_{i}_{\\ }^{0}G_{i} + a_{j}_{\\ }^{0}G_{j} + RTln\\left( {P_{i}}^{a_{i}}{P_{j}}^{a_{j}} \\right)$\n", + "\n", + "The equilibrium condition for the chemical reaction of an associate\n", + "forming from its constituents in an ideal gas phase is obtained as\n", + "\n", + "Eq. \u2011\n", + "$\\mathrm{\\Delta}_{f}G_{i_{a_{i}}j_{a_{j}}} + RTln\\frac{{P_{i}}^{a_{i}}{P_{j}}^{a_{j}}}{P_{i_{a_{i}}j_{a_{j}}}} = 0$\n", + "\n", + "For non-ideal phases, the mole fractions of various associates can be\n", + "calculated numerically through the minimization of Gibbs energy under\n", + "the constraints of and .\n" + ], + "metadata": {} + }, + { + "id": "a19845f0", + "cell_type": "markdown", + "source": [ + "### Ternary random solutions\n", + "\n", + "From , the Gibbs-Duhem equation of a ternary system consisting of\n", + "components $A$, $B$ and $C$ is written as\n", + "\n", + "Eq. \u2011\n", + "$0 = - SdT - Vd( - P) - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B} - {N_{C}d\\mu}_{C}$\n", + "\n", + "This equation represents a five-dimensional surface. It can be\n", + "visualized in a three-dimensional space with two of the five potentials\n", + "fixed. Usually $T$ and $P$ are kept constant as they are the natural\n", + "variables of $G$, and reduces to\n", + "\n", + "Eq. \u2011 $0 = - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B} - {N_{C}d\\mu}_{C}$\n", + "\n", + "A phase can thus be represented by a surface in the three-dimensional\n", + "space of $\\mu_{A}$, $\\mu_{B}$, and $\\mu_{C}$ at constant $T$ and $P$\n", + "with similar geometric appearance as .\n", + "\n", + "From , Gibbs energy of a ternary solution is written as\n", + "\n", + "Eq. \u2011\n", + "$G_{m} = x_{A}_{\\ }^{0}G_{A} + x_{B}_{\\ }^{0}G_{B} + x_{C}_{\\ }^{0}G_{C} + RT\\left( x_{A}l{nx}_{A} + x_{B}l{nx}_{B} + x_{C}l{nx}_{C} \\right) +_{\\ }^{E}G_{m}$\n", + "\n", + "When the mole fraction of one component approaches zero,\n", + "$_{\\ }^{E}G_{m}$ reduces to the excess Gibbs energy of mixing of the\n", + "binary systems of the remaining two components, represented by .\n", + "However, for a given composition of a ternary solution, there is no\n", + "unique way to assign the contributions from $_{\\ }^{E}G_{m}$ of each\n", + "binary to $_{\\ }^{E}G_{m}$ of the ternary solution because\n", + "$_{\\ }^{E}G_{m}$ of the ternary solution contains information of both\n", + "binary and ternary interactions. A variety of models is available in the\n", + "literature (see \\[1\\]). One intuitive approach would be to use the same\n", + "formula as that in the binary system, i.e. , with the mole fractions\n", + "substituted by the values in the ternary system, and $_{\\ }^{E}G_{m}$ of\n", + "a ternary solution may thus be defined as the following by including the\n", + "ternary interaction involving all three components,\n", + "\n", + "Eq. \u2011\n", + "$_{\\ }^{E}G_{m} = x_{A}x_{B}L_{AB} + x_{A}x_{C}L_{AC} + x_{B}x_{C}L_{BC} + x_{A}x_{B}x_{C}L_{ABC}$\n", + "\n", + "The chemical potential of a component is represented by . When all\n", + "interaction parameters in are constant, i.e. a ternary regular solution,\n", + "the chemical potential of component $A$ can be derived as\n", + "\n", + "Eq. \u2011\n", + "$\\mu_{A} = G_{A} =_{\\ }^{0}G_{A} + RTlnx_{A} + x_{B}L_{AB} + x_{C}L_{AC} -_{\\ }^{E}G_{m} =_{\\ }^{0}G_{A} + RTlnx_{A} + x_{B}{\\left( 1 - x_{A} \\right)L}_{AB} + x_{C}\\left( 1 - x_{A} \\right)L_{AC} - x_{B}x_{C}L_{BC} + x_{B}x_{C}\\left( 1 - 2x_{A} \\right)L_{ABC} =_{\\ }^{0}G_{A} + RTlnx_{B} + x_{B}^{2}L_{AB} + x_{C}^{2}L_{AC} + x_{B}x_{C}\\left( L_{AB} + L_{AC} - L_{BC} \\right) + x_{B}x_{C}\\left( 1 - 2x_{A} \\right)L_{ABC}$\n", + "\n", + "Similar equations can be derived for component $B$ and C with\n", + "$L_{AB} = L_{BA}$, $L_{AC} = L_{CA}$, and $L_{BC} = L_{CB}$. A schematic\n", + "molar Gibbs energy diagram at constant temperature and pressure is shown\n", + "in with all three binary systems having $L_{ij} < 0$ of similar values.\n", + "\n", + "Figure \u2011: Schematic ternary molar Gibbs energy diagram as a function of\n", + "compositions for given temperature and pressure\n", + "\n", + "To evaluate the stability of a ternary solution, one needs to calculate\n", + "the elements in the determinant shown in . Using the mole of component\n", + "$C$ as the independent molar quantity, the limit of stability is\n", + "expresses as\n", + "\n", + "Eq. \u2011 $G_{AA}G_{BB} - G_{AB}G_{BA} = 0$\n", + "\n", + "As an example, $G_{AA}$ is shown in the following equation, which must\n", + "be positive for the solution to be stable\n", + "\n", + "Eq. \u2011\n", + "${N\\left( \\frac{\\partial\\mu_{A}}{\\partial N_{A}} \\right)}_{T,P,N_{B},N_{C}} = NG_{AA} = \\frac{RT\\left( 1 - x_{A} \\right)}{x_{A}} - 2x_{B}^{2}L_{AB} - 2x_{C}^{2}L_{AC} - 2x_{B}x_{C}\\left( L_{AB} + L_{AC} - L_{BC} \\right) - 2x_{B}x_{C}\\left( 2 - 3x_{A} \\right)L_{ABC}$\n", + "\n", + "It is evident that any instability in binary systems with positive\n", + "interaction parameters extends into the ternary system. It can also be\n", + "seen that even if all binary interaction parameters are negative, i.e.\n", + "no instability in the binary systems, it is possible that becomes\n", + "negative for some combinations of the binary interaction parameters such\n", + "that $\\mathrm{\\Delta}L = L_{AB} + L_{AC} - L_{BC}$ becomes very positive\n", + "and overshadows the contributions due to $L_{AB}$ and $L_{AC}$, i.e.\n", + "$L_{BC}$ is more negative than $L_{AB}$ and $L_{AC}$ combined. In an\n", + "extreme case with $L_{AB} = L_{AC} = L_{ABC} = 0$ and $L_{BC} < 0$, i.e.\n", + "ideal solutions for the $A - B$ and $A - C$ binary systems, stable\n", + "solution in the $B - C$ binary system, and no additional ternary\n", + "interaction, reduces to\n", + "\n", + "Eq. \u2011\n", + "$N\\left( \\frac{\\partial\\mu_{A}}{\\partial N_{A}} \\right)_{T,P,N_{B},N_{C}} = \\frac{RT\\left( 1 - x_{A} \\right)}{x_{A}} + 2x_{B}x_{C}L_{BC}$\n", + "\n", + "Setting\n", + "$\\left( \\frac{\\partial\\mu_{A}}{\\partial N_{A}} \\right)_{T,P,N_{B},N_{C}} = 0$,\n", + "one obtains\n", + "\n", + "Eq. \u2011\n", + "$- \\frac{RT}{2L_{BC}} = \\frac{x_{A}x_{B}x_{C}}{1 - x_{A}} = \\frac{{\\left( 1 - x_{B} - x_{C} \\right)x}_{B}x_{C}}{x_{B} + x_{C}}$\n", + "\n", + "With $- \\frac{RT}{2L_{BC}}$ being positive due to $L_{BC} < 0$, there is\n", + "a parabola-shaped composition area in which the solution is unstable at\n", + "constant temperature and pressure. This is reasonable because the system\n", + "tends to maximize the number of B-C bonds due to its lower energy, which\n", + "competes with the entropy of mixing among the three elements and results\n", + "in segregation of B-C bonds, thus miscibility gap at low temperatures.\n", + "\n", + "To evaluate the ternary consolute point, the second derivatives for\n", + "component A and B are obtained as\n", + "\n", + "Eq. \u2011\n", + "${N\\left( \\frac{\\partial_{\\ }^{2}\\mu_{A}}{\\partial N_{A}^{2}} \\right)}_{T,P,N_{B},N_{C}} = \\frac{RT\\left( 1 - x_{A} \\right)}{x_{A}^{2}} + 4x_{B}^{2}L_{AB} + 4x_{C}^{2}L_{AC} + 4x_{B}x_{C}\\left( L_{AB} + L_{AC} - L_{BC} \\right) + 2x_{B}x_{C}\\left( 7 - 9x_{A} \\right)L_{ABC} = 0$\n", + "\n", + "Eq. \u2011\n", + "${N\\left( \\frac{\\partial_{\\ }^{2}\\mu_{B}}{\\partial N_{B}^{2}} \\right)}_{T,P,N_{A},N_{C}} = \\frac{RT\\left( 1 - x_{B} \\right)}{x_{B}^{2}} + 4x_{A}^{2}L_{AB} + 4x_{C}^{2}L_{BC} + 4x_{A}x_{C}\\left( L_{AB} + L_{BC} - L_{AC} \\right) + 2x_{A}x_{C}\\left( 7 - 9x_{B} \\right)L_{ABC} = 0$\n", + "\n", + "The consolute point can then be obtained using , and .\n", + "\n", + "It is observed in that $\\left( 1 - 2x_{A} \\right)L_{ABC} = 0$ at\n", + "$x_{A} = 0.5$, i.e. the ternary interaction parameter does not\n", + "contribute to the chemical potential of $A$. It is also observed in that\n", + "the contribution from the ternary interaction parameter changes sign at\n", + "$x_{i} = 2/3$ due to $\\left( 2 - 3x_{A} \\right)L_{ABC} = 0$.\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/laws_of_thermodynamics/combined_law_of_thermodynamics_and_equilibrium_conditions.ipynb b/src/psu410/src/laws_of_thermodynamics/combined_law_of_thermodynamics_and_equilibrium_conditions.ipynb new file mode 100644 index 0000000..dad65a4 --- /dev/null +++ b/src/psu410/src/laws_of_thermodynamics/combined_law_of_thermodynamics_and_equilibrium_conditions.ipynb @@ -0,0 +1,229 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "6a9e41f7", + "cell_type": "markdown", + "source": [ + "## Combined law of thermodynamics and equilibrium conditions\n", + "\n", + "For a system with an irreversible internal process taking place, the\n", + "entropy change in the system thus consists of three parts: the heat\n", + "exchange with the surrounding defined by , the entropy production due to\n", + "the internal process represented by , and the entropy of mass exchange\n", + "with the surrounding. The total entropy change of the system can thus be\n", + "written as follows\n", + "\n", + "Eq. \u2011 $dS = \\frac{dQ}{T} + d_{ip}S + \\sum_{}^{}{S_{i}dN_{i}}$\n", + "\n", + "where $S_{i}$ is the unit entropy of component *i* in the surroundings,\n", + "often called partial entropy of component *i*, and will be further\n", + "discussed in Chapter\n", + "\n", + "Combining and and re-arranging, one obtains\n", + "\n", + "Eq. \u2011 $dQ = TdS - Dd\\xi - \\sum_{}^{}{TS_{i}dN_{i}}$\n", + "\n", + "Inserting and into yields the combined law of thermodynamics from the\n", + "first and second laws of thermodynamics,\n", + "\n", + "Eq. \u2011\n", + "$dU = TdS - PdV + \\sum_{}^{}\\left( H_{i} - TS_{i} \\right){dN}_{i} - Dd\\xi$\n", + "\n", + "The internal energy of the system is thus a function of *S*, *V*,\n", + "*Ni* and *\u03be* of the system, which are called natural\n", + "variables of the internal energy, i.e. *U*(*S*,*V*,*Ni*,*\u03be*).\n", + "The other variables are dependent variables and can be represented by\n", + "partial derivatives of the internal energy with respect to their\n", + "respective natural variables with other natural variables kept constant\n", + "as shown below\n", + "\n", + "Eq. \u2011\n", + "$T = \\left( \\frac{\\partial U}{\\partial S} \\right)_{V,\\ N_{i},\\ \\xi}$\n", + "\n", + "Eq. \u2011\n", + "$- P = \\left( \\frac{\\partial U}{\\partial V} \\right)_{S,\\ N_{i},\\ \\xi}$\n", + "\n", + "Eq. \u2011\n", + "$\\mu_{i} = H_{i} - TS_{i} = \\left( \\frac{\\partial U}{\\partial N_{i}} \\right)_{S,\\ V,N_{j \\neq i},\\ \\xi} = U_{i}$\n", + "\n", + "Eq. \u2011\n", + "$- D = \\left( \\frac{\\partial U}{\\partial\\xi} \\right)_{S,\\ V,N_{i}\\ }$\n", + "\n", + "In , a new variable, $\\mu_{i}$, is introduced. It is called chemical\n", + "potential and defined as the internal energy change with respect of the\n", + "addition of the component *i* when the entropy, volume and the amount of\n", + "other components of the system are kept constant. It may be worth\n", + "pointing out that for a system at equilibrium, i.e. $d_{ip}S = 0$, and\n", + "with constant entropy, $dS = 0$, if it exchanges mass with the\n", + "surroundings, $dN_{i} \\neq 0$, the system must also exchange heat with\n", + "the surroundings at the same time in order to keep the entropy invariant\n", + "as demonstrated by .\n", + "\n", + "The pairs of the natural variables and their corresponding partial\n", + "derivatives are called conjugate variables, i.e. *S* and *T*, *V* and\n", + "*\u2013P*, *Ni* and $\\mu_{i}$, and *\u03be* and *\u2013D*. There are minus\n", + "sign in front of *P* and *D* as the increase of volume and the progress\n", + "of the internal process decrease the internal energy of the system. The\n", + "importance of this conjugate relation will be evident when various forms\n", + "of combined thermodynamic laws and various types of phase diagrams are\n", + "introduced in the book.\n", + "\n", + "The last pair of conjugate variables, *\u03be* and *\u2013D*, is worthy of further\n", + "discussion. Based on the second law of thermodynamics, i.e. , no\n", + "internal processes take place spontaneously if there is no entropy\n", + "productions, i.e. D\u22640 or *d\u03be*=0 and *D*\\>0. With D\u22640, there is no\n", + "driving for any internal processes, and the system is at a full\n", + "equilibrium state. The last term in drops off, and *\u03be* becomes a\n", + "dependent variable of the system and can be calculated from the\n", + "equilibrium conditions. With *d\u03be*=0 and *D*\\>0, the system is under a\n", + "constrained equilibrium or freezing-in condition when the internal\n", + "process is constrained not to take place, and *\u03be* remains to be an\n", + "independent variable of the system.\n", + "\n", + "These two cases represent the two branches of thermodynamics:\n", + "equilibrium, reversible thermodynamics and irreversible thermodynamics.\n", + "It is clear from the above discussions that these two branches are\n", + "identical if the internal energy is not only a function of *S*, *V*, and\n", + "*Ni* , but also any internal process variable *\u03be*. This means\n", + "that one should be able to evaluate the internal energy of a system for\n", + "any freezing-in equilibrium conditions in addition to the full\n", + "equilibrium condition. In the rest of the book, the freezing-in\n", + "equilibrium and full equilibrium are not differentiated unless\n", + "specified.\n", + "\n", + "As the mechanical work under hydrostatic pressure is very important in\n", + "experiments, let us define a new quantity called enthalpy as follows\n", + "\n", + "Eq. \u2011 $H = U + PV$\n", + "\n", + "Its differential form can be obtained from as\n", + "\n", + "Eq. \u2011 $dH = dU + d(PV) = dQ + VdP + \\sum_{}^{}H_{i}{dN}_{i}$\n", + "\n", + "There are two significant consequences of the above equation. First, for\n", + "a close system under constant pressure, i.e. ${dN}_{i} = dP = 0$, one\n", + "has $dH = dQ$. This implies that the enthalpy change in a system is\n", + "equal to the heat exchange between the system and the surrounding of the\n", + "system, which is why enthalpy and heat are often exchangeable in the\n", + "literature. Second, for an adiabatic system under constant pressure,\n", + "i.e. $dQ = dP = 0$, can be re-arranged to the following equation\n", + "\n", + "Eq. \u2011\n", + "$H_{i} = \\left( \\frac{\\partial H}{\\partial N_{i}} \\right)_{N_{j \\neq i,\\ \\ dQ = dP = 0}}$\n", + "\n", + "$H_{i}$ is thus the partial enthalpy of component *i* and will be\n", + "further discussed in Chapter . The chemical potential of component *i*\n", + "defined in is thus related to the partial enthalpy and partial entropy\n", + "of the component.\n", + "\n", + "To further define equilibrium conditions of a system, consider a\n", + "homogeneous system in a state of internal equilibrium, i.e. no\n", + "spontaneous internal processes are possible with $Dd\\xi = 0$, and\n", + "becomes\n", + "\n", + "Eq. \u2011\n", + "$dU = TdS - PdV + \\sum_{}^{}\\mu_{i}{dN}_{i} = \\sum_{}^{}{Y_{i}dX_{i}}$\n", + "\n", + "where *X* represents *S*, *V*, *Ni*, and *Y* their conjugate\n", + "variables *T*, *-P*, $\\mu_{i}$. The state of the system with *c*\n", + "independent components is completely determined by the *c+2* variables,\n", + "i.e. *S*, *V*, and *Ni* with *i* from 1 to *c*.\n", + "\n", + "To simplify the situation, let us limit the discussion to an isolated\n", + "equilibrium system, i.e. $dU = 0$, and conduct a virtual internal\n", + "experiment inside the system by moving an infinitesimal amount of\n", + "$X_{i}$, ${dX}_{i}$, with other $X_{j}$ kept constant, from one region\n", + "of the system to another region of the system as schematically shown in\n", + ".\n", + "\n", + "Figure \u2011: Virtual experiment for a system at equilibrium\n", + "\n", + "As the system is homogeneous and at equilibrium,\n", + "$- dX_{i}^{'} = dX_{i}^{\"} = dX_{i}$. The total change of the internal\n", + "energy of this internal process is the combination of the changes in the\n", + "two regions, i.e.\n", + "\n", + "Eq. \u2011\n", + "$dU = dU^{'} + dU^{\"} = Y_{i}^{'}dX_{i}^{'} + Y_{i}^{\"}dX_{i}^{\"} = \\left( - Y_{i}^{'} + Y_{i}^{\"} \\right)dX_{i} = 0$\n", + "\n", + "Therefore, $Y_{i}^{'} = Y_{i}^{\"}$ for *T*, *-P*, and $\\mu_{i}$,\n", + "indicating that *T*, *-P*, and $\\mu_{i}$ are homogeneous in the system,\n", + "respectively, and are thus named as potentials of the system.\n", + "Furthermore these potentials are independent of the size of the system\n", + "and are often referred to as intensive variables in the literature. On\n", + "the other hand, all *X:s*, i.e. *S*, *V*, and *Ni*, are\n", + "proportional to the size of the system and may be normalized with\n", + "respect to the size of the system, usually in terms of total moles,\n", + "\n", + "Eq. \u2011 $N = \\sum_{}^{}N_{i}$\n", + "\n", + "They are thus called molar quantities and often referred to as extensive\n", + "variables, and the respective normalized variables are molar entropy,\n", + "molar volume, and mole fractions, defined as follows\n", + "\n", + "Eq. \u2011 $S_{m} = \\frac{S}{N}$\n", + "\n", + "Eq. \u2011 $V_{m} = \\frac{V}{N}$\n", + "\n", + "Eq. \u2011 $x_{i} = \\frac{N_{i}}{N}$\n", + "\n", + "Consider a small subsystem in this homogeneous system at equilibrium and\n", + "let the subsystem grow in size. The entropy, volume, and mass enclosed\n", + "in the subsystem increase as follows\n", + "\n", + "Eq. \u2011 ${dS = S}_{m}dN$\n", + "\n", + "Eq. \u2011 $dV = V_{m}dN$\n", + "\n", + "Eq. \u2011 $dN_{i} = x_{i}dN$\n", + "\n", + "The corresponding change in the internal energy of the subsystem becomes\n", + "\n", + "Eq. \u2011\n", + "$dU = TdS - PdV + \\sum_{}^{}\\mu_{i}{dN}_{i} = \\left( TS_{m} - PV_{m} + \\sum_{}^{}\\mu_{i}x_{i} \\right)dN = U_{m}dN$\n", + "\n", + "By integration one obtains the integral form of the internal energy as\n", + "\n", + "Eq. \u2011\n", + "$U = TS - PV + \\sum_{}^{}\\mu_{i}N_{i} = \\left( TS_{m} - PV_{m} + \\sum_{}^{}\\mu_{i}x_{i} \\right)N = U_{m}N$\n", + "\n", + "Similarly, the molar enthalpy can be defined as follows\n", + "\n", + "Eq. \u2011\n", + "$H = U + PV = U_{m}N + PV_{m}N = \\left( U_{m} + PV_{m} \\right)N = H_{m}N$\n", + "\n", + "In case a potential is not homogeneous in a system, the system will not\n", + "be in a state of equilibrium. Let us consider the same virtual\n", + "experiment as shown in for an isolated system that is not in\n", + "equilibrium, i.e. by moving an infinitesimal amount of $X_{i}$,\n", + "${dX}_{i}$, with other $X_{j}$ kept constant, from one region of the\n", + "system to another region of the system with the two regions having\n", + "different potentials. The total internal energy change equals to zero as\n", + "the virtual experiment has $dU = 0$. Similarly, each region can be\n", + "considered to be homogeneous by itself, and one has\n", + "$- dX_{i}^{'} = dX_{i}^{\"} = dX_{i}$. The total internal energy change\n", + "in the system is thus the sum of these two regions plus the entropy\n", + "production due to the internal process with $d\\xi = dX_{i}$, i.e.\n", + "\n", + "Eq. \u2011\n", + "$dU = dU^{'} + dU^{\"} + Dd\\xi = Y_{i}^{'}dX_{i}^{'} + Y_{i}^{\"}dX_{i}^{\"} + Dd\\xi = \\left( - Y_{i}^{'} + Y_{i}^{\"} \\right)dX_{i} + Dd\\xi = 0$\n", + "\n", + "Consequently, one obtains the following\n", + "\n", + "*Eq. 1\u201131* $D = Y_{i}^{'} - Y_{i}^{\"}$\n", + "\n", + "The driving force thus represents the difference of the potential at the\n", + "two regions, and the internal process is to eliminate inhomogeneity of\n", + "the potential with the heat transfer from high temperature regions to\n", + "low temperature regions, volume shrink of low pressure regions (high\n", + "$\u2013P$) and volume expansion of high pressure regions (low $\u2013P$), and\n", + "transport of components from high chemical potential regions to low\n", + "chemical potential regions.\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/laws_of_thermodynamics/first_and_second_laws_of_thermodynamics.ipynb b/src/psu410/src/laws_of_thermodynamics/first_and_second_laws_of_thermodynamics.ipynb new file mode 100644 index 0000000..68bda34 --- /dev/null +++ b/src/psu410/src/laws_of_thermodynamics/first_and_second_laws_of_thermodynamics.ipynb @@ -0,0 +1,128 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "dd639118", + "cell_type": "markdown", + "source": [ + "## First and second laws of thermodynamics\n", + "\n", + "A system typically consists of many chemical components. The first law\n", + "of thermodynamics states that the exchanges of heat, work, and\n", + "individual components with its surroundings must obey the law of\n", + "conversation of energy. In the domain of materials science and\n", + "engineering, the energy of interest is at the atomic and molecular\n", + "levels. The energy at the higher and lower levels such as nuclear energy\n", + "and kinetic and potential energies of a rigid body are usually excluded\n", + "from the discussion of thermodynamics of materials.\n", + "\n", + "Let us consider a system receiving an amount of heat, *dQ*, and an\n", + "amount of work, *dW*, and an amount of each independent component *i*,\n", + "*dNi*, from the surroundings. Such a system is called an open\n", + "system in contrast to a closed system when *dNi*=0 for all\n", + "components, i.e. no exchange of mass between the system and the\n", + "surrounding. Other types of systems commonly defined in thermodynamics\n", + "include adiabatic systems without exchange of heat, i.e. *dQ*=0, and\n", + "isolated systems without exchange of any, i.e. *dQ*= *dW*=\n", + "*dNi*=0.\n", + "\n", + "The corresponding change of energy in the system, i.e. the internal\n", + "energy change, *dU*, is formulated in terms of the first law of\n", + "thermodynamics as follows,\n", + "\n", + "Eq. \u2011 $dU = dQ + dW + \\sum_{}^{}{H_{i}dN_{i}}$\n", + "\n", + "where $H_{i}$ is the unit energy of component *i* in the surroundings,\n", + "and the summation is for all components in the system, which can be\n", + "controlled independently from the surroundings, i.e. the independent\n", + "components of the system.\n", + "\n", + "It is self-evident that the left-hand side of refers to the change\n", + "inside the system, while its right-hand side is for the contributions\n", + "from the surroundings to the system. In principle, no matter how the\n", + "heat and mass are added, or how the work is done to the system, as far\n", + "as their summation is the same, the change of the internal energy will\n", + "be the same based on the first law of thermodynamics, indicating that\n", + "the system reaches the same state for a closed system. The internal\n", + "energy is thus a state function in a close system as it does not depend\n", + "on how the state is reached.\n", + "\n", + "On the other hand, for the purpose of easy mathematical treatment, a\n", + "reversible process can be considered for a closed system, in which the\n", + "initial state of the system can be restored reversibly without any\n", + "changes left to the surroundings. Therefore, the heat transferred and\n", + "the work done to the system are identical to the heat and work lost by\n", + "the surroundings and vice versa. The classic example of reversible\n", + "processes is the Carnot\u2019s cycle, which is shown in . It consists of four\n", + "reversible processes for a closed system. The four reversible processes\n", + "are compression at constant temperature *T1* (isothermal,),\n", + "compression without heat exchange (adiabatic) ending at *T2*,\n", + "isothermal expansion at *T2*, and adiabatic expansion ending\n", + "at *T1*.\n", + "\n", + "Figure \u2011: Schematics of the Carnot\u2019s cycle\n", + "\n", + "The Carnot\u2019s cycle involves a simple type of mechanical work, either\n", + "hydrostatic expansion or compression, with the work that the surrounding\n", + "does to the system represented by\n", + "\n", + "Eq. \u2011 $dW = - PdV$\n", + "\n", + "with *P* being the external pressure that the surrounding exerts on the\n", + "system and *V* the volume of the system. It is now necessary to\n", + "differentiate the external and internal variables for further discussion\n", + "with the former representing variables in the surroundings and the\n", + "latter representing variables in the system. For the isothermal\n", + "processes in the Carnot\u2019s cycle, the entropy change of the system, *dS*,\n", + "can be defined as the heat exchange divided by temperature\n", + "\n", + "Eq. \u2011 $dS = \\frac{dQ}{T}$\n", + "\n", + "In addition to processes involving heat, work, and mass exchanges\n", + "between the system and the surroundings, there could be internal\n", + "processes taking place inside the system. As the system cannot do work\n", + "to itself, the criterion for whether an internal process can occur\n", + "spontaneously must be related to the heat exchange, which is related to\n", + "the entropy change as shown by .\n", + "\n", + "It is a known fact that heat can spontaneously transfer from a higher\n", + "temperature (*T2*) region to a lower temperature\n", + "(*T1*) region inside a system if the heat conduction is\n", + "allowed, and this process is irreversible because heat cannot be\n", + "conducted from a low temperature region to a high temperature region by\n", + "itself. indicates that for the same amount of heat change, the entropy\n", + "change at *T1* is higher than that at *T2*, and\n", + "the heat conduction thus results in a positive entropy change in the\n", + "system, i.e.\n", + "\n", + "Eq. \u2011\n", + "$\\mathrm{\\Delta}S = - \\frac{dQ}{T_{2}} + \\frac{dQ}{T_{1}} = \\frac{dQ}{{T_{2}T}_{1}}\\left( {T_{2} - T}_{1} \\right) > 0$\n", + "\n", + "Consequently, the second law of thermodynamics is obtained, which states\n", + "that for an internal process to take place spontaneously or\n", + "irreversibly, this internal process (*ip*) must have a positive entropy\n", + "production, which can be written in a differential form as follows\n", + "\n", + "Eq. \u2011 $d_{ip}S > 0$\n", + "\n", + "From the definition of entropy change shown by , the amount of heat\n", + "produced by this irreversible internal process can be calculated as\n", + "follows\n", + "\n", + "Eq. \u2011 $d_{ip}Q = Td_{ip}S$\n", + "\n", + "Let us represent this internal process by *d\u03be* and define the driving\n", + "force for this internal process by *D*. The work done by this internal\n", + "process is thus *Dd\u03be*, which is released as heat, i.e.\n", + "\n", + "Eq. \u2011 $Dd\\xi = d_{ip}Q = Td_{ip}S$\n", + "\n", + "An irreversible process thus must have a positive driving force in order\n", + "for it to take place spontaneously.\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/laws_of_thermodynamics/gibbsduhem_equation.ipynb b/src/psu410/src/laws_of_thermodynamics/gibbsduhem_equation.ipynb new file mode 100644 index 0000000..c71fa13 --- /dev/null +++ b/src/psu410/src/laws_of_thermodynamics/gibbsduhem_equation.ipynb @@ -0,0 +1,91 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "8d6220ab", + "cell_type": "markdown", + "source": [ + "## Gibbs-Duhem equation\n", + "\n", + "In experiments, it is difficult to control *S* and *V* of a system in\n", + "comparison with their conjugate variables *T* and *-P*. It is thus\n", + "desirable to construct new functions to represent the system with *T*\n", + "and *-P* as natural variables of the functions. One of them is enthalpy\n", + "defined in , and other two can be defined as follows\n", + "\n", + "Eq. \u2011 $F = U - TS$\n", + "\n", + "Eq. \u2011 $G = U - TS + PV = \\sum_{}^{}\\mu_{i}N_{i} = H - TS = F + PV$\n", + "\n", + "with *F* and *G* called Helmholtz energy and Gibbs energy, respectively.\n", + "The middle part of is obtained using $U$ from . The corresponding\n", + "combined law of thermodynamics in terms of *H*, *F*, and *G* can be\n", + "obtained through the Legendre transformation of as\n", + "\n", + "Eq. \u2011 $dH = TdS - Vd( - P) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$\n", + "\n", + "Eq. \u2011 $dF = - SdT - PdV + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$\n", + "\n", + "Eq. \u2011 $dG = - SdT - Vd( - P) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$\n", + "\n", + "The independent variables in each of the above form are regarded as the\n", + "natural variables to the corresponding function. The integral forms of\n", + "all the functions can thus be written as the following with their\n", + "natural variables listed in the parenthesis\n", + "\n", + "Eq. \u2011 $U = U\\left( S,V,N_{i},\\xi \\right)$\n", + "\n", + "Eq. \u2011 $H = H\\left( S, - P,N_{i},\\xi \\right)$\n", + "\n", + "Eq. \u2011 $F = F\\left( T,V,N_{i},\\xi \\right)$\n", + "\n", + "Eq. \u2011 $G = G\\left( T, - P,N_{i},\\xi \\right)$\n", + "\n", + "By differentiating , one obtains\n", + "\n", + "Eq. \u2011\n", + "$dG = \\sum_{}^{}\\mu_{i}{dN}_{i} + \\sum_{}^{}{N_{i}d\\mu}_{i} = - SdT - Vd( - P) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$\n", + "\n", + "For a system at equilibrium, $Dd\\xi = 0$, re-arranging gives the\n", + "Gibbs-Duhem equation\n", + "\n", + "Eq. \u2011 $0 = - SdT - Vd( - P) - \\sum_{}^{}{N_{i}d\\mu}_{i}$\n", + "\n", + "This equation indicates that for a homogeneous system with *c*\n", + "independent components at equilibrium, there is a direct relation among\n", + "all the *c+2* potentials, and they are $c$ chemical potentials\n", + "($\\mu_{i}$), temperature, and pressure. Consequently, only *c+1*\n", + "potentials can change independently, and the remaining potential is\n", + "dependent on the other potentials. As discussed in connection with ,\n", + "there are $c + 2$ independent variables for an equilibrium system with\n", + "*c* independent components, where all of them are molar quantities.\n", + "\n", + "With the relationships between potentials and molar quantities defined\n", + "by to , one can switch between potentials and molar quantities as\n", + "natural variables of the system. For example, one can define a new free\n", + "energy function when the chemical potential of one component is\n", + "controlled from the surroundings instead of its content and obtain the\n", + "following combined first and second law of thermodynamics\n", + "\n", + "Eq. \u2011 $\\Phi = G - \\mu_{1}N_{1} = \\sum_{i = 2}^{c}{\\mu_{i}N_{i}}$\n", + "\n", + "Eq. \u2011\n", + "$d\\Phi = - SdT - Vd( - P) - N_{1}{d\\mu}_{1} + \\sum_{i = 2}^{c}{\\mu_{i}{dN}_{i}} - Dd\\xi$\n", + "\n", + "However, even though the $c + 2$ molar quantities are independent of\n", + "each other, indicates that not all the $c + 2$ potentials are\n", + "independent, i.e., if chemical potentials of all components are changed\n", + "to natural variables, one would obtain . Therefore, among the *c+2*\n", + "independent variables used to define the system, the maximum number of\n", + "independent potential is *c+1*, and at least one of the *c+2*\n", + "independent variables must be a molar quantity. This variable is usually\n", + "chosen to be the size of the system or the major element in the system.\n", + "The Gibbs-Duhem equation is used to derive Gibbs phase rule in\n", + "heterogeneous systems, which is discussed in Chapter of the book.\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/laws_of_thermodynamics/index.ipynb b/src/psu410/src/laws_of_thermodynamics/index.ipynb new file mode 100644 index 0000000..71f7bf7 --- /dev/null +++ b/src/psu410/src/laws_of_thermodynamics/index.ipynb @@ -0,0 +1,15 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "a942967d", + "cell_type": "markdown", + "source": [ + "# Laws of thermodynamics\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/laws_of_thermodynamics/stability_at_equilibrium_and_property_anomaly.ipynb b/src/psu410/src/laws_of_thermodynamics/stability_at_equilibrium_and_property_anomaly.ipynb new file mode 100644 index 0000000..687b1e7 --- /dev/null +++ b/src/psu410/src/laws_of_thermodynamics/stability_at_equilibrium_and_property_anomaly.ipynb @@ -0,0 +1,192 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "1e25979a", + "cell_type": "markdown", + "source": [ + "## Stability at equilibrium and property anomaly\n", + "\n", + "As shown by , potentials are homogenous for a homogeneous system in a\n", + "state of internal equilibrium. To study the stability of the equilibrium\n", + "state, one considers the entropy production due to a fluctuation of a\n", + "molar quantity as an internal process. Based on the second law of\n", + "thermodynamics, the driving force, as the first derivative of the\n", + "entropy production with respect to the internal process, is zero for\n", + "such a fluctuation at equilibrium, i.e. *D*=0, and the entropy of\n", + "production thus depends on the second derivative. It can be written as\n", + "follows\n", + "\n", + "Eq. \u2011\n", + "$Td_{ip}S = \\frac{\\partial_{ip}S}{\\partial\\xi}d\\xi + \\frac{1}{2}{\\frac{\\partial_{ip}^{2}S}{\\partial\\xi^{2}}(d\\xi)}^{2} = Dd\\xi - \\frac{1}{2}D_{2}(d\\xi)^{2}$\n", + "\n", + "with $D_{2} = - \\frac{\\partial_{ip}^{2}S}{\\partial\\xi^{2}}$. When\n", + "$\\frac{\\partial_{ip}^{2}S}{\\partial\\xi^{2}} < 0$ or $D_{2} > 0$ along\n", + "with $D = 0$, the fluctuation does not produce positive entropy of\n", + "production and can thus not develop further. The equilibrium state of\n", + "the system is therefore stable against the fluctuation. On the other\n", + "hand, when $\\frac{\\partial_{ip}^{2}S}{\\partial\\xi^{2}} > 0$ or\n", + "$D_{2} < 0$ along with $D = 0$, the fluctuation creates positive entropy\n", + "of production and can continue to grow. The equilibrium state of the\n", + "system is therefore unstable against the fluctuation. In connection with\n", + ", one can realize that for a system at stable equilibrium without heat\n", + "and mass exchanges with the surroundings, its entropy is at its maximum,\n", + "and there are no other internal processes, which could produce any more\n", + "entropy. This is schematically shown in .\n", + "\n", + "Figure \u2011: Schematic diagram showing maximum entropy\n", + "\n", + "Using , , and , the combined law of thermodynamics can be written as\n", + "\n", + "*Eq. 1\u201133*\n", + "$dU = \\sum_{}^{}{Y_{i}dX_{i}} - Dd\\xi + \\frac{1}{2}D_{2}(d\\xi)^{2}$\n", + "\n", + "Let us carry out the same virtual internal experiment shown in Chapter ,\n", + "i.e. moving an infinitesimal amount of *Xi* in a homogenous\n", + "system with other $X_{j}$ kept constant in an isolated system, i.e.\n", + "$dU = 0$ and $D = 0$. The internal energy change due to this internal\n", + "process is\n", + "\n", + "Eq. \u2011\n", + "$dU = \\frac{1}{2}D_{2}\\left\\{ \\left( dX_{i}^{'} \\right)^{2} + \\left( dX_{i}^{\"} \\right)^{2} \\right\\}$\n", + "\n", + "For a homogeneous system in a state of stable equilibrium with\n", + "$\\left( dX_{i}^{'} \\right)^{2} = \\left( dX_{i}^{\"} \\right)^{2} = \\left( dX_{i} \\right)^{2}$,\n", + "this internal process must result in an increase of internal energy,\n", + "$dU > 0$, and thus gives\n", + "\n", + "Eq. \u2011\n", + "$D_{2} = 2\\left( \\frac{\\partial^{2}U}{\\partial\\left( X_{i} \\right)^{2}} \\right)_{X_{j}} = 2\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}} > 0$\n", + "\n", + "shows that for a system to be stable, any pair of conjugate variables\n", + "must change in the same direction when other independent molar\n", + "quantities are kept constant. For the conjugate variables discussed so\n", + "far, it means that for a stable system, the addition of entropy\n", + "increases temperature with $\\frac{\\partial T}{\\partial S}$\\>0, the\n", + "volume decreases with pressure or increases with the negative of\n", + "pressure with $\\frac{\\partial( - P)}{\\partial V} > 0$, and the chemical\n", + "potential of a component increases with its amount, i.e.\n", + "$\\frac{\\partial\\mu_{i}}{\\partial N_{i}} > 0$, where the derivatives are\n", + "taken with all other molar quantities kept constant. The limit of\n", + "stability is reached when becomes zero, i.e.\n", + "\n", + "Eq. \u2011\n", + "$D_{2} = 2\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}} = 0$\n", + "\n", + "shows schematically the energy as a function of configurations including\n", + "three states: unstable, stable, and metastable. Both the stable and\n", + "metastable states have positive curvatures due to $D_{2} > 0$, while the\n", + "unstable state has a negative curvature due to $D_{2} < 0$. There is an\n", + "inflection point of $D_{2} = 0$ for a state between a stable or\n", + "metastable state with $D_{2} > 0$ and an unstable state with\n", + "$D_{2} < 0$. These two inflection points, called spinodal, represent the\n", + "limit of stability. The states between the two inflection points are\n", + "unstable, and other states are either stable or metastable. The two\n", + "inflection points can move apart from or close to each other depending\n", + "on the change of external conditions, i.e. the natural variables. One\n", + "extreme situation is when these two inflection points merge into one\n", + "point, and the instability occurs only at this particular point. It is\n", + "evident that all three states, stable, metastable, and unstable, also\n", + "merge into one point. This point is called critical or consolute point,\n", + "beyond which the instability no longer exists.\n", + "\n", + "Figure \u2011: Schematic diagram showing the stable and unstable equilibrium\n", + "states\n", + "\n", + "To mathematically define the consolute point, the third derivative needs\n", + "to be added to because both $D$ and $D_{2}$ vanish at this point, i.e.\n", + "\n", + "Eq. \u2011\n", + "$Td_{ip}S = \\frac{\\partial_{ip}S}{\\partial\\xi}d\\xi + \\frac{1}{2}{\\frac{\\partial_{ip}^{2}S}{\\partial\\xi^{2}}(d\\xi)}^{2} + \\frac{1}{6}{\\frac{\\partial_{ip}^{3}S}{\\partial\\xi^{3}}(d\\xi)}^{3} = Dd\\xi - \\frac{1}{2}D_{2}(d\\xi)^{2} + \\frac{1}{6}{D_{3}(d\\xi)}^{3}$\n", + "\n", + "Eq. \u2011\n", + "$dU = \\sum_{}^{}{Y_{i}dX_{i}} - Dd\\xi + \\frac{1}{2}D_{2}(d\\xi)^{2} - \\frac{1}{6}{D_{3}(d\\xi)}^{3}$\n", + "\n", + "At the consolute point, the third derivative also becomes zero, i.e.\n", + "\n", + "Eq. \u2011 $D_{3} = \\frac{\\partial_{ip}^{3}S}{\\partial\\xi^{3}}^{3} = 0$\n", + "\n", + "Let us further discuss the properties of the system in relation to the\n", + "critical point. By taking the inverse of the equation of the limit of\n", + "stability, , one obtains\n", + "\n", + "Eq. \u2011\n", + "$\\left( \\frac{\\partial X_{i}}{\\partial Y_{i}} \\right)_{X_{j}} = + \\ \\infty$\n", + "\n", + "i.e. all $X_{i}$ quantities diverge at the critical point. Therefore,\n", + "when a system approaches the critical point from its stable region, the\n", + "change of a molar quantity with respect to its conjugate potential\n", + "varies dramatically and becomes infinite at the critical point,\n", + "resulting in property anomalies in the system. In the unstable region,\n", + "the system would thus separate into stable subsystems and becomes\n", + "heterogeneous, and $X_{i}$:s change discontinuously between subsystems.\n", + "While in the stable region, the change of a molar quantity with respect\n", + "to its conjugate potential decreases as the system moves away from the\n", + "critical point and remains positive due to the stability criteria\n", + "denoted by .\n", + "\n", + "However, it is not clear how a molar quantity changes with respect to a\n", + "non-conjugate potential at the critical point. From the Maxwell\n", + "relation, one has\n", + "\n", + "Eq. \u2011\n", + "$\\left( \\frac{\\partial Y_{i}}{\\partial X_{j}} \\right)_{X_{k \\neq j}} = \\frac{\\partial^{2}U}{\\partial X_{i}\\partial X_{j}} = \\left( \\frac{\\partial Y_{j}}{\\partial X_{i}} \\right)_{X_{k \\neq i}}$\n", + "\n", + "Eq. \u2011\n", + "$\\left( \\frac{\\partial X_{j}}{\\partial Y_{i}} \\right)_{X_{k \\neq j}} = \\left( \\frac{\\partial X_{i}}{\\partial Y_{j}} \\right)_{X_{k \\neq i}}$\n", + "\n", + "Since all $X_{i}$:s diverge at the critical point, both derivatives in\n", + "should also go to infinite at the critical point. To investigate their\n", + "signs, let us carry out a virtual experiment similar to the one in\n", + "deriving the stability condition ( and ). In this case, two internal\n", + "processes are needed for moving two molar quantities simultaneously in\n", + "an isolated system, i.e.\n", + "\n", + "*Eq. 1\u201143*\n", + "$dU = - D_{\\xi_{1}}d\\xi_{1} - D_{\\xi_{2}}d\\xi_{2} + D_{\\xi_{1}\\xi_{2}}d\\xi_{1}d\\xi_{2} + \\frac{1}{2}D_{2\\xi_{1}}\\left( d\\xi_{1} \\right)^{2} + \\frac{1}{2}D_{2\\xi_{2}}\\left( d\\xi_{2} \\right)^{2}$\n", + "\n", + "Based on the above discussions, in a stable system at equilibrium with\n", + "$D_{\\xi_{1}} = D_{\\xi_{2}} = 0$, $D_{2\\xi_{1}} > 0$ and\n", + "$D_{2\\xi_{2}} > 0$, the sign of $D_{\\xi_{1}\\xi_{2}}$ cannot be\n", + "unambiguously determined in keeping the change of internal energy\n", + "positive, i.e. $dU > 0$. This indicates that the quantities in can be\n", + "either positive or negative in the stable region and become zero at the\n", + "critical point. By the same token, the quantities in can be either\n", + "positive or negative and become positive or negative infinite at the\n", + "critical point.\n", + "\n", + "A profound conclusion from this analysis is that in a stable system even\n", + "though a molar quantity always changes in the same direction as its\n", + "conjugate potential, the same molar quantity may change in the opposite\n", + "direction of a non-conjugate potential, resulting in additional\n", + "anomalies represented by Eq. 1\u201140. One example of is the thermal\n", + "expansion in a closed system, i.e. $dN_{i} = 0$, as follows\n", + "\n", + "Eq. \u2011\n", + "$\\left( \\frac{\\partial V}{\\partial T} \\right)_{S} = \\left( \\frac{\\partial S}{\\partial( - P)} \\right)_{V}$\n", + "\n", + "The left-hand side of can be understood as follows: with the increase of\n", + "temperature, the system regulates its pressure in order to keep the\n", + "entropy from increasing, which results in the volume change of the\n", + "system. The behavior of the system depends on whether the pressure\n", + "decreases or increases in order to maintain the entropy of the system\n", + "constant. If the pressure decreases to maintain the entropy of the\n", + "system constant, the volume would increase with the increase of\n", + "temperature, i.e. the left-hand side of the equation has a positive\n", + "sign, which is also shown by the right-hand side of the equation as the\n", + "changes of $S$ and $\u2013P$ have the same sign. That the volume increases\n", + "with temperature is the normal scenario. On the other hand, if the\n", + "pressure increases to maintain the entropy of the system constant, the\n", + "volume would decrease with the increase of temperature, resulting in a\n", + "negative sign for the left-hand side of the equation. This decrease of\n", + "volume with the increase of temperature is usually considered to be\n", + "anomalous, originated from the increase of entropy by the decrease of\n", + "$\u2013P$ or the increase of pressure. More discussions on entropy will\n", + "follow in Chapter 5.2.5 and Chapter 9.\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/phase_equilibria_in_heterogeneous_systems/general_condition_of_equilibrium.ipynb b/src/psu410/src/phase_equilibria_in_heterogeneous_systems/general_condition_of_equilibrium.ipynb new file mode 100644 index 0000000..f8957bb --- /dev/null +++ b/src/psu410/src/phase_equilibria_in_heterogeneous_systems/general_condition_of_equilibrium.ipynb @@ -0,0 +1,123 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "b85f238b", + "cell_type": "markdown", + "source": [ + "## General condition of equilibrium\n", + "\n", + "A system is heterogeneous when some properties have different values at\n", + "different portions of the system when the system is at equilibrium. Two\n", + "scenarios may exist where the variations of properties can be either\n", + "continuous or discontinuous. In the scenario of continuous variations,\n", + "the gradients of those variations must be coupled so that the system\n", + "remains at equilibrium. The number of independent variables is thus\n", + "reduced. These gradients must also be constrained along the boundaries\n", + "between the system and the surroundings. This type of constrained\n", + "equilibrium is not discussed in the book as it involves heterogeneous\n", + "boundary conditions between the system and the surroundings and depends\n", + "on the morphology of the system.\n", + "\n", + "In the second scenario with discontinuous variations, those properties\n", + "have different values in different portions of the system, but remain\n", + "homogenous within each portion. The system is in equilibrium as each\n", + "portion is in equilibrium with all other portions of the system. Those\n", + "homogeneous portions represent different phases in the system with the\n", + "properties in each phase being homogeneous at equilibrium. In the\n", + "previous chapter, it has been shown that all potentials are homogeneous\n", + "in a homogeneous system.\n", + "\n", + "For a heterogeneous system, the same conclusion can be obtained. If the\n", + "temperature is inhomogeneous, heat can be conducted from high\n", + "temperature locations to low temperature locations, and this process is\n", + "irreversible based on the second law of thermodynamics because it\n", + "increases the internal entropy of the system. If the pressure is\n", + "inhomogeneous, the amounts of lower molar volume phases will increase to\n", + "reduce the internal energy of the system. If the chemical potential of a\n", + "component is inhomogeneous, the chemical potential difference of the\n", + "component will drive that component to the locations with a lower\n", + "chemical potential so the internal energy of the system can be\n", + "decreased. Therefore, it can be concluded that all potentials are\n", + "homogeneous in a heterogeneous system at equilibrium, and the variables\n", + "that are not homogeneous are thus their conjugate molar quantities.\n", + "Under certain special circumstances to be discussed in later part of\n", + "this book, some molar quantities may also have the same values in\n", + "difference phases.\n", + "\n", + "In a system at equilibrium with $c$ independent components, there are\n", + "$c + 2$ pairs of conjugate variables based on or though more can be\n", + "added as shown by depending on experimental conditions. For simplicity,\n", + "most discussions in this book are limited to systems with $c + 2$ pairs\n", + "of conjugate variables unless otherwise specified, with number \u201c2\u201d\n", + "representing conjugate variables of $T - S$ and $( - P) - V$.\n", + "\n", + "For a system under constant temperature, pressure, and moles of each\n", + "independent component, the equilibrium condition derives from as\n", + "\n", + "Eq. \u2011\n", + "$dG = - SdT - Vd( - P) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi = - Dd\\xi = 0$\n", + "\n", + "Consequently, the equilibrium state is defined by the minimization of\n", + "Gibbs energy of the system at constant $T$, $P$ and $N_{i}$ because the\n", + "second derivatives need to be positive for the equilibrium system to be\n", + "stable as stipulated by . For heterogeneous systems with two or more\n", + "phases, Gibbs energy of the system is the weighted summation of Gibbs\n", + "energies of individual phases, i.e.\n", + "\n", + "Eq. \u2011 $\\frac{G}{N} = G_{m} = \\sum_{\\beta}^{}{f^{\\beta}G_{m}^{\\beta}}$\n", + "\n", + "where $f^{\\beta}$ and $G_{m}^{\\beta}$ are the mole fraction and molar\n", + "Gibbs energy of the phase $\\beta$, respectively, and the summation goes\n", + "over all phases in the system. $f^{\\beta}$ is equal to zero for phases\n", + "not present in the equilibrium state.\n", + "\n", + "The minimization of Gibbs energy of the system is carried out under the\n", + "following mass balance conditions\n", + "\n", + "Eq. \u2011\n", + "$x_{i} = \\sum_{\\beta}^{}{f^{\\beta}x_{i}^{\\beta}} = \\sum_{\\beta}^{}{f^{\\beta}\\frac{\\sum_{\\beta - t}^{}{a^{\\beta - t}\\sum_{k}^{}{i_{k}^{\\beta - t}y_{k}^{\\beta - t}}}}{\\sum_{\\beta - t}^{}{a^{\\beta - t}\\left( 1 - y_{va}^{\\beta - t} \\right)}}}$\n", + "\n", + "Eq. \u2011 $\\sum_{i}^{}x_{i} = 1$\n", + "\n", + "Eq. \u2011 $\\sum_{k}^{}y_{k}^{\\beta - t} = 1$\n", + "\n", + "where $a^{\\beta - t}$ and $y_{k}^{\\beta - t}$ are the number of site in\n", + "sublattice $t$ in the $\\beta$ phase and the corresponding site fraction\n", + "of species $k$ in the sublattice, respectively, and $i_{k}^{\\beta - t}$\n", + "is the stoichiometry of the component $i$ in the species $k$, as used in\n", + ". The summation in runs over species for each sublattice. For phases\n", + "containing ionic species, electroneutrality also needs to be maintained,\n", + "i.e. is applied to each phase. This minimization problem of Gibbs energy\n", + "under the constraint of mass conservation can be solved by means of a\n", + "range of algorithms. It should be noted that the mole fractions of\n", + "phases and site fractions of species are bounded between 0 and 1.\n", + "\n", + "This minimization procedure must result in that potentials are\n", + "homogeneous in the system as discussed above. Since the present book\n", + "deals with thermodynamics of materials, the chemical potential of each\n", + "component is of particular interest and must be homogenous in all phases\n", + "of the system at equilibrium, i.e.\n", + "\n", + "Eq. \u2011\n", + "$\\mu_{i}^{\\alpha} = \\mu_{i}^{\\beta} = \\mu_{i}^{\\gamma}\\ldots\\ldots$\n", + "\n", + "For phases in which the chemical potentials of individual components\n", + "cannot be evaluated due to stoichiometry, the combined chemical\n", + "potentials can be used to relate individual potentials as shown by and .\n", + "As proved in Chapter shown by , the chemical potential of a component in\n", + "a solution is represented by the intercept on the Gibbs energy axis by\n", + "the multi-dimensional tangent surface of Gibbs energy of the solution\n", + "plotted with respect to mole fractions of independent components. The\n", + "Gibbs energy functions of all phases in equilibrium must thus share the\n", + "same tangent surface. This is usually referred to as the common tangent\n", + "construction for phases at equilibrium. Any phase with its Gibbs energy\n", + "curve above the tangent surface is not stable under given compositions\n", + "of the system.\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/phase_equilibria_in_heterogeneous_systems/gibbs_phase_rule.ipynb b/src/psu410/src/phase_equilibria_in_heterogeneous_systems/gibbs_phase_rule.ipynb new file mode 100644 index 0000000..49b84eb --- /dev/null +++ b/src/psu410/src/phase_equilibria_in_heterogeneous_systems/gibbs_phase_rule.ipynb @@ -0,0 +1,64 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "2c9e5b25", + "cell_type": "markdown", + "source": [ + "## Gibbs phase rule\n", + "\n", + "The Gibbs-Duhem equation, i.e. , states that only $c + 1$ potentials are\n", + "independent in a homogeneous system with $c$ independent components and\n", + "the additional two variables of temperature and pressure. In a\n", + "heterogeneous system at equilibrium, this equation can be applied to\n", + "individual phases as each phase is homogeneous. Noting that each\n", + "potential has the same value in all phases at equilibrium, can be\n", + "written as follows for each individual phase, $\\ \\beta$, in the system\n", + "at equilibrium\n", + "\n", + "Eq. \u2011\n", + "$0 = - S^{\\beta}dT - V^{\\beta}d( - P) - \\sum_{}^{}{N_{i}^{\\beta}d\\mu}_{i}$\n", + "\n", + "For a system with $p$ phases at equilibrium, there are $p$ such\n", + "equations relating the potentials in the system. The number of\n", + "independent potentials thus becomes\n", + "\n", + "Eq. \u2011 $\\upsilon = c + 2 - p$\n", + "\n", + "is called Gibbs phase rule. It dictates the number of potentials that\n", + "can change independently for a given number of phases co-existing at\n", + "equilibrium, commonly called degree of freedom of the system at\n", + "equilibrium. It stipulates that the maximum number of phases can\n", + "co-exist in a system at equilibrium is obtained by setting\n", + "$\\upsilon = 0$, called an invariant equilibrium due to the zero degree\n", + "of freedom,\n", + "\n", + "Eq. \u2011 $p_{\\max} = c + 2$\n", + "\n", + "There are thus maximum three phases in a one-component system, four\n", + "phases in a binary system, five phases in a ternary system, and so on,\n", + "that can co-exist simultaneously at equilibrium with all potentials in\n", + "the system at fixed values. This should not be confused with the total\n", + "number of phases that could exist, but not co-exist in a system, which\n", + "of course are not limited by Gibbs phase rule.\n", + "\n", + "It should be emphasized that the degree of freedom, $\\upsilon$, is\n", + "referred to the number of potentials only, not to molar quantities of\n", + "the system because molar quantities are generally not homogeneous in a\n", + "heterogeneous system. For example, in a system at equilibrium with\n", + "$\\upsilon = 0$, the amount of each component can be varied, while\n", + "keeping the number of phases at $p_{\\max} = c + 2$. This can be done\n", + "through changing the amount of each phase in the system through the mass\n", + "balance equation, , without altering the composition of each phase and\n", + "thus the chemical potentials in the system. As mentioned at the\n", + "beginning of Chapter , the number of independent variables in a system\n", + "at equilibrium, i.e. the sum of independent potentials and independent\n", + "molar quantities, is $c + 2$ with the maximum number of independent\n", + "potentials determined by Gibbs phase rule, .\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/phase_equilibria_in_heterogeneous_systems/index.ipynb b/src/psu410/src/phase_equilibria_in_heterogeneous_systems/index.ipynb new file mode 100644 index 0000000..bd980a9 --- /dev/null +++ b/src/psu410/src/phase_equilibria_in_heterogeneous_systems/index.ipynb @@ -0,0 +1,15 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "f99b4808", + "cell_type": "markdown", + "source": [ + "# Phase equilibria in heterogeneous systems\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/phase_equilibria_in_heterogeneous_systems/molar_phase_diagrams.ipynb b/src/psu410/src/phase_equilibria_in_heterogeneous_systems/molar_phase_diagrams.ipynb new file mode 100644 index 0000000..bcdc3ca --- /dev/null +++ b/src/psu410/src/phase_equilibria_in_heterogeneous_systems/molar_phase_diagrams.ipynb @@ -0,0 +1,360 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "ff60c582", + "cell_type": "markdown", + "source": [ + "## Molar phase diagrams\n", + "\n", + "The potential phase diagrams discussed in Chapter present information on\n", + "which phases are in equilibrium under given values of potentials, but do\n", + "not have any information on the properties of phases in equilibrium. On\n", + "the other hand, there are direct relations between potentials and their\n", + "conjugate molar quantities for each phase at equilibrium depicted by to\n", + ". One may thus substitute the potentials by their conjugate molar\n", + "quantities in the potential phase diagrams as molar quantities provide\n", + "more information on the properties of phases and the system. This is\n", + "particularly true when chemical potentials are replaced by compositions\n", + "as the compositions of the system are often the variables controlled and\n", + "measured in experiments instead of chemical potentials.\n" + ], + "metadata": {} + }, + { + "id": "4a4a0253", + "cell_type": "markdown", + "source": [ + "### Phase diagrams with both potentials and molar quantities\n", + "\n", + "Based on Gibbs phase rule discussed in Chapters and , the\n", + "dimensionalities of phase regions in a potential phase diagram are given\n", + "by or . With every potential substituted by its conjugate molar\n", + "quantity, the dimensionalities of phase regions increase by one until\n", + "the phase region reaches the dimensionality of the phase diagram. The\n", + "axes of this phase diagram now consist of both potentials and molar\n", + "quantities. The dimensionality of a phase region can thus be represented\n", + "by the following equation based on\n", + "\n", + "Eq. \u2011\n", + "$\\upsilon_{m} = \\left( c - n_{s} \\right) + 2 - p + n_{m} \\leq \\left( c - n_{s} \\right) + 1$\n", + "\n", + "where $n_{m}$ is the number of molar axes. This equation is applicable\n", + "to phase regions with more than $n_{m} + 1$ phases. For phase regions\n", + "with $n_{m} + 1$ phases or fewer, the dimensionalities are the same as\n", + "the phase diagram, i.e. $\\upsilon_{m} = \\left( c - n_{s} \\right) + 1$,\n", + "and no longer vary with the number of molar quantity axes. When all\n", + "$\\left( c - n_{s} \\right) + 1$ potentials are substituted by their\n", + "conjugate molar quantities, one obtains a complete molar phase diagrams\n", + "to be discussed in Chapter , and all phase regions have the\n", + "same-dimensionality of $\\left( c - n_{s} \\right) + 1$.\n", + "\n", + "For the sake of graphic visualization, let us exam a two-dimensional\n", + "phase diagram of a one-component system. Topologically, it is equivalent\n", + "to a multi-component system with $n_{s} = c - 1$. In , three\n", + "two-dimensional phase diagrams are shown for pure Fe. In principle, one\n", + "can use any one of them to illustrate mixed potential and molar phase\n", + "diagrams. For the purpose of practical usefulness, one selects the\n", + "$T - ( - P)$ potential diagram as temperature and pressure are the two\n", + "typical variables controlled in experiments of one-component systems.\n", + "The conjugate molar quantities of $- P$ and $T$ are molar volume and\n", + "molar entropy, respectively. For stable phases, any pair of conjugate\n", + "variables changes in the same direction as illustrated by and , i.e. the\n", + "phase stable at higher $T$ has higher molar entropy, and the phase\n", + "stable at higher $\u2013P$, i.e. lower pressure, has higher molar volume. Let\n", + "us first substitute $\u2013P$ by $V_{m}$ as shown in a. The dimensionality of\n", + "a sing-phase region remains unchanged because of\n", + "$p = 1 < 2 = n_{m} + 1$. The dimensionality of two-phase regions is\n", + "changed from 1 to 2 due to $\\upsilon_{m} = 3 - 2 + 1 = 2$ from .\n", + "\n", + "Figure \u2011: T-Vm and Sm-(-P) phase diagrams of Fe\n", + "\n", + "As both phases in a two-phase equilibrium have the same temperature, all\n", + "tie-lines, depicted by dotted lines, are perpendicular to the\n", + "temperature axis in a. When the molar volume of the system changes from\n", + "one end of a tie-line to another end at a constant temperature, the mole\n", + "fraction of one phase increases from 0 to 1, and the mole fraction of\n", + "another phase decreases from 1 to 0. The tine-lines at various\n", + "temperatures combine together to form a two-dimensional two-phase\n", + "region. The two curves at the two ends of tie-lines represent the\n", + "boundaries between the single-phase and two-phase regions and are no\n", + "longer phase regions themselves. They are thus called phase boundaries.\n", + "\n", + "By the same token, by changing the temperature at a constant molar\n", + "volume of the system, the system will locate on different tie-lines with\n", + "the amounts of the two phases determined by the lever rule. It is thus\n", + "clear that the system maintains the two-phase equilibrium state with\n", + "both $T$ and $V_{m}$ changing independently inside the two-dimensional\n", + "two-phase region. This seems in contradiction to Gibbs phase rule being\n", + "$\\upsilon = 3 - p = 1$ from , but it is not because Gibbs phase rule\n", + "applies strictly to potential phase diagrams only, while the $T - V_{m}$\n", + "phase diagram has one of its axis being a molar quantity. As an\n", + "alternative, one may consider as a modified Gibbs phase rule in\n", + "describing the dimensionality of a phase region in a mixed potential and\n", + "molar phase diagram with $p \\geq n_{m} + 1$.\n", + "\n", + "The two three-phase equilibria in pure Fe are also represented by\n", + "tie-lines connecting all three phases. The dimensionality of three-phase\n", + "regions is $\\upsilon_{m} = 3 - 3 + 1 = 1$ from , and the three two-phase\n", + "tie-lines for a three-phase equilibrium thus overlap each other with\n", + "their three molar volumes on the same tie-line.\n", + "\n", + "Let us examine the two three-phase equilibria in more detail. In the\n", + "\u03b3/\u03b1/\u03b5 three-phase equilibrium at $T_{E} = 756.6\\ K$ and\n", + "$\\left( - P_{E} \\right) = - 1.046 \\bullet 10^{10}\\ Pa$, the molar\n", + "volumes of \u03b1, \u03b3, and \u03b5 are 6.837, 6.677, 6.582 $10^{- 6}\\ m^{3}/mol$,\n", + "respectively. There are two two-phase regions at higher temperatures and\n", + "one two-phase region at lower temperatures. This is also shown in the\n", + "potential phase diagram of a with two two-phase curves entering into and\n", + "one two-phase curve leaving from the three-phase equilibrium point with\n", + "decreasing temperature. Considering the system with the fixed molar\n", + "volume equal to that of the $\\gamma$ phase, i.e.\n", + "$V_{m}^{\\ }\\left( T_{E} \\right) = 6.677 \\bullet 10^{- 6}\\ m^{3}/mol$. At\n", + "$T > T_{E}$, the system is in the single \u03b3 phase region. With decrease\n", + "in temperature across $T_{E}$, it enters into the $\\alpha + \\varepsilon$\n", + "two-phase region. This transformation can be written as follows and is\n", + "called a eutectoid reaction\n", + "\n", + "Eq. \u2011 $\\gamma \\rightarrow \\alpha + \\varepsilon$\n", + "\n", + "This type of transformations is named as eutectic reaction if the high\n", + "temperature phase is a liquid phase. If the system molar volume is\n", + "between $V_{m}^{\\gamma}$ and $V_{m}^{\\alpha}$ at $756.6K$, with decrease\n", + "of temperature, the system first moves from the single \u03b3 phase region to\n", + "the $\\alpha + \\gamma$ two-phase region when the\n", + "$\\gamma/(\\alpha + \\gamma)$ phase boundary is crossed. When the\n", + "temperature reaches $T_{E}$, the eutectoid transformation takes place in\n", + "the remaining \u03b3 phase. The $\\alpha$ formed prior to the eutectoid\n", + "transformation is called proeutectoid $\\alpha$. By the same token, when\n", + "the system molar volume is between $V_{m}^{\\gamma}$ and\n", + "$V_{m}^{\\varepsilon}$ at $756.6K$, proeutectoid $\\varepsilon$ would form\n", + "followed by the eutectoid transformation with decrease in temperature.\n", + "\n", + "On the other hand, the $L/\\delta/\\gamma$ three-phase equilibrium at\n", + "$T_{P} = 1977.9\\ K$ and\n", + "$\\left( - P_{P} \\right) = - 5.111 \\bullet \\ 10^{9}\\ Pa$ has different\n", + "characteristics, where the subscript P will be defined shortly. There\n", + "are one two-phase equilibrium above and two two-phase equilibria below\n", + "the invariant temperature, shown in the potential phase diagram of a\n", + "with one two-phase curve entering into and two two-phase curves leaving\n", + "from the three-phase equilibrium point with decreasing temperature. The\n", + "molar volumes of $L$, $\\delta$, and $\\gamma$ at $T_{P}$ are 7.735,\n", + "7.542, and 7.498 $10^{- 6}\\ m^{3}/mol$, respectively. For $T > T_{P}$,\n", + "the two-phase region is $L + \\gamma$. If the system molar volume is\n", + "between 7.735 and 7.498 $10^{- 6}\\ m^{3}/mol$, i.e.\n", + "$V_{m}^{L}\\left( T_{P} \\right)$ and\n", + "$V_{m}^{\\gamma}\\left( T_{P} \\right)$, when the temperature reaches\n", + "$T_{P}$, $L$ and $\\gamma$ are combined to form $\\delta$ with the\n", + "transformation written as\n", + "\n", + "Eq. \u2011 $L + \\gamma \\rightarrow \\delta$\n", + "\n", + "This type of reactions is called as peritectic reaction or peritectoid\n", + "reaction when the liquid phase is replaced by a solid phase, denoted by\n", + "the subscript P. At $T < T_{P}$, one or both high temperature phases may\n", + "no longer be present in equilibrium depending on the value of the system\n", + "molar volume. For\n", + "$V_{m}^{\\ } = V_{m}^{\\delta}\\left( T_{P} \\right) = 7.542 \\bullet 10^{- 6}\\ m^{3}/mol$,\n", + "the peritectic reaction, , can come to completion with no $L$ and\n", + "$\\gamma$ left upon decreasing temperature. For\n", + "$V_{m}^{\\gamma}\\left( T_{P} \\right) = 7.498 < V_{m}^{\\ } < V_{m}^{\\delta}\\left( T_{P} \\right) = 7.542\\ \\left( 10^{- 6}\\ m^{3}/mol \\right)$,\n", + "the liquid phase is consumed, and the system enters the\n", + "$\\gamma + \\delta$ two-phase region. On the other hand, for\n", + "$V_{m}^{\\delta}\\left( T_{P} \\right) = 7.542 < V_{m}^{\\ } < V_{m}^{\\ L}\\left( T_{P} \\right) = 7.735\\ \\ \\left( 10^{- 6}\\ m^{3}/mol \\right)$,\n", + "the $\\gamma$ phase is consumed instead, and the system enters into the\n", + "$L + \\delta$ two-phase equilibrium region upon cooling.\n", + "\n", + "Let us now replace $T$ by $S_{m}$ to obtain the $( - P) - S_{m}$ phase\n", + "diagram shown in b. The morphology of this phase diagram is identical to\n", + "the $T - V_{m}$ phase diagram just discussed with all tie-lines\n", + "perpendicular to the pressure axis. The transformations at the two\n", + "three-phase equilibria with $( - P)$ decreasing or $P$ increasing are as\n", + "follows\n", + "\n", + "Eq. \u2011 $\\gamma + \\alpha \\rightarrow \\varepsilon$\n", + "\n", + "Eq. \u2011 $\\delta \\rightarrow L + \\gamma$\n", + "\n", + "To visualize two-dimensional phase diagrams of binary systems, one\n", + "usually keeps the pressure constant. One type of commonly used binary\n", + "phase diagram is the temperature-composition ($T - x$) phase diagram. As\n", + "an example, let us re-plot the $T - \\mu_{C}$ potential diagram shown in\n", + "into a $T - x_{C}$ mixed potential and molar phase diagram by replacing\n", + "the chemical potential of $C$ by its mole fraction. The $T - x_{C}$\n", + "phase diagram thus obtained is shown in . In this phase diagram there\n", + "are one peritectic reaction and two eutectic reactions as follows\n", + "\n", + "Eq. \u2011 $L + \\delta \\rightarrow \\gamma$\n", + "\n", + "Eq. \u2011 $L \\rightarrow \\gamma + C$\n", + "\n", + "Eq. 3\u201161 $\\gamma \\rightarrow \\alpha + C$\n", + "\n", + "Figure \u2011: $T - x_{C}\\ $ phase diagram of the Fe-C binary system\n", + "\n", + "In in Chapter , it was discussed the formation of miscibility gaps due\n", + "to repulsive interactions between components. One example is shown in\n", + "for the Al-Zn binary system in terms of both $T - \\mu_{Zn}$ potential\n", + "phase diagram and $\\ T - x_{Zn}\\ $ mixed potential and molar phase\n", + "diagram.\n", + "\n", + "Figure \u2011: $T - \\mu_{Zn}$ potential phase diagram (a) and\n", + "$\\ T - x_{Zn}\\ $ mixed potential and molar phase diagram (b) of the\n", + "Al-Zn binary system\n", + "\n", + "In ,there are one eutectic reaction and one eutectoid reaction as\n", + "follows\n", + "\n", + "Eq. \u2011 $L \\rightarrow fcc + hcp$\n", + "\n", + "Eq. \u2011 $fcc\\# 1 \\rightarrow fcc\\# 2 + hcp$\n", + "\n", + "The eutectoid reaction, , is also termed as monotectoid reaction because\n", + "the fcc phase appears on both sides of the reaction with different\n", + "compositions, $fcc\\# 1$ and $fcc\\# 2$, due to the miscibility gap. The\n", + "highest temperature of the miscibility gap is called the consolute point\n", + "as discussed in Chapter , which can be clearly seen in the\n", + "$T - \\mu_{Zn}$ potential phase diagram shown in . This is a critical\n", + "point, marking the limit of instability as shown in .\n", + "\n", + "When there is only one phase on either side of the reaction, i.e. both\n", + "phases have the same composition, the reaction is called a congruent\n", + "reaction. One example is shown in for the $T - x_{SiO_{2}}\\ $ mixed\n", + "potential and molar phase diagram of the CaO-SiO2\n", + "pseudo-binary system with two congruent reactions as follows\n", + "\n", + "Eq. 3\u201164 $L \\rightarrow CaSiO_{3}$\n", + "\n", + "Eq. 3\u201165 $L \\rightarrow {Ca}_{2}SiO_{4}$\n", + "\n", + "They are not invariant reactions based on the Gibbs phase rule. In , it\n", + "is noted that there are one miscibility gap in the liquid phase close to\n", + "the $SiO_{2}$ side, four eutectic reactions with one being monotectic\n", + "involving two liquid phases due to the miscibility gap, and three\n", + "peritectic reactions.\n", + "\n", + "Figure \u2011: $T - x_{SiO_{2}}\\ $ mixed potential and molar phase diagram of\n", + "the CaO-SiO2 pseudo-binary system.\n", + "\n", + "Let us generalize the above discussion to phase diagrams with\n", + "$\\left( c - n_{s} \\right) + 1$ axes. In such a phase diagram, the\n", + "maximum number of phases is given by as\n", + "$p_{\\max} = \\left( c - n_{s} \\right) + 2$. The number of phases on the\n", + "either side of an invariant reaction can vary from one phase to\n", + "$p_{\\max} - 1 = \\left( c - n_{s} \\right) + 1$ phases with the remaining\n", + "phases on the other side of the reaction, typically with the potential\n", + "decreasing from left to right. The invariant reaction with one phase on\n", + "the left of the reaction is named as eutectic, or eutectoid reaction\n", + "depending on if the phase on the left of the reaction is liquid or\n", + "solid. The rest invariant reactions are named as peritectic or\n", + "peritectoid reactions with or without a liquid phase.\n" + ], + "metadata": {} + }, + { + "id": "189dbf9e", + "cell_type": "markdown", + "source": [ + "### Phase diagrams with only molar quantities\n", + "\n", + "When all $\\left( c - n_{s} \\right) + 1$ potentials in a potential phase\n", + "diagrams are replaced by their conjugate molar quantities, one obtains a\n", + "molar phase diagram with molar quantities on all axes of the phase\n", + "diagram. For regions with the number of phases\n", + "$p \\leq n_{m} + 1 = \\left( c - n_{s} \\right) + 2 = p_{\\max}$ (see ), the\n", + "phase regions have the same-dimensionality as that of the phase diagram,\n", + "i.e. all phase regions have the same-dimensionality of\n", + "$\\left( c - n_{s} \\right) + 1$, and any geometric feature with\n", + "lower-dimensionalities, i.e. from 0 to $\\ \\left( c - n_{s} \\right)$, is\n", + "not phase regions, but phase boundaries between neighbouring phase\n", + "regions. For the sake of graphic visualization, the molar phase diagram\n", + "of pure Fe is shown in obtained by combining the two mixed phase\n", + "diagrams in .\n", + "\n", + "Figure \u2011: Molar phase diagram of Fe\n", + "\n", + "In this molar phase diagram, all one-, two-, and three-phase regions are\n", + "two-dimensional, the same as the dimensionality of the phase diagram. A\n", + "two-phase region is made up by tie-lines connecting the two phases in\n", + "equilibrium, while a three phase-region is surrounded by three two-phase\n", + "tie-lines, i.e. a tie-triangle. The amount of each phase in the\n", + "tie-triangle can be obtained using the lever rule represented by and .\n", + "As can be seen, phase boundaries between a one-phase region and a\n", + "two-phase region are one-dimensional. When the system crosses such a\n", + "phase boundary, the number of phases changes by one from two to one or\n", + "vice versa. Phase boundaries between two- and three-phase regions are\n", + "represented by two-phase tie-lines. When the system crosses such a phase\n", + "boundary, the number of phases also changes by one. The\n", + "lowest-dimensional phase boundaries are the points between one- and\n", + "three-phase regions that are zero-dimensional and the intercept of four\n", + "one-dimensional phase boundaries. When the system crosses such a phase\n", + "boundary, the number of phases changes by two.\n", + "\n", + "For multi-component systems, the phase relations cannot be directly\n", + "visualized. By representing the system equations in terms of equilibrium\n", + "conditions and level rules on a phase boundary using phases separately\n", + "from the two adjacent phase regions, Palatnik and Landau \\[4\\]\n", + "postulated that the difference between the number of unknowns and\n", + "equations gives the dimensionality of the phase boundary and derived the\n", + "following relationships\n", + "\n", + "Eq. \u2011 $D^{+} + D^{-} = r - b = c - n_{s} - b$\n", + "\n", + "where $D^{+}$ and $D^{-}$ are the numbers of phases added and removed\n", + "when the phase boundary is crossed, and $r$ and $b$ are the\n", + "dimensionalities of the phase diagram and the phase boundary,\n", + "respectively. They termed the as the contact rule, which is named as the\n", + "MPL boundary rule by Hillert \\[1\\].\n", + "\n", + "By the same token, is applicable to any phase boundary where the two\n", + "adjacent phase regions have the same-dimensionality of the phase\n", + "diagram, even in phase diagrams with a mixture of potentials and molar\n", + "quantities as the diagram axes. This can be understood because the\n", + "potentials are homogeneous in all phases in equilibrium on the phase\n", + "boundary. The phase boundary is thus equivalent to those in a complete\n", + "molar phase diagram with its number of components equal to the number of\n", + "molar axes in a mixed potential and molar phase diagrams minus one, i.e.\n", + "\n", + "Eq. \u2011 $c^{'} = n_{m} - 1 = c - n_{s}$\n", + "\n", + "The last part of stems from the discussion related to when all\n", + "$c^{'} + 1$ potentials are replaced by their conjugate molar quantities,\n", + "which is analogous to a molar phase diagram with $\\ c$ independent\n", + "components and $n_{s}$ potentials fixed.\n", + "\n", + "For a two-dimensional phase diagram with $r = 2$, the phase boundary can\n", + "be either zero- or one--dimensional. As shown in , the basic element of\n", + "a molar phase diagram is a joint of four one-dimensional phase boundary\n", + "lines. When a phase boundary line is crossed, the number of phases is\n", + "either increased or decreased by one. The joint of four one-dimensional\n", + "phase boundary lines is zero-dimensional. The number of phases differs\n", + "by two between the phase regions across the zero-dimensional joint. Two\n", + "scenarios are possible\n", + "\n", + "Two phases are added or removed, i.e. $D^{+} = 2$ and $D^{-} = 0$ or\n", + "$D^{+} = 0$ and $D^{-} = 2$, and the number of phases in the two phase\n", + "regions differs by two;\n", + "\n", + "One phase is added, one phase is removed, i.e. $D^{+} = D^{-} = 1$, and\n", + "the phase regions have the same number of phases.\n", + "\n", + "By combining the contact rules for both zero- and one-dimensional phase\n", + "boundaries, it is evident that the above two scenarios co-exist in a\n", + "joint of four-phase regions with two-phase regions following the first\n", + "scenario and other two-phase regions following the second scenario.\n", + "Based on the Schreinemakers\u2019 rule generalized by Hillert \\[5\\], each of\n", + "the two-phase regions with the same number of phases contains one\n", + "extrapolation of the phase boundaries, while the other two-phase regions\n", + "contain either zero or two extrapolations of the phase boundaries. This\n", + "can be observed for all the zero-dimensional phase boundaries in and is\n", + "further schematically illustrated in for general cases.\n", + "\n", + "Figure \u2011: Schematic molar phase diagram, demonstrating the\n", + "Schreinemakers\u2019 rule\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/phase_equilibria_in_heterogeneous_systems/potential_phase_diagrams.ipynb b/src/psu410/src/phase_equilibria_in_heterogeneous_systems/potential_phase_diagrams.ipynb new file mode 100644 index 0000000..094b6d2 --- /dev/null +++ b/src/psu410/src/phase_equilibria_in_heterogeneous_systems/potential_phase_diagrams.ipynb @@ -0,0 +1,678 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "33603d36", + "cell_type": "markdown", + "source": [ + "## Potential phase diagrams\n", + "\n", + "Gibbs phase rule can be further understood through the\n", + "$c + 2$-dimensional space of potentials consisting of $T$, $- P$, and\n", + "$\\mu_{i}$ with $i$ from 1 to $c$. Each phase is a $c + 1$-dimensional\n", + "feature in this $c + 2$-dimensional space characterized by . The\n", + "directions of this $c + 1$-dimensional feature are represented by their\n", + "molar quantities as shown by following equations,\n", + "\n", + "Eq. \u2011\n", + "$\\ \\left( \\frac{\\partial\\mu_{i}}{\\partial T} \\right)_{P,\\mu_{j \\neq i}} = - \\frac{S^{\\beta}}{N_{i}^{\\beta}} = - \\frac{S_{m}^{\\beta}}{x_{i}^{\\beta}}$\n", + "\n", + "Eq. \u2011\n", + "$\\ \\left( \\frac{\\partial\\mu_{i}}{\\partial( - P)} \\right)_{T,\\mu_{j \\neq i}} = - \\frac{V^{\\beta}}{N_{i}^{\\beta}} = - \\frac{V_{m}^{\\beta}}{x_{i}^{\\beta}}$\n", + "\n", + "Eq. \u2011\n", + "$\\ \\left( \\frac{\\partial( - P)}{\\partial T} \\right)_{\\mu_{i}} = - \\frac{S^{\\beta}}{V^{\\beta}} = - \\frac{S_{m}^{\\beta}}{V_{m}^{\\beta}}$\n", + "\n", + "As can be seen, all the direction derivatives are negative, indicating\n", + "that the $c + 1$-dimensional feature is convex. The intercept of any two\n", + "$c + 1$-dimensional features is thus a $c$-dimensional feature. On this\n", + "$c$-dimensional feature, these two phases are in equilibrium with each\n", + "other because each potential has the same value in both phases. This\n", + "feature thus represents a two-phase equilibrium. By the same token, the\n", + "intercept of any three $c + 1$-dimensional features is a\n", + "$c - 1$-dimensional feature in the $c + 2$-dimensional space of\n", + "potentials and represents a three-phase equilibrium. This continues\n", + "until the number of phases reaches $c + 2$ with all $c + 2$ potentials\n", + "completely determined, and the dimension of their intercepts becomes\n", + "zero.\n", + "\n", + "Those $c + 1$ to zero-dimensional geometrical features in the\n", + "$c + 2$-dimensional space of potentials thus denote one-phase,\n", + "two-phase, three-phase to $(c + 2)$-phase equilibria of the system with\n", + "the dimensionality of the feature and the number of phases in\n", + "equilibrium related by , i.e. Gibbs phase rule. Their arrangements in\n", + "the $c + 2$-dimensional space of potentials thus depict the phase\n", + "relations in the system and are commonly called phase diagrams. Since\n", + "all the diagram axes in the phase diagram discussed above are\n", + "potentials, the diagram is called potential phase diagram in order to\n", + "differentiate it from phase diagrams with some or all diagram axes being\n", + "their conjugate molar quantities. Both potential and molar phase\n", + "diagrams are discussed in this chapter.\n" + ], + "metadata": {} + }, + { + "id": "fa0ef81c", + "cell_type": "markdown", + "source": [ + "### Potential phase diagrams of one-component systems\n", + "\n", + "As physical vision of human being is limited to three-dimensions, only\n", + "one-component system can be completely visualized as shown in for one\n", + "phase where any two of the three potentials can change independently.\n", + "From Gibbs phase rule, when two phases are in equilibrium, only one\n", + "potential can vary freely if the two-phase equilibrium is to be\n", + "maintained. While three phases are in equilibrium, the degree of freedom\n", + "is zero, and all three potentials are fixed.\n", + "\n", + "For a two-phase equilibrium, two surfaces intersect each other as\n", + "depicted by the dashed line in . This two-phase equilibrium line is\n", + "obtained by applying to both phases in a one-component system. Since one\n", + "of the potentials is dependent on the other two, one can eliminate it by\n", + "dividing the equation by its conjugate molar quantity and subtracting\n", + "the two equations, resulting in the following three equations\n", + "\n", + "Eq. \u2011\n", + "$0 = \\left( \\frac{S^{\\alpha}}{N_{A}^{\\alpha}} - \\frac{S^{\\beta}}{N_{A}^{\\beta}} \\right)dT + \\left( \\frac{V^{\\alpha}}{N_{A}^{\\alpha}} - \\frac{V^{\\beta}}{N_{A}^{\\beta}} \\right)d( - P) = \\mathrm{\\Delta}S_{m}^{\\alpha\\beta}dT + \\mathrm{\\Delta}V_{m}^{\\alpha\\beta}d( - P)$\n", + "\n", + "Eq. \u2011\n", + "$0 = \\left( \\frac{S^{\\alpha}}{V^{\\alpha}} - \\frac{S^{\\beta}}{V^{\\beta}} \\right)dT + \\left( \\frac{N_{A}^{\\alpha}}{V^{\\alpha}} - \\frac{N_{A}^{\\beta}}{V^{\\beta}} \\right){d\\mu}_{A} = \\mathrm{\\Delta}\\left( \\frac{S_{m}}{V_{m}} \\right)^{\\alpha\\beta}dT + \\mathrm{\\Delta}\\left( \\frac{1}{V_{m}} \\right)^{\\alpha\\beta}{d\\mu}_{A}$\n", + "\n", + "Eq. \u2011\n", + "$0 = \\left( \\frac{V^{\\alpha}}{S^{\\alpha}} - \\frac{V^{\\beta}}{S^{\\beta}} \\right)d( - P) + \\left( \\frac{N_{A}^{\\alpha}}{S^{\\alpha}} - \\frac{N_{A}^{\\beta}}{S^{\\beta}} \\right){d\\mu}_{A} = \\mathrm{\\Delta}\\left( \\frac{V_{m}}{S_{m}} \\right)^{\\alpha\\beta}d( - P) + \\mathrm{\\Delta}\\left( \\frac{1}{S_{m}} \\right)^{\\alpha\\beta}{d\\mu}_{A}$\n", + "\n", + "Figure \u2011: Gibbs energy surfaces of two phases and their intersection,\n", + "representing the two-phase equilibrium.\n", + "\n", + "The directions of the two-phase equilibrium line can thus be obtained as\n", + "\n", + "Eq. \u2011\n", + "$\\frac{d( - P)}{dT} = - \\frac{\\mathrm{\\Delta}S_{m}^{\\alpha\\beta}}{\\mathrm{\\Delta}V_{m}^{\\alpha\\beta}}$\n", + "\n", + "Eq. \u2011\n", + "$\\frac{dT}{{d\\mu}_{A}} = - \\frac{\\mathrm{\\Delta}\\left( \\frac{1}{V_{m}} \\right)^{\\alpha\\beta}}{\\mathrm{\\Delta}\\left( \\frac{S_{m}}{V_{m}} \\right)^{\\alpha\\beta}}$\n", + "\n", + "Eq. \u2011\n", + "$\\frac{d( - P)}{{d\\mu}_{A}} = - \\frac{\\mathrm{\\Delta}\\left( \\frac{1}{S_{m}} \\right)^{\\alpha\\beta}}{\\mathrm{\\Delta}\\left( \\frac{V_{m}}{S_{m}} \\right)^{\\alpha\\beta}}$\n", + "\n", + "These three equations define the mathematical forms of the two-phase\n", + "equilibrium line in the two-dimensional spaces of $T - ( - P)$,\n", + "$\\mu_{A} - T$ and $\\mu_{A} - ( - P)$, respectively, and can thus be\n", + "plotted as two-dimensional diagrams. is commonly called\n", + "Clausius-Clapeyron equation in the literature. One may thus call all\n", + "three equations above as generalized Clausius-Clapeyron equations. At\n", + "equilibrium, the chemical potentials of the component in both phases are\n", + "equal to each other, so are their Gibbs energies. One thus has\n", + "\n", + "Eq. \u2011\n", + "$G_{m}^{\\alpha} - G_{m}^{\\beta} = 0 = \\mathrm{\\Delta}G_{m} = \\mathrm{\\Delta}H_{m}^{\\alpha\\beta} - T\\mathrm{\\Delta}S_{m}^{\\alpha\\beta}$\n", + "\n", + "The Clausius-Clapeyron equation, , can be re-written as\n", + "\n", + "Eq. \u2011\n", + "$\\frac{d( - P)}{dT} = - \\frac{\\mathrm{\\Delta}H_{m}^{\\alpha\\beta}}{T\\mathrm{\\Delta}V_{m}^{\\alpha\\beta}}$\n", + "\n", + "As an example, three potential phase diagrams of pure Fe are shown in .\n", + "There are four phases in the system, bcc, fcc, hcp, and liquid. In the\n", + "literature, the high temperature and low temperature bcc phases are\n", + "usually denoted by \u03b4 (high temperature) and \u03b1 (low temperature), the fcc\n", + "and hcp phases by \u03b3 and \u03b5, and the liquid phase by L, respectively. In\n", + "these figures, the two-dimensional areas are single-phase regions where\n", + "two potentials can change independently with the system remaining as\n", + "single-phase. The lines denote two-phase equilibrium regions where only\n", + "one potential can vary independently if the two-phase equilibrium is to\n", + "be maintained. The points where three two-phase equilibrium lines meet\n", + "represent the invariant three-phase equilibria with three potentials\n", + "fixed.\n", + "\n", + "Figure \u2011 $T - ( - P)$, $\\mu_{A} - T$ and $\\mu_{A} - ( - P)$ phase\n", + "diagrams of pure Fe\n", + "\n", + "Based on the discussions in Chapter , enthalpy and entropy of a phase\n", + "increase monotonically with temperature, and phases stable at higher\n", + "temperatures have higher enthalpy and entropy than phases stable at\n", + "lower temperatures. Consequently, the two-phase equilibrium lines in a\n", + "$T - ( - P)$ potential phase diagram have negative slopes if the phase\n", + "stable at higher temperatures also has larger molar volume than the\n", + "phase stable at lower temperatures (note that if $P$ is plotted instead\n", + "of $- P$, the slope is positive). This is the case for the two-phase\n", + "equilibrium lines of \u03b4/L, \u03b3/L, and \u03b3/\u03b4 at high temperatures, and \u03b5/\u03b3\n", + "shown in a. On the other hand, the two-phase equilibrium lines of \u03b1/\u03b5\n", + "and \u03b1/\u03b3 at low temperatures have positive slopes, indicating that \u03b5 and\n", + "\u03b3 have smaller molar volume than \u03b1 as \u03b5 and \u03b3 are more stable at higher\n", + "pressures than \u03b1 at constant temperatures. It is thus evident that the\n", + "phase stable at higher pressure can have either higher or lower entropy\n", + "than the phase stable at lower pressure, and the phase stable at higher\n", + "temperature can have either higher or lower volume than the phase stable\n", + "at lower temperature. This is the property anomaly discussed in Chapter\n", + ".\n", + "\n", + "Another useful example of potential phase diagram is the\n", + "pressure-temperature phase diagram of $H_{2}O$ shown in with three\n", + "phases: ice, water, and vapour. It is known that the solid ice has many\n", + "polymorphic structures at high pressures, which are not included in this\n", + "diagram. As in the pure Fe potential phase diagram discussed above, the\n", + "single-phase regions of ice, water, and vapour are represented by the\n", + "two-dimensional areas with two-degree of freedom based on the Gibbs\n", + "phase rule, the lines are for the two-phase regions of ice-water,\n", + "ice-vapour, and water-vapour, and the three-phase equilibrium has\n", + "zero-degree of freedom represented by a point at $273.16\\ K$ and\n", + "$611.73\\ Pa$.\n", + "\n", + "There are two features in which are different from those of Fe shown in\n", + "a. The first feature is that the slope of the liquid-solid two-phase\n", + "equilibrium line in has the opposite sign of that in a. This is because\n", + "solid ice has larger molar volume than liquid water, while the molar\n", + "volume of liquid Fe is larger than those of fcc-Fe and bcc-Fe. The\n", + "second feature is that the two-phase equilibrium line of water-vapour\n", + "ends at $647\\ K$ and $22.064 \\cdot 10^{6}\\ Pa$. Beyond this point, the\n", + "difference between vapour and water disappears when the pressure and\n", + "temperature are changed, i.e. it behaves like one phase. This point is a\n", + "critical point, as discussed in Chapter . However, it should be pointed\n", + "out that it does not represent an invariant reaction as the degree of\n", + "freedom based on the Gibbs phase rule is equal to one and not zero. On\n", + "the other hand, both the temperature and pressure of the critical point\n", + "are invariant due to the two constraints introduced by the limit of\n", + "stability of a single phase, i.e. the second and third derivatives of\n", + "temperature to entropy or pressure to volume are zero.\n", + "\n", + "Figure \u2011: P-T phase diagram of $H_{2}O$\n" + ], + "metadata": {} + }, + { + "id": "7b01cdb0", + "cell_type": "markdown", + "source": [ + "### Potential phase diagrams of two-component systems\n", + "\n", + "From Gibbs-Duhem equation (see ), a single phase in a two-component\n", + "system has three independent potentials, out of the four potentials of\n", + "$T$, $- P$, $\\mu_{A}$, and $\\mu_{B}$, and is a three-dimensional\n", + "geometric feature in a four-dimensional space. Alternatively, it can be\n", + "represented by a three-dimensional space constructed by the three\n", + "independent potentials. A two-phase equilibrium is thus a\n", + "two-dimensional surface in this three-dimensional space created by the\n", + "intercept of two three-dimensional spaces, and a three-phase equilibrium\n", + "is a one-dimensional line, and a four-phase equilibrium is a\n", + "zero-dimensional point. This is shown in for the Fe-C binary system\n", + "involving four phases: fcc, bcc, Fe3C and graphite. Since any\n", + "one of the four potentials can be chosen as the dependent one, four\n", + "three-dimensional potential phase diagrams are depicted in .\n", + "\n", + "Figure \u2011: Projected potential phase diagram of Fe-C system with fcc,\n", + "bcc, Fe3C, and graphite\n", + "\n", + "The two-phase equilibrium surfaces are obtained by choosing any one of\n", + "the four potentials as the dependent one and solving the Gibbs-Duhem\n", + "equations for both phases, resulting in following four equations\n", + "\n", + "Eq. \u2011\n", + "$0 = \\left( \\frac{S^{\\alpha}}{N_{A}^{\\alpha}} - \\frac{S^{\\beta}}{N_{A}^{\\beta}} \\right)dT + \\left( \\frac{V^{\\alpha}}{N_{A}^{\\alpha}} - \\frac{V^{\\beta}}{N_{A}^{\\beta}} \\right)d( - P) + \\left( \\frac{N_{B}^{\\alpha}}{N_{A}^{\\alpha}} - \\frac{N_{B}^{\\beta}}{N_{A}^{\\beta}} \\right){d\\mu}_{B} = \\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}dT + \\mathrm{\\Delta}V_{mA}^{\\alpha\\beta}d( - P) + \\mathrm{\\Delta}z_{B}^{\\alpha\\beta}{d\\mu}_{B}$\n", + "\n", + "Eq. \u2011\n", + "$0 = \\left( \\frac{S^{\\alpha}}{N_{B}^{\\alpha}} - \\frac{S^{\\beta}}{N_{B}^{\\beta}} \\right)dT + \\left( \\frac{V^{\\alpha}}{N_{B}^{\\alpha}} - \\frac{V^{\\beta}}{N_{B}^{\\beta}} \\right)d( - P) + \\left( \\frac{N_{B}^{\\alpha}}{N_{B}^{\\alpha}} - \\frac{N_{B}^{\\beta}}{N_{B}^{\\beta}} \\right){d\\mu}_{B} = \\mathrm{\\Delta}S_{mB}^{\\alpha\\beta}dT + \\mathrm{\\Delta}V_{mB}^{\\alpha\\beta}d( - P) + \\mathrm{\\Delta}z_{A}^{\\alpha\\beta}{d\\mu}_{A}$\n", + "\n", + "Eq. \u2011\n", + "$0 = \\left( \\frac{S^{\\alpha}}{V^{\\alpha}} - \\frac{S^{\\beta}}{V^{\\beta}} \\right)dT + \\left( \\frac{N_{A}^{\\alpha}}{V^{\\alpha}} - \\frac{N_{A}^{\\beta}}{V^{\\beta}} \\right){d\\mu}_{A} + \\left( \\frac{N_{B}^{\\alpha}}{V^{\\alpha}} - \\frac{N_{B}^{\\beta}}{V^{\\beta}} \\right){d\\mu}_{B} = \\mathrm{\\Delta}\\left( \\frac{S_{m}}{V_{m}} \\right)^{\\alpha\\beta}dT + \\mathrm{\\Delta}\\left( \\frac{1}{V_{mA}} \\right)^{\\alpha\\beta}{d\\mu}_{A} + \\mathrm{\\Delta}\\left( \\frac{1}{V_{mB}} \\right)^{\\alpha\\beta}{d\\mu}_{B}$\n", + "\n", + "Eq. \u2011\n", + "$0 = \\left( \\frac{V^{\\alpha}}{S^{\\alpha}} - \\frac{V^{\\beta}}{S^{\\beta}} \\right)d( - P) + \\left( \\frac{N_{A}^{\\alpha}}{S^{\\alpha}} - \\frac{N_{A}^{\\beta}}{S^{\\beta}} \\right){d\\mu}_{A} + \\left( \\frac{N_{B}^{\\alpha}}{S^{\\alpha}} - \\frac{N_{B}^{\\beta}}{S^{\\beta}} \\right){d\\mu}_{B} = \\mathrm{\\Delta}\\left( \\frac{V_{m}}{S_{m}} \\right)^{\\alpha\\beta}d( - P) + \\mathrm{\\Delta}\\left( \\frac{1}{S_{mA}} \\right)^{\\alpha\\beta}{d\\mu}_{A} + \\mathrm{\\Delta}\\left( \\frac{1}{S_{mB}} \\right)^{\\alpha\\beta}{d\\mu}_{B}$\n", + "\n", + "A three-phase equilibrium line is represented by the intercept of two\n", + "two-phase surfaces by applying any one of the above four equations to\n", + "two two-phase equilibria. Let us use as an example\n", + "\n", + "Eq. \u2011\n", + "$0 = \\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}dT + \\mathrm{\\Delta}V_{mA}^{\\alpha\\beta}d( - P) + {\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}{d\\mu}_{B}$\n", + "\n", + "Eq. \u2011\n", + "$0 = \\mathrm{\\Delta}S_{mA}^{\\alpha\\gamma}dT + \\mathrm{\\Delta}V_{mA}^{\\alpha\\gamma}d( - P) + \\mathrm{\\Delta}{z_{B}^{\\alpha\\gamma}d\\mu}_{B}$\n", + "\n", + "It is self-evident that the two-phase equilibrium surface between\n", + "$\\beta$ and $\\gamma$ is not independent and can be obtained by\n", + "subtraction of and\n", + "\n", + "Eq. \u2011\n", + "$0 = \\left( \\mathrm{\\Delta}S_{mA}^{\\alpha\\beta} - \\mathrm{\\Delta}S_{mA}^{\\alpha\\gamma} \\right)dT + \\left( \\mathrm{\\Delta}V_{mA}^{\\alpha\\beta} - \\mathrm{\\Delta}V_{mA}^{\\alpha\\gamma} \\right)d( - P) + \\left( {\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta} - \\mathrm{\\Delta}z_{B}^{\\alpha\\gamma} \\right){d\\mu}_{B} = \\mathrm{\\Delta}S_{mA}^{\\gamma\\beta}dT + \\mathrm{\\Delta}V_{mA}^{\\gamma\\beta}d( - P) + {\\mathrm{\\Delta}z}_{B}^{\\gamma\\beta}{d\\mu}_{B}$\n", + "\n", + "Eliminating one of three potentials in and , one can obtain three\n", + "equations for the three-phase equilibrium line\n", + "\n", + "Eq. \u2011\n", + "$\\frac{d( - P)}{dT} = - \\frac{\\frac{\\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}}{{\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}S_{mA}^{\\alpha\\gamma}}{\\mathrm{\\Delta}z_{B}^{\\alpha\\gamma}}}{\\frac{\\mathrm{\\Delta}V_{mA}^{\\alpha\\beta}}{{\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}V_{mA}^{\\alpha\\gamma}}{\\mathrm{\\Delta}z_{B}^{\\alpha\\gamma}}}$\n", + "\n", + "Eq. \u2011\n", + "$\\frac{dT}{{d\\mu}_{B}} = - \\frac{\\frac{{\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}}{\\mathrm{\\Delta}V_{mA}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}z_{B}^{\\alpha\\gamma}}{\\mathrm{\\Delta}V_{mA}^{\\alpha\\gamma}}}{\\frac{\\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}}{\\mathrm{\\Delta}V_{mA}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}S_{mA}^{\\alpha\\gamma}}{\\mathrm{\\Delta}V_{mA}^{\\alpha\\gamma}}}$\n", + "\n", + "Eq. \u2011\n", + "$\\frac{d( - P)}{{d\\mu}_{B}} = - \\frac{\\frac{{\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}}{\\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}z_{B}^{\\alpha\\gamma}}{\\mathrm{\\Delta}S_{mA}^{\\alpha\\gamma}}}{\\frac{\\mathrm{\\Delta}V_{mA}^{\\alpha\\beta}}{\\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}V_{mA}^{\\alpha\\gamma}}{\\mathrm{\\Delta}S_{mA}^{\\alpha\\gamma}}}$\n", + "\n", + "to can be referred as generalized Clausius-Clapeyron equations for\n", + "binary systems. Similar equations can be derived for\n", + "$T - ( - P) - \\mu_{A}$, $T - \\mu_{A} - \\mu_{B}$, and\n", + "$( - P) - \\mu_{A} - \\mu_{B}$ potential phase diagrams from to , and are\n", + "listed below\n", + "\n", + "- Generalized Clausius-Clapeyron equations for a three-phase equilibrium\n", + " in $T - ( - P) - \\mu_{A}$ potential phase diagrams\n", + "\n", + "Eq. \u2011\n", + "$\\frac{d( - P)}{dT} = - \\frac{\\frac{\\mathrm{\\Delta}S_{mB}^{\\alpha\\beta}}{{\\mathrm{\\Delta}z}_{A}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}S_{mB}^{\\alpha\\gamma}}{\\mathrm{\\Delta}z_{A}^{\\alpha\\gamma}}}{\\frac{\\mathrm{\\Delta}V_{mB}^{\\alpha\\beta}}{{\\mathrm{\\Delta}z}_{A}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}V_{mB}^{\\alpha\\gamma}}{\\mathrm{\\Delta}z_{A}^{\\alpha\\gamma}}}$\n", + "\n", + "Eq. \u2011\n", + "$\\frac{dT}{{d\\mu}_{A}} = - \\frac{\\frac{{\\mathrm{\\Delta}z}_{A}^{\\alpha\\beta}}{\\mathrm{\\Delta}V_{mB}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}z_{A}^{\\alpha\\gamma}}{\\mathrm{\\Delta}V_{mB}^{\\alpha\\gamma}}}{\\frac{\\mathrm{\\Delta}S_{mB}^{\\alpha\\beta}}{\\mathrm{\\Delta}V_{mB}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}S_{mB}^{\\alpha\\gamma}}{\\mathrm{\\Delta}V_{mB}^{\\alpha\\gamma}}}$\n", + "\n", + "Eq. \u2011\n", + "$\\frac{d( - P)}{{d\\mu}_{A}} = - \\frac{\\frac{{\\mathrm{\\Delta}z}_{A}^{\\alpha\\beta}}{\\mathrm{\\Delta}S_{mB}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}z_{A}^{\\alpha\\gamma}}{\\mathrm{\\Delta}S_{mB}^{\\alpha\\gamma}}}{\\frac{\\mathrm{\\Delta}V_{mB}^{\\alpha\\beta}}{\\mathrm{\\Delta}S_{mB}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}V_{mB}^{\\alpha\\gamma}}{\\mathrm{\\Delta}S_{mB}^{\\alpha\\gamma}}}$\n", + "\n", + "- Generalized Clausius-Clapeyron equations for a three-phase equilibrium\n", + " in $\\ T - \\mu_{A} - \\mu_{B}$ potential phase diagrams\n", + "\n", + "Eq. \u2011\n", + "$\\frac{dT}{{d\\mu}_{A}} = - \\frac{\\frac{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}}{\\frac{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}}$\n", + "\n", + "Eq. \u2011\n", + "$\\frac{dT}{{d\\mu}_{B}} = - \\frac{\\frac{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}}{\\frac{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}}$\n", + "\n", + "Eq. \u2011\n", + "$\\frac{{d\\mu}_{A}}{{d\\mu}_{B}} = - \\frac{\\frac{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\gamma}}}{\\frac{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\gamma}}}$\n", + "\n", + "- Generalized Clausius-Clapeyron equations for a three-phase equilibrium\n", + " in $( - P) - \\mu_{A} - \\mu_{B}$ potential phase diagrams\n", + "\n", + "Eq. \u2011\n", + "$\\frac{d( - P)}{{d\\mu}_{A}} = - \\frac{\\frac{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}}{\\frac{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}}$\n", + "\n", + "Eq. \u2011\n", + "$\\frac{d( - P)}{{d\\mu}_{B}} = - \\frac{\\frac{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}}{\\frac{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}}$\n", + "\n", + "Eq. \u2011\n", + "$\\frac{{d\\mu}_{A}}{{d\\mu}_{B}} = - \\frac{\\frac{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\gamma}}}{\\frac{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\gamma}}}$\n" + ], + "metadata": {} + }, + { + "id": "9749bc6c", + "cell_type": "markdown", + "source": [ + "### Projection and section of phase diagrams with potential and molar quantities\n", + "\n", + "As discussed in Chapter , projections of high-dimensional phase diagrams\n", + "usually cannot keep all information. However, there is one type of\n", + "widely used projection in the literature, i.e. the liquidus surface in\n", + "ternary systems under constant pressure with temperature and mole\n", + "fractions of two components as its axes. The projection along the\n", + "temperature axis reveals the composition regions for primary phases that\n", + "solidify from liquid upon cooling. These regions are separated by\n", + "univariant lines of three-phase equilibria. The projections along one of\n", + "the two mole fractions show the temperature as a function of composition\n", + "on the univariant three-phase equilibrium lines and also depict whether\n", + "a four-phase equilibrium is peritectic or eutectic. There are four\n", + "scenarios for the three univariant three-phase equilibrium lines to meet\n", + "at the four-phase equilibrium as depicted in and discussed individually\n", + "below.\n", + "\n", + "Figure \u2011: Schematic four options for three univariant three-phase\n", + "equilibrium lines to meet at the invariant four-phase equilibrium\n", + "\n", + "The first scenario is that with decreasing temperature, all three\n", + "univariant lines merge into the four-phase equilibrium. It indicates\n", + "that the liquid phase does not exist at temperatures below the\n", + "four-phase invariant reaction. This invariant reaction is thus a ternary\n", + "eutectic reaction with liquid completely transformed to three solid\n", + "phases upon cooling, i.e.\n", + "\n", + "Eq. \u2011 $L \\rightarrow \\alpha + \\beta + \\gamma$\n", + "\n", + "In the second scenario, two univariant lines merge into and one leaves\n", + "from the four-phase equilibrium with decreasing temperature. This means\n", + "that one solid phase at higher temperature is no longer stable at lower\n", + "temperature, and it must react with the liquid phase to form the\n", + "remaining two solid phases. The four-phase invariant reaction is thus\n", + "peritectic. The solid phase common to both univariant lines at high\n", + "temperatures reacts with the liquid phase. Assuming that this phase is\n", + "$\\alpha$, the four-phase invariant reaction becomes\n", + "\n", + "Eq. \u2011 $L + \\alpha \\rightarrow \\beta + \\gamma$\n", + "\n", + "In the third scenario, one univariant line points to and two leave from\n", + "the four-phase equilibrium with decreasing temperature. A new phase\n", + "forms at low temperatures from the three high temperature phases, e.g.\n", + "liquid, $\\alpha$, and $\\beta$, with the four-phase invariant reaction as\n", + "\n", + "Eq. \u2011 $L + \\alpha + \\beta \\rightarrow \\gamma$\n", + "\n", + "The fourth scenario is the inverse of the first scenario, indicating the\n", + "formation of liquid from solid phases upon cooling, i.e.\n", + "\n", + "Eq. \u2011 $\\alpha + \\beta + \\gamma \\rightarrow L$\n", + "\n", + "This case has not been observed in reality.\n", + "\n", + "As an example, the liquidus projections of the Al-Fe-Si ternary system\n", + "are shown in in two formats\\[6\\], i.e. (a) three-dimensional liquidus\n", + "surface with the isotherms showing the liquidus contours; (b)\n", + "conventional projection to the composition axis with the temperature\n", + "decrease shown by arrows; (c) projection to the temperature and weight\n", + "fraction of Si. The first to third scenarios of invariant reactions\n", + "discussed above can clearly be identified and listed in . It is evident\n", + "that c provides the easiest route to visualize the type of invariant\n", + "reactions as show by .\n", + "\n", + "Figure \u2011: Liquidus of the Al-Fe-Si ternary system\\[6\\], (a)\n", + "three-dimensional presentation of the liquidus; (b) projection to the\n", + "composition triangle with isotherms (dotted lines) superimposed and\n", + "their temperatures indicated close to the horizontal axis.\n", + "\n", + "Table \u2011: Invariant liquidus reactions of the Al-Fe-Si ternary system\n", + "with the composition of the liquid phase \\[6\\]\n", + "\n", + "| Reaction | T, \u00b0C | wFe, % | wSi, % |\n", + "|----------|-------|-------------------|-------------------|\n", + "| | 1178 | 77.9 | 21.0 |\n", + "| | 1155 | 49.0 | 0.16 |\n", + "| | 1127 | 52.7 | 2.81 |\n", + "| -H-L | 1076 | 41.6 | 44.0 |\n", + "| | 1073 | 51.0 | 7.19 |\n", + "| | 1050 | 53.8 | 18.4 |\n", + "| -H-L | 1019 | 34.8 | 44.4 |\n", + "| | 1004 | 49.0 | 12.6 |\n", + "| | 1000 | 46.7 | 13.2 |\n", + "| -L | 940 | 37.9 | 32.0 |\n", + "| | 921 | 33.7 | 20.1 |\n", + "| | 899 | 33.8 | 32.2 |\n", + "| | 884 | 30.8 | 26.0 |\n", + "| -L | 877 | 29.5 | 35.2 |\n", + "| | 851 | 23.3 | 21.6 |\n", + "| | 834 | 22.2 | 31.7 |\n", + "| | 825 | 22.1 | 25.7 |\n", + "| | 823 | 21.8 | 25.4 |\n", + "| | 715 | 6.64 | 10.8 |\n", + "| | 694 | 6.11 | 17.1 |\n", + "| | 680 | 4.68 | 11.6 |\n", + "| | 630 | 2.11 | 4.10 |\n", + "| | 616 | 1.76 | 6.56 |\n", + "| | 598 | 1.22 | 14.3 |\n", + "| | 575 | 0.73 | 12.7 |\n", + "\n", + "In contrast to projections, sectioning is used more often to understand\n", + "phase relations in multi-component systems. Sectioning of a potential\n", + "phase diagrams is relatively simple as the resulted phase diagram\n", + "behaves like a system with one component less. The same is true if\n", + "potentials are sectioned in phase diagrams with both potential and molar\n", + "quantities as the section is along the tie-lines of the fixed\n", + "potentials. As an example, shows the ternary Al-Fe-Si potential and\n", + "molar phase diagrams sectioned at T=1273K and P=1atm, commonly referred\n", + "to as isothermal section. It is evident that the geometric features of\n", + "both phase diagrams are identical to those of pure Fe shown in and ,\n", + "respectively, with one-, two-, and three-phase regions and corresponding\n", + "phase boundaries.\n", + "\n", + "Figure \u2011: Ternary isothermal section of the Al-Fe-Si ternary system at\n", + "T=1273K and P=1atm\n", + "\n", + "On the other hand, when the phase diagram is sectioned along a molar\n", + "quantity, it would usually not follow a tie-line because phases in\n", + "equilibrium usually have different values for the same molar quantity.\n", + "Consequently, there are no tie-lines inside such phase diagrams in\n", + "general, and any phase regions only show which phases are in equilibrium\n", + "with each other without any information on the values of molar\n", + "quantities of individual phases.\n", + "\n", + "This type of sectioning reduces both the dimensionalities of the phase\n", + "diagram and phase boundary by the same number, but does not alter the\n", + "number of phases in the adjacent phase regions. The contact rule, i.e. ,\n", + "thus remains valid and is applicable to phase regions with the\n", + "same-dimensionality as that of the sectioned phase diagram. Similarly,\n", + "the Schreinemakers\u2019 rule shown in is valid under the same conditions.\n", + "\n", + "For example, the two-dimensional phase diagram of the Mg-Al-Zn ternary\n", + "system sectioned with one atmospheric pressure and the weight fraction\n", + "of Zn fixed at 0.01 is shown in plotted with temperature and mole\n", + "fraction of Al \\[7\\]. This phase diagram is commonly called isopleth and\n", + "is generated by fixing one potential, the pressure, $P$, changing the\n", + "chemical potentials of Al and Zn to their conjugate molar quantities\n", + "represented by weight fractions of Al and Zn, and sectioning at\n", + "$w_{Zn} = 0.01$. From the discussions in Chapter , the phase regions\n", + "with the number of phases equal to three or fewer, i.e.\n", + "$p \\leq n_{m} + 1 = 3$, have the same-dimensionality as the phase\n", + "diagram, i.e. two-dimensional in the present case, and the phase\n", + "boundary rule is applicable. The maximum number of phases co-existing at\n", + "equilibrium is given by as following for the present case\n", + "\n", + "Eq. \u2011 $p_{\\max} = \\left( c - n_{s} \\right) + 2 = 3 - 1 + 2 = 4$\n", + "\n", + "This is because introducing molar quantities only increases the\n", + "dimensionality of phase regions and does not change the maximum number\n", + "of co-existing phases.\n", + "\n", + "The dimensionality of a four-phase region is calculated from as\n", + "\n", + "Eq. \u2011\n", + "$\\upsilon_{m} = \\left( c - n_{s} \\right) + 2 - p + n_{m} - n_{ms} = 3 - 1 + 2 - 4 + 2 - 1 = 1$\n", + "\n", + "where $n_{ms}$ is the number of sectioned molar quantities. Since the\n", + "dimensionality of a four-phase region is lower than that of the phase\n", + "diagram, the phase boundary rule cannot be applied directly. Such a\n", + "four-phase region, liquid+Mg+\u03b3+\u03c6, is shown in between three three-phase\n", + "regions of liquid+Mg+\u03b3, liquid+Mg+\u03c6, and Mg+\u03b3+\u03c6.\n", + "\n", + "Figure \u2011: Isopleth with the weight fraction of Zn fixed at 0.01 of the\n", + "Mg-Al-Zn ternary system.\n", + "\n", + "also displays information on what phases are in equilibrium for a given\n", + "alloy at various temperatures. One example is shown by the dotted\n", + "vertical line marking the weight fraction of Al being 0.09, a widely\n", + "used Mg alloy called AZ91. Various phases are present at different\n", + "temperature ranges, but the equilibrium phase fractions and phase\n", + "compositions are not shown in the figure as the tie-lines are not in the\n", + "plane of the phase diagram and have to be calculated at each temperature\n", + "individually. shows the amount of each phase of the AZ91 alloy as a\n", + "function of temperature with the dotted lines depicting the values under\n", + "the equilibrium condition and the solid lines depicting the values under\n", + "the so-called Scheil condition assuming no diffusion in solid phases and\n", + "infinitelyfast diffusion in liquid. Similarly, the composition of each\n", + "phase can also be plotted as shown in .\n", + "\n", + "Figure \u2011: Mole fraction of individual phases under equilibrium (dotted\n", + "curves) and Scheil (solid curves) conditions in the AZ91 alloy\n", + "\n", + "Figure \u2011: Mass fraction of Al and Zn in the Mg solid solution phase\n", + "under equilibrium (dotted curves) and Scheil (solid curves) conditions\n", + "in AZ91\n" + ], + "metadata": {} + }, + { + "id": "4299ee7b", + "cell_type": "markdown", + "source": [ + "### Section of potential phase diagrams\n", + "\n", + "Based on the Gibbs-Duhem equation (see ), a single-phase equilibrium in\n", + "a system with more than two independent components has more than three\n", + "independent potentials. There is no problem in representing them using\n", + "the mathematical formulas discussed so far, but it is not possible for\n", + "us to visualize graphically the full potential phase diagrams in\n", + "multi-component systems with more than two independent components. In\n", + "principles, there are two options. One option is to project the\n", + "multi-dimensional potential phase diagram into a two- or three-dimension\n", + "diagram, and another option is to section the multi-dimensional\n", + "potential phase diagram by fixing the values of some potentials.\n", + "\n", + "The projection approach is used for one-component systems in Chapter .\n", + "Since a two-phase equilibrium in a one-component potential phase diagram\n", + "is one-dimensional, the projection does not lose any information, and\n", + "the same is true for a three-phase equilibrium in a one-component\n", + "potential phase diagram. In a binary system, the projections of three-\n", + "and four-phase equilibria do not lose any information, while the\n", + "projections of two-phase equilibria become two-dimensional and cannot\n", + "retain all the information as the original-dimensionality of these\n", + "two-phase equilibria is three. Consequently, sectioning at fixed values\n", + "of some potentials is necessary in order to visualize the phase\n", + "relations in systems with two or more components. Gibbs phase rule shown\n", + "in and are thus modified to\n", + "\n", + "Eq. \u2011 $\\upsilon = c + 2 - p - n_{s} = \\left( c - n_{s} \\right) + 2 - p$\n", + "\n", + "Eq. \u2011 $p_{\\max} = \\left( c - n_{s} \\right) + 2$\n", + "\n", + "where $n_{s}$ is the number of potentials fixed in sectioning. As can be\n", + "seen in the last part of , the number of sectioning is equivalent to the\n", + "reduction of the effective number of independent components. Therefore,\n", + "any multi-component systems with $n_{s} = c - i$ behave like an\n", + "$i$-component system. The equations presented in Chapter and are thus\n", + "directly applicable to multi-component systems with $n_{s} = c - 1$ and\n", + "$n_{s} = c - 2$, respectively.\n", + "\n", + "A common practice in experiments is to fix pressure, temperature or\n", + "chemical potentials of volatile components as they are usually the\n", + "variables controlled experimentally. In a binary system, the potential\n", + "phase diagram at constant pressure can be represented by any two of the\n", + "three potentials, i.e. two chemical potentials and temperature, with the\n", + "remaining potential being dependent, and has the identical morphology as\n", + "a one-component system. The Gibbs-Duhem equation under such conditions\n", + "becomes\n", + "\n", + "Eq. \u2011 $0 = - SdT - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B}$\n", + "\n", + "The corresponding two-phase Clausius-Clapeyron equations are written as\n", + "\n", + "Eq. \u2011\n", + "$\\frac{dT}{{d\\mu}_{A}} = - \\frac{{\\mathrm{\\Delta}z}_{A}^{\\alpha\\beta}}{\\mathrm{\\Delta}S_{mB}^{\\alpha\\beta}}$\n", + "\n", + "Eq. \u2011\n", + "$\\frac{dT}{{d\\mu}_{B}} = - \\frac{{\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}}{\\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}}$\n", + "\n", + "Eq. \u2011\n", + "$\\frac{{d\\mu}_{A}}{{d\\mu}_{B}} = - \\frac{\\mathrm{\\Delta}\\left( \\frac{1}{S_{mB}^{}} \\right)^{\\alpha\\beta}}{\\mathrm{\\Delta}\\left( \\frac{1}{S_{mA}^{}} \\right)^{\\alpha\\beta}}$\n", + "\n", + "As an example, the $T$-$\\mu_{C}$ potential phase diagram for the Fe-C\n", + "binary system at one atmospheric pressure is shown in .\n", + "\n", + "Figure \u2011: $T$-$\\mu_{C}$ potential phase diagram for the Fe-C binary\n", + "system at $P = 1atm$\n", + "\n", + "In a ternary system, two potentials need to be fixed in order to obtain\n", + "two-dimensional potential phase diagrams. When the pressure and the\n", + "chemical potential of one species are fixed, the system behaves like a\n", + "binary system discussed above. When the system temperature and pressure\n", + "are fixed, the Gibbs-Duhem equation is written as\n", + "\n", + "Eq. \u2011 $0 = - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B} - {N_{C}d\\mu}_{C}$\n", + "\n", + "Taking the component A as the dependent element, the two-phase\n", + "Clausius-Clapeyron equation is simplified as\n", + "\n", + "Eq. \u2011\n", + "$\\frac{{d\\mu}_{B}}{{d\\mu}_{C}} = \\frac{d\\left( \\ln a_{B} \\right)}{d\\left( \\ln a_{C} \\right)} = - \\frac{\\mathrm{\\Delta}z_{C}^{\\alpha\\beta}}{{\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}}$\n", + "\n", + "When the two phases in equilibrium are stoichiometric phases, the\n", + "two-phase equilibrium is thus a straight line. For example, the Ti-O-Cl\n", + "potential phase diagram at 600\u00b0C and one atmospheric pressure is shown\n", + "in . Since both O and Cl are volatile components, their activities are\n", + "usually represented by their partial pressures with the pure\n", + "O2 and Cl2 gas as their respective reference\n", + "states at the given temperature and pressure.\n", + "\n", + "Figure \u2011: Ti-O-Cl potential phase diagram at 600\u00b0C and one atmospheric\n", + "pressure\n", + "\n", + "For systems with more than three components, the chemical potentials of\n", + "one or more components must be fixed in order to obtain a\n", + "two-dimensional potential phase diagram similar to the potential phase\n", + "diagrams discussed above.\n" + ], + "metadata": {} + }, + { + "id": "b614d254", + "cell_type": "markdown", + "source": [ + "### Tie-lines and lever rule\n", + "\n", + "It is self-evident from to Eq. 2\u20114 that while potentials are homogeneous\n", + "in all phases in a heterogeneous system at equilibrium, the molar\n", + "quantities usually have different values in individual phases. This is\n", + "also stipulated in various Clausius-Clapeyron equations such as to and\n", + "to . The difference in molar quantities thus increases the\n", + "dimensionality of the phase region by the number of potentials replaced\n", + "by their conjugate molar quantities. The maximum dimensionality of a\n", + "phase region is the dimensionality of the phase diagram under\n", + "consideration. This thus creates a finite space between phases in\n", + "equilibrium in the phase diagram with some axes being molar quantities.\n", + "\n", + "For an equilibrium system under constant $T$, $P$ and $N_{i}$, the\n", + "potentials in the system and their conjugate molar quantities in each\n", + "phase are all uniquely defined. In a phase diagram with one or more\n", + "potentials replaced by their conjugate molar quantities, two phases in\n", + "equilibrium in a system with $c$ independent components are connected by\n", + "a $c$ dimensional line in a *c*+2 dimensional space or\n", + "its$\\ $*c+1*-dimensional projection as discussed in Chapter . These\n", + "lines are called tie-lines and collectively represent a two-phase\n", + "equilibrium region. For a $k$-phase equilibrium, there are total\n", + "$C_{k}^{2} = \\frac{k(k - 1)}{2}$ tie-lines connecting every two phases\n", + "with $k - 1\\ $of them independent because the number of independent\n", + "tie-lines increases by one with each new phase added. For the invariant\n", + "equilibrium with zero degree of freedom, the number of phases in\n", + "equilibrium is $c + 2$ shown by containing\n", + "$C_{c + 2}^{2} = \\frac{(c + 2)(c + 1)}{2}$ tie-lines with $c + 1$ of\n", + "them being independent.\n", + "\n", + "Inside the space encapsulated by the tie-lines, the axis variables of\n", + "the phase diagram (a mixture of potentials and molar quantities) can be\n", + "changed independently without changing the phases in equilibrium and\n", + "their properties. Only the relative amounts of individual phases are\n", + "adjusted accordingly to maintain the conservation of the molar\n", + "quantities in the system specified by the molar quantity axes of the\n", + "phase diagram. The geometric feature circumscribing the space\n", + "encapsulated by the tie-lines no longer represents any phase regions,\n", + "but a boundary between the neighbouring phase regions. Its\n", + "characteristics will be discussed in more details in the next few\n", + "sections. As properties in each phase are homogeneous, the values of\n", + "molar quantities of a system are simply the sum of individual phases and\n", + "can be represented by the following equation\n", + "\n", + "Eq. \u2011 $A_{m} = \\sum_{\\alpha}^{}{f^{\\alpha}A_{m}^{\\alpha}}$\n", + "\n", + "where $A_{m}$ and $A_{m}^{\\alpha}$ represent the values of a molar\n", + "quantity of the system and the $\\alpha$ phase, respectively,\n", + "$f^{\\alpha}$ the mole fraction of the $\\alpha$ phase, and the summation\n", + "goes over all phases in equilibrium with each other. With\n", + "$\\sum_{\\alpha}^{}f^{\\alpha} = 1$, can be re-arranged into the following\n", + "equation\n", + "\n", + "Eq. \u2011\n", + "$\\sum_{\\alpha}^{}{f^{\\alpha}\\left( {A_{m} - A}_{m}^{\\alpha} \\right)} = 0$\n", + "\n", + "is commonly referred as lever rule. For a two-phase equilibrium of\n", + "$\\alpha$ and $\\beta$, it becomes\n", + "\n", + "Eq. \u2011\n", + "$f^{\\alpha} = \\frac{A_{m}^{\\beta} - A_{m}}{A_{m}^{\\beta} - A_{m}^{\\alpha}}$\n", + "\n", + "Eq. \u2011\n", + "$f^{\\beta} = \\frac{A_{m}^{\\alpha} - A_{m}}{A_{m}^{\\alpha} - A_{m}^{\\beta}}$\n", + "\n", + "For a phase diagram with the number of axes being\n", + "$n = \\left( c - n_{s} \\right) + 1$, the number of possible axes being\n", + "molar quantities is thus $k \\leq n$. There are thus $k$ equations\n", + "similar to with one for each molar quantity, $A_{mi}$, resulting in the\n", + "following $k + 1$ equations\n", + "\n", + "Eq. \u2011\n", + "$\\sum_{\\alpha}^{}{f^{\\alpha}\\left( {A_{mi} - A}_{mi}^{\\alpha} \\right)} = 0$\n", + "\n", + "Eq. \u2011 $1 - \\sum_{\\alpha}^{}f^{\\alpha} = 0$\n", + "\n", + "The summations in and go over the phases in equilibrium, and the amount\n", + "of each phase is obtained by solving these $k + 1$ equations\n", + "simultaneously along with the equilibrium conditions.\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/src/psu410/src/references/index.ipynb b/src/psu410/src/references/index.ipynb new file mode 100644 index 0000000..fe54700 --- /dev/null +++ b/src/psu410/src/references/index.ipynb @@ -0,0 +1,274 @@ +{ + "nbformat": 4, + "nbformat_minor": 5, + "metadata": {}, + "cells": [ + { + "id": "5624d039", + "cell_type": "markdown", + "source": [ + "# References\n", + "\n", + "1\\. M. 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Bull.*\n", + "**59** (2014) 1619-1623.\n" + ], + "metadata": {} + } + ] +} \ No newline at end of file From 34fd1b6eedf1bc00d30512d2ca4c78a49ea765a4 Mon Sep 17 00:00:00 2001 From: tonyfast Date: Mon, 8 Jan 2024 19:59:39 -0800 Subject: [PATCH 5/9] + --- src/nobook/bootstrap.ipynb | 241 ++++++++++++------------------------- 1 file changed, 80 insertions(+), 161 deletions(-) diff --git a/src/nobook/bootstrap.ipynb b/src/nobook/bootstrap.ipynb index 68b0a92..d4e385a 100644 --- a/src/nobook/bootstrap.ipynb +++ b/src/nobook/bootstrap.ipynb @@ -17,15 +17,15 @@ }, { "cell_type": "code", - "execution_count": 17, + "execution_count": 1, "id": "c4601326-6456-47e8-99c9-ae8c8f59e32b", "metadata": { "execution": { - "iopub.execute_input": "2024-01-09T03:51:36.901615Z", - "iopub.status.busy": "2024-01-09T03:51:36.901463Z", - "iopub.status.idle": "2024-01-09T03:51:36.906101Z", - "shell.execute_reply": "2024-01-09T03:51:36.905665Z", - "shell.execute_reply.started": "2024-01-09T03:51:36.901603Z" + "iopub.execute_input": "2024-01-09T03:52:37.061875Z", + "iopub.status.busy": "2024-01-09T03:52:37.061650Z", + "iopub.status.idle": "2024-01-09T03:52:37.400293Z", + "shell.execute_reply": "2024-01-09T03:52:37.399927Z", + "shell.execute_reply.started": "2024-01-09T03:52:37.061851Z" }, "tags": [] }, @@ -58,11 +58,11 @@ "id": "2f23bf91-3d5d-4bd5-8619-e05ac0df6c53", "metadata": { "execution": { - "iopub.execute_input": "2024-01-09T03:49:55.457320Z", - "iopub.status.busy": "2024-01-09T03:49:55.457169Z", - "iopub.status.idle": "2024-01-09T03:49:55.459361Z", - "shell.execute_reply": "2024-01-09T03:49:55.459058Z", - "shell.execute_reply.started": "2024-01-09T03:49:55.457309Z" + "iopub.execute_input": "2024-01-09T03:52:37.401586Z", + "iopub.status.busy": "2024-01-09T03:52:37.401275Z", + "iopub.status.idle": "2024-01-09T03:52:37.403939Z", + "shell.execute_reply": "2024-01-09T03:52:37.403613Z", + "shell.execute_reply.started": "2024-01-09T03:52:37.401567Z" }, "tags": [] }, @@ -96,11 +96,11 @@ "id": "bd674c8e-4a31-4544-b6b9-0aa2573385c8", "metadata": { "execution": { - "iopub.execute_input": "2024-01-09T03:49:55.460025Z", - "iopub.status.busy": "2024-01-09T03:49:55.459752Z", - "iopub.status.idle": "2024-01-09T03:49:55.462463Z", - "shell.execute_reply": "2024-01-09T03:49:55.462096Z", - "shell.execute_reply.started": "2024-01-09T03:49:55.460015Z" + "iopub.execute_input": "2024-01-09T03:52:37.404517Z", + "iopub.status.busy": "2024-01-09T03:52:37.404414Z", + "iopub.status.idle": "2024-01-09T03:52:37.406732Z", + "shell.execute_reply": "2024-01-09T03:52:37.406484Z", + "shell.execute_reply.started": "2024-01-09T03:52:37.404506Z" }, "tags": [] }, @@ -120,11 +120,11 @@ "id": "345b47f7-6fbf-40f5-a64f-5646db86dd2b", "metadata": { "execution": { - 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"iopub.execute_input": "2024-01-09T03:49:55.478861Z", - "iopub.status.busy": "2024-01-09T03:49:55.478774Z", - "iopub.status.idle": "2024-01-09T03:50:08.876085Z", - "shell.execute_reply": "2024-01-09T03:50:08.875809Z", - "shell.execute_reply.started": "2024-01-09T03:49:55.478853Z" + "iopub.execute_input": "2024-01-09T03:52:37.423040Z", + "iopub.status.busy": "2024-01-09T03:52:37.422946Z", + "iopub.status.idle": "2024-01-09T03:52:50.965098Z", + "shell.execute_reply": "2024-01-09T03:52:50.964729Z", + "shell.execute_reply.started": "2024-01-09T03:52:37.423031Z" }, "tags": [] }, @@ -298,7 +298,7 @@ "the table of contents, 9 primary chapters, and the references are captured ... ok\n", "\n", "----------------------------------------------------------------------\n", - "Ran 1 test in 13.391s\n", + "Ran 1 test in 13.535s\n", "\n", "OK\n" ] @@ -306,7 +306,7 @@ { "data": { "text/plain": [ - "" + "" ] }, "execution_count": 10, @@ -338,11 +338,11 @@ "id": "30ca19bc-ea48-47de-add9-59856cfc7529", "metadata": { "execution": { - 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" /home/tbone/Documents/nobook/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_stoichiometric_phases.ipynb,\n", - " /home/tbone/Documents/nobook/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_fept.ipynb,\n", - " /home/tbone/Documents/nobook/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_cerium.ipynb,\n", - " /home/tbone/Documents/nobook/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/concept_of_materials_genome.ipynb,\n", - " /home/tbone/Documents/nobook/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/index.ipynb,\n", - " /home/tbone/Documents/nobook/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/mms_model_applied_to_thermal_expansion.ipynb,\n", - " /home/tbone/Documents/nobook/psu410/src/experimental_data_for_thermodynamic_modeling/additional_methods.ipynb,\n", - " /home/tbone/Documents/nobook/psu410/src/experimental_data_for_thermodynamic_modeling/index.ipynb,\n", - " /home/tbone/Documents/nobook/psu410/src/experimental_data_for_thermodynamic_modeling/phase_equilibrium_data_from_experiments.ipynb,\n", - " /home/tbone/Documents/nobook/psu410/src/experimental_data_for_thermodynamic_modeling/thermodynamic_data_from_experiments.ipynb,\n", - " /home/tbone/Documents/nobook/psu410/src/firstprinciples_calculations_and_theory/bornoppenheimer_approximation.ipynb,\n", - " /home/tbone/Documents/nobook/psu410/src/firstprinciples_calculations_and_theory/firstprinciples_approaches_to_disordered_alloys.ipynb,\n", - " /home/tbone/Documents/nobook/psu410/src/firstprinciples_calculations_and_theory/index.ipynb,\n", - " /home/tbone/Documents/nobook/psu410/src/firstprinciples_calculations_and_theory/firstprinciples_formulation_of_thermodynamics.ipynb,\n", - " /home/tbone/Documents/nobook/psu410/src/firstprinciples_calculations_and_theory/lattice_dynamics.ipynb,\n", - " /home/tbone/Documents/nobook/psu410/src/firstprinciples_calculations_and_theory/nickel_as_the_prototype.ipynb,\n", - " /home/tbone/Documents/nobook/psu410/src/firstprinciples_calculations_and_theory/quantum_theory_for_the_motion_of_electrons.ipynb,\n", - " /home/tbone/Documents/nobook/psu410/src/gibbs_energy_function/binary_random_solutions.ipynb,\n", - " /home/tbone/Documents/nobook/psu410/src/gibbs_energy_function/elastic_magnetic_and_electric_contributions_to_free_energy.ipynb,\n", - " /home/tbone/Documents/nobook/psu410/src/gibbs_energy_function/index.ipynb,\n", - " /home/tbone/Documents/nobook/psu410/src/gibbs_energy_function/phases_with_fixed_compositions.ipynb,\n", - " /home/tbone/Documents/nobook/psu410/src/gibbs_energy_function/phases_with_variable_compositions_random_solutions.ipynb,\n", - " /home/tbone/Documents/nobook/psu410/src/gibbs_energy_function/phases_with_variable_compositions_solutions_with_ordering.ipynb,\n", - " /home/tbone/Documents/nobook/psu410/src/gibbs_energy_function/polymer_solutions_and_polymer_blends.ipynb,\n", - " /home/tbone/Documents/nobook/psu410/src/laws_of_thermodynamics/combined_law_of_thermodynamics_and_equilibrium_conditions.ipynb,\n", - " /home/tbone/Documents/nobook/psu410/src/laws_of_thermodynamics/first_and_second_laws_of_thermodynamics.ipynb,\n", - " /home/tbone/Documents/nobook/psu410/src/laws_of_thermodynamics/gibbsduhem_equation.ipynb,\n", - " /home/tbone/Documents/nobook/psu410/src/laws_of_thermodynamics/index.ipynb,\n", - " /home/tbone/Documents/nobook/psu410/src/laws_of_thermodynamics/stability_at_equilibrium_and_property_anomaly.ipynb,\n", - " /home/tbone/Documents/nobook/psu410/src/phase_equilibria_in_heterogeneous_systems/general_condition_of_equilibrium.ipynb,\n", - " /home/tbone/Documents/nobook/psu410/src/phase_equilibria_in_heterogeneous_systems/gibbs_phase_rule.ipynb,\n", - " /home/tbone/Documents/nobook/psu410/src/phase_equilibria_in_heterogeneous_systems/molar_phase_diagrams.ipynb,\n", - " /home/tbone/Documents/nobook/psu410/src/phase_equilibria_in_heterogeneous_systems/index.ipynb,\n", - " /home/tbone/Documents/nobook/psu410/src/phase_equilibria_in_heterogeneous_systems/potential_phase_diagrams.ipynb,\n", - " /home/tbone/Documents/nobook/psu410/src/references/index.ipynb],\n", - " dtype='object')" - ] - }, - "execution_count": 14, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - " get_notebooks(md).index" - ] - }, - { - "cell_type": "code", - "execution_count": 15, "id": "1f7b0f6b-6b56-48a5-8562-baa57317fa09", "metadata": { "execution": { - "iopub.execute_input": "2024-01-09T03:50:09.036339Z", - "iopub.status.busy": "2024-01-09T03:50:09.036212Z", - "iopub.status.idle": "2024-01-09T03:50:09.119950Z", - "shell.execute_reply": "2024-01-09T03:50:09.119540Z", - "shell.execute_reply.started": "2024-01-09T03:50:09.036330Z" + "iopub.execute_input": "2024-01-09T03:52:50.981865Z", + "iopub.status.busy": "2024-01-09T03:52:50.981711Z", + "iopub.status.idle": "2024-01-09T03:52:51.175071Z", + "shell.execute_reply": "2024-01-09T03:52:51.174638Z", + "shell.execute_reply.started": "2024-01-09T03:52:50.981851Z" }, "tags": [] }, @@ -584,7 +503,7 @@ " 14905]" ] }, - "execution_count": 15, + "execution_count": 14, "metadata": {}, "output_type": "execute_result" } @@ -598,15 +517,15 @@ }, { "cell_type": "code", - "execution_count": 16, + "execution_count": 15, "id": "ae617c09-6440-48a7-a015-2c60896fb1a1", "metadata": { "execution": { - "iopub.execute_input": "2024-01-09T03:50:09.120816Z", - "iopub.status.busy": "2024-01-09T03:50:09.120557Z", - "iopub.status.idle": "2024-01-09T03:50:09.130521Z", - "shell.execute_reply": "2024-01-09T03:50:09.130155Z", - "shell.execute_reply.started": "2024-01-09T03:50:09.120799Z" + "iopub.execute_input": "2024-01-09T03:52:51.175814Z", + "iopub.status.busy": "2024-01-09T03:52:51.175611Z", + "iopub.status.idle": "2024-01-09T03:52:51.186329Z", + "shell.execute_reply": "2024-01-09T03:52:51.185973Z", + "shell.execute_reply.started": "2024-01-09T03:52:51.175797Z" }, "tags": [] }, From 1b13897cefc569fa313929665b1bd9e89ced2120 Mon Sep 17 00:00:00 2001 From: tonyfast Date: Wed, 10 Jan 2024 12:07:36 -0800 Subject: [PATCH 6/9] cleaner bootstrapping --- src/nobook/bootstrap.ipynb | 4595 ++++++++++++----- ...lingham_diagram_and_buffered_systems.ipynb | 153 +- .../index.ipynb | 19 +- ...ential_and_integrated_driving_forces.ipynb | 93 +- ...te_and_chemical_transport_reactions_.ipynb | 150 +- .../trends_of_entropies_of_reactions.ipynb | 75 +- .../application_examples.ipynb | 39 +- ...queous_solution_and_pourbaix_diagram.ipynb | 254 +- ...erence_states_of_electrolyte_species.ipynb | 340 +- ...mical_cells_and_half_cell_potentials.ipynb | 39 +- ...ctions_and_electrochemical_reactions.ipynb | 456 +- .../index.ipynb | 24 +- .../importance_of_lattice_stability.ipynb | 112 +- .../index.ipynb | 50 +- .../modeling_of_pure_elements.ipynb | 67 +- .../modeling_of_random_solution_phases.ipynb | 54 +- ...ution_phases_with_longrange_ordering.ipynb | 161 +- .../modeling_of_stoichiometric_phases.ipynb | 33 +- .../application_to_cerium.ipynb | 200 +- .../application_to_fept.ipynb | 122 +- .../concept_of_materials_genome.ipynb | 57 +- .../index.ipynb | 23 +- ...s_model_applied_to_thermal_expansion.ipynb | 118 +- .../additional_methods.ipynb | 268 - .../index.ipynb | 25 +- ...se_equilibrium_data_from_experiments.ipynb | 63 +- .../thermodynamic_data_from_experiments.ipynb | 68 +- .../bornoppenheimer_approximation.ipynb | 388 -- ...ples_approaches_to_disordered_alloys.ipynb | 65 +- ...ciples_formulation_of_thermodynamics.ipynb | 345 +- .../index.ipynb | 51 +- .../lattice_dynamics.ipynb | 188 +- .../nickel_as_the_prototype.ipynb | 242 +- ...m_theory_for_the_motion_of_electrons.ipynb | 386 +- .../binary_random_solutions.ipynb | 164 - ...lectric_contributions_to_free_energy.ipynb | 122 +- .../src/gibbs_energy_function/index.ipynb | 206 +- .../phases_with_fixed_compositions.ipynb | 270 +- ...riable_compositions_random_solutions.ipynb | 138 +- ...compositions_solutions_with_ordering.ipynb | 30 +- ...polymer_solutions_and_polymer_blends.ipynb | 704 +-- ...odynamics_and_equilibrium_conditions.ipynb | 220 +- ...st_and_second_laws_of_thermodynamics.ipynb | 119 +- .../gibbsduhem_equation.ipynb | 82 +- .../src/laws_of_thermodynamics/index.ipynb | 6 +- ..._at_equilibrium_and_property_anomaly.ipynb | 183 +- .../general_condition_of_equilibrium.ipynb | 114 +- .../gibbs_phase_rule.ipynb | 55 +- .../index.ipynb | 6 +- .../molar_phase_diagrams.ipynb | 359 +- .../potential_phase_diagrams.ipynb | 663 +-- src/psu410/src/references/index.ipynb | 265 +- 52 files changed, 3471 insertions(+), 9558 deletions(-) delete mode 100644 src/psu410/src/experimental_data_for_thermodynamic_modeling/additional_methods.ipynb delete mode 100644 src/psu410/src/firstprinciples_calculations_and_theory/bornoppenheimer_approximation.ipynb delete mode 100644 src/psu410/src/gibbs_energy_function/binary_random_solutions.ipynb diff --git a/src/nobook/bootstrap.ipynb b/src/nobook/bootstrap.ipynb index d4e385a..a7cc8a2 100644 --- a/src/nobook/bootstrap.ipynb +++ b/src/nobook/bootstrap.ipynb @@ -5,14 +5,7 @@ "id": "2edc5efc-167c-4fb8-a2cd-57d9bad95ecc", "metadata": {}, "source": [ - "# bootstrapping interactive notebook content for _computational thermodynamics of materials_\n", - "\n", - "\n", - "1. table of contents\n", - "1. 9 primary sections or h1s\n", - "2. references\n", - "\n", - "each primary section will be exported to a separate package in the source." + "# bootstrapping interactive notebook content for _computational thermodynamics of materials_" ] }, { @@ -21,11 +14,11 @@ "id": "c4601326-6456-47e8-99c9-ae8c8f59e32b", "metadata": { "execution": { - "iopub.execute_input": "2024-01-09T03:52:37.061875Z", - "iopub.status.busy": "2024-01-09T03:52:37.061650Z", - "iopub.status.idle": "2024-01-09T03:52:37.400293Z", - "shell.execute_reply": "2024-01-09T03:52:37.399927Z", - "shell.execute_reply.started": "2024-01-09T03:52:37.061851Z" + "iopub.execute_input": "2024-01-10T20:05:46.590649Z", + "iopub.status.busy": "2024-01-10T20:05:46.590407Z", + "iopub.status.idle": "2024-01-10T20:05:46.925236Z", + "shell.execute_reply": "2024-01-10T20:05:46.924850Z", + "shell.execute_reply.started": "2024-01-10T20:05:46.590633Z" }, "tags": [] }, @@ -38,7 +31,7 @@ " INTERACTIVE = MAIN and not FILE\n", " singleton = functools.lru_cache(1)\n", " from pathlib import Path\n", - " HERE = Path(__file__).parent if \"__file__\" in globals() else Path(subprocess.check_output([\"pwd\"]).decode().strip())\n", + " HERE = Path(__file__).parent if FILE else Path(subprocess.check_output([\"pwd\"]).decode().strip())\n", " PSU = HERE.parent / \"psu410\" / \"src\" / \"psu410\"\n", "\n", " import nbformat.v4 as v4, anyio, json" @@ -46,7 +39,15 @@ }, { "cell_type": "markdown", - "id": "9c16ef6c-355b-4a6a-8ca7-94d77d434930", + "id": "54dc9414-fe0f-4a30-b914-77d3609250b6", + "metadata": {}, + "source": [ + "## loading the original textbook and data" + ] + }, + { + "cell_type": "markdown", + "id": "f7c68a07-a7f2-4620-8cfb-77228fcbfd7f", "metadata": {}, "source": [ "`get_docx` loads in the original document using `python-docx` data structures." @@ -58,19 +59,30 @@ "id": "2f23bf91-3d5d-4bd5-8619-e05ac0df6c53", "metadata": { "execution": { - "iopub.execute_input": "2024-01-09T03:52:37.401586Z", - "iopub.status.busy": "2024-01-09T03:52:37.401275Z", - "iopub.status.idle": "2024-01-09T03:52:37.403939Z", - "shell.execute_reply": "2024-01-09T03:52:37.403613Z", - "shell.execute_reply.started": "2024-01-09T03:52:37.401567Z" + "iopub.execute_input": "2024-01-10T20:05:46.926226Z", + "iopub.status.busy": "2024-01-10T20:05:46.926038Z", + "iopub.status.idle": "2024-01-10T20:05:47.054396Z", + "shell.execute_reply": "2024-01-10T20:05:47.054069Z", + "shell.execute_reply.started": "2024-01-10T20:05:46.926215Z" }, "tags": [] }, - "outputs": [], + "outputs": [ + { + "data": { + "text/plain": [ + "" + ] + }, + "execution_count": 2, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ - " @singleton\n", " def get_docx() -> \"docx.document.Document\":\n", - " return __import__(\"docx\").Document(\"2015-07-15-Textbook-Cambridge-U-P.docx\")" + " return __import__(\"docx\").Document(\"2015-07-15-Textbook-Cambridge-U-P.docx\")\n", + " INTERACTIVE and get_docx()" ] }, { @@ -87,6 +99,9 @@ "tags": [] }, "source": [ + "`get_paragraphs` represents the document a collections of `docx.Paragraph` like object.\n", + "we extract features about each row to use further in our translation of docx to notebooks.\n", + "\n", "the snippet discovers the style id's of each paragraph. this means we could remediate the the headings here and make them formal headings" ] }, @@ -96,196 +111,1705 @@ "id": "bd674c8e-4a31-4544-b6b9-0aa2573385c8", "metadata": { "execution": { - 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mdComputational Thermodynamics of Materials\\n\\nZ...# Laws of thermodynamics\\n## First and second laws of thermodynamics\\n\\n...## Combined law of thermodynamics and equilibr...## Stability at equilibrium and property anoma...## Gibbs-Duhem equation\\n\\nIn experiments, it ...# Gibbs energy function\\n\\nAs shown in through...## Phases with fixed compositions\\n\\nThe homog...## Phases with variable compositions: Random s...### Random solutions\\n\\nThe ideal Gibbs energy......### Galvanic protection\\n\\nA galvanic reaction...### Fuel cells\\n\\nFuel cells are devices to co...### Ion transport membranes\\n\\nIon transport m...### Electrical batteries\\n\\nBatteries utilize ...# Critical phenomena, thermal expansion, and M...## MMS model applied to thermal expansion\\n\\nA...## Application to cerium\\n\\nCerium (Ce) displa...## Application to Fe<sub>3</sub>Pt\\n\\nInvar wa...## Concept of Materials Genome®\\n\\n“A genome i...# References\\n\\n1\\. M. Hillert, *Phase Equilib...
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" + ], + "text/plain": [ + "package 0 \\\n", + "doc 0 \n", + "cell 0 \n", + "original 0 \n", + "docx \n", + "md Computational Thermodynamics of Materials\\n\\nZ... \n", + "\n", + "package 1 \\\n", + "doc 1 \n", + "cell 1 \n", + "original 100 \n", + "docx \n", + "md # Laws of thermodynamics\\n \n", + "\n", + "package \\\n", + "doc 2 \n", + "cell 2 \n", + "original 102 \n", + "docx \n", + "md ## First and second laws of thermodynamics\\n\\n... \n", + "\n", + "package \\\n", + "doc 3 \n", + "cell 3 \n", + "original 147 \n", + "docx \n", + "md ## Combined law of thermodynamics and equilibr... \n", + "\n", + "package \\\n", + "doc 4 \n", + "cell 4 \n", + "original 251 \n", + "docx \n", + "md ## Stability at equilibrium and property anoma... \n", + "\n", + "package \\\n", + "doc 5 \n", + "cell 5 \n", + "original 316 \n", + "docx \n", + "md ## Gibbs-Duhem equation\\n\\nIn experiments, it ... \n", + "\n", + "package 2 \\\n", + "doc 6 \n", + "cell 6 \n", + "original 359 \n", + "docx \n", + "md # Gibbs energy function\\n\\nAs shown in through... \n", + "\n", + "package \\\n", + "doc 7 \n", + "cell 7 \n", + "original 435 \n", + "docx \n", + "md ## Phases with fixed compositions\\n\\nThe homog... \n", + "\n", + "package \\\n", + "doc 8 \n", + "cell 8 \n", + "original 563 \n", + "docx \n", + "md ## Phases with variable compositions: Random s... \n", + "\n", + "package ... \\\n", + "doc ... \n", + "cell 9 ... \n", + "original 617 ... \n", + "docx ... \n", + "md ### Random solutions\\n\\nThe ideal Gibbs energy... ... \n", + "\n", + "package 8 \\\n", + "doc 42 \n", + "cell 87 \n", + "original 2959 \n", + "docx \n", + "md ### Galvanic protection\\n\\nA galvanic reaction... \n", + "\n", + "package \\\n", + "doc \n", + "cell 88 \n", + "original 2982 \n", + "docx \n", + "md ### Fuel cells\\n\\nFuel cells are devices to co... \n", + "\n", + "package \\\n", + "doc \n", + "cell 89 \n", + "original 3004 \n", + "docx \n", + "md ### Ion transport membranes\\n\\nIon transport m... \n", + "\n", + "package \\\n", + "doc \n", + "cell 90 \n", + "original 3022 \n", + "docx \n", + "md ### Electrical batteries\\n\\nBatteries utilize ... \n", + "\n", + "package 9 \\\n", + "doc 43 \n", + "cell 91 \n", + "original 3098 \n", + "docx \n", + "md # Critical phenomena, thermal expansion, and M... \n", + "\n", + "package \\\n", + "doc 44 \n", + "cell 92 \n", + "original 3102 \n", + "docx \n", + "md ## MMS model applied to thermal expansion\\n\\nA... \n", + "\n", + "package \\\n", + "doc 45 \n", + "cell 93 \n", + "original 3140 \n", + "docx \n", + "md ## Application to cerium\\n\\nCerium (Ce) displa... \n", + "\n", + "package \\\n", + "doc 46 \n", + "cell 94 \n", + "original 3184 \n", + "docx \n", + "md ## Application to Fe3Pt\\n\\nInvar wa... \n", + "\n", + "package \\\n", + "doc 47 \n", + "cell 95 \n", + "original 3208 \n", + "docx \n", + "md ## Concept of Materials Genome®\\n\\n“A genome i... \n", + "\n", + "package 10 \n", + "doc 48 \n", + "cell 96 \n", + "original 3213 \n", + "docx \n", + "md # References\\n\\n1\\. M. Hillert, *Phase Equilib... \n", + "\n", + "[2 rows x 97 columns]" + ] + }, + "execution_count": 4, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ + " async def get_documents(p):\n", + " compact = p.groupby([\"package\", \"doc\", \"cell\"]).apply(\n", + " lambda df: pandas.Series([\n", + " df.sort_values(\"original\").p.pipe(get_new_docx)\n", + " ], pandas.Index([df.original.min()], name=\"original\"))\n", + " ).to_frame(\"docx\")\n", + " return compact.assign(md=await asyncio.gather(*map(get_docx_markdown, compact.docx)))\n", + "\n", " def get_new_docx(elements: list[\"docx.Type\"]) -> \"docx.document.Document\":\n", + " \"creates a new `docx` data structure using only the specified elements.\"\n", " try: return (new := docx.Document())\n", - " finally: new._body._element.extend(x._element for x in elements)" + " finally: new._body._element.extend(x._element for x in elements) \n", + " \n", + " async def get_docx_markdown(document) -> str:\n", + " \"\"\"save docx to disc and return a stringified markdown version. this is async because it shells out everytime\"\"\"\n", + " document.save((tmp := tempfile.NamedTemporaryFile(suffix=\".docx\")).name)\n", + " return subprocess.check_output([PANDOC, \"--to\", PANDOC_TO, \"--from\", PANDOC_FROM, tmp.name]).decode()\n", + " \n", + " INTERACTIVE and (documents := await get_documents(p)).T" ] }, { "cell_type": "markdown", - "id": "147921f4-7ba8-45da-96af-a55fbe9b6568", + "id": "44b336b4-a228-47e4-ab5d-aac8f776f3e4", "metadata": {}, "source": [ - "`get_docx_markdown` will asynchronously shell out many times to compile markdown from a `docx` data structure." + "## composing the target files for the documents\n", + "\n", + "from the information in our dataframes we need to compose the target locations for the collected documents." + ] + }, + { + "cell_type": "markdown", + "id": "211b1dd8-278c-4496-a1fa-90cf81e8c705", + "metadata": {}, + "source": [ + "`get_contents` reduces our data to just the h1 and h2 headings in the document.\n", + "we use this to compose our virtual file system layout." ] }, { "cell_type": "code", - "execution_count": 6, - "id": "d4e75784-0945-47e0-9260-4116ca2e0e5d", + "execution_count": 5, + "id": "58d15767-aa89-4bd3-b6c5-66acc848ee7a", "metadata": { "execution": { - "iopub.execute_input": "2024-01-09T03:52:37.412409Z", - "iopub.status.busy": "2024-01-09T03:52:37.412318Z", - "iopub.status.idle": "2024-01-09T03:52:37.414444Z", - "shell.execute_reply": "2024-01-09T03:52:37.414199Z", - "shell.execute_reply.started": "2024-01-09T03:52:37.412400Z" + "iopub.execute_input": "2024-01-10T20:06:01.423322Z", + "iopub.status.busy": "2024-01-10T20:06:01.422951Z", + "iopub.status.idle": "2024-01-10T20:06:01.448552Z", + "shell.execute_reply": "2024-01-10T20:06:01.448076Z", + "shell.execute_reply.started": "2024-01-10T20:06:01.423304Z" }, "tags": [] }, - "outputs": [], + "outputs": [ + { + "data": { + "text/html": [ + "
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textLaws of thermodynamicsFirst and second laws of thermodynamicsCombined law of thermodynamics and equilibrium...Stability at equilibrium and property anomalyGibbs-Duhem equationGibbs energy functionPhases with fixed compositionsPhases with variable compositions: Random solu...Random solutionsBinary random solutions...Galvanic protectionFuel cellsIon transport membranesElectrical batteriesCritical phenomena, thermal expansion, and Mat...MMS model applied to thermal expansionApplication to ceriumApplication to Fe3PtConcept of Materials Genome®References
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package_namelaws_of_thermodynamicslaws_of_thermodynamicslaws_of_thermodynamicslaws_of_thermodynamicslaws_of_thermodynamicsgibbs_energy_functiongibbs_energy_functiongibbs_energy_functiongibbs_energy_functiongibbs_energy_function...applications_to_electrochemical_systemsapplications_to_electrochemical_systemsapplications_to_electrochemical_systemsapplications_to_electrochemical_systemscritical_phenomena_thermal_expansion_and_mater...critical_phenomena_thermal_expansion_and_mater...critical_phenomena_thermal_expansion_and_mater...critical_phenomena_thermal_expansion_and_mater...critical_phenomena_thermal_expansion_and_mater...references
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" + ], + "text/plain": [ + "package 1 \\\n", + "doc 1 \n", + "cell 1 \n", + "original 100 \n", + "p str:\n", - " document.save((tmp := tempfile.NamedTemporaryFile(suffix=\".docx\")).name)\n", - " return subprocess.check_output([PANDOC, \"--to\", PANDOC_TO, \"--from\", PANDOC_FROM, tmp.name]).decode()" + " def get_contents(p):\n", + " contents = p.dropna(subset=\"h\")\n", + " contents = contents.assign(text=contents.p.apply(operator.attrgetter(\"text\")))\n", + "\n", + " contents = contents.assign(\n", + " slug=contents[contents.h.le(2)].reindex_like(contents).text.ffill().apply(slugify),\n", + " )\n", + " contents = contents.assign(\n", + " package_name=contents[contents.h.eq(1)].slug.reindex_like(contents).ffill(),\n", + " document_name=contents[contents.h.le(2)].slug.reindex_like(contents).ffill(),\n", + " )\n", + " contents.loc[contents.package_name.eq(contents.document_name), \"document_name\"] = \"index\"\n", + " contents =contents.assign(target=PSU.parent / contents.package_name / contents.document_name.add(\".ipynb\"))\n", + " return contents.reset_index().set_index([\"package\", \"doc\"])\n", + " \n", + " def slugify(s): return pipe(s, str.lower, partial(re.sub, \"\\s\", \"_\"), partial(re.sub, \"[^\\.a-z\\_]\", \"\"))\n", + " \n", + " INTERACTIVE and (contents := get_contents(p)).T" ] }, { "cell_type": "markdown", - "id": "f3cb525f-d778-47cd-9141-cb7eab4149d6", + "id": "e8b52e2e-7167-4641-a63f-9dea80dd1ee0", "metadata": {}, "source": [ - "define chapters by the 10 h1s we found and the nearest heading to identify notebooks." + "the `tree` shows our predicted layout" ] }, { "cell_type": "code", - "execution_count": 7, - "id": "dfc37237-2240-4585-949d-0dbc13d1e7cc", + "execution_count": 6, + "id": "dc547412-dcc4-43a5-b56f-45dcf60fb11e", "metadata": { "execution": { - "iopub.execute_input": "2024-01-09T03:52:37.415458Z", - "iopub.status.busy": "2024-01-09T03:52:37.415297Z", - "iopub.status.idle": "2024-01-09T03:52:37.417922Z", - "shell.execute_reply": "2024-01-09T03:52:37.417669Z", - "shell.execute_reply.started": "2024-01-09T03:52:37.415449Z" + "iopub.execute_input": "2024-01-10T20:06:01.449273Z", + "iopub.status.busy": "2024-01-10T20:06:01.449111Z", + "iopub.status.idle": "2024-01-10T20:06:01.475960Z", + "shell.execute_reply": "2024-01-10T20:06:01.475623Z", + "shell.execute_reply.started": "2024-01-10T20:06:01.449258Z" }, "tags": [] }, - "outputs": [], + "outputs": [ + { + "data": { + "text/html": [ + "
psu410\n",
+       "├── applications_to_chemical_reactions\n",
+       "│   ├── index.ipynb\n",
+       "│   ├── internal_process_and_differential_and_integrated_driving_forces.ipynb\n",
+       "│   ├── ellingham_diagram_and_buffered_systems.ipynb\n",
+       "│   ├── trends_of_entropies_of_reactions.ipynb\n",
+       "│   └── maximum_reaction_rate_and_chemical_transport_reactions_.ipynb\n",
+       "├── applications_to_electrochemical_systems\n",
+       "│   ├── index.ipynb\n",
+       "│   ├── electrolyte_reactions_and_electrochemical_reactions.ipynb\n",
+       "│   ├── concentrations_activities_and_reference_states_of_electrolyte_species.ipynb\n",
+       "│   ├── electrochemical_cells_and_half_cell_potentials.ipynb\n",
+       "│   ├── aqueous_solution_and_pourbaix_diagram.ipynb\n",
+       "│   └── application_examples.ipynb\n",
+       "├── calpahd_modeling_of_thermodynamics\n",
+       "│   ├── index.ipynb\n",
+       "│   ├── importance_of_lattice_stability.ipynb\n",
+       "│   ├── modeling_of_pure_elements.ipynb\n",
+       "│   ├── modeling_of_stoichiometric_phases.ipynb\n",
+       "│   ├── modeling_of_random_solution_phases.ipynb\n",
+       "│   └── modeling_of_solution_phases_with_longrange_ordering.ipynb\n",
+       "├── critical_phenomena_thermal_expansion_and_materials_genome\n",
+       "│   ├── index.ipynb\n",
+       "│   ├── mms_model_applied_to_thermal_expansion.ipynb\n",
+       "│   ├── application_to_cerium.ipynb\n",
+       "│   ├── application_to_fept.ipynb\n",
+       "│   └── concept_of_materials_genome.ipynb\n",
+       "├── experimental_data_for_thermodynamic_modeling\n",
+       "│   ├── index.ipynb\n",
+       "│   ├── phase_equilibrium_data_from_experiments.ipynb\n",
+       "│   └── thermodynamic_data_from_experiments.ipynb\n",
+       "├── firstprinciples_calculations_and_theory\n",
+       "│   ├── index.ipynb\n",
+       "│   ├── nickel_as_the_prototype.ipynb\n",
+       "│   ├── firstprinciples_formulation_of_thermodynamics.ipynb\n",
+       "│   ├── quantum_theory_for_the_motion_of_electrons.ipynb\n",
+       "│   ├── lattice_dynamics.ipynb\n",
+       "│   └── firstprinciples_approaches_to_disordered_alloys.ipynb\n",
+       "├── gibbs_energy_function\n",
+       "│   ├── index.ipynb\n",
+       "│   ├── phases_with_fixed_compositions.ipynb\n",
+       "│   ├── phases_with_variable_compositions_random_solutions.ipynb\n",
+       "│   ├── phases_with_variable_compositions_solutions_with_ordering.ipynb\n",
+       "│   ├── polymer_solutions_and_polymer_blends.ipynb\n",
+       "│   └── elastic_magnetic_and_electric_contributions_to_free_energy.ipynb\n",
+       "├── laws_of_thermodynamics\n",
+       "│   ├── index.ipynb\n",
+       "│   ├── first_and_second_laws_of_thermodynamics.ipynb\n",
+       "│   ├── combined_law_of_thermodynamics_and_equilibrium_conditions.ipynb\n",
+       "│   ├── stability_at_equilibrium_and_property_anomaly.ipynb\n",
+       "│   └── gibbsduhem_equation.ipynb\n",
+       "├── phase_equilibria_in_heterogeneous_systems\n",
+       "│   ├── index.ipynb\n",
+       "│   ├── general_condition_of_equilibrium.ipynb\n",
+       "│   ├── gibbs_phase_rule.ipynb\n",
+       "│   ├── potential_phase_diagrams.ipynb\n",
+       "│   └── molar_phase_diagrams.ipynb\n",
+       "└── references\n",
+       "    └── index.ipynb\n",
+       "
\n" + ], + "text/plain": [ + "psu410\n", + "├── applications_to_chemical_reactions\n", + "│ ├── index.ipynb\n", + "│ ├── internal_process_and_differential_and_integrated_driving_forces.ipynb\n", + "│ ├── ellingham_diagram_and_buffered_systems.ipynb\n", + "│ ├── trends_of_entropies_of_reactions.ipynb\n", + "│ └── maximum_reaction_rate_and_chemical_transport_reactions_.ipynb\n", + "├── applications_to_electrochemical_systems\n", + "│ ├── index.ipynb\n", + "│ ├── electrolyte_reactions_and_electrochemical_reactions.ipynb\n", + "│ ├── concentrations_activities_and_reference_states_of_electrolyte_species.ipynb\n", + "│ ├── electrochemical_cells_and_half_cell_potentials.ipynb\n", + "│ ├── aqueous_solution_and_pourbaix_diagram.ipynb\n", + "│ └── application_examples.ipynb\n", + "├── calpahd_modeling_of_thermodynamics\n", + "│ ├── index.ipynb\n", + "│ ├── importance_of_lattice_stability.ipynb\n", + "│ ├── modeling_of_pure_elements.ipynb\n", + "│ ├── modeling_of_stoichiometric_phases.ipynb\n", + "│ ├── modeling_of_random_solution_phases.ipynb\n", + "│ └── modeling_of_solution_phases_with_longrange_ordering.ipynb\n", + "├── critical_phenomena_thermal_expansion_and_materials_genome\n", + "│ ├── index.ipynb\n", + "│ ├── mms_model_applied_to_thermal_expansion.ipynb\n", + "│ ├── application_to_cerium.ipynb\n", + "│ ├── application_to_fept.ipynb\n", + "│ └── concept_of_materials_genome.ipynb\n", + "├── experimental_data_for_thermodynamic_modeling\n", + "│ ├── index.ipynb\n", + "│ ├── phase_equilibrium_data_from_experiments.ipynb\n", + "│ └── thermodynamic_data_from_experiments.ipynb\n", + "├── firstprinciples_calculations_and_theory\n", + "│ ├── index.ipynb\n", + "│ ├── nickel_as_the_prototype.ipynb\n", + "│ ├── firstprinciples_formulation_of_thermodynamics.ipynb\n", + "│ ├── quantum_theory_for_the_motion_of_electrons.ipynb\n", + "│ ├── lattice_dynamics.ipynb\n", + "│ └── firstprinciples_approaches_to_disordered_alloys.ipynb\n", + "├── gibbs_energy_function\n", + "│ ├── index.ipynb\n", + "│ ├── phases_with_fixed_compositions.ipynb\n", + "│ ├── phases_with_variable_compositions_random_solutions.ipynb\n", + "│ ├── phases_with_variable_compositions_solutions_with_ordering.ipynb\n", + "│ ├── polymer_solutions_and_polymer_blends.ipynb\n", + "│ └── elastic_magnetic_and_electric_contributions_to_free_energy.ipynb\n", + "├── laws_of_thermodynamics\n", + "│ ├── index.ipynb\n", + "│ ├── first_and_second_laws_of_thermodynamics.ipynb\n", + "│ ├── combined_law_of_thermodynamics_and_equilibrium_conditions.ipynb\n", + "│ ├── stability_at_equilibrium_and_property_anomaly.ipynb\n", + "│ └── gibbsduhem_equation.ipynb\n", + "├── phase_equilibria_in_heterogeneous_systems\n", + "│ ├── index.ipynb\n", + "│ ├── general_condition_of_equilibrium.ipynb\n", + "│ ├── gibbs_phase_rule.ipynb\n", + "│ ├── potential_phase_diagrams.ipynb\n", + "│ └── molar_phase_diagrams.ipynb\n", + "└── references\n", + " └── index.ipynb\n" + ] + }, + "execution_count": 6, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ - " def get_organized_docx(df):\n", - " df = df.join(\n", - " df[df.style_id.str.match(\"^Heading1\")].p.apply(operator.attrgetter(\"text\")).rename(\"chapter\")\n", - " ).join(\n", - " df[df.style_id.str.match(\"^Heading\")].p.apply(operator.attrgetter(\"text\")).rename(\"heading\")\n", - " ).ffill().fillna(\"empty\").set_index([\"chapter\", \"heading\"])\n", - " df = df.groupby([\"chapter\", \"heading\"]).p.apply(get_new_docx).rename(\"docx\")\n", - " df.pop((\"empty\", \"empty\"))\n", - " df = df.to_frame().set_index(pandas.RangeIndex(0, len(df), name=\"order\"), append=True)\n", - " return df.docx" + " def contents_as_tree(contents):\n", + " \"\"\"`contents_as_tree` shows the prospective file layout of the contents.\"\"\"\n", + " import rich.tree\n", + " tree = rich.tree.Tree(\"psu410\")\n", + " for parent, files in contents.target.drop_duplicates().pipe(\n", + " lambda s: s.groupby(s.apply(operator.attrgetter(\"parent\")))\n", + " ):\n", + " tree.add(t := rich.tree.Tree(parent.name))\n", + " pipe(files, map(compose_left(operator.attrgetter(\"name\"), t.add)), list)\n", + " return tree\n", + "\n", + " INTERACTIVE and (tree := contents_as_tree(contents))" ] }, { - 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target/home/tbone/Documents/nobook/src/psu410/src/laws_of_thermodynamics/index.ipynb/home/tbone/Documents/nobook/src/psu410/src/laws_of_thermodynamics/first_and_second_laws_of_thermodynamics.ipynb/home/tbone/Documents/nobook/src/psu410/src/laws_of_thermodynamics/combined_law_of_thermodynamics_and_equilibrium_conditions.ipynb/home/tbone/Documents/nobook/src/psu410/src/laws_of_thermodynamics/stability_at_equilibrium_and_property_anomaly.ipynb/home/tbone/Documents/nobook/src/psu410/src/laws_of_thermodynamics/gibbsduhem_equation.ipynb/home/tbone/Documents/nobook/src/psu410/src/gibbs_energy_function/index.ipynb/home/tbone/Documents/nobook/src/psu410/src/gibbs_energy_function/phases_with_fixed_compositions.ipynb/home/tbone/Documents/nobook/src/psu410/src/gibbs_energy_function/phases_with_variable_compositions_random_solutions.ipynb/home/tbone/Documents/nobook/src/psu410/src/gibbs_energy_function/phases_with_variable_compositions_solutions_with_ordering.ipynb/home/tbone/Documents/nobook/src/psu410/src/gibbs_energy_function/polymer_solutions_and_polymer_blends.ipynb.../home/tbone/Documents/nobook/src/psu410/src/applications_to_electrochemical_systems/concentrations_activities_and_reference_states_of_electrolyte_species.ipynb/home/tbone/Documents/nobook/src/psu410/src/applications_to_electrochemical_systems/electrochemical_cells_and_half_cell_potentials.ipynb/home/tbone/Documents/nobook/src/psu410/src/applications_to_electrochemical_systems/aqueous_solution_and_pourbaix_diagram.ipynb/home/tbone/Documents/nobook/src/psu410/src/applications_to_electrochemical_systems/application_examples.ipynb/home/tbone/Documents/nobook/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/index.ipynb/home/tbone/Documents/nobook/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/mms_model_applied_to_thermal_expansion.ipynb/home/tbone/Documents/nobook/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_cerium.ipynb/home/tbone/Documents/nobook/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_fept.ipynb/home/tbone/Documents/nobook/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/concept_of_materials_genome.ipynb/home/tbone/Documents/nobook/src/psu410/src/references/index.ipynb
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" + ], + "text/plain": [ + "target /home/tbone/Documents/nobook/src/psu410/src/laws_of_thermodynamics/index.ipynb \\\n", + "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", + "\n", + "target /home/tbone/Documents/nobook/src/psu410/src/laws_of_thermodynamics/first_and_second_laws_of_thermodynamics.ipynb \\\n", + "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", + "\n", + "target /home/tbone/Documents/nobook/src/psu410/src/laws_of_thermodynamics/combined_law_of_thermodynamics_and_equilibrium_conditions.ipynb \\\n", + "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", + "\n", + "target /home/tbone/Documents/nobook/src/psu410/src/laws_of_thermodynamics/stability_at_equilibrium_and_property_anomaly.ipynb \\\n", + "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", + "\n", + "target /home/tbone/Documents/nobook/src/psu410/src/laws_of_thermodynamics/gibbsduhem_equation.ipynb \\\n", + "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", + "\n", + "target /home/tbone/Documents/nobook/src/psu410/src/gibbs_energy_function/index.ipynb \\\n", + "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", + "\n", + "target /home/tbone/Documents/nobook/src/psu410/src/gibbs_energy_function/phases_with_fixed_compositions.ipynb \\\n", + "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", + "\n", + "target /home/tbone/Documents/nobook/src/psu410/src/gibbs_energy_function/phases_with_variable_compositions_random_solutions.ipynb \\\n", + "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", + "\n", + "target /home/tbone/Documents/nobook/src/psu410/src/gibbs_energy_function/phases_with_variable_compositions_solutions_with_ordering.ipynb \\\n", + "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", + "\n", + "target /home/tbone/Documents/nobook/src/psu410/src/gibbs_energy_function/polymer_solutions_and_polymer_blends.ipynb \\\n", + "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", + "\n", + "target ... \\\n", + "nb ... \n", + "\n", + "target /home/tbone/Documents/nobook/src/psu410/src/applications_to_electrochemical_systems/concentrations_activities_and_reference_states_of_electrolyte_species.ipynb \\\n", + "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", + "\n", + "target /home/tbone/Documents/nobook/src/psu410/src/applications_to_electrochemical_systems/electrochemical_cells_and_half_cell_potentials.ipynb \\\n", + "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", + "\n", + "target /home/tbone/Documents/nobook/src/psu410/src/applications_to_electrochemical_systems/aqueous_solution_and_pourbaix_diagram.ipynb \\\n", + "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", + "\n", + "target /home/tbone/Documents/nobook/src/psu410/src/applications_to_electrochemical_systems/application_examples.ipynb \\\n", + "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", + "\n", + "target /home/tbone/Documents/nobook/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/index.ipynb \\\n", + "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", + "\n", + "target /home/tbone/Documents/nobook/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/mms_model_applied_to_thermal_expansion.ipynb \\\n", + "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", + "\n", + "target /home/tbone/Documents/nobook/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_cerium.ipynb \\\n", + "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", + "\n", + "target /home/tbone/Documents/nobook/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_fept.ipynb \\\n", + "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", + "\n", + "target /home/tbone/Documents/nobook/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/concept_of_materials_genome.ipynb \\\n", + "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", + "\n", + "target /home/tbone/Documents/nobook/src/psu410/src/references/index.ipynb \n", + "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", + "\n", + "[1 rows x 48 columns]" + ] + }, + "execution_count": 7, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + " def get_notebooks(documents, contents):\n", + " notebooks = documents.groupby([\"package\", \"doc\"]).apply(lambda df: v4.new_notebook(cells=[v4.new_markdown_cell(x) for x in df.md]))\n", + " return contents.drop_duplicates(subset=\"target\").join(notebooks.rename(\"nb\"))[[\"target\", \"nb\"]].set_index(\"target\")\n", + "\n", + " INTERACTIVE and (notebooks := get_notebooks(documents, contents)).T" + ] + }, + { + "cell_type": "markdown", + "id": "bd45abcb-1a65-4fd5-bfb4-01d27dcb6c75", + "metadata": {}, "source": [ - " def arun(co):\n", - " \"\"\"an async runner that patches nested event loops when working interactively.\"\"\"\n", - " try: return asyncio.get_running_loop().run_until_complete(co)\n", - " except RuntimeError: return __import__(\"nest_asyncio\").apply() or arun(co)" + "## testing" ] }, { "cell_type": "code", - "execution_count": 10, - "id": "ec3b7821-e291-4bbb-b651-82f36a925d43", + "execution_count": 8, + "id": "86f04370-5233-4f0f-b920-06f5af9b92cc", "metadata": { "execution": { - "iopub.execute_input": "2024-01-09T03:52:37.423040Z", - "iopub.status.busy": "2024-01-09T03:52:37.422946Z", - "iopub.status.idle": "2024-01-09T03:52:50.965098Z", - "shell.execute_reply": "2024-01-09T03:52:50.964729Z", - "shell.execute_reply.started": "2024-01-09T03:52:37.423031Z" + "iopub.execute_input": "2024-01-10T20:06:01.538030Z", + "iopub.status.busy": "2024-01-10T20:06:01.537884Z", + "iopub.status.idle": "2024-01-10T20:06:15.569073Z", + "shell.execute_reply": "2024-01-10T20:06:15.568741Z", + "shell.execute_reply.started": "2024-01-10T20:06:01.538004Z" }, "tags": [] }, @@ -298,7 +1822,7 @@ "the table of contents, 9 primary chapters, and the references are captured ... ok\n", "\n", "----------------------------------------------------------------------\n", - "Ran 1 test in 13.535s\n", + "Ran 1 test in 14.024s\n", "\n", "OK\n" ] @@ -306,143 +1830,57 @@ { "data": { "text/plain": [ - "" + "" ] }, - "execution_count": 10, + "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ " class Tests(unittest.TestCase):\n", + " \n", + " @classmethod\n", + " def arun(cls, co):\n", + " \"\"\"an async runner that patches nested event loops when working interactively.\"\"\"\n", + " try: return asyncio.get_running_loop().run_until_complete(co)\n", + " except RuntimeError: return __import__(\"nest_asyncio\").apply() or cls.arun(co)\n", + " \n", " def setUp(self):\n", - " self.md = arun(get_organized_markdown())\n", - " self.toc, self.sections = get_frames().values()\n", - " \n", - " # executing this test case in a notebook updates the global namespace\n", - " INTERACTIVE and globals().update(md=self.md, toc=self.toc, sections=self.sections)\n", + " self.paragraphs = get_paragraphs()\n", + " self.contents = get_contents(self.paragraphs)\n", + " self.documents = self.arun(get_documents(self.paragraphs))\n", + " INTERACTIVE and globals().update(self=self)\n", " \n", " def test_number_of_sections(self):\n", " \"\"\"the table of contents, 9 primary chapters, and the references are captured\"\"\"\n", " # all of the main headings are captured as Heading1s\n", - " assert self.sections.style_id.eq(\"Heading1\").sum() == 10\n", - " assert len(self.md.index.get_level_values(\"chapter\").unique()) == 10\n", + " assert self.paragraphs.style_id.eq(\"Heading1\").sum() == 10\n", + " assert len(self.documents.index.get_level_values(\"package\").unique()) == 11\n", " \n", " INTERACTIVE and unittest.main(argv=[\"discover\"], verbosity=2, exit=False)" ] }, - { - "cell_type": "code", - "execution_count": 11, - "id": "30ca19bc-ea48-47de-add9-59856cfc7529", - "metadata": { - "execution": { - "iopub.execute_input": "2024-01-09T03:52:50.965803Z", - "iopub.status.busy": "2024-01-09T03:52:50.965644Z", - "iopub.status.idle": "2024-01-09T03:52:50.968432Z", - "shell.execute_reply": "2024-01-09T03:52:50.968095Z", - "shell.execute_reply.started": "2024-01-09T03:52:50.965789Z" - }, - "tags": [] - }, - "outputs": [], - "source": [ - " def slugify_index(s): return pipe(s, str.lower, partial(re.sub, \"\\s\", \"_\"), partial(re.sub, \"[^\\.a-z\\_]\", \"\"))" - ] - }, { "cell_type": "markdown", - "id": "3666bff3-621a-44ba-a62f-2e9d5aa14a4c", - "metadata": { - "execution": { - "iopub.execute_input": "2024-01-09T02:31:44.903477Z", - "iopub.status.busy": "2024-01-09T02:31:44.903224Z", - "iopub.status.idle": "2024-01-09T02:31:44.906381Z", - "shell.execute_reply": "2024-01-09T02:31:44.905839Z", - "shell.execute_reply.started": "2024-01-09T02:31:44.903455Z" - }, - "tags": [] - }, - "source": [ - "`get_notebooks` processes a dataframe of markdown into a dataframe of notebooks." - ] - }, - { - "cell_type": "code", - "execution_count": 12, - "id": "03e3aca3-6205-49d3-b02c-a8acaee48457", - "metadata": { - "execution": { - "iopub.execute_input": "2024-01-09T03:52:50.969120Z", - "iopub.status.busy": "2024-01-09T03:52:50.968912Z", - "iopub.status.idle": "2024-01-09T03:52:50.974859Z", - "shell.execute_reply": "2024-01-09T03:52:50.974452Z", - "shell.execute_reply.started": "2024-01-09T03:52:50.969105Z" - }, - "tags": [] - }, - "outputs": [], - "source": [ - " def get_notebooks(md):\n", - " df = md.to_frame().assign(level=md.str.extract(\"^(#{1,})\")[0].apply(len).rename(\"level\"),)\n", - " df = df.join(\n", - " df.set_index(\"level\", append=True)[[]].assign(x=1).unstack().x[[1,2]].fillna(0).cumsum().apply(\n", - " tuple, axis=1\n", - " ).rename(\"group\")\n", - " ).groupby([\"chapter\", \"group\"]).apply(\n", - " lambda x: pandas.Series({\n", - " PSU.parent / slugify_index(x.name[0]) / slugify_index((\n", - " x.index[0][1] if x.level.iloc[0] > 1 else \"index\"\n", - " ) + \".ipynb\") : v4.new_notebook(cells=[\n", - " v4.new_markdown_cell(y.splitlines(True)) for y in x.markdown\n", - " ])\n", - " })\n", - " ).to_frame(\"nb\").reset_index([\"chapter\", \"group\"], drop=True)\n", - " return df" - ] - }, - { - "cell_type": "markdown", - "id": "6d11170a-85d2-4da9-84db-aa7617070287", + "id": "43ec7f7b-17c5-48d1-85db-f0348100e5f7", "metadata": {}, "source": [ - "an asynchronous method to write notebooks `json` to disc." + "### write the notebooks" ] }, { "cell_type": "code", - "execution_count": 13, - "id": "636a096d-aad3-4af3-86ae-15ee357fc90c", - "metadata": { - "execution": { - "iopub.execute_input": "2024-01-09T03:52:50.975634Z", - "iopub.status.busy": "2024-01-09T03:52:50.975379Z", - "iopub.status.idle": "2024-01-09T03:52:50.981198Z", - "shell.execute_reply": "2024-01-09T03:52:50.980865Z", - "shell.execute_reply.started": "2024-01-09T03:52:50.975619Z" - }, - "tags": [] - }, - "outputs": [], - "source": [ - " async def write_nb(target, data):\n", - " target = anyio.Path(target)\n", - " await target.parent.mkdir(exist_ok=True, parents=True)\n", - " return await target.write_text(json.dumps(data, indent=2))" - ] - }, - { - "cell_type": "code", - "execution_count": 14, - "id": "1f7b0f6b-6b56-48a5-8562-baa57317fa09", + "execution_count": 9, + "id": "bda27d6b-3be8-47d6-bb37-1c19c7e0153d", "metadata": { "execution": { - "iopub.execute_input": "2024-01-09T03:52:50.981865Z", - "iopub.status.busy": "2024-01-09T03:52:50.981711Z", - "iopub.status.idle": "2024-01-09T03:52:51.175071Z", - "shell.execute_reply": "2024-01-09T03:52:51.174638Z", - "shell.execute_reply.started": "2024-01-09T03:52:50.981851Z" + "iopub.execute_input": "2024-01-10T20:06:15.569666Z", + "iopub.status.busy": "2024-01-10T20:06:15.569538Z", + "iopub.status.idle": "2024-01-10T20:06:15.595378Z", + "shell.execute_reply": "2024-01-10T20:06:15.595105Z", + "shell.execute_reply.started": "2024-01-10T20:06:15.569656Z" }, "tags": [] }, @@ -450,82 +1888,95 @@ { "data": { "text/plain": [ - "[1193,\n", - " 10326,\n", - " 6567,\n", - " 10446,\n", - " 3959,\n", - " 369,\n", - " 1649,\n", - " 15674,\n", - " 20489,\n", - " 1546,\n", - " 26652,\n", - " 3254,\n", - " 7739,\n", - " 3906,\n", - " 3562,\n", - " 11518,\n", - " 2040,\n", - " 8518,\n", - " 14044,\n", - " 4168,\n", - " 1465,\n", - " 6998,\n", - " 17843,\n", - " 1668,\n", - " 3310,\n", - " 3214,\n", - " 17167,\n", - " 3183,\n", - " 3663,\n", - " 18933,\n", - " 9064,\n", - " 12942,\n", - " 18174,\n", - " 8852,\n", - " 8221,\n", - " 14162,\n", - " 18271,\n", - " 8662,\n", - " 267,\n", - " 45361,\n", - " 12929,\n", - " 7327,\n", - " 4717,\n", - " 230,\n", - " 12381,\n", - " 7680,\n", - " 3372,\n", - " 23097,\n", - " 249,\n", - " 49552,\n", - " 14905]" + "[212,\n", + " 5953,\n", + " 10343,\n", + " 10239,\n", + " 3787,\n", + " 9954,\n", + " 15085,\n", + " 28358,\n", + " 21795,\n", + " 3716,\n", + " 6811,\n", + " 231,\n", + " 6366,\n", + " 2766,\n", + " 27449,\n", + " 33614,\n", + " 1422,\n", + " 13475,\n", + " 6927,\n", + " 3105,\n", + " 15566,\n", + " 9650,\n", + " 10585,\n", + " 13885,\n", + " 12342,\n", + " 2708,\n", + " 6449,\n", + " 3156,\n", + " 1698,\n", + " 2968,\n", + " 9640,\n", + " 1019,\n", + " 5505,\n", + " 8544,\n", + " 3113,\n", + " 8700,\n", + " 1415,\n", + " 4134,\n", + " 3471,\n", + " 13581,\n", + " 12680,\n", + " 18648,\n", + " 1243,\n", + " 5636,\n", + " 11698,\n", + " 7108,\n", + " 3538,\n", + " 11779]" ] }, - "execution_count": 14, + "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ - " 10 and INTERACTIVE and await asyncio.gather(*(\n", - " write_nb(target, nb) for target, nb in get_notebooks(md).nb.items()\n", - " ))\n", - " " + " async def write_json(target, data):\n", + " target = anyio.Path(target)\n", + " await target.parent.mkdir(exist_ok=True, parents=True)\n", + " return await target.write_text(json.dumps(data, indent=2))\n", + " \n", + " async def write_nb(notebooks):\n", + " return await asyncio.gather(*(\n", + " write_json(target, nb) for target, nb in notebooks.nb.items()\n", + " )) \n", + "\n", + "\n", + " 10 and INTERACTIVE and await write_nb(notebooks)" + ] + }, + { + "cell_type": "markdown", + "id": "98838d88-c9bc-4387-9be4-15121804add8", + "metadata": {}, + "source": [ + "### show an interactive sample of the transformed markdown" ] }, { "cell_type": "code", - "execution_count": 15, + "execution_count": 10, "id": "ae617c09-6440-48a7-a015-2c60896fb1a1", "metadata": { "execution": { - "iopub.execute_input": "2024-01-09T03:52:51.175814Z", - "iopub.status.busy": "2024-01-09T03:52:51.175611Z", - "iopub.status.idle": "2024-01-09T03:52:51.186329Z", - "shell.execute_reply": "2024-01-09T03:52:51.185973Z", - "shell.execute_reply.started": "2024-01-09T03:52:51.175797Z" + "iopub.execute_input": "2024-01-10T20:06:15.595877Z", + "iopub.status.busy": "2024-01-10T20:06:15.595750Z", + "iopub.status.idle": "2024-01-10T20:06:15.604266Z", + "shell.execute_reply": "2024-01-10T20:06:15.603930Z", + "shell.execute_reply.started": "2024-01-10T20:06:15.595867Z" }, "tags": [] }, @@ -533,154 +1984,120 @@ { "data": { "text/markdown": [ - "## Ellingham diagram and buffered systems\n", - "\n", - "One type of chemical reactions is between a pure element in liquid or\n", - "solid states and its oxides involving one mole of oxygen gas, i.e.\n", - "\n", - "*Eq. 7‑7* $\\frac{4}{v_{M}}M + O_{2}(gas) = M_{4/v_{M}}O_{2}$\n", - "\n", - "with $v_{M}$ being the valence of the element $M$ in the oxide. Taking\n", - "the pure $M$ and the gaseous $O_{2}$ at the reaction temperature and one\n", - "atmospheric pressure as their respective reference states, both\n", - "activities of the pure $M$ solid or liquid phase and its oxide are\n", - "unity, and the activity of $O_{2}$ equals to its partial pressure in an\n", - "ideal gas. becomes\n", - "\n", - "*Eq. 7‑8*\n", - "$RTlnP_{O_{2}} = \\mathrm{\\Delta}_{\\ }^{0}G = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S$\n", - "\n", - "Based on , one can plot $\\mathrm{\\Delta}_{\\ }^{0}G$ as a function of\n", - "temperature for various oxidation reactions, which is called Ellingham\n", - "diagram. The intercept at $T = 0K$ is given by\n", - "$\\mathrm{\\Delta}_{\\ }^{0}H$, and the slope is represented by\n", - "$- \\mathrm{\\Delta}_{\\ }^{0}S$, depicted by the following equation\n", - "\n", - "*Eq. 7‑9*\n", - "$\\mathrm{\\Delta}_{\\ }^{0}S = S_{M_{4/v_{M}}O_{2}} - S_{O_{2}} - \\frac{4}{v_{M}}S_{M}$\n", - "\n", - "Since the entropy of one mole of $O_{2}$ gas is significantly larger\n", - "than those of the pure element and its oxide when they are in solid or\n", - "liquid states and the entropy difference between the pure element and\n", - "its oxide, the entropy of reaction is thus dominated by the reduction of\n", - "the entropy by one mole of $O_{2}$ gas. Consequently, the entropies of\n", - "reaction are approximately the same for most reactions when the pure\n", - "elements and their oxides are solid as seen in where most lines have\n", - "similar slopes.\n", - "\n", - "Figure ‑: Ellingham Diagram for a number of metal-oxide systems.\n", - "\n", - "For a chemical reaction on the Ellingham diagram at a given temperature,\n", - "$\\mathrm{\\Delta}_{\\ }^{0}G$ can be read from the y-axis of the diagram,\n", - "and the equilibrium partial pressure of $O_{2}$ gas can be calculated\n", - "using . Alternatively, one can plot the left part of for a given\n", - "$P_{O_{2}}$ as a function of temperature on the Ellingham diagram, i.e.\n", - "\n", - "*Eq. 7‑10* $\\mathrm{\\Delta}_{\\ }^{0}{G =}RTlnP_{O_{2}}$\n", - "\n", - "This results in straight lines, representing iso-partial-pressure lines\n", - "of $O_{2}$, with the intercept being zero at $T = 0K$ and slopes of\n", - "$RlnP_{O_{2}}$, which are negative for $P_{O_{2}}$ lower than one\n", - "atmospheric pressure (1atm). The values of $P_{O_{2}}$ are marked on the\n", - "secondary axis on the right of the Ellingham diagram. The intersection\n", - "of the isoactivitiy line and the equilibrium line of the chemical\n", - "reaction thus gives the relation of equilibrium temperature and\n", - "equilibrium partial pressure of $O_{2}$. This is demonstrated in .\n", - "\n", - "Figure ‑: Intersection of iso-partial-pressure lines of $O_{2}$ and\n", - "equilibrium lines in the Ellingham diagram\n", - "\n", - "For each chemical reaction in the Ellingham diagram, the three phases\n", - "are in equilibrium on the line represented by , i.e. metal, metal\n", - "oxides, and O2 gas. For conditions above the line, the value\n", - "of $P_{O_{2}}$ is larger than its equilibrium value, and the metal will\n", - "be oxidized. For conditions below the line, the value of $P_{O_{2}}$ is\n", - "lower than its equilibrium value, and the metal oxide will be reduced.\n", - "Therefore, the metal oxides in the upper part of the Ellingham diagram\n", - "can be reduced by the metals in the lower part of the diagram, and vice\n", - "versus, metals in the lower part of the diagram can be oxidized by the\n", - "metal oxides in the upper part of the diagram. For example, in , Ca can\n", - "reduce all oxides, and Cu2O is the least stable oxide.\n", - "\n", - "From the above Ellingham diagram, it is noted that equilibrium partial\n", - "pressures of $O_{2}$ are very low for most chemical reactions with many\n", - "of them lower than $10^{- 12}\\ atm$. One approach to obtain such a low\n", - "pressure is to use auxiliary reactions containing $O_{2}$ that are easy\n", - "to control and are independent of the equilibrium system of interest\n", - "except the sharing of the oxygen partial pressure. Two common auxiliary\n", - "reactions are the $H_{2}$/$H_{2}O$ and $CO$/$CO_{2}$ mixtures. For the\n", - "$H_{2}$/$H_{2}O$ mixture, the chemical reaction is\n", - "\n", - "*Eq. 7‑11* $2H_{2}(gas) + O_{2}(gas) = 2H_{2}O(gas)$\n", - "\n", - "The equilibrium oxygen partial pressure is obtained as\n", - "\n", - "*Eq. 7‑12*\n", - "$- RTln\\left\\{ {\\frac{1}{P_{O_{2}}}\\left( \\frac{P_{H_{2}O}}{P_{H_{2}}} \\right)}^{2} \\right\\} = \\mathrm{\\Delta}_{\\ }^{0}G = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S = - 498488\\ + 112.972T\\ (J)$\n", - "\n", - "where the $\\mathrm{\\Delta}_{\\ }^{0}H$ and $\\mathrm{\\Delta}_{\\ }^{0}S$\n", - "are taken from the substance thermodynamic database (SSUB4) compiled by\n", - "Scientific Group Thermodata Europe (SGTE) \\[59\\], which are slightly\n", - "dependent on temperature, and the values in are evaluated at 1273K using\n", - "Thermo-Calc \\[60\\]. At any given temperature, one has the following\n", - "relation\n", - "\n", - "*Eq. 7‑13*\n", - "$RTlnP_{O_{2}} = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S - 2RTln\\frac{P_{H_{2}}}{P_{H_{2}O}} = - 498488 + \\left( 112.972 - 2Rln\\frac{P_{H_{2}}}{P_{H_{2}O}} \\right)T$\n", - "\n", - "Its intercept at $T = 0K$ is given by\n", - "$\\mathrm{\\Delta}_{\\ }^{0}H = - 498,488\\ (J)$, and the slope by\n", - "$- \\mathrm{\\Delta}_{\\ }^{0}S - 2Rln\\left( \\frac{P_{H_{2}}}{P_{H_{2}O}} \\right) = 112.972 - 2Rln\\left( \\frac{P_{H_{2}}}{P_{H_{2}O}} \\right)$.\n", - "The values of the $\\frac{P_{H_{2}}}{P_{H_{2}O}}$ ratio are marked on\n", - "another secondary axis on the right of the Ellingham diagram. The\n", - "intersection of the iso-partial-pressure-ratio line and the equilibrium\n", - "line of the chemical reaction gives the relation of equilibrium\n", - "temperature and equilibrium partial pressure ratio\n", - "$\\frac{P_{H_{2}}}{P_{H_{2}O}}$ for desired $P_{O_{2}}$ of the chemical\n", - "equilibrium.\n", - "\n", - "For the $CO$/$CO_{2}$mixture, the chemical reaction is shown below\n", - "\n", - "*Eq. 7‑14* $2CO(gas) + O_{2}(gas) = 2CO_{2}(gas)$\n", - "\n", - "Similar, from the SSUB database, the following equation is obtained\n", - "\n", - "*Eq. 7‑15*\n", - "$- RTln\\left\\{ {\\frac{1}{P_{O_{2}}}\\left( \\frac{P_{{CO}_{2}}}{P_{CO}} \\right)}^{2} \\right\\} = \\mathrm{\\Delta}_{\\ }^{0}G = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S = - 562,927 - 172.020T\\ (J)$\n", - "\n", - "with $\\mathrm{\\Delta}_{\\ }^{0}H$ and $\\mathrm{\\Delta}_{\\ }^{0}S$\n", - "calculated at 1273K, which are also slightly temperature-dependent as in\n", - "the case of the $H_{2}$/$H_{2}O$ mixture. The iso-partial-pressure-ratio\n", - "lines are written as\n", - "\n", - "*Eq. 7‑16*\n", - "$RTlnP_{O_{2}} = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S - 2RTln\\frac{P_{CO}}{P_{{CO}_{2}}} = - 562,927 + \\left( 172.020 - 2Rln\\frac{P_{CO}}{P_{{CO}_{2}}} \\right)T\\ $\n", - "\n", - "Its intercept at $T = 0K$ is given by\n", - "$\\mathrm{\\Delta}_{\\ }^{0}H = - 562,927\\ (J)$, and the slope by\n", - "$172.020 - - 2Rln\\left( \\frac{P_{CO}}{P_{{CO}_{2}}} \\right)$. The values\n", - "of the $\\frac{P_{CO}}{P_{{CO}_{2}}}$ ratio are marked on the third\n", - "secondary axis on the right of the Ellingham diagram. The intersection\n", - "of the iso-partial-pressure-ratio line and the equilibrium line of the\n", - "chemical reaction gives the relation of equilibrium temperature and\n", - "equilibrium partial pressure ratio $\\frac{P_{CO}}{P_{{CO}_{2}}}$ for\n", - "desired $P_{O_{2}}$ of the chemical equilibrium.\n", - "\n", - "Therefore, one can use a mixture of the $H_{2}$/$H_{2}O$ or\n", - "$CO$/$CO_{2}$ to obtain the desired low $P_{O_{2}}$ values using and or\n", - "calculating from the SSUB database. For example,\n", - "$P_{O_{2}} = 10^{- 15}\\ atm$ at 1273K, the calculated values from the\n", - "SSUB database are $\\frac{P_{H_{2}}}{P_{H_{2}O}} \\approx 1.67$ and\n", - "$\\frac{P_{CO}}{P_{{CO}_{2}}} \\approx 2.78$, respectively, in which the\n", - "temperature dependences of $\\mathrm{\\Delta}_{\\ }^{0}H$ and\n", - "$\\mathrm{\\Delta}_{\\ }^{0}S$ and the many more gaseous species in the gas\n", - "phase are taken into account. On the other hand, the reading from the\n", - "Ellingham diagram gives $\\frac{P_{H_{2}}}{P_{H_{2}O}} \\approx 2.0$ and\n", - "$\\frac{P_{CO}}{P_{{CO}_{2}}} \\approx 2.4$, and the agreement with the\n", - "more accurate calculations above is remarkable keeping in mind the\n", - "uncertainties in graphically drawing the lines and reading both values\n", - "in the logarithmic scales from the diagram, indicating the robustness of\n", - "the Ellingham diagram.\n" + "## First and second laws of thermodynamics\n", + "\n", + "A system typically consists of many chemical components. The first law\n", + "of thermodynamics states that the exchanges of heat, work, and\n", + "individual components with its surroundings must obey the law of\n", + "conversation of energy. In the domain of materials science and\n", + "engineering, the energy of interest is at the atomic and molecular\n", + "levels. The energy at the higher and lower levels such as nuclear energy\n", + "and kinetic and potential energies of a rigid body are usually excluded\n", + "from the discussion of thermodynamics of materials.\n", + "\n", + "Let us consider a system receiving an amount of heat, *dQ*, and an\n", + "amount of work, *dW*, and an amount of each independent component *i*,\n", + "*dNi*, from the surroundings. Such a system is called an open\n", + "system in contrast to a closed system when *dNi*=0 for all\n", + "components, i.e. no exchange of mass between the system and the\n", + "surrounding. Other types of systems commonly defined in thermodynamics\n", + "include adiabatic systems without exchange of heat, i.e. *dQ*=0, and\n", + "isolated systems without exchange of any, i.e. *dQ*= *dW*=\n", + "*dNi*=0.\n", + "\n", + "The corresponding change of energy in the system, i.e. the internal\n", + "energy change, *dU*, is formulated in terms of the first law of\n", + "thermodynamics as follows,\n", + "\n", + "Eq. ‑ $dU = dQ + dW + \\sum_{}^{}{H_{i}dN_{i}}$\n", + "\n", + "where $H_{i}$ is the unit energy of component *i* in the surroundings,\n", + "and the summation is for all components in the system, which can be\n", + "controlled independently from the surroundings, i.e. the independent\n", + "components of the system.\n", + "\n", + "It is self-evident that the left-hand side of refers to the change\n", + "inside the system, while its right-hand side is for the contributions\n", + "from the surroundings to the system. In principle, no matter how the\n", + "heat and mass are added, or how the work is done to the system, as far\n", + "as their summation is the same, the change of the internal energy will\n", + "be the same based on the first law of thermodynamics, indicating that\n", + "the system reaches the same state for a closed system. The internal\n", + "energy is thus a state function in a close system as it does not depend\n", + "on how the state is reached.\n", + "\n", + "On the other hand, for the purpose of easy mathematical treatment, a\n", + "reversible process can be considered for a closed system, in which the\n", + "initial state of the system can be restored reversibly without any\n", + "changes left to the surroundings. Therefore, the heat transferred and\n", + "the work done to the system are identical to the heat and work lost by\n", + "the surroundings and vice versa. The classic example of reversible\n", + "processes is the Carnot’s cycle, which is shown in . It consists of four\n", + "reversible processes for a closed system. The four reversible processes\n", + "are compression at constant temperature *T1* (isothermal,),\n", + "compression without heat exchange (adiabatic) ending at *T2*,\n", + "isothermal expansion at *T2*, and adiabatic expansion ending\n", + "at *T1*.\n", + "\n", + "Figure ‑: Schematics of the Carnot’s cycle\n", + "\n", + "The Carnot’s cycle involves a simple type of mechanical work, either\n", + "hydrostatic expansion or compression, with the work that the surrounding\n", + "does to the system represented by\n", + "\n", + "Eq. ‑ $dW = - PdV$\n", + "\n", + "with *P* being the external pressure that the surrounding exerts on the\n", + "system and *V* the volume of the system. It is now necessary to\n", + "differentiate the external and internal variables for further discussion\n", + "with the former representing variables in the surroundings and the\n", + "latter representing variables in the system. For the isothermal\n", + "processes in the Carnot’s cycle, the entropy change of the system, *dS*,\n", + "can be defined as the heat exchange divided by temperature\n", + "\n", + "Eq. ‑ $dS = \\frac{dQ}{T}$\n", + "\n", + "In addition to processes involving heat, work, and mass exchanges\n", + "between the system and the surroundings, there could be internal\n", + "processes taking place inside the system. As the system cannot do work\n", + "to itself, the criterion for whether an internal process can occur\n", + "spontaneously must be related to the heat exchange, which is related to\n", + "the entropy change as shown by .\n", + "\n", + "It is a known fact that heat can spontaneously transfer from a higher\n", + "temperature (*T2*) region to a lower temperature\n", + "(*T1*) region inside a system if the heat conduction is\n", + "allowed, and this process is irreversible because heat cannot be\n", + "conducted from a low temperature region to a high temperature region by\n", + "itself. indicates that for the same amount of heat change, the entropy\n", + "change at *T1* is higher than that at *T2*, and\n", + "the heat conduction thus results in a positive entropy change in the\n", + "system, i.e.\n", + "\n", + "Eq. ‑\n", + "$\\mathrm{\\Delta}S = - \\frac{dQ}{T_{2}} + \\frac{dQ}{T_{1}} = \\frac{dQ}{{T_{2}T}_{1}}\\left( {T_{2} - T}_{1} \\right) > 0$\n", + "\n", + "Consequently, the second law of thermodynamics is obtained, which states\n", + "that for an internal process to take place spontaneously or\n", + "irreversibly, this internal process (*ip*) must have a positive entropy\n", + "production, which can be written in a differential form as follows\n", + "\n", + "Eq. ‑ $d_{ip}S > 0$\n", + "\n", + "From the definition of entropy change shown by , the amount of heat\n", + "produced by this irreversible internal process can be calculated as\n", + "follows\n", + "\n", + "Eq. ‑ $d_{ip}Q = Td_{ip}S$\n", + "\n", + "Let us represent this internal process by *dξ* and define the driving\n", + "force for this internal process by *D*. The work done by this internal\n", + "process is thus *Ddξ*, which is released as heat, i.e.\n", + "\n", + "Eq. ‑ $Dd\\xi = d_{ip}Q = Td_{ip}S$\n", + "\n", + "An irreversible process thus must have a positive driving force in order\n", + "for it to take place spontaneously.\n" ], "text/plain": [ "" @@ -692,331 +2109,221 @@ { "data": { "text/markdown": [ - "## Internal process and differential and integrated driving forces\n", - "\n", - "The driving force for an internal process can be defined as following\n", - "using $U$, $H$, $F$, or $G$ as discussed in Chapter and depending on\n", - "what system variables are kept constant\n", - "\n", - "*Eq. 7‑1*\n", - "$- D = \\left( \\frac{\\partial U}{\\partial\\xi} \\right)_{S,\\ V,N_{i}\\ } = \\left( \\frac{\\partial H}{\\partial\\xi} \\right)_{S,\\ P,N_{i}\\ } = \\left( \\frac{\\partial F}{\\partial\\xi} \\right)_{T,\\ V,N_{i}\\ } = \\left( \\frac{\\partial G}{\\partial\\xi} \\right)_{T,P,N_{i}\\ }$\n", - "\n", - "This can be termed as differential driving force as it relates the\n", - "derivative of energy with respect to an internal process so the change\n", - "is infinitesimally small and does not affect the properties of the\n", - "system significantly. For a system under constant $T$ and $P$, let us\n", - "consider an internal process for forming a new phase $\\alpha$ with the\n", - "composition $x_{i}^{\\alpha}$ and Gibbs energy\n", - "$G_{m}^{\\alpha}\\left( x_{i}^{\\alpha} \\right)$. The differential driving\n", - "force for such an internal process can thus be defined as\n", - "\n", - "*Eq. 7‑2*\n", - "$- D = G_{m}^{\\alpha}\\left( x_{i}^{\\alpha} \\right) - \\sum_{i}^{}x_{i}^{\\alpha}\\mu_{i} = \\sum_{i}^{}x_{i}^{\\alpha}\\left( \\mu_{i}^{\\alpha} - \\mu_{i} \\right)$\n", - "\n", - "where $\\mu_{i}$ is the chemical potential of component *i* in the\n", - "system. This may also be called as nucleation driving force as if the\n", - "$\\alpha$ phase is nucleating in the system. As discussed in Chapter ,\n", - "chemical potentials are the intercepts on the Gibbs energy axis by the\n", - "tangent plane of Gibbs energy. thus represents the distance between the\n", - "tangent planes of the original system and the $\\alpha$ phase at the\n", - "composition of the $\\alpha$ phase. Evidently, this distance is at its\n", - "maximum when the two tangent planes are parallel to each other, i.e. the\n", - "commonly called parallel tangent construction when evaluating nucleation\n", - "driving force.\n", - "\n", - "The situation is different for chemical reactions where the amount of\n", - "each component in reactants is the same as that in products, i.e. there\n", - "is a mass balance between reactants and products. The driving force for\n", - "a chemical reaction is defined by the Gibbs energy difference between\n", - "the reactants and products as if all the reactants are transferred to\n", - "the products. This driving force may thus be called integrated driving\n", - "force as it describes the energy difference of a system under two\n", - "different states, i.e.\n", - "\n", - "*Eq. 7‑3*\n", - "$- \\int_{}^{}{Dd\\xi} = \\mathrm{\\Delta}G = \\sum_{p}^{}{n_{p}G_{p}} - \\sum_{r}^{}{n_{r}G_{r}}$\n", - "\n", - "where superscripts $p$ and $r$ denote products and reactants, $n_{p}$\n", - "and $n_{r}$ their corresponding moles, $G_{p}$and $G_{r}$ the Gibbs\n", - "energies in per mole of formula of their respective stoichiometries as\n", - "written in the chemical reaction. Conventionally, the products and\n", - "reactants are represented by species or stoichiometric compounds rather\n", - "than individual phases. This is particularly evident when various\n", - "gaseous species are considered in a chemical reaction. Consequently,\n", - "$G_{p}$and $G_{r}$ in represent the chemical potentials of product and\n", - "reactant species. For species with a fixed composition, its chemical\n", - "potential is the same as its Gibbs energy as shown by . For a species in\n", - "a solution phase, its chemical potential is related to its activity as\n", - "shown in . can thus be further written as\n", - "\n", - "*Eq. 7‑4*\n", - "$\\mathrm{\\Delta}G = \\sum_{p}^{}{n_{p}_{\\ }^{0}G_{p}} - \\sum_{r}^{}{n_{r}_{\\ }^{0}G_{r}} + RTln\\frac{\\prod_{p}^{}\\left( a_{p} \\right)^{n_{p}}}{\\prod_{r}^{}\\left( a_{r} \\right)^{n_{r}}} = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{\\prod_{p}^{}\\left( a_{p} \\right)^{n_{p}}}{\\prod_{r}^{}\\left( a_{r} \\right)^{n_{r}}}$\n", - "\n", - "It is evident from that for stoichiometric phases in a chemical\n", - "reaction, their activities equal to one. At equilibrium, the integrated\n", - "driving force becomes zero, i.e. $\\mathrm{\\Delta}G = 0$, and is\n", - "re-arranged to become\n", - "\n", - "*Eq. 7‑5*\n", - "$- RTln\\frac{\\prod_{p}^{}\\left( a_{p} \\right)^{n_{p}}}{\\prod_{r}^{}\\left( a_{r} \\right)^{n_{r}}} = - RTlnK_{e} = \\mathrm{\\Delta}_{\\ }^{0}G = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S$\n", - "\n", - "where $K_{e}$ is often called reaction constant relating the activities\n", - "of products and reactants at equilibrium. In a system with\n", - "$\\frac{\\prod_{p}^{}\\left( a_{p} \\right)^{n_{p}}}{\\prod_{r}^{}\\left( a_{r} \\right)^{n_{r}}} > K_{e}$,\n", - "the chemical reaction goes to left, and the products are decomposed,\n", - "while\n", - "$\\frac{\\prod_{p}^{}\\left( a_{p} \\right)^{n_{p}}}{\\prod_{r}^{}\\left( a_{r} \\right)^{n_{r}}} < K_{e}$,\n", - "the chemical reaction goes to right, and the products are formed. is\n", - "often recast into the following form by dividing $–RT$ on both sides of\n", - "the equation\n", - "\n", - "*Eq. 7‑6*\n", - "$\\ln K_{e} = - \\frac{\\mathrm{\\Delta}_{\\ }^{0}H}{RT} + \\frac{\\mathrm{\\Delta}_{\\ }^{0}S}{R}$\n", - "\n", - "With $\\ln K_{e}$ plotted with respect to $1/T$, indicates that the slope\n", - "is $- \\mathrm{\\Delta}_{\\ }^{0}H$/R and the intercept on the y axis is\n", - "$\\mathrm{\\Delta}_{\\ }^{0}{S/R}$ as shown in .\n", - "\n", - "Figure ‑: $\\ln K_{e}$ plotted with respect to $1/T$ for several M-O\n", - "systems at 1 bar and $K_{e}$ represented by the partial pressure ratio\n", - "of CO2 and CO.\n" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/markdown": [ - "## Maximum reaction rate and chemical transport reactions \n", - "\n", - "Equilibrium thermodynamics can be used to calculate the maximum rates of\n", - "reaction that are possible in dynamically reacting systems, such as when\n", - "a corrosive gas passes over a heated sample. Other examples of such\n", - "reactions include the reduction of a metal oxide in flowing hydrogen,\n", - "the evaporation of a material in a vacuum, and the deposition of a thin\n", - "film by a chemical vapour deposition process. The basic assumption used\n", - "in calculating these maximum reaction rates is that local equilibrium\n", - "exists at the location of the considered reaction.\n", - "\n", - "This section examines several examples in which maximum reaction rates\n", - "are calculated. The system can typically be divided into three regions:\n", - "(1) the input region, which is usually near room temperature, (2) the\n", - "high temperature region in which the primary reaction of interest is\n", - "occurring, and (3) the exit region. Such a system is almost always at\n", - "constant pressure throughout the system. Since the three regions have\n", - "different temperatures, the key is to use the mass conservation of the\n", - "carrier gas in all three regions. With the input gas at $T = 298K$ and\n", - "$P = 1atm$, the volume of one mole of ideal gas is\n", - "$0.0244\\left( m^{3} \\right) = 24.4(liter)$. The number of moles of input\n", - "gas flowing through the system can be written as\n", - "\n", - "*Eq. 7‑17*\n", - "$n_{gas}^{0} = \\frac{Pf_{gas}}{RT} = \\frac{f_{gas}}{24.4} = 0.0409f_{gas}$\n", - "\n", - "where $f_{gas}$ is the input gas flow rate in liter per unit time at\n", - "$T = 298K$ and $P = 1atm$, and $R$ the gas constant.\n", - "\n", - "The first example is the evaporation of water in a flowing stream of dry\n", - "hydrogen. A schematic diagram of the system being considered is given in\n", - ". The goal of the calculation is to determine the maximum rate of\n", - "$H_{2}O(liquid)$ loss, $n_{H_{2}O}^{\\ }$, through vaporization in a\n", - "flowing stream of $H_{2}$, e.g. for generating a given $H_{2}O/H_{2}$\n", - "ratio for producing a given $O_{2}$ pressure in a high temperature\n", - "system as related to the Ellingham diagram discussed in Chapter . The\n", - "maximum rate is determined by saturating the $H_{2}$ with the\n", - "equilibrium vapour pressure of $H_{2}O(liquid)$ with the number of moles\n", - "of $H_{2}$ being $n_{H_{2}}^{0} = 0.0409f_{H_{2}}$ and $f_{H_{2}}$ being\n", - "the input flow rate of $H_{2}$ in liter per unit time.\n", - "\n", - "Figure ‑: Schematic diagram of vaporization of water in a flowing stream\n", - "of dry hydrogen\n", - "\n", - "In the first input region, the total number moles of gas is\n", - "$N = n_{H_{2}}^{0}$. In the second high temperature region with the\n", - "temperature and pressure being $T_{sys}$ and $P_{sys}$, the total number\n", - "moles of gas is $N = n_{H_{2}}^{0} + n_{H_{2}O}^{\\ }$, which is the same\n", - "for the third exit region with $T_{exit}$. To avoid condensation, one\n", - "needs to maintain $T_{exit} > T_{sys}$. The vapour pressure of $H_{2}O$\n", - "at $T_{sys}$, $P_{H_{2}O\\ }$, can be obtained from equilibrium\n", - "thermodynamic calculations. Since the $H_{2}$ and $H_{2}O$ are occupying\n", - "the same volume at the same temperature, the maximum number of moles\n", - "$H_{2}O$ can be calculated from the following relation based on the\n", - "ideal gas law\n", - "\n", - "*Eq. 7‑18*\n", - "$n_{H_{2}O}^{\\ } = \\frac{P_{H_{2}O}^{\\ }}{P_{H_{2}}^{\\ }}n_{H_{2}}^{0} = \\frac{P_{H_{2}O}^{\\ }}{P_{sys}^{\\ } - P_{H_{2}O}^{\\ }}n_{H_{2}}^{0} = \\frac{P_{H_{2}O}^{\\ }}{P_{sys}^{\\ } - P_{H_{2}O}^{\\ }}\\frac{f_{H_{2}}}{24.4}$\n", - "\n", - "For $P_{sys}^{\\ } = 101325Pa$, one can calculate $P_{H_{2}O}^{\\ }$ and\n", - "plot $\\frac{n_{H_{2}O}^{\\ }}{f_{H_{2}}}$ as a function of temperature\n", - "from the SSUB database as shown in . The corresponding partial pressure\n", - "ratio, $P_{H_{2}O}/P_{H_{2}}$ , is plotted in , which can be used to\n", - "calculate $P_{O_{2}}^{\\ }$ at any given temperatures. An example is\n", - "shown in with $T_{sys} = 348.15K$ and $P_{H_{2}O}/P_{H_{2}} = 0.607$.\n", - "\n", - "Figure ‑: Ratio of maximum number of moles $H_{2}O$ with respect to\n", - "hydrogen flow rate, $\\frac{n_{H_{2}O}^{\\ }}{f_{H_{2}}}$, plotted as a\n", - "function of temperature.\n", - "\n", - "Figure ‑: Partial pressure ratio, $P_{H_{2}O}/P_{H_{2}}$, corresponding\n", - "to .\n", - "\n", - "Figure ‑: Partial pressure of oxygen, $P_{O_{2}}^{\\ }$, as a function of\n", - "temperature with $T_{sys} = 348.15K$ and $P_{H_{2}O}/P_{H_{2}} = 0.607$.\n", - "\n", - "The second example is the corrosion of $SiO_{2}$(s) by flowing $H_{2}$\n", - "gas at high temperatures. Considering $T_{sys}^{\\ } = 1700K$,\n", - "$P_{sys}^{\\ } = 101325Pa$, and $f_{H_{2}} = 1\\ liter/min$, the system is\n", - "thus defined with 0.0409 moles of $H_{2}$ and an equilibrium between the\n", - "gas phase and the tridymite $SiO_{2}$. The equilibrium calculation gives\n", - "0.04092 moles of gas with its constitutions as\n", - "$H_{2}:H_{2}O:SiO = 0.998887:4.833 \\bullet 10^{- 4}:4.832 \\bullet 10^{- 4}$.\n", - "The corrosion rate of SiO2(s) is thus\n", - "$0.04092x4.832 \\bullet 10^{- 4} = 1.98 \\bullet 10^{- 5}mol/min = 1.19gram/min.$\n", - "\n", - "Another example is to consider that the flowing $CO$ gas of 298K and\n", - "1atm ($= 101325Pa$) at a rate of $1\\ liter/min$ passes through and\n", - "equilibrates with single phase $C(s)$ at 1500K. The equilibrium system\n", - "is defined by $T = 1500K$, $P = 1atm$, and 0.0409 moles of $CO$ with the\n", - "equilibrium between the gas phase and C(s). The equilibrium calculation\n", - "gives 0.040872 moles of gas phase with the mole fraction of $CO$ being\n", - "0.999327, resulting in the loss of *CO* or the deposition of *C(s)* at a\n", - "rate of\n", - "$0.0409 - 0.040872 \\bullet 0.999327 = 5.55 \\bullet 10^{- 5}mole/min$.\n", - "\n", - "A chemical transport reaction is a reaction in which a condensed phase\n", - "reacts with a gas phase to form vapour-phase products, which in turn\n", - "undergo the reverse reformation of the condensed phase. Two well-known\n", - "examples of such reactions are\n", - "\n", - "*Eq. 7‑19* $M(s) + \\frac{n}{2}I_{2}(g) = MI_{n}(g)$\n", - "\n", - "*Eq. 7‑20* $Ni(s) + 4CO(g) = Ni{(CO)}_{4}(g)$\n", - "\n", - "which are used in the purification of metals by the iodide process and\n", - "in the purification of nickel by the Mond-Langer process. In both\n", - "processes the forward reaction is favoured by lower temperatures and the\n", - "reverse reaction by higher temperatures, resulting in the deposition of\n", - "the metal. The most common technique for causing a chemical transport of\n", - "a condensed substance makes use of the temperature dependence of the\n", - "equilibrium constant. As was discussed previously, the enthalpy of\n", - "reaction, $\\mathrm{\\Delta}_{\\ }^{0}H$, determines the manner in which\n", - "$K_{e}$ changes with temperature (see ). The value of $K_{e}$ increases\n", - "with increasing *T* for $\\mathrm{\\Delta}_{\\ }^{0}H > 0$, $K_{e}$\n", - "decreases with increasing *T* for $\\mathrm{\\Delta}_{\\ }^{0}H < 0$, and\n", - "$K_{e}$ is independent of T for $\\mathrm{\\Delta}_{\\ }^{0}H = 0$. The\n", - "$\\mathrm{\\Delta}_{\\ }^{0}H$ and $\\mathrm{\\Delta}_{\\ }^{0}S$ values for\n", - "chemical transport reactions may be either positive or negative. For\n", - "reactions by both are positive, and for reactions by both are negative.\n", - "\n", - "In a typical experiment the starting solid is located at the point in a\n", - "temperature gradient that corresponds to the largest $K_{e}$ value for\n", - "the experimental condition. As the gaseous species migrate to other\n", - "locations in the system with temperatures corresponding to lower $K_{e}$\n", - "values, the reverse reaction occurs to satisfy the new equilibrium\n", - "requirements, and the solid phase is deposited. The dependence of\n", - "$K_{e}$ on $\\mathrm{\\Delta}_{\\ }^{0}H$ results in a material transport\n", - "from hot to cold for $\\mathrm{\\Delta}_{\\ }^{0}H > 0$ (the same as for\n", - "vaporization-condensation reactions), from cold to hot for\n", - "$\\mathrm{\\Delta}_{\\ }^{0}H < 0$, and no transport for\n", - "$\\mathrm{\\Delta}_{\\ }^{0}H = 0$.\n", - "\n", - "The success of a particular reaction in causing an appreciable transport\n", - "of a condensed phase depends mainly upon the partial pressure gradients\n", - "or concentration gradients of the gaseous species in the system. A\n", - "reaction whose equilibrium is extreme toward either the reactant side or\n", - "the product side will not give an appreciable transport of material. The\n", - "concentration gradients are too small in such a system. Reactions with\n", - "equilibrium constants near unity at the experimental temperatures\n", - "usually give the largest transport since small changes in $K_{e}$ cause\n", - "large changes in concentrations. The general condition required for\n", - "obtaining a $K_{e}$ value near unity at a reasonable temperature is that\n", - "$\\mathrm{\\Delta}_{\\ }^{0}H$ and $\\mathrm{\\Delta}_{\\ }^{0}S$ both have\n", - "the same sign, resulting from the equalities of .\n" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/markdown": [ - "## Trends of entropies of reactions\n", + "## Combined law of thermodynamics and equilibrium conditions\n", + "\n", + "For a system with an irreversible internal process taking place, the\n", + "entropy change in the system thus consists of three parts: the heat\n", + "exchange with the surrounding defined by , the entropy production due to\n", + "the internal process represented by , and the entropy of mass exchange\n", + "with the surrounding. The total entropy change of the system can thus be\n", + "written as follows\n", + "\n", + "Eq. ‑ $dS = \\frac{dQ}{T} + d_{ip}S + \\sum_{}^{}{S_{i}dN_{i}}$\n", + "\n", + "where $S_{i}$ is the unit entropy of component *i* in the surroundings,\n", + "often called partial entropy of component *i*, and will be further\n", + "discussed in Chapter\n", + "\n", + "Combining and and re-arranging, one obtains\n", + "\n", + "Eq. ‑ $dQ = TdS - Dd\\xi - \\sum_{}^{}{TS_{i}dN_{i}}$\n", + "\n", + "Inserting and into yields the combined law of thermodynamics from the\n", + "first and second laws of thermodynamics,\n", + "\n", + "Eq. ‑\n", + "$dU = TdS - PdV + \\sum_{}^{}\\left( H_{i} - TS_{i} \\right){dN}_{i} - Dd\\xi$\n", + "\n", + "The internal energy of the system is thus a function of *S*, *V*,\n", + "*Ni* and *ξ* of the system, which are called natural\n", + "variables of the internal energy, i.e. *U*(*S*,*V*,*Ni*,*ξ*).\n", + "The other variables are dependent variables and can be represented by\n", + "partial derivatives of the internal energy with respect to their\n", + "respective natural variables with other natural variables kept constant\n", + "as shown below\n", + "\n", + "Eq. ‑\n", + "$T = \\left( \\frac{\\partial U}{\\partial S} \\right)_{V,\\ N_{i},\\ \\xi}$\n", + "\n", + "Eq. ‑\n", + "$- P = \\left( \\frac{\\partial U}{\\partial V} \\right)_{S,\\ N_{i},\\ \\xi}$\n", + "\n", + "Eq. ‑\n", + "$\\mu_{i} = H_{i} - TS_{i} = \\left( \\frac{\\partial U}{\\partial N_{i}} \\right)_{S,\\ V,N_{j \\neq i},\\ \\xi} = U_{i}$\n", + "\n", + "Eq. ‑\n", + "$- D = \\left( \\frac{\\partial U}{\\partial\\xi} \\right)_{S,\\ V,N_{i}\\ }$\n", + "\n", + "In , a new variable, $\\mu_{i}$, is introduced. It is called chemical\n", + "potential and defined as the internal energy change with respect of the\n", + "addition of the component *i* when the entropy, volume and the amount of\n", + "other components of the system are kept constant. It may be worth\n", + "pointing out that for a system at equilibrium, i.e. $d_{ip}S = 0$, and\n", + "with constant entropy, $dS = 0$, if it exchanges mass with the\n", + "surroundings, $dN_{i} \\neq 0$, the system must also exchange heat with\n", + "the surroundings at the same time in order to keep the entropy invariant\n", + "as demonstrated by .\n", + "\n", + "The pairs of the natural variables and their corresponding partial\n", + "derivatives are called conjugate variables, i.e. *S* and *T*, *V* and\n", + "*–P*, *Ni* and $\\mu_{i}$, and *ξ* and *–D*. There are minus\n", + "sign in front of *P* and *D* as the increase of volume and the progress\n", + "of the internal process decrease the internal energy of the system. The\n", + "importance of this conjugate relation will be evident when various forms\n", + "of combined thermodynamic laws and various types of phase diagrams are\n", + "introduced in the book.\n", + "\n", + "The last pair of conjugate variables, *ξ* and *–D*, is worthy of further\n", + "discussion. Based on the second law of thermodynamics, i.e. , no\n", + "internal processes take place spontaneously if there is no entropy\n", + "productions, i.e. D≤0 or *dξ*=0 and *D*\\>0. With D≤0, there is no\n", + "driving for any internal processes, and the system is at a full\n", + "equilibrium state. The last term in drops off, and *ξ* becomes a\n", + "dependent variable of the system and can be calculated from the\n", + "equilibrium conditions. With *dξ*=0 and *D*\\>0, the system is under a\n", + "constrained equilibrium or freezing-in condition when the internal\n", + "process is constrained not to take place, and *ξ* remains to be an\n", + "independent variable of the system.\n", + "\n", + "These two cases represent the two branches of thermodynamics:\n", + "equilibrium, reversible thermodynamics and irreversible thermodynamics.\n", + "It is clear from the above discussions that these two branches are\n", + "identical if the internal energy is not only a function of *S*, *V*, and\n", + "*Ni* , but also any internal process variable *ξ*. This means\n", + "that one should be able to evaluate the internal energy of a system for\n", + "any freezing-in equilibrium conditions in addition to the full\n", + "equilibrium condition. In the rest of the book, the freezing-in\n", + "equilibrium and full equilibrium are not differentiated unless\n", + "specified.\n", + "\n", + "As the mechanical work under hydrostatic pressure is very important in\n", + "experiments, let us define a new quantity called enthalpy as follows\n", + "\n", + "Eq. ‑ $H = U + PV$\n", + "\n", + "Its differential form can be obtained from as\n", + "\n", + "Eq. ‑ $dH = dU + d(PV) = dQ + VdP + \\sum_{}^{}H_{i}{dN}_{i}$\n", + "\n", + "There are two significant consequences of the above equation. First, for\n", + "a close system under constant pressure, i.e. ${dN}_{i} = dP = 0$, one\n", + "has $dH = dQ$. This implies that the enthalpy change in a system is\n", + "equal to the heat exchange between the system and the surrounding of the\n", + "system, which is why enthalpy and heat are often exchangeable in the\n", + "literature. Second, for an adiabatic system under constant pressure,\n", + "i.e. $dQ = dP = 0$, can be re-arranged to the following equation\n", + "\n", + "Eq. ‑\n", + "$H_{i} = \\left( \\frac{\\partial H}{\\partial N_{i}} \\right)_{N_{j \\neq i,\\ \\ dQ = dP = 0}}$\n", + "\n", + "$H_{i}$ is thus the partial enthalpy of component *i* and will be\n", + "further discussed in Chapter . The chemical potential of component *i*\n", + "defined in is thus related to the partial enthalpy and partial entropy\n", + "of the component.\n", + "\n", + "To further define equilibrium conditions of a system, consider a\n", + "homogeneous system in a state of internal equilibrium, i.e. no\n", + "spontaneous internal processes are possible with $Dd\\xi = 0$, and\n", + "becomes\n", + "\n", + "Eq. ‑\n", + "$dU = TdS - PdV + \\sum_{}^{}\\mu_{i}{dN}_{i} = \\sum_{}^{}{Y_{i}dX_{i}}$\n", + "\n", + "where *X* represents *S*, *V*, *Ni*, and *Y* their conjugate\n", + "variables *T*, *-P*, $\\mu_{i}$. The state of the system with *c*\n", + "independent components is completely determined by the *c+2* variables,\n", + "i.e. *S*, *V*, and *Ni* with *i* from 1 to *c*.\n", + "\n", + "To simplify the situation, let us limit the discussion to an isolated\n", + "equilibrium system, i.e. $dU = 0$, and conduct a virtual internal\n", + "experiment inside the system by moving an infinitesimal amount of\n", + "$X_{i}$, ${dX}_{i}$, with other $X_{j}$ kept constant, from one region\n", + "of the system to another region of the system as schematically shown in\n", + ".\n", + "\n", + "Figure ‑: Virtual experiment for a system at equilibrium\n", "\n", - "The reaction entropy, $\\mathrm{\\Delta}_{\\ }^{0}S$ in , plays an\n", - "important role in determining equilibria of high-temperature reactions.\n", - "The most important single factor that determines the entropy of a\n", - "reaction is the net change in the number of moles of gas as briefly\n", - "mentioned in the discussion of the Ellingham diagram above. The reason\n", - "this is true can be explained as follows.\n", + "As the system is homogeneous and at equilibrium,\n", + "$- dX_{i}^{'} = dX_{i}^{\"} = dX_{i}$. The total change of the internal\n", + "energy of this internal process is the combination of the changes in the\n", + "two regions, i.e.\n", "\n", - "The entropy of a substance can be thought of as being the sum of four\n", - "parts: (i) translational, (ii) rotational, (iii) vibrational, and (iv)\n", - "electronic. The translational entropy of a gas is the largest entropy\n", - "term under most conditions. To the extent that the other contributions\n", - "cancel between reactants and products, the entropy of reaction is\n", - "determined by the change in the number of moles of gaseous molecules.\n", - "Based on the literature data or calculations from the SSUB database, the\n", - "net change in the number of moles of gas in a reaction results\n", - "approximately in an entropy of reaction of about 175±45 J/K/mole-gas at\n", - "298K for many halides and oxides. The chemical reactions of and\n", - "discussed above both reduce the gas by one mole, and their entropies of\n", - "reaction are -113 and -172 J/K at 1273K, and -89 and -173J/K at 298K,\n", - "respectively, indicating that the chemical reaction of is an exception\n", - "of the empirical rule. For chemical reactions shown in the Ellingham\n", - "diagram, their entropies of reaction follow this empirical rule pretty\n", - "well with some of them shown in calculated from the SGTE database.\n", + "Eq. ‑\n", + "$dU = dU^{'} + dU^{\"} = Y_{i}^{'}dX_{i}^{'} + Y_{i}^{\"}dX_{i}^{\"} = \\left( - Y_{i}^{'} + Y_{i}^{\"} \\right)dX_{i} = 0$\n", "\n", - "Table ‑: Entropies of reactions with gas at 298.15K, J/K\n", + "Therefore, $Y_{i}^{'} = Y_{i}^{\"}$ for *T*, *-P*, and $\\mu_{i}$,\n", + "indicating that *T*, *-P*, and $\\mu_{i}$ are homogeneous in the system,\n", + "respectively, and are thus named as potentials of the system.\n", + "Furthermore these potentials are independent of the size of the system\n", + "and are often referred to as intensive variables in the literature. On\n", + "the other hand, all *X:s*, i.e. *S*, *V*, and *Ni*, are\n", + "proportional to the size of the system and may be normalized with\n", + "respect to the size of the system, usually in terms of total moles,\n", "\n", - "Reaction: Si+O2 =SiO2 -182\n", + "Eq. ‑ $N = \\sum_{}^{}N_{i}$\n", "\n", - "Reaction: Ti+O2=TiO2 -185\n", + "They are thus called molar quantities and often referred to as extensive\n", + "variables, and the respective normalized variables are molar entropy,\n", + "molar volume, and mole fractions, defined as follows\n", "\n", - "Reaction: 2Mg+O2=2MgO -217\n", + "Eq. ‑ $S_{m} = \\frac{S}{N}$\n", "\n", - "Reaction: 2Ca+O2=2CaO -212\n", + "Eq. ‑ $V_{m} = \\frac{V}{N}$\n", "\n", - "Reaction: 2Mn+O2=2MnO -150\n", + "Eq. ‑ $x_{i} = \\frac{N_{i}}{N}$\n", "\n", - "Since the entropy of a reaction is primarily determined by the net\n", - "change in the number of moles of gas, the entropies for reactions\n", - "involving only condensed phases must be small. The entropies of fusion\n", - "of monatomic solids are usually in the range 8-15 J/K/mole-atom as shown\n", - "for some elements in . Most metals and many ionic salts have values that\n", - "lie in this range when given in terms of per mole of atom of material.\n", - "There are few exceptions such as silicon and boron shown in the table.\n", - "For solid-state reactions, the average values can be approximated as\n", - "0±8J/K/mole-atom as also shown in the table.\n", + "Consider a small subsystem in this homogeneous system at equilibrium and\n", + "let the subsystem grow in size. The entropy, volume, and mass enclosed\n", + "in the subsystem increase as follows\n", "\n", - "Table ‑: Entropies of reactions of condensed phases at 298.15K, J/K\n", + "Eq. ‑ ${dS = S}_{m}dN$\n", "\n", - "Reaction: Si(s)=Si(l) 29.762\n", + "Eq. ‑ $dV = V_{m}dN$\n", "\n", - "Reaction: Ti(s2)=Ti(l) 7.288\n", + "Eq. ‑ $dN_{i} = x_{i}dN$\n", "\n", - "Reaction: Mg(s2)=Mg(l) 9.184\n", + "The corresponding change in the internal energy of the subsystem becomes\n", "\n", - "Reaction: Ca(s2)=Ca(l) 7.659\n", + "Eq. ‑\n", + "$dU = TdS - PdV + \\sum_{}^{}\\mu_{i}{dN}_{i} = \\left( TS_{m} - PV_{m} + \\sum_{}^{}\\mu_{i}x_{i} \\right)dN = U_{m}dN$\n", "\n", - "Reaction: Mn(s2)=Mn(l) 11.443\n", + "By integration one obtains the integral form of the internal energy as\n", "\n", - "Reaction: W(s)=W(l) 14.158\n", + "Eq. ‑\n", + "$U = TS - PV + \\sum_{}^{}\\mu_{i}N_{i} = \\left( TS_{m} - PV_{m} + \\sum_{}^{}\\mu_{i}x_{i} \\right)N = U_{m}N$\n", "\n", - "Reaction: B(s)=B(l) 21.380\n", + "Similarly, the molar enthalpy can be defined as follows\n", "\n", - "Reaction: 3Fe+C=CFe3 17.060\n", + "Eq. ‑\n", + "$H = U + PV = U_{m}N + PV_{m}N = \\left( U_{m} + PV_{m} \\right)N = H_{m}N$\n", "\n", - "Reaction: S+Mn=MnS 13.909\n", + "In case a potential is not homogeneous in a system, the system will not\n", + "be in a state of equilibrium. Let us consider the same virtual\n", + "experiment as shown in for an isolated system that is not in\n", + "equilibrium, i.e. by moving an infinitesimal amount of $X_{i}$,\n", + "${dX}_{i}$, with other $X_{j}$ kept constant, from one region of the\n", + "system to another region of the system with the two regions having\n", + "different potentials. The total internal energy change equals to zero as\n", + "the virtual experiment has $dU = 0$. Similarly, each region can be\n", + "considered to be homogeneous by itself, and one has\n", + "$- dX_{i}^{'} = dX_{i}^{\"} = dX_{i}$. The total internal energy change\n", + "in the system is thus the sum of these two regions plus the entropy\n", + "production due to the internal process with $d\\xi = dX_{i}$, i.e.\n", "\n", - "Reaction: NiO+Fe2O3=Fe2NiO4\n", - "0.464\n" + "Eq. ‑\n", + "$dU = dU^{'} + dU^{\"} + Dd\\xi = Y_{i}^{'}dX_{i}^{'} + Y_{i}^{\"}dX_{i}^{\"} + Dd\\xi = \\left( - Y_{i}^{'} + Y_{i}^{\"} \\right)dX_{i} + Dd\\xi = 0$\n", + "\n", + "Consequently, one obtains the following\n", + "\n", + "*Eq. 1‑31* $D = Y_{i}^{'} - Y_{i}^{\"}$\n", + "\n", + "The driving force thus represents the difference of the potential at the\n", + "two regions, and the internal process is to eliminate inhomogeneity of\n", + "the potential with the heat transfer from high temperature regions to\n", + "low temperature regions, volume shrink of low pressure regions (high\n", + "$–P$) and volume expansion of high pressure regions (low $–P$), and\n", + "transport of components from high chemical potential regions to low\n", + "chemical potential regions.\n" ], "text/plain": [ "" @@ -1028,255 +2335,272 @@ { "data": { "text/markdown": [ - "## Aqueous solution and Pourbaix diagram\n", - "\n", - "The importance of aqueous solutions in all aspects of life is so well\n", - "known and needs not be discussed further. Since many electrochemical\n", - "processes involve electrolyte solutions in an aqueous solvent,\n", - "electrochemical processes including water, hydrogen, and/or oxygen are\n", - "discussed in more details. The hydrogen-oxygen cell can be described for\n", - "both acidic electrolytes and alkaline electrolytes. With acidic\n", - "electrolytes, H+ is in much higher concentrations than\n", - "OH-, and thus half-cell reactions with H+ as an\n", - "ionic transport species are more important than those involving\n", - "OH-. The reverse is true for alkaline electrolytes that\n", - "contain high OH-concentrations. Other than for nearly neutral\n", - "acid-base systems, either H+ or OH- dominates the\n", - "other by several orders of magnitude as can be seen from the value of\n", - "the 298 K dissociation constant for H2O:\n", - "\n", - "*Eq. 8‑35* H2O(l) = H+(aq) + OH-(aq)\n", - "\n", - "with the reaction constant being Ke =\n", - "\\[H+\\]\\[OH-\\] = 10-14 and\n", - "$\\mathrm{\\Delta}_{\\ }^{0}G$= -RT *ln* Ke *β*= +79,908 J. By\n", - "convention, one defines pH = - log \\[H+\\] and pOH = - log\n", - "\\[OH-\\], and then pH + pOH = 14.\n", - "\n", - "Under acidic electrolyte conditions of low pH (high \\[H+\\]\n", - "concentrations) the anode reaction in a hydrogen-oxygen cell is:\n", - "\n", - "*Eq. 8‑36* ½ H2(g) = H+(aq) + e-\n", - "\n", - "with ε1o = 0.0 V and\n", - "$\\mathrm{\\Delta}_{\\ }^{0}G_{1}$= 0 J. The corresponding cathode\n", - "(reduction) reaction is:\n", - "\n", - "*Eq. 8‑37* 2 H+(aq) + ½ O2(g) + 2 e- =\n", - "H2O(l)\n", - "\n", - "with ε2o = 1.229 V and\n", - "$\\mathrm{\\Delta}_{\\ }^{0}G_{2}$ = -2\\*1.229\\*96,485 J = -237,160 J. The\n", - "net cell reaction for acidic electrolytes is:\n", - "\n", - "*Eq. 8‑38* H2(g) + ½ O2(g) = H2O(l)\n", - "\n", - "with εocell = 1.229 V and\n", - "$\\mathrm{\\Delta}_{\\ }^{0}G_{cell}$ = -2\\*1.229\\*96,485 J = -237,160 J\n", - "\n", - "Under alkaline electrolyte conditions of high pH (high\n", - "\\[OH-\\] concentrations) the anode reaction in a\n", - "hydrogen-oxygen cell is:\n", - "\n", - "*Eq. 8‑39* 2 OH-(aq) + H2(g) = 2\n", - "H2O(l) + 2 e-\n", - "\n", - "with ε3o = 0.828 V and\n", - "$\\mathrm{\\Delta}_{\\ }^{0}G_{3}$= -2\\*0.828\\*96,485 J = -159,779 J. The\n", - "corresponding cathode (reduction) reaction is:\n", - "\n", - "*Eq. 8‑40* H2O(l) + ½ O2(g) + 2 e- = 2\n", - "OH-(aq)\n", - "\n", - "with ε4o = 0.401 V and\n", - "$\\mathrm{\\Delta}_{\\ }^{0}G_{4}$ = -2\\*0.401\\*96,485 J = -77,381 J. The\n", - "net cell reaction for alkaline electrolytes is:\n", - "\n", - "*Eq. 8‑41* H2(g) + ½ O2(g) = H2O(l)\n", - "\n", - "with εocell = 1.229 V and\n", - "$\\mathrm{\\Delta}_{\\ }^{0}G_{cell}$ = -2\\*1.229\\*96,485 J = -237,160 J.\n", - "\n", - "Plots of ε versus pH for a given chemical system have been typically\n", - "used to exhibit the stability relationships of ionic species and solid\n", - "phases in aqueous-based electrochemical systems. These graphs are often\n", - "called Pourbaix diagrams after the inventor and are at constant\n", - "temperature and constant pressure diagrams for a constant concentration,\n", - "usually for one metallic element. By convention, the ε in a Pourbaix\n", - "diagram corresponds to the potential for the cathode reduction reactions\n", - "in the electrochemical half-cell with electrons as reactants. Pourbaix\n", - "diagrams can be extended to multi-component materials when thermodynamic\n", - "properties of the components are available in both the materials and the\n", - "aqueous solution.\n", - "\n", - "An example of an ε versus pH diagram is shown in for the\n", - "Ni-H2O system at a 298K, 1 bar, and $c_{{Ni}^{2 +}} = 0.001$\n", - "molality. Three stability regions for Ni species are shown: Ni(s),\n", - "NiO(s), and \\[Ni2+\\]. The two dashed lines on this diagram\n", - "correspond to hydrogen reduction () and oxygen reduction () reactions,\n", - "respectively.\n", - "\n", - "Figure ‑: An ε versus pH, Pourbaix diagram for the Ni-H2O at\n", - "298K, 1 bar, and $c_{{Ni}^{2 +}} = 0.001$ molality.\n", - "\n", - "For the ε and pH conditions within the boundaries of the Ni(s) region,\n", - "no solid phase other than Ni(s) is stable, no ionic species with a\n", - "concentration of 1 molarity is stable, and no gas species with a\n", - "pressure of 1 bar is stable. Similar statements could be made about the\n", - "NiO(s) and \\[Ni2+\\] areas on the diagram. In the\n", - "\\[Ni2+\\] area, introduction of Ni(s) or NiO(s) into the\n", - "system would result in the dissolution of these solid phases since they\n", - "are not stable with respect to the \\[Ni2+\\] aqueous solution.\n", - "The corresponding chemical reactions proceed spontaneously to the right\n", - "as follows until the solid phases are consumed:\n", - "\n", - "*Eq. 8‑42* Ni(s) → Ni2+(1 molarity) + 2 e-\n", - "\n", - "*Eq. 8‑43* NiO(s) + 2 H+(aq) → Ni2+(1 molarity) +\n", - "H2O(l)\n", - "\n", - "No H+(aq) in involved in the first reaction, , so the\n", - "boundary line separating Ni(s) and Ni2+ is independent of pH.\n", - "No oxidation or reduction occurs in the second reaction, , i.e. no\n", - "electrons are reactants or products in the reaction, the boundary line\n", - "separating NiO(s) and Ni2+ is independent of ε.\n", - "\n", - "Note the convention that the ε is the potential for a cathode reduction\n", - "reaction, and boundary lines between two stability regions depict\n", - "conditions under which partial equilibrium occurs for the two species\n", - "for the ε and pH values at any point on these lines. For the boundary\n", - "line separating Ni(s) and Ni2+ in an ideal aqueous solution,\n", - "i.e. the reverse of , the following equation is obtained.\n", - "\n", - "*Eq. 8‑44* ε = εo = -0.268 V\n", - "\n", - "For the NiO(s)-Ni2+ boundary line of an ideal solution, the\n", - "reaction, , is a complete equilibrium, and thus the relationship is\n", - "\n", - "*Eq. 8‑45*\n", - "$0 = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{1}{\\left( c_{H^{+}} \\right)^{2}} = \\mathrm{\\Delta}_{\\ }^{0}G + 2 \\cdot 2.303 \\cdot RT \\cdot pH$\n", - "\n", - "*Eq. 8‑46*\n", - "$pH = - \\frac{\\mathrm{\\Delta}_{\\ }^{0}G}{2 \\cdot 2.303 \\cdot RT}$\n", - "\n", - "where $\\mathrm{\\Delta}_{\\ }^{0}G$ is obtained as follows and can be\n", - "calculated from the SSUB database and the standard potential of Ni,\n", + "## Stability at equilibrium and property anomaly\n", + "\n", + "As shown by , potentials are homogenous for a homogeneous system in a\n", + "state of internal equilibrium. To study the stability of the equilibrium\n", + "state, one considers the entropy production due to a fluctuation of a\n", + "molar quantity as an internal process. Based on the second law of\n", + "thermodynamics, the driving force, as the first derivative of the\n", + "entropy production with respect to the internal process, is zero for\n", + "such a fluctuation at equilibrium, i.e. *D*=0, and the entropy of\n", + "production thus depends on the second derivative. It can be written as\n", + "follows\n", + "\n", + "Eq. ‑\n", + "$Td_{ip}S = \\frac{\\partial_{ip}S}{\\partial\\xi}d\\xi + \\frac{1}{2}{\\frac{\\partial_{ip}^{2}S}{\\partial\\xi^{2}}(d\\xi)}^{2} = Dd\\xi - \\frac{1}{2}D_{2}(d\\xi)^{2}$\n", + "\n", + "with $D_{2} = - \\frac{\\partial_{ip}^{2}S}{\\partial\\xi^{2}}$. When\n", + "$\\frac{\\partial_{ip}^{2}S}{\\partial\\xi^{2}} < 0$ or $D_{2} > 0$ along\n", + "with $D = 0$, the fluctuation does not produce positive entropy of\n", + "production and can thus not develop further. The equilibrium state of\n", + "the system is therefore stable against the fluctuation. On the other\n", + "hand, when $\\frac{\\partial_{ip}^{2}S}{\\partial\\xi^{2}} > 0$ or\n", + "$D_{2} < 0$ along with $D = 0$, the fluctuation creates positive entropy\n", + "of production and can continue to grow. The equilibrium state of the\n", + "system is therefore unstable against the fluctuation. In connection with\n", + ", one can realize that for a system at stable equilibrium without heat\n", + "and mass exchanges with the surroundings, its entropy is at its maximum,\n", + "and there are no other internal processes, which could produce any more\n", + "entropy. This is schematically shown in .\n", + "\n", + "Figure ‑: Schematic diagram showing maximum entropy\n", + "\n", + "Using , , and , the combined law of thermodynamics can be written as\n", + "\n", + "*Eq. 1‑33*\n", + "$dU = \\sum_{}^{}{Y_{i}dX_{i}} - Dd\\xi + \\frac{1}{2}D_{2}(d\\xi)^{2}$\n", + "\n", + "Let us carry out the same virtual internal experiment shown in Chapter ,\n", + "i.e. moving an infinitesimal amount of *Xi* in a homogenous\n", + "system with other $X_{j}$ kept constant in an isolated system, i.e.\n", + "$dU = 0$ and $D = 0$. The internal energy change due to this internal\n", + "process is\n", + "\n", + "Eq. ‑\n", + "$dU = \\frac{1}{2}D_{2}\\left\\{ \\left( dX_{i}^{'} \\right)^{2} + \\left( dX_{i}^{\"} \\right)^{2} \\right\\}$\n", + "\n", + "For a homogeneous system in a state of stable equilibrium with\n", + "$\\left( dX_{i}^{'} \\right)^{2} = \\left( dX_{i}^{\"} \\right)^{2} = \\left( dX_{i} \\right)^{2}$,\n", + "this internal process must result in an increase of internal energy,\n", + "$dU > 0$, and thus gives\n", + "\n", + "Eq. ‑\n", + "$D_{2} = 2\\left( \\frac{\\partial^{2}U}{\\partial\\left( X_{i} \\right)^{2}} \\right)_{X_{j}} = 2\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}} > 0$\n", + "\n", + "shows that for a system to be stable, any pair of conjugate variables\n", + "must change in the same direction when other independent molar\n", + "quantities are kept constant. For the conjugate variables discussed so\n", + "far, it means that for a stable system, the addition of entropy\n", + "increases temperature with $\\frac{\\partial T}{\\partial S}$\\>0, the\n", + "volume decreases with pressure or increases with the negative of\n", + "pressure with $\\frac{\\partial( - P)}{\\partial V} > 0$, and the chemical\n", + "potential of a component increases with its amount, i.e.\n", + "$\\frac{\\partial\\mu_{i}}{\\partial N_{i}} > 0$, where the derivatives are\n", + "taken with all other molar quantities kept constant. The limit of\n", + "stability is reached when becomes zero, i.e.\n", + "\n", + "Eq. ‑\n", + "$D_{2} = 2\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}} = 0$\n", + "\n", + "shows schematically the energy as a function of configurations including\n", + "three states: unstable, stable, and metastable. Both the stable and\n", + "metastable states have positive curvatures due to $D_{2} > 0$, while the\n", + "unstable state has a negative curvature due to $D_{2} < 0$. There is an\n", + "inflection point of $D_{2} = 0$ for a state between a stable or\n", + "metastable state with $D_{2} > 0$ and an unstable state with\n", + "$D_{2} < 0$. These two inflection points, called spinodal, represent the\n", + "limit of stability. The states between the two inflection points are\n", + "unstable, and other states are either stable or metastable. The two\n", + "inflection points can move apart from or close to each other depending\n", + "on the change of external conditions, i.e. the natural variables. One\n", + "extreme situation is when these two inflection points merge into one\n", + "point, and the instability occurs only at this particular point. It is\n", + "evident that all three states, stable, metastable, and unstable, also\n", + "merge into one point. This point is called critical or consolute point,\n", + "beyond which the instability no longer exists.\n", + "\n", + "Figure ‑: Schematic diagram showing the stable and unstable equilibrium\n", + "states\n", + "\n", + "To mathematically define the consolute point, the third derivative needs\n", + "to be added to because both $D$ and $D_{2}$ vanish at this point, i.e.\n", + "\n", + "Eq. ‑\n", + "$Td_{ip}S = \\frac{\\partial_{ip}S}{\\partial\\xi}d\\xi + \\frac{1}{2}{\\frac{\\partial_{ip}^{2}S}{\\partial\\xi^{2}}(d\\xi)}^{2} + \\frac{1}{6}{\\frac{\\partial_{ip}^{3}S}{\\partial\\xi^{3}}(d\\xi)}^{3} = Dd\\xi - \\frac{1}{2}D_{2}(d\\xi)^{2} + \\frac{1}{6}{D_{3}(d\\xi)}^{3}$\n", + "\n", + "Eq. ‑\n", + "$dU = \\sum_{}^{}{Y_{i}dX_{i}} - Dd\\xi + \\frac{1}{2}D_{2}(d\\xi)^{2} - \\frac{1}{6}{D_{3}(d\\xi)}^{3}$\n", + "\n", + "At the consolute point, the third derivative also becomes zero, i.e.\n", + "\n", + "Eq. ‑ $D_{3} = \\frac{\\partial_{ip}^{3}S}{\\partial\\xi^{3}}^{3} = 0$\n", + "\n", + "Let us further discuss the properties of the system in relation to the\n", + "critical point. By taking the inverse of the equation of the limit of\n", + "stability, , one obtains\n", + "\n", + "Eq. ‑\n", + "$\\left( \\frac{\\partial X_{i}}{\\partial Y_{i}} \\right)_{X_{j}} = + \\ \\infty$\n", + "\n", + "i.e. all $X_{i}$ quantities diverge at the critical point. Therefore,\n", + "when a system approaches the critical point from its stable region, the\n", + "change of a molar quantity with respect to its conjugate potential\n", + "varies dramatically and becomes infinite at the critical point,\n", + "resulting in property anomalies in the system. In the unstable region,\n", + "the system would thus separate into stable subsystems and becomes\n", + "heterogeneous, and $X_{i}$:s change discontinuously between subsystems.\n", + "While in the stable region, the change of a molar quantity with respect\n", + "to its conjugate potential decreases as the system moves away from the\n", + "critical point and remains positive due to the stability criteria\n", + "denoted by .\n", + "\n", + "However, it is not clear how a molar quantity changes with respect to a\n", + "non-conjugate potential at the critical point. From the Maxwell\n", + "relation, one has\n", + "\n", + "Eq. ‑\n", + "$\\left( \\frac{\\partial Y_{i}}{\\partial X_{j}} \\right)_{X_{k \\neq j}} = \\frac{\\partial^{2}U}{\\partial X_{i}\\partial X_{j}} = \\left( \\frac{\\partial Y_{j}}{\\partial X_{i}} \\right)_{X_{k \\neq i}}$\n", + "\n", + "Eq. ‑\n", + "$\\left( \\frac{\\partial X_{j}}{\\partial Y_{i}} \\right)_{X_{k \\neq j}} = \\left( \\frac{\\partial X_{i}}{\\partial Y_{j}} \\right)_{X_{k \\neq i}}$\n", + "\n", + "Since all $X_{i}$:s diverge at the critical point, both derivatives in\n", + "should also go to infinite at the critical point. To investigate their\n", + "signs, let us carry out a virtual experiment similar to the one in\n", + "deriving the stability condition ( and ). In this case, two internal\n", + "processes are needed for moving two molar quantities simultaneously in\n", + "an isolated system, i.e.\n", + "\n", + "*Eq. 1‑43*\n", + "$dU = - D_{\\xi_{1}}d\\xi_{1} - D_{\\xi_{2}}d\\xi_{2} + D_{\\xi_{1}\\xi_{2}}d\\xi_{1}d\\xi_{2} + \\frac{1}{2}D_{2\\xi_{1}}\\left( d\\xi_{1} \\right)^{2} + \\frac{1}{2}D_{2\\xi_{2}}\\left( d\\xi_{2} \\right)^{2}$\n", + "\n", + "Based on the above discussions, in a stable system at equilibrium with\n", + "$D_{\\xi_{1}} = D_{\\xi_{2}} = 0$, $D_{2\\xi_{1}} > 0$ and\n", + "$D_{2\\xi_{2}} > 0$, the sign of $D_{\\xi_{1}\\xi_{2}}$ cannot be\n", + "unambiguously determined in keeping the change of internal energy\n", + "positive, i.e. $dU > 0$. This indicates that the quantities in can be\n", + "either positive or negative in the stable region and become zero at the\n", + "critical point. By the same token, the quantities in can be either\n", + "positive or negative and become positive or negative infinite at the\n", + "critical point.\n", + "\n", + "A profound conclusion from this analysis is that in a stable system even\n", + "though a molar quantity always changes in the same direction as its\n", + "conjugate potential, the same molar quantity may change in the opposite\n", + "direction of a non-conjugate potential, resulting in additional\n", + "anomalies represented by Eq. 1‑40. One example of is the thermal\n", + "expansion in a closed system, i.e. $dN_{i} = 0$, as follows\n", + "\n", + "Eq. ‑\n", + "$\\left( \\frac{\\partial V}{\\partial T} \\right)_{S} = \\left( \\frac{\\partial S}{\\partial( - P)} \\right)_{V}$\n", + "\n", + "The left-hand side of can be understood as follows: with the increase of\n", + "temperature, the system regulates its pressure in order to keep the\n", + "entropy from increasing, which results in the volume change of the\n", + "system. The behavior of the system depends on whether the pressure\n", + "decreases or increases in order to maintain the entropy of the system\n", + "constant. If the pressure decreases to maintain the entropy of the\n", + "system constant, the volume would increase with the increase of\n", + "temperature, i.e. the left-hand side of the equation has a positive\n", + "sign, which is also shown by the right-hand side of the equation as the\n", + "changes of $S$ and $–P$ have the same sign. That the volume increases\n", + "with temperature is the normal scenario. On the other hand, if the\n", + "pressure increases to maintain the entropy of the system constant, the\n", + "volume would decrease with the increase of temperature, resulting in a\n", + "negative sign for the left-hand side of the equation. This decrease of\n", + "volume with the increase of temperature is usually considered to be\n", + "anomalous, originated from the increase of entropy by the decrease of\n", + "$–P$ or the increase of pressure. More discussions on entropy will\n", + "follow in Chapter 5.2.5 and Chapter 9.\n" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/markdown": [ + "## Gibbs-Duhem equation\n", "\n", - "*Eq. 8‑47*\n", + "In experiments, it is difficult to control *S* and *V* of a system in\n", + "comparison with their conjugate variables *T* and *-P*. It is thus\n", + "desirable to construct new functions to represent the system with *T*\n", + "and *-P* as natural variables of the functions. One of them is enthalpy\n", + "defined in , and other two can be defined as follows\n", "\n", - "At a specified temperature, only one standard free energy and only one\n", - "equilibrium constant exists for this chemical reaction, and thus only\n", - "one specific value of $pH = 6.631$ exists for the reaction represented\n", - "by in this Pourbaix diagram.\n", + "Eq. ‑ $F = U - TS$\n", "\n", - "The diagonal line in represents the equilibrium between Ni(s) and NiO(s)\n", - "and is for a partial equilibrium reaction that is the sum of reactions\n", - "of and\n", + "Eq. ‑ $G = U - TS + PV = \\sum_{}^{}\\mu_{i}N_{i} = H - TS = F + PV$\n", "\n", - "*Eq. 8‑48* NiO(s) + 2 H+(aq) + 2 e- ═ Ni(s) +\n", - "H2O(l)\n", - "\n", - "The reduction of Ni from a divalent state in NiO to metallic Ni(s)\n", - "occurs, but the reaction also depends on the H+\n", - "concentration, the pH. The corresponding Gibbs energy and Nernst\n", - "equations are,\n", - "\n", - "*Eq. 8‑49*\n", - "$\\mathrm{\\Delta}G = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{1}{\\left( c_{H^{+}} \\right)^{2}} = - 23,939 + 2 \\cdot 2.303 \\cdot RT \\cdot pH$\n", - "\n", - "*Eq. 8‑50*\n", - "$\\varepsilon\\ = \\ \\varepsilon^{0}\\ - \\ \\frac{RT}{2f}\\ln\\frac{1}{\\left( c_{H^{+}} \\right)^{2}}\\ = 0.124\\ –\\ \\frac{2.303 \\cdot RT}{f}pH$\n", - "\n", - "where $\\mathrm{\\Delta}_{\\ }^{0}G$ can be calculated as follows\n", - "\n", - "*Eq. 8‑51*\n", - "\n", - "The two additional lines in correspond to the reduction reactions\n", - "related to H2 and O2 gases, i.e. the stability of\n", - "H2O. The lower one is for the reverse of under εo\n", - "= 0 and $P_{H_{2}} = 1$ with the Nernst equation being\n", - "\n", - "*Eq. 8‑52*\n", - "$\\varepsilon\\ = \\ \\varepsilon^{0}\\ - \\ \\frac{RT}{f}\\ln\\frac{\\left( P_{H_{2}} \\right)^{1/2}}{c_{H^{+}}}\\ = - \\frac{2.303 \\cdot RT}{f}pH$\n", - "\n", - "As the pH increases from 0, ε becomes more negative as is depicted. The\n", - "top dashed line corresponds to the oxygen reduction reaction represented\n", - "by under εo = 1.225 calculated from the aqueous solution\n", - "database in Thermo-Calc \\[60\\] and $P_{O_{2}} = 1$ with the Nernst\n", - "equation being\n", - "\n", - "Eq. 8‑53\n", - "$\\varepsilon\\ = \\ \\varepsilon^{0}\\ - \\ \\frac{RT}{2f}\\ln\\frac{\\left( P_{O_{2}} \\right)^{1/2}}{\\left( c_{H^{+}} \\right)^{2}}\\ = 1.225\\ –\\ \\frac{2.303 \\cdot RT}{f}pH$\n", + "with *F* and *G* called Helmholtz energy and Gibbs energy, respectively.\n", + "The middle part of is obtained using $U$ from . The corresponding\n", + "combined law of thermodynamics in terms of *H*, *F*, and *G* can be\n", + "obtained through the Legendre transformation of as\n", "\n", - "The dependence of ε on pH is identical for both reduction reaction and ,\n", - "and their intercepts at $pH = 0$ differ by their difference in their\n", - "εo values.\n", + "Eq. ‑ $dH = TdS - Vd( - P) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$\n", "\n", - "In this simple Pourbaix diagram of Ni in an ideal aqueous solution, all\n", - "boundary lines are straight because there is only one ionic species of\n", - "Ni in the aqueous solution, i.e. Ni2+. When there are more\n", - "than one ionic species in the aqueous solution, the boundary lines may\n", - "no longer be straight due to the competition between species. One\n", - "example is Cu with two main ionic species of Cu+2 and\n", - "CuOH+, and the reduction reaction between the metallic Cu and\n", - "the aqueous solution involves both two species, i.e.\n", + "Eq. ‑ $dF = - SdT - PdV + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$\n", "\n", - "*Eq. 8‑54*\n", - "${xCu}^{2 + \\ }\\ + \\ (1 - x){CuOH}^{+} + (1 - x)H^{+} + 2\\ e - \\ \\ = \\ \\ Cu(s) + {(1 - x)H}_{2}O$\n", + "Eq. ‑ $dG = - SdT - Vd( - P) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$\n", "\n", - "with\n", + "The independent variables in each of the above form are regarded as the\n", + "natural variables to the corresponding function. The integral forms of\n", + "all the functions can thus be written as the following with their\n", + "natural variables listed in the parenthesis\n", "\n", - "*Eq. 8‑55*\n", - "$\\mathrm{\\Delta}G = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{1}{\\left( c_{{Cu}^{2 +}} \\right)^{x}\\left( c_{{CuOH}^{+}}c_{H^{+}} \\right)^{1 - x}} = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{1}{\\left( c_{{Cu}^{2 +}} \\right)^{x}\\left( c_{{CuOH}^{+}} \\right)^{1 - x}} + 2.303(1 - x) \\cdot RT \\cdot pH$\n", + "Eq. ‑ $U = U\\left( S,V,N_{i},\\xi \\right)$\n", "\n", - "*Eq. 8‑56*\n", - "$\\varepsilon = \\varepsilon^{0} - \\frac{RT}{2f}\\ln\\frac{1}{\\left( c_{{Cu}^{2 +}} \\right)^{x}\\left( c_{{CuOH}^{+}} \\right)^{1 - x}} - \\frac{2.303(1 - x) \\cdot RT}{2f}pH$.\n", + "Eq. ‑ $H = H\\left( S, - P,N_{i},\\xi \\right)$\n", "\n", - "It is evident that both the slope and the intercept at $pH = 0$ are a\n", - "function of the concentration of ${CuOH}^{+}$, which is a function of\n", - "$pH$. Consequently, the boundary between the metallic Cu and the aqueous\n", - "solution is no longer a straight line as shown in .\n", + "Eq. ‑ $F = F\\left( T,V,N_{i},\\xi \\right)$\n", "\n", - "Figure ‑: An ε versus pH, Pourbaix diagram for the Cu-H2O\n", - "system at 298K, 1 bar, and $c_{Cu} = 0.001$ molality.\n", + "Eq. ‑ $G = G\\left( T, - P,N_{i},\\xi \\right)$\n", "\n", - "The concentrations of various species in the aqueous solution, i.e.\n", - "commonly called speciation, are plotted in , showing the change of\n", - "dominant species as a function of pH value.\n", + "By differentiating , one obtains\n", "\n", - "Figure ‑: Concentrations of ionic species in the aqueous solution at\n", - "$\\varepsilon = 0.3\\ V$ from .\n", + "Eq. ‑\n", + "$dG = \\sum_{}^{}\\mu_{i}{dN}_{i} + \\sum_{}^{}{N_{i}d\\mu}_{i} = - SdT - Vd( - P) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$\n", "\n", - "In Pourbaix diagrams for alloys with two or more elements, activities of\n", - "individual elements are to be used in calculating the potentials of\n", - "reduction reactions. Considering a Fe-Ni alloy with Fe2+ and\n", - "Ni2+ in the aqueous solution, the reduction reactions for Fe\n", - "and Ni can be written separately as\n", + "For a system at equilibrium, $Dd\\xi = 0$, re-arranging gives the\n", + "Gibbs-Duhem equation\n", "\n", - "*Eq. 8‑57* Ni2+(cNi) + 2 e- → Ni\n", - "(aNi in alloy)\n", + "Eq. ‑ $0 = - SdT - Vd( - P) - \\sum_{}^{}{N_{i}d\\mu}_{i}$\n", "\n", - "*Eq. 8‑58* Fe2+(cFe) + 2 e- → Fe\n", - "(aFe in alloy)\n", + "This equation indicates that for a homogeneous system with *c*\n", + "independent components at equilibrium, there is a direct relation among\n", + "all the *c+2* potentials, and they are $c$ chemical potentials\n", + "($\\mu_{i}$), temperature, and pressure. Consequently, only *c+1*\n", + "potentials can change independently, and the remaining potential is\n", + "dependent on the other potentials. As discussed in connection with ,\n", + "there are $c + 2$ independent variables for an equilibrium system with\n", + "*c* independent components, where all of them are molar quantities.\n", "\n", - "with their potentials as\n", + "With the relationships between potentials and molar quantities defined\n", + "by to , one can switch between potentials and molar quantities as\n", + "natural variables of the system. For example, one can define a new free\n", + "energy function when the chemical potential of one component is\n", + "controlled from the surroundings instead of its content and obtain the\n", + "following combined first and second law of thermodynamics\n", "\n", - "*Eq. 8‑59*\n", - "$\\varepsilon_{Ni}\\ = \\ \\varepsilon_{Ni}^{0} - \\frac{2.303RT}{2f}\\ln\\frac{a_{Ni}}{c_{Ni}} = \\ - 0.268 - \\frac{2.303RT}{2f}\\ln\\frac{a_{Ni}}{c_{Ni}}$\n", + "Eq. ‑ $\\Phi = G - \\mu_{1}N_{1} = \\sum_{i = 2}^{c}{\\mu_{i}N_{i}}$\n", "\n", - "*Eq. 8‑60*\n", - "$\\varepsilon_{Fe}\\ = \\ \\varepsilon_{Fe}^{0} - \\frac{2.303RT}{2f}\\ln\\frac{a_{Fe}}{c_{Fe}} = \\ - 0.441 - \\frac{2.303RT}{2f}\\ln\\frac{a_{Fe}}{c_{Fe}}$\n", + "Eq. ‑\n", + "$d\\Phi = - SdT - Vd( - P) - N_{1}{d\\mu}_{1} + \\sum_{i = 2}^{c}{\\mu_{i}{dN}_{i}} - Dd\\xi$\n", "\n", - "In principle, there are two scenarios for a given set of $a_{Ni}$ and\n", - "$a_{Fe}$ of the alloy. The first scenario is at the limit of a dilute\n", - "aqueous solution, i.e. $c_{Ni} = c_{Fe} = 0.001\\ $molarity,\n", - "$\\varepsilon_{Ni}$ and $\\varepsilon_{Fe}$ can be calculated, and the\n", - "element with the lower potential has the tendency to dissolve first,\n", - "which can result in the so-called dialloying effect. The second scenario\n", - "is for the equal potential, i.e. $\\varepsilon_{Ni} = \\varepsilon_{Fe}$\n", - "due to the externally imposed potential, and the equilibrium\n", - "concentrations of Fe+2 and Ni+2 can be calculated\n", - "from and .\n" + "However, even though the $c + 2$ molar quantities are independent of\n", + "each other, indicates that not all the $c + 2$ potentials are\n", + "independent, i.e., if chemical potentials of all components are changed\n", + "to natural variables, one would obtain . Therefore, among the *c+2*\n", + "independent variables used to define the system, the maximum number of\n", + "independent potential is *c+1*, and at least one of the *c+2*\n", + "independent variables must be a molar quantity. This variable is usually\n", + "chosen to be the size of the system or the major element in the system.\n", + "The Gibbs-Duhem equation is used to derive Gibbs phase rule in\n", + "heterogeneous systems, which is discussed in Chapter of the book.\n" ], "text/plain": [ "" @@ -1288,71 +2612,169 @@ { "data": { "text/markdown": [ - "## Concentrations, activities, and reference states of electrolyte species\n", - "\n", - "Thermodynamic descriptions of ionic species in solutions are different\n", - "from those of neutral species, which leads to a need for defining\n", - "concentration units, standard states, activities, and activity\n", - "coefficients of ionic solutions. In most studies of electrochemical\n", - "corrosion and electrodeposition, and in applied work of electrochemical\n", - "engineers, ionic species concentrations are given in units of molarity,\n", - "the number of moles of a species in a liter of solution (mol/l)\n", - "symbolically represented in equations by either *ci* or\n", - "\\[M+Z\\]. The other common concentration used for ionic\n", - "species is molality, which is defined as the number of moles of a\n", - "species in 1000g of solvent. For dilute aqueous solutions, molarity and\n", - "molality values are very similar.\n", - "\n", - "As discussed in Chapter , a practical definition of the activity of a\n", - "species *i* is the thermodynamic reactivity, or tendency to react, of\n", - "species *i* in the system of interest as compared to *i* in its\n", - "reference state form. The reference state of a species is typically\n", - "chosen as a specific chemical/physical state of the species at 1 atm\n", - "external pressure and the temperature of interest. Similarly, a typical\n", - "reference state for ionic species in aqueous solutions is the 1 molar\n", - "ideal solution at 1 bar external pressure and the temperature of\n", - "interest. If an electrolyte solution behaves ideally, then the activity\n", - "of species *i* in solution is\n", - "\n", - "*Eq. 8‑6*\n", - "$a_{i} = \\frac{c_{i}\\left( \\frac{mol}{l} \\right)}{c_{i}^{0}\\left( \\frac{mol}{l} \\right)} = \\frac{c_{i}\\left( \\frac{mol}{l} \\right)}{1\\left( \\frac{mol}{l} \\right)} = c_{i}(dimensionless)$\n", - "\n", - "where $c_{i}$ is the molar concentration of *i* in the solution divided\n", - "by $c_{i}^{0}$, the 1 molar reference state ideal solution\n", - "concentration. Thus, in ideal solutions, the activity of an electrolyte\n", - "species is numerically equal to its molar concentration. The above\n", - "treatment of ionic species is equivalent to the common practice of\n", - "depicting the activity of a gas by the value of its ideal gas partial\n", - "pressure in units of bar.\n", - "\n", - "The activity coefficient corrects for the nonideality of the species in\n", - "solution as defined in . If the solution is ideal, $\\gamma_{i} = 1$ for\n", - "all concentrations of a species in solution. For all solutions, one\n", - "expects $\\gamma_{i} \\rightarrow 1$ as $c_{i} \\rightarrow 1$. It is not\n", - "possible to measure $\\gamma_{i^{+}}$ or $\\gamma_{i^{-}}$ for individual\n", - "charged ions, only a geometric mean of the positive and negative ion\n", - "values. Consider the following ionic solution\n", - "\n", - "Eq. 8‑7\n", - "\n", - "Its chemical potential can be written as\n", - "\n", - "*Eq. 8‑8*\n", - "\n", - "Its geometric average or mean activity and activity coefficient are\n", - "defined as\n", + "# Gibbs energy function\n", "\n", - "*Eq. 8‑9*\n", + "As shown in through , all functions have $N_{i}$ and $\\xi$ as natural\n", + "variables while they differ in other two natural variables. In typical\n", + "materials-related experiments, temperature and pressure are the two\n", + "variables controlled. They are also the natural variables of Gibbs\n", + "energy. Consequently, Gibbs energy is the most widely used function in\n", + "thermodynamics of materials science. The rest of this book focuses on\n", + "Gibbs energy for this reason. In this chapter, the mathematical formulas\n", + "for Gibbs energy of phases with fixed and variable compositions are\n", + "discussed which are needed for quantitative calculations of Gibbs energy\n", + "under given values of its natural variables.\n", "\n", - "*Eq. 8‑10*\n", + "From , the molar Gibbs energy can be defined as\n", "\n", - "For example, one can define\n", - "$\\gamma_{\\pm} = \\left( \\gamma_{{Na}^{+}}\\gamma_{{Cl}^{-}} \\right)^{1/2}$\n", - "and\n", - "$\\gamma_{\\pm} = \\left( \\gamma_{{Al}^{3 +}}^{2}\\gamma_{{{SO}_{4}}^{2 -}}^{3} \\right)^{1/5}$\n", - "for NaCl and Al2(SO4)3, respectively.\n", - "For idea, weak electrolytes, $\\gamma_{\\pm} = 1$, and for non-ideal,\n", - "strong electrolytes, $\\gamma_{\\pm} \\neq 1$.\n" + "Eq. ‑\n", + "$G_{m}\\left( T,P,x_{i},\\xi \\right) = \\frac{G}{N} = \\sum_{}^{}\\mu_{i}x_{i}$\n", + "\n", + "The molar entropy, molar volume, chemical potential, and the driving\n", + "force can be obtained from as\n", + "\n", + "Eq. ‑\n", + "$S_{m} = \\frac{S}{N} = - \\frac{1}{N}\\left( \\frac{\\partial G}{\\partial T} \\right)_{P,\\ N_{i},\\ \\xi} = {- \\left( \\frac{\\partial G_{m}}{\\partial T} \\right)}_{P,\\ x_{i},\\ \\xi}$\n", + "\n", + "Eq. ‑\n", + "$V_{m} = \\frac{V}{N} = \\frac{1}{N}\\left( \\frac{\\partial G}{\\partial P} \\right)_{T,\\ N_{i},\\ \\xi} = \\left( \\frac{\\partial G_{m}}{\\partial P} \\right)_{T,\\ x_{i},\\ \\xi}$\n", + "\n", + "Eq. ‑\n", + "$\\mu_{i} = \\left( \\frac{\\partial G}{\\partial N_{i}} \\right)_{T,P,N_{j \\neq i},\\ \\xi}$\n", + "\n", + "Eq. ‑\n", + "$- D = \\left( \\frac{\\partial G}{\\partial\\xi} \\right)_{T,P,N_{i}\\ }$\n", + "\n", + "Based on , the molar enthalpy is written as\n", + "\n", + "Eq. ‑ $H_{m} = G_{m} + TS_{m}$\n", + "\n", + "Other physical properties of the system can also be represented by the\n", + "partial derivatives of Gibbs energy such as heat capacity, $C_{P}$,\n", + "volume thermal expansivity, $\\alpha_{V}$, isothermal compressibility,\n", + "$\\kappa_{T}$, as follows under constant pressure or temperature\n", + "\n", + "Eq. ‑\n", + "$C_{P} = \\left( \\frac{\\partial Q}{\\partial T} \\right)_{P} = \\left( \\frac{\\partial H}{\\partial T} \\right)_{P} = T\\left( \\frac{\\partial(G + TS)}{\\partial T} \\right)_{P} = T\\left( \\frac{\\partial S}{\\partial T} \\right)_{P} = - T\\left( \\frac{\\partial^{2}G}{\\partial T^{2}} \\right)_{P}$\n", + "\n", + "Eq. ‑\n", + "$\\alpha_{V} = \\frac{\\left( \\frac{\\partial V}{\\partial T} \\right)_{P}}{V} = \\frac{\\left( \\frac{\\left( \\partial G/\\partial( - P) \\right)_{T}}{\\partial T} \\right)_{P}}{\\left( \\partial G/\\partial( - P) \\right)_{T}} = \\frac{\\frac{\\partial^{2}G}{\\partial T\\partial( - P)}}{\\left( \\partial G/\\partial( - P) \\right)_{T}}$\n", + "\n", + "Eq. ‑\n", + "$\\kappa_{T} = \\frac{\\left( \\frac{\\partial V}{\\partial( - P)} \\right)_{T}}{V} = \\frac{\\left( \\frac{\\left( \\partial G/\\partial( - P) \\right)_{T}}{\\partial( - P)} \\right)_{T}}{\\left( \\partial G/\\partial( - P) \\right)_{T}} = \\frac{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}}{\\left( \\partial G/\\partial( - P) \\right)_{T}} = \\frac{1}{B}$\n", + "\n", + "where the $N_{i}$ and $\\xi$ are kept constant for all partial\n", + "derivatives, and $B$ is the bulk modulus.\n", + "\n", + "In , $G$ cannot be directly replaced by $G_{m}$ because *N* also depends\n", + "on *Ni*. The thermodynamic quantities under such conditions,\n", + "i.e. varying the amount of a component at constant temperature and\n", + "pressure, are called partial quantities which are introduced in Eq. 1‑8\n", + "for partial entropy and for partial enthalpy. This definition can be\n", + "extended to all molar quantities such as partial volume and partial\n", + "Gibbs energy. Partial quantities of a molar quantity, $A$, can thus be\n", + "defined in general as\n", + "\n", + "Eq. ‑\n", + "$A_{i} = \\left( \\frac{\\partial A}{\\partial N_{i}} \\right)_{T,P,N_{j \\neq i},\\ \\xi}$\n", + "\n", + "The general differential form of a molar quantity for a system at\n", + "equilibrium can be represented by its partial quantities as\n", + "\n", + "Eq. ‑\n", + "$dA = \\left( \\frac{\\partial A}{\\partial T} \\right)dT + \\left( \\frac{\\partial A}{\\partial P} \\right)dP + \\sum_{}^{}\\left( \\frac{\\partial A}{\\partial N_{i}} \\right){dN}_{i}$\n", + "\n", + "where the subscripts representing variables kept constant, i.e. the\n", + "remaining natural variables of Gibbs energy not in the denominator, are\n", + "omitted for simplicity. This will be done throughout the book unless\n", + "specified otherwise.\n", + "\n", + "Using the following relations: $A = NA_{m}$, $N = \\sum_{}^{}N_{j}$,\n", + "$x_{i} = N_{i}/N$,\n", + "$\\frac{{\\partial x}_{i}}{{\\partial N}_{i}} = \\left( 1 - x_{i} \\right)/N$,\n", + "and $\\frac{{\\partial x}_{k}}{{\\partial N}_{i}} = {- x}_{k}/N$, can be\n", + "derived as, under constant T and P,\n", + "\n", + "Eq. ‑\n", + "$A_{i} = A_{m} + N\\sum_{j = 1}^{c}{\\frac{\\partial A_{m}}{\\partial x_{j}}\\frac{\\partial x_{j}}{\\partial N_{i}}} = A_{m} + \\frac{\\partial A_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial A_{m}}{\\partial x_{j}}$\n", + "\n", + "where the summation is for all *c* components and the partial\n", + "derivatives are taken with other mole fractions kept constant. However,\n", + "mole fractions are not independent, but follow the relation\n", + "$\\sum_{}^{}x_{i} = 1$. Taking $x_{1} = 1 - \\sum_{j = 2}^{c}x_{j}$ as the\n", + "dependent mole fraction, can be rewritten as\n", + "\n", + "Eq. ‑\n", + "$A_{i} = A_{m} + \\left( \\frac{\\partial A_{m}}{\\partial x_{i}} - \\frac{\\partial A_{m}}{\\partial x_{1}} \\right) - \\sum_{j = 2}^{c}x_{j}\\left( \\frac{\\partial A_{m}}{\\partial x_{j}} - \\frac{\\partial A_{m}}{\\partial x_{1}} \\right)$\n", + "\n", + "The difference of the partial derivatives in the parenthesis in\n", + "represents the partial derivative of $A_{m}$ with respect to the mole\n", + "fraction of one component when the first component is selected as the\n", + "dependent component. Applying and to Gibbs energy, the partial Gibbs\n", + "energy or chemical potential of component $i$ is obtained as\n", + "\n", + "Eq. ‑\n", + "$\\mu_{i} = G_{i} = G_{m} + \\frac{\\partial G_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial G_{m}}{\\partial x_{j}} = G_{m} + \\left( \\frac{\\partial G_{m}}{\\partial x_{i}} - \\frac{\\partial G_{m}}{\\partial x_{1}} \\right) - \\sum_{j = 2}^{c}x_{j}\\left( \\frac{\\partial G_{m}}{\\partial x_{j}} - \\frac{\\partial G_{m}}{\\partial x_{1}} \\right)$\n", + "\n", + "The derivatives in the stability equation, , are defined with the molar\n", + "quantities kept constant. On the other hand, Gibbs energy has two\n", + "potentials, temperature and pressure, as natural variables instead. One\n", + "would thus need to compare the stability conditions when a variable kept\n", + "fixed is changed from a molar quantity to its conjugate potential. This\n", + "can be carried out through the use of Jacobians to change the\n", + "independent variables\n", + "\n", + "Eq. ‑\n", + "$\\frac{\\partial\\left( Y_{i},Y_{j} \\right)}{\\partial\\left( X_{i},X_{j} \\right)} = \\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{Y_{j}}\\left( \\frac{\\partial Y_{j}}{\\partial X_{j}} \\right)_{X_{i}} = \\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}}\\left( \\frac{\\partial Y_{j}}{\\partial X_{j}} \\right)_{X_{i}} - \\left( \\frac{\\partial Y_{i}}{\\partial X_{j}} \\right)_{X_{i}}\\left( \\frac{\\partial Y_{j}}{\\partial X_{i}} \\right)_{X_{j}}$\n", + "\n", + "For a stable system, both\n", + "$\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}}$ and\n", + "$\\left( \\frac{\\partial Y_{j}}{\\partial X_{j}} \\right)_{X_{i}}$ are\n", + "positive based on . Using the Maxwell relation shown by , one thus\n", + "obtains\n", + "\n", + "Eq. ‑\n", + "$\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}} - \\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{Y_{j}} = \\left( \\frac{\\partial Y_{i}}{\\partial X_{j}} \\right)_{X_{i}}\\left( \\frac{\\partial Y_{j}}{\\partial X_{i}} \\right)_{X_{j}}/\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}} \\geq 0$\n", + "\n", + "This means that\n", + "$\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{Y_{j}}$ will go\n", + "to zero before\n", + "$\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}}$ does. It\n", + "indicates that the stability condition becomes more restrictive when\n", + "potentials are kept constant in place of their conjugate molar\n", + "quantities. Based on the Gibbs-Duhem equation of , the maximum number of\n", + "independent potentials is *c+1*, and the last potential is dependent,\n", + "i.e.\n", + "\n", + "Eq. ‑\n", + "$\\left( \\frac{\\partial Y_{c + 2}}{\\partial X_{c + 2}} \\right)_{Y_{j \\leq c + 1}} = 0$\n", + "\n", + "Therefore, the limit of stability is determined when the derivative\n", + "becomes zero with one molar quantity kept constant, e.g.\n", + "\n", + "Eq. ‑\n", + "$\\left( \\frac{\\partial Y_{c + 1}}{\\partial X_{c + 1}} \\right)_{Y_{j < c + 1},X_{c + 2}} = 0$\n", + "\n", + "This is because this derivative reaches zero faster than any other\n", + "derivatives with more molar quantities kept constant. shows that all\n", + "molar quantities diverge at the limit of stability. The consolute point\n", + "is obtained with $c$ additional conditions as follows based on\n", + "\n", + "Eq. ‑\n", + "$\\left( \\frac{\\partial^{2}Y_{i}}{\\partial\\left( X_{i} \\right)^{2}} \\right)_{Y_{j \\leq c + 1, \\neq i},X_{c + 2}} = 0$\n", + "\n", + "Together with , all $c + 1$ independent potentials at the consolute\n", + "point can be determined. It is evident that the consolute point is a\n", + "zero-dimensional point in a two-dimensional space of independent\n", + "potentials in a one-component system. With the addition of a second\n", + "component to form a binary system, this consolute point in the\n", + "one-component system extends into a one-dimensional line. This line\n", + "represents the limit of stability of the binary system, and a consolute\n", + "point is located at the end of this line. It is thus evident that in a\n", + "system with $c$ independent components, the limit of stability is a\n", + "*c-1*-dimensional hypersurface in a space of $c + 1$ independent\n", + "potentials, while the consolute point is a zero-dimensional point in all\n", + "systems, which may be called the invariant critical point.\n" ], "text/plain": [ "" @@ -1364,156 +2786,271 @@ { "data": { "text/markdown": [ - "### Electrical batteries\n", - "\n", - "Batteries utilize electrochemical reactions to generate electricity for\n", - "various devices. The theoretic voltage of a battery can be calculated\n", - "from and , i.e.\n", - "\n", - "*Eq. 8‑73*\n", - "$\\varepsilon = - \\frac{\\mathrm{\\Delta}G}{zf} = \\varepsilon^{0} - \\frac{RTlnQ}{zf}$\n", - "\n", - "with $\\mathrm{\\Delta}G$ being the driving force of the net cell reaction\n", - "and $Q$ being the reaction activity quotient. The actual voltage of a\n", - "battery is lower than the theoretical one due to kinetic limitations of\n", - "cell reactions and resistance to ion diffusion through the electrolyte.\n", - "Based on whether the cell reactions are reversible or not, batteries\n", - "typically categorized as either primary disposable or secondary\n", - "rechargeable batteries. The net cell reactions in primary disposable\n", - "batteries are not easily reversible, and electrode materials may not\n", - "return to their original forms by applying a higher external potential\n", - "of opposite sign. Consequently, primary batteries cannot be reliably\n", - "recharged. On the other hand, the net cell reactions in secondary\n", - "batteries are easily reversible. Furthermore, two half-cells in\n", - "batteries may use different electrolytes with each half-cell enclosed in\n", - "a container and a separator permeable to conducting ions but not the\n", - "bulk of the electrolytes.\n", - "\n", - "One common primary battery is zinc-carbon battery with a zinc anode\n", - "cylinder and a carbon cathode central rod. The electrolytes are ammonium\n", - "or zinc chloride next to the zinc anode and a mixture of ammonium\n", - "chloride and manganese dioxide next to the carbon cathode. The half cell\n", - "and net reactions with ammonium chloride are as follows\n", + "## Phases with fixed compositions\n", + "\n", + "The homogeneous system discussed so far means that there is only one\n", + "phase in the system, i.e. a single-phase system. A phase with a fixed\n", + "composition can be a pure element or a stoichiometric compound. There is\n", + "thus only one independent component in the system. A stoichiometric\n", + "compound contains more than one element, but the relative amounts of\n", + "each element are fixed by the stoichiometry and cannot vary\n", + "independently, i.e., $dN_{i} = x_{i}dN$. The combined law of\n", + "thermodynamics becomes\n", + "\n", + "Eq. ‑\n", + "$dG = - SdT - Vd( - P) + \\left( \\sum_{}^{}{x_{i}\\mu_{i}} \\right)dN - Dd\\xi = - SdT - Vd( - P) + G_{m}dN - Dd\\xi$\n", + "\n", + "$G_{m}$ is the molar Gibbs energy of the stoichiometric compound and can\n", + "be regarded as the chemical potential of the stoichiometric\n", + "phase,$\\ \\alpha$,\n", + "\n", + "Eq. ‑ $G_{m} = \\mu^{\\alpha} = \\sum_{}^{}{x_{i}\\mu_{i}}$\n", + "\n", + "The chemical potential of individual components in the phase cannot be\n", + "defined because the amount of each component cannot be varied\n", + "independently. For a stoichiometric phase of $N$ moles of atoms at\n", + "equilibrium with $dG$=$Nd\\mu^{\\alpha} + \\mu^{\\alpha}dN$, reduces to\n", + "\n", + "Eq. ‑ $\\ 0 = - SdT - Vd( - P) - Nd\\mu^{\\alpha}$\n", + "\n", + "which is the Gibbs-Duhem equation, , applied to a stoichiometric phase.\n", + "It can be represented graphically by a surface in a three-dimensional\n", + "space composed of $\\ \\mu^{\\alpha}$, *T* and *–P*. The direction of the\n", + "surface is represented by the three partial directives between any two\n", + "of $\\ \\mu^{\\alpha}$, *T* and *–P* with the third one kept constant, i.e.\n", + "\n", + "Eq. ‑\n", + "$\\ \\left( \\frac{\\partial\\mu^{\\alpha}}{\\partial T} \\right)_{P} = - \\frac{S}{N} = - S_{m}$\n", + "\n", + "Eq. ‑\n", + "$\\ \\left( \\frac{\\partial\\mu^{\\alpha}}{\\partial( - P)} \\right)_{T} = - \\frac{V}{N} = - V_{m}$\n", + "\n", + "Eq. ‑\n", + "$\\ \\left( \\frac{\\partial( - P)}{\\partial T} \\right)_{\\mu^{\\alpha}} = - \\frac{S}{V} = - \\frac{S_{m}}{V_{m}}$\n", + "\n", + "Based on Nernst’s heat theorem, the entropy difference between two\n", + "crystals approaches zero when the temperature approaches absolute zero.\n", + "It is thus a common practice to put $S = 0$ for a crystal at 0 K. This\n", + "is usually referred as the third law of thermodynamics. From the\n", + "definition of entropy change by , $S$ or $S_{m}$ is always positive at\n", + "finite temperatures as the system or the crystal absorbs heat from the\n", + "surroundings to increase its temperature. $V$ or $V_{m}$ of a phase is a\n", + "well-defined physical quantity, and its absolute value can be given and\n", + "is always positive. The above three equations can be written in a\n", + "general form as\n", + "\n", + "Eq. ‑\n", + "$\\ \\left( \\frac{\\partial Y_{i}}{\\partial Y_{j}} \\right)_{Y_{k}} = - \\frac{X_{j}}{X_{i}} < 0$\n", + "\n", + "The surface thus has negative slopes in all its directions. The\n", + "curvature of the surface can be derived from\n", + "\n", + "Eq. ‑\n", + "$\\ \\left( \\frac{\\partial^{2}Y_{i}}{\\partial\\left( Y_{j} \\right)^{2}} \\right)_{Y_{k}} = - \\left( \\frac{\\partial\\left( \\frac{X_{j}}{X_{i}} \\right)}{\\partial Y_{j}} \\right)_{Y_{k}} = - \\frac{1}{X_{i}}\\left( \\frac{\\partial X_{j}}{\\partial Y_{j}} \\right)_{Y_{k}} + \\frac{X_{j}}{\\left( X_{i} \\right)^{2}}\\left( \\frac{\\partial X_{i}}{\\partial Y_{j}} \\right)_{Y_{k}} = - \\frac{1}{X_{i}}\\left\\lbrack \\left( \\frac{\\partial X_{j}}{\\partial Y_{j}} \\right)_{Y_{k}} - \\frac{X_{j}}{X_{i}}{\\left( \\frac{\\partial X_{i}}{\\partial Y_{i}} \\right)\\left( \\frac{\\partial Y_{i}}{\\partial Y_{j}} \\right)}_{Y_{k}} \\right\\rbrack = - \\frac{1}{X_{i}}\\left\\lbrack \\left( \\frac{\\partial X_{j}}{\\partial Y_{j}} \\right)_{Y_{k}} + \\left( \\frac{X_{j}}{X_{i}} \\right)^{2}\\left( \\frac{\\partial X_{i}}{\\partial Y_{i}} \\right)_{Y_{k}} \\right\\rbrack < 0$\n", + "\n", + "Both terms inside the last bracket are positive for a system in a state\n", + "of stable internal equilibrium, and the surface thus has a negative\n", + "curvature and is convex everywhere as shown in .\n", + "\n", + "Figure ‑: Gibbs energy of a one-component phase as a function of\n", + "temperature and negative pressure, showing the convex shape\n", + "\n", + "From experimental observations, it is known that\n", + "$S_{m}^{vapor} \\gg S_{m}^{liquid} > S_{m}^{solid}$. The curves of\n", + "$G_{m}$ or $\\mu^{\\alpha}$ plotted with respect to $T$ at constant $P$\n", + "would thus have the most negative slope for a vapour phase followed by\n", + "its liquid and solid phases. As an example, shows Gibbs energy of Zn in\n", + "its solid, liquid, and vapour forms as a function of $T$ at constant\n", + "$P = 1$ atmospheric pressure.\n", + "\n", + "Figure ‑: Molar Gibbs energy of Zn as a function of T at constant P\n", + "\n", + "Similarly it is common that\n", + "$V_{m}^{vapor} \\gg V_{m}^{liquid} > V_{m}^{solid}$, and the curves of\n", + "$G_{m}$ or $\\mu^{\\alpha}$ plotted with respect to $P$ at constant $T$\n", + "would thus have the most positive slope for a vapour phase followed by\n", + "its liquid and solid phases, though there are cases that\n", + "$V_{m}^{liquid} < V_{m}^{solid}$ such as those of water and ice. As an\n", + "example, shows Gibbs energy of Fe in its two solid (fcc and bcc),\n", + "liquid, and vapour forms as a function of $P$ at constant $T = 1273K$.\n", + "\n", + "Figure ‑: Molar Gibbs energy of Zn as a function of P at constant T\n", + "\n", + "The quantities measurable by experiments typically include temperature,\n", + "pressure, volume, composition, and amount of heat flow in the combined\n", + "law of thermodynamics discussed so far. By measuring the heat needed to\n", + "increase the temperature of a phase, the heat capacity of the phase is\n", + "obtained as shown by Eq. 2‑7. A typical heat capacity curve as a\n", + "function of temperature is shown in for fcc-Al, hcp-Mg, and an\n", + "intermetallic phase Al12Mg17.\n", + "\n", + "Figure ‑: Heat capacity of fcc-Al, hcp-Mg, and\n", + "Al12Mg17 as a function of temperature\n", + "\n", + "There are various theoretical models for the heat capacity under\n", + "constant volume to be discussed in Chapter 5 of this book, which is\n", + "defined as\n", "\n", - "*Eq. 8‑74* Zn + 2NH3 →\n", - "Zn(NH3)22+ + 2 e-\n", + "Eq. ‑\n", + "$C_{V} = \\left( \\frac{\\partial U}{\\partial T} \\right)_{V} = T\\left( \\frac{\\partial(F + TS)}{\\partial T} \\right)_{V} = T\\left( \\frac{\\partial S}{\\partial T} \\right)_{V} = - T\\left( \\frac{\\partial^{2}F}{\\partial T^{2}} \\right)_{V}$\n", "\n", - "*Eq. 8‑75* 2NH4Cl + 2MnO2 + 2 e- →\n", - "2NH3 + Mn2O3 +\n", - "H2O+2Cl−\n", + "To establish the relationship between $C_{P}$ defined by and $C_{V}$,\n", + "$U$ needs to be represented as a function of $T$ and $V$ in terms of $G$\n", + "and its derivatives with respect to Gibbs energy’s natural variables of\n", + "$T$ and $P$. It can be done as follows\n", "\n", - "*Eq. 8‑76* Zn + 2MnO2 + 2NH4Cl →\n", - "Mn2O3 +\n", - "Zn(NH3)2Cl2 + H2O.\n", + "Eq. ‑\n", + "$dV = \\frac{\\partial V}{\\partial T}dT + \\frac{\\partial V}{\\partial( - P)}d( - P) = - \\frac{\\partial^{2}G}{\\partial T( - P)}dT - \\frac{\\partial^{2}G}{\\partial( - P)^{2}}d( - P)$\n", "\n", - "The electric potential of the reaction is, treating all compounds as\n", - "stoichiometric compounds\n", + "Eq. ‑\n", + "$dU = \\frac{\\partial(G + TS - PV)}{\\partial T}dT + \\frac{\\partial(G + TS - PV)}{\\partial( - P)}d( - P) = - \\left( T\\frac{\\partial^{2}G}{\\partial T^{2}} - P\\frac{\\partial^{2}G}{\\partial T( - P)} \\right)dT—\\left( T\\frac{\\partial^{2}G}{\\partial T( - P)} + P\\frac{\\partial^{2}G}{\\partial( - P)^{2}} \\right)\\left( - \\frac{1}{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}}dV + \\frac{\\frac{\\partial^{2}G}{\\partial T( - P)}}{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}}dT \\right) = - \\left\\lbrack T\\frac{\\partial^{2}G}{\\partial T^{2}} - T\\frac{\\left( \\frac{\\partial^{2}G}{\\partial T( - P)} \\right)^{2}}{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}} \\right\\rbrack dT + \\left( - T\\frac{\\frac{\\partial^{2}G}{\\partial T( - P)}}{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}} + P \\right)dV$\n", "\n", - "Eq. 8‑77\n", - "$\\varepsilon = - \\frac{\\mathrm{\\Delta}G}{2f} = - \\frac{\\mathrm{\\Delta}^{0}G}{2f} = \\frac{1}{2f}\\left(_{\\ }^{0}G^{Zn} + 2_{\\ }^{0}G^{{MnO}_{2}} + 2_{\\ }^{0}G^{{NH}_{4}Cl} -_{\\ }^{0}G^{H_{2}O} -_{\\ }^{0}G^{Zn\\left( {NH}_{3} \\right)_{2}{Cl}_{2}} -_{\\ }^{0}G^{{{Mn}_{2}O}_{3}} \\right)$.\n", + "Eq. ‑\n", + "$C_{V} = C_{P} + T\\frac{\\left( \\frac{\\partial^{2}G}{\\partial T( - P)} \\right)^{2}}{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}} = C_{P} - \\frac{\\alpha_{V}^{2}VT}{\\kappa_{T}} = C_{P} - \\alpha_{V}^{2}BVT$\n", "\n", - "The Gibbs energy of $Zn\\left( {NH}_{3} \\right)_{2}{Cl}_{2}$ is not\n", - "available in current databases and has been recently estimated to be\n", - "−505,375 J/mole-formula \\[62\\]. The value of at 298.15K is thus obtained\n", - "as 1.67 V, which is pretty close to the actual operating voltage of the\n", - "battery around 1.5 V.\n", + "where the thermal expansion, $\\alpha_{V}$, and the compressibility or\n", + "bulk modulus, $\\kappa_{T}$ or $B$, are defined by and , respectively.\n", + "From the heat capacity, the enthalpy and entropy can be obtained by\n", + "integration of at a constant pressure\n", "\n", - "While with zinc chloride, the cell reactions and electric potential may\n", - "be written as\n", + "Eq. ‑\n", + "$S = S_{0} + \\int_{0}^{T}\\frac{C_{P}}{T}dT = S_{0} + \\int_{0}^{298.15}\\frac{C_{P}}{T}dT + \\int_{298.15}^{T}\\frac{C_{P}}{T}dT = S_{298.15} + \\int_{298.15}^{T}\\frac{C_{P}}{T}dT$\n", "\n", - "Eq. 8‑78 Zn + ZnCl2 + 2OH− → 2ZnOHCl + 2\n", - "e-\n", + "Eq. ‑\n", + "$H = H_{0} + \\int_{0}^{T}C_{P}dT = H_{0} + \\int_{0}^{298.15}C_{P}dT + \\int_{298.15}^{T}C_{P}dT = H_{298.15} + \\int_{298.15}^{T}C_{P}dT$\n", "\n", - "*Eq. 8‑79* MnO2 + H2O + e- → MnOOH +\n", - "OH-\n", + "In the above equations, two temperature ranges of integration are chosen\n", + "for practical applications as most processing procedures in the field of\n", + "materials science and engineering take place at temperatures above the\n", + "room temperature. Based on the third-law of thermodynamics, $S_{0} = 0$,\n", + "$S_{298.15}$ can be obtained by integration. On the other hand for\n", + "$H_{0} = U_{0} + PV$, one does not know the absolute value of the\n", + "internal energy and thus have to select a reference state for $H$. In\n", + "principle, the reference state can be arbitrarily chosen. A widely used\n", + "reference state in the thermodynamic modeling practice is to set\n", + "$H_{298.15}^{SER} = 0$ at ambient pressure for pure elements at their\n", + "respective stable structures at room temperature, called stable element\n", + "reference (SER) state with\n", "\n", - "*Eq. 8‑80* Zn + 2 MnO2 + ZnCl2 + 2 H2O\n", - "→ 2 MnOOH + 2 ZnOHCl\n", + "Eq. ‑\n", + "$G_{298.15}^{SER} = H_{298.15}^{SER} - TS_{298.15}^{SER} = - TS_{298.15}^{SER}$\n", "\n", - "*Eq. 8‑81*\n", - "$\\varepsilon = \\frac{1}{2f}\\left(_{\\ }^{0}G^{Zn} + 2_{\\ }^{0}G^{{MnO}_{2}} +_{\\ }^{0}G^{Zn{Cl}_{2}} + 2_{\\ }^{0}G^{H_{2}O} - 2_{\\ }^{0}G^{MnOOH} - 2_{\\ }^{0}G^{ZnOHCl} \\right)$\n", + "It is further noted that after defining $S_{298.15}$ and $H_{298.15}$,\n", + "one only needs the heat capacity at higher temperatures. This makes the\n", + "mathematical representation of heat capacity simpler due to a relatively\n", + "simple temperature dependence of heat capacity at higher temperatures in\n", + "comparison with the variation at lower temperatures. One common\n", + "expression for heat capacity at high temperatures and ambient pressure\n", + "is as follows\n", "\n", - "Secondary batteries can be recharged by applying an external electrical\n", - "potential, which reverses the net cell reaction that occur during\n", - "discharging. The oldest form of rechargeable battery is the lead-acid\n", - "batteries used in automotive, and the latest development is the\n", - "lithium-ion (Li-ion) batteries. A lead-acid battery typically uses Pb\n", - "and PbO2 as the cathode and anode electrodes and a 35% sulfuric acid and\n", - "65% water solution as the electrolyte. Its anode and cathode reactions\n", - "can be simplified as follows\n", + "Eq. ‑ $C_{P} = c + dT + \\frac{e}{T^{2}} + fT^{2}$\n", "\n", - "Eq. 8‑82 $Pb + SO_{4}^{2 -} = PbSO_{4} + 2e^{-}$\n", + "where c, d, e, and f are parameters fitted to experimental or theoretic\n", + "data and compiled in various handbooks.\n", "\n", - "Eq. 8‑83 $PbO_{2} + 4H^{+} + SO_{4}^{2 -} + 2e^{-} = PbSO_{4} + 2H_{2}O$\n", + "The corresponding $S$, $H$, and $G$ are obtained as\n", "\n", - "The net cell reaction is\n", + "Eq. ‑ $S = b^{'} + clnT + dT - \\frac{e}{{2T}^{2}} + \\frac{f}{2}T^{2}$\n", "\n", - "Eq. 8‑84 $Pb + PbO_{2} + 2H_{2}SO_{4}^{\\ } = 2PbSO_{4} + 2H_{2}O$.\n", + "Eq. ‑ $H = a + cT + \\frac{d}{2}T^{2} - \\frac{e}{T} + \\frac{f}{3}T^{3}$\n", "\n", - "Its electric potential is represented by the following equation\n", + "Eq. ‑\n", + "$G = H - TS = a - bT - cTlnT - \\frac{d}{2}T^{2} - \\frac{e}{2T} - \\frac{f}{6}T^{3}$\n", "\n", - "*Eq. 8‑85*\n", - "$\\varepsilon = - \\frac{1}{2f}\\left( 2_{\\ }^{0}G^{H_{2}O} + 2_{\\ }^{0}G^{{PbSO}_{4}} -_{\\ }^{0}G^{Pb} -_{\\ }^{0}G^{{PbO}_{2}} - 2_{\\ }^{0}G^{H_{2}SO_{4}} \\right)$\n", + "with $b = b^{'} - c$. The integration constants $b^{'}$ and $a$ are\n", + "evaluated from $S_{298.15}$ and $H_{298.15}$. As an example, the\n", + "enthalpy and entropy of Zn in solid (hcp), liquid, and gas forms are\n", + "plotted in and , respectively. The distances between any two curves in\n", + "and represent the enthalpy or entropy differences between the two\n", + "phases. It can be seen that the gas has much higher enthalpy and entropy\n", + "than the solid and liquid.\n", "\n", - "with the value at 298.15K being 2.651 V calculated from Thermo-Calc\n", - "\\[60\\]. During discharging, the reaction goes to right, and $PbSO_{4}$\n", - "is formed on both anode and cathode, while during charging, the reaction\n", - "goes to the left, and $Pb$ and $PbO_{2}$ are restored. In practical\n", - "applications, other ionic species such as ${H_{3}O}^{+}$ and\n", - "$HSO_{4}^{-}$ may form in the electrolyte, complicating the reactions\n", - "and affecting its potential.\n", + "Figure ‑: Enthalpy of Zn as a function of temperature at one atmospheric\n", + "pressure\n", "\n", - "In lithium ion batteries, lithium ions migrate in electrolytes between\n", - "electrodes made of intercalated lithium compounds during charging and\n", - "discharging. LiCoO2 and LiFePO4 are two of the\n", - "several cathode materials used in lithium ion batteries, and the anode\n", - "is typically made of carbon or metallic Li. The anode and cathode\n", - "reactions for LiCoO2 batteries can be written in simple forms as follows\n", + "Figure ‑: Entropy of Zn as a function of temperature at one atmospheric\n", + "pressure\n", "\n", - "Eq. 8‑86 ${Li}_{x}C_{6} = x{Li}^{+} + xe^{-} + 6C$\n", + "Similarly, one can add the pressure dependence into the Gibbs energy\n", + "function such as\n", "\n", - "Eq. 8‑87 $x{Li}^{+} + xe^{-} + {Li}_{1 - x}CoO_{2} = LiCoO_{2}$\n", - "\n", - "with the net reaction and electric potential being\n", + "Eq. ‑\n", + "$G = a - bT - cTlnT - \\frac{d}{2}T^{2} - \\frac{e}{2T} - \\frac{f}{6}T^{3} + gP + hTP + mP^{2}$\n", "\n", - "Eq. 8‑88 ${Li}_{x}C_{6} + {Li}_{1 - x}CoO_{2} = LiCoO_{2} + 6C$\n", + "where g, h, and m are parameters fitted to experimental or theoretic\n", + "data and compiled in various handbooks.\n", "\n", - "*Eq. 8‑89*\n", - "$\\varepsilon = - \\frac{1}{xf}\\left\\{ 6_{\\ }^{0}G^{C} +_{\\ }^{0}G^{{LiCoO}_{2}} - G^{{{Li}_{x}C}_{6}} - G^{{{Li}_{1 - x}CoO}_{2}} \\right\\} = - \\frac{1}{f}\\left\\{ \\left( \\mu_{Li}^{{Li}_{1 - x}CoO_{2}} - \\mu_{Li}^{{Li}_{x}C} \\right) - \\frac{1}{x}\\left( \\mu_{LiCoO_{2}}^{{Li}_{1 - x}CoO_{2}} -_{\\ }^{0}G^{{LiCoO}_{2}} \\right) \\right\\}$\n", + "The expression for $V$ can be derived as\n", "\n", - "The electric potential is a function of $x$. The value in the first\n", - "parenthesis in the above equation denotes the chemical potential\n", - "difference of Li between two electrodes, and the value in the second\n", - "parenthesis represents the chemical potential difference of\n", - "${LiCoO}_{2}$ between the states in the solution phase of\n", - "${Li}_{1 - x}CoO_{2}$ and by itself. Gibbs energies of ${{Li}_{x}C}_{6}$\n", - "and ${{Li}_{1 - x}CoO}_{2}$ need to be obtained as a function $x$ in\n", - "order to calculate the electric potential of the battery.\n", + "Eq. ‑ $V = g + hT + 2mP$\n", "\n", - "LiFePO4 uses metallic lithium as the anode with following\n", - "half-cell and net cell reactions\n", + "The Helmholtz energy can be expressed as a function of its natural\n", + "variables by solving $P\\ $ from\n", "\n", - "Eq. 8‑90 $xLi = x{Li}^{+} + xe^{-}$\n", + "Eq. ‑\n", + "$F = G - PV = a - bT - cTlnT - \\frac{d}{2}T^{2} - \\frac{e}{2T} - \\frac{f}{6}T^{3} - \\frac{(g + hT - V)^{2}}{4m}$\n", "\n", - "Eq. 8‑91 $x{Li}^{+} + xe^{-} + {Li}_{1 - x}FePO_{4} = LiFePO_{4}$\n", + "In the literature there are many models to represent the temperature and\n", + "pressure dependences of thermodynamic properties. The Gibbs energy\n", + "difference between a stoichiometric compound and the components at their\n", + "reference states of which the compound is composed,\n", + "${_{\\ }^{0}G}_{i}^{ref}$, is termed as Gibbs energy of formation, i.e.\n", + "\n", + "Eq. ‑ $\\mathrm{\\Delta}_{f}G = G - \\sum_{}^{}N_{i}{_{\\ }^{0}G}_{i}^{ref}$\n", "\n", - "Eq. 8‑92 $xLi + {Li}_{1 - x}FePO_{4} = LiFePO_{4}$.\n", + "with $N_{i}$ being the stoichiometry of the compound. Similarly,\n", + "enthalpy of formation, entropy of formation, and heat capacity of\n", + "formation with respect to components at their reference states,\n", + "$_{\\ }^{0}H_{i}^{ref}$, $_{\\ }^{0}S_{i}^{ref}$, and\n", + "$_{\\ }^{0}{C_{P}}_{i}^{ref}$, can be defined as\n", + "\n", + "Eq. ‑ $\\mathrm{\\Delta}_{f}H = H - \\sum_{}^{}{N_{i}_{\\ }^{0}H_{i}^{ref}}$\n", + "\n", + "Eq. ‑ $\\mathrm{\\Delta}_{f}S = S - \\sum_{}^{}{N_{i}_{\\ }^{0}S_{i}^{ref}}$\n", + "\n", + "Eq. ‑\n", + "$\\mathrm{\\Delta}_{f}C_{P} = C_{P} - \\sum_{}^{}{N_{i}_{\\ }^{0}{C_{P}}_{i}^{ref}}$\n", + "\n", + "It should be mentioned that one mole of a compound usually refers to one\n", + "mole of formula of stoichiometry of the compound. With a formula like\n", + "$A_{a}B_{b}C_{c}$, the compound is composed of total $(a + b + c)$ moles\n", + "of components. One should thus be very careful when dealing with\n", + "numerical values to be sure whether the data is in terms of per mole of\n", + "formula or per mole of components. At the same time the reference states\n", + "must be clearly defined. When the SER state defined in is selected as\n", + "the reference state, the above formation quantities are called standard\n", + "formation quantities such as standard enthalpy of formation.\n", + "\n", + "Since there are only two independent potentials in a one-component\n", + "system, its limit of stability can be evaluated with one potential kept\n", + "constant, i.e. either $T$ or $P$. Consequently, either Helmholtz energy\n", + "or enthalpy is to be used in deriving the limit of stability of a\n", + "homogeneous system. For the practical usefulness, let us use Helmholtz\n", + "energy because its natural variables of $T$ and *V* are measurable\n", + "quantities in typical experiments, while one of the natural variables of\n", + "enthalpy, entropy, is not. From and , the limit of stability for a\n", + "one-component system at constant temperature can be written as\n", "\n", - "Its electric potential is also a function of $x$, i.e.\n", + "Eq. ‑\n", + "$\\left( \\frac{\\partial( - P)}{\\partial V} \\right)_{T,N} = F_{VV} = \\frac{1}{V\\kappa_{T}} = \\frac{B}{V} = 0$\n", "\n", - "*Eq. 8‑93*\n", - "$\\varepsilon = - \\frac{1}{xf}\\left\\{_{\\ }^{0}G^{LiFePO_{4}} - x\\ ^{0}G^{Li} - G^{{Li}_{1 - x}FePO_{4}} \\right\\} = - \\frac{1}{f}\\left\\{ \\left( \\mu_{Li}^{{Li}_{1 - x}FePO_{4}} - \\ ^{0}\\mu_{Li} \\right) - \\frac{1}{x}\\left( \\mu_{LiFePO_{4}}^{{Li}_{1 - x}FePO_{4}} -_{\\ }^{0}G^{LiFePO_{4}} \\right) \\right\\}$\n", + "where the isothermal compressibility and bulk modulus, $\\kappa_{T}$ and\n", + "$B$, are defined in . The limit of stability is thus determined when the\n", + "isothermal compressibility diverges or the bulk modulus becomes zero\n", + "because $V$ has finite values at any temperatures. It is evident that\n", + "Helmholtz energy must have higher order dependence on volume than in for\n", + "a system with instability because $F_{VV}$ from is constant.\n", "\n", - "The value in the first parenthesis in the above equation denotes the\n", - "chemical potential difference of Li between two electrodes, and the\n", - "value in the second parenthesis represents the chemical potential\n", - "difference of $LiFePO_{4}$ between the states in the solution phase of\n", - "${Li}_{1 - x}FePO_{4}$ and by itself. Consequently, Gibbs energy of\n", - "${Li}_{1 - x}FePO_{4}$ needs to be obtained as a function $x$ in order\n", - "to calculate the electric potential of the battery. It is known that\n", - "there are several miscibility gaps in the $FePO_{4}$ and $LiFePO_{4}$\n", - "psuedo-binary system, in which the chemical potentials are constants, so\n", - "is the electric potential, resulting in stable battery output.\n" + "From , the consolute point is defined by\n", + "\n", + "Eq. ‑\n", + "$F_{VVV} = \\left( \\frac{\\partial^{2}( - P)}{\\partial V^{2}} \\right)_{T,N} = \\frac{\\partial\\left( \\frac{1}{V\\kappa_{T}} \\right)}{\\partial V} = - \\frac{1 + \\frac{V}{\\kappa_{T}}\\frac{\\partial\\kappa_{T}}{\\partial V}}{\\kappa_{T}V^{2}} = 0$\n", + "\n", + "Since $\\kappa_{T}$ becomes infinite at the limit of stability,\n", + "$\\frac{\\partial\\kappa_{T}}{\\partial V}$ approaches negative infinite\n", + "when the critical/consolute point is approached so that\n", + "$\\frac{V}{\\kappa_{T}}\\frac{\\partial\\kappa_{T}}{\\partial V} = - 1$ and\n", + "$F_{VVV} = 0$.\n" ], "text/plain": [ "" @@ -1525,82 +3062,115 @@ { "data": { "text/markdown": [ - "## Electrolyte reactions and electrochemical reactions\n", - "\n", - "Electrolytes that dissolve in a polar solvent such as water to produce\n", - "ionic species do not necessarily exhibit valence changes. A simple\n", - "example is the strong electrolyte NaCl(s) dissolving in water to produce\n", - "solvated ions\n", - "\n", - "*Eq. 8‑1* $NaCl(s)\\ \\ = \\ \\ {Na}^{+}(aq)\\ \\ + \\ \\ {Cl}^{–}(aq)$\n", - "\n", - "where the $(aq)$ indicates the ionic species in an aqueous solution. In\n", - "this system, the ion concentrations must become quite large before the\n", - "solution is saturated and can exist in equilibrium with $NaCl(s)$. Its\n", - "reaction constant, defined by , is shown as\n", - "$K_{e} = \\ \\ a_{{Na}^{+}}a_{{Cl}^{-}}$. If the product of the ion\n", - "activities is less than $K_{e}$, the solution is not saturated, and more\n", - "$NaCl(s)\\ \\ $ can be dissolved.\n", - "\n", - "The precipitation of $AgCl(s)$, a weak electrolyte, occurs quite readily\n", - "when ${Cl}^{–}$ ions are added to an aqueous solution containing\n", - "${Ag}^{+}(aq)$:\n", - "\n", - "*Eq. 8‑2* ${Ag}^{+}(aq)\\ + \\ \\ {Cl}^{–}(aq)\\ = AgCl(s)$\n", - "\n", - "The equilibrium constant for this reaction,\n", - "$K_{e} = \\ \\frac{1}{\\left( a_{{Ag}^{+}}a_{{Cl}^{-}} \\right)}$ is quite\n", - "large, so the equilibrium product of the ion activities, proportional to\n", - "their concentrations, is quite small. In the laboratory, the above\n", - "reaction could occur as a result of adding hydrochloric acid to a silver\n", - "nitrate solution. The accompanying $H^{+}(aq)$ \\[or ${H_{3}O}^{+}(aq)$\\]\n", - "and ${NO_{3}}^{-}(aq)$ ions in solution are not directly involved in the\n", - "silver chloride precipitation reaction so are not shown in reaction\n", - "represented by .\n", - "\n", - "The above ionic equilibria in the $AgCl(s) - H_{2}O$ system is not only\n", - "important for understanding this electrolyte system, but also critical\n", - "in electrochemical systems in which *Ag(s)* undergoes a valence change\n", - "at one electrode and reacts with a ${Cl}^{–}(aq)$ ion to produce\n", - "*AgCl(s)*, and an electron that is externally transported finite\n", - "distances to another electrode. The *oxidation* reaction occurs at the\n", - "Ag/AgCl electrode (*anode* half-cell reaction where electrons are\n", - "*added* into the system)\n", - "\n", - "*Eq. 8‑3* $Ag(s)\\ \\ + \\ {Cl}^{-}(aq)\\ \\ = \\ \\ AgCl(s)\\ \\ + \\ \\ e^{-}$\n", - "\n", - "A *reduction* reaction occurs at the other electrode (*cathode*\n", - "half-cell reaction where electrons are *consumed* by the reaction)\n", - "\n", - "*Eq. 8‑4* $\\frac{1}{2}{Cl}_{2}(g) + \\ \\ e^{-} = \\ {Cl}^{-}(aq)\\ \\ $\n", - "\n", - "The *net cell reaction* results in the formation AgCl(s) from its\n", - "elements\n", - "\n", - "*Eq. 8‑5* $Ag(s)\\ \\ + \\frac{1}{2}{Cl}_{2}\\ (g)\\ = \\ \\ AgCl(s)$\n", - "\n", - "Without knowledge of the physical system under which the reaction is\n", - "occurring, it would not be possible to know if reaction of was a result\n", - "of chlorine gas reacting directly with Ag(s), or if the reaction was\n", - "part of an electrochemical cell with a transport of electrons and ions\n", - "over finite distances. The addition of the two half-cell reactions gives\n", - "the *net cell reaction*, which does not show electrons as either\n", - "reactant or product species and may or may not include ionic species in\n", - "the reaction*.* A schematic of an electrochemical cell for the above\n", - "system is shown in Figure 8‑1.\n", - "\n", - "Figure ‑: Schematic diagram of an electrochemical cell consisting of a\n", - "chlorine electrode and a silver-silver chloride electrode.\n", - "\n", - "Oxidation and reduction can occur in electrolyte reactions without\n", - "creating an electrochemical cell. This is the case when chlorine gas\n", - "reacts directly with silver on a Ag(s) surface. Reaction of above is the\n", - "net reaction for this process, but the electrons produced from the\n", - "oxidation of Ag(s) are not transported over finite distances before\n", - "combining with Cl2(g) in its reduction to Cl–(aq).\n", - "No anode or cathode half-cell reactions exist in this system. The\n", - "electrons and ions involved in the reaction move only over atomic-scale\n", - "distances.\n" + "## Phases with variable compositions: Random solutions\n", + "\n", + "The combined law of thermodynamics and the Gibbs-Duhem equation of a\n", + "solution phase with variable compositions are shown by and ,\n", + "respectively. A phase can be represented by a *c+1*-dimensional surface\n", + "in a *c+2*-dimensional space of potentials based on the Gibbs-Duhem\n", + "equation. The directions and curvature of the surface are represented by\n", + "the partial derivatives shown by and secondary derivatives shown by ,\n", + "both being negative for a stable phase. To develop a mathematical\n", + "formula for Gibbs energy of a phase with variable compositions, one can\n", + "consider a phase as a mixture of independent components that the phase\n", + "is made of. Its Gibbs energy function can be postulated as the summation\n", + "of Gibbs energy of the independent components of the same structure of\n", + "the solution, $_{\\ }^{0}G_{i}$, plus the contribution due to the mixing,\n", + "$_{\\ }^{mixing}G$ or $_{\\ }^{M}G$\n", + "\n", + "Eq. ‑ $G = \\sum_{}^{}{N_{i}_{\\ }^{0}G_{i}} +_{\\ }^{M}G$\n", + "\n", + "Since the system size usually is not important in thermodynamics, its\n", + "properties are typically normalized to one mole with its composition\n", + "represented by mole fractions of components. The molar Gibbs energy is\n", + "obtained as shown below with the molar Gibbs energy of mixing separated\n", + "into two parts: ideal Gibbs energy of mixing assuming no chemical\n", + "interaction among components, $_{\\ }^{ideal}G_{m}$ or $_{\\ }^{I}G_{m}$,\n", + "and excess Gibbs energy of mixing due to chemical reaction among\n", + "components, $_{\\ }^{excess}G_{m}$ or $_{\\ }^{E}G_{m}$\n", + "\n", + "Eq. ‑\n", + "$G_{m} = \\sum_{}^{}{x_{i}_{\\ }^{0}G_{i}} +_{\\ }^{M}G_{m} = \\sum_{}^{}{x_{i}_{\\ }^{0}G_{i}} +_{\\ }^{I}G_{m} +_{\\ }^{E}G_{m}$\n", + "\n", + "From , the chemical potential of a component is thus\n", + "\n", + "Eq. ‑\n", + "$\\mu_{i} =_{\\ }^{0}G_{i} +_{\\ }^{I}{G_{m} +}_{\\ }^{E}G_{m} + \\frac{\\partial\\left(_{\\ }^{I}{G_{m} +}_{\\ }^{E}G_{m} \\right)}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial\\left(_{\\ }^{I}{G_{m} +}_{\\ }^{E}G_{m} \\right)}{\\partial x_{j}}$\n", + "\n", + "One may define the chemical activity of component *i*, $a_{i}^{\\ }$, as\n", + "follows\n", + "\n", + "Eq. ‑\n", + "$RTlna_{i}^{\\ } = \\mu_{i} -_{\\ }^{0}G_{i}^{\\ } =_{\\ }^{I}{G_{m} +}\\frac{\\partial_{\\ }^{I}G_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial_{\\ }^{I}G_{m}}{\\partial x_{j}} +_{\\ }^{E}G_{m} + \\frac{\\partial_{\\ }^{E}G_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial_{\\ }^{E}G_{m}}{\\partial x_{j}}$\n", + "\n", + "In this definition, the chemical activity or simply activity is\n", + "calculated with respect to the pure elements in the structure of the\n", + "solution for practical reasons as one would like to understand the\n", + "chemical potential difference of components in the solution and by\n", + "itself with the same structure. It should be noted that this reference\n", + "state for chemical activity is usually different from the SER reference\n", + "state defined in as the solution may have a different structure than\n", + "those of pure components in their SER states. On the other hand, the\n", + "activity under the SER reference state can be easily obtained by\n", + "replacing $_{\\ }^{0}G_{i}$ with ${_{\\ }^{0}G}_{i}^{SER}$ from . In\n", + "principle, one may choose any structure as the reference state for\n", + "activity to be useful for practical applications, i.e.\n", + "\n", + "Eq. ‑ $RTlna_{i}^{ref} = \\mu_{i} -_{\\ }^{0}G_{i}^{ref}$\n", + "\n", + "For example, the activity of a component in a liquid solution is defined\n", + "with respect to the pure component in its liquid form from , but can\n", + "also be referred to its SER state which is solid using . The following\n", + "sections will discuss in more details how components mix when they are\n", + "brought together including concepts such as random mixing, short-range\n", + "ordering, and long-range ordering.\n", + "\n", + "The limit of stability of a solution with respect to composition\n", + "fluctuation under constant *T*, *P*, and *N1* can be derived\n", + "as follows from and\n", + "\n", + "Eq. ‑\n", + "$\\left( \\frac{\\partial\\mu_{i}}{\\partial N_{i}} \\right)_{T,P,N_{j \\neq i},i > 1} > \\left( \\frac{\\partial\\mu_{i}}{\\partial N_{i}} \\right)_{T,P,N_{1},{\\mu_{2},N}_{j \\neq i},i,j > 2}\\ldots... > \\left( \\frac{\\partial\\mu_{c}}{\\partial N_{c}} \\right)_{T,P,N_{1},\\mu_{2}..\\mu_{c - 1}} = 0$\n", + "\n", + "The first term can be derived from as follows\n", + "\n", + "Eq. ‑\n", + "$\\left( \\frac{\\partial\\mu_{i}}{\\partial N_{i}} \\right)_{T,P,N_{j \\neq i},i > 1} = \\sum_{j = 1}^{c}\\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{i}\\partial x_{j}}\\frac{\\partial x_{j}}{\\partial N_{i}} - \\sum_{j = 1}^{c}{x_{j}\\sum_{k = 1}^{c}{\\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{j}\\partial x_{k}}\\frac{\\partial x_{k}}{\\partial N_{i}}}} = \\frac{1}{N}\\left( \\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{i}^{2}} - \\sum_{j = 1}^{c}{x_{j}\\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{j}^{2}}} - \\sum_{j = 1}^{c}{x_{j}\\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{i}\\partial x_{j}}} + \\sum_{j = 1}^{c}{\\sum_{k = 1}^{c}{{x_{j}x}_{k}\\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{j}\\partial x_{k}}}} \\right)$\n", + "\n", + "Denoting\n", + "$G_{ij} = \\left( \\frac{\\partial\\mu_{i}}{\\partial N_{j}} \\right)_{T,P,N_{k \\neq j}}$and\n", + "using to change the variables kept constant from molar quantities to\n", + "potentials one-by-one (see \\[1\\]), the limit of stability can be further\n", + "obtained as\n", + "\n", + "Eq. ‑\n", + "$\\left( \\frac{\\partial\\mu_{c}}{\\partial N_{c}} \\right)_{T,P,N_{1},\\mu_{2}..\\mu_{c - 1}} = \\frac{\\det\\left( G_{ij}:2 \\leq i,j \\leq c \\right)}{\\det\\left( G_{ij}:2 \\leq i,j \\leq c - 1 \\right)} = 0$\n", + "\n", + "where $\\det$ stands for determinant. indicates that\n", + "$\\det\\left( G_{ij}:2 \\leq i,j \\leq c \\right) = 0$ at the limit of\n", + "stability. Considering $x_{1} = 1 - \\sum_{j \\neq 1}^{}x_{j}$, let us\n", + "introduce\n", + "\n", + "Eq. ‑\n", + "$g_{i} = \\mu_{i} - \\mu_{1} = \\left( \\frac{\\partial G_{m}}{\\partial x_{i}} \\right)_{x_{k \\neq i}} - \\left( \\frac{\\partial G_{m}}{\\partial x_{1}} \\right)_{x_{k \\neq 1}}$\n", + "\n", + "and\n", + "\n", + "Eq. ‑\n", + "$g_{ij} = \\frac{\\partial g_{i}}{\\partial x_{j}} = \\frac{\\partial\\left( \\mu_{i} - \\mu_{1} \\right)}{\\partial x_{j}} = \\frac{\\partial^{2}G_{m}}{\\partial x_{i}\\partial x_{j}} - \\frac{\\partial^{2}G_{m}}{\\partial x_{1}\\partial x_{j}} - \\frac{\\partial^{2}G_{m}}{\\partial x_{i}\\partial x_{1}} + \\frac{\\partial^{2}G_{m}}{\\partial\\left( x_{1} \\right)^{2}}$\n", + "\n", + "The limit of stability can be re-written as\n", + "\n", + "Eq. ‑\n", + "$\\left( \\frac{\\partial\\left( \\mu_{c} - \\mu_{1} \\right)}{\\partial x_{c}} \\right)_{T,P,N,\\mu_{2} - \\mu_{1},\\ ...\\mu_{c - 1} - \\mu_{1}} = \\frac{\\det\\left( g_{ij}:2 \\leq i,j \\leq c \\right)}{\\det\\left( g_{ij}:2 \\leq i,j \\leq c - 1 \\right)} = 0$\n", + "\n", + "i.e. $\\det\\left( g_{ij}:2 \\leq i,j \\leq c \\right) = 0$. The consolute\n", + "point can be defined following\n", + "\n", + "Eq. ‑\n", + "$\\left( \\frac{\\partial^{2}\\mu_{c}}{\\partial\\left( N_{c} \\right)^{2}} \\right)_{T,P,N_{1},\\mu_{2}..\\mu_{c - 1}} = \\left( \\frac{\\partial^{2}\\left( \\mu_{c} - \\mu_{1} \\right)}{\\partial\\left( x_{c} \\right)^{2}} \\right)_{T,P,N,\\mu_{2} - \\mu_{1},\\ ...\\mu_{c - 1} - \\mu_{1}} = 0$\n", + "\n", + "No closed mathematic form has been published in the literature.\n" ], "text/plain": [ "" @@ -1612,52 +3182,103 @@ { "data": { "text/markdown": [ - "### Fuel cells\n", - "\n", - "Fuel cells are devices to convert chemical energy to electricity and\n", - "heat through electrochemical reactions with the fuel and oxygen supplied\n", - "to the anode and cathode, respectively. Typical ions migrating through\n", - "the electrolyte are $H^{+}$, ${OH}^{-}$, ${CO}_{3}^{2 -}$, and\n", - "$O^{2 -}$. In fuel cells with $H^{+}$ as migrating ions, H2\n", - "molecules are dissociated into $H^{+}$ on the anode, which are combined\n", - "with O2 on the cathode to form H2O and release\n", - "heat with the half cell and the net cell reactions in their simplest\n", - "form as shown by for the anode, for the cathode, and for the net cell,\n", - "respectively. Commonly used electrolytes are polymer and phosphoric\n", - "acid, and both anode and cathode reactions are facilitated by catalyst,\n", - "typically platinum. The thermodynamic limit of power which can be\n", - "generated by the fuel cell is represented by\n", - "\n", - "*Eq. 8‑63*\n", - "$w = - \\mathrm{\\Delta}G = - \\mathrm{\\Delta}_{\\ }^{0}G_{cell} + RTln\\left( P_{H_{2}}P_{O_{2}}^{1/2} \\right)$\n", - "\n", - "For fuel cells with anions as migrating ions, the anions are generated\n", - "on the cathode with H2O formed and heat generated on the\n", - "anode. Their representative cathode reactions are\n", - "\n", - "*Eq. 8‑64* $\\frac{1}{2}O_{2} + H_{2}O + 2e^{-}{= 2OH}^{-}$\n", - "\n", - "*Eq. 8‑65* ${\\frac{1}{2}O}_{2} + CO_{2} + 2e^{-} = {CO}_{3}^{2 -}$\n", - "\n", - "*Eq. 8‑66* $\\frac{1}{2}O_{2} + 2e^{-} = O^{2 -}$.\n", - "\n", - "The anode reaction for is the reaction represented by , operating at low\n", - "temperatures and using catalyst for both electrodes. The anode reactions\n", - "for and are\n", - "\n", - "*Eq. 8‑67* ${CO}_{3}^{2 -} + H_{2} = H_{2}O + CO_{2} + 2e^{-}$\n", - "\n", - "*Eq. 8‑68* $O^{2 -} + H_{2} = H_{2}O + 2e^{-}$\n", - "\n", - "respectively. To enable the diffusion of ${CO}_{3}^{2 -}$ and $O^{2 -}$\n", - "through the cathode and the electrolyte, both fuel cells are operated at\n", - "relative high temperatures, with the former typically in molten\n", - "carbonate solutions and the latter through solid oxides. Due to the high\n", - "operating temperatures, fuels are converted to hydrogen within the fuel\n", - "cell itself by a process called internal reforming, removing the need\n", - "for precious-metal catalyst and enabling the use of a variety of fuels.\n", - "The net cell reaction for all three fuel cells is the same as in the\n", - "case of $H^{+}$, represented by .\n" + "### Random solutions\n", + "\n", + "The ideal Gibbs energy of mixing represents an ideal solution in which\n", + "all sites are equivalent and the distributions of components on the\n", + "sites are completely random. The number of different configurations to\n", + "arrange all components is\n", + "\n", + "Eq. ‑ $w = \\frac{N!}{\\prod_{}^{}\\left( N_{i}! \\right)}$\n", + "\n", + "Based on Boltzmann’s relation from statistic thermodynamics when all\n", + "configurations have the same probability to be observed, the ideal\n", + "configurational molar entropy of mixing due to the distribution is\n", + "\n", + "Eq. ‑\n", + "$_{\\ }^{ideal}S_{m} =_{\\ }^{I}S_{m} = \\frac{Rlnw}{N} = R\\frac{lnN! - \\sum_{}^{}\\ln\\left( N_{i}! \\right)}{N} \\cong R\\frac{NlnN - \\sum_{}^{}{N_{i}l{nN}_{i}}}{N} = - R\\sum_{}^{}{x_{i}l{nx}_{i}}$\n", + "\n", + "where $R$ is the gas constant. represents the entropy difference between\n", + "the ideal solution and the states when individual components are by\n", + "themselves, i.e. the mechanical mixing of components. As $x_{i}$ is\n", + "smaller than unity, the entropy production to form an ideal solution\n", + "from pure components is thus positive, indicating that it is a\n", + "spontaneous process. In such an ideal solution, it is assumed that there\n", + "are no interactions between components, and the enthalpy of mixing is\n", + "thus zero as the internal energy and the volume of the system do not\n", + "change. The ideal Gibbs energy of mixing is written as\n", + "\n", + "Eq. ‑ $_{\\ }^{I}G = - T_{\\ }^{I}S_{m} = RT\\sum_{}^{}{x_{i}l{nx}_{i}}$\n", + "\n", + "The Gibbs energy of real solutions, i.e. , becomes\n", + "\n", + "Eq. ‑\n", + "$G_{m} = \\sum_{}^{}{x_{i}_{\\ }^{0}G_{i}} + RT\\sum_{}^{}{x_{i}l{nx}_{i}} +_{\\ }^{E}G_{m}$\n", + "\n", + "From , the chemical potential is obtained as\n", + "\n", + "Eq. ‑\n", + "$\\mu_{i} = G_{i} =_{\\ }^{0}G_{i} + RTlnx_{i} +_{\\ }^{E}G_{m} + \\frac{\\partial_{\\ }^{E}G_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial_{\\ }^{E}G_{m}}{\\partial x_{j}}$\n", + "\n", + "From the chemical activity of , the activity coefficient, $\\gamma_{i}$,\n", + "can be defined as follows\n", + "\n", + "Eq. ‑\n", + "$\\gamma_{i} = \\frac{a_{i}}{x_{i}} = \\frac{1}{x_{i}}\\exp\\frac{G_{i} -_{\\ }^{0}G_{i}\\ }{RT}$\n", + "\n", + "The solution is an ideal solution with $\\gamma_{i} = 1$, and is said to\n", + "be positively or negatively deviating from an ideal solution with\n", + "$\\gamma_{i} > 1$ or $\\gamma_{i} < 1$, respectively. The chemical\n", + "potential is related to the activity and activity coefficient by the\n", + "following equation\n", + "\n", + "Eq. ‑\n", + "$\\mu_{i} =_{\\ }^{0}G_{i} + RTlna_{i} =_{\\ }^{0}G_{i} + RTln\\gamma_{i}x_{i} =_{\\ }^{0}G_{i} + RTlnx_{i} + RTln\\gamma_{i}$\n", + "\n", + "Let us further exam in more details in order to better understand the\n", + "relation between $G_{m}$ and $\\mu_{i}$. The partial derivatives in\n", + "represent the directions of molar Gibbs energy in the composition space,\n", + "i.e. the tangents of molar Gibbs energy with respect to mole fractions\n", + "of independent components. Collectively, they define the\n", + "multi-dimensional tangent plane of molar Gibbs energy at the given\n", + "composition, $x_{i}^{0}$. The mathematical representation of this\n", + "tangent plane, $z_{G_{m}}$, is defined by its directional derivatives\n", + "and the distance from the point where the derivatives are taken,\n", + "\n", + "Eq. ‑\n", + "$z_{G_{m}} = G_{m}\\left( x_{i}^{0} \\right) + \\sum_{i = 1}^{c}{\\left( \\frac{\\partial G_{m}}{\\partial x_{i}} \\right)_{x_{i}^{0}}\\left( x_{i} - x_{i}^{0} \\right)}$\n", + "\n", + "The intercept of this tangent plane at each pure component axis, i.e.\n", + "$x_{i} = 1$ and $x_{j \\neq i} = 0$, is obtained as\n", + "\n", + "Eq. ‑\n", + "$z_{G_{m},x_{i} = 1} = G_{m}\\left( x_{i}^{0} \\right) + \\left( \\frac{\\partial G_{m}}{\\partial x_{i}} \\right)_{x_{i}^{\\ } = x_{i}^{0}} - \\sum_{j = 1}^{c}{x_{j}^{0}\\left( \\frac{\\partial G_{m}}{\\partial x_{j}} \\right)_{x_{i}^{\\ } = x_{i}^{0}}}$\n", + "\n", + "This is identical to at the point $x_{i}^{0}$. It is thus shown that the\n", + "chemical potential of a component in a solution is represented by the\n", + "intercept of the tangent plane of Gibbs energy of the solution on the\n", + "$G_{m}$ axis of the component. The distance between the intercept and\n", + "the Gibbs energy of the pure component in the same structure of the\n", + "solution is related to the chemical activity of the component as defined\n", + "by . On the other hand, it is evident that one can choose any other\n", + "structure of the pure element to define the chemical activity to compare\n", + "chemical potential of the component as shown by .\n", + "\n", + "The stability of a solution is evaluated following , and the derivatives\n", + "of chemical potential with respect to its moles, i.e. the elements in\n", + "the determinant, are obtained as follows from and ,\n", + "\n", + "Eq. ‑\n", + "$\\frac{N}{RT}\\frac{\\partial\\mu_{i}}{\\partial N_{i}} = \\frac{N}{RT}G_{ii} = \\frac{1 - x_{i}}{x_{i}} + \\frac{1}{\\gamma_{i}}\\left( \\frac{\\partial\\gamma_{i}}{\\partial x_{i}} - \\sum_{j = 1}^{c}{x_{j}\\frac{\\partial\\gamma_{i}}{\\partial x_{j}}} \\right)$\n", + "\n", + "Eq. ‑\n", + "$\\frac{N}{RT}\\frac{\\partial\\mu_{i}}{\\partial N_{k}} = \\frac{N}{RT}G_{ik} = - \\frac{x_{k}}{x_{i}} + \\frac{1}{\\gamma_{i}}\\left( 1 - \\sum_{j = 1}^{c}{x_{j}\\frac{\\partial\\gamma_{i}}{\\partial x_{j}}} \\right)$\n", + "\n", + "To further study Gibbs energy of solution phases, let us discuss the\n", + "details on the excess Gibbs energy of mixing. At this point, one can\n", + "start with lower-order systems with fewer components, i.e. two component\n", + "and three-component systems, noting that the Gibbs energy of phases with\n", + "one component is already presented in Chapter .\n" ], "text/plain": [ "" @@ -1669,156 +3290,282 @@ { "data": { "text/markdown": [ - "### Half cell potentials\n", - "\n", - "When electron current flows between electrodes, reactions are occurring\n", - "at the electrodes and concentration gradients causing polarization\n", - "develop around the electrodes. These gradients result in extraneous\n", - "potentials to occur at the electrodes. In such cases cell equilibrium is\n", - "not established and measured cell potentials are not those for true\n", - "partial equilibrium. If a cell is short-circuited with the electrodes\n", - "connected by a conductor, current will flow until the external potential\n", - "becomes zero, i.e. εext = 0, and equilibrium is established\n", - "with same conditions as non-electrochemical systems. If an external\n", - "potential, εext, is applied to the cell, chemical reactions\n", - "occur until the cell potential balances to εext, and no\n", - "current flows. This potential is called open-circuit voltage (OCV) in\n", - "the literature. It is important to realize that OCV includes all\n", - "reactions that occur on the electrode surface when the electrode is in\n", - "contact with the electrolyte, such as passivation discussed in Chapter .\n", - "Partial equilibrium in a cell is achieved when the cell potential is\n", - "balanced by an applied external potential. In such partial equilibrium\n", - "cases, equilibrium thermodynamic analyses can be used even though the\n", - "cell potential is not zero, i.e. εcell ≠ 0. This\n", - "differentiates electrochemical systems from other equilibrium systems\n", - "discussed previously.\n", - "\n", - "The number of electrons involved in a net cell reaction is important in\n", - "relating cell potential and the Gibbs energy change for the cell\n", - "reaction. As will be illustrated later in this section, this number\n", - "denotes the number of electrons involved in the half-cell reactions that\n", - "were added to yield the net cell reaction. The electrical work achieved\n", - "by the transport of an electrical charge through a cell potential can be\n", - "written as\n", - "\n", - "*Eq. 8‑20* $w = z\\ f\\ \\varepsilon$\n", - "\n", - "where *z* represents the moles of electrons in cell reaction, *f* the\n", - "Faraday constant equal to 96,485 J/V/mole-electron, and *ε* the\n", - "potential difference, often referred as electromotive force (emf) in the\n", - "literature. For a system at constant temperature, pressure, and\n", - "composition, this work is the same as the Gibbs energy difference\n", - "between the two electrodes, i.e.\n", - "\n", - "*Eq. 8‑21* $- \\Delta G = w = zf\\varepsilon$\n", - "\n", - "where the negative sign is because the system does work to the\n", - "surrounding when the Gibbs energy of the system is decreased. When the\n", - "applied external potential is larger than the cell potential, the\n", - "surrounding does work to the system, and a common example is the\n", - "charging of a battery. Thermodynamic relations discussed in previous\n", - "chapters can thus be directly applied to electrochemical systems with\n", - "some common equations shown in .\n", - "\n", - "Table ‑: Thermodynamic Equations for Electrochemical Cells\n", - "\n", - "*∆G = -z f ε*\n", - "\n", - "*∆S = - (∂∆G/∂T)P = + z f (∂ε/∂T)P*\n", - "\n", - "*∆H = \\[∂(∆G/T)/ ∂(1/T)\\]P = - z f \\[∂(ε/T)/∂(1/T)\\]P = z f \\[T(∂ε/∂T)P\n", - "– ε\\]*\n", - "\n", - "*∆CP = (∂∆H/∂T)P = T z f (∂2ε/∂T2)P*\n", - "\n", - "A half-cell reaction potential cannot be measured directly, only its\n", - "potential relative to another half-cell reaction. By convention, a\n", - "standard half-cell potential is measured relative to the standard\n", - "hydrogen half-cell reduction reaction at 25oC (298K) and 1\n", - "bar, which has a defined standard potential of zero volts,\n", - "\n", - "*Eq. 8‑22* H+(aq, a=1) + e- = 1/2 H2(g,\n", - "1 bar)\n", - "\n", - "with εo (H+/H2,g) = 0.00 volts. The\n", - "standard half-cell reduction reactions of metals at 25oC are\n", - "for the general reaction\n", - "\n", - "*Eq. 8‑23* Mz+(aq, a=1) + z e- = M(s)\n", - "\n", - "with εo (Mz+/M) volts. Half-cell reactions with\n", - "the most positive standard electrode potentials have a tendency to\n", - "spontaneously proceed toward reduction (cathode reactions). Half-cell\n", - "reactions with the most negative standard electrode potentials have a\n", - "tendency to spontaneously proceed toward oxidation (anode reactions).\n", - "\n", - "Consider, for example, a cell made up of a standard hydrogen electrode\n", - "and a standard zinc electrode with εo\n", - "(H+/H2,g) = 0.00 volts and εo\n", - "(Zn2+/Zn) = -0.762 volts. Thus, the H+ would tend\n", - "to be reduced, and the zinc metal would tend to be oxidized, and the\n", - "spontaneous reaction if all species had unit activities would be\n", - "\n", - "*Eq. 8‑24* 2 H+(aq, a=1)+ Zn = H2(1 bar) +\n", - "Zn2+(aq, a=1)\n", - "\n", - "with εocell = 0.762 volts and\n", - "$\\mathrm{\\Delta}_{\\ }^{0}G = - 2*96485*\\varepsilon_{cell}^{0}\\ $. The\n", - "cathode half-cell reaction would be the same as , while the anode\n", - "half-cell reaction would be\n", - "\n", - "*Eq. 8‑25* Zn = Zn2+(aq, a=1) + 2 e-\n", - "\n", - "When ion concentrations and H2 gas do not all have unit\n", - "activities, the Gibbs energy and cell potential of the cell reaction, ,\n", - "becomes\n", + "### Binary random solutions\n", + "\n", + "From , the Gibbs-Duhem equation of a binary system consisting of\n", + "components $A$ and $B$ is written as\n", + "\n", + "Eq. ‑ $0 = - SdT - Vd( - P) - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B}$\n", + "\n", + "This equation represents a four-dimensional surface. It is self-evident\n", + "that both and hold for stable binary solutions too, i.e. the directions\n", + "and the curvature of the surface are all negative. To visualize the\n", + "four-dimensional surface in the three-dimensional space, one needs to\n", + "fix one of the four potentials. As $T$ and $P$ are the natural variables\n", + "of Gibbs energy, they are usually chosen to be kept constant. One can\n", + "typically investigate behaviors of systems consisting of condensed\n", + "phases by varying the temperature at constant pressure. at constant\n", + "pressure thus becomes\n", "\n", - "*Eq. 8‑26*\n", - "$\\mathrm{\\Delta}G = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{a_{{Zn}^{2 +}}P_{H_{2}}}{\\left( a_{H^{+}} \\right)^{2}}$\n", + "Eq. ‑ $0 = - SdT - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B}$\n", "\n", - "*Eq. 8‑27*\n", - "$\\varepsilon_{cell} = \\varepsilon_{cell}^{0} - \\frac{RT}{zf}\\ln\\frac{a_{{Zn}^{2 +}}P_{H_{2}}}{\\left( a_{H^{+}} \\right)^{2}}$\n", + "Similar to and , the property of a phase can be represented by a\n", + "two-dimensional surface in the three-dimensional space composed of $T$,\n", + "$\\mu_{A}$, and $\\mu_{B}$ under constant $P$, keeping in mind the\n", + "following\n", "\n", - "The standard reduction potentials of some common metals at\n", - "25oC are given in .\n", + "Eq. ‑\n", + "$G_{m} = x_{A}\\mu_{A} + x_{B}\\mu_{B} = x_{A}_{\\ }^{0}G_{A} + x_{B}_{\\ }^{0}G_{B} + RT\\left( x_{A}l{nx}_{A} + x_{B}l{nx}_{B} \\right) +_{\\ }^{E}G_{m}$\n", "\n", - "Table ‑: Standard reduction potentials of some common metals\n", + "Since $_{\\ }^{E}G_{m}$ must be zero for pure components $A$ and $B$, it\n", + "needs to be in the following form\n", "\n", - "A cell reaction can be established by different half-cell reactions. For\n", - "example, the following reaction can be derived from two different cells\n", + "Eq. ‑ $_{\\ }^{E}G_{m} = x_{A}x_{B}L_{AB}$\n", "\n", - "*Eq. 8‑28* $3\\ {Fe}^{2 + \\ }\\ = \\ \\ 2\\ {Fe}^{3 +}\\ \\ + \\ \\ Fe(s)$\n", + "with $L_{AB}$ being a parameter denoting the interaction between\n", + "components $A$ and $B$, called interaction parameter. When $L_{AB} = 0$,\n", + "the solution is an ideal solution. When $L_{AB}$ is a non-zero constant\n", + "independent of temperature and composition, the solution is called a\n", + "regular solution. Its excess entropy and excess enthalpy of mixing are\n", + "obtained as\n", "\n", - "cell A\n", + "Eq. ‑ $_{\\ }^{E}S_{m} = \\frac{\\partial_{\\ }^{E}G_{m}}{\\partial T} = 0$\n", "\n", - "*Eq. 8‑29* $3{Fe}^{2 +} + \\ 6\\ e - \\ = \\ \\ 3Fe(s)$ εo1 =\n", - "-0.440 V\n", + "Eq. ‑\n", + "$_{\\ }^{E}H_{m} =_{\\ }^{E}G_{m} - T_{\\ }^{E}S_{m} = x_{A}x_{B}L_{AB}$\n", "\n", - "*Eq. 8‑30* $2\\ Fe(s)\\ \\ = \\ \\ 2\\ {Fe}^{3 +}\\ \\ + \\ 6\\ e -$\n", - "εo2 = +0.036 V\n", + "The chemical potential of component $A$ or $B$ in a binary regular\n", + "solution can be derived as\n", "\n", - "cell B\n", + "Eq. ‑\n", + "$\\mu_{i} =_{\\ }^{0}G_{i} + RTlnx_{i} + \\left( 1 - x_{i} \\right)^{2}L_{AB}$\n", "\n", - "*Eq. 8‑31* $2{Fe}^{2 +}\\ = \\ 2{Fe}^{3 + \\ \\ } + 2\\ e^{-}$\n", - "*εo4 = -0.772 V*\n", + "In a dilute solution with $x_{i} \\rightarrow 0$, one can have\n", "\n", - "*Eq. 8‑32* ${Fe}^{2 + \\ }\\ + \\ 2\\ e - \\ \\ = \\ \\ Fe(s)$ *εo5\n", - "= -0.440 V*\n", + "Eq. ‑\n", + "$RTln\\gamma_{i} = \\left( 1 - x_{i} \\right)^{2}L_{AB} \\approx L_{AB}$\n", "\n", - "Both give the same net reaction shown by , but with 6 and 2 electrons\n", - "and standard cell potentials being εocell A = -0.404 V and\n", - "εocell B = -1.212 V, respectively. However, the standard\n", - "Gibbs energies of both cells are the same, i.e.\n", + "Eq. ‑ $\\gamma_{i} = e^{\\frac{L_{AB}}{RT}}$\n", "\n", - "*Eq. 8‑33* $\\mathrm{\\Delta}_{\\ }^{0}G$*cell A = -6 f (-0.404)\n", - "= + 2.424 f*\n", + "The activity is thus proportional to its mole fraction, which is called\n", + "Henry’s law. By the same token, for the solvent, i.e.\n", + "$x_{i} \\rightarrow 1$,\n", "\n", - "*Eq. 8‑34* $\\mathrm{\\Delta}_{\\ }^{0}G$*cell B = -2 f (-1.212)\n", - "= + 2.424 f*\n", + "Eq. ‑ $RTln\\gamma_{i} = \\left( 1 - x_{i} \\right)^{2}L_{AB} \\approx 0$\n", "\n", - "It is shown that $\\mathrm{\\Delta}_{\\ }^{0}G$ values are independent of\n", - "half-cell reactions and depend only on the net reaction because the net\n", - "reaction is neutral in electron and balanced in mass.\n" + "which gives $\\gamma_{i} \\approx 1$, and its activity approaches its mole\n", + "fraction. This is called Raoult’s law.\n", + "\n", + "The stability of a binary solution is derived from as\n", + "\n", + "Eq. ‑\n", + "$\\left( \\frac{\\partial\\mu_{A}}{\\partial N_{A}} \\right)_{T,P,N_{B}} = \\left\\lbrack \\frac{RT}{x_{A}} - 2\\left( 1 - x_{A} \\right)L_{AB} \\right\\rbrack\\frac{1 - x_{A}}{N}$\n", + "\n", + "Eq. ‑\n", + "$\\left( \\frac{\\partial\\mu_{B}}{\\partial N_{B}} \\right)_{T,P,N_{A}} = \\left\\lbrack \\frac{RT}{x_{B}} - 2\\left( 1 - x_{B} \\right)L_{AB} \\right\\rbrack\\frac{1 - x_{B}}{N}$\n", + "\n", + "It should be noted that the two chemical potentials in a binary system\n", + "at constant temperature and pressure are dependent on each other due to\n", + "the Gibbs-Duhem equation shown in , i.e.\n", + "\n", + "Eq. ‑ $0 = - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B}$\n", + "\n", + "and the two chemical potentials depend on each other by the following\n", + "relation\n", + "\n", + "Eq. ‑\n", + "$\\left( \\frac{\\partial\\mu_{A}}{\\partial\\mu_{B}} \\right)_{T,P} = - \\frac{N_{B}}{N_{A}}$\n", + "\n", + "Therefore, at the limit of stability, both and go to zero at the same\n", + "time, which is obtained when\n", + "\n", + "Eq. ‑ $RT = 2{x_{A}x_{B}L}_{AB}$\n", + "\n", + "As the absolute temperature cannot be negative, has no solution for a\n", + "solution phase with $L_{AB} < 0$, i.e. the solution phase is stable with\n", + "respect to the composition fluctuation. For a solution with\n", + "$L_{AB} > 0$, its limit of stability is represented .\n", + "\n", + "A schematic molar Gibbs energy of a solution with $L_{AB} < 0$ at\n", + "constant temperature and pressure is shown in along with the ideal and\n", + "excess Gibbs energy of mixing. A tangent line on the molar Gibbs energy\n", + "of the solution is drawn, and its two intercepts at $x_{B} = 0$ and\n", + "$x_{B} = 1$ give the chemical potentials of components $A$ and $B$,\n", + "$\\mu_{A}$ and $\\mu_{B}$ by , respectively. It is evident that $\\mu_{A}$\n", + "and $\\mu_{B}$ are not independent on each other as they are two points\n", + "on the same straight line. This is a graphic representation of the\n", + "Gibbs-Duhem equation of . The chemical activity of component $B$ is also\n", + "depicted with the reference state being the pure B with the same\n", + "structure. As shown in , other structures of pure B can be selected as\n", + "the reference states of the chemical activity of component B, resulting\n", + "in the different distances to its chemical potential in the solution,\n", + "thus different values of its chemical activities. It is clear that this\n", + "change of reference state for chemical activity does not affect the\n", + "chemical potential of the component in the solution.\n", + "\n", + "Figure ‑: Schematic molar Gibbs energy diagram with $L_{AB} < 0$\n", + "\n", + "When $L_{AB} > 0$, represents a parabola in the $T - x_{i}$\n", + "two-dimensional coordinate, symmetric with respect to $x_{A}$ and\n", + "$x_{B}$, shown in , i.e. the spinodal of the solution. The consolute\n", + "point is obtained by applying to and letting equal to zero at the\n", + "consolute point\n", + "\n", + "Eq. ‑\n", + "$\\left( \\frac{\\partial^{2}\\mu_{A}}{\\partial N_{A}^{2}} \\right)_{T,P,N_{B}} = \\left\\lbrack - \\frac{RT}{x_{A}^{2}} + 2L_{AB} \\right\\rbrack\\left( \\frac{1 - x_{A}}{N} \\right)^{2} = 0$\n", + "\n", + "which gives\n", + "\n", + "Eq. ‑ $T_{cons} = 2x_{A}^{2}L_{AB}$\n", + "\n", + "Solving and , one obtains $x_{A} = x_{B} = 0.5$ and\n", + "\n", + "Eq. ‑ $T_{cons} = \\frac{L_{AB}}{2R}$\n", + "\n", + "Figure ‑: A Spinodal curve with $L_{AB} > 0$\n", + "\n", + "A schematic molar Gibbs energy diagram at temperatures below the\n", + "consolute point is shown in . It can be seen that part of the molar\n", + "Gibbs energy has negative curvature, and the solution becomes unstable.\n", + "The chemical potential thus does not change monotonically with respect\n", + "to composition and its derivative changes sign at the inflexion point.\n", + "\n", + "Figure ‑: Schematic molar Gibbs energy diagram with $L_{AB} > 0$\n", + "\n", + "For more complex solutions, $L_{AB}$ can be a function of temperature,\n", + "pressure, and compositions. In principle, the temperature and pressure\n", + "dependences can be treated by means of formula similar to . There are\n", + "various approaches in the literature to consider the composition\n", + "dependence of $L_{AB}$. The empirical Redlich-Kister polynomial stands\n", + "out as the one most widely used because it can be extrapolated to\n", + "ternary and multi-component systems consistently, which will be\n", + "discussed in Chapter .\n" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/markdown": [ + "### Ternary random solutions\n", + "\n", + "From , the Gibbs-Duhem equation of a ternary system consisting of\n", + "components $A$, $B$ and $C$ is written as\n", + "\n", + "Eq. ‑\n", + "$0 = - SdT - Vd( - P) - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B} - {N_{C}d\\mu}_{C}$\n", + "\n", + "This equation represents a five-dimensional surface. It can be\n", + "visualized in a three-dimensional space with two of the five potentials\n", + "fixed. Usually $T$ and $P$ are kept constant as they are the natural\n", + "variables of $G$, and reduces to\n", + "\n", + "Eq. ‑ $0 = - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B} - {N_{C}d\\mu}_{C}$\n", + "\n", + "A phase can thus be represented by a surface in the three-dimensional\n", + "space of $\\mu_{A}$, $\\mu_{B}$, and $\\mu_{C}$ at constant $T$ and $P$\n", + "with similar geometric appearance as .\n", + "\n", + "From , Gibbs energy of a ternary solution is written as\n", + "\n", + "Eq. ‑\n", + "$G_{m} = x_{A}_{\\ }^{0}G_{A} + x_{B}_{\\ }^{0}G_{B} + x_{C}_{\\ }^{0}G_{C} + RT\\left( x_{A}l{nx}_{A} + x_{B}l{nx}_{B} + x_{C}l{nx}_{C} \\right) +_{\\ }^{E}G_{m}$\n", + "\n", + "When the mole fraction of one component approaches zero,\n", + "$_{\\ }^{E}G_{m}$ reduces to the excess Gibbs energy of mixing of the\n", + "binary systems of the remaining two components, represented by .\n", + "However, for a given composition of a ternary solution, there is no\n", + "unique way to assign the contributions from $_{\\ }^{E}G_{m}$ of each\n", + "binary to $_{\\ }^{E}G_{m}$ of the ternary solution because\n", + "$_{\\ }^{E}G_{m}$ of the ternary solution contains information of both\n", + "binary and ternary interactions. A variety of models is available in the\n", + "literature (see \\[1\\]). One intuitive approach would be to use the same\n", + "formula as that in the binary system, i.e. , with the mole fractions\n", + "substituted by the values in the ternary system, and $_{\\ }^{E}G_{m}$ of\n", + "a ternary solution may thus be defined as the following by including the\n", + "ternary interaction involving all three components,\n", + "\n", + "Eq. ‑\n", + "$_{\\ }^{E}G_{m} = x_{A}x_{B}L_{AB} + x_{A}x_{C}L_{AC} + x_{B}x_{C}L_{BC} + x_{A}x_{B}x_{C}L_{ABC}$\n", + "\n", + "The chemical potential of a component is represented by . When all\n", + "interaction parameters in are constant, i.e. a ternary regular solution,\n", + "the chemical potential of component $A$ can be derived as\n", + "\n", + "Eq. ‑\n", + "$\\mu_{A} = G_{A} =_{\\ }^{0}G_{A} + RTlnx_{A} + x_{B}L_{AB} + x_{C}L_{AC} -_{\\ }^{E}G_{m} =_{\\ }^{0}G_{A} + RTlnx_{A} + x_{B}{\\left( 1 - x_{A} \\right)L}_{AB} + x_{C}\\left( 1 - x_{A} \\right)L_{AC} - x_{B}x_{C}L_{BC} + x_{B}x_{C}\\left( 1 - 2x_{A} \\right)L_{ABC} =_{\\ }^{0}G_{A} + RTlnx_{B} + x_{B}^{2}L_{AB} + x_{C}^{2}L_{AC} + x_{B}x_{C}\\left( L_{AB} + L_{AC} - L_{BC} \\right) + x_{B}x_{C}\\left( 1 - 2x_{A} \\right)L_{ABC}$\n", + "\n", + "Similar equations can be derived for component $B$ and C with\n", + "$L_{AB} = L_{BA}$, $L_{AC} = L_{CA}$, and $L_{BC} = L_{CB}$. A schematic\n", + "molar Gibbs energy diagram at constant temperature and pressure is shown\n", + "in with all three binary systems having $L_{ij} < 0$ of similar values.\n", + "\n", + "Figure ‑: Schematic ternary molar Gibbs energy diagram as a function of\n", + "compositions for given temperature and pressure\n", + "\n", + "To evaluate the stability of a ternary solution, one needs to calculate\n", + "the elements in the determinant shown in . Using the mole of component\n", + "$C$ as the independent molar quantity, the limit of stability is\n", + "expresses as\n", + "\n", + "Eq. ‑ $G_{AA}G_{BB} - G_{AB}G_{BA} = 0$\n", + "\n", + "As an example, $G_{AA}$ is shown in the following equation, which must\n", + "be positive for the solution to be stable\n", + "\n", + "Eq. ‑\n", + "${N\\left( \\frac{\\partial\\mu_{A}}{\\partial N_{A}} \\right)}_{T,P,N_{B},N_{C}} = NG_{AA} = \\frac{RT\\left( 1 - x_{A} \\right)}{x_{A}} - 2x_{B}^{2}L_{AB} - 2x_{C}^{2}L_{AC} - 2x_{B}x_{C}\\left( L_{AB} + L_{AC} - L_{BC} \\right) - 2x_{B}x_{C}\\left( 2 - 3x_{A} \\right)L_{ABC}$\n", + "\n", + "It is evident that any instability in binary systems with positive\n", + "interaction parameters extends into the ternary system. It can also be\n", + "seen that even if all binary interaction parameters are negative, i.e.\n", + "no instability in the binary systems, it is possible that becomes\n", + "negative for some combinations of the binary interaction parameters such\n", + "that $\\mathrm{\\Delta}L = L_{AB} + L_{AC} - L_{BC}$ becomes very positive\n", + "and overshadows the contributions due to $L_{AB}$ and $L_{AC}$, i.e.\n", + "$L_{BC}$ is more negative than $L_{AB}$ and $L_{AC}$ combined. In an\n", + "extreme case with $L_{AB} = L_{AC} = L_{ABC} = 0$ and $L_{BC} < 0$, i.e.\n", + "ideal solutions for the $A - B$ and $A - C$ binary systems, stable\n", + "solution in the $B - C$ binary system, and no additional ternary\n", + "interaction, reduces to\n", + "\n", + "Eq. ‑\n", + "$N\\left( \\frac{\\partial\\mu_{A}}{\\partial N_{A}} \\right)_{T,P,N_{B},N_{C}} = \\frac{RT\\left( 1 - x_{A} \\right)}{x_{A}} + 2x_{B}x_{C}L_{BC}$\n", + "\n", + "Setting\n", + "$\\left( \\frac{\\partial\\mu_{A}}{\\partial N_{A}} \\right)_{T,P,N_{B},N_{C}} = 0$,\n", + "one obtains\n", + "\n", + "Eq. ‑\n", + "$- \\frac{RT}{2L_{BC}} = \\frac{x_{A}x_{B}x_{C}}{1 - x_{A}} = \\frac{{\\left( 1 - x_{B} - x_{C} \\right)x}_{B}x_{C}}{x_{B} + x_{C}}$\n", + "\n", + "With $- \\frac{RT}{2L_{BC}}$ being positive due to $L_{BC} < 0$, there is\n", + "a parabola-shaped composition area in which the solution is unstable at\n", + "constant temperature and pressure. This is reasonable because the system\n", + "tends to maximize the number of B-C bonds due to its lower energy, which\n", + "competes with the entropy of mixing among the three elements and results\n", + "in segregation of B-C bonds, thus miscibility gap at low temperatures.\n", + "\n", + "To evaluate the ternary consolute point, the second derivatives for\n", + "component A and B are obtained as\n", + "\n", + "Eq. ‑\n", + "${N\\left( \\frac{\\partial_{\\ }^{2}\\mu_{A}}{\\partial N_{A}^{2}} \\right)}_{T,P,N_{B},N_{C}} = \\frac{RT\\left( 1 - x_{A} \\right)}{x_{A}^{2}} + 4x_{B}^{2}L_{AB} + 4x_{C}^{2}L_{AC} + 4x_{B}x_{C}\\left( L_{AB} + L_{AC} - L_{BC} \\right) + 2x_{B}x_{C}\\left( 7 - 9x_{A} \\right)L_{ABC} = 0$\n", + "\n", + "Eq. ‑\n", + "${N\\left( \\frac{\\partial_{\\ }^{2}\\mu_{B}}{\\partial N_{B}^{2}} \\right)}_{T,P,N_{A},N_{C}} = \\frac{RT\\left( 1 - x_{B} \\right)}{x_{B}^{2}} + 4x_{A}^{2}L_{AB} + 4x_{C}^{2}L_{BC} + 4x_{A}x_{C}\\left( L_{AB} + L_{BC} - L_{AC} \\right) + 2x_{A}x_{C}\\left( 7 - 9x_{B} \\right)L_{ABC} = 0$\n", + "\n", + "The consolute point can then be obtained using , and .\n", + "\n", + "It is observed in that $\\left( 1 - 2x_{A} \\right)L_{ABC} = 0$ at\n", + "$x_{A} = 0.5$, i.e. the ternary interaction parameter does not\n", + "contribute to the chemical potential of $A$. It is also observed in that\n", + "the contribution from the ternary interaction parameter changes sign at\n", + "$x_{i} = 2/3$ due to $\\left( 2 - 3x_{A} \\right)L_{ABC} = 0$.\n" ], "text/plain": [ "" @@ -1830,7 +3577,7 @@ ], "source": [ " if INTERACTIVE:\n", - " display(*md[md.str.contains(\"\\$\")].iloc[:10].apply(Markdown))" + " display(*documents.md[documents.md.str.contains(\"\\$\")].iloc[:10].apply(Markdown))" ] }, { diff --git a/src/psu410/src/applications_to_chemical_reactions/ellingham_diagram_and_buffered_systems.ipynb b/src/psu410/src/applications_to_chemical_reactions/ellingham_diagram_and_buffered_systems.ipynb index e53f112..c121add 100644 --- a/src/psu410/src/applications_to_chemical_reactions/ellingham_diagram_and_buffered_systems.ipynb +++ b/src/psu410/src/applications_to_chemical_reactions/ellingham_diagram_and_buffered_systems.ipynb @@ -4,158 +4,9 @@ "metadata": {}, "cells": [ { - "id": "899cf1bf", + "id": "93c0cda7", "cell_type": "markdown", - "source": [ - "## Ellingham diagram and buffered systems\n", - "\n", - "One type of chemical reactions is between a pure element in liquid or\n", - "solid states and its oxides involving one mole of oxygen gas, i.e.\n", - "\n", - "*Eq. 7\u20117* $\\frac{4}{v_{M}}M + O_{2}(gas) = M_{4/v_{M}}O_{2}$\n", - "\n", - "with $v_{M}$ being the valence of the element $M$ in the oxide. Taking\n", - "the pure $M$ and the gaseous $O_{2}$ at the reaction temperature and one\n", - "atmospheric pressure as their respective reference states, both\n", - "activities of the pure $M$ solid or liquid phase and its oxide are\n", - "unity, and the activity of $O_{2}$ equals to its partial pressure in an\n", - "ideal gas. becomes\n", - "\n", - "*Eq. 7\u20118*\n", - "$RTlnP_{O_{2}} = \\mathrm{\\Delta}_{\\ }^{0}G = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S$\n", - "\n", - "Based on , one can plot $\\mathrm{\\Delta}_{\\ }^{0}G$ as a function of\n", - "temperature for various oxidation reactions, which is called Ellingham\n", - "diagram. The intercept at $T = 0K$ is given by\n", - "$\\mathrm{\\Delta}_{\\ }^{0}H$, and the slope is represented by\n", - "$- \\mathrm{\\Delta}_{\\ }^{0}S$, depicted by the following equation\n", - "\n", - "*Eq. 7\u20119*\n", - "$\\mathrm{\\Delta}_{\\ }^{0}S = S_{M_{4/v_{M}}O_{2}} - S_{O_{2}} - \\frac{4}{v_{M}}S_{M}$\n", - "\n", - "Since the entropy of one mole of $O_{2}$ gas is significantly larger\n", - "than those of the pure element and its oxide when they are in solid or\n", - "liquid states and the entropy difference between the pure element and\n", - "its oxide, the entropy of reaction is thus dominated by the reduction of\n", - "the entropy by one mole of $O_{2}$ gas. Consequently, the entropies of\n", - "reaction are approximately the same for most reactions when the pure\n", - "elements and their oxides are solid as seen in where most lines have\n", - "similar slopes.\n", - "\n", - "Figure \u2011: Ellingham Diagram for a number of metal-oxide systems.\n", - "\n", - "For a chemical reaction on the Ellingham diagram at a given temperature,\n", - "$\\mathrm{\\Delta}_{\\ }^{0}G$ can be read from the y-axis of the diagram,\n", - "and the equilibrium partial pressure of $O_{2}$ gas can be calculated\n", - "using . Alternatively, one can plot the left part of for a given\n", - "$P_{O_{2}}$ as a function of temperature on the Ellingham diagram, i.e.\n", - "\n", - "*Eq. 7\u201110* $\\mathrm{\\Delta}_{\\ }^{0}{G =}RTlnP_{O_{2}}$\n", - "\n", - "This results in straight lines, representing iso-partial-pressure lines\n", - "of $O_{2}$, with the intercept being zero at $T = 0K$ and slopes of\n", - "$RlnP_{O_{2}}$, which are negative for $P_{O_{2}}$ lower than one\n", - "atmospheric pressure (1atm). The values of $P_{O_{2}}$ are marked on the\n", - "secondary axis on the right of the Ellingham diagram. The intersection\n", - "of the isoactivitiy line and the equilibrium line of the chemical\n", - "reaction thus gives the relation of equilibrium temperature and\n", - "equilibrium partial pressure of $O_{2}$. This is demonstrated in .\n", - "\n", - "Figure \u2011: Intersection of iso-partial-pressure lines of $O_{2}$ and\n", - "equilibrium lines in the Ellingham diagram\n", - "\n", - "For each chemical reaction in the Ellingham diagram, the three phases\n", - "are in equilibrium on the line represented by , i.e. metal, metal\n", - "oxides, and O2 gas. For conditions above the line, the value\n", - "of $P_{O_{2}}$ is larger than its equilibrium value, and the metal will\n", - "be oxidized. For conditions below the line, the value of $P_{O_{2}}$ is\n", - "lower than its equilibrium value, and the metal oxide will be reduced.\n", - "Therefore, the metal oxides in the upper part of the Ellingham diagram\n", - "can be reduced by the metals in the lower part of the diagram, and vice\n", - "versus, metals in the lower part of the diagram can be oxidized by the\n", - "metal oxides in the upper part of the diagram. For example, in , Ca can\n", - "reduce all oxides, and Cu2O is the least stable oxide.\n", - "\n", - "From the above Ellingham diagram, it is noted that equilibrium partial\n", - "pressures of $O_{2}$ are very low for most chemical reactions with many\n", - "of them lower than $10^{- 12}\\ atm$. One approach to obtain such a low\n", - "pressure is to use auxiliary reactions containing $O_{2}$ that are easy\n", - "to control and are independent of the equilibrium system of interest\n", - "except the sharing of the oxygen partial pressure. Two common auxiliary\n", - "reactions are the $H_{2}$/$H_{2}O$ and $CO$/$CO_{2}$ mixtures. For the\n", - "$H_{2}$/$H_{2}O$ mixture, the chemical reaction is\n", - "\n", - "*Eq. 7\u201111* $2H_{2}(gas) + O_{2}(gas) = 2H_{2}O(gas)$\n", - "\n", - "The equilibrium oxygen partial pressure is obtained as\n", - "\n", - "*Eq. 7\u201112*\n", - "$- RTln\\left\\{ {\\frac{1}{P_{O_{2}}}\\left( \\frac{P_{H_{2}O}}{P_{H_{2}}} \\right)}^{2} \\right\\} = \\mathrm{\\Delta}_{\\ }^{0}G = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S = - 498488\\ + 112.972T\\ (J)$\n", - "\n", - "where the $\\mathrm{\\Delta}_{\\ }^{0}H$ and $\\mathrm{\\Delta}_{\\ }^{0}S$\n", - "are taken from the substance thermodynamic database (SSUB4) compiled by\n", - "Scientific Group Thermodata Europe (SGTE) \\[59\\], which are slightly\n", - "dependent on temperature, and the values in are evaluated at 1273K using\n", - "Thermo-Calc \\[60\\]. At any given temperature, one has the following\n", - "relation\n", - "\n", - "*Eq. 7\u201113*\n", - "$RTlnP_{O_{2}} = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S - 2RTln\\frac{P_{H_{2}}}{P_{H_{2}O}} = - 498488 + \\left( 112.972 - 2Rln\\frac{P_{H_{2}}}{P_{H_{2}O}} \\right)T$\n", - "\n", - "Its intercept at $T = 0K$ is given by\n", - "$\\mathrm{\\Delta}_{\\ }^{0}H = - 498,488\\ (J)$, and the slope by\n", - "$- \\mathrm{\\Delta}_{\\ }^{0}S - 2Rln\\left( \\frac{P_{H_{2}}}{P_{H_{2}O}} \\right) = 112.972 - 2Rln\\left( \\frac{P_{H_{2}}}{P_{H_{2}O}} \\right)$.\n", - "The values of the $\\frac{P_{H_{2}}}{P_{H_{2}O}}$ ratio are marked on\n", - "another secondary axis on the right of the Ellingham diagram. The\n", - "intersection of the iso-partial-pressure-ratio line and the equilibrium\n", - "line of the chemical reaction gives the relation of equilibrium\n", - "temperature and equilibrium partial pressure ratio\n", - "$\\frac{P_{H_{2}}}{P_{H_{2}O}}$ for desired $P_{O_{2}}$ of the chemical\n", - "equilibrium.\n", - "\n", - "For the $CO$/$CO_{2}$mixture, the chemical reaction is shown below\n", - "\n", - "*Eq. 7\u201114* $2CO(gas) + O_{2}(gas) = 2CO_{2}(gas)$\n", - "\n", - "Similar, from the SSUB database, the following equation is obtained\n", - "\n", - "*Eq. 7\u201115*\n", - "$- RTln\\left\\{ {\\frac{1}{P_{O_{2}}}\\left( \\frac{P_{{CO}_{2}}}{P_{CO}} \\right)}^{2} \\right\\} = \\mathrm{\\Delta}_{\\ }^{0}G = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S = - 562,927 - 172.020T\\ (J)$\n", - "\n", - "with $\\mathrm{\\Delta}_{\\ }^{0}H$ and $\\mathrm{\\Delta}_{\\ }^{0}S$\n", - "calculated at 1273K, which are also slightly temperature-dependent as in\n", - "the case of the $H_{2}$/$H_{2}O$ mixture. The iso-partial-pressure-ratio\n", - "lines are written as\n", - "\n", - "*Eq. 7\u201116*\n", - "$RTlnP_{O_{2}} = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S - 2RTln\\frac{P_{CO}}{P_{{CO}_{2}}} = - 562,927 + \\left( 172.020 - 2Rln\\frac{P_{CO}}{P_{{CO}_{2}}} \\right)T\\ $\n", - "\n", - "Its intercept at $T = 0K$ is given by\n", - "$\\mathrm{\\Delta}_{\\ }^{0}H = - 562,927\\ (J)$, and the slope by\n", - "$172.020 - - 2Rln\\left( \\frac{P_{CO}}{P_{{CO}_{2}}} \\right)$. The values\n", - "of the $\\frac{P_{CO}}{P_{{CO}_{2}}}$ ratio are marked on the third\n", - "secondary axis on the right of the Ellingham diagram. The intersection\n", - "of the iso-partial-pressure-ratio line and the equilibrium line of the\n", - "chemical reaction gives the relation of equilibrium temperature and\n", - "equilibrium partial pressure ratio $\\frac{P_{CO}}{P_{{CO}_{2}}}$ for\n", - "desired $P_{O_{2}}$ of the chemical equilibrium.\n", - "\n", - "Therefore, one can use a mixture of the $H_{2}$/$H_{2}O$ or\n", - "$CO$/$CO_{2}$ to obtain the desired low $P_{O_{2}}$ values using and or\n", - "calculating from the SSUB database. For example,\n", - "$P_{O_{2}} = 10^{- 15}\\ atm$ at 1273K, the calculated values from the\n", - "SSUB database are $\\frac{P_{H_{2}}}{P_{H_{2}O}} \\approx 1.67$ and\n", - "$\\frac{P_{CO}}{P_{{CO}_{2}}} \\approx 2.78$, respectively, in which the\n", - "temperature dependences of $\\mathrm{\\Delta}_{\\ }^{0}H$ and\n", - "$\\mathrm{\\Delta}_{\\ }^{0}S$ and the many more gaseous species in the gas\n", - "phase are taken into account. On the other hand, the reading from the\n", - "Ellingham diagram gives $\\frac{P_{H_{2}}}{P_{H_{2}O}} \\approx 2.0$ and\n", - "$\\frac{P_{CO}}{P_{{CO}_{2}}} \\approx 2.4$, and the agreement with the\n", - "more accurate calculations above is remarkable keeping in mind the\n", - "uncertainties in graphically drawing the lines and reading both values\n", - "in the logarithmic scales from the diagram, indicating the robustness of\n", - "the Ellingham diagram.\n" - ], + "source": "## Ellingham diagram and buffered systems\n\nOne type of chemical reactions is between a pure element in liquid or\nsolid states and its oxides involving one mole of oxygen gas, i.e.\n\n*Eq. 7\u20117* $\\frac{4}{v_{M}}M + O_{2}(gas) = M_{4/v_{M}}O_{2}$\n\nwith $v_{M}$ being the valence of the element $M$ in the oxide. Taking\nthe pure $M$ and the gaseous $O_{2}$ at the reaction temperature and one\natmospheric pressure as their respective reference states, both\nactivities of the pure $M$ solid or liquid phase and its oxide are\nunity, and the activity of $O_{2}$ equals to its partial pressure in an\nideal gas. becomes\n\n*Eq. 7\u20118*\n$RTlnP_{O_{2}} = \\mathrm{\\Delta}_{\\ }^{0}G = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S$\n\nBased on , one can plot $\\mathrm{\\Delta}_{\\ }^{0}G$ as a function of\ntemperature for various oxidation reactions, which is called Ellingham\ndiagram. The intercept at $T = 0K$ is given by\n$\\mathrm{\\Delta}_{\\ }^{0}H$, and the slope is represented by\n$- \\mathrm{\\Delta}_{\\ }^{0}S$, depicted by the following equation\n\n*Eq. 7\u20119*\n$\\mathrm{\\Delta}_{\\ }^{0}S = S_{M_{4/v_{M}}O_{2}} - S_{O_{2}} - \\frac{4}{v_{M}}S_{M}$\n\nSince the entropy of one mole of $O_{2}$ gas is significantly larger\nthan those of the pure element and its oxide when they are in solid or\nliquid states and the entropy difference between the pure element and\nits oxide, the entropy of reaction is thus dominated by the reduction of\nthe entropy by one mole of $O_{2}$ gas. Consequently, the entropies of\nreaction are approximately the same for most reactions when the pure\nelements and their oxides are solid as seen in where most lines have\nsimilar slopes.\n\nFigure \u2011: Ellingham Diagram for a number of metal-oxide systems.\n\nFor a chemical reaction on the Ellingham diagram at a given temperature,\n$\\mathrm{\\Delta}_{\\ }^{0}G$ can be read from the y-axis of the diagram,\nand the equilibrium partial pressure of $O_{2}$ gas can be calculated\nusing . Alternatively, one can plot the left part of for a given\n$P_{O_{2}}$ as a function of temperature on the Ellingham diagram, i.e.\n\n*Eq. 7\u201110* $\\mathrm{\\Delta}_{\\ }^{0}{G =}RTlnP_{O_{2}}$\n\nThis results in straight lines, representing iso-partial-pressure lines\nof $O_{2}$, with the intercept being zero at $T = 0K$ and slopes of\n$RlnP_{O_{2}}$, which are negative for $P_{O_{2}}$ lower than one\natmospheric pressure (1atm). The values of $P_{O_{2}}$ are marked on the\nsecondary axis on the right of the Ellingham diagram. The intersection\nof the isoactivitiy line and the equilibrium line of the chemical\nreaction thus gives the relation of equilibrium temperature and\nequilibrium partial pressure of $O_{2}$. This is demonstrated in .\n\nFigure \u2011: Intersection of iso-partial-pressure lines of $O_{2}$ and\nequilibrium lines in the Ellingham diagram\n\nFor each chemical reaction in the Ellingham diagram, the three phases\nare in equilibrium on the line represented by , i.e. metal, metal\noxides, and O2 gas. For conditions above the line, the value\nof $P_{O_{2}}$ is larger than its equilibrium value, and the metal will\nbe oxidized. For conditions below the line, the value of $P_{O_{2}}$ is\nlower than its equilibrium value, and the metal oxide will be reduced.\nTherefore, the metal oxides in the upper part of the Ellingham diagram\ncan be reduced by the metals in the lower part of the diagram, and vice\nversus, metals in the lower part of the diagram can be oxidized by the\nmetal oxides in the upper part of the diagram. For example, in , Ca can\nreduce all oxides, and Cu2O is the least stable oxide.\n\nFrom the above Ellingham diagram, it is noted that equilibrium partial\npressures of $O_{2}$ are very low for most chemical reactions with many\nof them lower than $10^{- 12}\\ atm$. One approach to obtain such a low\npressure is to use auxiliary reactions containing $O_{2}$ that are easy\nto control and are independent of the equilibrium system of interest\nexcept the sharing of the oxygen partial pressure. Two common auxiliary\nreactions are the $H_{2}$/$H_{2}O$ and $CO$/$CO_{2}$ mixtures. For the\n$H_{2}$/$H_{2}O$ mixture, the chemical reaction is\n\n*Eq. 7\u201111* $2H_{2}(gas) + O_{2}(gas) = 2H_{2}O(gas)$\n\nThe equilibrium oxygen partial pressure is obtained as\n\n*Eq. 7\u201112*\n$- RTln\\left\\{ {\\frac{1}{P_{O_{2}}}\\left( \\frac{P_{H_{2}O}}{P_{H_{2}}} \\right)}^{2} \\right\\} = \\mathrm{\\Delta}_{\\ }^{0}G = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S = - 498488\\ + 112.972T\\ (J)$\n\nwhere the $\\mathrm{\\Delta}_{\\ }^{0}H$ and $\\mathrm{\\Delta}_{\\ }^{0}S$\nare taken from the substance thermodynamic database (SSUB4) compiled by\nScientific Group Thermodata Europe (SGTE) \\[59\\], which are slightly\ndependent on temperature, and the values in are evaluated at 1273K using\nThermo-Calc \\[60\\]. At any given temperature, one has the following\nrelation\n\n*Eq. 7\u201113*\n$RTlnP_{O_{2}} = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S - 2RTln\\frac{P_{H_{2}}}{P_{H_{2}O}} = - 498488 + \\left( 112.972 - 2Rln\\frac{P_{H_{2}}}{P_{H_{2}O}} \\right)T$\n\nIts intercept at $T = 0K$ is given by\n$\\mathrm{\\Delta}_{\\ }^{0}H = - 498,488\\ (J)$, and the slope by\n$- \\mathrm{\\Delta}_{\\ }^{0}S - 2Rln\\left( \\frac{P_{H_{2}}}{P_{H_{2}O}} \\right) = 112.972 - 2Rln\\left( \\frac{P_{H_{2}}}{P_{H_{2}O}} \\right)$.\nThe values of the $\\frac{P_{H_{2}}}{P_{H_{2}O}}$ ratio are marked on\nanother secondary axis on the right of the Ellingham diagram. The\nintersection of the iso-partial-pressure-ratio line and the equilibrium\nline of the chemical reaction gives the relation of equilibrium\ntemperature and equilibrium partial pressure ratio\n$\\frac{P_{H_{2}}}{P_{H_{2}O}}$ for desired $P_{O_{2}}$ of the chemical\nequilibrium.\n\nFor the $CO$/$CO_{2}$mixture, the chemical reaction is shown below\n\n*Eq. 7\u201114* $2CO(gas) + O_{2}(gas) = 2CO_{2}(gas)$\n\nSimilar, from the SSUB database, the following equation is obtained\n\n*Eq. 7\u201115*\n$- RTln\\left\\{ {\\frac{1}{P_{O_{2}}}\\left( \\frac{P_{{CO}_{2}}}{P_{CO}} \\right)}^{2} \\right\\} = \\mathrm{\\Delta}_{\\ }^{0}G = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S = - 562,927 - 172.020T\\ (J)$\n\nwith $\\mathrm{\\Delta}_{\\ }^{0}H$ and $\\mathrm{\\Delta}_{\\ }^{0}S$\ncalculated at 1273K, which are also slightly temperature-dependent as in\nthe case of the $H_{2}$/$H_{2}O$ mixture. The iso-partial-pressure-ratio\nlines are written as\n\n*Eq. 7\u201116*\n$RTlnP_{O_{2}} = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S - 2RTln\\frac{P_{CO}}{P_{{CO}_{2}}} = - 562,927 + \\left( 172.020 - 2Rln\\frac{P_{CO}}{P_{{CO}_{2}}} \\right)T\\ $\n\nIts intercept at $T = 0K$ is given by\n$\\mathrm{\\Delta}_{\\ }^{0}H = - 562,927\\ (J)$, and the slope by\n$172.020 - - 2Rln\\left( \\frac{P_{CO}}{P_{{CO}_{2}}} \\right)$. The values\nof the $\\frac{P_{CO}}{P_{{CO}_{2}}}$ ratio are marked on the third\nsecondary axis on the right of the Ellingham diagram. The intersection\nof the iso-partial-pressure-ratio line and the equilibrium line of the\nchemical reaction gives the relation of equilibrium temperature and\nequilibrium partial pressure ratio $\\frac{P_{CO}}{P_{{CO}_{2}}}$ for\ndesired $P_{O_{2}}$ of the chemical equilibrium.\n\nTherefore, one can use a mixture of the $H_{2}$/$H_{2}O$ or\n$CO$/$CO_{2}$ to obtain the desired low $P_{O_{2}}$ values using and or\ncalculating from the SSUB database. For example,\n$P_{O_{2}} = 10^{- 15}\\ atm$ at 1273K, the calculated values from the\nSSUB database are $\\frac{P_{H_{2}}}{P_{H_{2}O}} \\approx 1.67$ and\n$\\frac{P_{CO}}{P_{{CO}_{2}}} \\approx 2.78$, respectively, in which the\ntemperature dependences of $\\mathrm{\\Delta}_{\\ }^{0}H$ and\n$\\mathrm{\\Delta}_{\\ }^{0}S$ and the many more gaseous species in the gas\nphase are taken into account. On the other hand, the reading from the\nEllingham diagram gives $\\frac{P_{H_{2}}}{P_{H_{2}O}} \\approx 2.0$ and\n$\\frac{P_{CO}}{P_{{CO}_{2}}} \\approx 2.4$, and the agreement with the\nmore accurate calculations above is remarkable keeping in mind the\nuncertainties in graphically drawing the lines and reading both values\nin the logarithmic scales from the diagram, indicating the robustness of\nthe Ellingham diagram.\n", "metadata": {} } ] diff --git a/src/psu410/src/applications_to_chemical_reactions/index.ipynb b/src/psu410/src/applications_to_chemical_reactions/index.ipynb index bc79398..a3f41d1 100644 --- a/src/psu410/src/applications_to_chemical_reactions/index.ipynb +++ b/src/psu410/src/applications_to_chemical_reactions/index.ipynb @@ -4,24 +4,9 @@ "metadata": {}, "cells": [ { - "id": "ce48ca09", + "id": "b87385fe", "cell_type": "markdown", - "source": [ - "# Applications to chemical reactions\n", - "\n", - "A chemical reaction can be viewed as a framework dividing a system into\n", - "two closed subsystems: reactants and products. The phases and species of\n", - "reactants and products are selected from possible phases and species\n", - "that may form from the independent components of the system. A chemical\n", - "reaction can thus be considered as an internal process to transfer heat\n", - "and work between the two subsystems of reactants and products. It is\n", - "evident that this subset of phases and species only represents partial\n", - "equilibrium information of the system under given external conditions,\n", - "and more stable equilibrium states may exist by including more phases\n", - "and species with the global equilibrium depicted by phase diagrams\n", - "discussed in previous chapters with all known phases and species\n", - "included.\n" - ], + "source": "# Applications to chemical reactions\n\nA chemical reaction can be viewed as a framework dividing a system into\ntwo closed subsystems: reactants and products. The phases and species of\nreactants and products are selected from possible phases and species\nthat may form from the independent components of the system. A chemical\nreaction can thus be considered as an internal process to transfer heat\nand work between the two subsystems of reactants and products. It is\nevident that this subset of phases and species only represents partial\nequilibrium information of the system under given external conditions,\nand more stable equilibrium states may exist by including more phases\nand species with the global equilibrium depicted by phase diagrams\ndiscussed in previous chapters with all known phases and species\nincluded.\n", "metadata": {} } ] diff --git a/src/psu410/src/applications_to_chemical_reactions/internal_process_and_differential_and_integrated_driving_forces.ipynb b/src/psu410/src/applications_to_chemical_reactions/internal_process_and_differential_and_integrated_driving_forces.ipynb index 91f9412..a1f94a5 100644 --- a/src/psu410/src/applications_to_chemical_reactions/internal_process_and_differential_and_integrated_driving_forces.ipynb +++ b/src/psu410/src/applications_to_chemical_reactions/internal_process_and_differential_and_integrated_driving_forces.ipynb @@ -4,98 +4,9 @@ "metadata": {}, "cells": [ { - "id": "38ff033b", + "id": "215e22d0", "cell_type": "markdown", - "source": [ - "## Internal process and differential and integrated driving forces\n", - "\n", - "The driving force for an internal process can be defined as following\n", - "using $U$, $H$, $F$, or $G$ as discussed in Chapter and depending on\n", - "what system variables are kept constant\n", - "\n", - "*Eq. 7\u20111*\n", - "$- D = \\left( \\frac{\\partial U}{\\partial\\xi} \\right)_{S,\\ V,N_{i}\\ } = \\left( \\frac{\\partial H}{\\partial\\xi} \\right)_{S,\\ P,N_{i}\\ } = \\left( \\frac{\\partial F}{\\partial\\xi} \\right)_{T,\\ V,N_{i}\\ } = \\left( \\frac{\\partial G}{\\partial\\xi} \\right)_{T,P,N_{i}\\ }$\n", - "\n", - "This can be termed as differential driving force as it relates the\n", - "derivative of energy with respect to an internal process so the change\n", - "is infinitesimally small and does not affect the properties of the\n", - "system significantly. For a system under constant $T$ and $P$, let us\n", - "consider an internal process for forming a new phase $\\alpha$ with the\n", - "composition $x_{i}^{\\alpha}$ and Gibbs energy\n", - "$G_{m}^{\\alpha}\\left( x_{i}^{\\alpha} \\right)$. The differential driving\n", - "force for such an internal process can thus be defined as\n", - "\n", - "*Eq. 7\u20112*\n", - "$- D = G_{m}^{\\alpha}\\left( x_{i}^{\\alpha} \\right) - \\sum_{i}^{}x_{i}^{\\alpha}\\mu_{i} = \\sum_{i}^{}x_{i}^{\\alpha}\\left( \\mu_{i}^{\\alpha} - \\mu_{i} \\right)$\n", - "\n", - "where $\\mu_{i}$ is the chemical potential of component *i* in the\n", - "system. This may also be called as nucleation driving force as if the\n", - "$\\alpha$ phase is nucleating in the system. As discussed in Chapter ,\n", - "chemical potentials are the intercepts on the Gibbs energy axis by the\n", - "tangent plane of Gibbs energy. thus represents the distance between the\n", - "tangent planes of the original system and the $\\alpha$ phase at the\n", - "composition of the $\\alpha$ phase. Evidently, this distance is at its\n", - "maximum when the two tangent planes are parallel to each other, i.e. the\n", - "commonly called parallel tangent construction when evaluating nucleation\n", - "driving force.\n", - "\n", - "The situation is different for chemical reactions where the amount of\n", - "each component in reactants is the same as that in products, i.e. there\n", - "is a mass balance between reactants and products. The driving force for\n", - "a chemical reaction is defined by the Gibbs energy difference between\n", - "the reactants and products as if all the reactants are transferred to\n", - "the products. This driving force may thus be called integrated driving\n", - "force as it describes the energy difference of a system under two\n", - "different states, i.e.\n", - "\n", - "*Eq. 7\u20113*\n", - "$- \\int_{}^{}{Dd\\xi} = \\mathrm{\\Delta}G = \\sum_{p}^{}{n_{p}G_{p}} - \\sum_{r}^{}{n_{r}G_{r}}$\n", - "\n", - "where superscripts $p$ and $r$ denote products and reactants, $n_{p}$\n", - "and $n_{r}$ their corresponding moles, $G_{p}$and $G_{r}$ the Gibbs\n", - "energies in per mole of formula of their respective stoichiometries as\n", - "written in the chemical reaction. Conventionally, the products and\n", - "reactants are represented by species or stoichiometric compounds rather\n", - "than individual phases. This is particularly evident when various\n", - "gaseous species are considered in a chemical reaction. Consequently,\n", - "$G_{p}$and $G_{r}$ in represent the chemical potentials of product and\n", - "reactant species. For species with a fixed composition, its chemical\n", - "potential is the same as its Gibbs energy as shown by . For a species in\n", - "a solution phase, its chemical potential is related to its activity as\n", - "shown in . can thus be further written as\n", - "\n", - "*Eq. 7\u20114*\n", - "$\\mathrm{\\Delta}G = \\sum_{p}^{}{n_{p}_{\\ }^{0}G_{p}} - \\sum_{r}^{}{n_{r}_{\\ }^{0}G_{r}} + RTln\\frac{\\prod_{p}^{}\\left( a_{p} \\right)^{n_{p}}}{\\prod_{r}^{}\\left( a_{r} \\right)^{n_{r}}} = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{\\prod_{p}^{}\\left( a_{p} \\right)^{n_{p}}}{\\prod_{r}^{}\\left( a_{r} \\right)^{n_{r}}}$\n", - "\n", - "It is evident from that for stoichiometric phases in a chemical\n", - "reaction, their activities equal to one. At equilibrium, the integrated\n", - "driving force becomes zero, i.e. $\\mathrm{\\Delta}G = 0$, and is\n", - "re-arranged to become\n", - "\n", - "*Eq. 7\u20115*\n", - "$- RTln\\frac{\\prod_{p}^{}\\left( a_{p} \\right)^{n_{p}}}{\\prod_{r}^{}\\left( a_{r} \\right)^{n_{r}}} = - RTlnK_{e} = \\mathrm{\\Delta}_{\\ }^{0}G = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S$\n", - "\n", - "where $K_{e}$ is often called reaction constant relating the activities\n", - "of products and reactants at equilibrium. In a system with\n", - "$\\frac{\\prod_{p}^{}\\left( a_{p} \\right)^{n_{p}}}{\\prod_{r}^{}\\left( a_{r} \\right)^{n_{r}}} > K_{e}$,\n", - "the chemical reaction goes to left, and the products are decomposed,\n", - "while\n", - "$\\frac{\\prod_{p}^{}\\left( a_{p} \\right)^{n_{p}}}{\\prod_{r}^{}\\left( a_{r} \\right)^{n_{r}}} < K_{e}$,\n", - "the chemical reaction goes to right, and the products are formed. is\n", - "often recast into the following form by dividing $\u2013RT$ on both sides of\n", - "the equation\n", - "\n", - "*Eq. 7\u20116*\n", - "$\\ln K_{e} = - \\frac{\\mathrm{\\Delta}_{\\ }^{0}H}{RT} + \\frac{\\mathrm{\\Delta}_{\\ }^{0}S}{R}$\n", - "\n", - "With $\\ln K_{e}$ plotted with respect to $1/T$, indicates that the slope\n", - "is $- \\mathrm{\\Delta}_{\\ }^{0}H$/R and the intercept on the y axis is\n", - "$\\mathrm{\\Delta}_{\\ }^{0}{S/R}$ as shown in .\n", - "\n", - "Figure \u2011: $\\ln K_{e}$ plotted with respect to $1/T$ for several M-O\n", - "systems at 1 bar and $K_{e}$ represented by the partial pressure ratio\n", - "of CO2 and CO.\n" - ], + "source": "## Internal process and differential and integrated driving forces\n\nThe driving force for an internal process can be defined as following\nusing $U$, $H$, $F$, or $G$ as discussed in Chapter and depending on\nwhat system variables are kept constant\n\n*Eq. 7\u20111*\n$- D = \\left( \\frac{\\partial U}{\\partial\\xi} \\right)_{S,\\ V,N_{i}\\ } = \\left( \\frac{\\partial H}{\\partial\\xi} \\right)_{S,\\ P,N_{i}\\ } = \\left( \\frac{\\partial F}{\\partial\\xi} \\right)_{T,\\ V,N_{i}\\ } = \\left( \\frac{\\partial G}{\\partial\\xi} \\right)_{T,P,N_{i}\\ }$\n\nThis can be termed as differential driving force as it relates the\nderivative of energy with respect to an internal process so the change\nis infinitesimally small and does not affect the properties of the\nsystem significantly. For a system under constant $T$ and $P$, let us\nconsider an internal process for forming a new phase $\\alpha$ with the\ncomposition $x_{i}^{\\alpha}$ and Gibbs energy\n$G_{m}^{\\alpha}\\left( x_{i}^{\\alpha} \\right)$. The differential driving\nforce for such an internal process can thus be defined as\n\n*Eq. 7\u20112*\n$- D = G_{m}^{\\alpha}\\left( x_{i}^{\\alpha} \\right) - \\sum_{i}^{}x_{i}^{\\alpha}\\mu_{i} = \\sum_{i}^{}x_{i}^{\\alpha}\\left( \\mu_{i}^{\\alpha} - \\mu_{i} \\right)$\n\nwhere $\\mu_{i}$ is the chemical potential of component *i* in the\nsystem. This may also be called as nucleation driving force as if the\n$\\alpha$ phase is nucleating in the system. As discussed in Chapter ,\nchemical potentials are the intercepts on the Gibbs energy axis by the\ntangent plane of Gibbs energy. thus represents the distance between the\ntangent planes of the original system and the $\\alpha$ phase at the\ncomposition of the $\\alpha$ phase. Evidently, this distance is at its\nmaximum when the two tangent planes are parallel to each other, i.e. the\ncommonly called parallel tangent construction when evaluating nucleation\ndriving force.\n\nThe situation is different for chemical reactions where the amount of\neach component in reactants is the same as that in products, i.e. there\nis a mass balance between reactants and products. The driving force for\na chemical reaction is defined by the Gibbs energy difference between\nthe reactants and products as if all the reactants are transferred to\nthe products. This driving force may thus be called integrated driving\nforce as it describes the energy difference of a system under two\ndifferent states, i.e.\n\n*Eq. 7\u20113*\n$- \\int_{}^{}{Dd\\xi} = \\mathrm{\\Delta}G = \\sum_{p}^{}{n_{p}G_{p}} - \\sum_{r}^{}{n_{r}G_{r}}$\n\nwhere superscripts $p$ and $r$ denote products and reactants, $n_{p}$\nand $n_{r}$ their corresponding moles, $G_{p}$and $G_{r}$ the Gibbs\nenergies in per mole of formula of their respective stoichiometries as\nwritten in the chemical reaction. Conventionally, the products and\nreactants are represented by species or stoichiometric compounds rather\nthan individual phases. This is particularly evident when various\ngaseous species are considered in a chemical reaction. Consequently,\n$G_{p}$and $G_{r}$ in represent the chemical potentials of product and\nreactant species. For species with a fixed composition, its chemical\npotential is the same as its Gibbs energy as shown by . For a species in\na solution phase, its chemical potential is related to its activity as\nshown in . can thus be further written as\n\n*Eq. 7\u20114*\n$\\mathrm{\\Delta}G = \\sum_{p}^{}{n_{p}_{\\ }^{0}G_{p}} - \\sum_{r}^{}{n_{r}_{\\ }^{0}G_{r}} + RTln\\frac{\\prod_{p}^{}\\left( a_{p} \\right)^{n_{p}}}{\\prod_{r}^{}\\left( a_{r} \\right)^{n_{r}}} = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{\\prod_{p}^{}\\left( a_{p} \\right)^{n_{p}}}{\\prod_{r}^{}\\left( a_{r} \\right)^{n_{r}}}$\n\nIt is evident from that for stoichiometric phases in a chemical\nreaction, their activities equal to one. At equilibrium, the integrated\ndriving force becomes zero, i.e. $\\mathrm{\\Delta}G = 0$, and is\nre-arranged to become\n\n*Eq. 7\u20115*\n$- RTln\\frac{\\prod_{p}^{}\\left( a_{p} \\right)^{n_{p}}}{\\prod_{r}^{}\\left( a_{r} \\right)^{n_{r}}} = - RTlnK_{e} = \\mathrm{\\Delta}_{\\ }^{0}G = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S$\n\nwhere $K_{e}$ is often called reaction constant relating the activities\nof products and reactants at equilibrium. In a system with\n$\\frac{\\prod_{p}^{}\\left( a_{p} \\right)^{n_{p}}}{\\prod_{r}^{}\\left( a_{r} \\right)^{n_{r}}} > K_{e}$,\nthe chemical reaction goes to left, and the products are decomposed,\nwhile\n$\\frac{\\prod_{p}^{}\\left( a_{p} \\right)^{n_{p}}}{\\prod_{r}^{}\\left( a_{r} \\right)^{n_{r}}} < K_{e}$,\nthe chemical reaction goes to right, and the products are formed. is\noften recast into the following form by dividing $\u2013RT$ on both sides of\nthe equation\n\n*Eq. 7\u20116*\n$\\ln K_{e} = - \\frac{\\mathrm{\\Delta}_{\\ }^{0}H}{RT} + \\frac{\\mathrm{\\Delta}_{\\ }^{0}S}{R}$\n\nWith $\\ln K_{e}$ plotted with respect to $1/T$, indicates that the slope\nis $- \\mathrm{\\Delta}_{\\ }^{0}H$/R and the intercept on the y axis is\n$\\mathrm{\\Delta}_{\\ }^{0}{S/R}$ as shown in .\n\nFigure \u2011: $\\ln K_{e}$ plotted with respect to $1/T$ for several M-O\nsystems at 1 bar and $K_{e}$ represented by the partial pressure ratio\nof CO2 and CO.\n", "metadata": {} } ] diff --git a/src/psu410/src/applications_to_chemical_reactions/maximum_reaction_rate_and_chemical_transport_reactions_.ipynb b/src/psu410/src/applications_to_chemical_reactions/maximum_reaction_rate_and_chemical_transport_reactions_.ipynb index e93d0d8..a23ee15 100644 --- a/src/psu410/src/applications_to_chemical_reactions/maximum_reaction_rate_and_chemical_transport_reactions_.ipynb +++ b/src/psu410/src/applications_to_chemical_reactions/maximum_reaction_rate_and_chemical_transport_reactions_.ipynb @@ -4,155 +4,9 @@ "metadata": {}, "cells": [ { - "id": "2e089526", + "id": "1baa04de", "cell_type": "markdown", - "source": [ - "## Maximum reaction rate and chemical transport reactions \n", - "\n", - "Equilibrium thermodynamics can be used to calculate the maximum rates of\n", - "reaction that are possible in dynamically reacting systems, such as when\n", - "a corrosive gas passes over a heated sample. Other examples of such\n", - "reactions include the reduction of a metal oxide in flowing hydrogen,\n", - "the evaporation of a material in a vacuum, and the deposition of a thin\n", - "film by a chemical vapour deposition process. The basic assumption used\n", - "in calculating these maximum reaction rates is that local equilibrium\n", - "exists at the location of the considered reaction.\n", - "\n", - "This section examines several examples in which maximum reaction rates\n", - "are calculated. The system can typically be divided into three regions:\n", - "(1) the input region, which is usually near room temperature, (2) the\n", - "high temperature region in which the primary reaction of interest is\n", - "occurring, and (3) the exit region. Such a system is almost always at\n", - "constant pressure throughout the system. Since the three regions have\n", - "different temperatures, the key is to use the mass conservation of the\n", - "carrier gas in all three regions. With the input gas at $T = 298K$ and\n", - "$P = 1atm$, the volume of one mole of ideal gas is\n", - "$0.0244\\left( m^{3} \\right) = 24.4(liter)$. The number of moles of input\n", - "gas flowing through the system can be written as\n", - "\n", - "*Eq. 7\u201117*\n", - "$n_{gas}^{0} = \\frac{Pf_{gas}}{RT} = \\frac{f_{gas}}{24.4} = 0.0409f_{gas}$\n", - "\n", - "where $f_{gas}$ is the input gas flow rate in liter per unit time at\n", - "$T = 298K$ and $P = 1atm$, and $R$ the gas constant.\n", - "\n", - "The first example is the evaporation of water in a flowing stream of dry\n", - "hydrogen. A schematic diagram of the system being considered is given in\n", - ". The goal of the calculation is to determine the maximum rate of\n", - "$H_{2}O(liquid)$ loss, $n_{H_{2}O}^{\\ }$, through vaporization in a\n", - "flowing stream of $H_{2}$, e.g. for generating a given $H_{2}O/H_{2}$\n", - "ratio for producing a given $O_{2}$ pressure in a high temperature\n", - "system as related to the Ellingham diagram discussed in Chapter . The\n", - "maximum rate is determined by saturating the $H_{2}$ with the\n", - "equilibrium vapour pressure of $H_{2}O(liquid)$ with the number of moles\n", - "of $H_{2}$ being $n_{H_{2}}^{0} = 0.0409f_{H_{2}}$ and $f_{H_{2}}$ being\n", - "the input flow rate of $H_{2}$ in liter per unit time.\n", - "\n", - "Figure \u2011: Schematic diagram of vaporization of water in a flowing stream\n", - "of dry hydrogen\n", - "\n", - "In the first input region, the total number moles of gas is\n", - "$N = n_{H_{2}}^{0}$. In the second high temperature region with the\n", - "temperature and pressure being $T_{sys}$ and $P_{sys}$, the total number\n", - "moles of gas is $N = n_{H_{2}}^{0} + n_{H_{2}O}^{\\ }$, which is the same\n", - "for the third exit region with $T_{exit}$. To avoid condensation, one\n", - "needs to maintain $T_{exit} > T_{sys}$. The vapour pressure of $H_{2}O$\n", - "at $T_{sys}$, $P_{H_{2}O\\ }$, can be obtained from equilibrium\n", - "thermodynamic calculations. Since the $H_{2}$ and $H_{2}O$ are occupying\n", - "the same volume at the same temperature, the maximum number of moles\n", - "$H_{2}O$ can be calculated from the following relation based on the\n", - "ideal gas law\n", - "\n", - "*Eq. 7\u201118*\n", - "$n_{H_{2}O}^{\\ } = \\frac{P_{H_{2}O}^{\\ }}{P_{H_{2}}^{\\ }}n_{H_{2}}^{0} = \\frac{P_{H_{2}O}^{\\ }}{P_{sys}^{\\ } - P_{H_{2}O}^{\\ }}n_{H_{2}}^{0} = \\frac{P_{H_{2}O}^{\\ }}{P_{sys}^{\\ } - P_{H_{2}O}^{\\ }}\\frac{f_{H_{2}}}{24.4}$\n", - "\n", - "For $P_{sys}^{\\ } = 101325Pa$, one can calculate $P_{H_{2}O}^{\\ }$ and\n", - "plot $\\frac{n_{H_{2}O}^{\\ }}{f_{H_{2}}}$ as a function of temperature\n", - "from the SSUB database as shown in . The corresponding partial pressure\n", - "ratio, $P_{H_{2}O}/P_{H_{2}}$ , is plotted in , which can be used to\n", - "calculate $P_{O_{2}}^{\\ }$ at any given temperatures. An example is\n", - "shown in with $T_{sys} = 348.15K$ and $P_{H_{2}O}/P_{H_{2}} = 0.607$.\n", - "\n", - "Figure \u2011: Ratio of maximum number of moles $H_{2}O$ with respect to\n", - "hydrogen flow rate, $\\frac{n_{H_{2}O}^{\\ }}{f_{H_{2}}}$, plotted as a\n", - "function of temperature.\n", - "\n", - "Figure \u2011: Partial pressure ratio, $P_{H_{2}O}/P_{H_{2}}$, corresponding\n", - "to .\n", - "\n", - "Figure \u2011: Partial pressure of oxygen, $P_{O_{2}}^{\\ }$, as a function of\n", - "temperature with $T_{sys} = 348.15K$ and $P_{H_{2}O}/P_{H_{2}} = 0.607$.\n", - "\n", - "The second example is the corrosion of $SiO_{2}$(s) by flowing $H_{2}$\n", - "gas at high temperatures. Considering $T_{sys}^{\\ } = 1700K$,\n", - "$P_{sys}^{\\ } = 101325Pa$, and $f_{H_{2}} = 1\\ liter/min$, the system is\n", - "thus defined with 0.0409 moles of $H_{2}$ and an equilibrium between the\n", - "gas phase and the tridymite $SiO_{2}$. The equilibrium calculation gives\n", - "0.04092 moles of gas with its constitutions as\n", - "$H_{2}:H_{2}O:SiO = 0.998887:4.833 \\bullet 10^{- 4}:4.832 \\bullet 10^{- 4}$.\n", - "The corrosion rate of SiO2(s) is thus\n", - "$0.04092x4.832 \\bullet 10^{- 4} = 1.98 \\bullet 10^{- 5}mol/min = 1.19gram/min.$\n", - "\n", - "Another example is to consider that the flowing $CO$ gas of 298K and\n", - "1atm ($= 101325Pa$) at a rate of $1\\ liter/min$ passes through and\n", - "equilibrates with single phase $C(s)$ at 1500K. The equilibrium system\n", - "is defined by $T = 1500K$, $P = 1atm$, and 0.0409 moles of $CO$ with the\n", - "equilibrium between the gas phase and C(s). The equilibrium calculation\n", - "gives 0.040872 moles of gas phase with the mole fraction of $CO$ being\n", - "0.999327, resulting in the loss of *CO* or the deposition of *C(s)* at a\n", - "rate of\n", - "$0.0409 - 0.040872 \\bullet 0.999327 = 5.55 \\bullet 10^{- 5}mole/min$.\n", - "\n", - "A chemical transport reaction is a reaction in which a condensed phase\n", - "reacts with a gas phase to form vapour-phase products, which in turn\n", - "undergo the reverse reformation of the condensed phase. Two well-known\n", - "examples of such reactions are\n", - "\n", - "*Eq. 7\u201119* $M(s) + \\frac{n}{2}I_{2}(g) = MI_{n}(g)$\n", - "\n", - "*Eq. 7\u201120* $Ni(s) + 4CO(g) = Ni{(CO)}_{4}(g)$\n", - "\n", - "which are used in the purification of metals by the iodide process and\n", - "in the purification of nickel by the Mond-Langer process. In both\n", - "processes the forward reaction is favoured by lower temperatures and the\n", - "reverse reaction by higher temperatures, resulting in the deposition of\n", - "the metal. The most common technique for causing a chemical transport of\n", - "a condensed substance makes use of the temperature dependence of the\n", - "equilibrium constant. As was discussed previously, the enthalpy of\n", - "reaction, $\\mathrm{\\Delta}_{\\ }^{0}H$, determines the manner in which\n", - "$K_{e}$ changes with temperature (see ). The value of $K_{e}$ increases\n", - "with increasing *T* for $\\mathrm{\\Delta}_{\\ }^{0}H > 0$, $K_{e}$\n", - "decreases with increasing *T* for $\\mathrm{\\Delta}_{\\ }^{0}H < 0$, and\n", - "$K_{e}$ is independent of T for $\\mathrm{\\Delta}_{\\ }^{0}H = 0$. The\n", - "$\\mathrm{\\Delta}_{\\ }^{0}H$ and $\\mathrm{\\Delta}_{\\ }^{0}S$ values for\n", - "chemical transport reactions may be either positive or negative. For\n", - "reactions by both are positive, and for reactions by both are negative.\n", - "\n", - "In a typical experiment the starting solid is located at the point in a\n", - "temperature gradient that corresponds to the largest $K_{e}$ value for\n", - "the experimental condition. As the gaseous species migrate to other\n", - "locations in the system with temperatures corresponding to lower $K_{e}$\n", - "values, the reverse reaction occurs to satisfy the new equilibrium\n", - "requirements, and the solid phase is deposited. The dependence of\n", - "$K_{e}$ on $\\mathrm{\\Delta}_{\\ }^{0}H$ results in a material transport\n", - "from hot to cold for $\\mathrm{\\Delta}_{\\ }^{0}H > 0$ (the same as for\n", - "vaporization-condensation reactions), from cold to hot for\n", - "$\\mathrm{\\Delta}_{\\ }^{0}H < 0$, and no transport for\n", - "$\\mathrm{\\Delta}_{\\ }^{0}H = 0$.\n", - "\n", - "The success of a particular reaction in causing an appreciable transport\n", - "of a condensed phase depends mainly upon the partial pressure gradients\n", - "or concentration gradients of the gaseous species in the system. A\n", - "reaction whose equilibrium is extreme toward either the reactant side or\n", - "the product side will not give an appreciable transport of material. The\n", - "concentration gradients are too small in such a system. Reactions with\n", - "equilibrium constants near unity at the experimental temperatures\n", - "usually give the largest transport since small changes in $K_{e}$ cause\n", - "large changes in concentrations. The general condition required for\n", - "obtaining a $K_{e}$ value near unity at a reasonable temperature is that\n", - "$\\mathrm{\\Delta}_{\\ }^{0}H$ and $\\mathrm{\\Delta}_{\\ }^{0}S$ both have\n", - "the same sign, resulting from the equalities of .\n" - ], + "source": "## Maximum reaction rate and chemical transport reactions \n\nEquilibrium thermodynamics can be used to calculate the maximum rates of\nreaction that are possible in dynamically reacting systems, such as when\na corrosive gas passes over a heated sample. Other examples of such\nreactions include the reduction of a metal oxide in flowing hydrogen,\nthe evaporation of a material in a vacuum, and the deposition of a thin\nfilm by a chemical vapour deposition process. The basic assumption used\nin calculating these maximum reaction rates is that local equilibrium\nexists at the location of the considered reaction.\n\nThis section examines several examples in which maximum reaction rates\nare calculated. The system can typically be divided into three regions:\n(1) the input region, which is usually near room temperature, (2) the\nhigh temperature region in which the primary reaction of interest is\noccurring, and (3) the exit region. Such a system is almost always at\nconstant pressure throughout the system. Since the three regions have\ndifferent temperatures, the key is to use the mass conservation of the\ncarrier gas in all three regions. With the input gas at $T = 298K$ and\n$P = 1atm$, the volume of one mole of ideal gas is\n$0.0244\\left( m^{3} \\right) = 24.4(liter)$. The number of moles of input\ngas flowing through the system can be written as\n\n*Eq. 7\u201117*\n$n_{gas}^{0} = \\frac{Pf_{gas}}{RT} = \\frac{f_{gas}}{24.4} = 0.0409f_{gas}$\n\nwhere $f_{gas}$ is the input gas flow rate in liter per unit time at\n$T = 298K$ and $P = 1atm$, and $R$ the gas constant.\n\nThe first example is the evaporation of water in a flowing stream of dry\nhydrogen. A schematic diagram of the system being considered is given in\n. The goal of the calculation is to determine the maximum rate of\n$H_{2}O(liquid)$ loss, $n_{H_{2}O}^{\\ }$, through vaporization in a\nflowing stream of $H_{2}$, e.g. for generating a given $H_{2}O/H_{2}$\nratio for producing a given $O_{2}$ pressure in a high temperature\nsystem as related to the Ellingham diagram discussed in Chapter . The\nmaximum rate is determined by saturating the $H_{2}$ with the\nequilibrium vapour pressure of $H_{2}O(liquid)$ with the number of moles\nof $H_{2}$ being $n_{H_{2}}^{0} = 0.0409f_{H_{2}}$ and $f_{H_{2}}$ being\nthe input flow rate of $H_{2}$ in liter per unit time.\n\nFigure \u2011: Schematic diagram of vaporization of water in a flowing stream\nof dry hydrogen\n\nIn the first input region, the total number moles of gas is\n$N = n_{H_{2}}^{0}$. In the second high temperature region with the\ntemperature and pressure being $T_{sys}$ and $P_{sys}$, the total number\nmoles of gas is $N = n_{H_{2}}^{0} + n_{H_{2}O}^{\\ }$, which is the same\nfor the third exit region with $T_{exit}$. To avoid condensation, one\nneeds to maintain $T_{exit} > T_{sys}$. The vapour pressure of $H_{2}O$\nat $T_{sys}$, $P_{H_{2}O\\ }$, can be obtained from equilibrium\nthermodynamic calculations. Since the $H_{2}$ and $H_{2}O$ are occupying\nthe same volume at the same temperature, the maximum number of moles\n$H_{2}O$ can be calculated from the following relation based on the\nideal gas law\n\n*Eq. 7\u201118*\n$n_{H_{2}O}^{\\ } = \\frac{P_{H_{2}O}^{\\ }}{P_{H_{2}}^{\\ }}n_{H_{2}}^{0} = \\frac{P_{H_{2}O}^{\\ }}{P_{sys}^{\\ } - P_{H_{2}O}^{\\ }}n_{H_{2}}^{0} = \\frac{P_{H_{2}O}^{\\ }}{P_{sys}^{\\ } - P_{H_{2}O}^{\\ }}\\frac{f_{H_{2}}}{24.4}$\n\nFor $P_{sys}^{\\ } = 101325Pa$, one can calculate $P_{H_{2}O}^{\\ }$ and\nplot $\\frac{n_{H_{2}O}^{\\ }}{f_{H_{2}}}$ as a function of temperature\nfrom the SSUB database as shown in . The corresponding partial pressure\nratio, $P_{H_{2}O}/P_{H_{2}}$ , is plotted in , which can be used to\ncalculate $P_{O_{2}}^{\\ }$ at any given temperatures. An example is\nshown in with $T_{sys} = 348.15K$ and $P_{H_{2}O}/P_{H_{2}} = 0.607$.\n\nFigure \u2011: Ratio of maximum number of moles $H_{2}O$ with respect to\nhydrogen flow rate, $\\frac{n_{H_{2}O}^{\\ }}{f_{H_{2}}}$, plotted as a\nfunction of temperature.\n\nFigure \u2011: Partial pressure ratio, $P_{H_{2}O}/P_{H_{2}}$, corresponding\nto .\n\nFigure \u2011: Partial pressure of oxygen, $P_{O_{2}}^{\\ }$, as a function of\ntemperature with $T_{sys} = 348.15K$ and $P_{H_{2}O}/P_{H_{2}} = 0.607$.\n\nThe second example is the corrosion of $SiO_{2}$(s) by flowing $H_{2}$\ngas at high temperatures. Considering $T_{sys}^{\\ } = 1700K$,\n$P_{sys}^{\\ } = 101325Pa$, and $f_{H_{2}} = 1\\ liter/min$, the system is\nthus defined with 0.0409 moles of $H_{2}$ and an equilibrium between the\ngas phase and the tridymite $SiO_{2}$. The equilibrium calculation gives\n0.04092 moles of gas with its constitutions as\n$H_{2}:H_{2}O:SiO = 0.998887:4.833 \\bullet 10^{- 4}:4.832 \\bullet 10^{- 4}$.\nThe corrosion rate of SiO2(s) is thus\n$0.04092x4.832 \\bullet 10^{- 4} = 1.98 \\bullet 10^{- 5}mol/min = 1.19gram/min.$\n\nAnother example is to consider that the flowing $CO$ gas of 298K and\n1atm ($= 101325Pa$) at a rate of $1\\ liter/min$ passes through and\nequilibrates with single phase $C(s)$ at 1500K. The equilibrium system\nis defined by $T = 1500K$, $P = 1atm$, and 0.0409 moles of $CO$ with the\nequilibrium between the gas phase and C(s). The equilibrium calculation\ngives 0.040872 moles of gas phase with the mole fraction of $CO$ being\n0.999327, resulting in the loss of *CO* or the deposition of *C(s)* at a\nrate of\n$0.0409 - 0.040872 \\bullet 0.999327 = 5.55 \\bullet 10^{- 5}mole/min$.\n\nA chemical transport reaction is a reaction in which a condensed phase\nreacts with a gas phase to form vapour-phase products, which in turn\nundergo the reverse reformation of the condensed phase. Two well-known\nexamples of such reactions are\n\n*Eq. 7\u201119* $M(s) + \\frac{n}{2}I_{2}(g) = MI_{n}(g)$\n\n*Eq. 7\u201120* $Ni(s) + 4CO(g) = Ni{(CO)}_{4}(g)$\n\nwhich are used in the purification of metals by the iodide process and\nin the purification of nickel by the Mond-Langer process. In both\nprocesses the forward reaction is favoured by lower temperatures and the\nreverse reaction by higher temperatures, resulting in the deposition of\nthe metal. The most common technique for causing a chemical transport of\na condensed substance makes use of the temperature dependence of the\nequilibrium constant. As was discussed previously, the enthalpy of\nreaction, $\\mathrm{\\Delta}_{\\ }^{0}H$, determines the manner in which\n$K_{e}$ changes with temperature (see ). The value of $K_{e}$ increases\nwith increasing *T* for $\\mathrm{\\Delta}_{\\ }^{0}H > 0$, $K_{e}$\ndecreases with increasing *T* for $\\mathrm{\\Delta}_{\\ }^{0}H < 0$, and\n$K_{e}$ is independent of T for $\\mathrm{\\Delta}_{\\ }^{0}H = 0$. The\n$\\mathrm{\\Delta}_{\\ }^{0}H$ and $\\mathrm{\\Delta}_{\\ }^{0}S$ values for\nchemical transport reactions may be either positive or negative. For\nreactions by both are positive, and for reactions by both are negative.\n\nIn a typical experiment the starting solid is located at the point in a\ntemperature gradient that corresponds to the largest $K_{e}$ value for\nthe experimental condition. As the gaseous species migrate to other\nlocations in the system with temperatures corresponding to lower $K_{e}$\nvalues, the reverse reaction occurs to satisfy the new equilibrium\nrequirements, and the solid phase is deposited. The dependence of\n$K_{e}$ on $\\mathrm{\\Delta}_{\\ }^{0}H$ results in a material transport\nfrom hot to cold for $\\mathrm{\\Delta}_{\\ }^{0}H > 0$ (the same as for\nvaporization-condensation reactions), from cold to hot for\n$\\mathrm{\\Delta}_{\\ }^{0}H < 0$, and no transport for\n$\\mathrm{\\Delta}_{\\ }^{0}H = 0$.\n\nThe success of a particular reaction in causing an appreciable transport\nof a condensed phase depends mainly upon the partial pressure gradients\nor concentration gradients of the gaseous species in the system. A\nreaction whose equilibrium is extreme toward either the reactant side or\nthe product side will not give an appreciable transport of material. The\nconcentration gradients are too small in such a system. Reactions with\nequilibrium constants near unity at the experimental temperatures\nusually give the largest transport since small changes in $K_{e}$ cause\nlarge changes in concentrations. The general condition required for\nobtaining a $K_{e}$ value near unity at a reasonable temperature is that\n$\\mathrm{\\Delta}_{\\ }^{0}H$ and $\\mathrm{\\Delta}_{\\ }^{0}S$ both have\nthe same sign, resulting from the equalities of .\n", "metadata": {} } ] diff --git a/src/psu410/src/applications_to_chemical_reactions/trends_of_entropies_of_reactions.ipynb b/src/psu410/src/applications_to_chemical_reactions/trends_of_entropies_of_reactions.ipynb index 198ee50..6708fd4 100644 --- a/src/psu410/src/applications_to_chemical_reactions/trends_of_entropies_of_reactions.ipynb +++ b/src/psu410/src/applications_to_chemical_reactions/trends_of_entropies_of_reactions.ipynb @@ -4,80 +4,9 @@ "metadata": {}, "cells": [ { - "id": "f0c53519", + "id": "b51b2eff", "cell_type": "markdown", - "source": [ - "## Trends of entropies of reactions\n", - "\n", - "The reaction entropy, $\\mathrm{\\Delta}_{\\ }^{0}S$ in , plays an\n", - "important role in determining equilibria of high-temperature reactions.\n", - "The most important single factor that determines the entropy of a\n", - "reaction is the net change in the number of moles of gas as briefly\n", - "mentioned in the discussion of the Ellingham diagram above. The reason\n", - "this is true can be explained as follows.\n", - "\n", - "The entropy of a substance can be thought of as being the sum of four\n", - "parts: (i) translational, (ii) rotational, (iii) vibrational, and (iv)\n", - "electronic. The translational entropy of a gas is the largest entropy\n", - "term under most conditions. To the extent that the other contributions\n", - "cancel between reactants and products, the entropy of reaction is\n", - "determined by the change in the number of moles of gaseous molecules.\n", - "Based on the literature data or calculations from the SSUB database, the\n", - "net change in the number of moles of gas in a reaction results\n", - "approximately in an entropy of reaction of about 175\u00b145 J/K/mole-gas at\n", - "298K for many halides and oxides. The chemical reactions of and\n", - "discussed above both reduce the gas by one mole, and their entropies of\n", - "reaction are -113 and -172 J/K at 1273K, and -89 and -173J/K at 298K,\n", - "respectively, indicating that the chemical reaction of is an exception\n", - "of the empirical rule. For chemical reactions shown in the Ellingham\n", - "diagram, their entropies of reaction follow this empirical rule pretty\n", - "well with some of them shown in calculated from the SGTE database.\n", - "\n", - "Table \u2011: Entropies of reactions with gas at 298.15K, J/K\n", - "\n", - "Reaction: Si+O2 =SiO2 -182\n", - "\n", - "Reaction: Ti+O2=TiO2 -185\n", - "\n", - "Reaction: 2Mg+O2=2MgO -217\n", - "\n", - "Reaction: 2Ca+O2=2CaO -212\n", - "\n", - "Reaction: 2Mn+O2=2MnO -150\n", - "\n", - "Since the entropy of a reaction is primarily determined by the net\n", - "change in the number of moles of gas, the entropies for reactions\n", - "involving only condensed phases must be small. The entropies of fusion\n", - "of monatomic solids are usually in the range 8-15 J/K/mole-atom as shown\n", - "for some elements in . Most metals and many ionic salts have values that\n", - "lie in this range when given in terms of per mole of atom of material.\n", - "There are few exceptions such as silicon and boron shown in the table.\n", - "For solid-state reactions, the average values can be approximated as\n", - "0\u00b18J/K/mole-atom as also shown in the table.\n", - "\n", - "Table \u2011: Entropies of reactions of condensed phases at 298.15K, J/K\n", - "\n", - "Reaction: Si(s)=Si(l) 29.762\n", - "\n", - "Reaction: Ti(s2)=Ti(l) 7.288\n", - "\n", - "Reaction: Mg(s2)=Mg(l) 9.184\n", - "\n", - "Reaction: Ca(s2)=Ca(l) 7.659\n", - "\n", - "Reaction: Mn(s2)=Mn(l) 11.443\n", - "\n", - "Reaction: W(s)=W(l) 14.158\n", - "\n", - "Reaction: B(s)=B(l) 21.380\n", - "\n", - "Reaction: 3Fe+C=CFe3 17.060\n", - "\n", - "Reaction: S+Mn=MnS 13.909\n", - "\n", - "Reaction: NiO+Fe2O3=Fe2NiO4\n", - "0.464\n" - ], + "source": "## Trends of entropies of reactions\n\nThe reaction entropy, $\\mathrm{\\Delta}_{\\ }^{0}S$ in , plays an\nimportant role in determining equilibria of high-temperature reactions.\nThe most important single factor that determines the entropy of a\nreaction is the net change in the number of moles of gas as briefly\nmentioned in the discussion of the Ellingham diagram above. The reason\nthis is true can be explained as follows.\n\nThe entropy of a substance can be thought of as being the sum of four\nparts: (i) translational, (ii) rotational, (iii) vibrational, and (iv)\nelectronic. The translational entropy of a gas is the largest entropy\nterm under most conditions. To the extent that the other contributions\ncancel between reactants and products, the entropy of reaction is\ndetermined by the change in the number of moles of gaseous molecules.\nBased on the literature data or calculations from the SSUB database, the\nnet change in the number of moles of gas in a reaction results\napproximately in an entropy of reaction of about 175\u00b145 J/K/mole-gas at\n298K for many halides and oxides. The chemical reactions of and\ndiscussed above both reduce the gas by one mole, and their entropies of\nreaction are -113 and -172 J/K at 1273K, and -89 and -173J/K at 298K,\nrespectively, indicating that the chemical reaction of is an exception\nof the empirical rule. For chemical reactions shown in the Ellingham\ndiagram, their entropies of reaction follow this empirical rule pretty\nwell with some of them shown in calculated from the SGTE database.\n\nTable \u2011: Entropies of reactions with gas at 298.15K, J/K\n\nReaction: Si+O2 =SiO2 -182\n\nReaction: Ti+O2=TiO2 -185\n\nReaction: 2Mg+O2=2MgO -217\n\nReaction: 2Ca+O2=2CaO -212\n\nReaction: 2Mn+O2=2MnO -150\n\nSince the entropy of a reaction is primarily determined by the net\nchange in the number of moles of gas, the entropies for reactions\ninvolving only condensed phases must be small. The entropies of fusion\nof monatomic solids are usually in the range 8-15 J/K/mole-atom as shown\nfor some elements in . Most metals and many ionic salts have values that\nlie in this range when given in terms of per mole of atom of material.\nThere are few exceptions such as silicon and boron shown in the table.\nFor solid-state reactions, the average values can be approximated as\n0\u00b18J/K/mole-atom as also shown in the table.\n\nTable \u2011: Entropies of reactions of condensed phases at 298.15K, J/K\n\nReaction: Si(s)=Si(l) 29.762\n\nReaction: Ti(s2)=Ti(l) 7.288\n\nReaction: Mg(s2)=Mg(l) 9.184\n\nReaction: Ca(s2)=Ca(l) 7.659\n\nReaction: Mn(s2)=Mn(l) 11.443\n\nReaction: W(s)=W(l) 14.158\n\nReaction: B(s)=B(l) 21.380\n\nReaction: 3Fe+C=CFe3 17.060\n\nReaction: S+Mn=MnS 13.909\n\nReaction: NiO+Fe2O3=Fe2NiO4\n0.464\n", "metadata": {} } ] diff --git a/src/psu410/src/applications_to_electrochemical_systems/application_examples.ipynb b/src/psu410/src/applications_to_electrochemical_systems/application_examples.ipynb index 3fbf77a..34a46a0 100644 --- a/src/psu410/src/applications_to_electrochemical_systems/application_examples.ipynb +++ b/src/psu410/src/applications_to_electrochemical_systems/application_examples.ipynb @@ -4,14 +4,39 @@ "metadata": {}, "cells": [ { - "id": "cde26072", + "id": "9d0e901f", "cell_type": "markdown", - "source": [ - "## Application examples\n", - "\n", - "Among many applications of electrochemistry, several of them are briefly\n", - "discussed in this section.\n" - ], + "source": "## Application examples\n\nAmong many applications of electrochemistry, several of them are briefly\ndiscussed in this section.\n", + "metadata": {} + }, + { + "id": "16ed3744", + "cell_type": "markdown", + "source": "### Metastability and passivation\n\nOur modern industrial society is built on various metals such as Fe, Ni,\nAl, Ti, and Zr alloys which are reactive, but exhibit extraordinary\nkinetic stabilities in oxidizing environments due to the existence of a\nthin reaction product film on the surface. This film effectively\nisolates the metal from the corrosive environment, a phenomenon called\npassivation. One interesting experiment by Faraday in 1836 who reported\nthat iron corrodes freely in dilute nitric acid, while in concentrated\nnitric acid, no reaction apparently occurred. To understand this\nphenomenon, let us exam a simple, schematic Pourbaix diagram for the\niron-water system shown in .\n\nFigure \u2011: Schematic Pourbaix diagram for iron illustrating the\nresolution of the Faraday paradox in the corrosion of iron in nitric\nacid \\[61\\]. Lines (a) (b), and (c) correspond to the equilibria: (a)\n$H^{+} + e^{-} = 1/2H_{2}$; (b) $O_{2} + 4H^{+} + 4e^{-} = 2H_{2}O$; (c)\n$NO_{3}^{-} + 3H^{+} + 2e^{-} = HNO_{2} + H_{2}O$, respectively\n\nFor iron in deaerated acid solution, the partial anodic and cathodic\nreactions are Line 1 $\\left( Fe/{Fe}^{2 +} \\right)$ and Line (a),\nrespectively, resulting in a corrosion potential that lies between Lines\n$1$ and $a$. In oxygenated (aerated) solutions, the corrosion potential\nmay lie between Lines $1$ and $b$, because the reduction of oxygen is a\npossible (likely) cathodic reaction. Since dilute HNO3 is\nonly a weak oxidizing agent, the principal cathodic reaction was most\nlikely hydrogen evolution, and hence the corrosion potential is expected\nto lie between Lines $1$ and $a$ at relatively high pH, as shown. Since\nthe $Fe/{Fe}^{2 +}$ reaction is relatively fast compared with\n$H^{+}/H_{2}$ on iron, if the corrosion potential is situated below the\nextension of Line 2 (Fe/Fe3O4) into the\n${Fe}^{2 +}$ stability region, Fe3O4 cannot form\non the surface, even as a metastable phase.\n\nOn the other hand, concentrated HNO3 is a strong oxidizing\nagent due to the reaction:\n$NO_{3}^{-} + 3H^{+} + 2e^{-} = HNO_{2} + H_{2}O$, so that the corrosion\npotential can lie anywhere between Lines $1$ and $c$ at low pH. Since\nReaction (2) is likely to be fast, the corrosion potential will be high\nand certainly will be more positive than the extension of Line 2 into\nthe stability region for ${Fe}^{2 +}$ at low pH. Therefore,\nFe3O4 becomes metastable and can form between the\naqueous solution and iron. The thickness of this\nFe3O4 film depends on its dissolution rate into\nthe aqueous solution and its growth rate at the interface with iron,\nwhich depends on the diffusion of ionic species across the film. Its\nexistence results in passivity and the observed kinetic inactivity of\niron in this medium. When the potential becomes even more positive above\nthe extension of Line $3$, Fe2O3 may form on\nFe3O4 as an additional metastable phase, resulting\nin the commonly observed bilayer structure.\n", + "metadata": {} + }, + { + "id": "f6d149c7", + "cell_type": "markdown", + "source": "### Galvanic protection\n\nA galvanic reaction takes place between two different materials at the\ntwo respective electrodes, each with different tendency to hold on to\nelectrons. Consider the following electrochemical cell used to protect\nCu tanks against oxidation by using a \u201csacrificial\u201d Fe electrode\n\nanode solution cathode\n\nFe(s) \\| Fe2+ \\| SO4= \\|\nCu2+\\| Cu(s) \\|\n\nCathode reduction: Cu2+ + 2 e- = Cu \u03b5o\n(volts) = 0.34\n\nAnode oxidation: Fe(s) = Fe2+ + 2 e- \u03b5o (volts) =\n0.44\n\nNet reaction: Cu2+ + Fe =\\> Cu + Fe2+\n\u03b5o (volts) = 0.78\n\nIf the cell has a direct connection between the electrodes, i.e. it has\na short circuit: \u2206G \u2192 0 and thus \u03b5cell \u2192 0. Since the value\nof \u03b5ocell \\> 0 for the net cell reaction, the equilibrium\nconstant *Ke* \\> 1, which means\n\\[Fe2+\\]/\\[Cu2+\\] \\> 1. By assuming an ideal\nelectrolyte solution, the activities in *Ke* can be\nrepresented by concentrations (in molar concentration units), assuming\nthat solid Fe and Cu are present at unit activities. If the electrodes\nof Cu and Fe are short circuited while in contact with the same\n\"electrolyte solution\", the final equilibrium concentrations can be\ncalculated by the standard equation,\n\n*Eq. 8\u201161* *Ke* = \\[Fe2+\\]/\\[Cu2+\\] =\nexp (-\u2206oG/RT)\n\nor, using the Nernst equations,\n\n*Eq. 8\u201162* \u03b5 = \u03b5o -(RT/z f) ln (\\[Fe2+\\]/\\[Cu2+\\])\n= 0\n\nand the above standard cell potential:\n\\[Fe2+\\]/\\[Cu2+\\] = 2.4\u20221026\n\nWith this large ratio it can be seen that the tendency to produce\nCu+2 ions, i.e., the tendency to corrode the Cu(s), is\nextremely small if a sacrificial Fe electrode is configured in an\nelectrochemical cell with the Cu tank.\n", + "metadata": {} + }, + { + "id": "ac54ddda", + "cell_type": "markdown", + "source": "### Fuel cells\n\nFuel cells are devices to convert chemical energy to electricity and\nheat through electrochemical reactions with the fuel and oxygen supplied\nto the anode and cathode, respectively. Typical ions migrating through\nthe electrolyte are $H^{+}$, ${OH}^{-}$, ${CO}_{3}^{2 -}$, and\n$O^{2 -}$. In fuel cells with $H^{+}$ as migrating ions, H2\nmolecules are dissociated into $H^{+}$ on the anode, which are combined\nwith O2 on the cathode to form H2O and release\nheat with the half cell and the net cell reactions in their simplest\nform as shown by for the anode, for the cathode, and for the net cell,\nrespectively. Commonly used electrolytes are polymer and phosphoric\nacid, and both anode and cathode reactions are facilitated by catalyst,\ntypically platinum. The thermodynamic limit of power which can be\ngenerated by the fuel cell is represented by\n\n*Eq. 8\u201163*\n$w = - \\mathrm{\\Delta}G = - \\mathrm{\\Delta}_{\\ }^{0}G_{cell} + RTln\\left( P_{H_{2}}P_{O_{2}}^{1/2} \\right)$\n\nFor fuel cells with anions as migrating ions, the anions are generated\non the cathode with H2O formed and heat generated on the\nanode. Their representative cathode reactions are\n\n*Eq. 8\u201164* $\\frac{1}{2}O_{2} + H_{2}O + 2e^{-}{= 2OH}^{-}$\n\n*Eq. 8\u201165* ${\\frac{1}{2}O}_{2} + CO_{2} + 2e^{-} = {CO}_{3}^{2 -}$\n\n*Eq. 8\u201166* $\\frac{1}{2}O_{2} + 2e^{-} = O^{2 -}$.\n\nThe anode reaction for is the reaction represented by , operating at low\ntemperatures and using catalyst for both electrodes. The anode reactions\nfor and are\n\n*Eq. 8\u201167* ${CO}_{3}^{2 -} + H_{2} = H_{2}O + CO_{2} + 2e^{-}$\n\n*Eq. 8\u201168* $O^{2 -} + H_{2} = H_{2}O + 2e^{-}$\n\nrespectively. To enable the diffusion of ${CO}_{3}^{2 -}$ and $O^{2 -}$\nthrough the cathode and the electrolyte, both fuel cells are operated at\nrelative high temperatures, with the former typically in molten\ncarbonate solutions and the latter through solid oxides. Due to the high\noperating temperatures, fuels are converted to hydrogen within the fuel\ncell itself by a process called internal reforming, removing the need\nfor precious-metal catalyst and enabling the use of a variety of fuels.\nThe net cell reaction for all three fuel cells is the same as in the\ncase of $H^{+}$, represented by .\n", + "metadata": {} + }, + { + "id": "01a4c089", + "cell_type": "markdown", + "source": "### Ion transport membranes\n\nIon transport membranes (ITMs) are ceramic membranes conducting both\nelectrons and oxygen ions, but no other species. The chemical potential\ndifference of oxygen between two sides of a membrane provides the\ndriving force for oxygen to diffuse through the membrane. Commonly used\nITM oxides include perovskite and fluorite, with the chemical formula of\nABO3 and AO2, respectively, typically with more\nthan one elements in the A-site and / or the B-site to tailor the\nelectron and ionic conductivities. Key thermodynamic properties of ITM\noxides are their stability in service environments, vacancy\nconcentrations in the oxygen and cation sites, valances of cations. On\nthe high oxygen partial pressure side, the reaction is as the following\n\n*Eq. 8\u201169* $\\frac{1}{2}O_{2} + 2e^{-} = O^{2 -}$\n\nAt the same time the number of oxygen vacancy is reduced, resulting in a\nlower concentration of oxygen vacancy and higher oxygen activity in the\noxide on the high oxygen partial pressure side. On the low oxygen\npartial pressure side, the reaction is reversed to produce oxygen\nmolecules, i.e.\n\n*Eq. 8\u201170* $O^{2 -} = \\frac{1}{2}O_{2} + 2e^{-}$\n\nThis reaction results in higher oxygen vacancy concentration and lower\noxygen activity. At both sides, the charge neutrality is compensated by\nthe valance changes of cations, resulting in the electron flow in the\nopposite direction of oxygen diffusion. The ionic conductivity is\ndictated by the oxygen transportation across the membrane with the\ndriving force of the following net reaction\n\n*Eq. 8\u201171*\n$\\frac{1}{2}O_{2}\\left( P_{high} \\right) = \\frac{1}{2}O_{2}\\left( P_{low} \\right)$\n\n*Eq. 8\u201172*\n$\\mathrm{\\Delta}G = 0.5RTln\\left( \\frac{P_{low}}{P_{high}} \\right)$.\n\nThe oxygen transportation is closely related to the concentration of\noxygen vacancy in the membrane, which is obtained by minimizing the\nGibbs energy of the phase under given temperature and oxygen partial\npressure conditions. High vacancy concentrations can be obtained by\ncation dopants with lower valances or small energy differences between\nvarious valance states. However, at the same time, high vacancy\nconcentrations reduce the thermodynamic stability of the membrane, which\nmay result in its decomposition into more stable phases.\n", + "metadata": {} + }, + { + "id": "9b9a3496", + "cell_type": "markdown", + "source": "### Electrical batteries\n\nBatteries utilize electrochemical reactions to generate electricity for\nvarious devices. The theoretic voltage of a battery can be calculated\nfrom and , i.e.\n\n*Eq. 8\u201173*\n$\\varepsilon = - \\frac{\\mathrm{\\Delta}G}{zf} = \\varepsilon^{0} - \\frac{RTlnQ}{zf}$\n\nwith $\\mathrm{\\Delta}G$ being the driving force of the net cell reaction\nand $Q$ being the reaction activity quotient. The actual voltage of a\nbattery is lower than the theoretical one due to kinetic limitations of\ncell reactions and resistance to ion diffusion through the electrolyte.\nBased on whether the cell reactions are reversible or not, batteries\ntypically categorized as either primary disposable or secondary\nrechargeable batteries. The net cell reactions in primary disposable\nbatteries are not easily reversible, and electrode materials may not\nreturn to their original forms by applying a higher external potential\nof opposite sign. Consequently, primary batteries cannot be reliably\nrecharged. On the other hand, the net cell reactions in secondary\nbatteries are easily reversible. Furthermore, two half-cells in\nbatteries may use different electrolytes with each half-cell enclosed in\na container and a separator permeable to conducting ions but not the\nbulk of the electrolytes.\n\nOne common primary battery is zinc-carbon battery with a zinc anode\ncylinder and a carbon cathode central rod. The electrolytes are ammonium\nor zinc chloride next to the zinc anode and a mixture of ammonium\nchloride and manganese dioxide next to the carbon cathode. The half cell\nand net reactions with ammonium chloride are as follows\n\n*Eq. 8\u201174* Zn + 2NH3 \u2192\nZn(NH3)22+ + 2 e-\n\n*Eq. 8\u201175* 2NH4Cl + 2MnO2 + 2 e- \u2192\n2NH3 + Mn2O3 +\nH2O+2Cl\u2212\n\n*Eq. 8\u201176* Zn + 2MnO2 + 2NH4Cl \u2192\nMn2O3 +\nZn(NH3)2Cl2 + H2O.\n\nThe electric potential of the reaction is, treating all compounds as\nstoichiometric compounds\n\nEq. 8\u201177\n$\\varepsilon = - \\frac{\\mathrm{\\Delta}G}{2f} = - \\frac{\\mathrm{\\Delta}^{0}G}{2f} = \\frac{1}{2f}\\left(_{\\ }^{0}G^{Zn} + 2_{\\ }^{0}G^{{MnO}_{2}} + 2_{\\ }^{0}G^{{NH}_{4}Cl} -_{\\ }^{0}G^{H_{2}O} -_{\\ }^{0}G^{Zn\\left( {NH}_{3} \\right)_{2}{Cl}_{2}} -_{\\ }^{0}G^{{{Mn}_{2}O}_{3}} \\right)$.\n\nThe Gibbs energy of $Zn\\left( {NH}_{3} \\right)_{2}{Cl}_{2}$ is not\navailable in current databases and has been recently estimated to be\n\u2212505,375 J/mole-formula \\[62\\]. The value of at 298.15K is thus obtained\nas 1.67 V, which is pretty close to the actual operating voltage of the\nbattery around 1.5 V.\n\nWhile with zinc chloride, the cell reactions and electric potential may\nbe written as\n\nEq. 8\u201178 Zn + ZnCl2 + 2OH\u2212 \u2192 2ZnOHCl + 2\ne-\n\n*Eq. 8\u201179* MnO2 + H2O + e- \u2192 MnOOH +\nOH-\n\n*Eq. 8\u201180* Zn + 2 MnO2 + ZnCl2 + 2 H2O\n\u2192 2 MnOOH + 2 ZnOHCl\n\n*Eq. 8\u201181*\n$\\varepsilon = \\frac{1}{2f}\\left(_{\\ }^{0}G^{Zn} + 2_{\\ }^{0}G^{{MnO}_{2}} +_{\\ }^{0}G^{Zn{Cl}_{2}} + 2_{\\ }^{0}G^{H_{2}O} - 2_{\\ }^{0}G^{MnOOH} - 2_{\\ }^{0}G^{ZnOHCl} \\right)$\n\nSecondary batteries can be recharged by applying an external electrical\npotential, which reverses the net cell reaction that occur during\ndischarging. The oldest form of rechargeable battery is the lead-acid\nbatteries used in automotive, and the latest development is the\nlithium-ion (Li-ion) batteries. A lead-acid battery typically uses Pb\nand PbO2 as the cathode and anode electrodes and a 35% sulfuric acid and\n65% water solution as the electrolyte. Its anode and cathode reactions\ncan be simplified as follows\n\nEq. 8\u201182 $Pb + SO_{4}^{2 -} = PbSO_{4} + 2e^{-}$\n\nEq. 8\u201183 $PbO_{2} + 4H^{+} + SO_{4}^{2 -} + 2e^{-} = PbSO_{4} + 2H_{2}O$\n\nThe net cell reaction is\n\nEq. 8\u201184 $Pb + PbO_{2} + 2H_{2}SO_{4}^{\\ } = 2PbSO_{4} + 2H_{2}O$.\n\nIts electric potential is represented by the following equation\n\n*Eq. 8\u201185*\n$\\varepsilon = - \\frac{1}{2f}\\left( 2_{\\ }^{0}G^{H_{2}O} + 2_{\\ }^{0}G^{{PbSO}_{4}} -_{\\ }^{0}G^{Pb} -_{\\ }^{0}G^{{PbO}_{2}} - 2_{\\ }^{0}G^{H_{2}SO_{4}} \\right)$\n\nwith the value at 298.15K being 2.651 V calculated from Thermo-Calc\n\\[60\\]. During discharging, the reaction goes to right, and $PbSO_{4}$\nis formed on both anode and cathode, while during charging, the reaction\ngoes to the left, and $Pb$ and $PbO_{2}$ are restored. In practical\napplications, other ionic species such as ${H_{3}O}^{+}$ and\n$HSO_{4}^{-}$ may form in the electrolyte, complicating the reactions\nand affecting its potential.\n\nIn lithium ion batteries, lithium ions migrate in electrolytes between\nelectrodes made of intercalated lithium compounds during charging and\ndischarging. LiCoO2 and LiFePO4 are two of the\nseveral cathode materials used in lithium ion batteries, and the anode\nis typically made of carbon or metallic Li. The anode and cathode\nreactions for LiCoO2 batteries can be written in simple forms as follows\n\nEq. 8\u201186 ${Li}_{x}C_{6} = x{Li}^{+} + xe^{-} + 6C$\n\nEq. 8\u201187 $x{Li}^{+} + xe^{-} + {Li}_{1 - x}CoO_{2} = LiCoO_{2}$\n\nwith the net reaction and electric potential being\n\nEq. 8\u201188 ${Li}_{x}C_{6} + {Li}_{1 - x}CoO_{2} = LiCoO_{2} + 6C$\n\n*Eq. 8\u201189*\n$\\varepsilon = - \\frac{1}{xf}\\left\\{ 6_{\\ }^{0}G^{C} +_{\\ }^{0}G^{{LiCoO}_{2}} - G^{{{Li}_{x}C}_{6}} - G^{{{Li}_{1 - x}CoO}_{2}} \\right\\} = - \\frac{1}{f}\\left\\{ \\left( \\mu_{Li}^{{Li}_{1 - x}CoO_{2}} - \\mu_{Li}^{{Li}_{x}C} \\right) - \\frac{1}{x}\\left( \\mu_{LiCoO_{2}}^{{Li}_{1 - x}CoO_{2}} -_{\\ }^{0}G^{{LiCoO}_{2}} \\right) \\right\\}$\n\nThe electric potential is a function of $x$. The value in the first\nparenthesis in the above equation denotes the chemical potential\ndifference of Li between two electrodes, and the value in the second\nparenthesis represents the chemical potential difference of\n${LiCoO}_{2}$ between the states in the solution phase of\n${Li}_{1 - x}CoO_{2}$ and by itself. Gibbs energies of ${{Li}_{x}C}_{6}$\nand ${{Li}_{1 - x}CoO}_{2}$ need to be obtained as a function $x$ in\norder to calculate the electric potential of the battery.\n\nLiFePO4 uses metallic lithium as the anode with following\nhalf-cell and net cell reactions\n\nEq. 8\u201190 $xLi = x{Li}^{+} + xe^{-}$\n\nEq. 8\u201191 $x{Li}^{+} + xe^{-} + {Li}_{1 - x}FePO_{4} = LiFePO_{4}$\n\nEq. 8\u201192 $xLi + {Li}_{1 - x}FePO_{4} = LiFePO_{4}$.\n\nIts electric potential is also a function of $x$, i.e.\n\n*Eq. 8\u201193*\n$\\varepsilon = - \\frac{1}{xf}\\left\\{_{\\ }^{0}G^{LiFePO_{4}} - x\\ ^{0}G^{Li} - G^{{Li}_{1 - x}FePO_{4}} \\right\\} = - \\frac{1}{f}\\left\\{ \\left( \\mu_{Li}^{{Li}_{1 - x}FePO_{4}} - \\ ^{0}\\mu_{Li} \\right) - \\frac{1}{x}\\left( \\mu_{LiFePO_{4}}^{{Li}_{1 - x}FePO_{4}} -_{\\ }^{0}G^{LiFePO_{4}} \\right) \\right\\}$\n\nThe value in the first parenthesis in the above equation denotes the\nchemical potential difference of Li between two electrodes, and the\nvalue in the second parenthesis represents the chemical potential\ndifference of $LiFePO_{4}$ between the states in the solution phase of\n${Li}_{1 - x}FePO_{4}$ and by itself. Consequently, Gibbs energy of\n${Li}_{1 - x}FePO_{4}$ needs to be obtained as a function $x$ in order\nto calculate the electric potential of the battery. It is known that\nthere are several miscibility gaps in the $FePO_{4}$ and $LiFePO_{4}$\npsuedo-binary system, in which the chemical potentials are constants, so\nis the electric potential, resulting in stable battery output.\n", "metadata": {} } ] diff --git a/src/psu410/src/applications_to_electrochemical_systems/aqueous_solution_and_pourbaix_diagram.ipynb b/src/psu410/src/applications_to_electrochemical_systems/aqueous_solution_and_pourbaix_diagram.ipynb index ba52c57..08a2c2e 100644 --- a/src/psu410/src/applications_to_electrochemical_systems/aqueous_solution_and_pourbaix_diagram.ipynb +++ b/src/psu410/src/applications_to_electrochemical_systems/aqueous_solution_and_pourbaix_diagram.ipynb @@ -4,259 +4,9 @@ "metadata": {}, "cells": [ { - "id": "91f80ded", + "id": "ea5a58bd", "cell_type": "markdown", - "source": [ - "## Aqueous solution and Pourbaix diagram\n", - "\n", - "The importance of aqueous solutions in all aspects of life is so well\n", - "known and needs not be discussed further. Since many electrochemical\n", - "processes involve electrolyte solutions in an aqueous solvent,\n", - "electrochemical processes including water, hydrogen, and/or oxygen are\n", - "discussed in more details. The hydrogen-oxygen cell can be described for\n", - "both acidic electrolytes and alkaline electrolytes. With acidic\n", - "electrolytes, H+ is in much higher concentrations than\n", - "OH-, and thus half-cell reactions with H+ as an\n", - "ionic transport species are more important than those involving\n", - "OH-. The reverse is true for alkaline electrolytes that\n", - "contain high OH-concentrations. Other than for nearly neutral\n", - "acid-base systems, either H+ or OH- dominates the\n", - "other by several orders of magnitude as can be seen from the value of\n", - "the 298 K dissociation constant for H2O:\n", - "\n", - "*Eq. 8\u201135* H2O(l) = H+(aq) + OH-(aq)\n", - "\n", - "with the reaction constant being Ke =\n", - "\\[H+\\]\\[OH-\\] = 10-14 and\n", - "$\\mathrm{\\Delta}_{\\ }^{0}G$= -RT *ln* Ke *\u03b2*= +79,908 J. By\n", - "convention, one defines pH = - log \\[H+\\] and pOH = - log\n", - "\\[OH-\\], and then pH + pOH = 14.\n", - "\n", - "Under acidic electrolyte conditions of low pH (high \\[H+\\]\n", - "concentrations) the anode reaction in a hydrogen-oxygen cell is:\n", - "\n", - "*Eq. 8\u201136* \u00bd H2(g) = H+(aq) + e-\n", - "\n", - "with \u03b51o = 0.0 V and\n", - "$\\mathrm{\\Delta}_{\\ }^{0}G_{1}$= 0 J. The corresponding cathode\n", - "(reduction) reaction is:\n", - "\n", - "*Eq. 8\u201137* 2 H+(aq) + \u00bd O2(g) + 2 e- =\n", - "H2O(l)\n", - "\n", - "with \u03b52o = 1.229 V and\n", - "$\\mathrm{\\Delta}_{\\ }^{0}G_{2}$ = -2\\*1.229\\*96,485 J = -237,160 J. The\n", - "net cell reaction for acidic electrolytes is:\n", - "\n", - "*Eq. 8\u201138* H2(g) + \u00bd O2(g) = H2O(l)\n", - "\n", - "with \u03b5ocell = 1.229 V and\n", - "$\\mathrm{\\Delta}_{\\ }^{0}G_{cell}$ = -2\\*1.229\\*96,485 J = -237,160 J\n", - "\n", - "Under alkaline electrolyte conditions of high pH (high\n", - "\\[OH-\\] concentrations) the anode reaction in a\n", - "hydrogen-oxygen cell is:\n", - "\n", - "*Eq. 8\u201139* 2 OH-(aq) + H2(g) = 2\n", - "H2O(l) + 2 e-\n", - "\n", - "with \u03b53o = 0.828 V and\n", - "$\\mathrm{\\Delta}_{\\ }^{0}G_{3}$= -2\\*0.828\\*96,485 J = -159,779 J. The\n", - "corresponding cathode (reduction) reaction is:\n", - "\n", - "*Eq. 8\u201140* H2O(l) + \u00bd O2(g) + 2 e- = 2\n", - "OH-(aq)\n", - "\n", - "with \u03b54o = 0.401 V and\n", - "$\\mathrm{\\Delta}_{\\ }^{0}G_{4}$ = -2\\*0.401\\*96,485 J = -77,381 J. The\n", - "net cell reaction for alkaline electrolytes is:\n", - "\n", - "*Eq. 8\u201141* H2(g) + \u00bd O2(g) = H2O(l)\n", - "\n", - "with \u03b5ocell = 1.229 V and\n", - "$\\mathrm{\\Delta}_{\\ }^{0}G_{cell}$ = -2\\*1.229\\*96,485 J = -237,160 J.\n", - "\n", - "Plots of \u03b5 versus pH for a given chemical system have been typically\n", - "used to exhibit the stability relationships of ionic species and solid\n", - "phases in aqueous-based electrochemical systems. These graphs are often\n", - "called Pourbaix diagrams after the inventor and are at constant\n", - "temperature and constant pressure diagrams for a constant concentration,\n", - "usually for one metallic element. By convention, the \u03b5 in a Pourbaix\n", - "diagram corresponds to the potential for the cathode reduction reactions\n", - "in the electrochemical half-cell with electrons as reactants. Pourbaix\n", - "diagrams can be extended to multi-component materials when thermodynamic\n", - "properties of the components are available in both the materials and the\n", - "aqueous solution.\n", - "\n", - "An example of an \u03b5 versus pH diagram is shown in for the\n", - "Ni-H2O system at a 298K, 1 bar, and $c_{{Ni}^{2 +}} = 0.001$\n", - "molality. Three stability regions for Ni species are shown: Ni(s),\n", - "NiO(s), and \\[Ni2+\\]. The two dashed lines on this diagram\n", - "correspond to hydrogen reduction () and oxygen reduction () reactions,\n", - "respectively.\n", - "\n", - "Figure \u2011: An \u03b5 versus pH, Pourbaix diagram for the Ni-H2O at\n", - "298K, 1 bar, and $c_{{Ni}^{2 +}} = 0.001$ molality.\n", - "\n", - "For the \u03b5 and pH conditions within the boundaries of the Ni(s) region,\n", - "no solid phase other than Ni(s) is stable, no ionic species with a\n", - "concentration of 1 molarity is stable, and no gas species with a\n", - "pressure of 1 bar is stable. Similar statements could be made about the\n", - "NiO(s) and \\[Ni2+\\] areas on the diagram. In the\n", - "\\[Ni2+\\] area, introduction of Ni(s) or NiO(s) into the\n", - "system would result in the dissolution of these solid phases since they\n", - "are not stable with respect to the \\[Ni2+\\] aqueous solution.\n", - "The corresponding chemical reactions proceed spontaneously to the right\n", - "as follows until the solid phases are consumed:\n", - "\n", - "*Eq. 8\u201142* Ni(s) \u2192 Ni2+(1 molarity) + 2 e-\n", - "\n", - "*Eq. 8\u201143* NiO(s) + 2 H+(aq) \u2192 Ni2+(1 molarity) +\n", - "H2O(l)\n", - "\n", - "No H+(aq) in involved in the first reaction, , so the\n", - "boundary line separating Ni(s) and Ni2+ is independent of pH.\n", - "No oxidation or reduction occurs in the second reaction, , i.e. no\n", - "electrons are reactants or products in the reaction, the boundary line\n", - "separating NiO(s) and Ni2+ is independent of \u03b5.\n", - "\n", - "Note the convention that the \u03b5 is the potential for a cathode reduction\n", - "reaction, and boundary lines between two stability regions depict\n", - "conditions under which partial equilibrium occurs for the two species\n", - "for the \u03b5 and pH values at any point on these lines. For the boundary\n", - "line separating Ni(s) and Ni2+ in an ideal aqueous solution,\n", - "i.e. the reverse of , the following equation is obtained.\n", - "\n", - "*Eq. 8\u201144* \u03b5 = \u03b5o = -0.268 V\n", - "\n", - "For the NiO(s)-Ni2+ boundary line of an ideal solution, the\n", - "reaction, , is a complete equilibrium, and thus the relationship is\n", - "\n", - "*Eq. 8\u201145*\n", - "$0 = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{1}{\\left( c_{H^{+}} \\right)^{2}} = \\mathrm{\\Delta}_{\\ }^{0}G + 2 \\cdot 2.303 \\cdot RT \\cdot pH$\n", - "\n", - "*Eq. 8\u201146*\n", - "$pH = - \\frac{\\mathrm{\\Delta}_{\\ }^{0}G}{2 \\cdot 2.303 \\cdot RT}$\n", - "\n", - "where $\\mathrm{\\Delta}_{\\ }^{0}G$ is obtained as follows and can be\n", - "calculated from the SSUB database and the standard potential of Ni,\n", - "\n", - "*Eq. 8\u201147*\n", - "\n", - "At a specified temperature, only one standard free energy and only one\n", - "equilibrium constant exists for this chemical reaction, and thus only\n", - "one specific value of $pH = 6.631$ exists for the reaction represented\n", - "by in this Pourbaix diagram.\n", - "\n", - "The diagonal line in represents the equilibrium between Ni(s) and NiO(s)\n", - "and is for a partial equilibrium reaction that is the sum of reactions\n", - "of and\n", - "\n", - "*Eq. 8\u201148* NiO(s) + 2 H+(aq) + 2 e- \u2550 Ni(s) +\n", - "H2O(l)\n", - "\n", - "The reduction of Ni from a divalent state in NiO to metallic Ni(s)\n", - "occurs, but the reaction also depends on the H+\n", - "concentration, the pH. The corresponding Gibbs energy and Nernst\n", - "equations are,\n", - "\n", - "*Eq. 8\u201149*\n", - "$\\mathrm{\\Delta}G = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{1}{\\left( c_{H^{+}} \\right)^{2}} = - 23,939 + 2 \\cdot 2.303 \\cdot RT \\cdot pH$\n", - "\n", - "*Eq. 8\u201150*\n", - "$\\varepsilon\\ = \\ \\varepsilon^{0}\\ - \\ \\frac{RT}{2f}\\ln\\frac{1}{\\left( c_{H^{+}} \\right)^{2}}\\ = 0.124\\ \u2013\\ \\frac{2.303 \\cdot RT}{f}pH$\n", - "\n", - "where $\\mathrm{\\Delta}_{\\ }^{0}G$ can be calculated as follows\n", - "\n", - "*Eq. 8\u201151*\n", - "\n", - "The two additional lines in correspond to the reduction reactions\n", - "related to H2 and O2 gases, i.e. the stability of\n", - "H2O. The lower one is for the reverse of under \u03b5o\n", - "= 0 and $P_{H_{2}} = 1$ with the Nernst equation being\n", - "\n", - "*Eq. 8\u201152*\n", - "$\\varepsilon\\ = \\ \\varepsilon^{0}\\ - \\ \\frac{RT}{f}\\ln\\frac{\\left( P_{H_{2}} \\right)^{1/2}}{c_{H^{+}}}\\ = - \\frac{2.303 \\cdot RT}{f}pH$\n", - "\n", - "As the pH increases from 0, \u03b5 becomes more negative as is depicted. The\n", - "top dashed line corresponds to the oxygen reduction reaction represented\n", - "by under \u03b5o = 1.225 calculated from the aqueous solution\n", - "database in Thermo-Calc \\[60\\] and $P_{O_{2}} = 1$ with the Nernst\n", - "equation being\n", - "\n", - "Eq. 8\u201153\n", - "$\\varepsilon\\ = \\ \\varepsilon^{0}\\ - \\ \\frac{RT}{2f}\\ln\\frac{\\left( P_{O_{2}} \\right)^{1/2}}{\\left( c_{H^{+}} \\right)^{2}}\\ = 1.225\\ \u2013\\ \\frac{2.303 \\cdot RT}{f}pH$\n", - "\n", - "The dependence of \u03b5 on pH is identical for both reduction reaction and ,\n", - "and their intercepts at $pH = 0$ differ by their difference in their\n", - "\u03b5o values.\n", - "\n", - "In this simple Pourbaix diagram of Ni in an ideal aqueous solution, all\n", - "boundary lines are straight because there is only one ionic species of\n", - "Ni in the aqueous solution, i.e. Ni2+. When there are more\n", - "than one ionic species in the aqueous solution, the boundary lines may\n", - "no longer be straight due to the competition between species. One\n", - "example is Cu with two main ionic species of Cu+2 and\n", - "CuOH+, and the reduction reaction between the metallic Cu and\n", - "the aqueous solution involves both two species, i.e.\n", - "\n", - "*Eq. 8\u201154*\n", - "${xCu}^{2 + \\ }\\ + \\ (1 - x){CuOH}^{+} + (1 - x)H^{+} + 2\\ e - \\ \\ = \\ \\ Cu(s) + {(1 - x)H}_{2}O$\n", - "\n", - "with\n", - "\n", - "*Eq. 8\u201155*\n", - "$\\mathrm{\\Delta}G = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{1}{\\left( c_{{Cu}^{2 +}} \\right)^{x}\\left( c_{{CuOH}^{+}}c_{H^{+}} \\right)^{1 - x}} = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{1}{\\left( c_{{Cu}^{2 +}} \\right)^{x}\\left( c_{{CuOH}^{+}} \\right)^{1 - x}} + 2.303(1 - x) \\cdot RT \\cdot pH$\n", - "\n", - "*Eq. 8\u201156*\n", - "$\\varepsilon = \\varepsilon^{0} - \\frac{RT}{2f}\\ln\\frac{1}{\\left( c_{{Cu}^{2 +}} \\right)^{x}\\left( c_{{CuOH}^{+}} \\right)^{1 - x}} - \\frac{2.303(1 - x) \\cdot RT}{2f}pH$.\n", - "\n", - "It is evident that both the slope and the intercept at $pH = 0$ are a\n", - "function of the concentration of ${CuOH}^{+}$, which is a function of\n", - "$pH$. Consequently, the boundary between the metallic Cu and the aqueous\n", - "solution is no longer a straight line as shown in .\n", - "\n", - "Figure \u2011: An \u03b5 versus pH, Pourbaix diagram for the Cu-H2O\n", - "system at 298K, 1 bar, and $c_{Cu} = 0.001$ molality.\n", - "\n", - "The concentrations of various species in the aqueous solution, i.e.\n", - "commonly called speciation, are plotted in , showing the change of\n", - "dominant species as a function of pH value.\n", - "\n", - "Figure \u2011: Concentrations of ionic species in the aqueous solution at\n", - "$\\varepsilon = 0.3\\ V$ from .\n", - "\n", - "In Pourbaix diagrams for alloys with two or more elements, activities of\n", - "individual elements are to be used in calculating the potentials of\n", - "reduction reactions. Considering a Fe-Ni alloy with Fe2+ and\n", - "Ni2+ in the aqueous solution, the reduction reactions for Fe\n", - "and Ni can be written separately as\n", - "\n", - "*Eq. 8\u201157* Ni2+(cNi) + 2 e- \u2192 Ni\n", - "(aNi in alloy)\n", - "\n", - "*Eq. 8\u201158* Fe2+(cFe) + 2 e- \u2192 Fe\n", - "(aFe in alloy)\n", - "\n", - "with their potentials as\n", - "\n", - "*Eq. 8\u201159*\n", - "$\\varepsilon_{Ni}\\ = \\ \\varepsilon_{Ni}^{0} - \\frac{2.303RT}{2f}\\ln\\frac{a_{Ni}}{c_{Ni}} = \\ - 0.268 - \\frac{2.303RT}{2f}\\ln\\frac{a_{Ni}}{c_{Ni}}$\n", - "\n", - "*Eq. 8\u201160*\n", - "$\\varepsilon_{Fe}\\ = \\ \\varepsilon_{Fe}^{0} - \\frac{2.303RT}{2f}\\ln\\frac{a_{Fe}}{c_{Fe}} = \\ - 0.441 - \\frac{2.303RT}{2f}\\ln\\frac{a_{Fe}}{c_{Fe}}$\n", - "\n", - "In principle, there are two scenarios for a given set of $a_{Ni}$ and\n", - "$a_{Fe}$ of the alloy. The first scenario is at the limit of a dilute\n", - "aqueous solution, i.e. $c_{Ni} = c_{Fe} = 0.001\\ $molarity,\n", - "$\\varepsilon_{Ni}$ and $\\varepsilon_{Fe}$ can be calculated, and the\n", - "element with the lower potential has the tendency to dissolve first,\n", - "which can result in the so-called dialloying effect. The second scenario\n", - "is for the equal potential, i.e. $\\varepsilon_{Ni} = \\varepsilon_{Fe}$\n", - "due to the externally imposed potential, and the equilibrium\n", - "concentrations of Fe+2 and Ni+2 can be calculated\n", - "from and .\n" - ], + "source": "## Aqueous solution and Pourbaix diagram\n\nThe importance of aqueous solutions in all aspects of life is so well\nknown and needs not be discussed further. Since many electrochemical\nprocesses involve electrolyte solutions in an aqueous solvent,\nelectrochemical processes including water, hydrogen, and/or oxygen are\ndiscussed in more details. The hydrogen-oxygen cell can be described for\nboth acidic electrolytes and alkaline electrolytes. With acidic\nelectrolytes, H+ is in much higher concentrations than\nOH-, and thus half-cell reactions with H+ as an\nionic transport species are more important than those involving\nOH-. The reverse is true for alkaline electrolytes that\ncontain high OH-concentrations. Other than for nearly neutral\nacid-base systems, either H+ or OH- dominates the\nother by several orders of magnitude as can be seen from the value of\nthe 298 K dissociation constant for H2O:\n\n*Eq. 8\u201135* H2O(l) = H+(aq) + OH-(aq)\n\nwith the reaction constant being Ke =\n\\[H+\\]\\[OH-\\] = 10-14 and\n$\\mathrm{\\Delta}_{\\ }^{0}G$= -RT *ln* Ke *\u03b2*= +79,908 J. By\nconvention, one defines pH = - log \\[H+\\] and pOH = - log\n\\[OH-\\], and then pH + pOH = 14.\n\nUnder acidic electrolyte conditions of low pH (high \\[H+\\]\nconcentrations) the anode reaction in a hydrogen-oxygen cell is:\n\n*Eq. 8\u201136* \u00bd H2(g) = H+(aq) + e-\n\nwith \u03b51o = 0.0 V and\n$\\mathrm{\\Delta}_{\\ }^{0}G_{1}$= 0 J. The corresponding cathode\n(reduction) reaction is:\n\n*Eq. 8\u201137* 2 H+(aq) + \u00bd O2(g) + 2 e- =\nH2O(l)\n\nwith \u03b52o = 1.229 V and\n$\\mathrm{\\Delta}_{\\ }^{0}G_{2}$ = -2\\*1.229\\*96,485 J = -237,160 J. The\nnet cell reaction for acidic electrolytes is:\n\n*Eq. 8\u201138* H2(g) + \u00bd O2(g) = H2O(l)\n\nwith \u03b5ocell = 1.229 V and\n$\\mathrm{\\Delta}_{\\ }^{0}G_{cell}$ = -2\\*1.229\\*96,485 J = -237,160 J\n\nUnder alkaline electrolyte conditions of high pH (high\n\\[OH-\\] concentrations) the anode reaction in a\nhydrogen-oxygen cell is:\n\n*Eq. 8\u201139* 2 OH-(aq) + H2(g) = 2\nH2O(l) + 2 e-\n\nwith \u03b53o = 0.828 V and\n$\\mathrm{\\Delta}_{\\ }^{0}G_{3}$= -2\\*0.828\\*96,485 J = -159,779 J. The\ncorresponding cathode (reduction) reaction is:\n\n*Eq. 8\u201140* H2O(l) + \u00bd O2(g) + 2 e- = 2\nOH-(aq)\n\nwith \u03b54o = 0.401 V and\n$\\mathrm{\\Delta}_{\\ }^{0}G_{4}$ = -2\\*0.401\\*96,485 J = -77,381 J. The\nnet cell reaction for alkaline electrolytes is:\n\n*Eq. 8\u201141* H2(g) + \u00bd O2(g) = H2O(l)\n\nwith \u03b5ocell = 1.229 V and\n$\\mathrm{\\Delta}_{\\ }^{0}G_{cell}$ = -2\\*1.229\\*96,485 J = -237,160 J.\n\nPlots of \u03b5 versus pH for a given chemical system have been typically\nused to exhibit the stability relationships of ionic species and solid\nphases in aqueous-based electrochemical systems. These graphs are often\ncalled Pourbaix diagrams after the inventor and are at constant\ntemperature and constant pressure diagrams for a constant concentration,\nusually for one metallic element. By convention, the \u03b5 in a Pourbaix\ndiagram corresponds to the potential for the cathode reduction reactions\nin the electrochemical half-cell with electrons as reactants. Pourbaix\ndiagrams can be extended to multi-component materials when thermodynamic\nproperties of the components are available in both the materials and the\naqueous solution.\n\nAn example of an \u03b5 versus pH diagram is shown in for the\nNi-H2O system at a 298K, 1 bar, and $c_{{Ni}^{2 +}} = 0.001$\nmolality. Three stability regions for Ni species are shown: Ni(s),\nNiO(s), and \\[Ni2+\\]. The two dashed lines on this diagram\ncorrespond to hydrogen reduction () and oxygen reduction () reactions,\nrespectively.\n\nFigure \u2011: An \u03b5 versus pH, Pourbaix diagram for the Ni-H2O at\n298K, 1 bar, and $c_{{Ni}^{2 +}} = 0.001$ molality.\n\nFor the \u03b5 and pH conditions within the boundaries of the Ni(s) region,\nno solid phase other than Ni(s) is stable, no ionic species with a\nconcentration of 1 molarity is stable, and no gas species with a\npressure of 1 bar is stable. Similar statements could be made about the\nNiO(s) and \\[Ni2+\\] areas on the diagram. In the\n\\[Ni2+\\] area, introduction of Ni(s) or NiO(s) into the\nsystem would result in the dissolution of these solid phases since they\nare not stable with respect to the \\[Ni2+\\] aqueous solution.\nThe corresponding chemical reactions proceed spontaneously to the right\nas follows until the solid phases are consumed:\n\n*Eq. 8\u201142* Ni(s) \u2192 Ni2+(1 molarity) + 2 e-\n\n*Eq. 8\u201143* NiO(s) + 2 H+(aq) \u2192 Ni2+(1 molarity) +\nH2O(l)\n\nNo H+(aq) in involved in the first reaction, , so the\nboundary line separating Ni(s) and Ni2+ is independent of pH.\nNo oxidation or reduction occurs in the second reaction, , i.e. no\nelectrons are reactants or products in the reaction, the boundary line\nseparating NiO(s) and Ni2+ is independent of \u03b5.\n\nNote the convention that the \u03b5 is the potential for a cathode reduction\nreaction, and boundary lines between two stability regions depict\nconditions under which partial equilibrium occurs for the two species\nfor the \u03b5 and pH values at any point on these lines. For the boundary\nline separating Ni(s) and Ni2+ in an ideal aqueous solution,\ni.e. the reverse of , the following equation is obtained.\n\n*Eq. 8\u201144* \u03b5 = \u03b5o = -0.268 V\n\nFor the NiO(s)-Ni2+ boundary line of an ideal solution, the\nreaction, , is a complete equilibrium, and thus the relationship is\n\n*Eq. 8\u201145*\n$0 = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{1}{\\left( c_{H^{+}} \\right)^{2}} = \\mathrm{\\Delta}_{\\ }^{0}G + 2 \\cdot 2.303 \\cdot RT \\cdot pH$\n\n*Eq. 8\u201146*\n$pH = - \\frac{\\mathrm{\\Delta}_{\\ }^{0}G}{2 \\cdot 2.303 \\cdot RT}$\n\nwhere $\\mathrm{\\Delta}_{\\ }^{0}G$ is obtained as follows and can be\ncalculated from the SSUB database and the standard potential of Ni,\n\n*Eq. 8\u201147*\n\nAt a specified temperature, only one standard free energy and only one\nequilibrium constant exists for this chemical reaction, and thus only\none specific value of $pH = 6.631$ exists for the reaction represented\nby in this Pourbaix diagram.\n\nThe diagonal line in represents the equilibrium between Ni(s) and NiO(s)\nand is for a partial equilibrium reaction that is the sum of reactions\nof and\n\n*Eq. 8\u201148* NiO(s) + 2 H+(aq) + 2 e- \u2550 Ni(s) +\nH2O(l)\n\nThe reduction of Ni from a divalent state in NiO to metallic Ni(s)\noccurs, but the reaction also depends on the H+\nconcentration, the pH. The corresponding Gibbs energy and Nernst\nequations are,\n\n*Eq. 8\u201149*\n$\\mathrm{\\Delta}G = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{1}{\\left( c_{H^{+}} \\right)^{2}} = - 23,939 + 2 \\cdot 2.303 \\cdot RT \\cdot pH$\n\n*Eq. 8\u201150*\n$\\varepsilon\\ = \\ \\varepsilon^{0}\\ - \\ \\frac{RT}{2f}\\ln\\frac{1}{\\left( c_{H^{+}} \\right)^{2}}\\ = 0.124\\ \u2013\\ \\frac{2.303 \\cdot RT}{f}pH$\n\nwhere $\\mathrm{\\Delta}_{\\ }^{0}G$ can be calculated as follows\n\n*Eq. 8\u201151*\n\nThe two additional lines in correspond to the reduction reactions\nrelated to H2 and O2 gases, i.e. the stability of\nH2O. The lower one is for the reverse of under \u03b5o\n= 0 and $P_{H_{2}} = 1$ with the Nernst equation being\n\n*Eq. 8\u201152*\n$\\varepsilon\\ = \\ \\varepsilon^{0}\\ - \\ \\frac{RT}{f}\\ln\\frac{\\left( P_{H_{2}} \\right)^{1/2}}{c_{H^{+}}}\\ = - \\frac{2.303 \\cdot RT}{f}pH$\n\nAs the pH increases from 0, \u03b5 becomes more negative as is depicted. The\ntop dashed line corresponds to the oxygen reduction reaction represented\nby under \u03b5o = 1.225 calculated from the aqueous solution\ndatabase in Thermo-Calc \\[60\\] and $P_{O_{2}} = 1$ with the Nernst\nequation being\n\nEq. 8\u201153\n$\\varepsilon\\ = \\ \\varepsilon^{0}\\ - \\ \\frac{RT}{2f}\\ln\\frac{\\left( P_{O_{2}} \\right)^{1/2}}{\\left( c_{H^{+}} \\right)^{2}}\\ = 1.225\\ \u2013\\ \\frac{2.303 \\cdot RT}{f}pH$\n\nThe dependence of \u03b5 on pH is identical for both reduction reaction and ,\nand their intercepts at $pH = 0$ differ by their difference in their\n\u03b5o values.\n\nIn this simple Pourbaix diagram of Ni in an ideal aqueous solution, all\nboundary lines are straight because there is only one ionic species of\nNi in the aqueous solution, i.e. Ni2+. When there are more\nthan one ionic species in the aqueous solution, the boundary lines may\nno longer be straight due to the competition between species. One\nexample is Cu with two main ionic species of Cu+2 and\nCuOH+, and the reduction reaction between the metallic Cu and\nthe aqueous solution involves both two species, i.e.\n\n*Eq. 8\u201154*\n${xCu}^{2 + \\ }\\ + \\ (1 - x){CuOH}^{+} + (1 - x)H^{+} + 2\\ e - \\ \\ = \\ \\ Cu(s) + {(1 - x)H}_{2}O$\n\nwith\n\n*Eq. 8\u201155*\n$\\mathrm{\\Delta}G = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{1}{\\left( c_{{Cu}^{2 +}} \\right)^{x}\\left( c_{{CuOH}^{+}}c_{H^{+}} \\right)^{1 - x}} = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{1}{\\left( c_{{Cu}^{2 +}} \\right)^{x}\\left( c_{{CuOH}^{+}} \\right)^{1 - x}} + 2.303(1 - x) \\cdot RT \\cdot pH$\n\n*Eq. 8\u201156*\n$\\varepsilon = \\varepsilon^{0} - \\frac{RT}{2f}\\ln\\frac{1}{\\left( c_{{Cu}^{2 +}} \\right)^{x}\\left( c_{{CuOH}^{+}} \\right)^{1 - x}} - \\frac{2.303(1 - x) \\cdot RT}{2f}pH$.\n\nIt is evident that both the slope and the intercept at $pH = 0$ are a\nfunction of the concentration of ${CuOH}^{+}$, which is a function of\n$pH$. Consequently, the boundary between the metallic Cu and the aqueous\nsolution is no longer a straight line as shown in .\n\nFigure \u2011: An \u03b5 versus pH, Pourbaix diagram for the Cu-H2O\nsystem at 298K, 1 bar, and $c_{Cu} = 0.001$ molality.\n\nThe concentrations of various species in the aqueous solution, i.e.\ncommonly called speciation, are plotted in , showing the change of\ndominant species as a function of pH value.\n\nFigure \u2011: Concentrations of ionic species in the aqueous solution at\n$\\varepsilon = 0.3\\ V$ from .\n\nIn Pourbaix diagrams for alloys with two or more elements, activities of\nindividual elements are to be used in calculating the potentials of\nreduction reactions. Considering a Fe-Ni alloy with Fe2+ and\nNi2+ in the aqueous solution, the reduction reactions for Fe\nand Ni can be written separately as\n\n*Eq. 8\u201157* Ni2+(cNi) + 2 e- \u2192 Ni\n(aNi in alloy)\n\n*Eq. 8\u201158* Fe2+(cFe) + 2 e- \u2192 Fe\n(aFe in alloy)\n\nwith their potentials as\n\n*Eq. 8\u201159*\n$\\varepsilon_{Ni}\\ = \\ \\varepsilon_{Ni}^{0} - \\frac{2.303RT}{2f}\\ln\\frac{a_{Ni}}{c_{Ni}} = \\ - 0.268 - \\frac{2.303RT}{2f}\\ln\\frac{a_{Ni}}{c_{Ni}}$\n\n*Eq. 8\u201160*\n$\\varepsilon_{Fe}\\ = \\ \\varepsilon_{Fe}^{0} - \\frac{2.303RT}{2f}\\ln\\frac{a_{Fe}}{c_{Fe}} = \\ - 0.441 - \\frac{2.303RT}{2f}\\ln\\frac{a_{Fe}}{c_{Fe}}$\n\nIn principle, there are two scenarios for a given set of $a_{Ni}$ and\n$a_{Fe}$ of the alloy. The first scenario is at the limit of a dilute\naqueous solution, i.e. $c_{Ni} = c_{Fe} = 0.001\\ $molarity,\n$\\varepsilon_{Ni}$ and $\\varepsilon_{Fe}$ can be calculated, and the\nelement with the lower potential has the tendency to dissolve first,\nwhich can result in the so-called dialloying effect. The second scenario\nis for the equal potential, i.e. $\\varepsilon_{Ni} = \\varepsilon_{Fe}$\ndue to the externally imposed potential, and the equilibrium\nconcentrations of Fe+2 and Ni+2 can be calculated\nfrom and .\n", "metadata": {} } ] diff --git a/src/psu410/src/applications_to_electrochemical_systems/concentrations_activities_and_reference_states_of_electrolyte_species.ipynb b/src/psu410/src/applications_to_electrochemical_systems/concentrations_activities_and_reference_states_of_electrolyte_species.ipynb index ef38ce0..40bcc85 100644 --- a/src/psu410/src/applications_to_electrochemical_systems/concentrations_activities_and_reference_states_of_electrolyte_species.ipynb +++ b/src/psu410/src/applications_to_electrochemical_systems/concentrations_activities_and_reference_states_of_electrolyte_species.ipynb @@ -4,345 +4,9 @@ "metadata": {}, "cells": [ { - "id": "6bfb8286", + "id": "95503830", "cell_type": "markdown", - "source": [ - "## Concentrations, activities, and reference states of electrolyte species\n", - "\n", - "Thermodynamic descriptions of ionic species in solutions are different\n", - "from those of neutral species, which leads to a need for defining\n", - "concentration units, standard states, activities, and activity\n", - "coefficients of ionic solutions. In most studies of electrochemical\n", - "corrosion and electrodeposition, and in applied work of electrochemical\n", - "engineers, ionic species concentrations are given in units of molarity,\n", - "the number of moles of a species in a liter of solution (mol/l)\n", - "symbolically represented in equations by either *ci* or\n", - "\\[M+Z\\]. The other common concentration used for ionic\n", - "species is molality, which is defined as the number of moles of a\n", - "species in 1000g of solvent. For dilute aqueous solutions, molarity and\n", - "molality values are very similar.\n", - "\n", - "As discussed in Chapter , a practical definition of the activity of a\n", - "species *i* is the thermodynamic reactivity, or tendency to react, of\n", - "species *i* in the system of interest as compared to *i* in its\n", - "reference state form. The reference state of a species is typically\n", - "chosen as a specific chemical/physical state of the species at 1 atm\n", - "external pressure and the temperature of interest. Similarly, a typical\n", - "reference state for ionic species in aqueous solutions is the 1 molar\n", - "ideal solution at 1 bar external pressure and the temperature of\n", - "interest. If an electrolyte solution behaves ideally, then the activity\n", - "of species *i* in solution is\n", - "\n", - "*Eq. 8\u20116*\n", - "$a_{i} = \\frac{c_{i}\\left( \\frac{mol}{l} \\right)}{c_{i}^{0}\\left( \\frac{mol}{l} \\right)} = \\frac{c_{i}\\left( \\frac{mol}{l} \\right)}{1\\left( \\frac{mol}{l} \\right)} = c_{i}(dimensionless)$\n", - "\n", - "where $c_{i}$ is the molar concentration of *i* in the solution divided\n", - "by $c_{i}^{0}$, the 1 molar reference state ideal solution\n", - "concentration. Thus, in ideal solutions, the activity of an electrolyte\n", - "species is numerically equal to its molar concentration. The above\n", - "treatment of ionic species is equivalent to the common practice of\n", - "depicting the activity of a gas by the value of its ideal gas partial\n", - "pressure in units of bar.\n", - "\n", - "The activity coefficient corrects for the nonideality of the species in\n", - "solution as defined in . If the solution is ideal, $\\gamma_{i} = 1$ for\n", - "all concentrations of a species in solution. For all solutions, one\n", - "expects $\\gamma_{i} \\rightarrow 1$ as $c_{i} \\rightarrow 1$. It is not\n", - "possible to measure $\\gamma_{i^{+}}$ or $\\gamma_{i^{-}}$ for individual\n", - "charged ions, only a geometric mean of the positive and negative ion\n", - "values. Consider the following ionic solution\n", - "\n", - "Eq. 8\u20117\n", - "\n", - "Its chemical potential can be written as\n", - "\n", - "*Eq. 8\u20118*\n", - "\n", - "Its geometric average or mean activity and activity coefficient are\n", - "defined as\n", - "\n", - "*Eq. 8\u20119*\n", - "\n", - "*Eq. 8\u201110*\n", - "\n", - "For example, one can define\n", - "$\\gamma_{\\pm} = \\left( \\gamma_{{Na}^{+}}\\gamma_{{Cl}^{-}} \\right)^{1/2}$\n", - "and\n", - "$\\gamma_{\\pm} = \\left( \\gamma_{{Al}^{3 +}}^{2}\\gamma_{{{SO}_{4}}^{2 -}}^{3} \\right)^{1/5}$\n", - "for NaCl and Al2(SO4)3, respectively.\n", - "For idea, weak electrolytes, $\\gamma_{\\pm} = 1$, and for non-ideal,\n", - "strong electrolytes, $\\gamma_{\\pm} \\neq 1$.\n" - ], - "metadata": {} - }, - { - "id": "ce234b05", - "cell_type": "markdown", - "source": [ - "### Electrical batteries\n", - "\n", - "Batteries utilize electrochemical reactions to generate electricity for\n", - "various devices. The theoretic voltage of a battery can be calculated\n", - "from and , i.e.\n", - "\n", - "*Eq. 8\u201173*\n", - "$\\varepsilon = - \\frac{\\mathrm{\\Delta}G}{zf} = \\varepsilon^{0} - \\frac{RTlnQ}{zf}$\n", - "\n", - "with $\\mathrm{\\Delta}G$ being the driving force of the net cell reaction\n", - "and $Q$ being the reaction activity quotient. The actual voltage of a\n", - "battery is lower than the theoretical one due to kinetic limitations of\n", - "cell reactions and resistance to ion diffusion through the electrolyte.\n", - "Based on whether the cell reactions are reversible or not, batteries\n", - "typically categorized as either primary disposable or secondary\n", - "rechargeable batteries. The net cell reactions in primary disposable\n", - "batteries are not easily reversible, and electrode materials may not\n", - "return to their original forms by applying a higher external potential\n", - "of opposite sign. Consequently, primary batteries cannot be reliably\n", - "recharged. On the other hand, the net cell reactions in secondary\n", - "batteries are easily reversible. Furthermore, two half-cells in\n", - "batteries may use different electrolytes with each half-cell enclosed in\n", - "a container and a separator permeable to conducting ions but not the\n", - "bulk of the electrolytes.\n", - "\n", - "One common primary battery is zinc-carbon battery with a zinc anode\n", - "cylinder and a carbon cathode central rod. The electrolytes are ammonium\n", - "or zinc chloride next to the zinc anode and a mixture of ammonium\n", - "chloride and manganese dioxide next to the carbon cathode. The half cell\n", - "and net reactions with ammonium chloride are as follows\n", - "\n", - "*Eq. 8\u201174* Zn + 2NH3 \u2192\n", - "Zn(NH3)22+ + 2 e-\n", - "\n", - "*Eq. 8\u201175* 2NH4Cl + 2MnO2 + 2 e- \u2192\n", - "2NH3 + Mn2O3 +\n", - "H2O+2Cl\u2212\n", - "\n", - "*Eq. 8\u201176* Zn + 2MnO2 + 2NH4Cl \u2192\n", - "Mn2O3 +\n", - "Zn(NH3)2Cl2 + H2O.\n", - "\n", - "The electric potential of the reaction is, treating all compounds as\n", - "stoichiometric compounds\n", - "\n", - "Eq. 8\u201177\n", - "$\\varepsilon = - \\frac{\\mathrm{\\Delta}G}{2f} = - \\frac{\\mathrm{\\Delta}^{0}G}{2f} = \\frac{1}{2f}\\left(_{\\ }^{0}G^{Zn} + 2_{\\ }^{0}G^{{MnO}_{2}} + 2_{\\ }^{0}G^{{NH}_{4}Cl} -_{\\ }^{0}G^{H_{2}O} -_{\\ }^{0}G^{Zn\\left( {NH}_{3} \\right)_{2}{Cl}_{2}} -_{\\ }^{0}G^{{{Mn}_{2}O}_{3}} \\right)$.\n", - "\n", - "The Gibbs energy of $Zn\\left( {NH}_{3} \\right)_{2}{Cl}_{2}$ is not\n", - "available in current databases and has been recently estimated to be\n", - "\u2212505,375 J/mole-formula \\[62\\]. The value of at 298.15K is thus obtained\n", - "as 1.67 V, which is pretty close to the actual operating voltage of the\n", - "battery around 1.5 V.\n", - "\n", - "While with zinc chloride, the cell reactions and electric potential may\n", - "be written as\n", - "\n", - "Eq. 8\u201178 Zn + ZnCl2 + 2OH\u2212 \u2192 2ZnOHCl + 2\n", - "e-\n", - "\n", - "*Eq. 8\u201179* MnO2 + H2O + e- \u2192 MnOOH +\n", - "OH-\n", - "\n", - "*Eq. 8\u201180* Zn + 2 MnO2 + ZnCl2 + 2 H2O\n", - "\u2192 2 MnOOH + 2 ZnOHCl\n", - "\n", - "*Eq. 8\u201181*\n", - "$\\varepsilon = \\frac{1}{2f}\\left(_{\\ }^{0}G^{Zn} + 2_{\\ }^{0}G^{{MnO}_{2}} +_{\\ }^{0}G^{Zn{Cl}_{2}} + 2_{\\ }^{0}G^{H_{2}O} - 2_{\\ }^{0}G^{MnOOH} - 2_{\\ }^{0}G^{ZnOHCl} \\right)$\n", - "\n", - "Secondary batteries can be recharged by applying an external electrical\n", - "potential, which reverses the net cell reaction that occur during\n", - "discharging. The oldest form of rechargeable battery is the lead-acid\n", - "batteries used in automotive, and the latest development is the\n", - "lithium-ion (Li-ion) batteries. A lead-acid battery typically uses Pb\n", - "and PbO2 as the cathode and anode electrodes and a 35% sulfuric acid and\n", - "65% water solution as the electrolyte. Its anode and cathode reactions\n", - "can be simplified as follows\n", - "\n", - "Eq. 8\u201182 $Pb + SO_{4}^{2 -} = PbSO_{4} + 2e^{-}$\n", - "\n", - "Eq. 8\u201183 $PbO_{2} + 4H^{+} + SO_{4}^{2 -} + 2e^{-} = PbSO_{4} + 2H_{2}O$\n", - "\n", - "The net cell reaction is\n", - "\n", - "Eq. 8\u201184 $Pb + PbO_{2} + 2H_{2}SO_{4}^{\\ } = 2PbSO_{4} + 2H_{2}O$.\n", - "\n", - "Its electric potential is represented by the following equation\n", - "\n", - "*Eq. 8\u201185*\n", - "$\\varepsilon = - \\frac{1}{2f}\\left( 2_{\\ }^{0}G^{H_{2}O} + 2_{\\ }^{0}G^{{PbSO}_{4}} -_{\\ }^{0}G^{Pb} -_{\\ }^{0}G^{{PbO}_{2}} - 2_{\\ }^{0}G^{H_{2}SO_{4}} \\right)$\n", - "\n", - "with the value at 298.15K being 2.651 V calculated from Thermo-Calc\n", - "\\[60\\]. During discharging, the reaction goes to right, and $PbSO_{4}$\n", - "is formed on both anode and cathode, while during charging, the reaction\n", - "goes to the left, and $Pb$ and $PbO_{2}$ are restored. In practical\n", - "applications, other ionic species such as ${H_{3}O}^{+}$ and\n", - "$HSO_{4}^{-}$ may form in the electrolyte, complicating the reactions\n", - "and affecting its potential.\n", - "\n", - "In lithium ion batteries, lithium ions migrate in electrolytes between\n", - "electrodes made of intercalated lithium compounds during charging and\n", - "discharging. LiCoO2 and LiFePO4 are two of the\n", - "several cathode materials used in lithium ion batteries, and the anode\n", - "is typically made of carbon or metallic Li. The anode and cathode\n", - "reactions for LiCoO2 batteries can be written in simple forms as follows\n", - "\n", - "Eq. 8\u201186 ${Li}_{x}C_{6} = x{Li}^{+} + xe^{-} + 6C$\n", - "\n", - "Eq. 8\u201187 $x{Li}^{+} + xe^{-} + {Li}_{1 - x}CoO_{2} = LiCoO_{2}$\n", - "\n", - "with the net reaction and electric potential being\n", - "\n", - "Eq. 8\u201188 ${Li}_{x}C_{6} + {Li}_{1 - x}CoO_{2} = LiCoO_{2} + 6C$\n", - "\n", - "*Eq. 8\u201189*\n", - "$\\varepsilon = - \\frac{1}{xf}\\left\\{ 6_{\\ }^{0}G^{C} +_{\\ }^{0}G^{{LiCoO}_{2}} - G^{{{Li}_{x}C}_{6}} - G^{{{Li}_{1 - x}CoO}_{2}} \\right\\} = - \\frac{1}{f}\\left\\{ \\left( \\mu_{Li}^{{Li}_{1 - x}CoO_{2}} - \\mu_{Li}^{{Li}_{x}C} \\right) - \\frac{1}{x}\\left( \\mu_{LiCoO_{2}}^{{Li}_{1 - x}CoO_{2}} -_{\\ }^{0}G^{{LiCoO}_{2}} \\right) \\right\\}$\n", - "\n", - "The electric potential is a function of $x$. The value in the first\n", - "parenthesis in the above equation denotes the chemical potential\n", - "difference of Li between two electrodes, and the value in the second\n", - "parenthesis represents the chemical potential difference of\n", - "${LiCoO}_{2}$ between the states in the solution phase of\n", - "${Li}_{1 - x}CoO_{2}$ and by itself. Gibbs energies of ${{Li}_{x}C}_{6}$\n", - "and ${{Li}_{1 - x}CoO}_{2}$ need to be obtained as a function $x$ in\n", - "order to calculate the electric potential of the battery.\n", - "\n", - "LiFePO4 uses metallic lithium as the anode with following\n", - "half-cell and net cell reactions\n", - "\n", - "Eq. 8\u201190 $xLi = x{Li}^{+} + xe^{-}$\n", - "\n", - "Eq. 8\u201191 $x{Li}^{+} + xe^{-} + {Li}_{1 - x}FePO_{4} = LiFePO_{4}$\n", - "\n", - "Eq. 8\u201192 $xLi + {Li}_{1 - x}FePO_{4} = LiFePO_{4}$.\n", - "\n", - "Its electric potential is also a function of $x$, i.e.\n", - "\n", - "*Eq. 8\u201193*\n", - "$\\varepsilon = - \\frac{1}{xf}\\left\\{_{\\ }^{0}G^{LiFePO_{4}} - x\\ ^{0}G^{Li} - G^{{Li}_{1 - x}FePO_{4}} \\right\\} = - \\frac{1}{f}\\left\\{ \\left( \\mu_{Li}^{{Li}_{1 - x}FePO_{4}} - \\ ^{0}\\mu_{Li} \\right) - \\frac{1}{x}\\left( \\mu_{LiFePO_{4}}^{{Li}_{1 - x}FePO_{4}} -_{\\ }^{0}G^{LiFePO_{4}} \\right) \\right\\}$\n", - "\n", - "The value in the first parenthesis in the above equation denotes the\n", - "chemical potential difference of Li between two electrodes, and the\n", - "value in the second parenthesis represents the chemical potential\n", - "difference of $LiFePO_{4}$ between the states in the solution phase of\n", - "${Li}_{1 - x}FePO_{4}$ and by itself. Consequently, Gibbs energy of\n", - "${Li}_{1 - x}FePO_{4}$ needs to be obtained as a function $x$ in order\n", - "to calculate the electric potential of the battery. It is known that\n", - "there are several miscibility gaps in the $FePO_{4}$ and $LiFePO_{4}$\n", - "psuedo-binary system, in which the chemical potentials are constants, so\n", - "is the electric potential, resulting in stable battery output.\n" - ], - "metadata": {} - }, - { - "id": "16f36a6d", - "cell_type": "markdown", - "source": [ - "### Electrochemical cells\n", - "\n", - "A potential difference, i.e. voltage difference, can be generated\n", - "between the electrodes in a cell from differences in the potentials of\n", - "the half-cell reactions. This potential can originate from potential\n", - "differences of two chemically different half-cells (a *galvanic cell*),\n", - "or concentration differences in two otherwise identical half-cells (a\n", - "*concentration cell*). Each type of cell is illustrated below.\n", - "\n", - "The reaction between copper ions and zinc illustrated below represents\n", - "the *net cell reaction* of a *galvanic* cell in which the oxidation of\n", - "Zn(s) occurs at the *anode* electrode, and the reduction of\n", - "Cu+2 occurs at the *cathode* electrode\n", - "\n", - "*Eq. 8\u201111* Cu2+(aq) + Zn(s) = Cu(s) + Zn2+(aq).\n", - "\n", - "The reaction at each electrode, the *half-cell reaction*, includes ions\n", - "and electrons as reactant and/or product species. The anode, oxidation\n", - "reaction is represented by\n", - "\n", - "*Eq. 8\u201112* Zn(s) = Zn2+(aq) + 2 e-.\n", - "\n", - "Electrons are products of anode reactions and flow externally from anode\n", - "to cathode. By convention the activities of the electrons in an\n", - "equilibrium cell are taken as unity. The cathode, reduction reaction is\n", - "written as\n", - "\n", - "*Eq. 8\u201113* Cu2+(aq) + 2 e- = Cu(s)\n", - "\n", - "Electrons are reactants of cathode reactions and supplied by an external\n", - "flow from the anode. In addition to consuming electrons at the cathode\n", - "at the same rate as they are produced at the anode, charge balance is\n", - "maintained in the electrolyte by the generation of Zn2+ ions\n", - "at the same rate that Cu2+ ions are consumed. A schematic\n", - "diagram in illustrates the simple physical relationships in such an\n", - "electrochemical cell.\n", - "\n", - "Figure \u2011: Schematic diagram of a galvanic electrochemical cell\n", - "consisting of a zinc electrode and a copper electrode.\n", - "\n", - "A concentration cell in which an electrochemical potential is developed\n", - "because of concentration differences between otherwise equivalent anode\n", - "and cathode reactions is illustrated below. Such a cell can be produced\n", - "by the oxidation and reduction of copper at two separate electrodes as\n", - "is depicted in the following reactions\n", - "\n", - "*Eq. 8\u201114* Cu(s) = Cu2+(aq, ca) + 2 e- (anode)\n", - "\n", - "*Eq. 8\u201115* Cu2+(aq, cb) + 2 e- = Cu(s) (cathode)\n", - "\n", - "where ca and cb are the respective concentrations\n", - "of Cu2+ in the aqueous solutions at the anode and cathode,\n", - "and ca \\< cb. The net cell reaction is\n", - "\n", - "*Eq. 8\u201116* Cu2+(aq, cb) = Cu2+(aq,\n", - "ca)\n", - "\n", - "where the reaction occurs spontaneously to decrease cb and to\n", - "increase ca until the two concentrations become the same,\n", - "cb = ca. A schematic diagram of such a cell is\n", - "shown in .\n", - "\n", - "Figure \u2011: Schematic diagram of a concentration electrochemical cell\n", - "consisting of two copper electrodes.\n", - "\n", - "A semi-impermeable membrane, or a salt bridge, must exist in such a cell\n", - "to maintain charge balance. As Cu+2 ions are produced at the\n", - "anode and consumed at the cathode, the negatively charged ions in the\n", - "solution, for example Cl-, must be transferred from the\n", - "cathode region to the anode region of the cell to maintain electrically\n", - "neutral solutions.\n", - "\n", - "The above concentration cell provides a good example for illustrating a\n", - "standard notation for depicting an electrochemical cell. This cell can\n", - "be represented by .\n", - "\n", - "Figure \u2011: Standard notation of an electrochemical cell\n", - "\n", - "The anode where oxidation occurs is always denoted on the left, and the\n", - "cathode where reduction occurs is on the right. A *single line*\n", - "separating phases denotes an *interface* between two phases. The above\n", - "anode electrode and reaction of are symbolically represented by\n", - "\n", - "*Eq. 8\u201117* \\| Cu(s) \\| Cu2+(ca).\n", - "\n", - "The interface between the external conductor and Cu(s) is depicted by\n", - "the single line to the left of Cu(s), while the single line between the\n", - "Cu(s) and Cu2+(ca) depicts the interface between\n", - "the anode electrode and the electrolyte solution. Similarly, the cathode\n", - "electrode and reaction of are symbolically represented by\n", - "\n", - "*Eq. 8\u201118* Cu2+(cb) \\| Cu(s) \\|.\n", - "\n", - "A double line between the two copper ions in the notation denotes a\n", - "physical separation of two solution phases, the anode and cathode\n", - "electrolyte regions that exhibit different concentrations of copper ions\n", - "\n", - "*Eq. 8\u201119* Cu2+(ca) \\| \\|\n", - "Cu2+(cb).\n", - "\n", - "These solution phases are physically connected by a semi-impermeable\n", - "membrane or salt bridge that allows a common negative ion, for example\n", - "Cl-, of the solution phases to diffuse from one region to the\n", - "other in order to maintain charge balance as the cell reaction occurs.\n", - "The Cu2+ ions cannot be transported from one region to the\n", - "other.\n" - ], + "source": "## Concentrations, activities, and reference states of electrolyte species\n\nThermodynamic descriptions of ionic species in solutions are different\nfrom those of neutral species, which leads to a need for defining\nconcentration units, standard states, activities, and activity\ncoefficients of ionic solutions. In most studies of electrochemical\ncorrosion and electrodeposition, and in applied work of electrochemical\nengineers, ionic species concentrations are given in units of molarity,\nthe number of moles of a species in a liter of solution (mol/l)\nsymbolically represented in equations by either *ci* or\n\\[M+Z\\]. The other common concentration used for ionic\nspecies is molality, which is defined as the number of moles of a\nspecies in 1000g of solvent. For dilute aqueous solutions, molarity and\nmolality values are very similar.\n\nAs discussed in Chapter , a practical definition of the activity of a\nspecies *i* is the thermodynamic reactivity, or tendency to react, of\nspecies *i* in the system of interest as compared to *i* in its\nreference state form. The reference state of a species is typically\nchosen as a specific chemical/physical state of the species at 1 atm\nexternal pressure and the temperature of interest. Similarly, a typical\nreference state for ionic species in aqueous solutions is the 1 molar\nideal solution at 1 bar external pressure and the temperature of\ninterest. If an electrolyte solution behaves ideally, then the activity\nof species *i* in solution is\n\n*Eq. 8\u20116*\n$a_{i} = \\frac{c_{i}\\left( \\frac{mol}{l} \\right)}{c_{i}^{0}\\left( \\frac{mol}{l} \\right)} = \\frac{c_{i}\\left( \\frac{mol}{l} \\right)}{1\\left( \\frac{mol}{l} \\right)} = c_{i}(dimensionless)$\n\nwhere $c_{i}$ is the molar concentration of *i* in the solution divided\nby $c_{i}^{0}$, the 1 molar reference state ideal solution\nconcentration. Thus, in ideal solutions, the activity of an electrolyte\nspecies is numerically equal to its molar concentration. The above\ntreatment of ionic species is equivalent to the common practice of\ndepicting the activity of a gas by the value of its ideal gas partial\npressure in units of bar.\n\nThe activity coefficient corrects for the nonideality of the species in\nsolution as defined in . If the solution is ideal, $\\gamma_{i} = 1$ for\nall concentrations of a species in solution. For all solutions, one\nexpects $\\gamma_{i} \\rightarrow 1$ as $c_{i} \\rightarrow 1$. It is not\npossible to measure $\\gamma_{i^{+}}$ or $\\gamma_{i^{-}}$ for individual\ncharged ions, only a geometric mean of the positive and negative ion\nvalues. Consider the following ionic solution\n\nEq. 8\u20117\n\nIts chemical potential can be written as\n\n*Eq. 8\u20118*\n\nIts geometric average or mean activity and activity coefficient are\ndefined as\n\n*Eq. 8\u20119*\n\n*Eq. 8\u201110*\n\nFor example, one can define\n$\\gamma_{\\pm} = \\left( \\gamma_{{Na}^{+}}\\gamma_{{Cl}^{-}} \\right)^{1/2}$\nand\n$\\gamma_{\\pm} = \\left( \\gamma_{{Al}^{3 +}}^{2}\\gamma_{{{SO}_{4}}^{2 -}}^{3} \\right)^{1/5}$\nfor NaCl and Al2(SO4)3, respectively.\nFor idea, weak electrolytes, $\\gamma_{\\pm} = 1$, and for non-ideal,\nstrong electrolytes, $\\gamma_{\\pm} \\neq 1$.\n", "metadata": {} } ] diff --git a/src/psu410/src/applications_to_electrochemical_systems/electrochemical_cells_and_half_cell_potentials.ipynb b/src/psu410/src/applications_to_electrochemical_systems/electrochemical_cells_and_half_cell_potentials.ipynb index 5782fb8..1346408 100644 --- a/src/psu410/src/applications_to_electrochemical_systems/electrochemical_cells_and_half_cell_potentials.ipynb +++ b/src/psu410/src/applications_to_electrochemical_systems/electrochemical_cells_and_half_cell_potentials.ipynb @@ -4,32 +4,21 @@ "metadata": {}, "cells": [ { - "id": "7ac97a5b", + "id": "eab98886", "cell_type": "markdown", - "source": [ - "## Electrochemical cells and half cell potentials\n", - "\n", - "An electrochemical system must fulfill certain requirements in order to\n", - "apply equilibrium thermodynamic descriptions of the system, including\n", - "\n", - "- The cell must be reversible when slight changes in conditions\n", - " (potentials, concentrations, pressures, temperature) cause\n", - " electrochemical reactions and an external flow of electrons to occur\n", - " in the direction needed to re-establish equilibrium.\n", - "\n", - "- All non-electrochemical reactions in the system must be prevented as\n", - " such reactions would cause a shift in equilibrium without causing a\n", - " shift in cell potential and thus a driving force for external electron\n", - " flow.\n", - "\n", - "- Chemical reactions must occur only when an external current flows.\n", - " These finite distances for external electron transport can be a short\n", - " as grain size-dimensions in many corrosion reactions, or this\n", - " transport may be through an external electrical conductor connecting\n", - " the anode and cathode half-cell as in batteries.\n", - "\n", - "- Charge balance as well as mass balance is required of all reactions.\n" - ], + "source": "## Electrochemical cells and half cell potentials\n\nAn electrochemical system must fulfill certain requirements in order to\napply equilibrium thermodynamic descriptions of the system, including\n\n- The cell must be reversible when slight changes in conditions\n (potentials, concentrations, pressures, temperature) cause\n electrochemical reactions and an external flow of electrons to occur\n in the direction needed to re-establish equilibrium.\n\n- All non-electrochemical reactions in the system must be prevented as\n such reactions would cause a shift in equilibrium without causing a\n shift in cell potential and thus a driving force for external electron\n flow.\n\n- Chemical reactions must occur only when an external current flows.\n These finite distances for external electron transport can be a short\n as grain size-dimensions in many corrosion reactions, or this\n transport may be through an external electrical conductor connecting\n the anode and cathode half-cell as in batteries.\n\n- Charge balance as well as mass balance is required of all reactions.\n", + "metadata": {} + }, + { + "id": "984589b8", + "cell_type": "markdown", + "source": "### Electrochemical cells\n\nA potential difference, i.e. voltage difference, can be generated\nbetween the electrodes in a cell from differences in the potentials of\nthe half-cell reactions. This potential can originate from potential\ndifferences of two chemically different half-cells (a *galvanic cell*),\nor concentration differences in two otherwise identical half-cells (a\n*concentration cell*). Each type of cell is illustrated below.\n\nThe reaction between copper ions and zinc illustrated below represents\nthe *net cell reaction* of a *galvanic* cell in which the oxidation of\nZn(s) occurs at the *anode* electrode, and the reduction of\nCu+2 occurs at the *cathode* electrode\n\n*Eq. 8\u201111* Cu2+(aq) + Zn(s) = Cu(s) + Zn2+(aq).\n\nThe reaction at each electrode, the *half-cell reaction*, includes ions\nand electrons as reactant and/or product species. The anode, oxidation\nreaction is represented by\n\n*Eq. 8\u201112* Zn(s) = Zn2+(aq) + 2 e-.\n\nElectrons are products of anode reactions and flow externally from anode\nto cathode. By convention the activities of the electrons in an\nequilibrium cell are taken as unity. The cathode, reduction reaction is\nwritten as\n\n*Eq. 8\u201113* Cu2+(aq) + 2 e- = Cu(s)\n\nElectrons are reactants of cathode reactions and supplied by an external\nflow from the anode. In addition to consuming electrons at the cathode\nat the same rate as they are produced at the anode, charge balance is\nmaintained in the electrolyte by the generation of Zn2+ ions\nat the same rate that Cu2+ ions are consumed. A schematic\ndiagram in illustrates the simple physical relationships in such an\nelectrochemical cell.\n\nFigure \u2011: Schematic diagram of a galvanic electrochemical cell\nconsisting of a zinc electrode and a copper electrode.\n\nA concentration cell in which an electrochemical potential is developed\nbecause of concentration differences between otherwise equivalent anode\nand cathode reactions is illustrated below. Such a cell can be produced\nby the oxidation and reduction of copper at two separate electrodes as\nis depicted in the following reactions\n\n*Eq. 8\u201114* Cu(s) = Cu2+(aq, ca) + 2 e- (anode)\n\n*Eq. 8\u201115* Cu2+(aq, cb) + 2 e- = Cu(s) (cathode)\n\nwhere ca and cb are the respective concentrations\nof Cu2+ in the aqueous solutions at the anode and cathode,\nand ca \\< cb. The net cell reaction is\n\n*Eq. 8\u201116* Cu2+(aq, cb) = Cu2+(aq,\nca)\n\nwhere the reaction occurs spontaneously to decrease cb and to\nincrease ca until the two concentrations become the same,\ncb = ca. A schematic diagram of such a cell is\nshown in .\n\nFigure \u2011: Schematic diagram of a concentration electrochemical cell\nconsisting of two copper electrodes.\n\nA semi-impermeable membrane, or a salt bridge, must exist in such a cell\nto maintain charge balance. As Cu+2 ions are produced at the\nanode and consumed at the cathode, the negatively charged ions in the\nsolution, for example Cl-, must be transferred from the\ncathode region to the anode region of the cell to maintain electrically\nneutral solutions.\n\nThe above concentration cell provides a good example for illustrating a\nstandard notation for depicting an electrochemical cell. This cell can\nbe represented by .\n\nFigure \u2011: Standard notation of an electrochemical cell\n\nThe anode where oxidation occurs is always denoted on the left, and the\ncathode where reduction occurs is on the right. A *single line*\nseparating phases denotes an *interface* between two phases. The above\nanode electrode and reaction of are symbolically represented by\n\n*Eq. 8\u201117* \\| Cu(s) \\| Cu2+(ca).\n\nThe interface between the external conductor and Cu(s) is depicted by\nthe single line to the left of Cu(s), while the single line between the\nCu(s) and Cu2+(ca) depicts the interface between\nthe anode electrode and the electrolyte solution. Similarly, the cathode\nelectrode and reaction of are symbolically represented by\n\n*Eq. 8\u201118* Cu2+(cb) \\| Cu(s) \\|.\n\nA double line between the two copper ions in the notation denotes a\nphysical separation of two solution phases, the anode and cathode\nelectrolyte regions that exhibit different concentrations of copper ions\n\n*Eq. 8\u201119* Cu2+(ca) \\| \\|\nCu2+(cb).\n\nThese solution phases are physically connected by a semi-impermeable\nmembrane or salt bridge that allows a common negative ion, for example\nCl-, of the solution phases to diffuse from one region to the\nother in order to maintain charge balance as the cell reaction occurs.\nThe Cu2+ ions cannot be transported from one region to the\nother.\n", + "metadata": {} + }, + { + "id": "c8c84ce0", + "cell_type": "markdown", + "source": "### Half cell potentials\n\nWhen electron current flows between electrodes, reactions are occurring\nat the electrodes and concentration gradients causing polarization\ndevelop around the electrodes. These gradients result in extraneous\npotentials to occur at the electrodes. In such cases cell equilibrium is\nnot established and measured cell potentials are not those for true\npartial equilibrium. If a cell is short-circuited with the electrodes\nconnected by a conductor, current will flow until the external potential\nbecomes zero, i.e. \u03b5ext = 0, and equilibrium is established\nwith same conditions as non-electrochemical systems. If an external\npotential, \u03b5ext, is applied to the cell, chemical reactions\noccur until the cell potential balances to \u03b5ext, and no\ncurrent flows. This potential is called open-circuit voltage (OCV) in\nthe literature. It is important to realize that OCV includes all\nreactions that occur on the electrode surface when the electrode is in\ncontact with the electrolyte, such as passivation discussed in Chapter .\nPartial equilibrium in a cell is achieved when the cell potential is\nbalanced by an applied external potential. In such partial equilibrium\ncases, equilibrium thermodynamic analyses can be used even though the\ncell potential is not zero, i.e. \u03b5cell \u2260 0. This\ndifferentiates electrochemical systems from other equilibrium systems\ndiscussed previously.\n\nThe number of electrons involved in a net cell reaction is important in\nrelating cell potential and the Gibbs energy change for the cell\nreaction. As will be illustrated later in this section, this number\ndenotes the number of electrons involved in the half-cell reactions that\nwere added to yield the net cell reaction. The electrical work achieved\nby the transport of an electrical charge through a cell potential can be\nwritten as\n\n*Eq. 8\u201120* $w = z\\ f\\ \\varepsilon$\n\nwhere *z* represents the moles of electrons in cell reaction, *f* the\nFaraday constant equal to 96,485 J/V/mole-electron, and *\u03b5* the\npotential difference, often referred as electromotive force (emf) in the\nliterature. For a system at constant temperature, pressure, and\ncomposition, this work is the same as the Gibbs energy difference\nbetween the two electrodes, i.e.\n\n*Eq. 8\u201121* $- \\Delta G = w = zf\\varepsilon$\n\nwhere the negative sign is because the system does work to the\nsurrounding when the Gibbs energy of the system is decreased. When the\napplied external potential is larger than the cell potential, the\nsurrounding does work to the system, and a common example is the\ncharging of a battery. Thermodynamic relations discussed in previous\nchapters can thus be directly applied to electrochemical systems with\nsome common equations shown in .\n\nTable \u2011: Thermodynamic Equations for Electrochemical Cells\n\n*\u2206G = -z f \u03b5*\n\n*\u2206S = - (\u2202\u2206G/\u2202T)P = + z f (\u2202\u03b5/\u2202T)P*\n\n*\u2206H = \\[\u2202(\u2206G/T)/ \u2202(1/T)\\]P = - z f \\[\u2202(\u03b5/T)/\u2202(1/T)\\]P = z f \\[T(\u2202\u03b5/\u2202T)P\n\u2013 \u03b5\\]*\n\n*\u2206CP = (\u2202\u2206H/\u2202T)P = T z f (\u22022\u03b5/\u2202T2)P*\n\nA half-cell reaction potential cannot be measured directly, only its\npotential relative to another half-cell reaction. By convention, a\nstandard half-cell potential is measured relative to the standard\nhydrogen half-cell reduction reaction at 25oC (298K) and 1\nbar, which has a defined standard potential of zero volts,\n\n*Eq. 8\u201122* H+(aq, a=1) + e- = 1/2 H2(g,\n1 bar)\n\nwith \u03b5o (H+/H2,g) = 0.00 volts. The\nstandard half-cell reduction reactions of metals at 25oC are\nfor the general reaction\n\n*Eq. 8\u201123* Mz+(aq, a=1) + z e- = M(s)\n\nwith \u03b5o (Mz+/M) volts. Half-cell reactions with\nthe most positive standard electrode potentials have a tendency to\nspontaneously proceed toward reduction (cathode reactions). Half-cell\nreactions with the most negative standard electrode potentials have a\ntendency to spontaneously proceed toward oxidation (anode reactions).\n\nConsider, for example, a cell made up of a standard hydrogen electrode\nand a standard zinc electrode with \u03b5o\n(H+/H2,g) = 0.00 volts and \u03b5o\n(Zn2+/Zn) = -0.762 volts. Thus, the H+ would tend\nto be reduced, and the zinc metal would tend to be oxidized, and the\nspontaneous reaction if all species had unit activities would be\n\n*Eq. 8\u201124* 2 H+(aq, a=1)+ Zn = H2(1 bar) +\nZn2+(aq, a=1)\n\nwith \u03b5ocell = 0.762 volts and\n$\\mathrm{\\Delta}_{\\ }^{0}G = - 2*96485*\\varepsilon_{cell}^{0}\\ $. The\ncathode half-cell reaction would be the same as , while the anode\nhalf-cell reaction would be\n\n*Eq. 8\u201125* Zn = Zn2+(aq, a=1) + 2 e-\n\nWhen ion concentrations and H2 gas do not all have unit\nactivities, the Gibbs energy and cell potential of the cell reaction, ,\nbecomes\n\n*Eq. 8\u201126*\n$\\mathrm{\\Delta}G = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{a_{{Zn}^{2 +}}P_{H_{2}}}{\\left( a_{H^{+}} \\right)^{2}}$\n\n*Eq. 8\u201127*\n$\\varepsilon_{cell} = \\varepsilon_{cell}^{0} - \\frac{RT}{zf}\\ln\\frac{a_{{Zn}^{2 +}}P_{H_{2}}}{\\left( a_{H^{+}} \\right)^{2}}$\n\nThe standard reduction potentials of some common metals at\n25oC are given in .\n\nTable \u2011: Standard reduction potentials of some common metals\n\nA cell reaction can be established by different half-cell reactions. For\nexample, the following reaction can be derived from two different cells\n\n*Eq. 8\u201128* $3\\ {Fe}^{2 + \\ }\\ = \\ \\ 2\\ {Fe}^{3 +}\\ \\ + \\ \\ Fe(s)$\n\ncell A\n\n*Eq. 8\u201129* $3{Fe}^{2 +} + \\ 6\\ e - \\ = \\ \\ 3Fe(s)$ \u03b5o1 =\n-0.440 V\n\n*Eq. 8\u201130* $2\\ Fe(s)\\ \\ = \\ \\ 2\\ {Fe}^{3 +}\\ \\ + \\ 6\\ e -$\n\u03b5o2 = +0.036 V\n\ncell B\n\n*Eq. 8\u201131* $2{Fe}^{2 +}\\ = \\ 2{Fe}^{3 + \\ \\ } + 2\\ e^{-}$\n*\u03b5o4 = -0.772 V*\n\n*Eq. 8\u201132* ${Fe}^{2 + \\ }\\ + \\ 2\\ e - \\ \\ = \\ \\ Fe(s)$ *\u03b5o5\n= -0.440 V*\n\nBoth give the same net reaction shown by , but with 6 and 2 electrons\nand standard cell potentials being \u03b5ocell A = -0.404 V and\n\u03b5ocell B = -1.212 V, respectively. However, the standard\nGibbs energies of both cells are the same, i.e.\n\n*Eq. 8\u201133* $\\mathrm{\\Delta}_{\\ }^{0}G$*cell A = -6 f (-0.404)\n= + 2.424 f*\n\n*Eq. 8\u201134* $\\mathrm{\\Delta}_{\\ }^{0}G$*cell B = -2 f (-1.212)\n= + 2.424 f*\n\nIt is shown that $\\mathrm{\\Delta}_{\\ }^{0}G$ values are independent of\nhalf-cell reactions and depend only on the net reaction because the net\nreaction is neutral in electron and balanced in mass.\n", "metadata": {} } ] diff --git a/src/psu410/src/applications_to_electrochemical_systems/electrolyte_reactions_and_electrochemical_reactions.ipynb b/src/psu410/src/applications_to_electrochemical_systems/electrolyte_reactions_and_electrochemical_reactions.ipynb index 4c3982d..4c2aa3c 100644 --- a/src/psu410/src/applications_to_electrochemical_systems/electrolyte_reactions_and_electrochemical_reactions.ipynb +++ b/src/psu410/src/applications_to_electrochemical_systems/electrolyte_reactions_and_electrochemical_reactions.ipynb @@ -4,461 +4,9 @@ "metadata": {}, "cells": [ { - "id": "7a91931d", + "id": "ece8e64b", "cell_type": "markdown", - "source": [ - "## Electrolyte reactions and electrochemical reactions\n", - "\n", - "Electrolytes that dissolve in a polar solvent such as water to produce\n", - "ionic species do not necessarily exhibit valence changes. A simple\n", - "example is the strong electrolyte NaCl(s) dissolving in water to produce\n", - "solvated ions\n", - "\n", - "*Eq. 8\u20111* $NaCl(s)\\ \\ = \\ \\ {Na}^{+}(aq)\\ \\ + \\ \\ {Cl}^{\u2013}(aq)$\n", - "\n", - "where the $(aq)$ indicates the ionic species in an aqueous solution. In\n", - "this system, the ion concentrations must become quite large before the\n", - "solution is saturated and can exist in equilibrium with $NaCl(s)$. Its\n", - "reaction constant, defined by , is shown as\n", - "$K_{e} = \\ \\ a_{{Na}^{+}}a_{{Cl}^{-}}$. If the product of the ion\n", - "activities is less than $K_{e}$, the solution is not saturated, and more\n", - "$NaCl(s)\\ \\ $ can be dissolved.\n", - "\n", - "The precipitation of $AgCl(s)$, a weak electrolyte, occurs quite readily\n", - "when ${Cl}^{\u2013}$ ions are added to an aqueous solution containing\n", - "${Ag}^{+}(aq)$:\n", - "\n", - "*Eq. 8\u20112* ${Ag}^{+}(aq)\\ + \\ \\ {Cl}^{\u2013}(aq)\\ = AgCl(s)$\n", - "\n", - "The equilibrium constant for this reaction,\n", - "$K_{e} = \\ \\frac{1}{\\left( a_{{Ag}^{+}}a_{{Cl}^{-}} \\right)}$ is quite\n", - "large, so the equilibrium product of the ion activities, proportional to\n", - "their concentrations, is quite small. In the laboratory, the above\n", - "reaction could occur as a result of adding hydrochloric acid to a silver\n", - "nitrate solution. The accompanying $H^{+}(aq)$ \\[or ${H_{3}O}^{+}(aq)$\\]\n", - "and ${NO_{3}}^{-}(aq)$ ions in solution are not directly involved in the\n", - "silver chloride precipitation reaction so are not shown in reaction\n", - "represented by .\n", - "\n", - "The above ionic equilibria in the $AgCl(s) - H_{2}O$ system is not only\n", - "important for understanding this electrolyte system, but also critical\n", - "in electrochemical systems in which *Ag(s)* undergoes a valence change\n", - "at one electrode and reacts with a ${Cl}^{\u2013}(aq)$ ion to produce\n", - "*AgCl(s)*, and an electron that is externally transported finite\n", - "distances to another electrode. The *oxidation* reaction occurs at the\n", - "Ag/AgCl electrode (*anode* half-cell reaction where electrons are\n", - "*added* into the system)\n", - "\n", - "*Eq. 8\u20113* $Ag(s)\\ \\ + \\ {Cl}^{-}(aq)\\ \\ = \\ \\ AgCl(s)\\ \\ + \\ \\ e^{-}$\n", - "\n", - "A *reduction* reaction occurs at the other electrode (*cathode*\n", - "half-cell reaction where electrons are *consumed* by the reaction)\n", - "\n", - "*Eq. 8\u20114* $\\frac{1}{2}{Cl}_{2}(g) + \\ \\ e^{-} = \\ {Cl}^{-}(aq)\\ \\ $\n", - "\n", - "The *net cell reaction* results in the formation AgCl(s) from its\n", - "elements\n", - "\n", - "*Eq. 8\u20115* $Ag(s)\\ \\ + \\frac{1}{2}{Cl}_{2}\\ (g)\\ = \\ \\ AgCl(s)$\n", - "\n", - "Without knowledge of the physical system under which the reaction is\n", - "occurring, it would not be possible to know if reaction of was a result\n", - "of chlorine gas reacting directly with Ag(s), or if the reaction was\n", - "part of an electrochemical cell with a transport of electrons and ions\n", - "over finite distances. The addition of the two half-cell reactions gives\n", - "the *net cell reaction*, which does not show electrons as either\n", - "reactant or product species and may or may not include ionic species in\n", - "the reaction*.* A schematic of an electrochemical cell for the above\n", - "system is shown in Figure 8\u20111.\n", - "\n", - "Figure \u2011: Schematic diagram of an electrochemical cell consisting of a\n", - "chlorine electrode and a silver-silver chloride electrode.\n", - "\n", - "Oxidation and reduction can occur in electrolyte reactions without\n", - "creating an electrochemical cell. This is the case when chlorine gas\n", - "reacts directly with silver on a Ag(s) surface. Reaction of above is the\n", - "net reaction for this process, but the electrons produced from the\n", - "oxidation of Ag(s) are not transported over finite distances before\n", - "combining with Cl2(g) in its reduction to Cl\u2013(aq).\n", - "No anode or cathode half-cell reactions exist in this system. The\n", - "electrons and ions involved in the reaction move only over atomic-scale\n", - "distances.\n" - ], - "metadata": {} - }, - { - "id": "0b1b59f6", - "cell_type": "markdown", - "source": [ - "### Fuel cells\n", - "\n", - "Fuel cells are devices to convert chemical energy to electricity and\n", - "heat through electrochemical reactions with the fuel and oxygen supplied\n", - "to the anode and cathode, respectively. Typical ions migrating through\n", - "the electrolyte are $H^{+}$, ${OH}^{-}$, ${CO}_{3}^{2 -}$, and\n", - "$O^{2 -}$. In fuel cells with $H^{+}$ as migrating ions, H2\n", - "molecules are dissociated into $H^{+}$ on the anode, which are combined\n", - "with O2 on the cathode to form H2O and release\n", - "heat with the half cell and the net cell reactions in their simplest\n", - "form as shown by for the anode, for the cathode, and for the net cell,\n", - "respectively. Commonly used electrolytes are polymer and phosphoric\n", - "acid, and both anode and cathode reactions are facilitated by catalyst,\n", - "typically platinum. The thermodynamic limit of power which can be\n", - "generated by the fuel cell is represented by\n", - "\n", - "*Eq. 8\u201163*\n", - "$w = - \\mathrm{\\Delta}G = - \\mathrm{\\Delta}_{\\ }^{0}G_{cell} + RTln\\left( P_{H_{2}}P_{O_{2}}^{1/2} \\right)$\n", - "\n", - "For fuel cells with anions as migrating ions, the anions are generated\n", - "on the cathode with H2O formed and heat generated on the\n", - "anode. Their representative cathode reactions are\n", - "\n", - "*Eq. 8\u201164* $\\frac{1}{2}O_{2} + H_{2}O + 2e^{-}{= 2OH}^{-}$\n", - "\n", - "*Eq. 8\u201165* ${\\frac{1}{2}O}_{2} + CO_{2} + 2e^{-} = {CO}_{3}^{2 -}$\n", - "\n", - "*Eq. 8\u201166* $\\frac{1}{2}O_{2} + 2e^{-} = O^{2 -}$.\n", - "\n", - "The anode reaction for is the reaction represented by , operating at low\n", - "temperatures and using catalyst for both electrodes. The anode reactions\n", - "for and are\n", - "\n", - "*Eq. 8\u201167* ${CO}_{3}^{2 -} + H_{2} = H_{2}O + CO_{2} + 2e^{-}$\n", - "\n", - "*Eq. 8\u201168* $O^{2 -} + H_{2} = H_{2}O + 2e^{-}$\n", - "\n", - "respectively. To enable the diffusion of ${CO}_{3}^{2 -}$ and $O^{2 -}$\n", - "through the cathode and the electrolyte, both fuel cells are operated at\n", - "relative high temperatures, with the former typically in molten\n", - "carbonate solutions and the latter through solid oxides. Due to the high\n", - "operating temperatures, fuels are converted to hydrogen within the fuel\n", - "cell itself by a process called internal reforming, removing the need\n", - "for precious-metal catalyst and enabling the use of a variety of fuels.\n", - "The net cell reaction for all three fuel cells is the same as in the\n", - "case of $H^{+}$, represented by .\n" - ], - "metadata": {} - }, - { - "id": "560fe529", - "cell_type": "markdown", - "source": [ - "### Galvanic protection\n", - "\n", - "A galvanic reaction takes place between two different materials at the\n", - "two respective electrodes, each with different tendency to hold on to\n", - "electrons. Consider the following electrochemical cell used to protect\n", - "Cu tanks against oxidation by using a \u201csacrificial\u201d Fe electrode\n", - "\n", - "anode solution cathode\n", - "\n", - "Fe(s) \\| Fe2+ \\| SO4= \\|\n", - "Cu2+\\| Cu(s) \\|\n", - "\n", - "Cathode reduction: Cu2+ + 2 e- = Cu \u03b5o\n", - "(volts) = 0.34\n", - "\n", - "Anode oxidation: Fe(s) = Fe2+ + 2 e- \u03b5o (volts) =\n", - "0.44\n", - "\n", - "Net reaction: Cu2+ + Fe =\\> Cu + Fe2+\n", - "\u03b5o (volts) = 0.78\n", - "\n", - "If the cell has a direct connection between the electrodes, i.e. it has\n", - "a short circuit: \u2206G \u2192 0 and thus \u03b5cell \u2192 0. Since the value\n", - "of \u03b5ocell \\> 0 for the net cell reaction, the equilibrium\n", - "constant *Ke* \\> 1, which means\n", - "\\[Fe2+\\]/\\[Cu2+\\] \\> 1. By assuming an ideal\n", - "electrolyte solution, the activities in *Ke* can be\n", - "represented by concentrations (in molar concentration units), assuming\n", - "that solid Fe and Cu are present at unit activities. If the electrodes\n", - "of Cu and Fe are short circuited while in contact with the same\n", - "\"electrolyte solution\", the final equilibrium concentrations can be\n", - "calculated by the standard equation,\n", - "\n", - "*Eq. 8\u201161* *Ke* = \\[Fe2+\\]/\\[Cu2+\\] =\n", - "exp (-\u2206oG/RT)\n", - "\n", - "or, using the Nernst equations,\n", - "\n", - "*Eq. 8\u201162* \u03b5 = \u03b5o -(RT/z f) ln (\\[Fe2+\\]/\\[Cu2+\\])\n", - "= 0\n", - "\n", - "and the above standard cell potential:\n", - "\\[Fe2+\\]/\\[Cu2+\\] = 2.4\u20221026\n", - "\n", - "With this large ratio it can be seen that the tendency to produce\n", - "Cu+2 ions, i.e., the tendency to corrode the Cu(s), is\n", - "extremely small if a sacrificial Fe electrode is configured in an\n", - "electrochemical cell with the Cu tank.\n" - ], - "metadata": {} - }, - { - "id": "6dd6c6c3", - "cell_type": "markdown", - "source": [ - "### Half cell potentials\n", - "\n", - "When electron current flows between electrodes, reactions are occurring\n", - "at the electrodes and concentration gradients causing polarization\n", - "develop around the electrodes. These gradients result in extraneous\n", - "potentials to occur at the electrodes. In such cases cell equilibrium is\n", - "not established and measured cell potentials are not those for true\n", - "partial equilibrium. If a cell is short-circuited with the electrodes\n", - "connected by a conductor, current will flow until the external potential\n", - "becomes zero, i.e. \u03b5ext = 0, and equilibrium is established\n", - "with same conditions as non-electrochemical systems. If an external\n", - "potential, \u03b5ext, is applied to the cell, chemical reactions\n", - "occur until the cell potential balances to \u03b5ext, and no\n", - "current flows. This potential is called open-circuit voltage (OCV) in\n", - "the literature. It is important to realize that OCV includes all\n", - "reactions that occur on the electrode surface when the electrode is in\n", - "contact with the electrolyte, such as passivation discussed in Chapter .\n", - "Partial equilibrium in a cell is achieved when the cell potential is\n", - "balanced by an applied external potential. In such partial equilibrium\n", - "cases, equilibrium thermodynamic analyses can be used even though the\n", - "cell potential is not zero, i.e. \u03b5cell \u2260 0. This\n", - "differentiates electrochemical systems from other equilibrium systems\n", - "discussed previously.\n", - "\n", - "The number of electrons involved in a net cell reaction is important in\n", - "relating cell potential and the Gibbs energy change for the cell\n", - "reaction. As will be illustrated later in this section, this number\n", - "denotes the number of electrons involved in the half-cell reactions that\n", - "were added to yield the net cell reaction. The electrical work achieved\n", - "by the transport of an electrical charge through a cell potential can be\n", - "written as\n", - "\n", - "*Eq. 8\u201120* $w = z\\ f\\ \\varepsilon$\n", - "\n", - "where *z* represents the moles of electrons in cell reaction, *f* the\n", - "Faraday constant equal to 96,485 J/V/mole-electron, and *\u03b5* the\n", - "potential difference, often referred as electromotive force (emf) in the\n", - "literature. For a system at constant temperature, pressure, and\n", - "composition, this work is the same as the Gibbs energy difference\n", - "between the two electrodes, i.e.\n", - "\n", - "*Eq. 8\u201121* $- \\Delta G = w = zf\\varepsilon$\n", - "\n", - "where the negative sign is because the system does work to the\n", - "surrounding when the Gibbs energy of the system is decreased. When the\n", - "applied external potential is larger than the cell potential, the\n", - "surrounding does work to the system, and a common example is the\n", - "charging of a battery. Thermodynamic relations discussed in previous\n", - "chapters can thus be directly applied to electrochemical systems with\n", - "some common equations shown in .\n", - "\n", - "Table \u2011: Thermodynamic Equations for Electrochemical Cells\n", - "\n", - "*\u2206G = -z f \u03b5*\n", - "\n", - "*\u2206S = - (\u2202\u2206G/\u2202T)P = + z f (\u2202\u03b5/\u2202T)P*\n", - "\n", - "*\u2206H = \\[\u2202(\u2206G/T)/ \u2202(1/T)\\]P = - z f \\[\u2202(\u03b5/T)/\u2202(1/T)\\]P = z f \\[T(\u2202\u03b5/\u2202T)P\n", - "\u2013 \u03b5\\]*\n", - "\n", - "*\u2206CP = (\u2202\u2206H/\u2202T)P = T z f (\u22022\u03b5/\u2202T2)P*\n", - "\n", - "A half-cell reaction potential cannot be measured directly, only its\n", - "potential relative to another half-cell reaction. By convention, a\n", - "standard half-cell potential is measured relative to the standard\n", - "hydrogen half-cell reduction reaction at 25oC (298K) and 1\n", - "bar, which has a defined standard potential of zero volts,\n", - "\n", - "*Eq. 8\u201122* H+(aq, a=1) + e- = 1/2 H2(g,\n", - "1 bar)\n", - "\n", - "with \u03b5o (H+/H2,g) = 0.00 volts. The\n", - "standard half-cell reduction reactions of metals at 25oC are\n", - "for the general reaction\n", - "\n", - "*Eq. 8\u201123* Mz+(aq, a=1) + z e- = M(s)\n", - "\n", - "with \u03b5o (Mz+/M) volts. Half-cell reactions with\n", - "the most positive standard electrode potentials have a tendency to\n", - "spontaneously proceed toward reduction (cathode reactions). Half-cell\n", - "reactions with the most negative standard electrode potentials have a\n", - "tendency to spontaneously proceed toward oxidation (anode reactions).\n", - "\n", - "Consider, for example, a cell made up of a standard hydrogen electrode\n", - "and a standard zinc electrode with \u03b5o\n", - "(H+/H2,g) = 0.00 volts and \u03b5o\n", - "(Zn2+/Zn) = -0.762 volts. Thus, the H+ would tend\n", - "to be reduced, and the zinc metal would tend to be oxidized, and the\n", - "spontaneous reaction if all species had unit activities would be\n", - "\n", - "*Eq. 8\u201124* 2 H+(aq, a=1)+ Zn = H2(1 bar) +\n", - "Zn2+(aq, a=1)\n", - "\n", - "with \u03b5ocell = 0.762 volts and\n", - "$\\mathrm{\\Delta}_{\\ }^{0}G = - 2*96485*\\varepsilon_{cell}^{0}\\ $. The\n", - "cathode half-cell reaction would be the same as , while the anode\n", - "half-cell reaction would be\n", - "\n", - "*Eq. 8\u201125* Zn = Zn2+(aq, a=1) + 2 e-\n", - "\n", - "When ion concentrations and H2 gas do not all have unit\n", - "activities, the Gibbs energy and cell potential of the cell reaction, ,\n", - "becomes\n", - "\n", - "*Eq. 8\u201126*\n", - "$\\mathrm{\\Delta}G = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{a_{{Zn}^{2 +}}P_{H_{2}}}{\\left( a_{H^{+}} \\right)^{2}}$\n", - "\n", - "*Eq. 8\u201127*\n", - "$\\varepsilon_{cell} = \\varepsilon_{cell}^{0} - \\frac{RT}{zf}\\ln\\frac{a_{{Zn}^{2 +}}P_{H_{2}}}{\\left( a_{H^{+}} \\right)^{2}}$\n", - "\n", - "The standard reduction potentials of some common metals at\n", - "25oC are given in .\n", - "\n", - "Table \u2011: Standard reduction potentials of some common metals\n", - "\n", - "A cell reaction can be established by different half-cell reactions. For\n", - "example, the following reaction can be derived from two different cells\n", - "\n", - "*Eq. 8\u201128* $3\\ {Fe}^{2 + \\ }\\ = \\ \\ 2\\ {Fe}^{3 +}\\ \\ + \\ \\ Fe(s)$\n", - "\n", - "cell A\n", - "\n", - "*Eq. 8\u201129* $3{Fe}^{2 +} + \\ 6\\ e - \\ = \\ \\ 3Fe(s)$ \u03b5o1 =\n", - "-0.440 V\n", - "\n", - "*Eq. 8\u201130* $2\\ Fe(s)\\ \\ = \\ \\ 2\\ {Fe}^{3 +}\\ \\ + \\ 6\\ e -$\n", - "\u03b5o2 = +0.036 V\n", - "\n", - "cell B\n", - "\n", - "*Eq. 8\u201131* $2{Fe}^{2 +}\\ = \\ 2{Fe}^{3 + \\ \\ } + 2\\ e^{-}$\n", - "*\u03b5o4 = -0.772 V*\n", - "\n", - "*Eq. 8\u201132* ${Fe}^{2 + \\ }\\ + \\ 2\\ e - \\ \\ = \\ \\ Fe(s)$ *\u03b5o5\n", - "= -0.440 V*\n", - "\n", - "Both give the same net reaction shown by , but with 6 and 2 electrons\n", - "and standard cell potentials being \u03b5ocell A = -0.404 V and\n", - "\u03b5ocell B = -1.212 V, respectively. However, the standard\n", - "Gibbs energies of both cells are the same, i.e.\n", - "\n", - "*Eq. 8\u201133* $\\mathrm{\\Delta}_{\\ }^{0}G$*cell A = -6 f (-0.404)\n", - "= + 2.424 f*\n", - "\n", - "*Eq. 8\u201134* $\\mathrm{\\Delta}_{\\ }^{0}G$*cell B = -2 f (-1.212)\n", - "= + 2.424 f*\n", - "\n", - "It is shown that $\\mathrm{\\Delta}_{\\ }^{0}G$ values are independent of\n", - "half-cell reactions and depend only on the net reaction because the net\n", - "reaction is neutral in electron and balanced in mass.\n" - ], - "metadata": {} - }, - { - "id": "d098364d", - "cell_type": "markdown", - "source": [ - "### Ion transport membranes\n", - "\n", - "Ion transport membranes (ITMs) are ceramic membranes conducting both\n", - "electrons and oxygen ions, but no other species. The chemical potential\n", - "difference of oxygen between two sides of a membrane provides the\n", - "driving force for oxygen to diffuse through the membrane. Commonly used\n", - "ITM oxides include perovskite and fluorite, with the chemical formula of\n", - "ABO3 and AO2, respectively, typically with more\n", - "than one elements in the A-site and / or the B-site to tailor the\n", - "electron and ionic conductivities. Key thermodynamic properties of ITM\n", - "oxides are their stability in service environments, vacancy\n", - "concentrations in the oxygen and cation sites, valances of cations. On\n", - "the high oxygen partial pressure side, the reaction is as the following\n", - "\n", - "*Eq. 8\u201169* $\\frac{1}{2}O_{2} + 2e^{-} = O^{2 -}$\n", - "\n", - "At the same time the number of oxygen vacancy is reduced, resulting in a\n", - "lower concentration of oxygen vacancy and higher oxygen activity in the\n", - "oxide on the high oxygen partial pressure side. On the low oxygen\n", - "partial pressure side, the reaction is reversed to produce oxygen\n", - "molecules, i.e.\n", - "\n", - "*Eq. 8\u201170* $O^{2 -} = \\frac{1}{2}O_{2} + 2e^{-}$\n", - "\n", - "This reaction results in higher oxygen vacancy concentration and lower\n", - "oxygen activity. At both sides, the charge neutrality is compensated by\n", - "the valance changes of cations, resulting in the electron flow in the\n", - "opposite direction of oxygen diffusion. The ionic conductivity is\n", - "dictated by the oxygen transportation across the membrane with the\n", - "driving force of the following net reaction\n", - "\n", - "*Eq. 8\u201171*\n", - "$\\frac{1}{2}O_{2}\\left( P_{high} \\right) = \\frac{1}{2}O_{2}\\left( P_{low} \\right)$\n", - "\n", - "*Eq. 8\u201172*\n", - "$\\mathrm{\\Delta}G = 0.5RTln\\left( \\frac{P_{low}}{P_{high}} \\right)$.\n", - "\n", - "The oxygen transportation is closely related to the concentration of\n", - "oxygen vacancy in the membrane, which is obtained by minimizing the\n", - "Gibbs energy of the phase under given temperature and oxygen partial\n", - "pressure conditions. High vacancy concentrations can be obtained by\n", - "cation dopants with lower valances or small energy differences between\n", - "various valance states. However, at the same time, high vacancy\n", - "concentrations reduce the thermodynamic stability of the membrane, which\n", - "may result in its decomposition into more stable phases.\n" - ], - "metadata": {} - }, - { - "id": "728bc14f", - "cell_type": "markdown", - "source": [ - "### Metastability and passivation\n", - "\n", - "Our modern industrial society is built on various metals such as Fe, Ni,\n", - "Al, Ti, and Zr alloys which are reactive, but exhibit extraordinary\n", - "kinetic stabilities in oxidizing environments due to the existence of a\n", - "thin reaction product film on the surface. This film effectively\n", - "isolates the metal from the corrosive environment, a phenomenon called\n", - "passivation. One interesting experiment by Faraday in 1836 who reported\n", - "that iron corrodes freely in dilute nitric acid, while in concentrated\n", - "nitric acid, no reaction apparently occurred. To understand this\n", - "phenomenon, let us exam a simple, schematic Pourbaix diagram for the\n", - "iron-water system shown in .\n", - "\n", - "Figure \u2011: Schematic Pourbaix diagram for iron illustrating the\n", - "resolution of the Faraday paradox in the corrosion of iron in nitric\n", - "acid \\[61\\]. Lines (a) (b), and (c) correspond to the equilibria: (a)\n", - "$H^{+} + e^{-} = 1/2H_{2}$; (b) $O_{2} + 4H^{+} + 4e^{-} = 2H_{2}O$; (c)\n", - "$NO_{3}^{-} + 3H^{+} + 2e^{-} = HNO_{2} + H_{2}O$, respectively\n", - "\n", - "For iron in deaerated acid solution, the partial anodic and cathodic\n", - "reactions are Line 1 $\\left( Fe/{Fe}^{2 +} \\right)$ and Line (a),\n", - "respectively, resulting in a corrosion potential that lies between Lines\n", - "$1$ and $a$. In oxygenated (aerated) solutions, the corrosion potential\n", - "may lie between Lines $1$ and $b$, because the reduction of oxygen is a\n", - "possible (likely) cathodic reaction. Since dilute HNO3 is\n", - "only a weak oxidizing agent, the principal cathodic reaction was most\n", - "likely hydrogen evolution, and hence the corrosion potential is expected\n", - "to lie between Lines $1$ and $a$ at relatively high pH, as shown. Since\n", - "the $Fe/{Fe}^{2 +}$ reaction is relatively fast compared with\n", - "$H^{+}/H_{2}$ on iron, if the corrosion potential is situated below the\n", - "extension of Line 2 (Fe/Fe3O4) into the\n", - "${Fe}^{2 +}$ stability region, Fe3O4 cannot form\n", - "on the surface, even as a metastable phase.\n", - "\n", - "On the other hand, concentrated HNO3 is a strong oxidizing\n", - "agent due to the reaction:\n", - "$NO_{3}^{-} + 3H^{+} + 2e^{-} = HNO_{2} + H_{2}O$, so that the corrosion\n", - "potential can lie anywhere between Lines $1$ and $c$ at low pH. Since\n", - "Reaction (2) is likely to be fast, the corrosion potential will be high\n", - "and certainly will be more positive than the extension of Line 2 into\n", - "the stability region for ${Fe}^{2 +}$ at low pH. Therefore,\n", - "Fe3O4 becomes metastable and can form between the\n", - "aqueous solution and iron. The thickness of this\n", - "Fe3O4 film depends on its dissolution rate into\n", - "the aqueous solution and its growth rate at the interface with iron,\n", - "which depends on the diffusion of ionic species across the film. Its\n", - "existence results in passivity and the observed kinetic inactivity of\n", - "iron in this medium. When the potential becomes even more positive above\n", - "the extension of Line $3$, Fe2O3 may form on\n", - "Fe3O4 as an additional metastable phase, resulting\n", - "in the commonly observed bilayer structure.\n" - ], + "source": "## Electrolyte reactions and electrochemical reactions\n\nElectrolytes that dissolve in a polar solvent such as water to produce\nionic species do not necessarily exhibit valence changes. A simple\nexample is the strong electrolyte NaCl(s) dissolving in water to produce\nsolvated ions\n\n*Eq. 8\u20111* $NaCl(s)\\ \\ = \\ \\ {Na}^{+}(aq)\\ \\ + \\ \\ {Cl}^{\u2013}(aq)$\n\nwhere the $(aq)$ indicates the ionic species in an aqueous solution. In\nthis system, the ion concentrations must become quite large before the\nsolution is saturated and can exist in equilibrium with $NaCl(s)$. Its\nreaction constant, defined by , is shown as\n$K_{e} = \\ \\ a_{{Na}^{+}}a_{{Cl}^{-}}$. If the product of the ion\nactivities is less than $K_{e}$, the solution is not saturated, and more\n$NaCl(s)\\ \\ $ can be dissolved.\n\nThe precipitation of $AgCl(s)$, a weak electrolyte, occurs quite readily\nwhen ${Cl}^{\u2013}$ ions are added to an aqueous solution containing\n${Ag}^{+}(aq)$:\n\n*Eq. 8\u20112* ${Ag}^{+}(aq)\\ + \\ \\ {Cl}^{\u2013}(aq)\\ = AgCl(s)$\n\nThe equilibrium constant for this reaction,\n$K_{e} = \\ \\frac{1}{\\left( a_{{Ag}^{+}}a_{{Cl}^{-}} \\right)}$ is quite\nlarge, so the equilibrium product of the ion activities, proportional to\ntheir concentrations, is quite small. In the laboratory, the above\nreaction could occur as a result of adding hydrochloric acid to a silver\nnitrate solution. The accompanying $H^{+}(aq)$ \\[or ${H_{3}O}^{+}(aq)$\\]\nand ${NO_{3}}^{-}(aq)$ ions in solution are not directly involved in the\nsilver chloride precipitation reaction so are not shown in reaction\nrepresented by .\n\nThe above ionic equilibria in the $AgCl(s) - H_{2}O$ system is not only\nimportant for understanding this electrolyte system, but also critical\nin electrochemical systems in which *Ag(s)* undergoes a valence change\nat one electrode and reacts with a ${Cl}^{\u2013}(aq)$ ion to produce\n*AgCl(s)*, and an electron that is externally transported finite\ndistances to another electrode. The *oxidation* reaction occurs at the\nAg/AgCl electrode (*anode* half-cell reaction where electrons are\n*added* into the system)\n\n*Eq. 8\u20113* $Ag(s)\\ \\ + \\ {Cl}^{-}(aq)\\ \\ = \\ \\ AgCl(s)\\ \\ + \\ \\ e^{-}$\n\nA *reduction* reaction occurs at the other electrode (*cathode*\nhalf-cell reaction where electrons are *consumed* by the reaction)\n\n*Eq. 8\u20114* $\\frac{1}{2}{Cl}_{2}(g) + \\ \\ e^{-} = \\ {Cl}^{-}(aq)\\ \\ $\n\nThe *net cell reaction* results in the formation AgCl(s) from its\nelements\n\n*Eq. 8\u20115* $Ag(s)\\ \\ + \\frac{1}{2}{Cl}_{2}\\ (g)\\ = \\ \\ AgCl(s)$\n\nWithout knowledge of the physical system under which the reaction is\noccurring, it would not be possible to know if reaction of was a result\nof chlorine gas reacting directly with Ag(s), or if the reaction was\npart of an electrochemical cell with a transport of electrons and ions\nover finite distances. The addition of the two half-cell reactions gives\nthe *net cell reaction*, which does not show electrons as either\nreactant or product species and may or may not include ionic species in\nthe reaction*.* A schematic of an electrochemical cell for the above\nsystem is shown in Figure 8\u20111.\n\nFigure \u2011: Schematic diagram of an electrochemical cell consisting of a\nchlorine electrode and a silver-silver chloride electrode.\n\nOxidation and reduction can occur in electrolyte reactions without\ncreating an electrochemical cell. This is the case when chlorine gas\nreacts directly with silver on a Ag(s) surface. Reaction of above is the\nnet reaction for this process, but the electrons produced from the\noxidation of Ag(s) are not transported over finite distances before\ncombining with Cl2(g) in its reduction to Cl\u2013(aq).\nNo anode or cathode half-cell reactions exist in this system. The\nelectrons and ions involved in the reaction move only over atomic-scale\ndistances.\n", "metadata": {} } ] diff --git a/src/psu410/src/applications_to_electrochemical_systems/index.ipynb b/src/psu410/src/applications_to_electrochemical_systems/index.ipynb index 7f8042b..411a8f5 100644 --- a/src/psu410/src/applications_to_electrochemical_systems/index.ipynb +++ b/src/psu410/src/applications_to_electrochemical_systems/index.ipynb @@ -4,29 +4,9 @@ "metadata": {}, "cells": [ { - "id": "b425decf", + "id": "e01daf27", "cell_type": "markdown", - "source": [ - "# Applications to electrochemical systems\n", - "\n", - "The two basic types of chemical processes involving ions as reactant\n", - "and/or product species are electrolyte reactions and electrochemical\n", - "reactions. *Electrolyte reactions* are accompanied by the *atomic-scale\n", - "movement* of ionic species and possibly electrons. Chemical changes that\n", - "produce changes in valence and *electron and ion transport over finite\n", - "distances* constitute an area of science termed *electrochemistry*. The\n", - "latter chemical changes occur in an electrochemical cell comprised of\n", - "two electrodes, an anode and a cathode, which are coupled by an\n", - "electrolyte and an external electron conductor. Most thermodynamic\n", - "concepts and analyses described in previous chapters remain unchanged\n", - "when applied to electrochemistry, but the analysis of electrochemical\n", - "systems does require some new terminology, new definitions, and new\n", - "conventions. The primary focus of this chapter is on applications of\n", - "thermodynamics to electrochemical reactions that involve either aqueous\n", - "electrolyte solutions or solid state electrolytes. Since all\n", - "electrochemical systems include ionized species as reactant and/or\n", - "product species, electrolyte reactions will also be discussed.\n" - ], + "source": "# Applications to electrochemical systems\n\nThe two basic types of chemical processes involving ions as reactant\nand/or product species are electrolyte reactions and electrochemical\nreactions. *Electrolyte reactions* are accompanied by the *atomic-scale\nmovement* of ionic species and possibly electrons. Chemical changes that\nproduce changes in valence and *electron and ion transport over finite\ndistances* constitute an area of science termed *electrochemistry*. The\nlatter chemical changes occur in an electrochemical cell comprised of\ntwo electrodes, an anode and a cathode, which are coupled by an\nelectrolyte and an external electron conductor. Most thermodynamic\nconcepts and analyses described in previous chapters remain unchanged\nwhen applied to electrochemistry, but the analysis of electrochemical\nsystems does require some new terminology, new definitions, and new\nconventions. The primary focus of this chapter is on applications of\nthermodynamics to electrochemical reactions that involve either aqueous\nelectrolyte solutions or solid state electrolytes. Since all\nelectrochemical systems include ionized species as reactant and/or\nproduct species, electrolyte reactions will also be discussed.\n", "metadata": {} } ] diff --git a/src/psu410/src/calpahd_modeling_of_thermodynamics/importance_of_lattice_stability.ipynb b/src/psu410/src/calpahd_modeling_of_thermodynamics/importance_of_lattice_stability.ipynb index d25171d..3870d0e 100644 --- a/src/psu410/src/calpahd_modeling_of_thermodynamics/importance_of_lattice_stability.ipynb +++ b/src/psu410/src/calpahd_modeling_of_thermodynamics/importance_of_lattice_stability.ipynb @@ -4,117 +4,9 @@ "metadata": {}, "cells": [ { - "id": "ba6b61d7", + "id": "ba9e840b", "cell_type": "markdown", - "source": [ - "## Importance of lattice stability\n", - "\n", - "For modeling of Gibbs energy of individual phases, it is necessary to\n", - "define the values of $_{\\ }^{0}G_{i}$ in . However, the independent\n", - "component *i* may not be stable in the crystal structure of the phase\n", - "under consideration, so its Gibbs energy could not be obtained directly\n", - "from experiments and must be estimated with respect to the Gibbs energy\n", - "of their stable crystal structure. In the pioneering work by Kaufman\n", - "\\[47\\], this Gibbs energy difference was termed as lattice stability and\n", - "obtained through extrapolations in either temperature-pressure or\n", - "temperature-composition phase diagrams. It is evident from that the\n", - "values of $_{\\ }^{0}G_{i}$ and $_{\\ }^{M}G$ jointly contribute to the\n", - "Gibbs energy of the solution, and Kaufman had to simplify the treatment\n", - "of $_{\\ }^{M}G$ in order to show the importance of the concept of\n", - "lattice stability. Using ideal or regular solution models, Kaufman was\n", - "able to define the lattice stability for pure elements and remarkably\n", - "reproduce many features of binary phase diagrams by introducing the\n", - "interaction parameters afterwards.\n", - "\n", - "Over the years, there had been various revisions of lattice stability\n", - "values for common crystal structures \\[48\\], and every revision\n", - "necessitates the re-evaluation of interaction parameters in the solution\n", - "phase shown in . It was until the lattice stability values established\n", - "by the Scientific Group Thermodata Europe (SGTE) \\[52\\] that the\n", - "development of binary thermodynamic models using the same thermodynamic\n", - "models of pure elements became possible, and those binary models thus\n", - "developed in different groups around the world can be combined to create\n", - "thermodynamic models of ternary and multi-component systems. Clearly,\n", - "any further modifications of the SGTE pure element database will require\n", - "the re-modeling of all binary and ternary systems in which the models of\n", - "pure elements are changed. This challenge is briefly addressed in the\n", - "later part of this chapter.\n", - "\n", - "A less addressed issue is the Gibbs energy of end-members in\n", - "non-stoichiometric compounds, i.e. , where each sublattice contains only\n", - "one element. In case all sublattices are occupied by the same element,\n", - "it is the lattice stability of the elements in the structure of the\n", - "compound. Since the stable composition ranges of non-stoichiometric\n", - "compounds are typically small, the existing method cannot be used to\n", - "reliably evaluate the Gibbs energy of end-members, and currently there\n", - "is not a commonly accepted lattice stability database for compounds.\n", - "Most values used in the existing databases have been either roughly\n", - "estimated or computed from first-principles calculations. Such a\n", - "standard database is highly desirable in order to make various models of\n", - "compounds compatible.\n", - "\n", - "In an effort to compare the lattice stability from the CALPHAD models\n", - "and the first-principles calculations, Wang et al. \\[53\\] systematically\n", - "calculated the total energies of 78 pure elements at zero Kelvin in the\n", - "face-centered-cubic (fcc), body-centered-cubic (bcc), and\n", - "hexagonal-close-packed (hcp) crystal structures using the projector\n", - "augmented-wave (PAW) method within the generalized gradient\n", - "approximation (GGA). The calculated values are compared with the values\n", - "in the SGTE database as shown in and . For non-transition metal\n", - "elements, the differences between the SGTE data and the PAW-GGA data are\n", - "typically around 1\u223c2 kJ/mole-of-atoms or less, while for some transition\n", - "metal elements, the differences can be quite large, for example, as high\n", - "as about 54 kJ/mole-of-atoms for and about 40 kJ/mole-of-atoms for . and\n", - "present the differences between the PAW-GGA data and the SGTE data, for\n", - "elements from the Ti group to the Ni group, respectively.\n", - "\n", - "Table \u2011: Lattice stability Ebcc-fcc (kJ/mole-of-atoms).\n", - "\n", - "Table \u2011: Lattice stability Ehcp-fcc (kJ/mole-of-atoms).\n", - "\n", - "Figure \u2011: Lattice stability difference between bcc and fcc, , for\n", - "selected elements between PAW-GGA and SGTE\n", - "\n", - "Figure \u2011: Lattice stability difference between hcp and fcc, , for\n", - "selected elements between PAW-GGA and SGTE\n", - "\n", - "The large differences between the first-principles calculations and the\n", - "SGTE data could partly be attributed to the instability of the\n", - "higher-energy phases, the entropies of which at finite *T* become\n", - "abnormal. The lattice instabilities along the tetragonal transformation\n", - "path between fcc and bcc structures with the continuous change of the\n", - "c/a ratio defined in a bcc-based tetragonal lattice are demonstrated for\n", - "bcc Mo, Ta, W in and for fcc Al, Cu, Ni in . It is shown that the fcc\n", - "structure of bcc Mo, Ta, and W is a local maximum with respect to the\n", - "tetragonal transformation, and the higher the maximum is, the larger the\n", - "discrepancy between the SGTE data and the present PAW-GGA data, while\n", - "for fcc Al, Cu, Ni, the bcc structure is at a local maximum. Similarly,\n", - "the lattice instabilities along the tetragonal transformation path for\n", - "the hcp metals Ru and Os as shown in . The behavior of energy against\n", - "c/a ratio of these two hcp metals is very similar to those of fcc\n", - "elements.\n", - "\n", - "Figure \u2011: Total energy, , along the Bain deformation path between bcc\n", - "and fcc for Mo, Ta, and W.\n", - "\n", - "Figure \u2011: Total energy, , along the tetragonal transformation path\n", - "between bcc and fcc for Ni, Al, and Cu.\n", - "\n", - "Figure \u2011: Total energy, , along the tetragonal transformation path for\n", - "Ru and Os\n", - "\n", - "It can be concluded that a fcc structure for elements with bcc being the\n", - "ground state or a bcc structure for elements with fcc being the ground\n", - "state, is unstable with respect to the tetragonal transformation. For an\n", - "unstable structure, the harmonic description of its vibrational entropy\n", - "is thermodynamically incorrect since the potential surface seen by the\n", - "lattice ion can no longer be approximated by a parabola. If an unstable\n", - "structure of a pure element is stabilized at high temperatures, its\n", - "entropy has to be abnormal. The instability issue has been recently\n", - "addressed by ab initio molecular dynamics simulations at high\n", - "temperatures using W as an example \\[54\\], which is beyond the scope of\n", - "the book and thus not discussed here.\n" - ], + "source": "## Importance of lattice stability\n\nFor modeling of Gibbs energy of individual phases, it is necessary to\ndefine the values of $_{\\ }^{0}G_{i}$ in . However, the independent\ncomponent *i* may not be stable in the crystal structure of the phase\nunder consideration, so its Gibbs energy could not be obtained directly\nfrom experiments and must be estimated with respect to the Gibbs energy\nof their stable crystal structure. In the pioneering work by Kaufman\n\\[47\\], this Gibbs energy difference was termed as lattice stability and\nobtained through extrapolations in either temperature-pressure or\ntemperature-composition phase diagrams. It is evident from that the\nvalues of $_{\\ }^{0}G_{i}$ and $_{\\ }^{M}G$ jointly contribute to the\nGibbs energy of the solution, and Kaufman had to simplify the treatment\nof $_{\\ }^{M}G$ in order to show the importance of the concept of\nlattice stability. Using ideal or regular solution models, Kaufman was\nable to define the lattice stability for pure elements and remarkably\nreproduce many features of binary phase diagrams by introducing the\ninteraction parameters afterwards.\n\nOver the years, there had been various revisions of lattice stability\nvalues for common crystal structures \\[48\\], and every revision\nnecessitates the re-evaluation of interaction parameters in the solution\nphase shown in . It was until the lattice stability values established\nby the Scientific Group Thermodata Europe (SGTE) \\[52\\] that the\ndevelopment of binary thermodynamic models using the same thermodynamic\nmodels of pure elements became possible, and those binary models thus\ndeveloped in different groups around the world can be combined to create\nthermodynamic models of ternary and multi-component systems. Clearly,\nany further modifications of the SGTE pure element database will require\nthe re-modeling of all binary and ternary systems in which the models of\npure elements are changed. This challenge is briefly addressed in the\nlater part of this chapter.\n\nA less addressed issue is the Gibbs energy of end-members in\nnon-stoichiometric compounds, i.e. , where each sublattice contains only\none element. In case all sublattices are occupied by the same element,\nit is the lattice stability of the elements in the structure of the\ncompound. Since the stable composition ranges of non-stoichiometric\ncompounds are typically small, the existing method cannot be used to\nreliably evaluate the Gibbs energy of end-members, and currently there\nis not a commonly accepted lattice stability database for compounds.\nMost values used in the existing databases have been either roughly\nestimated or computed from first-principles calculations. Such a\nstandard database is highly desirable in order to make various models of\ncompounds compatible.\n\nIn an effort to compare the lattice stability from the CALPHAD models\nand the first-principles calculations, Wang et al. \\[53\\] systematically\ncalculated the total energies of 78 pure elements at zero Kelvin in the\nface-centered-cubic (fcc), body-centered-cubic (bcc), and\nhexagonal-close-packed (hcp) crystal structures using the projector\naugmented-wave (PAW) method within the generalized gradient\napproximation (GGA). The calculated values are compared with the values\nin the SGTE database as shown in and . For non-transition metal\nelements, the differences between the SGTE data and the PAW-GGA data are\ntypically around 1\u223c2 kJ/mole-of-atoms or less, while for some transition\nmetal elements, the differences can be quite large, for example, as high\nas about 54 kJ/mole-of-atoms for and about 40 kJ/mole-of-atoms for . and\npresent the differences between the PAW-GGA data and the SGTE data, for\nelements from the Ti group to the Ni group, respectively.\n\nTable \u2011: Lattice stability Ebcc-fcc (kJ/mole-of-atoms).\n\nTable \u2011: Lattice stability Ehcp-fcc (kJ/mole-of-atoms).\n\nFigure \u2011: Lattice stability difference between bcc and fcc, , for\nselected elements between PAW-GGA and SGTE\n\nFigure \u2011: Lattice stability difference between hcp and fcc, , for\nselected elements between PAW-GGA and SGTE\n\nThe large differences between the first-principles calculations and the\nSGTE data could partly be attributed to the instability of the\nhigher-energy phases, the entropies of which at finite *T* become\nabnormal. The lattice instabilities along the tetragonal transformation\npath between fcc and bcc structures with the continuous change of the\nc/a ratio defined in a bcc-based tetragonal lattice are demonstrated for\nbcc Mo, Ta, W in and for fcc Al, Cu, Ni in . It is shown that the fcc\nstructure of bcc Mo, Ta, and W is a local maximum with respect to the\ntetragonal transformation, and the higher the maximum is, the larger the\ndiscrepancy between the SGTE data and the present PAW-GGA data, while\nfor fcc Al, Cu, Ni, the bcc structure is at a local maximum. Similarly,\nthe lattice instabilities along the tetragonal transformation path for\nthe hcp metals Ru and Os as shown in . The behavior of energy against\nc/a ratio of these two hcp metals is very similar to those of fcc\nelements.\n\nFigure \u2011: Total energy, , along the Bain deformation path between bcc\nand fcc for Mo, Ta, and W.\n\nFigure \u2011: Total energy, , along the tetragonal transformation path\nbetween bcc and fcc for Ni, Al, and Cu.\n\nFigure \u2011: Total energy, , along the tetragonal transformation path for\nRu and Os\n\nIt can be concluded that a fcc structure for elements with bcc being the\nground state or a bcc structure for elements with fcc being the ground\nstate, is unstable with respect to the tetragonal transformation. For an\nunstable structure, the harmonic description of its vibrational entropy\nis thermodynamically incorrect since the potential surface seen by the\nlattice ion can no longer be approximated by a parabola. If an unstable\nstructure of a pure element is stabilized at high temperatures, its\nentropy has to be abnormal. The instability issue has been recently\naddressed by ab initio molecular dynamics simulations at high\ntemperatures using W as an example \\[54\\], which is beyond the scope of\nthe book and thus not discussed here.\n", "metadata": {} } ] diff --git a/src/psu410/src/calpahd_modeling_of_thermodynamics/index.ipynb b/src/psu410/src/calpahd_modeling_of_thermodynamics/index.ipynb index 4037370..affcd41 100644 --- a/src/psu410/src/calpahd_modeling_of_thermodynamics/index.ipynb +++ b/src/psu410/src/calpahd_modeling_of_thermodynamics/index.ipynb @@ -4,55 +4,9 @@ "metadata": {}, "cells": [ { - "id": "58cfeeff", + "id": "543d5219", "cell_type": "markdown", - "source": [ - "# CALPAHD modeling of thermodynamics\n", - "\n", - "CALPHAD modeling of thermodynamics was pioneered by Kaufman \\[47\\] and\n", - "has been reviewed in details by Saunders and Miodownik \\[48\\] and Lukas,\n", - "Fries and Sundman \\[49\\]. Information on features of software tools for\n", - "CALPHAD modeling can be found at two series of publications in the\n", - "CALPHAD journal \\[50-51\\]. The key feature of the CALPHAD method is the\n", - "modeling of Gibbs energy of individual phases using both thermodynamic\n", - "and phase equilibrium data. The main significances of the CALPHAD method\n", - "are as follows\n", - "\n", - "1. It enabled the development of the concept of lattice stability, i.e.\n", - " the energy difference of a pure element with stable and non-stable\n", - " crystal structures;\n", - "\n", - "2. The Gibbs energy expression of each phase covers the full\n", - " temperature, pressure, and composition spaces including both stable\n", - " and non-stable regions of the phase. This enables the evaluation of\n", - " the Gibbs energy of a system as a function of non-equilibrium state,\n", - " i.e. with \u03be as an independent variable;\n", - "\n", - "3. Thermodynamic data are usually obtained by measurements of heat such\n", - " as enthalpy of transition and heat capacity as discussed in Chapter\n", - " , which bear large uncertainties typically in the range of\n", - " kilojoules per mole-of-atom. On the other hand, phase equilibrium\n", - " data as discussed in Chapter , though more accurate, only contain\n", - " information on compositions of phases at equilibria, i.e., the\n", - " relative Gibbs energy of phases at equilibrium. The combination of\n", - " these two sets of data is foundational in CALPHAD modeling that\n", - " allows for accurate modeling of thermodynamic properties of\n", - " individual phases and reliable calculations of phase stability and\n", - " driving forces;\n", - "\n", - "4. It provides a framework to model thermodynamic properties of\n", - " multi-component systems of industrial importance, enabling the\n", - " computational materials design. It has also been extended to model a\n", - " range of properties of individual phases in multi-component systems\n", - " such as diffusion coefficients, elastic coefficients, and thermal\n", - " expansion, supplying input data for computational simulations of\n", - " phase transformations during materials processing.\n", - "\n", - "In this chapter, the basics of CALPHAD modeling of Gibbs energy of\n", - "individual phases are presented. For detailed implementations in various\n", - "software packages and modeling procedures, readers are referred to the\n", - "references listed above.\n" - ], + "source": "# CALPAHD modeling of thermodynamics\n\nCALPHAD modeling of thermodynamics was pioneered by Kaufman \\[47\\] and\nhas been reviewed in details by Saunders and Miodownik \\[48\\] and Lukas,\nFries and Sundman \\[49\\]. Information on features of software tools for\nCALPHAD modeling can be found at two series of publications in the\nCALPHAD journal \\[50-51\\]. The key feature of the CALPHAD method is the\nmodeling of Gibbs energy of individual phases using both thermodynamic\nand phase equilibrium data. The main significances of the CALPHAD method\nare as follows\n\n1. It enabled the development of the concept of lattice stability, i.e.\n the energy difference of a pure element with stable and non-stable\n crystal structures;\n\n2. The Gibbs energy expression of each phase covers the full\n temperature, pressure, and composition spaces including both stable\n and non-stable regions of the phase. This enables the evaluation of\n the Gibbs energy of a system as a function of non-equilibrium state,\n i.e. with \u03be as an independent variable;\n\n3. Thermodynamic data are usually obtained by measurements of heat such\n as enthalpy of transition and heat capacity as discussed in Chapter\n , which bear large uncertainties typically in the range of\n kilojoules per mole-of-atom. On the other hand, phase equilibrium\n data as discussed in Chapter , though more accurate, only contain\n information on compositions of phases at equilibria, i.e., the\n relative Gibbs energy of phases at equilibrium. The combination of\n these two sets of data is foundational in CALPHAD modeling that\n allows for accurate modeling of thermodynamic properties of\n individual phases and reliable calculations of phase stability and\n driving forces;\n\n4. It provides a framework to model thermodynamic properties of\n multi-component systems of industrial importance, enabling the\n computational materials design. It has also been extended to model a\n range of properties of individual phases in multi-component systems\n such as diffusion coefficients, elastic coefficients, and thermal\n expansion, supplying input data for computational simulations of\n phase transformations during materials processing.\n\nIn this chapter, the basics of CALPHAD modeling of Gibbs energy of\nindividual phases are presented. For detailed implementations in various\nsoftware packages and modeling procedures, readers are referred to the\nreferences listed above.\n", "metadata": {} } ] diff --git a/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_pure_elements.ipynb b/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_pure_elements.ipynb index 36b8aa5..be1152e 100644 --- a/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_pure_elements.ipynb +++ b/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_pure_elements.ipynb @@ -4,72 +4,9 @@ "metadata": {}, "cells": [ { - "id": "04834bb0", + "id": "54cc29d0", "cell_type": "markdown", - "source": [ - "## Modeling of pure elements\n", - "\n", - "In modeling the Gibbs energy of pure elements in the SER structure in\n", - "terms of , the coefficients in are evaluated using the heat capacity\n", - "data, $b^{'}$ in is evaluated by the value of $S_{298.15}$, and $a$ in\n", - "is evaluated by $H_{298.15}^{SER} = 0$, respectively. For the high\n", - "temperature phase, the enthalpy of transformation from the low\n", - "temperature phase to the high temperature phase,\n", - "${\\mathrm{\\Delta}H}_{trans}^{}$, can be measured by calorimetry methods\n", - "discussed in Chapter , and the entropy of transformation,\n", - "${\\mathrm{\\Delta}S}_{trans}^{}$, is then calculated using the\n", - "equilibrium condition of equal Gibbs energy of the two phases, i.e.\n", - "\n", - "*Eq. 6\u20111*\n", - "${\\mathrm{\\Delta}S}_{trans}^{} = \\frac{{\\mathrm{\\Delta}H}_{trans}^{}}{T_{trans}^{}}$\n", - "\n", - "where $T_{trans}^{}$ is the transition temperature.\n", - "${\\mathrm{\\Delta}H}_{trans}^{}$ and ${\\mathrm{\\Delta}S}_{trans}^{}$ are\n", - "then used to evaluate the integration constants, $b^{'}$ and $a$, in the\n", - "place of $S_{298.15}$ and $H_{298.15}^{SER}$ for the structure in the\n", - "SER state.\n", - "\n", - "This works well for the stable temperature range of each phase. However,\n", - "there is an issue in extrapolation above and below the melting\n", - "temperature (Tm). It is known that the heat capacity of the\n", - "solid phase, , increases with temperature, while that of the liquid\n", - "phase, , is typically constant. The extrapolation of the Gibbs energy of\n", - "a solid phase to above its melting temperature can result in the solid\n", - "phase becoming more stable than the liquid phase at high temperatures.\n", - "By the same token, the extrapolation of the Gibbs energy of a liquid\n", - "phase to below its melting temperature can result in the liquid phase\n", - "becoming more stable than the solid phase at low temperatures. To\n", - "address this problem, it is proposed by SGTE that the heat capacity of\n", - "the solid phase approaches that of the liquid at high temperatures, and\n", - "that of the liquid phase approaches that of the solid phase at low\n", - "temperatures using the following equations\n", - "\n", - "1. for solid at T\\>Tm\n", - "\n", - "*Eq. 6\u20112*\n", - "\n", - "*Eq. 6\u20113*\n", - "\n", - "2. for liquid at T\\m\n", - "\n", - "*Eq. 6\u20114*\n", - "\n", - "*Eq. 6\u20115*\n", - "\n", - "As an example, the heat capacity of solid fcc Al and liquid Al in the\n", - "SGTE pure element database is plotted in . It can be seen that the heat\n", - "capacity of fcc Al approaches that of liquid Al at high temperatures,\n", - "while the heat capacity of liquid Al approaches that of fcc Al at low\n", - "temperatures. It ensures that the liquid Al is stable at high\n", - "temperatures, and fcc Al is stable at low temperatures. However, this\n", - "simple model for liquid is often not satisfactory, in comparison with\n", - "available experimental data in supercooled liquid, particularly those\n", - "systems with glass transitions. New models are thus needed and are being\n", - "developed in the CALPHAD community.\n", - "\n", - "Figure \u2011: Heat capacity of fcc Al solid and liquid as a function of\n", - "temperature\n" - ], + "source": "## Modeling of pure elements\n\nIn modeling the Gibbs energy of pure elements in the SER structure in\nterms of , the coefficients in are evaluated using the heat capacity\ndata, $b^{'}$ in is evaluated by the value of $S_{298.15}$, and $a$ in\nis evaluated by $H_{298.15}^{SER} = 0$, respectively. For the high\ntemperature phase, the enthalpy of transformation from the low\ntemperature phase to the high temperature phase,\n${\\mathrm{\\Delta}H}_{trans}^{}$, can be measured by calorimetry methods\ndiscussed in Chapter , and the entropy of transformation,\n${\\mathrm{\\Delta}S}_{trans}^{}$, is then calculated using the\nequilibrium condition of equal Gibbs energy of the two phases, i.e.\n\n*Eq. 6\u20111*\n${\\mathrm{\\Delta}S}_{trans}^{} = \\frac{{\\mathrm{\\Delta}H}_{trans}^{}}{T_{trans}^{}}$\n\nwhere $T_{trans}^{}$ is the transition temperature.\n${\\mathrm{\\Delta}H}_{trans}^{}$ and ${\\mathrm{\\Delta}S}_{trans}^{}$ are\nthen used to evaluate the integration constants, $b^{'}$ and $a$, in the\nplace of $S_{298.15}$ and $H_{298.15}^{SER}$ for the structure in the\nSER state.\n\nThis works well for the stable temperature range of each phase. However,\nthere is an issue in extrapolation above and below the melting\ntemperature (Tm). It is known that the heat capacity of the\nsolid phase, , increases with temperature, while that of the liquid\nphase, , is typically constant. The extrapolation of the Gibbs energy of\na solid phase to above its melting temperature can result in the solid\nphase becoming more stable than the liquid phase at high temperatures.\nBy the same token, the extrapolation of the Gibbs energy of a liquid\nphase to below its melting temperature can result in the liquid phase\nbecoming more stable than the solid phase at low temperatures. To\naddress this problem, it is proposed by SGTE that the heat capacity of\nthe solid phase approaches that of the liquid at high temperatures, and\nthat of the liquid phase approaches that of the solid phase at low\ntemperatures using the following equations\n\n1. for solid at T\\>Tm\n\n*Eq. 6\u20112*\n\n*Eq. 6\u20113*\n\n2. for liquid at T\\m\n\n*Eq. 6\u20114*\n\n*Eq. 6\u20115*\n\nAs an example, the heat capacity of solid fcc Al and liquid Al in the\nSGTE pure element database is plotted in . It can be seen that the heat\ncapacity of fcc Al approaches that of liquid Al at high temperatures,\nwhile the heat capacity of liquid Al approaches that of fcc Al at low\ntemperatures. It ensures that the liquid Al is stable at high\ntemperatures, and fcc Al is stable at low temperatures. However, this\nsimple model for liquid is often not satisfactory, in comparison with\navailable experimental data in supercooled liquid, particularly those\nsystems with glass transitions. New models are thus needed and are being\ndeveloped in the CALPHAD community.\n\nFigure \u2011: Heat capacity of fcc Al solid and liquid as a function of\ntemperature\n", "metadata": {} } ] diff --git a/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_random_solution_phases.ipynb b/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_random_solution_phases.ipynb index 7789f29..642b5db 100644 --- a/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_random_solution_phases.ipynb +++ b/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_random_solution_phases.ipynb @@ -4,59 +4,9 @@ "metadata": {}, "cells": [ { - "id": "420083e9", + "id": "a262dc50", "cell_type": "markdown", - "source": [ - "## Modeling of random solution phases\n", - "\n", - "Depending on the degree of short-range ordering in a solution phase,\n", - "various Gibbs energy models are available as discussed in Chapter . When\n", - "the short-range ordering is weak, it can be accounted for by the\n", - "composition-dependence of excess Gibbs energy in a binary system in\n", - "terms of the Redlich-Kister polynomial as follows\n", - "\n", - "*Eq. 6\u20117*\n", - "\n", - "where the interaction parameters, , can be temperature dependent or even\n", - "have contributions from heat capacity in the form of when data is\n", - "available. shows that and are symmetrical with respect of composition,\n", - "while is asymmetrical. Their individual contributions to the excess\n", - "Gibbs energy are shown in with all interaction parameters being -30,000\n", - "J/mole-of-atoms.\n", - "\n", - "Figure \u2011: Contributions of interactions parameters to the excess Gibbs\n", - "energy\n", - "\n", - "It can be seen in that even though all interaction parameters are\n", - "negative, the asymmetrical feature of results in the curvature change in\n", - "the excess Gibbs energy as a function of composition. This indicates the\n", - "tendency to form a miscibility gap at low temperatures. The interaction\n", - "parameters are to be evaluated from the data of enthalpy, entropy, and\n", - "heat capacity of mixing. The experimental data on enthalpy of mixing are\n", - "available for the liquid phase in some systems, but typically very\n", - "limited for solid solution phases. The first-principles calculations can\n", - "predict the enthalpy, entropy, and heat capacity of mixing in solid\n", - "solution phases using the dilute solution approach with one solute atom\n", - "in a supercell and the CPA/CE/SQS approach for concentrated solutions as\n", - "discussed in Chapter . It demonstrates again that the interaction\n", - "parameters and the lattice stability jointly determine the Gibbs energy\n", - "of an individual phase. The change of lattice stability requires the\n", - "re-evaluation of interaction parameters.\n", - "\n", - "For individual phases with strong short-range ordering, quasichemical or\n", - "associated models can be used. As discussed in Chapter , with fixed\n", - "composition in the system, the amounts of various chemical bonds or\n", - "associates are related through the mass conservation in the system and\n", - "are calculated through the minimization of Gibbs energy of the phase\n", - "with given temperature, pressure, and the amount of each independent\n", - "components. The model parameters include the formation energy of bonds\n", - "or associates and interaction between various bonds or associates,\n", - "noting that pure elements can be considered as the simplest associates.\n", - "The interactions between pure elements can be predicted from\n", - "first-principles calculations as mentioned above, but currently there\n", - "are no efficient approaches to predict the interactions between\n", - "associates from first-principles calculations.\n" - ], + "source": "## Modeling of random solution phases\n\nDepending on the degree of short-range ordering in a solution phase,\nvarious Gibbs energy models are available as discussed in Chapter . When\nthe short-range ordering is weak, it can be accounted for by the\ncomposition-dependence of excess Gibbs energy in a binary system in\nterms of the Redlich-Kister polynomial as follows\n\n*Eq. 6\u20117*\n\nwhere the interaction parameters, , can be temperature dependent or even\nhave contributions from heat capacity in the form of when data is\navailable. shows that and are symmetrical with respect of composition,\nwhile is asymmetrical. Their individual contributions to the excess\nGibbs energy are shown in with all interaction parameters being -30,000\nJ/mole-of-atoms.\n\nFigure \u2011: Contributions of interactions parameters to the excess Gibbs\nenergy\n\nIt can be seen in that even though all interaction parameters are\nnegative, the asymmetrical feature of results in the curvature change in\nthe excess Gibbs energy as a function of composition. This indicates the\ntendency to form a miscibility gap at low temperatures. The interaction\nparameters are to be evaluated from the data of enthalpy, entropy, and\nheat capacity of mixing. The experimental data on enthalpy of mixing are\navailable for the liquid phase in some systems, but typically very\nlimited for solid solution phases. The first-principles calculations can\npredict the enthalpy, entropy, and heat capacity of mixing in solid\nsolution phases using the dilute solution approach with one solute atom\nin a supercell and the CPA/CE/SQS approach for concentrated solutions as\ndiscussed in Chapter . It demonstrates again that the interaction\nparameters and the lattice stability jointly determine the Gibbs energy\nof an individual phase. The change of lattice stability requires the\nre-evaluation of interaction parameters.\n\nFor individual phases with strong short-range ordering, quasichemical or\nassociated models can be used. As discussed in Chapter , with fixed\ncomposition in the system, the amounts of various chemical bonds or\nassociates are related through the mass conservation in the system and\nare calculated through the minimization of Gibbs energy of the phase\nwith given temperature, pressure, and the amount of each independent\ncomponents. The model parameters include the formation energy of bonds\nor associates and interaction between various bonds or associates,\nnoting that pure elements can be considered as the simplest associates.\nThe interactions between pure elements can be predicted from\nfirst-principles calculations as mentioned above, but currently there\nare no efficient approaches to predict the interactions between\nassociates from first-principles calculations.\n", "metadata": {} } ] diff --git a/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_solution_phases_with_longrange_ordering.ipynb b/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_solution_phases_with_longrange_ordering.ipynb index eeaf83f..0dfae13 100644 --- a/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_solution_phases_with_longrange_ordering.ipynb +++ b/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_solution_phases_with_longrange_ordering.ipynb @@ -4,166 +4,9 @@ "metadata": {}, "cells": [ { - "id": "73e38aa7", + "id": "de1cecb8", "cell_type": "markdown", - "source": [ - "## Modeling of solution phases with long-range ordering\n", - "\n", - "The commonly used Gibbs energy model is shown in Chapter with the\n", - "crystal lattice divided into sublattices, often referred to as the\n", - "compound energy formalism \\[55\\]. The Gibbs energy of end-members\n", - "represented by plays the same important role for solution phases with\n", - "sublattice as the lattice stability for random solution phases. The\n", - "end-members are modeled in the same way as the stoichiometric phases\n", - "discussed in Chapter . The enthalpy and entropy of mixing in each\n", - "sublattice can be predicted by first-principles calculations using the\n", - "dilute solution and SQS approaches discussed in Chapter and modeled in\n", - "the same way as the random solution discussed in Chapter .\n", - "\n", - "It is important to realize that with a simple two-sublattice model of\n", - "(A,B)a(C,D)b, the miscibility gap can easily form\n", - "even without any interaction parameters when the Gibbs energy of\n", - "end-members differ from each other significantly. The contribution of\n", - "end-members to the Gibbs energy of the phase, i.e. , is re-written as\n", - "follows and schematically shown in ,\n", - "\n", - "*Eq. 6\u20118*\n", - "\n", - "Figure \u2011: Schematic diagrams depicting (a) the concentration square with\n", - "the site fractions of B and D on the horizontal and vertical axes,\n", - "respectively, and (b) Gibbs energy reference plan for\n", - "(A,B)a(C,D)b, as represented by .\n", - "\n", - "From b, it is evident that there is a strong tendency to form a\n", - "miscibility gap between the composition sets of\n", - "(A)a(C)b and (B)a(D)b due to\n", - "their lower Gibbs energies than the other two end-members. Since it\n", - "would be rare for all four end-members to have their Gibbs energy values\n", - "equal, the miscibility gap in this type of phases is practically\n", - "inevitable at low temperatures. An example is shown in for the complex\n", - "titanium niobium carbonitride (Ti,Nb)(C,N). The solid lines parallel to\n", - "the direction from NbC to TiN are tie-lines. The Gibbs energy values of\n", - "TiC, TiN, NbC and NbN are -144495, -229236, -132324, and -179772\n", - "J/mole-of-atoms, respectively. The Gibbs energy value of TiN is\n", - "significantly lower than the other values, resulting in the tie-lines\n", - "originating from the TiN corner.\n", - "\n", - "Figure \u2011: Miscibility gap in (Ti,Nb)(C,N) at 1673K with the straight\n", - "lines in the middle of the plot being tie-lines.\n", - "\n", - "The order-disorder transitions can be similarly described with the\n", - "simplest case being a two-sublattice model of\n", - "(A,B)a(A,B)b. When the site fractions of A or B in\n", - "both sublattices are the same, it becomes a random solution model; when\n", - "they are different, the phase is partially ordered; and when there is\n", - "only one component in each sublattice, the phase is fully ordered as a\n", - "stoichiometric compound. The Gibbs energy of this phase is obtained from\n", - "as follows\n", - "\n", - "*Eq. 6\u20119*\n", - "\n", - "$G_{mf} = y_{A}^{I}y_{A}^{II}\\ _{\\ }^{0}G_{A:A} + y_{A}^{I}y_{B}^{II}\\ _{\\ }^{0}G_{A:B} + y_{B}^{I}y_{A}^{II}\\ _{\\ }^{0}G_{B:A} + y_{B}^{I}y_{B}^{II}\\ _{\\ }^{0}G_{B:B} + + aRT\\left( y_{A}^{I}\\ln y_{A}^{I} + y_{B}^{I}\\ln y_{B}^{I} \\right) + bRT\\left( y_{A}^{II}\\ln y_{A}^{II} + y_{B}^{II}\\ln y_{B}^{II} \\right) + y_{A}^{II}y_{A}^{I}y_{B}^{I}\\ L_{A,B:A} + y_{B}^{II}y_{A}^{I}y_{B}^{I}\\ L_{A,B:B} + y_{A}^{I}y_{A}^{II}y_{B}^{II}\\ L_{A:A,B} + y_{B}^{I}y_{A}^{II}y_{B}^{II}\\ L_{B:A,B} + y_{A}^{I}y_{B}^{I}\\ y_{A}^{II}y_{B}^{II}L_{A,B:A,B}$\n", - "\n", - "where the superscript denotes the sublattice, and column and comma\n", - "separate sublattices and interaction components, respectively. The\n", - "relationship between site fraction and over-all atomic fractions in such\n", - "a two-sublattice model can be represented by and schematically shown in\n", - ". The two red dashed lines represent the phase with\n", - "$x_{B} = \\frac{a}{a + b}$, but different a/b ratios. Along the red\n", - "dashed lines, the phase can adjust the site fraction to minimize its\n", - "Gibbs energy, i.e. it has one internal degree of freedom to be either\n", - "disordered on the blue diagonal line between A:A and B:B, or ordered at\n", - "anywhere else. The interplay of interaction parameters and site\n", - "fractions is depicted where $L_{A,B:A}$, $L_{A,B:B}$, $L_{A:A,B}$, and\n", - "$L_{B:A,B}$ affect the four sides, and $L_{A,B:A,B}$ influences the\n", - "center part.\n", - "\n", - "Figure \u2011: Schematic composition square of\n", - "(A,B)a(A,B)b.\n", - "\n", - "When fully disordered with $y_{A}^{I} = \\ y_{A}^{II} = x_{A}$ and\n", - "$y_{B}^{I} = y_{B}^{II} = x_{B}$, becomes\n", - "\n", - "*Eq. 6\u201110*\n", - "\n", - "$$G_{mf} = x_{A}\\left( 1 - x_{B} \\right)_{\\ }^{0}G_{A:A} + x_{A}x_{B}\\ _{\\ }^{0}G_{A:B} + x_{A}x_{B}\\ _{\\ }^{0}G_{B:A} + x_{B}\\left( 1 - x_{A} \\right)\\ _{\\ }^{0}G_{B:B} + + (a + b)RT\\left( x_{A}\\ln x_{A} + x_{B}\\ln x_{B} \\right) + \\ x_{A}x_{B}x_{A}L_{A,B:A} + x_{A}x_{B}x_{B}\\ L_{A,B:B} + x_{A}x_{B}x_{A}\\ L_{A:A,B} + x_{A}x_{B}x_{B}\\ L_{B:A,B} + x_{A}x_{A}x_{B}x_{B}L_{A,B:A,B} = x_{A}_{\\ }^{0}G_{A:A} + x_{B}\\ _{\\ }^{0}G_{B:B} + (a + b)RT\\left( x_{A}\\ln x_{A} + x_{B}\\ln x_{B} \\right) + x_{A}x_{B}\\left\\lbrack \\left(_{\\ }^{0}G_{A:B} +_{\\ }^{0}G_{B:A} -_{\\ }^{0}G_{A:A} -_{\\ }^{0}G_{B:B} \\right) + x_{A}\\left( L_{A,B:A} + L_{A:A,B} \\right) + x_{B}\\ \\left( L_{A,B:B} + L_{B:A,B} \\right) + x_{A}x_{B}L_{A,B:A,B} \\right\\rbrack = (a + b)\\left\\lbrack x_{A}_{\\ }^{0}G_{A} + x_{B}\\ _{\\ }^{0}G_{B} + RT\\left( x_{A}\\ln x_{A} + x_{B}\\ln x_{B} \\right) + x_{A}x_{B}L_{A,B} \\right\\rbrack$$\n", - "\n", - "with\n", - "\n", - "*Eq. 6\u201111* $_{\\ }^{0}G_{A:A} = (a + b)_{\\ }^{0}G_{A}$\n", - "\n", - "*Eq. 6\u201112* $_{\\ }^{0}G_{B:B} = (a + b)_{\\ }^{0}G_{B}$\n", - "\n", - "*Eq. 6\u201113*\n", - "\n", - "$$L_{A,B} = \\left\\lbrack \\left(_{\\ }^{0}G_{A:B} +_{\\ }^{0}G_{B:A} -_{\\ }^{0}G_{A:A} -_{\\ }^{0}G_{B:B} \\right) + x_{A}\\left( L_{A,B:A} + L_{A:A,B} \\right) + x_{B}\\ \\left( L_{A,B:B} + L_{B:A,B} \\right) + x_{A}x_{B}L_{A,B:A,B} \\right\\rbrack/(a + b)$$\n", - "\n", - "where $_{\\ }^{0}G_{A}$, $_{\\ }^{0}G_{B}$, and $L_{A,B}$ are the molar\n", - "Gibbs energy of pure A and B and the molar interaction parameter in the\n", - "disordered solid solution, respectively. It is evident that the\n", - "interatcation paramter $L_{A,B}$ is fully determined by the parameters\n", - "in the ordered phase, but the parameters in the ordered phase are not\n", - "uniquely determined by the interaction parameters in the disordered\n", - "phase.\n", - "\n", - "Due to crystal symmetry, some of the parameters in are related. For\n", - "example, in the BCC A2/B2 ordering with a=b=0.5, the BCC corner or\n", - "center lattice sites are favored by either one type of atoms, but the\n", - "two sublattices are equivalent crystallographically, resulting in\n", - "following relations\n", - "\n", - "*Eq. 6\u201114* $\\ _{\\ }^{0}G_{A:B} = \\ _{\\ }^{0}G_{B:A}$\n", - "\n", - "*Eq. 6\u201115* $L_{A,B:A} = L_{A:A,B}$\n", - "\n", - "*Eq. 6\u201116* $L_{A,B:B} = \\ L_{B:A,B}$\n", - "\n", - "For more complex ordering of BCC lattice such as B32, D03,\n", - "and L21 shown in with ideal compositions being AB,\n", - "A3B, and A2BC, respectively, more sublattices are\n", - "needed, noting that the L21 Heusler structure exists in\n", - "ternary systems only. To use one model to describe all ordering in the\n", - "BCC lattice, the minimum cluster is an irregular tetrahedron with four\n", - "sublattices as depicted in as discussed in the modeling of the Al-Fe\n", - "system \\[56\\]. In such a four sublattice model of\n", - "$(A,B)_{0.25}^{I}(A,B)_{0.25}^{II}(A,B)_{0.25}^{II}(A,B)_{0.25}^{IV}$,\n", - "the site fractions of A2, B2, B32, and D03, are represented\n", - "by $y_{i}^{I} = \\ y_{i}^{II} = y_{i}^{III} = \\ y_{i}^{IV}$,\n", - "$y_{i}^{I} = \\ y_{i}^{II} \\neq y_{i}^{III} = \\ y_{i}^{IV}$,\n", - "$y_{i}^{I} = \\ y_{i}^{III} \\neq y_{i}^{II} = \\ y_{i}^{IV}$, and\n", - "$y_{i}^{I} = \\ y_{i}^{II} \\neq y_{i}^{III} \\neq \\ y_{i}^{IV}$,\n", - "respectively. The site fractions of L21 are the same as those\n", - "of D03 except with at least three components.\n", - "\n", - "Figure \u2011: Atomic structures and four sublattice tetrahedrons of BCC\n", - "disordered and ordered phases.\n", - "\n", - "Another common ordering phenomenon is in the FCC lattice including the\n", - "disordered A1 structure and ordered L10 and L12\n", - "structures as shown in . In the L10 structure, the\n", - "neighboring (001) planes are favored by different atoms, respectively,\n", - "resulting in an ideal composition of AB. While in the L12\n", - "structure, the corners and faces are favored by different atoms,\n", - "respectively, resulting in an ideal composition of A3B. In a\n", - "four sublattice model of\n", - "$(A,B)_{0.25}^{I}(A,B)_{0.25}^{II}(A,B)_{0.25}^{II}(A,B)_{0.25}^{IV}$,\n", - "the site fractions of A1, L10 and L12 are\n", - "represented by $y_{i}^{I} = \\ y_{i}^{II} = y_{i}^{III} = \\ y_{i}^{IV}$,\n", - "$y_{i}^{I} = \\ y_{i}^{II} \\neq y_{i}^{III} = \\ y_{i}^{IV}$, and\n", - "$y_{i}^{I} = \\ y_{i}^{II} = y_{i}^{III} \\neq \\ y_{i}^{IV}$, respectively\n", - "\\[57\\].\n", - "\n", - "Figure \u2011: Atomic structures and four sublattice tetrahedrons of FCC\n", - "disordered and ordered phases.\n", - "\n", - "As mentioned above, the interaction parameters in each sublattice can be\n", - "predicted by first-principles calculations using the dilute solution and\n", - "SQS approaches when there is only one component in each of the remaining\n", - "sublattices. For the interaction involving two components in two or more\n", - "sublattices, i.e. $L_{A,B:A,B}$ in applicable to four sublattice models\n", - "\\[58\\], the energetics from cluster expansion (CE) approach discussed in\n", - "Chapter can be used to evaluate the interaction parameters.\n" - ], + "source": "## Modeling of solution phases with long-range ordering\n\nThe commonly used Gibbs energy model is shown in Chapter with the\ncrystal lattice divided into sublattices, often referred to as the\ncompound energy formalism \\[55\\]. The Gibbs energy of end-members\nrepresented by plays the same important role for solution phases with\nsublattice as the lattice stability for random solution phases. The\nend-members are modeled in the same way as the stoichiometric phases\ndiscussed in Chapter . The enthalpy and entropy of mixing in each\nsublattice can be predicted by first-principles calculations using the\ndilute solution and SQS approaches discussed in Chapter and modeled in\nthe same way as the random solution discussed in Chapter .\n\nIt is important to realize that with a simple two-sublattice model of\n(A,B)a(C,D)b, the miscibility gap can easily form\neven without any interaction parameters when the Gibbs energy of\nend-members differ from each other significantly. The contribution of\nend-members to the Gibbs energy of the phase, i.e. , is re-written as\nfollows and schematically shown in ,\n\n*Eq. 6\u20118*\n\nFigure \u2011: Schematic diagrams depicting (a) the concentration square with\nthe site fractions of B and D on the horizontal and vertical axes,\nrespectively, and (b) Gibbs energy reference plan for\n(A,B)a(C,D)b, as represented by .\n\nFrom b, it is evident that there is a strong tendency to form a\nmiscibility gap between the composition sets of\n(A)a(C)b and (B)a(D)b due to\ntheir lower Gibbs energies than the other two end-members. Since it\nwould be rare for all four end-members to have their Gibbs energy values\nequal, the miscibility gap in this type of phases is practically\ninevitable at low temperatures. An example is shown in for the complex\ntitanium niobium carbonitride (Ti,Nb)(C,N). The solid lines parallel to\nthe direction from NbC to TiN are tie-lines. The Gibbs energy values of\nTiC, TiN, NbC and NbN are -144495, -229236, -132324, and -179772\nJ/mole-of-atoms, respectively. The Gibbs energy value of TiN is\nsignificantly lower than the other values, resulting in the tie-lines\noriginating from the TiN corner.\n\nFigure \u2011: Miscibility gap in (Ti,Nb)(C,N) at 1673K with the straight\nlines in the middle of the plot being tie-lines.\n\nThe order-disorder transitions can be similarly described with the\nsimplest case being a two-sublattice model of\n(A,B)a(A,B)b. When the site fractions of A or B in\nboth sublattices are the same, it becomes a random solution model; when\nthey are different, the phase is partially ordered; and when there is\nonly one component in each sublattice, the phase is fully ordered as a\nstoichiometric compound. The Gibbs energy of this phase is obtained from\nas follows\n\n*Eq. 6\u20119*\n\n$G_{mf} = y_{A}^{I}y_{A}^{II}\\ _{\\ }^{0}G_{A:A} + y_{A}^{I}y_{B}^{II}\\ _{\\ }^{0}G_{A:B} + y_{B}^{I}y_{A}^{II}\\ _{\\ }^{0}G_{B:A} + y_{B}^{I}y_{B}^{II}\\ _{\\ }^{0}G_{B:B} + + aRT\\left( y_{A}^{I}\\ln y_{A}^{I} + y_{B}^{I}\\ln y_{B}^{I} \\right) + bRT\\left( y_{A}^{II}\\ln y_{A}^{II} + y_{B}^{II}\\ln y_{B}^{II} \\right) + y_{A}^{II}y_{A}^{I}y_{B}^{I}\\ L_{A,B:A} + y_{B}^{II}y_{A}^{I}y_{B}^{I}\\ L_{A,B:B} + y_{A}^{I}y_{A}^{II}y_{B}^{II}\\ L_{A:A,B} + y_{B}^{I}y_{A}^{II}y_{B}^{II}\\ L_{B:A,B} + y_{A}^{I}y_{B}^{I}\\ y_{A}^{II}y_{B}^{II}L_{A,B:A,B}$\n\nwhere the superscript denotes the sublattice, and column and comma\nseparate sublattices and interaction components, respectively. The\nrelationship between site fraction and over-all atomic fractions in such\na two-sublattice model can be represented by and schematically shown in\n. The two red dashed lines represent the phase with\n$x_{B} = \\frac{a}{a + b}$, but different a/b ratios. Along the red\ndashed lines, the phase can adjust the site fraction to minimize its\nGibbs energy, i.e. it has one internal degree of freedom to be either\ndisordered on the blue diagonal line between A:A and B:B, or ordered at\nanywhere else. The interplay of interaction parameters and site\nfractions is depicted where $L_{A,B:A}$, $L_{A,B:B}$, $L_{A:A,B}$, and\n$L_{B:A,B}$ affect the four sides, and $L_{A,B:A,B}$ influences the\ncenter part.\n\nFigure \u2011: Schematic composition square of\n(A,B)a(A,B)b.\n\nWhen fully disordered with $y_{A}^{I} = \\ y_{A}^{II} = x_{A}$ and\n$y_{B}^{I} = y_{B}^{II} = x_{B}$, becomes\n\n*Eq. 6\u201110*\n\n$$G_{mf} = x_{A}\\left( 1 - x_{B} \\right)_{\\ }^{0}G_{A:A} + x_{A}x_{B}\\ _{\\ }^{0}G_{A:B} + x_{A}x_{B}\\ _{\\ }^{0}G_{B:A} + x_{B}\\left( 1 - x_{A} \\right)\\ _{\\ }^{0}G_{B:B} + + (a + b)RT\\left( x_{A}\\ln x_{A} + x_{B}\\ln x_{B} \\right) + \\ x_{A}x_{B}x_{A}L_{A,B:A} + x_{A}x_{B}x_{B}\\ L_{A,B:B} + x_{A}x_{B}x_{A}\\ L_{A:A,B} + x_{A}x_{B}x_{B}\\ L_{B:A,B} + x_{A}x_{A}x_{B}x_{B}L_{A,B:A,B} = x_{A}_{\\ }^{0}G_{A:A} + x_{B}\\ _{\\ }^{0}G_{B:B} + (a + b)RT\\left( x_{A}\\ln x_{A} + x_{B}\\ln x_{B} \\right) + x_{A}x_{B}\\left\\lbrack \\left(_{\\ }^{0}G_{A:B} +_{\\ }^{0}G_{B:A} -_{\\ }^{0}G_{A:A} -_{\\ }^{0}G_{B:B} \\right) + x_{A}\\left( L_{A,B:A} + L_{A:A,B} \\right) + x_{B}\\ \\left( L_{A,B:B} + L_{B:A,B} \\right) + x_{A}x_{B}L_{A,B:A,B} \\right\\rbrack = (a + b)\\left\\lbrack x_{A}_{\\ }^{0}G_{A} + x_{B}\\ _{\\ }^{0}G_{B} + RT\\left( x_{A}\\ln x_{A} + x_{B}\\ln x_{B} \\right) + x_{A}x_{B}L_{A,B} \\right\\rbrack$$\n\nwith\n\n*Eq. 6\u201111* $_{\\ }^{0}G_{A:A} = (a + b)_{\\ }^{0}G_{A}$\n\n*Eq. 6\u201112* $_{\\ }^{0}G_{B:B} = (a + b)_{\\ }^{0}G_{B}$\n\n*Eq. 6\u201113*\n\n$$L_{A,B} = \\left\\lbrack \\left(_{\\ }^{0}G_{A:B} +_{\\ }^{0}G_{B:A} -_{\\ }^{0}G_{A:A} -_{\\ }^{0}G_{B:B} \\right) + x_{A}\\left( L_{A,B:A} + L_{A:A,B} \\right) + x_{B}\\ \\left( L_{A,B:B} + L_{B:A,B} \\right) + x_{A}x_{B}L_{A,B:A,B} \\right\\rbrack/(a + b)$$\n\nwhere $_{\\ }^{0}G_{A}$, $_{\\ }^{0}G_{B}$, and $L_{A,B}$ are the molar\nGibbs energy of pure A and B and the molar interaction parameter in the\ndisordered solid solution, respectively. It is evident that the\ninteratcation paramter $L_{A,B}$ is fully determined by the parameters\nin the ordered phase, but the parameters in the ordered phase are not\nuniquely determined by the interaction parameters in the disordered\nphase.\n\nDue to crystal symmetry, some of the parameters in are related. For\nexample, in the BCC A2/B2 ordering with a=b=0.5, the BCC corner or\ncenter lattice sites are favored by either one type of atoms, but the\ntwo sublattices are equivalent crystallographically, resulting in\nfollowing relations\n\n*Eq. 6\u201114* $\\ _{\\ }^{0}G_{A:B} = \\ _{\\ }^{0}G_{B:A}$\n\n*Eq. 6\u201115* $L_{A,B:A} = L_{A:A,B}$\n\n*Eq. 6\u201116* $L_{A,B:B} = \\ L_{B:A,B}$\n\nFor more complex ordering of BCC lattice such as B32, D03,\nand L21 shown in with ideal compositions being AB,\nA3B, and A2BC, respectively, more sublattices are\nneeded, noting that the L21 Heusler structure exists in\nternary systems only. To use one model to describe all ordering in the\nBCC lattice, the minimum cluster is an irregular tetrahedron with four\nsublattices as depicted in as discussed in the modeling of the Al-Fe\nsystem \\[56\\]. In such a four sublattice model of\n$(A,B)_{0.25}^{I}(A,B)_{0.25}^{II}(A,B)_{0.25}^{II}(A,B)_{0.25}^{IV}$,\nthe site fractions of A2, B2, B32, and D03, are represented\nby $y_{i}^{I} = \\ y_{i}^{II} = y_{i}^{III} = \\ y_{i}^{IV}$,\n$y_{i}^{I} = \\ y_{i}^{II} \\neq y_{i}^{III} = \\ y_{i}^{IV}$,\n$y_{i}^{I} = \\ y_{i}^{III} \\neq y_{i}^{II} = \\ y_{i}^{IV}$, and\n$y_{i}^{I} = \\ y_{i}^{II} \\neq y_{i}^{III} \\neq \\ y_{i}^{IV}$,\nrespectively. The site fractions of L21 are the same as those\nof D03 except with at least three components.\n\nFigure \u2011: Atomic structures and four sublattice tetrahedrons of BCC\ndisordered and ordered phases.\n\nAnother common ordering phenomenon is in the FCC lattice including the\ndisordered A1 structure and ordered L10 and L12\nstructures as shown in . In the L10 structure, the\nneighboring (001) planes are favored by different atoms, respectively,\nresulting in an ideal composition of AB. While in the L12\nstructure, the corners and faces are favored by different atoms,\nrespectively, resulting in an ideal composition of A3B. In a\nfour sublattice model of\n$(A,B)_{0.25}^{I}(A,B)_{0.25}^{II}(A,B)_{0.25}^{II}(A,B)_{0.25}^{IV}$,\nthe site fractions of A1, L10 and L12 are\nrepresented by $y_{i}^{I} = \\ y_{i}^{II} = y_{i}^{III} = \\ y_{i}^{IV}$,\n$y_{i}^{I} = \\ y_{i}^{II} \\neq y_{i}^{III} = \\ y_{i}^{IV}$, and\n$y_{i}^{I} = \\ y_{i}^{II} = y_{i}^{III} \\neq \\ y_{i}^{IV}$, respectively\n\\[57\\].\n\nFigure \u2011: Atomic structures and four sublattice tetrahedrons of FCC\ndisordered and ordered phases.\n\nAs mentioned above, the interaction parameters in each sublattice can be\npredicted by first-principles calculations using the dilute solution and\nSQS approaches when there is only one component in each of the remaining\nsublattices. For the interaction involving two components in two or more\nsublattices, i.e. $L_{A,B:A,B}$ in applicable to four sublattice models\n\\[58\\], the energetics from cluster expansion (CE) approach discussed in\nChapter can be used to evaluate the interaction parameters.\n", "metadata": {} } ] diff --git a/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_stoichiometric_phases.ipynb b/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_stoichiometric_phases.ipynb index cf55c4c..3c92f8e 100644 --- a/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_stoichiometric_phases.ipynb +++ b/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_stoichiometric_phases.ipynb @@ -4,38 +4,9 @@ "metadata": {}, "cells": [ { - "id": "289b5213", + "id": "e05f5889", "cell_type": "markdown", - "source": [ - "## Modeling of stoichiometric phases\n", - "\n", - "The Gibbs energy of a stoichiometric phase can be modeled in the same\n", - "way as that of pure elements discussed above using the data of heat\n", - "capacity, $S_{298.15}$, and enthalpy of formation at 298.15 K (). When\n", - "these data are not available from experiments, they can be predicted by\n", - "first-principles calculations as discussed Chapter . It should be\n", - "pointed out that constraints placed on heat capacity of stoichiometric\n", - "compounds above melting temperatures, i.e. and , have not been\n", - "rigorously implemented in the literature for such modeling, partially\n", - "because the heat capacity of the corresponding liquid solution is not\n", - "well established, and the heat capacity of a compound is often not\n", - "available.\n", - "\n", - "When the data of heat capacity is not available, a simple approach,\n", - "commonly referred to as Neumann\u2013Kopp Rule assuming that the heat\n", - "capacity of formation of is zero, can be used. The Gibbs energy of the\n", - "compound is written as\n", - "\n", - "*Eq. 6\u20116*\n", - "$G = \\sum_{}^{}N_{i}{_{\\ }^{0}G}_{i}^{ref} + \\mathrm{\\Delta}_{f}H - T\\mathrm{\\Delta}_{f}S$\n", - "\n", - "with $\\mathrm{\\Delta}_{f}H$ and $\\mathrm{\\Delta}_{f}S$ modeled as\n", - "constants. An additional draw back of is that the melting temperature of\n", - "the compound can be higher than those of pure elements, and the heat\n", - "capacity of the compound may thus be questionable at temperatures higher\n", - "than the melting temperatures of pure elements due to the non-physical\n", - "contributions from pure elements based on .\n" - ], + "source": "## Modeling of stoichiometric phases\n\nThe Gibbs energy of a stoichiometric phase can be modeled in the same\nway as that of pure elements discussed above using the data of heat\ncapacity, $S_{298.15}$, and enthalpy of formation at 298.15 K (). When\nthese data are not available from experiments, they can be predicted by\nfirst-principles calculations as discussed Chapter . It should be\npointed out that constraints placed on heat capacity of stoichiometric\ncompounds above melting temperatures, i.e. and , have not been\nrigorously implemented in the literature for such modeling, partially\nbecause the heat capacity of the corresponding liquid solution is not\nwell established, and the heat capacity of a compound is often not\navailable.\n\nWhen the data of heat capacity is not available, a simple approach,\ncommonly referred to as Neumann\u2013Kopp Rule assuming that the heat\ncapacity of formation of is zero, can be used. The Gibbs energy of the\ncompound is written as\n\n*Eq. 6\u20116*\n$G = \\sum_{}^{}N_{i}{_{\\ }^{0}G}_{i}^{ref} + \\mathrm{\\Delta}_{f}H - T\\mathrm{\\Delta}_{f}S$\n\nwith $\\mathrm{\\Delta}_{f}H$ and $\\mathrm{\\Delta}_{f}S$ modeled as\nconstants. An additional draw back of is that the melting temperature of\nthe compound can be higher than those of pure elements, and the heat\ncapacity of the compound may thus be questionable at temperatures higher\nthan the melting temperatures of pure elements due to the non-physical\ncontributions from pure elements based on .\n", "metadata": {} } ] diff --git a/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_cerium.ipynb b/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_cerium.ipynb index 7aab4c6..9208a11 100644 --- a/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_cerium.ipynb +++ b/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_cerium.ipynb @@ -4,205 +4,9 @@ "metadata": {}, "cells": [ { - "id": "a5a4b22a", + "id": "a71a60c2", "cell_type": "markdown", - "source": [ - "## Application to cerium\n", - "\n", - "Cerium (Ce) displays intriguing physical and chemical properties of\n", - "which the most fascinating is its first-order isostructural phase\n", - "transition. This involves a magnetic, high temperature/high volume\n", - "\u201c\u03b3-phase\u201d and a nonmagnetic, low temperature/low volume \u201c\u03b1-phase\u201d, both\n", - "in the same face-centered-cubic (fcc) lattice structure. At 298 K and\n", - "0.7 GPa, the \u03b3\u2192\u03b1 transition is accompanied by a 14-17% volume collapse.\n", - "The Ce phase transition has been studied extensively including our own\n", - "works with two or three microstates \\[55, 64\\].\n", - "\n", - "The simplest model for the system is to consider two microstates:\n", - "ferromagnetic and nonmagnetic. The first-principles calculations of the\n", - "free energy of two Ce microstates are problematic in the absence of\n", - "strong correlation of the *f*-electrons in the DFT Hamiltonian. The\n", - "relative stability of the nonmagnetic (\u201cdelocalized\u201d) Ce 4*f* state to\n", - "that of the magnetic (\u201clocalized\u201d) Ce 4*f* state is greatly\n", - "overestimated in the GGA \\[31-32\\] with spin polarization. The usual\n", - "approach to surmount this is the Dudarev DFT + U method \\[65\\] with the\n", - "on-site Coulomb and exchange interactions as described with a\n", - "Hartree-Fock approximation added to the DFT Hamiltonian. This method\n", - "offers the advantage that only the difference between the Hubbard U (due\n", - "to the energy increase from an electron addition to a specific site) and\n", - "the J (due to the screened exchange energy) need to be specified *a\n", - "priori*.\n", - "\n", - "Evaluation of numerous U \u2013 J values over a 1.0 \u2013 6.0 eV range revealed\n", - "that 1.6 eV gives the most consistent prediction of nonmagnetic Ce and\n", - "magnetic Ce energetics over the range of atomic volumes that includes\n", - "both microstates at 0 K. The energy-volume curves thus obtained is\n", - "plotted in , showing that the nonmagnetic microstate is the ground\n", - "state, and the equilibrium between the two microstates at 0 K is at the\n", - "negative pressure of -0.87GPa. Since and , NTE does not exist in the\n", - "system.\n", - "\n", - "Figure \u2011: Variation of cell energy (eV) with atomic volume\n", - "(\u00c53) for Ce computed with strong correlation based upon\n", - "Dudarev\u2019s method with U \u2013 J = 1.6 eV.\n", - "\n", - "To take into account the possible magnetic disordering in the\n", - "ferromagnetic microstate at finite temperatures, the following\n", - "contribution is added to the free energy of the ferromagnetic microstate\n", - "\n", - "Eq. 9\u20119\n", - "\n", - "where *MS* is the spin moment, and *l* = 3 the orbital\n", - "angular momentum of an *f*-electron. is a generalization of Hund\u2019s rule,\n", - "with total angular momentum . The Helmholtz energies thus obtained for\n", - "both microstates and the system are shown in at several temperatures\n", - "with the tie-lines included. In the figure, the blue, dot-dashed curves\n", - "are for the nonmagnetic microstate, red, solid curves for ferromagnetic\n", - "microstate, cyan shadows for entropy of mixing between two microstates,\n", - "and the circle in (e) is the critical point. The numbers below the black\n", - "dashed line, representing the common tangent curves, mark the transition\n", - "pressures. The 0-K static energies of the nonmagnetic microstate and the\n", - "magnetic microstate are also plotted in (a) using the solid circles and\n", - "dotted lines.\n", - "\n", - "Figure \u2011: Helmholtz energies at (a) 0 K; (b) 100 K; (c) 165 K; (d) 300\n", - "K; (e) 476 K; and (f) 600 K.\n", - "\n", - "The temperature vs volume phase diagram is plotted and compared with\n", - "available experimental data as shown in \\[66\\]. In this figure, the\n", - "volume (V) is normalized to its equilibrium volume (VN) at\n", - "atmospheric pressure and room temperature. In the pressure range of 2.25\n", - "\u2013 3.5 GPa, the system is within the single-phase region at all\n", - "temperatures considered as shown by the five continuous isobaric volumes\n", - "as a function of temperature. In this pressure range, normal thermal\n", - "expansion is observed at both low and high temperatures on each isobaric\n", - "curve where the probability of each microstate does not change\n", - "significantly with temperature. While in the middle temperature range on\n", - "each isobaric curve, the colossal positive thermal expansion (CPTE),\n", - "highlighted by the pink open diamond symbols, exists due to the fast\n", - "increase of probability of the metastable ferromagnetic microstate with\n", - "respect to temperature, i.e. , , and . This CPTE is much higher than the\n", - "individual positive thermal expansions of the stable and metastable\n", - "microstates, respectively.\n", - "\n", - "Figure \u2011: Calculated temperature-volume phase diagram of Ce.\n", - "\n", - "With decreasing pressure, the system reaches a critical point (green\n", - "circle) where the homogeneous single phase becomes unstable, represented\n", - "by and , and both entropy and volume change infinitely. At even lower\n", - "pressure, a miscibility gap forms, and the single phase separates into\n", - "two phases with the same fcc crystal structures, but different magnetic\n", - "spin structures. Inside the miscibility gap, the volume changes\n", - "discontinuously with respect to temperature by the so-called first-order\n", - "transition as shown by the isobaric curve at zero pressure, compared\n", - "well with experimental volume data (solid squares) under ambient\n", - "pressure.\n", - "\n", - "The fraction of the ferromagnetic microstate, *x*mag, in \u03b1-Ce\n", - "(blue) and \u03b3-Ce (red) calculated using is plotted in as a function of\n", - "pressure along the miscibility gap phase boundary. It can be seen that\n", - "the fraction of the ferromagnetic microstate in \u03b1-Ce increases with\n", - "increasing pressure while the fraction of the ferromagnetic microstate\n", - "in \u03b3-Ce decreases. At the critical point, the fraction of ferromagnetic\n", - "microstate is calculated to be 0.58. This is in qualitative agreement\n", - "with the 0.67 value (filled circle) estimated experimentally at the\n", - "critical point.\n", - "\n", - "Figure \u2011: Fraction of ferromagnetic microstate in \u03b1-Ce (blue) and \u03b3-Ce\n", - "(red) along the \u03b3-\u03b1 phase boundary.\n", - "\n", - "The relative volume, *V/VN*, as a function of pressure, is\n", - "plotted as the black solid lines in from 200 to 600 K at 50 K\n", - "increments. The blue and red solid lines correspond to \u03b1-Ce and \u03b3-Ce,\n", - "respectively. Symbols denote experimental data in the literature, except\n", - "the open green circle being the calculated critical point, in good\n", - "agreement with the computed isotherms. In the two-phase miscibility gap\n", - "region, the \u03b3\u2192\u03b1 volume collapse is again noted, with the magnitude of\n", - "the collapse increasing with decreasing *T*. This is shown explicitly by\n", - "the dashed vertical lines at *T* = 200, 250, 300, 350, 400, and 450 K.\n", - "For *T* \\> 476 K, the calculated isotherms show an anomalous slope\n", - "change which closely matches the behavior near *V/VN* = ~0.85\n", - "from experiment.\n", - "\n", - "Figure \u2011: Equation-of-states for Ce. The black solid lines represent the\n", - "calculated isotherms from 200 to 600 K at \u2206T = 50 K increments.\n", - "\n", - "A more complex model is to add the anti-ferromagnetic microstate. Thus\n", - "obtained E-V and Helmholtz energy curves at 0 K are shown in . The\n", - "equilibrium volume energies reveal that the energy of the\n", - "anti-ferromagnetic microstate at the equilibrium volume is close to that\n", - "of the nonmagnetic microstate but substantially lower than that of the\n", - "ferromagnetic microstate. It should be noted that the magnetic spin\n", - "disordering in the system is taken into account by the two magnetic\n", - "microstates, and the contribution denoted by the mean-field theory, i.e.\n", - ", should thus not be added to either magnetic microstate to avoid double\n", - "counting. The predicted critical point values are 546 K and 2.05 GPa,\n", - "closer to the experimental data than the previous predication with two\n", - "microstates as shown in the temperature-pressure phase diagram in in\n", - "comparison with experimental data.\n", - "\n", - "Figure \u2011: (a) dot-dashed line with **\u25cb** (red), dashed line **\u25d1**\n", - "(blue), and solid line with \u25cf (blue) represent the 0 K static total\n", - "energies for nonmagnetic (NM), anti-ferromagnetic (AFM), and\n", - "ferromagnetic (FM) microstates of Ce, respectively. (b) The solid lines\n", - "denote Helmholtz energy (per atom) from 0 to 600 K at \u2206T = 100 K; the\n", - "heavy dot-dashed (\u03b1-Ce, red) and solid (\u03b3-Ce, blue) looping curves\n", - "enclose the two-phase region with the light red dot-dashed lines\n", - "connecting the common tangents of each isotherm; the black dashed line\n", - "denotes zero pressure equilibrium state at given T; **\u25cb** (red) and \u25cf\n", - "(blue) emphasize the phase boundary at 300 K while **\u25d1** (green) is the\n", - "critical point.\n", - "\n", - "Figure \u2011: Calculated temperature-pressure phase diagram along with\n", - "experimental data.\n", - "\n", - "The calculated entropy changes are plotted in a in terms of lattice\n", - "vibration only (black dashed line), lattice vibration plus thermal\n", - "electron (black dot-dashed line), and lattice vibration plus thermal\n", - "electron and plus configuration coupling (solid blue). The black square\n", - "is the estimated vibrational entropy change at 0.7 GPa of \u03b3-Ce relative\n", - "to \u03b1-Ce, and other open (solid) symbols are from experimental\n", - "measurements of total entropy. Various contributions to the Helmholtz\n", - "energy along the \u03b3-\u03b1 phase boundary are plotted in b in terms of *T\u2206S*\n", - "(blue diamonds), *\u2206E* (green circles), and *P\u2206V* (red squares), shown\n", - "excellent agreement with experimental data.\n", - "\n", - "Figure \u2011: Calculated (a) entropy; (b) various contributions to Helmholtz\n", - "energy of Ce along with experimental data.\n", - "\n", - "The predicted fractions of three microstates as a function of\n", - "temperature and heat capacity at the critical pressure of 2.05 Pa are\n", - "shown in a and b, respectively. Near the critical point, the theory\n", - "predicts that the system is a mixture of the various microstates. a\n", - "depicts that for T \\< 300 K, the system consists mainly of the\n", - "nonmagnetic Ce state which results in \u03b1-Ce. For *T* \\> 300 K, the\n", - "thermal populations of the magnetic states increase with increasing\n", - "temperature. Finally, for *T* \\> 546 K (the critical point), 70% of the\n", - "system is composed of the antiferromagnetic Ce state with the remaining\n", - "30% consisting of the nonmagnetic and ferromagnetic Ce states. This is\n", - "in agreement with the common belief that \u03b3-Ce is magnetic with a\n", - "partially disordered local moment (paramagnetic) and that \u03b1-Ce is\n", - "nonmagnetic.\n", - "\n", - "b shows the predicted temperature evolution of contributions to heat\n", - "capacity: vibrational and magnetic (*Cf/T*), electronic\n", - "(*Cel*/*T*), and their sum (*Cf+el/T*) at 2.05\n", - "GPa. The theory suggests the following: (a) below ~500 K,\n", - "*Cf+el/T* shows an exponential temperature dependence due to\n", - "the statistic fluctuation among the non-magnetic, ferromagnetic, and\n", - "antiferromagnetic states; (b) a peak appears at ~500 K in the\n", - "*Cf+el/T* curve, which typically suggests the Schottky\n", - "anomaly; (c) the electronic specific heat coefficient\n", - "(*Cel*/*T*) is linear against *T*; (d) above *~*500 K the sum\n", - "of *Cf/T* and *Cel*/*T* renders\n", - "*Cf+el/T* temperature-independent.\n", - "\n", - "Figure \u2011: (a) Thermal populations of the nonmagnetic (red dot-dashed),\n", - "anti-ferromagnetic (green dashed), and ferromagnetic (blue solid) as a\n", - "function of temperature at the critical pressure of 2.05 GPa. (b)\n", - "Cel/T (black dashed line), Cf/T (black dot-dashed\n", - "line), and their sum Cf+el/T (blue solid line) at 2.05 GPa.\n" - ], + "source": "## Application to cerium\n\nCerium (Ce) displays intriguing physical and chemical properties of\nwhich the most fascinating is its first-order isostructural phase\ntransition. This involves a magnetic, high temperature/high volume\n\u201c\u03b3-phase\u201d and a nonmagnetic, low temperature/low volume \u201c\u03b1-phase\u201d, both\nin the same face-centered-cubic (fcc) lattice structure. At 298 K and\n0.7 GPa, the \u03b3\u2192\u03b1 transition is accompanied by a 14-17% volume collapse.\nThe Ce phase transition has been studied extensively including our own\nworks with two or three microstates \\[55, 64\\].\n\nThe simplest model for the system is to consider two microstates:\nferromagnetic and nonmagnetic. The first-principles calculations of the\nfree energy of two Ce microstates are problematic in the absence of\nstrong correlation of the *f*-electrons in the DFT Hamiltonian. The\nrelative stability of the nonmagnetic (\u201cdelocalized\u201d) Ce 4*f* state to\nthat of the magnetic (\u201clocalized\u201d) Ce 4*f* state is greatly\noverestimated in the GGA \\[31-32\\] with spin polarization. The usual\napproach to surmount this is the Dudarev DFT + U method \\[65\\] with the\non-site Coulomb and exchange interactions as described with a\nHartree-Fock approximation added to the DFT Hamiltonian. This method\noffers the advantage that only the difference between the Hubbard U (due\nto the energy increase from an electron addition to a specific site) and\nthe J (due to the screened exchange energy) need to be specified *a\npriori*.\n\nEvaluation of numerous U \u2013 J values over a 1.0 \u2013 6.0 eV range revealed\nthat 1.6 eV gives the most consistent prediction of nonmagnetic Ce and\nmagnetic Ce energetics over the range of atomic volumes that includes\nboth microstates at 0 K. The energy-volume curves thus obtained is\nplotted in , showing that the nonmagnetic microstate is the ground\nstate, and the equilibrium between the two microstates at 0 K is at the\nnegative pressure of -0.87GPa. Since and , NTE does not exist in the\nsystem.\n\nFigure \u2011: Variation of cell energy (eV) with atomic volume\n(\u00c53) for Ce computed with strong correlation based upon\nDudarev\u2019s method with U \u2013 J = 1.6 eV.\n\nTo take into account the possible magnetic disordering in the\nferromagnetic microstate at finite temperatures, the following\ncontribution is added to the free energy of the ferromagnetic microstate\n\nEq. 9\u20119\n\nwhere *MS* is the spin moment, and *l* = 3 the orbital\nangular momentum of an *f*-electron. is a generalization of Hund\u2019s rule,\nwith total angular momentum . The Helmholtz energies thus obtained for\nboth microstates and the system are shown in at several temperatures\nwith the tie-lines included. In the figure, the blue, dot-dashed curves\nare for the nonmagnetic microstate, red, solid curves for ferromagnetic\nmicrostate, cyan shadows for entropy of mixing between two microstates,\nand the circle in (e) is the critical point. The numbers below the black\ndashed line, representing the common tangent curves, mark the transition\npressures. The 0-K static energies of the nonmagnetic microstate and the\nmagnetic microstate are also plotted in (a) using the solid circles and\ndotted lines.\n\nFigure \u2011: Helmholtz energies at (a) 0 K; (b) 100 K; (c) 165 K; (d) 300\nK; (e) 476 K; and (f) 600 K.\n\nThe temperature vs volume phase diagram is plotted and compared with\navailable experimental data as shown in \\[66\\]. In this figure, the\nvolume (V) is normalized to its equilibrium volume (VN) at\natmospheric pressure and room temperature. In the pressure range of 2.25\n\u2013 3.5 GPa, the system is within the single-phase region at all\ntemperatures considered as shown by the five continuous isobaric volumes\nas a function of temperature. In this pressure range, normal thermal\nexpansion is observed at both low and high temperatures on each isobaric\ncurve where the probability of each microstate does not change\nsignificantly with temperature. While in the middle temperature range on\neach isobaric curve, the colossal positive thermal expansion (CPTE),\nhighlighted by the pink open diamond symbols, exists due to the fast\nincrease of probability of the metastable ferromagnetic microstate with\nrespect to temperature, i.e. , , and . This CPTE is much higher than the\nindividual positive thermal expansions of the stable and metastable\nmicrostates, respectively.\n\nFigure \u2011: Calculated temperature-volume phase diagram of Ce.\n\nWith decreasing pressure, the system reaches a critical point (green\ncircle) where the homogeneous single phase becomes unstable, represented\nby and , and both entropy and volume change infinitely. At even lower\npressure, a miscibility gap forms, and the single phase separates into\ntwo phases with the same fcc crystal structures, but different magnetic\nspin structures. Inside the miscibility gap, the volume changes\ndiscontinuously with respect to temperature by the so-called first-order\ntransition as shown by the isobaric curve at zero pressure, compared\nwell with experimental volume data (solid squares) under ambient\npressure.\n\nThe fraction of the ferromagnetic microstate, *x*mag, in \u03b1-Ce\n(blue) and \u03b3-Ce (red) calculated using is plotted in as a function of\npressure along the miscibility gap phase boundary. It can be seen that\nthe fraction of the ferromagnetic microstate in \u03b1-Ce increases with\nincreasing pressure while the fraction of the ferromagnetic microstate\nin \u03b3-Ce decreases. At the critical point, the fraction of ferromagnetic\nmicrostate is calculated to be 0.58. This is in qualitative agreement\nwith the 0.67 value (filled circle) estimated experimentally at the\ncritical point.\n\nFigure \u2011: Fraction of ferromagnetic microstate in \u03b1-Ce (blue) and \u03b3-Ce\n(red) along the \u03b3-\u03b1 phase boundary.\n\nThe relative volume, *V/VN*, as a function of pressure, is\nplotted as the black solid lines in from 200 to 600 K at 50 K\nincrements. The blue and red solid lines correspond to \u03b1-Ce and \u03b3-Ce,\nrespectively. Symbols denote experimental data in the literature, except\nthe open green circle being the calculated critical point, in good\nagreement with the computed isotherms. In the two-phase miscibility gap\nregion, the \u03b3\u2192\u03b1 volume collapse is again noted, with the magnitude of\nthe collapse increasing with decreasing *T*. This is shown explicitly by\nthe dashed vertical lines at *T* = 200, 250, 300, 350, 400, and 450 K.\nFor *T* \\> 476 K, the calculated isotherms show an anomalous slope\nchange which closely matches the behavior near *V/VN* = ~0.85\nfrom experiment.\n\nFigure \u2011: Equation-of-states for Ce. The black solid lines represent the\ncalculated isotherms from 200 to 600 K at \u2206T = 50 K increments.\n\nA more complex model is to add the anti-ferromagnetic microstate. Thus\nobtained E-V and Helmholtz energy curves at 0 K are shown in . The\nequilibrium volume energies reveal that the energy of the\nanti-ferromagnetic microstate at the equilibrium volume is close to that\nof the nonmagnetic microstate but substantially lower than that of the\nferromagnetic microstate. It should be noted that the magnetic spin\ndisordering in the system is taken into account by the two magnetic\nmicrostates, and the contribution denoted by the mean-field theory, i.e.\n, should thus not be added to either magnetic microstate to avoid double\ncounting. The predicted critical point values are 546 K and 2.05 GPa,\ncloser to the experimental data than the previous predication with two\nmicrostates as shown in the temperature-pressure phase diagram in in\ncomparison with experimental data.\n\nFigure \u2011: (a) dot-dashed line with **\u25cb** (red), dashed line **\u25d1**\n(blue), and solid line with \u25cf (blue) represent the 0 K static total\nenergies for nonmagnetic (NM), anti-ferromagnetic (AFM), and\nferromagnetic (FM) microstates of Ce, respectively. (b) The solid lines\ndenote Helmholtz energy (per atom) from 0 to 600 K at \u2206T = 100 K; the\nheavy dot-dashed (\u03b1-Ce, red) and solid (\u03b3-Ce, blue) looping curves\nenclose the two-phase region with the light red dot-dashed lines\nconnecting the common tangents of each isotherm; the black dashed line\ndenotes zero pressure equilibrium state at given T; **\u25cb** (red) and \u25cf\n(blue) emphasize the phase boundary at 300 K while **\u25d1** (green) is the\ncritical point.\n\nFigure \u2011: Calculated temperature-pressure phase diagram along with\nexperimental data.\n\nThe calculated entropy changes are plotted in a in terms of lattice\nvibration only (black dashed line), lattice vibration plus thermal\nelectron (black dot-dashed line), and lattice vibration plus thermal\nelectron and plus configuration coupling (solid blue). The black square\nis the estimated vibrational entropy change at 0.7 GPa of \u03b3-Ce relative\nto \u03b1-Ce, and other open (solid) symbols are from experimental\nmeasurements of total entropy. Various contributions to the Helmholtz\nenergy along the \u03b3-\u03b1 phase boundary are plotted in b in terms of *T\u2206S*\n(blue diamonds), *\u2206E* (green circles), and *P\u2206V* (red squares), shown\nexcellent agreement with experimental data.\n\nFigure \u2011: Calculated (a) entropy; (b) various contributions to Helmholtz\nenergy of Ce along with experimental data.\n\nThe predicted fractions of three microstates as a function of\ntemperature and heat capacity at the critical pressure of 2.05 Pa are\nshown in a and b, respectively. Near the critical point, the theory\npredicts that the system is a mixture of the various microstates. a\ndepicts that for T \\< 300 K, the system consists mainly of the\nnonmagnetic Ce state which results in \u03b1-Ce. For *T* \\> 300 K, the\nthermal populations of the magnetic states increase with increasing\ntemperature. Finally, for *T* \\> 546 K (the critical point), 70% of the\nsystem is composed of the antiferromagnetic Ce state with the remaining\n30% consisting of the nonmagnetic and ferromagnetic Ce states. This is\nin agreement with the common belief that \u03b3-Ce is magnetic with a\npartially disordered local moment (paramagnetic) and that \u03b1-Ce is\nnonmagnetic.\n\nb shows the predicted temperature evolution of contributions to heat\ncapacity: vibrational and magnetic (*Cf/T*), electronic\n(*Cel*/*T*), and their sum (*Cf+el/T*) at 2.05\nGPa. The theory suggests the following: (a) below ~500 K,\n*Cf+el/T* shows an exponential temperature dependence due to\nthe statistic fluctuation among the non-magnetic, ferromagnetic, and\nantiferromagnetic states; (b) a peak appears at ~500 K in the\n*Cf+el/T* curve, which typically suggests the Schottky\nanomaly; (c) the electronic specific heat coefficient\n(*Cel*/*T*) is linear against *T*; (d) above *~*500 K the sum\nof *Cf/T* and *Cel*/*T* renders\n*Cf+el/T* temperature-independent.\n\nFigure \u2011: (a) Thermal populations of the nonmagnetic (red dot-dashed),\nanti-ferromagnetic (green dashed), and ferromagnetic (blue solid) as a\nfunction of temperature at the critical pressure of 2.05 GPa. (b)\nCel/T (black dashed line), Cf/T (black dot-dashed\nline), and their sum Cf+el/T (blue solid line) at 2.05 GPa.\n", "metadata": {} } ] diff --git a/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_fept.ipynb b/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_fept.ipynb index 4a5e384..a88b4b5 100644 --- a/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_fept.ipynb +++ b/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_fept.ipynb @@ -4,127 +4,9 @@ "metadata": {}, "cells": [ { - "id": "844edfab", + "id": "09dbdce6", "cell_type": "markdown", - "source": [ - "## Application to Fe3Pt\n", - "\n", - "Invar was first discovered in the intermetallic\n", - "Fe65Ni35 alloy and is characterized by \u201canomalies\u201d\n", - "including thermal expansion, equation-of-state, elastic modulus, heat\n", - "capacity, magnetization, and Curie temperature. There are a number of\n", - "theoretical models for Invar such as the Weiss 2-\u03b3 model, the\n", - "non-collinear spin model, and the disordered local moment approach as\n", - "reviewed in Ref. \\[67\\]. In this section, the application of the MMS\n", - "model to the ordered L12 Fe3Pt is presented to\n", - "study the Invar anomaly at finite temperatures.\n", - "\n", - "For a supercell of 12 atoms with nine magnetic Fe atoms in\n", - "Fe3Pt, if only the up and down spins are considered, the\n", - "system contains 29 = 512 spin configurations, which are, by\n", - "symmetry, reduced to 37 non-equivalent ones. They are the microstates in\n", - "the MMS model, and FMC and SFC are used to represent the ferromagnetic\n", - "and spin flipping microstates, respectively. For the first-principles\n", - "calculations of each microstate, VASP package \\[14\\] within the\n", - "projector-augmented wave (PAW) method and the exchange-correlation part\n", - "of the density functional treated within the GGA of\n", - "Perdew-Burke-Ernzerhof (PBE) \\[32\\] are employed with details in Ref.\n", - "\\[67\\]. For the lattice vibration, the Debye-Gr\u00fcneisen approach\n", - "described in Chapter 5.2.4 is used.\n", - "\n", - "presents the first-principles 0 K total energies of 36 non-equivalent\n", - "SFCs as well as the FMC as a function of atomic volume. It can be seen\n", - "that there are a number of SFCs, whose energies are in the range of ~1\n", - "mRy/atom to that of the FMC. It is noted that all the SFCs studied\n", - "herein have the equilibrium averaged atomic volumes at least 1.8 %\n", - "smaller than that of the FMC, the 0 GPa ground state. In , the two\n", - "lowest energy SFCs have been labeled as SFC55 and SFC41 with their spin\n", - "arrangements very similar to the double layer antiferromagnetic state.\n", - "The nonmagnetic configuration has a very small atomic volume of 11.66\n", - "\u00c53/atom and much higher energy than both FMC and all SFCs,\n", - "and is thus not shown here.\n", - "\n", - "Figure \u2011: 0 K total energies. The heavy black line represents the FMC.\n", - "The symbols **\u25cb**, \u25d4, \u25d1, and \u25d5 with dashed lines indicate the minima of\n", - "the energy-volume curves of the SFCs with spin polarization rates of\n", - "1/9, 3/9, 5/9, and 7/9, respectively. The red **\u25cb** and \u25d4 with\n", - "dot-dashed lines mark the two lowest SFCs in energy.\n", - "\n", - "The normalized Helmholtz energies of all SFCs are plotted in with the\n", - "Helmholtz energy of FMC as the reference state, shown that the FMC has\n", - "the lowest Helmholtz energy at all temperatures considered. If only the\n", - "relative Helmholtz energies of microstates are considered, FMC should be\n", - "stable at all temperatures. However, the configurational entropy due to\n", - "the mixing among multiple microstates, i.e. , lowers the system free\n", - "energy by introducing the statistic probability of metastable SFCs.\n", - "\n", - "Figure \u2011: Normalized Helmholtz energy of all SFCs with respect to that\n", - "of FMC.\n", - "\n", - "Through the minimization of Helmholtz energy of , the\n", - "temperature-pressure and temperature-volume phase diagrams are obtained\n", - "and shown in \\[66\\]. A critical point at 141 K and 5.81 GPa is predicted\n", - "with *V* = 12.61 \u00c53. Below the critical point, it is a\n", - "two-phase miscibility gap (the shadow area) with the dominant\n", - "microstates being FMC and SFCs, respectively, and the transition between\n", - "them is first-order. Above the critical point, the macroscopically\n", - "homogeneous single phase is stable, and the phase transitions between\n", - "the ferromagnetic-dominant phase with large volumes and the SFC-dominant\n", - "paramagnetic phase with small volumes are of second-order. The\n", - "second-order transition pressures and volumes are determined by the\n", - "condition that the weighted Helmholtz energy counting all SFCs equals to\n", - "the Helmholtz energy counting only FMC.\n", - "\n", - "Figure \u2011: Calculated (a) temperature-pressure and (b) temperature-volume\n", - "phase diagrams of Fe3Pt.\n", - "\n", - "In a, the data points are the measured pressure dependence of the Curie\n", - "temperature, and the agreement between the measurements and predictions\n", - "is remarkable. It should be pointed out that the classical Weiss 2-\u03b3\n", - "model predicts only first-order phase transitions while the\n", - "non-collinear spin model yields only second-order phase transitions at\n", - "all temperatures. In b, four isobaric volume curves are also plotted\n", - "with the predicted NTE regions marked by the pink open diamonds and the\n", - "experimental volume data under ambient pressure superimposed, showing\n", - "excellent agreement. It also depicts the gigantic NTE around the\n", - "critical point on the isobaric curve at 7 GPa.\n", - "\n", - "indicates that the entropies of SFCs are larger than that of FMC so\n", - "their Helmholtz energy differences decrease with temperature. This is in\n", - "line with the origin of NTE in a single phase due to the statistic\n", - "existence of metastable microstates with lower volumes and higher\n", - "entropies than the stable state in a temperature range where their\n", - "probabilities change dramatically. plots the calculated thermal\n", - "populations of the FMC (black solid line) and that of the sum over all\n", - "SFCs (black dot-dashed line) under ambient pressure. The two major\n", - "contributions to the paramagnetic (PM) phase are from SFC55 and SFC41,\n", - "which are also plotted using red dashed and long dashed lines,\n", - "respectively. The system is dominated by the FMC at temperatures below\n", - "half of the transition temperature, and the populations of SFCs increase\n", - "monotonously at temperatures higher than half of the transition\n", - "temperature. As mentioned above, the transition temperature is defined\n", - "when the population of all SFCs is the same as that of FMC due to their\n", - "equal Helmholtz energy.\n", - "\n", - "Figure \u2011: Calculated thermal populations of FMC, SFC55, SFC41, and sum\n", - "of all SFCS, respectively.\n", - "\n", - "The predicted thermal volume expansion and the derived linear thermal\n", - "expansion coefficient (LTEC) under ambient pressure are plotted in . A\n", - "positive thermal expansion is predicted from 100 K to 288 K, followed by\n", - "a negative thermal expansion in the range of 289 ~ 449 K, and then a\n", - "positive thermal expansion again at \\>450 K, in excellent agreement with\n", - "experiments. The only disagreement between the predictions and\n", - "experiments occur at T \\< 100 K where the calculations did not reproduce\n", - "the negative thermal expansion for Fe3Pt. Large supercell or\n", - "more spin configurations may be needed.\n", - "\n", - "Figure \u2011: (a) Relative volume increase (V-\n", - "V300)/V300 with V300 being the\n", - "equilibrium volume at 300K and 0 GPa for the ordered Fe3Pt.\n", - "(b) Linear thermal expansion coefficient (LTEC) along with experimental\n", - "data (symbols) with details in Ref. \\[67\\].\n" - ], + "source": "## Application to Fe3Pt\n\nInvar was first discovered in the intermetallic\nFe65Ni35 alloy and is characterized by \u201canomalies\u201d\nincluding thermal expansion, equation-of-state, elastic modulus, heat\ncapacity, magnetization, and Curie temperature. There are a number of\ntheoretical models for Invar such as the Weiss 2-\u03b3 model, the\nnon-collinear spin model, and the disordered local moment approach as\nreviewed in Ref. \\[67\\]. In this section, the application of the MMS\nmodel to the ordered L12 Fe3Pt is presented to\nstudy the Invar anomaly at finite temperatures.\n\nFor a supercell of 12 atoms with nine magnetic Fe atoms in\nFe3Pt, if only the up and down spins are considered, the\nsystem contains 29 = 512 spin configurations, which are, by\nsymmetry, reduced to 37 non-equivalent ones. They are the microstates in\nthe MMS model, and FMC and SFC are used to represent the ferromagnetic\nand spin flipping microstates, respectively. For the first-principles\ncalculations of each microstate, VASP package \\[14\\] within the\nprojector-augmented wave (PAW) method and the exchange-correlation part\nof the density functional treated within the GGA of\nPerdew-Burke-Ernzerhof (PBE) \\[32\\] are employed with details in Ref.\n\\[67\\]. For the lattice vibration, the Debye-Gr\u00fcneisen approach\ndescribed in Chapter 5.2.4 is used.\n\npresents the first-principles 0 K total energies of 36 non-equivalent\nSFCs as well as the FMC as a function of atomic volume. It can be seen\nthat there are a number of SFCs, whose energies are in the range of ~1\nmRy/atom to that of the FMC. It is noted that all the SFCs studied\nherein have the equilibrium averaged atomic volumes at least 1.8 %\nsmaller than that of the FMC, the 0 GPa ground state. In , the two\nlowest energy SFCs have been labeled as SFC55 and SFC41 with their spin\narrangements very similar to the double layer antiferromagnetic state.\nThe nonmagnetic configuration has a very small atomic volume of 11.66\n\u00c53/atom and much higher energy than both FMC and all SFCs,\nand is thus not shown here.\n\nFigure \u2011: 0 K total energies. The heavy black line represents the FMC.\nThe symbols **\u25cb**, \u25d4, \u25d1, and \u25d5 with dashed lines indicate the minima of\nthe energy-volume curves of the SFCs with spin polarization rates of\n1/9, 3/9, 5/9, and 7/9, respectively. The red **\u25cb** and \u25d4 with\ndot-dashed lines mark the two lowest SFCs in energy.\n\nThe normalized Helmholtz energies of all SFCs are plotted in with the\nHelmholtz energy of FMC as the reference state, shown that the FMC has\nthe lowest Helmholtz energy at all temperatures considered. If only the\nrelative Helmholtz energies of microstates are considered, FMC should be\nstable at all temperatures. However, the configurational entropy due to\nthe mixing among multiple microstates, i.e. , lowers the system free\nenergy by introducing the statistic probability of metastable SFCs.\n\nFigure \u2011: Normalized Helmholtz energy of all SFCs with respect to that\nof FMC.\n\nThrough the minimization of Helmholtz energy of , the\ntemperature-pressure and temperature-volume phase diagrams are obtained\nand shown in \\[66\\]. A critical point at 141 K and 5.81 GPa is predicted\nwith *V* = 12.61 \u00c53. Below the critical point, it is a\ntwo-phase miscibility gap (the shadow area) with the dominant\nmicrostates being FMC and SFCs, respectively, and the transition between\nthem is first-order. Above the critical point, the macroscopically\nhomogeneous single phase is stable, and the phase transitions between\nthe ferromagnetic-dominant phase with large volumes and the SFC-dominant\nparamagnetic phase with small volumes are of second-order. The\nsecond-order transition pressures and volumes are determined by the\ncondition that the weighted Helmholtz energy counting all SFCs equals to\nthe Helmholtz energy counting only FMC.\n\nFigure \u2011: Calculated (a) temperature-pressure and (b) temperature-volume\nphase diagrams of Fe3Pt.\n\nIn a, the data points are the measured pressure dependence of the Curie\ntemperature, and the agreement between the measurements and predictions\nis remarkable. It should be pointed out that the classical Weiss 2-\u03b3\nmodel predicts only first-order phase transitions while the\nnon-collinear spin model yields only second-order phase transitions at\nall temperatures. In b, four isobaric volume curves are also plotted\nwith the predicted NTE regions marked by the pink open diamonds and the\nexperimental volume data under ambient pressure superimposed, showing\nexcellent agreement. It also depicts the gigantic NTE around the\ncritical point on the isobaric curve at 7 GPa.\n\nindicates that the entropies of SFCs are larger than that of FMC so\ntheir Helmholtz energy differences decrease with temperature. This is in\nline with the origin of NTE in a single phase due to the statistic\nexistence of metastable microstates with lower volumes and higher\nentropies than the stable state in a temperature range where their\nprobabilities change dramatically. plots the calculated thermal\npopulations of the FMC (black solid line) and that of the sum over all\nSFCs (black dot-dashed line) under ambient pressure. The two major\ncontributions to the paramagnetic (PM) phase are from SFC55 and SFC41,\nwhich are also plotted using red dashed and long dashed lines,\nrespectively. The system is dominated by the FMC at temperatures below\nhalf of the transition temperature, and the populations of SFCs increase\nmonotonously at temperatures higher than half of the transition\ntemperature. As mentioned above, the transition temperature is defined\nwhen the population of all SFCs is the same as that of FMC due to their\nequal Helmholtz energy.\n\nFigure \u2011: Calculated thermal populations of FMC, SFC55, SFC41, and sum\nof all SFCS, respectively.\n\nThe predicted thermal volume expansion and the derived linear thermal\nexpansion coefficient (LTEC) under ambient pressure are plotted in . A\npositive thermal expansion is predicted from 100 K to 288 K, followed by\na negative thermal expansion in the range of 289 ~ 449 K, and then a\npositive thermal expansion again at \\>450 K, in excellent agreement with\nexperiments. The only disagreement between the predictions and\nexperiments occur at T \\< 100 K where the calculations did not reproduce\nthe negative thermal expansion for Fe3Pt. Large supercell or\nmore spin configurations may be needed.\n\nFigure \u2011: (a) Relative volume increase (V-\nV300)/V300 with V300 being the\nequilibrium volume at 300K and 0 GPa for the ordered Fe3Pt.\n(b) Linear thermal expansion coefficient (LTEC) along with experimental\ndata (symbols) with details in Ref. \\[67\\].\n", "metadata": {} } ] diff --git a/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/concept_of_materials_genome.ipynb b/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/concept_of_materials_genome.ipynb index e7e53a2..336d2cc 100644 --- a/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/concept_of_materials_genome.ipynb +++ b/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/concept_of_materials_genome.ipynb @@ -4,62 +4,9 @@ "metadata": {}, "cells": [ { - "id": "93eed2fd", + "id": "e9947ec5", "cell_type": "markdown", - "source": [ - "## Concept of Materials Genome\u00ae\n", - "\n", - "\u201cA genome is a set of information encoded in the language of DNA that\n", - "serves as a blueprint for an organism\u2019s growth and development. The word\n", - "genome, when applied in nonbiological contexts, connotes a fundamental\n", - "building block toward a larger purpose\u201d \\[68\\]. Materials Genome\u00ae (a\n", - "trademark of Materials Genome, Inc., Pennsylvania, USA) thus concerns\n", - "the building blocks of materials. Most part of this book focuses on the\n", - "Gibbs/Helmholtz energies of individual phases as a function of its\n", - "natural variables, and the same in the CALPHAD modeling of\n", - "thermodynamics and other properties of individual phases.\n", - "Multi-component materials systems and their properties are built on the\n", - "individual phases and their properties. Individual phases are thus\n", - "naturally considered as building blocks of materials. Consequently, the\n", - "language of CALPHAD thermodynamics and kinetics contains the genomics of\n", - "materials by representing experimental and theoretical results in\n", - "databases to make them applicable to a much wider context than the\n", - "original experiments or calculations \\[69\\]. The change of individual\n", - "phases in terms of their properties, amounts, and interactions with\n", - "other phases with respect to external conditions thus determines the\n", - "performance of the materials.\n", - "\n", - "On the other hand, at critical points and beyond, phases lose their\n", - "individuality and form one macroscopically homogeneous system, and the\n", - "properties of the system change dramatically with respect to external\n", - "conditions. As shown in this chapter, these dramatically responses can\n", - "be predicted through the statistic competition of stable and metastable\n", - "microstates. From the thermodynamic point of view, under any given\n", - "conditions, one of the individual microstate has the lowest\n", - "Gibbs/Helmholtz energy and is stable, while all other microstates have\n", - "higher free energy and are metastable or unstable. These metastable or\n", - "even unstable microstates are brought into statistic existence in the\n", - "matrix of the stable microstate due to the entropy of mixing of the\n", - "stable and metastable configurations. Those microstates may thus be\n", - "considered as the building blocks of individual phases \\[70\\].\n", - "\n", - "It has demonstrated in this chapter that the properties of a\n", - "macroscopically homogeneous system with multiple microstates are not a\n", - "linear combination of properties of constituent microstates and depends\n", - "significantly on the change rate of statistic probability of microstates\n", - "with respect to external fields. This change rate is determined by the\n", - "free energy difference between the stable and metastable microstates and\n", - "its change with respect to external fields. As shown in and , this\n", - "change rate can be correlated to the distance of the system with respect\n", - "to the critical point in the system. At the critical point, there is a\n", - "mathematical singularity when the single phase becomes unstable. When\n", - "the macroscopically homogeneous single-phase system moves away from the\n", - "critical point, its properties become less and less dramatic, but always\n", - "remain a certain degree of anomaly. The properties of a system can thus\n", - "be dramatically altered and designed by changing the position of the\n", - "critical point through adjustments of chemical compositions and strain\n", - "energy in thin films.\n" - ], + "source": "## Concept of Materials Genome\u00ae\n\n\u201cA genome is a set of information encoded in the language of DNA that\nserves as a blueprint for an organism\u2019s growth and development. The word\ngenome, when applied in nonbiological contexts, connotes a fundamental\nbuilding block toward a larger purpose\u201d \\[68\\]. Materials Genome\u00ae (a\ntrademark of Materials Genome, Inc., Pennsylvania, USA) thus concerns\nthe building blocks of materials. Most part of this book focuses on the\nGibbs/Helmholtz energies of individual phases as a function of its\nnatural variables, and the same in the CALPHAD modeling of\nthermodynamics and other properties of individual phases.\nMulti-component materials systems and their properties are built on the\nindividual phases and their properties. Individual phases are thus\nnaturally considered as building blocks of materials. Consequently, the\nlanguage of CALPHAD thermodynamics and kinetics contains the genomics of\nmaterials by representing experimental and theoretical results in\ndatabases to make them applicable to a much wider context than the\noriginal experiments or calculations \\[69\\]. The change of individual\nphases in terms of their properties, amounts, and interactions with\nother phases with respect to external conditions thus determines the\nperformance of the materials.\n\nOn the other hand, at critical points and beyond, phases lose their\nindividuality and form one macroscopically homogeneous system, and the\nproperties of the system change dramatically with respect to external\nconditions. As shown in this chapter, these dramatically responses can\nbe predicted through the statistic competition of stable and metastable\nmicrostates. From the thermodynamic point of view, under any given\nconditions, one of the individual microstate has the lowest\nGibbs/Helmholtz energy and is stable, while all other microstates have\nhigher free energy and are metastable or unstable. These metastable or\neven unstable microstates are brought into statistic existence in the\nmatrix of the stable microstate due to the entropy of mixing of the\nstable and metastable configurations. Those microstates may thus be\nconsidered as the building blocks of individual phases \\[70\\].\n\nIt has demonstrated in this chapter that the properties of a\nmacroscopically homogeneous system with multiple microstates are not a\nlinear combination of properties of constituent microstates and depends\nsignificantly on the change rate of statistic probability of microstates\nwith respect to external fields. This change rate is determined by the\nfree energy difference between the stable and metastable microstates and\nits change with respect to external fields. As shown in and , this\nchange rate can be correlated to the distance of the system with respect\nto the critical point in the system. At the critical point, there is a\nmathematical singularity when the single phase becomes unstable. When\nthe macroscopically homogeneous single-phase system moves away from the\ncritical point, its properties become less and less dramatic, but always\nremain a certain degree of anomaly. The properties of a system can thus\nbe dramatically altered and designed by changing the position of the\ncritical point through adjustments of chemical compositions and strain\nenergy in thin films.\n", "metadata": {} } ] diff --git a/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/index.ipynb b/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/index.ipynb index 54cceed..f05c1a9 100644 --- a/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/index.ipynb +++ b/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/index.ipynb @@ -4,28 +4,9 @@ "metadata": {}, "cells": [ { - "id": "7ac53aa2", + "id": "5cac8904", "cell_type": "markdown", - "source": [ - "# Critical phenomena, thermal expansion, and Materials Genome\u00ae\n", - "\n", - "In Chapter , it was shown that all molar quantities of a homogeneous\n", - "system diverge at the critical point, i.e. the limit of stability,\n", - "including the additional ones shown in . As illustrated by , even though\n", - "each molar quantity changes in the same direction as its conjugate\n", - "potential, i.e. with the same sign, its dependence with respect to a\n", - "nonconjugate potentials can either be in the same sign or opposite\n", - "signs. It is often considered to be normal when they change in the same\n", - "direction, while abnormal when in different directions.\n", - "\n", - "In this chapter, the thermal expansion defined by , is used as an\n", - "example for detailed discussion of those extraordinary phenomena in the\n", - "context of a critical point based on the MMS model presented in Chapter\n", - ". The MMS model is first discussed in terms of thermal expansion and\n", - "then applied to elemental cerium (Ce) with the colossal positive thermal\n", - "expansion (CPTE) and Fe3Pt with negative thermal expansion\n", - "(NTE).\n" - ], + "source": "# Critical phenomena, thermal expansion, and Materials Genome\u00ae\n\nIn Chapter , it was shown that all molar quantities of a homogeneous\nsystem diverge at the critical point, i.e. the limit of stability,\nincluding the additional ones shown in . As illustrated by , even though\neach molar quantity changes in the same direction as its conjugate\npotential, i.e. with the same sign, its dependence with respect to a\nnonconjugate potentials can either be in the same sign or opposite\nsigns. It is often considered to be normal when they change in the same\ndirection, while abnormal when in different directions.\n\nIn this chapter, the thermal expansion defined by , is used as an\nexample for detailed discussion of those extraordinary phenomena in the\ncontext of a critical point based on the MMS model presented in Chapter\n. The MMS model is first discussed in terms of thermal expansion and\nthen applied to elemental cerium (Ce) with the colossal positive thermal\nexpansion (CPTE) and Fe3Pt with negative thermal expansion\n(NTE).\n", "metadata": {} } ] diff --git a/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/mms_model_applied_to_thermal_expansion.ipynb b/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/mms_model_applied_to_thermal_expansion.ipynb index faf4f8c..b057391 100644 --- a/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/mms_model_applied_to_thermal_expansion.ipynb +++ b/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/mms_model_applied_to_thermal_expansion.ipynb @@ -4,123 +4,9 @@ "metadata": {}, "cells": [ { - "id": "c636c3f0", + "id": "e1834dd0", "cell_type": "markdown", - "source": [ - "## MMS model applied to thermal expansion\n", - "\n", - "As shown in , the thermal expansion of a system can be positive, zero\n", - "and negative depending on the pressure dependence of entropy of the\n", - "system. Let us carry out a virtual experiment by analyzing a system\n", - "starting with one microstate only, *\u03b1*, when the metastable *\u03b2*\n", - "microstate has a higher entropy than the *\u03b1* microstate, i.e. and the\n", - "relative stability of the *\u03b2* microstate thus increases with\n", - "temperature. The cases with will be discussed after this starting with a\n", - "mixture of the *\u03b1* and *\u03b2* microstates.\n", - "\n", - "When a metastable microstate, *\u03b2*, is introduced by changing pressure\n", - "under constant temperature, based on and , the entropy change of the\n", - "system can be written as\n", - "\n", - "Eq. 9\u20111\n", - "\n", - "where represents the statistic probability of the microstate in the\n", - "system. With , this would results in a positive entropy change of , i.\n", - "e. , since . If this entropy increase is due to the decrease of\n", - "pressure, i.e. because volume and its conjugate potential (negative\n", - "pressure) change in the same direction, the volume thermal expansion of\n", - "the system is positive due to the increase of the population of the *\u03b2*\n", - "microstate with a larger volume. In this case,\n", - "$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}} > 0$, and the\n", - "volume and entropy of the two microstates change in the same direction.\n", - "\n", - "On the other hand, if this entropy increase is realized by increasing\n", - "pressure, i.e. , the volume thermal expansion of the system is negative\n", - "due to the increase of the population of the *\u03b2* microstate with a\n", - "smaller volume. In this case,\n", - "$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}} < 0$, and the\n", - "volume and entropy of the two microstates change in the opposite\n", - "directions.\n", - "\n", - "Therefore, the sign of\n", - "$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}}$ of two\n", - "microstates can be used as a criterion to determine whether a system\n", - "possesses NTE, i.e. positive\n", - "$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}}$ for positive\n", - "thermal expansion, and negative\n", - "$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}}$ for NTE. At a\n", - "critical point, the entropy change with respect to temperature is\n", - "infinite, resulting in either infinite positive or infinite negative\n", - "thermal expansion correspondingly. When the system moves away from the\n", - "critical point into the macroscopically homogeneous single-phase region,\n", - "the thermal expansion becomes less positive or negative. A number of\n", - "systems with\n", - "$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}} < 0$, thus\n", - "potentially NTE, are listed in the supplementary information of Ref.\n", - "\\[63\\].\n", - "\n", - "Now let us consider the case that the metastable *\u03b2* microstate has\n", - "lower entropy than the *\u03b1* microstate, i.e. , and the *\u03b2* microstate is\n", - "thus more stable at low temperatures. The system at higher temperatures\n", - "contains thus only the *\u03b1* microstate and has positive thermal\n", - "expansion. When the metastable *\u03b2* microstate is introduced, the sign of\n", - "the entropy change in can be either positive or negative because the\n", - "first term is negative and the second term is positive, and its sign\n", - "thus depends on the value of the entropy difference between two\n", - "microstates and the probability of the metastable *\u03b2* microstate. The\n", - "virtual experiment should thus be carried out in a system with the\n", - "highest MCE in , i.e. when the two microstates have the same free energy\n", - "and are in equilibrium with each other. From and , the system entropy\n", - "can be written as\n", - "\n", - "Eq. 9\u20112\n", - "\n", - "With the change of pressure, will either increase or decrease, and the\n", - "entropy of the system becomes\n", - "\n", - "Eq. 9\u20113\n", - "\n", - "The difference of and is obtained as\n", - "\n", - "Eq. 9\u20114\n", - "\n", - "The second term in is always negative, and the first term is also\n", - "negative if because . It is thus evident that if is increased by\n", - "decreasing pressure, the entropy of the system decreases, and the system\n", - "would possess negative thermal expansion because is negative, and the\n", - "entropy is reduced by the decrease of pressure. At the same time, and\n", - "$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}} < 0$, the latter\n", - "being the same condition for a negative thermal expansion as in the\n", - "first virtual experiment with . One can thus conclude that for a\n", - "two-phase equilibrium line with\n", - "$\\frac{dT}{dP} = \\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}} < 0$,\n", - "both phases can display negative thermal expansion. On the other hand,\n", - "if is increased by increasing pressure, the system would possess\n", - "positive thermal expansion because the entropy is reduced by the\n", - "increase of pressure.\n", - "\n", - "Furthermore, the thermal expansion of a system can be approximated as\n", - "follows using the rule of mixture of volumes\n", - "\n", - "Eq. 9\u20115\n", - "\n", - "Eq. 9\u20116\n", - "\n", - "where , , , , , and are the thermal expansion coefficients and volumes\n", - "of the system and the \u03b1 and \u03b2 microstates, respectively. For\n", - "simplification, let us assume both microstates have similar positive\n", - "thermal expansion, i.e. , and becomes\n", - "\n", - "Eq. 9\u20117\n", - "\n", - "shows that it is the combination of volume difference and that\n", - "determines the macroscopic thermal expansion. By setting , one obtains\n", - "\n", - "Eq. 9\u20118\n", - "\n", - "For , for $p^{\\beta} \\geq 0$, and for , at $p^{\\beta} \\rightarrow 0$ .\n", - "The readers are reminded that the sign of is the same as the sign of .\n" - ], + "source": "## MMS model applied to thermal expansion\n\nAs shown in , the thermal expansion of a system can be positive, zero\nand negative depending on the pressure dependence of entropy of the\nsystem. Let us carry out a virtual experiment by analyzing a system\nstarting with one microstate only, *\u03b1*, when the metastable *\u03b2*\nmicrostate has a higher entropy than the *\u03b1* microstate, i.e. and the\nrelative stability of the *\u03b2* microstate thus increases with\ntemperature. The cases with will be discussed after this starting with a\nmixture of the *\u03b1* and *\u03b2* microstates.\n\nWhen a metastable microstate, *\u03b2*, is introduced by changing pressure\nunder constant temperature, based on and , the entropy change of the\nsystem can be written as\n\nEq. 9\u20111\n\nwhere represents the statistic probability of the microstate in the\nsystem. With , this would results in a positive entropy change of , i.\ne. , since . If this entropy increase is due to the decrease of\npressure, i.e. because volume and its conjugate potential (negative\npressure) change in the same direction, the volume thermal expansion of\nthe system is positive due to the increase of the population of the *\u03b2*\nmicrostate with a larger volume. In this case,\n$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}} > 0$, and the\nvolume and entropy of the two microstates change in the same direction.\n\nOn the other hand, if this entropy increase is realized by increasing\npressure, i.e. , the volume thermal expansion of the system is negative\ndue to the increase of the population of the *\u03b2* microstate with a\nsmaller volume. In this case,\n$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}} < 0$, and the\nvolume and entropy of the two microstates change in the opposite\ndirections.\n\nTherefore, the sign of\n$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}}$ of two\nmicrostates can be used as a criterion to determine whether a system\npossesses NTE, i.e. positive\n$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}}$ for positive\nthermal expansion, and negative\n$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}}$ for NTE. At a\ncritical point, the entropy change with respect to temperature is\ninfinite, resulting in either infinite positive or infinite negative\nthermal expansion correspondingly. When the system moves away from the\ncritical point into the macroscopically homogeneous single-phase region,\nthe thermal expansion becomes less positive or negative. A number of\nsystems with\n$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}} < 0$, thus\npotentially NTE, are listed in the supplementary information of Ref.\n\\[63\\].\n\nNow let us consider the case that the metastable *\u03b2* microstate has\nlower entropy than the *\u03b1* microstate, i.e. , and the *\u03b2* microstate is\nthus more stable at low temperatures. The system at higher temperatures\ncontains thus only the *\u03b1* microstate and has positive thermal\nexpansion. When the metastable *\u03b2* microstate is introduced, the sign of\nthe entropy change in can be either positive or negative because the\nfirst term is negative and the second term is positive, and its sign\nthus depends on the value of the entropy difference between two\nmicrostates and the probability of the metastable *\u03b2* microstate. The\nvirtual experiment should thus be carried out in a system with the\nhighest MCE in , i.e. when the two microstates have the same free energy\nand are in equilibrium with each other. From and , the system entropy\ncan be written as\n\nEq. 9\u20112\n\nWith the change of pressure, will either increase or decrease, and the\nentropy of the system becomes\n\nEq. 9\u20113\n\nThe difference of and is obtained as\n\nEq. 9\u20114\n\nThe second term in is always negative, and the first term is also\nnegative if because . It is thus evident that if is increased by\ndecreasing pressure, the entropy of the system decreases, and the system\nwould possess negative thermal expansion because is negative, and the\nentropy is reduced by the decrease of pressure. At the same time, and\n$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}} < 0$, the latter\nbeing the same condition for a negative thermal expansion as in the\nfirst virtual experiment with . One can thus conclude that for a\ntwo-phase equilibrium line with\n$\\frac{dT}{dP} = \\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}} < 0$,\nboth phases can display negative thermal expansion. On the other hand,\nif is increased by increasing pressure, the system would possess\npositive thermal expansion because the entropy is reduced by the\nincrease of pressure.\n\nFurthermore, the thermal expansion of a system can be approximated as\nfollows using the rule of mixture of volumes\n\nEq. 9\u20115\n\nEq. 9\u20116\n\nwhere , , , , , and are the thermal expansion coefficients and volumes\nof the system and the \u03b1 and \u03b2 microstates, respectively. For\nsimplification, let us assume both microstates have similar positive\nthermal expansion, i.e. , and becomes\n\nEq. 9\u20117\n\nshows that it is the combination of volume difference and that\ndetermines the macroscopic thermal expansion. By setting , one obtains\n\nEq. 9\u20118\n\nFor , for $p^{\\beta} \\geq 0$, and for , at $p^{\\beta} \\rightarrow 0$ .\nThe readers are reminded that the sign of is the same as the sign of .\n", "metadata": {} } ] diff --git a/src/psu410/src/experimental_data_for_thermodynamic_modeling/additional_methods.ipynb b/src/psu410/src/experimental_data_for_thermodynamic_modeling/additional_methods.ipynb deleted file mode 100644 index ffb33f4..0000000 --- a/src/psu410/src/experimental_data_for_thermodynamic_modeling/additional_methods.ipynb +++ /dev/null @@ -1,268 +0,0 @@ -{ - "nbformat": 4, - "nbformat_minor": 5, - "metadata": {}, - "cells": [ - { - "id": "442c290a", - "cell_type": "markdown", - "source": [ - "### Additional methods\n", - "\n", - "Electrical resistivities of different phases are usually different. A\n", - "change of slope of electric resistivity as a function of composition or\n", - "temperature or pressure reflects a phase transformation. This technique\n", - "is simple and reliable.\n", - "\n", - "Magnetic transitions can be measured using a vibrating sample or\n", - "superconducting quantum interference device (SQUID) magnetometer by\n", - "determining the magnetic moment of a sample in presence of an applied\n", - "magnetic field. Magnetic field lines form closed loops, resulting in an\n", - "external dipolar and demagnetizing field in a finite-sized sample. The\n", - "effective field sensed by the sample is thus the difference between the\n", - "applied field and the demagnetizing field. The magnetic transition\n", - "temperature is evaluated from the Arrott plots where the ratio of\n", - "magnetic field over magnetization with a proper exponent is plotted with\n", - "respect to the magnetization with another proper exponent for a series\n", - "of temperatures. Those proper exponents result in parallel isotherm\n", - "lines, and the isotherm line intersecting the origin corresponds to the\n", - "magnetic transition temperature.\n", - "\n", - "Thin films with composition gradients, commonly referred to as\n", - "combinatorial libraries, can be used to study the phase relations\n", - "similar to diffusion couples/multiples though the results may differ due\n", - "to the effects of surface and potential interactions with the substrate.\n", - "\n", - "Phase relations at high pressures are measured in equipment using\n", - "diamond anvil cells (DAC) or multi-anvil devices. The high pressure is\n", - "realized by decreasing the area, i.e. the anvil culet size. Pressures up\n", - "to 100 GPa can be created in DAC with a culet size of 0.3mm for small\n", - "samples in the order of 0.2 to 0.4mm. For large samples, the\n", - "large-volume presses (LVP) technique is developed, typically using WC\n", - "with the pressure mostly limited to 30 GPa and the sample size ranging\n", - "from 1mm3 to 1cm3. The pressure can be measured by\n", - "either the ruby (Cr3+ doped Al2O3)\n", - "fluorescence line shift or the molar volume of a pressure marker by\n", - "X-ray diffraction. The samples in DAC apparatus are heated by laser or\n", - "resistive wire or a small heater around the samples, while high\n", - "temperatures in LVPs are achieved by resistive heaters. Crystal\n", - "structures are detected by in-situ X-ray or synchrotron diffraction.\n", - "Attentions need to be paid to temperature and pressure homogeneity and\n", - "the non-hydrostatic stresses, which are both better controlled in LVP\n", - "equipment.\n" - ], - "metadata": {} - }, - { - "id": "9d5856c5", - "cell_type": "markdown", - "source": [ - "### Combustion, direct reaction, and heat capacity calorimetry\n", - "\n", - "In combustion calorimetry, the sample is ignited and reacts with\n", - "reactive gases like oxygen or fluorine. To accurately calculate the\n", - "enthalpy of formation from the enthalpy of combustion, the reliable\n", - "characterization of reactants and reaction products is critical, such as\n", - "incomplete combustion, impurities in the reactants which are often\n", - "ill-defined, and more than one reaction gaseous species and condensed\n", - "phases. Combustion calorimeters are usually of isoperibol type around\n", - "room temperature in a water bath.\n", - "\n", - "The direct reaction calorimetry is similar to the combustion calorimetry\n", - "though at high temperatures in heat-flux or adiabatic environments. The\n", - "partial enthalpy of reaction can also be measured if the partial\n", - "pressure of volatile species can be controlled and measured. The key\n", - "factor for accurate results is that both the reactants and reaction\n", - "products are well characterized and the reaction goes to completion\n", - "quickly like in the combustion calorimetry. For reactive reactants,\n", - "special procedure is needed to avoid their loss before the reaction\n", - "takes place.\n", - "\n", - "Heat capacity is defined as the amount of heat needed to increase the\n", - "temperature by 1K as shown by , and its integration with respect to\n", - "temperature from 0K gives entropy as shown by . At low temperatures,\n", - "adiabatic calorimetry gives more accurate data of heat capacity though\n", - "it is time consuming and requires complex instruments. At high\n", - "temperatures, the efficient and less accurate DSC method is more widely\n", - "used.\n" - ], - "metadata": {} - }, - { - "id": "4c2375c8", - "cell_type": "markdown", - "source": [ - "### Diffusion couples/multiples\n", - "\n", - "The major challenge in the equilibrated material approach is to ensure\n", - "that the whole sample reaches equilibrium. On the other hand the\n", - "diffusion couple/multiple technique does not require the whole sample to\n", - "be in equilibrium and is based on the assumption that any two phases in\n", - "contact are in equilibrium with each other at the phase interface, and\n", - "the phase compositions can be obtained by extrapolation of concentration\n", - "profiles in the two phases to the phase interface. Since the total\n", - "system of a diffusion couple is not at equilibrium, many kinetic\n", - "phenomena related to diffusion can be studied in a diffusion couple,\n", - "such as interdiffusion coefficients, parabolic growth kinetics of\n", - "product layer thickness, diffusion path (represented by the local\n", - "overall compositions) in ternary and multi-component systems for\n", - "visualizing the microstructure of reaction zones, and other properties,\n", - "all as a function of composition, which are beyond the scope of the\n", - "present book.\n", - "\n", - "Typical diffusion couples are in solid-state with two materials brought\n", - "into intimate contact to allow diffusion of elements between the two\n", - "materials though solid-liquid diffusion couples are also used. The\n", - "contacting faces are commonly ground and polished flat, clamped together\n", - "using mechanical mechanisms, and annealed at high temperatures where\n", - "diffusion can take place to a significant degree in a matter of days,\n", - "weeks or months. The samples are then quenched to retain the\n", - "high-temperature equilibrium features. For metallic systems, diffusion\n", - "couples can also be prepared by eletrolytical and electroless plating\n", - "techniques. It is important to avoid the formation of liquid during\n", - "annealing as it ruins the sample geometry. Furthermore, good adherence\n", - "at the interfaces is critical for reliable data.\n", - "\n", - "Since diffusion couples are not in a fully equilibrium state, the\n", - "tie-lines between two phases at the phase interfaces need to be obtained\n", - "by extrapolations of concentration profiles in neighbouring phases. The\n", - "electron propagation in quantitative EPMA is typically in the range of\n", - "1-2\u03bcm, yielding an excitation volume of approximately 2-4\u03bcm diameter and\n", - "a requirement of reasonable layer widths of phases on both sides of the\n", - "interface for accurate extrapolation. Therefore, the reliable\n", - "composition of a single phase must be taken several micrometers away\n", - "from the interface. When steep concentration gradients exist near the\n", - "interfaces, the extrapolation may lead to large errors, and analytical\n", - "electron microscopy is then needed. Furthermore, the fluorescence\n", - "effects, where the primary excitation can be powerful enough to excite\n", - "other elements in the sample, resulting in enhanced X-ray production and\n", - "the need of proper corrections. For a new phase to become observable\n", - "experimentally, it must nucleate and grow to reach the resolution limit\n", - "of analytical tools. It is thus not uncommon that some known equilibrium\n", - "phases are not found, and some non-equilibrium phases form. One way to\n", - "circumvent this issue is to use incremental diffusion couples with\n", - "narrow concentrations next to the phase of interest so that only this\n", - "phase may be formed.\n", - "\n", - "For ternary and higher-order systems, a more efficient approach can be\n", - "used by placing a thin layer of the third alloy between two alloys. The\n", - "thin central layer is eventually consumed, and the diffusion path is not\n", - "fixed as in the semi-infinite diffusion couples. The phase compositions\n", - "change continuously with time as a result of the overlapping of two\n", - "quasi-equilibrated diffusion zones.\n", - "\n", - "A diffusion multiple contains three or more pure elements or alloys of\n", - "different compositions and is a sample with multiple diffusion couples\n", - "and diffusion triples in it. It is more efficient in terms of both\n", - "materials and time in comparison with equilibrated alloys and diffusion\n", - "couples. All alloy blocks are prepared individually and sealed in vacuum\n", - "in a cylinder which is also used as one alloy for the diffusion\n", - "multiple. The sealed cylinder with vacuum inside also serves as the can\n", - "for subsequent hot isostatic pressing to achieve intimate interfacial\n", - "contacts. The cylinder can then be cut into disks for further annealing\n", - "treatments. A broad range of design strategies is needed for complex\n", - "diffusion multiples along with automated plotting procedures due to the\n", - "large amounts of EMPA data. The major source of error lies in the\n", - "extraction of tie-lines from EPMA results due to very condensed\n", - "information in a small area and the deviation of scanned lines from\n", - "those perpendicular to the interface.\n", - "\n", - "In terms of local equilibrium characteristics of diffusion\n", - "couples/multiples, it is evident that in the equilibrated materials\n", - "approach, it may not be necessary to reach the full equilibrium for the\n", - "whole sample if one is only interested in the local equilibrium\n", - "compositions between two neighboring phases, which can even provide\n", - "information on metastable extensions of two phases if the two phases are\n", - "in a metastable equilibrium at the annealing temperature.\n" - ], - "metadata": {} - }, - { - "id": "1b1bfe27", - "cell_type": "markdown", - "source": [ - "### Equilibrated materials\n", - "\n", - "The most common method to determine phase equilibria is to use\n", - "equilibrated materials. This method typically involves material\n", - "preparation through high temperature melting or powder metallurgy,\n", - "homogenization heat treatment, isothermal or cooling/heating procedure,\n", - "and identifications of crystal structures and phase compositions. It is\n", - "important to avoid macro-inhomogeneity as it can be difficult remove the\n", - "inhomogeneity in subsequent treatments. It is also important to use\n", - "starting materials of highest purity and minimize the loss and\n", - "contamination of materials during the whole experiments using a\n", - "protective atmosphere of inert gas or vacuum. The typical melting\n", - "techniques include high temperature furnaces with crucibles, arc\n", - "melting, and induction melting. Attentions need to be paid for possible\n", - "reactions between materials and crucibles/containers, which can be\n", - "avoided by levitating the materials through electromagnetic fields or\n", - "other means. In addition to use pure elements as raw materials, master\n", - "alloys with well-controlled compositions are often utilized because the\n", - "compositions and melting properties of master alloys are usually much\n", - "closer to those of final materials than the pure elements. For materials\n", - "with very high melting temperature or volatile components, the powder\n", - "metallurgy method can be used where compacts are made, capsulated, and\n", - "sintered.\n", - "\n", - "Homogenization during subsequent heat treatment is achieved through\n", - "diffusion, in which time and temperature are two important parameters.\n", - "To accelerate the homogenization process, the heat treatment temperature\n", - "should be as close to the solidus temperature as possible taking into\n", - "account the composition inhomogeneity with variable solidus\n", - "temperatures. When there are phase transformations taking place during\n", - "the heat treatment, extra time is needed for the heat treatment.\n", - "\n", - "The phase boundaries are then determined through measurements of either\n", - "compositions of individual phases that are in equilibrium under constant\n", - "temperature, pressure/stress/strain, and electric/magnetic fields or\n", - "discontinuity in some physical properties of materials due to a phase\n", - "transition from the continuous change of temperature or\n", - "pressure/stress/strain or electric/magnetic fields. The measurement of\n", - "compositions is usually carried out at ambient conditions so it is\n", - "necessary that the phases are fully equilibrated at experimental\n", - "conditions, which requires rigorous verifications, and can be \u201cquenched\u201d\n", - "to ambient conditions to remain unaltered during \u201cquenching\u201d, at least\n", - "in terms of compositions. The compositions are typically measured by\n", - "scanning electron microscopy (SEM) equipped with energy dispersive\n", - "spectrometer (EDS) or wavelength dispersive spectroscopy (WDS), with a\n", - "micro-level spatial resolution and a better compositional resolution of\n", - "WDS than EDS. A dedicated SEM with WDS gives another important, widely\n", - "used composition measurement technique called electron probe\n", - "microanalysis (EMPA). For submicron-sized phases, analytical\n", - "transmission electron microscopy equipped with EDS can be used though\n", - "care must be taken to avoid interference from neighboring phases.\n", - "\n", - "To accurately identify phases in equilibrium under experimental\n", - "conditions, in-situ characterizations are necessary, which complicates\n", - "the experimentation. An alternative indirect method is to measure a\n", - "physical property that changes discontinuously or dramatically for a\n", - "first- or second-order phase transition, such as heat, volume, electric\n", - "conductivity, and magnetization. There are two widely used techniques in\n", - "measuring heat: differential thermal analysis (DTA) and differential\n", - "scanning calorimetry (DSC). Both attempt to measure the difference in\n", - "temperature with the same amount of power supplied between a sample and\n", - "an inert standard during heating or cooling. A DSC may also measure the\n", - "difference in the amount of power supplied to keep their temperatures\n", - "identical. The deviation of this difference from a baseline indicates a\n", - "phase transition in the sample and is plotted as a function of time or\n", - "temperature of either the sample or the inert standard. It is evident\n", - "that the major challenges in both DTA and DSC techniques are to reach\n", - "thermal equilibrium between the sample/standard and the instrument and\n", - "the thermodynamic equilibrium within the sample due to the continuous\n", - "heating or cooling. The thermal equilibrium can be improved or mitigated\n", - "through various methods such as sample shape and size, crucible\n", - "selection, mixture with the material used for the inert standard, and\n", - "with thermocouples in direct contact with the sample and the inert\n", - "standard. However, the thermodynamic equilibrium within the sample can\n", - "only be reached when the heating/cooling rate is comparable with the\n", - "rate of the phase transition in the sample, which is almost impossible\n", - "if the phase transition typically involves diffusion in solid phases.\n", - "Therefore, extreme care is needed in interpreting the temperature\n", - "determination and the amount of heat associated with the DTA/DSC curves\n", - "as discussed in detail in the reference \\[11\\].\n" - ], - "metadata": {} - } - ] -} \ No newline at end of file diff --git a/src/psu410/src/experimental_data_for_thermodynamic_modeling/index.ipynb b/src/psu410/src/experimental_data_for_thermodynamic_modeling/index.ipynb index 84a2a3f..cf88515 100644 --- a/src/psu410/src/experimental_data_for_thermodynamic_modeling/index.ipynb +++ b/src/psu410/src/experimental_data_for_thermodynamic_modeling/index.ipynb @@ -4,30 +4,9 @@ "metadata": {}, "cells": [ { - "id": "8fcdf199", + "id": "df761675", "cell_type": "markdown", - "source": [ - "# Experimental data for thermodynamic modeling\n", - "\n", - "The most widely used thermodynamic modeling technique is the CALPHAD\n", - "(CALculation of PHAse Diagram) method to be discussed in detail in\n", - "Chapter 6. The input data in the evaluations of thermodynamic model\n", - "parameters have been primarily from experiments and estimations until\n", - "first-principles calculations based on the density functional theory\n", - "\\[8\\] became a user tool in the later 1990\u2019s. Experimental data include\n", - "both thermodynamic and phase equilibrium data, while the\n", - "first-principles calculations, which only provide thermodynamic data of\n", - "individual phases, are discussed more extensively in Chapter .\n", - "\n", - "The three recently published books summarize the commonly used methods\n", - "on experimental measurements of thermodynamic properties of single \\[9\\]\n", - "and multiple phases \\[10\\] and phase diagrams \\[11\\]. They are briefly\n", - "discussed here, and readers are referred to these books for details. The\n", - "main techniques for crystal structure analysis include X-ray\n", - "diffraction, electron backscatter diffraction (EBSD), electron\n", - "diffraction in transmission electron microscopy, neutral scattering, and\n", - "synchrotron scattering, which are not discussed in this book.\n" - ], + "source": "# Experimental data for thermodynamic modeling\n\nThe most widely used thermodynamic modeling technique is the CALPHAD\n(CALculation of PHAse Diagram) method to be discussed in detail in\nChapter 6. The input data in the evaluations of thermodynamic model\nparameters have been primarily from experiments and estimations until\nfirst-principles calculations based on the density functional theory\n\\[8\\] became a user tool in the later 1990\u2019s. Experimental data include\nboth thermodynamic and phase equilibrium data, while the\nfirst-principles calculations, which only provide thermodynamic data of\nindividual phases, are discussed more extensively in Chapter .\n\nThe three recently published books summarize the commonly used methods\non experimental measurements of thermodynamic properties of single \\[9\\]\nand multiple phases \\[10\\] and phase diagrams \\[11\\]. They are briefly\ndiscussed here, and readers are referred to these books for details. The\nmain techniques for crystal structure analysis include X-ray\ndiffraction, electron backscatter diffraction (EBSD), electron\ndiffraction in transmission electron microscopy, neutral scattering, and\nsynchrotron scattering, which are not discussed in this book.\n", "metadata": {} } ] diff --git a/src/psu410/src/experimental_data_for_thermodynamic_modeling/phase_equilibrium_data_from_experiments.ipynb b/src/psu410/src/experimental_data_for_thermodynamic_modeling/phase_equilibrium_data_from_experiments.ipynb index ef58950..5420b59 100644 --- a/src/psu410/src/experimental_data_for_thermodynamic_modeling/phase_equilibrium_data_from_experiments.ipynb +++ b/src/psu410/src/experimental_data_for_thermodynamic_modeling/phase_equilibrium_data_from_experiments.ipynb @@ -4,58 +4,27 @@ "metadata": {}, "cells": [ { - "id": "18183ae3", + "id": "fb60ba1d", "cell_type": "markdown", - "source": [ - "## Phase equilibrium data from experiments\n" - ], + "source": "## Phase equilibrium data from experiments\n", "metadata": {} }, { - "id": "18e4172d", + "id": "977e7a47", "cell_type": "markdown", - "source": [ - "### Solution calorimetry\n", - "\n", - "The book edited by Marsh and O\u2019Hare \\[12\\] documented the detailed\n", - "experimental techniques used for solution calorimetry. In one\n", - "experiment, the enthalpy of solution of a single phase is measured in a\n", - "particular solvent. To convert this enthalpy of solution into enthalpy\n", - "of formation of the phase, a thermodynamic cycle is setup for a chemical\n", - "reaction to form this phase from either constitutive pure elements or\n", - "compounds. Therefore in another experiment, the enthalpy of solution of\n", - "constitutive pure elements or compounds is measured in the solvent as\n", - "identical as possible to that used in the first experiment. The\n", - "difference of the two enthalpies of solution thus gives the enthalpy of\n", - "formation of the single phase from its constitutive elements or\n", - "compounds at the temperature of the samples before they are dropped into\n", - "the solvent, usually at room temperature.\n", - "\n", - "The solvent can be aqueous solutions at ambient temperatures and\n", - "pressures or metallic/salt/oxide melts at high temperatures under either\n", - "adiabatic or isoperibol conditions. The adiabatic calorimetry measures\n", - "the temperature change of the solvent and is usually more accurate than\n", - "the isoperibol calorimetry that measures the heat generated during the\n", - "dissolution, though the adiabatic ones are with more complex\n", - "instruments. It is important that the choices of solvent and temperature\n", - "ensure the complete dissolution of all substances into the solvent to\n", - "form a homogeneous solution. Furthermore, the effect of dilution and of\n", - "changes in solvent composition needs to be considered in the calculation\n", - "of enthalpy of solution.\n", - "\n", - "A large number of solvents are used. For aqueous solvent, hydrofluoric\n", - "acid or mixtures of HF and HCl are often used. For oxides, buffer-type\n", - "systems are typical such as lead and alkali borates and alkali\n", - "tungstates/molybdates. For metallic phases, low melting metals such as\n", - "Sn, Bi, In, Pb, and Cd, sometimes Al and Cu, are used. Factors such as\n", - "solubility, dissolution kinetics, thermal history, stirring, heat flow,\n", - "particle size, and system size are to be optimized for accurate\n", - "measurements. To enhance solution kinetics, the compound to be studied\n", - "can be mixed with another element or compound so that the mixture can\n", - "form liquid in the solvent at higher reaction rate. In any case, it is\n", - "important to calibrate the system with pure elements and compounds of\n", - "known enthalpy of formation.\n" - ], + "source": "### Equilibrated materials\n\nThe most common method to determine phase equilibria is to use\nequilibrated materials. This method typically involves material\npreparation through high temperature melting or powder metallurgy,\nhomogenization heat treatment, isothermal or cooling/heating procedure,\nand identifications of crystal structures and phase compositions. It is\nimportant to avoid macro-inhomogeneity as it can be difficult remove the\ninhomogeneity in subsequent treatments. It is also important to use\nstarting materials of highest purity and minimize the loss and\ncontamination of materials during the whole experiments using a\nprotective atmosphere of inert gas or vacuum. The typical melting\ntechniques include high temperature furnaces with crucibles, arc\nmelting, and induction melting. Attentions need to be paid for possible\nreactions between materials and crucibles/containers, which can be\navoided by levitating the materials through electromagnetic fields or\nother means. In addition to use pure elements as raw materials, master\nalloys with well-controlled compositions are often utilized because the\ncompositions and melting properties of master alloys are usually much\ncloser to those of final materials than the pure elements. For materials\nwith very high melting temperature or volatile components, the powder\nmetallurgy method can be used where compacts are made, capsulated, and\nsintered.\n\nHomogenization during subsequent heat treatment is achieved through\ndiffusion, in which time and temperature are two important parameters.\nTo accelerate the homogenization process, the heat treatment temperature\nshould be as close to the solidus temperature as possible taking into\naccount the composition inhomogeneity with variable solidus\ntemperatures. When there are phase transformations taking place during\nthe heat treatment, extra time is needed for the heat treatment.\n\nThe phase boundaries are then determined through measurements of either\ncompositions of individual phases that are in equilibrium under constant\ntemperature, pressure/stress/strain, and electric/magnetic fields or\ndiscontinuity in some physical properties of materials due to a phase\ntransition from the continuous change of temperature or\npressure/stress/strain or electric/magnetic fields. The measurement of\ncompositions is usually carried out at ambient conditions so it is\nnecessary that the phases are fully equilibrated at experimental\nconditions, which requires rigorous verifications, and can be \u201cquenched\u201d\nto ambient conditions to remain unaltered during \u201cquenching\u201d, at least\nin terms of compositions. The compositions are typically measured by\nscanning electron microscopy (SEM) equipped with energy dispersive\nspectrometer (EDS) or wavelength dispersive spectroscopy (WDS), with a\nmicro-level spatial resolution and a better compositional resolution of\nWDS than EDS. A dedicated SEM with WDS gives another important, widely\nused composition measurement technique called electron probe\nmicroanalysis (EMPA). For submicron-sized phases, analytical\ntransmission electron microscopy equipped with EDS can be used though\ncare must be taken to avoid interference from neighboring phases.\n\nTo accurately identify phases in equilibrium under experimental\nconditions, in-situ characterizations are necessary, which complicates\nthe experimentation. An alternative indirect method is to measure a\nphysical property that changes discontinuously or dramatically for a\nfirst- or second-order phase transition, such as heat, volume, electric\nconductivity, and magnetization. There are two widely used techniques in\nmeasuring heat: differential thermal analysis (DTA) and differential\nscanning calorimetry (DSC). Both attempt to measure the difference in\ntemperature with the same amount of power supplied between a sample and\nan inert standard during heating or cooling. A DSC may also measure the\ndifference in the amount of power supplied to keep their temperatures\nidentical. The deviation of this difference from a baseline indicates a\nphase transition in the sample and is plotted as a function of time or\ntemperature of either the sample or the inert standard. It is evident\nthat the major challenges in both DTA and DSC techniques are to reach\nthermal equilibrium between the sample/standard and the instrument and\nthe thermodynamic equilibrium within the sample due to the continuous\nheating or cooling. The thermal equilibrium can be improved or mitigated\nthrough various methods such as sample shape and size, crucible\nselection, mixture with the material used for the inert standard, and\nwith thermocouples in direct contact with the sample and the inert\nstandard. However, the thermodynamic equilibrium within the sample can\nonly be reached when the heating/cooling rate is comparable with the\nrate of the phase transition in the sample, which is almost impossible\nif the phase transition typically involves diffusion in solid phases.\nTherefore, extreme care is needed in interpreting the temperature\ndetermination and the amount of heat associated with the DTA/DSC curves\nas discussed in detail in the reference \\[11\\].\n", + "metadata": {} + }, + { + "id": "b173f28f", + "cell_type": "markdown", + "source": "### Diffusion couples/multiples\n\nThe major challenge in the equilibrated material approach is to ensure\nthat the whole sample reaches equilibrium. On the other hand the\ndiffusion couple/multiple technique does not require the whole sample to\nbe in equilibrium and is based on the assumption that any two phases in\ncontact are in equilibrium with each other at the phase interface, and\nthe phase compositions can be obtained by extrapolation of concentration\nprofiles in the two phases to the phase interface. Since the total\nsystem of a diffusion couple is not at equilibrium, many kinetic\nphenomena related to diffusion can be studied in a diffusion couple,\nsuch as interdiffusion coefficients, parabolic growth kinetics of\nproduct layer thickness, diffusion path (represented by the local\noverall compositions) in ternary and multi-component systems for\nvisualizing the microstructure of reaction zones, and other properties,\nall as a function of composition, which are beyond the scope of the\npresent book.\n\nTypical diffusion couples are in solid-state with two materials brought\ninto intimate contact to allow diffusion of elements between the two\nmaterials though solid-liquid diffusion couples are also used. The\ncontacting faces are commonly ground and polished flat, clamped together\nusing mechanical mechanisms, and annealed at high temperatures where\ndiffusion can take place to a significant degree in a matter of days,\nweeks or months. The samples are then quenched to retain the\nhigh-temperature equilibrium features. For metallic systems, diffusion\ncouples can also be prepared by eletrolytical and electroless plating\ntechniques. It is important to avoid the formation of liquid during\nannealing as it ruins the sample geometry. Furthermore, good adherence\nat the interfaces is critical for reliable data.\n\nSince diffusion couples are not in a fully equilibrium state, the\ntie-lines between two phases at the phase interfaces need to be obtained\nby extrapolations of concentration profiles in neighbouring phases. The\nelectron propagation in quantitative EPMA is typically in the range of\n1-2\u03bcm, yielding an excitation volume of approximately 2-4\u03bcm diameter and\na requirement of reasonable layer widths of phases on both sides of the\ninterface for accurate extrapolation. Therefore, the reliable\ncomposition of a single phase must be taken several micrometers away\nfrom the interface. When steep concentration gradients exist near the\ninterfaces, the extrapolation may lead to large errors, and analytical\nelectron microscopy is then needed. Furthermore, the fluorescence\neffects, where the primary excitation can be powerful enough to excite\nother elements in the sample, resulting in enhanced X-ray production and\nthe need of proper corrections. For a new phase to become observable\nexperimentally, it must nucleate and grow to reach the resolution limit\nof analytical tools. It is thus not uncommon that some known equilibrium\nphases are not found, and some non-equilibrium phases form. One way to\ncircumvent this issue is to use incremental diffusion couples with\nnarrow concentrations next to the phase of interest so that only this\nphase may be formed.\n\nFor ternary and higher-order systems, a more efficient approach can be\nused by placing a thin layer of the third alloy between two alloys. The\nthin central layer is eventually consumed, and the diffusion path is not\nfixed as in the semi-infinite diffusion couples. The phase compositions\nchange continuously with time as a result of the overlapping of two\nquasi-equilibrated diffusion zones.\n\nA diffusion multiple contains three or more pure elements or alloys of\ndifferent compositions and is a sample with multiple diffusion couples\nand diffusion triples in it. It is more efficient in terms of both\nmaterials and time in comparison with equilibrated alloys and diffusion\ncouples. All alloy blocks are prepared individually and sealed in vacuum\nin a cylinder which is also used as one alloy for the diffusion\nmultiple. The sealed cylinder with vacuum inside also serves as the can\nfor subsequent hot isostatic pressing to achieve intimate interfacial\ncontacts. The cylinder can then be cut into disks for further annealing\ntreatments. A broad range of design strategies is needed for complex\ndiffusion multiples along with automated plotting procedures due to the\nlarge amounts of EMPA data. The major source of error lies in the\nextraction of tie-lines from EPMA results due to very condensed\ninformation in a small area and the deviation of scanned lines from\nthose perpendicular to the interface.\n\nIn terms of local equilibrium characteristics of diffusion\ncouples/multiples, it is evident that in the equilibrated materials\napproach, it may not be necessary to reach the full equilibrium for the\nwhole sample if one is only interested in the local equilibrium\ncompositions between two neighboring phases, which can even provide\ninformation on metastable extensions of two phases if the two phases are\nin a metastable equilibrium at the annealing temperature.\n", + "metadata": {} + }, + { + "id": "8e44cd11", + "cell_type": "markdown", + "source": "### Additional methods\n\nElectrical resistivities of different phases are usually different. A\nchange of slope of electric resistivity as a function of composition or\ntemperature or pressure reflects a phase transformation. This technique\nis simple and reliable.\n\nMagnetic transitions can be measured using a vibrating sample or\nsuperconducting quantum interference device (SQUID) magnetometer by\ndetermining the magnetic moment of a sample in presence of an applied\nmagnetic field. Magnetic field lines form closed loops, resulting in an\nexternal dipolar and demagnetizing field in a finite-sized sample. The\neffective field sensed by the sample is thus the difference between the\napplied field and the demagnetizing field. The magnetic transition\ntemperature is evaluated from the Arrott plots where the ratio of\nmagnetic field over magnetization with a proper exponent is plotted with\nrespect to the magnetization with another proper exponent for a series\nof temperatures. Those proper exponents result in parallel isotherm\nlines, and the isotherm line intersecting the origin corresponds to the\nmagnetic transition temperature.\n\nThin films with composition gradients, commonly referred to as\ncombinatorial libraries, can be used to study the phase relations\nsimilar to diffusion couples/multiples though the results may differ due\nto the effects of surface and potential interactions with the substrate.\n\nPhase relations at high pressures are measured in equipment using\ndiamond anvil cells (DAC) or multi-anvil devices. The high pressure is\nrealized by decreasing the area, i.e. the anvil culet size. Pressures up\nto 100 GPa can be created in DAC with a culet size of 0.3mm for small\nsamples in the order of 0.2 to 0.4mm. For large samples, the\nlarge-volume presses (LVP) technique is developed, typically using WC\nwith the pressure mostly limited to 30 GPa and the sample size ranging\nfrom 1mm3 to 1cm3. The pressure can be measured by\neither the ruby (Cr3+ doped Al2O3)\nfluorescence line shift or the molar volume of a pressure marker by\nX-ray diffraction. The samples in DAC apparatus are heated by laser or\nresistive wire or a small heater around the samples, while high\ntemperatures in LVPs are achieved by resistive heaters. Crystal\nstructures are detected by in-situ X-ray or synchrotron diffraction.\nAttentions need to be paid to temperature and pressure homogeneity and\nthe non-hydrostatic stresses, which are both better controlled in LVP\nequipment.\n", "metadata": {} } ] diff --git a/src/psu410/src/experimental_data_for_thermodynamic_modeling/thermodynamic_data_from_experiments.ipynb b/src/psu410/src/experimental_data_for_thermodynamic_modeling/thermodynamic_data_from_experiments.ipynb index d79ab67..32a09bc 100644 --- a/src/psu410/src/experimental_data_for_thermodynamic_modeling/thermodynamic_data_from_experiments.ipynb +++ b/src/psu410/src/experimental_data_for_thermodynamic_modeling/thermodynamic_data_from_experiments.ipynb @@ -4,63 +4,27 @@ "metadata": {}, "cells": [ { - "id": "2b11bcc5", + "id": "904c370b", "cell_type": "markdown", - "source": [ - "## Thermodynamic data from experiments\n", - "\n", - "Broadly speaking, thermodynamic data represents the values of Gibbs free\n", - "energy and all its first and second derivatives. For thermochemical\n", - "properties as the main topic of the book, calorimetric, electrochemical,\n", - "and vapor pressure methods are the primary techniques used with the\n", - "first for accurate measurement of enthalpy and entropy, and the latter\n", - "two for direct determination of Gibbs energy and activity. The\n", - "electrochemical method is discussed in Chapter . The calorimetric method\n", - "is divided into solution, combustion, direct reaction, and heat capacity\n", - "calorimetry, respectively, which all involve chemical reactions to be\n", - "discussed in detail in Chapter . The vapor pressure method involves the\n", - "equilibrium of volatile species between gas and samples and is divided\n", - "into Knudsen effusion and equilibration methods, respectively.\n" - ], + "source": "## Thermodynamic data from experiments\n\nBroadly speaking, thermodynamic data represents the values of Gibbs free\nenergy and all its first and second derivatives. For thermochemical\nproperties as the main topic of the book, calorimetric, electrochemical,\nand vapor pressure methods are the primary techniques used with the\nfirst for accurate measurement of enthalpy and entropy, and the latter\ntwo for direct determination of Gibbs energy and activity. The\nelectrochemical method is discussed in Chapter . The calorimetric method\nis divided into solution, combustion, direct reaction, and heat capacity\ncalorimetry, respectively, which all involve chemical reactions to be\ndiscussed in detail in Chapter . The vapor pressure method involves the\nequilibrium of volatile species between gas and samples and is divided\ninto Knudsen effusion and equilibration methods, respectively.\n", "metadata": {} }, { - "id": "ee7088c1", + "id": "666b22ae", "cell_type": "markdown", - "source": [ - "### Vapor pressure method\n", - "\n", - "In the Knudsen effusion method, a small amount of volatile species in\n", - "the gas phase effuses through a small orifice of 0.1 to 1 mm with\n", - "negligible influence of the equilibrium in the Knudsen cell. The vapor\n", - "is ionized and analyzed with a mass spectrometer. The partial pressure\n", - "of a species can be calculated from its ionization area and intensity\n", - "through a calibration factor determined by a reference material with\n", - "known partial pressure. For high temperature measurements, care must be\n", - "taken to avoid the reactions between cell and sample and fragmentation\n", - "of gas species on ionization. The typical vapor pressure range is\n", - "between 10-7 and 10 Pa.\n", - "\n", - "In various equilibration methods, the total vapor pressure is usually\n", - "measured directly using pressure gauges in the range of 10-7\n", - "and 100 kPa. Other direct or indirect methods include\n", - "\n", - "thermogravimetric method for measuring the vapor mass\n", - "\n", - "atomic absorption spectroscopy for gas composition\n", - "\n", - "measurement of sample composition equilibrated with a gas of\n", - "well-defined activity of the volatile species\n", - "\n", - "the dew point method in which the condensation temperature of the\n", - "volatile component is measured from the vapor equilibrated with the\n", - "sample at a higher temperature,\n", - "\n", - "the chemical transport method to be discussed in Chapter .\n", - "\n", - "The main error in all these methods is often due to inadequate\n", - "equilibration between vapor and sample.\n" - ], + "source": "### Solution calorimetry\n\nThe book edited by Marsh and O\u2019Hare \\[12\\] documented the detailed\nexperimental techniques used for solution calorimetry. In one\nexperiment, the enthalpy of solution of a single phase is measured in a\nparticular solvent. To convert this enthalpy of solution into enthalpy\nof formation of the phase, a thermodynamic cycle is setup for a chemical\nreaction to form this phase from either constitutive pure elements or\ncompounds. Therefore in another experiment, the enthalpy of solution of\nconstitutive pure elements or compounds is measured in the solvent as\nidentical as possible to that used in the first experiment. The\ndifference of the two enthalpies of solution thus gives the enthalpy of\nformation of the single phase from its constitutive elements or\ncompounds at the temperature of the samples before they are dropped into\nthe solvent, usually at room temperature.\n\nThe solvent can be aqueous solutions at ambient temperatures and\npressures or metallic/salt/oxide melts at high temperatures under either\nadiabatic or isoperibol conditions. The adiabatic calorimetry measures\nthe temperature change of the solvent and is usually more accurate than\nthe isoperibol calorimetry that measures the heat generated during the\ndissolution, though the adiabatic ones are with more complex\ninstruments. It is important that the choices of solvent and temperature\nensure the complete dissolution of all substances into the solvent to\nform a homogeneous solution. Furthermore, the effect of dilution and of\nchanges in solvent composition needs to be considered in the calculation\nof enthalpy of solution.\n\nA large number of solvents are used. For aqueous solvent, hydrofluoric\nacid or mixtures of HF and HCl are often used. For oxides, buffer-type\nsystems are typical such as lead and alkali borates and alkali\ntungstates/molybdates. For metallic phases, low melting metals such as\nSn, Bi, In, Pb, and Cd, sometimes Al and Cu, are used. Factors such as\nsolubility, dissolution kinetics, thermal history, stirring, heat flow,\nparticle size, and system size are to be optimized for accurate\nmeasurements. To enhance solution kinetics, the compound to be studied\ncan be mixed with another element or compound so that the mixture can\nform liquid in the solvent at higher reaction rate. In any case, it is\nimportant to calibrate the system with pure elements and compounds of\nknown enthalpy of formation.\n", + "metadata": {} + }, + { + "id": "75b2d203", + "cell_type": "markdown", + "source": "### Combustion, direct reaction, and heat capacity calorimetry\n\nIn combustion calorimetry, the sample is ignited and reacts with\nreactive gases like oxygen or fluorine. To accurately calculate the\nenthalpy of formation from the enthalpy of combustion, the reliable\ncharacterization of reactants and reaction products is critical, such as\nincomplete combustion, impurities in the reactants which are often\nill-defined, and more than one reaction gaseous species and condensed\nphases. Combustion calorimeters are usually of isoperibol type around\nroom temperature in a water bath.\n\nThe direct reaction calorimetry is similar to the combustion calorimetry\nthough at high temperatures in heat-flux or adiabatic environments. The\npartial enthalpy of reaction can also be measured if the partial\npressure of volatile species can be controlled and measured. The key\nfactor for accurate results is that both the reactants and reaction\nproducts are well characterized and the reaction goes to completion\nquickly like in the combustion calorimetry. For reactive reactants,\nspecial procedure is needed to avoid their loss before the reaction\ntakes place.\n\nHeat capacity is defined as the amount of heat needed to increase the\ntemperature by 1K as shown by , and its integration with respect to\ntemperature from 0K gives entropy as shown by . At low temperatures,\nadiabatic calorimetry gives more accurate data of heat capacity though\nit is time consuming and requires complex instruments. At high\ntemperatures, the efficient and less accurate DSC method is more widely\nused.\n", + "metadata": {} + }, + { + "id": "5fed3852", + "cell_type": "markdown", + "source": "### Vapor pressure method\n\nIn the Knudsen effusion method, a small amount of volatile species in\nthe gas phase effuses through a small orifice of 0.1 to 1 mm with\nnegligible influence of the equilibrium in the Knudsen cell. The vapor\nis ionized and analyzed with a mass spectrometer. The partial pressure\nof a species can be calculated from its ionization area and intensity\nthrough a calibration factor determined by a reference material with\nknown partial pressure. For high temperature measurements, care must be\ntaken to avoid the reactions between cell and sample and fragmentation\nof gas species on ionization. The typical vapor pressure range is\nbetween 10-7 and 10 Pa.\n\nIn various equilibration methods, the total vapor pressure is usually\nmeasured directly using pressure gauges in the range of 10-7\nand 100 kPa. Other direct or indirect methods include\n\nthermogravimetric method for measuring the vapor mass\n\natomic absorption spectroscopy for gas composition\n\nmeasurement of sample composition equilibrated with a gas of\nwell-defined activity of the volatile species\n\nthe dew point method in which the condensation temperature of the\nvolatile component is measured from the vapor equilibrated with the\nsample at a higher temperature,\n\nthe chemical transport method to be discussed in Chapter .\n\nThe main error in all these methods is often due to inadequate\nequilibration between vapor and sample.\n", "metadata": {} } ] diff --git a/src/psu410/src/firstprinciples_calculations_and_theory/bornoppenheimer_approximation.ipynb b/src/psu410/src/firstprinciples_calculations_and_theory/bornoppenheimer_approximation.ipynb deleted file mode 100644 index 37d7cdf..0000000 --- a/src/psu410/src/firstprinciples_calculations_and_theory/bornoppenheimer_approximation.ipynb +++ /dev/null @@ -1,388 +0,0 @@ -{ - "nbformat": 4, - "nbformat_minor": 5, - "metadata": {}, - "cells": [ - { - "id": "1ad8e47b", - "cell_type": "markdown", - "source": [ - "### Born-Oppenheimer Approximation\n", - "\n", - "For a time independent atomic system, it is often accurate enough to\n", - "write in or in terms of the electron coordinate **r** and nuclei\n", - "coordinate **R**,\n", - "\n", - "*Eq. 5\u201164* ,\n", - "\n", - "where *e* represents the electron charge and *i* and *j* label the\n", - "electrons, *I* and *J* the atomic nuclei, *ZI* the atomic\n", - "nuclear charge number of atom *I*, *e* the electron charge,\n", - "*me* the electron mass, *MI* the mass of atomic\n", - "nuclei *I*, the Laplace operator for electron *i*, and the Laplace\n", - "operator for atomic nuclei *I,* noting\n", - "\n", - "*Eq. 5\u201165* ,\n", - "\n", - "with respect to the Cartesian axis *x*, *y*, and *z*.\n", - "\n", - "Consider the fact that the electron mass is thousand times smaller than\n", - "the mass of the atomic nuclei, implying that the motions of the\n", - "electrons are much faster than the atomic nuclei, Born and Oppenheimer\n", - "proposed that the wave function of the whole system can be simply\n", - "approximated as the product of the electron wave function and the atomic\n", - "nuclei wave function as\n", - "\n", - "*Eq. 5\u201166* .\n", - "\n", - "With the auxiliary approximation of neglecting the dynamic coupling\n", - "between the motions of electrons and atomic nuclei, the Schr\u00f6dinger\n", - "equation for the motion of electrons becomes\n", - "\n", - "*Eq. 5\u201167* ,\n", - "\n", - "where\n", - "\n", - "*Eq. 5\u201168* ,\n", - "\n", - "and the Schr\u00f6dinger equation for the motion of atomic nuclei becomes\n", - "\n", - "*Eq. 5\u201169* ,\n", - "\n", - "where\n", - "\n", - "*Eq. 5\u201170* ,\n", - "\n", - "with\n", - "\n", - "*Eq. 5\u201171* ,\n", - "\n", - "where represents the complex conjugate of .\n" - ], - "metadata": {} - }, - { - "id": "63c06677", - "cell_type": "markdown", - "source": [ - "### Cluster Expansions\n", - "\n", - "Many properties of a solution phase such as energy are dependent on the\n", - "*configurations* - the arrangements of atoms on the lattice sites. In\n", - "cluster expansion \\[35, 42\\], the configuration dependence of properties\n", - "is formulated efficiently by a \u201clattice algebra\u201d which maps a\n", - "substitutional configuration into an Ising-like lattice model. Taking a\n", - "binary A1-*x*B*x* solution phase for instance, A\n", - "atoms are represented by the \u201cdown\u201d spins (*Si* = -1) and B\n", - "atoms are represented by the \u201cup\u201d spins (*Si* = +1) as\n", - "illustrated in . Using the cluster expansion technique, for a system\n", - "containing *N* atoms, the total energy of any alloy configuration *\u03c3* =\n", - "(*S*1, *S*2, \u2026, *SN*) can be\n", - "conveniently evaluated using the following Ising-like Hamiltonian:\n", - "\n", - "*Eq. 5\u2011147*\n", - "\n", - "where *J*\u2019s are the effective cluster interactions (ECI\u2019s); is a number\n", - "representing the atomic occupation at the lattice *i* under the\n", - "configuration *\u03c3*, which takes the values -1 and 1 for binary and -1, 0,\n", - "and 1 for ternary etc. In , the 2-site, 3-site, and 4-site correlations\n", - "are written as follows,\n", - "\n", - "Eq. 5\u2011148\n", - "\n", - "Eq. 5\u2011149\n", - "\n", - "Eq. 5\u2011150 .\n", - "\n", - "The expansion in would be exact as long as *all* the *n*-site\n", - "interactions are included. For a binary system, this can be observed by\n", - "the combination law that where is the number of *n*-site interactions.\n", - "However, in actual calculations, one never does an expansion to the\n", - "order *N* (containing 2*N* terms for binary) since it is too\n", - "long to be practical. In fact, since the interactions between widely\n", - "separated atoms are expected to be weaker than the interactions between\n", - "nearer atoms for most of the important properties, the expansion in is\n", - "usually truncated at certain distance to include only a few short-ranged\n", - "pair (2-site), triple (3-site), and up to the most, the quadruple\n", - "(4-site) interactions.\n", - "\n", - "Once a configuration is assigned, the *S*\u2019s are just the geometrical\n", - "factors. The common practice in the cluster expansion is that i) perform\n", - "first-principles calculations of a selected set of configurations\n", - "(around 20-100); ii) evaluate the values of *J*\u2019s using with the\n", - "energies from i); iii) use the fitted *J*\u2019s to predict the energy for a\n", - "desired set of configurations; and iv) make the ensemble average at a\n", - "given temperature for the energetics of the random alloys through Monte\n", - "Carlo simulations.\n" - ], - "metadata": {} - }, - { - "id": "69ae036f", - "cell_type": "markdown", - "source": [ - "### Debye-Gr\u00fcneisen approximation to the vibrational contribution \n", - "\n", - "Strictly speaking, the Debye theory is only accurate at very low\n", - "temperatures. It assumes a parabolic type of dependence of PDOS\u00a0on the\n", - "phonon frequency. This assumption is only correct at the scale of 10\u2019s K\n", - "because at low temperatures,\u00a0only the low frequency acoustic phonons are\n", - "activated which play the major role for the parabolic type of PDOS for\n", - "the low frequency range as shown in . That is why\u00a0there are two kinds of\n", - "Debye temperature: low-temperature Debye temperature and\n", - "high-temperature\u00a0Debye temperature. The low-temperature Debye\n", - "temperature can be strictly derived by fitting low temperature heat\n", - "capacity data. The high-temperature\u00a0Debye temperature is mostly a\n", - "phenomenological fitting parameter.\n", - "\n", - "The Debye model approximates the PDOS in by\n", - "\n", - "*Eq. 5\u201138* ,\n", - "\n", - "where is the so-called Debye cutoff frequency related to the Debye\n", - "temperature as\n", - "\n", - "*Eq. 5\u201139* .\n", - "\n", - "As the result, the vibrational contribution to Helmholtz energy under\n", - "the Debye approximation becomes,\n", - "\n", - "*Eq. 5\u201140*\n", - "\n", - "where *D*(*\u0398D/T*) is the Debye function given by .\n", - "\n", - "Under the Debye approximation, the formulation to calculate the entropy\n", - "becomes\n", - "\n", - "*Eq. 5\u201141* ,\n", - "\n", - "the formulation to calculate the internal energy becomes\n", - "\n", - "*Eq. 5\u201142* ,\n", - "\n", - "where\n", - "\n", - "*Eq. 5\u201143* ,\n", - "\n", - "and the formulation to calculate the heat capacity at constant volume\n", - "becomes\n", - "\n", - "*Eq. 5\u201144* .\n", - "\n", - "Here it is noted that is volume dependent which is often written in\n", - "terms of Gr\u00fcneisen constant:\n", - "\n", - "*Eq. 5\u201145* .\n", - "\n", - "It has been found that the dependence of on *V* usually is weak and\n", - "hence is often approximated as a constant. With , the formulation to\n", - "calculate the pressure becomes\n", - "\n", - "*Eq. 5\u201146* .\n", - "\n", - "An important result of the Debye approximation is that when , together\n", - "with , the heat capacity in is reduced to\n", - "\n", - "*Eq. 5\u201147* .\n", - "\n", - "This gives the Debye T3 law when the thermal electron\n", - "contribution is neglected. In the analysis of superconductor and heavy\n", - "Fermion materials, is often rewritten as\n", - "\n", - "*Eq. 5\u201148* ,\n", - "\n", - "which is more convenient for examining the heat capacity measured at\n", - "extremely low temperatures.\n" - ], - "metadata": {} - }, - { - "id": "72592df0", - "cell_type": "markdown", - "source": [ - "### Density functional theory (DFT) and 0 K Kohn-Sham equations\n", - "\n", - "The density functional theory advocates that the properties of a matter\n", - "are solely dictated by its electronic density distribution (or equally\n", - "say charge density), , in the real space. This is to say, that for an\n", - "arbitrary the total energy of the system, *E*, is always larger or equal\n", - "to a value, *E0*, called as the ground state energy:\n", - "\n", - "*Eq. 5\u201186* .\n", - "\n", - "In terms of variational principle, is equivalent to\n", - "\n", - "*Eq. 5\u201187* .\n", - "\n", - "Kohn and Sham \\[8\\] proposed to write the total energy as\n", - "\n", - "*Eq. 5\u201188* ,\n", - "\n", - "where represents the kinetic energy of the system, is the external\n", - "potential acting on the system, is the Hartree energy, and is the\n", - "so-called exchange-correlation energy with where and represent the\n", - "charge density of electrons with spin down and spin up, respectively .\n", - "Using\n", - "\n", - "*Eq. 5\u201189* ,\n", - "\n", - "together with the variational principle of , one can get the\n", - "one-electron Schr\u00f6dinger equation\n", - "\n", - "*Eq. 5\u201190*\n", - "\n", - "where *me* represents the mass of an electron and\n", - "\n", - "*Eq. 5\u201191*\n", - "\n", - "so that the total energy is obtained as\n", - "\n", - "*Eq. 5\u201192*\n", - "\n", - "The major challenge within DFT is that the accurate formulation of the\n", - "exchange-correlation energy is unknown. Except for the uniform electron\n", - "gas, no exact analytical form for the exchange-correlation energy has\n", - "yet been obtained. Therefore approximations must be made for the\n", - "exchange-correlation energy in calculating a realistic system. Until\n", - "now, the two most popular approximations are the local density\n", - "approximation (LDA) \\[30\\] and the generalized gradient approximation\n", - "(GGA) \\[31-32\\].\n", - "\n", - "The local density approximation (LDA) states that the\n", - "exchange-correlation energy is the same as that for a locally uniform\n", - "electron gas. In this case one can write *Vxc* as\n", - "\n", - "*Eq. 5\u201193* .\n", - "\n", - "Although this approximation is extremely simple, it works reasonably\n", - "well for many systems. The only remaining problem is to find an\n", - "approximate solution to . One of most employed parameterized expression\n", - "for is that by Perdew and Zunger \\[30\\].\n", - "\n", - "Many modern DFT codes use the more advanced generalized gradient\n", - "approximation (GGA) \\[31-32\\] to the exchange-correlation energy to\n", - "improve accuracy for certain physical properties. As the LDA\n", - "approximates the energy of the true density by the energy of a local\n", - "constant density, it fails in situations where the density undergoes\n", - "rapid changes such as in molecules. An improvement to this can be made\n", - "by considering the gradient of the electron density. Symbolically, this\n", - "can be written as\n", - "\n", - "*Eq. 5\u201194*\n", - "\n", - "The commonly used GGA is those due to Perdew et al. \\[31-32\\].\n" - ], - "metadata": {} - }, - { - "id": "68a79395", - "cell_type": "markdown", - "source": [ - "### Dynamical matrix and phonon mode\n", - "\n", - "Because of the periodicity of a crystal, one can make an initial guess\n", - "that the solutions of are elastic plane waves made of a collective\n", - "atomic vibrations \\[23, 33\\], from the harmonic approximation of ,\n", - "\n", - "*Eq. 5\u2011122* ,\n", - "\n", - "where *\u03c9* represents the frequency of the plane wave, and is a wave\n", - "vector designating the wave number and direction along which the plane\n", - "wave propagates. It should be pointed out that in is now independent of\n", - "the index *P.* That is, except for a phase factor, the atoms that are\n", - "equivalent by the translational symmetry among different primitive unit\n", - "cells will experience the same type of atomic motion, independent of the\n", - "positions of the primitive unit cell in the system. This is equivalent\n", - "to applying the periodic condition so that in obeys\n", - "\n", - "*Eq. 5\u2011123* .\n", - "\n", - "Note that is now independent of the index *P*.\n", - "\n", - "Furthermore, one wants to limit **q***t* in and to those\n", - "known as the exact wave vectors which represent a special set of points\n", - "in the reciprocal space that satisfy the condition\n", - "\n", - "*Eq. 5\u2011124* ,\n", - "\n", - "where is the Kronecker delta function. In fact, the number of equals to\n", - "the number of primitive unit cells contained in the system.\n", - "\n", - "Utilizing the translational invariance by which in \\[or in \\] depends on\n", - "*P* and *Q* only through the difference , the following Fourier\n", - "transformation can be employed to simplify\n", - "\n", - "*Eq. 5\u2011125* ,\n", - "\n", - "and one obtains\n", - "\n", - "*Eq. 5\u2011126* .\n", - "\n", - "The counterpart of with respect to is\n", - "\n", - "*Eq. 5\u2011127*\n", - "\n", - "is now an equation with 3*Na* degrees of freedom. At each ,\n", - "one can always find 3*Na* eigenvalues of (*i* = 1, \u2026,\n", - "3*Na*). The 3*Na* vibrations are often known as\n", - "phonon modes, noting again that *Na* is the number of atoms\n", - "in the primitive unit cell. Each of yields a set of which can be chosen\n", - "such that\n", - "\n", - "*Eq. 5\u2011128* ,\n", - "\n", - "where represents the complex conjugate of , and\n", - "\n", - "*Eq. 5\u2011129*\n", - "\n", - "Finally, for a solid, the summation over *Q* in can be abbreviated out,\n", - "resulting in, after transforming back to together with using\n", - "\n", - "*Eq. 5\u2011130* ,\n", - "\n", - "because of translational invariance by which depends on *P* and *Q* only\n", - "through the difference .\n", - "\n", - "In the case that a system is built by a parallelepiped multiplication of\n", - "the primitive unit cell with lattice vectors of **a***\u03b1*\n", - "(\u03b1=*x*, *y*, and *z*) in the form , in and takes the form\n", - "\n", - "*Eq. 5\u2011131* ,\n", - "\n", - "where = 0, 1, \u2026, , and is the primitive lattice vector in the reciprocal\n", - "space as\n", - "\n", - "*Eq. 5\u2011132* ,\n", - "\n", - "with *Va* representing the volume of the primitive unit cell\n", - "given as\n", - "\n", - "*Eq. 5\u2011133* .\n", - "\n", - "Combine and , it is easy to demonstrate\n", - "\n", - "*Eq. 5\u2011134* ,\n", - "\n", - "where is the Kronecker delta symbol.\n", - "\n", - "It can be shown that the **q** points defined in represent the exact\n", - "wave points. First, can be written as\n", - "\n", - "*Eq. 5\u2011135* ,\n", - "\n", - "where = 0, 1, \u2026, . Then, the left hand side of for the definition of\n", - "exact wave vector becomes\n", - "\n", - "*Eq. 5\u2011136* .\n", - "\n", - "Knowing the fact that\n", - "\n", - "*Eq. 5\u2011137* .\n", - "\n", - "Hence, it is proved the **q** points defined in are the exact wave\n", - "points.\n" - ], - "metadata": {} - } - ] -} \ No newline at end of file diff --git a/src/psu410/src/firstprinciples_calculations_and_theory/firstprinciples_approaches_to_disordered_alloys.ipynb b/src/psu410/src/firstprinciples_calculations_and_theory/firstprinciples_approaches_to_disordered_alloys.ipynb index 44553a1..b149d7d 100644 --- a/src/psu410/src/firstprinciples_calculations_and_theory/firstprinciples_approaches_to_disordered_alloys.ipynb +++ b/src/psu410/src/firstprinciples_calculations_and_theory/firstprinciples_approaches_to_disordered_alloys.ipynb @@ -4,52 +4,27 @@ "metadata": {}, "cells": [ { - "id": "92653dd4", + "id": "969becdf", "cell_type": "markdown", - "source": [ - "## First-principles approaches to disordered alloys\n", - "\n", - "First-principles calcuations discussed so far strictly rely on the exact\n", - "atomic positions in the unit cells. A brutal-force approach for random\n", - "solution phase would be to directly construct a large supercell and\n", - "randomly decorate the host lattice with different types of atoms. Such\n", - "an approach would necessarily require very large supercells to\n", - "adequately mimic the statistics of the random solutions. Since\n", - "first-principles methods are computationally constrained by the number\n", - "of atoms that one can treat, this brute-force approach is\n", - "computationally prohibitive. Take a binary\n", - "A1-*x*B*x* substitutional alloy as illustrated in\n", - "as an example, for a system containing *N* atoms, there can be\n", - "2*N* possible number of configurations, which is an\n", - "astronomically large number when *N* is large. It is an impossible task\n", - "to explore such a huge configuration space with available computing\n", - "resources.\n", - "\n", - "Figure \u2011. Mapping of a substitutional A1-xBx alloy\n", - "into an Ising-like lattice model \\[39-40\\].\n", - "\n", - "As a result, approximations must be made to the first-principles\n", - "calculations. At present, there are mainly three approaches to\n", - "calculating the disordered solution phases: the coherent potential\n", - "approximation (CPA) \\[41\\], the cluster expansion (CE) \\[42\\], and the\n", - "special quasirandom structures (SQS\u2019s) \\[43\\] approach.\n", - "\n", - "CPA \\[41\\] treats random alloys by considering the *average* occupations\n", - "of lattice sites in solving the Kohn-Sham equation. Since a mean-field\n", - "approach is employed, dependence of properties on the local environments\n", - "surrounding an atom is not treated explicitly in CPA. In a random\n", - "solution, there exists a distribution of local environments (e.g., in\n", - "bcc alloys, A or B surrounded by the various\n", - "A*X*B8-*X* coordination shells with *X* between 0\n", - "and 8), resulting in local environmentally-dependent quantities such as\n", - "charge transfer and local displacements of atoms from their ideal\n", - "lattice positions. Even in random A1-*x*B*x* solid\n", - "solutions, the average A-A, A-B and B-B bond lengths are generally\n", - "different. These effects can be considered by the CE and SQS approaches,\n", - "which are the focus of the next two subsections. In following\n", - "subsections, unless specifically noted, the formulism for the binary\n", - "system is discussed for the sake of simplicity.\n" - ], + "source": "## First-principles approaches to disordered alloys\n\nFirst-principles calcuations discussed so far strictly rely on the exact\natomic positions in the unit cells. A brutal-force approach for random\nsolution phase would be to directly construct a large supercell and\nrandomly decorate the host lattice with different types of atoms. Such\nan approach would necessarily require very large supercells to\nadequately mimic the statistics of the random solutions. Since\nfirst-principles methods are computationally constrained by the number\nof atoms that one can treat, this brute-force approach is\ncomputationally prohibitive. Take a binary\nA1-*x*B*x* substitutional alloy as illustrated in\nas an example, for a system containing *N* atoms, there can be\n2*N* possible number of configurations, which is an\nastronomically large number when *N* is large. It is an impossible task\nto explore such a huge configuration space with available computing\nresources.\n\nFigure \u2011. Mapping of a substitutional A1-xBx alloy\ninto an Ising-like lattice model \\[39-40\\].\n\nAs a result, approximations must be made to the first-principles\ncalculations. At present, there are mainly three approaches to\ncalculating the disordered solution phases: the coherent potential\napproximation (CPA) \\[41\\], the cluster expansion (CE) \\[42\\], and the\nspecial quasirandom structures (SQS\u2019s) \\[43\\] approach.\n\nCPA \\[41\\] treats random alloys by considering the *average* occupations\nof lattice sites in solving the Kohn-Sham equation. Since a mean-field\napproach is employed, dependence of properties on the local environments\nsurrounding an atom is not treated explicitly in CPA. In a random\nsolution, there exists a distribution of local environments (e.g., in\nbcc alloys, A or B surrounded by the various\nA*X*B8-*X* coordination shells with *X* between 0\nand 8), resulting in local environmentally-dependent quantities such as\ncharge transfer and local displacements of atoms from their ideal\nlattice positions. Even in random A1-*x*B*x* solid\nsolutions, the average A-A, A-B and B-B bond lengths are generally\ndifferent. These effects can be considered by the CE and SQS approaches,\nwhich are the focus of the next two subsections. In following\nsubsections, unless specifically noted, the formulism for the binary\nsystem is discussed for the sake of simplicity.\n", + "metadata": {} + }, + { + "id": "8789b01f", + "cell_type": "markdown", + "source": "### Cluster Expansions\n\nMany properties of a solution phase such as energy are dependent on the\n*configurations* - the arrangements of atoms on the lattice sites. In\ncluster expansion \\[35, 42\\], the configuration dependence of properties\nis formulated efficiently by a \u201clattice algebra\u201d which maps a\nsubstitutional configuration into an Ising-like lattice model. Taking a\nbinary A1-*x*B*x* solution phase for instance, A\natoms are represented by the \u201cdown\u201d spins (*Si* = -1) and B\natoms are represented by the \u201cup\u201d spins (*Si* = +1) as\nillustrated in . Using the cluster expansion technique, for a system\ncontaining *N* atoms, the total energy of any alloy configuration *\u03c3* =\n(*S*1, *S*2, \u2026, *SN*) can be\nconveniently evaluated using the following Ising-like Hamiltonian:\n\n*Eq. 5\u2011147*\n\nwhere *J*\u2019s are the effective cluster interactions (ECI\u2019s); is a number\nrepresenting the atomic occupation at the lattice *i* under the\nconfiguration *\u03c3*, which takes the values -1 and 1 for binary and -1, 0,\nand 1 for ternary etc. In , the 2-site, 3-site, and 4-site correlations\nare written as follows,\n\nEq. 5\u2011148\n\nEq. 5\u2011149\n\nEq. 5\u2011150 .\n\nThe expansion in would be exact as long as *all* the *n*-site\ninteractions are included. For a binary system, this can be observed by\nthe combination law that where is the number of *n*-site interactions.\nHowever, in actual calculations, one never does an expansion to the\norder *N* (containing 2*N* terms for binary) since it is too\nlong to be practical. In fact, since the interactions between widely\nseparated atoms are expected to be weaker than the interactions between\nnearer atoms for most of the important properties, the expansion in is\nusually truncated at certain distance to include only a few short-ranged\npair (2-site), triple (3-site), and up to the most, the quadruple\n(4-site) interactions.\n\nOnce a configuration is assigned, the *S*\u2019s are just the geometrical\nfactors. The common practice in the cluster expansion is that i) perform\nfirst-principles calculations of a selected set of configurations\n(around 20-100); ii) evaluate the values of *J*\u2019s using with the\nenergies from i); iii) use the fitted *J*\u2019s to predict the energy for a\ndesired set of configurations; and iv) make the ensemble average at a\ngiven temperature for the energetics of the random alloys through Monte\nCarlo simulations.\n", + "metadata": {} + }, + { + "id": "ff30bb64", + "cell_type": "markdown", + "source": "### Special Quasirandom Structures\n\nSQS\u2019s \\[43-44\\] are specially designed *small-unit-cell* periodic\nstructures with minimal number of atoms per unit cell, which closely\nmimic the most relevant, near-neighbor pair and multisite correlation\nfunctions of random substitutional alloys. The correlation functions are\nclassified by their *n*-site component \u201cfigures\u201d *f* = (*n,m*), where\nthe index *n* is called vortex for pair, triple, and quadruple\ncorrelations (*n* = 2, 3, and 4); *m* measures the correlation distance.\n\nIn the SQS approach, a distribution of distinct local environments is\nmaintained and their average corresponds to the random alloy. Thus, a\nsingle DFT calculation of an SQS can give many important alloy\nproperties (e.g. equilibrium bond lengths, charge transfer, formation\nenthalpies, etc.) that depend on the existence of those distinct local\nenvironments. The SQS approach has been used extensively to study the\nformation enthalpies, bond length distributions, density of states, band\ngaps and optical properties in semiconductor alloys. It is noted that\nthe CE approach can treat short-range ordering efficiently, while it is\nnot clear how the SQS approach can be used to represent short-range\nordering.\n\nThe key quantities in the SQS approach are the *n*-site correlation\nfunctions. Specifically, the 2-site correlation function corresponding\nto the 2-site component \u201cfigures\u201d (2,*m*) is\n\nEq. 5\u2011151\n\nwhere represents the total number of possible pairs with correlation\ndistance (neighboring distance) *Rij* being equal to *m*. The\n3-site correlation function corresponding to the 3-site component\n\u201cfigures\u201d (3,*m*) is\n\nEq. 5\u2011152\n\nwhere represents the total number of all possible 3-site \u201cfigures\u201d with\nthe correlation distance (size and shape) *Rijk* being equal\nto *m*. The 4-site correlation function corresponding to the 4-site\ncomponent \u201cfigures\u201d (4,*m*) is\n\nEq. 5\u2011153 ,\n\nwhere represents the total number of all possible 4-site \u201cfigures\u201d with\ncorrelation distance (size and shape) *Rijkl* being equal to\n*m*.\n\nWith a given supercell size *N*, the essential task of the SQS approach\nis to search through all configurations that approach as close as\npossible to the correlation functions of the perfectly random (*R*)\nstructure and for the binary system it is\n\nEq. 5\u2011154 .\n\nDescribing random alloys by small unit-cell periodical structures surely\nintroduces erroneous correlations beyond a certain distance. However,\nsince interactions between nearest neighbors are generally more\nimportant than interactions between more distant neighbors, SQS\u2019s can be\nconstructed in such a way that they exactly reproduce the correlation\nfunctions of a random alloy between the first few nearest neighbors,\ndeferring errors due to periodicity to more distant neighbors. The\npractical procedure could be that to find the structures that match the\n2-site correlation functions up to a given neighboring distance, and\nthen add the conditions matching the high order correlation functions up\nto certain correlation distance.\n\nAppendix B is a collection of the SQS\u2019s with a variety of compositions\nfor binary fcc, bcc, hcp, and L12 structures, for ternary\nfcc, bcc, and B2 structures, and Perovskite in the cubic ABO3\nstructure. The used format is that of VASP.\n", + "metadata": {} + }, + { + "id": "4cd57d86", + "cell_type": "markdown", + "source": "### Phonon calculations for SQS\n\nA somewhat more theoretically demanding application of the SQS approach\nis the calculation of the phonon dispersions of a random alloy.\nConsidering the fact that the size of an SQS cell in general is around\n8-32, phonon calculations based on SQS is doable, either using the SQS\ncell or its supercell, e.g., of it. However, one notes that while the\nphonon density of states can be calculated straightforwardly, the\ncalculations of the phonon dispersions run into a problem. That is,\nsince phonon calculation treats the SQS as primitive unit cell made of\nmore atoms than the primitive unit cell of the ideal lattice, the number\nof phonon dispersions derived from a regular phonon calculations is a\nlot greater than that measured for the random alloy. Say, one uses an\nSQS containing 16 atoms for an fcc solid solution, the regular phonon\ncalculations would produce phonon dispersions in comparison to that just\nthree phonon dispersions from measurement. By averaging over the force\nconstants of a SQS, the dynamical matrix can be calculated with respect\nto the wave vector space of the ideal lattice of the random alloys.\n\nOne consideration that must be taken into account is that the phonon\ndispersions measured from the inelastic neutron scattering experiments\nonly represent the averaged vibrations of the ideal lattice. For random\nalloys or phases with minor geometry distortion, it is suggested to\ncalculate the dynamical matrix by instead of *Eq. 5\u2011125* as (see \\[45\\])\n\n*Eq. 5\u2011155* ,\n\nwhere in the case of random alloy, represents the averaged atomic mass\nat the *j*th lattice site. The purpose of the summation over *Q* is to\naverage the effects of local distortions, making it possible of\ncomparing the calculated dispersions to the measured dispersions\nrepresenting the averaged vibrations of the ideal lattice. As a result,\nthe dimension of the SQS supercell dynamical matrix is thus reduced to\nmatch that of the primitive unit cell of the ideal lattice for the\ncalculation of the phonon frequencies. The calculational procedure is as\nfollows:\n\n1. Make an SQS supercell based on the primitive unit cell of ideal\n lattice to mimic the correlation functions of the random solution;\n\n2. Relax the SQS supercell with respect to the internal atomic\n positions while keeping the cell shape and volume fixed;\n\n3. Make the phonon supercell by further enlarging the SQS supercell and\n calculate the force constants; and\n\n4. Calculate the dynamical matrix , with the wave vector, **q**, being\n defined from the primitive unit cell of the ideal lattice, through\n the following Fourier transformation.\n\nThe calculated phonon dispersions, along the directions (00\u03be), (0\u03be\u03be),\nand (\u03be\u03be\u03be), are compared with the inelastic neutron scattering data in\nfor Cu3Au.\n\nFigure 5\u201113. Phonon dispersions for random Cu3Au. The solid\n(black) lines represent the present calculation and the open circles\nrepresent the inelastic neutron scattering data with details in Ref.\n\\[45\\]. The dashed (blue) lines represent the calculated results using\nthe ab initio transferable force-constant model by Dutta et al. \\[46\\].\n", "metadata": {} } ] diff --git a/src/psu410/src/firstprinciples_calculations_and_theory/firstprinciples_formulation_of_thermodynamics.ipynb b/src/psu410/src/firstprinciples_calculations_and_theory/firstprinciples_formulation_of_thermodynamics.ipynb index 4daeccb..aeefb19 100644 --- a/src/psu410/src/firstprinciples_calculations_and_theory/firstprinciples_formulation_of_thermodynamics.ipynb +++ b/src/psu410/src/firstprinciples_calculations_and_theory/firstprinciples_formulation_of_thermodynamics.ipynb @@ -4,352 +4,39 @@ "metadata": {}, "cells": [ { - "id": "771a08a9", + "id": "fc30e7f5", "cell_type": "markdown", - "source": [ - "## First-principles formulation of thermodynamics\n" - ], + "source": "## First-principles formulation of thermodynamics\n", "metadata": {} }, { - "id": "9f6cde7e", + "id": "a7e89071", "cell_type": "markdown", - "source": [ - "### Formation enthalpy of Ni3Al\n", - "\n", - "One can do the similar calculations for the elemental metal Al and the\n", - "compound Ni3Al which has the L12 structure,\n", - "following the same steps in the calculations of Ni. The formation\n", - "enthalpy in the unit of per mole atom can be calculated as\n", - "\n", - "*Eq. 5\u201115* ,\n", - "\n", - "where , , and represent the enthalpies of Ni3Al, Ni, and Al\n", - "in the energy unit of per mole atom, respectively. Plotted in is the\n", - "calculated formation enthalpy of Ni3Al (curve) as a function\n", - "of temperature from 0 to 1600 K at *P* = 0 compared with experimental\n", - "data (symbols) with details in Ref. \\[16\\].\n", - "\n", - ".\n", - "\n", - "Figure \u2011. Formation enthalpy of L12-Ni3Al with\n", - "respect to pure fcc Ni and Al.\n" - ], + "source": "### Helmholtz energy\n\nIn this chapter so far, all our discussions are limited to the case of a\nsystem which is made from a single microstate (microscopic state). Here,\nthe terminology microstate refers to the microscopic structure that is\ndistinguished by crystal structure, atom distributions in the lattice\nsites, and the arrangements of the local atomic spin and electronic\nangular momentum distributions among the lattice sites. From this\nsection on, the index \u03c3 is employed to label the microstate. For a solid\nmaterial at finite temperatures, a phase can be formed by a single\nmicrostate or a mixture of multiple microstates.\n\nLet us consider a canonical system made of *N* atoms with an average\natomic volume *V*. The study is limited to the motions of atomic nuclei\nand electrons. For such a system, one can use to index the energy\neigenvalues of the corresponding microscopic Hamiltonian associated with\nmicrostate *\u03c3*. The subscript **g** symbolically labels the different\nvibrational states for the motions of atomic nuclei and the subscript\n**n** symbolically labels the electronic states distinguished by the\ndifferent distributions of the electrons between the electronic valence\nand conduction bands. Neglecting electron-phonon coupling, one can\nassume that the contributions to between the vibrational and electronic\nstates are additive, so that\n\n*Eq. 5\u201116* ,\n\nwhere *Ec* is the static total energy of the microstate *\u03c3*.\nNote that in and represent the energetics of the vibrational state and\nthe electronic state respectively.\n\nOne then can formulate the canonical partition function of the\nmicrostate *\u03c3* at the given temperature *T* and volume *V* as\n\n*Eq. 5\u201117* ,\n\nwhere *\u03b2* = 1/*kBT* with *kB* being the\nBoltzmann\u2019s constant. As a result, with, the Helmholtz energy *F* per\natom for the microstate *\u03c3* is derived as follows:\n\n*Eq. 5\u201118* ,\n\nwhere the variable *N* has been abbreviated using\n\n*Eq. 5\u201119* ,\n\n*Eq. 5\u201120* ,\n\n*Eq. 5\u201121* .\n\nThe calculation of *Ec* is straightforward in most of the\nexisting first-principles codes as discussed earlier.\n", "metadata": {} }, { - "id": "6e3ebf45", + "id": "3902983f", "cell_type": "markdown", - "source": [ - "### Hartree-Fock approximation to solve Schr\u00f6dinger equation\n", - "\n", - "It was Hartree who first assumed that the electron wave function in can\n", - "be expressed as a product of a collection of *N* independent\n", - "one-electron wave functions, where *i* = 1, 2, \u2026, *N* with *N* being the\n", - "number of electrons in a system, in terms of the its space coordinate\n", - "**r** and spin state **s**. After that, Fock modified the Hartree\n", - "approximation by considering the fact that the wave function of a\n", - "multi-fermionic system should satisfy anti-symmetry requirements and\n", - "subsequently the Pauli exclusion principle that the total wave function\n", - "changes sign upon the exchange of fermions. Accordingly, the wave\n", - "function of *N* electrons system under the Hartree-Fock approximation is\n", - "expressed as the Slater determinant \\[29\\]\n", - "\n", - "*Eq. 5\u201172*\n", - "\n", - "For brevity, one can use the atomic unit that treats in , so that\n", - "\n", - "*Eq. 5\u201173*\n", - "\n", - "where , and\n", - "\n", - "*Eq. 5\u201174*\n", - "\n", - "Accordingly, the total energy of the system is expressed as\n", - "\n", - "*Eq. 5\u201175*\n", - "\n", - "where *Jij* is called as Coulomb/Hartree term\n", - "\n", - "*Eq. 5\u201176*\n", - "\n", - "where *Kij* is called as exchange term\n", - "\n", - "*Eq. 5\u201177* ,\n", - "\n", - "where = 1 if spin **s***i* and **s***j* points to\n", - "the same direction and = 0 if spin **s***i* and\n", - "**s***j* points to the opposite direction.\n", - "\n", - "By utilizing the variational condition of , one gets\n", - "\n", - "*Eq. 5\u201178* ,\n", - "\n", - "where is called one-electron energy, and\n", - "\n", - "*Eq. 5\u201179* ,\n", - "\n", - "with being the electronic charge density whose expression is\n", - "\n", - "*Eq. 5\u201180* ,\n", - "\n", - "*Eq. 5\u201181* .\n", - "\n", - "It should be especially noted here that to solve the Hartree-Fock\n", - "equation , the most time consuming part is due to the nonlocal exchange\n", - "term, knowing the fact that the being evaluated one-electron wave\n", - "function is also contained in the expression in the left hand side of by\n", - "means of .\n", - "\n", - "The configurational interaction method is the generalization of the\n", - "Hartree-Fock approximation. In such a case, *Y*, the number of\n", - "one-electron wave functions can be larger than the number of electrons,\n", - "*N*, in the system. Accordingly, from the number of one-electron wave\n", - "functions, , *y* = 1, 2, \u2026, *Y*, one can build the number of Stater\n", - "determinants, *M*, by the combinatorial mathematics that the maximum of\n", - "*M* can be\n", - "\n", - "*Eq. 5\u201182* .\n", - "\n", - "As a result, the wave function of a collection of *N* electron system\n", - "becomes the recombination of the *M* Stater determinants as\n", - "\n", - "*Eq. 5\u201183* ,\n", - "\n", - "where the coefficients is to be found from the multiple linear equation\n", - "\n", - "*Eq. 5\u201184* .\n", - "\n", - "The matrix element in *Eq. 5\u201184* is determined by the integral\n", - "\n", - "*Eq. 5\u201185* .\n" - ], + "source": "### Mermin statistics to the thermal electronic contribution \n\nFor the calculation of *Fel* in , the most computationally\nflexible approach is to use the Mermin statistics \\[28\\] by which\n\n*Eq. 5\u201122*\n\nwhere the bare electronic entropy *Sel* takes the form, after\nreplacing the summation in over the electronic states with integration\n\n*Eq. 5\u201123* ,\n\nby means of utilizing , the electronic density of states *n*. *f* in is\nthe Fermi distribution that takes the form\n\n*Eq. 5\u201124* .\n\nNote that in is the electronic chemical potential that is strongly\ntemperature dependent. At each given *T*, should be carefully calculated\nkeeping the number of electrons unchanged in solving the following\nequation:\n\n*Eq. 5\u201125* ,\n\nnoting that is the Fermi energy calculated at 0 K. With respect to , the\nthermal electronic energy *Eel*, due to the thermal electron\nexcitations, can be calculated through\n\n*Eq. 5\u201126* .\n\nAt low temperature, , , and are reduced to\n\n*Eq. 5\u201127*\n\nwhere *\u03bb* is the so-called electronic specific heat coefficient\ncalculated as\n\n*Eq. 5\u201128* ,\n\nwhere is the electronic density of states at the Fermi level, and\n\n*Eq. 5\u201129*\n\n*Eq. 5\u201130*\n\nFrom , one can easily derive the electronic contribution to the specific\nheat at low temperature as\n\n*Eq. 5\u201131* .\n\nUsually, the dependence of on *V* is weak. Therefore for a normal\nconductor (except for the heavy Fermion metal or superconductor related\nmaterials), at low temperatures, the electronic contribution to the heat\ncapacity is linear against *T*. also indicates that for insulators, by\nmeans of , the electronic contribution to the heat capacity is zero\nsince for insulators .\n", "metadata": {} }, { - "id": "1249586e", + "id": "927b0b2a", "cell_type": "markdown", - "source": [ - "### Helmholtz energy\n", - "\n", - "In this chapter so far, all our discussions are limited to the case of a\n", - "system which is made from a single microstate (microscopic state). Here,\n", - "the terminology microstate refers to the microscopic structure that is\n", - "distinguished by crystal structure, atom distributions in the lattice\n", - "sites, and the arrangements of the local atomic spin and electronic\n", - "angular momentum distributions among the lattice sites. From this\n", - "section on, the index \u03c3 is employed to label the microstate. For a solid\n", - "material at finite temperatures, a phase can be formed by a single\n", - "microstate or a mixture of multiple microstates.\n", - "\n", - "Let us consider a canonical system made of *N* atoms with an average\n", - "atomic volume *V*. The study is limited to the motions of atomic nuclei\n", - "and electrons. For such a system, one can use to index the energy\n", - "eigenvalues of the corresponding microscopic Hamiltonian associated with\n", - "microstate *\u03c3*. The subscript **g** symbolically labels the different\n", - "vibrational states for the motions of atomic nuclei and the subscript\n", - "**n** symbolically labels the electronic states distinguished by the\n", - "different distributions of the electrons between the electronic valence\n", - "and conduction bands. Neglecting electron-phonon coupling, one can\n", - "assume that the contributions to between the vibrational and electronic\n", - "states are additive, so that\n", - "\n", - "*Eq. 5\u201116* ,\n", - "\n", - "where *Ec* is the static total energy of the microstate *\u03c3*.\n", - "Note that in and represent the energetics of the vibrational state and\n", - "the electronic state respectively.\n", - "\n", - "One then can formulate the canonical partition function of the\n", - "microstate *\u03c3* at the given temperature *T* and volume *V* as\n", - "\n", - "*Eq. 5\u201117* ,\n", - "\n", - "where *\u03b2* = 1/*kBT* with *kB* being the\n", - "Boltzmann\u2019s constant. As a result, with, the Helmholtz energy *F* per\n", - "atom for the microstate *\u03c3* is derived as follows:\n", - "\n", - "*Eq. 5\u201118* ,\n", - "\n", - "where the variable *N* has been abbreviated using\n", - "\n", - "*Eq. 5\u201119* ,\n", - "\n", - "*Eq. 5\u201120* ,\n", - "\n", - "*Eq. 5\u201121* .\n", - "\n", - "The calculation of *Ec* is straightforward in most of the\n", - "existing first-principles codes as discussed earlier.\n" - ], + "source": "### Vibrational contribution by phonon theory\n\nUnder the harmonic/quasiharmonic approximation, the lattice dynamics or\nphonon theory is currently the most established method. It truncates the\ninteraction potential up to the second order. In such a case, in can be\nexpressed in terms of phonon frequency as follows\n\n*Eq. 5\u201132* ,\n\nwhere the label **g** has the meanings of (*g1*,\n*g2*, \u2026, *g3N*) that *gj* can take any\ninteger values from zero to infinite.\n\nAs a result, is reduced to\n\n*Eq. 5\u201133* ,\n\nor equivalently,\n\n*Eq. 5\u201134* ,\n\nwhere an integration has been used to replace the summation in by means\nof introducing a function, , named the phonon density of states (PDOS)\nwhose integration over *\u03c9* equals to three per atom.\n\nAccordingly, the formulation to calculate the entropy becomes\n\n*Eq. 5\u201135* ,\n\nthe formulation to calculate the internal energy becomes\n\n*Eq. 5\u201136* ,\n\nand the formulation to calculate the heat capacity at constant volume\nbecomes\n\n*Eq. 5\u201137* .\n", "metadata": {} }, { - "id": "0887f571", + "id": "81df2df2", "cell_type": "markdown", - "source": [ - "### Helmholtz energy and quasiharmonic approximation\n", - "\n", - "At present, the most rigorous method to predict the thermodynamic\n", - "properties of a material at finite temperatures is the phonon approach.\n", - "In such an approach, the microscopic Hamiltonian is expanded up to the\n", - "second order. All the thermodynamic quantities are calculated using\n", - "formulations derived from the statistical physics without further\n", - "approximation. The great importance of the phonon theory is that all the\n", - "input parameters can be obtained by means of first-principles\n", - "calculations without using any phenomenological parameters.\n", - "\n", - "Let us consider a system with an averaged atomic volume *V*. Neglecting\n", - "the electron phonon coupling, it is a well demonstrated procedure \\[16\\]\n", - "to decompose the Helmholtz energy *F*(*V*,*T*) of the system at\n", - "temperature *T* into three additive contributions as follows\n", - "\n", - "*Eq. 5\u20111* ,\n", - "\n", - "where *Ec* is the static total energy which is the total\n", - "energy of the system at 0 K without any atomic vibrations,\n", - "*Fvib* is the vibrational contribution due to the lattice\n", - "ions, and *Fel* is the electronic contribution due to the\n", - "thermal electronic excitation at finite temperature which can become\n", - "important for metals at high temperature.\n", - "\n", - "The terminology of \u201cquasiharmonic approximation\u201d arises from the fact\n", - "that for a given volume, *Fvib*(*V*,*T*) is calculated under\n", - "the harmonic approximation and the anharmonic effects are carried out\n", - "solely through volume dependence of the phonon frequency. The easiest\n", - "computational implementation of is to first independently calculate the\n", - "Helmholtz energy at several selected volumes near the equilibrium volume\n", - "and then use the numerical interpolation to find the Helmholtz energy at\n", - "an arbitrary volume. The volume interval is usually at the scale of 3~5%\n", - "of the equilibrium volume. Too small volume interval can result in\n", - "numerical instability due to the numerical uncertainties in the static\n", - "total energy calculation, in particular, when one numerically computes\n", - "the first- and especially the second-order derivatives of the Helmholtz\n", - "energy in deriving the thermodynamic quantities. It should be noted that\n", - "whenever available, analytic formulas should be used instead of the\n", - "numerical second-order derivative to avoid numerical errors. For\n", - "instance, when the phonon approach is employed, the constant volume heat\n", - "capacity has the analytic expression in terms of phonon density of\n", - "states.\n", - "\n", - "Nickel metal adopts the fcc structure at ambient conditions and the\n", - "primitive unit cell contains one atom. Almost all the existing\n", - "first-principles codes have the function to calculate the static total\n", - "energy. The static total energy *Ec* in should be calculated\n", - "using the primitive unit cell. As the Helmholtz energy is to be\n", - "calculated at several volumes, a good practice is to plot the calculated\n", - "static total energy points together with the interpolated energy curve\n", - "to examine the convergence of the static total energy calculation. Since\n", - "the first-principles method often employs the self-consistent technique,\n", - "it could occur that calculations at certain volumes may not convergent,\n", - "which should be fixed by trying the various algebraic schemes provided\n", - "in most of the existing codes. Furthermore, since certain calculations\n", - "involve the second order derivative of the Helmholtz energy, a minor\n", - "uncertainty along the static total energy curve can result in large\n", - "deviation for the calculated properties such as thermal expansion\n", - "coefficient and bulk modulus. In that case, a reasonable solution is to\n", - "smoothen the static total energy using the modified Birch-Murnaghan\n", - "equation of states (EOS) \\[17-18\\]\n", - "\n", - "*Eq. 5\u20112* .\n", - "\n", - "Plotted in is the calculated static total energy of the elemental metal\n", - "Ni with the circles representing the calculated values and the curve\n", - "representing that by EOS fitting.\n", - "\n", - "Figure \u2011. Static total energy of nickel.\n", - "\n", - "The vibrational contribution to the Helmholtz energy by phonon theory\n", - "can be computed by \\[19\\]\n", - "\n", - "*Eq. 5\u20113* ,\n", - "\n", - "where is the Boltzmann\u2019s constant, *\u03c9* represents the phonon frequency,\n", - "and is the phonon density of states. It is recommended that is\n", - "calculated at the same volume set at which the static total static\n", - "energies are calculated.\n", - "\n", - "For the present prototype of Ni, the supercell method for the\n", - "calculation of has been employed. The procedure is follows:\n", - "\n", - "1. Make supercell by enlarging the primitive unit cell according to the\n", - " > defined neighbor interaction distance; Employ the first-principles\n", - " > code (VASP, \\[13-14\\] in this work) to calculate the interatomic\n", - " > force constants.\n", - "\n", - "2. Assign the mesh in the wave vector (**q**) space; Make the dynamical\n", - " > matrix at each **q** point; Diagonalize the dynamical matrix to\n", - " > find out the phonon frequencies at each **q** point; And finally\n", - " > collect all the phonon frequencies for all **q** points. The\n", - " > detailed formulation for phonon calculations is given in Chapter .\n", - "\n", - "For the phonon calculations, one can use the open source code YPHON\n", - "\\[15\\] by the present authors. Other choices can be the free ATAT code\n", - "\\[20\\] or the free PHON code \\[21\\]. For the calculation of the phonon\n", - "density of states, we have made a supercell containing 64 atoms which is\n", - "a 4\u00d74\u00d74 supercell of the primitive unit cell. Plotted in is the\n", - "calculated phonon density of states using YPHON code at the calculated\n", - "static equilibrium volume compared with the measured data at 10 K \\[22\\]\n", - "(symbols).\n", - "\n", - "Figure \u2011. Phonon density of states of nickel.\n", - "\n", - "For a first-principles thermodynamic calculation, an important step to\n", - "avoid possible calculation errors is to examine the phonon dispersions\n", - "first. Phonon dispersion \\[23\\] depicts the evolution of phonon\n", - "frequencies along the designated direction for a crystal. Phonon\n", - "dispersion can be measured rather accurately by inelastic neutron\n", - "scattering \\[24-26\\] or inelastic x-ray scattering \\[27\\] experiment.\n", - "Plotted in are the calculated phonon dispersions (curves) along the\n", - "\\[00\u03b6\\], \\[0\u03b61\\], \\[0\u03b6\u03b6\\], and \\[\u03b6\u03b6\u03b6\\] directions of Ni using YPHON code\n", - "compared with the neutron scattering data at 296 K (symbols) with\n", - "details in Ref. \\[16\\].\n", - "\n", - "Figure \u2011. Phonon dispersions of nickel. The solid lines represent\n", - "results calculated using a supercell containing 256 atoms which is 4\u00d74\u00d74\n", - "supercell of the conventional cubic unit cell. The dot-dashed lines\n", - "represent results calculated using a supercell containing 64 atoms which\n", - "is 4\u00d74\u00d74 supercell of the primitive unit cell.\n", - "\n", - "For the calculation of *Fel* in , the most computationally\n", - "convenient approach is to use the Mermin statistics as follows\n", - "\n", - "*Eq. 5\u20114* ,\n", - "\n", - "where is the thermal electronic energy, and *Sel* is the bare\n", - "electronic entropy. Both the calculations of and *Sel* need\n", - "the electronic density of states (EDOS) as input. The electronic density\n", - "of states can be obtained during the step of the static total energy\n", - "calculation. The detailed formulations for and *Sel* are\n", - "given in Chapter . Since Ni is magnetic, the EDOS of Ni can be\n", - "partitioned into those of spin up and spin down due to the spin freedom\n", - "of electron. The calculated EDOS for Ni is shown in where the solid,\n", - "dot-dashed, and dashed lines represent the total, spin up, and spin down\n", - "EDOS with the Fermi energy set to zero.\n", - "\n", - "Figure \u2011. Electronic density of states of nickel. That due to spin up is\n", - "plotted as positive value and that due to spin down is plotted as\n", - "negative value purely for the clarity of the figure. The \u201ctotal\u201d is the\n", - "sum of the absolute values of those of spin up and spin down.\n", - "\n", - "The calculated temperature evolution of Helmholtz energy as a function\n", - "of volume for Ni are illustrated in . The circles represent the\n", - "calculated static total energies. The solid curves represent the\n", - "Helmholtz energy curves from 0 to 1600 K at a temperature increment of\n", - "100 K as displayed from top to bottom in . The dashed line marks the\n", - "evolution of the equilibrium volume at *P*=0 with increasing\n", - "temperature. It is noted that Helmholtz energy always decreases with\n", - "increasing temperature due to the entropy term of \u2013*TS*. Note that the\n", - "at 0 K the Helmholtz energy is higher than the static total energy due\n", - "to the zero point vibrational energy as can be seen when *T* \u21920 which\n", - "reduces to\n", - "\n", - "*Eq. 5\u20115* ,\n", - "\n", - "which is positive.\n", - "\n", - "Figure \u2011. Temperature evolution of the Helmholtz energy for nickel.\n" - ], + "source": "### Debye-Gr\u00fcneisen approximation to the vibrational contribution \n\nStrictly speaking, the Debye theory is only accurate at very low\ntemperatures. It assumes a parabolic type of dependence of PDOS\u00a0on the\nphonon frequency. This assumption is only correct at the scale of 10\u2019s K\nbecause at low temperatures,\u00a0only the low frequency acoustic phonons are\nactivated which play the major role for the parabolic type of PDOS for\nthe low frequency range as shown in . That is why\u00a0there are two kinds of\nDebye temperature: low-temperature Debye temperature and\nhigh-temperature\u00a0Debye temperature. The low-temperature Debye\ntemperature can be strictly derived by fitting low temperature heat\ncapacity data. The high-temperature\u00a0Debye temperature is mostly a\nphenomenological fitting parameter.\n\nThe Debye model approximates the PDOS in by\n\n*Eq. 5\u201138* ,\n\nwhere is the so-called Debye cutoff frequency related to the Debye\ntemperature as\n\n*Eq. 5\u201139* .\n\nAs the result, the vibrational contribution to Helmholtz energy under\nthe Debye approximation becomes,\n\n*Eq. 5\u201140*\n\nwhere *D*(*\u0398D/T*) is the Debye function given by .\n\nUnder the Debye approximation, the formulation to calculate the entropy\nbecomes\n\n*Eq. 5\u201141* ,\n\nthe formulation to calculate the internal energy becomes\n\n*Eq. 5\u201142* ,\n\nwhere\n\n*Eq. 5\u201143* ,\n\nand the formulation to calculate the heat capacity at constant volume\nbecomes\n\n*Eq. 5\u201144* .\n\nHere it is noted that is volume dependent which is often written in\nterms of Gr\u00fcneisen constant:\n\n*Eq. 5\u201145* .\n\nIt has been found that the dependence of on *V* usually is weak and\nhence is often approximated as a constant. With , the formulation to\ncalculate the pressure becomes\n\n*Eq. 5\u201146* .\n\nAn important result of the Debye approximation is that when , together\nwith , the heat capacity in is reduced to\n\n*Eq. 5\u201147* .\n\nThis gives the Debye T3 law when the thermal electron\ncontribution is neglected. In the analysis of superconductor and heavy\nFermion materials, is often rewritten as\n\n*Eq. 5\u201148* ,\n\nwhich is more convenient for examining the heat capacity measured at\nextremely low temperatures.\n", + "metadata": {} + }, + { + "id": "ea721412", + "cell_type": "markdown", + "source": "### System with multiple microstates (MMS model)\n\nFor a system made of multiple microstates, the total partition function\nis the summation over the partition functions of all microstates, in ,\nas\n\n*Eq. 5\u201149* ,\n\nwhere is the multiplicity of the microstate *\u03c3.* It is immediately\napparent that is the thermal population of the microstate *\u03c3*.\nFurthermore, with , one obtains\n\n*Eq. 5\u201150* .\n\nrelates the total Helmholtz energy, *F(N,V,T)*, of a system with mixing\namong multiple microstates and the Helmholtz energy, , of individual\nmicrostates. An important result of is the configurational entropy due\nto the mixing among multiple microstates, named as microstate\nconfigurational entropy (MCE) in this book,\n\n*Eq. 5\u201151* ,\n\nwhich makes the entropy of a system with mixing among multiple\nmicrostates as\n\n*Eq. 5\u201152* .\n\nSimilarly, one can get the heat capacity at constant volume of a system\nwith mixing among multiple microstates as\n\n*Eq. 5\u201153* ,\n\nwhere the configurational contributions to the heat capacity due to the\nmixing among multiple microstates is\n\n*Eq. 5\u201154* .\n\nMoreover, the isothermal bulk modulus of a system with mixing among\nmultiple microstates can be also computed similarly\n\n*Eq. 5\u201155* ,\n\nWith\n\n*Eq. 5\u201156*\n\nwith being the partial pressure of the microstate *\u03c3*. This multiple\nmicrostate model (MMS model) is used in Chapter to quantitatively\npredict thermal expansion anomalies.\n", "metadata": {} } ] diff --git a/src/psu410/src/firstprinciples_calculations_and_theory/index.ipynb b/src/psu410/src/firstprinciples_calculations_and_theory/index.ipynb index dbc3fa2..4b99ed7 100644 --- a/src/psu410/src/firstprinciples_calculations_and_theory/index.ipynb +++ b/src/psu410/src/firstprinciples_calculations_and_theory/index.ipynb @@ -4,56 +4,9 @@ "metadata": {}, "cells": [ { - "id": "121ad3f6", + "id": "cdf1b9b7", "cell_type": "markdown", - "source": [ - "# First-principles calculations and theory\n", - "\n", - "In the previous chapter, the experimental techniques in obtaining the\n", - "thermochemical and phase equilibrium data that are the inputs for the\n", - "thermodynamic modeling of a system were summarized. However,\n", - "experimental data are not always available. This is due to the fact that\n", - "i) the experiments are expensive, especially true in developing new\n", - "materials; and ii) the experiments cannot reliably access the non-stable\n", - "phases in most cases. The alternative approach is to predict the\n", - "thermochemical data by first-principles calculations. The prediction of\n", - "the material properties, without using phenomenological parameters, is\n", - "the basic spirit of first-principles calculations. In particular, the\n", - "steady increase of both computer power and the efficiency of\n", - "computational methods have made the first-principles predictions of most\n", - "thermodynamic properties possible, including both enthalpy and entropy\n", - "as a function of temperature, volume, and/or pressure.\n", - "\n", - "By its definition, the term of \u201cfirst-principles\u201d represents a\n", - "philosophy that the prediction is to be based on a basic, fundamental\n", - "proposition or assumption that cannot be deduced from any other\n", - "proposition or assumption. This implies that the computational\n", - "formulations are based on the most fundamental theory of quantum\n", - "mechanics - Schr\u00f6dinger equation or density functional theory and the\n", - "inputs to the calculations must be based on well-defined physical\n", - "constants \u2013 the nuclear and electronic charges. In another word, once\n", - "the atomic species of an assigned material are known, the theory should\n", - "predict the energy of all possible crystalline structures, without\n", - "invoking any phenomenological fitting parameters.\n", - "\n", - "This chapter organized in the sequence from thermodynamic calculations\n", - "to fundamental theory to help those readers who are more interested in\n", - "realistic calculations using existing computer codes. The detailed\n", - "theoretical sections are presented to follow the section of\n", - "thermodynamic calculations for those readers who are also interested in\n", - "the derivation of the formulations used in the thermodynamic\n", - "calculations. The following subsections are arranged accordingly in the\n", - "order: (i) examples the commonly adopted calculation procedures for\n", - "thermodynamic properties using the elemental metal nickel as the main\n", - "prototype; (ii) derivation of the Helmholtz energy expression under the\n", - "first-principles framework; (iii) introduction of the solution to the\n", - "electronic Schr\u00f6dinger equation within the two well developed frameworks\n", - "\u2013 the quantum chemistry approach and the density functional theory; (iv)\n", - "detailed description of the procedure on how to solve the Schr\u00f6dinger\n", - "equation for the motions of the atomic nuclei by means of lattice\n", - "dynamics. The relation between the Helmholtz energy and Gibbs energy is\n", - "shown by .\n" - ], + "source": "# First-principles calculations and theory\n\nIn the previous chapter, the experimental techniques in obtaining the\nthermochemical and phase equilibrium data that are the inputs for the\nthermodynamic modeling of a system were summarized. However,\nexperimental data are not always available. This is due to the fact that\ni) the experiments are expensive, especially true in developing new\nmaterials; and ii) the experiments cannot reliably access the non-stable\nphases in most cases. The alternative approach is to predict the\nthermochemical data by first-principles calculations. The prediction of\nthe material properties, without using phenomenological parameters, is\nthe basic spirit of first-principles calculations. In particular, the\nsteady increase of both computer power and the efficiency of\ncomputational methods have made the first-principles predictions of most\nthermodynamic properties possible, including both enthalpy and entropy\nas a function of temperature, volume, and/or pressure.\n\nBy its definition, the term of \u201cfirst-principles\u201d represents a\nphilosophy that the prediction is to be based on a basic, fundamental\nproposition or assumption that cannot be deduced from any other\nproposition or assumption. This implies that the computational\nformulations are based on the most fundamental theory of quantum\nmechanics - Schr\u00f6dinger equation or density functional theory and the\ninputs to the calculations must be based on well-defined physical\nconstants \u2013 the nuclear and electronic charges. In another word, once\nthe atomic species of an assigned material are known, the theory should\npredict the energy of all possible crystalline structures, without\ninvoking any phenomenological fitting parameters.\n\nThis chapter organized in the sequence from thermodynamic calculations\nto fundamental theory to help those readers who are more interested in\nrealistic calculations using existing computer codes. The detailed\ntheoretical sections are presented to follow the section of\nthermodynamic calculations for those readers who are also interested in\nthe derivation of the formulations used in the thermodynamic\ncalculations. The following subsections are arranged accordingly in the\norder: (i) examples the commonly adopted calculation procedures for\nthermodynamic properties using the elemental metal nickel as the main\nprototype; (ii) derivation of the Helmholtz energy expression under the\nfirst-principles framework; (iii) introduction of the solution to the\nelectronic Schr\u00f6dinger equation within the two well developed frameworks\n\u2013 the quantum chemistry approach and the density functional theory; (iv)\ndetailed description of the procedure on how to solve the Schr\u00f6dinger\nequation for the motions of the atomic nuclei by means of lattice\ndynamics. The relation between the Helmholtz energy and Gibbs energy is\nshown by .\n", "metadata": {} } ] diff --git a/src/psu410/src/firstprinciples_calculations_and_theory/lattice_dynamics.ipynb b/src/psu410/src/firstprinciples_calculations_and_theory/lattice_dynamics.ipynb index 4893825..8e333aa 100644 --- a/src/psu410/src/firstprinciples_calculations_and_theory/lattice_dynamics.ipynb +++ b/src/psu410/src/firstprinciples_calculations_and_theory/lattice_dynamics.ipynb @@ -4,185 +4,33 @@ "metadata": {}, "cells": [ { - "id": "18f2b40f", + "id": "f1788709", "cell_type": "markdown", - "source": [ - "## Lattice Dynamics\n" - ], + "source": "## Lattice Dynamics\n", "metadata": {} }, { - "id": "33b80c96", + "id": "0e6d239c", "cell_type": "markdown", - "source": [ - "### Linear-response method vs supercell method\n", - "\n", - "The problem of lattice vibration for a solid is now transformed into\n", - "computing the dynamical matrix in . The first-principles solution of the\n", - "problem is currently divided into two categories: the linear-response\n", - "method \\[34\\] and the supercell method \\[35\\]. In the linear-response\n", - "method, utilizing the electronic linear response upon the undistorted\n", - "crystals \\[36\\], the evaluations of the dynamical matrix can be\n", - "performed through the density-functional perturbation theory \\[34\\]\n", - "without the approximation of the cutoff in neighboring interaction.\n", - "\n", - "Compared with the linear-response method, the supercell method is\n", - "conceptually simple and is easy to implement computationally. The\n", - "supercell method adopts the frozen phonon approximation through which\n", - "the changes in total energy or forces are calculated in the direct space\n", - "by displacing the atoms from their equilibrium positions. The advantage\n", - "of the supercell method is that the phonon frequencies at the exact wave\n", - "vectors, which are commensurable with the supercell, are calculated\n", - "exactly with no further approximation \\[37\\]. The shortcoming of the\n", - "supercell method is that it is often limited by the size of the\n", - "supercell that can be handled with current computing resources.\n", - "\n", - "In supercell approach, inaccuracies are thought to arise from the\n", - "truncation of the force constants \\[34-35\\]. This is only partially\n", - "true. In the supercell method, the calculated represents the cumulative\n", - "contributions of the atom indexed by *k* and *P* in the supercell and\n", - "all its images by translational transformation of the supercell in the\n", - "whole space. Let **L**\u03b1 represent the lattice vectors of the\n", - "supercell, then\n", - "\n", - "*Eq. 5\u2011138* .\n", - "\n", - "For the exact wave vectors in , one has\n", - "\n", - "*Eq. 5\u2011139* **,**\n", - "\n", - "where is an integer. Replacing in with in , one obtains\n", - "\n", - "Eq. 5\u2011140 .\n", - "\n", - "Therefore, the phonon frequencies calculated at the exact wave vectors\n", - "by the cumulative force constants approach are exact, and the supercell\n", - "size will not lead to errors in the calculated phonon frequencies\n", - "\\[37\\].\n", - "\n", - "Generally speaking, if a supercell contains *Nc* primitive\n", - "unit cells, one can always find *Nc* corresponding exact wave\n", - "vectors in the **q** space. In most linear response calculations, the\n", - "common choice of a 4\u00d74\u00d74 **q**-mesh is exactly equivalent to the 4\u00d74\u00d74\n", - "supercell in the real-space. Furthermore, since the supercell approach\n", - "does not impose any approximation on the electronic response to the\n", - "distortion of the lattice geometry, the effects of electron-phonon\n", - "interactions can be accounted for by the supercell method.\n", - "\n", - "In the supercell method, due to the impose of periodic condition, the\n", - "calculated force constant in the real-space cannot account for the\n", - "effects of the vibration-induced electric field for the polar materials.\n", - "It has been demonstrated that such an effect adds an additional term to\n", - "dynamical matrix in the reciprocal-space in the form,\n", - "\n", - "*Eq. 5\u2011141* ,\n", - "\n", - "where ***Z**\\* (j)* represents the Born effective charge\n", - "tensor of the *j*th atom in the primitive unit cell and\n", - "**\u03b5\u221e** the high frequency static dielectric tensor, i.e., the\n", - "contribution to the dielectric permittivity tensor from the electronic\n", - "polarization. As a result for the polar materials, the matrix element at\n", - "wave vector **q**\u21920 of the dynamical matrix in , by means of , should\n", - "have the form\n", - "\n", - "*Eq. 5\u2011142* .\n", - "\n", - "It can be demonstrated \\[38\\] that this is equivalent to replacing the\n", - "real space force constant in by\n", - "\n", - "*Eq. 5\u2011143* .\n", - "\n", - "The present implementation of the first-principles method in calculating\n", - "the phonon frequencies are mostly limited by the supercell size when\n", - "using or the number of exact wave vector points when using . A supercell\n", - "cell built on the primitive unit cell or exact wave vector mesh is\n", - "usually the common limit. If only the phonon frequencies derived from or\n", - "are used, can be rather unsmooth which will result in inaccurate\n", - "thermodynamic properties when it is used with . The mixed-space approach\n", - "\\[38\\] can circumvent this bottleneck by using the Fourier interpolation\n", - "to calculate the dynamical matrix for an arbitrary wave vector **q** as\n", - "\n", - "*Eq. 5\u2011144* ,\n", - "\n", - "with the help of , , and for a polar materials. Note that the term is\n", - "for polar materials only.\n", - "\n", - "The can be calculated as\n", - "\n", - "*Eq. 5\u2011145* ,\n", - "\n", - "where the function is usually taken as the Gaussian type\n", - "\n", - "*Eq. 5\u2011146* ,\n", - "\n", - "where is an adjustable damping (broadening) parameter whose role is to\n", - "smear the curve. in is the number of used **q** points. Empirically, a\n", - "**q** mesh of is accurate enough for most purposes and doable with the\n", - "YPHON code \\[15\\] efficiently, which is discussed in the Appendix A.\n" - ], + "source": "### Quantum theory for motion of atomic nuclei\n\nFor the convenience of discussion, the following convention of notations\nare used: *\u03b1* and *\u03b2* to label the Cartesian axes which is either *x*,\n*y*, or *z*, *j* and *k* to label atoms in the primitive unit cell,\n*mj* the atomic mass of the *j*th atom in the\nprimitive unit cell, **r**(*j*) the position of the *j*th\natom in the primitive unit cell, *P* and *Q* the index of the primitive\nunit cell in the system, **R**(*P*) the position of the *P*th\nprimitive unit cell in the system, and *V* the averaged volume of the\nprimitive unit cell.\n\nThe quantum theory for motion of atomic nuclei replicates closely the\nquantum theory for motion of electrons. That is to solve the wave\nfunction for the motions of the atomic nuclei for a Schr\u00f6dinger\u2019s\nequation with the potential being the total electronic energy derived\nfrom . The symbol is replaced by to represent the static total\nelectronic energy with **R** representing the static equilibrium\npositions of the atomic nuclei, **u** the displacements of atomic nuclei\naway from their static equilibrium positions, and *\u03c3* the additional\ndegree of freedom such as electronic states. The Schr\u00f6dinger\u2019s equation\nfor the motion of atomic nuclei is then\n\n*Eq. 5\u2011103* ,\n\nwhere\n\n*Eq. 5\u2011104* ,\n\nwith representing the kinetic energy operator\n\n*Eq. 5\u2011105* ,\n\nwhere *Nc* is the numbers of primitive unit cells in the\nsystem, *Na* is the numbers of atoms in the primitive unit\ncell, is the Plack constant, represents the *\u03b1*th Cartesian\ncomponent of **u** for the atom at *j*th lattice site in the\nprimitive unit cell and the *P*th primitive unit cell in the\nsystem.\n\nThe harmonic approximation \\[23, 33\\] truncates the into the second\norder in its Taylor\u2019s series\n\n*Eq. 5\u2011106* ,\n\nwhere is the real-space interatomic force constant. With the\napproximation of , it can be demonstrated that finding the solution of\nis equivalent to finding the vibrational frequencies of a classical\nsystem with *NcNa* particles for small mechanical\noscillations.\n\nLet us rewrite as\n\n*Eq. 5\u2011107* ,\n\nwhere\n\n*Eq. 5\u2011108* , and\n\n*Eq. 5\u2011109* .\n\nAccordingly, the kinetic energy operator in becomes\n\n*Eq. 5\u2011110* .\n", "metadata": {} }, { - "id": "156976ea", + "id": "668ad0ec", "cell_type": "markdown", - "source": [ - "### Mermin statistics to the thermal electronic contribution \n", - "\n", - "For the calculation of *Fel* in , the most computationally\n", - "flexible approach is to use the Mermin statistics \\[28\\] by which\n", - "\n", - "*Eq. 5\u201122*\n", - "\n", - "where the bare electronic entropy *Sel* takes the form, after\n", - "replacing the summation in over the electronic states with integration\n", - "\n", - "*Eq. 5\u201123* ,\n", - "\n", - "by means of utilizing , the electronic density of states *n*. *f* in is\n", - "the Fermi distribution that takes the form\n", - "\n", - "*Eq. 5\u201124* .\n", - "\n", - "Note that in is the electronic chemical potential that is strongly\n", - "temperature dependent. At each given *T*, should be carefully calculated\n", - "keeping the number of electrons unchanged in solving the following\n", - "equation:\n", - "\n", - "*Eq. 5\u201125* ,\n", - "\n", - "noting that is the Fermi energy calculated at 0 K. With respect to , the\n", - "thermal electronic energy *Eel*, due to the thermal electron\n", - "excitations, can be calculated through\n", - "\n", - "*Eq. 5\u201126* .\n", - "\n", - "At low temperature, , , and are reduced to\n", - "\n", - "*Eq. 5\u201127*\n", - "\n", - "where *\u03bb* is the so-called electronic specific heat coefficient\n", - "calculated as\n", - "\n", - "*Eq. 5\u201128* ,\n", - "\n", - "where is the electronic density of states at the Fermi level, and\n", - "\n", - "*Eq. 5\u201129*\n", - "\n", - "*Eq. 5\u201130*\n", - "\n", - "From , one can easily derive the electronic contribution to the specific\n", - "heat at low temperature as\n", - "\n", - "*Eq. 5\u201131* .\n", - "\n", - "Usually, the dependence of on *V* is weak. Therefore for a normal\n", - "conductor (except for the heavy Fermion metal or superconductor related\n", - "materials), at low temperatures, the electronic contribution to the heat\n", - "capacity is linear against *T*. also indicates that for insulators, by\n", - "means of , the electronic contribution to the heat capacity is zero\n", - "since for insulators .\n" - ], + "source": "### Normal coordinates, eigenenergetics, and phonons\n\nOne way to simplify the solution to the Schr\u00f6dinger\u2019s equation for\nmotion of atomic nuclei is to follow the study of the vibrations of the\natoms in a crystal \u2013 lattice dynamics. The essential step in lattice\ndynamics is to transform the problem of the correlated motions of 3*N*\nparticle into the problem of 3*N* independent harmonics. For this\npurpose, one can introduce a set of new coordinates ( *i*=1, 2, \u2026,\n3*N*), known as normal coordinates, by the transformation\n\n*Eq. 5\u2011111* ,\n\nwhere is the transformation coefficient that can be determined by\nsolving 3*N* simultaneous equations\n\n*Eq. 5\u2011112* ,\n\nwhere is to be determined so that one can find 3*N* , which are, not all\nzero. The equations are linear and homogeneous. Follow the basic theorem\nin linear algebra that, to find the nonzero solutions of the equations,\nthe determinant formed by must equal zero\n\n*Eq. 5\u2011113* ,\n\nwhere is the Kronecker delta symbol. Since is an equation with 3*N*\ndegrees, one can always find 3*N* values of (*i* = 1, \u2026, 3*N*). Each of\nyields a set of which can be chosen such that\n\n*Eq. 5\u2011114* ,\n\nwhere represents the Kronecker delta symbol and\n\n*Eq. 5\u2011115*\n\nThen with the normal coordinates defined in and utilizing , defined in\nis obtained as\n\n*Eq. 5\u2011116* .\n\nWith this equation, is simplified by the following process\n\n*Eq. 5\u2011117* ,\n\nnoting that in the above process and have been utilized.\n\nFurthermore, using , the kinetic energy operator in can be transformed\nas follows\n\n*Eq. 5\u2011118* ,\n\nnoting that in the above process and are utilized again.\n\nAs a result, under the harmonic approximation, the Hamiltonian in is\nsimplified as\n\n*Eq. 5\u2011119*\n\nwhich represents a quantum system containing 3*N* independent harmonics.\nCorresponding to the each of the *\u03c9i*, the quantum theory\ntells that the eigenenergy of a harmonics has the form\n\n*Eq. 5\u2011120* ,\n\nwith *gi* = 0, 1, \u2026, \u221e. Such a type of energetics behaves\nlike a Boson particle with energy and forms the concept of phonon.\n\nFurthermore, a state of the whole system is to be specified by the set\nof 3*N* independent quantum numbers **g** = (*g1*,\n*g1*, \u2026, *g3N*,). Finally, the energetics of a\nstate of the system formed by 3*N* independent harmonics, introduced by\n, is obtained with the summation of the energies of 3*N* independent\nharmonics as\n\n*Eq. 5\u2011121* .\n\nThis concludes the rationale by which how is derived.\n", + "metadata": {} + }, + { + "id": "a7415cda", + "cell_type": "markdown", + "source": "### Dynamical matrix and phonon mode\n\nBecause of the periodicity of a crystal, one can make an initial guess\nthat the solutions of are elastic plane waves made of a collective\natomic vibrations \\[23, 33\\], from the harmonic approximation of ,\n\n*Eq. 5\u2011122* ,\n\nwhere *\u03c9* represents the frequency of the plane wave, and is a wave\nvector designating the wave number and direction along which the plane\nwave propagates. It should be pointed out that in is now independent of\nthe index *P.* That is, except for a phase factor, the atoms that are\nequivalent by the translational symmetry among different primitive unit\ncells will experience the same type of atomic motion, independent of the\npositions of the primitive unit cell in the system. This is equivalent\nto applying the periodic condition so that in obeys\n\n*Eq. 5\u2011123* .\n\nNote that is now independent of the index *P*.\n\nFurthermore, one wants to limit **q***t* in and to those\nknown as the exact wave vectors which represent a special set of points\nin the reciprocal space that satisfy the condition\n\n*Eq. 5\u2011124* ,\n\nwhere is the Kronecker delta function. In fact, the number of equals to\nthe number of primitive unit cells contained in the system.\n\nUtilizing the translational invariance by which in \\[or in \\] depends on\n*P* and *Q* only through the difference , the following Fourier\ntransformation can be employed to simplify\n\n*Eq. 5\u2011125* ,\n\nand one obtains\n\n*Eq. 5\u2011126* .\n\nThe counterpart of with respect to is\n\n*Eq. 5\u2011127*\n\nis now an equation with 3*Na* degrees of freedom. At each ,\none can always find 3*Na* eigenvalues of (*i* = 1, \u2026,\n3*Na*). The 3*Na* vibrations are often known as\nphonon modes, noting again that *Na* is the number of atoms\nin the primitive unit cell. Each of yields a set of which can be chosen\nsuch that\n\n*Eq. 5\u2011128* ,\n\nwhere represents the complex conjugate of , and\n\n*Eq. 5\u2011129*\n\nFinally, for a solid, the summation over *Q* in can be abbreviated out,\nresulting in, after transforming back to together with using\n\n*Eq. 5\u2011130* ,\n\nbecause of translational invariance by which depends on *P* and *Q* only\nthrough the difference .\n\nIn the case that a system is built by a parallelepiped multiplication of\nthe primitive unit cell with lattice vectors of **a***\u03b1*\n(\u03b1=*x*, *y*, and *z*) in the form , in and takes the form\n\n*Eq. 5\u2011131* ,\n\nwhere = 0, 1, \u2026, , and is the primitive lattice vector in the reciprocal\nspace as\n\n*Eq. 5\u2011132* ,\n\nwith *Va* representing the volume of the primitive unit cell\ngiven as\n\n*Eq. 5\u2011133* .\n\nCombine and , it is easy to demonstrate\n\n*Eq. 5\u2011134* ,\n\nwhere is the Kronecker delta symbol.\n\nIt can be shown that the **q** points defined in represent the exact\nwave points. First, can be written as\n\n*Eq. 5\u2011135* ,\n\nwhere = 0, 1, \u2026, . Then, the left hand side of for the definition of\nexact wave vector becomes\n\n*Eq. 5\u2011136* .\n\nKnowing the fact that\n\n*Eq. 5\u2011137* .\n\nHence, it is proved the **q** points defined in are the exact wave\npoints.\n", + "metadata": {} + }, + { + "id": "c3c81656", + "cell_type": "markdown", + "source": "### Linear-response method vs supercell method\n\nThe problem of lattice vibration for a solid is now transformed into\ncomputing the dynamical matrix in . The first-principles solution of the\nproblem is currently divided into two categories: the linear-response\nmethod \\[34\\] and the supercell method \\[35\\]. In the linear-response\nmethod, utilizing the electronic linear response upon the undistorted\ncrystals \\[36\\], the evaluations of the dynamical matrix can be\nperformed through the density-functional perturbation theory \\[34\\]\nwithout the approximation of the cutoff in neighboring interaction.\n\nCompared with the linear-response method, the supercell method is\nconceptually simple and is easy to implement computationally. The\nsupercell method adopts the frozen phonon approximation through which\nthe changes in total energy or forces are calculated in the direct space\nby displacing the atoms from their equilibrium positions. The advantage\nof the supercell method is that the phonon frequencies at the exact wave\nvectors, which are commensurable with the supercell, are calculated\nexactly with no further approximation \\[37\\]. The shortcoming of the\nsupercell method is that it is often limited by the size of the\nsupercell that can be handled with current computing resources.\n\nIn supercell approach, inaccuracies are thought to arise from the\ntruncation of the force constants \\[34-35\\]. This is only partially\ntrue. In the supercell method, the calculated represents the cumulative\ncontributions of the atom indexed by *k* and *P* in the supercell and\nall its images by translational transformation of the supercell in the\nwhole space. Let **L**\u03b1 represent the lattice vectors of the\nsupercell, then\n\n*Eq. 5\u2011138* .\n\nFor the exact wave vectors in , one has\n\n*Eq. 5\u2011139* **,**\n\nwhere is an integer. Replacing in with in , one obtains\n\nEq. 5\u2011140 .\n\nTherefore, the phonon frequencies calculated at the exact wave vectors\nby the cumulative force constants approach are exact, and the supercell\nsize will not lead to errors in the calculated phonon frequencies\n\\[37\\].\n\nGenerally speaking, if a supercell contains *Nc* primitive\nunit cells, one can always find *Nc* corresponding exact wave\nvectors in the **q** space. In most linear response calculations, the\ncommon choice of a 4\u00d74\u00d74 **q**-mesh is exactly equivalent to the 4\u00d74\u00d74\nsupercell in the real-space. Furthermore, since the supercell approach\ndoes not impose any approximation on the electronic response to the\ndistortion of the lattice geometry, the effects of electron-phonon\ninteractions can be accounted for by the supercell method.\n\nIn the supercell method, due to the impose of periodic condition, the\ncalculated force constant in the real-space cannot account for the\neffects of the vibration-induced electric field for the polar materials.\nIt has been demonstrated that such an effect adds an additional term to\ndynamical matrix in the reciprocal-space in the form,\n\n*Eq. 5\u2011141* ,\n\nwhere ***Z**\\* (j)* represents the Born effective charge\ntensor of the *j*th atom in the primitive unit cell and\n**\u03b5\u221e** the high frequency static dielectric tensor, i.e., the\ncontribution to the dielectric permittivity tensor from the electronic\npolarization. As a result for the polar materials, the matrix element at\nwave vector **q**\u21920 of the dynamical matrix in , by means of , should\nhave the form\n\n*Eq. 5\u2011142* .\n\nIt can be demonstrated \\[38\\] that this is equivalent to replacing the\nreal space force constant in by\n\n*Eq. 5\u2011143* .\n\nThe present implementation of the first-principles method in calculating\nthe phonon frequencies are mostly limited by the supercell size when\nusing or the number of exact wave vector points when using . A supercell\ncell built on the primitive unit cell or exact wave vector mesh is\nusually the common limit. If only the phonon frequencies derived from or\nare used, can be rather unsmooth which will result in inaccurate\nthermodynamic properties when it is used with . The mixed-space approach\n\\[38\\] can circumvent this bottleneck by using the Fourier interpolation\nto calculate the dynamical matrix for an arbitrary wave vector **q** as\n\n*Eq. 5\u2011144* ,\n\nwith the help of , , and for a polar materials. Note that the term is\nfor polar materials only.\n\nThe can be calculated as\n\n*Eq. 5\u2011145* ,\n\nwhere the function is usually taken as the Gaussian type\n\n*Eq. 5\u2011146* ,\n\nwhere is an adjustable damping (broadening) parameter whose role is to\nsmear the curve. in is the number of used **q** points. Empirically, a\n**q** mesh of is accurate enough for most purposes and doable with the\nYPHON code \\[15\\] efficiently, which is discussed in the Appendix A.\n", "metadata": {} } ] diff --git a/src/psu410/src/firstprinciples_calculations_and_theory/nickel_as_the_prototype.ipynb b/src/psu410/src/firstprinciples_calculations_and_theory/nickel_as_the_prototype.ipynb index e25d418..4951f56 100644 --- a/src/psu410/src/firstprinciples_calculations_and_theory/nickel_as_the_prototype.ipynb +++ b/src/psu410/src/firstprinciples_calculations_and_theory/nickel_as_the_prototype.ipynb @@ -4,253 +4,27 @@ "metadata": {}, "cells": [ { - "id": "55dfbde7", + "id": "a54c87ec", "cell_type": "markdown", - "source": [ - "## Nickel as the prototype\n", - "\n", - "This section exemplifies the step-by-step procedure in calculating the\n", - "thermal properties within the framework of first-principles phonon\n", - "approach, using the elemental metal Ni as the prototype. The calculation\n", - "of the formation enthalpy of Ni3Al is given at the end. The\n", - "calculation in this section is limited to the case for the ferromagnetic\n", - "phase, i.e. single microstate, implying that no configurational mixtures\n", - "or magnetic phase transitions is considered which is discussed in\n", - "Chapter 5.2.5 and Chapter 6.\n", - "\n", - "The Vienna Ab-initio Simulation Package (VASP) \\[13-14\\] has been\n", - "employed for electronic calculations, and the YPHON code \\[15\\] has been\n", - "employed for phonon calculations. VASP is a code based on the\n", - "pseudopotential approach to the density functional theory using plane\n", - "wave function as the basis set, by which only the valence electrons are\n", - "handled explicitly and the core electrons are approximated by an\n", - "effective pseudopotential. The same energy cutoff values, which\n", - "determine the number of plane waves in the expansion of electronic wave\n", - "function, have been used for Ni, Al, and Ni3Al. The rationale\n", - "for the derivations of the formulations used in this section is to be\n", - "given in Chapter for readers who want to have an in-depth understanding\n", - "of the physics behind the used formulations.\n" - ], + "source": "## Nickel as the prototype\n\nThis section exemplifies the step-by-step procedure in calculating the\nthermal properties within the framework of first-principles phonon\napproach, using the elemental metal Ni as the prototype. The calculation\nof the formation enthalpy of Ni3Al is given at the end. The\ncalculation in this section is limited to the case for the ferromagnetic\nphase, i.e. single microstate, implying that no configurational mixtures\nor magnetic phase transitions is considered which is discussed in\nChapter 5.2.5 and Chapter 6.\n\nThe Vienna Ab-initio Simulation Package (VASP) \\[13-14\\] has been\nemployed for electronic calculations, and the YPHON code \\[15\\] has been\nemployed for phonon calculations. VASP is a code based on the\npseudopotential approach to the density functional theory using plane\nwave function as the basis set, by which only the valence electrons are\nhandled explicitly and the core electrons are approximated by an\neffective pseudopotential. The same energy cutoff values, which\ndetermine the number of plane waves in the expansion of electronic wave\nfunction, have been used for Ni, Al, and Ni3Al. The rationale\nfor the derivations of the formulations used in this section is to be\ngiven in Chapter for readers who want to have an in-depth understanding\nof the physics behind the used formulations.\n", "metadata": {} }, { - "id": "5c3e2e16", + "id": "16d7283b", "cell_type": "markdown", - "source": [ - "### Normal coordinates, eigenenergetics, and phonons\n", - "\n", - "One way to simplify the solution to the Schr\u00f6dinger\u2019s equation for\n", - "motion of atomic nuclei is to follow the study of the vibrations of the\n", - "atoms in a crystal \u2013 lattice dynamics. The essential step in lattice\n", - "dynamics is to transform the problem of the correlated motions of 3*N*\n", - "particle into the problem of 3*N* independent harmonics. For this\n", - "purpose, one can introduce a set of new coordinates ( *i*=1, 2, \u2026,\n", - "3*N*), known as normal coordinates, by the transformation\n", - "\n", - "*Eq. 5\u2011111* ,\n", - "\n", - "where is the transformation coefficient that can be determined by\n", - "solving 3*N* simultaneous equations\n", - "\n", - "*Eq. 5\u2011112* ,\n", - "\n", - "where is to be determined so that one can find 3*N* , which are, not all\n", - "zero. The equations are linear and homogeneous. Follow the basic theorem\n", - "in linear algebra that, to find the nonzero solutions of the equations,\n", - "the determinant formed by must equal zero\n", - "\n", - "*Eq. 5\u2011113* ,\n", - "\n", - "where is the Kronecker delta symbol. Since is an equation with 3*N*\n", - "degrees, one can always find 3*N* values of (*i* = 1, \u2026, 3*N*). Each of\n", - "yields a set of which can be chosen such that\n", - "\n", - "*Eq. 5\u2011114* ,\n", - "\n", - "where represents the Kronecker delta symbol and\n", - "\n", - "*Eq. 5\u2011115*\n", - "\n", - "Then with the normal coordinates defined in and utilizing , defined in\n", - "is obtained as\n", - "\n", - "*Eq. 5\u2011116* .\n", - "\n", - "With this equation, is simplified by the following process\n", - "\n", - "*Eq. 5\u2011117* ,\n", - "\n", - "noting that in the above process and have been utilized.\n", - "\n", - "Furthermore, using , the kinetic energy operator in can be transformed\n", - "as follows\n", - "\n", - "*Eq. 5\u2011118* ,\n", - "\n", - "noting that in the above process and are utilized again.\n", - "\n", - "As a result, under the harmonic approximation, the Hamiltonian in is\n", - "simplified as\n", - "\n", - "*Eq. 5\u2011119*\n", - "\n", - "which represents a quantum system containing 3*N* independent harmonics.\n", - "Corresponding to the each of the *\u03c9i*, the quantum theory\n", - "tells that the eigenenergy of a harmonics has the form\n", - "\n", - "*Eq. 5\u2011120* ,\n", - "\n", - "with *gi* = 0, 1, \u2026, \u221e. Such a type of energetics behaves\n", - "like a Boson particle with energy and forms the concept of phonon.\n", - "\n", - "Furthermore, a state of the whole system is to be specified by the set\n", - "of 3*N* independent quantum numbers **g** = (*g1*,\n", - "*g1*, \u2026, *g3N*,). Finally, the energetics of a\n", - "state of the system formed by 3*N* independent harmonics, introduced by\n", - ", is obtained with the summation of the energies of 3*N* independent\n", - "harmonics as\n", - "\n", - "*Eq. 5\u2011121* .\n", - "\n", - "This concludes the rationale by which how is derived.\n" - ], + "source": "### Helmholtz energy and quasiharmonic approximation\n\nAt present, the most rigorous method to predict the thermodynamic\nproperties of a material at finite temperatures is the phonon approach.\nIn such an approach, the microscopic Hamiltonian is expanded up to the\nsecond order. All the thermodynamic quantities are calculated using\nformulations derived from the statistical physics without further\napproximation. The great importance of the phonon theory is that all the\ninput parameters can be obtained by means of first-principles\ncalculations without using any phenomenological parameters.\n\nLet us consider a system with an averaged atomic volume *V*. Neglecting\nthe electron phonon coupling, it is a well demonstrated procedure \\[16\\]\nto decompose the Helmholtz energy *F*(*V*,*T*) of the system at\ntemperature *T* into three additive contributions as follows\n\n*Eq. 5\u20111* ,\n\nwhere *Ec* is the static total energy which is the total\nenergy of the system at 0 K without any atomic vibrations,\n*Fvib* is the vibrational contribution due to the lattice\nions, and *Fel* is the electronic contribution due to the\nthermal electronic excitation at finite temperature which can become\nimportant for metals at high temperature.\n\nThe terminology of \u201cquasiharmonic approximation\u201d arises from the fact\nthat for a given volume, *Fvib*(*V*,*T*) is calculated under\nthe harmonic approximation and the anharmonic effects are carried out\nsolely through volume dependence of the phonon frequency. The easiest\ncomputational implementation of is to first independently calculate the\nHelmholtz energy at several selected volumes near the equilibrium volume\nand then use the numerical interpolation to find the Helmholtz energy at\nan arbitrary volume. The volume interval is usually at the scale of 3~5%\nof the equilibrium volume. Too small volume interval can result in\nnumerical instability due to the numerical uncertainties in the static\ntotal energy calculation, in particular, when one numerically computes\nthe first- and especially the second-order derivatives of the Helmholtz\nenergy in deriving the thermodynamic quantities. It should be noted that\nwhenever available, analytic formulas should be used instead of the\nnumerical second-order derivative to avoid numerical errors. For\ninstance, when the phonon approach is employed, the constant volume heat\ncapacity has the analytic expression in terms of phonon density of\nstates.\n\nNickel metal adopts the fcc structure at ambient conditions and the\nprimitive unit cell contains one atom. Almost all the existing\nfirst-principles codes have the function to calculate the static total\nenergy. The static total energy *Ec* in should be calculated\nusing the primitive unit cell. As the Helmholtz energy is to be\ncalculated at several volumes, a good practice is to plot the calculated\nstatic total energy points together with the interpolated energy curve\nto examine the convergence of the static total energy calculation. Since\nthe first-principles method often employs the self-consistent technique,\nit could occur that calculations at certain volumes may not convergent,\nwhich should be fixed by trying the various algebraic schemes provided\nin most of the existing codes. Furthermore, since certain calculations\ninvolve the second order derivative of the Helmholtz energy, a minor\nuncertainty along the static total energy curve can result in large\ndeviation for the calculated properties such as thermal expansion\ncoefficient and bulk modulus. In that case, a reasonable solution is to\nsmoothen the static total energy using the modified Birch-Murnaghan\nequation of states (EOS) \\[17-18\\]\n\n*Eq. 5\u20112* .\n\nPlotted in is the calculated static total energy of the elemental metal\nNi with the circles representing the calculated values and the curve\nrepresenting that by EOS fitting.\n\nFigure \u2011. Static total energy of nickel.\n\nThe vibrational contribution to the Helmholtz energy by phonon theory\ncan be computed by \\[19\\]\n\n*Eq. 5\u20113* ,\n\nwhere is the Boltzmann\u2019s constant, *\u03c9* represents the phonon frequency,\nand is the phonon density of states. It is recommended that is\ncalculated at the same volume set at which the static total static\nenergies are calculated.\n\nFor the present prototype of Ni, the supercell method for the\ncalculation of has been employed. The procedure is follows:\n\n1. Make supercell by enlarging the primitive unit cell according to the\n > defined neighbor interaction distance; Employ the first-principles\n > code (VASP, \\[13-14\\] in this work) to calculate the interatomic\n > force constants.\n\n2. Assign the mesh in the wave vector (**q**) space; Make the dynamical\n > matrix at each **q** point; Diagonalize the dynamical matrix to\n > find out the phonon frequencies at each **q** point; And finally\n > collect all the phonon frequencies for all **q** points. The\n > detailed formulation for phonon calculations is given in Chapter .\n\nFor the phonon calculations, one can use the open source code YPHON\n\\[15\\] by the present authors. Other choices can be the free ATAT code\n\\[20\\] or the free PHON code \\[21\\]. For the calculation of the phonon\ndensity of states, we have made a supercell containing 64 atoms which is\na 4\u00d74\u00d74 supercell of the primitive unit cell. Plotted in is the\ncalculated phonon density of states using YPHON code at the calculated\nstatic equilibrium volume compared with the measured data at 10 K \\[22\\]\n(symbols).\n\nFigure \u2011. Phonon density of states of nickel.\n\nFor a first-principles thermodynamic calculation, an important step to\navoid possible calculation errors is to examine the phonon dispersions\nfirst. Phonon dispersion \\[23\\] depicts the evolution of phonon\nfrequencies along the designated direction for a crystal. Phonon\ndispersion can be measured rather accurately by inelastic neutron\nscattering \\[24-26\\] or inelastic x-ray scattering \\[27\\] experiment.\nPlotted in are the calculated phonon dispersions (curves) along the\n\\[00\u03b6\\], \\[0\u03b61\\], \\[0\u03b6\u03b6\\], and \\[\u03b6\u03b6\u03b6\\] directions of Ni using YPHON code\ncompared with the neutron scattering data at 296 K (symbols) with\ndetails in Ref. \\[16\\].\n\nFigure \u2011. Phonon dispersions of nickel. The solid lines represent\nresults calculated using a supercell containing 256 atoms which is 4\u00d74\u00d74\nsupercell of the conventional cubic unit cell. The dot-dashed lines\nrepresent results calculated using a supercell containing 64 atoms which\nis 4\u00d74\u00d74 supercell of the primitive unit cell.\n\nFor the calculation of *Fel* in , the most computationally\nconvenient approach is to use the Mermin statistics as follows\n\n*Eq. 5\u20114* ,\n\nwhere is the thermal electronic energy, and *Sel* is the bare\nelectronic entropy. Both the calculations of and *Sel* need\nthe electronic density of states (EDOS) as input. The electronic density\nof states can be obtained during the step of the static total energy\ncalculation. The detailed formulations for and *Sel* are\ngiven in Chapter . Since Ni is magnetic, the EDOS of Ni can be\npartitioned into those of spin up and spin down due to the spin freedom\nof electron. The calculated EDOS for Ni is shown in where the solid,\ndot-dashed, and dashed lines represent the total, spin up, and spin down\nEDOS with the Fermi energy set to zero.\n\nFigure \u2011. Electronic density of states of nickel. That due to spin up is\nplotted as positive value and that due to spin down is plotted as\nnegative value purely for the clarity of the figure. The \u201ctotal\u201d is the\nsum of the absolute values of those of spin up and spin down.\n\nThe calculated temperature evolution of Helmholtz energy as a function\nof volume for Ni are illustrated in . The circles represent the\ncalculated static total energies. The solid curves represent the\nHelmholtz energy curves from 0 to 1600 K at a temperature increment of\n100 K as displayed from top to bottom in . The dashed line marks the\nevolution of the equilibrium volume at *P*=0 with increasing\ntemperature. It is noted that Helmholtz energy always decreases with\nincreasing temperature due to the entropy term of \u2013*TS*. Note that the\nat 0 K the Helmholtz energy is higher than the static total energy due\nto the zero point vibrational energy as can be seen when *T* \u21920 which\nreduces to\n\n*Eq. 5\u20115* ,\n\nwhich is positive.\n\nFigure \u2011. Temperature evolution of the Helmholtz energy for nickel.\n", "metadata": {} }, { - "id": "5cba0f25", + "id": "60b7be55", "cell_type": "markdown", - "source": [ - "### Phonon calculations for SQS\n", - "\n", - "A somewhat more theoretically demanding application of the SQS approach\n", - "is the calculation of the phonon dispersions of a random alloy.\n", - "Considering the fact that the size of an SQS cell in general is around\n", - "8-32, phonon calculations based on SQS is doable, either using the SQS\n", - "cell or its supercell, e.g., of it. However, one notes that while the\n", - "phonon density of states can be calculated straightforwardly, the\n", - "calculations of the phonon dispersions run into a problem. That is,\n", - "since phonon calculation treats the SQS as primitive unit cell made of\n", - "more atoms than the primitive unit cell of the ideal lattice, the number\n", - "of phonon dispersions derived from a regular phonon calculations is a\n", - "lot greater than that measured for the random alloy. Say, one uses an\n", - "SQS containing 16 atoms for an fcc solid solution, the regular phonon\n", - "calculations would produce phonon dispersions in comparison to that just\n", - "three phonon dispersions from measurement. By averaging over the force\n", - "constants of a SQS, the dynamical matrix can be calculated with respect\n", - "to the wave vector space of the ideal lattice of the random alloys.\n", - "\n", - "One consideration that must be taken into account is that the phonon\n", - "dispersions measured from the inelastic neutron scattering experiments\n", - "only represent the averaged vibrations of the ideal lattice. For random\n", - "alloys or phases with minor geometry distortion, it is suggested to\n", - "calculate the dynamical matrix by instead of *Eq. 5\u2011125* as (see \\[45\\])\n", - "\n", - "*Eq. 5\u2011155* ,\n", - "\n", - "where in the case of random alloy, represents the averaged atomic mass\n", - "at the *j*th lattice site. The purpose of the summation over *Q* is to\n", - "average the effects of local distortions, making it possible of\n", - "comparing the calculated dispersions to the measured dispersions\n", - "representing the averaged vibrations of the ideal lattice. As a result,\n", - "the dimension of the SQS supercell dynamical matrix is thus reduced to\n", - "match that of the primitive unit cell of the ideal lattice for the\n", - "calculation of the phonon frequencies. The calculational procedure is as\n", - "follows:\n", - "\n", - "1. Make an SQS supercell based on the primitive unit cell of ideal\n", - " lattice to mimic the correlation functions of the random solution;\n", - "\n", - "2. Relax the SQS supercell with respect to the internal atomic\n", - " positions while keeping the cell shape and volume fixed;\n", - "\n", - "3. Make the phonon supercell by further enlarging the SQS supercell and\n", - " calculate the force constants; and\n", - "\n", - "4. Calculate the dynamical matrix , with the wave vector, **q**, being\n", - " defined from the primitive unit cell of the ideal lattice, through\n", - " the following Fourier transformation.\n", - "\n", - "The calculated phonon dispersions, along the directions (00\u03be), (0\u03be\u03be),\n", - "and (\u03be\u03be\u03be), are compared with the inelastic neutron scattering data in\n", - "for Cu3Au.\n", - "\n", - "Figure 5\u201113. Phonon dispersions for random Cu3Au. The solid\n", - "(black) lines represent the present calculation and the open circles\n", - "represent the inelastic neutron scattering data with details in Ref.\n", - "\\[45\\]. The dashed (blue) lines represent the calculated results using\n", - "the ab initio transferable force-constant model by Dutta et al. \\[46\\].\n" - ], + "source": "### Volume, entropy, enthalpy, thermal expansion, bulk modulus, and heat capacity \n\nThe equilibrium volume *Veq* (*P*,*T*) at given *T* and *P*\ncan be obtained by finding the root of the following equation\n\n*Eq. 5\u20116* .\n\nThe dashed line in illustrates *Veq* (*P*,*T*) as a function\nof *T* from 0 to 1600 K at *P* = 0 for Ni.\n\nThe entropy can be calculated through *F* by\n\n*Eq. 5\u20117* .\n\nPlotted in is the calculated entropy (curve) of Ni as a function of\ntemperature from 0 to 1600 K at *P* = 0 compared with the recommended\ndata (symbols) with details in Ref. \\[16\\].\n\nFigure \u2011. Entropy of nickel as a function of temperature.\n\nBased on *F* and *S*, the enthalpy at given *P* and *T* can be computed\nas\n\n*Eq. 5\u20118* .\n\nPlotted in is the calculated enthalpy (curve) of Ni as function of\ntemperature from 0 to 1600 K at *P* = 0 comparing with the recommended\ndata (open circles) with details in Ref. \\[16\\]\n\nFigure \u2011. Enthalpy of nickel as a function of temperature.\n\nWith the equilibrium volume *Veq* (*P*,*T*) calculated by ,\nthe volume thermal expansion coefficient defined by can be calculated by\n\n*Eq. 5\u20119* .\n\nPlotted in is the calculated thermal expansion coefficient (curve) of\nnickel as function of temperature from 0 to 1600 K at *P* = 0 comparing\nwith experimental data (symbols) with details in Ref. \\[16\\]\n\nFigure \u2011. Volume thermal expansion coefficient of nickel as a function\nof temperature.\n\nThe bulk modulus of a material represents the substance's resistance to\nuniform compression. Depending on how the temperature varies during\ncompression, a distinction should be made between the isothermal bulk\nmodulus (constant temperature) and adiabatic bulk modulus (constant\nentropy or no heat transfer). As a matter of fact, most of the\nexperimental data are adiabatic whereas most of the published\ntheoretical data are isothermal.\n\nThe isothermal bulk modulus, as defined in terms of Gibbs energy shown\nby , can be calculated by\n\n*Eq. 5\u201110*\n\nBased on the isothermal bulk modulus, the adiabatic bulk modulus can be\ncalculated by\n\n*Eq. 5\u201111* ,\n\nwhere *CP* and *CV* represent the constant\npressure heat capacity and constant volume heat capacity respectively.\nPlotted in is the calculated bulk moduli (curves) of Ni as function of\ntemperature from 0 to 1600 K at *P* = 0. The experimental data are from\nultrasonic measurements (symbols, see Ref. \\[28\\] for more details) that\nare therefore adiabatic bulk moduli calculated based on the measured\nadiabatic elastic constants using the relation\n\n*Eq. 5\u201112* .\n\nFigure \u2011. Bulk moduli of nickel as a function of temperature. Solid\nline: adiabatic; Dashed line: isothermal.\n\nThe heat capacity at constant volume, as defined in terms of Gibbs\nenergy shown by Eq. 2\u201128, can be calculated by\n\n*Eq. 5\u201113* ,\n\nwhere represents the internal energy. The heat capacity at constant\npressure (see ) can then be calculated as\n\n*Eq. 5\u201114* ,\n\nutilizing the calculated thermal expansion coefficient in and bulk\nmodulus in .\n\nIt can be advocated that thermal expansion makes the difference between\nthe heat capacity at constant volume and the heat capacity at constant\npressure. The calculated contributions to the heat capacity of Ni as\nfunction of temperature from 0 to 1600 K at *P* = 0 are illustrated in\nwhere the lattice vibration and the thermal electron contributions have\nbeen separated out. From , it is observed a large difference between the\ncalcualted *CP* (solid line) and the experimental data\n(symbols, see Ref. \\[18\\] for more details) at 600 K due to the magnetic\nphase transition which has not been considered in the calculation. It\nshould be pointed out that for Ni the thermal electronic contribution to\nthe heat capacity (dashed line in ) is substantial at high temperatures.\n\nFigure \u2011. Heat capacity of nickel as a function of temperature.\nrepresents the calculated lattice vibration contribution; represents the\ncalculated thermal electronic contribution.\n", "metadata": {} }, { - "id": "687b1264", + "id": "6b534057", "cell_type": "markdown", - "source": [ - "### Quantum theory for motion of atomic nuclei\n", - "\n", - "For the convenience of discussion, the following convention of notations\n", - "are used: *\u03b1* and *\u03b2* to label the Cartesian axes which is either *x*,\n", - "*y*, or *z*, *j* and *k* to label atoms in the primitive unit cell,\n", - "*mj* the atomic mass of the *j*th atom in the\n", - "primitive unit cell, **r**(*j*) the position of the *j*th\n", - "atom in the primitive unit cell, *P* and *Q* the index of the primitive\n", - "unit cell in the system, **R**(*P*) the position of the *P*th\n", - "primitive unit cell in the system, and *V* the averaged volume of the\n", - "primitive unit cell.\n", - "\n", - "The quantum theory for motion of atomic nuclei replicates closely the\n", - "quantum theory for motion of electrons. That is to solve the wave\n", - "function for the motions of the atomic nuclei for a Schr\u00f6dinger\u2019s\n", - "equation with the potential being the total electronic energy derived\n", - "from . The symbol is replaced by to represent the static total\n", - "electronic energy with **R** representing the static equilibrium\n", - "positions of the atomic nuclei, **u** the displacements of atomic nuclei\n", - "away from their static equilibrium positions, and *\u03c3* the additional\n", - "degree of freedom such as electronic states. The Schr\u00f6dinger\u2019s equation\n", - "for the motion of atomic nuclei is then\n", - "\n", - "*Eq. 5\u2011103* ,\n", - "\n", - "where\n", - "\n", - "*Eq. 5\u2011104* ,\n", - "\n", - "with representing the kinetic energy operator\n", - "\n", - "*Eq. 5\u2011105* ,\n", - "\n", - "where *Nc* is the numbers of primitive unit cells in the\n", - "system, *Na* is the numbers of atoms in the primitive unit\n", - "cell, is the Plack constant, represents the *\u03b1*th Cartesian\n", - "component of **u** for the atom at *j*th lattice site in the\n", - "primitive unit cell and the *P*th primitive unit cell in the\n", - "system.\n", - "\n", - "The harmonic approximation \\[23, 33\\] truncates the into the second\n", - "order in its Taylor\u2019s series\n", - "\n", - "*Eq. 5\u2011106* ,\n", - "\n", - "where is the real-space interatomic force constant. With the\n", - "approximation of , it can be demonstrated that finding the solution of\n", - "is equivalent to finding the vibrational frequencies of a classical\n", - "system with *NcNa* particles for small mechanical\n", - "oscillations.\n", - "\n", - "Let us rewrite as\n", - "\n", - "*Eq. 5\u2011107* ,\n", - "\n", - "where\n", - "\n", - "*Eq. 5\u2011108* , and\n", - "\n", - "*Eq. 5\u2011109* .\n", - "\n", - "Accordingly, the kinetic energy operator in becomes\n", - "\n", - "*Eq. 5\u2011110* .\n" - ], + "source": "### Formation enthalpy of Ni3Al\n\nOne can do the similar calculations for the elemental metal Al and the\ncompound Ni3Al which has the L12 structure,\nfollowing the same steps in the calculations of Ni. The formation\nenthalpy in the unit of per mole atom can be calculated as\n\n*Eq. 5\u201115* ,\n\nwhere , , and represent the enthalpies of Ni3Al, Ni, and Al\nin the energy unit of per mole atom, respectively. Plotted in is the\ncalculated formation enthalpy of Ni3Al (curve) as a function\nof temperature from 0 to 1600 K at *P* = 0 compared with experimental\ndata (symbols) with details in Ref. \\[16\\].\n\n.\n\nFigure \u2011. Formation enthalpy of L12-Ni3Al with\nrespect to pure fcc Ni and Al.\n", "metadata": {} } ] diff --git a/src/psu410/src/firstprinciples_calculations_and_theory/quantum_theory_for_the_motion_of_electrons.ipynb b/src/psu410/src/firstprinciples_calculations_and_theory/quantum_theory_for_the_motion_of_electrons.ipynb index 2ae4fe6..8b75f2a 100644 --- a/src/psu410/src/firstprinciples_calculations_and_theory/quantum_theory_for_the_motion_of_electrons.ipynb +++ b/src/psu410/src/firstprinciples_calculations_and_theory/quantum_theory_for_the_motion_of_electrons.ipynb @@ -4,401 +4,39 @@ "metadata": {}, "cells": [ { - "id": "6f45af32", + "id": "2c82690a", "cell_type": "markdown", - "source": [ - "## Quantum theory for the motion of electrons\n" - ], + "source": "## Quantum theory for the motion of electrons\n", "metadata": {} }, { - "id": "9128fb36", + "id": "6f383f95", "cell_type": "markdown", - "source": [ - "### Schr\u00f6dinger equation\n", - "\n", - "The Schr\u00f6dinger equation is typically written as follows\n", - "\n", - "*Eq. 5\u201157* ,\n", - "\n", - "where is the reduced Planck constant, **X** an abbreviation of the space\n", - "coordinates and spin states of the multiple particle system, the energy\n", - "operator called Hamiltonian. When is independent of time *t*, one can\n", - "separate the coordinate **X** from the time *t* in finding the solution\n", - "of by writing\n", - "\n", - "*Eq. 5\u201158* ,\n", - "\n", - "by which the stationary solutions of can be expressed through letting\n", - "\n", - "*Eq. 5\u201159* ,\n", - "\n", - "resulting in\n", - "\n", - "*Eq. 5\u201160*\n", - "\n", - "*Eq. 5\u201161*\n", - "\n", - "Note that is the frequency of de Broglie matter wave.\n", - "\n", - "For any trial function \u039b(**X**) (in the Hilbert space) for \u03a8(**X**), the\n", - "variational principle tells us that the energy of the system always has\n", - "a lower bound through a ground state with energy *E0*, as\n", - "\n", - "*Eq. 5\u201162* ,\n", - "\n", - "where represents the complex conjugate of \u039b(**X**), resulting in\n", - "\n", - "*Eq. 5\u201163*\n", - "\n", - "which is known as the Rayleigh-Ritz variational principle,\n" - ], + "source": "### Schr\u00f6dinger equation\n\nThe Schr\u00f6dinger equation is typically written as follows\n\n*Eq. 5\u201157* ,\n\nwhere is the reduced Planck constant, **X** an abbreviation of the space\ncoordinates and spin states of the multiple particle system, the energy\noperator called Hamiltonian. When is independent of time *t*, one can\nseparate the coordinate **X** from the time *t* in finding the solution\nof by writing\n\n*Eq. 5\u201158* ,\n\nby which the stationary solutions of can be expressed through letting\n\n*Eq. 5\u201159* ,\n\nresulting in\n\n*Eq. 5\u201160*\n\n*Eq. 5\u201161*\n\nNote that is the frequency of de Broglie matter wave.\n\nFor any trial function \u039b(**X**) (in the Hilbert space) for \u03a8(**X**), the\nvariational principle tells us that the energy of the system always has\na lower bound through a ground state with energy *E0*, as\n\n*Eq. 5\u201162* ,\n\nwhere represents the complex conjugate of \u039b(**X**), resulting in\n\n*Eq. 5\u201163*\n\nwhich is known as the Rayleigh-Ritz variational principle,\n", "metadata": {} }, { - "id": "c21545db", + "id": "23677eac", "cell_type": "markdown", - "source": [ - "#### Solving the Kohn-Sham Equations for a Solid\n", - "\n", - "For a solid, is still a mathematical challenge with infinite number of\n", - "one-electron wave functions to be solved and therefore cannot be solved\n", - "directly in the real space. To reduce the dimension of the problem, one\n", - "can choose to solve the equation at a specific point in the reciprocal\n", - "space. According to Bloch\u2019s theorem, the wave function for a solid can\n", - "be written as the product of a wavelike part, , and a cell periodic\n", - "part,\n", - "\n", - "*Eq. 5\u201195*\n", - "\n", - "where can be expressed as a sum of a finite number of plane waves whose\n", - "wave vectors are reciprocal lattice vectors of the crystal\n", - "\n", - "*Eq. 5\u201196* .\n", - "\n", - "so that\n", - "\n", - "*Eq. 5\u201197*\n", - "\n", - "where the band index *j* is used to number the eigenenergy and the\n", - "eigenvector at a given . The number of plane waves is determined by the\n", - "following equation\n", - "\n", - "*Eq. 5\u201198*\n", - "\n", - "where is called energy cutoff.\n", - "\n", - "Utilize the obtained wave functions, the charge density can be\n", - "calculated with the Brillouin zone integration\n", - "\n", - "*Eq. 5\u201199*\n", - "\n", - "Where\n", - "\n", - "*Eq. 5\u2011100*\n", - "\n", - "where the parameter is to make the integration over the charge density\n", - "within the primitive unit cell equal to the number of electrons, *N,* in\n", - "the primitive unit cell. Numerically, the integration can be\n", - "approximated by summation over a set of discrete *k*-mesh as\n", - "\n", - "*Eq. 5\u2011101*\n", - "\n", - "where *N*BZ represents the number of points in the first\n", - "Brillouin zone in the *k*-mesh. When a solid possesses symmetry, the\n", - "summation in the above equation can be further reduced to the summation\n", - "over the irreducible Brillouin zone (IBZ).\n", - "\n", - "*Eq. 5\u2011102*\n", - "\n", - "where is a weight factor that represents the number of points that are\n", - "equivalent to by space group symmetry.\n" - ], + "source": "### Born-Oppenheimer Approximation\n\nFor a time independent atomic system, it is often accurate enough to\nwrite in or in terms of the electron coordinate **r** and nuclei\ncoordinate **R**,\n\n*Eq. 5\u201164* ,\n\nwhere *e* represents the electron charge and *i* and *j* label the\nelectrons, *I* and *J* the atomic nuclei, *ZI* the atomic\nnuclear charge number of atom *I*, *e* the electron charge,\n*me* the electron mass, *MI* the mass of atomic\nnuclei *I*, the Laplace operator for electron *i*, and the Laplace\noperator for atomic nuclei *I,* noting\n\n*Eq. 5\u201165* ,\n\nwith respect to the Cartesian axis *x*, *y*, and *z*.\n\nConsider the fact that the electron mass is thousand times smaller than\nthe mass of the atomic nuclei, implying that the motions of the\nelectrons are much faster than the atomic nuclei, Born and Oppenheimer\nproposed that the wave function of the whole system can be simply\napproximated as the product of the electron wave function and the atomic\nnuclei wave function as\n\n*Eq. 5\u201166* .\n\nWith the auxiliary approximation of neglecting the dynamic coupling\nbetween the motions of electrons and atomic nuclei, the Schr\u00f6dinger\nequation for the motion of electrons becomes\n\n*Eq. 5\u201167* ,\n\nwhere\n\n*Eq. 5\u201168* ,\n\nand the Schr\u00f6dinger equation for the motion of atomic nuclei becomes\n\n*Eq. 5\u201169* ,\n\nwhere\n\n*Eq. 5\u201170* ,\n\nwith\n\n*Eq. 5\u201171* ,\n\nwhere represents the complex conjugate of .\n", "metadata": {} }, { - "id": "028b08b6", + "id": "f30dcaab", "cell_type": "markdown", - "source": [ - "### Special Quasirandom Structures\n", - "\n", - "SQS\u2019s \\[43-44\\] are specially designed *small-unit-cell* periodic\n", - "structures with minimal number of atoms per unit cell, which closely\n", - "mimic the most relevant, near-neighbor pair and multisite correlation\n", - "functions of random substitutional alloys. The correlation functions are\n", - "classified by their *n*-site component \u201cfigures\u201d *f* = (*n,m*), where\n", - "the index *n* is called vortex for pair, triple, and quadruple\n", - "correlations (*n* = 2, 3, and 4); *m* measures the correlation distance.\n", - "\n", - "In the SQS approach, a distribution of distinct local environments is\n", - "maintained and their average corresponds to the random alloy. Thus, a\n", - "single DFT calculation of an SQS can give many important alloy\n", - "properties (e.g. equilibrium bond lengths, charge transfer, formation\n", - "enthalpies, etc.) that depend on the existence of those distinct local\n", - "environments. The SQS approach has been used extensively to study the\n", - "formation enthalpies, bond length distributions, density of states, band\n", - "gaps and optical properties in semiconductor alloys. It is noted that\n", - "the CE approach can treat short-range ordering efficiently, while it is\n", - "not clear how the SQS approach can be used to represent short-range\n", - "ordering.\n", - "\n", - "The key quantities in the SQS approach are the *n*-site correlation\n", - "functions. Specifically, the 2-site correlation function corresponding\n", - "to the 2-site component \u201cfigures\u201d (2,*m*) is\n", - "\n", - "Eq. 5\u2011151\n", - "\n", - "where represents the total number of possible pairs with correlation\n", - "distance (neighboring distance) *Rij* being equal to *m*. The\n", - "3-site correlation function corresponding to the 3-site component\n", - "\u201cfigures\u201d (3,*m*) is\n", - "\n", - "Eq. 5\u2011152\n", - "\n", - "where represents the total number of all possible 3-site \u201cfigures\u201d with\n", - "the correlation distance (size and shape) *Rijk* being equal\n", - "to *m*. The 4-site correlation function corresponding to the 4-site\n", - "component \u201cfigures\u201d (4,*m*) is\n", - "\n", - "Eq. 5\u2011153 ,\n", - "\n", - "where represents the total number of all possible 4-site \u201cfigures\u201d with\n", - "correlation distance (size and shape) *Rijkl* being equal to\n", - "*m*.\n", - "\n", - "With a given supercell size *N*, the essential task of the SQS approach\n", - "is to search through all configurations that approach as close as\n", - "possible to the correlation functions of the perfectly random (*R*)\n", - "structure and for the binary system it is\n", - "\n", - "Eq. 5\u2011154 .\n", - "\n", - "Describing random alloys by small unit-cell periodical structures surely\n", - "introduces erroneous correlations beyond a certain distance. However,\n", - "since interactions between nearest neighbors are generally more\n", - "important than interactions between more distant neighbors, SQS\u2019s can be\n", - "constructed in such a way that they exactly reproduce the correlation\n", - "functions of a random alloy between the first few nearest neighbors,\n", - "deferring errors due to periodicity to more distant neighbors. The\n", - "practical procedure could be that to find the structures that match the\n", - "2-site correlation functions up to a given neighboring distance, and\n", - "then add the conditions matching the high order correlation functions up\n", - "to certain correlation distance.\n", - "\n", - "Appendix B is a collection of the SQS\u2019s with a variety of compositions\n", - "for binary fcc, bcc, hcp, and L12 structures, for ternary\n", - "fcc, bcc, and B2 structures, and Perovskite in the cubic ABO3\n", - "structure. The used format is that of VASP.\n" - ], + "source": "### Hartree-Fock approximation to solve Schr\u00f6dinger equation\n\nIt was Hartree who first assumed that the electron wave function in can\nbe expressed as a product of a collection of *N* independent\none-electron wave functions, where *i* = 1, 2, \u2026, *N* with *N* being the\nnumber of electrons in a system, in terms of the its space coordinate\n**r** and spin state **s**. After that, Fock modified the Hartree\napproximation by considering the fact that the wave function of a\nmulti-fermionic system should satisfy anti-symmetry requirements and\nsubsequently the Pauli exclusion principle that the total wave function\nchanges sign upon the exchange of fermions. Accordingly, the wave\nfunction of *N* electrons system under the Hartree-Fock approximation is\nexpressed as the Slater determinant \\[29\\]\n\n*Eq. 5\u201172*\n\nFor brevity, one can use the atomic unit that treats in , so that\n\n*Eq. 5\u201173*\n\nwhere , and\n\n*Eq. 5\u201174*\n\nAccordingly, the total energy of the system is expressed as\n\n*Eq. 5\u201175*\n\nwhere *Jij* is called as Coulomb/Hartree term\n\n*Eq. 5\u201176*\n\nwhere *Kij* is called as exchange term\n\n*Eq. 5\u201177* ,\n\nwhere = 1 if spin **s***i* and **s***j* points to\nthe same direction and = 0 if spin **s***i* and\n**s***j* points to the opposite direction.\n\nBy utilizing the variational condition of , one gets\n\n*Eq. 5\u201178* ,\n\nwhere is called one-electron energy, and\n\n*Eq. 5\u201179* ,\n\nwith being the electronic charge density whose expression is\n\n*Eq. 5\u201180* ,\n\n*Eq. 5\u201181* .\n\nIt should be especially noted here that to solve the Hartree-Fock\nequation , the most time consuming part is due to the nonlocal exchange\nterm, knowing the fact that the being evaluated one-electron wave\nfunction is also contained in the expression in the left hand side of by\nmeans of .\n\nThe configurational interaction method is the generalization of the\nHartree-Fock approximation. In such a case, *Y*, the number of\none-electron wave functions can be larger than the number of electrons,\n*N*, in the system. Accordingly, from the number of one-electron wave\nfunctions, , *y* = 1, 2, \u2026, *Y*, one can build the number of Stater\ndeterminants, *M*, by the combinatorial mathematics that the maximum of\n*M* can be\n\n*Eq. 5\u201182* .\n\nAs a result, the wave function of a collection of *N* electron system\nbecomes the recombination of the *M* Stater determinants as\n\n*Eq. 5\u201183* ,\n\nwhere the coefficients is to be found from the multiple linear equation\n\n*Eq. 5\u201184* .\n\nThe matrix element in *Eq. 5\u201184* is determined by the integral\n\n*Eq. 5\u201185* .\n", "metadata": {} }, { - "id": "3e725123", + "id": "6be3ff7e", "cell_type": "markdown", - "source": [ - "### System with multiple microstates (MMS model)\n", - "\n", - "For a system made of multiple microstates, the total partition function\n", - "is the summation over the partition functions of all microstates, in ,\n", - "as\n", - "\n", - "*Eq. 5\u201149* ,\n", - "\n", - "where is the multiplicity of the microstate *\u03c3.* It is immediately\n", - "apparent that is the thermal population of the microstate *\u03c3*.\n", - "Furthermore, with , one obtains\n", - "\n", - "*Eq. 5\u201150* .\n", - "\n", - "relates the total Helmholtz energy, *F(N,V,T)*, of a system with mixing\n", - "among multiple microstates and the Helmholtz energy, , of individual\n", - "microstates. An important result of is the configurational entropy due\n", - "to the mixing among multiple microstates, named as microstate\n", - "configurational entropy (MCE) in this book,\n", - "\n", - "*Eq. 5\u201151* ,\n", - "\n", - "which makes the entropy of a system with mixing among multiple\n", - "microstates as\n", - "\n", - "*Eq. 5\u201152* .\n", - "\n", - "Similarly, one can get the heat capacity at constant volume of a system\n", - "with mixing among multiple microstates as\n", - "\n", - "*Eq. 5\u201153* ,\n", - "\n", - "where the configurational contributions to the heat capacity due to the\n", - "mixing among multiple microstates is\n", - "\n", - "*Eq. 5\u201154* .\n", - "\n", - "Moreover, the isothermal bulk modulus of a system with mixing among\n", - "multiple microstates can be also computed similarly\n", - "\n", - "*Eq. 5\u201155* ,\n", - "\n", - "With\n", - "\n", - "*Eq. 5\u201156*\n", - "\n", - "with being the partial pressure of the microstate *\u03c3*. This multiple\n", - "microstate model (MMS model) is used in Chapter to quantitatively\n", - "predict thermal expansion anomalies.\n" - ], + "source": "### Density functional theory (DFT) and 0 K Kohn-Sham equations\n\nThe density functional theory advocates that the properties of a matter\nare solely dictated by its electronic density distribution (or equally\nsay charge density), , in the real space. This is to say, that for an\narbitrary the total energy of the system, *E*, is always larger or equal\nto a value, *E0*, called as the ground state energy:\n\n*Eq. 5\u201186* .\n\nIn terms of variational principle, is equivalent to\n\n*Eq. 5\u201187* .\n\nKohn and Sham \\[8\\] proposed to write the total energy as\n\n*Eq. 5\u201188* ,\n\nwhere represents the kinetic energy of the system, is the external\npotential acting on the system, is the Hartree energy, and is the\nso-called exchange-correlation energy with where and represent the\ncharge density of electrons with spin down and spin up, respectively .\nUsing\n\n*Eq. 5\u201189* ,\n\ntogether with the variational principle of , one can get the\none-electron Schr\u00f6dinger equation\n\n*Eq. 5\u201190*\n\nwhere *me* represents the mass of an electron and\n\n*Eq. 5\u201191*\n\nso that the total energy is obtained as\n\n*Eq. 5\u201192*\n\nThe major challenge within DFT is that the accurate formulation of the\nexchange-correlation energy is unknown. Except for the uniform electron\ngas, no exact analytical form for the exchange-correlation energy has\nyet been obtained. Therefore approximations must be made for the\nexchange-correlation energy in calculating a realistic system. Until\nnow, the two most popular approximations are the local density\napproximation (LDA) \\[30\\] and the generalized gradient approximation\n(GGA) \\[31-32\\].\n\nThe local density approximation (LDA) states that the\nexchange-correlation energy is the same as that for a locally uniform\nelectron gas. In this case one can write *Vxc* as\n\n*Eq. 5\u201193* .\n\nAlthough this approximation is extremely simple, it works reasonably\nwell for many systems. The only remaining problem is to find an\napproximate solution to . One of most employed parameterized expression\nfor is that by Perdew and Zunger \\[30\\].\n\nMany modern DFT codes use the more advanced generalized gradient\napproximation (GGA) \\[31-32\\] to the exchange-correlation energy to\nimprove accuracy for certain physical properties. As the LDA\napproximates the energy of the true density by the energy of a local\nconstant density, it fails in situations where the density undergoes\nrapid changes such as in molecules. An improvement to this can be made\nby considering the gradient of the electron density. Symbolically, this\ncan be written as\n\n*Eq. 5\u201194*\n\nThe commonly used GGA is those due to Perdew et al. \\[31-32\\].\n", "metadata": {} }, { - "id": "f2f7173c", + "id": "9d69a63e", "cell_type": "markdown", - "source": [ - "### Vibrational contribution by phonon theory\n", - "\n", - "Under the harmonic/quasiharmonic approximation, the lattice dynamics or\n", - "phonon theory is currently the most established method. It truncates the\n", - "interaction potential up to the second order. In such a case, in can be\n", - "expressed in terms of phonon frequency as follows\n", - "\n", - "*Eq. 5\u201132* ,\n", - "\n", - "where the label **g** has the meanings of (*g1*,\n", - "*g2*, \u2026, *g3N*) that *gj* can take any\n", - "integer values from zero to infinite.\n", - "\n", - "As a result, is reduced to\n", - "\n", - "*Eq. 5\u201133* ,\n", - "\n", - "or equivalently,\n", - "\n", - "*Eq. 5\u201134* ,\n", - "\n", - "where an integration has been used to replace the summation in by means\n", - "of introducing a function, , named the phonon density of states (PDOS)\n", - "whose integration over *\u03c9* equals to three per atom.\n", - "\n", - "Accordingly, the formulation to calculate the entropy becomes\n", - "\n", - "*Eq. 5\u201135* ,\n", - "\n", - "the formulation to calculate the internal energy becomes\n", - "\n", - "*Eq. 5\u201136* ,\n", - "\n", - "and the formulation to calculate the heat capacity at constant volume\n", - "becomes\n", - "\n", - "*Eq. 5\u201137* .\n" - ], - "metadata": {} - }, - { - "id": "533ec4a5", - "cell_type": "markdown", - "source": [ - "### Volume, entropy, enthalpy, thermal expansion, bulk modulus, and heat capacity \n", - "\n", - "The equilibrium volume *Veq* (*P*,*T*) at given *T* and *P*\n", - "can be obtained by finding the root of the following equation\n", - "\n", - "*Eq. 5\u20116* .\n", - "\n", - "The dashed line in illustrates *Veq* (*P*,*T*) as a function\n", - "of *T* from 0 to 1600 K at *P* = 0 for Ni.\n", - "\n", - "The entropy can be calculated through *F* by\n", - "\n", - "*Eq. 5\u20117* .\n", - "\n", - "Plotted in is the calculated entropy (curve) of Ni as a function of\n", - "temperature from 0 to 1600 K at *P* = 0 compared with the recommended\n", - "data (symbols) with details in Ref. \\[16\\].\n", - "\n", - "Figure \u2011. Entropy of nickel as a function of temperature.\n", - "\n", - "Based on *F* and *S*, the enthalpy at given *P* and *T* can be computed\n", - "as\n", - "\n", - "*Eq. 5\u20118* .\n", - "\n", - "Plotted in is the calculated enthalpy (curve) of Ni as function of\n", - "temperature from 0 to 1600 K at *P* = 0 comparing with the recommended\n", - "data (open circles) with details in Ref. \\[16\\]\n", - "\n", - "Figure \u2011. Enthalpy of nickel as a function of temperature.\n", - "\n", - "With the equilibrium volume *Veq* (*P*,*T*) calculated by ,\n", - "the volume thermal expansion coefficient defined by can be calculated by\n", - "\n", - "*Eq. 5\u20119* .\n", - "\n", - "Plotted in is the calculated thermal expansion coefficient (curve) of\n", - "nickel as function of temperature from 0 to 1600 K at *P* = 0 comparing\n", - "with experimental data (symbols) with details in Ref. \\[16\\]\n", - "\n", - "Figure \u2011. Volume thermal expansion coefficient of nickel as a function\n", - "of temperature.\n", - "\n", - "The bulk modulus of a material represents the substance's resistance to\n", - "uniform compression. Depending on how the temperature varies during\n", - "compression, a distinction should be made between the isothermal bulk\n", - "modulus (constant temperature) and adiabatic bulk modulus (constant\n", - "entropy or no heat transfer). As a matter of fact, most of the\n", - "experimental data are adiabatic whereas most of the published\n", - "theoretical data are isothermal.\n", - "\n", - "The isothermal bulk modulus, as defined in terms of Gibbs energy shown\n", - "by , can be calculated by\n", - "\n", - "*Eq. 5\u201110*\n", - "\n", - "Based on the isothermal bulk modulus, the adiabatic bulk modulus can be\n", - "calculated by\n", - "\n", - "*Eq. 5\u201111* ,\n", - "\n", - "where *CP* and *CV* represent the constant\n", - "pressure heat capacity and constant volume heat capacity respectively.\n", - "Plotted in is the calculated bulk moduli (curves) of Ni as function of\n", - "temperature from 0 to 1600 K at *P* = 0. The experimental data are from\n", - "ultrasonic measurements (symbols, see Ref. \\[28\\] for more details) that\n", - "are therefore adiabatic bulk moduli calculated based on the measured\n", - "adiabatic elastic constants using the relation\n", - "\n", - "*Eq. 5\u201112* .\n", - "\n", - "Figure \u2011. Bulk moduli of nickel as a function of temperature. Solid\n", - "line: adiabatic; Dashed line: isothermal.\n", - "\n", - "The heat capacity at constant volume, as defined in terms of Gibbs\n", - "energy shown by Eq. 2\u201128, can be calculated by\n", - "\n", - "*Eq. 5\u201113* ,\n", - "\n", - "where represents the internal energy. The heat capacity at constant\n", - "pressure (see ) can then be calculated as\n", - "\n", - "*Eq. 5\u201114* ,\n", - "\n", - "utilizing the calculated thermal expansion coefficient in and bulk\n", - "modulus in .\n", - "\n", - "It can be advocated that thermal expansion makes the difference between\n", - "the heat capacity at constant volume and the heat capacity at constant\n", - "pressure. The calculated contributions to the heat capacity of Ni as\n", - "function of temperature from 0 to 1600 K at *P* = 0 are illustrated in\n", - "where the lattice vibration and the thermal electron contributions have\n", - "been separated out. From , it is observed a large difference between the\n", - "calcualted *CP* (solid line) and the experimental data\n", - "(symbols, see Ref. \\[18\\] for more details) at 600 K due to the magnetic\n", - "phase transition which has not been considered in the calculation. It\n", - "should be pointed out that for Ni the thermal electronic contribution to\n", - "the heat capacity (dashed line in ) is substantial at high temperatures.\n", - "\n", - "Figure \u2011. Heat capacity of nickel as a function of temperature.\n", - "represents the calculated lattice vibration contribution; represents the\n", - "calculated thermal electronic contribution.\n" - ], + "source": "#### Solving the Kohn-Sham Equations for a Solid\n\nFor a solid, is still a mathematical challenge with infinite number of\none-electron wave functions to be solved and therefore cannot be solved\ndirectly in the real space. To reduce the dimension of the problem, one\ncan choose to solve the equation at a specific point in the reciprocal\nspace. According to Bloch\u2019s theorem, the wave function for a solid can\nbe written as the product of a wavelike part, , and a cell periodic\npart,\n\n*Eq. 5\u201195*\n\nwhere can be expressed as a sum of a finite number of plane waves whose\nwave vectors are reciprocal lattice vectors of the crystal\n\n*Eq. 5\u201196* .\n\nso that\n\n*Eq. 5\u201197*\n\nwhere the band index *j* is used to number the eigenenergy and the\neigenvector at a given . The number of plane waves is determined by the\nfollowing equation\n\n*Eq. 5\u201198*\n\nwhere is called energy cutoff.\n\nUtilize the obtained wave functions, the charge density can be\ncalculated with the Brillouin zone integration\n\n*Eq. 5\u201199*\n\nWhere\n\n*Eq. 5\u2011100*\n\nwhere the parameter is to make the integration over the charge density\nwithin the primitive unit cell equal to the number of electrons, *N,* in\nthe primitive unit cell. Numerically, the integration can be\napproximated by summation over a set of discrete *k*-mesh as\n\n*Eq. 5\u2011101*\n\nwhere *N*BZ represents the number of points in the first\nBrillouin zone in the *k*-mesh. When a solid possesses symmetry, the\nsummation in the above equation can be further reduced to the summation\nover the irreducible Brillouin zone (IBZ).\n\n*Eq. 5\u2011102*\n\nwhere is a weight factor that represents the number of points that are\nequivalent to by space group symmetry.\n", "metadata": {} } ] diff --git a/src/psu410/src/gibbs_energy_function/binary_random_solutions.ipynb b/src/psu410/src/gibbs_energy_function/binary_random_solutions.ipynb deleted file mode 100644 index b8cb041..0000000 --- a/src/psu410/src/gibbs_energy_function/binary_random_solutions.ipynb +++ /dev/null @@ -1,164 +0,0 @@ -{ - "nbformat": 4, - "nbformat_minor": 5, - "metadata": {}, - "cells": [ - { - "id": "ecc765e4", - "cell_type": "markdown", - "source": [ - "### Binary random solutions\n", - "\n", - "From , the Gibbs-Duhem equation of a binary system consisting of\n", - "components $A$ and $B$ is written as\n", - "\n", - "Eq. \u2011 $0 = - SdT - Vd( - P) - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B}$\n", - "\n", - "This equation represents a four-dimensional surface. It is self-evident\n", - "that both and hold for stable binary solutions too, i.e. the directions\n", - "and the curvature of the surface are all negative. To visualize the\n", - "four-dimensional surface in the three-dimensional space, one needs to\n", - "fix one of the four potentials. As $T$ and $P$ are the natural variables\n", - "of Gibbs energy, they are usually chosen to be kept constant. One can\n", - "typically investigate behaviors of systems consisting of condensed\n", - "phases by varying the temperature at constant pressure. at constant\n", - "pressure thus becomes\n", - "\n", - "Eq. \u2011 $0 = - SdT - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B}$\n", - "\n", - "Similar to and , the property of a phase can be represented by a\n", - "two-dimensional surface in the three-dimensional space composed of $T$,\n", - "$\\mu_{A}$, and $\\mu_{B}$ under constant $P$, keeping in mind the\n", - "following\n", - "\n", - "Eq. \u2011\n", - "$G_{m} = x_{A}\\mu_{A} + x_{B}\\mu_{B} = x_{A}_{\\ }^{0}G_{A} + x_{B}_{\\ }^{0}G_{B} + RT\\left( x_{A}l{nx}_{A} + x_{B}l{nx}_{B} \\right) +_{\\ }^{E}G_{m}$\n", - "\n", - "Since $_{\\ }^{E}G_{m}$ must be zero for pure components $A$ and $B$, it\n", - "needs to be in the following form\n", - "\n", - "Eq. \u2011 $_{\\ }^{E}G_{m} = x_{A}x_{B}L_{AB}$\n", - "\n", - "with $L_{AB}$ being a parameter denoting the interaction between\n", - "components $A$ and $B$, called interaction parameter. When $L_{AB} = 0$,\n", - "the solution is an ideal solution. When $L_{AB}$ is a non-zero constant\n", - "independent of temperature and composition, the solution is called a\n", - "regular solution. Its excess entropy and excess enthalpy of mixing are\n", - "obtained as\n", - "\n", - "Eq. \u2011 $_{\\ }^{E}S_{m} = \\frac{\\partial_{\\ }^{E}G_{m}}{\\partial T} = 0$\n", - "\n", - "Eq. \u2011\n", - "$_{\\ }^{E}H_{m} =_{\\ }^{E}G_{m} - T_{\\ }^{E}S_{m} = x_{A}x_{B}L_{AB}$\n", - "\n", - "The chemical potential of component $A$ or $B$ in a binary regular\n", - "solution can be derived as\n", - "\n", - "Eq. \u2011\n", - "$\\mu_{i} =_{\\ }^{0}G_{i} + RTlnx_{i} + \\left( 1 - x_{i} \\right)^{2}L_{AB}$\n", - "\n", - "In a dilute solution with $x_{i} \\rightarrow 0$, one can have\n", - "\n", - "Eq. \u2011\n", - "$RTln\\gamma_{i} = \\left( 1 - x_{i} \\right)^{2}L_{AB} \\approx L_{AB}$\n", - "\n", - "Eq. \u2011 $\\gamma_{i} = e^{\\frac{L_{AB}}{RT}}$\n", - "\n", - "The activity is thus proportional to its mole fraction, which is called\n", - "Henry\u2019s law. By the same token, for the solvent, i.e.\n", - "$x_{i} \\rightarrow 1$,\n", - "\n", - "Eq. \u2011 $RTln\\gamma_{i} = \\left( 1 - x_{i} \\right)^{2}L_{AB} \\approx 0$\n", - "\n", - "which gives $\\gamma_{i} \\approx 1$, and its activity approaches its mole\n", - "fraction. This is called Raoult\u2019s law.\n", - "\n", - "The stability of a binary solution is derived from as\n", - "\n", - "Eq. \u2011\n", - "$\\left( \\frac{\\partial\\mu_{A}}{\\partial N_{A}} \\right)_{T,P,N_{B}} = \\left\\lbrack \\frac{RT}{x_{A}} - 2\\left( 1 - x_{A} \\right)L_{AB} \\right\\rbrack\\frac{1 - x_{A}}{N}$\n", - "\n", - "Eq. \u2011\n", - "$\\left( \\frac{\\partial\\mu_{B}}{\\partial N_{B}} \\right)_{T,P,N_{A}} = \\left\\lbrack \\frac{RT}{x_{B}} - 2\\left( 1 - x_{B} \\right)L_{AB} \\right\\rbrack\\frac{1 - x_{B}}{N}$\n", - "\n", - "It should be noted that the two chemical potentials in a binary system\n", - "at constant temperature and pressure are dependent on each other due to\n", - "the Gibbs-Duhem equation shown in , i.e.\n", - "\n", - "Eq. \u2011 $0 = - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B}$\n", - "\n", - "and the two chemical potentials depend on each other by the following\n", - "relation\n", - "\n", - "Eq. \u2011\n", - "$\\left( \\frac{\\partial\\mu_{A}}{\\partial\\mu_{B}} \\right)_{T,P} = - \\frac{N_{B}}{N_{A}}$\n", - "\n", - "Therefore, at the limit of stability, both and go to zero at the same\n", - "time, which is obtained when\n", - "\n", - "Eq. \u2011 $RT = 2{x_{A}x_{B}L}_{AB}$\n", - "\n", - "As the absolute temperature cannot be negative, has no solution for a\n", - "solution phase with $L_{AB} < 0$, i.e. the solution phase is stable with\n", - "respect to the composition fluctuation. For a solution with\n", - "$L_{AB} > 0$, its limit of stability is represented .\n", - "\n", - "A schematic molar Gibbs energy of a solution with $L_{AB} < 0$ at\n", - "constant temperature and pressure is shown in along with the ideal and\n", - "excess Gibbs energy of mixing. A tangent line on the molar Gibbs energy\n", - "of the solution is drawn, and its two intercepts at $x_{B} = 0$ and\n", - "$x_{B} = 1$ give the chemical potentials of components $A$ and $B$,\n", - "$\\mu_{A}$ and $\\mu_{B}$ by , respectively. It is evident that $\\mu_{A}$\n", - "and $\\mu_{B}$ are not independent on each other as they are two points\n", - "on the same straight line. This is a graphic representation of the\n", - "Gibbs-Duhem equation of . The chemical activity of component $B$ is also\n", - "depicted with the reference state being the pure B with the same\n", - "structure. As shown in , other structures of pure B can be selected as\n", - "the reference states of the chemical activity of component B, resulting\n", - "in the different distances to its chemical potential in the solution,\n", - "thus different values of its chemical activities. It is clear that this\n", - "change of reference state for chemical activity does not affect the\n", - "chemical potential of the component in the solution.\n", - "\n", - "Figure \u2011: Schematic molar Gibbs energy diagram with $L_{AB} < 0$\n", - "\n", - "When $L_{AB} > 0$, represents a parabola in the $T - x_{i}$\n", - "two-dimensional coordinate, symmetric with respect to $x_{A}$ and\n", - "$x_{B}$, shown in , i.e. the spinodal of the solution. The consolute\n", - "point is obtained by applying to and letting equal to zero at the\n", - "consolute point\n", - "\n", - "Eq. \u2011\n", - "$\\left( \\frac{\\partial^{2}\\mu_{A}}{\\partial N_{A}^{2}} \\right)_{T,P,N_{B}} = \\left\\lbrack - \\frac{RT}{x_{A}^{2}} + 2L_{AB} \\right\\rbrack\\left( \\frac{1 - x_{A}}{N} \\right)^{2} = 0$\n", - "\n", - "which gives\n", - "\n", - "Eq. \u2011 $T_{cons} = 2x_{A}^{2}L_{AB}$\n", - "\n", - "Solving and , one obtains $x_{A} = x_{B} = 0.5$ and\n", - "\n", - "Eq. \u2011 $T_{cons} = \\frac{L_{AB}}{2R}$\n", - "\n", - "Figure \u2011: A Spinodal curve with $L_{AB} > 0$\n", - "\n", - "A schematic molar Gibbs energy diagram at temperatures below the\n", - "consolute point is shown in . It can be seen that part of the molar\n", - "Gibbs energy has negative curvature, and the solution becomes unstable.\n", - "The chemical potential thus does not change monotonically with respect\n", - "to composition and its derivative changes sign at the inflexion point.\n", - "\n", - "Figure \u2011: Schematic molar Gibbs energy diagram with $L_{AB} > 0$\n", - "\n", - "For more complex solutions, $L_{AB}$ can be a function of temperature,\n", - "pressure, and compositions. In principle, the temperature and pressure\n", - "dependences can be treated by means of formula similar to . There are\n", - "various approaches in the literature to consider the composition\n", - "dependence of $L_{AB}$. The empirical Redlich-Kister polynomial stands\n", - "out as the one most widely used because it can be extrapolated to\n", - "ternary and multi-component systems consistently, which will be\n", - "discussed in Chapter .\n" - ], - "metadata": {} - } - ] -} \ No newline at end of file diff --git a/src/psu410/src/gibbs_energy_function/elastic_magnetic_and_electric_contributions_to_free_energy.ipynb b/src/psu410/src/gibbs_energy_function/elastic_magnetic_and_electric_contributions_to_free_energy.ipynb index 1ccae10..31c4e42 100644 --- a/src/psu410/src/gibbs_energy_function/elastic_magnetic_and_electric_contributions_to_free_energy.ipynb +++ b/src/psu410/src/gibbs_energy_function/elastic_magnetic_and_electric_contributions_to_free_energy.ipynb @@ -4,127 +4,9 @@ "metadata": {}, "cells": [ { - "id": "93e921a2", + "id": "5bcbb7fe", "cell_type": "markdown", - "source": [ - "## Elastic, magnetic, and electric contributions to free energy\n", - "\n", - "Chapters and focus on the thermal and hydrostatical pressure\n", - "contributions to Gibbs energy, which are the two prime variables\n", - "affecting phase stability in typical experimental environments. However,\n", - "there are other internal and external contributions, which are\n", - "particularly important for crystalline phases. Two important internal\n", - "contributions are from magnetic and electric polarizations of materials\n", - "with the corresponding external contributions due to the magnetic and\n", - "electric fields. Furthermore, for non-hydrostatic pressing of solid\n", - "phases, the $PV$ term in the combined law is to be replaced by elastic\n", - "energy calculated from elastic stress and elastic strain. The\n", - "corresponding works done to a system are as follows \\[3\\]\n", - "\n", - "Eq. \u2011\n", - "$dW_{elastic} = - V\\sum_{i,j,k,l}^{}{\\sigma_{ij}{d\\varepsilon}_{kl}}$\n", - "\n", - "Eq. \u2011 $dW_{magnetic} = - V\\sum_{i}^{}{H_{i}{dB}_{i}}$\n", - "\n", - "Eq. \u2011 $dW_{electric} = - V\\sum_{i}^{}{E_{i}{dD}_{i}}$\n", - "\n", - "where $i,j,\\ k,\\ l = 1,2,3$, $\\sigma_{ij}$ and $\\varepsilon_{kl}$ are\n", - "the components of stress and strain; $H_{i}$ and $B_{i}$ are the\n", - "components of magnetic field and magnetic induction; $E_{i}$ and $D_{i}$\n", - "are the components of electric field and electric displacement; and $V$\n", - "is the volume of the crystal. The negative sign in front of the equation\n", - "is due to the fact that the system does work to the surroundings when it\n", - "expands its volume due to the strain, magnetic induction, and electric\n", - "displacement.\n", - "\n", - "Using the combined law of thermodynamics, can thus be re-written as\n", - "follows\n", - "\n", - "Eq. \u2011\n", - "$dU = TdS - V\\left( \\sum_{i,j,k,l}^{}{\\sigma_{ij}{d\\varepsilon}_{kl}} + \\sum_{i}^{}{H_{i}{dB}_{i}} + \\sum_{i}^{}{E_{i}{dD}_{i}} \\right) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$\n", - "\n", - "The Legendre transformation similar to Helmholtz energy and Gibbs\n", - "energy, and , can be made to obtain the following characteristic free\n", - "energy functions\n", - "\n", - "Eq. \u2011 $dF = d(U - TS)$\n", - "\n", - "$$= - SdT - V\\left( \\sum_{i,j,k,l}^{}{\\sigma_{ij}{d\\varepsilon}_{kl}} + \\sum_{i}^{}{H_{i}{dB}_{i}} + \\sum_{i}^{}{E_{i}{dD}_{i}} \\right) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$$\n", - "\n", - "Eq. \u2011 $dF_{H} = d\\left( U - TS + \\sum_{i}^{}{H_{i}B_{i}} \\right)$\n", - "\n", - "$$= - SdT - V\\left( \\sum_{i,j,k,l}^{}{\\sigma_{ij}{d\\varepsilon}_{kl}} - \\sum_{i}^{}{B_{i}dH_{i}} + \\sum_{i}^{}{E_{i}{dD}_{i}} \\right) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$$\n", - "\n", - "Eq. \u2011 $dF_{E} = d\\left( U - TS + \\sum_{i}^{}{E_{i}D_{i}} \\right)$\n", - "\n", - "$$= - SdT - V\\left( \\sum_{i,j,k,l}^{}{\\sigma_{ij}{d\\varepsilon}_{kl}} + \\sum_{i}^{}{H_{i}{dB}_{i}} - \\sum_{i}^{}{D_{i}{dE}_{i}} \\right) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$$\n", - "\n", - "Eq. \u2011\n", - "$dF_{EH} = d\\left( U - TS + \\sum_{i}^{}{H_{i}B_{i}} + \\sum_{i}^{}{E_{i}D_{i}} \\right)$\n", - "\n", - "$$= - SdT - V\\left( \\sum_{i,j,k,l}^{}{\\sigma_{ij}{d\\varepsilon}_{kl}} - \\sum_{i}^{}{B_{i}{dH}_{i}} - \\sum_{i}^{}{D_{i}{dE}_{i}} \\right) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$$\n", - "\n", - "Eq. \u2011\n", - "$dG = d\\left( U - TS + \\sum_{i,j,k,l}^{}{\\sigma_{ij}\\varepsilon_{kl}} + \\sum_{i}^{}{H_{i}B_{i}} + \\sum_{i}^{}{E_{i}D_{i}} \\right)$\n", - "\n", - "$$= - SdT + V\\left( \\sum_{i,j,k,l}^{}{\\varepsilon_{ij}d\\sigma}_{kl} + \\sum_{i}^{}{B_{i}dH_{i}} + \\sum_{i}^{}{D_{i}{dE}_{i}} \\right) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$$\n", - "\n", - "From above equations, it can be seen that the natural variables of\n", - "various free energies are\n", - "$F\\left( T,\\ \\varepsilon_{ij},\\ B_{i},\\ D_{i},\\ N_{i},\\ \\xi \\right)$,\n", - "$F_{H}\\left( T,\\ \\varepsilon_{ij},\\ H_{i},\\ D_{i},\\ N_{i},\\ \\xi \\right)$,\n", - "$F_{E}\\left( T,\\ \\varepsilon_{ij},\\ B_{i},\\ E_{i},\\ N_{i},\\ \\xi \\right)$,\n", - "$F_{EH}\\left( T,\\ \\varepsilon_{ij},\\ H_{i},\\ E_{i},\\ N_{i},\\ \\xi \\right)$,\n", - "and $G\\left( T,\\ \\sigma_{ij},\\ H_{i},\\ E_{i},\\ N_{i},\\ \\xi \\right)$.\n", - "Clearly, there can be more combinations when the components of\n", - "$\\varepsilon_{ij}$, $D_{i}$, and $B_{i}$ are partially replaced by their\n", - "conjugate potentials. The free energies listed above are useful\n", - "depending on how the system is constrained by the surroundings. For\n", - "practical applications, the elastic, magnetic, and electric properties\n", - "are usually considered for phases with fixed compositions, and at\n", - "equilibrium can then be written as\n", - "\n", - "Eq. \u2011\n", - "$dG = - SdT + V\\left( \\sum_{i,j,k.l}^{}{\\varepsilon_{ij}d\\sigma}_{kl} + \\sum_{i}^{}{B_{i}dH_{i}} + \\sum_{i}^{}{D_{i}{dE}_{i}} \\right) + \\mu dN$\n", - "\n", - "The corresponding Gibbs-Duhem equation follows as below\n", - "\n", - "Eq. \u2011\n", - "$0 = - SdT + V\\left( \\sum_{i,j,k,l}^{}{\\varepsilon_{ij}d\\sigma}_{kl} + \\sum_{i}^{}{B_{i}dH_{i}} + \\sum_{i}^{}{D_{i}{dE}_{i}} \\right) - Nd\\mu$\n", - "\n", - "The general differential form of a molar quantity can be extended from\n", - "as follows as a one-component system\n", - "\n", - "Eq. \u2011\n", - "\n", - "Eq. \u2011\n", - "\n", - "Eq. \u2011\n", - "\n", - "Eq. \u2011\n", - "\n", - "The first derivatives in to are the second directives of Gibbs energy\n", - "with respect to its natural variables, i.e. potentials, and have their\n", - "respective nomenclatures as shown in . The limit of stability follows\n", - "and can be re-written as\n", - "\n", - "Eq. \u2011\n", - "$\\left( \\frac{\\partial X_{i}}{\\partial Y_{i}} \\right)_{N,Y_{j}} = \\infty$\n", - "\n", - "This means that the derivatives in to , i.e. the quantities in , diverge\n", - "at the limit of stability.\n", - "\n", - "Table \u2011: Physical quantities related to the first derivatives in to .\n", - "The table is symmetric due to the Maxwell relations related to the\n", - "second directives of Gibbs energy with respect to its natural variables.\n", - "\n", - "| | T | $$\\sigma_{kl}$$ | $$E_{k}$$ | $$H_{k}$$ |\n", - "|----------------------|------------------------------------|------------------------------------|---------------------------------------|---------------------------------------|\n", - "| S | C/T, heat capacity | $\\alpha_{kl}$, piezocaloric effect | $p_{k}$, electrocaloric effect | $m_{k}$, magnetocaloric effect |\n", - "| $$\\varepsilon_{ij}$$ | $\\alpha_{ij}$, thermal expansion | $s_{ijkl}$, elastic compliance | $d_{ijk}$, converse piezoelectricity | $q_{ijk}$, piezomagnetic moduli |\n", - "| $$D_{i}$$ | $p_{i}$, pyroelectric coefficients | $d_{ikl}$, piezoelectric moduli | $k_{ik}$, permittivities | $a_{ik}$, magnetoelectric coefficient |\n", - "| $$B_{i}$$ | $m_{i}$, pyromagnetic coefficient | $q_{ikl}$, piezomagnetic moduli | $a_{ik}$, magnetoelectric coefficient | $\\mu_{ik}$, permeability |\n" - ], + "source": "## Elastic, magnetic, and electric contributions to free energy\n\nChapters and focus on the thermal and hydrostatical pressure\ncontributions to Gibbs energy, which are the two prime variables\naffecting phase stability in typical experimental environments. However,\nthere are other internal and external contributions, which are\nparticularly important for crystalline phases. Two important internal\ncontributions are from magnetic and electric polarizations of materials\nwith the corresponding external contributions due to the magnetic and\nelectric fields. Furthermore, for non-hydrostatic pressing of solid\nphases, the $PV$ term in the combined law is to be replaced by elastic\nenergy calculated from elastic stress and elastic strain. The\ncorresponding works done to a system are as follows \\[3\\]\n\nEq. \u2011\n$dW_{elastic} = - V\\sum_{i,j,k,l}^{}{\\sigma_{ij}{d\\varepsilon}_{kl}}$\n\nEq. \u2011 $dW_{magnetic} = - V\\sum_{i}^{}{H_{i}{dB}_{i}}$\n\nEq. \u2011 $dW_{electric} = - V\\sum_{i}^{}{E_{i}{dD}_{i}}$\n\nwhere $i,j,\\ k,\\ l = 1,2,3$, $\\sigma_{ij}$ and $\\varepsilon_{kl}$ are\nthe components of stress and strain; $H_{i}$ and $B_{i}$ are the\ncomponents of magnetic field and magnetic induction; $E_{i}$ and $D_{i}$\nare the components of electric field and electric displacement; and $V$\nis the volume of the crystal. The negative sign in front of the equation\nis due to the fact that the system does work to the surroundings when it\nexpands its volume due to the strain, magnetic induction, and electric\ndisplacement.\n\nUsing the combined law of thermodynamics, can thus be re-written as\nfollows\n\nEq. \u2011\n$dU = TdS - V\\left( \\sum_{i,j,k,l}^{}{\\sigma_{ij}{d\\varepsilon}_{kl}} + \\sum_{i}^{}{H_{i}{dB}_{i}} + \\sum_{i}^{}{E_{i}{dD}_{i}} \\right) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$\n\nThe Legendre transformation similar to Helmholtz energy and Gibbs\nenergy, and , can be made to obtain the following characteristic free\nenergy functions\n\nEq. \u2011 $dF = d(U - TS)$\n\n$$= - SdT - V\\left( \\sum_{i,j,k,l}^{}{\\sigma_{ij}{d\\varepsilon}_{kl}} + \\sum_{i}^{}{H_{i}{dB}_{i}} + \\sum_{i}^{}{E_{i}{dD}_{i}} \\right) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$$\n\nEq. \u2011 $dF_{H} = d\\left( U - TS + \\sum_{i}^{}{H_{i}B_{i}} \\right)$\n\n$$= - SdT - V\\left( \\sum_{i,j,k,l}^{}{\\sigma_{ij}{d\\varepsilon}_{kl}} - \\sum_{i}^{}{B_{i}dH_{i}} + \\sum_{i}^{}{E_{i}{dD}_{i}} \\right) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$$\n\nEq. \u2011 $dF_{E} = d\\left( U - TS + \\sum_{i}^{}{E_{i}D_{i}} \\right)$\n\n$$= - SdT - V\\left( \\sum_{i,j,k,l}^{}{\\sigma_{ij}{d\\varepsilon}_{kl}} + \\sum_{i}^{}{H_{i}{dB}_{i}} - \\sum_{i}^{}{D_{i}{dE}_{i}} \\right) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$$\n\nEq. \u2011\n$dF_{EH} = d\\left( U - TS + \\sum_{i}^{}{H_{i}B_{i}} + \\sum_{i}^{}{E_{i}D_{i}} \\right)$\n\n$$= - SdT - V\\left( \\sum_{i,j,k,l}^{}{\\sigma_{ij}{d\\varepsilon}_{kl}} - \\sum_{i}^{}{B_{i}{dH}_{i}} - \\sum_{i}^{}{D_{i}{dE}_{i}} \\right) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$$\n\nEq. \u2011\n$dG = d\\left( U - TS + \\sum_{i,j,k,l}^{}{\\sigma_{ij}\\varepsilon_{kl}} + \\sum_{i}^{}{H_{i}B_{i}} + \\sum_{i}^{}{E_{i}D_{i}} \\right)$\n\n$$= - SdT + V\\left( \\sum_{i,j,k,l}^{}{\\varepsilon_{ij}d\\sigma}_{kl} + \\sum_{i}^{}{B_{i}dH_{i}} + \\sum_{i}^{}{D_{i}{dE}_{i}} \\right) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$$\n\nFrom above equations, it can be seen that the natural variables of\nvarious free energies are\n$F\\left( T,\\ \\varepsilon_{ij},\\ B_{i},\\ D_{i},\\ N_{i},\\ \\xi \\right)$,\n$F_{H}\\left( T,\\ \\varepsilon_{ij},\\ H_{i},\\ D_{i},\\ N_{i},\\ \\xi \\right)$,\n$F_{E}\\left( T,\\ \\varepsilon_{ij},\\ B_{i},\\ E_{i},\\ N_{i},\\ \\xi \\right)$,\n$F_{EH}\\left( T,\\ \\varepsilon_{ij},\\ H_{i},\\ E_{i},\\ N_{i},\\ \\xi \\right)$,\nand $G\\left( T,\\ \\sigma_{ij},\\ H_{i},\\ E_{i},\\ N_{i},\\ \\xi \\right)$.\nClearly, there can be more combinations when the components of\n$\\varepsilon_{ij}$, $D_{i}$, and $B_{i}$ are partially replaced by their\nconjugate potentials. The free energies listed above are useful\ndepending on how the system is constrained by the surroundings. For\npractical applications, the elastic, magnetic, and electric properties\nare usually considered for phases with fixed compositions, and at\nequilibrium can then be written as\n\nEq. \u2011\n$dG = - SdT + V\\left( \\sum_{i,j,k.l}^{}{\\varepsilon_{ij}d\\sigma}_{kl} + \\sum_{i}^{}{B_{i}dH_{i}} + \\sum_{i}^{}{D_{i}{dE}_{i}} \\right) + \\mu dN$\n\nThe corresponding Gibbs-Duhem equation follows as below\n\nEq. \u2011\n$0 = - SdT + V\\left( \\sum_{i,j,k,l}^{}{\\varepsilon_{ij}d\\sigma}_{kl} + \\sum_{i}^{}{B_{i}dH_{i}} + \\sum_{i}^{}{D_{i}{dE}_{i}} \\right) - Nd\\mu$\n\nThe general differential form of a molar quantity can be extended from\nas follows as a one-component system\n\nEq. \u2011\n\nEq. \u2011\n\nEq. \u2011\n\nEq. \u2011\n\nThe first derivatives in to are the second directives of Gibbs energy\nwith respect to its natural variables, i.e. potentials, and have their\nrespective nomenclatures as shown in . The limit of stability follows\nand can be re-written as\n\nEq. \u2011\n$\\left( \\frac{\\partial X_{i}}{\\partial Y_{i}} \\right)_{N,Y_{j}} = \\infty$\n\nThis means that the derivatives in to , i.e. the quantities in , diverge\nat the limit of stability.\n\nTable \u2011: Physical quantities related to the first derivatives in to .\nThe table is symmetric due to the Maxwell relations related to the\nsecond directives of Gibbs energy with respect to its natural variables.\n\n| | T | $$\\sigma_{kl}$$ | $$E_{k}$$ | $$H_{k}$$ |\n|----------------------|------------------------------------|------------------------------------|---------------------------------------|---------------------------------------|\n| S | C/T, heat capacity | $\\alpha_{kl}$, piezocaloric effect | $p_{k}$, electrocaloric effect | $m_{k}$, magnetocaloric effect |\n| $$\\varepsilon_{ij}$$ | $\\alpha_{ij}$, thermal expansion | $s_{ijkl}$, elastic compliance | $d_{ijk}$, converse piezoelectricity | $q_{ijk}$, piezomagnetic moduli |\n| $$D_{i}$$ | $p_{i}$, pyroelectric coefficients | $d_{ikl}$, piezoelectric moduli | $k_{ik}$, permittivities | $a_{ik}$, magnetoelectric coefficient |\n| $$B_{i}$$ | $m_{i}$, pyromagnetic coefficient | $q_{ikl}$, piezomagnetic moduli | $a_{ik}$, magnetoelectric coefficient | $\\mu_{ik}$, permeability |\n", "metadata": {} } ] diff --git a/src/psu410/src/gibbs_energy_function/index.ipynb b/src/psu410/src/gibbs_energy_function/index.ipynb index 92a6983..377709e 100644 --- a/src/psu410/src/gibbs_energy_function/index.ipynb +++ b/src/psu410/src/gibbs_energy_function/index.ipynb @@ -4,211 +4,9 @@ "metadata": {}, "cells": [ { - "id": "94948523", + "id": "dbdfb491", "cell_type": "markdown", - "source": [ - "# Gibbs energy function\n", - "\n", - "As shown in through , all functions have $N_{i}$ and $\\xi$ as natural\n", - "variables while they differ in other two natural variables. In typical\n", - "materials-related experiments, temperature and pressure are the two\n", - "variables controlled. They are also the natural variables of Gibbs\n", - "energy. Consequently, Gibbs energy is the most widely used function in\n", - "thermodynamics of materials science. The rest of this book focuses on\n", - "Gibbs energy for this reason. In this chapter, the mathematical formulas\n", - "for Gibbs energy of phases with fixed and variable compositions are\n", - "discussed which are needed for quantitative calculations of Gibbs energy\n", - "under given values of its natural variables.\n", - "\n", - "From , the molar Gibbs energy can be defined as\n", - "\n", - "Eq. \u2011\n", - "$G_{m}\\left( T,P,x_{i},\\xi \\right) = \\frac{G}{N} = \\sum_{}^{}\\mu_{i}x_{i}$\n", - "\n", - "The molar entropy, molar volume, chemical potential, and the driving\n", - "force can be obtained from as\n", - "\n", - "Eq. \u2011\n", - "$S_{m} = \\frac{S}{N} = - \\frac{1}{N}\\left( \\frac{\\partial G}{\\partial T} \\right)_{P,\\ N_{i},\\ \\xi} = {- \\left( \\frac{\\partial G_{m}}{\\partial T} \\right)}_{P,\\ x_{i},\\ \\xi}$\n", - "\n", - "Eq. \u2011\n", - "$V_{m} = \\frac{V}{N} = \\frac{1}{N}\\left( \\frac{\\partial G}{\\partial P} \\right)_{T,\\ N_{i},\\ \\xi} = \\left( \\frac{\\partial G_{m}}{\\partial P} \\right)_{T,\\ x_{i},\\ \\xi}$\n", - "\n", - "Eq. \u2011\n", - "$\\mu_{i} = \\left( \\frac{\\partial G}{\\partial N_{i}} \\right)_{T,P,N_{j \\neq i},\\ \\xi}$\n", - "\n", - "Eq. \u2011\n", - "$- D = \\left( \\frac{\\partial G}{\\partial\\xi} \\right)_{T,P,N_{i}\\ }$\n", - "\n", - "Based on , the molar enthalpy is written as\n", - "\n", - "Eq. \u2011 $H_{m} = G_{m} + TS_{m}$\n", - "\n", - "Other physical properties of the system can also be represented by the\n", - "partial derivatives of Gibbs energy such as heat capacity, $C_{P}$,\n", - "volume thermal expansivity, $\\alpha_{V}$, isothermal compressibility,\n", - "$\\kappa_{T}$, as follows under constant pressure or temperature\n", - "\n", - "Eq. \u2011\n", - "$C_{P} = \\left( \\frac{\\partial Q}{\\partial T} \\right)_{P} = \\left( \\frac{\\partial H}{\\partial T} \\right)_{P} = T\\left( \\frac{\\partial(G + TS)}{\\partial T} \\right)_{P} = T\\left( \\frac{\\partial S}{\\partial T} \\right)_{P} = - T\\left( \\frac{\\partial^{2}G}{\\partial T^{2}} \\right)_{P}$\n", - "\n", - "Eq. \u2011\n", - "$\\alpha_{V} = \\frac{\\left( \\frac{\\partial V}{\\partial T} \\right)_{P}}{V} = \\frac{\\left( \\frac{\\left( \\partial G/\\partial( - P) \\right)_{T}}{\\partial T} \\right)_{P}}{\\left( \\partial G/\\partial( - P) \\right)_{T}} = \\frac{\\frac{\\partial^{2}G}{\\partial T\\partial( - P)}}{\\left( \\partial G/\\partial( - P) \\right)_{T}}$\n", - "\n", - "Eq. \u2011\n", - "$\\kappa_{T} = \\frac{\\left( \\frac{\\partial V}{\\partial( - P)} \\right)_{T}}{V} = \\frac{\\left( \\frac{\\left( \\partial G/\\partial( - P) \\right)_{T}}{\\partial( - P)} \\right)_{T}}{\\left( \\partial G/\\partial( - P) \\right)_{T}} = \\frac{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}}{\\left( \\partial G/\\partial( - P) \\right)_{T}} = \\frac{1}{B}$\n", - "\n", - "where the $N_{i}$ and $\\xi$ are kept constant for all partial\n", - "derivatives, and $B$ is the bulk modulus.\n", - "\n", - "In , $G$ cannot be directly replaced by $G_{m}$ because *N* also depends\n", - "on *Ni*. The thermodynamic quantities under such conditions,\n", - "i.e. varying the amount of a component at constant temperature and\n", - "pressure, are called partial quantities which are introduced in Eq. 1\u20118\n", - "for partial entropy and for partial enthalpy. This definition can be\n", - "extended to all molar quantities such as partial volume and partial\n", - "Gibbs energy. Partial quantities of a molar quantity, $A$, can thus be\n", - "defined in general as\n", - "\n", - "Eq. \u2011\n", - "$A_{i} = \\left( \\frac{\\partial A}{\\partial N_{i}} \\right)_{T,P,N_{j \\neq i},\\ \\xi}$\n", - "\n", - "The general differential form of a molar quantity for a system at\n", - "equilibrium can be represented by its partial quantities as\n", - "\n", - "Eq. \u2011\n", - "$dA = \\left( \\frac{\\partial A}{\\partial T} \\right)dT + \\left( \\frac{\\partial A}{\\partial P} \\right)dP + \\sum_{}^{}\\left( \\frac{\\partial A}{\\partial N_{i}} \\right){dN}_{i}$\n", - "\n", - "where the subscripts representing variables kept constant, i.e. the\n", - "remaining natural variables of Gibbs energy not in the denominator, are\n", - "omitted for simplicity. This will be done throughout the book unless\n", - "specified otherwise.\n", - "\n", - "Using the following relations: $A = NA_{m}$, $N = \\sum_{}^{}N_{j}$,\n", - "$x_{i} = N_{i}/N$,\n", - "$\\frac{{\\partial x}_{i}}{{\\partial N}_{i}} = \\left( 1 - x_{i} \\right)/N$,\n", - "and $\\frac{{\\partial x}_{k}}{{\\partial N}_{i}} = {- x}_{k}/N$, can be\n", - "derived as, under constant T and P,\n", - "\n", - "Eq. \u2011\n", - "$A_{i} = A_{m} + N\\sum_{j = 1}^{c}{\\frac{\\partial A_{m}}{\\partial x_{j}}\\frac{\\partial x_{j}}{\\partial N_{i}}} = A_{m} + \\frac{\\partial A_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial A_{m}}{\\partial x_{j}}$\n", - "\n", - "where the summation is for all *c* components and the partial\n", - "derivatives are taken with other mole fractions kept constant. However,\n", - "mole fractions are not independent, but follow the relation\n", - "$\\sum_{}^{}x_{i} = 1$. Taking $x_{1} = 1 - \\sum_{j = 2}^{c}x_{j}$ as the\n", - "dependent mole fraction, can be rewritten as\n", - "\n", - "Eq. \u2011\n", - "$A_{i} = A_{m} + \\left( \\frac{\\partial A_{m}}{\\partial x_{i}} - \\frac{\\partial A_{m}}{\\partial x_{1}} \\right) - \\sum_{j = 2}^{c}x_{j}\\left( \\frac{\\partial A_{m}}{\\partial x_{j}} - \\frac{\\partial A_{m}}{\\partial x_{1}} \\right)$\n", - "\n", - "The difference of the partial derivatives in the parenthesis in\n", - "represents the partial derivative of $A_{m}$ with respect to the mole\n", - "fraction of one component when the first component is selected as the\n", - "dependent component. Applying and to Gibbs energy, the partial Gibbs\n", - "energy or chemical potential of component $i$ is obtained as\n", - "\n", - "Eq. \u2011\n", - "$\\mu_{i} = G_{i} = G_{m} + \\frac{\\partial G_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial G_{m}}{\\partial x_{j}} = G_{m} + \\left( \\frac{\\partial G_{m}}{\\partial x_{i}} - \\frac{\\partial G_{m}}{\\partial x_{1}} \\right) - \\sum_{j = 2}^{c}x_{j}\\left( \\frac{\\partial G_{m}}{\\partial x_{j}} - \\frac{\\partial G_{m}}{\\partial x_{1}} \\right)$\n", - "\n", - "The derivatives in the stability equation, , are defined with the molar\n", - "quantities kept constant. On the other hand, Gibbs energy has two\n", - "potentials, temperature and pressure, as natural variables instead. One\n", - "would thus need to compare the stability conditions when a variable kept\n", - "fixed is changed from a molar quantity to its conjugate potential. This\n", - "can be carried out through the use of Jacobians to change the\n", - "independent variables\n", - "\n", - "Eq. \u2011\n", - "$\\frac{\\partial\\left( Y_{i},Y_{j} \\right)}{\\partial\\left( X_{i},X_{j} \\right)} = \\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{Y_{j}}\\left( \\frac{\\partial Y_{j}}{\\partial X_{j}} \\right)_{X_{i}} = \\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}}\\left( \\frac{\\partial Y_{j}}{\\partial X_{j}} \\right)_{X_{i}} - \\left( \\frac{\\partial Y_{i}}{\\partial X_{j}} \\right)_{X_{i}}\\left( \\frac{\\partial Y_{j}}{\\partial X_{i}} \\right)_{X_{j}}$\n", - "\n", - "For a stable system, both\n", - "$\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}}$ and\n", - "$\\left( \\frac{\\partial Y_{j}}{\\partial X_{j}} \\right)_{X_{i}}$ are\n", - "positive based on . Using the Maxwell relation shown by , one thus\n", - "obtains\n", - "\n", - "Eq. \u2011\n", - "$\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}} - \\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{Y_{j}} = \\left( \\frac{\\partial Y_{i}}{\\partial X_{j}} \\right)_{X_{i}}\\left( \\frac{\\partial Y_{j}}{\\partial X_{i}} \\right)_{X_{j}}/\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}} \\geq 0$\n", - "\n", - "This means that\n", - "$\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{Y_{j}}$ will go\n", - "to zero before\n", - "$\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}}$ does. It\n", - "indicates that the stability condition becomes more restrictive when\n", - "potentials are kept constant in place of their conjugate molar\n", - "quantities. Based on the Gibbs-Duhem equation of , the maximum number of\n", - "independent potentials is *c+1*, and the last potential is dependent,\n", - "i.e.\n", - "\n", - "Eq. \u2011\n", - "$\\left( \\frac{\\partial Y_{c + 2}}{\\partial X_{c + 2}} \\right)_{Y_{j \\leq c + 1}} = 0$\n", - "\n", - "Therefore, the limit of stability is determined when the derivative\n", - "becomes zero with one molar quantity kept constant, e.g.\n", - "\n", - "Eq. \u2011\n", - "$\\left( \\frac{\\partial Y_{c + 1}}{\\partial X_{c + 1}} \\right)_{Y_{j < c + 1},X_{c + 2}} = 0$\n", - "\n", - "This is because this derivative reaches zero faster than any other\n", - "derivatives with more molar quantities kept constant. shows that all\n", - "molar quantities diverge at the limit of stability. The consolute point\n", - "is obtained with $c$ additional conditions as follows based on\n", - "\n", - "Eq. \u2011\n", - "$\\left( \\frac{\\partial^{2}Y_{i}}{\\partial\\left( X_{i} \\right)^{2}} \\right)_{Y_{j \\leq c + 1, \\neq i},X_{c + 2}} = 0$\n", - "\n", - "Together with , all $c + 1$ independent potentials at the consolute\n", - "point can be determined. It is evident that the consolute point is a\n", - "zero-dimensional point in a two-dimensional space of independent\n", - "potentials in a one-component system. With the addition of a second\n", - "component to form a binary system, this consolute point in the\n", - "one-component system extends into a one-dimensional line. This line\n", - "represents the limit of stability of the binary system, and a consolute\n", - "point is located at the end of this line. It is thus evident that in a\n", - "system with $c$ independent components, the limit of stability is a\n", - "*c-1*-dimensional hypersurface in a space of $c + 1$ independent\n", - "potentials, while the consolute point is a zero-dimensional point in all\n", - "systems, which may be called the invariant critical point.\n" - ], - "metadata": {} - }, - { - "id": "2252bace", - "cell_type": "markdown", - "source": [ - "### Multi-component random solutions\n", - "\n", - "Similar to a ternary solution, the excess Gibbs energy of mixing of a\n", - "multi-component solution can be written as\n", - "\n", - "Eq. \u2011\n", - "$_{\\ }^{E}G_{m} = \\sum_{i}^{}{\\sum_{j}^{}{x_{i}x_{j}L_{ij}}} + \\sum_{i}^{}{\\sum_{j}^{}{\\sum_{k}^{}{x_{i}x_{j}x_{k}L_{ijk}}}}$\n", - "\n", - "In principle, one can add interaction parameters for quaternary and\n", - "higher order systems, but their contributions to Gibbs energy are\n", - "relatively minor because the major contributions have already been taken\n", - "into account by the binary and ternary interactions. It is anticipated\n", - "that not only the interaction parameters of four or more components are\n", - "small, but also the multiplication of mole fractions in front the\n", - "interaction parameters diminishes their contribution to the Gibbs energy\n", - "even further.\n", - "\n", - "Under the condition that all interaction parameters are constant, the\n", - "chemical potential of a component in a multi-component system with\n", - "binary and ternary interaction parameters can be extended from as\n", - "\n", - "Eq. \u2011\n", - "$\\mu_{i} =_{\\ }^{0}G_{i} + RTlnx_{i} + \\sum_{j \\neq i}^{}{x_{j}^{2}L_{ij}} + \\sum_{k > j}^{}{\\sum_{j \\neq i}^{}{x_{j}x_{k}}}\\left\\lbrack L_{ij} + L_{ik} - L_{jk} + \\left( 1 - 2x_{i} \\right)L_{ijk} \\right\\rbrack$\n", - "\n", - "The stability of the solution can also be extended from as\n", - "\n", - "Eq. \u2011\n", - "${N\\left( \\frac{\\partial\\mu_{i}}{\\partial N_{i}} \\right)}_{T,P,N_{j \\neq i}} = NG_{ii} = \\frac{RT\\left( 1 - x_{i} \\right)}{x_{i}} - 2\\sum_{j \\neq i}^{}{x_{j}^{2}L_{ij}} - 2\\sum_{k > j}^{}{\\sum_{j \\neq i}^{}{x_{j}x_{k}}}\\left\\lbrack L_{ij} + L_{ik} - L_{jk} + \\left( 2 - 3x_{i} \\right)L_{ijk} \\right\\rbrack$\n", - "\n", - "The limit of stability of a multi-component random solution can be\n", - "represented by or .\n" - ], + "source": "# Gibbs energy function\n\nAs shown in through , all functions have $N_{i}$ and $\\xi$ as natural\nvariables while they differ in other two natural variables. In typical\nmaterials-related experiments, temperature and pressure are the two\nvariables controlled. They are also the natural variables of Gibbs\nenergy. Consequently, Gibbs energy is the most widely used function in\nthermodynamics of materials science. The rest of this book focuses on\nGibbs energy for this reason. In this chapter, the mathematical formulas\nfor Gibbs energy of phases with fixed and variable compositions are\ndiscussed which are needed for quantitative calculations of Gibbs energy\nunder given values of its natural variables.\n\nFrom , the molar Gibbs energy can be defined as\n\nEq. \u2011\n$G_{m}\\left( T,P,x_{i},\\xi \\right) = \\frac{G}{N} = \\sum_{}^{}\\mu_{i}x_{i}$\n\nThe molar entropy, molar volume, chemical potential, and the driving\nforce can be obtained from as\n\nEq. \u2011\n$S_{m} = \\frac{S}{N} = - \\frac{1}{N}\\left( \\frac{\\partial G}{\\partial T} \\right)_{P,\\ N_{i},\\ \\xi} = {- \\left( \\frac{\\partial G_{m}}{\\partial T} \\right)}_{P,\\ x_{i},\\ \\xi}$\n\nEq. \u2011\n$V_{m} = \\frac{V}{N} = \\frac{1}{N}\\left( \\frac{\\partial G}{\\partial P} \\right)_{T,\\ N_{i},\\ \\xi} = \\left( \\frac{\\partial G_{m}}{\\partial P} \\right)_{T,\\ x_{i},\\ \\xi}$\n\nEq. \u2011\n$\\mu_{i} = \\left( \\frac{\\partial G}{\\partial N_{i}} \\right)_{T,P,N_{j \\neq i},\\ \\xi}$\n\nEq. \u2011\n$- D = \\left( \\frac{\\partial G}{\\partial\\xi} \\right)_{T,P,N_{i}\\ }$\n\nBased on , the molar enthalpy is written as\n\nEq. \u2011 $H_{m} = G_{m} + TS_{m}$\n\nOther physical properties of the system can also be represented by the\npartial derivatives of Gibbs energy such as heat capacity, $C_{P}$,\nvolume thermal expansivity, $\\alpha_{V}$, isothermal compressibility,\n$\\kappa_{T}$, as follows under constant pressure or temperature\n\nEq. \u2011\n$C_{P} = \\left( \\frac{\\partial Q}{\\partial T} \\right)_{P} = \\left( \\frac{\\partial H}{\\partial T} \\right)_{P} = T\\left( \\frac{\\partial(G + TS)}{\\partial T} \\right)_{P} = T\\left( \\frac{\\partial S}{\\partial T} \\right)_{P} = - T\\left( \\frac{\\partial^{2}G}{\\partial T^{2}} \\right)_{P}$\n\nEq. \u2011\n$\\alpha_{V} = \\frac{\\left( \\frac{\\partial V}{\\partial T} \\right)_{P}}{V} = \\frac{\\left( \\frac{\\left( \\partial G/\\partial( - P) \\right)_{T}}{\\partial T} \\right)_{P}}{\\left( \\partial G/\\partial( - P) \\right)_{T}} = \\frac{\\frac{\\partial^{2}G}{\\partial T\\partial( - P)}}{\\left( \\partial G/\\partial( - P) \\right)_{T}}$\n\nEq. \u2011\n$\\kappa_{T} = \\frac{\\left( \\frac{\\partial V}{\\partial( - P)} \\right)_{T}}{V} = \\frac{\\left( \\frac{\\left( \\partial G/\\partial( - P) \\right)_{T}}{\\partial( - P)} \\right)_{T}}{\\left( \\partial G/\\partial( - P) \\right)_{T}} = \\frac{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}}{\\left( \\partial G/\\partial( - P) \\right)_{T}} = \\frac{1}{B}$\n\nwhere the $N_{i}$ and $\\xi$ are kept constant for all partial\nderivatives, and $B$ is the bulk modulus.\n\nIn , $G$ cannot be directly replaced by $G_{m}$ because *N* also depends\non *Ni*. The thermodynamic quantities under such conditions,\ni.e. varying the amount of a component at constant temperature and\npressure, are called partial quantities which are introduced in Eq. 1\u20118\nfor partial entropy and for partial enthalpy. This definition can be\nextended to all molar quantities such as partial volume and partial\nGibbs energy. Partial quantities of a molar quantity, $A$, can thus be\ndefined in general as\n\nEq. \u2011\n$A_{i} = \\left( \\frac{\\partial A}{\\partial N_{i}} \\right)_{T,P,N_{j \\neq i},\\ \\xi}$\n\nThe general differential form of a molar quantity for a system at\nequilibrium can be represented by its partial quantities as\n\nEq. \u2011\n$dA = \\left( \\frac{\\partial A}{\\partial T} \\right)dT + \\left( \\frac{\\partial A}{\\partial P} \\right)dP + \\sum_{}^{}\\left( \\frac{\\partial A}{\\partial N_{i}} \\right){dN}_{i}$\n\nwhere the subscripts representing variables kept constant, i.e. the\nremaining natural variables of Gibbs energy not in the denominator, are\nomitted for simplicity. This will be done throughout the book unless\nspecified otherwise.\n\nUsing the following relations: $A = NA_{m}$, $N = \\sum_{}^{}N_{j}$,\n$x_{i} = N_{i}/N$,\n$\\frac{{\\partial x}_{i}}{{\\partial N}_{i}} = \\left( 1 - x_{i} \\right)/N$,\nand $\\frac{{\\partial x}_{k}}{{\\partial N}_{i}} = {- x}_{k}/N$, can be\nderived as, under constant T and P,\n\nEq. \u2011\n$A_{i} = A_{m} + N\\sum_{j = 1}^{c}{\\frac{\\partial A_{m}}{\\partial x_{j}}\\frac{\\partial x_{j}}{\\partial N_{i}}} = A_{m} + \\frac{\\partial A_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial A_{m}}{\\partial x_{j}}$\n\nwhere the summation is for all *c* components and the partial\nderivatives are taken with other mole fractions kept constant. However,\nmole fractions are not independent, but follow the relation\n$\\sum_{}^{}x_{i} = 1$. Taking $x_{1} = 1 - \\sum_{j = 2}^{c}x_{j}$ as the\ndependent mole fraction, can be rewritten as\n\nEq. \u2011\n$A_{i} = A_{m} + \\left( \\frac{\\partial A_{m}}{\\partial x_{i}} - \\frac{\\partial A_{m}}{\\partial x_{1}} \\right) - \\sum_{j = 2}^{c}x_{j}\\left( \\frac{\\partial A_{m}}{\\partial x_{j}} - \\frac{\\partial A_{m}}{\\partial x_{1}} \\right)$\n\nThe difference of the partial derivatives in the parenthesis in\nrepresents the partial derivative of $A_{m}$ with respect to the mole\nfraction of one component when the first component is selected as the\ndependent component. Applying and to Gibbs energy, the partial Gibbs\nenergy or chemical potential of component $i$ is obtained as\n\nEq. \u2011\n$\\mu_{i} = G_{i} = G_{m} + \\frac{\\partial G_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial G_{m}}{\\partial x_{j}} = G_{m} + \\left( \\frac{\\partial G_{m}}{\\partial x_{i}} - \\frac{\\partial G_{m}}{\\partial x_{1}} \\right) - \\sum_{j = 2}^{c}x_{j}\\left( \\frac{\\partial G_{m}}{\\partial x_{j}} - \\frac{\\partial G_{m}}{\\partial x_{1}} \\right)$\n\nThe derivatives in the stability equation, , are defined with the molar\nquantities kept constant. On the other hand, Gibbs energy has two\npotentials, temperature and pressure, as natural variables instead. One\nwould thus need to compare the stability conditions when a variable kept\nfixed is changed from a molar quantity to its conjugate potential. This\ncan be carried out through the use of Jacobians to change the\nindependent variables\n\nEq. \u2011\n$\\frac{\\partial\\left( Y_{i},Y_{j} \\right)}{\\partial\\left( X_{i},X_{j} \\right)} = \\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{Y_{j}}\\left( \\frac{\\partial Y_{j}}{\\partial X_{j}} \\right)_{X_{i}} = \\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}}\\left( \\frac{\\partial Y_{j}}{\\partial X_{j}} \\right)_{X_{i}} - \\left( \\frac{\\partial Y_{i}}{\\partial X_{j}} \\right)_{X_{i}}\\left( \\frac{\\partial Y_{j}}{\\partial X_{i}} \\right)_{X_{j}}$\n\nFor a stable system, both\n$\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}}$ and\n$\\left( \\frac{\\partial Y_{j}}{\\partial X_{j}} \\right)_{X_{i}}$ are\npositive based on . Using the Maxwell relation shown by , one thus\nobtains\n\nEq. \u2011\n$\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}} - \\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{Y_{j}} = \\left( \\frac{\\partial Y_{i}}{\\partial X_{j}} \\right)_{X_{i}}\\left( \\frac{\\partial Y_{j}}{\\partial X_{i}} \\right)_{X_{j}}/\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}} \\geq 0$\n\nThis means that\n$\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{Y_{j}}$ will go\nto zero before\n$\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}}$ does. It\nindicates that the stability condition becomes more restrictive when\npotentials are kept constant in place of their conjugate molar\nquantities. Based on the Gibbs-Duhem equation of , the maximum number of\nindependent potentials is *c+1*, and the last potential is dependent,\ni.e.\n\nEq. \u2011\n$\\left( \\frac{\\partial Y_{c + 2}}{\\partial X_{c + 2}} \\right)_{Y_{j \\leq c + 1}} = 0$\n\nTherefore, the limit of stability is determined when the derivative\nbecomes zero with one molar quantity kept constant, e.g.\n\nEq. \u2011\n$\\left( \\frac{\\partial Y_{c + 1}}{\\partial X_{c + 1}} \\right)_{Y_{j < c + 1},X_{c + 2}} = 0$\n\nThis is because this derivative reaches zero faster than any other\nderivatives with more molar quantities kept constant. shows that all\nmolar quantities diverge at the limit of stability. The consolute point\nis obtained with $c$ additional conditions as follows based on\n\nEq. \u2011\n$\\left( \\frac{\\partial^{2}Y_{i}}{\\partial\\left( X_{i} \\right)^{2}} \\right)_{Y_{j \\leq c + 1, \\neq i},X_{c + 2}} = 0$\n\nTogether with , all $c + 1$ independent potentials at the consolute\npoint can be determined. It is evident that the consolute point is a\nzero-dimensional point in a two-dimensional space of independent\npotentials in a one-component system. With the addition of a second\ncomponent to form a binary system, this consolute point in the\none-component system extends into a one-dimensional line. This line\nrepresents the limit of stability of the binary system, and a consolute\npoint is located at the end of this line. It is thus evident that in a\nsystem with $c$ independent components, the limit of stability is a\n*c-1*-dimensional hypersurface in a space of $c + 1$ independent\npotentials, while the consolute point is a zero-dimensional point in all\nsystems, which may be called the invariant critical point.\n", "metadata": {} } ] diff --git a/src/psu410/src/gibbs_energy_function/phases_with_fixed_compositions.ipynb b/src/psu410/src/gibbs_energy_function/phases_with_fixed_compositions.ipynb index 030a892..b3c1ab7 100644 --- a/src/psu410/src/gibbs_energy_function/phases_with_fixed_compositions.ipynb +++ b/src/psu410/src/gibbs_energy_function/phases_with_fixed_compositions.ipynb @@ -4,275 +4,9 @@ "metadata": {}, "cells": [ { - "id": "286da7ba", + "id": "cdda5cf8", "cell_type": "markdown", - "source": [ - "## Phases with fixed compositions\n", - "\n", - "The homogeneous system discussed so far means that there is only one\n", - "phase in the system, i.e. a single-phase system. A phase with a fixed\n", - "composition can be a pure element or a stoichiometric compound. There is\n", - "thus only one independent component in the system. A stoichiometric\n", - "compound contains more than one element, but the relative amounts of\n", - "each element are fixed by the stoichiometry and cannot vary\n", - "independently, i.e., $dN_{i} = x_{i}dN$. The combined law of\n", - "thermodynamics becomes\n", - "\n", - "Eq. \u2011\n", - "$dG = - SdT - Vd( - P) + \\left( \\sum_{}^{}{x_{i}\\mu_{i}} \\right)dN - Dd\\xi = - SdT - Vd( - P) + G_{m}dN - Dd\\xi$\n", - "\n", - "$G_{m}$ is the molar Gibbs energy of the stoichiometric compound and can\n", - "be regarded as the chemical potential of the stoichiometric\n", - "phase,$\\ \\alpha$,\n", - "\n", - "Eq. \u2011 $G_{m} = \\mu^{\\alpha} = \\sum_{}^{}{x_{i}\\mu_{i}}$\n", - "\n", - "The chemical potential of individual components in the phase cannot be\n", - "defined because the amount of each component cannot be varied\n", - "independently. For a stoichiometric phase of $N$ moles of atoms at\n", - "equilibrium with $dG$=$Nd\\mu^{\\alpha} + \\mu^{\\alpha}dN$, reduces to\n", - "\n", - "Eq. \u2011 $\\ 0 = - SdT - Vd( - P) - Nd\\mu^{\\alpha}$\n", - "\n", - "which is the Gibbs-Duhem equation, , applied to a stoichiometric phase.\n", - "It can be represented graphically by a surface in a three-dimensional\n", - "space composed of $\\ \\mu^{\\alpha}$, *T* and *\u2013P*. The direction of the\n", - "surface is represented by the three partial directives between any two\n", - "of $\\ \\mu^{\\alpha}$, *T* and *\u2013P* with the third one kept constant, i.e.\n", - "\n", - "Eq. \u2011\n", - "$\\ \\left( \\frac{\\partial\\mu^{\\alpha}}{\\partial T} \\right)_{P} = - \\frac{S}{N} = - S_{m}$\n", - "\n", - "Eq. \u2011\n", - "$\\ \\left( \\frac{\\partial\\mu^{\\alpha}}{\\partial( - P)} \\right)_{T} = - \\frac{V}{N} = - V_{m}$\n", - "\n", - "Eq. \u2011\n", - "$\\ \\left( \\frac{\\partial( - P)}{\\partial T} \\right)_{\\mu^{\\alpha}} = - \\frac{S}{V} = - \\frac{S_{m}}{V_{m}}$\n", - "\n", - "Based on Nernst\u2019s heat theorem, the entropy difference between two\n", - "crystals approaches zero when the temperature approaches absolute zero.\n", - "It is thus a common practice to put $S = 0$ for a crystal at 0 K. This\n", - "is usually referred as the third law of thermodynamics. From the\n", - "definition of entropy change by , $S$ or $S_{m}$ is always positive at\n", - "finite temperatures as the system or the crystal absorbs heat from the\n", - "surroundings to increase its temperature. $V$ or $V_{m}$ of a phase is a\n", - "well-defined physical quantity, and its absolute value can be given and\n", - "is always positive. The above three equations can be written in a\n", - "general form as\n", - "\n", - "Eq. \u2011\n", - "$\\ \\left( \\frac{\\partial Y_{i}}{\\partial Y_{j}} \\right)_{Y_{k}} = - \\frac{X_{j}}{X_{i}} < 0$\n", - "\n", - "The surface thus has negative slopes in all its directions. The\n", - "curvature of the surface can be derived from\n", - "\n", - "Eq. \u2011\n", - "$\\ \\left( \\frac{\\partial^{2}Y_{i}}{\\partial\\left( Y_{j} \\right)^{2}} \\right)_{Y_{k}} = - \\left( \\frac{\\partial\\left( \\frac{X_{j}}{X_{i}} \\right)}{\\partial Y_{j}} \\right)_{Y_{k}} = - \\frac{1}{X_{i}}\\left( \\frac{\\partial X_{j}}{\\partial Y_{j}} \\right)_{Y_{k}} + \\frac{X_{j}}{\\left( X_{i} \\right)^{2}}\\left( \\frac{\\partial X_{i}}{\\partial Y_{j}} \\right)_{Y_{k}} = - \\frac{1}{X_{i}}\\left\\lbrack \\left( \\frac{\\partial X_{j}}{\\partial Y_{j}} \\right)_{Y_{k}} - \\frac{X_{j}}{X_{i}}{\\left( \\frac{\\partial X_{i}}{\\partial Y_{i}} \\right)\\left( \\frac{\\partial Y_{i}}{\\partial Y_{j}} \\right)}_{Y_{k}} \\right\\rbrack = - \\frac{1}{X_{i}}\\left\\lbrack \\left( \\frac{\\partial X_{j}}{\\partial Y_{j}} \\right)_{Y_{k}} + \\left( \\frac{X_{j}}{X_{i}} \\right)^{2}\\left( \\frac{\\partial X_{i}}{\\partial Y_{i}} \\right)_{Y_{k}} \\right\\rbrack < 0$\n", - "\n", - "Both terms inside the last bracket are positive for a system in a state\n", - "of stable internal equilibrium, and the surface thus has a negative\n", - "curvature and is convex everywhere as shown in .\n", - "\n", - "Figure \u2011: Gibbs energy of a one-component phase as a function of\n", - "temperature and negative pressure, showing the convex shape\n", - "\n", - "From experimental observations, it is known that\n", - "$S_{m}^{vapor} \\gg S_{m}^{liquid} > S_{m}^{solid}$. The curves of\n", - "$G_{m}$ or $\\mu^{\\alpha}$ plotted with respect to $T$ at constant $P$\n", - "would thus have the most negative slope for a vapour phase followed by\n", - "its liquid and solid phases. As an example, shows Gibbs energy of Zn in\n", - "its solid, liquid, and vapour forms as a function of $T$ at constant\n", - "$P = 1$ atmospheric pressure.\n", - "\n", - "Figure \u2011: Molar Gibbs energy of Zn as a function of T at constant P\n", - "\n", - "Similarly it is common that\n", - "$V_{m}^{vapor} \\gg V_{m}^{liquid} > V_{m}^{solid}$, and the curves of\n", - "$G_{m}$ or $\\mu^{\\alpha}$ plotted with respect to $P$ at constant $T$\n", - "would thus have the most positive slope for a vapour phase followed by\n", - "its liquid and solid phases, though there are cases that\n", - "$V_{m}^{liquid} < V_{m}^{solid}$ such as those of water and ice. As an\n", - "example, shows Gibbs energy of Fe in its two solid (fcc and bcc),\n", - "liquid, and vapour forms as a function of $P$ at constant $T = 1273K$.\n", - "\n", - "Figure \u2011: Molar Gibbs energy of Zn as a function of P at constant T\n", - "\n", - "The quantities measurable by experiments typically include temperature,\n", - "pressure, volume, composition, and amount of heat flow in the combined\n", - "law of thermodynamics discussed so far. By measuring the heat needed to\n", - "increase the temperature of a phase, the heat capacity of the phase is\n", - "obtained as shown by Eq. 2\u20117. A typical heat capacity curve as a\n", - "function of temperature is shown in for fcc-Al, hcp-Mg, and an\n", - "intermetallic phase Al12Mg17.\n", - "\n", - "Figure \u2011: Heat capacity of fcc-Al, hcp-Mg, and\n", - "Al12Mg17 as a function of temperature\n", - "\n", - "There are various theoretical models for the heat capacity under\n", - "constant volume to be discussed in Chapter 5 of this book, which is\n", - "defined as\n", - "\n", - "Eq. \u2011\n", - "$C_{V} = \\left( \\frac{\\partial U}{\\partial T} \\right)_{V} = T\\left( \\frac{\\partial(F + TS)}{\\partial T} \\right)_{V} = T\\left( \\frac{\\partial S}{\\partial T} \\right)_{V} = - T\\left( \\frac{\\partial^{2}F}{\\partial T^{2}} \\right)_{V}$\n", - "\n", - "To establish the relationship between $C_{P}$ defined by and $C_{V}$,\n", - "$U$ needs to be represented as a function of $T$ and $V$ in terms of $G$\n", - "and its derivatives with respect to Gibbs energy\u2019s natural variables of\n", - "$T$ and $P$. It can be done as follows\n", - "\n", - "Eq. \u2011\n", - "$dV = \\frac{\\partial V}{\\partial T}dT + \\frac{\\partial V}{\\partial( - P)}d( - P) = - \\frac{\\partial^{2}G}{\\partial T( - P)}dT - \\frac{\\partial^{2}G}{\\partial( - P)^{2}}d( - P)$\n", - "\n", - "Eq. \u2011\n", - "$dU = \\frac{\\partial(G + TS - PV)}{\\partial T}dT + \\frac{\\partial(G + TS - PV)}{\\partial( - P)}d( - P) = - \\left( T\\frac{\\partial^{2}G}{\\partial T^{2}} - P\\frac{\\partial^{2}G}{\\partial T( - P)} \\right)dT\u2014\\left( T\\frac{\\partial^{2}G}{\\partial T( - P)} + P\\frac{\\partial^{2}G}{\\partial( - P)^{2}} \\right)\\left( - \\frac{1}{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}}dV + \\frac{\\frac{\\partial^{2}G}{\\partial T( - P)}}{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}}dT \\right) = - \\left\\lbrack T\\frac{\\partial^{2}G}{\\partial T^{2}} - T\\frac{\\left( \\frac{\\partial^{2}G}{\\partial T( - P)} \\right)^{2}}{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}} \\right\\rbrack dT + \\left( - T\\frac{\\frac{\\partial^{2}G}{\\partial T( - P)}}{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}} + P \\right)dV$\n", - "\n", - "Eq. \u2011\n", - "$C_{V} = C_{P} + T\\frac{\\left( \\frac{\\partial^{2}G}{\\partial T( - P)} \\right)^{2}}{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}} = C_{P} - \\frac{\\alpha_{V}^{2}VT}{\\kappa_{T}} = C_{P} - \\alpha_{V}^{2}BVT$\n", - "\n", - "where the thermal expansion, $\\alpha_{V}$, and the compressibility or\n", - "bulk modulus, $\\kappa_{T}$ or $B$, are defined by and , respectively.\n", - "From the heat capacity, the enthalpy and entropy can be obtained by\n", - "integration of at a constant pressure\n", - "\n", - "Eq. \u2011\n", - "$S = S_{0} + \\int_{0}^{T}\\frac{C_{P}}{T}dT = S_{0} + \\int_{0}^{298.15}\\frac{C_{P}}{T}dT + \\int_{298.15}^{T}\\frac{C_{P}}{T}dT = S_{298.15} + \\int_{298.15}^{T}\\frac{C_{P}}{T}dT$\n", - "\n", - "Eq. \u2011\n", - "$H = H_{0} + \\int_{0}^{T}C_{P}dT = H_{0} + \\int_{0}^{298.15}C_{P}dT + \\int_{298.15}^{T}C_{P}dT = H_{298.15} + \\int_{298.15}^{T}C_{P}dT$\n", - "\n", - "In the above equations, two temperature ranges of integration are chosen\n", - "for practical applications as most processing procedures in the field of\n", - "materials science and engineering take place at temperatures above the\n", - "room temperature. Based on the third-law of thermodynamics, $S_{0} = 0$,\n", - "$S_{298.15}$ can be obtained by integration. On the other hand for\n", - "$H_{0} = U_{0} + PV$, one does not know the absolute value of the\n", - "internal energy and thus have to select a reference state for $H$. In\n", - "principle, the reference state can be arbitrarily chosen. A widely used\n", - "reference state in the thermodynamic modeling practice is to set\n", - "$H_{298.15}^{SER} = 0$ at ambient pressure for pure elements at their\n", - "respective stable structures at room temperature, called stable element\n", - "reference (SER) state with\n", - "\n", - "Eq. \u2011\n", - "$G_{298.15}^{SER} = H_{298.15}^{SER} - TS_{298.15}^{SER} = - TS_{298.15}^{SER}$\n", - "\n", - "It is further noted that after defining $S_{298.15}$ and $H_{298.15}$,\n", - "one only needs the heat capacity at higher temperatures. This makes the\n", - "mathematical representation of heat capacity simpler due to a relatively\n", - "simple temperature dependence of heat capacity at higher temperatures in\n", - "comparison with the variation at lower temperatures. One common\n", - "expression for heat capacity at high temperatures and ambient pressure\n", - "is as follows\n", - "\n", - "Eq. \u2011 $C_{P} = c + dT + \\frac{e}{T^{2}} + fT^{2}$\n", - "\n", - "where c, d, e, and f are parameters fitted to experimental or theoretic\n", - "data and compiled in various handbooks.\n", - "\n", - "The corresponding $S$, $H$, and $G$ are obtained as\n", - "\n", - "Eq. \u2011 $S = b^{'} + clnT + dT - \\frac{e}{{2T}^{2}} + \\frac{f}{2}T^{2}$\n", - "\n", - "Eq. \u2011 $H = a + cT + \\frac{d}{2}T^{2} - \\frac{e}{T} + \\frac{f}{3}T^{3}$\n", - "\n", - "Eq. \u2011\n", - "$G = H - TS = a - bT - cTlnT - \\frac{d}{2}T^{2} - \\frac{e}{2T} - \\frac{f}{6}T^{3}$\n", - "\n", - "with $b = b^{'} - c$. The integration constants $b^{'}$ and $a$ are\n", - "evaluated from $S_{298.15}$ and $H_{298.15}$. As an example, the\n", - "enthalpy and entropy of Zn in solid (hcp), liquid, and gas forms are\n", - "plotted in and , respectively. The distances between any two curves in\n", - "and represent the enthalpy or entropy differences between the two\n", - "phases. It can be seen that the gas has much higher enthalpy and entropy\n", - "than the solid and liquid.\n", - "\n", - "Figure \u2011: Enthalpy of Zn as a function of temperature at one atmospheric\n", - "pressure\n", - "\n", - "Figure \u2011: Entropy of Zn as a function of temperature at one atmospheric\n", - "pressure\n", - "\n", - "Similarly, one can add the pressure dependence into the Gibbs energy\n", - "function such as\n", - "\n", - "Eq. \u2011\n", - "$G = a - bT - cTlnT - \\frac{d}{2}T^{2} - \\frac{e}{2T} - \\frac{f}{6}T^{3} + gP + hTP + mP^{2}$\n", - "\n", - "where g, h, and m are parameters fitted to experimental or theoretic\n", - "data and compiled in various handbooks.\n", - "\n", - "The expression for $V$ can be derived as\n", - "\n", - "Eq. \u2011 $V = g + hT + 2mP$\n", - "\n", - "The Helmholtz energy can be expressed as a function of its natural\n", - "variables by solving $P\\ $ from\n", - "\n", - "Eq. \u2011\n", - "$F = G - PV = a - bT - cTlnT - \\frac{d}{2}T^{2} - \\frac{e}{2T} - \\frac{f}{6}T^{3} - \\frac{(g + hT - V)^{2}}{4m}$\n", - "\n", - "In the literature there are many models to represent the temperature and\n", - "pressure dependences of thermodynamic properties. The Gibbs energy\n", - "difference between a stoichiometric compound and the components at their\n", - "reference states of which the compound is composed,\n", - "${_{\\ }^{0}G}_{i}^{ref}$, is termed as Gibbs energy of formation, i.e.\n", - "\n", - "Eq. \u2011 $\\mathrm{\\Delta}_{f}G = G - \\sum_{}^{}N_{i}{_{\\ }^{0}G}_{i}^{ref}$\n", - "\n", - "with $N_{i}$ being the stoichiometry of the compound. Similarly,\n", - "enthalpy of formation, entropy of formation, and heat capacity of\n", - "formation with respect to components at their reference states,\n", - "$_{\\ }^{0}H_{i}^{ref}$, $_{\\ }^{0}S_{i}^{ref}$, and\n", - "$_{\\ }^{0}{C_{P}}_{i}^{ref}$, can be defined as\n", - "\n", - "Eq. \u2011 $\\mathrm{\\Delta}_{f}H = H - \\sum_{}^{}{N_{i}_{\\ }^{0}H_{i}^{ref}}$\n", - "\n", - "Eq. \u2011 $\\mathrm{\\Delta}_{f}S = S - \\sum_{}^{}{N_{i}_{\\ }^{0}S_{i}^{ref}}$\n", - "\n", - "Eq. \u2011\n", - "$\\mathrm{\\Delta}_{f}C_{P} = C_{P} - \\sum_{}^{}{N_{i}_{\\ }^{0}{C_{P}}_{i}^{ref}}$\n", - "\n", - "It should be mentioned that one mole of a compound usually refers to one\n", - "mole of formula of stoichiometry of the compound. With a formula like\n", - "$A_{a}B_{b}C_{c}$, the compound is composed of total $(a + b + c)$ moles\n", - "of components. One should thus be very careful when dealing with\n", - "numerical values to be sure whether the data is in terms of per mole of\n", - "formula or per mole of components. At the same time the reference states\n", - "must be clearly defined. When the SER state defined in is selected as\n", - "the reference state, the above formation quantities are called standard\n", - "formation quantities such as standard enthalpy of formation.\n", - "\n", - "Since there are only two independent potentials in a one-component\n", - "system, its limit of stability can be evaluated with one potential kept\n", - "constant, i.e. either $T$ or $P$. Consequently, either Helmholtz energy\n", - "or enthalpy is to be used in deriving the limit of stability of a\n", - "homogeneous system. For the practical usefulness, let us use Helmholtz\n", - "energy because its natural variables of $T$ and *V* are measurable\n", - "quantities in typical experiments, while one of the natural variables of\n", - "enthalpy, entropy, is not. From and , the limit of stability for a\n", - "one-component system at constant temperature can be written as\n", - "\n", - "Eq. \u2011\n", - "$\\left( \\frac{\\partial( - P)}{\\partial V} \\right)_{T,N} = F_{VV} = \\frac{1}{V\\kappa_{T}} = \\frac{B}{V} = 0$\n", - "\n", - "where the isothermal compressibility and bulk modulus, $\\kappa_{T}$ and\n", - "$B$, are defined in . The limit of stability is thus determined when the\n", - "isothermal compressibility diverges or the bulk modulus becomes zero\n", - "because $V$ has finite values at any temperatures. It is evident that\n", - "Helmholtz energy must have higher order dependence on volume than in for\n", - "a system with instability because $F_{VV}$ from is constant.\n", - "\n", - "From , the consolute point is defined by\n", - "\n", - "Eq. \u2011\n", - "$F_{VVV} = \\left( \\frac{\\partial^{2}( - P)}{\\partial V^{2}} \\right)_{T,N} = \\frac{\\partial\\left( \\frac{1}{V\\kappa_{T}} \\right)}{\\partial V} = - \\frac{1 + \\frac{V}{\\kappa_{T}}\\frac{\\partial\\kappa_{T}}{\\partial V}}{\\kappa_{T}V^{2}} = 0$\n", - "\n", - "Since $\\kappa_{T}$ becomes infinite at the limit of stability,\n", - "$\\frac{\\partial\\kappa_{T}}{\\partial V}$ approaches negative infinite\n", - "when the critical/consolute point is approached so that\n", - "$\\frac{V}{\\kappa_{T}}\\frac{\\partial\\kappa_{T}}{\\partial V} = - 1$ and\n", - "$F_{VVV} = 0$.\n" - ], + "source": "## Phases with fixed compositions\n\nThe homogeneous system discussed so far means that there is only one\nphase in the system, i.e. a single-phase system. A phase with a fixed\ncomposition can be a pure element or a stoichiometric compound. There is\nthus only one independent component in the system. A stoichiometric\ncompound contains more than one element, but the relative amounts of\neach element are fixed by the stoichiometry and cannot vary\nindependently, i.e., $dN_{i} = x_{i}dN$. The combined law of\nthermodynamics becomes\n\nEq. \u2011\n$dG = - SdT - Vd( - P) + \\left( \\sum_{}^{}{x_{i}\\mu_{i}} \\right)dN - Dd\\xi = - SdT - Vd( - P) + G_{m}dN - Dd\\xi$\n\n$G_{m}$ is the molar Gibbs energy of the stoichiometric compound and can\nbe regarded as the chemical potential of the stoichiometric\nphase,$\\ \\alpha$,\n\nEq. \u2011 $G_{m} = \\mu^{\\alpha} = \\sum_{}^{}{x_{i}\\mu_{i}}$\n\nThe chemical potential of individual components in the phase cannot be\ndefined because the amount of each component cannot be varied\nindependently. For a stoichiometric phase of $N$ moles of atoms at\nequilibrium with $dG$=$Nd\\mu^{\\alpha} + \\mu^{\\alpha}dN$, reduces to\n\nEq. \u2011 $\\ 0 = - SdT - Vd( - P) - Nd\\mu^{\\alpha}$\n\nwhich is the Gibbs-Duhem equation, , applied to a stoichiometric phase.\nIt can be represented graphically by a surface in a three-dimensional\nspace composed of $\\ \\mu^{\\alpha}$, *T* and *\u2013P*. The direction of the\nsurface is represented by the three partial directives between any two\nof $\\ \\mu^{\\alpha}$, *T* and *\u2013P* with the third one kept constant, i.e.\n\nEq. \u2011\n$\\ \\left( \\frac{\\partial\\mu^{\\alpha}}{\\partial T} \\right)_{P} = - \\frac{S}{N} = - S_{m}$\n\nEq. \u2011\n$\\ \\left( \\frac{\\partial\\mu^{\\alpha}}{\\partial( - P)} \\right)_{T} = - \\frac{V}{N} = - V_{m}$\n\nEq. \u2011\n$\\ \\left( \\frac{\\partial( - P)}{\\partial T} \\right)_{\\mu^{\\alpha}} = - \\frac{S}{V} = - \\frac{S_{m}}{V_{m}}$\n\nBased on Nernst\u2019s heat theorem, the entropy difference between two\ncrystals approaches zero when the temperature approaches absolute zero.\nIt is thus a common practice to put $S = 0$ for a crystal at 0 K. This\nis usually referred as the third law of thermodynamics. From the\ndefinition of entropy change by , $S$ or $S_{m}$ is always positive at\nfinite temperatures as the system or the crystal absorbs heat from the\nsurroundings to increase its temperature. $V$ or $V_{m}$ of a phase is a\nwell-defined physical quantity, and its absolute value can be given and\nis always positive. The above three equations can be written in a\ngeneral form as\n\nEq. \u2011\n$\\ \\left( \\frac{\\partial Y_{i}}{\\partial Y_{j}} \\right)_{Y_{k}} = - \\frac{X_{j}}{X_{i}} < 0$\n\nThe surface thus has negative slopes in all its directions. The\ncurvature of the surface can be derived from\n\nEq. \u2011\n$\\ \\left( \\frac{\\partial^{2}Y_{i}}{\\partial\\left( Y_{j} \\right)^{2}} \\right)_{Y_{k}} = - \\left( \\frac{\\partial\\left( \\frac{X_{j}}{X_{i}} \\right)}{\\partial Y_{j}} \\right)_{Y_{k}} = - \\frac{1}{X_{i}}\\left( \\frac{\\partial X_{j}}{\\partial Y_{j}} \\right)_{Y_{k}} + \\frac{X_{j}}{\\left( X_{i} \\right)^{2}}\\left( \\frac{\\partial X_{i}}{\\partial Y_{j}} \\right)_{Y_{k}} = - \\frac{1}{X_{i}}\\left\\lbrack \\left( \\frac{\\partial X_{j}}{\\partial Y_{j}} \\right)_{Y_{k}} - \\frac{X_{j}}{X_{i}}{\\left( \\frac{\\partial X_{i}}{\\partial Y_{i}} \\right)\\left( \\frac{\\partial Y_{i}}{\\partial Y_{j}} \\right)}_{Y_{k}} \\right\\rbrack = - \\frac{1}{X_{i}}\\left\\lbrack \\left( \\frac{\\partial X_{j}}{\\partial Y_{j}} \\right)_{Y_{k}} + \\left( \\frac{X_{j}}{X_{i}} \\right)^{2}\\left( \\frac{\\partial X_{i}}{\\partial Y_{i}} \\right)_{Y_{k}} \\right\\rbrack < 0$\n\nBoth terms inside the last bracket are positive for a system in a state\nof stable internal equilibrium, and the surface thus has a negative\ncurvature and is convex everywhere as shown in .\n\nFigure \u2011: Gibbs energy of a one-component phase as a function of\ntemperature and negative pressure, showing the convex shape\n\nFrom experimental observations, it is known that\n$S_{m}^{vapor} \\gg S_{m}^{liquid} > S_{m}^{solid}$. The curves of\n$G_{m}$ or $\\mu^{\\alpha}$ plotted with respect to $T$ at constant $P$\nwould thus have the most negative slope for a vapour phase followed by\nits liquid and solid phases. As an example, shows Gibbs energy of Zn in\nits solid, liquid, and vapour forms as a function of $T$ at constant\n$P = 1$ atmospheric pressure.\n\nFigure \u2011: Molar Gibbs energy of Zn as a function of T at constant P\n\nSimilarly it is common that\n$V_{m}^{vapor} \\gg V_{m}^{liquid} > V_{m}^{solid}$, and the curves of\n$G_{m}$ or $\\mu^{\\alpha}$ plotted with respect to $P$ at constant $T$\nwould thus have the most positive slope for a vapour phase followed by\nits liquid and solid phases, though there are cases that\n$V_{m}^{liquid} < V_{m}^{solid}$ such as those of water and ice. As an\nexample, shows Gibbs energy of Fe in its two solid (fcc and bcc),\nliquid, and vapour forms as a function of $P$ at constant $T = 1273K$.\n\nFigure \u2011: Molar Gibbs energy of Zn as a function of P at constant T\n\nThe quantities measurable by experiments typically include temperature,\npressure, volume, composition, and amount of heat flow in the combined\nlaw of thermodynamics discussed so far. By measuring the heat needed to\nincrease the temperature of a phase, the heat capacity of the phase is\nobtained as shown by Eq. 2\u20117. A typical heat capacity curve as a\nfunction of temperature is shown in for fcc-Al, hcp-Mg, and an\nintermetallic phase Al12Mg17.\n\nFigure \u2011: Heat capacity of fcc-Al, hcp-Mg, and\nAl12Mg17 as a function of temperature\n\nThere are various theoretical models for the heat capacity under\nconstant volume to be discussed in Chapter 5 of this book, which is\ndefined as\n\nEq. \u2011\n$C_{V} = \\left( \\frac{\\partial U}{\\partial T} \\right)_{V} = T\\left( \\frac{\\partial(F + TS)}{\\partial T} \\right)_{V} = T\\left( \\frac{\\partial S}{\\partial T} \\right)_{V} = - T\\left( \\frac{\\partial^{2}F}{\\partial T^{2}} \\right)_{V}$\n\nTo establish the relationship between $C_{P}$ defined by and $C_{V}$,\n$U$ needs to be represented as a function of $T$ and $V$ in terms of $G$\nand its derivatives with respect to Gibbs energy\u2019s natural variables of\n$T$ and $P$. It can be done as follows\n\nEq. \u2011\n$dV = \\frac{\\partial V}{\\partial T}dT + \\frac{\\partial V}{\\partial( - P)}d( - P) = - \\frac{\\partial^{2}G}{\\partial T( - P)}dT - \\frac{\\partial^{2}G}{\\partial( - P)^{2}}d( - P)$\n\nEq. \u2011\n$dU = \\frac{\\partial(G + TS - PV)}{\\partial T}dT + \\frac{\\partial(G + TS - PV)}{\\partial( - P)}d( - P) = - \\left( T\\frac{\\partial^{2}G}{\\partial T^{2}} - P\\frac{\\partial^{2}G}{\\partial T( - P)} \\right)dT\u2014\\left( T\\frac{\\partial^{2}G}{\\partial T( - P)} + P\\frac{\\partial^{2}G}{\\partial( - P)^{2}} \\right)\\left( - \\frac{1}{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}}dV + \\frac{\\frac{\\partial^{2}G}{\\partial T( - P)}}{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}}dT \\right) = - \\left\\lbrack T\\frac{\\partial^{2}G}{\\partial T^{2}} - T\\frac{\\left( \\frac{\\partial^{2}G}{\\partial T( - P)} \\right)^{2}}{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}} \\right\\rbrack dT + \\left( - T\\frac{\\frac{\\partial^{2}G}{\\partial T( - P)}}{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}} + P \\right)dV$\n\nEq. \u2011\n$C_{V} = C_{P} + T\\frac{\\left( \\frac{\\partial^{2}G}{\\partial T( - P)} \\right)^{2}}{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}} = C_{P} - \\frac{\\alpha_{V}^{2}VT}{\\kappa_{T}} = C_{P} - \\alpha_{V}^{2}BVT$\n\nwhere the thermal expansion, $\\alpha_{V}$, and the compressibility or\nbulk modulus, $\\kappa_{T}$ or $B$, are defined by and , respectively.\nFrom the heat capacity, the enthalpy and entropy can be obtained by\nintegration of at a constant pressure\n\nEq. \u2011\n$S = S_{0} + \\int_{0}^{T}\\frac{C_{P}}{T}dT = S_{0} + \\int_{0}^{298.15}\\frac{C_{P}}{T}dT + \\int_{298.15}^{T}\\frac{C_{P}}{T}dT = S_{298.15} + \\int_{298.15}^{T}\\frac{C_{P}}{T}dT$\n\nEq. \u2011\n$H = H_{0} + \\int_{0}^{T}C_{P}dT = H_{0} + \\int_{0}^{298.15}C_{P}dT + \\int_{298.15}^{T}C_{P}dT = H_{298.15} + \\int_{298.15}^{T}C_{P}dT$\n\nIn the above equations, two temperature ranges of integration are chosen\nfor practical applications as most processing procedures in the field of\nmaterials science and engineering take place at temperatures above the\nroom temperature. Based on the third-law of thermodynamics, $S_{0} = 0$,\n$S_{298.15}$ can be obtained by integration. On the other hand for\n$H_{0} = U_{0} + PV$, one does not know the absolute value of the\ninternal energy and thus have to select a reference state for $H$. In\nprinciple, the reference state can be arbitrarily chosen. A widely used\nreference state in the thermodynamic modeling practice is to set\n$H_{298.15}^{SER} = 0$ at ambient pressure for pure elements at their\nrespective stable structures at room temperature, called stable element\nreference (SER) state with\n\nEq. \u2011\n$G_{298.15}^{SER} = H_{298.15}^{SER} - TS_{298.15}^{SER} = - TS_{298.15}^{SER}$\n\nIt is further noted that after defining $S_{298.15}$ and $H_{298.15}$,\none only needs the heat capacity at higher temperatures. This makes the\nmathematical representation of heat capacity simpler due to a relatively\nsimple temperature dependence of heat capacity at higher temperatures in\ncomparison with the variation at lower temperatures. One common\nexpression for heat capacity at high temperatures and ambient pressure\nis as follows\n\nEq. \u2011 $C_{P} = c + dT + \\frac{e}{T^{2}} + fT^{2}$\n\nwhere c, d, e, and f are parameters fitted to experimental or theoretic\ndata and compiled in various handbooks.\n\nThe corresponding $S$, $H$, and $G$ are obtained as\n\nEq. \u2011 $S = b^{'} + clnT + dT - \\frac{e}{{2T}^{2}} + \\frac{f}{2}T^{2}$\n\nEq. \u2011 $H = a + cT + \\frac{d}{2}T^{2} - \\frac{e}{T} + \\frac{f}{3}T^{3}$\n\nEq. \u2011\n$G = H - TS = a - bT - cTlnT - \\frac{d}{2}T^{2} - \\frac{e}{2T} - \\frac{f}{6}T^{3}$\n\nwith $b = b^{'} - c$. The integration constants $b^{'}$ and $a$ are\nevaluated from $S_{298.15}$ and $H_{298.15}$. As an example, the\nenthalpy and entropy of Zn in solid (hcp), liquid, and gas forms are\nplotted in and , respectively. The distances between any two curves in\nand represent the enthalpy or entropy differences between the two\nphases. It can be seen that the gas has much higher enthalpy and entropy\nthan the solid and liquid.\n\nFigure \u2011: Enthalpy of Zn as a function of temperature at one atmospheric\npressure\n\nFigure \u2011: Entropy of Zn as a function of temperature at one atmospheric\npressure\n\nSimilarly, one can add the pressure dependence into the Gibbs energy\nfunction such as\n\nEq. \u2011\n$G = a - bT - cTlnT - \\frac{d}{2}T^{2} - \\frac{e}{2T} - \\frac{f}{6}T^{3} + gP + hTP + mP^{2}$\n\nwhere g, h, and m are parameters fitted to experimental or theoretic\ndata and compiled in various handbooks.\n\nThe expression for $V$ can be derived as\n\nEq. \u2011 $V = g + hT + 2mP$\n\nThe Helmholtz energy can be expressed as a function of its natural\nvariables by solving $P\\ $ from\n\nEq. \u2011\n$F = G - PV = a - bT - cTlnT - \\frac{d}{2}T^{2} - \\frac{e}{2T} - \\frac{f}{6}T^{3} - \\frac{(g + hT - V)^{2}}{4m}$\n\nIn the literature there are many models to represent the temperature and\npressure dependences of thermodynamic properties. The Gibbs energy\ndifference between a stoichiometric compound and the components at their\nreference states of which the compound is composed,\n${_{\\ }^{0}G}_{i}^{ref}$, is termed as Gibbs energy of formation, i.e.\n\nEq. \u2011 $\\mathrm{\\Delta}_{f}G = G - \\sum_{}^{}N_{i}{_{\\ }^{0}G}_{i}^{ref}$\n\nwith $N_{i}$ being the stoichiometry of the compound. Similarly,\nenthalpy of formation, entropy of formation, and heat capacity of\nformation with respect to components at their reference states,\n$_{\\ }^{0}H_{i}^{ref}$, $_{\\ }^{0}S_{i}^{ref}$, and\n$_{\\ }^{0}{C_{P}}_{i}^{ref}$, can be defined as\n\nEq. \u2011 $\\mathrm{\\Delta}_{f}H = H - \\sum_{}^{}{N_{i}_{\\ }^{0}H_{i}^{ref}}$\n\nEq. \u2011 $\\mathrm{\\Delta}_{f}S = S - \\sum_{}^{}{N_{i}_{\\ }^{0}S_{i}^{ref}}$\n\nEq. \u2011\n$\\mathrm{\\Delta}_{f}C_{P} = C_{P} - \\sum_{}^{}{N_{i}_{\\ }^{0}{C_{P}}_{i}^{ref}}$\n\nIt should be mentioned that one mole of a compound usually refers to one\nmole of formula of stoichiometry of the compound. With a formula like\n$A_{a}B_{b}C_{c}$, the compound is composed of total $(a + b + c)$ moles\nof components. One should thus be very careful when dealing with\nnumerical values to be sure whether the data is in terms of per mole of\nformula or per mole of components. At the same time the reference states\nmust be clearly defined. When the SER state defined in is selected as\nthe reference state, the above formation quantities are called standard\nformation quantities such as standard enthalpy of formation.\n\nSince there are only two independent potentials in a one-component\nsystem, its limit of stability can be evaluated with one potential kept\nconstant, i.e. either $T$ or $P$. Consequently, either Helmholtz energy\nor enthalpy is to be used in deriving the limit of stability of a\nhomogeneous system. For the practical usefulness, let us use Helmholtz\nenergy because its natural variables of $T$ and *V* are measurable\nquantities in typical experiments, while one of the natural variables of\nenthalpy, entropy, is not. From and , the limit of stability for a\none-component system at constant temperature can be written as\n\nEq. \u2011\n$\\left( \\frac{\\partial( - P)}{\\partial V} \\right)_{T,N} = F_{VV} = \\frac{1}{V\\kappa_{T}} = \\frac{B}{V} = 0$\n\nwhere the isothermal compressibility and bulk modulus, $\\kappa_{T}$ and\n$B$, are defined in . The limit of stability is thus determined when the\nisothermal compressibility diverges or the bulk modulus becomes zero\nbecause $V$ has finite values at any temperatures. It is evident that\nHelmholtz energy must have higher order dependence on volume than in for\na system with instability because $F_{VV}$ from is constant.\n\nFrom , the consolute point is defined by\n\nEq. \u2011\n$F_{VVV} = \\left( \\frac{\\partial^{2}( - P)}{\\partial V^{2}} \\right)_{T,N} = \\frac{\\partial\\left( \\frac{1}{V\\kappa_{T}} \\right)}{\\partial V} = - \\frac{1 + \\frac{V}{\\kappa_{T}}\\frac{\\partial\\kappa_{T}}{\\partial V}}{\\kappa_{T}V^{2}} = 0$\n\nSince $\\kappa_{T}$ becomes infinite at the limit of stability,\n$\\frac{\\partial\\kappa_{T}}{\\partial V}$ approaches negative infinite\nwhen the critical/consolute point is approached so that\n$\\frac{V}{\\kappa_{T}}\\frac{\\partial\\kappa_{T}}{\\partial V} = - 1$ and\n$F_{VVV} = 0$.\n", "metadata": {} } ] diff --git a/src/psu410/src/gibbs_energy_function/phases_with_variable_compositions_random_solutions.ipynb b/src/psu410/src/gibbs_energy_function/phases_with_variable_compositions_random_solutions.ipynb index 36dd8fe..eabca3e 100644 --- a/src/psu410/src/gibbs_energy_function/phases_with_variable_compositions_random_solutions.ipynb +++ b/src/psu410/src/gibbs_energy_function/phases_with_variable_compositions_random_solutions.ipynb @@ -4,119 +4,33 @@ "metadata": {}, "cells": [ { - "id": "2236d2f8", + "id": "469ee6b7", "cell_type": "markdown", - "source": [ - "## Phases with variable compositions: Random solutions\n", - "\n", - "The combined law of thermodynamics and the Gibbs-Duhem equation of a\n", - "solution phase with variable compositions are shown by and ,\n", - "respectively. A phase can be represented by a *c+1*-dimensional surface\n", - "in a *c+2*-dimensional space of potentials based on the Gibbs-Duhem\n", - "equation. The directions and curvature of the surface are represented by\n", - "the partial derivatives shown by and secondary derivatives shown by ,\n", - "both being negative for a stable phase. To develop a mathematical\n", - "formula for Gibbs energy of a phase with variable compositions, one can\n", - "consider a phase as a mixture of independent components that the phase\n", - "is made of. Its Gibbs energy function can be postulated as the summation\n", - "of Gibbs energy of the independent components of the same structure of\n", - "the solution, $_{\\ }^{0}G_{i}$, plus the contribution due to the mixing,\n", - "$_{\\ }^{mixing}G$ or $_{\\ }^{M}G$\n", - "\n", - "Eq. \u2011 $G = \\sum_{}^{}{N_{i}_{\\ }^{0}G_{i}} +_{\\ }^{M}G$\n", - "\n", - "Since the system size usually is not important in thermodynamics, its\n", - "properties are typically normalized to one mole with its composition\n", - "represented by mole fractions of components. The molar Gibbs energy is\n", - "obtained as shown below with the molar Gibbs energy of mixing separated\n", - "into two parts: ideal Gibbs energy of mixing assuming no chemical\n", - "interaction among components, $_{\\ }^{ideal}G_{m}$ or $_{\\ }^{I}G_{m}$,\n", - "and excess Gibbs energy of mixing due to chemical reaction among\n", - "components, $_{\\ }^{excess}G_{m}$ or $_{\\ }^{E}G_{m}$\n", - "\n", - "Eq. \u2011\n", - "$G_{m} = \\sum_{}^{}{x_{i}_{\\ }^{0}G_{i}} +_{\\ }^{M}G_{m} = \\sum_{}^{}{x_{i}_{\\ }^{0}G_{i}} +_{\\ }^{I}G_{m} +_{\\ }^{E}G_{m}$\n", - "\n", - "From , the chemical potential of a component is thus\n", - "\n", - "Eq. \u2011\n", - "$\\mu_{i} =_{\\ }^{0}G_{i} +_{\\ }^{I}{G_{m} +}_{\\ }^{E}G_{m} + \\frac{\\partial\\left(_{\\ }^{I}{G_{m} +}_{\\ }^{E}G_{m} \\right)}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial\\left(_{\\ }^{I}{G_{m} +}_{\\ }^{E}G_{m} \\right)}{\\partial x_{j}}$\n", - "\n", - "One may define the chemical activity of component *i*, $a_{i}^{\\ }$, as\n", - "follows\n", - "\n", - "Eq. \u2011\n", - "$RTlna_{i}^{\\ } = \\mu_{i} -_{\\ }^{0}G_{i}^{\\ } =_{\\ }^{I}{G_{m} +}\\frac{\\partial_{\\ }^{I}G_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial_{\\ }^{I}G_{m}}{\\partial x_{j}} +_{\\ }^{E}G_{m} + \\frac{\\partial_{\\ }^{E}G_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial_{\\ }^{E}G_{m}}{\\partial x_{j}}$\n", - "\n", - "In this definition, the chemical activity or simply activity is\n", - "calculated with respect to the pure elements in the structure of the\n", - "solution for practical reasons as one would like to understand the\n", - "chemical potential difference of components in the solution and by\n", - "itself with the same structure. It should be noted that this reference\n", - "state for chemical activity is usually different from the SER reference\n", - "state defined in as the solution may have a different structure than\n", - "those of pure components in their SER states. On the other hand, the\n", - "activity under the SER reference state can be easily obtained by\n", - "replacing $_{\\ }^{0}G_{i}$ with ${_{\\ }^{0}G}_{i}^{SER}$ from . In\n", - "principle, one may choose any structure as the reference state for\n", - "activity to be useful for practical applications, i.e.\n", - "\n", - "Eq. \u2011 $RTlna_{i}^{ref} = \\mu_{i} -_{\\ }^{0}G_{i}^{ref}$\n", - "\n", - "For example, the activity of a component in a liquid solution is defined\n", - "with respect to the pure component in its liquid form from , but can\n", - "also be referred to its SER state which is solid using . The following\n", - "sections will discuss in more details how components mix when they are\n", - "brought together including concepts such as random mixing, short-range\n", - "ordering, and long-range ordering.\n", - "\n", - "The limit of stability of a solution with respect to composition\n", - "fluctuation under constant *T*, *P*, and *N1* can be derived\n", - "as follows from and\n", - "\n", - "Eq. \u2011\n", - "$\\left( \\frac{\\partial\\mu_{i}}{\\partial N_{i}} \\right)_{T,P,N_{j \\neq i},i > 1} > \\left( \\frac{\\partial\\mu_{i}}{\\partial N_{i}} \\right)_{T,P,N_{1},{\\mu_{2},N}_{j \\neq i},i,j > 2}\\ldots... > \\left( \\frac{\\partial\\mu_{c}}{\\partial N_{c}} \\right)_{T,P,N_{1},\\mu_{2}..\\mu_{c - 1}} = 0$\n", - "\n", - "The first term can be derived from as follows\n", - "\n", - "Eq. \u2011\n", - "$\\left( \\frac{\\partial\\mu_{i}}{\\partial N_{i}} \\right)_{T,P,N_{j \\neq i},i > 1} = \\sum_{j = 1}^{c}\\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{i}\\partial x_{j}}\\frac{\\partial x_{j}}{\\partial N_{i}} - \\sum_{j = 1}^{c}{x_{j}\\sum_{k = 1}^{c}{\\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{j}\\partial x_{k}}\\frac{\\partial x_{k}}{\\partial N_{i}}}} = \\frac{1}{N}\\left( \\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{i}^{2}} - \\sum_{j = 1}^{c}{x_{j}\\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{j}^{2}}} - \\sum_{j = 1}^{c}{x_{j}\\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{i}\\partial x_{j}}} + \\sum_{j = 1}^{c}{\\sum_{k = 1}^{c}{{x_{j}x}_{k}\\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{j}\\partial x_{k}}}} \\right)$\n", - "\n", - "Denoting\n", - "$G_{ij} = \\left( \\frac{\\partial\\mu_{i}}{\\partial N_{j}} \\right)_{T,P,N_{k \\neq j}}$and\n", - "using to change the variables kept constant from molar quantities to\n", - "potentials one-by-one (see \\[1\\]), the limit of stability can be further\n", - "obtained as\n", - "\n", - "Eq. \u2011\n", - "$\\left( \\frac{\\partial\\mu_{c}}{\\partial N_{c}} \\right)_{T,P,N_{1},\\mu_{2}..\\mu_{c - 1}} = \\frac{\\det\\left( G_{ij}:2 \\leq i,j \\leq c \\right)}{\\det\\left( G_{ij}:2 \\leq i,j \\leq c - 1 \\right)} = 0$\n", - "\n", - "where $\\det$ stands for determinant. indicates that\n", - "$\\det\\left( G_{ij}:2 \\leq i,j \\leq c \\right) = 0$ at the limit of\n", - "stability. Considering $x_{1} = 1 - \\sum_{j \\neq 1}^{}x_{j}$, let us\n", - "introduce\n", - "\n", - "Eq. \u2011\n", - "$g_{i} = \\mu_{i} - \\mu_{1} = \\left( \\frac{\\partial G_{m}}{\\partial x_{i}} \\right)_{x_{k \\neq i}} - \\left( \\frac{\\partial G_{m}}{\\partial x_{1}} \\right)_{x_{k \\neq 1}}$\n", - "\n", - "and\n", - "\n", - "Eq. \u2011\n", - "$g_{ij} = \\frac{\\partial g_{i}}{\\partial x_{j}} = \\frac{\\partial\\left( \\mu_{i} - \\mu_{1} \\right)}{\\partial x_{j}} = \\frac{\\partial^{2}G_{m}}{\\partial x_{i}\\partial x_{j}} - \\frac{\\partial^{2}G_{m}}{\\partial x_{1}\\partial x_{j}} - \\frac{\\partial^{2}G_{m}}{\\partial x_{i}\\partial x_{1}} + \\frac{\\partial^{2}G_{m}}{\\partial\\left( x_{1} \\right)^{2}}$\n", - "\n", - "The limit of stability can be re-written as\n", - "\n", - "Eq. \u2011\n", - "$\\left( \\frac{\\partial\\left( \\mu_{c} - \\mu_{1} \\right)}{\\partial x_{c}} \\right)_{T,P,N,\\mu_{2} - \\mu_{1},\\ ...\\mu_{c - 1} - \\mu_{1}} = \\frac{\\det\\left( g_{ij}:2 \\leq i,j \\leq c \\right)}{\\det\\left( g_{ij}:2 \\leq i,j \\leq c - 1 \\right)} = 0$\n", - "\n", - "i.e. $\\det\\left( g_{ij}:2 \\leq i,j \\leq c \\right) = 0$. The consolute\n", - "point can be defined following\n", - "\n", - "Eq. \u2011\n", - "$\\left( \\frac{\\partial^{2}\\mu_{c}}{\\partial\\left( N_{c} \\right)^{2}} \\right)_{T,P,N_{1},\\mu_{2}..\\mu_{c - 1}} = \\left( \\frac{\\partial^{2}\\left( \\mu_{c} - \\mu_{1} \\right)}{\\partial\\left( x_{c} \\right)^{2}} \\right)_{T,P,N,\\mu_{2} - \\mu_{1},\\ ...\\mu_{c - 1} - \\mu_{1}} = 0$\n", - "\n", - "No closed mathematic form has been published in the literature.\n" - ], + "source": "## Phases with variable compositions: Random solutions\n\nThe combined law of thermodynamics and the Gibbs-Duhem equation of a\nsolution phase with variable compositions are shown by and ,\nrespectively. A phase can be represented by a *c+1*-dimensional surface\nin a *c+2*-dimensional space of potentials based on the Gibbs-Duhem\nequation. The directions and curvature of the surface are represented by\nthe partial derivatives shown by and secondary derivatives shown by ,\nboth being negative for a stable phase. To develop a mathematical\nformula for Gibbs energy of a phase with variable compositions, one can\nconsider a phase as a mixture of independent components that the phase\nis made of. Its Gibbs energy function can be postulated as the summation\nof Gibbs energy of the independent components of the same structure of\nthe solution, $_{\\ }^{0}G_{i}$, plus the contribution due to the mixing,\n$_{\\ }^{mixing}G$ or $_{\\ }^{M}G$\n\nEq. \u2011 $G = \\sum_{}^{}{N_{i}_{\\ }^{0}G_{i}} +_{\\ }^{M}G$\n\nSince the system size usually is not important in thermodynamics, its\nproperties are typically normalized to one mole with its composition\nrepresented by mole fractions of components. The molar Gibbs energy is\nobtained as shown below with the molar Gibbs energy of mixing separated\ninto two parts: ideal Gibbs energy of mixing assuming no chemical\ninteraction among components, $_{\\ }^{ideal}G_{m}$ or $_{\\ }^{I}G_{m}$,\nand excess Gibbs energy of mixing due to chemical reaction among\ncomponents, $_{\\ }^{excess}G_{m}$ or $_{\\ }^{E}G_{m}$\n\nEq. \u2011\n$G_{m} = \\sum_{}^{}{x_{i}_{\\ }^{0}G_{i}} +_{\\ }^{M}G_{m} = \\sum_{}^{}{x_{i}_{\\ }^{0}G_{i}} +_{\\ }^{I}G_{m} +_{\\ }^{E}G_{m}$\n\nFrom , the chemical potential of a component is thus\n\nEq. \u2011\n$\\mu_{i} =_{\\ }^{0}G_{i} +_{\\ }^{I}{G_{m} +}_{\\ }^{E}G_{m} + \\frac{\\partial\\left(_{\\ }^{I}{G_{m} +}_{\\ }^{E}G_{m} \\right)}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial\\left(_{\\ }^{I}{G_{m} +}_{\\ }^{E}G_{m} \\right)}{\\partial x_{j}}$\n\nOne may define the chemical activity of component *i*, $a_{i}^{\\ }$, as\nfollows\n\nEq. \u2011\n$RTlna_{i}^{\\ } = \\mu_{i} -_{\\ }^{0}G_{i}^{\\ } =_{\\ }^{I}{G_{m} +}\\frac{\\partial_{\\ }^{I}G_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial_{\\ }^{I}G_{m}}{\\partial x_{j}} +_{\\ }^{E}G_{m} + \\frac{\\partial_{\\ }^{E}G_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial_{\\ }^{E}G_{m}}{\\partial x_{j}}$\n\nIn this definition, the chemical activity or simply activity is\ncalculated with respect to the pure elements in the structure of the\nsolution for practical reasons as one would like to understand the\nchemical potential difference of components in the solution and by\nitself with the same structure. It should be noted that this reference\nstate for chemical activity is usually different from the SER reference\nstate defined in as the solution may have a different structure than\nthose of pure components in their SER states. On the other hand, the\nactivity under the SER reference state can be easily obtained by\nreplacing $_{\\ }^{0}G_{i}$ with ${_{\\ }^{0}G}_{i}^{SER}$ from . In\nprinciple, one may choose any structure as the reference state for\nactivity to be useful for practical applications, i.e.\n\nEq. \u2011 $RTlna_{i}^{ref} = \\mu_{i} -_{\\ }^{0}G_{i}^{ref}$\n\nFor example, the activity of a component in a liquid solution is defined\nwith respect to the pure component in its liquid form from , but can\nalso be referred to its SER state which is solid using . The following\nsections will discuss in more details how components mix when they are\nbrought together including concepts such as random mixing, short-range\nordering, and long-range ordering.\n\nThe limit of stability of a solution with respect to composition\nfluctuation under constant *T*, *P*, and *N1* can be derived\nas follows from and\n\nEq. \u2011\n$\\left( \\frac{\\partial\\mu_{i}}{\\partial N_{i}} \\right)_{T,P,N_{j \\neq i},i > 1} > \\left( \\frac{\\partial\\mu_{i}}{\\partial N_{i}} \\right)_{T,P,N_{1},{\\mu_{2},N}_{j \\neq i},i,j > 2}\\ldots... > \\left( \\frac{\\partial\\mu_{c}}{\\partial N_{c}} \\right)_{T,P,N_{1},\\mu_{2}..\\mu_{c - 1}} = 0$\n\nThe first term can be derived from as follows\n\nEq. \u2011\n$\\left( \\frac{\\partial\\mu_{i}}{\\partial N_{i}} \\right)_{T,P,N_{j \\neq i},i > 1} = \\sum_{j = 1}^{c}\\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{i}\\partial x_{j}}\\frac{\\partial x_{j}}{\\partial N_{i}} - \\sum_{j = 1}^{c}{x_{j}\\sum_{k = 1}^{c}{\\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{j}\\partial x_{k}}\\frac{\\partial x_{k}}{\\partial N_{i}}}} = \\frac{1}{N}\\left( \\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{i}^{2}} - \\sum_{j = 1}^{c}{x_{j}\\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{j}^{2}}} - \\sum_{j = 1}^{c}{x_{j}\\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{i}\\partial x_{j}}} + \\sum_{j = 1}^{c}{\\sum_{k = 1}^{c}{{x_{j}x}_{k}\\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{j}\\partial x_{k}}}} \\right)$\n\nDenoting\n$G_{ij} = \\left( \\frac{\\partial\\mu_{i}}{\\partial N_{j}} \\right)_{T,P,N_{k \\neq j}}$and\nusing to change the variables kept constant from molar quantities to\npotentials one-by-one (see \\[1\\]), the limit of stability can be further\nobtained as\n\nEq. \u2011\n$\\left( \\frac{\\partial\\mu_{c}}{\\partial N_{c}} \\right)_{T,P,N_{1},\\mu_{2}..\\mu_{c - 1}} = \\frac{\\det\\left( G_{ij}:2 \\leq i,j \\leq c \\right)}{\\det\\left( G_{ij}:2 \\leq i,j \\leq c - 1 \\right)} = 0$\n\nwhere $\\det$ stands for determinant. indicates that\n$\\det\\left( G_{ij}:2 \\leq i,j \\leq c \\right) = 0$ at the limit of\nstability. Considering $x_{1} = 1 - \\sum_{j \\neq 1}^{}x_{j}$, let us\nintroduce\n\nEq. \u2011\n$g_{i} = \\mu_{i} - \\mu_{1} = \\left( \\frac{\\partial G_{m}}{\\partial x_{i}} \\right)_{x_{k \\neq i}} - \\left( \\frac{\\partial G_{m}}{\\partial x_{1}} \\right)_{x_{k \\neq 1}}$\n\nand\n\nEq. \u2011\n$g_{ij} = \\frac{\\partial g_{i}}{\\partial x_{j}} = \\frac{\\partial\\left( \\mu_{i} - \\mu_{1} \\right)}{\\partial x_{j}} = \\frac{\\partial^{2}G_{m}}{\\partial x_{i}\\partial x_{j}} - \\frac{\\partial^{2}G_{m}}{\\partial x_{1}\\partial x_{j}} - \\frac{\\partial^{2}G_{m}}{\\partial x_{i}\\partial x_{1}} + \\frac{\\partial^{2}G_{m}}{\\partial\\left( x_{1} \\right)^{2}}$\n\nThe limit of stability can be re-written as\n\nEq. \u2011\n$\\left( \\frac{\\partial\\left( \\mu_{c} - \\mu_{1} \\right)}{\\partial x_{c}} \\right)_{T,P,N,\\mu_{2} - \\mu_{1},\\ ...\\mu_{c - 1} - \\mu_{1}} = \\frac{\\det\\left( g_{ij}:2 \\leq i,j \\leq c \\right)}{\\det\\left( g_{ij}:2 \\leq i,j \\leq c - 1 \\right)} = 0$\n\ni.e. $\\det\\left( g_{ij}:2 \\leq i,j \\leq c \\right) = 0$. The consolute\npoint can be defined following\n\nEq. \u2011\n$\\left( \\frac{\\partial^{2}\\mu_{c}}{\\partial\\left( N_{c} \\right)^{2}} \\right)_{T,P,N_{1},\\mu_{2}..\\mu_{c - 1}} = \\left( \\frac{\\partial^{2}\\left( \\mu_{c} - \\mu_{1} \\right)}{\\partial\\left( x_{c} \\right)^{2}} \\right)_{T,P,N,\\mu_{2} - \\mu_{1},\\ ...\\mu_{c - 1} - \\mu_{1}} = 0$\n\nNo closed mathematic form has been published in the literature.\n", + "metadata": {} + }, + { + "id": "051f94e5", + "cell_type": "markdown", + "source": "### Random solutions\n\nThe ideal Gibbs energy of mixing represents an ideal solution in which\nall sites are equivalent and the distributions of components on the\nsites are completely random. The number of different configurations to\narrange all components is\n\nEq. \u2011 $w = \\frac{N!}{\\prod_{}^{}\\left( N_{i}! \\right)}$\n\nBased on Boltzmann\u2019s relation from statistic thermodynamics when all\nconfigurations have the same probability to be observed, the ideal\nconfigurational molar entropy of mixing due to the distribution is\n\nEq. \u2011\n$_{\\ }^{ideal}S_{m} =_{\\ }^{I}S_{m} = \\frac{Rlnw}{N} = R\\frac{lnN! - \\sum_{}^{}\\ln\\left( N_{i}! \\right)}{N} \\cong R\\frac{NlnN - \\sum_{}^{}{N_{i}l{nN}_{i}}}{N} = - R\\sum_{}^{}{x_{i}l{nx}_{i}}$\n\nwhere $R$ is the gas constant. represents the entropy difference between\nthe ideal solution and the states when individual components are by\nthemselves, i.e. the mechanical mixing of components. As $x_{i}$ is\nsmaller than unity, the entropy production to form an ideal solution\nfrom pure components is thus positive, indicating that it is a\nspontaneous process. In such an ideal solution, it is assumed that there\nare no interactions between components, and the enthalpy of mixing is\nthus zero as the internal energy and the volume of the system do not\nchange. The ideal Gibbs energy of mixing is written as\n\nEq. \u2011 $_{\\ }^{I}G = - T_{\\ }^{I}S_{m} = RT\\sum_{}^{}{x_{i}l{nx}_{i}}$\n\nThe Gibbs energy of real solutions, i.e. , becomes\n\nEq. \u2011\n$G_{m} = \\sum_{}^{}{x_{i}_{\\ }^{0}G_{i}} + RT\\sum_{}^{}{x_{i}l{nx}_{i}} +_{\\ }^{E}G_{m}$\n\nFrom , the chemical potential is obtained as\n\nEq. \u2011\n$\\mu_{i} = G_{i} =_{\\ }^{0}G_{i} + RTlnx_{i} +_{\\ }^{E}G_{m} + \\frac{\\partial_{\\ }^{E}G_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial_{\\ }^{E}G_{m}}{\\partial x_{j}}$\n\nFrom the chemical activity of , the activity coefficient, $\\gamma_{i}$,\ncan be defined as follows\n\nEq. \u2011\n$\\gamma_{i} = \\frac{a_{i}}{x_{i}} = \\frac{1}{x_{i}}\\exp\\frac{G_{i} -_{\\ }^{0}G_{i}\\ }{RT}$\n\nThe solution is an ideal solution with $\\gamma_{i} = 1$, and is said to\nbe positively or negatively deviating from an ideal solution with\n$\\gamma_{i} > 1$ or $\\gamma_{i} < 1$, respectively. The chemical\npotential is related to the activity and activity coefficient by the\nfollowing equation\n\nEq. \u2011\n$\\mu_{i} =_{\\ }^{0}G_{i} + RTlna_{i} =_{\\ }^{0}G_{i} + RTln\\gamma_{i}x_{i} =_{\\ }^{0}G_{i} + RTlnx_{i} + RTln\\gamma_{i}$\n\nLet us further exam in more details in order to better understand the\nrelation between $G_{m}$ and $\\mu_{i}$. The partial derivatives in\nrepresent the directions of molar Gibbs energy in the composition space,\ni.e. the tangents of molar Gibbs energy with respect to mole fractions\nof independent components. Collectively, they define the\nmulti-dimensional tangent plane of molar Gibbs energy at the given\ncomposition, $x_{i}^{0}$. The mathematical representation of this\ntangent plane, $z_{G_{m}}$, is defined by its directional derivatives\nand the distance from the point where the derivatives are taken,\n\nEq. \u2011\n$z_{G_{m}} = G_{m}\\left( x_{i}^{0} \\right) + \\sum_{i = 1}^{c}{\\left( \\frac{\\partial G_{m}}{\\partial x_{i}} \\right)_{x_{i}^{0}}\\left( x_{i} - x_{i}^{0} \\right)}$\n\nThe intercept of this tangent plane at each pure component axis, i.e.\n$x_{i} = 1$ and $x_{j \\neq i} = 0$, is obtained as\n\nEq. \u2011\n$z_{G_{m},x_{i} = 1} = G_{m}\\left( x_{i}^{0} \\right) + \\left( \\frac{\\partial G_{m}}{\\partial x_{i}} \\right)_{x_{i}^{\\ } = x_{i}^{0}} - \\sum_{j = 1}^{c}{x_{j}^{0}\\left( \\frac{\\partial G_{m}}{\\partial x_{j}} \\right)_{x_{i}^{\\ } = x_{i}^{0}}}$\n\nThis is identical to at the point $x_{i}^{0}$. It is thus shown that the\nchemical potential of a component in a solution is represented by the\nintercept of the tangent plane of Gibbs energy of the solution on the\n$G_{m}$ axis of the component. The distance between the intercept and\nthe Gibbs energy of the pure component in the same structure of the\nsolution is related to the chemical activity of the component as defined\nby . On the other hand, it is evident that one can choose any other\nstructure of the pure element to define the chemical activity to compare\nchemical potential of the component as shown by .\n\nThe stability of a solution is evaluated following , and the derivatives\nof chemical potential with respect to its moles, i.e. the elements in\nthe determinant, are obtained as follows from and ,\n\nEq. \u2011\n$\\frac{N}{RT}\\frac{\\partial\\mu_{i}}{\\partial N_{i}} = \\frac{N}{RT}G_{ii} = \\frac{1 - x_{i}}{x_{i}} + \\frac{1}{\\gamma_{i}}\\left( \\frac{\\partial\\gamma_{i}}{\\partial x_{i}} - \\sum_{j = 1}^{c}{x_{j}\\frac{\\partial\\gamma_{i}}{\\partial x_{j}}} \\right)$\n\nEq. \u2011\n$\\frac{N}{RT}\\frac{\\partial\\mu_{i}}{\\partial N_{k}} = \\frac{N}{RT}G_{ik} = - \\frac{x_{k}}{x_{i}} + \\frac{1}{\\gamma_{i}}\\left( 1 - \\sum_{j = 1}^{c}{x_{j}\\frac{\\partial\\gamma_{i}}{\\partial x_{j}}} \\right)$\n\nTo further study Gibbs energy of solution phases, let us discuss the\ndetails on the excess Gibbs energy of mixing. At this point, one can\nstart with lower-order systems with fewer components, i.e. two component\nand three-component systems, noting that the Gibbs energy of phases with\none component is already presented in Chapter .\n", + "metadata": {} + }, + { + "id": "b159479c", + "cell_type": "markdown", + "source": "### Binary random solutions\n\nFrom , the Gibbs-Duhem equation of a binary system consisting of\ncomponents $A$ and $B$ is written as\n\nEq. \u2011 $0 = - SdT - Vd( - P) - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B}$\n\nThis equation represents a four-dimensional surface. It is self-evident\nthat both and hold for stable binary solutions too, i.e. the directions\nand the curvature of the surface are all negative. To visualize the\nfour-dimensional surface in the three-dimensional space, one needs to\nfix one of the four potentials. As $T$ and $P$ are the natural variables\nof Gibbs energy, they are usually chosen to be kept constant. One can\ntypically investigate behaviors of systems consisting of condensed\nphases by varying the temperature at constant pressure. at constant\npressure thus becomes\n\nEq. \u2011 $0 = - SdT - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B}$\n\nSimilar to and , the property of a phase can be represented by a\ntwo-dimensional surface in the three-dimensional space composed of $T$,\n$\\mu_{A}$, and $\\mu_{B}$ under constant $P$, keeping in mind the\nfollowing\n\nEq. \u2011\n$G_{m} = x_{A}\\mu_{A} + x_{B}\\mu_{B} = x_{A}_{\\ }^{0}G_{A} + x_{B}_{\\ }^{0}G_{B} + RT\\left( x_{A}l{nx}_{A} + x_{B}l{nx}_{B} \\right) +_{\\ }^{E}G_{m}$\n\nSince $_{\\ }^{E}G_{m}$ must be zero for pure components $A$ and $B$, it\nneeds to be in the following form\n\nEq. \u2011 $_{\\ }^{E}G_{m} = x_{A}x_{B}L_{AB}$\n\nwith $L_{AB}$ being a parameter denoting the interaction between\ncomponents $A$ and $B$, called interaction parameter. When $L_{AB} = 0$,\nthe solution is an ideal solution. When $L_{AB}$ is a non-zero constant\nindependent of temperature and composition, the solution is called a\nregular solution. Its excess entropy and excess enthalpy of mixing are\nobtained as\n\nEq. \u2011 $_{\\ }^{E}S_{m} = \\frac{\\partial_{\\ }^{E}G_{m}}{\\partial T} = 0$\n\nEq. \u2011\n$_{\\ }^{E}H_{m} =_{\\ }^{E}G_{m} - T_{\\ }^{E}S_{m} = x_{A}x_{B}L_{AB}$\n\nThe chemical potential of component $A$ or $B$ in a binary regular\nsolution can be derived as\n\nEq. \u2011\n$\\mu_{i} =_{\\ }^{0}G_{i} + RTlnx_{i} + \\left( 1 - x_{i} \\right)^{2}L_{AB}$\n\nIn a dilute solution with $x_{i} \\rightarrow 0$, one can have\n\nEq. \u2011\n$RTln\\gamma_{i} = \\left( 1 - x_{i} \\right)^{2}L_{AB} \\approx L_{AB}$\n\nEq. \u2011 $\\gamma_{i} = e^{\\frac{L_{AB}}{RT}}$\n\nThe activity is thus proportional to its mole fraction, which is called\nHenry\u2019s law. By the same token, for the solvent, i.e.\n$x_{i} \\rightarrow 1$,\n\nEq. \u2011 $RTln\\gamma_{i} = \\left( 1 - x_{i} \\right)^{2}L_{AB} \\approx 0$\n\nwhich gives $\\gamma_{i} \\approx 1$, and its activity approaches its mole\nfraction. This is called Raoult\u2019s law.\n\nThe stability of a binary solution is derived from as\n\nEq. \u2011\n$\\left( \\frac{\\partial\\mu_{A}}{\\partial N_{A}} \\right)_{T,P,N_{B}} = \\left\\lbrack \\frac{RT}{x_{A}} - 2\\left( 1 - x_{A} \\right)L_{AB} \\right\\rbrack\\frac{1 - x_{A}}{N}$\n\nEq. \u2011\n$\\left( \\frac{\\partial\\mu_{B}}{\\partial N_{B}} \\right)_{T,P,N_{A}} = \\left\\lbrack \\frac{RT}{x_{B}} - 2\\left( 1 - x_{B} \\right)L_{AB} \\right\\rbrack\\frac{1 - x_{B}}{N}$\n\nIt should be noted that the two chemical potentials in a binary system\nat constant temperature and pressure are dependent on each other due to\nthe Gibbs-Duhem equation shown in , i.e.\n\nEq. \u2011 $0 = - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B}$\n\nand the two chemical potentials depend on each other by the following\nrelation\n\nEq. \u2011\n$\\left( \\frac{\\partial\\mu_{A}}{\\partial\\mu_{B}} \\right)_{T,P} = - \\frac{N_{B}}{N_{A}}$\n\nTherefore, at the limit of stability, both and go to zero at the same\ntime, which is obtained when\n\nEq. \u2011 $RT = 2{x_{A}x_{B}L}_{AB}$\n\nAs the absolute temperature cannot be negative, has no solution for a\nsolution phase with $L_{AB} < 0$, i.e. the solution phase is stable with\nrespect to the composition fluctuation. For a solution with\n$L_{AB} > 0$, its limit of stability is represented .\n\nA schematic molar Gibbs energy of a solution with $L_{AB} < 0$ at\nconstant temperature and pressure is shown in along with the ideal and\nexcess Gibbs energy of mixing. A tangent line on the molar Gibbs energy\nof the solution is drawn, and its two intercepts at $x_{B} = 0$ and\n$x_{B} = 1$ give the chemical potentials of components $A$ and $B$,\n$\\mu_{A}$ and $\\mu_{B}$ by , respectively. It is evident that $\\mu_{A}$\nand $\\mu_{B}$ are not independent on each other as they are two points\non the same straight line. This is a graphic representation of the\nGibbs-Duhem equation of . The chemical activity of component $B$ is also\ndepicted with the reference state being the pure B with the same\nstructure. As shown in , other structures of pure B can be selected as\nthe reference states of the chemical activity of component B, resulting\nin the different distances to its chemical potential in the solution,\nthus different values of its chemical activities. It is clear that this\nchange of reference state for chemical activity does not affect the\nchemical potential of the component in the solution.\n\nFigure \u2011: Schematic molar Gibbs energy diagram with $L_{AB} < 0$\n\nWhen $L_{AB} > 0$, represents a parabola in the $T - x_{i}$\ntwo-dimensional coordinate, symmetric with respect to $x_{A}$ and\n$x_{B}$, shown in , i.e. the spinodal of the solution. The consolute\npoint is obtained by applying to and letting equal to zero at the\nconsolute point\n\nEq. \u2011\n$\\left( \\frac{\\partial^{2}\\mu_{A}}{\\partial N_{A}^{2}} \\right)_{T,P,N_{B}} = \\left\\lbrack - \\frac{RT}{x_{A}^{2}} + 2L_{AB} \\right\\rbrack\\left( \\frac{1 - x_{A}}{N} \\right)^{2} = 0$\n\nwhich gives\n\nEq. \u2011 $T_{cons} = 2x_{A}^{2}L_{AB}$\n\nSolving and , one obtains $x_{A} = x_{B} = 0.5$ and\n\nEq. \u2011 $T_{cons} = \\frac{L_{AB}}{2R}$\n\nFigure \u2011: A Spinodal curve with $L_{AB} > 0$\n\nA schematic molar Gibbs energy diagram at temperatures below the\nconsolute point is shown in . It can be seen that part of the molar\nGibbs energy has negative curvature, and the solution becomes unstable.\nThe chemical potential thus does not change monotonically with respect\nto composition and its derivative changes sign at the inflexion point.\n\nFigure \u2011: Schematic molar Gibbs energy diagram with $L_{AB} > 0$\n\nFor more complex solutions, $L_{AB}$ can be a function of temperature,\npressure, and compositions. In principle, the temperature and pressure\ndependences can be treated by means of formula similar to . There are\nvarious approaches in the literature to consider the composition\ndependence of $L_{AB}$. The empirical Redlich-Kister polynomial stands\nout as the one most widely used because it can be extrapolated to\nternary and multi-component systems consistently, which will be\ndiscussed in Chapter .\n", + "metadata": {} + }, + { + "id": "4a5ace2b", + "cell_type": "markdown", + "source": "### Ternary random solutions\n\nFrom , the Gibbs-Duhem equation of a ternary system consisting of\ncomponents $A$, $B$ and $C$ is written as\n\nEq. \u2011\n$0 = - SdT - Vd( - P) - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B} - {N_{C}d\\mu}_{C}$\n\nThis equation represents a five-dimensional surface. It can be\nvisualized in a three-dimensional space with two of the five potentials\nfixed. Usually $T$ and $P$ are kept constant as they are the natural\nvariables of $G$, and reduces to\n\nEq. \u2011 $0 = - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B} - {N_{C}d\\mu}_{C}$\n\nA phase can thus be represented by a surface in the three-dimensional\nspace of $\\mu_{A}$, $\\mu_{B}$, and $\\mu_{C}$ at constant $T$ and $P$\nwith similar geometric appearance as .\n\nFrom , Gibbs energy of a ternary solution is written as\n\nEq. \u2011\n$G_{m} = x_{A}_{\\ }^{0}G_{A} + x_{B}_{\\ }^{0}G_{B} + x_{C}_{\\ }^{0}G_{C} + RT\\left( x_{A}l{nx}_{A} + x_{B}l{nx}_{B} + x_{C}l{nx}_{C} \\right) +_{\\ }^{E}G_{m}$\n\nWhen the mole fraction of one component approaches zero,\n$_{\\ }^{E}G_{m}$ reduces to the excess Gibbs energy of mixing of the\nbinary systems of the remaining two components, represented by .\nHowever, for a given composition of a ternary solution, there is no\nunique way to assign the contributions from $_{\\ }^{E}G_{m}$ of each\nbinary to $_{\\ }^{E}G_{m}$ of the ternary solution because\n$_{\\ }^{E}G_{m}$ of the ternary solution contains information of both\nbinary and ternary interactions. A variety of models is available in the\nliterature (see \\[1\\]). One intuitive approach would be to use the same\nformula as that in the binary system, i.e. , with the mole fractions\nsubstituted by the values in the ternary system, and $_{\\ }^{E}G_{m}$ of\na ternary solution may thus be defined as the following by including the\nternary interaction involving all three components,\n\nEq. \u2011\n$_{\\ }^{E}G_{m} = x_{A}x_{B}L_{AB} + x_{A}x_{C}L_{AC} + x_{B}x_{C}L_{BC} + x_{A}x_{B}x_{C}L_{ABC}$\n\nThe chemical potential of a component is represented by . When all\ninteraction parameters in are constant, i.e. a ternary regular solution,\nthe chemical potential of component $A$ can be derived as\n\nEq. \u2011\n$\\mu_{A} = G_{A} =_{\\ }^{0}G_{A} + RTlnx_{A} + x_{B}L_{AB} + x_{C}L_{AC} -_{\\ }^{E}G_{m} =_{\\ }^{0}G_{A} + RTlnx_{A} + x_{B}{\\left( 1 - x_{A} \\right)L}_{AB} + x_{C}\\left( 1 - x_{A} \\right)L_{AC} - x_{B}x_{C}L_{BC} + x_{B}x_{C}\\left( 1 - 2x_{A} \\right)L_{ABC} =_{\\ }^{0}G_{A} + RTlnx_{B} + x_{B}^{2}L_{AB} + x_{C}^{2}L_{AC} + x_{B}x_{C}\\left( L_{AB} + L_{AC} - L_{BC} \\right) + x_{B}x_{C}\\left( 1 - 2x_{A} \\right)L_{ABC}$\n\nSimilar equations can be derived for component $B$ and C with\n$L_{AB} = L_{BA}$, $L_{AC} = L_{CA}$, and $L_{BC} = L_{CB}$. A schematic\nmolar Gibbs energy diagram at constant temperature and pressure is shown\nin with all three binary systems having $L_{ij} < 0$ of similar values.\n\nFigure \u2011: Schematic ternary molar Gibbs energy diagram as a function of\ncompositions for given temperature and pressure\n\nTo evaluate the stability of a ternary solution, one needs to calculate\nthe elements in the determinant shown in . Using the mole of component\n$C$ as the independent molar quantity, the limit of stability is\nexpresses as\n\nEq. \u2011 $G_{AA}G_{BB} - G_{AB}G_{BA} = 0$\n\nAs an example, $G_{AA}$ is shown in the following equation, which must\nbe positive for the solution to be stable\n\nEq. \u2011\n${N\\left( \\frac{\\partial\\mu_{A}}{\\partial N_{A}} \\right)}_{T,P,N_{B},N_{C}} = NG_{AA} = \\frac{RT\\left( 1 - x_{A} \\right)}{x_{A}} - 2x_{B}^{2}L_{AB} - 2x_{C}^{2}L_{AC} - 2x_{B}x_{C}\\left( L_{AB} + L_{AC} - L_{BC} \\right) - 2x_{B}x_{C}\\left( 2 - 3x_{A} \\right)L_{ABC}$\n\nIt is evident that any instability in binary systems with positive\ninteraction parameters extends into the ternary system. It can also be\nseen that even if all binary interaction parameters are negative, i.e.\nno instability in the binary systems, it is possible that becomes\nnegative for some combinations of the binary interaction parameters such\nthat $\\mathrm{\\Delta}L = L_{AB} + L_{AC} - L_{BC}$ becomes very positive\nand overshadows the contributions due to $L_{AB}$ and $L_{AC}$, i.e.\n$L_{BC}$ is more negative than $L_{AB}$ and $L_{AC}$ combined. In an\nextreme case with $L_{AB} = L_{AC} = L_{ABC} = 0$ and $L_{BC} < 0$, i.e.\nideal solutions for the $A - B$ and $A - C$ binary systems, stable\nsolution in the $B - C$ binary system, and no additional ternary\ninteraction, reduces to\n\nEq. \u2011\n$N\\left( \\frac{\\partial\\mu_{A}}{\\partial N_{A}} \\right)_{T,P,N_{B},N_{C}} = \\frac{RT\\left( 1 - x_{A} \\right)}{x_{A}} + 2x_{B}x_{C}L_{BC}$\n\nSetting\n$\\left( \\frac{\\partial\\mu_{A}}{\\partial N_{A}} \\right)_{T,P,N_{B},N_{C}} = 0$,\none obtains\n\nEq. \u2011\n$- \\frac{RT}{2L_{BC}} = \\frac{x_{A}x_{B}x_{C}}{1 - x_{A}} = \\frac{{\\left( 1 - x_{B} - x_{C} \\right)x}_{B}x_{C}}{x_{B} + x_{C}}$\n\nWith $- \\frac{RT}{2L_{BC}}$ being positive due to $L_{BC} < 0$, there is\na parabola-shaped composition area in which the solution is unstable at\nconstant temperature and pressure. This is reasonable because the system\ntends to maximize the number of B-C bonds due to its lower energy, which\ncompetes with the entropy of mixing among the three elements and results\nin segregation of B-C bonds, thus miscibility gap at low temperatures.\n\nTo evaluate the ternary consolute point, the second derivatives for\ncomponent A and B are obtained as\n\nEq. \u2011\n${N\\left( \\frac{\\partial_{\\ }^{2}\\mu_{A}}{\\partial N_{A}^{2}} \\right)}_{T,P,N_{B},N_{C}} = \\frac{RT\\left( 1 - x_{A} \\right)}{x_{A}^{2}} + 4x_{B}^{2}L_{AB} + 4x_{C}^{2}L_{AC} + 4x_{B}x_{C}\\left( L_{AB} + L_{AC} - L_{BC} \\right) + 2x_{B}x_{C}\\left( 7 - 9x_{A} \\right)L_{ABC} = 0$\n\nEq. \u2011\n${N\\left( \\frac{\\partial_{\\ }^{2}\\mu_{B}}{\\partial N_{B}^{2}} \\right)}_{T,P,N_{A},N_{C}} = \\frac{RT\\left( 1 - x_{B} \\right)}{x_{B}^{2}} + 4x_{A}^{2}L_{AB} + 4x_{C}^{2}L_{BC} + 4x_{A}x_{C}\\left( L_{AB} + L_{BC} - L_{AC} \\right) + 2x_{A}x_{C}\\left( 7 - 9x_{B} \\right)L_{ABC} = 0$\n\nThe consolute point can then be obtained using , and .\n\nIt is observed in that $\\left( 1 - 2x_{A} \\right)L_{ABC} = 0$ at\n$x_{A} = 0.5$, i.e. the ternary interaction parameter does not\ncontribute to the chemical potential of $A$. It is also observed in that\nthe contribution from the ternary interaction parameter changes sign at\n$x_{i} = 2/3$ due to $\\left( 2 - 3x_{A} \\right)L_{ABC} = 0$.\n", + "metadata": {} + }, + { + "id": "fd9ad1e0", + "cell_type": "markdown", + "source": "### Multi-component random solutions\n\nSimilar to a ternary solution, the excess Gibbs energy of mixing of a\nmulti-component solution can be written as\n\nEq. \u2011\n$_{\\ }^{E}G_{m} = \\sum_{i}^{}{\\sum_{j}^{}{x_{i}x_{j}L_{ij}}} + \\sum_{i}^{}{\\sum_{j}^{}{\\sum_{k}^{}{x_{i}x_{j}x_{k}L_{ijk}}}}$\n\nIn principle, one can add interaction parameters for quaternary and\nhigher order systems, but their contributions to Gibbs energy are\nrelatively minor because the major contributions have already been taken\ninto account by the binary and ternary interactions. It is anticipated\nthat not only the interaction parameters of four or more components are\nsmall, but also the multiplication of mole fractions in front the\ninteraction parameters diminishes their contribution to the Gibbs energy\neven further.\n\nUnder the condition that all interaction parameters are constant, the\nchemical potential of a component in a multi-component system with\nbinary and ternary interaction parameters can be extended from as\n\nEq. \u2011\n$\\mu_{i} =_{\\ }^{0}G_{i} + RTlnx_{i} + \\sum_{j \\neq i}^{}{x_{j}^{2}L_{ij}} + \\sum_{k > j}^{}{\\sum_{j \\neq i}^{}{x_{j}x_{k}}}\\left\\lbrack L_{ij} + L_{ik} - L_{jk} + \\left( 1 - 2x_{i} \\right)L_{ijk} \\right\\rbrack$\n\nThe stability of the solution can also be extended from as\n\nEq. \u2011\n${N\\left( \\frac{\\partial\\mu_{i}}{\\partial N_{i}} \\right)}_{T,P,N_{j \\neq i}} = NG_{ii} = \\frac{RT\\left( 1 - x_{i} \\right)}{x_{i}} - 2\\sum_{j \\neq i}^{}{x_{j}^{2}L_{ij}} - 2\\sum_{k > j}^{}{\\sum_{j \\neq i}^{}{x_{j}x_{k}}}\\left\\lbrack L_{ij} + L_{ik} - L_{jk} + \\left( 2 - 3x_{i} \\right)L_{ijk} \\right\\rbrack$\n\nThe limit of stability of a multi-component random solution can be\nrepresented by or .\n", "metadata": {} } ] diff --git a/src/psu410/src/gibbs_energy_function/phases_with_variable_compositions_solutions_with_ordering.ipynb b/src/psu410/src/gibbs_energy_function/phases_with_variable_compositions_solutions_with_ordering.ipynb index a0c94f1..c4936f0 100644 --- a/src/psu410/src/gibbs_energy_function/phases_with_variable_compositions_solutions_with_ordering.ipynb +++ b/src/psu410/src/gibbs_energy_function/phases_with_variable_compositions_solutions_with_ordering.ipynb @@ -4,11 +4,33 @@ "metadata": {}, "cells": [ { - "id": "04ebd1df", + "id": "30fe313a", "cell_type": "markdown", - "source": [ - "## Phases with variable compositions: Solutions with ordering\n" - ], + "source": "## Phases with variable compositions: Solutions with ordering\n", + "metadata": {} + }, + { + "id": "ff7e47a3", + "cell_type": "markdown", + "source": "### Solutions with short-range ordering\n\nThe order in a system can be measured by correlation functions which\ndescribes how various components are correlated in space. For\nsimplicity, let us consider only the pairs between nearest neighbors\nwith the correlation function represented by the pair probability of\nnearest neighbor bonds between two components. In a random solution, the\nprobability to find nearest neighbor bonds between two components $i$\nand $j$ is\n\nEq. \u2011 $p_{ij} = x_{i}x_{j}$\n\nWhen $p_{ij} \\neq x_{i}x_{j}$, the nearest neighbors of component $i$\nare not occupied randomly by component $j$, rather certain components\nare favored, resulting in short-range ordering or local clustering in\nthe solution. When short-range ordering develops throughout the\nsolution, long-range ordering takes place, and each component has its\nown primary sites in the solution, to be discussed in Chapter . There\nare relations between bond probabilities and mole fractions of\ncomponents due to the mass balance as follows, with the assumption of\n$p_{ij} = p_{ji}$\n\nEq. \u2011 $\\sum_{i}^{}{\\sum_{j}^{}p_{ij}} = 1$\n\nEq. \u2011 $x_{i} = \\sum_{j}^{}p_{ij}$\n\nFor small deviations from a random solution, one can consider the\nformation of $i - j$ bonds from $i - i\\ $ and $j - j$ bonds and the\nideal mixing of three types of bonds, similar to a typical Ising model.\nThe bond reaction can be written as\n\nEq. \u2011 $(i - i\\ )bonds + (j - j)\\ bonds = 2\\ (i - j)\\ bonds$\n\nwith the Gibbs energy of reaction being\n\nEq. \u2011\n${\\mathrm{\\Delta}G}_{ij} = 2G_{ij} - \\left( G_{ii} + G_{jj} \\right)\\ $\n\nGibbs energy of the solution per mole of atom is thus represented by the\nbond energies and the ideal mixing of bonds plus non-ideal interactions\nbetween pairs,\n\nEq. \u2011\n$G_{m} = \\sum_{i}^{}{\\sum_{j}^{}{p_{ij}G_{ij}}} + \\frac{Z}{2}RT\\sum_{i}^{}{\\sum_{j}^{}{p_{ij}\\ln p_{ij}}} +_{\\ }^{E}G_{m}$\n\nwith $G_{ij}$ being the molar bond energy between components $i$ and\n$j$, $Z$ the number of bonds per atom which is divided by two in the\nequation due to two atoms needed to form one bond, and\n$_{\\ }^{E}G_{m} = \\sum_{}^{}{p_{ij}p_{kl}I_{ijkl}}$ the excess Gibbs\nenergy of mixing between bonds. This approach proposed by Guggenheim\n\\[2\\] is called quasi-chemical method as it is based on the chemical\nreaction shown by .\n\nHowever, the entropy of mixing in does not reduce to ideal entropy of\nmixing for a solution without short-range ordering as defined by . An\napproximated correction may be added for small degree of short-range\nordering as follows\n\nEq. \u2011\n$G_{m} = \\sum_{i}^{}{\\sum_{j}^{}{p_{ij}G_{ij}}} + \\frac{Z}{2}RT\\sum_{i}^{}{\\sum_{j}^{}{p_{ij}\\ln\\frac{p_{ij}}{x_{i}x_{j}}}} + RT\\sum_{}^{}{x_{i}l{nx}_{i}} +_{\\ }^{E}G_{m}$\n\nFor a random solution defined by , becomes\n\nEq. \u2011\n$G_{m} = \\sum_{}^{}{x_{i}_{\\ }^{0}G_{i}} + RT\\sum_{}^{}{x_{i}l{nx}_{i}} + \\sum_{}^{}{x_{i}x_{j}{\\mathrm{\\Delta}G}_{ij}} +_{\\ }^{E}G_{m}$\n\nwith $_{\\ }^{0}G_{i} = G_{ii}$, ${\\mathrm{\\Delta}G}_{ij}$ from Eq. 2\u2011107\nrepresenting the interaction parameter between components $i$ and $j$,\nand $_{\\ }^{E}G_{m} = \\sum_{}^{}{x_{i}{x_{j}x_{k}x}_{l}I_{ijkl}}$\ndenoting the higher order interactions, in comparison with .\n\nWhen short-range ordering exists in a solution, one typically uses the\nlaw of mass reaction for the chemical reaction represented by to define\nthe equilibrium among all bonds, i.e.\n\nEq. \u2011\n$\\frac{\\left( p_{ij} \\right)^{2}}{p_{ii}p_{jj}} = e^{- \\frac{{\\mathrm{\\Delta}G}_{ij}}{kT}}$\n\nHowever, this is under the assumption that the chemical activities of\nall bonds can be represented by their respective probabilities, which is\nonly true for an ideal solution even excluding dilute solutions due to\nthe Henry\u2019s law shown by . Preferrably, the bond probabilities can be\nobtained by calculating the driving force for the fluctuation of bond\nprobabilities under constant temperature, pressure, and amount of each\ncomponent along with the constraints defined by and and equating the\ndriving force to zero, i.e.\n\nEq. \u2011\n${\\frac{1}{N}\\left( \\frac{\\partial G}{\\partial\\xi} \\right)}_{T,P,N_{n}} = \\left( \\frac{\\partial G_{m}}{\\partial p_{ij}} \\right)_{T,P,N_{n}} = \\left( \\frac{\\partial G_{m}}{\\partial p_{ij}} \\right)_{T,P,x_{n},p_{kl \\neq ij}} - \\sum_{kl \\neq ij}^{}\\left( \\frac{\\partial G_{m}}{\\partial p_{kl}} \\right)_{T,P,x_{n},p_{op \\neq kl}} + \\left( \\frac{\\partial G_{m}}{\\partial x_{i}} \\right)_{T,P,x_{q \\neq i},p_{kl}} + \\left( \\frac{\\partial G_{m}}{\\partial x_{j}} \\right)_{T,P,x_{q \\neq j},p_{kl}} = 0$\n\nwhere $\\frac{\\partial p_{kl}}{\\partial p_{ij}} = - 1$ and\n$\\frac{\\partial x_{i}}{\\partial p_{ij}} = \\frac{\\partial x_{j}}{\\partial p_{ij}} = 1$\nare used from and . Numerical values of $p_{ij}$ can be obtained by\nminimization of Gibbs energy under constraints given by and .\n\nThe chemical potential of independent component $i$ is defined as in and\ncan be represented by the following equation\n\nEq. \u2011\n$\\mu_{i} = G_{m} + \\frac{\\partial G_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial G_{m}}{\\partial x_{j}} + 2\\sum_{j = 1}^{c}\\frac{\\partial G_{m}}{\\partial p_{ij}} - \\frac{\\partial G_{m}}{\\partial p_{ii}} - 2\\sum_{j = 1}^{c}x_{j}\\sum_{k = 1}^{c}\\frac{\\partial G_{m}}{\\partial p_{jk}} + \\sum_{j = 1}^{c}x_{j}\\frac{\\partial G_{m}}{\\partial p_{jj}}$\n\nThe stability of the solution can be derived similar to .\n\nWhen the bonding between components becomes very strong, distinctive new\ncomponents may form. They are not independent components and often\ncalled associates. Both the independent and dependent components are\ncollectively called species. The formation of an associate\n$i_{a_{i}}j_{b_{j}}$ consisting of $a_{i}$ mole of $i$ and $a_{j}$ mole\n$j$ can be written as\n\nEq. \u2011 $a_{i}i + a_{j}j = i_{a_{i}}j_{a_{j}}$\n\nGibbs energy of the associate follows the same format of that of a\nstoichiometric phase as shown by ,\n\nEq. \u2011\n${_{\\ }^{0}G}_{i_{a_{i}}j_{a_{j}}} = \\sum_{}^{}a_{i}{_{\\ }^{0}G}_{i}^{SER} + \\mathrm{\\Delta}_{f}G_{i_{a_{i}}j_{a_{j}}}$\n\nGibbs energy of the solution is obtained by extending to all species\n\nEq. \u2011\n$G_{m} = \\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}_{\\ }^{0}G_{i_{a_{i}}j_{a_{j}}}} + RT\\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}\\ln y_{i_{a_{i}}j_{a_{j}}}} +_{\\ }^{E}G_{m}$\n\nwhere $y_{i_{a_{i}}j_{a_{j}}}$ is the mole fraction of species\n$i_{a_{i}}j_{a_{j}}$ in the solution with $a_{i} = 1$ and $a_{j} = 0$\nfor component $i$ and $a_{i} = 0$ and $a_{j} = 1$ for component $j$. The\nequilibrium amount of each associate $i_{a_{i}}j_{a_{j}}$ is obtained in\ncombination of mass balance and the zero driving force for the variation\nof the amount of the associate similar to , i.e.\n\nEq. \u2011 $\\sum_{i}^{}{\\sum_{j}^{}y_{i_{a_{i}}j_{a_{j}}}} = 1$\n\nEq. \u2011 $x_{i} = \\sum_{}^{}{a_{i}y_{i_{a_{i}}j_{a_{j}}}}$\n\nEq. \u2011\n$\\left( \\frac{{\\partial G}_{m}}{\\partial y_{i_{a_{i}}j_{a_{j}}}} \\right)_{x_{i}} = 0$\n\nAssociates are particularly plentiful in a gas phase, and their amounts\nare significantly affected by pressure. For an ideal gas phase with\n$_{\\ }^{E}G_{m} = 0$ and $PV_{m} = RT$, the effect of pressure is added\nas follows\n\nEq. \u2011\n$G_{m} = \\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}_{\\ }^{0}G_{i_{a_{i}}j_{a_{j}}}} + RT\\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}\\ln y_{i_{a_{i}}j_{a_{j}}}} + \\int_{P_{0}}^{P}{V_{m}dP} = \\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}_{\\ }^{0}G_{i_{a_{i}}j_{a_{j}}}} + RT\\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}\\ln y_{i_{a_{i}}j_{a_{j}}}} + RT\\ln\\frac{P}{P_{0}}$\n\nwhere $P$ is the total pressure, and $P_{0}$ the reference pressure at\nwhich $_{\\ }^{0}G_{i_{a_{i}}j_{a_{j}}}$ is defined, usually chosen to be\none atmospheric pressure. thus becomes\n\nEq. \u2011\n$G_{m} = \\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}_{\\ }^{0}{G_{i_{a_{i}}j_{a_{j}}}(P = 1atm)}} + RT\\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}\\ln y_{i_{a_{i}}j_{a_{j}}}} + RTlnP$\n\nwith the unit of the total pressure $P$ being in atmospheric pressure\n(atm). The chemical potential of species $i_{a_{i}}j_{a_{j}}$ is equal\nto\n\nEq. \u2011\n$\\mu_{i_{a_{i}}j_{a_{j}}} =_{\\ }^{0}{G_{i_{a_{i}}j_{a_{j}}}(P = 1atm)} + RTlny_{i_{a_{i}}j_{a_{j}}}P =_{\\ }^{0}{G_{i_{a_{i}}j_{a_{j}}}(P = 1atm)} + RTlnP_{i_{a_{i}}j_{a_{j}}}$\n\nwhere $P_{i_{a_{i}}j_{a_{j}}}$ is the partial pressure of species\n$i_{a_{i}}j_{a_{j}}$ defined as\n\nEq. \u2011 $P_{i_{a_{i}}j_{a_{j}}} = y_{i_{a_{i}}j_{a_{j}}}P$\n\nCombining and with , the relation between chemical potentials of an\nassociate and its constituents is expressed as\n\nEq. \u2011\n$\\mu_{i_{a_{i}}j_{a_{j}}} = a_{i}\\mu_{i} + {a_{j}\\mu}_{j} = a_{i}_{\\ }^{0}G_{i} + a_{j}_{\\ }^{0}G_{j} + RTln\\left( {P_{i}}^{a_{i}}{P_{j}}^{a_{j}} \\right)$\n\nThe equilibrium condition for the chemical reaction of an associate\nforming from its constituents in an ideal gas phase is obtained as\n\nEq. \u2011\n$\\mathrm{\\Delta}_{f}G_{i_{a_{i}}j_{a_{j}}} + RTln\\frac{{P_{i}}^{a_{i}}{P_{j}}^{a_{j}}}{P_{i_{a_{i}}j_{a_{j}}}} = 0$\n\nFor non-ideal phases, the mole fractions of various associates can be\ncalculated numerically through the minimization of Gibbs energy under\nthe constraints of and .\n", + "metadata": {} + }, + { + "id": "b63ee9dd", + "cell_type": "markdown", + "source": "### Solutions with long-range ordering\n\nSo far, solutions in which a component can occupy any site in a phase\nare discussed. In many phases, this is not the case. For example, in the\nfcc solid solution of Fe and C, Fe atoms take the fcc lattice sites, and\nC atoms occupy the interstitial sites between the fcc lattice sites.\nTherefore, Fe atoms do not mix with C atoms on the fcc lattice sites,\nand they rather develop long-rang ordering by occupying their own\ndistinct sites in the phase. Long-range ordering can also develop when\nshort-range ordering extends to the whole lattice. New formula for the\nGibbs energy of mixing is needed by considering the details how\ncomponents are distributed and mixed in various sites in a phase.\n\nOne way to group various sites in a phase is based on equivalent\ncrystallographic positions in a phase, i.e. Wyckoff positions. Various\nsets of equivalent positions divide the lattice into sub-sets of\nlattices. Each set of equivalent positions forms a sublattice. The\ndistributions of components on each sublattice can be represented by\nmole fractions of components in the sublattice, commonly referred to as\nsite fractions and defined as\n\nEq. \u2011 $y_{i}^{t} = \\frac{N_{i}^{t}}{\\sum_{j}^{}N_{j}^{t}}$\n\nEq. \u2011 $\\sum_{i}^{}y_{i}^{t} = 1$\n\nwhere the superscript $t$ denotes the sublattice in which the component\nresides, and the summation is for all species in sublattice $t$\nincluding vacancy. Site fractions and mole fractions are related through\nthe mass balance as follows\n\nEq. \u2011\n$x_{i} = \\frac{\\sum_{}^{}{a^{t}y_{i}^{t}}}{\\sum_{}^{}{a^{t}\\left( 1 - y_{va}^{t} \\right)}}$\n\nwhere $a^{t}$ and $y_{va}^{t}$ are the number of sites and the site\nfraction of vacancy in the sublattice *t*.\n\nRandom solutions form when each component enters all sublattices\nequally. Mole fractions and site fractions thus become identical.\nSolutions with both substitutional and interstitial component, like the\nfcc Fe-C solution mentioned above, can be represented by two\nsublattices. Stoichiometric compounds have its site fractions equal to\nunity in each sublattice. When site fractions in a compound deviate from\nunity, the compound is no longer stoichiometric and develops a\ncomposition range of homogeneity. When the composition range is small,\nthe deviations are often referred to as defects. Since many properties\nof a compound are determined by defects, a distinct field of defect\nchemistry exists, predominantly for charged species. As will be\ndemonstrated in Chapter and the rest of the book, defects can be treated\nas an integral part of the thermodynamics of a phase with more than one\nsublattices.\n\nLet us consider a case where there is only one component in each\nsublattice, which represents one possible stoichiometric composition of\nthe phase and is often called an end-member of the phase. The Gibbs\nenergy of an end-member is the same as that of a phase with a fixed\ncomposition shown by or or . By re-arranging , Gibbs energy of an\nend-member, $_{\\ }^{0}G_{em}$, is obtained as\n\nEq. \u2011\n$_{\\ }^{0}G_{em} = \\sum_{t}^{}{a^{t}{_{\\ }^{0}G}_{i}^{t,ref}} + \\mathrm{\\Delta}_{f}G_{em}$\n\nwhere ${_{\\ }^{0}G}_{i}^{t,ref}$ represents the Gibbs energy of\ncomponent $i$ in a given reference state which occupies sublattice $t$\nin the end-member. For vacancy, $_{\\ }^{0}G_{Va} = 0$ is defined. The\ncontribution of each end-member to the Gibbs energy of the phase is the\nproduct of site fraction of each component in their respective\nsublattices and the Gibbs energy of the end-member per mole of formula\nunit (*mf*), i.e.\n\nEq. \u2011\n$_{\\ }^{0}G_{mf} = \\sum_{em}^{}\\left( \\prod_{t}^{}y_{i}^{t}_{\\ }^{0}G_{em} \\right)$\n\nThe ideal mixing in each sublattice is similar to that in a random\nsolution with mole fractions substituted by site fractions. The excess\nGibbs energy of mixing consists of two contributions: (i) the mixing in\none sublattice with all other sublattices containing only one component\nin each sublattice and (ii) the mixing simultaneously in more than one\nsublattices. The Gibbs energy of a solution phase with multi-sublattices\ncan thus be written in terms of per mole of formula unit as\n\nEq. \u2011\n$G_{mf} =_{\\ }^{0}G_{mf} + RT\\sum_{t}^{}{a^{t}\\sum_{i}^{}{y_{i}^{t}\\ln y_{i}^{t}}} +_{\\ }^{E}G_{mf}$\n\nwith $_{\\ }^{E}G_{mf}$ given by\n\nEq. \u2011\n$_{\\ }^{E}G_{mf} = \\sum_{t}^{}{\\prod_{s \\neq t}^{}y_{l}^{s}\\sum_{i > j}^{}{\\sum_{j}^{}{y_{i}^{t}y_{j}^{t}L_{i,j:l}^{t}}}} + \\sum_{t}^{}{\\prod_{s \\neq t}^{}y_{l}^{s}\\sum_{i > j}^{}{\\sum_{j > k}^{}{\\sum_{k}^{}{y_{i}^{t}y_{j}^{t}y_{k}^{t}L_{i,j,k:l}^{t}}}}} + \\sum_{t}^{}{\\prod_{s \\neq t,u}^{}y_{l}^{s}\\sum_{i > j}^{}{\\sum_{j > k}^{}{\\sum_{k}^{}{y_{i}^{t}y_{j}^{t}y_{m}^{u}y_{n}^{u}L_{i,j:m,n:l}^{t}}}}}$\n\nThe first term in represents the binary interaction between component\n$i$ and $j$ in sublattice $t$ with sublattice $s$ occupied by component\n$l$ with comma separating interacting components in one sublattice and\ncolon separating sublattices. The product,\n$\\prod_{s \\neq t}^{}y_{l}^{s}$, runs through all other sublattices with\none component in each sublattice, except sublattice $t$ in which the\ninteraction is considered. The second term denotes the ternary\ninteraction among $i$, $j$, and $k$ in sublattice $t$ with sublattice\n$s$ occupied by component $l$. The third term depicts the interactions\nsimultaneously in both sublattice $t$ and $u$, and the product runs\nthrough all other sublattices with one component in each sublattice,\nexcept sublattice $t$ and $u$ in which the interactions are considered.\nThe third term thus partially reflects the short-range ordering among\ncomponents between two sublattices. In principles, high-order\ninteraction parameters such as quaternary, quinary, and multiple\nsublattice interaction parameters could be added, but their\ncontributions to $_{\\ }^{E}G_{mf}$ are small due to the physical\ninsignificance of co-location of four or five components indicated by\nthe product of their site fractions in front of the interaction\nparameters.\n\nIn , the chemical potential of a stoichiometric compound was defined in\nterms of a summation of chemical potential of individual components in\nthe compound because the relative amounts of components are constrained\nby the stoichiometry of the compound and chemical potentials of\nindividual components can not vary independently. By the same token, the\nchemical potential of an end-member in a solution can be written as\n\nEq. \u2011 $\\mu_{em} = G_{em} = \\sum_{t}^{}a^{t}\\mu_{i}^{t}$\n\nwhere $\\mu_{i}^{t}$ is the chemical potential of component $i$ that\noccupies the sublattice $t$ in the end-member, and can be derived using\n\nEq. \u2011\n$\\mu_{i}^{t} = a^{t}_{\\ }^{0}G_{i}^{t,ref} + a^{t}RTlny_{i}^{t} +_{\\ }^{E}G_{mf}^{\\ } + \\frac{\\partial_{\\ }^{E}G_{mf}^{\\ }}{\\partial y_{i}^{t}} - \\sum_{j}^{\\ }y_{j}^{t}\\frac{\\partial_{\\ }^{E}G_{mf}^{\\ }}{\\partial y_{j}^{t}}$\n\nFor constant interaction parameters in , for chemical potential reduces\nto the following expression from\n\nEq. \u2011\n$\\mu_{i}^{t} = a^{t}_{\\ }^{0}G_{i}^{t} + a^{t}RTlny_{i}^{t} + \\sum_{j \\neq i}^{}{\\left( y_{i}^{t} \\right)^{2}L}_{i,j}^{t} + \\sum_{k > j}^{}{\\sum_{j \\neq i}^{}{y_{j}^{t}y_{k}^{t}\\left\\lbrack L_{i,j}^{t} + L_{i,k}^{t} - L_{j,k}^{t} + \\left( 1 - 2y_{i}^{t} \\right)L_{i,j,k}^{t} \\right\\rbrack}}$\n\nThe stability of the solution is defined by\n$\\frac{\\partial\\mu_{em}}{\\partial N_{em}}$ with $N_{em}$ being the moles\nof the end-member in the solution given by\n$N_{em} = N\\prod_{u}^{}y_{j}^{u}$ and\n\nEq. \u2011\n$\\frac{\\partial N_{em}}{\\partial y_{j}^{u}} = \\frac{N\\prod_{t}^{}y_{i}^{t}}{y_{j}^{u}\\left( 1 - \\prod_{t}^{}y_{i}^{t} \\right)}$\n\nFollowing , one obtains\n\nEq. \u2011\n$\\frac{\\partial\\mu_{em}}{\\partial N_{em}} = \\sum_{u}^{}{\\frac{\\partial\\left( \\sum_{t}^{}a^{t}\\mu_{i}^{t} \\right)}{\\partial y_{j}^{u}}\\frac{\\partial y_{j}^{u}}{\\partial N_{em}}} = \\frac{\\left( 1 - \\prod_{t}^{}y_{j}^{t} \\right)}{N\\prod_{t}^{}y_{j}^{t}}\\sum_{u}^{}{a^{u}y_{i}^{u}\\frac{\\partial\\mu_{i}^{u}}{\\partial y_{i}^{u}}} = \\frac{\\left( 1 - \\prod_{t}^{}y_{j}^{t} \\right)}{N\\prod_{t}^{}y_{j}^{t}}\\sum_{u}^{}{a^{u}y_{i}^{u}\\left\\{ \\frac{RT\\left( 1 - y_{i}^{u} \\right)}{y_{i}^{u}} - 2\\sum_{j \\neq i}^{}\\left( y_{j}^{u} \\right)^{2}L_{i,j}^{u} - 2\\sum_{k > j}^{}{\\sum_{j \\neq i}^{}{y_{j}^{u}y_{k}^{u}\\left\\lbrack L_{i,j}^{u} + L_{i,k}^{u} - L_{j,k}^{u} + \\left( 2 - 3y_{i}^{u} \\right)L_{i,j,k}^{u} \\right\\rbrack}} \\right\\}}$\n\nIt is self-evident from that a site fraction is only uniquely defined\nfrom the mole fraction of the component when the component enters into\none sublattice only and does not form any associates. Therefore, in\ngeneral, the distribution of components on sublattices and different\nkinds of molecules can only be obtained by equilibrium calculations, and\nthe thermodynamic properties for such a phase can thus not be\nrepresented in a closed form using mole fractions of independent\ncomponents. This was demonstrated in Chapter when short-range ordering\nexists in solution phases, where the energy minimization procedure was\nused to obtain the distribution of components on different kinds of\nbonds and the amounts of individual associates.\n", + "metadata": {} + }, + { + "id": "578e73db", + "cell_type": "markdown", + "source": "### Solutions with both short-range and long-range ordering\n\nThe short-range ordering in a solution with long-rang ordering can take\nplace in each sublattice or between two sublattices. The short-range\nordering in one sublattice can be treated similarly as in Chapter 2.3.1\nwith mole fractions substituted by site fractions. In case that\nassociates form, the relation between mole fractions and site fractions\nbecomes more complicated as follows\n\nEq. \u2011\n$x_{i} = \\frac{\\sum_{}^{}{a^{t}\\sum_{k}^{}{i_{k}y}_{k}^{t}}}{\\sum_{}^{}{a^{t}\\left( 1 - y_{va}^{t} \\right)}}$\n\nwhere the summation for $k$ goes over all associates in sublattice $t$\ncontaining component $i$.\n\nThe short-range ordering between two-sublattices indicates that a\ncomponent in one sublattice has different interactions with different\ncomponents in another sublattice. This results in local ordering of one\ncomponent around another component in two neighboring sublattices. Such\nlocal ordering involves interactions between two sublattices shown as\nthe third term in .\n", + "metadata": {} + }, + { + "id": "180919dd", + "cell_type": "markdown", + "source": "### Solutions with charged species\n\nOne special type of solutions with both short-range and long-range\nordering is solutions with charged species, i.e. ionic solutions, plus\nelectrons and holes. There is an additional constraint on species\nconcentrations to maintain the charge neutrality of such solutions, i.e.\n\nEq. \u2011 $0 = \\sum_{t}^{}{\\sum_{i}^{}{a^{t}y_{i}^{t}v_{i}^{t}}}$\n\nwhere $v_{i}^{t}$ is the valance of species $i$ in sublattice $t$\nincluding its sign which is positive for cations, negative for anions,\nand zero for neutral species. The conventional defect chemistry theory\nis typically based on the ideal mass action laws and applicable to a\nsingle set of defects and at very low defect concentrations, i.e. in the\nlimit of ideal solutions. For interacting defects, their concentrations\nshould be replaced by their activities, which can be obtained from\nthermodynamic principles similarly as discussed in previous sections. It\nshould be emphasized that in addition to formation of many more charged\nspecies, one component may have different valences, particularly the\ntransition metals. Consequently, there can be many more species in an\nionic phase than the number of independent components in the system, and\ntheir concentrations can be found by equilibrium calculations as\ndiscussed in Chapter .\n", "metadata": {} } ] diff --git a/src/psu410/src/gibbs_energy_function/polymer_solutions_and_polymer_blends.ipynb b/src/psu410/src/gibbs_energy_function/polymer_solutions_and_polymer_blends.ipynb index 327d655..3ebed0f 100644 --- a/src/psu410/src/gibbs_energy_function/polymer_solutions_and_polymer_blends.ipynb +++ b/src/psu410/src/gibbs_energy_function/polymer_solutions_and_polymer_blends.ipynb @@ -4,709 +4,9 @@ "metadata": {}, "cells": [ { - "id": "184c02c4", + "id": "0aedfb02", "cell_type": "markdown", - "source": [ - "## Polymer solutions and polymer blends\n", - "\n", - "A polymer solution is a mixture between polymer molecules and solvents,\n", - "while a polymer blend is a mixture between different polymer molecules.\n", - "A polymer molecule consists of the same repeating units of one or more\n", - "monomers, which can be an atom or a small molecule. The number of\n", - "repeating units is called the degree of polymerization and can be as\n", - "large as 104\u2013105. It defines the molecular mass,\n", - "i.e. the mass of one polymer molecule. There are three typical\n", - "architectures of polymerization: a linear chain, a branched chain, and a\n", - "cross-linked polymer. Nearly all polymers are mixtures of molecules with\n", - "a different degree of polymerization with a molecular mass distribution,\n", - "complicating the modeling of their thermodynamic properties because of\n", - "the dependence of properties on molecular mass.\n", - "\n", - "Gibbs energy functions of polymers with a single molecular mass can be\n", - "treated similarly as in Chapter . For a polymer solution, the ideal\n", - "entropy of mixing is quite different from that of atomically random\n", - "solutions discussed in Chapter because the monomers in a polymer\n", - "molecule are connected to each other and cannot move freely. One common\n", - "approach to calculate the ideal entropy of mixing is to evoke a lattice\n", - "model and assume that one monomer occupies a lattice site with a fixed\n", - "volume. The number of translational states of a single molecule is equal\n", - "to the number of lattice sites available. In a homogeneous solution, the\n", - "total number of lattice sites available is\n", - "\n", - "Eq. \u2011 $n = \\sum_{i}^{}{m_{i}n_{i}}$\n", - "\n", - "where $n_{i}$ and $m_{i}$ are the number of molecule $i$ and the number\n", - "of lattice sites per molecule $i$, respectively. While in its pure\n", - "state, i.e. before mixing, the number of states of molecule $i$ in terms\n", - "of the number of lattice sites is\n", - "\n", - "Eq. \u2011 $w_{i} = m_{i}n_{i} = n\\varphi_{i}$\n", - "\n", - "where $\\varphi_{i}$ is the volume fraction of molecule $i$ in the\n", - "solution. The entropy change per molecule $i$ is thus\n", - "\n", - "Eq. \u2011\n", - "$S_{i} = kln(n) - klnw_{i} = kln\\frac{1}{\\varphi_{i}} = - kln\\varphi_{i}$\n", - "\n", - "The total entropy of mixing is the summation for all molecules,\n", - "normalized to per mole of lattice site\n", - "\n", - "Eq. \u2011\n", - "$_{}^{I}S_{m} = \\frac{N_{a}}{n}\\sum_{i}^{}{n_{i}S_{i}} = - R\\sum_{i}^{}{\\frac{\\varphi_{i}}{m_{i}}\\ln\\varphi_{i}}$\n", - "\n", - "where $N_{a}$ is the Avogadro number. When $m_{i} = 1$ for all\n", - "molecules, Eq. 2\u2011143 reduces to . Since $m_{i}$ values are typically\n", - "very large numbers for polymers, the entropy of mixing in polymer\n", - "solutions and blends is thus significantly lower than those in\n", - "non-polymer solutions as shown schematically in for binary systems with\n", - "various $m_{i}$ values.\n", - "\n", - "Figure \u2011: Schematic entropy of mixing in solutions with the numbers of\n", - "lattice sites per molecule shown.\n", - "\n", - "Similar to , Gibbs energy of a multi-component random polymer\n", - "solution/blend can be written as\n", - "\n", - "Eq. \u2011\n", - "$G_{m} = \\sum_{}^{}\\frac{\\varphi_{i}}{m_{i}}_{\\ }^{0}G_{im} + RT\\left( \\sum_{}^{}{\\frac{\\varphi_{i}}{m_{i}}\\ln\\varphi_{i}} + \\sum_{}^{}{\\varphi_{i}\\varphi_{j}\\chi_{ij}} \\right)$\n", - "\n", - "where $_{\\ }^{0}G_{im}$ is the Gibbs energy of molecular $i$ per mole of\n", - "lattice site, and $\\chi_{ij}$ the unitless interaction parameter between\n", - "molecule $i$ and $j$. Other equations shown in Chapter can be derived\n", - "similarly too. It is to be noted that is very similar to the\n", - "Flory\u2013Huggins solution equation widely used in the polymer community.\n" - ], - "metadata": {} - }, - { - "id": "8736a1bd", - "cell_type": "markdown", - "source": [ - "### Random solutions\n", - "\n", - "The ideal Gibbs energy of mixing represents an ideal solution in which\n", - "all sites are equivalent and the distributions of components on the\n", - "sites are completely random. The number of different configurations to\n", - "arrange all components is\n", - "\n", - "Eq. \u2011 $w = \\frac{N!}{\\prod_{}^{}\\left( N_{i}! \\right)}$\n", - "\n", - "Based on Boltzmann\u2019s relation from statistic thermodynamics when all\n", - "configurations have the same probability to be observed, the ideal\n", - "configurational molar entropy of mixing due to the distribution is\n", - "\n", - "Eq. \u2011\n", - "$_{\\ }^{ideal}S_{m} =_{\\ }^{I}S_{m} = \\frac{Rlnw}{N} = R\\frac{lnN! - \\sum_{}^{}\\ln\\left( N_{i}! \\right)}{N} \\cong R\\frac{NlnN - \\sum_{}^{}{N_{i}l{nN}_{i}}}{N} = - R\\sum_{}^{}{x_{i}l{nx}_{i}}$\n", - "\n", - "where $R$ is the gas constant. represents the entropy difference between\n", - "the ideal solution and the states when individual components are by\n", - "themselves, i.e. the mechanical mixing of components. As $x_{i}$ is\n", - "smaller than unity, the entropy production to form an ideal solution\n", - "from pure components is thus positive, indicating that it is a\n", - "spontaneous process. In such an ideal solution, it is assumed that there\n", - "are no interactions between components, and the enthalpy of mixing is\n", - "thus zero as the internal energy and the volume of the system do not\n", - "change. The ideal Gibbs energy of mixing is written as\n", - "\n", - "Eq. \u2011 $_{\\ }^{I}G = - T_{\\ }^{I}S_{m} = RT\\sum_{}^{}{x_{i}l{nx}_{i}}$\n", - "\n", - "The Gibbs energy of real solutions, i.e. , becomes\n", - "\n", - "Eq. \u2011\n", - "$G_{m} = \\sum_{}^{}{x_{i}_{\\ }^{0}G_{i}} + RT\\sum_{}^{}{x_{i}l{nx}_{i}} +_{\\ }^{E}G_{m}$\n", - "\n", - "From , the chemical potential is obtained as\n", - "\n", - "Eq. \u2011\n", - "$\\mu_{i} = G_{i} =_{\\ }^{0}G_{i} + RTlnx_{i} +_{\\ }^{E}G_{m} + \\frac{\\partial_{\\ }^{E}G_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial_{\\ }^{E}G_{m}}{\\partial x_{j}}$\n", - "\n", - "From the chemical activity of , the activity coefficient, $\\gamma_{i}$,\n", - "can be defined as follows\n", - "\n", - "Eq. \u2011\n", - "$\\gamma_{i} = \\frac{a_{i}}{x_{i}} = \\frac{1}{x_{i}}\\exp\\frac{G_{i} -_{\\ }^{0}G_{i}\\ }{RT}$\n", - "\n", - "The solution is an ideal solution with $\\gamma_{i} = 1$, and is said to\n", - "be positively or negatively deviating from an ideal solution with\n", - "$\\gamma_{i} > 1$ or $\\gamma_{i} < 1$, respectively. The chemical\n", - "potential is related to the activity and activity coefficient by the\n", - "following equation\n", - "\n", - "Eq. \u2011\n", - "$\\mu_{i} =_{\\ }^{0}G_{i} + RTlna_{i} =_{\\ }^{0}G_{i} + RTln\\gamma_{i}x_{i} =_{\\ }^{0}G_{i} + RTlnx_{i} + RTln\\gamma_{i}$\n", - "\n", - "Let us further exam in more details in order to better understand the\n", - "relation between $G_{m}$ and $\\mu_{i}$. The partial derivatives in\n", - "represent the directions of molar Gibbs energy in the composition space,\n", - "i.e. the tangents of molar Gibbs energy with respect to mole fractions\n", - "of independent components. Collectively, they define the\n", - "multi-dimensional tangent plane of molar Gibbs energy at the given\n", - "composition, $x_{i}^{0}$. The mathematical representation of this\n", - "tangent plane, $z_{G_{m}}$, is defined by its directional derivatives\n", - "and the distance from the point where the derivatives are taken,\n", - "\n", - "Eq. \u2011\n", - "$z_{G_{m}} = G_{m}\\left( x_{i}^{0} \\right) + \\sum_{i = 1}^{c}{\\left( \\frac{\\partial G_{m}}{\\partial x_{i}} \\right)_{x_{i}^{0}}\\left( x_{i} - x_{i}^{0} \\right)}$\n", - "\n", - "The intercept of this tangent plane at each pure component axis, i.e.\n", - "$x_{i} = 1$ and $x_{j \\neq i} = 0$, is obtained as\n", - "\n", - "Eq. \u2011\n", - "$z_{G_{m},x_{i} = 1} = G_{m}\\left( x_{i}^{0} \\right) + \\left( \\frac{\\partial G_{m}}{\\partial x_{i}} \\right)_{x_{i}^{\\ } = x_{i}^{0}} - \\sum_{j = 1}^{c}{x_{j}^{0}\\left( \\frac{\\partial G_{m}}{\\partial x_{j}} \\right)_{x_{i}^{\\ } = x_{i}^{0}}}$\n", - "\n", - "This is identical to at the point $x_{i}^{0}$. It is thus shown that the\n", - "chemical potential of a component in a solution is represented by the\n", - "intercept of the tangent plane of Gibbs energy of the solution on the\n", - "$G_{m}$ axis of the component. The distance between the intercept and\n", - "the Gibbs energy of the pure component in the same structure of the\n", - "solution is related to the chemical activity of the component as defined\n", - "by . On the other hand, it is evident that one can choose any other\n", - "structure of the pure element to define the chemical activity to compare\n", - "chemical potential of the component as shown by .\n", - "\n", - "The stability of a solution is evaluated following , and the derivatives\n", - "of chemical potential with respect to its moles, i.e. the elements in\n", - "the determinant, are obtained as follows from and ,\n", - "\n", - "Eq. \u2011\n", - "$\\frac{N}{RT}\\frac{\\partial\\mu_{i}}{\\partial N_{i}} = \\frac{N}{RT}G_{ii} = \\frac{1 - x_{i}}{x_{i}} + \\frac{1}{\\gamma_{i}}\\left( \\frac{\\partial\\gamma_{i}}{\\partial x_{i}} - \\sum_{j = 1}^{c}{x_{j}\\frac{\\partial\\gamma_{i}}{\\partial x_{j}}} \\right)$\n", - "\n", - "Eq. \u2011\n", - "$\\frac{N}{RT}\\frac{\\partial\\mu_{i}}{\\partial N_{k}} = \\frac{N}{RT}G_{ik} = - \\frac{x_{k}}{x_{i}} + \\frac{1}{\\gamma_{i}}\\left( 1 - \\sum_{j = 1}^{c}{x_{j}\\frac{\\partial\\gamma_{i}}{\\partial x_{j}}} \\right)$\n", - "\n", - "To further study Gibbs energy of solution phases, let us discuss the\n", - "details on the excess Gibbs energy of mixing. At this point, one can\n", - "start with lower-order systems with fewer components, i.e. two component\n", - "and three-component systems, noting that the Gibbs energy of phases with\n", - "one component is already presented in Chapter .\n" - ], - "metadata": {} - }, - { - "id": "084be49b", - "cell_type": "markdown", - "source": [ - "### Solutions with both short-range and long-range ordering\n", - "\n", - "The short-range ordering in a solution with long-rang ordering can take\n", - "place in each sublattice or between two sublattices. The short-range\n", - "ordering in one sublattice can be treated similarly as in Chapter 2.3.1\n", - "with mole fractions substituted by site fractions. In case that\n", - "associates form, the relation between mole fractions and site fractions\n", - "becomes more complicated as follows\n", - "\n", - "Eq. \u2011\n", - "$x_{i} = \\frac{\\sum_{}^{}{a^{t}\\sum_{k}^{}{i_{k}y}_{k}^{t}}}{\\sum_{}^{}{a^{t}\\left( 1 - y_{va}^{t} \\right)}}$\n", - "\n", - "where the summation for $k$ goes over all associates in sublattice $t$\n", - "containing component $i$.\n", - "\n", - "The short-range ordering between two-sublattices indicates that a\n", - "component in one sublattice has different interactions with different\n", - "components in another sublattice. This results in local ordering of one\n", - "component around another component in two neighboring sublattices. Such\n", - "local ordering involves interactions between two sublattices shown as\n", - "the third term in .\n" - ], - "metadata": {} - }, - { - "id": "8c40d224", - "cell_type": "markdown", - "source": [ - "### Solutions with charged species\n", - "\n", - "One special type of solutions with both short-range and long-range\n", - "ordering is solutions with charged species, i.e. ionic solutions, plus\n", - "electrons and holes. There is an additional constraint on species\n", - "concentrations to maintain the charge neutrality of such solutions, i.e.\n", - "\n", - "Eq. \u2011 $0 = \\sum_{t}^{}{\\sum_{i}^{}{a^{t}y_{i}^{t}v_{i}^{t}}}$\n", - "\n", - "where $v_{i}^{t}$ is the valance of species $i$ in sublattice $t$\n", - "including its sign which is positive for cations, negative for anions,\n", - "and zero for neutral species. The conventional defect chemistry theory\n", - "is typically based on the ideal mass action laws and applicable to a\n", - "single set of defects and at very low defect concentrations, i.e. in the\n", - "limit of ideal solutions. For interacting defects, their concentrations\n", - "should be replaced by their activities, which can be obtained from\n", - "thermodynamic principles similarly as discussed in previous sections. It\n", - "should be emphasized that in addition to formation of many more charged\n", - "species, one component may have different valences, particularly the\n", - "transition metals. Consequently, there can be many more species in an\n", - "ionic phase than the number of independent components in the system, and\n", - "their concentrations can be found by equilibrium calculations as\n", - "discussed in Chapter .\n" - ], - "metadata": {} - }, - { - "id": "be4c0142", - "cell_type": "markdown", - "source": [ - "### Solutions with long-range ordering\n", - "\n", - "So far, solutions in which a component can occupy any site in a phase\n", - "are discussed. In many phases, this is not the case. For example, in the\n", - "fcc solid solution of Fe and C, Fe atoms take the fcc lattice sites, and\n", - "C atoms occupy the interstitial sites between the fcc lattice sites.\n", - "Therefore, Fe atoms do not mix with C atoms on the fcc lattice sites,\n", - "and they rather develop long-rang ordering by occupying their own\n", - "distinct sites in the phase. Long-range ordering can also develop when\n", - "short-range ordering extends to the whole lattice. New formula for the\n", - "Gibbs energy of mixing is needed by considering the details how\n", - "components are distributed and mixed in various sites in a phase.\n", - "\n", - "One way to group various sites in a phase is based on equivalent\n", - "crystallographic positions in a phase, i.e. Wyckoff positions. Various\n", - "sets of equivalent positions divide the lattice into sub-sets of\n", - "lattices. Each set of equivalent positions forms a sublattice. The\n", - "distributions of components on each sublattice can be represented by\n", - "mole fractions of components in the sublattice, commonly referred to as\n", - "site fractions and defined as\n", - "\n", - "Eq. \u2011 $y_{i}^{t} = \\frac{N_{i}^{t}}{\\sum_{j}^{}N_{j}^{t}}$\n", - "\n", - "Eq. \u2011 $\\sum_{i}^{}y_{i}^{t} = 1$\n", - "\n", - "where the superscript $t$ denotes the sublattice in which the component\n", - "resides, and the summation is for all species in sublattice $t$\n", - "including vacancy. Site fractions and mole fractions are related through\n", - "the mass balance as follows\n", - "\n", - "Eq. \u2011\n", - "$x_{i} = \\frac{\\sum_{}^{}{a^{t}y_{i}^{t}}}{\\sum_{}^{}{a^{t}\\left( 1 - y_{va}^{t} \\right)}}$\n", - "\n", - "where $a^{t}$ and $y_{va}^{t}$ are the number of sites and the site\n", - "fraction of vacancy in the sublattice *t*.\n", - "\n", - "Random solutions form when each component enters all sublattices\n", - "equally. Mole fractions and site fractions thus become identical.\n", - "Solutions with both substitutional and interstitial component, like the\n", - "fcc Fe-C solution mentioned above, can be represented by two\n", - "sublattices. Stoichiometric compounds have its site fractions equal to\n", - "unity in each sublattice. When site fractions in a compound deviate from\n", - "unity, the compound is no longer stoichiometric and develops a\n", - "composition range of homogeneity. When the composition range is small,\n", - "the deviations are often referred to as defects. Since many properties\n", - "of a compound are determined by defects, a distinct field of defect\n", - "chemistry exists, predominantly for charged species. As will be\n", - "demonstrated in Chapter and the rest of the book, defects can be treated\n", - "as an integral part of the thermodynamics of a phase with more than one\n", - "sublattices.\n", - "\n", - "Let us consider a case where there is only one component in each\n", - "sublattice, which represents one possible stoichiometric composition of\n", - "the phase and is often called an end-member of the phase. The Gibbs\n", - "energy of an end-member is the same as that of a phase with a fixed\n", - "composition shown by or or . By re-arranging , Gibbs energy of an\n", - "end-member, $_{\\ }^{0}G_{em}$, is obtained as\n", - "\n", - "Eq. \u2011\n", - "$_{\\ }^{0}G_{em} = \\sum_{t}^{}{a^{t}{_{\\ }^{0}G}_{i}^{t,ref}} + \\mathrm{\\Delta}_{f}G_{em}$\n", - "\n", - "where ${_{\\ }^{0}G}_{i}^{t,ref}$ represents the Gibbs energy of\n", - "component $i$ in a given reference state which occupies sublattice $t$\n", - "in the end-member. For vacancy, $_{\\ }^{0}G_{Va} = 0$ is defined. The\n", - "contribution of each end-member to the Gibbs energy of the phase is the\n", - "product of site fraction of each component in their respective\n", - "sublattices and the Gibbs energy of the end-member per mole of formula\n", - "unit (*mf*), i.e.\n", - "\n", - "Eq. \u2011\n", - "$_{\\ }^{0}G_{mf} = \\sum_{em}^{}\\left( \\prod_{t}^{}y_{i}^{t}_{\\ }^{0}G_{em} \\right)$\n", - "\n", - "The ideal mixing in each sublattice is similar to that in a random\n", - "solution with mole fractions substituted by site fractions. The excess\n", - "Gibbs energy of mixing consists of two contributions: (i) the mixing in\n", - "one sublattice with all other sublattices containing only one component\n", - "in each sublattice and (ii) the mixing simultaneously in more than one\n", - "sublattices. The Gibbs energy of a solution phase with multi-sublattices\n", - "can thus be written in terms of per mole of formula unit as\n", - "\n", - "Eq. \u2011\n", - "$G_{mf} =_{\\ }^{0}G_{mf} + RT\\sum_{t}^{}{a^{t}\\sum_{i}^{}{y_{i}^{t}\\ln y_{i}^{t}}} +_{\\ }^{E}G_{mf}$\n", - "\n", - "with $_{\\ }^{E}G_{mf}$ given by\n", - "\n", - "Eq. \u2011\n", - "$_{\\ }^{E}G_{mf} = \\sum_{t}^{}{\\prod_{s \\neq t}^{}y_{l}^{s}\\sum_{i > j}^{}{\\sum_{j}^{}{y_{i}^{t}y_{j}^{t}L_{i,j:l}^{t}}}} + \\sum_{t}^{}{\\prod_{s \\neq t}^{}y_{l}^{s}\\sum_{i > j}^{}{\\sum_{j > k}^{}{\\sum_{k}^{}{y_{i}^{t}y_{j}^{t}y_{k}^{t}L_{i,j,k:l}^{t}}}}} + \\sum_{t}^{}{\\prod_{s \\neq t,u}^{}y_{l}^{s}\\sum_{i > j}^{}{\\sum_{j > k}^{}{\\sum_{k}^{}{y_{i}^{t}y_{j}^{t}y_{m}^{u}y_{n}^{u}L_{i,j:m,n:l}^{t}}}}}$\n", - "\n", - "The first term in represents the binary interaction between component\n", - "$i$ and $j$ in sublattice $t$ with sublattice $s$ occupied by component\n", - "$l$ with comma separating interacting components in one sublattice and\n", - "colon separating sublattices. The product,\n", - "$\\prod_{s \\neq t}^{}y_{l}^{s}$, runs through all other sublattices with\n", - "one component in each sublattice, except sublattice $t$ in which the\n", - "interaction is considered. The second term denotes the ternary\n", - "interaction among $i$, $j$, and $k$ in sublattice $t$ with sublattice\n", - "$s$ occupied by component $l$. The third term depicts the interactions\n", - "simultaneously in both sublattice $t$ and $u$, and the product runs\n", - "through all other sublattices with one component in each sublattice,\n", - "except sublattice $t$ and $u$ in which the interactions are considered.\n", - "The third term thus partially reflects the short-range ordering among\n", - "components between two sublattices. In principles, high-order\n", - "interaction parameters such as quaternary, quinary, and multiple\n", - "sublattice interaction parameters could be added, but their\n", - "contributions to $_{\\ }^{E}G_{mf}$ are small due to the physical\n", - "insignificance of co-location of four or five components indicated by\n", - "the product of their site fractions in front of the interaction\n", - "parameters.\n", - "\n", - "In , the chemical potential of a stoichiometric compound was defined in\n", - "terms of a summation of chemical potential of individual components in\n", - "the compound because the relative amounts of components are constrained\n", - "by the stoichiometry of the compound and chemical potentials of\n", - "individual components can not vary independently. By the same token, the\n", - "chemical potential of an end-member in a solution can be written as\n", - "\n", - "Eq. \u2011 $\\mu_{em} = G_{em} = \\sum_{t}^{}a^{t}\\mu_{i}^{t}$\n", - "\n", - "where $\\mu_{i}^{t}$ is the chemical potential of component $i$ that\n", - "occupies the sublattice $t$ in the end-member, and can be derived using\n", - "\n", - "Eq. \u2011\n", - "$\\mu_{i}^{t} = a^{t}_{\\ }^{0}G_{i}^{t,ref} + a^{t}RTlny_{i}^{t} +_{\\ }^{E}G_{mf}^{\\ } + \\frac{\\partial_{\\ }^{E}G_{mf}^{\\ }}{\\partial y_{i}^{t}} - \\sum_{j}^{\\ }y_{j}^{t}\\frac{\\partial_{\\ }^{E}G_{mf}^{\\ }}{\\partial y_{j}^{t}}$\n", - "\n", - "For constant interaction parameters in , for chemical potential reduces\n", - "to the following expression from\n", - "\n", - "Eq. \u2011\n", - "$\\mu_{i}^{t} = a^{t}_{\\ }^{0}G_{i}^{t} + a^{t}RTlny_{i}^{t} + \\sum_{j \\neq i}^{}{\\left( y_{i}^{t} \\right)^{2}L}_{i,j}^{t} + \\sum_{k > j}^{}{\\sum_{j \\neq i}^{}{y_{j}^{t}y_{k}^{t}\\left\\lbrack L_{i,j}^{t} + L_{i,k}^{t} - L_{j,k}^{t} + \\left( 1 - 2y_{i}^{t} \\right)L_{i,j,k}^{t} \\right\\rbrack}}$\n", - "\n", - "The stability of the solution is defined by\n", - "$\\frac{\\partial\\mu_{em}}{\\partial N_{em}}$ with $N_{em}$ being the moles\n", - "of the end-member in the solution given by\n", - "$N_{em} = N\\prod_{u}^{}y_{j}^{u}$ and\n", - "\n", - "Eq. \u2011\n", - "$\\frac{\\partial N_{em}}{\\partial y_{j}^{u}} = \\frac{N\\prod_{t}^{}y_{i}^{t}}{y_{j}^{u}\\left( 1 - \\prod_{t}^{}y_{i}^{t} \\right)}$\n", - "\n", - "Following , one obtains\n", - "\n", - "Eq. \u2011\n", - "$\\frac{\\partial\\mu_{em}}{\\partial N_{em}} = \\sum_{u}^{}{\\frac{\\partial\\left( \\sum_{t}^{}a^{t}\\mu_{i}^{t} \\right)}{\\partial y_{j}^{u}}\\frac{\\partial y_{j}^{u}}{\\partial N_{em}}} = \\frac{\\left( 1 - \\prod_{t}^{}y_{j}^{t} \\right)}{N\\prod_{t}^{}y_{j}^{t}}\\sum_{u}^{}{a^{u}y_{i}^{u}\\frac{\\partial\\mu_{i}^{u}}{\\partial y_{i}^{u}}} = \\frac{\\left( 1 - \\prod_{t}^{}y_{j}^{t} \\right)}{N\\prod_{t}^{}y_{j}^{t}}\\sum_{u}^{}{a^{u}y_{i}^{u}\\left\\{ \\frac{RT\\left( 1 - y_{i}^{u} \\right)}{y_{i}^{u}} - 2\\sum_{j \\neq i}^{}\\left( y_{j}^{u} \\right)^{2}L_{i,j}^{u} - 2\\sum_{k > j}^{}{\\sum_{j \\neq i}^{}{y_{j}^{u}y_{k}^{u}\\left\\lbrack L_{i,j}^{u} + L_{i,k}^{u} - L_{j,k}^{u} + \\left( 2 - 3y_{i}^{u} \\right)L_{i,j,k}^{u} \\right\\rbrack}} \\right\\}}$\n", - "\n", - "It is self-evident from that a site fraction is only uniquely defined\n", - "from the mole fraction of the component when the component enters into\n", - "one sublattice only and does not form any associates. Therefore, in\n", - "general, the distribution of components on sublattices and different\n", - "kinds of molecules can only be obtained by equilibrium calculations, and\n", - "the thermodynamic properties for such a phase can thus not be\n", - "represented in a closed form using mole fractions of independent\n", - "components. This was demonstrated in Chapter when short-range ordering\n", - "exists in solution phases, where the energy minimization procedure was\n", - "used to obtain the distribution of components on different kinds of\n", - "bonds and the amounts of individual associates.\n" - ], - "metadata": {} - }, - { - "id": "2b814171", - "cell_type": "markdown", - "source": [ - "### Solutions with short-range ordering\n", - "\n", - "The order in a system can be measured by correlation functions which\n", - "describes how various components are correlated in space. For\n", - "simplicity, let us consider only the pairs between nearest neighbors\n", - "with the correlation function represented by the pair probability of\n", - "nearest neighbor bonds between two components. In a random solution, the\n", - "probability to find nearest neighbor bonds between two components $i$\n", - "and $j$ is\n", - "\n", - "Eq. \u2011 $p_{ij} = x_{i}x_{j}$\n", - "\n", - "When $p_{ij} \\neq x_{i}x_{j}$, the nearest neighbors of component $i$\n", - "are not occupied randomly by component $j$, rather certain components\n", - "are favored, resulting in short-range ordering or local clustering in\n", - "the solution. When short-range ordering develops throughout the\n", - "solution, long-range ordering takes place, and each component has its\n", - "own primary sites in the solution, to be discussed in Chapter . There\n", - "are relations between bond probabilities and mole fractions of\n", - "components due to the mass balance as follows, with the assumption of\n", - "$p_{ij} = p_{ji}$\n", - "\n", - "Eq. \u2011 $\\sum_{i}^{}{\\sum_{j}^{}p_{ij}} = 1$\n", - "\n", - "Eq. \u2011 $x_{i} = \\sum_{j}^{}p_{ij}$\n", - "\n", - "For small deviations from a random solution, one can consider the\n", - "formation of $i - j$ bonds from $i - i\\ $ and $j - j$ bonds and the\n", - "ideal mixing of three types of bonds, similar to a typical Ising model.\n", - "The bond reaction can be written as\n", - "\n", - "Eq. \u2011 $(i - i\\ )bonds + (j - j)\\ bonds = 2\\ (i - j)\\ bonds$\n", - "\n", - "with the Gibbs energy of reaction being\n", - "\n", - "Eq. \u2011\n", - "${\\mathrm{\\Delta}G}_{ij} = 2G_{ij} - \\left( G_{ii} + G_{jj} \\right)\\ $\n", - "\n", - "Gibbs energy of the solution per mole of atom is thus represented by the\n", - "bond energies and the ideal mixing of bonds plus non-ideal interactions\n", - "between pairs,\n", - "\n", - "Eq. \u2011\n", - "$G_{m} = \\sum_{i}^{}{\\sum_{j}^{}{p_{ij}G_{ij}}} + \\frac{Z}{2}RT\\sum_{i}^{}{\\sum_{j}^{}{p_{ij}\\ln p_{ij}}} +_{\\ }^{E}G_{m}$\n", - "\n", - "with $G_{ij}$ being the molar bond energy between components $i$ and\n", - "$j$, $Z$ the number of bonds per atom which is divided by two in the\n", - "equation due to two atoms needed to form one bond, and\n", - "$_{\\ }^{E}G_{m} = \\sum_{}^{}{p_{ij}p_{kl}I_{ijkl}}$ the excess Gibbs\n", - "energy of mixing between bonds. This approach proposed by Guggenheim\n", - "\\[2\\] is called quasi-chemical method as it is based on the chemical\n", - "reaction shown by .\n", - "\n", - "However, the entropy of mixing in does not reduce to ideal entropy of\n", - "mixing for a solution without short-range ordering as defined by . An\n", - "approximated correction may be added for small degree of short-range\n", - "ordering as follows\n", - "\n", - "Eq. \u2011\n", - "$G_{m} = \\sum_{i}^{}{\\sum_{j}^{}{p_{ij}G_{ij}}} + \\frac{Z}{2}RT\\sum_{i}^{}{\\sum_{j}^{}{p_{ij}\\ln\\frac{p_{ij}}{x_{i}x_{j}}}} + RT\\sum_{}^{}{x_{i}l{nx}_{i}} +_{\\ }^{E}G_{m}$\n", - "\n", - "For a random solution defined by , becomes\n", - "\n", - "Eq. \u2011\n", - "$G_{m} = \\sum_{}^{}{x_{i}_{\\ }^{0}G_{i}} + RT\\sum_{}^{}{x_{i}l{nx}_{i}} + \\sum_{}^{}{x_{i}x_{j}{\\mathrm{\\Delta}G}_{ij}} +_{\\ }^{E}G_{m}$\n", - "\n", - "with $_{\\ }^{0}G_{i} = G_{ii}$, ${\\mathrm{\\Delta}G}_{ij}$ from Eq. 2\u2011107\n", - "representing the interaction parameter between components $i$ and $j$,\n", - "and $_{\\ }^{E}G_{m} = \\sum_{}^{}{x_{i}{x_{j}x_{k}x}_{l}I_{ijkl}}$\n", - "denoting the higher order interactions, in comparison with .\n", - "\n", - "When short-range ordering exists in a solution, one typically uses the\n", - "law of mass reaction for the chemical reaction represented by to define\n", - "the equilibrium among all bonds, i.e.\n", - "\n", - "Eq. \u2011\n", - "$\\frac{\\left( p_{ij} \\right)^{2}}{p_{ii}p_{jj}} = e^{- \\frac{{\\mathrm{\\Delta}G}_{ij}}{kT}}$\n", - "\n", - "However, this is under the assumption that the chemical activities of\n", - "all bonds can be represented by their respective probabilities, which is\n", - "only true for an ideal solution even excluding dilute solutions due to\n", - "the Henry\u2019s law shown by . Preferrably, the bond probabilities can be\n", - "obtained by calculating the driving force for the fluctuation of bond\n", - "probabilities under constant temperature, pressure, and amount of each\n", - "component along with the constraints defined by and and equating the\n", - "driving force to zero, i.e.\n", - "\n", - "Eq. \u2011\n", - "${\\frac{1}{N}\\left( \\frac{\\partial G}{\\partial\\xi} \\right)}_{T,P,N_{n}} = \\left( \\frac{\\partial G_{m}}{\\partial p_{ij}} \\right)_{T,P,N_{n}} = \\left( \\frac{\\partial G_{m}}{\\partial p_{ij}} \\right)_{T,P,x_{n},p_{kl \\neq ij}} - \\sum_{kl \\neq ij}^{}\\left( \\frac{\\partial G_{m}}{\\partial p_{kl}} \\right)_{T,P,x_{n},p_{op \\neq kl}} + \\left( \\frac{\\partial G_{m}}{\\partial x_{i}} \\right)_{T,P,x_{q \\neq i},p_{kl}} + \\left( \\frac{\\partial G_{m}}{\\partial x_{j}} \\right)_{T,P,x_{q \\neq j},p_{kl}} = 0$\n", - "\n", - "where $\\frac{\\partial p_{kl}}{\\partial p_{ij}} = - 1$ and\n", - "$\\frac{\\partial x_{i}}{\\partial p_{ij}} = \\frac{\\partial x_{j}}{\\partial p_{ij}} = 1$\n", - "are used from and . Numerical values of $p_{ij}$ can be obtained by\n", - "minimization of Gibbs energy under constraints given by and .\n", - "\n", - "The chemical potential of independent component $i$ is defined as in and\n", - "can be represented by the following equation\n", - "\n", - "Eq. \u2011\n", - "$\\mu_{i} = G_{m} + \\frac{\\partial G_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial G_{m}}{\\partial x_{j}} + 2\\sum_{j = 1}^{c}\\frac{\\partial G_{m}}{\\partial p_{ij}} - \\frac{\\partial G_{m}}{\\partial p_{ii}} - 2\\sum_{j = 1}^{c}x_{j}\\sum_{k = 1}^{c}\\frac{\\partial G_{m}}{\\partial p_{jk}} + \\sum_{j = 1}^{c}x_{j}\\frac{\\partial G_{m}}{\\partial p_{jj}}$\n", - "\n", - "The stability of the solution can be derived similar to .\n", - "\n", - "When the bonding between components becomes very strong, distinctive new\n", - "components may form. They are not independent components and often\n", - "called associates. Both the independent and dependent components are\n", - "collectively called species. The formation of an associate\n", - "$i_{a_{i}}j_{b_{j}}$ consisting of $a_{i}$ mole of $i$ and $a_{j}$ mole\n", - "$j$ can be written as\n", - "\n", - "Eq. \u2011 $a_{i}i + a_{j}j = i_{a_{i}}j_{a_{j}}$\n", - "\n", - "Gibbs energy of the associate follows the same format of that of a\n", - "stoichiometric phase as shown by ,\n", - "\n", - "Eq. \u2011\n", - "${_{\\ }^{0}G}_{i_{a_{i}}j_{a_{j}}} = \\sum_{}^{}a_{i}{_{\\ }^{0}G}_{i}^{SER} + \\mathrm{\\Delta}_{f}G_{i_{a_{i}}j_{a_{j}}}$\n", - "\n", - "Gibbs energy of the solution is obtained by extending to all species\n", - "\n", - "Eq. \u2011\n", - "$G_{m} = \\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}_{\\ }^{0}G_{i_{a_{i}}j_{a_{j}}}} + RT\\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}\\ln y_{i_{a_{i}}j_{a_{j}}}} +_{\\ }^{E}G_{m}$\n", - "\n", - "where $y_{i_{a_{i}}j_{a_{j}}}$ is the mole fraction of species\n", - "$i_{a_{i}}j_{a_{j}}$ in the solution with $a_{i} = 1$ and $a_{j} = 0$\n", - "for component $i$ and $a_{i} = 0$ and $a_{j} = 1$ for component $j$. The\n", - "equilibrium amount of each associate $i_{a_{i}}j_{a_{j}}$ is obtained in\n", - "combination of mass balance and the zero driving force for the variation\n", - "of the amount of the associate similar to , i.e.\n", - "\n", - "Eq. \u2011 $\\sum_{i}^{}{\\sum_{j}^{}y_{i_{a_{i}}j_{a_{j}}}} = 1$\n", - "\n", - "Eq. \u2011 $x_{i} = \\sum_{}^{}{a_{i}y_{i_{a_{i}}j_{a_{j}}}}$\n", - "\n", - "Eq. \u2011\n", - "$\\left( \\frac{{\\partial G}_{m}}{\\partial y_{i_{a_{i}}j_{a_{j}}}} \\right)_{x_{i}} = 0$\n", - "\n", - "Associates are particularly plentiful in a gas phase, and their amounts\n", - "are significantly affected by pressure. For an ideal gas phase with\n", - "$_{\\ }^{E}G_{m} = 0$ and $PV_{m} = RT$, the effect of pressure is added\n", - "as follows\n", - "\n", - "Eq. \u2011\n", - "$G_{m} = \\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}_{\\ }^{0}G_{i_{a_{i}}j_{a_{j}}}} + RT\\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}\\ln y_{i_{a_{i}}j_{a_{j}}}} + \\int_{P_{0}}^{P}{V_{m}dP} = \\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}_{\\ }^{0}G_{i_{a_{i}}j_{a_{j}}}} + RT\\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}\\ln y_{i_{a_{i}}j_{a_{j}}}} + RT\\ln\\frac{P}{P_{0}}$\n", - "\n", - "where $P$ is the total pressure, and $P_{0}$ the reference pressure at\n", - "which $_{\\ }^{0}G_{i_{a_{i}}j_{a_{j}}}$ is defined, usually chosen to be\n", - "one atmospheric pressure. thus becomes\n", - "\n", - "Eq. \u2011\n", - "$G_{m} = \\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}_{\\ }^{0}{G_{i_{a_{i}}j_{a_{j}}}(P = 1atm)}} + RT\\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}\\ln y_{i_{a_{i}}j_{a_{j}}}} + RTlnP$\n", - "\n", - "with the unit of the total pressure $P$ being in atmospheric pressure\n", - "(atm). The chemical potential of species $i_{a_{i}}j_{a_{j}}$ is equal\n", - "to\n", - "\n", - "Eq. \u2011\n", - "$\\mu_{i_{a_{i}}j_{a_{j}}} =_{\\ }^{0}{G_{i_{a_{i}}j_{a_{j}}}(P = 1atm)} + RTlny_{i_{a_{i}}j_{a_{j}}}P =_{\\ }^{0}{G_{i_{a_{i}}j_{a_{j}}}(P = 1atm)} + RTlnP_{i_{a_{i}}j_{a_{j}}}$\n", - "\n", - "where $P_{i_{a_{i}}j_{a_{j}}}$ is the partial pressure of species\n", - "$i_{a_{i}}j_{a_{j}}$ defined as\n", - "\n", - "Eq. \u2011 $P_{i_{a_{i}}j_{a_{j}}} = y_{i_{a_{i}}j_{a_{j}}}P$\n", - "\n", - "Combining and with , the relation between chemical potentials of an\n", - "associate and its constituents is expressed as\n", - "\n", - "Eq. \u2011\n", - "$\\mu_{i_{a_{i}}j_{a_{j}}} = a_{i}\\mu_{i} + {a_{j}\\mu}_{j} = a_{i}_{\\ }^{0}G_{i} + a_{j}_{\\ }^{0}G_{j} + RTln\\left( {P_{i}}^{a_{i}}{P_{j}}^{a_{j}} \\right)$\n", - "\n", - "The equilibrium condition for the chemical reaction of an associate\n", - "forming from its constituents in an ideal gas phase is obtained as\n", - "\n", - "Eq. \u2011\n", - "$\\mathrm{\\Delta}_{f}G_{i_{a_{i}}j_{a_{j}}} + RTln\\frac{{P_{i}}^{a_{i}}{P_{j}}^{a_{j}}}{P_{i_{a_{i}}j_{a_{j}}}} = 0$\n", - "\n", - "For non-ideal phases, the mole fractions of various associates can be\n", - "calculated numerically through the minimization of Gibbs energy under\n", - "the constraints of and .\n" - ], - "metadata": {} - }, - { - "id": "a19845f0", - "cell_type": "markdown", - "source": [ - "### Ternary random solutions\n", - "\n", - "From , the Gibbs-Duhem equation of a ternary system consisting of\n", - "components $A$, $B$ and $C$ is written as\n", - "\n", - "Eq. \u2011\n", - "$0 = - SdT - Vd( - P) - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B} - {N_{C}d\\mu}_{C}$\n", - "\n", - "This equation represents a five-dimensional surface. It can be\n", - "visualized in a three-dimensional space with two of the five potentials\n", - "fixed. Usually $T$ and $P$ are kept constant as they are the natural\n", - "variables of $G$, and reduces to\n", - "\n", - "Eq. \u2011 $0 = - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B} - {N_{C}d\\mu}_{C}$\n", - "\n", - "A phase can thus be represented by a surface in the three-dimensional\n", - "space of $\\mu_{A}$, $\\mu_{B}$, and $\\mu_{C}$ at constant $T$ and $P$\n", - "with similar geometric appearance as .\n", - "\n", - "From , Gibbs energy of a ternary solution is written as\n", - "\n", - "Eq. \u2011\n", - "$G_{m} = x_{A}_{\\ }^{0}G_{A} + x_{B}_{\\ }^{0}G_{B} + x_{C}_{\\ }^{0}G_{C} + RT\\left( x_{A}l{nx}_{A} + x_{B}l{nx}_{B} + x_{C}l{nx}_{C} \\right) +_{\\ }^{E}G_{m}$\n", - "\n", - "When the mole fraction of one component approaches zero,\n", - "$_{\\ }^{E}G_{m}$ reduces to the excess Gibbs energy of mixing of the\n", - "binary systems of the remaining two components, represented by .\n", - "However, for a given composition of a ternary solution, there is no\n", - "unique way to assign the contributions from $_{\\ }^{E}G_{m}$ of each\n", - "binary to $_{\\ }^{E}G_{m}$ of the ternary solution because\n", - "$_{\\ }^{E}G_{m}$ of the ternary solution contains information of both\n", - "binary and ternary interactions. A variety of models is available in the\n", - "literature (see \\[1\\]). One intuitive approach would be to use the same\n", - "formula as that in the binary system, i.e. , with the mole fractions\n", - "substituted by the values in the ternary system, and $_{\\ }^{E}G_{m}$ of\n", - "a ternary solution may thus be defined as the following by including the\n", - "ternary interaction involving all three components,\n", - "\n", - "Eq. \u2011\n", - "$_{\\ }^{E}G_{m} = x_{A}x_{B}L_{AB} + x_{A}x_{C}L_{AC} + x_{B}x_{C}L_{BC} + x_{A}x_{B}x_{C}L_{ABC}$\n", - "\n", - "The chemical potential of a component is represented by . When all\n", - "interaction parameters in are constant, i.e. a ternary regular solution,\n", - "the chemical potential of component $A$ can be derived as\n", - "\n", - "Eq. \u2011\n", - "$\\mu_{A} = G_{A} =_{\\ }^{0}G_{A} + RTlnx_{A} + x_{B}L_{AB} + x_{C}L_{AC} -_{\\ }^{E}G_{m} =_{\\ }^{0}G_{A} + RTlnx_{A} + x_{B}{\\left( 1 - x_{A} \\right)L}_{AB} + x_{C}\\left( 1 - x_{A} \\right)L_{AC} - x_{B}x_{C}L_{BC} + x_{B}x_{C}\\left( 1 - 2x_{A} \\right)L_{ABC} =_{\\ }^{0}G_{A} + RTlnx_{B} + x_{B}^{2}L_{AB} + x_{C}^{2}L_{AC} + x_{B}x_{C}\\left( L_{AB} + L_{AC} - L_{BC} \\right) + x_{B}x_{C}\\left( 1 - 2x_{A} \\right)L_{ABC}$\n", - "\n", - "Similar equations can be derived for component $B$ and C with\n", - "$L_{AB} = L_{BA}$, $L_{AC} = L_{CA}$, and $L_{BC} = L_{CB}$. A schematic\n", - "molar Gibbs energy diagram at constant temperature and pressure is shown\n", - "in with all three binary systems having $L_{ij} < 0$ of similar values.\n", - "\n", - "Figure \u2011: Schematic ternary molar Gibbs energy diagram as a function of\n", - "compositions for given temperature and pressure\n", - "\n", - "To evaluate the stability of a ternary solution, one needs to calculate\n", - "the elements in the determinant shown in . Using the mole of component\n", - "$C$ as the independent molar quantity, the limit of stability is\n", - "expresses as\n", - "\n", - "Eq. \u2011 $G_{AA}G_{BB} - G_{AB}G_{BA} = 0$\n", - "\n", - "As an example, $G_{AA}$ is shown in the following equation, which must\n", - "be positive for the solution to be stable\n", - "\n", - "Eq. \u2011\n", - "${N\\left( \\frac{\\partial\\mu_{A}}{\\partial N_{A}} \\right)}_{T,P,N_{B},N_{C}} = NG_{AA} = \\frac{RT\\left( 1 - x_{A} \\right)}{x_{A}} - 2x_{B}^{2}L_{AB} - 2x_{C}^{2}L_{AC} - 2x_{B}x_{C}\\left( L_{AB} + L_{AC} - L_{BC} \\right) - 2x_{B}x_{C}\\left( 2 - 3x_{A} \\right)L_{ABC}$\n", - "\n", - "It is evident that any instability in binary systems with positive\n", - "interaction parameters extends into the ternary system. It can also be\n", - "seen that even if all binary interaction parameters are negative, i.e.\n", - "no instability in the binary systems, it is possible that becomes\n", - "negative for some combinations of the binary interaction parameters such\n", - "that $\\mathrm{\\Delta}L = L_{AB} + L_{AC} - L_{BC}$ becomes very positive\n", - "and overshadows the contributions due to $L_{AB}$ and $L_{AC}$, i.e.\n", - "$L_{BC}$ is more negative than $L_{AB}$ and $L_{AC}$ combined. In an\n", - "extreme case with $L_{AB} = L_{AC} = L_{ABC} = 0$ and $L_{BC} < 0$, i.e.\n", - "ideal solutions for the $A - B$ and $A - C$ binary systems, stable\n", - "solution in the $B - C$ binary system, and no additional ternary\n", - "interaction, reduces to\n", - "\n", - "Eq. \u2011\n", - "$N\\left( \\frac{\\partial\\mu_{A}}{\\partial N_{A}} \\right)_{T,P,N_{B},N_{C}} = \\frac{RT\\left( 1 - x_{A} \\right)}{x_{A}} + 2x_{B}x_{C}L_{BC}$\n", - "\n", - "Setting\n", - "$\\left( \\frac{\\partial\\mu_{A}}{\\partial N_{A}} \\right)_{T,P,N_{B},N_{C}} = 0$,\n", - "one obtains\n", - "\n", - "Eq. \u2011\n", - "$- \\frac{RT}{2L_{BC}} = \\frac{x_{A}x_{B}x_{C}}{1 - x_{A}} = \\frac{{\\left( 1 - x_{B} - x_{C} \\right)x}_{B}x_{C}}{x_{B} + x_{C}}$\n", - "\n", - "With $- \\frac{RT}{2L_{BC}}$ being positive due to $L_{BC} < 0$, there is\n", - "a parabola-shaped composition area in which the solution is unstable at\n", - "constant temperature and pressure. This is reasonable because the system\n", - "tends to maximize the number of B-C bonds due to its lower energy, which\n", - "competes with the entropy of mixing among the three elements and results\n", - "in segregation of B-C bonds, thus miscibility gap at low temperatures.\n", - "\n", - "To evaluate the ternary consolute point, the second derivatives for\n", - "component A and B are obtained as\n", - "\n", - "Eq. \u2011\n", - "${N\\left( \\frac{\\partial_{\\ }^{2}\\mu_{A}}{\\partial N_{A}^{2}} \\right)}_{T,P,N_{B},N_{C}} = \\frac{RT\\left( 1 - x_{A} \\right)}{x_{A}^{2}} + 4x_{B}^{2}L_{AB} + 4x_{C}^{2}L_{AC} + 4x_{B}x_{C}\\left( L_{AB} + L_{AC} - L_{BC} \\right) + 2x_{B}x_{C}\\left( 7 - 9x_{A} \\right)L_{ABC} = 0$\n", - "\n", - "Eq. \u2011\n", - "${N\\left( \\frac{\\partial_{\\ }^{2}\\mu_{B}}{\\partial N_{B}^{2}} \\right)}_{T,P,N_{A},N_{C}} = \\frac{RT\\left( 1 - x_{B} \\right)}{x_{B}^{2}} + 4x_{A}^{2}L_{AB} + 4x_{C}^{2}L_{BC} + 4x_{A}x_{C}\\left( L_{AB} + L_{BC} - L_{AC} \\right) + 2x_{A}x_{C}\\left( 7 - 9x_{B} \\right)L_{ABC} = 0$\n", - "\n", - "The consolute point can then be obtained using , and .\n", - "\n", - "It is observed in that $\\left( 1 - 2x_{A} \\right)L_{ABC} = 0$ at\n", - "$x_{A} = 0.5$, i.e. the ternary interaction parameter does not\n", - "contribute to the chemical potential of $A$. It is also observed in that\n", - "the contribution from the ternary interaction parameter changes sign at\n", - "$x_{i} = 2/3$ due to $\\left( 2 - 3x_{A} \\right)L_{ABC} = 0$.\n" - ], + "source": "## Polymer solutions and polymer blends\n\nA polymer solution is a mixture between polymer molecules and solvents,\nwhile a polymer blend is a mixture between different polymer molecules.\nA polymer molecule consists of the same repeating units of one or more\nmonomers, which can be an atom or a small molecule. The number of\nrepeating units is called the degree of polymerization and can be as\nlarge as 104\u2013105. It defines the molecular mass,\ni.e. the mass of one polymer molecule. There are three typical\narchitectures of polymerization: a linear chain, a branched chain, and a\ncross-linked polymer. Nearly all polymers are mixtures of molecules with\na different degree of polymerization with a molecular mass distribution,\ncomplicating the modeling of their thermodynamic properties because of\nthe dependence of properties on molecular mass.\n\nGibbs energy functions of polymers with a single molecular mass can be\ntreated similarly as in Chapter . For a polymer solution, the ideal\nentropy of mixing is quite different from that of atomically random\nsolutions discussed in Chapter because the monomers in a polymer\nmolecule are connected to each other and cannot move freely. One common\napproach to calculate the ideal entropy of mixing is to evoke a lattice\nmodel and assume that one monomer occupies a lattice site with a fixed\nvolume. The number of translational states of a single molecule is equal\nto the number of lattice sites available. In a homogeneous solution, the\ntotal number of lattice sites available is\n\nEq. \u2011 $n = \\sum_{i}^{}{m_{i}n_{i}}$\n\nwhere $n_{i}$ and $m_{i}$ are the number of molecule $i$ and the number\nof lattice sites per molecule $i$, respectively. While in its pure\nstate, i.e. before mixing, the number of states of molecule $i$ in terms\nof the number of lattice sites is\n\nEq. \u2011 $w_{i} = m_{i}n_{i} = n\\varphi_{i}$\n\nwhere $\\varphi_{i}$ is the volume fraction of molecule $i$ in the\nsolution. The entropy change per molecule $i$ is thus\n\nEq. \u2011\n$S_{i} = kln(n) - klnw_{i} = kln\\frac{1}{\\varphi_{i}} = - kln\\varphi_{i}$\n\nThe total entropy of mixing is the summation for all molecules,\nnormalized to per mole of lattice site\n\nEq. \u2011\n$_{}^{I}S_{m} = \\frac{N_{a}}{n}\\sum_{i}^{}{n_{i}S_{i}} = - R\\sum_{i}^{}{\\frac{\\varphi_{i}}{m_{i}}\\ln\\varphi_{i}}$\n\nwhere $N_{a}$ is the Avogadro number. When $m_{i} = 1$ for all\nmolecules, Eq. 2\u2011143 reduces to . Since $m_{i}$ values are typically\nvery large numbers for polymers, the entropy of mixing in polymer\nsolutions and blends is thus significantly lower than those in\nnon-polymer solutions as shown schematically in for binary systems with\nvarious $m_{i}$ values.\n\nFigure \u2011: Schematic entropy of mixing in solutions with the numbers of\nlattice sites per molecule shown.\n\nSimilar to , Gibbs energy of a multi-component random polymer\nsolution/blend can be written as\n\nEq. \u2011\n$G_{m} = \\sum_{}^{}\\frac{\\varphi_{i}}{m_{i}}_{\\ }^{0}G_{im} + RT\\left( \\sum_{}^{}{\\frac{\\varphi_{i}}{m_{i}}\\ln\\varphi_{i}} + \\sum_{}^{}{\\varphi_{i}\\varphi_{j}\\chi_{ij}} \\right)$\n\nwhere $_{\\ }^{0}G_{im}$ is the Gibbs energy of molecular $i$ per mole of\nlattice site, and $\\chi_{ij}$ the unitless interaction parameter between\nmolecule $i$ and $j$. Other equations shown in Chapter can be derived\nsimilarly too. It is to be noted that is very similar to the\nFlory\u2013Huggins solution equation widely used in the polymer community.\n", "metadata": {} } ] diff --git a/src/psu410/src/laws_of_thermodynamics/combined_law_of_thermodynamics_and_equilibrium_conditions.ipynb b/src/psu410/src/laws_of_thermodynamics/combined_law_of_thermodynamics_and_equilibrium_conditions.ipynb index dad65a4..7b460eb 100644 --- a/src/psu410/src/laws_of_thermodynamics/combined_law_of_thermodynamics_and_equilibrium_conditions.ipynb +++ b/src/psu410/src/laws_of_thermodynamics/combined_law_of_thermodynamics_and_equilibrium_conditions.ipynb @@ -4,225 +4,9 @@ "metadata": {}, "cells": [ { - "id": "6a9e41f7", + "id": "b6dde096", "cell_type": "markdown", - "source": [ - "## Combined law of thermodynamics and equilibrium conditions\n", - "\n", - "For a system with an irreversible internal process taking place, the\n", - "entropy change in the system thus consists of three parts: the heat\n", - "exchange with the surrounding defined by , the entropy production due to\n", - "the internal process represented by , and the entropy of mass exchange\n", - "with the surrounding. The total entropy change of the system can thus be\n", - "written as follows\n", - "\n", - "Eq. \u2011 $dS = \\frac{dQ}{T} + d_{ip}S + \\sum_{}^{}{S_{i}dN_{i}}$\n", - "\n", - "where $S_{i}$ is the unit entropy of component *i* in the surroundings,\n", - "often called partial entropy of component *i*, and will be further\n", - "discussed in Chapter\n", - "\n", - "Combining and and re-arranging, one obtains\n", - "\n", - "Eq. \u2011 $dQ = TdS - Dd\\xi - \\sum_{}^{}{TS_{i}dN_{i}}$\n", - "\n", - "Inserting and into yields the combined law of thermodynamics from the\n", - "first and second laws of thermodynamics,\n", - "\n", - "Eq. \u2011\n", - "$dU = TdS - PdV + \\sum_{}^{}\\left( H_{i} - TS_{i} \\right){dN}_{i} - Dd\\xi$\n", - "\n", - "The internal energy of the system is thus a function of *S*, *V*,\n", - "*Ni* and *\u03be* of the system, which are called natural\n", - "variables of the internal energy, i.e. *U*(*S*,*V*,*Ni*,*\u03be*).\n", - "The other variables are dependent variables and can be represented by\n", - "partial derivatives of the internal energy with respect to their\n", - "respective natural variables with other natural variables kept constant\n", - "as shown below\n", - "\n", - "Eq. \u2011\n", - "$T = \\left( \\frac{\\partial U}{\\partial S} \\right)_{V,\\ N_{i},\\ \\xi}$\n", - "\n", - "Eq. \u2011\n", - "$- P = \\left( \\frac{\\partial U}{\\partial V} \\right)_{S,\\ N_{i},\\ \\xi}$\n", - "\n", - "Eq. \u2011\n", - "$\\mu_{i} = H_{i} - TS_{i} = \\left( \\frac{\\partial U}{\\partial N_{i}} \\right)_{S,\\ V,N_{j \\neq i},\\ \\xi} = U_{i}$\n", - "\n", - "Eq. \u2011\n", - "$- D = \\left( \\frac{\\partial U}{\\partial\\xi} \\right)_{S,\\ V,N_{i}\\ }$\n", - "\n", - "In , a new variable, $\\mu_{i}$, is introduced. It is called chemical\n", - "potential and defined as the internal energy change with respect of the\n", - "addition of the component *i* when the entropy, volume and the amount of\n", - "other components of the system are kept constant. It may be worth\n", - "pointing out that for a system at equilibrium, i.e. $d_{ip}S = 0$, and\n", - "with constant entropy, $dS = 0$, if it exchanges mass with the\n", - "surroundings, $dN_{i} \\neq 0$, the system must also exchange heat with\n", - "the surroundings at the same time in order to keep the entropy invariant\n", - "as demonstrated by .\n", - "\n", - "The pairs of the natural variables and their corresponding partial\n", - "derivatives are called conjugate variables, i.e. *S* and *T*, *V* and\n", - "*\u2013P*, *Ni* and $\\mu_{i}$, and *\u03be* and *\u2013D*. There are minus\n", - "sign in front of *P* and *D* as the increase of volume and the progress\n", - "of the internal process decrease the internal energy of the system. The\n", - "importance of this conjugate relation will be evident when various forms\n", - "of combined thermodynamic laws and various types of phase diagrams are\n", - "introduced in the book.\n", - "\n", - "The last pair of conjugate variables, *\u03be* and *\u2013D*, is worthy of further\n", - "discussion. Based on the second law of thermodynamics, i.e. , no\n", - "internal processes take place spontaneously if there is no entropy\n", - "productions, i.e. D\u22640 or *d\u03be*=0 and *D*\\>0. With D\u22640, there is no\n", - "driving for any internal processes, and the system is at a full\n", - "equilibrium state. The last term in drops off, and *\u03be* becomes a\n", - "dependent variable of the system and can be calculated from the\n", - "equilibrium conditions. With *d\u03be*=0 and *D*\\>0, the system is under a\n", - "constrained equilibrium or freezing-in condition when the internal\n", - "process is constrained not to take place, and *\u03be* remains to be an\n", - "independent variable of the system.\n", - "\n", - "These two cases represent the two branches of thermodynamics:\n", - "equilibrium, reversible thermodynamics and irreversible thermodynamics.\n", - "It is clear from the above discussions that these two branches are\n", - "identical if the internal energy is not only a function of *S*, *V*, and\n", - "*Ni* , but also any internal process variable *\u03be*. This means\n", - "that one should be able to evaluate the internal energy of a system for\n", - "any freezing-in equilibrium conditions in addition to the full\n", - "equilibrium condition. In the rest of the book, the freezing-in\n", - "equilibrium and full equilibrium are not differentiated unless\n", - "specified.\n", - "\n", - "As the mechanical work under hydrostatic pressure is very important in\n", - "experiments, let us define a new quantity called enthalpy as follows\n", - "\n", - "Eq. \u2011 $H = U + PV$\n", - "\n", - "Its differential form can be obtained from as\n", - "\n", - "Eq. \u2011 $dH = dU + d(PV) = dQ + VdP + \\sum_{}^{}H_{i}{dN}_{i}$\n", - "\n", - "There are two significant consequences of the above equation. First, for\n", - "a close system under constant pressure, i.e. ${dN}_{i} = dP = 0$, one\n", - "has $dH = dQ$. This implies that the enthalpy change in a system is\n", - "equal to the heat exchange between the system and the surrounding of the\n", - "system, which is why enthalpy and heat are often exchangeable in the\n", - "literature. Second, for an adiabatic system under constant pressure,\n", - "i.e. $dQ = dP = 0$, can be re-arranged to the following equation\n", - "\n", - "Eq. \u2011\n", - "$H_{i} = \\left( \\frac{\\partial H}{\\partial N_{i}} \\right)_{N_{j \\neq i,\\ \\ dQ = dP = 0}}$\n", - "\n", - "$H_{i}$ is thus the partial enthalpy of component *i* and will be\n", - "further discussed in Chapter . The chemical potential of component *i*\n", - "defined in is thus related to the partial enthalpy and partial entropy\n", - "of the component.\n", - "\n", - "To further define equilibrium conditions of a system, consider a\n", - "homogeneous system in a state of internal equilibrium, i.e. no\n", - "spontaneous internal processes are possible with $Dd\\xi = 0$, and\n", - "becomes\n", - "\n", - "Eq. \u2011\n", - "$dU = TdS - PdV + \\sum_{}^{}\\mu_{i}{dN}_{i} = \\sum_{}^{}{Y_{i}dX_{i}}$\n", - "\n", - "where *X* represents *S*, *V*, *Ni*, and *Y* their conjugate\n", - "variables *T*, *-P*, $\\mu_{i}$. The state of the system with *c*\n", - "independent components is completely determined by the *c+2* variables,\n", - "i.e. *S*, *V*, and *Ni* with *i* from 1 to *c*.\n", - "\n", - "To simplify the situation, let us limit the discussion to an isolated\n", - "equilibrium system, i.e. $dU = 0$, and conduct a virtual internal\n", - "experiment inside the system by moving an infinitesimal amount of\n", - "$X_{i}$, ${dX}_{i}$, with other $X_{j}$ kept constant, from one region\n", - "of the system to another region of the system as schematically shown in\n", - ".\n", - "\n", - "Figure \u2011: Virtual experiment for a system at equilibrium\n", - "\n", - "As the system is homogeneous and at equilibrium,\n", - "$- dX_{i}^{'} = dX_{i}^{\"} = dX_{i}$. The total change of the internal\n", - "energy of this internal process is the combination of the changes in the\n", - "two regions, i.e.\n", - "\n", - "Eq. \u2011\n", - "$dU = dU^{'} + dU^{\"} = Y_{i}^{'}dX_{i}^{'} + Y_{i}^{\"}dX_{i}^{\"} = \\left( - Y_{i}^{'} + Y_{i}^{\"} \\right)dX_{i} = 0$\n", - "\n", - "Therefore, $Y_{i}^{'} = Y_{i}^{\"}$ for *T*, *-P*, and $\\mu_{i}$,\n", - "indicating that *T*, *-P*, and $\\mu_{i}$ are homogeneous in the system,\n", - "respectively, and are thus named as potentials of the system.\n", - "Furthermore these potentials are independent of the size of the system\n", - "and are often referred to as intensive variables in the literature. On\n", - "the other hand, all *X:s*, i.e. *S*, *V*, and *Ni*, are\n", - "proportional to the size of the system and may be normalized with\n", - "respect to the size of the system, usually in terms of total moles,\n", - "\n", - "Eq. \u2011 $N = \\sum_{}^{}N_{i}$\n", - "\n", - "They are thus called molar quantities and often referred to as extensive\n", - "variables, and the respective normalized variables are molar entropy,\n", - "molar volume, and mole fractions, defined as follows\n", - "\n", - "Eq. \u2011 $S_{m} = \\frac{S}{N}$\n", - "\n", - "Eq. \u2011 $V_{m} = \\frac{V}{N}$\n", - "\n", - "Eq. \u2011 $x_{i} = \\frac{N_{i}}{N}$\n", - "\n", - "Consider a small subsystem in this homogeneous system at equilibrium and\n", - "let the subsystem grow in size. The entropy, volume, and mass enclosed\n", - "in the subsystem increase as follows\n", - "\n", - "Eq. \u2011 ${dS = S}_{m}dN$\n", - "\n", - "Eq. \u2011 $dV = V_{m}dN$\n", - "\n", - "Eq. \u2011 $dN_{i} = x_{i}dN$\n", - "\n", - "The corresponding change in the internal energy of the subsystem becomes\n", - "\n", - "Eq. \u2011\n", - "$dU = TdS - PdV + \\sum_{}^{}\\mu_{i}{dN}_{i} = \\left( TS_{m} - PV_{m} + \\sum_{}^{}\\mu_{i}x_{i} \\right)dN = U_{m}dN$\n", - "\n", - "By integration one obtains the integral form of the internal energy as\n", - "\n", - "Eq. \u2011\n", - "$U = TS - PV + \\sum_{}^{}\\mu_{i}N_{i} = \\left( TS_{m} - PV_{m} + \\sum_{}^{}\\mu_{i}x_{i} \\right)N = U_{m}N$\n", - "\n", - "Similarly, the molar enthalpy can be defined as follows\n", - "\n", - "Eq. \u2011\n", - "$H = U + PV = U_{m}N + PV_{m}N = \\left( U_{m} + PV_{m} \\right)N = H_{m}N$\n", - "\n", - "In case a potential is not homogeneous in a system, the system will not\n", - "be in a state of equilibrium. Let us consider the same virtual\n", - "experiment as shown in for an isolated system that is not in\n", - "equilibrium, i.e. by moving an infinitesimal amount of $X_{i}$,\n", - "${dX}_{i}$, with other $X_{j}$ kept constant, from one region of the\n", - "system to another region of the system with the two regions having\n", - "different potentials. The total internal energy change equals to zero as\n", - "the virtual experiment has $dU = 0$. Similarly, each region can be\n", - "considered to be homogeneous by itself, and one has\n", - "$- dX_{i}^{'} = dX_{i}^{\"} = dX_{i}$. The total internal energy change\n", - "in the system is thus the sum of these two regions plus the entropy\n", - "production due to the internal process with $d\\xi = dX_{i}$, i.e.\n", - "\n", - "Eq. \u2011\n", - "$dU = dU^{'} + dU^{\"} + Dd\\xi = Y_{i}^{'}dX_{i}^{'} + Y_{i}^{\"}dX_{i}^{\"} + Dd\\xi = \\left( - Y_{i}^{'} + Y_{i}^{\"} \\right)dX_{i} + Dd\\xi = 0$\n", - "\n", - "Consequently, one obtains the following\n", - "\n", - "*Eq. 1\u201131* $D = Y_{i}^{'} - Y_{i}^{\"}$\n", - "\n", - "The driving force thus represents the difference of the potential at the\n", - "two regions, and the internal process is to eliminate inhomogeneity of\n", - "the potential with the heat transfer from high temperature regions to\n", - "low temperature regions, volume shrink of low pressure regions (high\n", - "$\u2013P$) and volume expansion of high pressure regions (low $\u2013P$), and\n", - "transport of components from high chemical potential regions to low\n", - "chemical potential regions.\n" - ], + "source": "## Combined law of thermodynamics and equilibrium conditions\n\nFor a system with an irreversible internal process taking place, the\nentropy change in the system thus consists of three parts: the heat\nexchange with the surrounding defined by , the entropy production due to\nthe internal process represented by , and the entropy of mass exchange\nwith the surrounding. The total entropy change of the system can thus be\nwritten as follows\n\nEq. \u2011 $dS = \\frac{dQ}{T} + d_{ip}S + \\sum_{}^{}{S_{i}dN_{i}}$\n\nwhere $S_{i}$ is the unit entropy of component *i* in the surroundings,\noften called partial entropy of component *i*, and will be further\ndiscussed in Chapter\n\nCombining and and re-arranging, one obtains\n\nEq. \u2011 $dQ = TdS - Dd\\xi - \\sum_{}^{}{TS_{i}dN_{i}}$\n\nInserting and into yields the combined law of thermodynamics from the\nfirst and second laws of thermodynamics,\n\nEq. \u2011\n$dU = TdS - PdV + \\sum_{}^{}\\left( H_{i} - TS_{i} \\right){dN}_{i} - Dd\\xi$\n\nThe internal energy of the system is thus a function of *S*, *V*,\n*Ni* and *\u03be* of the system, which are called natural\nvariables of the internal energy, i.e. *U*(*S*,*V*,*Ni*,*\u03be*).\nThe other variables are dependent variables and can be represented by\npartial derivatives of the internal energy with respect to their\nrespective natural variables with other natural variables kept constant\nas shown below\n\nEq. \u2011\n$T = \\left( \\frac{\\partial U}{\\partial S} \\right)_{V,\\ N_{i},\\ \\xi}$\n\nEq. \u2011\n$- P = \\left( \\frac{\\partial U}{\\partial V} \\right)_{S,\\ N_{i},\\ \\xi}$\n\nEq. \u2011\n$\\mu_{i} = H_{i} - TS_{i} = \\left( \\frac{\\partial U}{\\partial N_{i}} \\right)_{S,\\ V,N_{j \\neq i},\\ \\xi} = U_{i}$\n\nEq. \u2011\n$- D = \\left( \\frac{\\partial U}{\\partial\\xi} \\right)_{S,\\ V,N_{i}\\ }$\n\nIn , a new variable, $\\mu_{i}$, is introduced. It is called chemical\npotential and defined as the internal energy change with respect of the\naddition of the component *i* when the entropy, volume and the amount of\nother components of the system are kept constant. It may be worth\npointing out that for a system at equilibrium, i.e. $d_{ip}S = 0$, and\nwith constant entropy, $dS = 0$, if it exchanges mass with the\nsurroundings, $dN_{i} \\neq 0$, the system must also exchange heat with\nthe surroundings at the same time in order to keep the entropy invariant\nas demonstrated by .\n\nThe pairs of the natural variables and their corresponding partial\nderivatives are called conjugate variables, i.e. *S* and *T*, *V* and\n*\u2013P*, *Ni* and $\\mu_{i}$, and *\u03be* and *\u2013D*. There are minus\nsign in front of *P* and *D* as the increase of volume and the progress\nof the internal process decrease the internal energy of the system. The\nimportance of this conjugate relation will be evident when various forms\nof combined thermodynamic laws and various types of phase diagrams are\nintroduced in the book.\n\nThe last pair of conjugate variables, *\u03be* and *\u2013D*, is worthy of further\ndiscussion. Based on the second law of thermodynamics, i.e. , no\ninternal processes take place spontaneously if there is no entropy\nproductions, i.e. D\u22640 or *d\u03be*=0 and *D*\\>0. With D\u22640, there is no\ndriving for any internal processes, and the system is at a full\nequilibrium state. The last term in drops off, and *\u03be* becomes a\ndependent variable of the system and can be calculated from the\nequilibrium conditions. With *d\u03be*=0 and *D*\\>0, the system is under a\nconstrained equilibrium or freezing-in condition when the internal\nprocess is constrained not to take place, and *\u03be* remains to be an\nindependent variable of the system.\n\nThese two cases represent the two branches of thermodynamics:\nequilibrium, reversible thermodynamics and irreversible thermodynamics.\nIt is clear from the above discussions that these two branches are\nidentical if the internal energy is not only a function of *S*, *V*, and\n*Ni* , but also any internal process variable *\u03be*. This means\nthat one should be able to evaluate the internal energy of a system for\nany freezing-in equilibrium conditions in addition to the full\nequilibrium condition. In the rest of the book, the freezing-in\nequilibrium and full equilibrium are not differentiated unless\nspecified.\n\nAs the mechanical work under hydrostatic pressure is very important in\nexperiments, let us define a new quantity called enthalpy as follows\n\nEq. \u2011 $H = U + PV$\n\nIts differential form can be obtained from as\n\nEq. \u2011 $dH = dU + d(PV) = dQ + VdP + \\sum_{}^{}H_{i}{dN}_{i}$\n\nThere are two significant consequences of the above equation. First, for\na close system under constant pressure, i.e. ${dN}_{i} = dP = 0$, one\nhas $dH = dQ$. This implies that the enthalpy change in a system is\nequal to the heat exchange between the system and the surrounding of the\nsystem, which is why enthalpy and heat are often exchangeable in the\nliterature. Second, for an adiabatic system under constant pressure,\ni.e. $dQ = dP = 0$, can be re-arranged to the following equation\n\nEq. \u2011\n$H_{i} = \\left( \\frac{\\partial H}{\\partial N_{i}} \\right)_{N_{j \\neq i,\\ \\ dQ = dP = 0}}$\n\n$H_{i}$ is thus the partial enthalpy of component *i* and will be\nfurther discussed in Chapter . The chemical potential of component *i*\ndefined in is thus related to the partial enthalpy and partial entropy\nof the component.\n\nTo further define equilibrium conditions of a system, consider a\nhomogeneous system in a state of internal equilibrium, i.e. no\nspontaneous internal processes are possible with $Dd\\xi = 0$, and\nbecomes\n\nEq. \u2011\n$dU = TdS - PdV + \\sum_{}^{}\\mu_{i}{dN}_{i} = \\sum_{}^{}{Y_{i}dX_{i}}$\n\nwhere *X* represents *S*, *V*, *Ni*, and *Y* their conjugate\nvariables *T*, *-P*, $\\mu_{i}$. The state of the system with *c*\nindependent components is completely determined by the *c+2* variables,\ni.e. *S*, *V*, and *Ni* with *i* from 1 to *c*.\n\nTo simplify the situation, let us limit the discussion to an isolated\nequilibrium system, i.e. $dU = 0$, and conduct a virtual internal\nexperiment inside the system by moving an infinitesimal amount of\n$X_{i}$, ${dX}_{i}$, with other $X_{j}$ kept constant, from one region\nof the system to another region of the system as schematically shown in\n.\n\nFigure \u2011: Virtual experiment for a system at equilibrium\n\nAs the system is homogeneous and at equilibrium,\n$- dX_{i}^{'} = dX_{i}^{\"} = dX_{i}$. The total change of the internal\nenergy of this internal process is the combination of the changes in the\ntwo regions, i.e.\n\nEq. \u2011\n$dU = dU^{'} + dU^{\"} = Y_{i}^{'}dX_{i}^{'} + Y_{i}^{\"}dX_{i}^{\"} = \\left( - Y_{i}^{'} + Y_{i}^{\"} \\right)dX_{i} = 0$\n\nTherefore, $Y_{i}^{'} = Y_{i}^{\"}$ for *T*, *-P*, and $\\mu_{i}$,\nindicating that *T*, *-P*, and $\\mu_{i}$ are homogeneous in the system,\nrespectively, and are thus named as potentials of the system.\nFurthermore these potentials are independent of the size of the system\nand are often referred to as intensive variables in the literature. On\nthe other hand, all *X:s*, i.e. *S*, *V*, and *Ni*, are\nproportional to the size of the system and may be normalized with\nrespect to the size of the system, usually in terms of total moles,\n\nEq. \u2011 $N = \\sum_{}^{}N_{i}$\n\nThey are thus called molar quantities and often referred to as extensive\nvariables, and the respective normalized variables are molar entropy,\nmolar volume, and mole fractions, defined as follows\n\nEq. \u2011 $S_{m} = \\frac{S}{N}$\n\nEq. \u2011 $V_{m} = \\frac{V}{N}$\n\nEq. \u2011 $x_{i} = \\frac{N_{i}}{N}$\n\nConsider a small subsystem in this homogeneous system at equilibrium and\nlet the subsystem grow in size. The entropy, volume, and mass enclosed\nin the subsystem increase as follows\n\nEq. \u2011 ${dS = S}_{m}dN$\n\nEq. \u2011 $dV = V_{m}dN$\n\nEq. \u2011 $dN_{i} = x_{i}dN$\n\nThe corresponding change in the internal energy of the subsystem becomes\n\nEq. \u2011\n$dU = TdS - PdV + \\sum_{}^{}\\mu_{i}{dN}_{i} = \\left( TS_{m} - PV_{m} + \\sum_{}^{}\\mu_{i}x_{i} \\right)dN = U_{m}dN$\n\nBy integration one obtains the integral form of the internal energy as\n\nEq. \u2011\n$U = TS - PV + \\sum_{}^{}\\mu_{i}N_{i} = \\left( TS_{m} - PV_{m} + \\sum_{}^{}\\mu_{i}x_{i} \\right)N = U_{m}N$\n\nSimilarly, the molar enthalpy can be defined as follows\n\nEq. \u2011\n$H = U + PV = U_{m}N + PV_{m}N = \\left( U_{m} + PV_{m} \\right)N = H_{m}N$\n\nIn case a potential is not homogeneous in a system, the system will not\nbe in a state of equilibrium. Let us consider the same virtual\nexperiment as shown in for an isolated system that is not in\nequilibrium, i.e. by moving an infinitesimal amount of $X_{i}$,\n${dX}_{i}$, with other $X_{j}$ kept constant, from one region of the\nsystem to another region of the system with the two regions having\ndifferent potentials. The total internal energy change equals to zero as\nthe virtual experiment has $dU = 0$. Similarly, each region can be\nconsidered to be homogeneous by itself, and one has\n$- dX_{i}^{'} = dX_{i}^{\"} = dX_{i}$. The total internal energy change\nin the system is thus the sum of these two regions plus the entropy\nproduction due to the internal process with $d\\xi = dX_{i}$, i.e.\n\nEq. \u2011\n$dU = dU^{'} + dU^{\"} + Dd\\xi = Y_{i}^{'}dX_{i}^{'} + Y_{i}^{\"}dX_{i}^{\"} + Dd\\xi = \\left( - Y_{i}^{'} + Y_{i}^{\"} \\right)dX_{i} + Dd\\xi = 0$\n\nConsequently, one obtains the following\n\n*Eq. 1\u201131* $D = Y_{i}^{'} - Y_{i}^{\"}$\n\nThe driving force thus represents the difference of the potential at the\ntwo regions, and the internal process is to eliminate inhomogeneity of\nthe potential with the heat transfer from high temperature regions to\nlow temperature regions, volume shrink of low pressure regions (high\n$\u2013P$) and volume expansion of high pressure regions (low $\u2013P$), and\ntransport of components from high chemical potential regions to low\nchemical potential regions.\n", "metadata": {} } ] diff --git a/src/psu410/src/laws_of_thermodynamics/first_and_second_laws_of_thermodynamics.ipynb b/src/psu410/src/laws_of_thermodynamics/first_and_second_laws_of_thermodynamics.ipynb index 68bda34..18870a4 100644 --- a/src/psu410/src/laws_of_thermodynamics/first_and_second_laws_of_thermodynamics.ipynb +++ b/src/psu410/src/laws_of_thermodynamics/first_and_second_laws_of_thermodynamics.ipynb @@ -4,124 +4,9 @@ "metadata": {}, "cells": [ { - "id": "dd639118", + "id": "57a35cb6", "cell_type": "markdown", - "source": [ - "## First and second laws of thermodynamics\n", - "\n", - "A system typically consists of many chemical components. The first law\n", - "of thermodynamics states that the exchanges of heat, work, and\n", - "individual components with its surroundings must obey the law of\n", - "conversation of energy. In the domain of materials science and\n", - "engineering, the energy of interest is at the atomic and molecular\n", - "levels. The energy at the higher and lower levels such as nuclear energy\n", - "and kinetic and potential energies of a rigid body are usually excluded\n", - "from the discussion of thermodynamics of materials.\n", - "\n", - "Let us consider a system receiving an amount of heat, *dQ*, and an\n", - "amount of work, *dW*, and an amount of each independent component *i*,\n", - "*dNi*, from the surroundings. Such a system is called an open\n", - "system in contrast to a closed system when *dNi*=0 for all\n", - "components, i.e. no exchange of mass between the system and the\n", - "surrounding. Other types of systems commonly defined in thermodynamics\n", - "include adiabatic systems without exchange of heat, i.e. *dQ*=0, and\n", - "isolated systems without exchange of any, i.e. *dQ*= *dW*=\n", - "*dNi*=0.\n", - "\n", - "The corresponding change of energy in the system, i.e. the internal\n", - "energy change, *dU*, is formulated in terms of the first law of\n", - "thermodynamics as follows,\n", - "\n", - "Eq. \u2011 $dU = dQ + dW + \\sum_{}^{}{H_{i}dN_{i}}$\n", - "\n", - "where $H_{i}$ is the unit energy of component *i* in the surroundings,\n", - "and the summation is for all components in the system, which can be\n", - "controlled independently from the surroundings, i.e. the independent\n", - "components of the system.\n", - "\n", - "It is self-evident that the left-hand side of refers to the change\n", - "inside the system, while its right-hand side is for the contributions\n", - "from the surroundings to the system. In principle, no matter how the\n", - "heat and mass are added, or how the work is done to the system, as far\n", - "as their summation is the same, the change of the internal energy will\n", - "be the same based on the first law of thermodynamics, indicating that\n", - "the system reaches the same state for a closed system. The internal\n", - "energy is thus a state function in a close system as it does not depend\n", - "on how the state is reached.\n", - "\n", - "On the other hand, for the purpose of easy mathematical treatment, a\n", - "reversible process can be considered for a closed system, in which the\n", - "initial state of the system can be restored reversibly without any\n", - "changes left to the surroundings. Therefore, the heat transferred and\n", - "the work done to the system are identical to the heat and work lost by\n", - "the surroundings and vice versa. The classic example of reversible\n", - "processes is the Carnot\u2019s cycle, which is shown in . It consists of four\n", - "reversible processes for a closed system. The four reversible processes\n", - "are compression at constant temperature *T1* (isothermal,),\n", - "compression without heat exchange (adiabatic) ending at *T2*,\n", - "isothermal expansion at *T2*, and adiabatic expansion ending\n", - "at *T1*.\n", - "\n", - "Figure \u2011: Schematics of the Carnot\u2019s cycle\n", - "\n", - "The Carnot\u2019s cycle involves a simple type of mechanical work, either\n", - "hydrostatic expansion or compression, with the work that the surrounding\n", - "does to the system represented by\n", - "\n", - "Eq. \u2011 $dW = - PdV$\n", - "\n", - "with *P* being the external pressure that the surrounding exerts on the\n", - "system and *V* the volume of the system. It is now necessary to\n", - "differentiate the external and internal variables for further discussion\n", - "with the former representing variables in the surroundings and the\n", - "latter representing variables in the system. For the isothermal\n", - "processes in the Carnot\u2019s cycle, the entropy change of the system, *dS*,\n", - "can be defined as the heat exchange divided by temperature\n", - "\n", - "Eq. \u2011 $dS = \\frac{dQ}{T}$\n", - "\n", - "In addition to processes involving heat, work, and mass exchanges\n", - "between the system and the surroundings, there could be internal\n", - "processes taking place inside the system. As the system cannot do work\n", - "to itself, the criterion for whether an internal process can occur\n", - "spontaneously must be related to the heat exchange, which is related to\n", - "the entropy change as shown by .\n", - "\n", - "It is a known fact that heat can spontaneously transfer from a higher\n", - "temperature (*T2*) region to a lower temperature\n", - "(*T1*) region inside a system if the heat conduction is\n", - "allowed, and this process is irreversible because heat cannot be\n", - "conducted from a low temperature region to a high temperature region by\n", - "itself. indicates that for the same amount of heat change, the entropy\n", - "change at *T1* is higher than that at *T2*, and\n", - "the heat conduction thus results in a positive entropy change in the\n", - "system, i.e.\n", - "\n", - "Eq. \u2011\n", - "$\\mathrm{\\Delta}S = - \\frac{dQ}{T_{2}} + \\frac{dQ}{T_{1}} = \\frac{dQ}{{T_{2}T}_{1}}\\left( {T_{2} - T}_{1} \\right) > 0$\n", - "\n", - "Consequently, the second law of thermodynamics is obtained, which states\n", - "that for an internal process to take place spontaneously or\n", - "irreversibly, this internal process (*ip*) must have a positive entropy\n", - "production, which can be written in a differential form as follows\n", - "\n", - "Eq. \u2011 $d_{ip}S > 0$\n", - "\n", - "From the definition of entropy change shown by , the amount of heat\n", - "produced by this irreversible internal process can be calculated as\n", - "follows\n", - "\n", - "Eq. \u2011 $d_{ip}Q = Td_{ip}S$\n", - "\n", - "Let us represent this internal process by *d\u03be* and define the driving\n", - "force for this internal process by *D*. The work done by this internal\n", - "process is thus *Dd\u03be*, which is released as heat, i.e.\n", - "\n", - "Eq. \u2011 $Dd\\xi = d_{ip}Q = Td_{ip}S$\n", - "\n", - "An irreversible process thus must have a positive driving force in order\n", - "for it to take place spontaneously.\n" - ], + "source": "## First and second laws of thermodynamics\n\nA system typically consists of many chemical components. The first law\nof thermodynamics states that the exchanges of heat, work, and\nindividual components with its surroundings must obey the law of\nconversation of energy. In the domain of materials science and\nengineering, the energy of interest is at the atomic and molecular\nlevels. The energy at the higher and lower levels such as nuclear energy\nand kinetic and potential energies of a rigid body are usually excluded\nfrom the discussion of thermodynamics of materials.\n\nLet us consider a system receiving an amount of heat, *dQ*, and an\namount of work, *dW*, and an amount of each independent component *i*,\n*dNi*, from the surroundings. Such a system is called an open\nsystem in contrast to a closed system when *dNi*=0 for all\ncomponents, i.e. no exchange of mass between the system and the\nsurrounding. Other types of systems commonly defined in thermodynamics\ninclude adiabatic systems without exchange of heat, i.e. *dQ*=0, and\nisolated systems without exchange of any, i.e. *dQ*= *dW*=\n*dNi*=0.\n\nThe corresponding change of energy in the system, i.e. the internal\nenergy change, *dU*, is formulated in terms of the first law of\nthermodynamics as follows,\n\nEq. \u2011 $dU = dQ + dW + \\sum_{}^{}{H_{i}dN_{i}}$\n\nwhere $H_{i}$ is the unit energy of component *i* in the surroundings,\nand the summation is for all components in the system, which can be\ncontrolled independently from the surroundings, i.e. the independent\ncomponents of the system.\n\nIt is self-evident that the left-hand side of refers to the change\ninside the system, while its right-hand side is for the contributions\nfrom the surroundings to the system. In principle, no matter how the\nheat and mass are added, or how the work is done to the system, as far\nas their summation is the same, the change of the internal energy will\nbe the same based on the first law of thermodynamics, indicating that\nthe system reaches the same state for a closed system. The internal\nenergy is thus a state function in a close system as it does not depend\non how the state is reached.\n\nOn the other hand, for the purpose of easy mathematical treatment, a\nreversible process can be considered for a closed system, in which the\ninitial state of the system can be restored reversibly without any\nchanges left to the surroundings. Therefore, the heat transferred and\nthe work done to the system are identical to the heat and work lost by\nthe surroundings and vice versa. The classic example of reversible\nprocesses is the Carnot\u2019s cycle, which is shown in . It consists of four\nreversible processes for a closed system. The four reversible processes\nare compression at constant temperature *T1* (isothermal,),\ncompression without heat exchange (adiabatic) ending at *T2*,\nisothermal expansion at *T2*, and adiabatic expansion ending\nat *T1*.\n\nFigure \u2011: Schematics of the Carnot\u2019s cycle\n\nThe Carnot\u2019s cycle involves a simple type of mechanical work, either\nhydrostatic expansion or compression, with the work that the surrounding\ndoes to the system represented by\n\nEq. \u2011 $dW = - PdV$\n\nwith *P* being the external pressure that the surrounding exerts on the\nsystem and *V* the volume of the system. It is now necessary to\ndifferentiate the external and internal variables for further discussion\nwith the former representing variables in the surroundings and the\nlatter representing variables in the system. For the isothermal\nprocesses in the Carnot\u2019s cycle, the entropy change of the system, *dS*,\ncan be defined as the heat exchange divided by temperature\n\nEq. \u2011 $dS = \\frac{dQ}{T}$\n\nIn addition to processes involving heat, work, and mass exchanges\nbetween the system and the surroundings, there could be internal\nprocesses taking place inside the system. As the system cannot do work\nto itself, the criterion for whether an internal process can occur\nspontaneously must be related to the heat exchange, which is related to\nthe entropy change as shown by .\n\nIt is a known fact that heat can spontaneously transfer from a higher\ntemperature (*T2*) region to a lower temperature\n(*T1*) region inside a system if the heat conduction is\nallowed, and this process is irreversible because heat cannot be\nconducted from a low temperature region to a high temperature region by\nitself. indicates that for the same amount of heat change, the entropy\nchange at *T1* is higher than that at *T2*, and\nthe heat conduction thus results in a positive entropy change in the\nsystem, i.e.\n\nEq. \u2011\n$\\mathrm{\\Delta}S = - \\frac{dQ}{T_{2}} + \\frac{dQ}{T_{1}} = \\frac{dQ}{{T_{2}T}_{1}}\\left( {T_{2} - T}_{1} \\right) > 0$\n\nConsequently, the second law of thermodynamics is obtained, which states\nthat for an internal process to take place spontaneously or\nirreversibly, this internal process (*ip*) must have a positive entropy\nproduction, which can be written in a differential form as follows\n\nEq. \u2011 $d_{ip}S > 0$\n\nFrom the definition of entropy change shown by , the amount of heat\nproduced by this irreversible internal process can be calculated as\nfollows\n\nEq. \u2011 $d_{ip}Q = Td_{ip}S$\n\nLet us represent this internal process by *d\u03be* and define the driving\nforce for this internal process by *D*. The work done by this internal\nprocess is thus *Dd\u03be*, which is released as heat, i.e.\n\nEq. \u2011 $Dd\\xi = d_{ip}Q = Td_{ip}S$\n\nAn irreversible process thus must have a positive driving force in order\nfor it to take place spontaneously.\n", "metadata": {} } ] diff --git a/src/psu410/src/laws_of_thermodynamics/gibbsduhem_equation.ipynb b/src/psu410/src/laws_of_thermodynamics/gibbsduhem_equation.ipynb index c71fa13..24d1f86 100644 --- a/src/psu410/src/laws_of_thermodynamics/gibbsduhem_equation.ipynb +++ b/src/psu410/src/laws_of_thermodynamics/gibbsduhem_equation.ipynb @@ -4,87 +4,9 @@ "metadata": {}, "cells": [ { - "id": "8d6220ab", + "id": "34832f69", "cell_type": "markdown", - "source": [ - "## Gibbs-Duhem equation\n", - "\n", - "In experiments, it is difficult to control *S* and *V* of a system in\n", - "comparison with their conjugate variables *T* and *-P*. It is thus\n", - "desirable to construct new functions to represent the system with *T*\n", - "and *-P* as natural variables of the functions. One of them is enthalpy\n", - "defined in , and other two can be defined as follows\n", - "\n", - "Eq. \u2011 $F = U - TS$\n", - "\n", - "Eq. \u2011 $G = U - TS + PV = \\sum_{}^{}\\mu_{i}N_{i} = H - TS = F + PV$\n", - "\n", - "with *F* and *G* called Helmholtz energy and Gibbs energy, respectively.\n", - "The middle part of is obtained using $U$ from . The corresponding\n", - "combined law of thermodynamics in terms of *H*, *F*, and *G* can be\n", - "obtained through the Legendre transformation of as\n", - "\n", - "Eq. \u2011 $dH = TdS - Vd( - P) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$\n", - "\n", - "Eq. \u2011 $dF = - SdT - PdV + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$\n", - "\n", - "Eq. \u2011 $dG = - SdT - Vd( - P) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$\n", - "\n", - "The independent variables in each of the above form are regarded as the\n", - "natural variables to the corresponding function. The integral forms of\n", - "all the functions can thus be written as the following with their\n", - "natural variables listed in the parenthesis\n", - "\n", - "Eq. \u2011 $U = U\\left( S,V,N_{i},\\xi \\right)$\n", - "\n", - "Eq. \u2011 $H = H\\left( S, - P,N_{i},\\xi \\right)$\n", - "\n", - "Eq. \u2011 $F = F\\left( T,V,N_{i},\\xi \\right)$\n", - "\n", - "Eq. \u2011 $G = G\\left( T, - P,N_{i},\\xi \\right)$\n", - "\n", - "By differentiating , one obtains\n", - "\n", - "Eq. \u2011\n", - "$dG = \\sum_{}^{}\\mu_{i}{dN}_{i} + \\sum_{}^{}{N_{i}d\\mu}_{i} = - SdT - Vd( - P) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$\n", - "\n", - "For a system at equilibrium, $Dd\\xi = 0$, re-arranging gives the\n", - "Gibbs-Duhem equation\n", - "\n", - "Eq. \u2011 $0 = - SdT - Vd( - P) - \\sum_{}^{}{N_{i}d\\mu}_{i}$\n", - "\n", - "This equation indicates that for a homogeneous system with *c*\n", - "independent components at equilibrium, there is a direct relation among\n", - "all the *c+2* potentials, and they are $c$ chemical potentials\n", - "($\\mu_{i}$), temperature, and pressure. Consequently, only *c+1*\n", - "potentials can change independently, and the remaining potential is\n", - "dependent on the other potentials. As discussed in connection with ,\n", - "there are $c + 2$ independent variables for an equilibrium system with\n", - "*c* independent components, where all of them are molar quantities.\n", - "\n", - "With the relationships between potentials and molar quantities defined\n", - "by to , one can switch between potentials and molar quantities as\n", - "natural variables of the system. For example, one can define a new free\n", - "energy function when the chemical potential of one component is\n", - "controlled from the surroundings instead of its content and obtain the\n", - "following combined first and second law of thermodynamics\n", - "\n", - "Eq. \u2011 $\\Phi = G - \\mu_{1}N_{1} = \\sum_{i = 2}^{c}{\\mu_{i}N_{i}}$\n", - "\n", - "Eq. \u2011\n", - "$d\\Phi = - SdT - Vd( - P) - N_{1}{d\\mu}_{1} + \\sum_{i = 2}^{c}{\\mu_{i}{dN}_{i}} - Dd\\xi$\n", - "\n", - "However, even though the $c + 2$ molar quantities are independent of\n", - "each other, indicates that not all the $c + 2$ potentials are\n", - "independent, i.e., if chemical potentials of all components are changed\n", - "to natural variables, one would obtain . Therefore, among the *c+2*\n", - "independent variables used to define the system, the maximum number of\n", - "independent potential is *c+1*, and at least one of the *c+2*\n", - "independent variables must be a molar quantity. This variable is usually\n", - "chosen to be the size of the system or the major element in the system.\n", - "The Gibbs-Duhem equation is used to derive Gibbs phase rule in\n", - "heterogeneous systems, which is discussed in Chapter of the book.\n" - ], + "source": "## Gibbs-Duhem equation\n\nIn experiments, it is difficult to control *S* and *V* of a system in\ncomparison with their conjugate variables *T* and *-P*. It is thus\ndesirable to construct new functions to represent the system with *T*\nand *-P* as natural variables of the functions. One of them is enthalpy\ndefined in , and other two can be defined as follows\n\nEq. \u2011 $F = U - TS$\n\nEq. \u2011 $G = U - TS + PV = \\sum_{}^{}\\mu_{i}N_{i} = H - TS = F + PV$\n\nwith *F* and *G* called Helmholtz energy and Gibbs energy, respectively.\nThe middle part of is obtained using $U$ from . The corresponding\ncombined law of thermodynamics in terms of *H*, *F*, and *G* can be\nobtained through the Legendre transformation of as\n\nEq. \u2011 $dH = TdS - Vd( - P) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$\n\nEq. \u2011 $dF = - SdT - PdV + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$\n\nEq. \u2011 $dG = - SdT - Vd( - P) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$\n\nThe independent variables in each of the above form are regarded as the\nnatural variables to the corresponding function. The integral forms of\nall the functions can thus be written as the following with their\nnatural variables listed in the parenthesis\n\nEq. \u2011 $U = U\\left( S,V,N_{i},\\xi \\right)$\n\nEq. \u2011 $H = H\\left( S, - P,N_{i},\\xi \\right)$\n\nEq. \u2011 $F = F\\left( T,V,N_{i},\\xi \\right)$\n\nEq. \u2011 $G = G\\left( T, - P,N_{i},\\xi \\right)$\n\nBy differentiating , one obtains\n\nEq. \u2011\n$dG = \\sum_{}^{}\\mu_{i}{dN}_{i} + \\sum_{}^{}{N_{i}d\\mu}_{i} = - SdT - Vd( - P) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$\n\nFor a system at equilibrium, $Dd\\xi = 0$, re-arranging gives the\nGibbs-Duhem equation\n\nEq. \u2011 $0 = - SdT - Vd( - P) - \\sum_{}^{}{N_{i}d\\mu}_{i}$\n\nThis equation indicates that for a homogeneous system with *c*\nindependent components at equilibrium, there is a direct relation among\nall the *c+2* potentials, and they are $c$ chemical potentials\n($\\mu_{i}$), temperature, and pressure. Consequently, only *c+1*\npotentials can change independently, and the remaining potential is\ndependent on the other potentials. As discussed in connection with ,\nthere are $c + 2$ independent variables for an equilibrium system with\n*c* independent components, where all of them are molar quantities.\n\nWith the relationships between potentials and molar quantities defined\nby to , one can switch between potentials and molar quantities as\nnatural variables of the system. For example, one can define a new free\nenergy function when the chemical potential of one component is\ncontrolled from the surroundings instead of its content and obtain the\nfollowing combined first and second law of thermodynamics\n\nEq. \u2011 $\\Phi = G - \\mu_{1}N_{1} = \\sum_{i = 2}^{c}{\\mu_{i}N_{i}}$\n\nEq. \u2011\n$d\\Phi = - SdT - Vd( - P) - N_{1}{d\\mu}_{1} + \\sum_{i = 2}^{c}{\\mu_{i}{dN}_{i}} - Dd\\xi$\n\nHowever, even though the $c + 2$ molar quantities are independent of\neach other, indicates that not all the $c + 2$ potentials are\nindependent, i.e., if chemical potentials of all components are changed\nto natural variables, one would obtain . Therefore, among the *c+2*\nindependent variables used to define the system, the maximum number of\nindependent potential is *c+1*, and at least one of the *c+2*\nindependent variables must be a molar quantity. This variable is usually\nchosen to be the size of the system or the major element in the system.\nThe Gibbs-Duhem equation is used to derive Gibbs phase rule in\nheterogeneous systems, which is discussed in Chapter of the book.\n", "metadata": {} } ] diff --git a/src/psu410/src/laws_of_thermodynamics/index.ipynb b/src/psu410/src/laws_of_thermodynamics/index.ipynb index 71f7bf7..9d399cd 100644 --- a/src/psu410/src/laws_of_thermodynamics/index.ipynb +++ b/src/psu410/src/laws_of_thermodynamics/index.ipynb @@ -4,11 +4,9 @@ "metadata": {}, "cells": [ { - "id": "a942967d", + "id": "2d5be24d", "cell_type": "markdown", - "source": [ - "# Laws of thermodynamics\n" - ], + "source": "# Laws of thermodynamics\n", "metadata": {} } ] diff --git a/src/psu410/src/laws_of_thermodynamics/stability_at_equilibrium_and_property_anomaly.ipynb b/src/psu410/src/laws_of_thermodynamics/stability_at_equilibrium_and_property_anomaly.ipynb index 687b1e7..e9c049e 100644 --- a/src/psu410/src/laws_of_thermodynamics/stability_at_equilibrium_and_property_anomaly.ipynb +++ b/src/psu410/src/laws_of_thermodynamics/stability_at_equilibrium_and_property_anomaly.ipynb @@ -4,188 +4,9 @@ "metadata": {}, "cells": [ { - "id": "1e25979a", + "id": "2aeff033", "cell_type": "markdown", - "source": [ - "## Stability at equilibrium and property anomaly\n", - "\n", - "As shown by , potentials are homogenous for a homogeneous system in a\n", - "state of internal equilibrium. To study the stability of the equilibrium\n", - "state, one considers the entropy production due to a fluctuation of a\n", - "molar quantity as an internal process. Based on the second law of\n", - "thermodynamics, the driving force, as the first derivative of the\n", - "entropy production with respect to the internal process, is zero for\n", - "such a fluctuation at equilibrium, i.e. *D*=0, and the entropy of\n", - "production thus depends on the second derivative. It can be written as\n", - "follows\n", - "\n", - "Eq. \u2011\n", - "$Td_{ip}S = \\frac{\\partial_{ip}S}{\\partial\\xi}d\\xi + \\frac{1}{2}{\\frac{\\partial_{ip}^{2}S}{\\partial\\xi^{2}}(d\\xi)}^{2} = Dd\\xi - \\frac{1}{2}D_{2}(d\\xi)^{2}$\n", - "\n", - "with $D_{2} = - \\frac{\\partial_{ip}^{2}S}{\\partial\\xi^{2}}$. When\n", - "$\\frac{\\partial_{ip}^{2}S}{\\partial\\xi^{2}} < 0$ or $D_{2} > 0$ along\n", - "with $D = 0$, the fluctuation does not produce positive entropy of\n", - "production and can thus not develop further. The equilibrium state of\n", - "the system is therefore stable against the fluctuation. On the other\n", - "hand, when $\\frac{\\partial_{ip}^{2}S}{\\partial\\xi^{2}} > 0$ or\n", - "$D_{2} < 0$ along with $D = 0$, the fluctuation creates positive entropy\n", - "of production and can continue to grow. The equilibrium state of the\n", - "system is therefore unstable against the fluctuation. In connection with\n", - ", one can realize that for a system at stable equilibrium without heat\n", - "and mass exchanges with the surroundings, its entropy is at its maximum,\n", - "and there are no other internal processes, which could produce any more\n", - "entropy. This is schematically shown in .\n", - "\n", - "Figure \u2011: Schematic diagram showing maximum entropy\n", - "\n", - "Using , , and , the combined law of thermodynamics can be written as\n", - "\n", - "*Eq. 1\u201133*\n", - "$dU = \\sum_{}^{}{Y_{i}dX_{i}} - Dd\\xi + \\frac{1}{2}D_{2}(d\\xi)^{2}$\n", - "\n", - "Let us carry out the same virtual internal experiment shown in Chapter ,\n", - "i.e. moving an infinitesimal amount of *Xi* in a homogenous\n", - "system with other $X_{j}$ kept constant in an isolated system, i.e.\n", - "$dU = 0$ and $D = 0$. The internal energy change due to this internal\n", - "process is\n", - "\n", - "Eq. \u2011\n", - "$dU = \\frac{1}{2}D_{2}\\left\\{ \\left( dX_{i}^{'} \\right)^{2} + \\left( dX_{i}^{\"} \\right)^{2} \\right\\}$\n", - "\n", - "For a homogeneous system in a state of stable equilibrium with\n", - "$\\left( dX_{i}^{'} \\right)^{2} = \\left( dX_{i}^{\"} \\right)^{2} = \\left( dX_{i} \\right)^{2}$,\n", - "this internal process must result in an increase of internal energy,\n", - "$dU > 0$, and thus gives\n", - "\n", - "Eq. \u2011\n", - "$D_{2} = 2\\left( \\frac{\\partial^{2}U}{\\partial\\left( X_{i} \\right)^{2}} \\right)_{X_{j}} = 2\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}} > 0$\n", - "\n", - "shows that for a system to be stable, any pair of conjugate variables\n", - "must change in the same direction when other independent molar\n", - "quantities are kept constant. For the conjugate variables discussed so\n", - "far, it means that for a stable system, the addition of entropy\n", - "increases temperature with $\\frac{\\partial T}{\\partial S}$\\>0, the\n", - "volume decreases with pressure or increases with the negative of\n", - "pressure with $\\frac{\\partial( - P)}{\\partial V} > 0$, and the chemical\n", - "potential of a component increases with its amount, i.e.\n", - "$\\frac{\\partial\\mu_{i}}{\\partial N_{i}} > 0$, where the derivatives are\n", - "taken with all other molar quantities kept constant. The limit of\n", - "stability is reached when becomes zero, i.e.\n", - "\n", - "Eq. \u2011\n", - "$D_{2} = 2\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}} = 0$\n", - "\n", - "shows schematically the energy as a function of configurations including\n", - "three states: unstable, stable, and metastable. Both the stable and\n", - "metastable states have positive curvatures due to $D_{2} > 0$, while the\n", - "unstable state has a negative curvature due to $D_{2} < 0$. There is an\n", - "inflection point of $D_{2} = 0$ for a state between a stable or\n", - "metastable state with $D_{2} > 0$ and an unstable state with\n", - "$D_{2} < 0$. These two inflection points, called spinodal, represent the\n", - "limit of stability. The states between the two inflection points are\n", - "unstable, and other states are either stable or metastable. The two\n", - "inflection points can move apart from or close to each other depending\n", - "on the change of external conditions, i.e. the natural variables. One\n", - "extreme situation is when these two inflection points merge into one\n", - "point, and the instability occurs only at this particular point. It is\n", - "evident that all three states, stable, metastable, and unstable, also\n", - "merge into one point. This point is called critical or consolute point,\n", - "beyond which the instability no longer exists.\n", - "\n", - "Figure \u2011: Schematic diagram showing the stable and unstable equilibrium\n", - "states\n", - "\n", - "To mathematically define the consolute point, the third derivative needs\n", - "to be added to because both $D$ and $D_{2}$ vanish at this point, i.e.\n", - "\n", - "Eq. \u2011\n", - "$Td_{ip}S = \\frac{\\partial_{ip}S}{\\partial\\xi}d\\xi + \\frac{1}{2}{\\frac{\\partial_{ip}^{2}S}{\\partial\\xi^{2}}(d\\xi)}^{2} + \\frac{1}{6}{\\frac{\\partial_{ip}^{3}S}{\\partial\\xi^{3}}(d\\xi)}^{3} = Dd\\xi - \\frac{1}{2}D_{2}(d\\xi)^{2} + \\frac{1}{6}{D_{3}(d\\xi)}^{3}$\n", - "\n", - "Eq. \u2011\n", - "$dU = \\sum_{}^{}{Y_{i}dX_{i}} - Dd\\xi + \\frac{1}{2}D_{2}(d\\xi)^{2} - \\frac{1}{6}{D_{3}(d\\xi)}^{3}$\n", - "\n", - "At the consolute point, the third derivative also becomes zero, i.e.\n", - "\n", - "Eq. \u2011 $D_{3} = \\frac{\\partial_{ip}^{3}S}{\\partial\\xi^{3}}^{3} = 0$\n", - "\n", - "Let us further discuss the properties of the system in relation to the\n", - "critical point. By taking the inverse of the equation of the limit of\n", - "stability, , one obtains\n", - "\n", - "Eq. \u2011\n", - "$\\left( \\frac{\\partial X_{i}}{\\partial Y_{i}} \\right)_{X_{j}} = + \\ \\infty$\n", - "\n", - "i.e. all $X_{i}$ quantities diverge at the critical point. Therefore,\n", - "when a system approaches the critical point from its stable region, the\n", - "change of a molar quantity with respect to its conjugate potential\n", - "varies dramatically and becomes infinite at the critical point,\n", - "resulting in property anomalies in the system. In the unstable region,\n", - "the system would thus separate into stable subsystems and becomes\n", - "heterogeneous, and $X_{i}$:s change discontinuously between subsystems.\n", - "While in the stable region, the change of a molar quantity with respect\n", - "to its conjugate potential decreases as the system moves away from the\n", - "critical point and remains positive due to the stability criteria\n", - "denoted by .\n", - "\n", - "However, it is not clear how a molar quantity changes with respect to a\n", - "non-conjugate potential at the critical point. From the Maxwell\n", - "relation, one has\n", - "\n", - "Eq. \u2011\n", - "$\\left( \\frac{\\partial Y_{i}}{\\partial X_{j}} \\right)_{X_{k \\neq j}} = \\frac{\\partial^{2}U}{\\partial X_{i}\\partial X_{j}} = \\left( \\frac{\\partial Y_{j}}{\\partial X_{i}} \\right)_{X_{k \\neq i}}$\n", - "\n", - "Eq. \u2011\n", - "$\\left( \\frac{\\partial X_{j}}{\\partial Y_{i}} \\right)_{X_{k \\neq j}} = \\left( \\frac{\\partial X_{i}}{\\partial Y_{j}} \\right)_{X_{k \\neq i}}$\n", - "\n", - "Since all $X_{i}$:s diverge at the critical point, both derivatives in\n", - "should also go to infinite at the critical point. To investigate their\n", - "signs, let us carry out a virtual experiment similar to the one in\n", - "deriving the stability condition ( and ). In this case, two internal\n", - "processes are needed for moving two molar quantities simultaneously in\n", - "an isolated system, i.e.\n", - "\n", - "*Eq. 1\u201143*\n", - "$dU = - D_{\\xi_{1}}d\\xi_{1} - D_{\\xi_{2}}d\\xi_{2} + D_{\\xi_{1}\\xi_{2}}d\\xi_{1}d\\xi_{2} + \\frac{1}{2}D_{2\\xi_{1}}\\left( d\\xi_{1} \\right)^{2} + \\frac{1}{2}D_{2\\xi_{2}}\\left( d\\xi_{2} \\right)^{2}$\n", - "\n", - "Based on the above discussions, in a stable system at equilibrium with\n", - "$D_{\\xi_{1}} = D_{\\xi_{2}} = 0$, $D_{2\\xi_{1}} > 0$ and\n", - "$D_{2\\xi_{2}} > 0$, the sign of $D_{\\xi_{1}\\xi_{2}}$ cannot be\n", - "unambiguously determined in keeping the change of internal energy\n", - "positive, i.e. $dU > 0$. This indicates that the quantities in can be\n", - "either positive or negative in the stable region and become zero at the\n", - "critical point. By the same token, the quantities in can be either\n", - "positive or negative and become positive or negative infinite at the\n", - "critical point.\n", - "\n", - "A profound conclusion from this analysis is that in a stable system even\n", - "though a molar quantity always changes in the same direction as its\n", - "conjugate potential, the same molar quantity may change in the opposite\n", - "direction of a non-conjugate potential, resulting in additional\n", - "anomalies represented by Eq. 1\u201140. One example of is the thermal\n", - "expansion in a closed system, i.e. $dN_{i} = 0$, as follows\n", - "\n", - "Eq. \u2011\n", - "$\\left( \\frac{\\partial V}{\\partial T} \\right)_{S} = \\left( \\frac{\\partial S}{\\partial( - P)} \\right)_{V}$\n", - "\n", - "The left-hand side of can be understood as follows: with the increase of\n", - "temperature, the system regulates its pressure in order to keep the\n", - "entropy from increasing, which results in the volume change of the\n", - "system. The behavior of the system depends on whether the pressure\n", - "decreases or increases in order to maintain the entropy of the system\n", - "constant. If the pressure decreases to maintain the entropy of the\n", - "system constant, the volume would increase with the increase of\n", - "temperature, i.e. the left-hand side of the equation has a positive\n", - "sign, which is also shown by the right-hand side of the equation as the\n", - "changes of $S$ and $\u2013P$ have the same sign. That the volume increases\n", - "with temperature is the normal scenario. On the other hand, if the\n", - "pressure increases to maintain the entropy of the system constant, the\n", - "volume would decrease with the increase of temperature, resulting in a\n", - "negative sign for the left-hand side of the equation. This decrease of\n", - "volume with the increase of temperature is usually considered to be\n", - "anomalous, originated from the increase of entropy by the decrease of\n", - "$\u2013P$ or the increase of pressure. More discussions on entropy will\n", - "follow in Chapter 5.2.5 and Chapter 9.\n" - ], + "source": "## Stability at equilibrium and property anomaly\n\nAs shown by , potentials are homogenous for a homogeneous system in a\nstate of internal equilibrium. To study the stability of the equilibrium\nstate, one considers the entropy production due to a fluctuation of a\nmolar quantity as an internal process. Based on the second law of\nthermodynamics, the driving force, as the first derivative of the\nentropy production with respect to the internal process, is zero for\nsuch a fluctuation at equilibrium, i.e. *D*=0, and the entropy of\nproduction thus depends on the second derivative. It can be written as\nfollows\n\nEq. \u2011\n$Td_{ip}S = \\frac{\\partial_{ip}S}{\\partial\\xi}d\\xi + \\frac{1}{2}{\\frac{\\partial_{ip}^{2}S}{\\partial\\xi^{2}}(d\\xi)}^{2} = Dd\\xi - \\frac{1}{2}D_{2}(d\\xi)^{2}$\n\nwith $D_{2} = - \\frac{\\partial_{ip}^{2}S}{\\partial\\xi^{2}}$. When\n$\\frac{\\partial_{ip}^{2}S}{\\partial\\xi^{2}} < 0$ or $D_{2} > 0$ along\nwith $D = 0$, the fluctuation does not produce positive entropy of\nproduction and can thus not develop further. The equilibrium state of\nthe system is therefore stable against the fluctuation. On the other\nhand, when $\\frac{\\partial_{ip}^{2}S}{\\partial\\xi^{2}} > 0$ or\n$D_{2} < 0$ along with $D = 0$, the fluctuation creates positive entropy\nof production and can continue to grow. The equilibrium state of the\nsystem is therefore unstable against the fluctuation. In connection with\n, one can realize that for a system at stable equilibrium without heat\nand mass exchanges with the surroundings, its entropy is at its maximum,\nand there are no other internal processes, which could produce any more\nentropy. This is schematically shown in .\n\nFigure \u2011: Schematic diagram showing maximum entropy\n\nUsing , , and , the combined law of thermodynamics can be written as\n\n*Eq. 1\u201133*\n$dU = \\sum_{}^{}{Y_{i}dX_{i}} - Dd\\xi + \\frac{1}{2}D_{2}(d\\xi)^{2}$\n\nLet us carry out the same virtual internal experiment shown in Chapter ,\ni.e. moving an infinitesimal amount of *Xi* in a homogenous\nsystem with other $X_{j}$ kept constant in an isolated system, i.e.\n$dU = 0$ and $D = 0$. The internal energy change due to this internal\nprocess is\n\nEq. \u2011\n$dU = \\frac{1}{2}D_{2}\\left\\{ \\left( dX_{i}^{'} \\right)^{2} + \\left( dX_{i}^{\"} \\right)^{2} \\right\\}$\n\nFor a homogeneous system in a state of stable equilibrium with\n$\\left( dX_{i}^{'} \\right)^{2} = \\left( dX_{i}^{\"} \\right)^{2} = \\left( dX_{i} \\right)^{2}$,\nthis internal process must result in an increase of internal energy,\n$dU > 0$, and thus gives\n\nEq. \u2011\n$D_{2} = 2\\left( \\frac{\\partial^{2}U}{\\partial\\left( X_{i} \\right)^{2}} \\right)_{X_{j}} = 2\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}} > 0$\n\nshows that for a system to be stable, any pair of conjugate variables\nmust change in the same direction when other independent molar\nquantities are kept constant. For the conjugate variables discussed so\nfar, it means that for a stable system, the addition of entropy\nincreases temperature with $\\frac{\\partial T}{\\partial S}$\\>0, the\nvolume decreases with pressure or increases with the negative of\npressure with $\\frac{\\partial( - P)}{\\partial V} > 0$, and the chemical\npotential of a component increases with its amount, i.e.\n$\\frac{\\partial\\mu_{i}}{\\partial N_{i}} > 0$, where the derivatives are\ntaken with all other molar quantities kept constant. The limit of\nstability is reached when becomes zero, i.e.\n\nEq. \u2011\n$D_{2} = 2\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}} = 0$\n\nshows schematically the energy as a function of configurations including\nthree states: unstable, stable, and metastable. Both the stable and\nmetastable states have positive curvatures due to $D_{2} > 0$, while the\nunstable state has a negative curvature due to $D_{2} < 0$. There is an\ninflection point of $D_{2} = 0$ for a state between a stable or\nmetastable state with $D_{2} > 0$ and an unstable state with\n$D_{2} < 0$. These two inflection points, called spinodal, represent the\nlimit of stability. The states between the two inflection points are\nunstable, and other states are either stable or metastable. The two\ninflection points can move apart from or close to each other depending\non the change of external conditions, i.e. the natural variables. One\nextreme situation is when these two inflection points merge into one\npoint, and the instability occurs only at this particular point. It is\nevident that all three states, stable, metastable, and unstable, also\nmerge into one point. This point is called critical or consolute point,\nbeyond which the instability no longer exists.\n\nFigure \u2011: Schematic diagram showing the stable and unstable equilibrium\nstates\n\nTo mathematically define the consolute point, the third derivative needs\nto be added to because both $D$ and $D_{2}$ vanish at this point, i.e.\n\nEq. \u2011\n$Td_{ip}S = \\frac{\\partial_{ip}S}{\\partial\\xi}d\\xi + \\frac{1}{2}{\\frac{\\partial_{ip}^{2}S}{\\partial\\xi^{2}}(d\\xi)}^{2} + \\frac{1}{6}{\\frac{\\partial_{ip}^{3}S}{\\partial\\xi^{3}}(d\\xi)}^{3} = Dd\\xi - \\frac{1}{2}D_{2}(d\\xi)^{2} + \\frac{1}{6}{D_{3}(d\\xi)}^{3}$\n\nEq. \u2011\n$dU = \\sum_{}^{}{Y_{i}dX_{i}} - Dd\\xi + \\frac{1}{2}D_{2}(d\\xi)^{2} - \\frac{1}{6}{D_{3}(d\\xi)}^{3}$\n\nAt the consolute point, the third derivative also becomes zero, i.e.\n\nEq. \u2011 $D_{3} = \\frac{\\partial_{ip}^{3}S}{\\partial\\xi^{3}}^{3} = 0$\n\nLet us further discuss the properties of the system in relation to the\ncritical point. By taking the inverse of the equation of the limit of\nstability, , one obtains\n\nEq. \u2011\n$\\left( \\frac{\\partial X_{i}}{\\partial Y_{i}} \\right)_{X_{j}} = + \\ \\infty$\n\ni.e. all $X_{i}$ quantities diverge at the critical point. Therefore,\nwhen a system approaches the critical point from its stable region, the\nchange of a molar quantity with respect to its conjugate potential\nvaries dramatically and becomes infinite at the critical point,\nresulting in property anomalies in the system. In the unstable region,\nthe system would thus separate into stable subsystems and becomes\nheterogeneous, and $X_{i}$:s change discontinuously between subsystems.\nWhile in the stable region, the change of a molar quantity with respect\nto its conjugate potential decreases as the system moves away from the\ncritical point and remains positive due to the stability criteria\ndenoted by .\n\nHowever, it is not clear how a molar quantity changes with respect to a\nnon-conjugate potential at the critical point. From the Maxwell\nrelation, one has\n\nEq. \u2011\n$\\left( \\frac{\\partial Y_{i}}{\\partial X_{j}} \\right)_{X_{k \\neq j}} = \\frac{\\partial^{2}U}{\\partial X_{i}\\partial X_{j}} = \\left( \\frac{\\partial Y_{j}}{\\partial X_{i}} \\right)_{X_{k \\neq i}}$\n\nEq. \u2011\n$\\left( \\frac{\\partial X_{j}}{\\partial Y_{i}} \\right)_{X_{k \\neq j}} = \\left( \\frac{\\partial X_{i}}{\\partial Y_{j}} \\right)_{X_{k \\neq i}}$\n\nSince all $X_{i}$:s diverge at the critical point, both derivatives in\nshould also go to infinite at the critical point. To investigate their\nsigns, let us carry out a virtual experiment similar to the one in\nderiving the stability condition ( and ). In this case, two internal\nprocesses are needed for moving two molar quantities simultaneously in\nan isolated system, i.e.\n\n*Eq. 1\u201143*\n$dU = - D_{\\xi_{1}}d\\xi_{1} - D_{\\xi_{2}}d\\xi_{2} + D_{\\xi_{1}\\xi_{2}}d\\xi_{1}d\\xi_{2} + \\frac{1}{2}D_{2\\xi_{1}}\\left( d\\xi_{1} \\right)^{2} + \\frac{1}{2}D_{2\\xi_{2}}\\left( d\\xi_{2} \\right)^{2}$\n\nBased on the above discussions, in a stable system at equilibrium with\n$D_{\\xi_{1}} = D_{\\xi_{2}} = 0$, $D_{2\\xi_{1}} > 0$ and\n$D_{2\\xi_{2}} > 0$, the sign of $D_{\\xi_{1}\\xi_{2}}$ cannot be\nunambiguously determined in keeping the change of internal energy\npositive, i.e. $dU > 0$. This indicates that the quantities in can be\neither positive or negative in the stable region and become zero at the\ncritical point. By the same token, the quantities in can be either\npositive or negative and become positive or negative infinite at the\ncritical point.\n\nA profound conclusion from this analysis is that in a stable system even\nthough a molar quantity always changes in the same direction as its\nconjugate potential, the same molar quantity may change in the opposite\ndirection of a non-conjugate potential, resulting in additional\nanomalies represented by Eq. 1\u201140. One example of is the thermal\nexpansion in a closed system, i.e. $dN_{i} = 0$, as follows\n\nEq. \u2011\n$\\left( \\frac{\\partial V}{\\partial T} \\right)_{S} = \\left( \\frac{\\partial S}{\\partial( - P)} \\right)_{V}$\n\nThe left-hand side of can be understood as follows: with the increase of\ntemperature, the system regulates its pressure in order to keep the\nentropy from increasing, which results in the volume change of the\nsystem. The behavior of the system depends on whether the pressure\ndecreases or increases in order to maintain the entropy of the system\nconstant. If the pressure decreases to maintain the entropy of the\nsystem constant, the volume would increase with the increase of\ntemperature, i.e. the left-hand side of the equation has a positive\nsign, which is also shown by the right-hand side of the equation as the\nchanges of $S$ and $\u2013P$ have the same sign. That the volume increases\nwith temperature is the normal scenario. On the other hand, if the\npressure increases to maintain the entropy of the system constant, the\nvolume would decrease with the increase of temperature, resulting in a\nnegative sign for the left-hand side of the equation. This decrease of\nvolume with the increase of temperature is usually considered to be\nanomalous, originated from the increase of entropy by the decrease of\n$\u2013P$ or the increase of pressure. More discussions on entropy will\nfollow in Chapter 5.2.5 and Chapter 9.\n", "metadata": {} } ] diff --git a/src/psu410/src/phase_equilibria_in_heterogeneous_systems/general_condition_of_equilibrium.ipynb b/src/psu410/src/phase_equilibria_in_heterogeneous_systems/general_condition_of_equilibrium.ipynb index f8957bb..62ed602 100644 --- a/src/psu410/src/phase_equilibria_in_heterogeneous_systems/general_condition_of_equilibrium.ipynb +++ b/src/psu410/src/phase_equilibria_in_heterogeneous_systems/general_condition_of_equilibrium.ipynb @@ -4,119 +4,9 @@ "metadata": {}, "cells": [ { - "id": "b85f238b", + "id": "448bcda8", "cell_type": "markdown", - "source": [ - "## General condition of equilibrium\n", - "\n", - "A system is heterogeneous when some properties have different values at\n", - "different portions of the system when the system is at equilibrium. Two\n", - "scenarios may exist where the variations of properties can be either\n", - "continuous or discontinuous. In the scenario of continuous variations,\n", - "the gradients of those variations must be coupled so that the system\n", - "remains at equilibrium. The number of independent variables is thus\n", - "reduced. These gradients must also be constrained along the boundaries\n", - "between the system and the surroundings. This type of constrained\n", - "equilibrium is not discussed in the book as it involves heterogeneous\n", - "boundary conditions between the system and the surroundings and depends\n", - "on the morphology of the system.\n", - "\n", - "In the second scenario with discontinuous variations, those properties\n", - "have different values in different portions of the system, but remain\n", - "homogenous within each portion. The system is in equilibrium as each\n", - "portion is in equilibrium with all other portions of the system. Those\n", - "homogeneous portions represent different phases in the system with the\n", - "properties in each phase being homogeneous at equilibrium. In the\n", - "previous chapter, it has been shown that all potentials are homogeneous\n", - "in a homogeneous system.\n", - "\n", - "For a heterogeneous system, the same conclusion can be obtained. If the\n", - "temperature is inhomogeneous, heat can be conducted from high\n", - "temperature locations to low temperature locations, and this process is\n", - "irreversible based on the second law of thermodynamics because it\n", - "increases the internal entropy of the system. If the pressure is\n", - "inhomogeneous, the amounts of lower molar volume phases will increase to\n", - "reduce the internal energy of the system. If the chemical potential of a\n", - "component is inhomogeneous, the chemical potential difference of the\n", - "component will drive that component to the locations with a lower\n", - "chemical potential so the internal energy of the system can be\n", - "decreased. Therefore, it can be concluded that all potentials are\n", - "homogeneous in a heterogeneous system at equilibrium, and the variables\n", - "that are not homogeneous are thus their conjugate molar quantities.\n", - "Under certain special circumstances to be discussed in later part of\n", - "this book, some molar quantities may also have the same values in\n", - "difference phases.\n", - "\n", - "In a system at equilibrium with $c$ independent components, there are\n", - "$c + 2$ pairs of conjugate variables based on or though more can be\n", - "added as shown by depending on experimental conditions. For simplicity,\n", - "most discussions in this book are limited to systems with $c + 2$ pairs\n", - "of conjugate variables unless otherwise specified, with number \u201c2\u201d\n", - "representing conjugate variables of $T - S$ and $( - P) - V$.\n", - "\n", - "For a system under constant temperature, pressure, and moles of each\n", - "independent component, the equilibrium condition derives from as\n", - "\n", - "Eq. \u2011\n", - "$dG = - SdT - Vd( - P) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi = - Dd\\xi = 0$\n", - "\n", - "Consequently, the equilibrium state is defined by the minimization of\n", - "Gibbs energy of the system at constant $T$, $P$ and $N_{i}$ because the\n", - "second derivatives need to be positive for the equilibrium system to be\n", - "stable as stipulated by . For heterogeneous systems with two or more\n", - "phases, Gibbs energy of the system is the weighted summation of Gibbs\n", - "energies of individual phases, i.e.\n", - "\n", - "Eq. \u2011 $\\frac{G}{N} = G_{m} = \\sum_{\\beta}^{}{f^{\\beta}G_{m}^{\\beta}}$\n", - "\n", - "where $f^{\\beta}$ and $G_{m}^{\\beta}$ are the mole fraction and molar\n", - "Gibbs energy of the phase $\\beta$, respectively, and the summation goes\n", - "over all phases in the system. $f^{\\beta}$ is equal to zero for phases\n", - "not present in the equilibrium state.\n", - "\n", - "The minimization of Gibbs energy of the system is carried out under the\n", - "following mass balance conditions\n", - "\n", - "Eq. \u2011\n", - "$x_{i} = \\sum_{\\beta}^{}{f^{\\beta}x_{i}^{\\beta}} = \\sum_{\\beta}^{}{f^{\\beta}\\frac{\\sum_{\\beta - t}^{}{a^{\\beta - t}\\sum_{k}^{}{i_{k}^{\\beta - t}y_{k}^{\\beta - t}}}}{\\sum_{\\beta - t}^{}{a^{\\beta - t}\\left( 1 - y_{va}^{\\beta - t} \\right)}}}$\n", - "\n", - "Eq. \u2011 $\\sum_{i}^{}x_{i} = 1$\n", - "\n", - "Eq. \u2011 $\\sum_{k}^{}y_{k}^{\\beta - t} = 1$\n", - "\n", - "where $a^{\\beta - t}$ and $y_{k}^{\\beta - t}$ are the number of site in\n", - "sublattice $t$ in the $\\beta$ phase and the corresponding site fraction\n", - "of species $k$ in the sublattice, respectively, and $i_{k}^{\\beta - t}$\n", - "is the stoichiometry of the component $i$ in the species $k$, as used in\n", - ". The summation in runs over species for each sublattice. For phases\n", - "containing ionic species, electroneutrality also needs to be maintained,\n", - "i.e. is applied to each phase. This minimization problem of Gibbs energy\n", - "under the constraint of mass conservation can be solved by means of a\n", - "range of algorithms. It should be noted that the mole fractions of\n", - "phases and site fractions of species are bounded between 0 and 1.\n", - "\n", - "This minimization procedure must result in that potentials are\n", - "homogeneous in the system as discussed above. Since the present book\n", - "deals with thermodynamics of materials, the chemical potential of each\n", - "component is of particular interest and must be homogenous in all phases\n", - "of the system at equilibrium, i.e.\n", - "\n", - "Eq. \u2011\n", - "$\\mu_{i}^{\\alpha} = \\mu_{i}^{\\beta} = \\mu_{i}^{\\gamma}\\ldots\\ldots$\n", - "\n", - "For phases in which the chemical potentials of individual components\n", - "cannot be evaluated due to stoichiometry, the combined chemical\n", - "potentials can be used to relate individual potentials as shown by and .\n", - "As proved in Chapter shown by , the chemical potential of a component in\n", - "a solution is represented by the intercept on the Gibbs energy axis by\n", - "the multi-dimensional tangent surface of Gibbs energy of the solution\n", - "plotted with respect to mole fractions of independent components. The\n", - "Gibbs energy functions of all phases in equilibrium must thus share the\n", - "same tangent surface. This is usually referred to as the common tangent\n", - "construction for phases at equilibrium. Any phase with its Gibbs energy\n", - "curve above the tangent surface is not stable under given compositions\n", - "of the system.\n" - ], + "source": "## General condition of equilibrium\n\nA system is heterogeneous when some properties have different values at\ndifferent portions of the system when the system is at equilibrium. Two\nscenarios may exist where the variations of properties can be either\ncontinuous or discontinuous. In the scenario of continuous variations,\nthe gradients of those variations must be coupled so that the system\nremains at equilibrium. The number of independent variables is thus\nreduced. These gradients must also be constrained along the boundaries\nbetween the system and the surroundings. This type of constrained\nequilibrium is not discussed in the book as it involves heterogeneous\nboundary conditions between the system and the surroundings and depends\non the morphology of the system.\n\nIn the second scenario with discontinuous variations, those properties\nhave different values in different portions of the system, but remain\nhomogenous within each portion. The system is in equilibrium as each\nportion is in equilibrium with all other portions of the system. Those\nhomogeneous portions represent different phases in the system with the\nproperties in each phase being homogeneous at equilibrium. In the\nprevious chapter, it has been shown that all potentials are homogeneous\nin a homogeneous system.\n\nFor a heterogeneous system, the same conclusion can be obtained. If the\ntemperature is inhomogeneous, heat can be conducted from high\ntemperature locations to low temperature locations, and this process is\nirreversible based on the second law of thermodynamics because it\nincreases the internal entropy of the system. If the pressure is\ninhomogeneous, the amounts of lower molar volume phases will increase to\nreduce the internal energy of the system. If the chemical potential of a\ncomponent is inhomogeneous, the chemical potential difference of the\ncomponent will drive that component to the locations with a lower\nchemical potential so the internal energy of the system can be\ndecreased. Therefore, it can be concluded that all potentials are\nhomogeneous in a heterogeneous system at equilibrium, and the variables\nthat are not homogeneous are thus their conjugate molar quantities.\nUnder certain special circumstances to be discussed in later part of\nthis book, some molar quantities may also have the same values in\ndifference phases.\n\nIn a system at equilibrium with $c$ independent components, there are\n$c + 2$ pairs of conjugate variables based on or though more can be\nadded as shown by depending on experimental conditions. For simplicity,\nmost discussions in this book are limited to systems with $c + 2$ pairs\nof conjugate variables unless otherwise specified, with number \u201c2\u201d\nrepresenting conjugate variables of $T - S$ and $( - P) - V$.\n\nFor a system under constant temperature, pressure, and moles of each\nindependent component, the equilibrium condition derives from as\n\nEq. \u2011\n$dG = - SdT - Vd( - P) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi = - Dd\\xi = 0$\n\nConsequently, the equilibrium state is defined by the minimization of\nGibbs energy of the system at constant $T$, $P$ and $N_{i}$ because the\nsecond derivatives need to be positive for the equilibrium system to be\nstable as stipulated by . For heterogeneous systems with two or more\nphases, Gibbs energy of the system is the weighted summation of Gibbs\nenergies of individual phases, i.e.\n\nEq. \u2011 $\\frac{G}{N} = G_{m} = \\sum_{\\beta}^{}{f^{\\beta}G_{m}^{\\beta}}$\n\nwhere $f^{\\beta}$ and $G_{m}^{\\beta}$ are the mole fraction and molar\nGibbs energy of the phase $\\beta$, respectively, and the summation goes\nover all phases in the system. $f^{\\beta}$ is equal to zero for phases\nnot present in the equilibrium state.\n\nThe minimization of Gibbs energy of the system is carried out under the\nfollowing mass balance conditions\n\nEq. \u2011\n$x_{i} = \\sum_{\\beta}^{}{f^{\\beta}x_{i}^{\\beta}} = \\sum_{\\beta}^{}{f^{\\beta}\\frac{\\sum_{\\beta - t}^{}{a^{\\beta - t}\\sum_{k}^{}{i_{k}^{\\beta - t}y_{k}^{\\beta - t}}}}{\\sum_{\\beta - t}^{}{a^{\\beta - t}\\left( 1 - y_{va}^{\\beta - t} \\right)}}}$\n\nEq. \u2011 $\\sum_{i}^{}x_{i} = 1$\n\nEq. \u2011 $\\sum_{k}^{}y_{k}^{\\beta - t} = 1$\n\nwhere $a^{\\beta - t}$ and $y_{k}^{\\beta - t}$ are the number of site in\nsublattice $t$ in the $\\beta$ phase and the corresponding site fraction\nof species $k$ in the sublattice, respectively, and $i_{k}^{\\beta - t}$\nis the stoichiometry of the component $i$ in the species $k$, as used in\n. The summation in runs over species for each sublattice. For phases\ncontaining ionic species, electroneutrality also needs to be maintained,\ni.e. is applied to each phase. This minimization problem of Gibbs energy\nunder the constraint of mass conservation can be solved by means of a\nrange of algorithms. It should be noted that the mole fractions of\nphases and site fractions of species are bounded between 0 and 1.\n\nThis minimization procedure must result in that potentials are\nhomogeneous in the system as discussed above. Since the present book\ndeals with thermodynamics of materials, the chemical potential of each\ncomponent is of particular interest and must be homogenous in all phases\nof the system at equilibrium, i.e.\n\nEq. \u2011\n$\\mu_{i}^{\\alpha} = \\mu_{i}^{\\beta} = \\mu_{i}^{\\gamma}\\ldots\\ldots$\n\nFor phases in which the chemical potentials of individual components\ncannot be evaluated due to stoichiometry, the combined chemical\npotentials can be used to relate individual potentials as shown by and .\nAs proved in Chapter shown by , the chemical potential of a component in\na solution is represented by the intercept on the Gibbs energy axis by\nthe multi-dimensional tangent surface of Gibbs energy of the solution\nplotted with respect to mole fractions of independent components. The\nGibbs energy functions of all phases in equilibrium must thus share the\nsame tangent surface. This is usually referred to as the common tangent\nconstruction for phases at equilibrium. Any phase with its Gibbs energy\ncurve above the tangent surface is not stable under given compositions\nof the system.\n", "metadata": {} } ] diff --git a/src/psu410/src/phase_equilibria_in_heterogeneous_systems/gibbs_phase_rule.ipynb b/src/psu410/src/phase_equilibria_in_heterogeneous_systems/gibbs_phase_rule.ipynb index 49b84eb..89e0c82 100644 --- a/src/psu410/src/phase_equilibria_in_heterogeneous_systems/gibbs_phase_rule.ipynb +++ b/src/psu410/src/phase_equilibria_in_heterogeneous_systems/gibbs_phase_rule.ipynb @@ -4,60 +4,9 @@ "metadata": {}, "cells": [ { - "id": "2c9e5b25", + "id": "bb986421", "cell_type": "markdown", - "source": [ - "## Gibbs phase rule\n", - "\n", - "The Gibbs-Duhem equation, i.e. , states that only $c + 1$ potentials are\n", - "independent in a homogeneous system with $c$ independent components and\n", - "the additional two variables of temperature and pressure. In a\n", - "heterogeneous system at equilibrium, this equation can be applied to\n", - "individual phases as each phase is homogeneous. Noting that each\n", - "potential has the same value in all phases at equilibrium, can be\n", - "written as follows for each individual phase, $\\ \\beta$, in the system\n", - "at equilibrium\n", - "\n", - "Eq. \u2011\n", - "$0 = - S^{\\beta}dT - V^{\\beta}d( - P) - \\sum_{}^{}{N_{i}^{\\beta}d\\mu}_{i}$\n", - "\n", - "For a system with $p$ phases at equilibrium, there are $p$ such\n", - "equations relating the potentials in the system. The number of\n", - "independent potentials thus becomes\n", - "\n", - "Eq. \u2011 $\\upsilon = c + 2 - p$\n", - "\n", - "is called Gibbs phase rule. It dictates the number of potentials that\n", - "can change independently for a given number of phases co-existing at\n", - "equilibrium, commonly called degree of freedom of the system at\n", - "equilibrium. It stipulates that the maximum number of phases can\n", - "co-exist in a system at equilibrium is obtained by setting\n", - "$\\upsilon = 0$, called an invariant equilibrium due to the zero degree\n", - "of freedom,\n", - "\n", - "Eq. \u2011 $p_{\\max} = c + 2$\n", - "\n", - "There are thus maximum three phases in a one-component system, four\n", - "phases in a binary system, five phases in a ternary system, and so on,\n", - "that can co-exist simultaneously at equilibrium with all potentials in\n", - "the system at fixed values. This should not be confused with the total\n", - "number of phases that could exist, but not co-exist in a system, which\n", - "of course are not limited by Gibbs phase rule.\n", - "\n", - "It should be emphasized that the degree of freedom, $\\upsilon$, is\n", - "referred to the number of potentials only, not to molar quantities of\n", - "the system because molar quantities are generally not homogeneous in a\n", - "heterogeneous system. For example, in a system at equilibrium with\n", - "$\\upsilon = 0$, the amount of each component can be varied, while\n", - "keeping the number of phases at $p_{\\max} = c + 2$. This can be done\n", - "through changing the amount of each phase in the system through the mass\n", - "balance equation, , without altering the composition of each phase and\n", - "thus the chemical potentials in the system. As mentioned at the\n", - "beginning of Chapter , the number of independent variables in a system\n", - "at equilibrium, i.e. the sum of independent potentials and independent\n", - "molar quantities, is $c + 2$ with the maximum number of independent\n", - "potentials determined by Gibbs phase rule, .\n" - ], + "source": "## Gibbs phase rule\n\nThe Gibbs-Duhem equation, i.e. , states that only $c + 1$ potentials are\nindependent in a homogeneous system with $c$ independent components and\nthe additional two variables of temperature and pressure. In a\nheterogeneous system at equilibrium, this equation can be applied to\nindividual phases as each phase is homogeneous. Noting that each\npotential has the same value in all phases at equilibrium, can be\nwritten as follows for each individual phase, $\\ \\beta$, in the system\nat equilibrium\n\nEq. \u2011\n$0 = - S^{\\beta}dT - V^{\\beta}d( - P) - \\sum_{}^{}{N_{i}^{\\beta}d\\mu}_{i}$\n\nFor a system with $p$ phases at equilibrium, there are $p$ such\nequations relating the potentials in the system. The number of\nindependent potentials thus becomes\n\nEq. \u2011 $\\upsilon = c + 2 - p$\n\nis called Gibbs phase rule. It dictates the number of potentials that\ncan change independently for a given number of phases co-existing at\nequilibrium, commonly called degree of freedom of the system at\nequilibrium. It stipulates that the maximum number of phases can\nco-exist in a system at equilibrium is obtained by setting\n$\\upsilon = 0$, called an invariant equilibrium due to the zero degree\nof freedom,\n\nEq. \u2011 $p_{\\max} = c + 2$\n\nThere are thus maximum three phases in a one-component system, four\nphases in a binary system, five phases in a ternary system, and so on,\nthat can co-exist simultaneously at equilibrium with all potentials in\nthe system at fixed values. This should not be confused with the total\nnumber of phases that could exist, but not co-exist in a system, which\nof course are not limited by Gibbs phase rule.\n\nIt should be emphasized that the degree of freedom, $\\upsilon$, is\nreferred to the number of potentials only, not to molar quantities of\nthe system because molar quantities are generally not homogeneous in a\nheterogeneous system. For example, in a system at equilibrium with\n$\\upsilon = 0$, the amount of each component can be varied, while\nkeeping the number of phases at $p_{\\max} = c + 2$. This can be done\nthrough changing the amount of each phase in the system through the mass\nbalance equation, , without altering the composition of each phase and\nthus the chemical potentials in the system. As mentioned at the\nbeginning of Chapter , the number of independent variables in a system\nat equilibrium, i.e. the sum of independent potentials and independent\nmolar quantities, is $c + 2$ with the maximum number of independent\npotentials determined by Gibbs phase rule, .\n", "metadata": {} } ] diff --git a/src/psu410/src/phase_equilibria_in_heterogeneous_systems/index.ipynb b/src/psu410/src/phase_equilibria_in_heterogeneous_systems/index.ipynb index bd980a9..dc56fcd 100644 --- a/src/psu410/src/phase_equilibria_in_heterogeneous_systems/index.ipynb +++ b/src/psu410/src/phase_equilibria_in_heterogeneous_systems/index.ipynb @@ -4,11 +4,9 @@ "metadata": {}, "cells": [ { - "id": "f99b4808", + "id": "84575b0c", "cell_type": "markdown", - "source": [ - "# Phase equilibria in heterogeneous systems\n" - ], + "source": "# Phase equilibria in heterogeneous systems\n", "metadata": {} } ] diff --git a/src/psu410/src/phase_equilibria_in_heterogeneous_systems/molar_phase_diagrams.ipynb b/src/psu410/src/phase_equilibria_in_heterogeneous_systems/molar_phase_diagrams.ipynb index bcdc3ca..112a8ec 100644 --- a/src/psu410/src/phase_equilibria_in_heterogeneous_systems/molar_phase_diagrams.ipynb +++ b/src/psu410/src/phase_equilibria_in_heterogeneous_systems/molar_phase_diagrams.ipynb @@ -4,356 +4,33 @@ "metadata": {}, "cells": [ { - "id": "ff60c582", + "id": "40a62bf1", "cell_type": "markdown", - "source": [ - "## Molar phase diagrams\n", - "\n", - "The potential phase diagrams discussed in Chapter present information on\n", - "which phases are in equilibrium under given values of potentials, but do\n", - "not have any information on the properties of phases in equilibrium. On\n", - "the other hand, there are direct relations between potentials and their\n", - "conjugate molar quantities for each phase at equilibrium depicted by to\n", - ". One may thus substitute the potentials by their conjugate molar\n", - "quantities in the potential phase diagrams as molar quantities provide\n", - "more information on the properties of phases and the system. This is\n", - "particularly true when chemical potentials are replaced by compositions\n", - "as the compositions of the system are often the variables controlled and\n", - "measured in experiments instead of chemical potentials.\n" - ], + "source": "## Molar phase diagrams\n\nThe potential phase diagrams discussed in Chapter present information on\nwhich phases are in equilibrium under given values of potentials, but do\nnot have any information on the properties of phases in equilibrium. On\nthe other hand, there are direct relations between potentials and their\nconjugate molar quantities for each phase at equilibrium depicted by to\n. One may thus substitute the potentials by their conjugate molar\nquantities in the potential phase diagrams as molar quantities provide\nmore information on the properties of phases and the system. This is\nparticularly true when chemical potentials are replaced by compositions\nas the compositions of the system are often the variables controlled and\nmeasured in experiments instead of chemical potentials.\n", "metadata": {} }, { - "id": "4a4a0253", + "id": "f8421ad2", "cell_type": "markdown", - "source": [ - "### Phase diagrams with both potentials and molar quantities\n", - "\n", - "Based on Gibbs phase rule discussed in Chapters and , the\n", - "dimensionalities of phase regions in a potential phase diagram are given\n", - "by or . With every potential substituted by its conjugate molar\n", - "quantity, the dimensionalities of phase regions increase by one until\n", - "the phase region reaches the dimensionality of the phase diagram. The\n", - "axes of this phase diagram now consist of both potentials and molar\n", - "quantities. The dimensionality of a phase region can thus be represented\n", - "by the following equation based on\n", - "\n", - "Eq. \u2011\n", - "$\\upsilon_{m} = \\left( c - n_{s} \\right) + 2 - p + n_{m} \\leq \\left( c - n_{s} \\right) + 1$\n", - "\n", - "where $n_{m}$ is the number of molar axes. This equation is applicable\n", - "to phase regions with more than $n_{m} + 1$ phases. For phase regions\n", - "with $n_{m} + 1$ phases or fewer, the dimensionalities are the same as\n", - "the phase diagram, i.e. $\\upsilon_{m} = \\left( c - n_{s} \\right) + 1$,\n", - "and no longer vary with the number of molar quantity axes. When all\n", - "$\\left( c - n_{s} \\right) + 1$ potentials are substituted by their\n", - "conjugate molar quantities, one obtains a complete molar phase diagrams\n", - "to be discussed in Chapter , and all phase regions have the\n", - "same-dimensionality of $\\left( c - n_{s} \\right) + 1$.\n", - "\n", - "For the sake of graphic visualization, let us exam a two-dimensional\n", - "phase diagram of a one-component system. Topologically, it is equivalent\n", - "to a multi-component system with $n_{s} = c - 1$. In , three\n", - "two-dimensional phase diagrams are shown for pure Fe. In principle, one\n", - "can use any one of them to illustrate mixed potential and molar phase\n", - "diagrams. For the purpose of practical usefulness, one selects the\n", - "$T - ( - P)$ potential diagram as temperature and pressure are the two\n", - "typical variables controlled in experiments of one-component systems.\n", - "The conjugate molar quantities of $- P$ and $T$ are molar volume and\n", - "molar entropy, respectively. For stable phases, any pair of conjugate\n", - "variables changes in the same direction as illustrated by and , i.e. the\n", - "phase stable at higher $T$ has higher molar entropy, and the phase\n", - "stable at higher $\u2013P$, i.e. lower pressure, has higher molar volume. Let\n", - "us first substitute $\u2013P$ by $V_{m}$ as shown in a. The dimensionality of\n", - "a sing-phase region remains unchanged because of\n", - "$p = 1 < 2 = n_{m} + 1$. The dimensionality of two-phase regions is\n", - "changed from 1 to 2 due to $\\upsilon_{m} = 3 - 2 + 1 = 2$ from .\n", - "\n", - "Figure \u2011: T-Vm and Sm-(-P) phase diagrams of Fe\n", - "\n", - "As both phases in a two-phase equilibrium have the same temperature, all\n", - "tie-lines, depicted by dotted lines, are perpendicular to the\n", - "temperature axis in a. When the molar volume of the system changes from\n", - "one end of a tie-line to another end at a constant temperature, the mole\n", - "fraction of one phase increases from 0 to 1, and the mole fraction of\n", - "another phase decreases from 1 to 0. The tine-lines at various\n", - "temperatures combine together to form a two-dimensional two-phase\n", - "region. The two curves at the two ends of tie-lines represent the\n", - "boundaries between the single-phase and two-phase regions and are no\n", - "longer phase regions themselves. They are thus called phase boundaries.\n", - "\n", - "By the same token, by changing the temperature at a constant molar\n", - "volume of the system, the system will locate on different tie-lines with\n", - "the amounts of the two phases determined by the lever rule. It is thus\n", - "clear that the system maintains the two-phase equilibrium state with\n", - "both $T$ and $V_{m}$ changing independently inside the two-dimensional\n", - "two-phase region. This seems in contradiction to Gibbs phase rule being\n", - "$\\upsilon = 3 - p = 1$ from , but it is not because Gibbs phase rule\n", - "applies strictly to potential phase diagrams only, while the $T - V_{m}$\n", - "phase diagram has one of its axis being a molar quantity. As an\n", - "alternative, one may consider as a modified Gibbs phase rule in\n", - "describing the dimensionality of a phase region in a mixed potential and\n", - "molar phase diagram with $p \\geq n_{m} + 1$.\n", - "\n", - "The two three-phase equilibria in pure Fe are also represented by\n", - "tie-lines connecting all three phases. The dimensionality of three-phase\n", - "regions is $\\upsilon_{m} = 3 - 3 + 1 = 1$ from , and the three two-phase\n", - "tie-lines for a three-phase equilibrium thus overlap each other with\n", - "their three molar volumes on the same tie-line.\n", - "\n", - "Let us examine the two three-phase equilibria in more detail. In the\n", - "\u03b3/\u03b1/\u03b5 three-phase equilibrium at $T_{E} = 756.6\\ K$ and\n", - "$\\left( - P_{E} \\right) = - 1.046 \\bullet 10^{10}\\ Pa$, the molar\n", - "volumes of \u03b1, \u03b3, and \u03b5 are 6.837, 6.677, 6.582 $10^{- 6}\\ m^{3}/mol$,\n", - "respectively. There are two two-phase regions at higher temperatures and\n", - "one two-phase region at lower temperatures. This is also shown in the\n", - "potential phase diagram of a with two two-phase curves entering into and\n", - "one two-phase curve leaving from the three-phase equilibrium point with\n", - "decreasing temperature. Considering the system with the fixed molar\n", - "volume equal to that of the $\\gamma$ phase, i.e.\n", - "$V_{m}^{\\ }\\left( T_{E} \\right) = 6.677 \\bullet 10^{- 6}\\ m^{3}/mol$. At\n", - "$T > T_{E}$, the system is in the single \u03b3 phase region. With decrease\n", - "in temperature across $T_{E}$, it enters into the $\\alpha + \\varepsilon$\n", - "two-phase region. This transformation can be written as follows and is\n", - "called a eutectoid reaction\n", - "\n", - "Eq. \u2011 $\\gamma \\rightarrow \\alpha + \\varepsilon$\n", - "\n", - "This type of transformations is named as eutectic reaction if the high\n", - "temperature phase is a liquid phase. If the system molar volume is\n", - "between $V_{m}^{\\gamma}$ and $V_{m}^{\\alpha}$ at $756.6K$, with decrease\n", - "of temperature, the system first moves from the single \u03b3 phase region to\n", - "the $\\alpha + \\gamma$ two-phase region when the\n", - "$\\gamma/(\\alpha + \\gamma)$ phase boundary is crossed. When the\n", - "temperature reaches $T_{E}$, the eutectoid transformation takes place in\n", - "the remaining \u03b3 phase. The $\\alpha$ formed prior to the eutectoid\n", - "transformation is called proeutectoid $\\alpha$. By the same token, when\n", - "the system molar volume is between $V_{m}^{\\gamma}$ and\n", - "$V_{m}^{\\varepsilon}$ at $756.6K$, proeutectoid $\\varepsilon$ would form\n", - "followed by the eutectoid transformation with decrease in temperature.\n", - "\n", - "On the other hand, the $L/\\delta/\\gamma$ three-phase equilibrium at\n", - "$T_{P} = 1977.9\\ K$ and\n", - "$\\left( - P_{P} \\right) = - 5.111 \\bullet \\ 10^{9}\\ Pa$ has different\n", - "characteristics, where the subscript P will be defined shortly. There\n", - "are one two-phase equilibrium above and two two-phase equilibria below\n", - "the invariant temperature, shown in the potential phase diagram of a\n", - "with one two-phase curve entering into and two two-phase curves leaving\n", - "from the three-phase equilibrium point with decreasing temperature. The\n", - "molar volumes of $L$, $\\delta$, and $\\gamma$ at $T_{P}$ are 7.735,\n", - "7.542, and 7.498 $10^{- 6}\\ m^{3}/mol$, respectively. For $T > T_{P}$,\n", - "the two-phase region is $L + \\gamma$. If the system molar volume is\n", - "between 7.735 and 7.498 $10^{- 6}\\ m^{3}/mol$, i.e.\n", - "$V_{m}^{L}\\left( T_{P} \\right)$ and\n", - "$V_{m}^{\\gamma}\\left( T_{P} \\right)$, when the temperature reaches\n", - "$T_{P}$, $L$ and $\\gamma$ are combined to form $\\delta$ with the\n", - "transformation written as\n", - "\n", - "Eq. \u2011 $L + \\gamma \\rightarrow \\delta$\n", - "\n", - "This type of reactions is called as peritectic reaction or peritectoid\n", - "reaction when the liquid phase is replaced by a solid phase, denoted by\n", - "the subscript P. At $T < T_{P}$, one or both high temperature phases may\n", - "no longer be present in equilibrium depending on the value of the system\n", - "molar volume. For\n", - "$V_{m}^{\\ } = V_{m}^{\\delta}\\left( T_{P} \\right) = 7.542 \\bullet 10^{- 6}\\ m^{3}/mol$,\n", - "the peritectic reaction, , can come to completion with no $L$ and\n", - "$\\gamma$ left upon decreasing temperature. For\n", - "$V_{m}^{\\gamma}\\left( T_{P} \\right) = 7.498 < V_{m}^{\\ } < V_{m}^{\\delta}\\left( T_{P} \\right) = 7.542\\ \\left( 10^{- 6}\\ m^{3}/mol \\right)$,\n", - "the liquid phase is consumed, and the system enters the\n", - "$\\gamma + \\delta$ two-phase region. On the other hand, for\n", - "$V_{m}^{\\delta}\\left( T_{P} \\right) = 7.542 < V_{m}^{\\ } < V_{m}^{\\ L}\\left( T_{P} \\right) = 7.735\\ \\ \\left( 10^{- 6}\\ m^{3}/mol \\right)$,\n", - "the $\\gamma$ phase is consumed instead, and the system enters into the\n", - "$L + \\delta$ two-phase equilibrium region upon cooling.\n", - "\n", - "Let us now replace $T$ by $S_{m}$ to obtain the $( - P) - S_{m}$ phase\n", - "diagram shown in b. The morphology of this phase diagram is identical to\n", - "the $T - V_{m}$ phase diagram just discussed with all tie-lines\n", - "perpendicular to the pressure axis. The transformations at the two\n", - "three-phase equilibria with $( - P)$ decreasing or $P$ increasing are as\n", - "follows\n", - "\n", - "Eq. \u2011 $\\gamma + \\alpha \\rightarrow \\varepsilon$\n", - "\n", - "Eq. \u2011 $\\delta \\rightarrow L + \\gamma$\n", - "\n", - "To visualize two-dimensional phase diagrams of binary systems, one\n", - "usually keeps the pressure constant. One type of commonly used binary\n", - "phase diagram is the temperature-composition ($T - x$) phase diagram. As\n", - "an example, let us re-plot the $T - \\mu_{C}$ potential diagram shown in\n", - "into a $T - x_{C}$ mixed potential and molar phase diagram by replacing\n", - "the chemical potential of $C$ by its mole fraction. The $T - x_{C}$\n", - "phase diagram thus obtained is shown in . In this phase diagram there\n", - "are one peritectic reaction and two eutectic reactions as follows\n", - "\n", - "Eq. \u2011 $L + \\delta \\rightarrow \\gamma$\n", - "\n", - "Eq. \u2011 $L \\rightarrow \\gamma + C$\n", - "\n", - "Eq. 3\u201161 $\\gamma \\rightarrow \\alpha + C$\n", - "\n", - "Figure \u2011: $T - x_{C}\\ $ phase diagram of the Fe-C binary system\n", - "\n", - "In in Chapter , it was discussed the formation of miscibility gaps due\n", - "to repulsive interactions between components. One example is shown in\n", - "for the Al-Zn binary system in terms of both $T - \\mu_{Zn}$ potential\n", - "phase diagram and $\\ T - x_{Zn}\\ $ mixed potential and molar phase\n", - "diagram.\n", - "\n", - "Figure \u2011: $T - \\mu_{Zn}$ potential phase diagram (a) and\n", - "$\\ T - x_{Zn}\\ $ mixed potential and molar phase diagram (b) of the\n", - "Al-Zn binary system\n", - "\n", - "In ,there are one eutectic reaction and one eutectoid reaction as\n", - "follows\n", - "\n", - "Eq. \u2011 $L \\rightarrow fcc + hcp$\n", - "\n", - "Eq. \u2011 $fcc\\# 1 \\rightarrow fcc\\# 2 + hcp$\n", - "\n", - "The eutectoid reaction, , is also termed as monotectoid reaction because\n", - "the fcc phase appears on both sides of the reaction with different\n", - "compositions, $fcc\\# 1$ and $fcc\\# 2$, due to the miscibility gap. The\n", - "highest temperature of the miscibility gap is called the consolute point\n", - "as discussed in Chapter , which can be clearly seen in the\n", - "$T - \\mu_{Zn}$ potential phase diagram shown in . This is a critical\n", - "point, marking the limit of instability as shown in .\n", - "\n", - "When there is only one phase on either side of the reaction, i.e. both\n", - "phases have the same composition, the reaction is called a congruent\n", - "reaction. One example is shown in for the $T - x_{SiO_{2}}\\ $ mixed\n", - "potential and molar phase diagram of the CaO-SiO2\n", - "pseudo-binary system with two congruent reactions as follows\n", - "\n", - "Eq. 3\u201164 $L \\rightarrow CaSiO_{3}$\n", - "\n", - "Eq. 3\u201165 $L \\rightarrow {Ca}_{2}SiO_{4}$\n", - "\n", - "They are not invariant reactions based on the Gibbs phase rule. In , it\n", - "is noted that there are one miscibility gap in the liquid phase close to\n", - "the $SiO_{2}$ side, four eutectic reactions with one being monotectic\n", - "involving two liquid phases due to the miscibility gap, and three\n", - "peritectic reactions.\n", - "\n", - "Figure \u2011: $T - x_{SiO_{2}}\\ $ mixed potential and molar phase diagram of\n", - "the CaO-SiO2 pseudo-binary system.\n", - "\n", - "Let us generalize the above discussion to phase diagrams with\n", - "$\\left( c - n_{s} \\right) + 1$ axes. In such a phase diagram, the\n", - "maximum number of phases is given by as\n", - "$p_{\\max} = \\left( c - n_{s} \\right) + 2$. The number of phases on the\n", - "either side of an invariant reaction can vary from one phase to\n", - "$p_{\\max} - 1 = \\left( c - n_{s} \\right) + 1$ phases with the remaining\n", - "phases on the other side of the reaction, typically with the potential\n", - "decreasing from left to right. The invariant reaction with one phase on\n", - "the left of the reaction is named as eutectic, or eutectoid reaction\n", - "depending on if the phase on the left of the reaction is liquid or\n", - "solid. The rest invariant reactions are named as peritectic or\n", - "peritectoid reactions with or without a liquid phase.\n" - ], + "source": "### Tie-lines and lever rule\n\nIt is self-evident from to Eq. 2\u20114 that while potentials are homogeneous\nin all phases in a heterogeneous system at equilibrium, the molar\nquantities usually have different values in individual phases. This is\nalso stipulated in various Clausius-Clapeyron equations such as to and\nto . The difference in molar quantities thus increases the\ndimensionality of the phase region by the number of potentials replaced\nby their conjugate molar quantities. The maximum dimensionality of a\nphase region is the dimensionality of the phase diagram under\nconsideration. This thus creates a finite space between phases in\nequilibrium in the phase diagram with some axes being molar quantities.\n\nFor an equilibrium system under constant $T$, $P$ and $N_{i}$, the\npotentials in the system and their conjugate molar quantities in each\nphase are all uniquely defined. In a phase diagram with one or more\npotentials replaced by their conjugate molar quantities, two phases in\nequilibrium in a system with $c$ independent components are connected by\na $c$ dimensional line in a *c*+2 dimensional space or\nits$\\ $*c+1*-dimensional projection as discussed in Chapter . These\nlines are called tie-lines and collectively represent a two-phase\nequilibrium region. For a $k$-phase equilibrium, there are total\n$C_{k}^{2} = \\frac{k(k - 1)}{2}$ tie-lines connecting every two phases\nwith $k - 1\\ $of them independent because the number of independent\ntie-lines increases by one with each new phase added. For the invariant\nequilibrium with zero degree of freedom, the number of phases in\nequilibrium is $c + 2$ shown by containing\n$C_{c + 2}^{2} = \\frac{(c + 2)(c + 1)}{2}$ tie-lines with $c + 1$ of\nthem being independent.\n\nInside the space encapsulated by the tie-lines, the axis variables of\nthe phase diagram (a mixture of potentials and molar quantities) can be\nchanged independently without changing the phases in equilibrium and\ntheir properties. Only the relative amounts of individual phases are\nadjusted accordingly to maintain the conservation of the molar\nquantities in the system specified by the molar quantity axes of the\nphase diagram. The geometric feature circumscribing the space\nencapsulated by the tie-lines no longer represents any phase regions,\nbut a boundary between the neighbouring phase regions. Its\ncharacteristics will be discussed in more details in the next few\nsections. As properties in each phase are homogeneous, the values of\nmolar quantities of a system are simply the sum of individual phases and\ncan be represented by the following equation\n\nEq. \u2011 $A_{m} = \\sum_{\\alpha}^{}{f^{\\alpha}A_{m}^{\\alpha}}$\n\nwhere $A_{m}$ and $A_{m}^{\\alpha}$ represent the values of a molar\nquantity of the system and the $\\alpha$ phase, respectively,\n$f^{\\alpha}$ the mole fraction of the $\\alpha$ phase, and the summation\ngoes over all phases in equilibrium with each other. With\n$\\sum_{\\alpha}^{}f^{\\alpha} = 1$, can be re-arranged into the following\nequation\n\nEq. \u2011\n$\\sum_{\\alpha}^{}{f^{\\alpha}\\left( {A_{m} - A}_{m}^{\\alpha} \\right)} = 0$\n\nis commonly referred as lever rule. For a two-phase equilibrium of\n$\\alpha$ and $\\beta$, it becomes\n\nEq. \u2011\n$f^{\\alpha} = \\frac{A_{m}^{\\beta} - A_{m}}{A_{m}^{\\beta} - A_{m}^{\\alpha}}$\n\nEq. \u2011\n$f^{\\beta} = \\frac{A_{m}^{\\alpha} - A_{m}}{A_{m}^{\\alpha} - A_{m}^{\\beta}}$\n\nFor a phase diagram with the number of axes being\n$n = \\left( c - n_{s} \\right) + 1$, the number of possible axes being\nmolar quantities is thus $k \\leq n$. There are thus $k$ equations\nsimilar to with one for each molar quantity, $A_{mi}$, resulting in the\nfollowing $k + 1$ equations\n\nEq. \u2011\n$\\sum_{\\alpha}^{}{f^{\\alpha}\\left( {A_{mi} - A}_{mi}^{\\alpha} \\right)} = 0$\n\nEq. \u2011 $1 - \\sum_{\\alpha}^{}f^{\\alpha} = 0$\n\nThe summations in and go over the phases in equilibrium, and the amount\nof each phase is obtained by solving these $k + 1$ equations\nsimultaneously along with the equilibrium conditions.\n", "metadata": {} }, { - "id": "189dbf9e", + "id": "c5827dc1", "cell_type": "markdown", - "source": [ - "### Phase diagrams with only molar quantities\n", - "\n", - "When all $\\left( c - n_{s} \\right) + 1$ potentials in a potential phase\n", - "diagrams are replaced by their conjugate molar quantities, one obtains a\n", - "molar phase diagram with molar quantities on all axes of the phase\n", - "diagram. For regions with the number of phases\n", - "$p \\leq n_{m} + 1 = \\left( c - n_{s} \\right) + 2 = p_{\\max}$ (see ), the\n", - "phase regions have the same-dimensionality as that of the phase diagram,\n", - "i.e. all phase regions have the same-dimensionality of\n", - "$\\left( c - n_{s} \\right) + 1$, and any geometric feature with\n", - "lower-dimensionalities, i.e. from 0 to $\\ \\left( c - n_{s} \\right)$, is\n", - "not phase regions, but phase boundaries between neighbouring phase\n", - "regions. For the sake of graphic visualization, the molar phase diagram\n", - "of pure Fe is shown in obtained by combining the two mixed phase\n", - "diagrams in .\n", - "\n", - "Figure \u2011: Molar phase diagram of Fe\n", - "\n", - "In this molar phase diagram, all one-, two-, and three-phase regions are\n", - "two-dimensional, the same as the dimensionality of the phase diagram. A\n", - "two-phase region is made up by tie-lines connecting the two phases in\n", - "equilibrium, while a three phase-region is surrounded by three two-phase\n", - "tie-lines, i.e. a tie-triangle. The amount of each phase in the\n", - "tie-triangle can be obtained using the lever rule represented by and .\n", - "As can be seen, phase boundaries between a one-phase region and a\n", - "two-phase region are one-dimensional. When the system crosses such a\n", - "phase boundary, the number of phases changes by one from two to one or\n", - "vice versa. Phase boundaries between two- and three-phase regions are\n", - "represented by two-phase tie-lines. When the system crosses such a phase\n", - "boundary, the number of phases also changes by one. The\n", - "lowest-dimensional phase boundaries are the points between one- and\n", - "three-phase regions that are zero-dimensional and the intercept of four\n", - "one-dimensional phase boundaries. When the system crosses such a phase\n", - "boundary, the number of phases changes by two.\n", - "\n", - "For multi-component systems, the phase relations cannot be directly\n", - "visualized. By representing the system equations in terms of equilibrium\n", - "conditions and level rules on a phase boundary using phases separately\n", - "from the two adjacent phase regions, Palatnik and Landau \\[4\\]\n", - "postulated that the difference between the number of unknowns and\n", - "equations gives the dimensionality of the phase boundary and derived the\n", - "following relationships\n", - "\n", - "Eq. \u2011 $D^{+} + D^{-} = r - b = c - n_{s} - b$\n", - "\n", - "where $D^{+}$ and $D^{-}$ are the numbers of phases added and removed\n", - "when the phase boundary is crossed, and $r$ and $b$ are the\n", - "dimensionalities of the phase diagram and the phase boundary,\n", - "respectively. They termed the as the contact rule, which is named as the\n", - "MPL boundary rule by Hillert \\[1\\].\n", - "\n", - "By the same token, is applicable to any phase boundary where the two\n", - "adjacent phase regions have the same-dimensionality of the phase\n", - "diagram, even in phase diagrams with a mixture of potentials and molar\n", - "quantities as the diagram axes. This can be understood because the\n", - "potentials are homogeneous in all phases in equilibrium on the phase\n", - "boundary. The phase boundary is thus equivalent to those in a complete\n", - "molar phase diagram with its number of components equal to the number of\n", - "molar axes in a mixed potential and molar phase diagrams minus one, i.e.\n", - "\n", - "Eq. \u2011 $c^{'} = n_{m} - 1 = c - n_{s}$\n", - "\n", - "The last part of stems from the discussion related to when all\n", - "$c^{'} + 1$ potentials are replaced by their conjugate molar quantities,\n", - "which is analogous to a molar phase diagram with $\\ c$ independent\n", - "components and $n_{s}$ potentials fixed.\n", - "\n", - "For a two-dimensional phase diagram with $r = 2$, the phase boundary can\n", - "be either zero- or one--dimensional. As shown in , the basic element of\n", - "a molar phase diagram is a joint of four one-dimensional phase boundary\n", - "lines. When a phase boundary line is crossed, the number of phases is\n", - "either increased or decreased by one. The joint of four one-dimensional\n", - "phase boundary lines is zero-dimensional. The number of phases differs\n", - "by two between the phase regions across the zero-dimensional joint. Two\n", - "scenarios are possible\n", - "\n", - "Two phases are added or removed, i.e. $D^{+} = 2$ and $D^{-} = 0$ or\n", - "$D^{+} = 0$ and $D^{-} = 2$, and the number of phases in the two phase\n", - "regions differs by two;\n", - "\n", - "One phase is added, one phase is removed, i.e. $D^{+} = D^{-} = 1$, and\n", - "the phase regions have the same number of phases.\n", - "\n", - "By combining the contact rules for both zero- and one-dimensional phase\n", - "boundaries, it is evident that the above two scenarios co-exist in a\n", - "joint of four-phase regions with two-phase regions following the first\n", - "scenario and other two-phase regions following the second scenario.\n", - "Based on the Schreinemakers\u2019 rule generalized by Hillert \\[5\\], each of\n", - "the two-phase regions with the same number of phases contains one\n", - "extrapolation of the phase boundaries, while the other two-phase regions\n", - "contain either zero or two extrapolations of the phase boundaries. This\n", - "can be observed for all the zero-dimensional phase boundaries in and is\n", - "further schematically illustrated in for general cases.\n", - "\n", - "Figure \u2011: Schematic molar phase diagram, demonstrating the\n", - "Schreinemakers\u2019 rule\n" - ], + "source": "### Phase diagrams with both potentials and molar quantities\n\nBased on Gibbs phase rule discussed in Chapters and , the\ndimensionalities of phase regions in a potential phase diagram are given\nby or . With every potential substituted by its conjugate molar\nquantity, the dimensionalities of phase regions increase by one until\nthe phase region reaches the dimensionality of the phase diagram. The\naxes of this phase diagram now consist of both potentials and molar\nquantities. The dimensionality of a phase region can thus be represented\nby the following equation based on\n\nEq. \u2011\n$\\upsilon_{m} = \\left( c - n_{s} \\right) + 2 - p + n_{m} \\leq \\left( c - n_{s} \\right) + 1$\n\nwhere $n_{m}$ is the number of molar axes. This equation is applicable\nto phase regions with more than $n_{m} + 1$ phases. For phase regions\nwith $n_{m} + 1$ phases or fewer, the dimensionalities are the same as\nthe phase diagram, i.e. $\\upsilon_{m} = \\left( c - n_{s} \\right) + 1$,\nand no longer vary with the number of molar quantity axes. When all\n$\\left( c - n_{s} \\right) + 1$ potentials are substituted by their\nconjugate molar quantities, one obtains a complete molar phase diagrams\nto be discussed in Chapter , and all phase regions have the\nsame-dimensionality of $\\left( c - n_{s} \\right) + 1$.\n\nFor the sake of graphic visualization, let us exam a two-dimensional\nphase diagram of a one-component system. Topologically, it is equivalent\nto a multi-component system with $n_{s} = c - 1$. In , three\ntwo-dimensional phase diagrams are shown for pure Fe. In principle, one\ncan use any one of them to illustrate mixed potential and molar phase\ndiagrams. For the purpose of practical usefulness, one selects the\n$T - ( - P)$ potential diagram as temperature and pressure are the two\ntypical variables controlled in experiments of one-component systems.\nThe conjugate molar quantities of $- P$ and $T$ are molar volume and\nmolar entropy, respectively. For stable phases, any pair of conjugate\nvariables changes in the same direction as illustrated by and , i.e. the\nphase stable at higher $T$ has higher molar entropy, and the phase\nstable at higher $\u2013P$, i.e. lower pressure, has higher molar volume. Let\nus first substitute $\u2013P$ by $V_{m}$ as shown in a. The dimensionality of\na sing-phase region remains unchanged because of\n$p = 1 < 2 = n_{m} + 1$. The dimensionality of two-phase regions is\nchanged from 1 to 2 due to $\\upsilon_{m} = 3 - 2 + 1 = 2$ from .\n\nFigure \u2011: T-Vm and Sm-(-P) phase diagrams of Fe\n\nAs both phases in a two-phase equilibrium have the same temperature, all\ntie-lines, depicted by dotted lines, are perpendicular to the\ntemperature axis in a. When the molar volume of the system changes from\none end of a tie-line to another end at a constant temperature, the mole\nfraction of one phase increases from 0 to 1, and the mole fraction of\nanother phase decreases from 1 to 0. The tine-lines at various\ntemperatures combine together to form a two-dimensional two-phase\nregion. The two curves at the two ends of tie-lines represent the\nboundaries between the single-phase and two-phase regions and are no\nlonger phase regions themselves. They are thus called phase boundaries.\n\nBy the same token, by changing the temperature at a constant molar\nvolume of the system, the system will locate on different tie-lines with\nthe amounts of the two phases determined by the lever rule. It is thus\nclear that the system maintains the two-phase equilibrium state with\nboth $T$ and $V_{m}$ changing independently inside the two-dimensional\ntwo-phase region. This seems in contradiction to Gibbs phase rule being\n$\\upsilon = 3 - p = 1$ from , but it is not because Gibbs phase rule\napplies strictly to potential phase diagrams only, while the $T - V_{m}$\nphase diagram has one of its axis being a molar quantity. As an\nalternative, one may consider as a modified Gibbs phase rule in\ndescribing the dimensionality of a phase region in a mixed potential and\nmolar phase diagram with $p \\geq n_{m} + 1$.\n\nThe two three-phase equilibria in pure Fe are also represented by\ntie-lines connecting all three phases. The dimensionality of three-phase\nregions is $\\upsilon_{m} = 3 - 3 + 1 = 1$ from , and the three two-phase\ntie-lines for a three-phase equilibrium thus overlap each other with\ntheir three molar volumes on the same tie-line.\n\nLet us examine the two three-phase equilibria in more detail. In the\n\u03b3/\u03b1/\u03b5 three-phase equilibrium at $T_{E} = 756.6\\ K$ and\n$\\left( - P_{E} \\right) = - 1.046 \\bullet 10^{10}\\ Pa$, the molar\nvolumes of \u03b1, \u03b3, and \u03b5 are 6.837, 6.677, 6.582 $10^{- 6}\\ m^{3}/mol$,\nrespectively. There are two two-phase regions at higher temperatures and\none two-phase region at lower temperatures. This is also shown in the\npotential phase diagram of a with two two-phase curves entering into and\none two-phase curve leaving from the three-phase equilibrium point with\ndecreasing temperature. Considering the system with the fixed molar\nvolume equal to that of the $\\gamma$ phase, i.e.\n$V_{m}^{\\ }\\left( T_{E} \\right) = 6.677 \\bullet 10^{- 6}\\ m^{3}/mol$. At\n$T > T_{E}$, the system is in the single \u03b3 phase region. With decrease\nin temperature across $T_{E}$, it enters into the $\\alpha + \\varepsilon$\ntwo-phase region. This transformation can be written as follows and is\ncalled a eutectoid reaction\n\nEq. \u2011 $\\gamma \\rightarrow \\alpha + \\varepsilon$\n\nThis type of transformations is named as eutectic reaction if the high\ntemperature phase is a liquid phase. If the system molar volume is\nbetween $V_{m}^{\\gamma}$ and $V_{m}^{\\alpha}$ at $756.6K$, with decrease\nof temperature, the system first moves from the single \u03b3 phase region to\nthe $\\alpha + \\gamma$ two-phase region when the\n$\\gamma/(\\alpha + \\gamma)$ phase boundary is crossed. When the\ntemperature reaches $T_{E}$, the eutectoid transformation takes place in\nthe remaining \u03b3 phase. The $\\alpha$ formed prior to the eutectoid\ntransformation is called proeutectoid $\\alpha$. By the same token, when\nthe system molar volume is between $V_{m}^{\\gamma}$ and\n$V_{m}^{\\varepsilon}$ at $756.6K$, proeutectoid $\\varepsilon$ would form\nfollowed by the eutectoid transformation with decrease in temperature.\n\nOn the other hand, the $L/\\delta/\\gamma$ three-phase equilibrium at\n$T_{P} = 1977.9\\ K$ and\n$\\left( - P_{P} \\right) = - 5.111 \\bullet \\ 10^{9}\\ Pa$ has different\ncharacteristics, where the subscript P will be defined shortly. There\nare one two-phase equilibrium above and two two-phase equilibria below\nthe invariant temperature, shown in the potential phase diagram of a\nwith one two-phase curve entering into and two two-phase curves leaving\nfrom the three-phase equilibrium point with decreasing temperature. The\nmolar volumes of $L$, $\\delta$, and $\\gamma$ at $T_{P}$ are 7.735,\n7.542, and 7.498 $10^{- 6}\\ m^{3}/mol$, respectively. For $T > T_{P}$,\nthe two-phase region is $L + \\gamma$. If the system molar volume is\nbetween 7.735 and 7.498 $10^{- 6}\\ m^{3}/mol$, i.e.\n$V_{m}^{L}\\left( T_{P} \\right)$ and\n$V_{m}^{\\gamma}\\left( T_{P} \\right)$, when the temperature reaches\n$T_{P}$, $L$ and $\\gamma$ are combined to form $\\delta$ with the\ntransformation written as\n\nEq. \u2011 $L + \\gamma \\rightarrow \\delta$\n\nThis type of reactions is called as peritectic reaction or peritectoid\nreaction when the liquid phase is replaced by a solid phase, denoted by\nthe subscript P. At $T < T_{P}$, one or both high temperature phases may\nno longer be present in equilibrium depending on the value of the system\nmolar volume. For\n$V_{m}^{\\ } = V_{m}^{\\delta}\\left( T_{P} \\right) = 7.542 \\bullet 10^{- 6}\\ m^{3}/mol$,\nthe peritectic reaction, , can come to completion with no $L$ and\n$\\gamma$ left upon decreasing temperature. For\n$V_{m}^{\\gamma}\\left( T_{P} \\right) = 7.498 < V_{m}^{\\ } < V_{m}^{\\delta}\\left( T_{P} \\right) = 7.542\\ \\left( 10^{- 6}\\ m^{3}/mol \\right)$,\nthe liquid phase is consumed, and the system enters the\n$\\gamma + \\delta$ two-phase region. On the other hand, for\n$V_{m}^{\\delta}\\left( T_{P} \\right) = 7.542 < V_{m}^{\\ } < V_{m}^{\\ L}\\left( T_{P} \\right) = 7.735\\ \\ \\left( 10^{- 6}\\ m^{3}/mol \\right)$,\nthe $\\gamma$ phase is consumed instead, and the system enters into the\n$L + \\delta$ two-phase equilibrium region upon cooling.\n\nLet us now replace $T$ by $S_{m}$ to obtain the $( - P) - S_{m}$ phase\ndiagram shown in b. The morphology of this phase diagram is identical to\nthe $T - V_{m}$ phase diagram just discussed with all tie-lines\nperpendicular to the pressure axis. The transformations at the two\nthree-phase equilibria with $( - P)$ decreasing or $P$ increasing are as\nfollows\n\nEq. \u2011 $\\gamma + \\alpha \\rightarrow \\varepsilon$\n\nEq. \u2011 $\\delta \\rightarrow L + \\gamma$\n\nTo visualize two-dimensional phase diagrams of binary systems, one\nusually keeps the pressure constant. One type of commonly used binary\nphase diagram is the temperature-composition ($T - x$) phase diagram. As\nan example, let us re-plot the $T - \\mu_{C}$ potential diagram shown in\ninto a $T - x_{C}$ mixed potential and molar phase diagram by replacing\nthe chemical potential of $C$ by its mole fraction. The $T - x_{C}$\nphase diagram thus obtained is shown in . In this phase diagram there\nare one peritectic reaction and two eutectic reactions as follows\n\nEq. \u2011 $L + \\delta \\rightarrow \\gamma$\n\nEq. \u2011 $L \\rightarrow \\gamma + C$\n\nEq. 3\u201161 $\\gamma \\rightarrow \\alpha + C$\n\nFigure \u2011: $T - x_{C}\\ $ phase diagram of the Fe-C binary system\n\nIn in Chapter , it was discussed the formation of miscibility gaps due\nto repulsive interactions between components. One example is shown in\nfor the Al-Zn binary system in terms of both $T - \\mu_{Zn}$ potential\nphase diagram and $\\ T - x_{Zn}\\ $ mixed potential and molar phase\ndiagram.\n\nFigure \u2011: $T - \\mu_{Zn}$ potential phase diagram (a) and\n$\\ T - x_{Zn}\\ $ mixed potential and molar phase diagram (b) of the\nAl-Zn binary system\n\nIn ,there are one eutectic reaction and one eutectoid reaction as\nfollows\n\nEq. \u2011 $L \\rightarrow fcc + hcp$\n\nEq. \u2011 $fcc\\# 1 \\rightarrow fcc\\# 2 + hcp$\n\nThe eutectoid reaction, , is also termed as monotectoid reaction because\nthe fcc phase appears on both sides of the reaction with different\ncompositions, $fcc\\# 1$ and $fcc\\# 2$, due to the miscibility gap. The\nhighest temperature of the miscibility gap is called the consolute point\nas discussed in Chapter , which can be clearly seen in the\n$T - \\mu_{Zn}$ potential phase diagram shown in . This is a critical\npoint, marking the limit of instability as shown in .\n\nWhen there is only one phase on either side of the reaction, i.e. both\nphases have the same composition, the reaction is called a congruent\nreaction. One example is shown in for the $T - x_{SiO_{2}}\\ $ mixed\npotential and molar phase diagram of the CaO-SiO2\npseudo-binary system with two congruent reactions as follows\n\nEq. 3\u201164 $L \\rightarrow CaSiO_{3}$\n\nEq. 3\u201165 $L \\rightarrow {Ca}_{2}SiO_{4}$\n\nThey are not invariant reactions based on the Gibbs phase rule. In , it\nis noted that there are one miscibility gap in the liquid phase close to\nthe $SiO_{2}$ side, four eutectic reactions with one being monotectic\ninvolving two liquid phases due to the miscibility gap, and three\nperitectic reactions.\n\nFigure \u2011: $T - x_{SiO_{2}}\\ $ mixed potential and molar phase diagram of\nthe CaO-SiO2 pseudo-binary system.\n\nLet us generalize the above discussion to phase diagrams with\n$\\left( c - n_{s} \\right) + 1$ axes. In such a phase diagram, the\nmaximum number of phases is given by as\n$p_{\\max} = \\left( c - n_{s} \\right) + 2$. The number of phases on the\neither side of an invariant reaction can vary from one phase to\n$p_{\\max} - 1 = \\left( c - n_{s} \\right) + 1$ phases with the remaining\nphases on the other side of the reaction, typically with the potential\ndecreasing from left to right. The invariant reaction with one phase on\nthe left of the reaction is named as eutectic, or eutectoid reaction\ndepending on if the phase on the left of the reaction is liquid or\nsolid. The rest invariant reactions are named as peritectic or\nperitectoid reactions with or without a liquid phase.\n", + "metadata": {} + }, + { + "id": "21db7899", + "cell_type": "markdown", + "source": "### Phase diagrams with only molar quantities\n\nWhen all $\\left( c - n_{s} \\right) + 1$ potentials in a potential phase\ndiagrams are replaced by their conjugate molar quantities, one obtains a\nmolar phase diagram with molar quantities on all axes of the phase\ndiagram. For regions with the number of phases\n$p \\leq n_{m} + 1 = \\left( c - n_{s} \\right) + 2 = p_{\\max}$ (see ), the\nphase regions have the same-dimensionality as that of the phase diagram,\ni.e. all phase regions have the same-dimensionality of\n$\\left( c - n_{s} \\right) + 1$, and any geometric feature with\nlower-dimensionalities, i.e. from 0 to $\\ \\left( c - n_{s} \\right)$, is\nnot phase regions, but phase boundaries between neighbouring phase\nregions. For the sake of graphic visualization, the molar phase diagram\nof pure Fe is shown in obtained by combining the two mixed phase\ndiagrams in .\n\nFigure \u2011: Molar phase diagram of Fe\n\nIn this molar phase diagram, all one-, two-, and three-phase regions are\ntwo-dimensional, the same as the dimensionality of the phase diagram. A\ntwo-phase region is made up by tie-lines connecting the two phases in\nequilibrium, while a three phase-region is surrounded by three two-phase\ntie-lines, i.e. a tie-triangle. The amount of each phase in the\ntie-triangle can be obtained using the lever rule represented by and .\nAs can be seen, phase boundaries between a one-phase region and a\ntwo-phase region are one-dimensional. When the system crosses such a\nphase boundary, the number of phases changes by one from two to one or\nvice versa. Phase boundaries between two- and three-phase regions are\nrepresented by two-phase tie-lines. When the system crosses such a phase\nboundary, the number of phases also changes by one. The\nlowest-dimensional phase boundaries are the points between one- and\nthree-phase regions that are zero-dimensional and the intercept of four\none-dimensional phase boundaries. When the system crosses such a phase\nboundary, the number of phases changes by two.\n\nFor multi-component systems, the phase relations cannot be directly\nvisualized. By representing the system equations in terms of equilibrium\nconditions and level rules on a phase boundary using phases separately\nfrom the two adjacent phase regions, Palatnik and Landau \\[4\\]\npostulated that the difference between the number of unknowns and\nequations gives the dimensionality of the phase boundary and derived the\nfollowing relationships\n\nEq. \u2011 $D^{+} + D^{-} = r - b = c - n_{s} - b$\n\nwhere $D^{+}$ and $D^{-}$ are the numbers of phases added and removed\nwhen the phase boundary is crossed, and $r$ and $b$ are the\ndimensionalities of the phase diagram and the phase boundary,\nrespectively. They termed the as the contact rule, which is named as the\nMPL boundary rule by Hillert \\[1\\].\n\nBy the same token, is applicable to any phase boundary where the two\nadjacent phase regions have the same-dimensionality of the phase\ndiagram, even in phase diagrams with a mixture of potentials and molar\nquantities as the diagram axes. This can be understood because the\npotentials are homogeneous in all phases in equilibrium on the phase\nboundary. The phase boundary is thus equivalent to those in a complete\nmolar phase diagram with its number of components equal to the number of\nmolar axes in a mixed potential and molar phase diagrams minus one, i.e.\n\nEq. \u2011 $c^{'} = n_{m} - 1 = c - n_{s}$\n\nThe last part of stems from the discussion related to when all\n$c^{'} + 1$ potentials are replaced by their conjugate molar quantities,\nwhich is analogous to a molar phase diagram with $\\ c$ independent\ncomponents and $n_{s}$ potentials fixed.\n\nFor a two-dimensional phase diagram with $r = 2$, the phase boundary can\nbe either zero- or one--dimensional. As shown in , the basic element of\na molar phase diagram is a joint of four one-dimensional phase boundary\nlines. When a phase boundary line is crossed, the number of phases is\neither increased or decreased by one. The joint of four one-dimensional\nphase boundary lines is zero-dimensional. The number of phases differs\nby two between the phase regions across the zero-dimensional joint. Two\nscenarios are possible\n\nTwo phases are added or removed, i.e. $D^{+} = 2$ and $D^{-} = 0$ or\n$D^{+} = 0$ and $D^{-} = 2$, and the number of phases in the two phase\nregions differs by two;\n\nOne phase is added, one phase is removed, i.e. $D^{+} = D^{-} = 1$, and\nthe phase regions have the same number of phases.\n\nBy combining the contact rules for both zero- and one-dimensional phase\nboundaries, it is evident that the above two scenarios co-exist in a\njoint of four-phase regions with two-phase regions following the first\nscenario and other two-phase regions following the second scenario.\nBased on the Schreinemakers\u2019 rule generalized by Hillert \\[5\\], each of\nthe two-phase regions with the same number of phases contains one\nextrapolation of the phase boundaries, while the other two-phase regions\ncontain either zero or two extrapolations of the phase boundaries. This\ncan be observed for all the zero-dimensional phase boundaries in and is\nfurther schematically illustrated in for general cases.\n\nFigure \u2011: Schematic molar phase diagram, demonstrating the\nSchreinemakers\u2019 rule\n", + "metadata": {} + }, + { + "id": "a5607756", + "cell_type": "markdown", + "source": "### Projection and section of phase diagrams with potential and molar quantities\n\nAs discussed in Chapter , projections of high-dimensional phase diagrams\nusually cannot keep all information. However, there is one type of\nwidely used projection in the literature, i.e. the liquidus surface in\nternary systems under constant pressure with temperature and mole\nfractions of two components as its axes. The projection along the\ntemperature axis reveals the composition regions for primary phases that\nsolidify from liquid upon cooling. These regions are separated by\nunivariant lines of three-phase equilibria. The projections along one of\nthe two mole fractions show the temperature as a function of composition\non the univariant three-phase equilibrium lines and also depict whether\na four-phase equilibrium is peritectic or eutectic. There are four\nscenarios for the three univariant three-phase equilibrium lines to meet\nat the four-phase equilibrium as depicted in and discussed individually\nbelow.\n\nFigure \u2011: Schematic four options for three univariant three-phase\nequilibrium lines to meet at the invariant four-phase equilibrium\n\nThe first scenario is that with decreasing temperature, all three\nunivariant lines merge into the four-phase equilibrium. It indicates\nthat the liquid phase does not exist at temperatures below the\nfour-phase invariant reaction. This invariant reaction is thus a ternary\neutectic reaction with liquid completely transformed to three solid\nphases upon cooling, i.e.\n\nEq. \u2011 $L \\rightarrow \\alpha + \\beta + \\gamma$\n\nIn the second scenario, two univariant lines merge into and one leaves\nfrom the four-phase equilibrium with decreasing temperature. This means\nthat one solid phase at higher temperature is no longer stable at lower\ntemperature, and it must react with the liquid phase to form the\nremaining two solid phases. The four-phase invariant reaction is thus\nperitectic. The solid phase common to both univariant lines at high\ntemperatures reacts with the liquid phase. Assuming that this phase is\n$\\alpha$, the four-phase invariant reaction becomes\n\nEq. \u2011 $L + \\alpha \\rightarrow \\beta + \\gamma$\n\nIn the third scenario, one univariant line points to and two leave from\nthe four-phase equilibrium with decreasing temperature. A new phase\nforms at low temperatures from the three high temperature phases, e.g.\nliquid, $\\alpha$, and $\\beta$, with the four-phase invariant reaction as\n\nEq. \u2011 $L + \\alpha + \\beta \\rightarrow \\gamma$\n\nThe fourth scenario is the inverse of the first scenario, indicating the\nformation of liquid from solid phases upon cooling, i.e.\n\nEq. \u2011 $\\alpha + \\beta + \\gamma \\rightarrow L$\n\nThis case has not been observed in reality.\n\nAs an example, the liquidus projections of the Al-Fe-Si ternary system\nare shown in in two formats\\[6\\], i.e. (a) three-dimensional liquidus\nsurface with the isotherms showing the liquidus contours; (b)\nconventional projection to the composition axis with the temperature\ndecrease shown by arrows; (c) projection to the temperature and weight\nfraction of Si. The first to third scenarios of invariant reactions\ndiscussed above can clearly be identified and listed in . It is evident\nthat c provides the easiest route to visualize the type of invariant\nreactions as show by .\n\nFigure \u2011: Liquidus of the Al-Fe-Si ternary system\\[6\\], (a)\nthree-dimensional presentation of the liquidus; (b) projection to the\ncomposition triangle with isotherms (dotted lines) superimposed and\ntheir temperatures indicated close to the horizontal axis.\n\nTable \u2011: Invariant liquidus reactions of the Al-Fe-Si ternary system\nwith the composition of the liquid phase \\[6\\]\n\n| Reaction | T, \u00b0C | wFe, % | wSi, % |\n|----------|-------|-------------------|-------------------|\n| | 1178 | 77.9 | 21.0 |\n| | 1155 | 49.0 | 0.16 |\n| | 1127 | 52.7 | 2.81 |\n| -H-L | 1076 | 41.6 | 44.0 |\n| | 1073 | 51.0 | 7.19 |\n| | 1050 | 53.8 | 18.4 |\n| -H-L | 1019 | 34.8 | 44.4 |\n| | 1004 | 49.0 | 12.6 |\n| | 1000 | 46.7 | 13.2 |\n| -L | 940 | 37.9 | 32.0 |\n| | 921 | 33.7 | 20.1 |\n| | 899 | 33.8 | 32.2 |\n| | 884 | 30.8 | 26.0 |\n| -L | 877 | 29.5 | 35.2 |\n| | 851 | 23.3 | 21.6 |\n| | 834 | 22.2 | 31.7 |\n| | 825 | 22.1 | 25.7 |\n| | 823 | 21.8 | 25.4 |\n| | 715 | 6.64 | 10.8 |\n| | 694 | 6.11 | 17.1 |\n| | 680 | 4.68 | 11.6 |\n| | 630 | 2.11 | 4.10 |\n| | 616 | 1.76 | 6.56 |\n| | 598 | 1.22 | 14.3 |\n| | 575 | 0.73 | 12.7 |\n\nIn contrast to projections, sectioning is used more often to understand\nphase relations in multi-component systems. Sectioning of a potential\nphase diagrams is relatively simple as the resulted phase diagram\nbehaves like a system with one component less. The same is true if\npotentials are sectioned in phase diagrams with both potential and molar\nquantities as the section is along the tie-lines of the fixed\npotentials. As an example, shows the ternary Al-Fe-Si potential and\nmolar phase diagrams sectioned at T=1273K and P=1atm, commonly referred\nto as isothermal section. It is evident that the geometric features of\nboth phase diagrams are identical to those of pure Fe shown in and ,\nrespectively, with one-, two-, and three-phase regions and corresponding\nphase boundaries.\n\nFigure \u2011: Ternary isothermal section of the Al-Fe-Si ternary system at\nT=1273K and P=1atm\n\nOn the other hand, when the phase diagram is sectioned along a molar\nquantity, it would usually not follow a tie-line because phases in\nequilibrium usually have different values for the same molar quantity.\nConsequently, there are no tie-lines inside such phase diagrams in\ngeneral, and any phase regions only show which phases are in equilibrium\nwith each other without any information on the values of molar\nquantities of individual phases.\n\nThis type of sectioning reduces both the dimensionalities of the phase\ndiagram and phase boundary by the same number, but does not alter the\nnumber of phases in the adjacent phase regions. The contact rule, i.e. ,\nthus remains valid and is applicable to phase regions with the\nsame-dimensionality as that of the sectioned phase diagram. Similarly,\nthe Schreinemakers\u2019 rule shown in is valid under the same conditions.\n\nFor example, the two-dimensional phase diagram of the Mg-Al-Zn ternary\nsystem sectioned with one atmospheric pressure and the weight fraction\nof Zn fixed at 0.01 is shown in plotted with temperature and mole\nfraction of Al \\[7\\]. This phase diagram is commonly called isopleth and\nis generated by fixing one potential, the pressure, $P$, changing the\nchemical potentials of Al and Zn to their conjugate molar quantities\nrepresented by weight fractions of Al and Zn, and sectioning at\n$w_{Zn} = 0.01$. From the discussions in Chapter , the phase regions\nwith the number of phases equal to three or fewer, i.e.\n$p \\leq n_{m} + 1 = 3$, have the same-dimensionality as the phase\ndiagram, i.e. two-dimensional in the present case, and the phase\nboundary rule is applicable. The maximum number of phases co-existing at\nequilibrium is given by as following for the present case\n\nEq. \u2011 $p_{\\max} = \\left( c - n_{s} \\right) + 2 = 3 - 1 + 2 = 4$\n\nThis is because introducing molar quantities only increases the\ndimensionality of phase regions and does not change the maximum number\nof co-existing phases.\n\nThe dimensionality of a four-phase region is calculated from as\n\nEq. \u2011\n$\\upsilon_{m} = \\left( c - n_{s} \\right) + 2 - p + n_{m} - n_{ms} = 3 - 1 + 2 - 4 + 2 - 1 = 1$\n\nwhere $n_{ms}$ is the number of sectioned molar quantities. Since the\ndimensionality of a four-phase region is lower than that of the phase\ndiagram, the phase boundary rule cannot be applied directly. Such a\nfour-phase region, liquid+Mg+\u03b3+\u03c6, is shown in between three three-phase\nregions of liquid+Mg+\u03b3, liquid+Mg+\u03c6, and Mg+\u03b3+\u03c6.\n\nFigure \u2011: Isopleth with the weight fraction of Zn fixed at 0.01 of the\nMg-Al-Zn ternary system.\n\nalso displays information on what phases are in equilibrium for a given\nalloy at various temperatures. One example is shown by the dotted\nvertical line marking the weight fraction of Al being 0.09, a widely\nused Mg alloy called AZ91. Various phases are present at different\ntemperature ranges, but the equilibrium phase fractions and phase\ncompositions are not shown in the figure as the tie-lines are not in the\nplane of the phase diagram and have to be calculated at each temperature\nindividually. shows the amount of each phase of the AZ91 alloy as a\nfunction of temperature with the dotted lines depicting the values under\nthe equilibrium condition and the solid lines depicting the values under\nthe so-called Scheil condition assuming no diffusion in solid phases and\ninfinitelyfast diffusion in liquid. Similarly, the composition of each\nphase can also be plotted as shown in .\n\nFigure \u2011: Mole fraction of individual phases under equilibrium (dotted\ncurves) and Scheil (solid curves) conditions in the AZ91 alloy\n\nFigure \u2011: Mass fraction of Al and Zn in the Mg solid solution phase\nunder equilibrium (dotted curves) and Scheil (solid curves) conditions\nin AZ91\n", "metadata": {} } ] diff --git a/src/psu410/src/phase_equilibria_in_heterogeneous_systems/potential_phase_diagrams.ipynb b/src/psu410/src/phase_equilibria_in_heterogeneous_systems/potential_phase_diagrams.ipynb index 094b6d2..17228c2 100644 --- a/src/psu410/src/phase_equilibria_in_heterogeneous_systems/potential_phase_diagrams.ipynb +++ b/src/psu410/src/phase_equilibria_in_heterogeneous_systems/potential_phase_diagrams.ipynb @@ -4,674 +4,27 @@ "metadata": {}, "cells": [ { - "id": "33603d36", + "id": "167e045a", "cell_type": "markdown", - "source": [ - "## Potential phase diagrams\n", - "\n", - "Gibbs phase rule can be further understood through the\n", - "$c + 2$-dimensional space of potentials consisting of $T$, $- P$, and\n", - "$\\mu_{i}$ with $i$ from 1 to $c$. Each phase is a $c + 1$-dimensional\n", - "feature in this $c + 2$-dimensional space characterized by . The\n", - "directions of this $c + 1$-dimensional feature are represented by their\n", - "molar quantities as shown by following equations,\n", - "\n", - "Eq. \u2011\n", - "$\\ \\left( \\frac{\\partial\\mu_{i}}{\\partial T} \\right)_{P,\\mu_{j \\neq i}} = - \\frac{S^{\\beta}}{N_{i}^{\\beta}} = - \\frac{S_{m}^{\\beta}}{x_{i}^{\\beta}}$\n", - "\n", - "Eq. \u2011\n", - "$\\ \\left( \\frac{\\partial\\mu_{i}}{\\partial( - P)} \\right)_{T,\\mu_{j \\neq i}} = - \\frac{V^{\\beta}}{N_{i}^{\\beta}} = - \\frac{V_{m}^{\\beta}}{x_{i}^{\\beta}}$\n", - "\n", - "Eq. \u2011\n", - "$\\ \\left( \\frac{\\partial( - P)}{\\partial T} \\right)_{\\mu_{i}} = - \\frac{S^{\\beta}}{V^{\\beta}} = - \\frac{S_{m}^{\\beta}}{V_{m}^{\\beta}}$\n", - "\n", - "As can be seen, all the direction derivatives are negative, indicating\n", - "that the $c + 1$-dimensional feature is convex. The intercept of any two\n", - "$c + 1$-dimensional features is thus a $c$-dimensional feature. On this\n", - "$c$-dimensional feature, these two phases are in equilibrium with each\n", - "other because each potential has the same value in both phases. This\n", - "feature thus represents a two-phase equilibrium. By the same token, the\n", - "intercept of any three $c + 1$-dimensional features is a\n", - "$c - 1$-dimensional feature in the $c + 2$-dimensional space of\n", - "potentials and represents a three-phase equilibrium. This continues\n", - "until the number of phases reaches $c + 2$ with all $c + 2$ potentials\n", - "completely determined, and the dimension of their intercepts becomes\n", - "zero.\n", - "\n", - "Those $c + 1$ to zero-dimensional geometrical features in the\n", - "$c + 2$-dimensional space of potentials thus denote one-phase,\n", - "two-phase, three-phase to $(c + 2)$-phase equilibria of the system with\n", - "the dimensionality of the feature and the number of phases in\n", - "equilibrium related by , i.e. Gibbs phase rule. Their arrangements in\n", - "the $c + 2$-dimensional space of potentials thus depict the phase\n", - "relations in the system and are commonly called phase diagrams. Since\n", - "all the diagram axes in the phase diagram discussed above are\n", - "potentials, the diagram is called potential phase diagram in order to\n", - "differentiate it from phase diagrams with some or all diagram axes being\n", - "their conjugate molar quantities. Both potential and molar phase\n", - "diagrams are discussed in this chapter.\n" - ], + "source": "## Potential phase diagrams\n\nGibbs phase rule can be further understood through the\n$c + 2$-dimensional space of potentials consisting of $T$, $- P$, and\n$\\mu_{i}$ with $i$ from 1 to $c$. Each phase is a $c + 1$-dimensional\nfeature in this $c + 2$-dimensional space characterized by . The\ndirections of this $c + 1$-dimensional feature are represented by their\nmolar quantities as shown by following equations,\n\nEq. \u2011\n$\\ \\left( \\frac{\\partial\\mu_{i}}{\\partial T} \\right)_{P,\\mu_{j \\neq i}} = - \\frac{S^{\\beta}}{N_{i}^{\\beta}} = - \\frac{S_{m}^{\\beta}}{x_{i}^{\\beta}}$\n\nEq. \u2011\n$\\ \\left( \\frac{\\partial\\mu_{i}}{\\partial( - P)} \\right)_{T,\\mu_{j \\neq i}} = - \\frac{V^{\\beta}}{N_{i}^{\\beta}} = - \\frac{V_{m}^{\\beta}}{x_{i}^{\\beta}}$\n\nEq. \u2011\n$\\ \\left( \\frac{\\partial( - P)}{\\partial T} \\right)_{\\mu_{i}} = - \\frac{S^{\\beta}}{V^{\\beta}} = - \\frac{S_{m}^{\\beta}}{V_{m}^{\\beta}}$\n\nAs can be seen, all the direction derivatives are negative, indicating\nthat the $c + 1$-dimensional feature is convex. The intercept of any two\n$c + 1$-dimensional features is thus a $c$-dimensional feature. On this\n$c$-dimensional feature, these two phases are in equilibrium with each\nother because each potential has the same value in both phases. This\nfeature thus represents a two-phase equilibrium. By the same token, the\nintercept of any three $c + 1$-dimensional features is a\n$c - 1$-dimensional feature in the $c + 2$-dimensional space of\npotentials and represents a three-phase equilibrium. This continues\nuntil the number of phases reaches $c + 2$ with all $c + 2$ potentials\ncompletely determined, and the dimension of their intercepts becomes\nzero.\n\nThose $c + 1$ to zero-dimensional geometrical features in the\n$c + 2$-dimensional space of potentials thus denote one-phase,\ntwo-phase, three-phase to $(c + 2)$-phase equilibria of the system with\nthe dimensionality of the feature and the number of phases in\nequilibrium related by , i.e. Gibbs phase rule. Their arrangements in\nthe $c + 2$-dimensional space of potentials thus depict the phase\nrelations in the system and are commonly called phase diagrams. Since\nall the diagram axes in the phase diagram discussed above are\npotentials, the diagram is called potential phase diagram in order to\ndifferentiate it from phase diagrams with some or all diagram axes being\ntheir conjugate molar quantities. Both potential and molar phase\ndiagrams are discussed in this chapter.\n", "metadata": {} }, { - "id": "fa0ef81c", + "id": "6b91a7a1", "cell_type": "markdown", - "source": [ - "### Potential phase diagrams of one-component systems\n", - "\n", - "As physical vision of human being is limited to three-dimensions, only\n", - "one-component system can be completely visualized as shown in for one\n", - "phase where any two of the three potentials can change independently.\n", - "From Gibbs phase rule, when two phases are in equilibrium, only one\n", - "potential can vary freely if the two-phase equilibrium is to be\n", - "maintained. While three phases are in equilibrium, the degree of freedom\n", - "is zero, and all three potentials are fixed.\n", - "\n", - "For a two-phase equilibrium, two surfaces intersect each other as\n", - "depicted by the dashed line in . This two-phase equilibrium line is\n", - "obtained by applying to both phases in a one-component system. Since one\n", - "of the potentials is dependent on the other two, one can eliminate it by\n", - "dividing the equation by its conjugate molar quantity and subtracting\n", - "the two equations, resulting in the following three equations\n", - "\n", - "Eq. \u2011\n", - "$0 = \\left( \\frac{S^{\\alpha}}{N_{A}^{\\alpha}} - \\frac{S^{\\beta}}{N_{A}^{\\beta}} \\right)dT + \\left( \\frac{V^{\\alpha}}{N_{A}^{\\alpha}} - \\frac{V^{\\beta}}{N_{A}^{\\beta}} \\right)d( - P) = \\mathrm{\\Delta}S_{m}^{\\alpha\\beta}dT + \\mathrm{\\Delta}V_{m}^{\\alpha\\beta}d( - P)$\n", - "\n", - "Eq. \u2011\n", - "$0 = \\left( \\frac{S^{\\alpha}}{V^{\\alpha}} - \\frac{S^{\\beta}}{V^{\\beta}} \\right)dT + \\left( \\frac{N_{A}^{\\alpha}}{V^{\\alpha}} - \\frac{N_{A}^{\\beta}}{V^{\\beta}} \\right){d\\mu}_{A} = \\mathrm{\\Delta}\\left( \\frac{S_{m}}{V_{m}} \\right)^{\\alpha\\beta}dT + \\mathrm{\\Delta}\\left( \\frac{1}{V_{m}} \\right)^{\\alpha\\beta}{d\\mu}_{A}$\n", - "\n", - "Eq. \u2011\n", - "$0 = \\left( \\frac{V^{\\alpha}}{S^{\\alpha}} - \\frac{V^{\\beta}}{S^{\\beta}} \\right)d( - P) + \\left( \\frac{N_{A}^{\\alpha}}{S^{\\alpha}} - \\frac{N_{A}^{\\beta}}{S^{\\beta}} \\right){d\\mu}_{A} = \\mathrm{\\Delta}\\left( \\frac{V_{m}}{S_{m}} \\right)^{\\alpha\\beta}d( - P) + \\mathrm{\\Delta}\\left( \\frac{1}{S_{m}} \\right)^{\\alpha\\beta}{d\\mu}_{A}$\n", - "\n", - "Figure \u2011: Gibbs energy surfaces of two phases and their intersection,\n", - "representing the two-phase equilibrium.\n", - "\n", - "The directions of the two-phase equilibrium line can thus be obtained as\n", - "\n", - "Eq. \u2011\n", - "$\\frac{d( - P)}{dT} = - \\frac{\\mathrm{\\Delta}S_{m}^{\\alpha\\beta}}{\\mathrm{\\Delta}V_{m}^{\\alpha\\beta}}$\n", - "\n", - "Eq. \u2011\n", - "$\\frac{dT}{{d\\mu}_{A}} = - \\frac{\\mathrm{\\Delta}\\left( \\frac{1}{V_{m}} \\right)^{\\alpha\\beta}}{\\mathrm{\\Delta}\\left( \\frac{S_{m}}{V_{m}} \\right)^{\\alpha\\beta}}$\n", - "\n", - "Eq. \u2011\n", - "$\\frac{d( - P)}{{d\\mu}_{A}} = - \\frac{\\mathrm{\\Delta}\\left( \\frac{1}{S_{m}} \\right)^{\\alpha\\beta}}{\\mathrm{\\Delta}\\left( \\frac{V_{m}}{S_{m}} \\right)^{\\alpha\\beta}}$\n", - "\n", - "These three equations define the mathematical forms of the two-phase\n", - "equilibrium line in the two-dimensional spaces of $T - ( - P)$,\n", - "$\\mu_{A} - T$ and $\\mu_{A} - ( - P)$, respectively, and can thus be\n", - "plotted as two-dimensional diagrams. is commonly called\n", - "Clausius-Clapeyron equation in the literature. One may thus call all\n", - "three equations above as generalized Clausius-Clapeyron equations. At\n", - "equilibrium, the chemical potentials of the component in both phases are\n", - "equal to each other, so are their Gibbs energies. One thus has\n", - "\n", - "Eq. \u2011\n", - "$G_{m}^{\\alpha} - G_{m}^{\\beta} = 0 = \\mathrm{\\Delta}G_{m} = \\mathrm{\\Delta}H_{m}^{\\alpha\\beta} - T\\mathrm{\\Delta}S_{m}^{\\alpha\\beta}$\n", - "\n", - "The Clausius-Clapeyron equation, , can be re-written as\n", - "\n", - "Eq. \u2011\n", - "$\\frac{d( - P)}{dT} = - \\frac{\\mathrm{\\Delta}H_{m}^{\\alpha\\beta}}{T\\mathrm{\\Delta}V_{m}^{\\alpha\\beta}}$\n", - "\n", - "As an example, three potential phase diagrams of pure Fe are shown in .\n", - "There are four phases in the system, bcc, fcc, hcp, and liquid. In the\n", - "literature, the high temperature and low temperature bcc phases are\n", - "usually denoted by \u03b4 (high temperature) and \u03b1 (low temperature), the fcc\n", - "and hcp phases by \u03b3 and \u03b5, and the liquid phase by L, respectively. In\n", - "these figures, the two-dimensional areas are single-phase regions where\n", - "two potentials can change independently with the system remaining as\n", - "single-phase. The lines denote two-phase equilibrium regions where only\n", - "one potential can vary independently if the two-phase equilibrium is to\n", - "be maintained. The points where three two-phase equilibrium lines meet\n", - "represent the invariant three-phase equilibria with three potentials\n", - "fixed.\n", - "\n", - "Figure \u2011 $T - ( - P)$, $\\mu_{A} - T$ and $\\mu_{A} - ( - P)$ phase\n", - "diagrams of pure Fe\n", - "\n", - "Based on the discussions in Chapter , enthalpy and entropy of a phase\n", - "increase monotonically with temperature, and phases stable at higher\n", - "temperatures have higher enthalpy and entropy than phases stable at\n", - "lower temperatures. Consequently, the two-phase equilibrium lines in a\n", - "$T - ( - P)$ potential phase diagram have negative slopes if the phase\n", - "stable at higher temperatures also has larger molar volume than the\n", - "phase stable at lower temperatures (note that if $P$ is plotted instead\n", - "of $- P$, the slope is positive). This is the case for the two-phase\n", - "equilibrium lines of \u03b4/L, \u03b3/L, and \u03b3/\u03b4 at high temperatures, and \u03b5/\u03b3\n", - "shown in a. On the other hand, the two-phase equilibrium lines of \u03b1/\u03b5\n", - "and \u03b1/\u03b3 at low temperatures have positive slopes, indicating that \u03b5 and\n", - "\u03b3 have smaller molar volume than \u03b1 as \u03b5 and \u03b3 are more stable at higher\n", - "pressures than \u03b1 at constant temperatures. It is thus evident that the\n", - "phase stable at higher pressure can have either higher or lower entropy\n", - "than the phase stable at lower pressure, and the phase stable at higher\n", - "temperature can have either higher or lower volume than the phase stable\n", - "at lower temperature. This is the property anomaly discussed in Chapter\n", - ".\n", - "\n", - "Another useful example of potential phase diagram is the\n", - "pressure-temperature phase diagram of $H_{2}O$ shown in with three\n", - "phases: ice, water, and vapour. It is known that the solid ice has many\n", - "polymorphic structures at high pressures, which are not included in this\n", - "diagram. As in the pure Fe potential phase diagram discussed above, the\n", - "single-phase regions of ice, water, and vapour are represented by the\n", - "two-dimensional areas with two-degree of freedom based on the Gibbs\n", - "phase rule, the lines are for the two-phase regions of ice-water,\n", - "ice-vapour, and water-vapour, and the three-phase equilibrium has\n", - "zero-degree of freedom represented by a point at $273.16\\ K$ and\n", - "$611.73\\ Pa$.\n", - "\n", - "There are two features in which are different from those of Fe shown in\n", - "a. The first feature is that the slope of the liquid-solid two-phase\n", - "equilibrium line in has the opposite sign of that in a. This is because\n", - "solid ice has larger molar volume than liquid water, while the molar\n", - "volume of liquid Fe is larger than those of fcc-Fe and bcc-Fe. The\n", - "second feature is that the two-phase equilibrium line of water-vapour\n", - "ends at $647\\ K$ and $22.064 \\cdot 10^{6}\\ Pa$. Beyond this point, the\n", - "difference between vapour and water disappears when the pressure and\n", - "temperature are changed, i.e. it behaves like one phase. This point is a\n", - "critical point, as discussed in Chapter . However, it should be pointed\n", - "out that it does not represent an invariant reaction as the degree of\n", - "freedom based on the Gibbs phase rule is equal to one and not zero. On\n", - "the other hand, both the temperature and pressure of the critical point\n", - "are invariant due to the two constraints introduced by the limit of\n", - "stability of a single phase, i.e. the second and third derivatives of\n", - "temperature to entropy or pressure to volume are zero.\n", - "\n", - "Figure \u2011: P-T phase diagram of $H_{2}O$\n" - ], + "source": "### Potential phase diagrams of one-component systems\n\nAs physical vision of human being is limited to three-dimensions, only\none-component system can be completely visualized as shown in for one\nphase where any two of the three potentials can change independently.\nFrom Gibbs phase rule, when two phases are in equilibrium, only one\npotential can vary freely if the two-phase equilibrium is to be\nmaintained. While three phases are in equilibrium, the degree of freedom\nis zero, and all three potentials are fixed.\n\nFor a two-phase equilibrium, two surfaces intersect each other as\ndepicted by the dashed line in . This two-phase equilibrium line is\nobtained by applying to both phases in a one-component system. Since one\nof the potentials is dependent on the other two, one can eliminate it by\ndividing the equation by its conjugate molar quantity and subtracting\nthe two equations, resulting in the following three equations\n\nEq. \u2011\n$0 = \\left( \\frac{S^{\\alpha}}{N_{A}^{\\alpha}} - \\frac{S^{\\beta}}{N_{A}^{\\beta}} \\right)dT + \\left( \\frac{V^{\\alpha}}{N_{A}^{\\alpha}} - \\frac{V^{\\beta}}{N_{A}^{\\beta}} \\right)d( - P) = \\mathrm{\\Delta}S_{m}^{\\alpha\\beta}dT + \\mathrm{\\Delta}V_{m}^{\\alpha\\beta}d( - P)$\n\nEq. \u2011\n$0 = \\left( \\frac{S^{\\alpha}}{V^{\\alpha}} - \\frac{S^{\\beta}}{V^{\\beta}} \\right)dT + \\left( \\frac{N_{A}^{\\alpha}}{V^{\\alpha}} - \\frac{N_{A}^{\\beta}}{V^{\\beta}} \\right){d\\mu}_{A} = \\mathrm{\\Delta}\\left( \\frac{S_{m}}{V_{m}} \\right)^{\\alpha\\beta}dT + \\mathrm{\\Delta}\\left( \\frac{1}{V_{m}} \\right)^{\\alpha\\beta}{d\\mu}_{A}$\n\nEq. \u2011\n$0 = \\left( \\frac{V^{\\alpha}}{S^{\\alpha}} - \\frac{V^{\\beta}}{S^{\\beta}} \\right)d( - P) + \\left( \\frac{N_{A}^{\\alpha}}{S^{\\alpha}} - \\frac{N_{A}^{\\beta}}{S^{\\beta}} \\right){d\\mu}_{A} = \\mathrm{\\Delta}\\left( \\frac{V_{m}}{S_{m}} \\right)^{\\alpha\\beta}d( - P) + \\mathrm{\\Delta}\\left( \\frac{1}{S_{m}} \\right)^{\\alpha\\beta}{d\\mu}_{A}$\n\nFigure \u2011: Gibbs energy surfaces of two phases and their intersection,\nrepresenting the two-phase equilibrium.\n\nThe directions of the two-phase equilibrium line can thus be obtained as\n\nEq. \u2011\n$\\frac{d( - P)}{dT} = - \\frac{\\mathrm{\\Delta}S_{m}^{\\alpha\\beta}}{\\mathrm{\\Delta}V_{m}^{\\alpha\\beta}}$\n\nEq. \u2011\n$\\frac{dT}{{d\\mu}_{A}} = - \\frac{\\mathrm{\\Delta}\\left( \\frac{1}{V_{m}} \\right)^{\\alpha\\beta}}{\\mathrm{\\Delta}\\left( \\frac{S_{m}}{V_{m}} \\right)^{\\alpha\\beta}}$\n\nEq. \u2011\n$\\frac{d( - P)}{{d\\mu}_{A}} = - \\frac{\\mathrm{\\Delta}\\left( \\frac{1}{S_{m}} \\right)^{\\alpha\\beta}}{\\mathrm{\\Delta}\\left( \\frac{V_{m}}{S_{m}} \\right)^{\\alpha\\beta}}$\n\nThese three equations define the mathematical forms of the two-phase\nequilibrium line in the two-dimensional spaces of $T - ( - P)$,\n$\\mu_{A} - T$ and $\\mu_{A} - ( - P)$, respectively, and can thus be\nplotted as two-dimensional diagrams. is commonly called\nClausius-Clapeyron equation in the literature. One may thus call all\nthree equations above as generalized Clausius-Clapeyron equations. At\nequilibrium, the chemical potentials of the component in both phases are\nequal to each other, so are their Gibbs energies. One thus has\n\nEq. \u2011\n$G_{m}^{\\alpha} - G_{m}^{\\beta} = 0 = \\mathrm{\\Delta}G_{m} = \\mathrm{\\Delta}H_{m}^{\\alpha\\beta} - T\\mathrm{\\Delta}S_{m}^{\\alpha\\beta}$\n\nThe Clausius-Clapeyron equation, , can be re-written as\n\nEq. \u2011\n$\\frac{d( - P)}{dT} = - \\frac{\\mathrm{\\Delta}H_{m}^{\\alpha\\beta}}{T\\mathrm{\\Delta}V_{m}^{\\alpha\\beta}}$\n\nAs an example, three potential phase diagrams of pure Fe are shown in .\nThere are four phases in the system, bcc, fcc, hcp, and liquid. In the\nliterature, the high temperature and low temperature bcc phases are\nusually denoted by \u03b4 (high temperature) and \u03b1 (low temperature), the fcc\nand hcp phases by \u03b3 and \u03b5, and the liquid phase by L, respectively. In\nthese figures, the two-dimensional areas are single-phase regions where\ntwo potentials can change independently with the system remaining as\nsingle-phase. The lines denote two-phase equilibrium regions where only\none potential can vary independently if the two-phase equilibrium is to\nbe maintained. The points where three two-phase equilibrium lines meet\nrepresent the invariant three-phase equilibria with three potentials\nfixed.\n\nFigure \u2011 $T - ( - P)$, $\\mu_{A} - T$ and $\\mu_{A} - ( - P)$ phase\ndiagrams of pure Fe\n\nBased on the discussions in Chapter , enthalpy and entropy of a phase\nincrease monotonically with temperature, and phases stable at higher\ntemperatures have higher enthalpy and entropy than phases stable at\nlower temperatures. Consequently, the two-phase equilibrium lines in a\n$T - ( - P)$ potential phase diagram have negative slopes if the phase\nstable at higher temperatures also has larger molar volume than the\nphase stable at lower temperatures (note that if $P$ is plotted instead\nof $- P$, the slope is positive). This is the case for the two-phase\nequilibrium lines of \u03b4/L, \u03b3/L, and \u03b3/\u03b4 at high temperatures, and \u03b5/\u03b3\nshown in a. On the other hand, the two-phase equilibrium lines of \u03b1/\u03b5\nand \u03b1/\u03b3 at low temperatures have positive slopes, indicating that \u03b5 and\n\u03b3 have smaller molar volume than \u03b1 as \u03b5 and \u03b3 are more stable at higher\npressures than \u03b1 at constant temperatures. It is thus evident that the\nphase stable at higher pressure can have either higher or lower entropy\nthan the phase stable at lower pressure, and the phase stable at higher\ntemperature can have either higher or lower volume than the phase stable\nat lower temperature. This is the property anomaly discussed in Chapter\n.\n\nAnother useful example of potential phase diagram is the\npressure-temperature phase diagram of $H_{2}O$ shown in with three\nphases: ice, water, and vapour. It is known that the solid ice has many\npolymorphic structures at high pressures, which are not included in this\ndiagram. As in the pure Fe potential phase diagram discussed above, the\nsingle-phase regions of ice, water, and vapour are represented by the\ntwo-dimensional areas with two-degree of freedom based on the Gibbs\nphase rule, the lines are for the two-phase regions of ice-water,\nice-vapour, and water-vapour, and the three-phase equilibrium has\nzero-degree of freedom represented by a point at $273.16\\ K$ and\n$611.73\\ Pa$.\n\nThere are two features in which are different from those of Fe shown in\na. The first feature is that the slope of the liquid-solid two-phase\nequilibrium line in has the opposite sign of that in a. This is because\nsolid ice has larger molar volume than liquid water, while the molar\nvolume of liquid Fe is larger than those of fcc-Fe and bcc-Fe. The\nsecond feature is that the two-phase equilibrium line of water-vapour\nends at $647\\ K$ and $22.064 \\cdot 10^{6}\\ Pa$. Beyond this point, the\ndifference between vapour and water disappears when the pressure and\ntemperature are changed, i.e. it behaves like one phase. This point is a\ncritical point, as discussed in Chapter . However, it should be pointed\nout that it does not represent an invariant reaction as the degree of\nfreedom based on the Gibbs phase rule is equal to one and not zero. On\nthe other hand, both the temperature and pressure of the critical point\nare invariant due to the two constraints introduced by the limit of\nstability of a single phase, i.e. the second and third derivatives of\ntemperature to entropy or pressure to volume are zero.\n\nFigure \u2011: P-T phase diagram of $H_{2}O$\n", "metadata": {} }, { - "id": "7b01cdb0", + "id": "9f0752e8", "cell_type": "markdown", - "source": [ - "### Potential phase diagrams of two-component systems\n", - "\n", - "From Gibbs-Duhem equation (see ), a single phase in a two-component\n", - "system has three independent potentials, out of the four potentials of\n", - "$T$, $- P$, $\\mu_{A}$, and $\\mu_{B}$, and is a three-dimensional\n", - "geometric feature in a four-dimensional space. Alternatively, it can be\n", - "represented by a three-dimensional space constructed by the three\n", - "independent potentials. A two-phase equilibrium is thus a\n", - "two-dimensional surface in this three-dimensional space created by the\n", - "intercept of two three-dimensional spaces, and a three-phase equilibrium\n", - "is a one-dimensional line, and a four-phase equilibrium is a\n", - "zero-dimensional point. This is shown in for the Fe-C binary system\n", - "involving four phases: fcc, bcc, Fe3C and graphite. Since any\n", - "one of the four potentials can be chosen as the dependent one, four\n", - "three-dimensional potential phase diagrams are depicted in .\n", - "\n", - "Figure \u2011: Projected potential phase diagram of Fe-C system with fcc,\n", - "bcc, Fe3C, and graphite\n", - "\n", - "The two-phase equilibrium surfaces are obtained by choosing any one of\n", - "the four potentials as the dependent one and solving the Gibbs-Duhem\n", - "equations for both phases, resulting in following four equations\n", - "\n", - "Eq. \u2011\n", - "$0 = \\left( \\frac{S^{\\alpha}}{N_{A}^{\\alpha}} - \\frac{S^{\\beta}}{N_{A}^{\\beta}} \\right)dT + \\left( \\frac{V^{\\alpha}}{N_{A}^{\\alpha}} - \\frac{V^{\\beta}}{N_{A}^{\\beta}} \\right)d( - P) + \\left( \\frac{N_{B}^{\\alpha}}{N_{A}^{\\alpha}} - \\frac{N_{B}^{\\beta}}{N_{A}^{\\beta}} \\right){d\\mu}_{B} = \\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}dT + \\mathrm{\\Delta}V_{mA}^{\\alpha\\beta}d( - P) + \\mathrm{\\Delta}z_{B}^{\\alpha\\beta}{d\\mu}_{B}$\n", - "\n", - "Eq. \u2011\n", - "$0 = \\left( \\frac{S^{\\alpha}}{N_{B}^{\\alpha}} - \\frac{S^{\\beta}}{N_{B}^{\\beta}} \\right)dT + \\left( \\frac{V^{\\alpha}}{N_{B}^{\\alpha}} - \\frac{V^{\\beta}}{N_{B}^{\\beta}} \\right)d( - P) + \\left( \\frac{N_{B}^{\\alpha}}{N_{B}^{\\alpha}} - \\frac{N_{B}^{\\beta}}{N_{B}^{\\beta}} \\right){d\\mu}_{B} = \\mathrm{\\Delta}S_{mB}^{\\alpha\\beta}dT + \\mathrm{\\Delta}V_{mB}^{\\alpha\\beta}d( - P) + \\mathrm{\\Delta}z_{A}^{\\alpha\\beta}{d\\mu}_{A}$\n", - "\n", - "Eq. \u2011\n", - "$0 = \\left( \\frac{S^{\\alpha}}{V^{\\alpha}} - \\frac{S^{\\beta}}{V^{\\beta}} \\right)dT + \\left( \\frac{N_{A}^{\\alpha}}{V^{\\alpha}} - \\frac{N_{A}^{\\beta}}{V^{\\beta}} \\right){d\\mu}_{A} + \\left( \\frac{N_{B}^{\\alpha}}{V^{\\alpha}} - \\frac{N_{B}^{\\beta}}{V^{\\beta}} \\right){d\\mu}_{B} = \\mathrm{\\Delta}\\left( \\frac{S_{m}}{V_{m}} \\right)^{\\alpha\\beta}dT + \\mathrm{\\Delta}\\left( \\frac{1}{V_{mA}} \\right)^{\\alpha\\beta}{d\\mu}_{A} + \\mathrm{\\Delta}\\left( \\frac{1}{V_{mB}} \\right)^{\\alpha\\beta}{d\\mu}_{B}$\n", - "\n", - "Eq. \u2011\n", - "$0 = \\left( \\frac{V^{\\alpha}}{S^{\\alpha}} - \\frac{V^{\\beta}}{S^{\\beta}} \\right)d( - P) + \\left( \\frac{N_{A}^{\\alpha}}{S^{\\alpha}} - \\frac{N_{A}^{\\beta}}{S^{\\beta}} \\right){d\\mu}_{A} + \\left( \\frac{N_{B}^{\\alpha}}{S^{\\alpha}} - \\frac{N_{B}^{\\beta}}{S^{\\beta}} \\right){d\\mu}_{B} = \\mathrm{\\Delta}\\left( \\frac{V_{m}}{S_{m}} \\right)^{\\alpha\\beta}d( - P) + \\mathrm{\\Delta}\\left( \\frac{1}{S_{mA}} \\right)^{\\alpha\\beta}{d\\mu}_{A} + \\mathrm{\\Delta}\\left( \\frac{1}{S_{mB}} \\right)^{\\alpha\\beta}{d\\mu}_{B}$\n", - "\n", - "A three-phase equilibrium line is represented by the intercept of two\n", - "two-phase surfaces by applying any one of the above four equations to\n", - "two two-phase equilibria. Let us use as an example\n", - "\n", - "Eq. \u2011\n", - "$0 = \\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}dT + \\mathrm{\\Delta}V_{mA}^{\\alpha\\beta}d( - P) + {\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}{d\\mu}_{B}$\n", - "\n", - "Eq. \u2011\n", - "$0 = \\mathrm{\\Delta}S_{mA}^{\\alpha\\gamma}dT + \\mathrm{\\Delta}V_{mA}^{\\alpha\\gamma}d( - P) + \\mathrm{\\Delta}{z_{B}^{\\alpha\\gamma}d\\mu}_{B}$\n", - "\n", - "It is self-evident that the two-phase equilibrium surface between\n", - "$\\beta$ and $\\gamma$ is not independent and can be obtained by\n", - "subtraction of and\n", - "\n", - "Eq. \u2011\n", - "$0 = \\left( \\mathrm{\\Delta}S_{mA}^{\\alpha\\beta} - \\mathrm{\\Delta}S_{mA}^{\\alpha\\gamma} \\right)dT + \\left( \\mathrm{\\Delta}V_{mA}^{\\alpha\\beta} - \\mathrm{\\Delta}V_{mA}^{\\alpha\\gamma} \\right)d( - P) + \\left( {\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta} - \\mathrm{\\Delta}z_{B}^{\\alpha\\gamma} \\right){d\\mu}_{B} = \\mathrm{\\Delta}S_{mA}^{\\gamma\\beta}dT + \\mathrm{\\Delta}V_{mA}^{\\gamma\\beta}d( - P) + {\\mathrm{\\Delta}z}_{B}^{\\gamma\\beta}{d\\mu}_{B}$\n", - "\n", - "Eliminating one of three potentials in and , one can obtain three\n", - "equations for the three-phase equilibrium line\n", - "\n", - "Eq. \u2011\n", - "$\\frac{d( - P)}{dT} = - \\frac{\\frac{\\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}}{{\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}S_{mA}^{\\alpha\\gamma}}{\\mathrm{\\Delta}z_{B}^{\\alpha\\gamma}}}{\\frac{\\mathrm{\\Delta}V_{mA}^{\\alpha\\beta}}{{\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}V_{mA}^{\\alpha\\gamma}}{\\mathrm{\\Delta}z_{B}^{\\alpha\\gamma}}}$\n", - "\n", - "Eq. \u2011\n", - "$\\frac{dT}{{d\\mu}_{B}} = - \\frac{\\frac{{\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}}{\\mathrm{\\Delta}V_{mA}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}z_{B}^{\\alpha\\gamma}}{\\mathrm{\\Delta}V_{mA}^{\\alpha\\gamma}}}{\\frac{\\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}}{\\mathrm{\\Delta}V_{mA}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}S_{mA}^{\\alpha\\gamma}}{\\mathrm{\\Delta}V_{mA}^{\\alpha\\gamma}}}$\n", - "\n", - "Eq. \u2011\n", - "$\\frac{d( - P)}{{d\\mu}_{B}} = - \\frac{\\frac{{\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}}{\\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}z_{B}^{\\alpha\\gamma}}{\\mathrm{\\Delta}S_{mA}^{\\alpha\\gamma}}}{\\frac{\\mathrm{\\Delta}V_{mA}^{\\alpha\\beta}}{\\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}V_{mA}^{\\alpha\\gamma}}{\\mathrm{\\Delta}S_{mA}^{\\alpha\\gamma}}}$\n", - "\n", - "to can be referred as generalized Clausius-Clapeyron equations for\n", - "binary systems. Similar equations can be derived for\n", - "$T - ( - P) - \\mu_{A}$, $T - \\mu_{A} - \\mu_{B}$, and\n", - "$( - P) - \\mu_{A} - \\mu_{B}$ potential phase diagrams from to , and are\n", - "listed below\n", - "\n", - "- Generalized Clausius-Clapeyron equations for a three-phase equilibrium\n", - " in $T - ( - P) - \\mu_{A}$ potential phase diagrams\n", - "\n", - "Eq. \u2011\n", - "$\\frac{d( - P)}{dT} = - \\frac{\\frac{\\mathrm{\\Delta}S_{mB}^{\\alpha\\beta}}{{\\mathrm{\\Delta}z}_{A}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}S_{mB}^{\\alpha\\gamma}}{\\mathrm{\\Delta}z_{A}^{\\alpha\\gamma}}}{\\frac{\\mathrm{\\Delta}V_{mB}^{\\alpha\\beta}}{{\\mathrm{\\Delta}z}_{A}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}V_{mB}^{\\alpha\\gamma}}{\\mathrm{\\Delta}z_{A}^{\\alpha\\gamma}}}$\n", - "\n", - "Eq. \u2011\n", - "$\\frac{dT}{{d\\mu}_{A}} = - \\frac{\\frac{{\\mathrm{\\Delta}z}_{A}^{\\alpha\\beta}}{\\mathrm{\\Delta}V_{mB}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}z_{A}^{\\alpha\\gamma}}{\\mathrm{\\Delta}V_{mB}^{\\alpha\\gamma}}}{\\frac{\\mathrm{\\Delta}S_{mB}^{\\alpha\\beta}}{\\mathrm{\\Delta}V_{mB}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}S_{mB}^{\\alpha\\gamma}}{\\mathrm{\\Delta}V_{mB}^{\\alpha\\gamma}}}$\n", - "\n", - "Eq. \u2011\n", - "$\\frac{d( - P)}{{d\\mu}_{A}} = - \\frac{\\frac{{\\mathrm{\\Delta}z}_{A}^{\\alpha\\beta}}{\\mathrm{\\Delta}S_{mB}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}z_{A}^{\\alpha\\gamma}}{\\mathrm{\\Delta}S_{mB}^{\\alpha\\gamma}}}{\\frac{\\mathrm{\\Delta}V_{mB}^{\\alpha\\beta}}{\\mathrm{\\Delta}S_{mB}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}V_{mB}^{\\alpha\\gamma}}{\\mathrm{\\Delta}S_{mB}^{\\alpha\\gamma}}}$\n", - "\n", - "- Generalized Clausius-Clapeyron equations for a three-phase equilibrium\n", - " in $\\ T - \\mu_{A} - \\mu_{B}$ potential phase diagrams\n", - "\n", - "Eq. \u2011\n", - "$\\frac{dT}{{d\\mu}_{A}} = - \\frac{\\frac{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}}{\\frac{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}}$\n", - "\n", - "Eq. \u2011\n", - "$\\frac{dT}{{d\\mu}_{B}} = - \\frac{\\frac{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}}{\\frac{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}}$\n", - "\n", - "Eq. \u2011\n", - "$\\frac{{d\\mu}_{A}}{{d\\mu}_{B}} = - \\frac{\\frac{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\gamma}}}{\\frac{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\gamma}}}$\n", - "\n", - "- Generalized Clausius-Clapeyron equations for a three-phase equilibrium\n", - " in $( - P) - \\mu_{A} - \\mu_{B}$ potential phase diagrams\n", - "\n", - "Eq. \u2011\n", - "$\\frac{d( - P)}{{d\\mu}_{A}} = - \\frac{\\frac{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}}{\\frac{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}}$\n", - "\n", - "Eq. \u2011\n", - "$\\frac{d( - P)}{{d\\mu}_{B}} = - \\frac{\\frac{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}}{\\frac{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}}$\n", - "\n", - "Eq. \u2011\n", - "$\\frac{{d\\mu}_{A}}{{d\\mu}_{B}} = - \\frac{\\frac{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\gamma}}}{\\frac{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\gamma}}}$\n" - ], + "source": "### Potential phase diagrams of two-component systems\n\nFrom Gibbs-Duhem equation (see ), a single phase in a two-component\nsystem has three independent potentials, out of the four potentials of\n$T$, $- P$, $\\mu_{A}$, and $\\mu_{B}$, and is a three-dimensional\ngeometric feature in a four-dimensional space. Alternatively, it can be\nrepresented by a three-dimensional space constructed by the three\nindependent potentials. A two-phase equilibrium is thus a\ntwo-dimensional surface in this three-dimensional space created by the\nintercept of two three-dimensional spaces, and a three-phase equilibrium\nis a one-dimensional line, and a four-phase equilibrium is a\nzero-dimensional point. This is shown in for the Fe-C binary system\ninvolving four phases: fcc, bcc, Fe3C and graphite. Since any\none of the four potentials can be chosen as the dependent one, four\nthree-dimensional potential phase diagrams are depicted in .\n\nFigure \u2011: Projected potential phase diagram of Fe-C system with fcc,\nbcc, Fe3C, and graphite\n\nThe two-phase equilibrium surfaces are obtained by choosing any one of\nthe four potentials as the dependent one and solving the Gibbs-Duhem\nequations for both phases, resulting in following four equations\n\nEq. \u2011\n$0 = \\left( \\frac{S^{\\alpha}}{N_{A}^{\\alpha}} - \\frac{S^{\\beta}}{N_{A}^{\\beta}} \\right)dT + \\left( \\frac{V^{\\alpha}}{N_{A}^{\\alpha}} - \\frac{V^{\\beta}}{N_{A}^{\\beta}} \\right)d( - P) + \\left( \\frac{N_{B}^{\\alpha}}{N_{A}^{\\alpha}} - \\frac{N_{B}^{\\beta}}{N_{A}^{\\beta}} \\right){d\\mu}_{B} = \\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}dT + \\mathrm{\\Delta}V_{mA}^{\\alpha\\beta}d( - P) + \\mathrm{\\Delta}z_{B}^{\\alpha\\beta}{d\\mu}_{B}$\n\nEq. \u2011\n$0 = \\left( \\frac{S^{\\alpha}}{N_{B}^{\\alpha}} - \\frac{S^{\\beta}}{N_{B}^{\\beta}} \\right)dT + \\left( \\frac{V^{\\alpha}}{N_{B}^{\\alpha}} - \\frac{V^{\\beta}}{N_{B}^{\\beta}} \\right)d( - P) + \\left( \\frac{N_{B}^{\\alpha}}{N_{B}^{\\alpha}} - \\frac{N_{B}^{\\beta}}{N_{B}^{\\beta}} \\right){d\\mu}_{B} = \\mathrm{\\Delta}S_{mB}^{\\alpha\\beta}dT + \\mathrm{\\Delta}V_{mB}^{\\alpha\\beta}d( - P) + \\mathrm{\\Delta}z_{A}^{\\alpha\\beta}{d\\mu}_{A}$\n\nEq. \u2011\n$0 = \\left( \\frac{S^{\\alpha}}{V^{\\alpha}} - \\frac{S^{\\beta}}{V^{\\beta}} \\right)dT + \\left( \\frac{N_{A}^{\\alpha}}{V^{\\alpha}} - \\frac{N_{A}^{\\beta}}{V^{\\beta}} \\right){d\\mu}_{A} + \\left( \\frac{N_{B}^{\\alpha}}{V^{\\alpha}} - \\frac{N_{B}^{\\beta}}{V^{\\beta}} \\right){d\\mu}_{B} = \\mathrm{\\Delta}\\left( \\frac{S_{m}}{V_{m}} \\right)^{\\alpha\\beta}dT + \\mathrm{\\Delta}\\left( \\frac{1}{V_{mA}} \\right)^{\\alpha\\beta}{d\\mu}_{A} + \\mathrm{\\Delta}\\left( \\frac{1}{V_{mB}} \\right)^{\\alpha\\beta}{d\\mu}_{B}$\n\nEq. \u2011\n$0 = \\left( \\frac{V^{\\alpha}}{S^{\\alpha}} - \\frac{V^{\\beta}}{S^{\\beta}} \\right)d( - P) + \\left( \\frac{N_{A}^{\\alpha}}{S^{\\alpha}} - \\frac{N_{A}^{\\beta}}{S^{\\beta}} \\right){d\\mu}_{A} + \\left( \\frac{N_{B}^{\\alpha}}{S^{\\alpha}} - \\frac{N_{B}^{\\beta}}{S^{\\beta}} \\right){d\\mu}_{B} = \\mathrm{\\Delta}\\left( \\frac{V_{m}}{S_{m}} \\right)^{\\alpha\\beta}d( - P) + \\mathrm{\\Delta}\\left( \\frac{1}{S_{mA}} \\right)^{\\alpha\\beta}{d\\mu}_{A} + \\mathrm{\\Delta}\\left( \\frac{1}{S_{mB}} \\right)^{\\alpha\\beta}{d\\mu}_{B}$\n\nA three-phase equilibrium line is represented by the intercept of two\ntwo-phase surfaces by applying any one of the above four equations to\ntwo two-phase equilibria. Let us use as an example\n\nEq. \u2011\n$0 = \\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}dT + \\mathrm{\\Delta}V_{mA}^{\\alpha\\beta}d( - P) + {\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}{d\\mu}_{B}$\n\nEq. \u2011\n$0 = \\mathrm{\\Delta}S_{mA}^{\\alpha\\gamma}dT + \\mathrm{\\Delta}V_{mA}^{\\alpha\\gamma}d( - P) + \\mathrm{\\Delta}{z_{B}^{\\alpha\\gamma}d\\mu}_{B}$\n\nIt is self-evident that the two-phase equilibrium surface between\n$\\beta$ and $\\gamma$ is not independent and can be obtained by\nsubtraction of and\n\nEq. \u2011\n$0 = \\left( \\mathrm{\\Delta}S_{mA}^{\\alpha\\beta} - \\mathrm{\\Delta}S_{mA}^{\\alpha\\gamma} \\right)dT + \\left( \\mathrm{\\Delta}V_{mA}^{\\alpha\\beta} - \\mathrm{\\Delta}V_{mA}^{\\alpha\\gamma} \\right)d( - P) + \\left( {\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta} - \\mathrm{\\Delta}z_{B}^{\\alpha\\gamma} \\right){d\\mu}_{B} = \\mathrm{\\Delta}S_{mA}^{\\gamma\\beta}dT + \\mathrm{\\Delta}V_{mA}^{\\gamma\\beta}d( - P) + {\\mathrm{\\Delta}z}_{B}^{\\gamma\\beta}{d\\mu}_{B}$\n\nEliminating one of three potentials in and , one can obtain three\nequations for the three-phase equilibrium line\n\nEq. \u2011\n$\\frac{d( - P)}{dT} = - \\frac{\\frac{\\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}}{{\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}S_{mA}^{\\alpha\\gamma}}{\\mathrm{\\Delta}z_{B}^{\\alpha\\gamma}}}{\\frac{\\mathrm{\\Delta}V_{mA}^{\\alpha\\beta}}{{\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}V_{mA}^{\\alpha\\gamma}}{\\mathrm{\\Delta}z_{B}^{\\alpha\\gamma}}}$\n\nEq. \u2011\n$\\frac{dT}{{d\\mu}_{B}} = - \\frac{\\frac{{\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}}{\\mathrm{\\Delta}V_{mA}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}z_{B}^{\\alpha\\gamma}}{\\mathrm{\\Delta}V_{mA}^{\\alpha\\gamma}}}{\\frac{\\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}}{\\mathrm{\\Delta}V_{mA}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}S_{mA}^{\\alpha\\gamma}}{\\mathrm{\\Delta}V_{mA}^{\\alpha\\gamma}}}$\n\nEq. \u2011\n$\\frac{d( - P)}{{d\\mu}_{B}} = - \\frac{\\frac{{\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}}{\\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}z_{B}^{\\alpha\\gamma}}{\\mathrm{\\Delta}S_{mA}^{\\alpha\\gamma}}}{\\frac{\\mathrm{\\Delta}V_{mA}^{\\alpha\\beta}}{\\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}V_{mA}^{\\alpha\\gamma}}{\\mathrm{\\Delta}S_{mA}^{\\alpha\\gamma}}}$\n\nto can be referred as generalized Clausius-Clapeyron equations for\nbinary systems. Similar equations can be derived for\n$T - ( - P) - \\mu_{A}$, $T - \\mu_{A} - \\mu_{B}$, and\n$( - P) - \\mu_{A} - \\mu_{B}$ potential phase diagrams from to , and are\nlisted below\n\n- Generalized Clausius-Clapeyron equations for a three-phase equilibrium\n in $T - ( - P) - \\mu_{A}$ potential phase diagrams\n\nEq. \u2011\n$\\frac{d( - P)}{dT} = - \\frac{\\frac{\\mathrm{\\Delta}S_{mB}^{\\alpha\\beta}}{{\\mathrm{\\Delta}z}_{A}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}S_{mB}^{\\alpha\\gamma}}{\\mathrm{\\Delta}z_{A}^{\\alpha\\gamma}}}{\\frac{\\mathrm{\\Delta}V_{mB}^{\\alpha\\beta}}{{\\mathrm{\\Delta}z}_{A}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}V_{mB}^{\\alpha\\gamma}}{\\mathrm{\\Delta}z_{A}^{\\alpha\\gamma}}}$\n\nEq. \u2011\n$\\frac{dT}{{d\\mu}_{A}} = - \\frac{\\frac{{\\mathrm{\\Delta}z}_{A}^{\\alpha\\beta}}{\\mathrm{\\Delta}V_{mB}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}z_{A}^{\\alpha\\gamma}}{\\mathrm{\\Delta}V_{mB}^{\\alpha\\gamma}}}{\\frac{\\mathrm{\\Delta}S_{mB}^{\\alpha\\beta}}{\\mathrm{\\Delta}V_{mB}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}S_{mB}^{\\alpha\\gamma}}{\\mathrm{\\Delta}V_{mB}^{\\alpha\\gamma}}}$\n\nEq. \u2011\n$\\frac{d( - P)}{{d\\mu}_{A}} = - \\frac{\\frac{{\\mathrm{\\Delta}z}_{A}^{\\alpha\\beta}}{\\mathrm{\\Delta}S_{mB}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}z_{A}^{\\alpha\\gamma}}{\\mathrm{\\Delta}S_{mB}^{\\alpha\\gamma}}}{\\frac{\\mathrm{\\Delta}V_{mB}^{\\alpha\\beta}}{\\mathrm{\\Delta}S_{mB}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}V_{mB}^{\\alpha\\gamma}}{\\mathrm{\\Delta}S_{mB}^{\\alpha\\gamma}}}$\n\n- Generalized Clausius-Clapeyron equations for a three-phase equilibrium\n in $\\ T - \\mu_{A} - \\mu_{B}$ potential phase diagrams\n\nEq. \u2011\n$\\frac{dT}{{d\\mu}_{A}} = - \\frac{\\frac{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}}{\\frac{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}}$\n\nEq. \u2011\n$\\frac{dT}{{d\\mu}_{B}} = - \\frac{\\frac{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}}{\\frac{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}}$\n\nEq. \u2011\n$\\frac{{d\\mu}_{A}}{{d\\mu}_{B}} = - \\frac{\\frac{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\gamma}}}{\\frac{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\gamma}}}$\n\n- Generalized Clausius-Clapeyron equations for a three-phase equilibrium\n in $( - P) - \\mu_{A} - \\mu_{B}$ potential phase diagrams\n\nEq. \u2011\n$\\frac{d( - P)}{{d\\mu}_{A}} = - \\frac{\\frac{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}}{\\frac{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}}$\n\nEq. \u2011\n$\\frac{d( - P)}{{d\\mu}_{B}} = - \\frac{\\frac{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}}{\\frac{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}}$\n\nEq. \u2011\n$\\frac{{d\\mu}_{A}}{{d\\mu}_{B}} = - \\frac{\\frac{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\gamma}}}{\\frac{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\gamma}}}$\n", "metadata": {} }, { - "id": "9749bc6c", + "id": "0b41a83f", "cell_type": "markdown", - "source": [ - "### Projection and section of phase diagrams with potential and molar quantities\n", - "\n", - "As discussed in Chapter , projections of high-dimensional phase diagrams\n", - "usually cannot keep all information. However, there is one type of\n", - "widely used projection in the literature, i.e. the liquidus surface in\n", - "ternary systems under constant pressure with temperature and mole\n", - "fractions of two components as its axes. The projection along the\n", - "temperature axis reveals the composition regions for primary phases that\n", - "solidify from liquid upon cooling. These regions are separated by\n", - "univariant lines of three-phase equilibria. The projections along one of\n", - "the two mole fractions show the temperature as a function of composition\n", - "on the univariant three-phase equilibrium lines and also depict whether\n", - "a four-phase equilibrium is peritectic or eutectic. There are four\n", - "scenarios for the three univariant three-phase equilibrium lines to meet\n", - "at the four-phase equilibrium as depicted in and discussed individually\n", - "below.\n", - "\n", - "Figure \u2011: Schematic four options for three univariant three-phase\n", - "equilibrium lines to meet at the invariant four-phase equilibrium\n", - "\n", - "The first scenario is that with decreasing temperature, all three\n", - "univariant lines merge into the four-phase equilibrium. It indicates\n", - "that the liquid phase does not exist at temperatures below the\n", - "four-phase invariant reaction. This invariant reaction is thus a ternary\n", - "eutectic reaction with liquid completely transformed to three solid\n", - "phases upon cooling, i.e.\n", - "\n", - "Eq. \u2011 $L \\rightarrow \\alpha + \\beta + \\gamma$\n", - "\n", - "In the second scenario, two univariant lines merge into and one leaves\n", - "from the four-phase equilibrium with decreasing temperature. This means\n", - "that one solid phase at higher temperature is no longer stable at lower\n", - "temperature, and it must react with the liquid phase to form the\n", - "remaining two solid phases. The four-phase invariant reaction is thus\n", - "peritectic. The solid phase common to both univariant lines at high\n", - "temperatures reacts with the liquid phase. Assuming that this phase is\n", - "$\\alpha$, the four-phase invariant reaction becomes\n", - "\n", - "Eq. \u2011 $L + \\alpha \\rightarrow \\beta + \\gamma$\n", - "\n", - "In the third scenario, one univariant line points to and two leave from\n", - "the four-phase equilibrium with decreasing temperature. A new phase\n", - "forms at low temperatures from the three high temperature phases, e.g.\n", - "liquid, $\\alpha$, and $\\beta$, with the four-phase invariant reaction as\n", - "\n", - "Eq. \u2011 $L + \\alpha + \\beta \\rightarrow \\gamma$\n", - "\n", - "The fourth scenario is the inverse of the first scenario, indicating the\n", - "formation of liquid from solid phases upon cooling, i.e.\n", - "\n", - "Eq. \u2011 $\\alpha + \\beta + \\gamma \\rightarrow L$\n", - "\n", - "This case has not been observed in reality.\n", - "\n", - "As an example, the liquidus projections of the Al-Fe-Si ternary system\n", - "are shown in in two formats\\[6\\], i.e. (a) three-dimensional liquidus\n", - "surface with the isotherms showing the liquidus contours; (b)\n", - "conventional projection to the composition axis with the temperature\n", - "decrease shown by arrows; (c) projection to the temperature and weight\n", - "fraction of Si. The first to third scenarios of invariant reactions\n", - "discussed above can clearly be identified and listed in . It is evident\n", - "that c provides the easiest route to visualize the type of invariant\n", - "reactions as show by .\n", - "\n", - "Figure \u2011: Liquidus of the Al-Fe-Si ternary system\\[6\\], (a)\n", - "three-dimensional presentation of the liquidus; (b) projection to the\n", - "composition triangle with isotherms (dotted lines) superimposed and\n", - "their temperatures indicated close to the horizontal axis.\n", - "\n", - "Table \u2011: Invariant liquidus reactions of the Al-Fe-Si ternary system\n", - "with the composition of the liquid phase \\[6\\]\n", - "\n", - "| Reaction | T, \u00b0C | wFe, % | wSi, % |\n", - "|----------|-------|-------------------|-------------------|\n", - "| | 1178 | 77.9 | 21.0 |\n", - "| | 1155 | 49.0 | 0.16 |\n", - "| | 1127 | 52.7 | 2.81 |\n", - "| -H-L | 1076 | 41.6 | 44.0 |\n", - "| | 1073 | 51.0 | 7.19 |\n", - "| | 1050 | 53.8 | 18.4 |\n", - "| -H-L | 1019 | 34.8 | 44.4 |\n", - "| | 1004 | 49.0 | 12.6 |\n", - "| | 1000 | 46.7 | 13.2 |\n", - "| -L | 940 | 37.9 | 32.0 |\n", - "| | 921 | 33.7 | 20.1 |\n", - "| | 899 | 33.8 | 32.2 |\n", - "| | 884 | 30.8 | 26.0 |\n", - "| -L | 877 | 29.5 | 35.2 |\n", - "| | 851 | 23.3 | 21.6 |\n", - "| | 834 | 22.2 | 31.7 |\n", - "| | 825 | 22.1 | 25.7 |\n", - "| | 823 | 21.8 | 25.4 |\n", - "| | 715 | 6.64 | 10.8 |\n", - "| | 694 | 6.11 | 17.1 |\n", - "| | 680 | 4.68 | 11.6 |\n", - "| | 630 | 2.11 | 4.10 |\n", - "| | 616 | 1.76 | 6.56 |\n", - "| | 598 | 1.22 | 14.3 |\n", - "| | 575 | 0.73 | 12.7 |\n", - "\n", - "In contrast to projections, sectioning is used more often to understand\n", - "phase relations in multi-component systems. Sectioning of a potential\n", - "phase diagrams is relatively simple as the resulted phase diagram\n", - "behaves like a system with one component less. The same is true if\n", - "potentials are sectioned in phase diagrams with both potential and molar\n", - "quantities as the section is along the tie-lines of the fixed\n", - "potentials. As an example, shows the ternary Al-Fe-Si potential and\n", - "molar phase diagrams sectioned at T=1273K and P=1atm, commonly referred\n", - "to as isothermal section. It is evident that the geometric features of\n", - "both phase diagrams are identical to those of pure Fe shown in and ,\n", - "respectively, with one-, two-, and three-phase regions and corresponding\n", - "phase boundaries.\n", - "\n", - "Figure \u2011: Ternary isothermal section of the Al-Fe-Si ternary system at\n", - "T=1273K and P=1atm\n", - "\n", - "On the other hand, when the phase diagram is sectioned along a molar\n", - "quantity, it would usually not follow a tie-line because phases in\n", - "equilibrium usually have different values for the same molar quantity.\n", - "Consequently, there are no tie-lines inside such phase diagrams in\n", - "general, and any phase regions only show which phases are in equilibrium\n", - "with each other without any information on the values of molar\n", - "quantities of individual phases.\n", - "\n", - "This type of sectioning reduces both the dimensionalities of the phase\n", - "diagram and phase boundary by the same number, but does not alter the\n", - "number of phases in the adjacent phase regions. The contact rule, i.e. ,\n", - "thus remains valid and is applicable to phase regions with the\n", - "same-dimensionality as that of the sectioned phase diagram. Similarly,\n", - "the Schreinemakers\u2019 rule shown in is valid under the same conditions.\n", - "\n", - "For example, the two-dimensional phase diagram of the Mg-Al-Zn ternary\n", - "system sectioned with one atmospheric pressure and the weight fraction\n", - "of Zn fixed at 0.01 is shown in plotted with temperature and mole\n", - "fraction of Al \\[7\\]. This phase diagram is commonly called isopleth and\n", - "is generated by fixing one potential, the pressure, $P$, changing the\n", - "chemical potentials of Al and Zn to their conjugate molar quantities\n", - "represented by weight fractions of Al and Zn, and sectioning at\n", - "$w_{Zn} = 0.01$. From the discussions in Chapter , the phase regions\n", - "with the number of phases equal to three or fewer, i.e.\n", - "$p \\leq n_{m} + 1 = 3$, have the same-dimensionality as the phase\n", - "diagram, i.e. two-dimensional in the present case, and the phase\n", - "boundary rule is applicable. The maximum number of phases co-existing at\n", - "equilibrium is given by as following for the present case\n", - "\n", - "Eq. \u2011 $p_{\\max} = \\left( c - n_{s} \\right) + 2 = 3 - 1 + 2 = 4$\n", - "\n", - "This is because introducing molar quantities only increases the\n", - "dimensionality of phase regions and does not change the maximum number\n", - "of co-existing phases.\n", - "\n", - "The dimensionality of a four-phase region is calculated from as\n", - "\n", - "Eq. \u2011\n", - "$\\upsilon_{m} = \\left( c - n_{s} \\right) + 2 - p + n_{m} - n_{ms} = 3 - 1 + 2 - 4 + 2 - 1 = 1$\n", - "\n", - "where $n_{ms}$ is the number of sectioned molar quantities. Since the\n", - "dimensionality of a four-phase region is lower than that of the phase\n", - "diagram, the phase boundary rule cannot be applied directly. Such a\n", - "four-phase region, liquid+Mg+\u03b3+\u03c6, is shown in between three three-phase\n", - "regions of liquid+Mg+\u03b3, liquid+Mg+\u03c6, and Mg+\u03b3+\u03c6.\n", - "\n", - "Figure \u2011: Isopleth with the weight fraction of Zn fixed at 0.01 of the\n", - "Mg-Al-Zn ternary system.\n", - "\n", - "also displays information on what phases are in equilibrium for a given\n", - "alloy at various temperatures. One example is shown by the dotted\n", - "vertical line marking the weight fraction of Al being 0.09, a widely\n", - "used Mg alloy called AZ91. Various phases are present at different\n", - "temperature ranges, but the equilibrium phase fractions and phase\n", - "compositions are not shown in the figure as the tie-lines are not in the\n", - "plane of the phase diagram and have to be calculated at each temperature\n", - "individually. shows the amount of each phase of the AZ91 alloy as a\n", - "function of temperature with the dotted lines depicting the values under\n", - "the equilibrium condition and the solid lines depicting the values under\n", - "the so-called Scheil condition assuming no diffusion in solid phases and\n", - "infinitelyfast diffusion in liquid. Similarly, the composition of each\n", - "phase can also be plotted as shown in .\n", - "\n", - "Figure \u2011: Mole fraction of individual phases under equilibrium (dotted\n", - "curves) and Scheil (solid curves) conditions in the AZ91 alloy\n", - "\n", - "Figure \u2011: Mass fraction of Al and Zn in the Mg solid solution phase\n", - "under equilibrium (dotted curves) and Scheil (solid curves) conditions\n", - "in AZ91\n" - ], - "metadata": {} - }, - { - "id": "4299ee7b", - "cell_type": "markdown", - "source": [ - "### Section of potential phase diagrams\n", - "\n", - "Based on the Gibbs-Duhem equation (see ), a single-phase equilibrium in\n", - "a system with more than two independent components has more than three\n", - "independent potentials. There is no problem in representing them using\n", - "the mathematical formulas discussed so far, but it is not possible for\n", - "us to visualize graphically the full potential phase diagrams in\n", - "multi-component systems with more than two independent components. In\n", - "principles, there are two options. One option is to project the\n", - "multi-dimensional potential phase diagram into a two- or three-dimension\n", - "diagram, and another option is to section the multi-dimensional\n", - "potential phase diagram by fixing the values of some potentials.\n", - "\n", - "The projection approach is used for one-component systems in Chapter .\n", - "Since a two-phase equilibrium in a one-component potential phase diagram\n", - "is one-dimensional, the projection does not lose any information, and\n", - "the same is true for a three-phase equilibrium in a one-component\n", - "potential phase diagram. In a binary system, the projections of three-\n", - "and four-phase equilibria do not lose any information, while the\n", - "projections of two-phase equilibria become two-dimensional and cannot\n", - "retain all the information as the original-dimensionality of these\n", - "two-phase equilibria is three. Consequently, sectioning at fixed values\n", - "of some potentials is necessary in order to visualize the phase\n", - "relations in systems with two or more components. Gibbs phase rule shown\n", - "in and are thus modified to\n", - "\n", - "Eq. \u2011 $\\upsilon = c + 2 - p - n_{s} = \\left( c - n_{s} \\right) + 2 - p$\n", - "\n", - "Eq. \u2011 $p_{\\max} = \\left( c - n_{s} \\right) + 2$\n", - "\n", - "where $n_{s}$ is the number of potentials fixed in sectioning. As can be\n", - "seen in the last part of , the number of sectioning is equivalent to the\n", - "reduction of the effective number of independent components. Therefore,\n", - "any multi-component systems with $n_{s} = c - i$ behave like an\n", - "$i$-component system. The equations presented in Chapter and are thus\n", - "directly applicable to multi-component systems with $n_{s} = c - 1$ and\n", - "$n_{s} = c - 2$, respectively.\n", - "\n", - "A common practice in experiments is to fix pressure, temperature or\n", - "chemical potentials of volatile components as they are usually the\n", - "variables controlled experimentally. In a binary system, the potential\n", - "phase diagram at constant pressure can be represented by any two of the\n", - "three potentials, i.e. two chemical potentials and temperature, with the\n", - "remaining potential being dependent, and has the identical morphology as\n", - "a one-component system. The Gibbs-Duhem equation under such conditions\n", - "becomes\n", - "\n", - "Eq. \u2011 $0 = - SdT - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B}$\n", - "\n", - "The corresponding two-phase Clausius-Clapeyron equations are written as\n", - "\n", - "Eq. \u2011\n", - "$\\frac{dT}{{d\\mu}_{A}} = - \\frac{{\\mathrm{\\Delta}z}_{A}^{\\alpha\\beta}}{\\mathrm{\\Delta}S_{mB}^{\\alpha\\beta}}$\n", - "\n", - "Eq. \u2011\n", - "$\\frac{dT}{{d\\mu}_{B}} = - \\frac{{\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}}{\\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}}$\n", - "\n", - "Eq. \u2011\n", - "$\\frac{{d\\mu}_{A}}{{d\\mu}_{B}} = - \\frac{\\mathrm{\\Delta}\\left( \\frac{1}{S_{mB}^{}} \\right)^{\\alpha\\beta}}{\\mathrm{\\Delta}\\left( \\frac{1}{S_{mA}^{}} \\right)^{\\alpha\\beta}}$\n", - "\n", - "As an example, the $T$-$\\mu_{C}$ potential phase diagram for the Fe-C\n", - "binary system at one atmospheric pressure is shown in .\n", - "\n", - "Figure \u2011: $T$-$\\mu_{C}$ potential phase diagram for the Fe-C binary\n", - "system at $P = 1atm$\n", - "\n", - "In a ternary system, two potentials need to be fixed in order to obtain\n", - "two-dimensional potential phase diagrams. When the pressure and the\n", - "chemical potential of one species are fixed, the system behaves like a\n", - "binary system discussed above. When the system temperature and pressure\n", - "are fixed, the Gibbs-Duhem equation is written as\n", - "\n", - "Eq. \u2011 $0 = - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B} - {N_{C}d\\mu}_{C}$\n", - "\n", - "Taking the component A as the dependent element, the two-phase\n", - "Clausius-Clapeyron equation is simplified as\n", - "\n", - "Eq. \u2011\n", - "$\\frac{{d\\mu}_{B}}{{d\\mu}_{C}} = \\frac{d\\left( \\ln a_{B} \\right)}{d\\left( \\ln a_{C} \\right)} = - \\frac{\\mathrm{\\Delta}z_{C}^{\\alpha\\beta}}{{\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}}$\n", - "\n", - "When the two phases in equilibrium are stoichiometric phases, the\n", - "two-phase equilibrium is thus a straight line. For example, the Ti-O-Cl\n", - "potential phase diagram at 600\u00b0C and one atmospheric pressure is shown\n", - "in . Since both O and Cl are volatile components, their activities are\n", - "usually represented by their partial pressures with the pure\n", - "O2 and Cl2 gas as their respective reference\n", - "states at the given temperature and pressure.\n", - "\n", - "Figure \u2011: Ti-O-Cl potential phase diagram at 600\u00b0C and one atmospheric\n", - "pressure\n", - "\n", - "For systems with more than three components, the chemical potentials of\n", - "one or more components must be fixed in order to obtain a\n", - "two-dimensional potential phase diagram similar to the potential phase\n", - "diagrams discussed above.\n" - ], - "metadata": {} - }, - { - "id": "b614d254", - "cell_type": "markdown", - "source": [ - "### Tie-lines and lever rule\n", - "\n", - "It is self-evident from to Eq. 2\u20114 that while potentials are homogeneous\n", - "in all phases in a heterogeneous system at equilibrium, the molar\n", - "quantities usually have different values in individual phases. This is\n", - "also stipulated in various Clausius-Clapeyron equations such as to and\n", - "to . The difference in molar quantities thus increases the\n", - "dimensionality of the phase region by the number of potentials replaced\n", - "by their conjugate molar quantities. The maximum dimensionality of a\n", - "phase region is the dimensionality of the phase diagram under\n", - "consideration. This thus creates a finite space between phases in\n", - "equilibrium in the phase diagram with some axes being molar quantities.\n", - "\n", - "For an equilibrium system under constant $T$, $P$ and $N_{i}$, the\n", - "potentials in the system and their conjugate molar quantities in each\n", - "phase are all uniquely defined. In a phase diagram with one or more\n", - "potentials replaced by their conjugate molar quantities, two phases in\n", - "equilibrium in a system with $c$ independent components are connected by\n", - "a $c$ dimensional line in a *c*+2 dimensional space or\n", - "its$\\ $*c+1*-dimensional projection as discussed in Chapter . These\n", - "lines are called tie-lines and collectively represent a two-phase\n", - "equilibrium region. For a $k$-phase equilibrium, there are total\n", - "$C_{k}^{2} = \\frac{k(k - 1)}{2}$ tie-lines connecting every two phases\n", - "with $k - 1\\ $of them independent because the number of independent\n", - "tie-lines increases by one with each new phase added. For the invariant\n", - "equilibrium with zero degree of freedom, the number of phases in\n", - "equilibrium is $c + 2$ shown by containing\n", - "$C_{c + 2}^{2} = \\frac{(c + 2)(c + 1)}{2}$ tie-lines with $c + 1$ of\n", - "them being independent.\n", - "\n", - "Inside the space encapsulated by the tie-lines, the axis variables of\n", - "the phase diagram (a mixture of potentials and molar quantities) can be\n", - "changed independently without changing the phases in equilibrium and\n", - "their properties. Only the relative amounts of individual phases are\n", - "adjusted accordingly to maintain the conservation of the molar\n", - "quantities in the system specified by the molar quantity axes of the\n", - "phase diagram. The geometric feature circumscribing the space\n", - "encapsulated by the tie-lines no longer represents any phase regions,\n", - "but a boundary between the neighbouring phase regions. Its\n", - "characteristics will be discussed in more details in the next few\n", - "sections. As properties in each phase are homogeneous, the values of\n", - "molar quantities of a system are simply the sum of individual phases and\n", - "can be represented by the following equation\n", - "\n", - "Eq. \u2011 $A_{m} = \\sum_{\\alpha}^{}{f^{\\alpha}A_{m}^{\\alpha}}$\n", - "\n", - "where $A_{m}$ and $A_{m}^{\\alpha}$ represent the values of a molar\n", - "quantity of the system and the $\\alpha$ phase, respectively,\n", - "$f^{\\alpha}$ the mole fraction of the $\\alpha$ phase, and the summation\n", - "goes over all phases in equilibrium with each other. With\n", - "$\\sum_{\\alpha}^{}f^{\\alpha} = 1$, can be re-arranged into the following\n", - "equation\n", - "\n", - "Eq. \u2011\n", - "$\\sum_{\\alpha}^{}{f^{\\alpha}\\left( {A_{m} - A}_{m}^{\\alpha} \\right)} = 0$\n", - "\n", - "is commonly referred as lever rule. For a two-phase equilibrium of\n", - "$\\alpha$ and $\\beta$, it becomes\n", - "\n", - "Eq. \u2011\n", - "$f^{\\alpha} = \\frac{A_{m}^{\\beta} - A_{m}}{A_{m}^{\\beta} - A_{m}^{\\alpha}}$\n", - "\n", - "Eq. \u2011\n", - "$f^{\\beta} = \\frac{A_{m}^{\\alpha} - A_{m}}{A_{m}^{\\alpha} - A_{m}^{\\beta}}$\n", - "\n", - "For a phase diagram with the number of axes being\n", - "$n = \\left( c - n_{s} \\right) + 1$, the number of possible axes being\n", - "molar quantities is thus $k \\leq n$. There are thus $k$ equations\n", - "similar to with one for each molar quantity, $A_{mi}$, resulting in the\n", - "following $k + 1$ equations\n", - "\n", - "Eq. \u2011\n", - "$\\sum_{\\alpha}^{}{f^{\\alpha}\\left( {A_{mi} - A}_{mi}^{\\alpha} \\right)} = 0$\n", - "\n", - "Eq. \u2011 $1 - \\sum_{\\alpha}^{}f^{\\alpha} = 0$\n", - "\n", - "The summations in and go over the phases in equilibrium, and the amount\n", - "of each phase is obtained by solving these $k + 1$ equations\n", - "simultaneously along with the equilibrium conditions.\n" - ], + "source": "### Section of potential phase diagrams\n\nBased on the Gibbs-Duhem equation (see ), a single-phase equilibrium in\na system with more than two independent components has more than three\nindependent potentials. There is no problem in representing them using\nthe mathematical formulas discussed so far, but it is not possible for\nus to visualize graphically the full potential phase diagrams in\nmulti-component systems with more than two independent components. In\nprinciples, there are two options. One option is to project the\nmulti-dimensional potential phase diagram into a two- or three-dimension\ndiagram, and another option is to section the multi-dimensional\npotential phase diagram by fixing the values of some potentials.\n\nThe projection approach is used for one-component systems in Chapter .\nSince a two-phase equilibrium in a one-component potential phase diagram\nis one-dimensional, the projection does not lose any information, and\nthe same is true for a three-phase equilibrium in a one-component\npotential phase diagram. In a binary system, the projections of three-\nand four-phase equilibria do not lose any information, while the\nprojections of two-phase equilibria become two-dimensional and cannot\nretain all the information as the original-dimensionality of these\ntwo-phase equilibria is three. Consequently, sectioning at fixed values\nof some potentials is necessary in order to visualize the phase\nrelations in systems with two or more components. Gibbs phase rule shown\nin and are thus modified to\n\nEq. \u2011 $\\upsilon = c + 2 - p - n_{s} = \\left( c - n_{s} \\right) + 2 - p$\n\nEq. \u2011 $p_{\\max} = \\left( c - n_{s} \\right) + 2$\n\nwhere $n_{s}$ is the number of potentials fixed in sectioning. As can be\nseen in the last part of , the number of sectioning is equivalent to the\nreduction of the effective number of independent components. Therefore,\nany multi-component systems with $n_{s} = c - i$ behave like an\n$i$-component system. The equations presented in Chapter and are thus\ndirectly applicable to multi-component systems with $n_{s} = c - 1$ and\n$n_{s} = c - 2$, respectively.\n\nA common practice in experiments is to fix pressure, temperature or\nchemical potentials of volatile components as they are usually the\nvariables controlled experimentally. In a binary system, the potential\nphase diagram at constant pressure can be represented by any two of the\nthree potentials, i.e. two chemical potentials and temperature, with the\nremaining potential being dependent, and has the identical morphology as\na one-component system. The Gibbs-Duhem equation under such conditions\nbecomes\n\nEq. \u2011 $0 = - SdT - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B}$\n\nThe corresponding two-phase Clausius-Clapeyron equations are written as\n\nEq. \u2011\n$\\frac{dT}{{d\\mu}_{A}} = - \\frac{{\\mathrm{\\Delta}z}_{A}^{\\alpha\\beta}}{\\mathrm{\\Delta}S_{mB}^{\\alpha\\beta}}$\n\nEq. \u2011\n$\\frac{dT}{{d\\mu}_{B}} = - \\frac{{\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}}{\\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}}$\n\nEq. \u2011\n$\\frac{{d\\mu}_{A}}{{d\\mu}_{B}} = - \\frac{\\mathrm{\\Delta}\\left( \\frac{1}{S_{mB}^{}} \\right)^{\\alpha\\beta}}{\\mathrm{\\Delta}\\left( \\frac{1}{S_{mA}^{}} \\right)^{\\alpha\\beta}}$\n\nAs an example, the $T$-$\\mu_{C}$ potential phase diagram for the Fe-C\nbinary system at one atmospheric pressure is shown in .\n\nFigure \u2011: $T$-$\\mu_{C}$ potential phase diagram for the Fe-C binary\nsystem at $P = 1atm$\n\nIn a ternary system, two potentials need to be fixed in order to obtain\ntwo-dimensional potential phase diagrams. When the pressure and the\nchemical potential of one species are fixed, the system behaves like a\nbinary system discussed above. When the system temperature and pressure\nare fixed, the Gibbs-Duhem equation is written as\n\nEq. \u2011 $0 = - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B} - {N_{C}d\\mu}_{C}$\n\nTaking the component A as the dependent element, the two-phase\nClausius-Clapeyron equation is simplified as\n\nEq. \u2011\n$\\frac{{d\\mu}_{B}}{{d\\mu}_{C}} = \\frac{d\\left( \\ln a_{B} \\right)}{d\\left( \\ln a_{C} \\right)} = - \\frac{\\mathrm{\\Delta}z_{C}^{\\alpha\\beta}}{{\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}}$\n\nWhen the two phases in equilibrium are stoichiometric phases, the\ntwo-phase equilibrium is thus a straight line. For example, the Ti-O-Cl\npotential phase diagram at 600\u00b0C and one atmospheric pressure is shown\nin . Since both O and Cl are volatile components, their activities are\nusually represented by their partial pressures with the pure\nO2 and Cl2 gas as their respective reference\nstates at the given temperature and pressure.\n\nFigure \u2011: Ti-O-Cl potential phase diagram at 600\u00b0C and one atmospheric\npressure\n\nFor systems with more than three components, the chemical potentials of\none or more components must be fixed in order to obtain a\ntwo-dimensional potential phase diagram similar to the potential phase\ndiagrams discussed above.\n", "metadata": {} } ] diff --git a/src/psu410/src/references/index.ipynb b/src/psu410/src/references/index.ipynb index fe54700..8760a2d 100644 --- a/src/psu410/src/references/index.ipynb +++ b/src/psu410/src/references/index.ipynb @@ -4,270 +4,9 @@ "metadata": {}, "cells": [ { - "id": "5624d039", + "id": "96a5d429", "cell_type": "markdown", - "source": [ - "# References\n", - "\n", - "1\\. M. 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Bernstein, *Computer Calculation of Phase\n", - "Diagram,* Academic Press Inc., New York, 1970.\n", - "\n", - "48\\. N. Saunders and A. P. Miodownik, *CALPHAD (Calculation of Phase\n", - "Diagrams): A Comprehensive Guide,* Pergamon, Oxford; New York, 1998.\n", - "\n", - "49\\. H. L. Lukas, S. G. Fries and B. Sundman, *Computational\n", - "Thermodynamics: The CALPHAD Method,* Cambridge University Press, 2007.\n", - "\n", - "50\\. \"Software for CALPHAD modeling\", **26(2)** (2002).\n", - "\n", - "51\\. \"Software for CALPHAD modeling\", **33(2)** (2009).\n", - "\n", - "52\\. A. T. Dinsdale, \"SGTE Data for Pure Elements\", *CALPHAD* **15**\n", - "(1991) 317-425.\n", - "\n", - "53\\. Y. Wang, S. Curtarolo, C. Jiang, R. Arroyave, T. Wang, G. Ceder, L.\n", - "Q. Chen and Z. K. Liu, \"Ab initio lattice stability in comparison with\n", - "CALPHAD lattice stability\", *Calphad-Comput. Coupling Ph. Diagrams\n", - "Thermochem.* **28** (2004) 79-90.\n", - "\n", - "54\\. V. Ozolins, \"First-Principles Calculations of Free Energies of\n", - "Unstable Phases: The Case of fcc W\", *Phys. Rev. Lett.* **102** (2009)\n", - "065702.\n", - "\n", - "55\\. Y. Wang, L. G. Hector, H. Zhang, S. L. Shang, L. Q. Chen and Z. K.\n", - "Liu, \"Thermodynamics of the Ce gamma-alpha transition:\n", - "Density-functional study\", *Phys. Rev. B* **78** (2008) 104113.\n", - "\n", - "56\\. B. Sundman, I. Ohnuma, N. Dupin, U. R. Kattner and S. G. Fries, \"An\n", - "assessment of the entire Al-Fe system including D0(3) ordering\", *Acta\n", - "Mater.* **57** (2009) 2896-2908.\n", - "\n", - "57\\. A. Kusoffsky, N. Dupin and B. Sundman, \"On the compound energy\n", - "formalism applied to fcc ordering\", *Calphad-Comput. Coupling Ph.\n", - "Diagrams Thermochem.* **25** (2001) 549-565.\n", - "\n", - "58\\. T. Abe and B. Sundman, \"A description of the effect of short range\n", - "ordering in the compound energy formalism\", *Calphad-Comput. Coupling\n", - "Ph. Diagrams Thermochem.* **27** (2003) 403-408.\n", - "\n", - "59\\. Scientific Group Thermodata Europe (SGTE), *Thermodynamic\n", - "Properties of Inorganic Materials*. Lehrstuhl f\u00fcr Theoretische\n", - "H\u00fcttenkunde, Ed. Landolt-Boernstein New Series, Group IV, Springer,\n", - "Verlag Berlin Heidelberg, 1999, vol. 19.\n", - "\n", - "60\\. J. O. Andersson, T. Helander, L. H. Hoglund, P. F. Shi and B.\n", - "Sundman, \"THERMO-CALC & DICTRA, computational tools for materials\n", - "science\", **26** (2002) 273-312.\n", - "\n", - "61\\. D. D. Macdonald, \"Passivity - the key to our metals-based\n", - "civilization\", *Pure Appl. Chem.* **71** (1999) 951-978.\n", - "\n", - "62\\. J. Larcin, W. C. Maskell and F. L. Tye, \"Leclanche cell\n", - "investigations .1. Zn(NH3)(2)Cl-2 solubility and the formation of ZnCl2\n", - "center dot 4Zn(OH)(2)center dot H2O\", *Electrochim. Acta* **42** (1997)\n", - "2649-2658.\n", - "\n", - "63\\. Z. K. Liu, Y. Wang and S. L. Shang, \"Origin of negative thermal\n", - "expansion phenomenon in solids\", *Scr. Mater.* **65** (2011) 664-667.\n", - "\n", - "64\\. Y. Wang, L. G. Hector, H. Zhang, S. L. Shang, L. Q. Chen and Z. K.\n", - "Liu, \"A thermodynamic framework for a system with itinerant-electron\n", - "magnetism\", *J. Phys.-Condes. Matter* **21** (2009) 326003.\n", - "\n", - "65\\. S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys and A.\n", - "P. Sutton, \"Electron-energy-loss spectra and the structural stability of\n", - "nickel oxide: An LSDA+U study\", *Phys. Rev. B* **57** (1998) 1505-1509.\n", - "\n", - "66\\. Z.-K. Liu, Y. Wang and S. Shang, \"Thermal Expansion Anomaly\n", - "Regulated by Entropy\", **4** (2014).\n", - "\n", - "67\\. Y. Wang, S. L. Shang, H. Zhang, L. Q. Chen and Z. K. Liu,\n", - "\"Thermodynamic fluctuations in magnetic states: Fe3Pt as a prototype\",\n", - "*Phil. Mag. Lett.* **90** (2010) 851-859.\n", - "\n", - "68\\. National Science and Technology Council, \u201cMaterials Genome\n", - "Initiative for Global Competitiveness\u201d,\n", - "*[http://www.whitehouse.gov/sites/default/files/microsites/ostp/materials_genome_initiative-final.pdf]()\n", - "Office of Science and Technology Policy, Washington DC*, June, 2011.\n", - "\n", - "69\\. L. Kaufman and J. Agren, \"CALPHAD, first and second generation -\n", - "Birth of the materials genome\", **70** (2014) 3-6.\n", - "\n", - "70\\. Z. K. Liu, \"Perspective on Materials Genome\u00ae\", *Chin. Sci. Bull.*\n", - "**59** (2014) 1619-1623.\n" - ], + "source": "# References\n\n1\\. M. Hillert, *Phase Equilibria, Phase Diagrams and Phase\nTransformations,* Cambridge University Press, Cambridge, ed. 2nd, 2007.\n\n2\\. E. A. Guggenheim, *Mixtures,* Clarendon Press, Oxford, 1952.\n\n3\\. J. F. Nye, *Physical properties of crystals : their representation\nby tensors and matrices* Clarendon, Oxford, New York, 1985.\n\n4\\. L. S. Palatnik and A. I. 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Fries, \"An\nassessment of the entire Al-Fe system including D0(3) ordering\", *Acta\nMater.* **57** (2009) 2896-2908.\n\n57\\. A. Kusoffsky, N. Dupin and B. Sundman, \"On the compound energy\nformalism applied to fcc ordering\", *Calphad-Comput. Coupling Ph.\nDiagrams Thermochem.* **25** (2001) 549-565.\n\n58\\. T. Abe and B. Sundman, \"A description of the effect of short range\nordering in the compound energy formalism\", *Calphad-Comput. Coupling\nPh. Diagrams Thermochem.* **27** (2003) 403-408.\n\n59\\. Scientific Group Thermodata Europe (SGTE), *Thermodynamic\nProperties of Inorganic Materials*. Lehrstuhl f\u00fcr Theoretische\nH\u00fcttenkunde, Ed. Landolt-Boernstein New Series, Group IV, Springer,\nVerlag Berlin Heidelberg, 1999, vol. 19.\n\n60\\. J. O. Andersson, T. Helander, L. H. Hoglund, P. F. Shi and B.\nSundman, \"THERMO-CALC & DICTRA, computational tools for materials\nscience\", **26** (2002) 273-312.\n\n61\\. D. D. Macdonald, \"Passivity - the key to our metals-based\ncivilization\", *Pure Appl. Chem.* **71** (1999) 951-978.\n\n62\\. J. Larcin, W. C. Maskell and F. L. Tye, \"Leclanche cell\ninvestigations .1. Zn(NH3)(2)Cl-2 solubility and the formation of ZnCl2\ncenter dot 4Zn(OH)(2)center dot H2O\", *Electrochim. Acta* **42** (1997)\n2649-2658.\n\n63\\. Z. K. Liu, Y. Wang and S. L. Shang, \"Origin of negative thermal\nexpansion phenomenon in solids\", *Scr. Mater.* **65** (2011) 664-667.\n\n64\\. Y. Wang, L. G. Hector, H. Zhang, S. L. Shang, L. Q. Chen and Z. K.\nLiu, \"A thermodynamic framework for a system with itinerant-electron\nmagnetism\", *J. Phys.-Condes. Matter* **21** (2009) 326003.\n\n65\\. S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys and A.\nP. Sutton, \"Electron-energy-loss spectra and the structural stability of\nnickel oxide: An LSDA+U study\", *Phys. Rev. B* **57** (1998) 1505-1509.\n\n66\\. Z.-K. Liu, Y. Wang and S. Shang, \"Thermal Expansion Anomaly\nRegulated by Entropy\", **4** (2014).\n\n67\\. Y. Wang, S. L. Shang, H. Zhang, L. Q. Chen and Z. K. Liu,\n\"Thermodynamic fluctuations in magnetic states: Fe3Pt as a prototype\",\n*Phil. Mag. Lett.* **90** (2010) 851-859.\n\n68\\. National Science and Technology Council, \u201cMaterials Genome\nInitiative for Global Competitiveness\u201d,\n*[http://www.whitehouse.gov/sites/default/files/microsites/ostp/materials_genome_initiative-final.pdf]()\nOffice of Science and Technology Policy, Washington DC*, June, 2011.\n\n69\\. L. Kaufman and J. Agren, \"CALPHAD, first and second generation -\nBirth of the materials genome\", **70** (2014) 3-6.\n\n70\\. Z. K. Liu, \"Perspective on Materials Genome\u00ae\", *Chin. Sci. Bull.*\n**59** (2014) 1619-1623.\n", "metadata": {} } ] From 5bfe0c02305c1f65ecb59d1e77a69552568f94f8 Mon Sep 17 00:00:00 2001 From: tonyfast Date: Wed, 10 Jan 2024 12:17:05 -0800 Subject: [PATCH 7/9] split lines in the markdown sources --- src/nobook/bootstrap.ipynb | 444 ++++++------- ...lingham_diagram_and_buffered_systems.ipynb | 153 ++++- .../index.ipynb | 19 +- ...ential_and_integrated_driving_forces.ipynb | 93 ++- ...te_and_chemical_transport_reactions_.ipynb | 150 ++++- .../trends_of_entropies_of_reactions.ipynb | 75 ++- .../application_examples.ipynb | 374 ++++++++++- ...queous_solution_and_pourbaix_diagram.ipynb | 254 ++++++- ...erence_states_of_electrolyte_species.ipynb | 70 +- ...mical_cells_and_half_cell_potentials.ipynb | 293 ++++++++- ...ctions_and_electrochemical_reactions.ipynb | 81 ++- .../index.ipynb | 24 +- .../importance_of_lattice_stability.ipynb | 112 +++- .../index.ipynb | 50 +- .../modeling_of_pure_elements.ipynb | 67 +- .../modeling_of_random_solution_phases.ipynb | 54 +- ...ution_phases_with_longrange_ordering.ipynb | 161 ++++- .../modeling_of_stoichiometric_phases.ipynb | 33 +- .../application_to_cerium.ipynb | 200 +++++- .../application_to_fept.ipynb | 122 +++- .../concept_of_materials_genome.ipynb | 57 +- .../index.ipynb | 23 +- ...s_model_applied_to_thermal_expansion.ipynb | 118 +++- .../index.ipynb | 25 +- ...se_equilibrium_data_from_experiments.ipynb | 226 ++++++- .../thermodynamic_data_from_experiments.ipynb | 134 +++- ...ples_approaches_to_disordered_alloys.ipynb | 239 ++++++- ...ciples_formulation_of_thermodynamics.ipynb | 296 ++++++++- .../index.ipynb | 51 +- .../lattice_dynamics.ipynb | 373 ++++++++++- .../nickel_as_the_prototype.ipynb | 325 ++++++++- ...m_theory_for_the_motion_of_electrons.ipynb | 326 ++++++++- ...lectric_contributions_to_free_energy.ipynb | 122 +++- .../src/gibbs_energy_function/index.ipynb | 168 ++++- .../phases_with_fixed_compositions.ipynb | 270 +++++++- ...riable_compositions_random_solutions.ipynb | 527 ++++++++++++++- ...compositions_solutions_with_ordering.ipynb | 403 +++++++++++- ...polymer_solutions_and_polymer_blends.ipynb | 73 +- ...odynamics_and_equilibrium_conditions.ipynb | 220 ++++++- ...st_and_second_laws_of_thermodynamics.ipynb | 119 +++- .../gibbsduhem_equation.ipynb | 82 ++- .../src/laws_of_thermodynamics/index.ipynb | 6 +- ..._at_equilibrium_and_property_anomaly.ipynb | 183 +++++- .../general_condition_of_equilibrium.ipynb | 114 +++- .../gibbs_phase_rule.ipynb | 55 +- .../index.ipynb | 6 +- .../molar_phase_diagrams.ipynb | 621 +++++++++++++++++- .../potential_phase_diagrams.ipynb | 385 ++++++++++- src/psu410/src/references/index.ipynb | 265 +++++++- 49 files changed, 8227 insertions(+), 414 deletions(-) diff --git a/src/nobook/bootstrap.ipynb b/src/nobook/bootstrap.ipynb index a7cc8a2..c4f1f8b 100644 --- a/src/nobook/bootstrap.ipynb +++ b/src/nobook/bootstrap.ipynb @@ -14,11 +14,11 @@ "id": "c4601326-6456-47e8-99c9-ae8c8f59e32b", "metadata": { "execution": { - 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"docx \n", + "docx \n", "md Computational Thermodynamics of Materials\\n\\nZ... \n", "\n", "package 1 \\\n", "doc 1 \n", "cell 1 \n", "original 100 \n", - "docx \n", + "docx \n", "md # Laws of thermodynamics\\n \n", "\n", "package \\\n", "doc 2 \n", "cell 2 \n", "original 102 \n", - "docx \n", + "docx \n", "md ## First and second laws of thermodynamics\\n\\n... \n", "\n", "package \\\n", "doc 3 \n", "cell 3 \n", "original 147 \n", - "docx \n", + "docx \n", "md ## Combined law of thermodynamics and equilibr... \n", "\n", "package \\\n", "doc 4 \n", "cell 4 \n", "original 251 \n", - "docx \n", + "docx \n", "md ## Stability at equilibrium and property anoma... \n", "\n", "package \\\n", "doc 5 \n", "cell 5 \n", "original 316 \n", - "docx \n", + "docx \n", "md ## Gibbs-Duhem equation\\n\\nIn experiments, it ... \n", "\n", "package 2 \\\n", "doc 6 \n", "cell 6 \n", "original 359 \n", - "docx \n", + "docx \n", "md # Gibbs energy function\\n\\nAs shown in through... \n", "\n", "package \\\n", "doc 7 \n", "cell 7 \n", "original 435 \n", - "docx \n", + "docx \n", "md ## Phases with fixed compositions\\n\\nThe homog... \n", "\n", "package \\\n", "doc 8 \n", "cell 8 \n", "original 563 \n", - "docx \n", + "docx \n", "md ## Phases with variable compositions: Random s... \n", "\n", "package ... \\\n", "doc ... \n", "cell 9 ... \n", "original 617 ... \n", - "docx ... \n", + "docx ... \n", "md ### Random solutions\\n\\nThe ideal Gibbs energy... ... \n", "\n", "package 8 \\\n", "doc 42 \n", "cell 87 \n", "original 2959 \n", - "docx \n", + "docx \n", "md ### Galvanic protection\\n\\nA galvanic reaction... \n", "\n", "package \\\n", "doc \n", "cell 88 \n", "original 2982 \n", - "docx \n", + "docx \n", "md ### Fuel cells\\n\\nFuel cells are devices to co... \n", "\n", "package \\\n", "doc \n", "cell 89 \n", "original 3004 \n", - "docx \n", + "docx \n", "md ### Ion transport membranes\\n\\nIon transport m... \n", "\n", "package \\\n", "doc \n", "cell 90 \n", "original 3022 \n", - "docx \n", + "docx \n", "md ### Electrical batteries\\n\\nBatteries utilize ... \n", "\n", "package 9 \\\n", "doc 43 \n", "cell 91 \n", "original 3098 \n", - "docx \n", + "docx \n", "md # Critical phenomena, thermal expansion, and M... \n", "\n", "package \\\n", "doc 44 \n", "cell 92 \n", "original 3102 \n", - "docx \n", + "docx \n", "md ## MMS model applied to thermal expansion\\n\\nA... \n", "\n", "package \\\n", "doc 45 \n", "cell 93 \n", "original 3140 \n", - "docx \n", + "docx \n", "md ## Application to cerium\\n\\nCerium (Ce) displa... \n", "\n", "package \\\n", "doc 46 \n", "cell 94 \n", "original 3184 \n", - "docx \n", + "docx \n", "md ## Application to Fe3Pt\\n\\nInvar wa... \n", "\n", "package \\\n", "doc 47 \n", "cell 95 \n", "original 3208 \n", - "docx \n", + "docx \n", "md ## Concept of Materials Genome®\\n\\n“A genome i... \n", "\n", "package 10 \n", "doc 48 \n", "cell 96 \n", "original 3213 \n", - "docx \n", + "docx \n", "md # References\\n\\n1\\. M. 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"iopub.execute_input": "2024-01-10T20:06:15.595877Z", - "iopub.status.busy": "2024-01-10T20:06:15.595750Z", - "iopub.status.idle": "2024-01-10T20:06:15.604266Z", - "shell.execute_reply": "2024-01-10T20:06:15.603930Z", - "shell.execute_reply.started": "2024-01-10T20:06:15.595867Z" + "iopub.execute_input": "2024-01-10T20:16:37.462885Z", + "iopub.status.busy": "2024-01-10T20:16:37.462699Z", + "iopub.status.idle": "2024-01-10T20:16:37.471213Z", + "shell.execute_reply": "2024-01-10T20:16:37.470863Z", + "shell.execute_reply.started": "2024-01-10T20:16:37.462876Z" }, "tags": [] }, diff --git a/src/psu410/src/applications_to_chemical_reactions/ellingham_diagram_and_buffered_systems.ipynb b/src/psu410/src/applications_to_chemical_reactions/ellingham_diagram_and_buffered_systems.ipynb index c121add..0ed9998 100644 --- a/src/psu410/src/applications_to_chemical_reactions/ellingham_diagram_and_buffered_systems.ipynb +++ b/src/psu410/src/applications_to_chemical_reactions/ellingham_diagram_and_buffered_systems.ipynb @@ -4,9 +4,158 @@ "metadata": {}, "cells": [ { - "id": "93c0cda7", + "id": "e079ac39", "cell_type": "markdown", - "source": "## Ellingham diagram and buffered systems\n\nOne type of chemical reactions is between a pure element in liquid or\nsolid states and its oxides involving one mole of oxygen gas, i.e.\n\n*Eq. 7\u20117* $\\frac{4}{v_{M}}M + O_{2}(gas) = M_{4/v_{M}}O_{2}$\n\nwith $v_{M}$ being the valence of the element $M$ in the oxide. Taking\nthe pure $M$ and the gaseous $O_{2}$ at the reaction temperature and one\natmospheric pressure as their respective reference states, both\nactivities of the pure $M$ solid or liquid phase and its oxide are\nunity, and the activity of $O_{2}$ equals to its partial pressure in an\nideal gas. becomes\n\n*Eq. 7\u20118*\n$RTlnP_{O_{2}} = \\mathrm{\\Delta}_{\\ }^{0}G = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S$\n\nBased on , one can plot $\\mathrm{\\Delta}_{\\ }^{0}G$ as a function of\ntemperature for various oxidation reactions, which is called Ellingham\ndiagram. The intercept at $T = 0K$ is given by\n$\\mathrm{\\Delta}_{\\ }^{0}H$, and the slope is represented by\n$- \\mathrm{\\Delta}_{\\ }^{0}S$, depicted by the following equation\n\n*Eq. 7\u20119*\n$\\mathrm{\\Delta}_{\\ }^{0}S = S_{M_{4/v_{M}}O_{2}} - S_{O_{2}} - \\frac{4}{v_{M}}S_{M}$\n\nSince the entropy of one mole of $O_{2}$ gas is significantly larger\nthan those of the pure element and its oxide when they are in solid or\nliquid states and the entropy difference between the pure element and\nits oxide, the entropy of reaction is thus dominated by the reduction of\nthe entropy by one mole of $O_{2}$ gas. Consequently, the entropies of\nreaction are approximately the same for most reactions when the pure\nelements and their oxides are solid as seen in where most lines have\nsimilar slopes.\n\nFigure \u2011: Ellingham Diagram for a number of metal-oxide systems.\n\nFor a chemical reaction on the Ellingham diagram at a given temperature,\n$\\mathrm{\\Delta}_{\\ }^{0}G$ can be read from the y-axis of the diagram,\nand the equilibrium partial pressure of $O_{2}$ gas can be calculated\nusing . Alternatively, one can plot the left part of for a given\n$P_{O_{2}}$ as a function of temperature on the Ellingham diagram, i.e.\n\n*Eq. 7\u201110* $\\mathrm{\\Delta}_{\\ }^{0}{G =}RTlnP_{O_{2}}$\n\nThis results in straight lines, representing iso-partial-pressure lines\nof $O_{2}$, with the intercept being zero at $T = 0K$ and slopes of\n$RlnP_{O_{2}}$, which are negative for $P_{O_{2}}$ lower than one\natmospheric pressure (1atm). The values of $P_{O_{2}}$ are marked on the\nsecondary axis on the right of the Ellingham diagram. The intersection\nof the isoactivitiy line and the equilibrium line of the chemical\nreaction thus gives the relation of equilibrium temperature and\nequilibrium partial pressure of $O_{2}$. This is demonstrated in .\n\nFigure \u2011: Intersection of iso-partial-pressure lines of $O_{2}$ and\nequilibrium lines in the Ellingham diagram\n\nFor each chemical reaction in the Ellingham diagram, the three phases\nare in equilibrium on the line represented by , i.e. metal, metal\noxides, and O2 gas. For conditions above the line, the value\nof $P_{O_{2}}$ is larger than its equilibrium value, and the metal will\nbe oxidized. For conditions below the line, the value of $P_{O_{2}}$ is\nlower than its equilibrium value, and the metal oxide will be reduced.\nTherefore, the metal oxides in the upper part of the Ellingham diagram\ncan be reduced by the metals in the lower part of the diagram, and vice\nversus, metals in the lower part of the diagram can be oxidized by the\nmetal oxides in the upper part of the diagram. For example, in , Ca can\nreduce all oxides, and Cu2O is the least stable oxide.\n\nFrom the above Ellingham diagram, it is noted that equilibrium partial\npressures of $O_{2}$ are very low for most chemical reactions with many\nof them lower than $10^{- 12}\\ atm$. One approach to obtain such a low\npressure is to use auxiliary reactions containing $O_{2}$ that are easy\nto control and are independent of the equilibrium system of interest\nexcept the sharing of the oxygen partial pressure. Two common auxiliary\nreactions are the $H_{2}$/$H_{2}O$ and $CO$/$CO_{2}$ mixtures. For the\n$H_{2}$/$H_{2}O$ mixture, the chemical reaction is\n\n*Eq. 7\u201111* $2H_{2}(gas) + O_{2}(gas) = 2H_{2}O(gas)$\n\nThe equilibrium oxygen partial pressure is obtained as\n\n*Eq. 7\u201112*\n$- RTln\\left\\{ {\\frac{1}{P_{O_{2}}}\\left( \\frac{P_{H_{2}O}}{P_{H_{2}}} \\right)}^{2} \\right\\} = \\mathrm{\\Delta}_{\\ }^{0}G = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S = - 498488\\ + 112.972T\\ (J)$\n\nwhere the $\\mathrm{\\Delta}_{\\ }^{0}H$ and $\\mathrm{\\Delta}_{\\ }^{0}S$\nare taken from the substance thermodynamic database (SSUB4) compiled by\nScientific Group Thermodata Europe (SGTE) \\[59\\], which are slightly\ndependent on temperature, and the values in are evaluated at 1273K using\nThermo-Calc \\[60\\]. At any given temperature, one has the following\nrelation\n\n*Eq. 7\u201113*\n$RTlnP_{O_{2}} = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S - 2RTln\\frac{P_{H_{2}}}{P_{H_{2}O}} = - 498488 + \\left( 112.972 - 2Rln\\frac{P_{H_{2}}}{P_{H_{2}O}} \\right)T$\n\nIts intercept at $T = 0K$ is given by\n$\\mathrm{\\Delta}_{\\ }^{0}H = - 498,488\\ (J)$, and the slope by\n$- \\mathrm{\\Delta}_{\\ }^{0}S - 2Rln\\left( \\frac{P_{H_{2}}}{P_{H_{2}O}} \\right) = 112.972 - 2Rln\\left( \\frac{P_{H_{2}}}{P_{H_{2}O}} \\right)$.\nThe values of the $\\frac{P_{H_{2}}}{P_{H_{2}O}}$ ratio are marked on\nanother secondary axis on the right of the Ellingham diagram. The\nintersection of the iso-partial-pressure-ratio line and the equilibrium\nline of the chemical reaction gives the relation of equilibrium\ntemperature and equilibrium partial pressure ratio\n$\\frac{P_{H_{2}}}{P_{H_{2}O}}$ for desired $P_{O_{2}}$ of the chemical\nequilibrium.\n\nFor the $CO$/$CO_{2}$mixture, the chemical reaction is shown below\n\n*Eq. 7\u201114* $2CO(gas) + O_{2}(gas) = 2CO_{2}(gas)$\n\nSimilar, from the SSUB database, the following equation is obtained\n\n*Eq. 7\u201115*\n$- RTln\\left\\{ {\\frac{1}{P_{O_{2}}}\\left( \\frac{P_{{CO}_{2}}}{P_{CO}} \\right)}^{2} \\right\\} = \\mathrm{\\Delta}_{\\ }^{0}G = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S = - 562,927 - 172.020T\\ (J)$\n\nwith $\\mathrm{\\Delta}_{\\ }^{0}H$ and $\\mathrm{\\Delta}_{\\ }^{0}S$\ncalculated at 1273K, which are also slightly temperature-dependent as in\nthe case of the $H_{2}$/$H_{2}O$ mixture. The iso-partial-pressure-ratio\nlines are written as\n\n*Eq. 7\u201116*\n$RTlnP_{O_{2}} = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S - 2RTln\\frac{P_{CO}}{P_{{CO}_{2}}} = - 562,927 + \\left( 172.020 - 2Rln\\frac{P_{CO}}{P_{{CO}_{2}}} \\right)T\\ $\n\nIts intercept at $T = 0K$ is given by\n$\\mathrm{\\Delta}_{\\ }^{0}H = - 562,927\\ (J)$, and the slope by\n$172.020 - - 2Rln\\left( \\frac{P_{CO}}{P_{{CO}_{2}}} \\right)$. The values\nof the $\\frac{P_{CO}}{P_{{CO}_{2}}}$ ratio are marked on the third\nsecondary axis on the right of the Ellingham diagram. The intersection\nof the iso-partial-pressure-ratio line and the equilibrium line of the\nchemical reaction gives the relation of equilibrium temperature and\nequilibrium partial pressure ratio $\\frac{P_{CO}}{P_{{CO}_{2}}}$ for\ndesired $P_{O_{2}}$ of the chemical equilibrium.\n\nTherefore, one can use a mixture of the $H_{2}$/$H_{2}O$ or\n$CO$/$CO_{2}$ to obtain the desired low $P_{O_{2}}$ values using and or\ncalculating from the SSUB database. For example,\n$P_{O_{2}} = 10^{- 15}\\ atm$ at 1273K, the calculated values from the\nSSUB database are $\\frac{P_{H_{2}}}{P_{H_{2}O}} \\approx 1.67$ and\n$\\frac{P_{CO}}{P_{{CO}_{2}}} \\approx 2.78$, respectively, in which the\ntemperature dependences of $\\mathrm{\\Delta}_{\\ }^{0}H$ and\n$\\mathrm{\\Delta}_{\\ }^{0}S$ and the many more gaseous species in the gas\nphase are taken into account. On the other hand, the reading from the\nEllingham diagram gives $\\frac{P_{H_{2}}}{P_{H_{2}O}} \\approx 2.0$ and\n$\\frac{P_{CO}}{P_{{CO}_{2}}} \\approx 2.4$, and the agreement with the\nmore accurate calculations above is remarkable keeping in mind the\nuncertainties in graphically drawing the lines and reading both values\nin the logarithmic scales from the diagram, indicating the robustness of\nthe Ellingham diagram.\n", + "source": [ + "## Ellingham diagram and buffered systems\n", + "\n", + "One type of chemical reactions is between a pure element in liquid or\n", + "solid states and its oxides involving one mole of oxygen gas, i.e.\n", + "\n", + "*Eq. 7\u20117* $\\frac{4}{v_{M}}M + O_{2}(gas) = M_{4/v_{M}}O_{2}$\n", + "\n", + "with $v_{M}$ being the valence of the element $M$ in the oxide. Taking\n", + "the pure $M$ and the gaseous $O_{2}$ at the reaction temperature and one\n", + "atmospheric pressure as their respective reference states, both\n", + "activities of the pure $M$ solid or liquid phase and its oxide are\n", + "unity, and the activity of $O_{2}$ equals to its partial pressure in an\n", + "ideal gas. becomes\n", + "\n", + "*Eq. 7\u20118*\n", + "$RTlnP_{O_{2}} = \\mathrm{\\Delta}_{\\ }^{0}G = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S$\n", + "\n", + "Based on , one can plot $\\mathrm{\\Delta}_{\\ }^{0}G$ as a function of\n", + "temperature for various oxidation reactions, which is called Ellingham\n", + "diagram. The intercept at $T = 0K$ is given by\n", + "$\\mathrm{\\Delta}_{\\ }^{0}H$, and the slope is represented by\n", + "$- \\mathrm{\\Delta}_{\\ }^{0}S$, depicted by the following equation\n", + "\n", + "*Eq. 7\u20119*\n", + "$\\mathrm{\\Delta}_{\\ }^{0}S = S_{M_{4/v_{M}}O_{2}} - S_{O_{2}} - \\frac{4}{v_{M}}S_{M}$\n", + "\n", + "Since the entropy of one mole of $O_{2}$ gas is significantly larger\n", + "than those of the pure element and its oxide when they are in solid or\n", + "liquid states and the entropy difference between the pure element and\n", + "its oxide, the entropy of reaction is thus dominated by the reduction of\n", + "the entropy by one mole of $O_{2}$ gas. Consequently, the entropies of\n", + "reaction are approximately the same for most reactions when the pure\n", + "elements and their oxides are solid as seen in where most lines have\n", + "similar slopes.\n", + "\n", + "Figure \u2011: Ellingham Diagram for a number of metal-oxide systems.\n", + "\n", + "For a chemical reaction on the Ellingham diagram at a given temperature,\n", + "$\\mathrm{\\Delta}_{\\ }^{0}G$ can be read from the y-axis of the diagram,\n", + "and the equilibrium partial pressure of $O_{2}$ gas can be calculated\n", + "using . Alternatively, one can plot the left part of for a given\n", + "$P_{O_{2}}$ as a function of temperature on the Ellingham diagram, i.e.\n", + "\n", + "*Eq. 7\u201110* $\\mathrm{\\Delta}_{\\ }^{0}{G =}RTlnP_{O_{2}}$\n", + "\n", + "This results in straight lines, representing iso-partial-pressure lines\n", + "of $O_{2}$, with the intercept being zero at $T = 0K$ and slopes of\n", + "$RlnP_{O_{2}}$, which are negative for $P_{O_{2}}$ lower than one\n", + "atmospheric pressure (1atm). The values of $P_{O_{2}}$ are marked on the\n", + "secondary axis on the right of the Ellingham diagram. The intersection\n", + "of the isoactivitiy line and the equilibrium line of the chemical\n", + "reaction thus gives the relation of equilibrium temperature and\n", + "equilibrium partial pressure of $O_{2}$. This is demonstrated in .\n", + "\n", + "Figure \u2011: Intersection of iso-partial-pressure lines of $O_{2}$ and\n", + "equilibrium lines in the Ellingham diagram\n", + "\n", + "For each chemical reaction in the Ellingham diagram, the three phases\n", + "are in equilibrium on the line represented by , i.e. metal, metal\n", + "oxides, and O2 gas. For conditions above the line, the value\n", + "of $P_{O_{2}}$ is larger than its equilibrium value, and the metal will\n", + "be oxidized. For conditions below the line, the value of $P_{O_{2}}$ is\n", + "lower than its equilibrium value, and the metal oxide will be reduced.\n", + "Therefore, the metal oxides in the upper part of the Ellingham diagram\n", + "can be reduced by the metals in the lower part of the diagram, and vice\n", + "versus, metals in the lower part of the diagram can be oxidized by the\n", + "metal oxides in the upper part of the diagram. For example, in , Ca can\n", + "reduce all oxides, and Cu2O is the least stable oxide.\n", + "\n", + "From the above Ellingham diagram, it is noted that equilibrium partial\n", + "pressures of $O_{2}$ are very low for most chemical reactions with many\n", + "of them lower than $10^{- 12}\\ atm$. One approach to obtain such a low\n", + "pressure is to use auxiliary reactions containing $O_{2}$ that are easy\n", + "to control and are independent of the equilibrium system of interest\n", + "except the sharing of the oxygen partial pressure. Two common auxiliary\n", + "reactions are the $H_{2}$/$H_{2}O$ and $CO$/$CO_{2}$ mixtures. For the\n", + "$H_{2}$/$H_{2}O$ mixture, the chemical reaction is\n", + "\n", + "*Eq. 7\u201111* $2H_{2}(gas) + O_{2}(gas) = 2H_{2}O(gas)$\n", + "\n", + "The equilibrium oxygen partial pressure is obtained as\n", + "\n", + "*Eq. 7\u201112*\n", + "$- RTln\\left\\{ {\\frac{1}{P_{O_{2}}}\\left( \\frac{P_{H_{2}O}}{P_{H_{2}}} \\right)}^{2} \\right\\} = \\mathrm{\\Delta}_{\\ }^{0}G = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S = - 498488\\ + 112.972T\\ (J)$\n", + "\n", + "where the $\\mathrm{\\Delta}_{\\ }^{0}H$ and $\\mathrm{\\Delta}_{\\ }^{0}S$\n", + "are taken from the substance thermodynamic database (SSUB4) compiled by\n", + "Scientific Group Thermodata Europe (SGTE) \\[59\\], which are slightly\n", + "dependent on temperature, and the values in are evaluated at 1273K using\n", + "Thermo-Calc \\[60\\]. At any given temperature, one has the following\n", + "relation\n", + "\n", + "*Eq. 7\u201113*\n", + "$RTlnP_{O_{2}} = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S - 2RTln\\frac{P_{H_{2}}}{P_{H_{2}O}} = - 498488 + \\left( 112.972 - 2Rln\\frac{P_{H_{2}}}{P_{H_{2}O}} \\right)T$\n", + "\n", + "Its intercept at $T = 0K$ is given by\n", + "$\\mathrm{\\Delta}_{\\ }^{0}H = - 498,488\\ (J)$, and the slope by\n", + "$- \\mathrm{\\Delta}_{\\ }^{0}S - 2Rln\\left( \\frac{P_{H_{2}}}{P_{H_{2}O}} \\right) = 112.972 - 2Rln\\left( \\frac{P_{H_{2}}}{P_{H_{2}O}} \\right)$.\n", + "The values of the $\\frac{P_{H_{2}}}{P_{H_{2}O}}$ ratio are marked on\n", + "another secondary axis on the right of the Ellingham diagram. The\n", + "intersection of the iso-partial-pressure-ratio line and the equilibrium\n", + "line of the chemical reaction gives the relation of equilibrium\n", + "temperature and equilibrium partial pressure ratio\n", + "$\\frac{P_{H_{2}}}{P_{H_{2}O}}$ for desired $P_{O_{2}}$ of the chemical\n", + "equilibrium.\n", + "\n", + "For the $CO$/$CO_{2}$mixture, the chemical reaction is shown below\n", + "\n", + "*Eq. 7\u201114* $2CO(gas) + O_{2}(gas) = 2CO_{2}(gas)$\n", + "\n", + "Similar, from the SSUB database, the following equation is obtained\n", + "\n", + "*Eq. 7\u201115*\n", + "$- RTln\\left\\{ {\\frac{1}{P_{O_{2}}}\\left( \\frac{P_{{CO}_{2}}}{P_{CO}} \\right)}^{2} \\right\\} = \\mathrm{\\Delta}_{\\ }^{0}G = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S = - 562,927 - 172.020T\\ (J)$\n", + "\n", + "with $\\mathrm{\\Delta}_{\\ }^{0}H$ and $\\mathrm{\\Delta}_{\\ }^{0}S$\n", + "calculated at 1273K, which are also slightly temperature-dependent as in\n", + "the case of the $H_{2}$/$H_{2}O$ mixture. The iso-partial-pressure-ratio\n", + "lines are written as\n", + "\n", + "*Eq. 7\u201116*\n", + "$RTlnP_{O_{2}} = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S - 2RTln\\frac{P_{CO}}{P_{{CO}_{2}}} = - 562,927 + \\left( 172.020 - 2Rln\\frac{P_{CO}}{P_{{CO}_{2}}} \\right)T\\ $\n", + "\n", + "Its intercept at $T = 0K$ is given by\n", + "$\\mathrm{\\Delta}_{\\ }^{0}H = - 562,927\\ (J)$, and the slope by\n", + "$172.020 - - 2Rln\\left( \\frac{P_{CO}}{P_{{CO}_{2}}} \\right)$. The values\n", + "of the $\\frac{P_{CO}}{P_{{CO}_{2}}}$ ratio are marked on the third\n", + "secondary axis on the right of the Ellingham diagram. The intersection\n", + "of the iso-partial-pressure-ratio line and the equilibrium line of the\n", + "chemical reaction gives the relation of equilibrium temperature and\n", + "equilibrium partial pressure ratio $\\frac{P_{CO}}{P_{{CO}_{2}}}$ for\n", + "desired $P_{O_{2}}$ of the chemical equilibrium.\n", + "\n", + "Therefore, one can use a mixture of the $H_{2}$/$H_{2}O$ or\n", + "$CO$/$CO_{2}$ to obtain the desired low $P_{O_{2}}$ values using and or\n", + "calculating from the SSUB database. For example,\n", + "$P_{O_{2}} = 10^{- 15}\\ atm$ at 1273K, the calculated values from the\n", + "SSUB database are $\\frac{P_{H_{2}}}{P_{H_{2}O}} \\approx 1.67$ and\n", + "$\\frac{P_{CO}}{P_{{CO}_{2}}} \\approx 2.78$, respectively, in which the\n", + "temperature dependences of $\\mathrm{\\Delta}_{\\ }^{0}H$ and\n", + "$\\mathrm{\\Delta}_{\\ }^{0}S$ and the many more gaseous species in the gas\n", + "phase are taken into account. On the other hand, the reading from the\n", + "Ellingham diagram gives $\\frac{P_{H_{2}}}{P_{H_{2}O}} \\approx 2.0$ and\n", + "$\\frac{P_{CO}}{P_{{CO}_{2}}} \\approx 2.4$, and the agreement with the\n", + "more accurate calculations above is remarkable keeping in mind the\n", + "uncertainties in graphically drawing the lines and reading both values\n", + "in the logarithmic scales from the diagram, indicating the robustness of\n", + "the Ellingham diagram.\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/applications_to_chemical_reactions/index.ipynb b/src/psu410/src/applications_to_chemical_reactions/index.ipynb index a3f41d1..978853a 100644 --- a/src/psu410/src/applications_to_chemical_reactions/index.ipynb +++ b/src/psu410/src/applications_to_chemical_reactions/index.ipynb @@ -4,9 +4,24 @@ "metadata": {}, "cells": [ { - "id": "b87385fe", + "id": "3d15fa98", "cell_type": "markdown", - "source": "# Applications to chemical reactions\n\nA chemical reaction can be viewed as a framework dividing a system into\ntwo closed subsystems: reactants and products. The phases and species of\nreactants and products are selected from possible phases and species\nthat may form from the independent components of the system. A chemical\nreaction can thus be considered as an internal process to transfer heat\nand work between the two subsystems of reactants and products. It is\nevident that this subset of phases and species only represents partial\nequilibrium information of the system under given external conditions,\nand more stable equilibrium states may exist by including more phases\nand species with the global equilibrium depicted by phase diagrams\ndiscussed in previous chapters with all known phases and species\nincluded.\n", + "source": [ + "# Applications to chemical reactions\n", + "\n", + "A chemical reaction can be viewed as a framework dividing a system into\n", + "two closed subsystems: reactants and products. The phases and species of\n", + "reactants and products are selected from possible phases and species\n", + "that may form from the independent components of the system. A chemical\n", + "reaction can thus be considered as an internal process to transfer heat\n", + "and work between the two subsystems of reactants and products. It is\n", + "evident that this subset of phases and species only represents partial\n", + "equilibrium information of the system under given external conditions,\n", + "and more stable equilibrium states may exist by including more phases\n", + "and species with the global equilibrium depicted by phase diagrams\n", + "discussed in previous chapters with all known phases and species\n", + "included.\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/applications_to_chemical_reactions/internal_process_and_differential_and_integrated_driving_forces.ipynb b/src/psu410/src/applications_to_chemical_reactions/internal_process_and_differential_and_integrated_driving_forces.ipynb index a1f94a5..1ec0188 100644 --- a/src/psu410/src/applications_to_chemical_reactions/internal_process_and_differential_and_integrated_driving_forces.ipynb +++ b/src/psu410/src/applications_to_chemical_reactions/internal_process_and_differential_and_integrated_driving_forces.ipynb @@ -4,9 +4,98 @@ "metadata": {}, "cells": [ { - "id": "215e22d0", + "id": "15a6786f", "cell_type": "markdown", - "source": "## Internal process and differential and integrated driving forces\n\nThe driving force for an internal process can be defined as following\nusing $U$, $H$, $F$, or $G$ as discussed in Chapter and depending on\nwhat system variables are kept constant\n\n*Eq. 7\u20111*\n$- D = \\left( \\frac{\\partial U}{\\partial\\xi} \\right)_{S,\\ V,N_{i}\\ } = \\left( \\frac{\\partial H}{\\partial\\xi} \\right)_{S,\\ P,N_{i}\\ } = \\left( \\frac{\\partial F}{\\partial\\xi} \\right)_{T,\\ V,N_{i}\\ } = \\left( \\frac{\\partial G}{\\partial\\xi} \\right)_{T,P,N_{i}\\ }$\n\nThis can be termed as differential driving force as it relates the\nderivative of energy with respect to an internal process so the change\nis infinitesimally small and does not affect the properties of the\nsystem significantly. For a system under constant $T$ and $P$, let us\nconsider an internal process for forming a new phase $\\alpha$ with the\ncomposition $x_{i}^{\\alpha}$ and Gibbs energy\n$G_{m}^{\\alpha}\\left( x_{i}^{\\alpha} \\right)$. The differential driving\nforce for such an internal process can thus be defined as\n\n*Eq. 7\u20112*\n$- D = G_{m}^{\\alpha}\\left( x_{i}^{\\alpha} \\right) - \\sum_{i}^{}x_{i}^{\\alpha}\\mu_{i} = \\sum_{i}^{}x_{i}^{\\alpha}\\left( \\mu_{i}^{\\alpha} - \\mu_{i} \\right)$\n\nwhere $\\mu_{i}$ is the chemical potential of component *i* in the\nsystem. This may also be called as nucleation driving force as if the\n$\\alpha$ phase is nucleating in the system. As discussed in Chapter ,\nchemical potentials are the intercepts on the Gibbs energy axis by the\ntangent plane of Gibbs energy. thus represents the distance between the\ntangent planes of the original system and the $\\alpha$ phase at the\ncomposition of the $\\alpha$ phase. Evidently, this distance is at its\nmaximum when the two tangent planes are parallel to each other, i.e. the\ncommonly called parallel tangent construction when evaluating nucleation\ndriving force.\n\nThe situation is different for chemical reactions where the amount of\neach component in reactants is the same as that in products, i.e. there\nis a mass balance between reactants and products. The driving force for\na chemical reaction is defined by the Gibbs energy difference between\nthe reactants and products as if all the reactants are transferred to\nthe products. This driving force may thus be called integrated driving\nforce as it describes the energy difference of a system under two\ndifferent states, i.e.\n\n*Eq. 7\u20113*\n$- \\int_{}^{}{Dd\\xi} = \\mathrm{\\Delta}G = \\sum_{p}^{}{n_{p}G_{p}} - \\sum_{r}^{}{n_{r}G_{r}}$\n\nwhere superscripts $p$ and $r$ denote products and reactants, $n_{p}$\nand $n_{r}$ their corresponding moles, $G_{p}$and $G_{r}$ the Gibbs\nenergies in per mole of formula of their respective stoichiometries as\nwritten in the chemical reaction. Conventionally, the products and\nreactants are represented by species or stoichiometric compounds rather\nthan individual phases. This is particularly evident when various\ngaseous species are considered in a chemical reaction. Consequently,\n$G_{p}$and $G_{r}$ in represent the chemical potentials of product and\nreactant species. For species with a fixed composition, its chemical\npotential is the same as its Gibbs energy as shown by . For a species in\na solution phase, its chemical potential is related to its activity as\nshown in . can thus be further written as\n\n*Eq. 7\u20114*\n$\\mathrm{\\Delta}G = \\sum_{p}^{}{n_{p}_{\\ }^{0}G_{p}} - \\sum_{r}^{}{n_{r}_{\\ }^{0}G_{r}} + RTln\\frac{\\prod_{p}^{}\\left( a_{p} \\right)^{n_{p}}}{\\prod_{r}^{}\\left( a_{r} \\right)^{n_{r}}} = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{\\prod_{p}^{}\\left( a_{p} \\right)^{n_{p}}}{\\prod_{r}^{}\\left( a_{r} \\right)^{n_{r}}}$\n\nIt is evident from that for stoichiometric phases in a chemical\nreaction, their activities equal to one. At equilibrium, the integrated\ndriving force becomes zero, i.e. $\\mathrm{\\Delta}G = 0$, and is\nre-arranged to become\n\n*Eq. 7\u20115*\n$- RTln\\frac{\\prod_{p}^{}\\left( a_{p} \\right)^{n_{p}}}{\\prod_{r}^{}\\left( a_{r} \\right)^{n_{r}}} = - RTlnK_{e} = \\mathrm{\\Delta}_{\\ }^{0}G = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S$\n\nwhere $K_{e}$ is often called reaction constant relating the activities\nof products and reactants at equilibrium. In a system with\n$\\frac{\\prod_{p}^{}\\left( a_{p} \\right)^{n_{p}}}{\\prod_{r}^{}\\left( a_{r} \\right)^{n_{r}}} > K_{e}$,\nthe chemical reaction goes to left, and the products are decomposed,\nwhile\n$\\frac{\\prod_{p}^{}\\left( a_{p} \\right)^{n_{p}}}{\\prod_{r}^{}\\left( a_{r} \\right)^{n_{r}}} < K_{e}$,\nthe chemical reaction goes to right, and the products are formed. is\noften recast into the following form by dividing $\u2013RT$ on both sides of\nthe equation\n\n*Eq. 7\u20116*\n$\\ln K_{e} = - \\frac{\\mathrm{\\Delta}_{\\ }^{0}H}{RT} + \\frac{\\mathrm{\\Delta}_{\\ }^{0}S}{R}$\n\nWith $\\ln K_{e}$ plotted with respect to $1/T$, indicates that the slope\nis $- \\mathrm{\\Delta}_{\\ }^{0}H$/R and the intercept on the y axis is\n$\\mathrm{\\Delta}_{\\ }^{0}{S/R}$ as shown in .\n\nFigure \u2011: $\\ln K_{e}$ plotted with respect to $1/T$ for several M-O\nsystems at 1 bar and $K_{e}$ represented by the partial pressure ratio\nof CO2 and CO.\n", + "source": [ + "## Internal process and differential and integrated driving forces\n", + "\n", + "The driving force for an internal process can be defined as following\n", + "using $U$, $H$, $F$, or $G$ as discussed in Chapter and depending on\n", + "what system variables are kept constant\n", + "\n", + "*Eq. 7\u20111*\n", + "$- D = \\left( \\frac{\\partial U}{\\partial\\xi} \\right)_{S,\\ V,N_{i}\\ } = \\left( \\frac{\\partial H}{\\partial\\xi} \\right)_{S,\\ P,N_{i}\\ } = \\left( \\frac{\\partial F}{\\partial\\xi} \\right)_{T,\\ V,N_{i}\\ } = \\left( \\frac{\\partial G}{\\partial\\xi} \\right)_{T,P,N_{i}\\ }$\n", + "\n", + "This can be termed as differential driving force as it relates the\n", + "derivative of energy with respect to an internal process so the change\n", + "is infinitesimally small and does not affect the properties of the\n", + "system significantly. For a system under constant $T$ and $P$, let us\n", + "consider an internal process for forming a new phase $\\alpha$ with the\n", + "composition $x_{i}^{\\alpha}$ and Gibbs energy\n", + "$G_{m}^{\\alpha}\\left( x_{i}^{\\alpha} \\right)$. The differential driving\n", + "force for such an internal process can thus be defined as\n", + "\n", + "*Eq. 7\u20112*\n", + "$- D = G_{m}^{\\alpha}\\left( x_{i}^{\\alpha} \\right) - \\sum_{i}^{}x_{i}^{\\alpha}\\mu_{i} = \\sum_{i}^{}x_{i}^{\\alpha}\\left( \\mu_{i}^{\\alpha} - \\mu_{i} \\right)$\n", + "\n", + "where $\\mu_{i}$ is the chemical potential of component *i* in the\n", + "system. This may also be called as nucleation driving force as if the\n", + "$\\alpha$ phase is nucleating in the system. As discussed in Chapter ,\n", + "chemical potentials are the intercepts on the Gibbs energy axis by the\n", + "tangent plane of Gibbs energy. thus represents the distance between the\n", + "tangent planes of the original system and the $\\alpha$ phase at the\n", + "composition of the $\\alpha$ phase. Evidently, this distance is at its\n", + "maximum when the two tangent planes are parallel to each other, i.e. the\n", + "commonly called parallel tangent construction when evaluating nucleation\n", + "driving force.\n", + "\n", + "The situation is different for chemical reactions where the amount of\n", + "each component in reactants is the same as that in products, i.e. there\n", + "is a mass balance between reactants and products. The driving force for\n", + "a chemical reaction is defined by the Gibbs energy difference between\n", + "the reactants and products as if all the reactants are transferred to\n", + "the products. This driving force may thus be called integrated driving\n", + "force as it describes the energy difference of a system under two\n", + "different states, i.e.\n", + "\n", + "*Eq. 7\u20113*\n", + "$- \\int_{}^{}{Dd\\xi} = \\mathrm{\\Delta}G = \\sum_{p}^{}{n_{p}G_{p}} - \\sum_{r}^{}{n_{r}G_{r}}$\n", + "\n", + "where superscripts $p$ and $r$ denote products and reactants, $n_{p}$\n", + "and $n_{r}$ their corresponding moles, $G_{p}$and $G_{r}$ the Gibbs\n", + "energies in per mole of formula of their respective stoichiometries as\n", + "written in the chemical reaction. Conventionally, the products and\n", + "reactants are represented by species or stoichiometric compounds rather\n", + "than individual phases. This is particularly evident when various\n", + "gaseous species are considered in a chemical reaction. Consequently,\n", + "$G_{p}$and $G_{r}$ in represent the chemical potentials of product and\n", + "reactant species. For species with a fixed composition, its chemical\n", + "potential is the same as its Gibbs energy as shown by . For a species in\n", + "a solution phase, its chemical potential is related to its activity as\n", + "shown in . can thus be further written as\n", + "\n", + "*Eq. 7\u20114*\n", + "$\\mathrm{\\Delta}G = \\sum_{p}^{}{n_{p}_{\\ }^{0}G_{p}} - \\sum_{r}^{}{n_{r}_{\\ }^{0}G_{r}} + RTln\\frac{\\prod_{p}^{}\\left( a_{p} \\right)^{n_{p}}}{\\prod_{r}^{}\\left( a_{r} \\right)^{n_{r}}} = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{\\prod_{p}^{}\\left( a_{p} \\right)^{n_{p}}}{\\prod_{r}^{}\\left( a_{r} \\right)^{n_{r}}}$\n", + "\n", + "It is evident from that for stoichiometric phases in a chemical\n", + "reaction, their activities equal to one. At equilibrium, the integrated\n", + "driving force becomes zero, i.e. $\\mathrm{\\Delta}G = 0$, and is\n", + "re-arranged to become\n", + "\n", + "*Eq. 7\u20115*\n", + "$- RTln\\frac{\\prod_{p}^{}\\left( a_{p} \\right)^{n_{p}}}{\\prod_{r}^{}\\left( a_{r} \\right)^{n_{r}}} = - RTlnK_{e} = \\mathrm{\\Delta}_{\\ }^{0}G = \\mathrm{\\Delta}_{\\ }^{0}H - T\\mathrm{\\Delta}_{\\ }^{0}S$\n", + "\n", + "where $K_{e}$ is often called reaction constant relating the activities\n", + "of products and reactants at equilibrium. In a system with\n", + "$\\frac{\\prod_{p}^{}\\left( a_{p} \\right)^{n_{p}}}{\\prod_{r}^{}\\left( a_{r} \\right)^{n_{r}}} > K_{e}$,\n", + "the chemical reaction goes to left, and the products are decomposed,\n", + "while\n", + "$\\frac{\\prod_{p}^{}\\left( a_{p} \\right)^{n_{p}}}{\\prod_{r}^{}\\left( a_{r} \\right)^{n_{r}}} < K_{e}$,\n", + "the chemical reaction goes to right, and the products are formed. is\n", + "often recast into the following form by dividing $\u2013RT$ on both sides of\n", + "the equation\n", + "\n", + "*Eq. 7\u20116*\n", + "$\\ln K_{e} = - \\frac{\\mathrm{\\Delta}_{\\ }^{0}H}{RT} + \\frac{\\mathrm{\\Delta}_{\\ }^{0}S}{R}$\n", + "\n", + "With $\\ln K_{e}$ plotted with respect to $1/T$, indicates that the slope\n", + "is $- \\mathrm{\\Delta}_{\\ }^{0}H$/R and the intercept on the y axis is\n", + "$\\mathrm{\\Delta}_{\\ }^{0}{S/R}$ as shown in .\n", + "\n", + "Figure \u2011: $\\ln K_{e}$ plotted with respect to $1/T$ for several M-O\n", + "systems at 1 bar and $K_{e}$ represented by the partial pressure ratio\n", + "of CO2 and CO.\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/applications_to_chemical_reactions/maximum_reaction_rate_and_chemical_transport_reactions_.ipynb b/src/psu410/src/applications_to_chemical_reactions/maximum_reaction_rate_and_chemical_transport_reactions_.ipynb index a23ee15..e745a0a 100644 --- a/src/psu410/src/applications_to_chemical_reactions/maximum_reaction_rate_and_chemical_transport_reactions_.ipynb +++ b/src/psu410/src/applications_to_chemical_reactions/maximum_reaction_rate_and_chemical_transport_reactions_.ipynb @@ -4,9 +4,155 @@ "metadata": {}, "cells": [ { - "id": "1baa04de", + "id": "2abbd0a0", "cell_type": "markdown", - "source": "## Maximum reaction rate and chemical transport reactions \n\nEquilibrium thermodynamics can be used to calculate the maximum rates of\nreaction that are possible in dynamically reacting systems, such as when\na corrosive gas passes over a heated sample. Other examples of such\nreactions include the reduction of a metal oxide in flowing hydrogen,\nthe evaporation of a material in a vacuum, and the deposition of a thin\nfilm by a chemical vapour deposition process. The basic assumption used\nin calculating these maximum reaction rates is that local equilibrium\nexists at the location of the considered reaction.\n\nThis section examines several examples in which maximum reaction rates\nare calculated. The system can typically be divided into three regions:\n(1) the input region, which is usually near room temperature, (2) the\nhigh temperature region in which the primary reaction of interest is\noccurring, and (3) the exit region. Such a system is almost always at\nconstant pressure throughout the system. Since the three regions have\ndifferent temperatures, the key is to use the mass conservation of the\ncarrier gas in all three regions. With the input gas at $T = 298K$ and\n$P = 1atm$, the volume of one mole of ideal gas is\n$0.0244\\left( m^{3} \\right) = 24.4(liter)$. The number of moles of input\ngas flowing through the system can be written as\n\n*Eq. 7\u201117*\n$n_{gas}^{0} = \\frac{Pf_{gas}}{RT} = \\frac{f_{gas}}{24.4} = 0.0409f_{gas}$\n\nwhere $f_{gas}$ is the input gas flow rate in liter per unit time at\n$T = 298K$ and $P = 1atm$, and $R$ the gas constant.\n\nThe first example is the evaporation of water in a flowing stream of dry\nhydrogen. A schematic diagram of the system being considered is given in\n. The goal of the calculation is to determine the maximum rate of\n$H_{2}O(liquid)$ loss, $n_{H_{2}O}^{\\ }$, through vaporization in a\nflowing stream of $H_{2}$, e.g. for generating a given $H_{2}O/H_{2}$\nratio for producing a given $O_{2}$ pressure in a high temperature\nsystem as related to the Ellingham diagram discussed in Chapter . The\nmaximum rate is determined by saturating the $H_{2}$ with the\nequilibrium vapour pressure of $H_{2}O(liquid)$ with the number of moles\nof $H_{2}$ being $n_{H_{2}}^{0} = 0.0409f_{H_{2}}$ and $f_{H_{2}}$ being\nthe input flow rate of $H_{2}$ in liter per unit time.\n\nFigure \u2011: Schematic diagram of vaporization of water in a flowing stream\nof dry hydrogen\n\nIn the first input region, the total number moles of gas is\n$N = n_{H_{2}}^{0}$. In the second high temperature region with the\ntemperature and pressure being $T_{sys}$ and $P_{sys}$, the total number\nmoles of gas is $N = n_{H_{2}}^{0} + n_{H_{2}O}^{\\ }$, which is the same\nfor the third exit region with $T_{exit}$. To avoid condensation, one\nneeds to maintain $T_{exit} > T_{sys}$. The vapour pressure of $H_{2}O$\nat $T_{sys}$, $P_{H_{2}O\\ }$, can be obtained from equilibrium\nthermodynamic calculations. Since the $H_{2}$ and $H_{2}O$ are occupying\nthe same volume at the same temperature, the maximum number of moles\n$H_{2}O$ can be calculated from the following relation based on the\nideal gas law\n\n*Eq. 7\u201118*\n$n_{H_{2}O}^{\\ } = \\frac{P_{H_{2}O}^{\\ }}{P_{H_{2}}^{\\ }}n_{H_{2}}^{0} = \\frac{P_{H_{2}O}^{\\ }}{P_{sys}^{\\ } - P_{H_{2}O}^{\\ }}n_{H_{2}}^{0} = \\frac{P_{H_{2}O}^{\\ }}{P_{sys}^{\\ } - P_{H_{2}O}^{\\ }}\\frac{f_{H_{2}}}{24.4}$\n\nFor $P_{sys}^{\\ } = 101325Pa$, one can calculate $P_{H_{2}O}^{\\ }$ and\nplot $\\frac{n_{H_{2}O}^{\\ }}{f_{H_{2}}}$ as a function of temperature\nfrom the SSUB database as shown in . The corresponding partial pressure\nratio, $P_{H_{2}O}/P_{H_{2}}$ , is plotted in , which can be used to\ncalculate $P_{O_{2}}^{\\ }$ at any given temperatures. An example is\nshown in with $T_{sys} = 348.15K$ and $P_{H_{2}O}/P_{H_{2}} = 0.607$.\n\nFigure \u2011: Ratio of maximum number of moles $H_{2}O$ with respect to\nhydrogen flow rate, $\\frac{n_{H_{2}O}^{\\ }}{f_{H_{2}}}$, plotted as a\nfunction of temperature.\n\nFigure \u2011: Partial pressure ratio, $P_{H_{2}O}/P_{H_{2}}$, corresponding\nto .\n\nFigure \u2011: Partial pressure of oxygen, $P_{O_{2}}^{\\ }$, as a function of\ntemperature with $T_{sys} = 348.15K$ and $P_{H_{2}O}/P_{H_{2}} = 0.607$.\n\nThe second example is the corrosion of $SiO_{2}$(s) by flowing $H_{2}$\ngas at high temperatures. Considering $T_{sys}^{\\ } = 1700K$,\n$P_{sys}^{\\ } = 101325Pa$, and $f_{H_{2}} = 1\\ liter/min$, the system is\nthus defined with 0.0409 moles of $H_{2}$ and an equilibrium between the\ngas phase and the tridymite $SiO_{2}$. The equilibrium calculation gives\n0.04092 moles of gas with its constitutions as\n$H_{2}:H_{2}O:SiO = 0.998887:4.833 \\bullet 10^{- 4}:4.832 \\bullet 10^{- 4}$.\nThe corrosion rate of SiO2(s) is thus\n$0.04092x4.832 \\bullet 10^{- 4} = 1.98 \\bullet 10^{- 5}mol/min = 1.19gram/min.$\n\nAnother example is to consider that the flowing $CO$ gas of 298K and\n1atm ($= 101325Pa$) at a rate of $1\\ liter/min$ passes through and\nequilibrates with single phase $C(s)$ at 1500K. The equilibrium system\nis defined by $T = 1500K$, $P = 1atm$, and 0.0409 moles of $CO$ with the\nequilibrium between the gas phase and C(s). The equilibrium calculation\ngives 0.040872 moles of gas phase with the mole fraction of $CO$ being\n0.999327, resulting in the loss of *CO* or the deposition of *C(s)* at a\nrate of\n$0.0409 - 0.040872 \\bullet 0.999327 = 5.55 \\bullet 10^{- 5}mole/min$.\n\nA chemical transport reaction is a reaction in which a condensed phase\nreacts with a gas phase to form vapour-phase products, which in turn\nundergo the reverse reformation of the condensed phase. Two well-known\nexamples of such reactions are\n\n*Eq. 7\u201119* $M(s) + \\frac{n}{2}I_{2}(g) = MI_{n}(g)$\n\n*Eq. 7\u201120* $Ni(s) + 4CO(g) = Ni{(CO)}_{4}(g)$\n\nwhich are used in the purification of metals by the iodide process and\nin the purification of nickel by the Mond-Langer process. In both\nprocesses the forward reaction is favoured by lower temperatures and the\nreverse reaction by higher temperatures, resulting in the deposition of\nthe metal. The most common technique for causing a chemical transport of\na condensed substance makes use of the temperature dependence of the\nequilibrium constant. As was discussed previously, the enthalpy of\nreaction, $\\mathrm{\\Delta}_{\\ }^{0}H$, determines the manner in which\n$K_{e}$ changes with temperature (see ). The value of $K_{e}$ increases\nwith increasing *T* for $\\mathrm{\\Delta}_{\\ }^{0}H > 0$, $K_{e}$\ndecreases with increasing *T* for $\\mathrm{\\Delta}_{\\ }^{0}H < 0$, and\n$K_{e}$ is independent of T for $\\mathrm{\\Delta}_{\\ }^{0}H = 0$. The\n$\\mathrm{\\Delta}_{\\ }^{0}H$ and $\\mathrm{\\Delta}_{\\ }^{0}S$ values for\nchemical transport reactions may be either positive or negative. For\nreactions by both are positive, and for reactions by both are negative.\n\nIn a typical experiment the starting solid is located at the point in a\ntemperature gradient that corresponds to the largest $K_{e}$ value for\nthe experimental condition. As the gaseous species migrate to other\nlocations in the system with temperatures corresponding to lower $K_{e}$\nvalues, the reverse reaction occurs to satisfy the new equilibrium\nrequirements, and the solid phase is deposited. The dependence of\n$K_{e}$ on $\\mathrm{\\Delta}_{\\ }^{0}H$ results in a material transport\nfrom hot to cold for $\\mathrm{\\Delta}_{\\ }^{0}H > 0$ (the same as for\nvaporization-condensation reactions), from cold to hot for\n$\\mathrm{\\Delta}_{\\ }^{0}H < 0$, and no transport for\n$\\mathrm{\\Delta}_{\\ }^{0}H = 0$.\n\nThe success of a particular reaction in causing an appreciable transport\nof a condensed phase depends mainly upon the partial pressure gradients\nor concentration gradients of the gaseous species in the system. A\nreaction whose equilibrium is extreme toward either the reactant side or\nthe product side will not give an appreciable transport of material. The\nconcentration gradients are too small in such a system. Reactions with\nequilibrium constants near unity at the experimental temperatures\nusually give the largest transport since small changes in $K_{e}$ cause\nlarge changes in concentrations. The general condition required for\nobtaining a $K_{e}$ value near unity at a reasonable temperature is that\n$\\mathrm{\\Delta}_{\\ }^{0}H$ and $\\mathrm{\\Delta}_{\\ }^{0}S$ both have\nthe same sign, resulting from the equalities of .\n", + "source": [ + "## Maximum reaction rate and chemical transport reactions \n", + "\n", + "Equilibrium thermodynamics can be used to calculate the maximum rates of\n", + "reaction that are possible in dynamically reacting systems, such as when\n", + "a corrosive gas passes over a heated sample. Other examples of such\n", + "reactions include the reduction of a metal oxide in flowing hydrogen,\n", + "the evaporation of a material in a vacuum, and the deposition of a thin\n", + "film by a chemical vapour deposition process. The basic assumption used\n", + "in calculating these maximum reaction rates is that local equilibrium\n", + "exists at the location of the considered reaction.\n", + "\n", + "This section examines several examples in which maximum reaction rates\n", + "are calculated. The system can typically be divided into three regions:\n", + "(1) the input region, which is usually near room temperature, (2) the\n", + "high temperature region in which the primary reaction of interest is\n", + "occurring, and (3) the exit region. Such a system is almost always at\n", + "constant pressure throughout the system. Since the three regions have\n", + "different temperatures, the key is to use the mass conservation of the\n", + "carrier gas in all three regions. With the input gas at $T = 298K$ and\n", + "$P = 1atm$, the volume of one mole of ideal gas is\n", + "$0.0244\\left( m^{3} \\right) = 24.4(liter)$. The number of moles of input\n", + "gas flowing through the system can be written as\n", + "\n", + "*Eq. 7\u201117*\n", + "$n_{gas}^{0} = \\frac{Pf_{gas}}{RT} = \\frac{f_{gas}}{24.4} = 0.0409f_{gas}$\n", + "\n", + "where $f_{gas}$ is the input gas flow rate in liter per unit time at\n", + "$T = 298K$ and $P = 1atm$, and $R$ the gas constant.\n", + "\n", + "The first example is the evaporation of water in a flowing stream of dry\n", + "hydrogen. A schematic diagram of the system being considered is given in\n", + ". The goal of the calculation is to determine the maximum rate of\n", + "$H_{2}O(liquid)$ loss, $n_{H_{2}O}^{\\ }$, through vaporization in a\n", + "flowing stream of $H_{2}$, e.g. for generating a given $H_{2}O/H_{2}$\n", + "ratio for producing a given $O_{2}$ pressure in a high temperature\n", + "system as related to the Ellingham diagram discussed in Chapter . The\n", + "maximum rate is determined by saturating the $H_{2}$ with the\n", + "equilibrium vapour pressure of $H_{2}O(liquid)$ with the number of moles\n", + "of $H_{2}$ being $n_{H_{2}}^{0} = 0.0409f_{H_{2}}$ and $f_{H_{2}}$ being\n", + "the input flow rate of $H_{2}$ in liter per unit time.\n", + "\n", + "Figure \u2011: Schematic diagram of vaporization of water in a flowing stream\n", + "of dry hydrogen\n", + "\n", + "In the first input region, the total number moles of gas is\n", + "$N = n_{H_{2}}^{0}$. In the second high temperature region with the\n", + "temperature and pressure being $T_{sys}$ and $P_{sys}$, the total number\n", + "moles of gas is $N = n_{H_{2}}^{0} + n_{H_{2}O}^{\\ }$, which is the same\n", + "for the third exit region with $T_{exit}$. To avoid condensation, one\n", + "needs to maintain $T_{exit} > T_{sys}$. The vapour pressure of $H_{2}O$\n", + "at $T_{sys}$, $P_{H_{2}O\\ }$, can be obtained from equilibrium\n", + "thermodynamic calculations. Since the $H_{2}$ and $H_{2}O$ are occupying\n", + "the same volume at the same temperature, the maximum number of moles\n", + "$H_{2}O$ can be calculated from the following relation based on the\n", + "ideal gas law\n", + "\n", + "*Eq. 7\u201118*\n", + "$n_{H_{2}O}^{\\ } = \\frac{P_{H_{2}O}^{\\ }}{P_{H_{2}}^{\\ }}n_{H_{2}}^{0} = \\frac{P_{H_{2}O}^{\\ }}{P_{sys}^{\\ } - P_{H_{2}O}^{\\ }}n_{H_{2}}^{0} = \\frac{P_{H_{2}O}^{\\ }}{P_{sys}^{\\ } - P_{H_{2}O}^{\\ }}\\frac{f_{H_{2}}}{24.4}$\n", + "\n", + "For $P_{sys}^{\\ } = 101325Pa$, one can calculate $P_{H_{2}O}^{\\ }$ and\n", + "plot $\\frac{n_{H_{2}O}^{\\ }}{f_{H_{2}}}$ as a function of temperature\n", + "from the SSUB database as shown in . The corresponding partial pressure\n", + "ratio, $P_{H_{2}O}/P_{H_{2}}$ , is plotted in , which can be used to\n", + "calculate $P_{O_{2}}^{\\ }$ at any given temperatures. An example is\n", + "shown in with $T_{sys} = 348.15K$ and $P_{H_{2}O}/P_{H_{2}} = 0.607$.\n", + "\n", + "Figure \u2011: Ratio of maximum number of moles $H_{2}O$ with respect to\n", + "hydrogen flow rate, $\\frac{n_{H_{2}O}^{\\ }}{f_{H_{2}}}$, plotted as a\n", + "function of temperature.\n", + "\n", + "Figure \u2011: Partial pressure ratio, $P_{H_{2}O}/P_{H_{2}}$, corresponding\n", + "to .\n", + "\n", + "Figure \u2011: Partial pressure of oxygen, $P_{O_{2}}^{\\ }$, as a function of\n", + "temperature with $T_{sys} = 348.15K$ and $P_{H_{2}O}/P_{H_{2}} = 0.607$.\n", + "\n", + "The second example is the corrosion of $SiO_{2}$(s) by flowing $H_{2}$\n", + "gas at high temperatures. Considering $T_{sys}^{\\ } = 1700K$,\n", + "$P_{sys}^{\\ } = 101325Pa$, and $f_{H_{2}} = 1\\ liter/min$, the system is\n", + "thus defined with 0.0409 moles of $H_{2}$ and an equilibrium between the\n", + "gas phase and the tridymite $SiO_{2}$. The equilibrium calculation gives\n", + "0.04092 moles of gas with its constitutions as\n", + "$H_{2}:H_{2}O:SiO = 0.998887:4.833 \\bullet 10^{- 4}:4.832 \\bullet 10^{- 4}$.\n", + "The corrosion rate of SiO2(s) is thus\n", + "$0.04092x4.832 \\bullet 10^{- 4} = 1.98 \\bullet 10^{- 5}mol/min = 1.19gram/min.$\n", + "\n", + "Another example is to consider that the flowing $CO$ gas of 298K and\n", + "1atm ($= 101325Pa$) at a rate of $1\\ liter/min$ passes through and\n", + "equilibrates with single phase $C(s)$ at 1500K. The equilibrium system\n", + "is defined by $T = 1500K$, $P = 1atm$, and 0.0409 moles of $CO$ with the\n", + "equilibrium between the gas phase and C(s). The equilibrium calculation\n", + "gives 0.040872 moles of gas phase with the mole fraction of $CO$ being\n", + "0.999327, resulting in the loss of *CO* or the deposition of *C(s)* at a\n", + "rate of\n", + "$0.0409 - 0.040872 \\bullet 0.999327 = 5.55 \\bullet 10^{- 5}mole/min$.\n", + "\n", + "A chemical transport reaction is a reaction in which a condensed phase\n", + "reacts with a gas phase to form vapour-phase products, which in turn\n", + "undergo the reverse reformation of the condensed phase. Two well-known\n", + "examples of such reactions are\n", + "\n", + "*Eq. 7\u201119* $M(s) + \\frac{n}{2}I_{2}(g) = MI_{n}(g)$\n", + "\n", + "*Eq. 7\u201120* $Ni(s) + 4CO(g) = Ni{(CO)}_{4}(g)$\n", + "\n", + "which are used in the purification of metals by the iodide process and\n", + "in the purification of nickel by the Mond-Langer process. In both\n", + "processes the forward reaction is favoured by lower temperatures and the\n", + "reverse reaction by higher temperatures, resulting in the deposition of\n", + "the metal. The most common technique for causing a chemical transport of\n", + "a condensed substance makes use of the temperature dependence of the\n", + "equilibrium constant. As was discussed previously, the enthalpy of\n", + "reaction, $\\mathrm{\\Delta}_{\\ }^{0}H$, determines the manner in which\n", + "$K_{e}$ changes with temperature (see ). The value of $K_{e}$ increases\n", + "with increasing *T* for $\\mathrm{\\Delta}_{\\ }^{0}H > 0$, $K_{e}$\n", + "decreases with increasing *T* for $\\mathrm{\\Delta}_{\\ }^{0}H < 0$, and\n", + "$K_{e}$ is independent of T for $\\mathrm{\\Delta}_{\\ }^{0}H = 0$. The\n", + "$\\mathrm{\\Delta}_{\\ }^{0}H$ and $\\mathrm{\\Delta}_{\\ }^{0}S$ values for\n", + "chemical transport reactions may be either positive or negative. For\n", + "reactions by both are positive, and for reactions by both are negative.\n", + "\n", + "In a typical experiment the starting solid is located at the point in a\n", + "temperature gradient that corresponds to the largest $K_{e}$ value for\n", + "the experimental condition. As the gaseous species migrate to other\n", + "locations in the system with temperatures corresponding to lower $K_{e}$\n", + "values, the reverse reaction occurs to satisfy the new equilibrium\n", + "requirements, and the solid phase is deposited. The dependence of\n", + "$K_{e}$ on $\\mathrm{\\Delta}_{\\ }^{0}H$ results in a material transport\n", + "from hot to cold for $\\mathrm{\\Delta}_{\\ }^{0}H > 0$ (the same as for\n", + "vaporization-condensation reactions), from cold to hot for\n", + "$\\mathrm{\\Delta}_{\\ }^{0}H < 0$, and no transport for\n", + "$\\mathrm{\\Delta}_{\\ }^{0}H = 0$.\n", + "\n", + "The success of a particular reaction in causing an appreciable transport\n", + "of a condensed phase depends mainly upon the partial pressure gradients\n", + "or concentration gradients of the gaseous species in the system. A\n", + "reaction whose equilibrium is extreme toward either the reactant side or\n", + "the product side will not give an appreciable transport of material. The\n", + "concentration gradients are too small in such a system. Reactions with\n", + "equilibrium constants near unity at the experimental temperatures\n", + "usually give the largest transport since small changes in $K_{e}$ cause\n", + "large changes in concentrations. The general condition required for\n", + "obtaining a $K_{e}$ value near unity at a reasonable temperature is that\n", + "$\\mathrm{\\Delta}_{\\ }^{0}H$ and $\\mathrm{\\Delta}_{\\ }^{0}S$ both have\n", + "the same sign, resulting from the equalities of .\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/applications_to_chemical_reactions/trends_of_entropies_of_reactions.ipynb b/src/psu410/src/applications_to_chemical_reactions/trends_of_entropies_of_reactions.ipynb index 6708fd4..e01968a 100644 --- a/src/psu410/src/applications_to_chemical_reactions/trends_of_entropies_of_reactions.ipynb +++ b/src/psu410/src/applications_to_chemical_reactions/trends_of_entropies_of_reactions.ipynb @@ -4,9 +4,80 @@ "metadata": {}, "cells": [ { - "id": "b51b2eff", + "id": "c11d5e72", "cell_type": "markdown", - "source": "## Trends of entropies of reactions\n\nThe reaction entropy, $\\mathrm{\\Delta}_{\\ }^{0}S$ in , plays an\nimportant role in determining equilibria of high-temperature reactions.\nThe most important single factor that determines the entropy of a\nreaction is the net change in the number of moles of gas as briefly\nmentioned in the discussion of the Ellingham diagram above. The reason\nthis is true can be explained as follows.\n\nThe entropy of a substance can be thought of as being the sum of four\nparts: (i) translational, (ii) rotational, (iii) vibrational, and (iv)\nelectronic. The translational entropy of a gas is the largest entropy\nterm under most conditions. To the extent that the other contributions\ncancel between reactants and products, the entropy of reaction is\ndetermined by the change in the number of moles of gaseous molecules.\nBased on the literature data or calculations from the SSUB database, the\nnet change in the number of moles of gas in a reaction results\napproximately in an entropy of reaction of about 175\u00b145 J/K/mole-gas at\n298K for many halides and oxides. The chemical reactions of and\ndiscussed above both reduce the gas by one mole, and their entropies of\nreaction are -113 and -172 J/K at 1273K, and -89 and -173J/K at 298K,\nrespectively, indicating that the chemical reaction of is an exception\nof the empirical rule. For chemical reactions shown in the Ellingham\ndiagram, their entropies of reaction follow this empirical rule pretty\nwell with some of them shown in calculated from the SGTE database.\n\nTable \u2011: Entropies of reactions with gas at 298.15K, J/K\n\nReaction: Si+O2 =SiO2 -182\n\nReaction: Ti+O2=TiO2 -185\n\nReaction: 2Mg+O2=2MgO -217\n\nReaction: 2Ca+O2=2CaO -212\n\nReaction: 2Mn+O2=2MnO -150\n\nSince the entropy of a reaction is primarily determined by the net\nchange in the number of moles of gas, the entropies for reactions\ninvolving only condensed phases must be small. The entropies of fusion\nof monatomic solids are usually in the range 8-15 J/K/mole-atom as shown\nfor some elements in . Most metals and many ionic salts have values that\nlie in this range when given in terms of per mole of atom of material.\nThere are few exceptions such as silicon and boron shown in the table.\nFor solid-state reactions, the average values can be approximated as\n0\u00b18J/K/mole-atom as also shown in the table.\n\nTable \u2011: Entropies of reactions of condensed phases at 298.15K, J/K\n\nReaction: Si(s)=Si(l) 29.762\n\nReaction: Ti(s2)=Ti(l) 7.288\n\nReaction: Mg(s2)=Mg(l) 9.184\n\nReaction: Ca(s2)=Ca(l) 7.659\n\nReaction: Mn(s2)=Mn(l) 11.443\n\nReaction: W(s)=W(l) 14.158\n\nReaction: B(s)=B(l) 21.380\n\nReaction: 3Fe+C=CFe3 17.060\n\nReaction: S+Mn=MnS 13.909\n\nReaction: NiO+Fe2O3=Fe2NiO4\n0.464\n", + "source": [ + "## Trends of entropies of reactions\n", + "\n", + "The reaction entropy, $\\mathrm{\\Delta}_{\\ }^{0}S$ in , plays an\n", + "important role in determining equilibria of high-temperature reactions.\n", + "The most important single factor that determines the entropy of a\n", + "reaction is the net change in the number of moles of gas as briefly\n", + "mentioned in the discussion of the Ellingham diagram above. The reason\n", + "this is true can be explained as follows.\n", + "\n", + "The entropy of a substance can be thought of as being the sum of four\n", + "parts: (i) translational, (ii) rotational, (iii) vibrational, and (iv)\n", + "electronic. The translational entropy of a gas is the largest entropy\n", + "term under most conditions. To the extent that the other contributions\n", + "cancel between reactants and products, the entropy of reaction is\n", + "determined by the change in the number of moles of gaseous molecules.\n", + "Based on the literature data or calculations from the SSUB database, the\n", + "net change in the number of moles of gas in a reaction results\n", + "approximately in an entropy of reaction of about 175\u00b145 J/K/mole-gas at\n", + "298K for many halides and oxides. The chemical reactions of and\n", + "discussed above both reduce the gas by one mole, and their entropies of\n", + "reaction are -113 and -172 J/K at 1273K, and -89 and -173J/K at 298K,\n", + "respectively, indicating that the chemical reaction of is an exception\n", + "of the empirical rule. For chemical reactions shown in the Ellingham\n", + "diagram, their entropies of reaction follow this empirical rule pretty\n", + "well with some of them shown in calculated from the SGTE database.\n", + "\n", + "Table \u2011: Entropies of reactions with gas at 298.15K, J/K\n", + "\n", + "Reaction: Si+O2 =SiO2 -182\n", + "\n", + "Reaction: Ti+O2=TiO2 -185\n", + "\n", + "Reaction: 2Mg+O2=2MgO -217\n", + "\n", + "Reaction: 2Ca+O2=2CaO -212\n", + "\n", + "Reaction: 2Mn+O2=2MnO -150\n", + "\n", + "Since the entropy of a reaction is primarily determined by the net\n", + "change in the number of moles of gas, the entropies for reactions\n", + "involving only condensed phases must be small. The entropies of fusion\n", + "of monatomic solids are usually in the range 8-15 J/K/mole-atom as shown\n", + "for some elements in . Most metals and many ionic salts have values that\n", + "lie in this range when given in terms of per mole of atom of material.\n", + "There are few exceptions such as silicon and boron shown in the table.\n", + "For solid-state reactions, the average values can be approximated as\n", + "0\u00b18J/K/mole-atom as also shown in the table.\n", + "\n", + "Table \u2011: Entropies of reactions of condensed phases at 298.15K, J/K\n", + "\n", + "Reaction: Si(s)=Si(l) 29.762\n", + "\n", + "Reaction: Ti(s2)=Ti(l) 7.288\n", + "\n", + "Reaction: Mg(s2)=Mg(l) 9.184\n", + "\n", + "Reaction: Ca(s2)=Ca(l) 7.659\n", + "\n", + "Reaction: Mn(s2)=Mn(l) 11.443\n", + "\n", + "Reaction: W(s)=W(l) 14.158\n", + "\n", + "Reaction: B(s)=B(l) 21.380\n", + "\n", + "Reaction: 3Fe+C=CFe3 17.060\n", + "\n", + "Reaction: S+Mn=MnS 13.909\n", + "\n", + "Reaction: NiO+Fe2O3=Fe2NiO4\n", + "0.464\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/applications_to_electrochemical_systems/application_examples.ipynb b/src/psu410/src/applications_to_electrochemical_systems/application_examples.ipynb index 34a46a0..0b5c281 100644 --- a/src/psu410/src/applications_to_electrochemical_systems/application_examples.ipynb +++ b/src/psu410/src/applications_to_electrochemical_systems/application_examples.ipynb @@ -4,39 +4,389 @@ "metadata": {}, "cells": [ { - "id": "9d0e901f", + "id": "bb41f564", "cell_type": "markdown", - "source": "## Application examples\n\nAmong many applications of electrochemistry, several of them are briefly\ndiscussed in this section.\n", + "source": [ + "## Application examples\n", + "\n", + "Among many applications of electrochemistry, several of them are briefly\n", + "discussed in this section.\n" + ], "metadata": {} }, { - "id": "16ed3744", + "id": "bc3b8e1c", "cell_type": "markdown", - "source": "### Metastability and passivation\n\nOur modern industrial society is built on various metals such as Fe, Ni,\nAl, Ti, and Zr alloys which are reactive, but exhibit extraordinary\nkinetic stabilities in oxidizing environments due to the existence of a\nthin reaction product film on the surface. This film effectively\nisolates the metal from the corrosive environment, a phenomenon called\npassivation. One interesting experiment by Faraday in 1836 who reported\nthat iron corrodes freely in dilute nitric acid, while in concentrated\nnitric acid, no reaction apparently occurred. To understand this\nphenomenon, let us exam a simple, schematic Pourbaix diagram for the\niron-water system shown in .\n\nFigure \u2011: Schematic Pourbaix diagram for iron illustrating the\nresolution of the Faraday paradox in the corrosion of iron in nitric\nacid \\[61\\]. Lines (a) (b), and (c) correspond to the equilibria: (a)\n$H^{+} + e^{-} = 1/2H_{2}$; (b) $O_{2} + 4H^{+} + 4e^{-} = 2H_{2}O$; (c)\n$NO_{3}^{-} + 3H^{+} + 2e^{-} = HNO_{2} + H_{2}O$, respectively\n\nFor iron in deaerated acid solution, the partial anodic and cathodic\nreactions are Line 1 $\\left( Fe/{Fe}^{2 +} \\right)$ and Line (a),\nrespectively, resulting in a corrosion potential that lies between Lines\n$1$ and $a$. In oxygenated (aerated) solutions, the corrosion potential\nmay lie between Lines $1$ and $b$, because the reduction of oxygen is a\npossible (likely) cathodic reaction. Since dilute HNO3 is\nonly a weak oxidizing agent, the principal cathodic reaction was most\nlikely hydrogen evolution, and hence the corrosion potential is expected\nto lie between Lines $1$ and $a$ at relatively high pH, as shown. Since\nthe $Fe/{Fe}^{2 +}$ reaction is relatively fast compared with\n$H^{+}/H_{2}$ on iron, if the corrosion potential is situated below the\nextension of Line 2 (Fe/Fe3O4) into the\n${Fe}^{2 +}$ stability region, Fe3O4 cannot form\non the surface, even as a metastable phase.\n\nOn the other hand, concentrated HNO3 is a strong oxidizing\nagent due to the reaction:\n$NO_{3}^{-} + 3H^{+} + 2e^{-} = HNO_{2} + H_{2}O$, so that the corrosion\npotential can lie anywhere between Lines $1$ and $c$ at low pH. Since\nReaction (2) is likely to be fast, the corrosion potential will be high\nand certainly will be more positive than the extension of Line 2 into\nthe stability region for ${Fe}^{2 +}$ at low pH. Therefore,\nFe3O4 becomes metastable and can form between the\naqueous solution and iron. The thickness of this\nFe3O4 film depends on its dissolution rate into\nthe aqueous solution and its growth rate at the interface with iron,\nwhich depends on the diffusion of ionic species across the film. Its\nexistence results in passivity and the observed kinetic inactivity of\niron in this medium. When the potential becomes even more positive above\nthe extension of Line $3$, Fe2O3 may form on\nFe3O4 as an additional metastable phase, resulting\nin the commonly observed bilayer structure.\n", + "source": [ + "### Metastability and passivation\n", + "\n", + "Our modern industrial society is built on various metals such as Fe, Ni,\n", + "Al, Ti, and Zr alloys which are reactive, but exhibit extraordinary\n", + "kinetic stabilities in oxidizing environments due to the existence of a\n", + "thin reaction product film on the surface. This film effectively\n", + "isolates the metal from the corrosive environment, a phenomenon called\n", + "passivation. One interesting experiment by Faraday in 1836 who reported\n", + "that iron corrodes freely in dilute nitric acid, while in concentrated\n", + "nitric acid, no reaction apparently occurred. To understand this\n", + "phenomenon, let us exam a simple, schematic Pourbaix diagram for the\n", + "iron-water system shown in .\n", + "\n", + "Figure \u2011: Schematic Pourbaix diagram for iron illustrating the\n", + "resolution of the Faraday paradox in the corrosion of iron in nitric\n", + "acid \\[61\\]. Lines (a) (b), and (c) correspond to the equilibria: (a)\n", + "$H^{+} + e^{-} = 1/2H_{2}$; (b) $O_{2} + 4H^{+} + 4e^{-} = 2H_{2}O$; (c)\n", + "$NO_{3}^{-} + 3H^{+} + 2e^{-} = HNO_{2} + H_{2}O$, respectively\n", + "\n", + "For iron in deaerated acid solution, the partial anodic and cathodic\n", + "reactions are Line 1 $\\left( Fe/{Fe}^{2 +} \\right)$ and Line (a),\n", + "respectively, resulting in a corrosion potential that lies between Lines\n", + "$1$ and $a$. In oxygenated (aerated) solutions, the corrosion potential\n", + "may lie between Lines $1$ and $b$, because the reduction of oxygen is a\n", + "possible (likely) cathodic reaction. Since dilute HNO3 is\n", + "only a weak oxidizing agent, the principal cathodic reaction was most\n", + "likely hydrogen evolution, and hence the corrosion potential is expected\n", + "to lie between Lines $1$ and $a$ at relatively high pH, as shown. Since\n", + "the $Fe/{Fe}^{2 +}$ reaction is relatively fast compared with\n", + "$H^{+}/H_{2}$ on iron, if the corrosion potential is situated below the\n", + "extension of Line 2 (Fe/Fe3O4) into the\n", + "${Fe}^{2 +}$ stability region, Fe3O4 cannot form\n", + "on the surface, even as a metastable phase.\n", + "\n", + "On the other hand, concentrated HNO3 is a strong oxidizing\n", + "agent due to the reaction:\n", + "$NO_{3}^{-} + 3H^{+} + 2e^{-} = HNO_{2} + H_{2}O$, so that the corrosion\n", + "potential can lie anywhere between Lines $1$ and $c$ at low pH. Since\n", + "Reaction (2) is likely to be fast, the corrosion potential will be high\n", + "and certainly will be more positive than the extension of Line 2 into\n", + "the stability region for ${Fe}^{2 +}$ at low pH. Therefore,\n", + "Fe3O4 becomes metastable and can form between the\n", + "aqueous solution and iron. The thickness of this\n", + "Fe3O4 film depends on its dissolution rate into\n", + "the aqueous solution and its growth rate at the interface with iron,\n", + "which depends on the diffusion of ionic species across the film. Its\n", + "existence results in passivity and the observed kinetic inactivity of\n", + "iron in this medium. When the potential becomes even more positive above\n", + "the extension of Line $3$, Fe2O3 may form on\n", + "Fe3O4 as an additional metastable phase, resulting\n", + "in the commonly observed bilayer structure.\n" + ], "metadata": {} }, { - "id": "f6d149c7", + "id": "c4f37fc5", "cell_type": "markdown", - "source": "### Galvanic protection\n\nA galvanic reaction takes place between two different materials at the\ntwo respective electrodes, each with different tendency to hold on to\nelectrons. Consider the following electrochemical cell used to protect\nCu tanks against oxidation by using a \u201csacrificial\u201d Fe electrode\n\nanode solution cathode\n\nFe(s) \\| Fe2+ \\| SO4= \\|\nCu2+\\| Cu(s) \\|\n\nCathode reduction: Cu2+ + 2 e- = Cu \u03b5o\n(volts) = 0.34\n\nAnode oxidation: Fe(s) = Fe2+ + 2 e- \u03b5o (volts) =\n0.44\n\nNet reaction: Cu2+ + Fe =\\> Cu + Fe2+\n\u03b5o (volts) = 0.78\n\nIf the cell has a direct connection between the electrodes, i.e. it has\na short circuit: \u2206G \u2192 0 and thus \u03b5cell \u2192 0. Since the value\nof \u03b5ocell \\> 0 for the net cell reaction, the equilibrium\nconstant *Ke* \\> 1, which means\n\\[Fe2+\\]/\\[Cu2+\\] \\> 1. By assuming an ideal\nelectrolyte solution, the activities in *Ke* can be\nrepresented by concentrations (in molar concentration units), assuming\nthat solid Fe and Cu are present at unit activities. If the electrodes\nof Cu and Fe are short circuited while in contact with the same\n\"electrolyte solution\", the final equilibrium concentrations can be\ncalculated by the standard equation,\n\n*Eq. 8\u201161* *Ke* = \\[Fe2+\\]/\\[Cu2+\\] =\nexp (-\u2206oG/RT)\n\nor, using the Nernst equations,\n\n*Eq. 8\u201162* \u03b5 = \u03b5o -(RT/z f) ln (\\[Fe2+\\]/\\[Cu2+\\])\n= 0\n\nand the above standard cell potential:\n\\[Fe2+\\]/\\[Cu2+\\] = 2.4\u20221026\n\nWith this large ratio it can be seen that the tendency to produce\nCu+2 ions, i.e., the tendency to corrode the Cu(s), is\nextremely small if a sacrificial Fe electrode is configured in an\nelectrochemical cell with the Cu tank.\n", + "source": [ + "### Galvanic protection\n", + "\n", + "A galvanic reaction takes place between two different materials at the\n", + "two respective electrodes, each with different tendency to hold on to\n", + "electrons. Consider the following electrochemical cell used to protect\n", + "Cu tanks against oxidation by using a \u201csacrificial\u201d Fe electrode\n", + "\n", + "anode solution cathode\n", + "\n", + "Fe(s) \\| Fe2+ \\| SO4= \\|\n", + "Cu2+\\| Cu(s) \\|\n", + "\n", + "Cathode reduction: Cu2+ + 2 e- = Cu \u03b5o\n", + "(volts) = 0.34\n", + "\n", + "Anode oxidation: Fe(s) = Fe2+ + 2 e- \u03b5o (volts) =\n", + "0.44\n", + "\n", + "Net reaction: Cu2+ + Fe =\\> Cu + Fe2+\n", + "\u03b5o (volts) = 0.78\n", + "\n", + "If the cell has a direct connection between the electrodes, i.e. it has\n", + "a short circuit: \u2206G \u2192 0 and thus \u03b5cell \u2192 0. Since the value\n", + "of \u03b5ocell \\> 0 for the net cell reaction, the equilibrium\n", + "constant *Ke* \\> 1, which means\n", + "\\[Fe2+\\]/\\[Cu2+\\] \\> 1. By assuming an ideal\n", + "electrolyte solution, the activities in *Ke* can be\n", + "represented by concentrations (in molar concentration units), assuming\n", + "that solid Fe and Cu are present at unit activities. If the electrodes\n", + "of Cu and Fe are short circuited while in contact with the same\n", + "\"electrolyte solution\", the final equilibrium concentrations can be\n", + "calculated by the standard equation,\n", + "\n", + "*Eq. 8\u201161* *Ke* = \\[Fe2+\\]/\\[Cu2+\\] =\n", + "exp (-\u2206oG/RT)\n", + "\n", + "or, using the Nernst equations,\n", + "\n", + "*Eq. 8\u201162* \u03b5 = \u03b5o -(RT/z f) ln (\\[Fe2+\\]/\\[Cu2+\\])\n", + "= 0\n", + "\n", + "and the above standard cell potential:\n", + "\\[Fe2+\\]/\\[Cu2+\\] = 2.4\u20221026\n", + "\n", + "With this large ratio it can be seen that the tendency to produce\n", + "Cu+2 ions, i.e., the tendency to corrode the Cu(s), is\n", + "extremely small if a sacrificial Fe electrode is configured in an\n", + "electrochemical cell with the Cu tank.\n" + ], "metadata": {} }, { - "id": "ac54ddda", + "id": "80535ac5", "cell_type": "markdown", - "source": "### Fuel cells\n\nFuel cells are devices to convert chemical energy to electricity and\nheat through electrochemical reactions with the fuel and oxygen supplied\nto the anode and cathode, respectively. Typical ions migrating through\nthe electrolyte are $H^{+}$, ${OH}^{-}$, ${CO}_{3}^{2 -}$, and\n$O^{2 -}$. In fuel cells with $H^{+}$ as migrating ions, H2\nmolecules are dissociated into $H^{+}$ on the anode, which are combined\nwith O2 on the cathode to form H2O and release\nheat with the half cell and the net cell reactions in their simplest\nform as shown by for the anode, for the cathode, and for the net cell,\nrespectively. Commonly used electrolytes are polymer and phosphoric\nacid, and both anode and cathode reactions are facilitated by catalyst,\ntypically platinum. The thermodynamic limit of power which can be\ngenerated by the fuel cell is represented by\n\n*Eq. 8\u201163*\n$w = - \\mathrm{\\Delta}G = - \\mathrm{\\Delta}_{\\ }^{0}G_{cell} + RTln\\left( P_{H_{2}}P_{O_{2}}^{1/2} \\right)$\n\nFor fuel cells with anions as migrating ions, the anions are generated\non the cathode with H2O formed and heat generated on the\nanode. Their representative cathode reactions are\n\n*Eq. 8\u201164* $\\frac{1}{2}O_{2} + H_{2}O + 2e^{-}{= 2OH}^{-}$\n\n*Eq. 8\u201165* ${\\frac{1}{2}O}_{2} + CO_{2} + 2e^{-} = {CO}_{3}^{2 -}$\n\n*Eq. 8\u201166* $\\frac{1}{2}O_{2} + 2e^{-} = O^{2 -}$.\n\nThe anode reaction for is the reaction represented by , operating at low\ntemperatures and using catalyst for both electrodes. The anode reactions\nfor and are\n\n*Eq. 8\u201167* ${CO}_{3}^{2 -} + H_{2} = H_{2}O + CO_{2} + 2e^{-}$\n\n*Eq. 8\u201168* $O^{2 -} + H_{2} = H_{2}O + 2e^{-}$\n\nrespectively. To enable the diffusion of ${CO}_{3}^{2 -}$ and $O^{2 -}$\nthrough the cathode and the electrolyte, both fuel cells are operated at\nrelative high temperatures, with the former typically in molten\ncarbonate solutions and the latter through solid oxides. Due to the high\noperating temperatures, fuels are converted to hydrogen within the fuel\ncell itself by a process called internal reforming, removing the need\nfor precious-metal catalyst and enabling the use of a variety of fuels.\nThe net cell reaction for all three fuel cells is the same as in the\ncase of $H^{+}$, represented by .\n", + "source": [ + "### Fuel cells\n", + "\n", + "Fuel cells are devices to convert chemical energy to electricity and\n", + "heat through electrochemical reactions with the fuel and oxygen supplied\n", + "to the anode and cathode, respectively. Typical ions migrating through\n", + "the electrolyte are $H^{+}$, ${OH}^{-}$, ${CO}_{3}^{2 -}$, and\n", + "$O^{2 -}$. In fuel cells with $H^{+}$ as migrating ions, H2\n", + "molecules are dissociated into $H^{+}$ on the anode, which are combined\n", + "with O2 on the cathode to form H2O and release\n", + "heat with the half cell and the net cell reactions in their simplest\n", + "form as shown by for the anode, for the cathode, and for the net cell,\n", + "respectively. Commonly used electrolytes are polymer and phosphoric\n", + "acid, and both anode and cathode reactions are facilitated by catalyst,\n", + "typically platinum. The thermodynamic limit of power which can be\n", + "generated by the fuel cell is represented by\n", + "\n", + "*Eq. 8\u201163*\n", + "$w = - \\mathrm{\\Delta}G = - \\mathrm{\\Delta}_{\\ }^{0}G_{cell} + RTln\\left( P_{H_{2}}P_{O_{2}}^{1/2} \\right)$\n", + "\n", + "For fuel cells with anions as migrating ions, the anions are generated\n", + "on the cathode with H2O formed and heat generated on the\n", + "anode. Their representative cathode reactions are\n", + "\n", + "*Eq. 8\u201164* $\\frac{1}{2}O_{2} + H_{2}O + 2e^{-}{= 2OH}^{-}$\n", + "\n", + "*Eq. 8\u201165* ${\\frac{1}{2}O}_{2} + CO_{2} + 2e^{-} = {CO}_{3}^{2 -}$\n", + "\n", + "*Eq. 8\u201166* $\\frac{1}{2}O_{2} + 2e^{-} = O^{2 -}$.\n", + "\n", + "The anode reaction for is the reaction represented by , operating at low\n", + "temperatures and using catalyst for both electrodes. The anode reactions\n", + "for and are\n", + "\n", + "*Eq. 8\u201167* ${CO}_{3}^{2 -} + H_{2} = H_{2}O + CO_{2} + 2e^{-}$\n", + "\n", + "*Eq. 8\u201168* $O^{2 -} + H_{2} = H_{2}O + 2e^{-}$\n", + "\n", + "respectively. To enable the diffusion of ${CO}_{3}^{2 -}$ and $O^{2 -}$\n", + "through the cathode and the electrolyte, both fuel cells are operated at\n", + "relative high temperatures, with the former typically in molten\n", + "carbonate solutions and the latter through solid oxides. Due to the high\n", + "operating temperatures, fuels are converted to hydrogen within the fuel\n", + "cell itself by a process called internal reforming, removing the need\n", + "for precious-metal catalyst and enabling the use of a variety of fuels.\n", + "The net cell reaction for all three fuel cells is the same as in the\n", + "case of $H^{+}$, represented by .\n" + ], "metadata": {} }, { - "id": "01a4c089", + "id": "8fcd81c6", "cell_type": "markdown", - "source": "### Ion transport membranes\n\nIon transport membranes (ITMs) are ceramic membranes conducting both\nelectrons and oxygen ions, but no other species. The chemical potential\ndifference of oxygen between two sides of a membrane provides the\ndriving force for oxygen to diffuse through the membrane. Commonly used\nITM oxides include perovskite and fluorite, with the chemical formula of\nABO3 and AO2, respectively, typically with more\nthan one elements in the A-site and / or the B-site to tailor the\nelectron and ionic conductivities. Key thermodynamic properties of ITM\noxides are their stability in service environments, vacancy\nconcentrations in the oxygen and cation sites, valances of cations. On\nthe high oxygen partial pressure side, the reaction is as the following\n\n*Eq. 8\u201169* $\\frac{1}{2}O_{2} + 2e^{-} = O^{2 -}$\n\nAt the same time the number of oxygen vacancy is reduced, resulting in a\nlower concentration of oxygen vacancy and higher oxygen activity in the\noxide on the high oxygen partial pressure side. On the low oxygen\npartial pressure side, the reaction is reversed to produce oxygen\nmolecules, i.e.\n\n*Eq. 8\u201170* $O^{2 -} = \\frac{1}{2}O_{2} + 2e^{-}$\n\nThis reaction results in higher oxygen vacancy concentration and lower\noxygen activity. At both sides, the charge neutrality is compensated by\nthe valance changes of cations, resulting in the electron flow in the\nopposite direction of oxygen diffusion. The ionic conductivity is\ndictated by the oxygen transportation across the membrane with the\ndriving force of the following net reaction\n\n*Eq. 8\u201171*\n$\\frac{1}{2}O_{2}\\left( P_{high} \\right) = \\frac{1}{2}O_{2}\\left( P_{low} \\right)$\n\n*Eq. 8\u201172*\n$\\mathrm{\\Delta}G = 0.5RTln\\left( \\frac{P_{low}}{P_{high}} \\right)$.\n\nThe oxygen transportation is closely related to the concentration of\noxygen vacancy in the membrane, which is obtained by minimizing the\nGibbs energy of the phase under given temperature and oxygen partial\npressure conditions. High vacancy concentrations can be obtained by\ncation dopants with lower valances or small energy differences between\nvarious valance states. However, at the same time, high vacancy\nconcentrations reduce the thermodynamic stability of the membrane, which\nmay result in its decomposition into more stable phases.\n", + "source": [ + "### Ion transport membranes\n", + "\n", + "Ion transport membranes (ITMs) are ceramic membranes conducting both\n", + "electrons and oxygen ions, but no other species. The chemical potential\n", + "difference of oxygen between two sides of a membrane provides the\n", + "driving force for oxygen to diffuse through the membrane. Commonly used\n", + "ITM oxides include perovskite and fluorite, with the chemical formula of\n", + "ABO3 and AO2, respectively, typically with more\n", + "than one elements in the A-site and / or the B-site to tailor the\n", + "electron and ionic conductivities. Key thermodynamic properties of ITM\n", + "oxides are their stability in service environments, vacancy\n", + "concentrations in the oxygen and cation sites, valances of cations. On\n", + "the high oxygen partial pressure side, the reaction is as the following\n", + "\n", + "*Eq. 8\u201169* $\\frac{1}{2}O_{2} + 2e^{-} = O^{2 -}$\n", + "\n", + "At the same time the number of oxygen vacancy is reduced, resulting in a\n", + "lower concentration of oxygen vacancy and higher oxygen activity in the\n", + "oxide on the high oxygen partial pressure side. On the low oxygen\n", + "partial pressure side, the reaction is reversed to produce oxygen\n", + "molecules, i.e.\n", + "\n", + "*Eq. 8\u201170* $O^{2 -} = \\frac{1}{2}O_{2} + 2e^{-}$\n", + "\n", + "This reaction results in higher oxygen vacancy concentration and lower\n", + "oxygen activity. At both sides, the charge neutrality is compensated by\n", + "the valance changes of cations, resulting in the electron flow in the\n", + "opposite direction of oxygen diffusion. The ionic conductivity is\n", + "dictated by the oxygen transportation across the membrane with the\n", + "driving force of the following net reaction\n", + "\n", + "*Eq. 8\u201171*\n", + "$\\frac{1}{2}O_{2}\\left( P_{high} \\right) = \\frac{1}{2}O_{2}\\left( P_{low} \\right)$\n", + "\n", + "*Eq. 8\u201172*\n", + "$\\mathrm{\\Delta}G = 0.5RTln\\left( \\frac{P_{low}}{P_{high}} \\right)$.\n", + "\n", + "The oxygen transportation is closely related to the concentration of\n", + "oxygen vacancy in the membrane, which is obtained by minimizing the\n", + "Gibbs energy of the phase under given temperature and oxygen partial\n", + "pressure conditions. High vacancy concentrations can be obtained by\n", + "cation dopants with lower valances or small energy differences between\n", + "various valance states. However, at the same time, high vacancy\n", + "concentrations reduce the thermodynamic stability of the membrane, which\n", + "may result in its decomposition into more stable phases.\n" + ], "metadata": {} }, { - "id": "9b9a3496", + "id": "b28ad146", "cell_type": "markdown", - "source": "### Electrical batteries\n\nBatteries utilize electrochemical reactions to generate electricity for\nvarious devices. The theoretic voltage of a battery can be calculated\nfrom and , i.e.\n\n*Eq. 8\u201173*\n$\\varepsilon = - \\frac{\\mathrm{\\Delta}G}{zf} = \\varepsilon^{0} - \\frac{RTlnQ}{zf}$\n\nwith $\\mathrm{\\Delta}G$ being the driving force of the net cell reaction\nand $Q$ being the reaction activity quotient. The actual voltage of a\nbattery is lower than the theoretical one due to kinetic limitations of\ncell reactions and resistance to ion diffusion through the electrolyte.\nBased on whether the cell reactions are reversible or not, batteries\ntypically categorized as either primary disposable or secondary\nrechargeable batteries. The net cell reactions in primary disposable\nbatteries are not easily reversible, and electrode materials may not\nreturn to their original forms by applying a higher external potential\nof opposite sign. Consequently, primary batteries cannot be reliably\nrecharged. On the other hand, the net cell reactions in secondary\nbatteries are easily reversible. Furthermore, two half-cells in\nbatteries may use different electrolytes with each half-cell enclosed in\na container and a separator permeable to conducting ions but not the\nbulk of the electrolytes.\n\nOne common primary battery is zinc-carbon battery with a zinc anode\ncylinder and a carbon cathode central rod. The electrolytes are ammonium\nor zinc chloride next to the zinc anode and a mixture of ammonium\nchloride and manganese dioxide next to the carbon cathode. The half cell\nand net reactions with ammonium chloride are as follows\n\n*Eq. 8\u201174* Zn + 2NH3 \u2192\nZn(NH3)22+ + 2 e-\n\n*Eq. 8\u201175* 2NH4Cl + 2MnO2 + 2 e- \u2192\n2NH3 + Mn2O3 +\nH2O+2Cl\u2212\n\n*Eq. 8\u201176* Zn + 2MnO2 + 2NH4Cl \u2192\nMn2O3 +\nZn(NH3)2Cl2 + H2O.\n\nThe electric potential of the reaction is, treating all compounds as\nstoichiometric compounds\n\nEq. 8\u201177\n$\\varepsilon = - \\frac{\\mathrm{\\Delta}G}{2f} = - \\frac{\\mathrm{\\Delta}^{0}G}{2f} = \\frac{1}{2f}\\left(_{\\ }^{0}G^{Zn} + 2_{\\ }^{0}G^{{MnO}_{2}} + 2_{\\ }^{0}G^{{NH}_{4}Cl} -_{\\ }^{0}G^{H_{2}O} -_{\\ }^{0}G^{Zn\\left( {NH}_{3} \\right)_{2}{Cl}_{2}} -_{\\ }^{0}G^{{{Mn}_{2}O}_{3}} \\right)$.\n\nThe Gibbs energy of $Zn\\left( {NH}_{3} \\right)_{2}{Cl}_{2}$ is not\navailable in current databases and has been recently estimated to be\n\u2212505,375 J/mole-formula \\[62\\]. The value of at 298.15K is thus obtained\nas 1.67 V, which is pretty close to the actual operating voltage of the\nbattery around 1.5 V.\n\nWhile with zinc chloride, the cell reactions and electric potential may\nbe written as\n\nEq. 8\u201178 Zn + ZnCl2 + 2OH\u2212 \u2192 2ZnOHCl + 2\ne-\n\n*Eq. 8\u201179* MnO2 + H2O + e- \u2192 MnOOH +\nOH-\n\n*Eq. 8\u201180* Zn + 2 MnO2 + ZnCl2 + 2 H2O\n\u2192 2 MnOOH + 2 ZnOHCl\n\n*Eq. 8\u201181*\n$\\varepsilon = \\frac{1}{2f}\\left(_{\\ }^{0}G^{Zn} + 2_{\\ }^{0}G^{{MnO}_{2}} +_{\\ }^{0}G^{Zn{Cl}_{2}} + 2_{\\ }^{0}G^{H_{2}O} - 2_{\\ }^{0}G^{MnOOH} - 2_{\\ }^{0}G^{ZnOHCl} \\right)$\n\nSecondary batteries can be recharged by applying an external electrical\npotential, which reverses the net cell reaction that occur during\ndischarging. The oldest form of rechargeable battery is the lead-acid\nbatteries used in automotive, and the latest development is the\nlithium-ion (Li-ion) batteries. A lead-acid battery typically uses Pb\nand PbO2 as the cathode and anode electrodes and a 35% sulfuric acid and\n65% water solution as the electrolyte. Its anode and cathode reactions\ncan be simplified as follows\n\nEq. 8\u201182 $Pb + SO_{4}^{2 -} = PbSO_{4} + 2e^{-}$\n\nEq. 8\u201183 $PbO_{2} + 4H^{+} + SO_{4}^{2 -} + 2e^{-} = PbSO_{4} + 2H_{2}O$\n\nThe net cell reaction is\n\nEq. 8\u201184 $Pb + PbO_{2} + 2H_{2}SO_{4}^{\\ } = 2PbSO_{4} + 2H_{2}O$.\n\nIts electric potential is represented by the following equation\n\n*Eq. 8\u201185*\n$\\varepsilon = - \\frac{1}{2f}\\left( 2_{\\ }^{0}G^{H_{2}O} + 2_{\\ }^{0}G^{{PbSO}_{4}} -_{\\ }^{0}G^{Pb} -_{\\ }^{0}G^{{PbO}_{2}} - 2_{\\ }^{0}G^{H_{2}SO_{4}} \\right)$\n\nwith the value at 298.15K being 2.651 V calculated from Thermo-Calc\n\\[60\\]. During discharging, the reaction goes to right, and $PbSO_{4}$\nis formed on both anode and cathode, while during charging, the reaction\ngoes to the left, and $Pb$ and $PbO_{2}$ are restored. In practical\napplications, other ionic species such as ${H_{3}O}^{+}$ and\n$HSO_{4}^{-}$ may form in the electrolyte, complicating the reactions\nand affecting its potential.\n\nIn lithium ion batteries, lithium ions migrate in electrolytes between\nelectrodes made of intercalated lithium compounds during charging and\ndischarging. LiCoO2 and LiFePO4 are two of the\nseveral cathode materials used in lithium ion batteries, and the anode\nis typically made of carbon or metallic Li. The anode and cathode\nreactions for LiCoO2 batteries can be written in simple forms as follows\n\nEq. 8\u201186 ${Li}_{x}C_{6} = x{Li}^{+} + xe^{-} + 6C$\n\nEq. 8\u201187 $x{Li}^{+} + xe^{-} + {Li}_{1 - x}CoO_{2} = LiCoO_{2}$\n\nwith the net reaction and electric potential being\n\nEq. 8\u201188 ${Li}_{x}C_{6} + {Li}_{1 - x}CoO_{2} = LiCoO_{2} + 6C$\n\n*Eq. 8\u201189*\n$\\varepsilon = - \\frac{1}{xf}\\left\\{ 6_{\\ }^{0}G^{C} +_{\\ }^{0}G^{{LiCoO}_{2}} - G^{{{Li}_{x}C}_{6}} - G^{{{Li}_{1 - x}CoO}_{2}} \\right\\} = - \\frac{1}{f}\\left\\{ \\left( \\mu_{Li}^{{Li}_{1 - x}CoO_{2}} - \\mu_{Li}^{{Li}_{x}C} \\right) - \\frac{1}{x}\\left( \\mu_{LiCoO_{2}}^{{Li}_{1 - x}CoO_{2}} -_{\\ }^{0}G^{{LiCoO}_{2}} \\right) \\right\\}$\n\nThe electric potential is a function of $x$. The value in the first\nparenthesis in the above equation denotes the chemical potential\ndifference of Li between two electrodes, and the value in the second\nparenthesis represents the chemical potential difference of\n${LiCoO}_{2}$ between the states in the solution phase of\n${Li}_{1 - x}CoO_{2}$ and by itself. Gibbs energies of ${{Li}_{x}C}_{6}$\nand ${{Li}_{1 - x}CoO}_{2}$ need to be obtained as a function $x$ in\norder to calculate the electric potential of the battery.\n\nLiFePO4 uses metallic lithium as the anode with following\nhalf-cell and net cell reactions\n\nEq. 8\u201190 $xLi = x{Li}^{+} + xe^{-}$\n\nEq. 8\u201191 $x{Li}^{+} + xe^{-} + {Li}_{1 - x}FePO_{4} = LiFePO_{4}$\n\nEq. 8\u201192 $xLi + {Li}_{1 - x}FePO_{4} = LiFePO_{4}$.\n\nIts electric potential is also a function of $x$, i.e.\n\n*Eq. 8\u201193*\n$\\varepsilon = - \\frac{1}{xf}\\left\\{_{\\ }^{0}G^{LiFePO_{4}} - x\\ ^{0}G^{Li} - G^{{Li}_{1 - x}FePO_{4}} \\right\\} = - \\frac{1}{f}\\left\\{ \\left( \\mu_{Li}^{{Li}_{1 - x}FePO_{4}} - \\ ^{0}\\mu_{Li} \\right) - \\frac{1}{x}\\left( \\mu_{LiFePO_{4}}^{{Li}_{1 - x}FePO_{4}} -_{\\ }^{0}G^{LiFePO_{4}} \\right) \\right\\}$\n\nThe value in the first parenthesis in the above equation denotes the\nchemical potential difference of Li between two electrodes, and the\nvalue in the second parenthesis represents the chemical potential\ndifference of $LiFePO_{4}$ between the states in the solution phase of\n${Li}_{1 - x}FePO_{4}$ and by itself. Consequently, Gibbs energy of\n${Li}_{1 - x}FePO_{4}$ needs to be obtained as a function $x$ in order\nto calculate the electric potential of the battery. It is known that\nthere are several miscibility gaps in the $FePO_{4}$ and $LiFePO_{4}$\npsuedo-binary system, in which the chemical potentials are constants, so\nis the electric potential, resulting in stable battery output.\n", + "source": [ + "### Electrical batteries\n", + "\n", + "Batteries utilize electrochemical reactions to generate electricity for\n", + "various devices. The theoretic voltage of a battery can be calculated\n", + "from and , i.e.\n", + "\n", + "*Eq. 8\u201173*\n", + "$\\varepsilon = - \\frac{\\mathrm{\\Delta}G}{zf} = \\varepsilon^{0} - \\frac{RTlnQ}{zf}$\n", + "\n", + "with $\\mathrm{\\Delta}G$ being the driving force of the net cell reaction\n", + "and $Q$ being the reaction activity quotient. The actual voltage of a\n", + "battery is lower than the theoretical one due to kinetic limitations of\n", + "cell reactions and resistance to ion diffusion through the electrolyte.\n", + "Based on whether the cell reactions are reversible or not, batteries\n", + "typically categorized as either primary disposable or secondary\n", + "rechargeable batteries. The net cell reactions in primary disposable\n", + "batteries are not easily reversible, and electrode materials may not\n", + "return to their original forms by applying a higher external potential\n", + "of opposite sign. Consequently, primary batteries cannot be reliably\n", + "recharged. On the other hand, the net cell reactions in secondary\n", + "batteries are easily reversible. Furthermore, two half-cells in\n", + "batteries may use different electrolytes with each half-cell enclosed in\n", + "a container and a separator permeable to conducting ions but not the\n", + "bulk of the electrolytes.\n", + "\n", + "One common primary battery is zinc-carbon battery with a zinc anode\n", + "cylinder and a carbon cathode central rod. The electrolytes are ammonium\n", + "or zinc chloride next to the zinc anode and a mixture of ammonium\n", + "chloride and manganese dioxide next to the carbon cathode. The half cell\n", + "and net reactions with ammonium chloride are as follows\n", + "\n", + "*Eq. 8\u201174* Zn + 2NH3 \u2192\n", + "Zn(NH3)22+ + 2 e-\n", + "\n", + "*Eq. 8\u201175* 2NH4Cl + 2MnO2 + 2 e- \u2192\n", + "2NH3 + Mn2O3 +\n", + "H2O+2Cl\u2212\n", + "\n", + "*Eq. 8\u201176* Zn + 2MnO2 + 2NH4Cl \u2192\n", + "Mn2O3 +\n", + "Zn(NH3)2Cl2 + H2O.\n", + "\n", + "The electric potential of the reaction is, treating all compounds as\n", + "stoichiometric compounds\n", + "\n", + "Eq. 8\u201177\n", + "$\\varepsilon = - \\frac{\\mathrm{\\Delta}G}{2f} = - \\frac{\\mathrm{\\Delta}^{0}G}{2f} = \\frac{1}{2f}\\left(_{\\ }^{0}G^{Zn} + 2_{\\ }^{0}G^{{MnO}_{2}} + 2_{\\ }^{0}G^{{NH}_{4}Cl} -_{\\ }^{0}G^{H_{2}O} -_{\\ }^{0}G^{Zn\\left( {NH}_{3} \\right)_{2}{Cl}_{2}} -_{\\ }^{0}G^{{{Mn}_{2}O}_{3}} \\right)$.\n", + "\n", + "The Gibbs energy of $Zn\\left( {NH}_{3} \\right)_{2}{Cl}_{2}$ is not\n", + "available in current databases and has been recently estimated to be\n", + "\u2212505,375 J/mole-formula \\[62\\]. The value of at 298.15K is thus obtained\n", + "as 1.67 V, which is pretty close to the actual operating voltage of the\n", + "battery around 1.5 V.\n", + "\n", + "While with zinc chloride, the cell reactions and electric potential may\n", + "be written as\n", + "\n", + "Eq. 8\u201178 Zn + ZnCl2 + 2OH\u2212 \u2192 2ZnOHCl + 2\n", + "e-\n", + "\n", + "*Eq. 8\u201179* MnO2 + H2O + e- \u2192 MnOOH +\n", + "OH-\n", + "\n", + "*Eq. 8\u201180* Zn + 2 MnO2 + ZnCl2 + 2 H2O\n", + "\u2192 2 MnOOH + 2 ZnOHCl\n", + "\n", + "*Eq. 8\u201181*\n", + "$\\varepsilon = \\frac{1}{2f}\\left(_{\\ }^{0}G^{Zn} + 2_{\\ }^{0}G^{{MnO}_{2}} +_{\\ }^{0}G^{Zn{Cl}_{2}} + 2_{\\ }^{0}G^{H_{2}O} - 2_{\\ }^{0}G^{MnOOH} - 2_{\\ }^{0}G^{ZnOHCl} \\right)$\n", + "\n", + "Secondary batteries can be recharged by applying an external electrical\n", + "potential, which reverses the net cell reaction that occur during\n", + "discharging. The oldest form of rechargeable battery is the lead-acid\n", + "batteries used in automotive, and the latest development is the\n", + "lithium-ion (Li-ion) batteries. A lead-acid battery typically uses Pb\n", + "and PbO2 as the cathode and anode electrodes and a 35% sulfuric acid and\n", + "65% water solution as the electrolyte. Its anode and cathode reactions\n", + "can be simplified as follows\n", + "\n", + "Eq. 8\u201182 $Pb + SO_{4}^{2 -} = PbSO_{4} + 2e^{-}$\n", + "\n", + "Eq. 8\u201183 $PbO_{2} + 4H^{+} + SO_{4}^{2 -} + 2e^{-} = PbSO_{4} + 2H_{2}O$\n", + "\n", + "The net cell reaction is\n", + "\n", + "Eq. 8\u201184 $Pb + PbO_{2} + 2H_{2}SO_{4}^{\\ } = 2PbSO_{4} + 2H_{2}O$.\n", + "\n", + "Its electric potential is represented by the following equation\n", + "\n", + "*Eq. 8\u201185*\n", + "$\\varepsilon = - \\frac{1}{2f}\\left( 2_{\\ }^{0}G^{H_{2}O} + 2_{\\ }^{0}G^{{PbSO}_{4}} -_{\\ }^{0}G^{Pb} -_{\\ }^{0}G^{{PbO}_{2}} - 2_{\\ }^{0}G^{H_{2}SO_{4}} \\right)$\n", + "\n", + "with the value at 298.15K being 2.651 V calculated from Thermo-Calc\n", + "\\[60\\]. During discharging, the reaction goes to right, and $PbSO_{4}$\n", + "is formed on both anode and cathode, while during charging, the reaction\n", + "goes to the left, and $Pb$ and $PbO_{2}$ are restored. In practical\n", + "applications, other ionic species such as ${H_{3}O}^{+}$ and\n", + "$HSO_{4}^{-}$ may form in the electrolyte, complicating the reactions\n", + "and affecting its potential.\n", + "\n", + "In lithium ion batteries, lithium ions migrate in electrolytes between\n", + "electrodes made of intercalated lithium compounds during charging and\n", + "discharging. LiCoO2 and LiFePO4 are two of the\n", + "several cathode materials used in lithium ion batteries, and the anode\n", + "is typically made of carbon or metallic Li. The anode and cathode\n", + "reactions for LiCoO2 batteries can be written in simple forms as follows\n", + "\n", + "Eq. 8\u201186 ${Li}_{x}C_{6} = x{Li}^{+} + xe^{-} + 6C$\n", + "\n", + "Eq. 8\u201187 $x{Li}^{+} + xe^{-} + {Li}_{1 - x}CoO_{2} = LiCoO_{2}$\n", + "\n", + "with the net reaction and electric potential being\n", + "\n", + "Eq. 8\u201188 ${Li}_{x}C_{6} + {Li}_{1 - x}CoO_{2} = LiCoO_{2} + 6C$\n", + "\n", + "*Eq. 8\u201189*\n", + "$\\varepsilon = - \\frac{1}{xf}\\left\\{ 6_{\\ }^{0}G^{C} +_{\\ }^{0}G^{{LiCoO}_{2}} - G^{{{Li}_{x}C}_{6}} - G^{{{Li}_{1 - x}CoO}_{2}} \\right\\} = - \\frac{1}{f}\\left\\{ \\left( \\mu_{Li}^{{Li}_{1 - x}CoO_{2}} - \\mu_{Li}^{{Li}_{x}C} \\right) - \\frac{1}{x}\\left( \\mu_{LiCoO_{2}}^{{Li}_{1 - x}CoO_{2}} -_{\\ }^{0}G^{{LiCoO}_{2}} \\right) \\right\\}$\n", + "\n", + "The electric potential is a function of $x$. The value in the first\n", + "parenthesis in the above equation denotes the chemical potential\n", + "difference of Li between two electrodes, and the value in the second\n", + "parenthesis represents the chemical potential difference of\n", + "${LiCoO}_{2}$ between the states in the solution phase of\n", + "${Li}_{1 - x}CoO_{2}$ and by itself. Gibbs energies of ${{Li}_{x}C}_{6}$\n", + "and ${{Li}_{1 - x}CoO}_{2}$ need to be obtained as a function $x$ in\n", + "order to calculate the electric potential of the battery.\n", + "\n", + "LiFePO4 uses metallic lithium as the anode with following\n", + "half-cell and net cell reactions\n", + "\n", + "Eq. 8\u201190 $xLi = x{Li}^{+} + xe^{-}$\n", + "\n", + "Eq. 8\u201191 $x{Li}^{+} + xe^{-} + {Li}_{1 - x}FePO_{4} = LiFePO_{4}$\n", + "\n", + "Eq. 8\u201192 $xLi + {Li}_{1 - x}FePO_{4} = LiFePO_{4}$.\n", + "\n", + "Its electric potential is also a function of $x$, i.e.\n", + "\n", + "*Eq. 8\u201193*\n", + "$\\varepsilon = - \\frac{1}{xf}\\left\\{_{\\ }^{0}G^{LiFePO_{4}} - x\\ ^{0}G^{Li} - G^{{Li}_{1 - x}FePO_{4}} \\right\\} = - \\frac{1}{f}\\left\\{ \\left( \\mu_{Li}^{{Li}_{1 - x}FePO_{4}} - \\ ^{0}\\mu_{Li} \\right) - \\frac{1}{x}\\left( \\mu_{LiFePO_{4}}^{{Li}_{1 - x}FePO_{4}} -_{\\ }^{0}G^{LiFePO_{4}} \\right) \\right\\}$\n", + "\n", + "The value in the first parenthesis in the above equation denotes the\n", + "chemical potential difference of Li between two electrodes, and the\n", + "value in the second parenthesis represents the chemical potential\n", + "difference of $LiFePO_{4}$ between the states in the solution phase of\n", + "${Li}_{1 - x}FePO_{4}$ and by itself. Consequently, Gibbs energy of\n", + "${Li}_{1 - x}FePO_{4}$ needs to be obtained as a function $x$ in order\n", + "to calculate the electric potential of the battery. It is known that\n", + "there are several miscibility gaps in the $FePO_{4}$ and $LiFePO_{4}$\n", + "psuedo-binary system, in which the chemical potentials are constants, so\n", + "is the electric potential, resulting in stable battery output.\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/applications_to_electrochemical_systems/aqueous_solution_and_pourbaix_diagram.ipynb b/src/psu410/src/applications_to_electrochemical_systems/aqueous_solution_and_pourbaix_diagram.ipynb index 08a2c2e..711457e 100644 --- a/src/psu410/src/applications_to_electrochemical_systems/aqueous_solution_and_pourbaix_diagram.ipynb +++ b/src/psu410/src/applications_to_electrochemical_systems/aqueous_solution_and_pourbaix_diagram.ipynb @@ -4,9 +4,259 @@ "metadata": {}, "cells": [ { - "id": "ea5a58bd", + "id": "fc524093", "cell_type": "markdown", - "source": "## Aqueous solution and Pourbaix diagram\n\nThe importance of aqueous solutions in all aspects of life is so well\nknown and needs not be discussed further. Since many electrochemical\nprocesses involve electrolyte solutions in an aqueous solvent,\nelectrochemical processes including water, hydrogen, and/or oxygen are\ndiscussed in more details. The hydrogen-oxygen cell can be described for\nboth acidic electrolytes and alkaline electrolytes. With acidic\nelectrolytes, H+ is in much higher concentrations than\nOH-, and thus half-cell reactions with H+ as an\nionic transport species are more important than those involving\nOH-. The reverse is true for alkaline electrolytes that\ncontain high OH-concentrations. Other than for nearly neutral\nacid-base systems, either H+ or OH- dominates the\nother by several orders of magnitude as can be seen from the value of\nthe 298 K dissociation constant for H2O:\n\n*Eq. 8\u201135* H2O(l) = H+(aq) + OH-(aq)\n\nwith the reaction constant being Ke =\n\\[H+\\]\\[OH-\\] = 10-14 and\n$\\mathrm{\\Delta}_{\\ }^{0}G$= -RT *ln* Ke *\u03b2*= +79,908 J. By\nconvention, one defines pH = - log \\[H+\\] and pOH = - log\n\\[OH-\\], and then pH + pOH = 14.\n\nUnder acidic electrolyte conditions of low pH (high \\[H+\\]\nconcentrations) the anode reaction in a hydrogen-oxygen cell is:\n\n*Eq. 8\u201136* \u00bd H2(g) = H+(aq) + e-\n\nwith \u03b51o = 0.0 V and\n$\\mathrm{\\Delta}_{\\ }^{0}G_{1}$= 0 J. The corresponding cathode\n(reduction) reaction is:\n\n*Eq. 8\u201137* 2 H+(aq) + \u00bd O2(g) + 2 e- =\nH2O(l)\n\nwith \u03b52o = 1.229 V and\n$\\mathrm{\\Delta}_{\\ }^{0}G_{2}$ = -2\\*1.229\\*96,485 J = -237,160 J. The\nnet cell reaction for acidic electrolytes is:\n\n*Eq. 8\u201138* H2(g) + \u00bd O2(g) = H2O(l)\n\nwith \u03b5ocell = 1.229 V and\n$\\mathrm{\\Delta}_{\\ }^{0}G_{cell}$ = -2\\*1.229\\*96,485 J = -237,160 J\n\nUnder alkaline electrolyte conditions of high pH (high\n\\[OH-\\] concentrations) the anode reaction in a\nhydrogen-oxygen cell is:\n\n*Eq. 8\u201139* 2 OH-(aq) + H2(g) = 2\nH2O(l) + 2 e-\n\nwith \u03b53o = 0.828 V and\n$\\mathrm{\\Delta}_{\\ }^{0}G_{3}$= -2\\*0.828\\*96,485 J = -159,779 J. The\ncorresponding cathode (reduction) reaction is:\n\n*Eq. 8\u201140* H2O(l) + \u00bd O2(g) + 2 e- = 2\nOH-(aq)\n\nwith \u03b54o = 0.401 V and\n$\\mathrm{\\Delta}_{\\ }^{0}G_{4}$ = -2\\*0.401\\*96,485 J = -77,381 J. The\nnet cell reaction for alkaline electrolytes is:\n\n*Eq. 8\u201141* H2(g) + \u00bd O2(g) = H2O(l)\n\nwith \u03b5ocell = 1.229 V and\n$\\mathrm{\\Delta}_{\\ }^{0}G_{cell}$ = -2\\*1.229\\*96,485 J = -237,160 J.\n\nPlots of \u03b5 versus pH for a given chemical system have been typically\nused to exhibit the stability relationships of ionic species and solid\nphases in aqueous-based electrochemical systems. These graphs are often\ncalled Pourbaix diagrams after the inventor and are at constant\ntemperature and constant pressure diagrams for a constant concentration,\nusually for one metallic element. By convention, the \u03b5 in a Pourbaix\ndiagram corresponds to the potential for the cathode reduction reactions\nin the electrochemical half-cell with electrons as reactants. Pourbaix\ndiagrams can be extended to multi-component materials when thermodynamic\nproperties of the components are available in both the materials and the\naqueous solution.\n\nAn example of an \u03b5 versus pH diagram is shown in for the\nNi-H2O system at a 298K, 1 bar, and $c_{{Ni}^{2 +}} = 0.001$\nmolality. Three stability regions for Ni species are shown: Ni(s),\nNiO(s), and \\[Ni2+\\]. The two dashed lines on this diagram\ncorrespond to hydrogen reduction () and oxygen reduction () reactions,\nrespectively.\n\nFigure \u2011: An \u03b5 versus pH, Pourbaix diagram for the Ni-H2O at\n298K, 1 bar, and $c_{{Ni}^{2 +}} = 0.001$ molality.\n\nFor the \u03b5 and pH conditions within the boundaries of the Ni(s) region,\nno solid phase other than Ni(s) is stable, no ionic species with a\nconcentration of 1 molarity is stable, and no gas species with a\npressure of 1 bar is stable. Similar statements could be made about the\nNiO(s) and \\[Ni2+\\] areas on the diagram. In the\n\\[Ni2+\\] area, introduction of Ni(s) or NiO(s) into the\nsystem would result in the dissolution of these solid phases since they\nare not stable with respect to the \\[Ni2+\\] aqueous solution.\nThe corresponding chemical reactions proceed spontaneously to the right\nas follows until the solid phases are consumed:\n\n*Eq. 8\u201142* Ni(s) \u2192 Ni2+(1 molarity) + 2 e-\n\n*Eq. 8\u201143* NiO(s) + 2 H+(aq) \u2192 Ni2+(1 molarity) +\nH2O(l)\n\nNo H+(aq) in involved in the first reaction, , so the\nboundary line separating Ni(s) and Ni2+ is independent of pH.\nNo oxidation or reduction occurs in the second reaction, , i.e. no\nelectrons are reactants or products in the reaction, the boundary line\nseparating NiO(s) and Ni2+ is independent of \u03b5.\n\nNote the convention that the \u03b5 is the potential for a cathode reduction\nreaction, and boundary lines between two stability regions depict\nconditions under which partial equilibrium occurs for the two species\nfor the \u03b5 and pH values at any point on these lines. For the boundary\nline separating Ni(s) and Ni2+ in an ideal aqueous solution,\ni.e. the reverse of , the following equation is obtained.\n\n*Eq. 8\u201144* \u03b5 = \u03b5o = -0.268 V\n\nFor the NiO(s)-Ni2+ boundary line of an ideal solution, the\nreaction, , is a complete equilibrium, and thus the relationship is\n\n*Eq. 8\u201145*\n$0 = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{1}{\\left( c_{H^{+}} \\right)^{2}} = \\mathrm{\\Delta}_{\\ }^{0}G + 2 \\cdot 2.303 \\cdot RT \\cdot pH$\n\n*Eq. 8\u201146*\n$pH = - \\frac{\\mathrm{\\Delta}_{\\ }^{0}G}{2 \\cdot 2.303 \\cdot RT}$\n\nwhere $\\mathrm{\\Delta}_{\\ }^{0}G$ is obtained as follows and can be\ncalculated from the SSUB database and the standard potential of Ni,\n\n*Eq. 8\u201147*\n\nAt a specified temperature, only one standard free energy and only one\nequilibrium constant exists for this chemical reaction, and thus only\none specific value of $pH = 6.631$ exists for the reaction represented\nby in this Pourbaix diagram.\n\nThe diagonal line in represents the equilibrium between Ni(s) and NiO(s)\nand is for a partial equilibrium reaction that is the sum of reactions\nof and\n\n*Eq. 8\u201148* NiO(s) + 2 H+(aq) + 2 e- \u2550 Ni(s) +\nH2O(l)\n\nThe reduction of Ni from a divalent state in NiO to metallic Ni(s)\noccurs, but the reaction also depends on the H+\nconcentration, the pH. The corresponding Gibbs energy and Nernst\nequations are,\n\n*Eq. 8\u201149*\n$\\mathrm{\\Delta}G = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{1}{\\left( c_{H^{+}} \\right)^{2}} = - 23,939 + 2 \\cdot 2.303 \\cdot RT \\cdot pH$\n\n*Eq. 8\u201150*\n$\\varepsilon\\ = \\ \\varepsilon^{0}\\ - \\ \\frac{RT}{2f}\\ln\\frac{1}{\\left( c_{H^{+}} \\right)^{2}}\\ = 0.124\\ \u2013\\ \\frac{2.303 \\cdot RT}{f}pH$\n\nwhere $\\mathrm{\\Delta}_{\\ }^{0}G$ can be calculated as follows\n\n*Eq. 8\u201151*\n\nThe two additional lines in correspond to the reduction reactions\nrelated to H2 and O2 gases, i.e. the stability of\nH2O. The lower one is for the reverse of under \u03b5o\n= 0 and $P_{H_{2}} = 1$ with the Nernst equation being\n\n*Eq. 8\u201152*\n$\\varepsilon\\ = \\ \\varepsilon^{0}\\ - \\ \\frac{RT}{f}\\ln\\frac{\\left( P_{H_{2}} \\right)^{1/2}}{c_{H^{+}}}\\ = - \\frac{2.303 \\cdot RT}{f}pH$\n\nAs the pH increases from 0, \u03b5 becomes more negative as is depicted. The\ntop dashed line corresponds to the oxygen reduction reaction represented\nby under \u03b5o = 1.225 calculated from the aqueous solution\ndatabase in Thermo-Calc \\[60\\] and $P_{O_{2}} = 1$ with the Nernst\nequation being\n\nEq. 8\u201153\n$\\varepsilon\\ = \\ \\varepsilon^{0}\\ - \\ \\frac{RT}{2f}\\ln\\frac{\\left( P_{O_{2}} \\right)^{1/2}}{\\left( c_{H^{+}} \\right)^{2}}\\ = 1.225\\ \u2013\\ \\frac{2.303 \\cdot RT}{f}pH$\n\nThe dependence of \u03b5 on pH is identical for both reduction reaction and ,\nand their intercepts at $pH = 0$ differ by their difference in their\n\u03b5o values.\n\nIn this simple Pourbaix diagram of Ni in an ideal aqueous solution, all\nboundary lines are straight because there is only one ionic species of\nNi in the aqueous solution, i.e. Ni2+. When there are more\nthan one ionic species in the aqueous solution, the boundary lines may\nno longer be straight due to the competition between species. One\nexample is Cu with two main ionic species of Cu+2 and\nCuOH+, and the reduction reaction between the metallic Cu and\nthe aqueous solution involves both two species, i.e.\n\n*Eq. 8\u201154*\n${xCu}^{2 + \\ }\\ + \\ (1 - x){CuOH}^{+} + (1 - x)H^{+} + 2\\ e - \\ \\ = \\ \\ Cu(s) + {(1 - x)H}_{2}O$\n\nwith\n\n*Eq. 8\u201155*\n$\\mathrm{\\Delta}G = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{1}{\\left( c_{{Cu}^{2 +}} \\right)^{x}\\left( c_{{CuOH}^{+}}c_{H^{+}} \\right)^{1 - x}} = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{1}{\\left( c_{{Cu}^{2 +}} \\right)^{x}\\left( c_{{CuOH}^{+}} \\right)^{1 - x}} + 2.303(1 - x) \\cdot RT \\cdot pH$\n\n*Eq. 8\u201156*\n$\\varepsilon = \\varepsilon^{0} - \\frac{RT}{2f}\\ln\\frac{1}{\\left( c_{{Cu}^{2 +}} \\right)^{x}\\left( c_{{CuOH}^{+}} \\right)^{1 - x}} - \\frac{2.303(1 - x) \\cdot RT}{2f}pH$.\n\nIt is evident that both the slope and the intercept at $pH = 0$ are a\nfunction of the concentration of ${CuOH}^{+}$, which is a function of\n$pH$. Consequently, the boundary between the metallic Cu and the aqueous\nsolution is no longer a straight line as shown in .\n\nFigure \u2011: An \u03b5 versus pH, Pourbaix diagram for the Cu-H2O\nsystem at 298K, 1 bar, and $c_{Cu} = 0.001$ molality.\n\nThe concentrations of various species in the aqueous solution, i.e.\ncommonly called speciation, are plotted in , showing the change of\ndominant species as a function of pH value.\n\nFigure \u2011: Concentrations of ionic species in the aqueous solution at\n$\\varepsilon = 0.3\\ V$ from .\n\nIn Pourbaix diagrams for alloys with two or more elements, activities of\nindividual elements are to be used in calculating the potentials of\nreduction reactions. Considering a Fe-Ni alloy with Fe2+ and\nNi2+ in the aqueous solution, the reduction reactions for Fe\nand Ni can be written separately as\n\n*Eq. 8\u201157* Ni2+(cNi) + 2 e- \u2192 Ni\n(aNi in alloy)\n\n*Eq. 8\u201158* Fe2+(cFe) + 2 e- \u2192 Fe\n(aFe in alloy)\n\nwith their potentials as\n\n*Eq. 8\u201159*\n$\\varepsilon_{Ni}\\ = \\ \\varepsilon_{Ni}^{0} - \\frac{2.303RT}{2f}\\ln\\frac{a_{Ni}}{c_{Ni}} = \\ - 0.268 - \\frac{2.303RT}{2f}\\ln\\frac{a_{Ni}}{c_{Ni}}$\n\n*Eq. 8\u201160*\n$\\varepsilon_{Fe}\\ = \\ \\varepsilon_{Fe}^{0} - \\frac{2.303RT}{2f}\\ln\\frac{a_{Fe}}{c_{Fe}} = \\ - 0.441 - \\frac{2.303RT}{2f}\\ln\\frac{a_{Fe}}{c_{Fe}}$\n\nIn principle, there are two scenarios for a given set of $a_{Ni}$ and\n$a_{Fe}$ of the alloy. The first scenario is at the limit of a dilute\naqueous solution, i.e. $c_{Ni} = c_{Fe} = 0.001\\ $molarity,\n$\\varepsilon_{Ni}$ and $\\varepsilon_{Fe}$ can be calculated, and the\nelement with the lower potential has the tendency to dissolve first,\nwhich can result in the so-called dialloying effect. The second scenario\nis for the equal potential, i.e. $\\varepsilon_{Ni} = \\varepsilon_{Fe}$\ndue to the externally imposed potential, and the equilibrium\nconcentrations of Fe+2 and Ni+2 can be calculated\nfrom and .\n", + "source": [ + "## Aqueous solution and Pourbaix diagram\n", + "\n", + "The importance of aqueous solutions in all aspects of life is so well\n", + "known and needs not be discussed further. Since many electrochemical\n", + "processes involve electrolyte solutions in an aqueous solvent,\n", + "electrochemical processes including water, hydrogen, and/or oxygen are\n", + "discussed in more details. The hydrogen-oxygen cell can be described for\n", + "both acidic electrolytes and alkaline electrolytes. With acidic\n", + "electrolytes, H+ is in much higher concentrations than\n", + "OH-, and thus half-cell reactions with H+ as an\n", + "ionic transport species are more important than those involving\n", + "OH-. The reverse is true for alkaline electrolytes that\n", + "contain high OH-concentrations. Other than for nearly neutral\n", + "acid-base systems, either H+ or OH- dominates the\n", + "other by several orders of magnitude as can be seen from the value of\n", + "the 298 K dissociation constant for H2O:\n", + "\n", + "*Eq. 8\u201135* H2O(l) = H+(aq) + OH-(aq)\n", + "\n", + "with the reaction constant being Ke =\n", + "\\[H+\\]\\[OH-\\] = 10-14 and\n", + "$\\mathrm{\\Delta}_{\\ }^{0}G$= -RT *ln* Ke *\u03b2*= +79,908 J. By\n", + "convention, one defines pH = - log \\[H+\\] and pOH = - log\n", + "\\[OH-\\], and then pH + pOH = 14.\n", + "\n", + "Under acidic electrolyte conditions of low pH (high \\[H+\\]\n", + "concentrations) the anode reaction in a hydrogen-oxygen cell is:\n", + "\n", + "*Eq. 8\u201136* \u00bd H2(g) = H+(aq) + e-\n", + "\n", + "with \u03b51o = 0.0 V and\n", + "$\\mathrm{\\Delta}_{\\ }^{0}G_{1}$= 0 J. The corresponding cathode\n", + "(reduction) reaction is:\n", + "\n", + "*Eq. 8\u201137* 2 H+(aq) + \u00bd O2(g) + 2 e- =\n", + "H2O(l)\n", + "\n", + "with \u03b52o = 1.229 V and\n", + "$\\mathrm{\\Delta}_{\\ }^{0}G_{2}$ = -2\\*1.229\\*96,485 J = -237,160 J. The\n", + "net cell reaction for acidic electrolytes is:\n", + "\n", + "*Eq. 8\u201138* H2(g) + \u00bd O2(g) = H2O(l)\n", + "\n", + "with \u03b5ocell = 1.229 V and\n", + "$\\mathrm{\\Delta}_{\\ }^{0}G_{cell}$ = -2\\*1.229\\*96,485 J = -237,160 J\n", + "\n", + "Under alkaline electrolyte conditions of high pH (high\n", + "\\[OH-\\] concentrations) the anode reaction in a\n", + "hydrogen-oxygen cell is:\n", + "\n", + "*Eq. 8\u201139* 2 OH-(aq) + H2(g) = 2\n", + "H2O(l) + 2 e-\n", + "\n", + "with \u03b53o = 0.828 V and\n", + "$\\mathrm{\\Delta}_{\\ }^{0}G_{3}$= -2\\*0.828\\*96,485 J = -159,779 J. The\n", + "corresponding cathode (reduction) reaction is:\n", + "\n", + "*Eq. 8\u201140* H2O(l) + \u00bd O2(g) + 2 e- = 2\n", + "OH-(aq)\n", + "\n", + "with \u03b54o = 0.401 V and\n", + "$\\mathrm{\\Delta}_{\\ }^{0}G_{4}$ = -2\\*0.401\\*96,485 J = -77,381 J. The\n", + "net cell reaction for alkaline electrolytes is:\n", + "\n", + "*Eq. 8\u201141* H2(g) + \u00bd O2(g) = H2O(l)\n", + "\n", + "with \u03b5ocell = 1.229 V and\n", + "$\\mathrm{\\Delta}_{\\ }^{0}G_{cell}$ = -2\\*1.229\\*96,485 J = -237,160 J.\n", + "\n", + "Plots of \u03b5 versus pH for a given chemical system have been typically\n", + "used to exhibit the stability relationships of ionic species and solid\n", + "phases in aqueous-based electrochemical systems. These graphs are often\n", + "called Pourbaix diagrams after the inventor and are at constant\n", + "temperature and constant pressure diagrams for a constant concentration,\n", + "usually for one metallic element. By convention, the \u03b5 in a Pourbaix\n", + "diagram corresponds to the potential for the cathode reduction reactions\n", + "in the electrochemical half-cell with electrons as reactants. Pourbaix\n", + "diagrams can be extended to multi-component materials when thermodynamic\n", + "properties of the components are available in both the materials and the\n", + "aqueous solution.\n", + "\n", + "An example of an \u03b5 versus pH diagram is shown in for the\n", + "Ni-H2O system at a 298K, 1 bar, and $c_{{Ni}^{2 +}} = 0.001$\n", + "molality. Three stability regions for Ni species are shown: Ni(s),\n", + "NiO(s), and \\[Ni2+\\]. The two dashed lines on this diagram\n", + "correspond to hydrogen reduction () and oxygen reduction () reactions,\n", + "respectively.\n", + "\n", + "Figure \u2011: An \u03b5 versus pH, Pourbaix diagram for the Ni-H2O at\n", + "298K, 1 bar, and $c_{{Ni}^{2 +}} = 0.001$ molality.\n", + "\n", + "For the \u03b5 and pH conditions within the boundaries of the Ni(s) region,\n", + "no solid phase other than Ni(s) is stable, no ionic species with a\n", + "concentration of 1 molarity is stable, and no gas species with a\n", + "pressure of 1 bar is stable. Similar statements could be made about the\n", + "NiO(s) and \\[Ni2+\\] areas on the diagram. In the\n", + "\\[Ni2+\\] area, introduction of Ni(s) or NiO(s) into the\n", + "system would result in the dissolution of these solid phases since they\n", + "are not stable with respect to the \\[Ni2+\\] aqueous solution.\n", + "The corresponding chemical reactions proceed spontaneously to the right\n", + "as follows until the solid phases are consumed:\n", + "\n", + "*Eq. 8\u201142* Ni(s) \u2192 Ni2+(1 molarity) + 2 e-\n", + "\n", + "*Eq. 8\u201143* NiO(s) + 2 H+(aq) \u2192 Ni2+(1 molarity) +\n", + "H2O(l)\n", + "\n", + "No H+(aq) in involved in the first reaction, , so the\n", + "boundary line separating Ni(s) and Ni2+ is independent of pH.\n", + "No oxidation or reduction occurs in the second reaction, , i.e. no\n", + "electrons are reactants or products in the reaction, the boundary line\n", + "separating NiO(s) and Ni2+ is independent of \u03b5.\n", + "\n", + "Note the convention that the \u03b5 is the potential for a cathode reduction\n", + "reaction, and boundary lines between two stability regions depict\n", + "conditions under which partial equilibrium occurs for the two species\n", + "for the \u03b5 and pH values at any point on these lines. For the boundary\n", + "line separating Ni(s) and Ni2+ in an ideal aqueous solution,\n", + "i.e. the reverse of , the following equation is obtained.\n", + "\n", + "*Eq. 8\u201144* \u03b5 = \u03b5o = -0.268 V\n", + "\n", + "For the NiO(s)-Ni2+ boundary line of an ideal solution, the\n", + "reaction, , is a complete equilibrium, and thus the relationship is\n", + "\n", + "*Eq. 8\u201145*\n", + "$0 = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{1}{\\left( c_{H^{+}} \\right)^{2}} = \\mathrm{\\Delta}_{\\ }^{0}G + 2 \\cdot 2.303 \\cdot RT \\cdot pH$\n", + "\n", + "*Eq. 8\u201146*\n", + "$pH = - \\frac{\\mathrm{\\Delta}_{\\ }^{0}G}{2 \\cdot 2.303 \\cdot RT}$\n", + "\n", + "where $\\mathrm{\\Delta}_{\\ }^{0}G$ is obtained as follows and can be\n", + "calculated from the SSUB database and the standard potential of Ni,\n", + "\n", + "*Eq. 8\u201147*\n", + "\n", + "At a specified temperature, only one standard free energy and only one\n", + "equilibrium constant exists for this chemical reaction, and thus only\n", + "one specific value of $pH = 6.631$ exists for the reaction represented\n", + "by in this Pourbaix diagram.\n", + "\n", + "The diagonal line in represents the equilibrium between Ni(s) and NiO(s)\n", + "and is for a partial equilibrium reaction that is the sum of reactions\n", + "of and\n", + "\n", + "*Eq. 8\u201148* NiO(s) + 2 H+(aq) + 2 e- \u2550 Ni(s) +\n", + "H2O(l)\n", + "\n", + "The reduction of Ni from a divalent state in NiO to metallic Ni(s)\n", + "occurs, but the reaction also depends on the H+\n", + "concentration, the pH. The corresponding Gibbs energy and Nernst\n", + "equations are,\n", + "\n", + "*Eq. 8\u201149*\n", + "$\\mathrm{\\Delta}G = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{1}{\\left( c_{H^{+}} \\right)^{2}} = - 23,939 + 2 \\cdot 2.303 \\cdot RT \\cdot pH$\n", + "\n", + "*Eq. 8\u201150*\n", + "$\\varepsilon\\ = \\ \\varepsilon^{0}\\ - \\ \\frac{RT}{2f}\\ln\\frac{1}{\\left( c_{H^{+}} \\right)^{2}}\\ = 0.124\\ \u2013\\ \\frac{2.303 \\cdot RT}{f}pH$\n", + "\n", + "where $\\mathrm{\\Delta}_{\\ }^{0}G$ can be calculated as follows\n", + "\n", + "*Eq. 8\u201151*\n", + "\n", + "The two additional lines in correspond to the reduction reactions\n", + "related to H2 and O2 gases, i.e. the stability of\n", + "H2O. The lower one is for the reverse of under \u03b5o\n", + "= 0 and $P_{H_{2}} = 1$ with the Nernst equation being\n", + "\n", + "*Eq. 8\u201152*\n", + "$\\varepsilon\\ = \\ \\varepsilon^{0}\\ - \\ \\frac{RT}{f}\\ln\\frac{\\left( P_{H_{2}} \\right)^{1/2}}{c_{H^{+}}}\\ = - \\frac{2.303 \\cdot RT}{f}pH$\n", + "\n", + "As the pH increases from 0, \u03b5 becomes more negative as is depicted. The\n", + "top dashed line corresponds to the oxygen reduction reaction represented\n", + "by under \u03b5o = 1.225 calculated from the aqueous solution\n", + "database in Thermo-Calc \\[60\\] and $P_{O_{2}} = 1$ with the Nernst\n", + "equation being\n", + "\n", + "Eq. 8\u201153\n", + "$\\varepsilon\\ = \\ \\varepsilon^{0}\\ - \\ \\frac{RT}{2f}\\ln\\frac{\\left( P_{O_{2}} \\right)^{1/2}}{\\left( c_{H^{+}} \\right)^{2}}\\ = 1.225\\ \u2013\\ \\frac{2.303 \\cdot RT}{f}pH$\n", + "\n", + "The dependence of \u03b5 on pH is identical for both reduction reaction and ,\n", + "and their intercepts at $pH = 0$ differ by their difference in their\n", + "\u03b5o values.\n", + "\n", + "In this simple Pourbaix diagram of Ni in an ideal aqueous solution, all\n", + "boundary lines are straight because there is only one ionic species of\n", + "Ni in the aqueous solution, i.e. Ni2+. When there are more\n", + "than one ionic species in the aqueous solution, the boundary lines may\n", + "no longer be straight due to the competition between species. One\n", + "example is Cu with two main ionic species of Cu+2 and\n", + "CuOH+, and the reduction reaction between the metallic Cu and\n", + "the aqueous solution involves both two species, i.e.\n", + "\n", + "*Eq. 8\u201154*\n", + "${xCu}^{2 + \\ }\\ + \\ (1 - x){CuOH}^{+} + (1 - x)H^{+} + 2\\ e - \\ \\ = \\ \\ Cu(s) + {(1 - x)H}_{2}O$\n", + "\n", + "with\n", + "\n", + "*Eq. 8\u201155*\n", + "$\\mathrm{\\Delta}G = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{1}{\\left( c_{{Cu}^{2 +}} \\right)^{x}\\left( c_{{CuOH}^{+}}c_{H^{+}} \\right)^{1 - x}} = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{1}{\\left( c_{{Cu}^{2 +}} \\right)^{x}\\left( c_{{CuOH}^{+}} \\right)^{1 - x}} + 2.303(1 - x) \\cdot RT \\cdot pH$\n", + "\n", + "*Eq. 8\u201156*\n", + "$\\varepsilon = \\varepsilon^{0} - \\frac{RT}{2f}\\ln\\frac{1}{\\left( c_{{Cu}^{2 +}} \\right)^{x}\\left( c_{{CuOH}^{+}} \\right)^{1 - x}} - \\frac{2.303(1 - x) \\cdot RT}{2f}pH$.\n", + "\n", + "It is evident that both the slope and the intercept at $pH = 0$ are a\n", + "function of the concentration of ${CuOH}^{+}$, which is a function of\n", + "$pH$. Consequently, the boundary between the metallic Cu and the aqueous\n", + "solution is no longer a straight line as shown in .\n", + "\n", + "Figure \u2011: An \u03b5 versus pH, Pourbaix diagram for the Cu-H2O\n", + "system at 298K, 1 bar, and $c_{Cu} = 0.001$ molality.\n", + "\n", + "The concentrations of various species in the aqueous solution, i.e.\n", + "commonly called speciation, are plotted in , showing the change of\n", + "dominant species as a function of pH value.\n", + "\n", + "Figure \u2011: Concentrations of ionic species in the aqueous solution at\n", + "$\\varepsilon = 0.3\\ V$ from .\n", + "\n", + "In Pourbaix diagrams for alloys with two or more elements, activities of\n", + "individual elements are to be used in calculating the potentials of\n", + "reduction reactions. Considering a Fe-Ni alloy with Fe2+ and\n", + "Ni2+ in the aqueous solution, the reduction reactions for Fe\n", + "and Ni can be written separately as\n", + "\n", + "*Eq. 8\u201157* Ni2+(cNi) + 2 e- \u2192 Ni\n", + "(aNi in alloy)\n", + "\n", + "*Eq. 8\u201158* Fe2+(cFe) + 2 e- \u2192 Fe\n", + "(aFe in alloy)\n", + "\n", + "with their potentials as\n", + "\n", + "*Eq. 8\u201159*\n", + "$\\varepsilon_{Ni}\\ = \\ \\varepsilon_{Ni}^{0} - \\frac{2.303RT}{2f}\\ln\\frac{a_{Ni}}{c_{Ni}} = \\ - 0.268 - \\frac{2.303RT}{2f}\\ln\\frac{a_{Ni}}{c_{Ni}}$\n", + "\n", + "*Eq. 8\u201160*\n", + "$\\varepsilon_{Fe}\\ = \\ \\varepsilon_{Fe}^{0} - \\frac{2.303RT}{2f}\\ln\\frac{a_{Fe}}{c_{Fe}} = \\ - 0.441 - \\frac{2.303RT}{2f}\\ln\\frac{a_{Fe}}{c_{Fe}}$\n", + "\n", + "In principle, there are two scenarios for a given set of $a_{Ni}$ and\n", + "$a_{Fe}$ of the alloy. The first scenario is at the limit of a dilute\n", + "aqueous solution, i.e. $c_{Ni} = c_{Fe} = 0.001\\ $molarity,\n", + "$\\varepsilon_{Ni}$ and $\\varepsilon_{Fe}$ can be calculated, and the\n", + "element with the lower potential has the tendency to dissolve first,\n", + "which can result in the so-called dialloying effect. The second scenario\n", + "is for the equal potential, i.e. $\\varepsilon_{Ni} = \\varepsilon_{Fe}$\n", + "due to the externally imposed potential, and the equilibrium\n", + "concentrations of Fe+2 and Ni+2 can be calculated\n", + "from and .\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/applications_to_electrochemical_systems/concentrations_activities_and_reference_states_of_electrolyte_species.ipynb b/src/psu410/src/applications_to_electrochemical_systems/concentrations_activities_and_reference_states_of_electrolyte_species.ipynb index 40bcc85..fec8d3f 100644 --- a/src/psu410/src/applications_to_electrochemical_systems/concentrations_activities_and_reference_states_of_electrolyte_species.ipynb +++ b/src/psu410/src/applications_to_electrochemical_systems/concentrations_activities_and_reference_states_of_electrolyte_species.ipynb @@ -4,9 +4,75 @@ "metadata": {}, "cells": [ { - "id": "95503830", + "id": "0564f0db", "cell_type": "markdown", - "source": "## Concentrations, activities, and reference states of electrolyte species\n\nThermodynamic descriptions of ionic species in solutions are different\nfrom those of neutral species, which leads to a need for defining\nconcentration units, standard states, activities, and activity\ncoefficients of ionic solutions. In most studies of electrochemical\ncorrosion and electrodeposition, and in applied work of electrochemical\nengineers, ionic species concentrations are given in units of molarity,\nthe number of moles of a species in a liter of solution (mol/l)\nsymbolically represented in equations by either *ci* or\n\\[M+Z\\]. The other common concentration used for ionic\nspecies is molality, which is defined as the number of moles of a\nspecies in 1000g of solvent. For dilute aqueous solutions, molarity and\nmolality values are very similar.\n\nAs discussed in Chapter , a practical definition of the activity of a\nspecies *i* is the thermodynamic reactivity, or tendency to react, of\nspecies *i* in the system of interest as compared to *i* in its\nreference state form. The reference state of a species is typically\nchosen as a specific chemical/physical state of the species at 1 atm\nexternal pressure and the temperature of interest. Similarly, a typical\nreference state for ionic species in aqueous solutions is the 1 molar\nideal solution at 1 bar external pressure and the temperature of\ninterest. If an electrolyte solution behaves ideally, then the activity\nof species *i* in solution is\n\n*Eq. 8\u20116*\n$a_{i} = \\frac{c_{i}\\left( \\frac{mol}{l} \\right)}{c_{i}^{0}\\left( \\frac{mol}{l} \\right)} = \\frac{c_{i}\\left( \\frac{mol}{l} \\right)}{1\\left( \\frac{mol}{l} \\right)} = c_{i}(dimensionless)$\n\nwhere $c_{i}$ is the molar concentration of *i* in the solution divided\nby $c_{i}^{0}$, the 1 molar reference state ideal solution\nconcentration. Thus, in ideal solutions, the activity of an electrolyte\nspecies is numerically equal to its molar concentration. The above\ntreatment of ionic species is equivalent to the common practice of\ndepicting the activity of a gas by the value of its ideal gas partial\npressure in units of bar.\n\nThe activity coefficient corrects for the nonideality of the species in\nsolution as defined in . If the solution is ideal, $\\gamma_{i} = 1$ for\nall concentrations of a species in solution. For all solutions, one\nexpects $\\gamma_{i} \\rightarrow 1$ as $c_{i} \\rightarrow 1$. It is not\npossible to measure $\\gamma_{i^{+}}$ or $\\gamma_{i^{-}}$ for individual\ncharged ions, only a geometric mean of the positive and negative ion\nvalues. Consider the following ionic solution\n\nEq. 8\u20117\n\nIts chemical potential can be written as\n\n*Eq. 8\u20118*\n\nIts geometric average or mean activity and activity coefficient are\ndefined as\n\n*Eq. 8\u20119*\n\n*Eq. 8\u201110*\n\nFor example, one can define\n$\\gamma_{\\pm} = \\left( \\gamma_{{Na}^{+}}\\gamma_{{Cl}^{-}} \\right)^{1/2}$\nand\n$\\gamma_{\\pm} = \\left( \\gamma_{{Al}^{3 +}}^{2}\\gamma_{{{SO}_{4}}^{2 -}}^{3} \\right)^{1/5}$\nfor NaCl and Al2(SO4)3, respectively.\nFor idea, weak electrolytes, $\\gamma_{\\pm} = 1$, and for non-ideal,\nstrong electrolytes, $\\gamma_{\\pm} \\neq 1$.\n", + "source": [ + "## Concentrations, activities, and reference states of electrolyte species\n", + "\n", + "Thermodynamic descriptions of ionic species in solutions are different\n", + "from those of neutral species, which leads to a need for defining\n", + "concentration units, standard states, activities, and activity\n", + "coefficients of ionic solutions. In most studies of electrochemical\n", + "corrosion and electrodeposition, and in applied work of electrochemical\n", + "engineers, ionic species concentrations are given in units of molarity,\n", + "the number of moles of a species in a liter of solution (mol/l)\n", + "symbolically represented in equations by either *ci* or\n", + "\\[M+Z\\]. The other common concentration used for ionic\n", + "species is molality, which is defined as the number of moles of a\n", + "species in 1000g of solvent. For dilute aqueous solutions, molarity and\n", + "molality values are very similar.\n", + "\n", + "As discussed in Chapter , a practical definition of the activity of a\n", + "species *i* is the thermodynamic reactivity, or tendency to react, of\n", + "species *i* in the system of interest as compared to *i* in its\n", + "reference state form. The reference state of a species is typically\n", + "chosen as a specific chemical/physical state of the species at 1 atm\n", + "external pressure and the temperature of interest. Similarly, a typical\n", + "reference state for ionic species in aqueous solutions is the 1 molar\n", + "ideal solution at 1 bar external pressure and the temperature of\n", + "interest. If an electrolyte solution behaves ideally, then the activity\n", + "of species *i* in solution is\n", + "\n", + "*Eq. 8\u20116*\n", + "$a_{i} = \\frac{c_{i}\\left( \\frac{mol}{l} \\right)}{c_{i}^{0}\\left( \\frac{mol}{l} \\right)} = \\frac{c_{i}\\left( \\frac{mol}{l} \\right)}{1\\left( \\frac{mol}{l} \\right)} = c_{i}(dimensionless)$\n", + "\n", + "where $c_{i}$ is the molar concentration of *i* in the solution divided\n", + "by $c_{i}^{0}$, the 1 molar reference state ideal solution\n", + "concentration. Thus, in ideal solutions, the activity of an electrolyte\n", + "species is numerically equal to its molar concentration. The above\n", + "treatment of ionic species is equivalent to the common practice of\n", + "depicting the activity of a gas by the value of its ideal gas partial\n", + "pressure in units of bar.\n", + "\n", + "The activity coefficient corrects for the nonideality of the species in\n", + "solution as defined in . If the solution is ideal, $\\gamma_{i} = 1$ for\n", + "all concentrations of a species in solution. For all solutions, one\n", + "expects $\\gamma_{i} \\rightarrow 1$ as $c_{i} \\rightarrow 1$. It is not\n", + "possible to measure $\\gamma_{i^{+}}$ or $\\gamma_{i^{-}}$ for individual\n", + "charged ions, only a geometric mean of the positive and negative ion\n", + "values. Consider the following ionic solution\n", + "\n", + "Eq. 8\u20117\n", + "\n", + "Its chemical potential can be written as\n", + "\n", + "*Eq. 8\u20118*\n", + "\n", + "Its geometric average or mean activity and activity coefficient are\n", + "defined as\n", + "\n", + "*Eq. 8\u20119*\n", + "\n", + "*Eq. 8\u201110*\n", + "\n", + "For example, one can define\n", + "$\\gamma_{\\pm} = \\left( \\gamma_{{Na}^{+}}\\gamma_{{Cl}^{-}} \\right)^{1/2}$\n", + "and\n", + "$\\gamma_{\\pm} = \\left( \\gamma_{{Al}^{3 +}}^{2}\\gamma_{{{SO}_{4}}^{2 -}}^{3} \\right)^{1/5}$\n", + "for NaCl and Al2(SO4)3, respectively.\n", + "For idea, weak electrolytes, $\\gamma_{\\pm} = 1$, and for non-ideal,\n", + "strong electrolytes, $\\gamma_{\\pm} \\neq 1$.\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/applications_to_electrochemical_systems/electrochemical_cells_and_half_cell_potentials.ipynb b/src/psu410/src/applications_to_electrochemical_systems/electrochemical_cells_and_half_cell_potentials.ipynb index 1346408..e04a471 100644 --- a/src/psu410/src/applications_to_electrochemical_systems/electrochemical_cells_and_half_cell_potentials.ipynb +++ b/src/psu410/src/applications_to_electrochemical_systems/electrochemical_cells_and_half_cell_potentials.ipynb @@ -4,21 +4,302 @@ "metadata": {}, "cells": [ { - "id": "eab98886", + "id": "e3027b69", "cell_type": "markdown", - "source": "## Electrochemical cells and half cell potentials\n\nAn electrochemical system must fulfill certain requirements in order to\napply equilibrium thermodynamic descriptions of the system, including\n\n- The cell must be reversible when slight changes in conditions\n (potentials, concentrations, pressures, temperature) cause\n electrochemical reactions and an external flow of electrons to occur\n in the direction needed to re-establish equilibrium.\n\n- All non-electrochemical reactions in the system must be prevented as\n such reactions would cause a shift in equilibrium without causing a\n shift in cell potential and thus a driving force for external electron\n flow.\n\n- Chemical reactions must occur only when an external current flows.\n These finite distances for external electron transport can be a short\n as grain size-dimensions in many corrosion reactions, or this\n transport may be through an external electrical conductor connecting\n the anode and cathode half-cell as in batteries.\n\n- Charge balance as well as mass balance is required of all reactions.\n", + "source": [ + "## Electrochemical cells and half cell potentials\n", + "\n", + "An electrochemical system must fulfill certain requirements in order to\n", + "apply equilibrium thermodynamic descriptions of the system, including\n", + "\n", + "- The cell must be reversible when slight changes in conditions\n", + " (potentials, concentrations, pressures, temperature) cause\n", + " electrochemical reactions and an external flow of electrons to occur\n", + " in the direction needed to re-establish equilibrium.\n", + "\n", + "- All non-electrochemical reactions in the system must be prevented as\n", + " such reactions would cause a shift in equilibrium without causing a\n", + " shift in cell potential and thus a driving force for external electron\n", + " flow.\n", + "\n", + "- Chemical reactions must occur only when an external current flows.\n", + " These finite distances for external electron transport can be a short\n", + " as grain size-dimensions in many corrosion reactions, or this\n", + " transport may be through an external electrical conductor connecting\n", + " the anode and cathode half-cell as in batteries.\n", + "\n", + "- Charge balance as well as mass balance is required of all reactions.\n" + ], "metadata": {} }, { - "id": "984589b8", + "id": "bc5e76b6", "cell_type": "markdown", - "source": "### Electrochemical cells\n\nA potential difference, i.e. voltage difference, can be generated\nbetween the electrodes in a cell from differences in the potentials of\nthe half-cell reactions. This potential can originate from potential\ndifferences of two chemically different half-cells (a *galvanic cell*),\nor concentration differences in two otherwise identical half-cells (a\n*concentration cell*). Each type of cell is illustrated below.\n\nThe reaction between copper ions and zinc illustrated below represents\nthe *net cell reaction* of a *galvanic* cell in which the oxidation of\nZn(s) occurs at the *anode* electrode, and the reduction of\nCu+2 occurs at the *cathode* electrode\n\n*Eq. 8\u201111* Cu2+(aq) + Zn(s) = Cu(s) + Zn2+(aq).\n\nThe reaction at each electrode, the *half-cell reaction*, includes ions\nand electrons as reactant and/or product species. The anode, oxidation\nreaction is represented by\n\n*Eq. 8\u201112* Zn(s) = Zn2+(aq) + 2 e-.\n\nElectrons are products of anode reactions and flow externally from anode\nto cathode. By convention the activities of the electrons in an\nequilibrium cell are taken as unity. The cathode, reduction reaction is\nwritten as\n\n*Eq. 8\u201113* Cu2+(aq) + 2 e- = Cu(s)\n\nElectrons are reactants of cathode reactions and supplied by an external\nflow from the anode. In addition to consuming electrons at the cathode\nat the same rate as they are produced at the anode, charge balance is\nmaintained in the electrolyte by the generation of Zn2+ ions\nat the same rate that Cu2+ ions are consumed. A schematic\ndiagram in illustrates the simple physical relationships in such an\nelectrochemical cell.\n\nFigure \u2011: Schematic diagram of a galvanic electrochemical cell\nconsisting of a zinc electrode and a copper electrode.\n\nA concentration cell in which an electrochemical potential is developed\nbecause of concentration differences between otherwise equivalent anode\nand cathode reactions is illustrated below. Such a cell can be produced\nby the oxidation and reduction of copper at two separate electrodes as\nis depicted in the following reactions\n\n*Eq. 8\u201114* Cu(s) = Cu2+(aq, ca) + 2 e- (anode)\n\n*Eq. 8\u201115* Cu2+(aq, cb) + 2 e- = Cu(s) (cathode)\n\nwhere ca and cb are the respective concentrations\nof Cu2+ in the aqueous solutions at the anode and cathode,\nand ca \\< cb. The net cell reaction is\n\n*Eq. 8\u201116* Cu2+(aq, cb) = Cu2+(aq,\nca)\n\nwhere the reaction occurs spontaneously to decrease cb and to\nincrease ca until the two concentrations become the same,\ncb = ca. A schematic diagram of such a cell is\nshown in .\n\nFigure \u2011: Schematic diagram of a concentration electrochemical cell\nconsisting of two copper electrodes.\n\nA semi-impermeable membrane, or a salt bridge, must exist in such a cell\nto maintain charge balance. As Cu+2 ions are produced at the\nanode and consumed at the cathode, the negatively charged ions in the\nsolution, for example Cl-, must be transferred from the\ncathode region to the anode region of the cell to maintain electrically\nneutral solutions.\n\nThe above concentration cell provides a good example for illustrating a\nstandard notation for depicting an electrochemical cell. This cell can\nbe represented by .\n\nFigure \u2011: Standard notation of an electrochemical cell\n\nThe anode where oxidation occurs is always denoted on the left, and the\ncathode where reduction occurs is on the right. A *single line*\nseparating phases denotes an *interface* between two phases. The above\nanode electrode and reaction of are symbolically represented by\n\n*Eq. 8\u201117* \\| Cu(s) \\| Cu2+(ca).\n\nThe interface between the external conductor and Cu(s) is depicted by\nthe single line to the left of Cu(s), while the single line between the\nCu(s) and Cu2+(ca) depicts the interface between\nthe anode electrode and the electrolyte solution. Similarly, the cathode\nelectrode and reaction of are symbolically represented by\n\n*Eq. 8\u201118* Cu2+(cb) \\| Cu(s) \\|.\n\nA double line between the two copper ions in the notation denotes a\nphysical separation of two solution phases, the anode and cathode\nelectrolyte regions that exhibit different concentrations of copper ions\n\n*Eq. 8\u201119* Cu2+(ca) \\| \\|\nCu2+(cb).\n\nThese solution phases are physically connected by a semi-impermeable\nmembrane or salt bridge that allows a common negative ion, for example\nCl-, of the solution phases to diffuse from one region to the\nother in order to maintain charge balance as the cell reaction occurs.\nThe Cu2+ ions cannot be transported from one region to the\nother.\n", + "source": [ + "### Electrochemical cells\n", + "\n", + "A potential difference, i.e. voltage difference, can be generated\n", + "between the electrodes in a cell from differences in the potentials of\n", + "the half-cell reactions. This potential can originate from potential\n", + "differences of two chemically different half-cells (a *galvanic cell*),\n", + "or concentration differences in two otherwise identical half-cells (a\n", + "*concentration cell*). Each type of cell is illustrated below.\n", + "\n", + "The reaction between copper ions and zinc illustrated below represents\n", + "the *net cell reaction* of a *galvanic* cell in which the oxidation of\n", + "Zn(s) occurs at the *anode* electrode, and the reduction of\n", + "Cu+2 occurs at the *cathode* electrode\n", + "\n", + "*Eq. 8\u201111* Cu2+(aq) + Zn(s) = Cu(s) + Zn2+(aq).\n", + "\n", + "The reaction at each electrode, the *half-cell reaction*, includes ions\n", + "and electrons as reactant and/or product species. The anode, oxidation\n", + "reaction is represented by\n", + "\n", + "*Eq. 8\u201112* Zn(s) = Zn2+(aq) + 2 e-.\n", + "\n", + "Electrons are products of anode reactions and flow externally from anode\n", + "to cathode. By convention the activities of the electrons in an\n", + "equilibrium cell are taken as unity. The cathode, reduction reaction is\n", + "written as\n", + "\n", + "*Eq. 8\u201113* Cu2+(aq) + 2 e- = Cu(s)\n", + "\n", + "Electrons are reactants of cathode reactions and supplied by an external\n", + "flow from the anode. In addition to consuming electrons at the cathode\n", + "at the same rate as they are produced at the anode, charge balance is\n", + "maintained in the electrolyte by the generation of Zn2+ ions\n", + "at the same rate that Cu2+ ions are consumed. A schematic\n", + "diagram in illustrates the simple physical relationships in such an\n", + "electrochemical cell.\n", + "\n", + "Figure \u2011: Schematic diagram of a galvanic electrochemical cell\n", + "consisting of a zinc electrode and a copper electrode.\n", + "\n", + "A concentration cell in which an electrochemical potential is developed\n", + "because of concentration differences between otherwise equivalent anode\n", + "and cathode reactions is illustrated below. Such a cell can be produced\n", + "by the oxidation and reduction of copper at two separate electrodes as\n", + "is depicted in the following reactions\n", + "\n", + "*Eq. 8\u201114* Cu(s) = Cu2+(aq, ca) + 2 e- (anode)\n", + "\n", + "*Eq. 8\u201115* Cu2+(aq, cb) + 2 e- = Cu(s) (cathode)\n", + "\n", + "where ca and cb are the respective concentrations\n", + "of Cu2+ in the aqueous solutions at the anode and cathode,\n", + "and ca \\< cb. The net cell reaction is\n", + "\n", + "*Eq. 8\u201116* Cu2+(aq, cb) = Cu2+(aq,\n", + "ca)\n", + "\n", + "where the reaction occurs spontaneously to decrease cb and to\n", + "increase ca until the two concentrations become the same,\n", + "cb = ca. A schematic diagram of such a cell is\n", + "shown in .\n", + "\n", + "Figure \u2011: Schematic diagram of a concentration electrochemical cell\n", + "consisting of two copper electrodes.\n", + "\n", + "A semi-impermeable membrane, or a salt bridge, must exist in such a cell\n", + "to maintain charge balance. As Cu+2 ions are produced at the\n", + "anode and consumed at the cathode, the negatively charged ions in the\n", + "solution, for example Cl-, must be transferred from the\n", + "cathode region to the anode region of the cell to maintain electrically\n", + "neutral solutions.\n", + "\n", + "The above concentration cell provides a good example for illustrating a\n", + "standard notation for depicting an electrochemical cell. This cell can\n", + "be represented by .\n", + "\n", + "Figure \u2011: Standard notation of an electrochemical cell\n", + "\n", + "The anode where oxidation occurs is always denoted on the left, and the\n", + "cathode where reduction occurs is on the right. A *single line*\n", + "separating phases denotes an *interface* between two phases. The above\n", + "anode electrode and reaction of are symbolically represented by\n", + "\n", + "*Eq. 8\u201117* \\| Cu(s) \\| Cu2+(ca).\n", + "\n", + "The interface between the external conductor and Cu(s) is depicted by\n", + "the single line to the left of Cu(s), while the single line between the\n", + "Cu(s) and Cu2+(ca) depicts the interface between\n", + "the anode electrode and the electrolyte solution. Similarly, the cathode\n", + "electrode and reaction of are symbolically represented by\n", + "\n", + "*Eq. 8\u201118* Cu2+(cb) \\| Cu(s) \\|.\n", + "\n", + "A double line between the two copper ions in the notation denotes a\n", + "physical separation of two solution phases, the anode and cathode\n", + "electrolyte regions that exhibit different concentrations of copper ions\n", + "\n", + "*Eq. 8\u201119* Cu2+(ca) \\| \\|\n", + "Cu2+(cb).\n", + "\n", + "These solution phases are physically connected by a semi-impermeable\n", + "membrane or salt bridge that allows a common negative ion, for example\n", + "Cl-, of the solution phases to diffuse from one region to the\n", + "other in order to maintain charge balance as the cell reaction occurs.\n", + "The Cu2+ ions cannot be transported from one region to the\n", + "other.\n" + ], "metadata": {} }, { - "id": "c8c84ce0", + "id": "d4493883", "cell_type": "markdown", - "source": "### Half cell potentials\n\nWhen electron current flows between electrodes, reactions are occurring\nat the electrodes and concentration gradients causing polarization\ndevelop around the electrodes. These gradients result in extraneous\npotentials to occur at the electrodes. In such cases cell equilibrium is\nnot established and measured cell potentials are not those for true\npartial equilibrium. If a cell is short-circuited with the electrodes\nconnected by a conductor, current will flow until the external potential\nbecomes zero, i.e. \u03b5ext = 0, and equilibrium is established\nwith same conditions as non-electrochemical systems. If an external\npotential, \u03b5ext, is applied to the cell, chemical reactions\noccur until the cell potential balances to \u03b5ext, and no\ncurrent flows. This potential is called open-circuit voltage (OCV) in\nthe literature. It is important to realize that OCV includes all\nreactions that occur on the electrode surface when the electrode is in\ncontact with the electrolyte, such as passivation discussed in Chapter .\nPartial equilibrium in a cell is achieved when the cell potential is\nbalanced by an applied external potential. In such partial equilibrium\ncases, equilibrium thermodynamic analyses can be used even though the\ncell potential is not zero, i.e. \u03b5cell \u2260 0. This\ndifferentiates electrochemical systems from other equilibrium systems\ndiscussed previously.\n\nThe number of electrons involved in a net cell reaction is important in\nrelating cell potential and the Gibbs energy change for the cell\nreaction. As will be illustrated later in this section, this number\ndenotes the number of electrons involved in the half-cell reactions that\nwere added to yield the net cell reaction. The electrical work achieved\nby the transport of an electrical charge through a cell potential can be\nwritten as\n\n*Eq. 8\u201120* $w = z\\ f\\ \\varepsilon$\n\nwhere *z* represents the moles of electrons in cell reaction, *f* the\nFaraday constant equal to 96,485 J/V/mole-electron, and *\u03b5* the\npotential difference, often referred as electromotive force (emf) in the\nliterature. For a system at constant temperature, pressure, and\ncomposition, this work is the same as the Gibbs energy difference\nbetween the two electrodes, i.e.\n\n*Eq. 8\u201121* $- \\Delta G = w = zf\\varepsilon$\n\nwhere the negative sign is because the system does work to the\nsurrounding when the Gibbs energy of the system is decreased. When the\napplied external potential is larger than the cell potential, the\nsurrounding does work to the system, and a common example is the\ncharging of a battery. Thermodynamic relations discussed in previous\nchapters can thus be directly applied to electrochemical systems with\nsome common equations shown in .\n\nTable \u2011: Thermodynamic Equations for Electrochemical Cells\n\n*\u2206G = -z f \u03b5*\n\n*\u2206S = - (\u2202\u2206G/\u2202T)P = + z f (\u2202\u03b5/\u2202T)P*\n\n*\u2206H = \\[\u2202(\u2206G/T)/ \u2202(1/T)\\]P = - z f \\[\u2202(\u03b5/T)/\u2202(1/T)\\]P = z f \\[T(\u2202\u03b5/\u2202T)P\n\u2013 \u03b5\\]*\n\n*\u2206CP = (\u2202\u2206H/\u2202T)P = T z f (\u22022\u03b5/\u2202T2)P*\n\nA half-cell reaction potential cannot be measured directly, only its\npotential relative to another half-cell reaction. By convention, a\nstandard half-cell potential is measured relative to the standard\nhydrogen half-cell reduction reaction at 25oC (298K) and 1\nbar, which has a defined standard potential of zero volts,\n\n*Eq. 8\u201122* H+(aq, a=1) + e- = 1/2 H2(g,\n1 bar)\n\nwith \u03b5o (H+/H2,g) = 0.00 volts. The\nstandard half-cell reduction reactions of metals at 25oC are\nfor the general reaction\n\n*Eq. 8\u201123* Mz+(aq, a=1) + z e- = M(s)\n\nwith \u03b5o (Mz+/M) volts. Half-cell reactions with\nthe most positive standard electrode potentials have a tendency to\nspontaneously proceed toward reduction (cathode reactions). Half-cell\nreactions with the most negative standard electrode potentials have a\ntendency to spontaneously proceed toward oxidation (anode reactions).\n\nConsider, for example, a cell made up of a standard hydrogen electrode\nand a standard zinc electrode with \u03b5o\n(H+/H2,g) = 0.00 volts and \u03b5o\n(Zn2+/Zn) = -0.762 volts. Thus, the H+ would tend\nto be reduced, and the zinc metal would tend to be oxidized, and the\nspontaneous reaction if all species had unit activities would be\n\n*Eq. 8\u201124* 2 H+(aq, a=1)+ Zn = H2(1 bar) +\nZn2+(aq, a=1)\n\nwith \u03b5ocell = 0.762 volts and\n$\\mathrm{\\Delta}_{\\ }^{0}G = - 2*96485*\\varepsilon_{cell}^{0}\\ $. The\ncathode half-cell reaction would be the same as , while the anode\nhalf-cell reaction would be\n\n*Eq. 8\u201125* Zn = Zn2+(aq, a=1) + 2 e-\n\nWhen ion concentrations and H2 gas do not all have unit\nactivities, the Gibbs energy and cell potential of the cell reaction, ,\nbecomes\n\n*Eq. 8\u201126*\n$\\mathrm{\\Delta}G = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{a_{{Zn}^{2 +}}P_{H_{2}}}{\\left( a_{H^{+}} \\right)^{2}}$\n\n*Eq. 8\u201127*\n$\\varepsilon_{cell} = \\varepsilon_{cell}^{0} - \\frac{RT}{zf}\\ln\\frac{a_{{Zn}^{2 +}}P_{H_{2}}}{\\left( a_{H^{+}} \\right)^{2}}$\n\nThe standard reduction potentials of some common metals at\n25oC are given in .\n\nTable \u2011: Standard reduction potentials of some common metals\n\nA cell reaction can be established by different half-cell reactions. For\nexample, the following reaction can be derived from two different cells\n\n*Eq. 8\u201128* $3\\ {Fe}^{2 + \\ }\\ = \\ \\ 2\\ {Fe}^{3 +}\\ \\ + \\ \\ Fe(s)$\n\ncell A\n\n*Eq. 8\u201129* $3{Fe}^{2 +} + \\ 6\\ e - \\ = \\ \\ 3Fe(s)$ \u03b5o1 =\n-0.440 V\n\n*Eq. 8\u201130* $2\\ Fe(s)\\ \\ = \\ \\ 2\\ {Fe}^{3 +}\\ \\ + \\ 6\\ e -$\n\u03b5o2 = +0.036 V\n\ncell B\n\n*Eq. 8\u201131* $2{Fe}^{2 +}\\ = \\ 2{Fe}^{3 + \\ \\ } + 2\\ e^{-}$\n*\u03b5o4 = -0.772 V*\n\n*Eq. 8\u201132* ${Fe}^{2 + \\ }\\ + \\ 2\\ e - \\ \\ = \\ \\ Fe(s)$ *\u03b5o5\n= -0.440 V*\n\nBoth give the same net reaction shown by , but with 6 and 2 electrons\nand standard cell potentials being \u03b5ocell A = -0.404 V and\n\u03b5ocell B = -1.212 V, respectively. However, the standard\nGibbs energies of both cells are the same, i.e.\n\n*Eq. 8\u201133* $\\mathrm{\\Delta}_{\\ }^{0}G$*cell A = -6 f (-0.404)\n= + 2.424 f*\n\n*Eq. 8\u201134* $\\mathrm{\\Delta}_{\\ }^{0}G$*cell B = -2 f (-1.212)\n= + 2.424 f*\n\nIt is shown that $\\mathrm{\\Delta}_{\\ }^{0}G$ values are independent of\nhalf-cell reactions and depend only on the net reaction because the net\nreaction is neutral in electron and balanced in mass.\n", + "source": [ + "### Half cell potentials\n", + "\n", + "When electron current flows between electrodes, reactions are occurring\n", + "at the electrodes and concentration gradients causing polarization\n", + "develop around the electrodes. These gradients result in extraneous\n", + "potentials to occur at the electrodes. In such cases cell equilibrium is\n", + "not established and measured cell potentials are not those for true\n", + "partial equilibrium. If a cell is short-circuited with the electrodes\n", + "connected by a conductor, current will flow until the external potential\n", + "becomes zero, i.e. \u03b5ext = 0, and equilibrium is established\n", + "with same conditions as non-electrochemical systems. If an external\n", + "potential, \u03b5ext, is applied to the cell, chemical reactions\n", + "occur until the cell potential balances to \u03b5ext, and no\n", + "current flows. This potential is called open-circuit voltage (OCV) in\n", + "the literature. It is important to realize that OCV includes all\n", + "reactions that occur on the electrode surface when the electrode is in\n", + "contact with the electrolyte, such as passivation discussed in Chapter .\n", + "Partial equilibrium in a cell is achieved when the cell potential is\n", + "balanced by an applied external potential. In such partial equilibrium\n", + "cases, equilibrium thermodynamic analyses can be used even though the\n", + "cell potential is not zero, i.e. \u03b5cell \u2260 0. This\n", + "differentiates electrochemical systems from other equilibrium systems\n", + "discussed previously.\n", + "\n", + "The number of electrons involved in a net cell reaction is important in\n", + "relating cell potential and the Gibbs energy change for the cell\n", + "reaction. As will be illustrated later in this section, this number\n", + "denotes the number of electrons involved in the half-cell reactions that\n", + "were added to yield the net cell reaction. The electrical work achieved\n", + "by the transport of an electrical charge through a cell potential can be\n", + "written as\n", + "\n", + "*Eq. 8\u201120* $w = z\\ f\\ \\varepsilon$\n", + "\n", + "where *z* represents the moles of electrons in cell reaction, *f* the\n", + "Faraday constant equal to 96,485 J/V/mole-electron, and *\u03b5* the\n", + "potential difference, often referred as electromotive force (emf) in the\n", + "literature. For a system at constant temperature, pressure, and\n", + "composition, this work is the same as the Gibbs energy difference\n", + "between the two electrodes, i.e.\n", + "\n", + "*Eq. 8\u201121* $- \\Delta G = w = zf\\varepsilon$\n", + "\n", + "where the negative sign is because the system does work to the\n", + "surrounding when the Gibbs energy of the system is decreased. When the\n", + "applied external potential is larger than the cell potential, the\n", + "surrounding does work to the system, and a common example is the\n", + "charging of a battery. Thermodynamic relations discussed in previous\n", + "chapters can thus be directly applied to electrochemical systems with\n", + "some common equations shown in .\n", + "\n", + "Table \u2011: Thermodynamic Equations for Electrochemical Cells\n", + "\n", + "*\u2206G = -z f \u03b5*\n", + "\n", + "*\u2206S = - (\u2202\u2206G/\u2202T)P = + z f (\u2202\u03b5/\u2202T)P*\n", + "\n", + "*\u2206H = \\[\u2202(\u2206G/T)/ \u2202(1/T)\\]P = - z f \\[\u2202(\u03b5/T)/\u2202(1/T)\\]P = z f \\[T(\u2202\u03b5/\u2202T)P\n", + "\u2013 \u03b5\\]*\n", + "\n", + "*\u2206CP = (\u2202\u2206H/\u2202T)P = T z f (\u22022\u03b5/\u2202T2)P*\n", + "\n", + "A half-cell reaction potential cannot be measured directly, only its\n", + "potential relative to another half-cell reaction. By convention, a\n", + "standard half-cell potential is measured relative to the standard\n", + "hydrogen half-cell reduction reaction at 25oC (298K) and 1\n", + "bar, which has a defined standard potential of zero volts,\n", + "\n", + "*Eq. 8\u201122* H+(aq, a=1) + e- = 1/2 H2(g,\n", + "1 bar)\n", + "\n", + "with \u03b5o (H+/H2,g) = 0.00 volts. The\n", + "standard half-cell reduction reactions of metals at 25oC are\n", + "for the general reaction\n", + "\n", + "*Eq. 8\u201123* Mz+(aq, a=1) + z e- = M(s)\n", + "\n", + "with \u03b5o (Mz+/M) volts. Half-cell reactions with\n", + "the most positive standard electrode potentials have a tendency to\n", + "spontaneously proceed toward reduction (cathode reactions). Half-cell\n", + "reactions with the most negative standard electrode potentials have a\n", + "tendency to spontaneously proceed toward oxidation (anode reactions).\n", + "\n", + "Consider, for example, a cell made up of a standard hydrogen electrode\n", + "and a standard zinc electrode with \u03b5o\n", + "(H+/H2,g) = 0.00 volts and \u03b5o\n", + "(Zn2+/Zn) = -0.762 volts. Thus, the H+ would tend\n", + "to be reduced, and the zinc metal would tend to be oxidized, and the\n", + "spontaneous reaction if all species had unit activities would be\n", + "\n", + "*Eq. 8\u201124* 2 H+(aq, a=1)+ Zn = H2(1 bar) +\n", + "Zn2+(aq, a=1)\n", + "\n", + "with \u03b5ocell = 0.762 volts and\n", + "$\\mathrm{\\Delta}_{\\ }^{0}G = - 2*96485*\\varepsilon_{cell}^{0}\\ $. The\n", + "cathode half-cell reaction would be the same as , while the anode\n", + "half-cell reaction would be\n", + "\n", + "*Eq. 8\u201125* Zn = Zn2+(aq, a=1) + 2 e-\n", + "\n", + "When ion concentrations and H2 gas do not all have unit\n", + "activities, the Gibbs energy and cell potential of the cell reaction, ,\n", + "becomes\n", + "\n", + "*Eq. 8\u201126*\n", + "$\\mathrm{\\Delta}G = \\mathrm{\\Delta}_{\\ }^{0}G + RTln\\frac{a_{{Zn}^{2 +}}P_{H_{2}}}{\\left( a_{H^{+}} \\right)^{2}}$\n", + "\n", + "*Eq. 8\u201127*\n", + "$\\varepsilon_{cell} = \\varepsilon_{cell}^{0} - \\frac{RT}{zf}\\ln\\frac{a_{{Zn}^{2 +}}P_{H_{2}}}{\\left( a_{H^{+}} \\right)^{2}}$\n", + "\n", + "The standard reduction potentials of some common metals at\n", + "25oC are given in .\n", + "\n", + "Table \u2011: Standard reduction potentials of some common metals\n", + "\n", + "A cell reaction can be established by different half-cell reactions. For\n", + "example, the following reaction can be derived from two different cells\n", + "\n", + "*Eq. 8\u201128* $3\\ {Fe}^{2 + \\ }\\ = \\ \\ 2\\ {Fe}^{3 +}\\ \\ + \\ \\ Fe(s)$\n", + "\n", + "cell A\n", + "\n", + "*Eq. 8\u201129* $3{Fe}^{2 +} + \\ 6\\ e - \\ = \\ \\ 3Fe(s)$ \u03b5o1 =\n", + "-0.440 V\n", + "\n", + "*Eq. 8\u201130* $2\\ Fe(s)\\ \\ = \\ \\ 2\\ {Fe}^{3 +}\\ \\ + \\ 6\\ e -$\n", + "\u03b5o2 = +0.036 V\n", + "\n", + "cell B\n", + "\n", + "*Eq. 8\u201131* $2{Fe}^{2 +}\\ = \\ 2{Fe}^{3 + \\ \\ } + 2\\ e^{-}$\n", + "*\u03b5o4 = -0.772 V*\n", + "\n", + "*Eq. 8\u201132* ${Fe}^{2 + \\ }\\ + \\ 2\\ e - \\ \\ = \\ \\ Fe(s)$ *\u03b5o5\n", + "= -0.440 V*\n", + "\n", + "Both give the same net reaction shown by , but with 6 and 2 electrons\n", + "and standard cell potentials being \u03b5ocell A = -0.404 V and\n", + "\u03b5ocell B = -1.212 V, respectively. However, the standard\n", + "Gibbs energies of both cells are the same, i.e.\n", + "\n", + "*Eq. 8\u201133* $\\mathrm{\\Delta}_{\\ }^{0}G$*cell A = -6 f (-0.404)\n", + "= + 2.424 f*\n", + "\n", + "*Eq. 8\u201134* $\\mathrm{\\Delta}_{\\ }^{0}G$*cell B = -2 f (-1.212)\n", + "= + 2.424 f*\n", + "\n", + "It is shown that $\\mathrm{\\Delta}_{\\ }^{0}G$ values are independent of\n", + "half-cell reactions and depend only on the net reaction because the net\n", + "reaction is neutral in electron and balanced in mass.\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/applications_to_electrochemical_systems/electrolyte_reactions_and_electrochemical_reactions.ipynb b/src/psu410/src/applications_to_electrochemical_systems/electrolyte_reactions_and_electrochemical_reactions.ipynb index 4c2aa3c..bf7faa3 100644 --- a/src/psu410/src/applications_to_electrochemical_systems/electrolyte_reactions_and_electrochemical_reactions.ipynb +++ b/src/psu410/src/applications_to_electrochemical_systems/electrolyte_reactions_and_electrochemical_reactions.ipynb @@ -4,9 +4,86 @@ "metadata": {}, "cells": [ { - "id": "ece8e64b", + "id": "d68abe36", "cell_type": "markdown", - "source": "## Electrolyte reactions and electrochemical reactions\n\nElectrolytes that dissolve in a polar solvent such as water to produce\nionic species do not necessarily exhibit valence changes. A simple\nexample is the strong electrolyte NaCl(s) dissolving in water to produce\nsolvated ions\n\n*Eq. 8\u20111* $NaCl(s)\\ \\ = \\ \\ {Na}^{+}(aq)\\ \\ + \\ \\ {Cl}^{\u2013}(aq)$\n\nwhere the $(aq)$ indicates the ionic species in an aqueous solution. In\nthis system, the ion concentrations must become quite large before the\nsolution is saturated and can exist in equilibrium with $NaCl(s)$. Its\nreaction constant, defined by , is shown as\n$K_{e} = \\ \\ a_{{Na}^{+}}a_{{Cl}^{-}}$. If the product of the ion\nactivities is less than $K_{e}$, the solution is not saturated, and more\n$NaCl(s)\\ \\ $ can be dissolved.\n\nThe precipitation of $AgCl(s)$, a weak electrolyte, occurs quite readily\nwhen ${Cl}^{\u2013}$ ions are added to an aqueous solution containing\n${Ag}^{+}(aq)$:\n\n*Eq. 8\u20112* ${Ag}^{+}(aq)\\ + \\ \\ {Cl}^{\u2013}(aq)\\ = AgCl(s)$\n\nThe equilibrium constant for this reaction,\n$K_{e} = \\ \\frac{1}{\\left( a_{{Ag}^{+}}a_{{Cl}^{-}} \\right)}$ is quite\nlarge, so the equilibrium product of the ion activities, proportional to\ntheir concentrations, is quite small. In the laboratory, the above\nreaction could occur as a result of adding hydrochloric acid to a silver\nnitrate solution. The accompanying $H^{+}(aq)$ \\[or ${H_{3}O}^{+}(aq)$\\]\nand ${NO_{3}}^{-}(aq)$ ions in solution are not directly involved in the\nsilver chloride precipitation reaction so are not shown in reaction\nrepresented by .\n\nThe above ionic equilibria in the $AgCl(s) - H_{2}O$ system is not only\nimportant for understanding this electrolyte system, but also critical\nin electrochemical systems in which *Ag(s)* undergoes a valence change\nat one electrode and reacts with a ${Cl}^{\u2013}(aq)$ ion to produce\n*AgCl(s)*, and an electron that is externally transported finite\ndistances to another electrode. The *oxidation* reaction occurs at the\nAg/AgCl electrode (*anode* half-cell reaction where electrons are\n*added* into the system)\n\n*Eq. 8\u20113* $Ag(s)\\ \\ + \\ {Cl}^{-}(aq)\\ \\ = \\ \\ AgCl(s)\\ \\ + \\ \\ e^{-}$\n\nA *reduction* reaction occurs at the other electrode (*cathode*\nhalf-cell reaction where electrons are *consumed* by the reaction)\n\n*Eq. 8\u20114* $\\frac{1}{2}{Cl}_{2}(g) + \\ \\ e^{-} = \\ {Cl}^{-}(aq)\\ \\ $\n\nThe *net cell reaction* results in the formation AgCl(s) from its\nelements\n\n*Eq. 8\u20115* $Ag(s)\\ \\ + \\frac{1}{2}{Cl}_{2}\\ (g)\\ = \\ \\ AgCl(s)$\n\nWithout knowledge of the physical system under which the reaction is\noccurring, it would not be possible to know if reaction of was a result\nof chlorine gas reacting directly with Ag(s), or if the reaction was\npart of an electrochemical cell with a transport of electrons and ions\nover finite distances. The addition of the two half-cell reactions gives\nthe *net cell reaction*, which does not show electrons as either\nreactant or product species and may or may not include ionic species in\nthe reaction*.* A schematic of an electrochemical cell for the above\nsystem is shown in Figure 8\u20111.\n\nFigure \u2011: Schematic diagram of an electrochemical cell consisting of a\nchlorine electrode and a silver-silver chloride electrode.\n\nOxidation and reduction can occur in electrolyte reactions without\ncreating an electrochemical cell. This is the case when chlorine gas\nreacts directly with silver on a Ag(s) surface. Reaction of above is the\nnet reaction for this process, but the electrons produced from the\noxidation of Ag(s) are not transported over finite distances before\ncombining with Cl2(g) in its reduction to Cl\u2013(aq).\nNo anode or cathode half-cell reactions exist in this system. The\nelectrons and ions involved in the reaction move only over atomic-scale\ndistances.\n", + "source": [ + "## Electrolyte reactions and electrochemical reactions\n", + "\n", + "Electrolytes that dissolve in a polar solvent such as water to produce\n", + "ionic species do not necessarily exhibit valence changes. A simple\n", + "example is the strong electrolyte NaCl(s) dissolving in water to produce\n", + "solvated ions\n", + "\n", + "*Eq. 8\u20111* $NaCl(s)\\ \\ = \\ \\ {Na}^{+}(aq)\\ \\ + \\ \\ {Cl}^{\u2013}(aq)$\n", + "\n", + "where the $(aq)$ indicates the ionic species in an aqueous solution. In\n", + "this system, the ion concentrations must become quite large before the\n", + "solution is saturated and can exist in equilibrium with $NaCl(s)$. Its\n", + "reaction constant, defined by , is shown as\n", + "$K_{e} = \\ \\ a_{{Na}^{+}}a_{{Cl}^{-}}$. If the product of the ion\n", + "activities is less than $K_{e}$, the solution is not saturated, and more\n", + "$NaCl(s)\\ \\ $ can be dissolved.\n", + "\n", + "The precipitation of $AgCl(s)$, a weak electrolyte, occurs quite readily\n", + "when ${Cl}^{\u2013}$ ions are added to an aqueous solution containing\n", + "${Ag}^{+}(aq)$:\n", + "\n", + "*Eq. 8\u20112* ${Ag}^{+}(aq)\\ + \\ \\ {Cl}^{\u2013}(aq)\\ = AgCl(s)$\n", + "\n", + "The equilibrium constant for this reaction,\n", + "$K_{e} = \\ \\frac{1}{\\left( a_{{Ag}^{+}}a_{{Cl}^{-}} \\right)}$ is quite\n", + "large, so the equilibrium product of the ion activities, proportional to\n", + "their concentrations, is quite small. In the laboratory, the above\n", + "reaction could occur as a result of adding hydrochloric acid to a silver\n", + "nitrate solution. The accompanying $H^{+}(aq)$ \\[or ${H_{3}O}^{+}(aq)$\\]\n", + "and ${NO_{3}}^{-}(aq)$ ions in solution are not directly involved in the\n", + "silver chloride precipitation reaction so are not shown in reaction\n", + "represented by .\n", + "\n", + "The above ionic equilibria in the $AgCl(s) - H_{2}O$ system is not only\n", + "important for understanding this electrolyte system, but also critical\n", + "in electrochemical systems in which *Ag(s)* undergoes a valence change\n", + "at one electrode and reacts with a ${Cl}^{\u2013}(aq)$ ion to produce\n", + "*AgCl(s)*, and an electron that is externally transported finite\n", + "distances to another electrode. The *oxidation* reaction occurs at the\n", + "Ag/AgCl electrode (*anode* half-cell reaction where electrons are\n", + "*added* into the system)\n", + "\n", + "*Eq. 8\u20113* $Ag(s)\\ \\ + \\ {Cl}^{-}(aq)\\ \\ = \\ \\ AgCl(s)\\ \\ + \\ \\ e^{-}$\n", + "\n", + "A *reduction* reaction occurs at the other electrode (*cathode*\n", + "half-cell reaction where electrons are *consumed* by the reaction)\n", + "\n", + "*Eq. 8\u20114* $\\frac{1}{2}{Cl}_{2}(g) + \\ \\ e^{-} = \\ {Cl}^{-}(aq)\\ \\ $\n", + "\n", + "The *net cell reaction* results in the formation AgCl(s) from its\n", + "elements\n", + "\n", + "*Eq. 8\u20115* $Ag(s)\\ \\ + \\frac{1}{2}{Cl}_{2}\\ (g)\\ = \\ \\ AgCl(s)$\n", + "\n", + "Without knowledge of the physical system under which the reaction is\n", + "occurring, it would not be possible to know if reaction of was a result\n", + "of chlorine gas reacting directly with Ag(s), or if the reaction was\n", + "part of an electrochemical cell with a transport of electrons and ions\n", + "over finite distances. The addition of the two half-cell reactions gives\n", + "the *net cell reaction*, which does not show electrons as either\n", + "reactant or product species and may or may not include ionic species in\n", + "the reaction*.* A schematic of an electrochemical cell for the above\n", + "system is shown in Figure 8\u20111.\n", + "\n", + "Figure \u2011: Schematic diagram of an electrochemical cell consisting of a\n", + "chlorine electrode and a silver-silver chloride electrode.\n", + "\n", + "Oxidation and reduction can occur in electrolyte reactions without\n", + "creating an electrochemical cell. This is the case when chlorine gas\n", + "reacts directly with silver on a Ag(s) surface. Reaction of above is the\n", + "net reaction for this process, but the electrons produced from the\n", + "oxidation of Ag(s) are not transported over finite distances before\n", + "combining with Cl2(g) in its reduction to Cl\u2013(aq).\n", + "No anode or cathode half-cell reactions exist in this system. The\n", + "electrons and ions involved in the reaction move only over atomic-scale\n", + "distances.\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/applications_to_electrochemical_systems/index.ipynb b/src/psu410/src/applications_to_electrochemical_systems/index.ipynb index 411a8f5..edbd9ee 100644 --- a/src/psu410/src/applications_to_electrochemical_systems/index.ipynb +++ b/src/psu410/src/applications_to_electrochemical_systems/index.ipynb @@ -4,9 +4,29 @@ "metadata": {}, "cells": [ { - "id": "e01daf27", + "id": "1f40160b", "cell_type": "markdown", - "source": "# Applications to electrochemical systems\n\nThe two basic types of chemical processes involving ions as reactant\nand/or product species are electrolyte reactions and electrochemical\nreactions. *Electrolyte reactions* are accompanied by the *atomic-scale\nmovement* of ionic species and possibly electrons. Chemical changes that\nproduce changes in valence and *electron and ion transport over finite\ndistances* constitute an area of science termed *electrochemistry*. The\nlatter chemical changes occur in an electrochemical cell comprised of\ntwo electrodes, an anode and a cathode, which are coupled by an\nelectrolyte and an external electron conductor. Most thermodynamic\nconcepts and analyses described in previous chapters remain unchanged\nwhen applied to electrochemistry, but the analysis of electrochemical\nsystems does require some new terminology, new definitions, and new\nconventions. The primary focus of this chapter is on applications of\nthermodynamics to electrochemical reactions that involve either aqueous\nelectrolyte solutions or solid state electrolytes. Since all\nelectrochemical systems include ionized species as reactant and/or\nproduct species, electrolyte reactions will also be discussed.\n", + "source": [ + "# Applications to electrochemical systems\n", + "\n", + "The two basic types of chemical processes involving ions as reactant\n", + "and/or product species are electrolyte reactions and electrochemical\n", + "reactions. *Electrolyte reactions* are accompanied by the *atomic-scale\n", + "movement* of ionic species and possibly electrons. Chemical changes that\n", + "produce changes in valence and *electron and ion transport over finite\n", + "distances* constitute an area of science termed *electrochemistry*. The\n", + "latter chemical changes occur in an electrochemical cell comprised of\n", + "two electrodes, an anode and a cathode, which are coupled by an\n", + "electrolyte and an external electron conductor. Most thermodynamic\n", + "concepts and analyses described in previous chapters remain unchanged\n", + "when applied to electrochemistry, but the analysis of electrochemical\n", + "systems does require some new terminology, new definitions, and new\n", + "conventions. The primary focus of this chapter is on applications of\n", + "thermodynamics to electrochemical reactions that involve either aqueous\n", + "electrolyte solutions or solid state electrolytes. Since all\n", + "electrochemical systems include ionized species as reactant and/or\n", + "product species, electrolyte reactions will also be discussed.\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/calpahd_modeling_of_thermodynamics/importance_of_lattice_stability.ipynb b/src/psu410/src/calpahd_modeling_of_thermodynamics/importance_of_lattice_stability.ipynb index 3870d0e..2f091d2 100644 --- a/src/psu410/src/calpahd_modeling_of_thermodynamics/importance_of_lattice_stability.ipynb +++ b/src/psu410/src/calpahd_modeling_of_thermodynamics/importance_of_lattice_stability.ipynb @@ -4,9 +4,117 @@ "metadata": {}, "cells": [ { - "id": "ba9e840b", + "id": "4d2c637c", "cell_type": "markdown", - "source": "## Importance of lattice stability\n\nFor modeling of Gibbs energy of individual phases, it is necessary to\ndefine the values of $_{\\ }^{0}G_{i}$ in . However, the independent\ncomponent *i* may not be stable in the crystal structure of the phase\nunder consideration, so its Gibbs energy could not be obtained directly\nfrom experiments and must be estimated with respect to the Gibbs energy\nof their stable crystal structure. In the pioneering work by Kaufman\n\\[47\\], this Gibbs energy difference was termed as lattice stability and\nobtained through extrapolations in either temperature-pressure or\ntemperature-composition phase diagrams. It is evident from that the\nvalues of $_{\\ }^{0}G_{i}$ and $_{\\ }^{M}G$ jointly contribute to the\nGibbs energy of the solution, and Kaufman had to simplify the treatment\nof $_{\\ }^{M}G$ in order to show the importance of the concept of\nlattice stability. Using ideal or regular solution models, Kaufman was\nable to define the lattice stability for pure elements and remarkably\nreproduce many features of binary phase diagrams by introducing the\ninteraction parameters afterwards.\n\nOver the years, there had been various revisions of lattice stability\nvalues for common crystal structures \\[48\\], and every revision\nnecessitates the re-evaluation of interaction parameters in the solution\nphase shown in . It was until the lattice stability values established\nby the Scientific Group Thermodata Europe (SGTE) \\[52\\] that the\ndevelopment of binary thermodynamic models using the same thermodynamic\nmodels of pure elements became possible, and those binary models thus\ndeveloped in different groups around the world can be combined to create\nthermodynamic models of ternary and multi-component systems. Clearly,\nany further modifications of the SGTE pure element database will require\nthe re-modeling of all binary and ternary systems in which the models of\npure elements are changed. This challenge is briefly addressed in the\nlater part of this chapter.\n\nA less addressed issue is the Gibbs energy of end-members in\nnon-stoichiometric compounds, i.e. , where each sublattice contains only\none element. In case all sublattices are occupied by the same element,\nit is the lattice stability of the elements in the structure of the\ncompound. Since the stable composition ranges of non-stoichiometric\ncompounds are typically small, the existing method cannot be used to\nreliably evaluate the Gibbs energy of end-members, and currently there\nis not a commonly accepted lattice stability database for compounds.\nMost values used in the existing databases have been either roughly\nestimated or computed from first-principles calculations. Such a\nstandard database is highly desirable in order to make various models of\ncompounds compatible.\n\nIn an effort to compare the lattice stability from the CALPHAD models\nand the first-principles calculations, Wang et al. \\[53\\] systematically\ncalculated the total energies of 78 pure elements at zero Kelvin in the\nface-centered-cubic (fcc), body-centered-cubic (bcc), and\nhexagonal-close-packed (hcp) crystal structures using the projector\naugmented-wave (PAW) method within the generalized gradient\napproximation (GGA). The calculated values are compared with the values\nin the SGTE database as shown in and . For non-transition metal\nelements, the differences between the SGTE data and the PAW-GGA data are\ntypically around 1\u223c2 kJ/mole-of-atoms or less, while for some transition\nmetal elements, the differences can be quite large, for example, as high\nas about 54 kJ/mole-of-atoms for and about 40 kJ/mole-of-atoms for . and\npresent the differences between the PAW-GGA data and the SGTE data, for\nelements from the Ti group to the Ni group, respectively.\n\nTable \u2011: Lattice stability Ebcc-fcc (kJ/mole-of-atoms).\n\nTable \u2011: Lattice stability Ehcp-fcc (kJ/mole-of-atoms).\n\nFigure \u2011: Lattice stability difference between bcc and fcc, , for\nselected elements between PAW-GGA and SGTE\n\nFigure \u2011: Lattice stability difference between hcp and fcc, , for\nselected elements between PAW-GGA and SGTE\n\nThe large differences between the first-principles calculations and the\nSGTE data could partly be attributed to the instability of the\nhigher-energy phases, the entropies of which at finite *T* become\nabnormal. The lattice instabilities along the tetragonal transformation\npath between fcc and bcc structures with the continuous change of the\nc/a ratio defined in a bcc-based tetragonal lattice are demonstrated for\nbcc Mo, Ta, W in and for fcc Al, Cu, Ni in . It is shown that the fcc\nstructure of bcc Mo, Ta, and W is a local maximum with respect to the\ntetragonal transformation, and the higher the maximum is, the larger the\ndiscrepancy between the SGTE data and the present PAW-GGA data, while\nfor fcc Al, Cu, Ni, the bcc structure is at a local maximum. Similarly,\nthe lattice instabilities along the tetragonal transformation path for\nthe hcp metals Ru and Os as shown in . The behavior of energy against\nc/a ratio of these two hcp metals is very similar to those of fcc\nelements.\n\nFigure \u2011: Total energy, , along the Bain deformation path between bcc\nand fcc for Mo, Ta, and W.\n\nFigure \u2011: Total energy, , along the tetragonal transformation path\nbetween bcc and fcc for Ni, Al, and Cu.\n\nFigure \u2011: Total energy, , along the tetragonal transformation path for\nRu and Os\n\nIt can be concluded that a fcc structure for elements with bcc being the\nground state or a bcc structure for elements with fcc being the ground\nstate, is unstable with respect to the tetragonal transformation. For an\nunstable structure, the harmonic description of its vibrational entropy\nis thermodynamically incorrect since the potential surface seen by the\nlattice ion can no longer be approximated by a parabola. If an unstable\nstructure of a pure element is stabilized at high temperatures, its\nentropy has to be abnormal. The instability issue has been recently\naddressed by ab initio molecular dynamics simulations at high\ntemperatures using W as an example \\[54\\], which is beyond the scope of\nthe book and thus not discussed here.\n", + "source": [ + "## Importance of lattice stability\n", + "\n", + "For modeling of Gibbs energy of individual phases, it is necessary to\n", + "define the values of $_{\\ }^{0}G_{i}$ in . However, the independent\n", + "component *i* may not be stable in the crystal structure of the phase\n", + "under consideration, so its Gibbs energy could not be obtained directly\n", + "from experiments and must be estimated with respect to the Gibbs energy\n", + "of their stable crystal structure. In the pioneering work by Kaufman\n", + "\\[47\\], this Gibbs energy difference was termed as lattice stability and\n", + "obtained through extrapolations in either temperature-pressure or\n", + "temperature-composition phase diagrams. It is evident from that the\n", + "values of $_{\\ }^{0}G_{i}$ and $_{\\ }^{M}G$ jointly contribute to the\n", + "Gibbs energy of the solution, and Kaufman had to simplify the treatment\n", + "of $_{\\ }^{M}G$ in order to show the importance of the concept of\n", + "lattice stability. Using ideal or regular solution models, Kaufman was\n", + "able to define the lattice stability for pure elements and remarkably\n", + "reproduce many features of binary phase diagrams by introducing the\n", + "interaction parameters afterwards.\n", + "\n", + "Over the years, there had been various revisions of lattice stability\n", + "values for common crystal structures \\[48\\], and every revision\n", + "necessitates the re-evaluation of interaction parameters in the solution\n", + "phase shown in . It was until the lattice stability values established\n", + "by the Scientific Group Thermodata Europe (SGTE) \\[52\\] that the\n", + "development of binary thermodynamic models using the same thermodynamic\n", + "models of pure elements became possible, and those binary models thus\n", + "developed in different groups around the world can be combined to create\n", + "thermodynamic models of ternary and multi-component systems. Clearly,\n", + "any further modifications of the SGTE pure element database will require\n", + "the re-modeling of all binary and ternary systems in which the models of\n", + "pure elements are changed. This challenge is briefly addressed in the\n", + "later part of this chapter.\n", + "\n", + "A less addressed issue is the Gibbs energy of end-members in\n", + "non-stoichiometric compounds, i.e. , where each sublattice contains only\n", + "one element. In case all sublattices are occupied by the same element,\n", + "it is the lattice stability of the elements in the structure of the\n", + "compound. Since the stable composition ranges of non-stoichiometric\n", + "compounds are typically small, the existing method cannot be used to\n", + "reliably evaluate the Gibbs energy of end-members, and currently there\n", + "is not a commonly accepted lattice stability database for compounds.\n", + "Most values used in the existing databases have been either roughly\n", + "estimated or computed from first-principles calculations. Such a\n", + "standard database is highly desirable in order to make various models of\n", + "compounds compatible.\n", + "\n", + "In an effort to compare the lattice stability from the CALPHAD models\n", + "and the first-principles calculations, Wang et al. \\[53\\] systematically\n", + "calculated the total energies of 78 pure elements at zero Kelvin in the\n", + "face-centered-cubic (fcc), body-centered-cubic (bcc), and\n", + "hexagonal-close-packed (hcp) crystal structures using the projector\n", + "augmented-wave (PAW) method within the generalized gradient\n", + "approximation (GGA). The calculated values are compared with the values\n", + "in the SGTE database as shown in and . For non-transition metal\n", + "elements, the differences between the SGTE data and the PAW-GGA data are\n", + "typically around 1\u223c2 kJ/mole-of-atoms or less, while for some transition\n", + "metal elements, the differences can be quite large, for example, as high\n", + "as about 54 kJ/mole-of-atoms for and about 40 kJ/mole-of-atoms for . and\n", + "present the differences between the PAW-GGA data and the SGTE data, for\n", + "elements from the Ti group to the Ni group, respectively.\n", + "\n", + "Table \u2011: Lattice stability Ebcc-fcc (kJ/mole-of-atoms).\n", + "\n", + "Table \u2011: Lattice stability Ehcp-fcc (kJ/mole-of-atoms).\n", + "\n", + "Figure \u2011: Lattice stability difference between bcc and fcc, , for\n", + "selected elements between PAW-GGA and SGTE\n", + "\n", + "Figure \u2011: Lattice stability difference between hcp and fcc, , for\n", + "selected elements between PAW-GGA and SGTE\n", + "\n", + "The large differences between the first-principles calculations and the\n", + "SGTE data could partly be attributed to the instability of the\n", + "higher-energy phases, the entropies of which at finite *T* become\n", + "abnormal. The lattice instabilities along the tetragonal transformation\n", + "path between fcc and bcc structures with the continuous change of the\n", + "c/a ratio defined in a bcc-based tetragonal lattice are demonstrated for\n", + "bcc Mo, Ta, W in and for fcc Al, Cu, Ni in . It is shown that the fcc\n", + "structure of bcc Mo, Ta, and W is a local maximum with respect to the\n", + "tetragonal transformation, and the higher the maximum is, the larger the\n", + "discrepancy between the SGTE data and the present PAW-GGA data, while\n", + "for fcc Al, Cu, Ni, the bcc structure is at a local maximum. Similarly,\n", + "the lattice instabilities along the tetragonal transformation path for\n", + "the hcp metals Ru and Os as shown in . The behavior of energy against\n", + "c/a ratio of these two hcp metals is very similar to those of fcc\n", + "elements.\n", + "\n", + "Figure \u2011: Total energy, , along the Bain deformation path between bcc\n", + "and fcc for Mo, Ta, and W.\n", + "\n", + "Figure \u2011: Total energy, , along the tetragonal transformation path\n", + "between bcc and fcc for Ni, Al, and Cu.\n", + "\n", + "Figure \u2011: Total energy, , along the tetragonal transformation path for\n", + "Ru and Os\n", + "\n", + "It can be concluded that a fcc structure for elements with bcc being the\n", + "ground state or a bcc structure for elements with fcc being the ground\n", + "state, is unstable with respect to the tetragonal transformation. For an\n", + "unstable structure, the harmonic description of its vibrational entropy\n", + "is thermodynamically incorrect since the potential surface seen by the\n", + "lattice ion can no longer be approximated by a parabola. If an unstable\n", + "structure of a pure element is stabilized at high temperatures, its\n", + "entropy has to be abnormal. The instability issue has been recently\n", + "addressed by ab initio molecular dynamics simulations at high\n", + "temperatures using W as an example \\[54\\], which is beyond the scope of\n", + "the book and thus not discussed here.\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/calpahd_modeling_of_thermodynamics/index.ipynb b/src/psu410/src/calpahd_modeling_of_thermodynamics/index.ipynb index affcd41..3a4e34d 100644 --- a/src/psu410/src/calpahd_modeling_of_thermodynamics/index.ipynb +++ b/src/psu410/src/calpahd_modeling_of_thermodynamics/index.ipynb @@ -4,9 +4,55 @@ "metadata": {}, "cells": [ { - "id": "543d5219", + "id": "ca65cc0e", "cell_type": "markdown", - "source": "# CALPAHD modeling of thermodynamics\n\nCALPHAD modeling of thermodynamics was pioneered by Kaufman \\[47\\] and\nhas been reviewed in details by Saunders and Miodownik \\[48\\] and Lukas,\nFries and Sundman \\[49\\]. Information on features of software tools for\nCALPHAD modeling can be found at two series of publications in the\nCALPHAD journal \\[50-51\\]. The key feature of the CALPHAD method is the\nmodeling of Gibbs energy of individual phases using both thermodynamic\nand phase equilibrium data. The main significances of the CALPHAD method\nare as follows\n\n1. It enabled the development of the concept of lattice stability, i.e.\n the energy difference of a pure element with stable and non-stable\n crystal structures;\n\n2. The Gibbs energy expression of each phase covers the full\n temperature, pressure, and composition spaces including both stable\n and non-stable regions of the phase. This enables the evaluation of\n the Gibbs energy of a system as a function of non-equilibrium state,\n i.e. with \u03be as an independent variable;\n\n3. Thermodynamic data are usually obtained by measurements of heat such\n as enthalpy of transition and heat capacity as discussed in Chapter\n , which bear large uncertainties typically in the range of\n kilojoules per mole-of-atom. On the other hand, phase equilibrium\n data as discussed in Chapter , though more accurate, only contain\n information on compositions of phases at equilibria, i.e., the\n relative Gibbs energy of phases at equilibrium. The combination of\n these two sets of data is foundational in CALPHAD modeling that\n allows for accurate modeling of thermodynamic properties of\n individual phases and reliable calculations of phase stability and\n driving forces;\n\n4. It provides a framework to model thermodynamic properties of\n multi-component systems of industrial importance, enabling the\n computational materials design. It has also been extended to model a\n range of properties of individual phases in multi-component systems\n such as diffusion coefficients, elastic coefficients, and thermal\n expansion, supplying input data for computational simulations of\n phase transformations during materials processing.\n\nIn this chapter, the basics of CALPHAD modeling of Gibbs energy of\nindividual phases are presented. For detailed implementations in various\nsoftware packages and modeling procedures, readers are referred to the\nreferences listed above.\n", + "source": [ + "# CALPAHD modeling of thermodynamics\n", + "\n", + "CALPHAD modeling of thermodynamics was pioneered by Kaufman \\[47\\] and\n", + "has been reviewed in details by Saunders and Miodownik \\[48\\] and Lukas,\n", + "Fries and Sundman \\[49\\]. Information on features of software tools for\n", + "CALPHAD modeling can be found at two series of publications in the\n", + "CALPHAD journal \\[50-51\\]. The key feature of the CALPHAD method is the\n", + "modeling of Gibbs energy of individual phases using both thermodynamic\n", + "and phase equilibrium data. The main significances of the CALPHAD method\n", + "are as follows\n", + "\n", + "1. It enabled the development of the concept of lattice stability, i.e.\n", + " the energy difference of a pure element with stable and non-stable\n", + " crystal structures;\n", + "\n", + "2. The Gibbs energy expression of each phase covers the full\n", + " temperature, pressure, and composition spaces including both stable\n", + " and non-stable regions of the phase. This enables the evaluation of\n", + " the Gibbs energy of a system as a function of non-equilibrium state,\n", + " i.e. with \u03be as an independent variable;\n", + "\n", + "3. Thermodynamic data are usually obtained by measurements of heat such\n", + " as enthalpy of transition and heat capacity as discussed in Chapter\n", + " , which bear large uncertainties typically in the range of\n", + " kilojoules per mole-of-atom. On the other hand, phase equilibrium\n", + " data as discussed in Chapter , though more accurate, only contain\n", + " information on compositions of phases at equilibria, i.e., the\n", + " relative Gibbs energy of phases at equilibrium. The combination of\n", + " these two sets of data is foundational in CALPHAD modeling that\n", + " allows for accurate modeling of thermodynamic properties of\n", + " individual phases and reliable calculations of phase stability and\n", + " driving forces;\n", + "\n", + "4. It provides a framework to model thermodynamic properties of\n", + " multi-component systems of industrial importance, enabling the\n", + " computational materials design. It has also been extended to model a\n", + " range of properties of individual phases in multi-component systems\n", + " such as diffusion coefficients, elastic coefficients, and thermal\n", + " expansion, supplying input data for computational simulations of\n", + " phase transformations during materials processing.\n", + "\n", + "In this chapter, the basics of CALPHAD modeling of Gibbs energy of\n", + "individual phases are presented. For detailed implementations in various\n", + "software packages and modeling procedures, readers are referred to the\n", + "references listed above.\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_pure_elements.ipynb b/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_pure_elements.ipynb index be1152e..87c5f77 100644 --- a/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_pure_elements.ipynb +++ b/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_pure_elements.ipynb @@ -4,9 +4,72 @@ "metadata": {}, "cells": [ { - "id": "54cc29d0", + "id": "ba726078", "cell_type": "markdown", - "source": "## Modeling of pure elements\n\nIn modeling the Gibbs energy of pure elements in the SER structure in\nterms of , the coefficients in are evaluated using the heat capacity\ndata, $b^{'}$ in is evaluated by the value of $S_{298.15}$, and $a$ in\nis evaluated by $H_{298.15}^{SER} = 0$, respectively. For the high\ntemperature phase, the enthalpy of transformation from the low\ntemperature phase to the high temperature phase,\n${\\mathrm{\\Delta}H}_{trans}^{}$, can be measured by calorimetry methods\ndiscussed in Chapter , and the entropy of transformation,\n${\\mathrm{\\Delta}S}_{trans}^{}$, is then calculated using the\nequilibrium condition of equal Gibbs energy of the two phases, i.e.\n\n*Eq. 6\u20111*\n${\\mathrm{\\Delta}S}_{trans}^{} = \\frac{{\\mathrm{\\Delta}H}_{trans}^{}}{T_{trans}^{}}$\n\nwhere $T_{trans}^{}$ is the transition temperature.\n${\\mathrm{\\Delta}H}_{trans}^{}$ and ${\\mathrm{\\Delta}S}_{trans}^{}$ are\nthen used to evaluate the integration constants, $b^{'}$ and $a$, in the\nplace of $S_{298.15}$ and $H_{298.15}^{SER}$ for the structure in the\nSER state.\n\nThis works well for the stable temperature range of each phase. However,\nthere is an issue in extrapolation above and below the melting\ntemperature (Tm). It is known that the heat capacity of the\nsolid phase, , increases with temperature, while that of the liquid\nphase, , is typically constant. The extrapolation of the Gibbs energy of\na solid phase to above its melting temperature can result in the solid\nphase becoming more stable than the liquid phase at high temperatures.\nBy the same token, the extrapolation of the Gibbs energy of a liquid\nphase to below its melting temperature can result in the liquid phase\nbecoming more stable than the solid phase at low temperatures. To\naddress this problem, it is proposed by SGTE that the heat capacity of\nthe solid phase approaches that of the liquid at high temperatures, and\nthat of the liquid phase approaches that of the solid phase at low\ntemperatures using the following equations\n\n1. for solid at T\\>Tm\n\n*Eq. 6\u20112*\n\n*Eq. 6\u20113*\n\n2. for liquid at T\\m\n\n*Eq. 6\u20114*\n\n*Eq. 6\u20115*\n\nAs an example, the heat capacity of solid fcc Al and liquid Al in the\nSGTE pure element database is plotted in . It can be seen that the heat\ncapacity of fcc Al approaches that of liquid Al at high temperatures,\nwhile the heat capacity of liquid Al approaches that of fcc Al at low\ntemperatures. It ensures that the liquid Al is stable at high\ntemperatures, and fcc Al is stable at low temperatures. However, this\nsimple model for liquid is often not satisfactory, in comparison with\navailable experimental data in supercooled liquid, particularly those\nsystems with glass transitions. New models are thus needed and are being\ndeveloped in the CALPHAD community.\n\nFigure \u2011: Heat capacity of fcc Al solid and liquid as a function of\ntemperature\n", + "source": [ + "## Modeling of pure elements\n", + "\n", + "In modeling the Gibbs energy of pure elements in the SER structure in\n", + "terms of , the coefficients in are evaluated using the heat capacity\n", + "data, $b^{'}$ in is evaluated by the value of $S_{298.15}$, and $a$ in\n", + "is evaluated by $H_{298.15}^{SER} = 0$, respectively. For the high\n", + "temperature phase, the enthalpy of transformation from the low\n", + "temperature phase to the high temperature phase,\n", + "${\\mathrm{\\Delta}H}_{trans}^{}$, can be measured by calorimetry methods\n", + "discussed in Chapter , and the entropy of transformation,\n", + "${\\mathrm{\\Delta}S}_{trans}^{}$, is then calculated using the\n", + "equilibrium condition of equal Gibbs energy of the two phases, i.e.\n", + "\n", + "*Eq. 6\u20111*\n", + "${\\mathrm{\\Delta}S}_{trans}^{} = \\frac{{\\mathrm{\\Delta}H}_{trans}^{}}{T_{trans}^{}}$\n", + "\n", + "where $T_{trans}^{}$ is the transition temperature.\n", + "${\\mathrm{\\Delta}H}_{trans}^{}$ and ${\\mathrm{\\Delta}S}_{trans}^{}$ are\n", + "then used to evaluate the integration constants, $b^{'}$ and $a$, in the\n", + "place of $S_{298.15}$ and $H_{298.15}^{SER}$ for the structure in the\n", + "SER state.\n", + "\n", + "This works well for the stable temperature range of each phase. However,\n", + "there is an issue in extrapolation above and below the melting\n", + "temperature (Tm). It is known that the heat capacity of the\n", + "solid phase, , increases with temperature, while that of the liquid\n", + "phase, , is typically constant. The extrapolation of the Gibbs energy of\n", + "a solid phase to above its melting temperature can result in the solid\n", + "phase becoming more stable than the liquid phase at high temperatures.\n", + "By the same token, the extrapolation of the Gibbs energy of a liquid\n", + "phase to below its melting temperature can result in the liquid phase\n", + "becoming more stable than the solid phase at low temperatures. To\n", + "address this problem, it is proposed by SGTE that the heat capacity of\n", + "the solid phase approaches that of the liquid at high temperatures, and\n", + "that of the liquid phase approaches that of the solid phase at low\n", + "temperatures using the following equations\n", + "\n", + "1. for solid at T\\>Tm\n", + "\n", + "*Eq. 6\u20112*\n", + "\n", + "*Eq. 6\u20113*\n", + "\n", + "2. for liquid at T\\m\n", + "\n", + "*Eq. 6\u20114*\n", + "\n", + "*Eq. 6\u20115*\n", + "\n", + "As an example, the heat capacity of solid fcc Al and liquid Al in the\n", + "SGTE pure element database is plotted in . It can be seen that the heat\n", + "capacity of fcc Al approaches that of liquid Al at high temperatures,\n", + "while the heat capacity of liquid Al approaches that of fcc Al at low\n", + "temperatures. It ensures that the liquid Al is stable at high\n", + "temperatures, and fcc Al is stable at low temperatures. However, this\n", + "simple model for liquid is often not satisfactory, in comparison with\n", + "available experimental data in supercooled liquid, particularly those\n", + "systems with glass transitions. New models are thus needed and are being\n", + "developed in the CALPHAD community.\n", + "\n", + "Figure \u2011: Heat capacity of fcc Al solid and liquid as a function of\n", + "temperature\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_random_solution_phases.ipynb b/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_random_solution_phases.ipynb index 642b5db..c493ede 100644 --- a/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_random_solution_phases.ipynb +++ b/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_random_solution_phases.ipynb @@ -4,9 +4,59 @@ "metadata": {}, "cells": [ { - "id": "a262dc50", + "id": "1375de02", "cell_type": "markdown", - "source": "## Modeling of random solution phases\n\nDepending on the degree of short-range ordering in a solution phase,\nvarious Gibbs energy models are available as discussed in Chapter . When\nthe short-range ordering is weak, it can be accounted for by the\ncomposition-dependence of excess Gibbs energy in a binary system in\nterms of the Redlich-Kister polynomial as follows\n\n*Eq. 6\u20117*\n\nwhere the interaction parameters, , can be temperature dependent or even\nhave contributions from heat capacity in the form of when data is\navailable. shows that and are symmetrical with respect of composition,\nwhile is asymmetrical. Their individual contributions to the excess\nGibbs energy are shown in with all interaction parameters being -30,000\nJ/mole-of-atoms.\n\nFigure \u2011: Contributions of interactions parameters to the excess Gibbs\nenergy\n\nIt can be seen in that even though all interaction parameters are\nnegative, the asymmetrical feature of results in the curvature change in\nthe excess Gibbs energy as a function of composition. This indicates the\ntendency to form a miscibility gap at low temperatures. The interaction\nparameters are to be evaluated from the data of enthalpy, entropy, and\nheat capacity of mixing. The experimental data on enthalpy of mixing are\navailable for the liquid phase in some systems, but typically very\nlimited for solid solution phases. The first-principles calculations can\npredict the enthalpy, entropy, and heat capacity of mixing in solid\nsolution phases using the dilute solution approach with one solute atom\nin a supercell and the CPA/CE/SQS approach for concentrated solutions as\ndiscussed in Chapter . It demonstrates again that the interaction\nparameters and the lattice stability jointly determine the Gibbs energy\nof an individual phase. The change of lattice stability requires the\nre-evaluation of interaction parameters.\n\nFor individual phases with strong short-range ordering, quasichemical or\nassociated models can be used. As discussed in Chapter , with fixed\ncomposition in the system, the amounts of various chemical bonds or\nassociates are related through the mass conservation in the system and\nare calculated through the minimization of Gibbs energy of the phase\nwith given temperature, pressure, and the amount of each independent\ncomponents. The model parameters include the formation energy of bonds\nor associates and interaction between various bonds or associates,\nnoting that pure elements can be considered as the simplest associates.\nThe interactions between pure elements can be predicted from\nfirst-principles calculations as mentioned above, but currently there\nare no efficient approaches to predict the interactions between\nassociates from first-principles calculations.\n", + "source": [ + "## Modeling of random solution phases\n", + "\n", + "Depending on the degree of short-range ordering in a solution phase,\n", + "various Gibbs energy models are available as discussed in Chapter . When\n", + "the short-range ordering is weak, it can be accounted for by the\n", + "composition-dependence of excess Gibbs energy in a binary system in\n", + "terms of the Redlich-Kister polynomial as follows\n", + "\n", + "*Eq. 6\u20117*\n", + "\n", + "where the interaction parameters, , can be temperature dependent or even\n", + "have contributions from heat capacity in the form of when data is\n", + "available. shows that and are symmetrical with respect of composition,\n", + "while is asymmetrical. Their individual contributions to the excess\n", + "Gibbs energy are shown in with all interaction parameters being -30,000\n", + "J/mole-of-atoms.\n", + "\n", + "Figure \u2011: Contributions of interactions parameters to the excess Gibbs\n", + "energy\n", + "\n", + "It can be seen in that even though all interaction parameters are\n", + "negative, the asymmetrical feature of results in the curvature change in\n", + "the excess Gibbs energy as a function of composition. This indicates the\n", + "tendency to form a miscibility gap at low temperatures. The interaction\n", + "parameters are to be evaluated from the data of enthalpy, entropy, and\n", + "heat capacity of mixing. The experimental data on enthalpy of mixing are\n", + "available for the liquid phase in some systems, but typically very\n", + "limited for solid solution phases. The first-principles calculations can\n", + "predict the enthalpy, entropy, and heat capacity of mixing in solid\n", + "solution phases using the dilute solution approach with one solute atom\n", + "in a supercell and the CPA/CE/SQS approach for concentrated solutions as\n", + "discussed in Chapter . It demonstrates again that the interaction\n", + "parameters and the lattice stability jointly determine the Gibbs energy\n", + "of an individual phase. The change of lattice stability requires the\n", + "re-evaluation of interaction parameters.\n", + "\n", + "For individual phases with strong short-range ordering, quasichemical or\n", + "associated models can be used. As discussed in Chapter , with fixed\n", + "composition in the system, the amounts of various chemical bonds or\n", + "associates are related through the mass conservation in the system and\n", + "are calculated through the minimization of Gibbs energy of the phase\n", + "with given temperature, pressure, and the amount of each independent\n", + "components. The model parameters include the formation energy of bonds\n", + "or associates and interaction between various bonds or associates,\n", + "noting that pure elements can be considered as the simplest associates.\n", + "The interactions between pure elements can be predicted from\n", + "first-principles calculations as mentioned above, but currently there\n", + "are no efficient approaches to predict the interactions between\n", + "associates from first-principles calculations.\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_solution_phases_with_longrange_ordering.ipynb b/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_solution_phases_with_longrange_ordering.ipynb index 0dfae13..cd2e24e 100644 --- a/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_solution_phases_with_longrange_ordering.ipynb +++ b/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_solution_phases_with_longrange_ordering.ipynb @@ -4,9 +4,166 @@ "metadata": {}, "cells": [ { - "id": "de1cecb8", + "id": "7036eab8", "cell_type": "markdown", - "source": "## Modeling of solution phases with long-range ordering\n\nThe commonly used Gibbs energy model is shown in Chapter with the\ncrystal lattice divided into sublattices, often referred to as the\ncompound energy formalism \\[55\\]. The Gibbs energy of end-members\nrepresented by plays the same important role for solution phases with\nsublattice as the lattice stability for random solution phases. The\nend-members are modeled in the same way as the stoichiometric phases\ndiscussed in Chapter . The enthalpy and entropy of mixing in each\nsublattice can be predicted by first-principles calculations using the\ndilute solution and SQS approaches discussed in Chapter and modeled in\nthe same way as the random solution discussed in Chapter .\n\nIt is important to realize that with a simple two-sublattice model of\n(A,B)a(C,D)b, the miscibility gap can easily form\neven without any interaction parameters when the Gibbs energy of\nend-members differ from each other significantly. The contribution of\nend-members to the Gibbs energy of the phase, i.e. , is re-written as\nfollows and schematically shown in ,\n\n*Eq. 6\u20118*\n\nFigure \u2011: Schematic diagrams depicting (a) the concentration square with\nthe site fractions of B and D on the horizontal and vertical axes,\nrespectively, and (b) Gibbs energy reference plan for\n(A,B)a(C,D)b, as represented by .\n\nFrom b, it is evident that there is a strong tendency to form a\nmiscibility gap between the composition sets of\n(A)a(C)b and (B)a(D)b due to\ntheir lower Gibbs energies than the other two end-members. Since it\nwould be rare for all four end-members to have their Gibbs energy values\nequal, the miscibility gap in this type of phases is practically\ninevitable at low temperatures. An example is shown in for the complex\ntitanium niobium carbonitride (Ti,Nb)(C,N). The solid lines parallel to\nthe direction from NbC to TiN are tie-lines. The Gibbs energy values of\nTiC, TiN, NbC and NbN are -144495, -229236, -132324, and -179772\nJ/mole-of-atoms, respectively. The Gibbs energy value of TiN is\nsignificantly lower than the other values, resulting in the tie-lines\noriginating from the TiN corner.\n\nFigure \u2011: Miscibility gap in (Ti,Nb)(C,N) at 1673K with the straight\nlines in the middle of the plot being tie-lines.\n\nThe order-disorder transitions can be similarly described with the\nsimplest case being a two-sublattice model of\n(A,B)a(A,B)b. When the site fractions of A or B in\nboth sublattices are the same, it becomes a random solution model; when\nthey are different, the phase is partially ordered; and when there is\nonly one component in each sublattice, the phase is fully ordered as a\nstoichiometric compound. The Gibbs energy of this phase is obtained from\nas follows\n\n*Eq. 6\u20119*\n\n$G_{mf} = y_{A}^{I}y_{A}^{II}\\ _{\\ }^{0}G_{A:A} + y_{A}^{I}y_{B}^{II}\\ _{\\ }^{0}G_{A:B} + y_{B}^{I}y_{A}^{II}\\ _{\\ }^{0}G_{B:A} + y_{B}^{I}y_{B}^{II}\\ _{\\ }^{0}G_{B:B} + + aRT\\left( y_{A}^{I}\\ln y_{A}^{I} + y_{B}^{I}\\ln y_{B}^{I} \\right) + bRT\\left( y_{A}^{II}\\ln y_{A}^{II} + y_{B}^{II}\\ln y_{B}^{II} \\right) + y_{A}^{II}y_{A}^{I}y_{B}^{I}\\ L_{A,B:A} + y_{B}^{II}y_{A}^{I}y_{B}^{I}\\ L_{A,B:B} + y_{A}^{I}y_{A}^{II}y_{B}^{II}\\ L_{A:A,B} + y_{B}^{I}y_{A}^{II}y_{B}^{II}\\ L_{B:A,B} + y_{A}^{I}y_{B}^{I}\\ y_{A}^{II}y_{B}^{II}L_{A,B:A,B}$\n\nwhere the superscript denotes the sublattice, and column and comma\nseparate sublattices and interaction components, respectively. The\nrelationship between site fraction and over-all atomic fractions in such\na two-sublattice model can be represented by and schematically shown in\n. The two red dashed lines represent the phase with\n$x_{B} = \\frac{a}{a + b}$, but different a/b ratios. Along the red\ndashed lines, the phase can adjust the site fraction to minimize its\nGibbs energy, i.e. it has one internal degree of freedom to be either\ndisordered on the blue diagonal line between A:A and B:B, or ordered at\nanywhere else. The interplay of interaction parameters and site\nfractions is depicted where $L_{A,B:A}$, $L_{A,B:B}$, $L_{A:A,B}$, and\n$L_{B:A,B}$ affect the four sides, and $L_{A,B:A,B}$ influences the\ncenter part.\n\nFigure \u2011: Schematic composition square of\n(A,B)a(A,B)b.\n\nWhen fully disordered with $y_{A}^{I} = \\ y_{A}^{II} = x_{A}$ and\n$y_{B}^{I} = y_{B}^{II} = x_{B}$, becomes\n\n*Eq. 6\u201110*\n\n$$G_{mf} = x_{A}\\left( 1 - x_{B} \\right)_{\\ }^{0}G_{A:A} + x_{A}x_{B}\\ _{\\ }^{0}G_{A:B} + x_{A}x_{B}\\ _{\\ }^{0}G_{B:A} + x_{B}\\left( 1 - x_{A} \\right)\\ _{\\ }^{0}G_{B:B} + + (a + b)RT\\left( x_{A}\\ln x_{A} + x_{B}\\ln x_{B} \\right) + \\ x_{A}x_{B}x_{A}L_{A,B:A} + x_{A}x_{B}x_{B}\\ L_{A,B:B} + x_{A}x_{B}x_{A}\\ L_{A:A,B} + x_{A}x_{B}x_{B}\\ L_{B:A,B} + x_{A}x_{A}x_{B}x_{B}L_{A,B:A,B} = x_{A}_{\\ }^{0}G_{A:A} + x_{B}\\ _{\\ }^{0}G_{B:B} + (a + b)RT\\left( x_{A}\\ln x_{A} + x_{B}\\ln x_{B} \\right) + x_{A}x_{B}\\left\\lbrack \\left(_{\\ }^{0}G_{A:B} +_{\\ }^{0}G_{B:A} -_{\\ }^{0}G_{A:A} -_{\\ }^{0}G_{B:B} \\right) + x_{A}\\left( L_{A,B:A} + L_{A:A,B} \\right) + x_{B}\\ \\left( L_{A,B:B} + L_{B:A,B} \\right) + x_{A}x_{B}L_{A,B:A,B} \\right\\rbrack = (a + b)\\left\\lbrack x_{A}_{\\ }^{0}G_{A} + x_{B}\\ _{\\ }^{0}G_{B} + RT\\left( x_{A}\\ln x_{A} + x_{B}\\ln x_{B} \\right) + x_{A}x_{B}L_{A,B} \\right\\rbrack$$\n\nwith\n\n*Eq. 6\u201111* $_{\\ }^{0}G_{A:A} = (a + b)_{\\ }^{0}G_{A}$\n\n*Eq. 6\u201112* $_{\\ }^{0}G_{B:B} = (a + b)_{\\ }^{0}G_{B}$\n\n*Eq. 6\u201113*\n\n$$L_{A,B} = \\left\\lbrack \\left(_{\\ }^{0}G_{A:B} +_{\\ }^{0}G_{B:A} -_{\\ }^{0}G_{A:A} -_{\\ }^{0}G_{B:B} \\right) + x_{A}\\left( L_{A,B:A} + L_{A:A,B} \\right) + x_{B}\\ \\left( L_{A,B:B} + L_{B:A,B} \\right) + x_{A}x_{B}L_{A,B:A,B} \\right\\rbrack/(a + b)$$\n\nwhere $_{\\ }^{0}G_{A}$, $_{\\ }^{0}G_{B}$, and $L_{A,B}$ are the molar\nGibbs energy of pure A and B and the molar interaction parameter in the\ndisordered solid solution, respectively. It is evident that the\ninteratcation paramter $L_{A,B}$ is fully determined by the parameters\nin the ordered phase, but the parameters in the ordered phase are not\nuniquely determined by the interaction parameters in the disordered\nphase.\n\nDue to crystal symmetry, some of the parameters in are related. For\nexample, in the BCC A2/B2 ordering with a=b=0.5, the BCC corner or\ncenter lattice sites are favored by either one type of atoms, but the\ntwo sublattices are equivalent crystallographically, resulting in\nfollowing relations\n\n*Eq. 6\u201114* $\\ _{\\ }^{0}G_{A:B} = \\ _{\\ }^{0}G_{B:A}$\n\n*Eq. 6\u201115* $L_{A,B:A} = L_{A:A,B}$\n\n*Eq. 6\u201116* $L_{A,B:B} = \\ L_{B:A,B}$\n\nFor more complex ordering of BCC lattice such as B32, D03,\nand L21 shown in with ideal compositions being AB,\nA3B, and A2BC, respectively, more sublattices are\nneeded, noting that the L21 Heusler structure exists in\nternary systems only. To use one model to describe all ordering in the\nBCC lattice, the minimum cluster is an irregular tetrahedron with four\nsublattices as depicted in as discussed in the modeling of the Al-Fe\nsystem \\[56\\]. In such a four sublattice model of\n$(A,B)_{0.25}^{I}(A,B)_{0.25}^{II}(A,B)_{0.25}^{II}(A,B)_{0.25}^{IV}$,\nthe site fractions of A2, B2, B32, and D03, are represented\nby $y_{i}^{I} = \\ y_{i}^{II} = y_{i}^{III} = \\ y_{i}^{IV}$,\n$y_{i}^{I} = \\ y_{i}^{II} \\neq y_{i}^{III} = \\ y_{i}^{IV}$,\n$y_{i}^{I} = \\ y_{i}^{III} \\neq y_{i}^{II} = \\ y_{i}^{IV}$, and\n$y_{i}^{I} = \\ y_{i}^{II} \\neq y_{i}^{III} \\neq \\ y_{i}^{IV}$,\nrespectively. The site fractions of L21 are the same as those\nof D03 except with at least three components.\n\nFigure \u2011: Atomic structures and four sublattice tetrahedrons of BCC\ndisordered and ordered phases.\n\nAnother common ordering phenomenon is in the FCC lattice including the\ndisordered A1 structure and ordered L10 and L12\nstructures as shown in . In the L10 structure, the\nneighboring (001) planes are favored by different atoms, respectively,\nresulting in an ideal composition of AB. While in the L12\nstructure, the corners and faces are favored by different atoms,\nrespectively, resulting in an ideal composition of A3B. In a\nfour sublattice model of\n$(A,B)_{0.25}^{I}(A,B)_{0.25}^{II}(A,B)_{0.25}^{II}(A,B)_{0.25}^{IV}$,\nthe site fractions of A1, L10 and L12 are\nrepresented by $y_{i}^{I} = \\ y_{i}^{II} = y_{i}^{III} = \\ y_{i}^{IV}$,\n$y_{i}^{I} = \\ y_{i}^{II} \\neq y_{i}^{III} = \\ y_{i}^{IV}$, and\n$y_{i}^{I} = \\ y_{i}^{II} = y_{i}^{III} \\neq \\ y_{i}^{IV}$, respectively\n\\[57\\].\n\nFigure \u2011: Atomic structures and four sublattice tetrahedrons of FCC\ndisordered and ordered phases.\n\nAs mentioned above, the interaction parameters in each sublattice can be\npredicted by first-principles calculations using the dilute solution and\nSQS approaches when there is only one component in each of the remaining\nsublattices. For the interaction involving two components in two or more\nsublattices, i.e. $L_{A,B:A,B}$ in applicable to four sublattice models\n\\[58\\], the energetics from cluster expansion (CE) approach discussed in\nChapter can be used to evaluate the interaction parameters.\n", + "source": [ + "## Modeling of solution phases with long-range ordering\n", + "\n", + "The commonly used Gibbs energy model is shown in Chapter with the\n", + "crystal lattice divided into sublattices, often referred to as the\n", + "compound energy formalism \\[55\\]. The Gibbs energy of end-members\n", + "represented by plays the same important role for solution phases with\n", + "sublattice as the lattice stability for random solution phases. The\n", + "end-members are modeled in the same way as the stoichiometric phases\n", + "discussed in Chapter . The enthalpy and entropy of mixing in each\n", + "sublattice can be predicted by first-principles calculations using the\n", + "dilute solution and SQS approaches discussed in Chapter and modeled in\n", + "the same way as the random solution discussed in Chapter .\n", + "\n", + "It is important to realize that with a simple two-sublattice model of\n", + "(A,B)a(C,D)b, the miscibility gap can easily form\n", + "even without any interaction parameters when the Gibbs energy of\n", + "end-members differ from each other significantly. The contribution of\n", + "end-members to the Gibbs energy of the phase, i.e. , is re-written as\n", + "follows and schematically shown in ,\n", + "\n", + "*Eq. 6\u20118*\n", + "\n", + "Figure \u2011: Schematic diagrams depicting (a) the concentration square with\n", + "the site fractions of B and D on the horizontal and vertical axes,\n", + "respectively, and (b) Gibbs energy reference plan for\n", + "(A,B)a(C,D)b, as represented by .\n", + "\n", + "From b, it is evident that there is a strong tendency to form a\n", + "miscibility gap between the composition sets of\n", + "(A)a(C)b and (B)a(D)b due to\n", + "their lower Gibbs energies than the other two end-members. Since it\n", + "would be rare for all four end-members to have their Gibbs energy values\n", + "equal, the miscibility gap in this type of phases is practically\n", + "inevitable at low temperatures. An example is shown in for the complex\n", + "titanium niobium carbonitride (Ti,Nb)(C,N). The solid lines parallel to\n", + "the direction from NbC to TiN are tie-lines. The Gibbs energy values of\n", + "TiC, TiN, NbC and NbN are -144495, -229236, -132324, and -179772\n", + "J/mole-of-atoms, respectively. The Gibbs energy value of TiN is\n", + "significantly lower than the other values, resulting in the tie-lines\n", + "originating from the TiN corner.\n", + "\n", + "Figure \u2011: Miscibility gap in (Ti,Nb)(C,N) at 1673K with the straight\n", + "lines in the middle of the plot being tie-lines.\n", + "\n", + "The order-disorder transitions can be similarly described with the\n", + "simplest case being a two-sublattice model of\n", + "(A,B)a(A,B)b. When the site fractions of A or B in\n", + "both sublattices are the same, it becomes a random solution model; when\n", + "they are different, the phase is partially ordered; and when there is\n", + "only one component in each sublattice, the phase is fully ordered as a\n", + "stoichiometric compound. The Gibbs energy of this phase is obtained from\n", + "as follows\n", + "\n", + "*Eq. 6\u20119*\n", + "\n", + "$G_{mf} = y_{A}^{I}y_{A}^{II}\\ _{\\ }^{0}G_{A:A} + y_{A}^{I}y_{B}^{II}\\ _{\\ }^{0}G_{A:B} + y_{B}^{I}y_{A}^{II}\\ _{\\ }^{0}G_{B:A} + y_{B}^{I}y_{B}^{II}\\ _{\\ }^{0}G_{B:B} + + aRT\\left( y_{A}^{I}\\ln y_{A}^{I} + y_{B}^{I}\\ln y_{B}^{I} \\right) + bRT\\left( y_{A}^{II}\\ln y_{A}^{II} + y_{B}^{II}\\ln y_{B}^{II} \\right) + y_{A}^{II}y_{A}^{I}y_{B}^{I}\\ L_{A,B:A} + y_{B}^{II}y_{A}^{I}y_{B}^{I}\\ L_{A,B:B} + y_{A}^{I}y_{A}^{II}y_{B}^{II}\\ L_{A:A,B} + y_{B}^{I}y_{A}^{II}y_{B}^{II}\\ L_{B:A,B} + y_{A}^{I}y_{B}^{I}\\ y_{A}^{II}y_{B}^{II}L_{A,B:A,B}$\n", + "\n", + "where the superscript denotes the sublattice, and column and comma\n", + "separate sublattices and interaction components, respectively. The\n", + "relationship between site fraction and over-all atomic fractions in such\n", + "a two-sublattice model can be represented by and schematically shown in\n", + ". The two red dashed lines represent the phase with\n", + "$x_{B} = \\frac{a}{a + b}$, but different a/b ratios. Along the red\n", + "dashed lines, the phase can adjust the site fraction to minimize its\n", + "Gibbs energy, i.e. it has one internal degree of freedom to be either\n", + "disordered on the blue diagonal line between A:A and B:B, or ordered at\n", + "anywhere else. The interplay of interaction parameters and site\n", + "fractions is depicted where $L_{A,B:A}$, $L_{A,B:B}$, $L_{A:A,B}$, and\n", + "$L_{B:A,B}$ affect the four sides, and $L_{A,B:A,B}$ influences the\n", + "center part.\n", + "\n", + "Figure \u2011: Schematic composition square of\n", + "(A,B)a(A,B)b.\n", + "\n", + "When fully disordered with $y_{A}^{I} = \\ y_{A}^{II} = x_{A}$ and\n", + "$y_{B}^{I} = y_{B}^{II} = x_{B}$, becomes\n", + "\n", + "*Eq. 6\u201110*\n", + "\n", + "$$G_{mf} = x_{A}\\left( 1 - x_{B} \\right)_{\\ }^{0}G_{A:A} + x_{A}x_{B}\\ _{\\ }^{0}G_{A:B} + x_{A}x_{B}\\ _{\\ }^{0}G_{B:A} + x_{B}\\left( 1 - x_{A} \\right)\\ _{\\ }^{0}G_{B:B} + + (a + b)RT\\left( x_{A}\\ln x_{A} + x_{B}\\ln x_{B} \\right) + \\ x_{A}x_{B}x_{A}L_{A,B:A} + x_{A}x_{B}x_{B}\\ L_{A,B:B} + x_{A}x_{B}x_{A}\\ L_{A:A,B} + x_{A}x_{B}x_{B}\\ L_{B:A,B} + x_{A}x_{A}x_{B}x_{B}L_{A,B:A,B} = x_{A}_{\\ }^{0}G_{A:A} + x_{B}\\ _{\\ }^{0}G_{B:B} + (a + b)RT\\left( x_{A}\\ln x_{A} + x_{B}\\ln x_{B} \\right) + x_{A}x_{B}\\left\\lbrack \\left(_{\\ }^{0}G_{A:B} +_{\\ }^{0}G_{B:A} -_{\\ }^{0}G_{A:A} -_{\\ }^{0}G_{B:B} \\right) + x_{A}\\left( L_{A,B:A} + L_{A:A,B} \\right) + x_{B}\\ \\left( L_{A,B:B} + L_{B:A,B} \\right) + x_{A}x_{B}L_{A,B:A,B} \\right\\rbrack = (a + b)\\left\\lbrack x_{A}_{\\ }^{0}G_{A} + x_{B}\\ _{\\ }^{0}G_{B} + RT\\left( x_{A}\\ln x_{A} + x_{B}\\ln x_{B} \\right) + x_{A}x_{B}L_{A,B} \\right\\rbrack$$\n", + "\n", + "with\n", + "\n", + "*Eq. 6\u201111* $_{\\ }^{0}G_{A:A} = (a + b)_{\\ }^{0}G_{A}$\n", + "\n", + "*Eq. 6\u201112* $_{\\ }^{0}G_{B:B} = (a + b)_{\\ }^{0}G_{B}$\n", + "\n", + "*Eq. 6\u201113*\n", + "\n", + "$$L_{A,B} = \\left\\lbrack \\left(_{\\ }^{0}G_{A:B} +_{\\ }^{0}G_{B:A} -_{\\ }^{0}G_{A:A} -_{\\ }^{0}G_{B:B} \\right) + x_{A}\\left( L_{A,B:A} + L_{A:A,B} \\right) + x_{B}\\ \\left( L_{A,B:B} + L_{B:A,B} \\right) + x_{A}x_{B}L_{A,B:A,B} \\right\\rbrack/(a + b)$$\n", + "\n", + "where $_{\\ }^{0}G_{A}$, $_{\\ }^{0}G_{B}$, and $L_{A,B}$ are the molar\n", + "Gibbs energy of pure A and B and the molar interaction parameter in the\n", + "disordered solid solution, respectively. It is evident that the\n", + "interatcation paramter $L_{A,B}$ is fully determined by the parameters\n", + "in the ordered phase, but the parameters in the ordered phase are not\n", + "uniquely determined by the interaction parameters in the disordered\n", + "phase.\n", + "\n", + "Due to crystal symmetry, some of the parameters in are related. For\n", + "example, in the BCC A2/B2 ordering with a=b=0.5, the BCC corner or\n", + "center lattice sites are favored by either one type of atoms, but the\n", + "two sublattices are equivalent crystallographically, resulting in\n", + "following relations\n", + "\n", + "*Eq. 6\u201114* $\\ _{\\ }^{0}G_{A:B} = \\ _{\\ }^{0}G_{B:A}$\n", + "\n", + "*Eq. 6\u201115* $L_{A,B:A} = L_{A:A,B}$\n", + "\n", + "*Eq. 6\u201116* $L_{A,B:B} = \\ L_{B:A,B}$\n", + "\n", + "For more complex ordering of BCC lattice such as B32, D03,\n", + "and L21 shown in with ideal compositions being AB,\n", + "A3B, and A2BC, respectively, more sublattices are\n", + "needed, noting that the L21 Heusler structure exists in\n", + "ternary systems only. To use one model to describe all ordering in the\n", + "BCC lattice, the minimum cluster is an irregular tetrahedron with four\n", + "sublattices as depicted in as discussed in the modeling of the Al-Fe\n", + "system \\[56\\]. In such a four sublattice model of\n", + "$(A,B)_{0.25}^{I}(A,B)_{0.25}^{II}(A,B)_{0.25}^{II}(A,B)_{0.25}^{IV}$,\n", + "the site fractions of A2, B2, B32, and D03, are represented\n", + "by $y_{i}^{I} = \\ y_{i}^{II} = y_{i}^{III} = \\ y_{i}^{IV}$,\n", + "$y_{i}^{I} = \\ y_{i}^{II} \\neq y_{i}^{III} = \\ y_{i}^{IV}$,\n", + "$y_{i}^{I} = \\ y_{i}^{III} \\neq y_{i}^{II} = \\ y_{i}^{IV}$, and\n", + "$y_{i}^{I} = \\ y_{i}^{II} \\neq y_{i}^{III} \\neq \\ y_{i}^{IV}$,\n", + "respectively. The site fractions of L21 are the same as those\n", + "of D03 except with at least three components.\n", + "\n", + "Figure \u2011: Atomic structures and four sublattice tetrahedrons of BCC\n", + "disordered and ordered phases.\n", + "\n", + "Another common ordering phenomenon is in the FCC lattice including the\n", + "disordered A1 structure and ordered L10 and L12\n", + "structures as shown in . In the L10 structure, the\n", + "neighboring (001) planes are favored by different atoms, respectively,\n", + "resulting in an ideal composition of AB. While in the L12\n", + "structure, the corners and faces are favored by different atoms,\n", + "respectively, resulting in an ideal composition of A3B. In a\n", + "four sublattice model of\n", + "$(A,B)_{0.25}^{I}(A,B)_{0.25}^{II}(A,B)_{0.25}^{II}(A,B)_{0.25}^{IV}$,\n", + "the site fractions of A1, L10 and L12 are\n", + "represented by $y_{i}^{I} = \\ y_{i}^{II} = y_{i}^{III} = \\ y_{i}^{IV}$,\n", + "$y_{i}^{I} = \\ y_{i}^{II} \\neq y_{i}^{III} = \\ y_{i}^{IV}$, and\n", + "$y_{i}^{I} = \\ y_{i}^{II} = y_{i}^{III} \\neq \\ y_{i}^{IV}$, respectively\n", + "\\[57\\].\n", + "\n", + "Figure \u2011: Atomic structures and four sublattice tetrahedrons of FCC\n", + "disordered and ordered phases.\n", + "\n", + "As mentioned above, the interaction parameters in each sublattice can be\n", + "predicted by first-principles calculations using the dilute solution and\n", + "SQS approaches when there is only one component in each of the remaining\n", + "sublattices. For the interaction involving two components in two or more\n", + "sublattices, i.e. $L_{A,B:A,B}$ in applicable to four sublattice models\n", + "\\[58\\], the energetics from cluster expansion (CE) approach discussed in\n", + "Chapter can be used to evaluate the interaction parameters.\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_stoichiometric_phases.ipynb b/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_stoichiometric_phases.ipynb index 3c92f8e..038e9c9 100644 --- a/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_stoichiometric_phases.ipynb +++ b/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_stoichiometric_phases.ipynb @@ -4,9 +4,38 @@ "metadata": {}, "cells": [ { - "id": "e05f5889", + "id": "a6ae79da", "cell_type": "markdown", - "source": "## Modeling of stoichiometric phases\n\nThe Gibbs energy of a stoichiometric phase can be modeled in the same\nway as that of pure elements discussed above using the data of heat\ncapacity, $S_{298.15}$, and enthalpy of formation at 298.15 K (). When\nthese data are not available from experiments, they can be predicted by\nfirst-principles calculations as discussed Chapter . It should be\npointed out that constraints placed on heat capacity of stoichiometric\ncompounds above melting temperatures, i.e. and , have not been\nrigorously implemented in the literature for such modeling, partially\nbecause the heat capacity of the corresponding liquid solution is not\nwell established, and the heat capacity of a compound is often not\navailable.\n\nWhen the data of heat capacity is not available, a simple approach,\ncommonly referred to as Neumann\u2013Kopp Rule assuming that the heat\ncapacity of formation of is zero, can be used. The Gibbs energy of the\ncompound is written as\n\n*Eq. 6\u20116*\n$G = \\sum_{}^{}N_{i}{_{\\ }^{0}G}_{i}^{ref} + \\mathrm{\\Delta}_{f}H - T\\mathrm{\\Delta}_{f}S$\n\nwith $\\mathrm{\\Delta}_{f}H$ and $\\mathrm{\\Delta}_{f}S$ modeled as\nconstants. An additional draw back of is that the melting temperature of\nthe compound can be higher than those of pure elements, and the heat\ncapacity of the compound may thus be questionable at temperatures higher\nthan the melting temperatures of pure elements due to the non-physical\ncontributions from pure elements based on .\n", + "source": [ + "## Modeling of stoichiometric phases\n", + "\n", + "The Gibbs energy of a stoichiometric phase can be modeled in the same\n", + "way as that of pure elements discussed above using the data of heat\n", + "capacity, $S_{298.15}$, and enthalpy of formation at 298.15 K (). When\n", + "these data are not available from experiments, they can be predicted by\n", + "first-principles calculations as discussed Chapter . It should be\n", + "pointed out that constraints placed on heat capacity of stoichiometric\n", + "compounds above melting temperatures, i.e. and , have not been\n", + "rigorously implemented in the literature for such modeling, partially\n", + "because the heat capacity of the corresponding liquid solution is not\n", + "well established, and the heat capacity of a compound is often not\n", + "available.\n", + "\n", + "When the data of heat capacity is not available, a simple approach,\n", + "commonly referred to as Neumann\u2013Kopp Rule assuming that the heat\n", + "capacity of formation of is zero, can be used. The Gibbs energy of the\n", + "compound is written as\n", + "\n", + "*Eq. 6\u20116*\n", + "$G = \\sum_{}^{}N_{i}{_{\\ }^{0}G}_{i}^{ref} + \\mathrm{\\Delta}_{f}H - T\\mathrm{\\Delta}_{f}S$\n", + "\n", + "with $\\mathrm{\\Delta}_{f}H$ and $\\mathrm{\\Delta}_{f}S$ modeled as\n", + "constants. An additional draw back of is that the melting temperature of\n", + "the compound can be higher than those of pure elements, and the heat\n", + "capacity of the compound may thus be questionable at temperatures higher\n", + "than the melting temperatures of pure elements due to the non-physical\n", + "contributions from pure elements based on .\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_cerium.ipynb b/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_cerium.ipynb index 9208a11..45ebd7e 100644 --- a/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_cerium.ipynb +++ b/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_cerium.ipynb @@ -4,9 +4,205 @@ "metadata": {}, "cells": [ { - "id": "a71a60c2", + "id": "e6bcdb88", "cell_type": "markdown", - "source": "## Application to cerium\n\nCerium (Ce) displays intriguing physical and chemical properties of\nwhich the most fascinating is its first-order isostructural phase\ntransition. This involves a magnetic, high temperature/high volume\n\u201c\u03b3-phase\u201d and a nonmagnetic, low temperature/low volume \u201c\u03b1-phase\u201d, both\nin the same face-centered-cubic (fcc) lattice structure. At 298 K and\n0.7 GPa, the \u03b3\u2192\u03b1 transition is accompanied by a 14-17% volume collapse.\nThe Ce phase transition has been studied extensively including our own\nworks with two or three microstates \\[55, 64\\].\n\nThe simplest model for the system is to consider two microstates:\nferromagnetic and nonmagnetic. The first-principles calculations of the\nfree energy of two Ce microstates are problematic in the absence of\nstrong correlation of the *f*-electrons in the DFT Hamiltonian. The\nrelative stability of the nonmagnetic (\u201cdelocalized\u201d) Ce 4*f* state to\nthat of the magnetic (\u201clocalized\u201d) Ce 4*f* state is greatly\noverestimated in the GGA \\[31-32\\] with spin polarization. The usual\napproach to surmount this is the Dudarev DFT + U method \\[65\\] with the\non-site Coulomb and exchange interactions as described with a\nHartree-Fock approximation added to the DFT Hamiltonian. This method\noffers the advantage that only the difference between the Hubbard U (due\nto the energy increase from an electron addition to a specific site) and\nthe J (due to the screened exchange energy) need to be specified *a\npriori*.\n\nEvaluation of numerous U \u2013 J values over a 1.0 \u2013 6.0 eV range revealed\nthat 1.6 eV gives the most consistent prediction of nonmagnetic Ce and\nmagnetic Ce energetics over the range of atomic volumes that includes\nboth microstates at 0 K. The energy-volume curves thus obtained is\nplotted in , showing that the nonmagnetic microstate is the ground\nstate, and the equilibrium between the two microstates at 0 K is at the\nnegative pressure of -0.87GPa. Since and , NTE does not exist in the\nsystem.\n\nFigure \u2011: Variation of cell energy (eV) with atomic volume\n(\u00c53) for Ce computed with strong correlation based upon\nDudarev\u2019s method with U \u2013 J = 1.6 eV.\n\nTo take into account the possible magnetic disordering in the\nferromagnetic microstate at finite temperatures, the following\ncontribution is added to the free energy of the ferromagnetic microstate\n\nEq. 9\u20119\n\nwhere *MS* is the spin moment, and *l* = 3 the orbital\nangular momentum of an *f*-electron. is a generalization of Hund\u2019s rule,\nwith total angular momentum . The Helmholtz energies thus obtained for\nboth microstates and the system are shown in at several temperatures\nwith the tie-lines included. In the figure, the blue, dot-dashed curves\nare for the nonmagnetic microstate, red, solid curves for ferromagnetic\nmicrostate, cyan shadows for entropy of mixing between two microstates,\nand the circle in (e) is the critical point. The numbers below the black\ndashed line, representing the common tangent curves, mark the transition\npressures. The 0-K static energies of the nonmagnetic microstate and the\nmagnetic microstate are also plotted in (a) using the solid circles and\ndotted lines.\n\nFigure \u2011: Helmholtz energies at (a) 0 K; (b) 100 K; (c) 165 K; (d) 300\nK; (e) 476 K; and (f) 600 K.\n\nThe temperature vs volume phase diagram is plotted and compared with\navailable experimental data as shown in \\[66\\]. In this figure, the\nvolume (V) is normalized to its equilibrium volume (VN) at\natmospheric pressure and room temperature. In the pressure range of 2.25\n\u2013 3.5 GPa, the system is within the single-phase region at all\ntemperatures considered as shown by the five continuous isobaric volumes\nas a function of temperature. In this pressure range, normal thermal\nexpansion is observed at both low and high temperatures on each isobaric\ncurve where the probability of each microstate does not change\nsignificantly with temperature. While in the middle temperature range on\neach isobaric curve, the colossal positive thermal expansion (CPTE),\nhighlighted by the pink open diamond symbols, exists due to the fast\nincrease of probability of the metastable ferromagnetic microstate with\nrespect to temperature, i.e. , , and . This CPTE is much higher than the\nindividual positive thermal expansions of the stable and metastable\nmicrostates, respectively.\n\nFigure \u2011: Calculated temperature-volume phase diagram of Ce.\n\nWith decreasing pressure, the system reaches a critical point (green\ncircle) where the homogeneous single phase becomes unstable, represented\nby and , and both entropy and volume change infinitely. At even lower\npressure, a miscibility gap forms, and the single phase separates into\ntwo phases with the same fcc crystal structures, but different magnetic\nspin structures. Inside the miscibility gap, the volume changes\ndiscontinuously with respect to temperature by the so-called first-order\ntransition as shown by the isobaric curve at zero pressure, compared\nwell with experimental volume data (solid squares) under ambient\npressure.\n\nThe fraction of the ferromagnetic microstate, *x*mag, in \u03b1-Ce\n(blue) and \u03b3-Ce (red) calculated using is plotted in as a function of\npressure along the miscibility gap phase boundary. It can be seen that\nthe fraction of the ferromagnetic microstate in \u03b1-Ce increases with\nincreasing pressure while the fraction of the ferromagnetic microstate\nin \u03b3-Ce decreases. At the critical point, the fraction of ferromagnetic\nmicrostate is calculated to be 0.58. This is in qualitative agreement\nwith the 0.67 value (filled circle) estimated experimentally at the\ncritical point.\n\nFigure \u2011: Fraction of ferromagnetic microstate in \u03b1-Ce (blue) and \u03b3-Ce\n(red) along the \u03b3-\u03b1 phase boundary.\n\nThe relative volume, *V/VN*, as a function of pressure, is\nplotted as the black solid lines in from 200 to 600 K at 50 K\nincrements. The blue and red solid lines correspond to \u03b1-Ce and \u03b3-Ce,\nrespectively. Symbols denote experimental data in the literature, except\nthe open green circle being the calculated critical point, in good\nagreement with the computed isotherms. In the two-phase miscibility gap\nregion, the \u03b3\u2192\u03b1 volume collapse is again noted, with the magnitude of\nthe collapse increasing with decreasing *T*. This is shown explicitly by\nthe dashed vertical lines at *T* = 200, 250, 300, 350, 400, and 450 K.\nFor *T* \\> 476 K, the calculated isotherms show an anomalous slope\nchange which closely matches the behavior near *V/VN* = ~0.85\nfrom experiment.\n\nFigure \u2011: Equation-of-states for Ce. The black solid lines represent the\ncalculated isotherms from 200 to 600 K at \u2206T = 50 K increments.\n\nA more complex model is to add the anti-ferromagnetic microstate. Thus\nobtained E-V and Helmholtz energy curves at 0 K are shown in . The\nequilibrium volume energies reveal that the energy of the\nanti-ferromagnetic microstate at the equilibrium volume is close to that\nof the nonmagnetic microstate but substantially lower than that of the\nferromagnetic microstate. It should be noted that the magnetic spin\ndisordering in the system is taken into account by the two magnetic\nmicrostates, and the contribution denoted by the mean-field theory, i.e.\n, should thus not be added to either magnetic microstate to avoid double\ncounting. The predicted critical point values are 546 K and 2.05 GPa,\ncloser to the experimental data than the previous predication with two\nmicrostates as shown in the temperature-pressure phase diagram in in\ncomparison with experimental data.\n\nFigure \u2011: (a) dot-dashed line with **\u25cb** (red), dashed line **\u25d1**\n(blue), and solid line with \u25cf (blue) represent the 0 K static total\nenergies for nonmagnetic (NM), anti-ferromagnetic (AFM), and\nferromagnetic (FM) microstates of Ce, respectively. (b) The solid lines\ndenote Helmholtz energy (per atom) from 0 to 600 K at \u2206T = 100 K; the\nheavy dot-dashed (\u03b1-Ce, red) and solid (\u03b3-Ce, blue) looping curves\nenclose the two-phase region with the light red dot-dashed lines\nconnecting the common tangents of each isotherm; the black dashed line\ndenotes zero pressure equilibrium state at given T; **\u25cb** (red) and \u25cf\n(blue) emphasize the phase boundary at 300 K while **\u25d1** (green) is the\ncritical point.\n\nFigure \u2011: Calculated temperature-pressure phase diagram along with\nexperimental data.\n\nThe calculated entropy changes are plotted in a in terms of lattice\nvibration only (black dashed line), lattice vibration plus thermal\nelectron (black dot-dashed line), and lattice vibration plus thermal\nelectron and plus configuration coupling (solid blue). The black square\nis the estimated vibrational entropy change at 0.7 GPa of \u03b3-Ce relative\nto \u03b1-Ce, and other open (solid) symbols are from experimental\nmeasurements of total entropy. Various contributions to the Helmholtz\nenergy along the \u03b3-\u03b1 phase boundary are plotted in b in terms of *T\u2206S*\n(blue diamonds), *\u2206E* (green circles), and *P\u2206V* (red squares), shown\nexcellent agreement with experimental data.\n\nFigure \u2011: Calculated (a) entropy; (b) various contributions to Helmholtz\nenergy of Ce along with experimental data.\n\nThe predicted fractions of three microstates as a function of\ntemperature and heat capacity at the critical pressure of 2.05 Pa are\nshown in a and b, respectively. Near the critical point, the theory\npredicts that the system is a mixture of the various microstates. a\ndepicts that for T \\< 300 K, the system consists mainly of the\nnonmagnetic Ce state which results in \u03b1-Ce. For *T* \\> 300 K, the\nthermal populations of the magnetic states increase with increasing\ntemperature. Finally, for *T* \\> 546 K (the critical point), 70% of the\nsystem is composed of the antiferromagnetic Ce state with the remaining\n30% consisting of the nonmagnetic and ferromagnetic Ce states. This is\nin agreement with the common belief that \u03b3-Ce is magnetic with a\npartially disordered local moment (paramagnetic) and that \u03b1-Ce is\nnonmagnetic.\n\nb shows the predicted temperature evolution of contributions to heat\ncapacity: vibrational and magnetic (*Cf/T*), electronic\n(*Cel*/*T*), and their sum (*Cf+el/T*) at 2.05\nGPa. The theory suggests the following: (a) below ~500 K,\n*Cf+el/T* shows an exponential temperature dependence due to\nthe statistic fluctuation among the non-magnetic, ferromagnetic, and\nantiferromagnetic states; (b) a peak appears at ~500 K in the\n*Cf+el/T* curve, which typically suggests the Schottky\nanomaly; (c) the electronic specific heat coefficient\n(*Cel*/*T*) is linear against *T*; (d) above *~*500 K the sum\nof *Cf/T* and *Cel*/*T* renders\n*Cf+el/T* temperature-independent.\n\nFigure \u2011: (a) Thermal populations of the nonmagnetic (red dot-dashed),\nanti-ferromagnetic (green dashed), and ferromagnetic (blue solid) as a\nfunction of temperature at the critical pressure of 2.05 GPa. (b)\nCel/T (black dashed line), Cf/T (black dot-dashed\nline), and their sum Cf+el/T (blue solid line) at 2.05 GPa.\n", + "source": [ + "## Application to cerium\n", + "\n", + "Cerium (Ce) displays intriguing physical and chemical properties of\n", + "which the most fascinating is its first-order isostructural phase\n", + "transition. This involves a magnetic, high temperature/high volume\n", + "\u201c\u03b3-phase\u201d and a nonmagnetic, low temperature/low volume \u201c\u03b1-phase\u201d, both\n", + "in the same face-centered-cubic (fcc) lattice structure. At 298 K and\n", + "0.7 GPa, the \u03b3\u2192\u03b1 transition is accompanied by a 14-17% volume collapse.\n", + "The Ce phase transition has been studied extensively including our own\n", + "works with two or three microstates \\[55, 64\\].\n", + "\n", + "The simplest model for the system is to consider two microstates:\n", + "ferromagnetic and nonmagnetic. The first-principles calculations of the\n", + "free energy of two Ce microstates are problematic in the absence of\n", + "strong correlation of the *f*-electrons in the DFT Hamiltonian. The\n", + "relative stability of the nonmagnetic (\u201cdelocalized\u201d) Ce 4*f* state to\n", + "that of the magnetic (\u201clocalized\u201d) Ce 4*f* state is greatly\n", + "overestimated in the GGA \\[31-32\\] with spin polarization. The usual\n", + "approach to surmount this is the Dudarev DFT + U method \\[65\\] with the\n", + "on-site Coulomb and exchange interactions as described with a\n", + "Hartree-Fock approximation added to the DFT Hamiltonian. This method\n", + "offers the advantage that only the difference between the Hubbard U (due\n", + "to the energy increase from an electron addition to a specific site) and\n", + "the J (due to the screened exchange energy) need to be specified *a\n", + "priori*.\n", + "\n", + "Evaluation of numerous U \u2013 J values over a 1.0 \u2013 6.0 eV range revealed\n", + "that 1.6 eV gives the most consistent prediction of nonmagnetic Ce and\n", + "magnetic Ce energetics over the range of atomic volumes that includes\n", + "both microstates at 0 K. The energy-volume curves thus obtained is\n", + "plotted in , showing that the nonmagnetic microstate is the ground\n", + "state, and the equilibrium between the two microstates at 0 K is at the\n", + "negative pressure of -0.87GPa. Since and , NTE does not exist in the\n", + "system.\n", + "\n", + "Figure \u2011: Variation of cell energy (eV) with atomic volume\n", + "(\u00c53) for Ce computed with strong correlation based upon\n", + "Dudarev\u2019s method with U \u2013 J = 1.6 eV.\n", + "\n", + "To take into account the possible magnetic disordering in the\n", + "ferromagnetic microstate at finite temperatures, the following\n", + "contribution is added to the free energy of the ferromagnetic microstate\n", + "\n", + "Eq. 9\u20119\n", + "\n", + "where *MS* is the spin moment, and *l* = 3 the orbital\n", + "angular momentum of an *f*-electron. is a generalization of Hund\u2019s rule,\n", + "with total angular momentum . The Helmholtz energies thus obtained for\n", + "both microstates and the system are shown in at several temperatures\n", + "with the tie-lines included. In the figure, the blue, dot-dashed curves\n", + "are for the nonmagnetic microstate, red, solid curves for ferromagnetic\n", + "microstate, cyan shadows for entropy of mixing between two microstates,\n", + "and the circle in (e) is the critical point. The numbers below the black\n", + "dashed line, representing the common tangent curves, mark the transition\n", + "pressures. The 0-K static energies of the nonmagnetic microstate and the\n", + "magnetic microstate are also plotted in (a) using the solid circles and\n", + "dotted lines.\n", + "\n", + "Figure \u2011: Helmholtz energies at (a) 0 K; (b) 100 K; (c) 165 K; (d) 300\n", + "K; (e) 476 K; and (f) 600 K.\n", + "\n", + "The temperature vs volume phase diagram is plotted and compared with\n", + "available experimental data as shown in \\[66\\]. In this figure, the\n", + "volume (V) is normalized to its equilibrium volume (VN) at\n", + "atmospheric pressure and room temperature. In the pressure range of 2.25\n", + "\u2013 3.5 GPa, the system is within the single-phase region at all\n", + "temperatures considered as shown by the five continuous isobaric volumes\n", + "as a function of temperature. In this pressure range, normal thermal\n", + "expansion is observed at both low and high temperatures on each isobaric\n", + "curve where the probability of each microstate does not change\n", + "significantly with temperature. While in the middle temperature range on\n", + "each isobaric curve, the colossal positive thermal expansion (CPTE),\n", + "highlighted by the pink open diamond symbols, exists due to the fast\n", + "increase of probability of the metastable ferromagnetic microstate with\n", + "respect to temperature, i.e. , , and . This CPTE is much higher than the\n", + "individual positive thermal expansions of the stable and metastable\n", + "microstates, respectively.\n", + "\n", + "Figure \u2011: Calculated temperature-volume phase diagram of Ce.\n", + "\n", + "With decreasing pressure, the system reaches a critical point (green\n", + "circle) where the homogeneous single phase becomes unstable, represented\n", + "by and , and both entropy and volume change infinitely. At even lower\n", + "pressure, a miscibility gap forms, and the single phase separates into\n", + "two phases with the same fcc crystal structures, but different magnetic\n", + "spin structures. Inside the miscibility gap, the volume changes\n", + "discontinuously with respect to temperature by the so-called first-order\n", + "transition as shown by the isobaric curve at zero pressure, compared\n", + "well with experimental volume data (solid squares) under ambient\n", + "pressure.\n", + "\n", + "The fraction of the ferromagnetic microstate, *x*mag, in \u03b1-Ce\n", + "(blue) and \u03b3-Ce (red) calculated using is plotted in as a function of\n", + "pressure along the miscibility gap phase boundary. It can be seen that\n", + "the fraction of the ferromagnetic microstate in \u03b1-Ce increases with\n", + "increasing pressure while the fraction of the ferromagnetic microstate\n", + "in \u03b3-Ce decreases. At the critical point, the fraction of ferromagnetic\n", + "microstate is calculated to be 0.58. This is in qualitative agreement\n", + "with the 0.67 value (filled circle) estimated experimentally at the\n", + "critical point.\n", + "\n", + "Figure \u2011: Fraction of ferromagnetic microstate in \u03b1-Ce (blue) and \u03b3-Ce\n", + "(red) along the \u03b3-\u03b1 phase boundary.\n", + "\n", + "The relative volume, *V/VN*, as a function of pressure, is\n", + "plotted as the black solid lines in from 200 to 600 K at 50 K\n", + "increments. The blue and red solid lines correspond to \u03b1-Ce and \u03b3-Ce,\n", + "respectively. Symbols denote experimental data in the literature, except\n", + "the open green circle being the calculated critical point, in good\n", + "agreement with the computed isotherms. In the two-phase miscibility gap\n", + "region, the \u03b3\u2192\u03b1 volume collapse is again noted, with the magnitude of\n", + "the collapse increasing with decreasing *T*. This is shown explicitly by\n", + "the dashed vertical lines at *T* = 200, 250, 300, 350, 400, and 450 K.\n", + "For *T* \\> 476 K, the calculated isotherms show an anomalous slope\n", + "change which closely matches the behavior near *V/VN* = ~0.85\n", + "from experiment.\n", + "\n", + "Figure \u2011: Equation-of-states for Ce. The black solid lines represent the\n", + "calculated isotherms from 200 to 600 K at \u2206T = 50 K increments.\n", + "\n", + "A more complex model is to add the anti-ferromagnetic microstate. Thus\n", + "obtained E-V and Helmholtz energy curves at 0 K are shown in . The\n", + "equilibrium volume energies reveal that the energy of the\n", + "anti-ferromagnetic microstate at the equilibrium volume is close to that\n", + "of the nonmagnetic microstate but substantially lower than that of the\n", + "ferromagnetic microstate. It should be noted that the magnetic spin\n", + "disordering in the system is taken into account by the two magnetic\n", + "microstates, and the contribution denoted by the mean-field theory, i.e.\n", + ", should thus not be added to either magnetic microstate to avoid double\n", + "counting. The predicted critical point values are 546 K and 2.05 GPa,\n", + "closer to the experimental data than the previous predication with two\n", + "microstates as shown in the temperature-pressure phase diagram in in\n", + "comparison with experimental data.\n", + "\n", + "Figure \u2011: (a) dot-dashed line with **\u25cb** (red), dashed line **\u25d1**\n", + "(blue), and solid line with \u25cf (blue) represent the 0 K static total\n", + "energies for nonmagnetic (NM), anti-ferromagnetic (AFM), and\n", + "ferromagnetic (FM) microstates of Ce, respectively. (b) The solid lines\n", + "denote Helmholtz energy (per atom) from 0 to 600 K at \u2206T = 100 K; the\n", + "heavy dot-dashed (\u03b1-Ce, red) and solid (\u03b3-Ce, blue) looping curves\n", + "enclose the two-phase region with the light red dot-dashed lines\n", + "connecting the common tangents of each isotherm; the black dashed line\n", + "denotes zero pressure equilibrium state at given T; **\u25cb** (red) and \u25cf\n", + "(blue) emphasize the phase boundary at 300 K while **\u25d1** (green) is the\n", + "critical point.\n", + "\n", + "Figure \u2011: Calculated temperature-pressure phase diagram along with\n", + "experimental data.\n", + "\n", + "The calculated entropy changes are plotted in a in terms of lattice\n", + "vibration only (black dashed line), lattice vibration plus thermal\n", + "electron (black dot-dashed line), and lattice vibration plus thermal\n", + "electron and plus configuration coupling (solid blue). The black square\n", + "is the estimated vibrational entropy change at 0.7 GPa of \u03b3-Ce relative\n", + "to \u03b1-Ce, and other open (solid) symbols are from experimental\n", + "measurements of total entropy. Various contributions to the Helmholtz\n", + "energy along the \u03b3-\u03b1 phase boundary are plotted in b in terms of *T\u2206S*\n", + "(blue diamonds), *\u2206E* (green circles), and *P\u2206V* (red squares), shown\n", + "excellent agreement with experimental data.\n", + "\n", + "Figure \u2011: Calculated (a) entropy; (b) various contributions to Helmholtz\n", + "energy of Ce along with experimental data.\n", + "\n", + "The predicted fractions of three microstates as a function of\n", + "temperature and heat capacity at the critical pressure of 2.05 Pa are\n", + "shown in a and b, respectively. Near the critical point, the theory\n", + "predicts that the system is a mixture of the various microstates. a\n", + "depicts that for T \\< 300 K, the system consists mainly of the\n", + "nonmagnetic Ce state which results in \u03b1-Ce. For *T* \\> 300 K, the\n", + "thermal populations of the magnetic states increase with increasing\n", + "temperature. Finally, for *T* \\> 546 K (the critical point), 70% of the\n", + "system is composed of the antiferromagnetic Ce state with the remaining\n", + "30% consisting of the nonmagnetic and ferromagnetic Ce states. This is\n", + "in agreement with the common belief that \u03b3-Ce is magnetic with a\n", + "partially disordered local moment (paramagnetic) and that \u03b1-Ce is\n", + "nonmagnetic.\n", + "\n", + "b shows the predicted temperature evolution of contributions to heat\n", + "capacity: vibrational and magnetic (*Cf/T*), electronic\n", + "(*Cel*/*T*), and their sum (*Cf+el/T*) at 2.05\n", + "GPa. The theory suggests the following: (a) below ~500 K,\n", + "*Cf+el/T* shows an exponential temperature dependence due to\n", + "the statistic fluctuation among the non-magnetic, ferromagnetic, and\n", + "antiferromagnetic states; (b) a peak appears at ~500 K in the\n", + "*Cf+el/T* curve, which typically suggests the Schottky\n", + "anomaly; (c) the electronic specific heat coefficient\n", + "(*Cel*/*T*) is linear against *T*; (d) above *~*500 K the sum\n", + "of *Cf/T* and *Cel*/*T* renders\n", + "*Cf+el/T* temperature-independent.\n", + "\n", + "Figure \u2011: (a) Thermal populations of the nonmagnetic (red dot-dashed),\n", + "anti-ferromagnetic (green dashed), and ferromagnetic (blue solid) as a\n", + "function of temperature at the critical pressure of 2.05 GPa. (b)\n", + "Cel/T (black dashed line), Cf/T (black dot-dashed\n", + "line), and their sum Cf+el/T (blue solid line) at 2.05 GPa.\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_fept.ipynb b/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_fept.ipynb index a88b4b5..7686fa4 100644 --- a/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_fept.ipynb +++ b/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_fept.ipynb @@ -4,9 +4,127 @@ "metadata": {}, "cells": [ { - "id": "09dbdce6", + "id": "8f719255", "cell_type": "markdown", - "source": "## Application to Fe3Pt\n\nInvar was first discovered in the intermetallic\nFe65Ni35 alloy and is characterized by \u201canomalies\u201d\nincluding thermal expansion, equation-of-state, elastic modulus, heat\ncapacity, magnetization, and Curie temperature. There are a number of\ntheoretical models for Invar such as the Weiss 2-\u03b3 model, the\nnon-collinear spin model, and the disordered local moment approach as\nreviewed in Ref. \\[67\\]. In this section, the application of the MMS\nmodel to the ordered L12 Fe3Pt is presented to\nstudy the Invar anomaly at finite temperatures.\n\nFor a supercell of 12 atoms with nine magnetic Fe atoms in\nFe3Pt, if only the up and down spins are considered, the\nsystem contains 29 = 512 spin configurations, which are, by\nsymmetry, reduced to 37 non-equivalent ones. They are the microstates in\nthe MMS model, and FMC and SFC are used to represent the ferromagnetic\nand spin flipping microstates, respectively. For the first-principles\ncalculations of each microstate, VASP package \\[14\\] within the\nprojector-augmented wave (PAW) method and the exchange-correlation part\nof the density functional treated within the GGA of\nPerdew-Burke-Ernzerhof (PBE) \\[32\\] are employed with details in Ref.\n\\[67\\]. For the lattice vibration, the Debye-Gr\u00fcneisen approach\ndescribed in Chapter 5.2.4 is used.\n\npresents the first-principles 0 K total energies of 36 non-equivalent\nSFCs as well as the FMC as a function of atomic volume. It can be seen\nthat there are a number of SFCs, whose energies are in the range of ~1\nmRy/atom to that of the FMC. It is noted that all the SFCs studied\nherein have the equilibrium averaged atomic volumes at least 1.8 %\nsmaller than that of the FMC, the 0 GPa ground state. In , the two\nlowest energy SFCs have been labeled as SFC55 and SFC41 with their spin\narrangements very similar to the double layer antiferromagnetic state.\nThe nonmagnetic configuration has a very small atomic volume of 11.66\n\u00c53/atom and much higher energy than both FMC and all SFCs,\nand is thus not shown here.\n\nFigure \u2011: 0 K total energies. The heavy black line represents the FMC.\nThe symbols **\u25cb**, \u25d4, \u25d1, and \u25d5 with dashed lines indicate the minima of\nthe energy-volume curves of the SFCs with spin polarization rates of\n1/9, 3/9, 5/9, and 7/9, respectively. The red **\u25cb** and \u25d4 with\ndot-dashed lines mark the two lowest SFCs in energy.\n\nThe normalized Helmholtz energies of all SFCs are plotted in with the\nHelmholtz energy of FMC as the reference state, shown that the FMC has\nthe lowest Helmholtz energy at all temperatures considered. If only the\nrelative Helmholtz energies of microstates are considered, FMC should be\nstable at all temperatures. However, the configurational entropy due to\nthe mixing among multiple microstates, i.e. , lowers the system free\nenergy by introducing the statistic probability of metastable SFCs.\n\nFigure \u2011: Normalized Helmholtz energy of all SFCs with respect to that\nof FMC.\n\nThrough the minimization of Helmholtz energy of , the\ntemperature-pressure and temperature-volume phase diagrams are obtained\nand shown in \\[66\\]. A critical point at 141 K and 5.81 GPa is predicted\nwith *V* = 12.61 \u00c53. Below the critical point, it is a\ntwo-phase miscibility gap (the shadow area) with the dominant\nmicrostates being FMC and SFCs, respectively, and the transition between\nthem is first-order. Above the critical point, the macroscopically\nhomogeneous single phase is stable, and the phase transitions between\nthe ferromagnetic-dominant phase with large volumes and the SFC-dominant\nparamagnetic phase with small volumes are of second-order. The\nsecond-order transition pressures and volumes are determined by the\ncondition that the weighted Helmholtz energy counting all SFCs equals to\nthe Helmholtz energy counting only FMC.\n\nFigure \u2011: Calculated (a) temperature-pressure and (b) temperature-volume\nphase diagrams of Fe3Pt.\n\nIn a, the data points are the measured pressure dependence of the Curie\ntemperature, and the agreement between the measurements and predictions\nis remarkable. It should be pointed out that the classical Weiss 2-\u03b3\nmodel predicts only first-order phase transitions while the\nnon-collinear spin model yields only second-order phase transitions at\nall temperatures. In b, four isobaric volume curves are also plotted\nwith the predicted NTE regions marked by the pink open diamonds and the\nexperimental volume data under ambient pressure superimposed, showing\nexcellent agreement. It also depicts the gigantic NTE around the\ncritical point on the isobaric curve at 7 GPa.\n\nindicates that the entropies of SFCs are larger than that of FMC so\ntheir Helmholtz energy differences decrease with temperature. This is in\nline with the origin of NTE in a single phase due to the statistic\nexistence of metastable microstates with lower volumes and higher\nentropies than the stable state in a temperature range where their\nprobabilities change dramatically. plots the calculated thermal\npopulations of the FMC (black solid line) and that of the sum over all\nSFCs (black dot-dashed line) under ambient pressure. The two major\ncontributions to the paramagnetic (PM) phase are from SFC55 and SFC41,\nwhich are also plotted using red dashed and long dashed lines,\nrespectively. The system is dominated by the FMC at temperatures below\nhalf of the transition temperature, and the populations of SFCs increase\nmonotonously at temperatures higher than half of the transition\ntemperature. As mentioned above, the transition temperature is defined\nwhen the population of all SFCs is the same as that of FMC due to their\nequal Helmholtz energy.\n\nFigure \u2011: Calculated thermal populations of FMC, SFC55, SFC41, and sum\nof all SFCS, respectively.\n\nThe predicted thermal volume expansion and the derived linear thermal\nexpansion coefficient (LTEC) under ambient pressure are plotted in . A\npositive thermal expansion is predicted from 100 K to 288 K, followed by\na negative thermal expansion in the range of 289 ~ 449 K, and then a\npositive thermal expansion again at \\>450 K, in excellent agreement with\nexperiments. The only disagreement between the predictions and\nexperiments occur at T \\< 100 K where the calculations did not reproduce\nthe negative thermal expansion for Fe3Pt. Large supercell or\nmore spin configurations may be needed.\n\nFigure \u2011: (a) Relative volume increase (V-\nV300)/V300 with V300 being the\nequilibrium volume at 300K and 0 GPa for the ordered Fe3Pt.\n(b) Linear thermal expansion coefficient (LTEC) along with experimental\ndata (symbols) with details in Ref. \\[67\\].\n", + "source": [ + "## Application to Fe3Pt\n", + "\n", + "Invar was first discovered in the intermetallic\n", + "Fe65Ni35 alloy and is characterized by \u201canomalies\u201d\n", + "including thermal expansion, equation-of-state, elastic modulus, heat\n", + "capacity, magnetization, and Curie temperature. There are a number of\n", + "theoretical models for Invar such as the Weiss 2-\u03b3 model, the\n", + "non-collinear spin model, and the disordered local moment approach as\n", + "reviewed in Ref. \\[67\\]. In this section, the application of the MMS\n", + "model to the ordered L12 Fe3Pt is presented to\n", + "study the Invar anomaly at finite temperatures.\n", + "\n", + "For a supercell of 12 atoms with nine magnetic Fe atoms in\n", + "Fe3Pt, if only the up and down spins are considered, the\n", + "system contains 29 = 512 spin configurations, which are, by\n", + "symmetry, reduced to 37 non-equivalent ones. They are the microstates in\n", + "the MMS model, and FMC and SFC are used to represent the ferromagnetic\n", + "and spin flipping microstates, respectively. For the first-principles\n", + "calculations of each microstate, VASP package \\[14\\] within the\n", + "projector-augmented wave (PAW) method and the exchange-correlation part\n", + "of the density functional treated within the GGA of\n", + "Perdew-Burke-Ernzerhof (PBE) \\[32\\] are employed with details in Ref.\n", + "\\[67\\]. For the lattice vibration, the Debye-Gr\u00fcneisen approach\n", + "described in Chapter 5.2.4 is used.\n", + "\n", + "presents the first-principles 0 K total energies of 36 non-equivalent\n", + "SFCs as well as the FMC as a function of atomic volume. It can be seen\n", + "that there are a number of SFCs, whose energies are in the range of ~1\n", + "mRy/atom to that of the FMC. It is noted that all the SFCs studied\n", + "herein have the equilibrium averaged atomic volumes at least 1.8 %\n", + "smaller than that of the FMC, the 0 GPa ground state. In , the two\n", + "lowest energy SFCs have been labeled as SFC55 and SFC41 with their spin\n", + "arrangements very similar to the double layer antiferromagnetic state.\n", + "The nonmagnetic configuration has a very small atomic volume of 11.66\n", + "\u00c53/atom and much higher energy than both FMC and all SFCs,\n", + "and is thus not shown here.\n", + "\n", + "Figure \u2011: 0 K total energies. The heavy black line represents the FMC.\n", + "The symbols **\u25cb**, \u25d4, \u25d1, and \u25d5 with dashed lines indicate the minima of\n", + "the energy-volume curves of the SFCs with spin polarization rates of\n", + "1/9, 3/9, 5/9, and 7/9, respectively. The red **\u25cb** and \u25d4 with\n", + "dot-dashed lines mark the two lowest SFCs in energy.\n", + "\n", + "The normalized Helmholtz energies of all SFCs are plotted in with the\n", + "Helmholtz energy of FMC as the reference state, shown that the FMC has\n", + "the lowest Helmholtz energy at all temperatures considered. If only the\n", + "relative Helmholtz energies of microstates are considered, FMC should be\n", + "stable at all temperatures. However, the configurational entropy due to\n", + "the mixing among multiple microstates, i.e. , lowers the system free\n", + "energy by introducing the statistic probability of metastable SFCs.\n", + "\n", + "Figure \u2011: Normalized Helmholtz energy of all SFCs with respect to that\n", + "of FMC.\n", + "\n", + "Through the minimization of Helmholtz energy of , the\n", + "temperature-pressure and temperature-volume phase diagrams are obtained\n", + "and shown in \\[66\\]. A critical point at 141 K and 5.81 GPa is predicted\n", + "with *V* = 12.61 \u00c53. Below the critical point, it is a\n", + "two-phase miscibility gap (the shadow area) with the dominant\n", + "microstates being FMC and SFCs, respectively, and the transition between\n", + "them is first-order. Above the critical point, the macroscopically\n", + "homogeneous single phase is stable, and the phase transitions between\n", + "the ferromagnetic-dominant phase with large volumes and the SFC-dominant\n", + "paramagnetic phase with small volumes are of second-order. The\n", + "second-order transition pressures and volumes are determined by the\n", + "condition that the weighted Helmholtz energy counting all SFCs equals to\n", + "the Helmholtz energy counting only FMC.\n", + "\n", + "Figure \u2011: Calculated (a) temperature-pressure and (b) temperature-volume\n", + "phase diagrams of Fe3Pt.\n", + "\n", + "In a, the data points are the measured pressure dependence of the Curie\n", + "temperature, and the agreement between the measurements and predictions\n", + "is remarkable. It should be pointed out that the classical Weiss 2-\u03b3\n", + "model predicts only first-order phase transitions while the\n", + "non-collinear spin model yields only second-order phase transitions at\n", + "all temperatures. In b, four isobaric volume curves are also plotted\n", + "with the predicted NTE regions marked by the pink open diamonds and the\n", + "experimental volume data under ambient pressure superimposed, showing\n", + "excellent agreement. It also depicts the gigantic NTE around the\n", + "critical point on the isobaric curve at 7 GPa.\n", + "\n", + "indicates that the entropies of SFCs are larger than that of FMC so\n", + "their Helmholtz energy differences decrease with temperature. This is in\n", + "line with the origin of NTE in a single phase due to the statistic\n", + "existence of metastable microstates with lower volumes and higher\n", + "entropies than the stable state in a temperature range where their\n", + "probabilities change dramatically. plots the calculated thermal\n", + "populations of the FMC (black solid line) and that of the sum over all\n", + "SFCs (black dot-dashed line) under ambient pressure. The two major\n", + "contributions to the paramagnetic (PM) phase are from SFC55 and SFC41,\n", + "which are also plotted using red dashed and long dashed lines,\n", + "respectively. The system is dominated by the FMC at temperatures below\n", + "half of the transition temperature, and the populations of SFCs increase\n", + "monotonously at temperatures higher than half of the transition\n", + "temperature. As mentioned above, the transition temperature is defined\n", + "when the population of all SFCs is the same as that of FMC due to their\n", + "equal Helmholtz energy.\n", + "\n", + "Figure \u2011: Calculated thermal populations of FMC, SFC55, SFC41, and sum\n", + "of all SFCS, respectively.\n", + "\n", + "The predicted thermal volume expansion and the derived linear thermal\n", + "expansion coefficient (LTEC) under ambient pressure are plotted in . A\n", + "positive thermal expansion is predicted from 100 K to 288 K, followed by\n", + "a negative thermal expansion in the range of 289 ~ 449 K, and then a\n", + "positive thermal expansion again at \\>450 K, in excellent agreement with\n", + "experiments. The only disagreement between the predictions and\n", + "experiments occur at T \\< 100 K where the calculations did not reproduce\n", + "the negative thermal expansion for Fe3Pt. Large supercell or\n", + "more spin configurations may be needed.\n", + "\n", + "Figure \u2011: (a) Relative volume increase (V-\n", + "V300)/V300 with V300 being the\n", + "equilibrium volume at 300K and 0 GPa for the ordered Fe3Pt.\n", + "(b) Linear thermal expansion coefficient (LTEC) along with experimental\n", + "data (symbols) with details in Ref. \\[67\\].\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/concept_of_materials_genome.ipynb b/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/concept_of_materials_genome.ipynb index 336d2cc..29bd85b 100644 --- a/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/concept_of_materials_genome.ipynb +++ b/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/concept_of_materials_genome.ipynb @@ -4,9 +4,62 @@ "metadata": {}, "cells": [ { - "id": "e9947ec5", + "id": "2e83b1bc", "cell_type": "markdown", - "source": "## Concept of Materials Genome\u00ae\n\n\u201cA genome is a set of information encoded in the language of DNA that\nserves as a blueprint for an organism\u2019s growth and development. The word\ngenome, when applied in nonbiological contexts, connotes a fundamental\nbuilding block toward a larger purpose\u201d \\[68\\]. Materials Genome\u00ae (a\ntrademark of Materials Genome, Inc., Pennsylvania, USA) thus concerns\nthe building blocks of materials. Most part of this book focuses on the\nGibbs/Helmholtz energies of individual phases as a function of its\nnatural variables, and the same in the CALPHAD modeling of\nthermodynamics and other properties of individual phases.\nMulti-component materials systems and their properties are built on the\nindividual phases and their properties. Individual phases are thus\nnaturally considered as building blocks of materials. Consequently, the\nlanguage of CALPHAD thermodynamics and kinetics contains the genomics of\nmaterials by representing experimental and theoretical results in\ndatabases to make them applicable to a much wider context than the\noriginal experiments or calculations \\[69\\]. The change of individual\nphases in terms of their properties, amounts, and interactions with\nother phases with respect to external conditions thus determines the\nperformance of the materials.\n\nOn the other hand, at critical points and beyond, phases lose their\nindividuality and form one macroscopically homogeneous system, and the\nproperties of the system change dramatically with respect to external\nconditions. As shown in this chapter, these dramatically responses can\nbe predicted through the statistic competition of stable and metastable\nmicrostates. From the thermodynamic point of view, under any given\nconditions, one of the individual microstate has the lowest\nGibbs/Helmholtz energy and is stable, while all other microstates have\nhigher free energy and are metastable or unstable. These metastable or\neven unstable microstates are brought into statistic existence in the\nmatrix of the stable microstate due to the entropy of mixing of the\nstable and metastable configurations. Those microstates may thus be\nconsidered as the building blocks of individual phases \\[70\\].\n\nIt has demonstrated in this chapter that the properties of a\nmacroscopically homogeneous system with multiple microstates are not a\nlinear combination of properties of constituent microstates and depends\nsignificantly on the change rate of statistic probability of microstates\nwith respect to external fields. This change rate is determined by the\nfree energy difference between the stable and metastable microstates and\nits change with respect to external fields. As shown in and , this\nchange rate can be correlated to the distance of the system with respect\nto the critical point in the system. At the critical point, there is a\nmathematical singularity when the single phase becomes unstable. When\nthe macroscopically homogeneous single-phase system moves away from the\ncritical point, its properties become less and less dramatic, but always\nremain a certain degree of anomaly. The properties of a system can thus\nbe dramatically altered and designed by changing the position of the\ncritical point through adjustments of chemical compositions and strain\nenergy in thin films.\n", + "source": [ + "## Concept of Materials Genome\u00ae\n", + "\n", + "\u201cA genome is a set of information encoded in the language of DNA that\n", + "serves as a blueprint for an organism\u2019s growth and development. The word\n", + "genome, when applied in nonbiological contexts, connotes a fundamental\n", + "building block toward a larger purpose\u201d \\[68\\]. Materials Genome\u00ae (a\n", + "trademark of Materials Genome, Inc., Pennsylvania, USA) thus concerns\n", + "the building blocks of materials. Most part of this book focuses on the\n", + "Gibbs/Helmholtz energies of individual phases as a function of its\n", + "natural variables, and the same in the CALPHAD modeling of\n", + "thermodynamics and other properties of individual phases.\n", + "Multi-component materials systems and their properties are built on the\n", + "individual phases and their properties. Individual phases are thus\n", + "naturally considered as building blocks of materials. Consequently, the\n", + "language of CALPHAD thermodynamics and kinetics contains the genomics of\n", + "materials by representing experimental and theoretical results in\n", + "databases to make them applicable to a much wider context than the\n", + "original experiments or calculations \\[69\\]. The change of individual\n", + "phases in terms of their properties, amounts, and interactions with\n", + "other phases with respect to external conditions thus determines the\n", + "performance of the materials.\n", + "\n", + "On the other hand, at critical points and beyond, phases lose their\n", + "individuality and form one macroscopically homogeneous system, and the\n", + "properties of the system change dramatically with respect to external\n", + "conditions. As shown in this chapter, these dramatically responses can\n", + "be predicted through the statistic competition of stable and metastable\n", + "microstates. From the thermodynamic point of view, under any given\n", + "conditions, one of the individual microstate has the lowest\n", + "Gibbs/Helmholtz energy and is stable, while all other microstates have\n", + "higher free energy and are metastable or unstable. These metastable or\n", + "even unstable microstates are brought into statistic existence in the\n", + "matrix of the stable microstate due to the entropy of mixing of the\n", + "stable and metastable configurations. Those microstates may thus be\n", + "considered as the building blocks of individual phases \\[70\\].\n", + "\n", + "It has demonstrated in this chapter that the properties of a\n", + "macroscopically homogeneous system with multiple microstates are not a\n", + "linear combination of properties of constituent microstates and depends\n", + "significantly on the change rate of statistic probability of microstates\n", + "with respect to external fields. This change rate is determined by the\n", + "free energy difference between the stable and metastable microstates and\n", + "its change with respect to external fields. As shown in and , this\n", + "change rate can be correlated to the distance of the system with respect\n", + "to the critical point in the system. At the critical point, there is a\n", + "mathematical singularity when the single phase becomes unstable. When\n", + "the macroscopically homogeneous single-phase system moves away from the\n", + "critical point, its properties become less and less dramatic, but always\n", + "remain a certain degree of anomaly. The properties of a system can thus\n", + "be dramatically altered and designed by changing the position of the\n", + "critical point through adjustments of chemical compositions and strain\n", + "energy in thin films.\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/index.ipynb b/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/index.ipynb index f05c1a9..3eb3c04 100644 --- a/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/index.ipynb +++ b/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/index.ipynb @@ -4,9 +4,28 @@ "metadata": {}, "cells": [ { - "id": "5cac8904", + "id": "bb30430b", "cell_type": "markdown", - "source": "# Critical phenomena, thermal expansion, and Materials Genome\u00ae\n\nIn Chapter , it was shown that all molar quantities of a homogeneous\nsystem diverge at the critical point, i.e. the limit of stability,\nincluding the additional ones shown in . As illustrated by , even though\neach molar quantity changes in the same direction as its conjugate\npotential, i.e. with the same sign, its dependence with respect to a\nnonconjugate potentials can either be in the same sign or opposite\nsigns. It is often considered to be normal when they change in the same\ndirection, while abnormal when in different directions.\n\nIn this chapter, the thermal expansion defined by , is used as an\nexample for detailed discussion of those extraordinary phenomena in the\ncontext of a critical point based on the MMS model presented in Chapter\n. The MMS model is first discussed in terms of thermal expansion and\nthen applied to elemental cerium (Ce) with the colossal positive thermal\nexpansion (CPTE) and Fe3Pt with negative thermal expansion\n(NTE).\n", + "source": [ + "# Critical phenomena, thermal expansion, and Materials Genome\u00ae\n", + "\n", + "In Chapter , it was shown that all molar quantities of a homogeneous\n", + "system diverge at the critical point, i.e. the limit of stability,\n", + "including the additional ones shown in . As illustrated by , even though\n", + "each molar quantity changes in the same direction as its conjugate\n", + "potential, i.e. with the same sign, its dependence with respect to a\n", + "nonconjugate potentials can either be in the same sign or opposite\n", + "signs. It is often considered to be normal when they change in the same\n", + "direction, while abnormal when in different directions.\n", + "\n", + "In this chapter, the thermal expansion defined by , is used as an\n", + "example for detailed discussion of those extraordinary phenomena in the\n", + "context of a critical point based on the MMS model presented in Chapter\n", + ". The MMS model is first discussed in terms of thermal expansion and\n", + "then applied to elemental cerium (Ce) with the colossal positive thermal\n", + "expansion (CPTE) and Fe3Pt with negative thermal expansion\n", + "(NTE).\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/mms_model_applied_to_thermal_expansion.ipynb b/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/mms_model_applied_to_thermal_expansion.ipynb index b057391..2928154 100644 --- a/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/mms_model_applied_to_thermal_expansion.ipynb +++ b/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/mms_model_applied_to_thermal_expansion.ipynb @@ -4,9 +4,123 @@ "metadata": {}, "cells": [ { - "id": "e1834dd0", + "id": "840ab7ad", "cell_type": "markdown", - "source": "## MMS model applied to thermal expansion\n\nAs shown in , the thermal expansion of a system can be positive, zero\nand negative depending on the pressure dependence of entropy of the\nsystem. Let us carry out a virtual experiment by analyzing a system\nstarting with one microstate only, *\u03b1*, when the metastable *\u03b2*\nmicrostate has a higher entropy than the *\u03b1* microstate, i.e. and the\nrelative stability of the *\u03b2* microstate thus increases with\ntemperature. The cases with will be discussed after this starting with a\nmixture of the *\u03b1* and *\u03b2* microstates.\n\nWhen a metastable microstate, *\u03b2*, is introduced by changing pressure\nunder constant temperature, based on and , the entropy change of the\nsystem can be written as\n\nEq. 9\u20111\n\nwhere represents the statistic probability of the microstate in the\nsystem. With , this would results in a positive entropy change of , i.\ne. , since . If this entropy increase is due to the decrease of\npressure, i.e. because volume and its conjugate potential (negative\npressure) change in the same direction, the volume thermal expansion of\nthe system is positive due to the increase of the population of the *\u03b2*\nmicrostate with a larger volume. In this case,\n$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}} > 0$, and the\nvolume and entropy of the two microstates change in the same direction.\n\nOn the other hand, if this entropy increase is realized by increasing\npressure, i.e. , the volume thermal expansion of the system is negative\ndue to the increase of the population of the *\u03b2* microstate with a\nsmaller volume. In this case,\n$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}} < 0$, and the\nvolume and entropy of the two microstates change in the opposite\ndirections.\n\nTherefore, the sign of\n$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}}$ of two\nmicrostates can be used as a criterion to determine whether a system\npossesses NTE, i.e. positive\n$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}}$ for positive\nthermal expansion, and negative\n$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}}$ for NTE. At a\ncritical point, the entropy change with respect to temperature is\ninfinite, resulting in either infinite positive or infinite negative\nthermal expansion correspondingly. When the system moves away from the\ncritical point into the macroscopically homogeneous single-phase region,\nthe thermal expansion becomes less positive or negative. A number of\nsystems with\n$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}} < 0$, thus\npotentially NTE, are listed in the supplementary information of Ref.\n\\[63\\].\n\nNow let us consider the case that the metastable *\u03b2* microstate has\nlower entropy than the *\u03b1* microstate, i.e. , and the *\u03b2* microstate is\nthus more stable at low temperatures. The system at higher temperatures\ncontains thus only the *\u03b1* microstate and has positive thermal\nexpansion. When the metastable *\u03b2* microstate is introduced, the sign of\nthe entropy change in can be either positive or negative because the\nfirst term is negative and the second term is positive, and its sign\nthus depends on the value of the entropy difference between two\nmicrostates and the probability of the metastable *\u03b2* microstate. The\nvirtual experiment should thus be carried out in a system with the\nhighest MCE in , i.e. when the two microstates have the same free energy\nand are in equilibrium with each other. From and , the system entropy\ncan be written as\n\nEq. 9\u20112\n\nWith the change of pressure, will either increase or decrease, and the\nentropy of the system becomes\n\nEq. 9\u20113\n\nThe difference of and is obtained as\n\nEq. 9\u20114\n\nThe second term in is always negative, and the first term is also\nnegative if because . It is thus evident that if is increased by\ndecreasing pressure, the entropy of the system decreases, and the system\nwould possess negative thermal expansion because is negative, and the\nentropy is reduced by the decrease of pressure. At the same time, and\n$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}} < 0$, the latter\nbeing the same condition for a negative thermal expansion as in the\nfirst virtual experiment with . One can thus conclude that for a\ntwo-phase equilibrium line with\n$\\frac{dT}{dP} = \\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}} < 0$,\nboth phases can display negative thermal expansion. On the other hand,\nif is increased by increasing pressure, the system would possess\npositive thermal expansion because the entropy is reduced by the\nincrease of pressure.\n\nFurthermore, the thermal expansion of a system can be approximated as\nfollows using the rule of mixture of volumes\n\nEq. 9\u20115\n\nEq. 9\u20116\n\nwhere , , , , , and are the thermal expansion coefficients and volumes\nof the system and the \u03b1 and \u03b2 microstates, respectively. For\nsimplification, let us assume both microstates have similar positive\nthermal expansion, i.e. , and becomes\n\nEq. 9\u20117\n\nshows that it is the combination of volume difference and that\ndetermines the macroscopic thermal expansion. By setting , one obtains\n\nEq. 9\u20118\n\nFor , for $p^{\\beta} \\geq 0$, and for , at $p^{\\beta} \\rightarrow 0$ .\nThe readers are reminded that the sign of is the same as the sign of .\n", + "source": [ + "## MMS model applied to thermal expansion\n", + "\n", + "As shown in , the thermal expansion of a system can be positive, zero\n", + "and negative depending on the pressure dependence of entropy of the\n", + "system. Let us carry out a virtual experiment by analyzing a system\n", + "starting with one microstate only, *\u03b1*, when the metastable *\u03b2*\n", + "microstate has a higher entropy than the *\u03b1* microstate, i.e. and the\n", + "relative stability of the *\u03b2* microstate thus increases with\n", + "temperature. The cases with will be discussed after this starting with a\n", + "mixture of the *\u03b1* and *\u03b2* microstates.\n", + "\n", + "When a metastable microstate, *\u03b2*, is introduced by changing pressure\n", + "under constant temperature, based on and , the entropy change of the\n", + "system can be written as\n", + "\n", + "Eq. 9\u20111\n", + "\n", + "where represents the statistic probability of the microstate in the\n", + "system. With , this would results in a positive entropy change of , i.\n", + "e. , since . If this entropy increase is due to the decrease of\n", + "pressure, i.e. because volume and its conjugate potential (negative\n", + "pressure) change in the same direction, the volume thermal expansion of\n", + "the system is positive due to the increase of the population of the *\u03b2*\n", + "microstate with a larger volume. In this case,\n", + "$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}} > 0$, and the\n", + "volume and entropy of the two microstates change in the same direction.\n", + "\n", + "On the other hand, if this entropy increase is realized by increasing\n", + "pressure, i.e. , the volume thermal expansion of the system is negative\n", + "due to the increase of the population of the *\u03b2* microstate with a\n", + "smaller volume. In this case,\n", + "$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}} < 0$, and the\n", + "volume and entropy of the two microstates change in the opposite\n", + "directions.\n", + "\n", + "Therefore, the sign of\n", + "$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}}$ of two\n", + "microstates can be used as a criterion to determine whether a system\n", + "possesses NTE, i.e. positive\n", + "$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}}$ for positive\n", + "thermal expansion, and negative\n", + "$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}}$ for NTE. At a\n", + "critical point, the entropy change with respect to temperature is\n", + "infinite, resulting in either infinite positive or infinite negative\n", + "thermal expansion correspondingly. When the system moves away from the\n", + "critical point into the macroscopically homogeneous single-phase region,\n", + "the thermal expansion becomes less positive or negative. A number of\n", + "systems with\n", + "$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}} < 0$, thus\n", + "potentially NTE, are listed in the supplementary information of Ref.\n", + "\\[63\\].\n", + "\n", + "Now let us consider the case that the metastable *\u03b2* microstate has\n", + "lower entropy than the *\u03b1* microstate, i.e. , and the *\u03b2* microstate is\n", + "thus more stable at low temperatures. The system at higher temperatures\n", + "contains thus only the *\u03b1* microstate and has positive thermal\n", + "expansion. When the metastable *\u03b2* microstate is introduced, the sign of\n", + "the entropy change in can be either positive or negative because the\n", + "first term is negative and the second term is positive, and its sign\n", + "thus depends on the value of the entropy difference between two\n", + "microstates and the probability of the metastable *\u03b2* microstate. The\n", + "virtual experiment should thus be carried out in a system with the\n", + "highest MCE in , i.e. when the two microstates have the same free energy\n", + "and are in equilibrium with each other. From and , the system entropy\n", + "can be written as\n", + "\n", + "Eq. 9\u20112\n", + "\n", + "With the change of pressure, will either increase or decrease, and the\n", + "entropy of the system becomes\n", + "\n", + "Eq. 9\u20113\n", + "\n", + "The difference of and is obtained as\n", + "\n", + "Eq. 9\u20114\n", + "\n", + "The second term in is always negative, and the first term is also\n", + "negative if because . It is thus evident that if is increased by\n", + "decreasing pressure, the entropy of the system decreases, and the system\n", + "would possess negative thermal expansion because is negative, and the\n", + "entropy is reduced by the decrease of pressure. At the same time, and\n", + "$\\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}} < 0$, the latter\n", + "being the same condition for a negative thermal expansion as in the\n", + "first virtual experiment with . One can thus conclude that for a\n", + "two-phase equilibrium line with\n", + "$\\frac{dT}{dP} = \\frac{\\Delta V^{\\alpha\\beta}}{\\Delta S^{\\alpha\\beta}} < 0$,\n", + "both phases can display negative thermal expansion. On the other hand,\n", + "if is increased by increasing pressure, the system would possess\n", + "positive thermal expansion because the entropy is reduced by the\n", + "increase of pressure.\n", + "\n", + "Furthermore, the thermal expansion of a system can be approximated as\n", + "follows using the rule of mixture of volumes\n", + "\n", + "Eq. 9\u20115\n", + "\n", + "Eq. 9\u20116\n", + "\n", + "where , , , , , and are the thermal expansion coefficients and volumes\n", + "of the system and the \u03b1 and \u03b2 microstates, respectively. For\n", + "simplification, let us assume both microstates have similar positive\n", + "thermal expansion, i.e. , and becomes\n", + "\n", + "Eq. 9\u20117\n", + "\n", + "shows that it is the combination of volume difference and that\n", + "determines the macroscopic thermal expansion. By setting , one obtains\n", + "\n", + "Eq. 9\u20118\n", + "\n", + "For , for $p^{\\beta} \\geq 0$, and for , at $p^{\\beta} \\rightarrow 0$ .\n", + "The readers are reminded that the sign of is the same as the sign of .\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/experimental_data_for_thermodynamic_modeling/index.ipynb b/src/psu410/src/experimental_data_for_thermodynamic_modeling/index.ipynb index cf88515..9d04047 100644 --- a/src/psu410/src/experimental_data_for_thermodynamic_modeling/index.ipynb +++ b/src/psu410/src/experimental_data_for_thermodynamic_modeling/index.ipynb @@ -4,9 +4,30 @@ "metadata": {}, "cells": [ { - "id": "df761675", + "id": "14788efe", "cell_type": "markdown", - "source": "# Experimental data for thermodynamic modeling\n\nThe most widely used thermodynamic modeling technique is the CALPHAD\n(CALculation of PHAse Diagram) method to be discussed in detail in\nChapter 6. The input data in the evaluations of thermodynamic model\nparameters have been primarily from experiments and estimations until\nfirst-principles calculations based on the density functional theory\n\\[8\\] became a user tool in the later 1990\u2019s. Experimental data include\nboth thermodynamic and phase equilibrium data, while the\nfirst-principles calculations, which only provide thermodynamic data of\nindividual phases, are discussed more extensively in Chapter .\n\nThe three recently published books summarize the commonly used methods\non experimental measurements of thermodynamic properties of single \\[9\\]\nand multiple phases \\[10\\] and phase diagrams \\[11\\]. They are briefly\ndiscussed here, and readers are referred to these books for details. The\nmain techniques for crystal structure analysis include X-ray\ndiffraction, electron backscatter diffraction (EBSD), electron\ndiffraction in transmission electron microscopy, neutral scattering, and\nsynchrotron scattering, which are not discussed in this book.\n", + "source": [ + "# Experimental data for thermodynamic modeling\n", + "\n", + "The most widely used thermodynamic modeling technique is the CALPHAD\n", + "(CALculation of PHAse Diagram) method to be discussed in detail in\n", + "Chapter 6. The input data in the evaluations of thermodynamic model\n", + "parameters have been primarily from experiments and estimations until\n", + "first-principles calculations based on the density functional theory\n", + "\\[8\\] became a user tool in the later 1990\u2019s. Experimental data include\n", + "both thermodynamic and phase equilibrium data, while the\n", + "first-principles calculations, which only provide thermodynamic data of\n", + "individual phases, are discussed more extensively in Chapter .\n", + "\n", + "The three recently published books summarize the commonly used methods\n", + "on experimental measurements of thermodynamic properties of single \\[9\\]\n", + "and multiple phases \\[10\\] and phase diagrams \\[11\\]. They are briefly\n", + "discussed here, and readers are referred to these books for details. The\n", + "main techniques for crystal structure analysis include X-ray\n", + "diffraction, electron backscatter diffraction (EBSD), electron\n", + "diffraction in transmission electron microscopy, neutral scattering, and\n", + "synchrotron scattering, which are not discussed in this book.\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/experimental_data_for_thermodynamic_modeling/phase_equilibrium_data_from_experiments.ipynb b/src/psu410/src/experimental_data_for_thermodynamic_modeling/phase_equilibrium_data_from_experiments.ipynb index 5420b59..d0e1bcb 100644 --- a/src/psu410/src/experimental_data_for_thermodynamic_modeling/phase_equilibrium_data_from_experiments.ipynb +++ b/src/psu410/src/experimental_data_for_thermodynamic_modeling/phase_equilibrium_data_from_experiments.ipynb @@ -4,27 +4,237 @@ "metadata": {}, "cells": [ { - "id": "fb60ba1d", + "id": "fbbb8610", "cell_type": "markdown", - "source": "## Phase equilibrium data from experiments\n", + "source": [ + "## Phase equilibrium data from experiments\n" + ], "metadata": {} }, { - "id": "977e7a47", + "id": "cae88515", "cell_type": "markdown", - "source": "### Equilibrated materials\n\nThe most common method to determine phase equilibria is to use\nequilibrated materials. This method typically involves material\npreparation through high temperature melting or powder metallurgy,\nhomogenization heat treatment, isothermal or cooling/heating procedure,\nand identifications of crystal structures and phase compositions. It is\nimportant to avoid macro-inhomogeneity as it can be difficult remove the\ninhomogeneity in subsequent treatments. It is also important to use\nstarting materials of highest purity and minimize the loss and\ncontamination of materials during the whole experiments using a\nprotective atmosphere of inert gas or vacuum. The typical melting\ntechniques include high temperature furnaces with crucibles, arc\nmelting, and induction melting. Attentions need to be paid for possible\nreactions between materials and crucibles/containers, which can be\navoided by levitating the materials through electromagnetic fields or\nother means. In addition to use pure elements as raw materials, master\nalloys with well-controlled compositions are often utilized because the\ncompositions and melting properties of master alloys are usually much\ncloser to those of final materials than the pure elements. For materials\nwith very high melting temperature or volatile components, the powder\nmetallurgy method can be used where compacts are made, capsulated, and\nsintered.\n\nHomogenization during subsequent heat treatment is achieved through\ndiffusion, in which time and temperature are two important parameters.\nTo accelerate the homogenization process, the heat treatment temperature\nshould be as close to the solidus temperature as possible taking into\naccount the composition inhomogeneity with variable solidus\ntemperatures. When there are phase transformations taking place during\nthe heat treatment, extra time is needed for the heat treatment.\n\nThe phase boundaries are then determined through measurements of either\ncompositions of individual phases that are in equilibrium under constant\ntemperature, pressure/stress/strain, and electric/magnetic fields or\ndiscontinuity in some physical properties of materials due to a phase\ntransition from the continuous change of temperature or\npressure/stress/strain or electric/magnetic fields. The measurement of\ncompositions is usually carried out at ambient conditions so it is\nnecessary that the phases are fully equilibrated at experimental\nconditions, which requires rigorous verifications, and can be \u201cquenched\u201d\nto ambient conditions to remain unaltered during \u201cquenching\u201d, at least\nin terms of compositions. The compositions are typically measured by\nscanning electron microscopy (SEM) equipped with energy dispersive\nspectrometer (EDS) or wavelength dispersive spectroscopy (WDS), with a\nmicro-level spatial resolution and a better compositional resolution of\nWDS than EDS. A dedicated SEM with WDS gives another important, widely\nused composition measurement technique called electron probe\nmicroanalysis (EMPA). For submicron-sized phases, analytical\ntransmission electron microscopy equipped with EDS can be used though\ncare must be taken to avoid interference from neighboring phases.\n\nTo accurately identify phases in equilibrium under experimental\nconditions, in-situ characterizations are necessary, which complicates\nthe experimentation. An alternative indirect method is to measure a\nphysical property that changes discontinuously or dramatically for a\nfirst- or second-order phase transition, such as heat, volume, electric\nconductivity, and magnetization. There are two widely used techniques in\nmeasuring heat: differential thermal analysis (DTA) and differential\nscanning calorimetry (DSC). Both attempt to measure the difference in\ntemperature with the same amount of power supplied between a sample and\nan inert standard during heating or cooling. A DSC may also measure the\ndifference in the amount of power supplied to keep their temperatures\nidentical. The deviation of this difference from a baseline indicates a\nphase transition in the sample and is plotted as a function of time or\ntemperature of either the sample or the inert standard. It is evident\nthat the major challenges in both DTA and DSC techniques are to reach\nthermal equilibrium between the sample/standard and the instrument and\nthe thermodynamic equilibrium within the sample due to the continuous\nheating or cooling. The thermal equilibrium can be improved or mitigated\nthrough various methods such as sample shape and size, crucible\nselection, mixture with the material used for the inert standard, and\nwith thermocouples in direct contact with the sample and the inert\nstandard. However, the thermodynamic equilibrium within the sample can\nonly be reached when the heating/cooling rate is comparable with the\nrate of the phase transition in the sample, which is almost impossible\nif the phase transition typically involves diffusion in solid phases.\nTherefore, extreme care is needed in interpreting the temperature\ndetermination and the amount of heat associated with the DTA/DSC curves\nas discussed in detail in the reference \\[11\\].\n", + "source": [ + "### Equilibrated materials\n", + "\n", + "The most common method to determine phase equilibria is to use\n", + "equilibrated materials. This method typically involves material\n", + "preparation through high temperature melting or powder metallurgy,\n", + "homogenization heat treatment, isothermal or cooling/heating procedure,\n", + "and identifications of crystal structures and phase compositions. It is\n", + "important to avoid macro-inhomogeneity as it can be difficult remove the\n", + "inhomogeneity in subsequent treatments. It is also important to use\n", + "starting materials of highest purity and minimize the loss and\n", + "contamination of materials during the whole experiments using a\n", + "protective atmosphere of inert gas or vacuum. The typical melting\n", + "techniques include high temperature furnaces with crucibles, arc\n", + "melting, and induction melting. Attentions need to be paid for possible\n", + "reactions between materials and crucibles/containers, which can be\n", + "avoided by levitating the materials through electromagnetic fields or\n", + "other means. In addition to use pure elements as raw materials, master\n", + "alloys with well-controlled compositions are often utilized because the\n", + "compositions and melting properties of master alloys are usually much\n", + "closer to those of final materials than the pure elements. For materials\n", + "with very high melting temperature or volatile components, the powder\n", + "metallurgy method can be used where compacts are made, capsulated, and\n", + "sintered.\n", + "\n", + "Homogenization during subsequent heat treatment is achieved through\n", + "diffusion, in which time and temperature are two important parameters.\n", + "To accelerate the homogenization process, the heat treatment temperature\n", + "should be as close to the solidus temperature as possible taking into\n", + "account the composition inhomogeneity with variable solidus\n", + "temperatures. When there are phase transformations taking place during\n", + "the heat treatment, extra time is needed for the heat treatment.\n", + "\n", + "The phase boundaries are then determined through measurements of either\n", + "compositions of individual phases that are in equilibrium under constant\n", + "temperature, pressure/stress/strain, and electric/magnetic fields or\n", + "discontinuity in some physical properties of materials due to a phase\n", + "transition from the continuous change of temperature or\n", + "pressure/stress/strain or electric/magnetic fields. The measurement of\n", + "compositions is usually carried out at ambient conditions so it is\n", + "necessary that the phases are fully equilibrated at experimental\n", + "conditions, which requires rigorous verifications, and can be \u201cquenched\u201d\n", + "to ambient conditions to remain unaltered during \u201cquenching\u201d, at least\n", + "in terms of compositions. The compositions are typically measured by\n", + "scanning electron microscopy (SEM) equipped with energy dispersive\n", + "spectrometer (EDS) or wavelength dispersive spectroscopy (WDS), with a\n", + "micro-level spatial resolution and a better compositional resolution of\n", + "WDS than EDS. A dedicated SEM with WDS gives another important, widely\n", + "used composition measurement technique called electron probe\n", + "microanalysis (EMPA). For submicron-sized phases, analytical\n", + "transmission electron microscopy equipped with EDS can be used though\n", + "care must be taken to avoid interference from neighboring phases.\n", + "\n", + "To accurately identify phases in equilibrium under experimental\n", + "conditions, in-situ characterizations are necessary, which complicates\n", + "the experimentation. An alternative indirect method is to measure a\n", + "physical property that changes discontinuously or dramatically for a\n", + "first- or second-order phase transition, such as heat, volume, electric\n", + "conductivity, and magnetization. There are two widely used techniques in\n", + "measuring heat: differential thermal analysis (DTA) and differential\n", + "scanning calorimetry (DSC). Both attempt to measure the difference in\n", + "temperature with the same amount of power supplied between a sample and\n", + "an inert standard during heating or cooling. A DSC may also measure the\n", + "difference in the amount of power supplied to keep their temperatures\n", + "identical. The deviation of this difference from a baseline indicates a\n", + "phase transition in the sample and is plotted as a function of time or\n", + "temperature of either the sample or the inert standard. It is evident\n", + "that the major challenges in both DTA and DSC techniques are to reach\n", + "thermal equilibrium between the sample/standard and the instrument and\n", + "the thermodynamic equilibrium within the sample due to the continuous\n", + "heating or cooling. The thermal equilibrium can be improved or mitigated\n", + "through various methods such as sample shape and size, crucible\n", + "selection, mixture with the material used for the inert standard, and\n", + "with thermocouples in direct contact with the sample and the inert\n", + "standard. However, the thermodynamic equilibrium within the sample can\n", + "only be reached when the heating/cooling rate is comparable with the\n", + "rate of the phase transition in the sample, which is almost impossible\n", + "if the phase transition typically involves diffusion in solid phases.\n", + "Therefore, extreme care is needed in interpreting the temperature\n", + "determination and the amount of heat associated with the DTA/DSC curves\n", + "as discussed in detail in the reference \\[11\\].\n" + ], "metadata": {} }, { - "id": "b173f28f", + "id": "dc62bc68", "cell_type": "markdown", - "source": "### Diffusion couples/multiples\n\nThe major challenge in the equilibrated material approach is to ensure\nthat the whole sample reaches equilibrium. On the other hand the\ndiffusion couple/multiple technique does not require the whole sample to\nbe in equilibrium and is based on the assumption that any two phases in\ncontact are in equilibrium with each other at the phase interface, and\nthe phase compositions can be obtained by extrapolation of concentration\nprofiles in the two phases to the phase interface. Since the total\nsystem of a diffusion couple is not at equilibrium, many kinetic\nphenomena related to diffusion can be studied in a diffusion couple,\nsuch as interdiffusion coefficients, parabolic growth kinetics of\nproduct layer thickness, diffusion path (represented by the local\noverall compositions) in ternary and multi-component systems for\nvisualizing the microstructure of reaction zones, and other properties,\nall as a function of composition, which are beyond the scope of the\npresent book.\n\nTypical diffusion couples are in solid-state with two materials brought\ninto intimate contact to allow diffusion of elements between the two\nmaterials though solid-liquid diffusion couples are also used. The\ncontacting faces are commonly ground and polished flat, clamped together\nusing mechanical mechanisms, and annealed at high temperatures where\ndiffusion can take place to a significant degree in a matter of days,\nweeks or months. The samples are then quenched to retain the\nhigh-temperature equilibrium features. For metallic systems, diffusion\ncouples can also be prepared by eletrolytical and electroless plating\ntechniques. It is important to avoid the formation of liquid during\nannealing as it ruins the sample geometry. Furthermore, good adherence\nat the interfaces is critical for reliable data.\n\nSince diffusion couples are not in a fully equilibrium state, the\ntie-lines between two phases at the phase interfaces need to be obtained\nby extrapolations of concentration profiles in neighbouring phases. The\nelectron propagation in quantitative EPMA is typically in the range of\n1-2\u03bcm, yielding an excitation volume of approximately 2-4\u03bcm diameter and\na requirement of reasonable layer widths of phases on both sides of the\ninterface for accurate extrapolation. Therefore, the reliable\ncomposition of a single phase must be taken several micrometers away\nfrom the interface. When steep concentration gradients exist near the\ninterfaces, the extrapolation may lead to large errors, and analytical\nelectron microscopy is then needed. Furthermore, the fluorescence\neffects, where the primary excitation can be powerful enough to excite\nother elements in the sample, resulting in enhanced X-ray production and\nthe need of proper corrections. For a new phase to become observable\nexperimentally, it must nucleate and grow to reach the resolution limit\nof analytical tools. It is thus not uncommon that some known equilibrium\nphases are not found, and some non-equilibrium phases form. One way to\ncircumvent this issue is to use incremental diffusion couples with\nnarrow concentrations next to the phase of interest so that only this\nphase may be formed.\n\nFor ternary and higher-order systems, a more efficient approach can be\nused by placing a thin layer of the third alloy between two alloys. The\nthin central layer is eventually consumed, and the diffusion path is not\nfixed as in the semi-infinite diffusion couples. The phase compositions\nchange continuously with time as a result of the overlapping of two\nquasi-equilibrated diffusion zones.\n\nA diffusion multiple contains three or more pure elements or alloys of\ndifferent compositions and is a sample with multiple diffusion couples\nand diffusion triples in it. It is more efficient in terms of both\nmaterials and time in comparison with equilibrated alloys and diffusion\ncouples. All alloy blocks are prepared individually and sealed in vacuum\nin a cylinder which is also used as one alloy for the diffusion\nmultiple. The sealed cylinder with vacuum inside also serves as the can\nfor subsequent hot isostatic pressing to achieve intimate interfacial\ncontacts. The cylinder can then be cut into disks for further annealing\ntreatments. A broad range of design strategies is needed for complex\ndiffusion multiples along with automated plotting procedures due to the\nlarge amounts of EMPA data. The major source of error lies in the\nextraction of tie-lines from EPMA results due to very condensed\ninformation in a small area and the deviation of scanned lines from\nthose perpendicular to the interface.\n\nIn terms of local equilibrium characteristics of diffusion\ncouples/multiples, it is evident that in the equilibrated materials\napproach, it may not be necessary to reach the full equilibrium for the\nwhole sample if one is only interested in the local equilibrium\ncompositions between two neighboring phases, which can even provide\ninformation on metastable extensions of two phases if the two phases are\nin a metastable equilibrium at the annealing temperature.\n", + "source": [ + "### Diffusion couples/multiples\n", + "\n", + "The major challenge in the equilibrated material approach is to ensure\n", + "that the whole sample reaches equilibrium. On the other hand the\n", + "diffusion couple/multiple technique does not require the whole sample to\n", + "be in equilibrium and is based on the assumption that any two phases in\n", + "contact are in equilibrium with each other at the phase interface, and\n", + "the phase compositions can be obtained by extrapolation of concentration\n", + "profiles in the two phases to the phase interface. Since the total\n", + "system of a diffusion couple is not at equilibrium, many kinetic\n", + "phenomena related to diffusion can be studied in a diffusion couple,\n", + "such as interdiffusion coefficients, parabolic growth kinetics of\n", + "product layer thickness, diffusion path (represented by the local\n", + "overall compositions) in ternary and multi-component systems for\n", + "visualizing the microstructure of reaction zones, and other properties,\n", + "all as a function of composition, which are beyond the scope of the\n", + "present book.\n", + "\n", + "Typical diffusion couples are in solid-state with two materials brought\n", + "into intimate contact to allow diffusion of elements between the two\n", + "materials though solid-liquid diffusion couples are also used. The\n", + "contacting faces are commonly ground and polished flat, clamped together\n", + "using mechanical mechanisms, and annealed at high temperatures where\n", + "diffusion can take place to a significant degree in a matter of days,\n", + "weeks or months. The samples are then quenched to retain the\n", + "high-temperature equilibrium features. For metallic systems, diffusion\n", + "couples can also be prepared by eletrolytical and electroless plating\n", + "techniques. It is important to avoid the formation of liquid during\n", + "annealing as it ruins the sample geometry. Furthermore, good adherence\n", + "at the interfaces is critical for reliable data.\n", + "\n", + "Since diffusion couples are not in a fully equilibrium state, the\n", + "tie-lines between two phases at the phase interfaces need to be obtained\n", + "by extrapolations of concentration profiles in neighbouring phases. The\n", + "electron propagation in quantitative EPMA is typically in the range of\n", + "1-2\u03bcm, yielding an excitation volume of approximately 2-4\u03bcm diameter and\n", + "a requirement of reasonable layer widths of phases on both sides of the\n", + "interface for accurate extrapolation. Therefore, the reliable\n", + "composition of a single phase must be taken several micrometers away\n", + "from the interface. When steep concentration gradients exist near the\n", + "interfaces, the extrapolation may lead to large errors, and analytical\n", + "electron microscopy is then needed. Furthermore, the fluorescence\n", + "effects, where the primary excitation can be powerful enough to excite\n", + "other elements in the sample, resulting in enhanced X-ray production and\n", + "the need of proper corrections. For a new phase to become observable\n", + "experimentally, it must nucleate and grow to reach the resolution limit\n", + "of analytical tools. It is thus not uncommon that some known equilibrium\n", + "phases are not found, and some non-equilibrium phases form. One way to\n", + "circumvent this issue is to use incremental diffusion couples with\n", + "narrow concentrations next to the phase of interest so that only this\n", + "phase may be formed.\n", + "\n", + "For ternary and higher-order systems, a more efficient approach can be\n", + "used by placing a thin layer of the third alloy between two alloys. The\n", + "thin central layer is eventually consumed, and the diffusion path is not\n", + "fixed as in the semi-infinite diffusion couples. The phase compositions\n", + "change continuously with time as a result of the overlapping of two\n", + "quasi-equilibrated diffusion zones.\n", + "\n", + "A diffusion multiple contains three or more pure elements or alloys of\n", + "different compositions and is a sample with multiple diffusion couples\n", + "and diffusion triples in it. It is more efficient in terms of both\n", + "materials and time in comparison with equilibrated alloys and diffusion\n", + "couples. All alloy blocks are prepared individually and sealed in vacuum\n", + "in a cylinder which is also used as one alloy for the diffusion\n", + "multiple. The sealed cylinder with vacuum inside also serves as the can\n", + "for subsequent hot isostatic pressing to achieve intimate interfacial\n", + "contacts. The cylinder can then be cut into disks for further annealing\n", + "treatments. A broad range of design strategies is needed for complex\n", + "diffusion multiples along with automated plotting procedures due to the\n", + "large amounts of EMPA data. The major source of error lies in the\n", + "extraction of tie-lines from EPMA results due to very condensed\n", + "information in a small area and the deviation of scanned lines from\n", + "those perpendicular to the interface.\n", + "\n", + "In terms of local equilibrium characteristics of diffusion\n", + "couples/multiples, it is evident that in the equilibrated materials\n", + "approach, it may not be necessary to reach the full equilibrium for the\n", + "whole sample if one is only interested in the local equilibrium\n", + "compositions between two neighboring phases, which can even provide\n", + "information on metastable extensions of two phases if the two phases are\n", + "in a metastable equilibrium at the annealing temperature.\n" + ], "metadata": {} }, { - "id": "8e44cd11", + "id": "1da46570", "cell_type": "markdown", - "source": "### Additional methods\n\nElectrical resistivities of different phases are usually different. A\nchange of slope of electric resistivity as a function of composition or\ntemperature or pressure reflects a phase transformation. This technique\nis simple and reliable.\n\nMagnetic transitions can be measured using a vibrating sample or\nsuperconducting quantum interference device (SQUID) magnetometer by\ndetermining the magnetic moment of a sample in presence of an applied\nmagnetic field. Magnetic field lines form closed loops, resulting in an\nexternal dipolar and demagnetizing field in a finite-sized sample. The\neffective field sensed by the sample is thus the difference between the\napplied field and the demagnetizing field. The magnetic transition\ntemperature is evaluated from the Arrott plots where the ratio of\nmagnetic field over magnetization with a proper exponent is plotted with\nrespect to the magnetization with another proper exponent for a series\nof temperatures. Those proper exponents result in parallel isotherm\nlines, and the isotherm line intersecting the origin corresponds to the\nmagnetic transition temperature.\n\nThin films with composition gradients, commonly referred to as\ncombinatorial libraries, can be used to study the phase relations\nsimilar to diffusion couples/multiples though the results may differ due\nto the effects of surface and potential interactions with the substrate.\n\nPhase relations at high pressures are measured in equipment using\ndiamond anvil cells (DAC) or multi-anvil devices. The high pressure is\nrealized by decreasing the area, i.e. the anvil culet size. Pressures up\nto 100 GPa can be created in DAC with a culet size of 0.3mm for small\nsamples in the order of 0.2 to 0.4mm. For large samples, the\nlarge-volume presses (LVP) technique is developed, typically using WC\nwith the pressure mostly limited to 30 GPa and the sample size ranging\nfrom 1mm3 to 1cm3. The pressure can be measured by\neither the ruby (Cr3+ doped Al2O3)\nfluorescence line shift or the molar volume of a pressure marker by\nX-ray diffraction. The samples in DAC apparatus are heated by laser or\nresistive wire or a small heater around the samples, while high\ntemperatures in LVPs are achieved by resistive heaters. Crystal\nstructures are detected by in-situ X-ray or synchrotron diffraction.\nAttentions need to be paid to temperature and pressure homogeneity and\nthe non-hydrostatic stresses, which are both better controlled in LVP\nequipment.\n", + "source": [ + "### Additional methods\n", + "\n", + "Electrical resistivities of different phases are usually different. A\n", + "change of slope of electric resistivity as a function of composition or\n", + "temperature or pressure reflects a phase transformation. This technique\n", + "is simple and reliable.\n", + "\n", + "Magnetic transitions can be measured using a vibrating sample or\n", + "superconducting quantum interference device (SQUID) magnetometer by\n", + "determining the magnetic moment of a sample in presence of an applied\n", + "magnetic field. Magnetic field lines form closed loops, resulting in an\n", + "external dipolar and demagnetizing field in a finite-sized sample. The\n", + "effective field sensed by the sample is thus the difference between the\n", + "applied field and the demagnetizing field. The magnetic transition\n", + "temperature is evaluated from the Arrott plots where the ratio of\n", + "magnetic field over magnetization with a proper exponent is plotted with\n", + "respect to the magnetization with another proper exponent for a series\n", + "of temperatures. Those proper exponents result in parallel isotherm\n", + "lines, and the isotherm line intersecting the origin corresponds to the\n", + "magnetic transition temperature.\n", + "\n", + "Thin films with composition gradients, commonly referred to as\n", + "combinatorial libraries, can be used to study the phase relations\n", + "similar to diffusion couples/multiples though the results may differ due\n", + "to the effects of surface and potential interactions with the substrate.\n", + "\n", + "Phase relations at high pressures are measured in equipment using\n", + "diamond anvil cells (DAC) or multi-anvil devices. The high pressure is\n", + "realized by decreasing the area, i.e. the anvil culet size. Pressures up\n", + "to 100 GPa can be created in DAC with a culet size of 0.3mm for small\n", + "samples in the order of 0.2 to 0.4mm. For large samples, the\n", + "large-volume presses (LVP) technique is developed, typically using WC\n", + "with the pressure mostly limited to 30 GPa and the sample size ranging\n", + "from 1mm3 to 1cm3. The pressure can be measured by\n", + "either the ruby (Cr3+ doped Al2O3)\n", + "fluorescence line shift or the molar volume of a pressure marker by\n", + "X-ray diffraction. The samples in DAC apparatus are heated by laser or\n", + "resistive wire or a small heater around the samples, while high\n", + "temperatures in LVPs are achieved by resistive heaters. Crystal\n", + "structures are detected by in-situ X-ray or synchrotron diffraction.\n", + "Attentions need to be paid to temperature and pressure homogeneity and\n", + "the non-hydrostatic stresses, which are both better controlled in LVP\n", + "equipment.\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/experimental_data_for_thermodynamic_modeling/thermodynamic_data_from_experiments.ipynb b/src/psu410/src/experimental_data_for_thermodynamic_modeling/thermodynamic_data_from_experiments.ipynb index 32a09bc..3a98bce 100644 --- a/src/psu410/src/experimental_data_for_thermodynamic_modeling/thermodynamic_data_from_experiments.ipynb +++ b/src/psu410/src/experimental_data_for_thermodynamic_modeling/thermodynamic_data_from_experiments.ipynb @@ -4,27 +4,145 @@ "metadata": {}, "cells": [ { - "id": "904c370b", + "id": "b922c877", "cell_type": "markdown", - "source": "## Thermodynamic data from experiments\n\nBroadly speaking, thermodynamic data represents the values of Gibbs free\nenergy and all its first and second derivatives. For thermochemical\nproperties as the main topic of the book, calorimetric, electrochemical,\nand vapor pressure methods are the primary techniques used with the\nfirst for accurate measurement of enthalpy and entropy, and the latter\ntwo for direct determination of Gibbs energy and activity. The\nelectrochemical method is discussed in Chapter . The calorimetric method\nis divided into solution, combustion, direct reaction, and heat capacity\ncalorimetry, respectively, which all involve chemical reactions to be\ndiscussed in detail in Chapter . The vapor pressure method involves the\nequilibrium of volatile species between gas and samples and is divided\ninto Knudsen effusion and equilibration methods, respectively.\n", + "source": [ + "## Thermodynamic data from experiments\n", + "\n", + "Broadly speaking, thermodynamic data represents the values of Gibbs free\n", + "energy and all its first and second derivatives. For thermochemical\n", + "properties as the main topic of the book, calorimetric, electrochemical,\n", + "and vapor pressure methods are the primary techniques used with the\n", + "first for accurate measurement of enthalpy and entropy, and the latter\n", + "two for direct determination of Gibbs energy and activity. The\n", + "electrochemical method is discussed in Chapter . The calorimetric method\n", + "is divided into solution, combustion, direct reaction, and heat capacity\n", + "calorimetry, respectively, which all involve chemical reactions to be\n", + "discussed in detail in Chapter . The vapor pressure method involves the\n", + "equilibrium of volatile species between gas and samples and is divided\n", + "into Knudsen effusion and equilibration methods, respectively.\n" + ], "metadata": {} }, { - "id": "666b22ae", + "id": "6407a501", "cell_type": "markdown", - "source": "### Solution calorimetry\n\nThe book edited by Marsh and O\u2019Hare \\[12\\] documented the detailed\nexperimental techniques used for solution calorimetry. In one\nexperiment, the enthalpy of solution of a single phase is measured in a\nparticular solvent. To convert this enthalpy of solution into enthalpy\nof formation of the phase, a thermodynamic cycle is setup for a chemical\nreaction to form this phase from either constitutive pure elements or\ncompounds. Therefore in another experiment, the enthalpy of solution of\nconstitutive pure elements or compounds is measured in the solvent as\nidentical as possible to that used in the first experiment. The\ndifference of the two enthalpies of solution thus gives the enthalpy of\nformation of the single phase from its constitutive elements or\ncompounds at the temperature of the samples before they are dropped into\nthe solvent, usually at room temperature.\n\nThe solvent can be aqueous solutions at ambient temperatures and\npressures or metallic/salt/oxide melts at high temperatures under either\nadiabatic or isoperibol conditions. The adiabatic calorimetry measures\nthe temperature change of the solvent and is usually more accurate than\nthe isoperibol calorimetry that measures the heat generated during the\ndissolution, though the adiabatic ones are with more complex\ninstruments. It is important that the choices of solvent and temperature\nensure the complete dissolution of all substances into the solvent to\nform a homogeneous solution. Furthermore, the effect of dilution and of\nchanges in solvent composition needs to be considered in the calculation\nof enthalpy of solution.\n\nA large number of solvents are used. For aqueous solvent, hydrofluoric\nacid or mixtures of HF and HCl are often used. For oxides, buffer-type\nsystems are typical such as lead and alkali borates and alkali\ntungstates/molybdates. For metallic phases, low melting metals such as\nSn, Bi, In, Pb, and Cd, sometimes Al and Cu, are used. Factors such as\nsolubility, dissolution kinetics, thermal history, stirring, heat flow,\nparticle size, and system size are to be optimized for accurate\nmeasurements. To enhance solution kinetics, the compound to be studied\ncan be mixed with another element or compound so that the mixture can\nform liquid in the solvent at higher reaction rate. In any case, it is\nimportant to calibrate the system with pure elements and compounds of\nknown enthalpy of formation.\n", + "source": [ + "### Solution calorimetry\n", + "\n", + "The book edited by Marsh and O\u2019Hare \\[12\\] documented the detailed\n", + "experimental techniques used for solution calorimetry. In one\n", + "experiment, the enthalpy of solution of a single phase is measured in a\n", + "particular solvent. To convert this enthalpy of solution into enthalpy\n", + "of formation of the phase, a thermodynamic cycle is setup for a chemical\n", + "reaction to form this phase from either constitutive pure elements or\n", + "compounds. Therefore in another experiment, the enthalpy of solution of\n", + "constitutive pure elements or compounds is measured in the solvent as\n", + "identical as possible to that used in the first experiment. The\n", + "difference of the two enthalpies of solution thus gives the enthalpy of\n", + "formation of the single phase from its constitutive elements or\n", + "compounds at the temperature of the samples before they are dropped into\n", + "the solvent, usually at room temperature.\n", + "\n", + "The solvent can be aqueous solutions at ambient temperatures and\n", + "pressures or metallic/salt/oxide melts at high temperatures under either\n", + "adiabatic or isoperibol conditions. The adiabatic calorimetry measures\n", + "the temperature change of the solvent and is usually more accurate than\n", + "the isoperibol calorimetry that measures the heat generated during the\n", + "dissolution, though the adiabatic ones are with more complex\n", + "instruments. It is important that the choices of solvent and temperature\n", + "ensure the complete dissolution of all substances into the solvent to\n", + "form a homogeneous solution. Furthermore, the effect of dilution and of\n", + "changes in solvent composition needs to be considered in the calculation\n", + "of enthalpy of solution.\n", + "\n", + "A large number of solvents are used. For aqueous solvent, hydrofluoric\n", + "acid or mixtures of HF and HCl are often used. For oxides, buffer-type\n", + "systems are typical such as lead and alkali borates and alkali\n", + "tungstates/molybdates. For metallic phases, low melting metals such as\n", + "Sn, Bi, In, Pb, and Cd, sometimes Al and Cu, are used. Factors such as\n", + "solubility, dissolution kinetics, thermal history, stirring, heat flow,\n", + "particle size, and system size are to be optimized for accurate\n", + "measurements. To enhance solution kinetics, the compound to be studied\n", + "can be mixed with another element or compound so that the mixture can\n", + "form liquid in the solvent at higher reaction rate. In any case, it is\n", + "important to calibrate the system with pure elements and compounds of\n", + "known enthalpy of formation.\n" + ], "metadata": {} }, { - "id": "75b2d203", + "id": "37c416cc", "cell_type": "markdown", - "source": "### Combustion, direct reaction, and heat capacity calorimetry\n\nIn combustion calorimetry, the sample is ignited and reacts with\nreactive gases like oxygen or fluorine. To accurately calculate the\nenthalpy of formation from the enthalpy of combustion, the reliable\ncharacterization of reactants and reaction products is critical, such as\nincomplete combustion, impurities in the reactants which are often\nill-defined, and more than one reaction gaseous species and condensed\nphases. Combustion calorimeters are usually of isoperibol type around\nroom temperature in a water bath.\n\nThe direct reaction calorimetry is similar to the combustion calorimetry\nthough at high temperatures in heat-flux or adiabatic environments. The\npartial enthalpy of reaction can also be measured if the partial\npressure of volatile species can be controlled and measured. The key\nfactor for accurate results is that both the reactants and reaction\nproducts are well characterized and the reaction goes to completion\nquickly like in the combustion calorimetry. For reactive reactants,\nspecial procedure is needed to avoid their loss before the reaction\ntakes place.\n\nHeat capacity is defined as the amount of heat needed to increase the\ntemperature by 1K as shown by , and its integration with respect to\ntemperature from 0K gives entropy as shown by . At low temperatures,\nadiabatic calorimetry gives more accurate data of heat capacity though\nit is time consuming and requires complex instruments. At high\ntemperatures, the efficient and less accurate DSC method is more widely\nused.\n", + "source": [ + "### Combustion, direct reaction, and heat capacity calorimetry\n", + "\n", + "In combustion calorimetry, the sample is ignited and reacts with\n", + "reactive gases like oxygen or fluorine. To accurately calculate the\n", + "enthalpy of formation from the enthalpy of combustion, the reliable\n", + "characterization of reactants and reaction products is critical, such as\n", + "incomplete combustion, impurities in the reactants which are often\n", + "ill-defined, and more than one reaction gaseous species and condensed\n", + "phases. Combustion calorimeters are usually of isoperibol type around\n", + "room temperature in a water bath.\n", + "\n", + "The direct reaction calorimetry is similar to the combustion calorimetry\n", + "though at high temperatures in heat-flux or adiabatic environments. The\n", + "partial enthalpy of reaction can also be measured if the partial\n", + "pressure of volatile species can be controlled and measured. The key\n", + "factor for accurate results is that both the reactants and reaction\n", + "products are well characterized and the reaction goes to completion\n", + "quickly like in the combustion calorimetry. For reactive reactants,\n", + "special procedure is needed to avoid their loss before the reaction\n", + "takes place.\n", + "\n", + "Heat capacity is defined as the amount of heat needed to increase the\n", + "temperature by 1K as shown by , and its integration with respect to\n", + "temperature from 0K gives entropy as shown by . At low temperatures,\n", + "adiabatic calorimetry gives more accurate data of heat capacity though\n", + "it is time consuming and requires complex instruments. At high\n", + "temperatures, the efficient and less accurate DSC method is more widely\n", + "used.\n" + ], "metadata": {} }, { - "id": "5fed3852", + "id": "6205849e", "cell_type": "markdown", - "source": "### Vapor pressure method\n\nIn the Knudsen effusion method, a small amount of volatile species in\nthe gas phase effuses through a small orifice of 0.1 to 1 mm with\nnegligible influence of the equilibrium in the Knudsen cell. The vapor\nis ionized and analyzed with a mass spectrometer. The partial pressure\nof a species can be calculated from its ionization area and intensity\nthrough a calibration factor determined by a reference material with\nknown partial pressure. For high temperature measurements, care must be\ntaken to avoid the reactions between cell and sample and fragmentation\nof gas species on ionization. The typical vapor pressure range is\nbetween 10-7 and 10 Pa.\n\nIn various equilibration methods, the total vapor pressure is usually\nmeasured directly using pressure gauges in the range of 10-7\nand 100 kPa. Other direct or indirect methods include\n\nthermogravimetric method for measuring the vapor mass\n\natomic absorption spectroscopy for gas composition\n\nmeasurement of sample composition equilibrated with a gas of\nwell-defined activity of the volatile species\n\nthe dew point method in which the condensation temperature of the\nvolatile component is measured from the vapor equilibrated with the\nsample at a higher temperature,\n\nthe chemical transport method to be discussed in Chapter .\n\nThe main error in all these methods is often due to inadequate\nequilibration between vapor and sample.\n", + "source": [ + "### Vapor pressure method\n", + "\n", + "In the Knudsen effusion method, a small amount of volatile species in\n", + "the gas phase effuses through a small orifice of 0.1 to 1 mm with\n", + "negligible influence of the equilibrium in the Knudsen cell. The vapor\n", + "is ionized and analyzed with a mass spectrometer. The partial pressure\n", + "of a species can be calculated from its ionization area and intensity\n", + "through a calibration factor determined by a reference material with\n", + "known partial pressure. For high temperature measurements, care must be\n", + "taken to avoid the reactions between cell and sample and fragmentation\n", + "of gas species on ionization. The typical vapor pressure range is\n", + "between 10-7 and 10 Pa.\n", + "\n", + "In various equilibration methods, the total vapor pressure is usually\n", + "measured directly using pressure gauges in the range of 10-7\n", + "and 100 kPa. Other direct or indirect methods include\n", + "\n", + "thermogravimetric method for measuring the vapor mass\n", + "\n", + "atomic absorption spectroscopy for gas composition\n", + "\n", + "measurement of sample composition equilibrated with a gas of\n", + "well-defined activity of the volatile species\n", + "\n", + "the dew point method in which the condensation temperature of the\n", + "volatile component is measured from the vapor equilibrated with the\n", + "sample at a higher temperature,\n", + "\n", + "the chemical transport method to be discussed in Chapter .\n", + "\n", + "The main error in all these methods is often due to inadequate\n", + "equilibration between vapor and sample.\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/firstprinciples_calculations_and_theory/firstprinciples_approaches_to_disordered_alloys.ipynb b/src/psu410/src/firstprinciples_calculations_and_theory/firstprinciples_approaches_to_disordered_alloys.ipynb index b149d7d..1da27c5 100644 --- a/src/psu410/src/firstprinciples_calculations_and_theory/firstprinciples_approaches_to_disordered_alloys.ipynb +++ b/src/psu410/src/firstprinciples_calculations_and_theory/firstprinciples_approaches_to_disordered_alloys.ipynb @@ -4,27 +4,250 @@ "metadata": {}, "cells": [ { - "id": "969becdf", + "id": "0a55a5c4", "cell_type": "markdown", - "source": "## First-principles approaches to disordered alloys\n\nFirst-principles calcuations discussed so far strictly rely on the exact\natomic positions in the unit cells. A brutal-force approach for random\nsolution phase would be to directly construct a large supercell and\nrandomly decorate the host lattice with different types of atoms. Such\nan approach would necessarily require very large supercells to\nadequately mimic the statistics of the random solutions. Since\nfirst-principles methods are computationally constrained by the number\nof atoms that one can treat, this brute-force approach is\ncomputationally prohibitive. Take a binary\nA1-*x*B*x* substitutional alloy as illustrated in\nas an example, for a system containing *N* atoms, there can be\n2*N* possible number of configurations, which is an\nastronomically large number when *N* is large. It is an impossible task\nto explore such a huge configuration space with available computing\nresources.\n\nFigure \u2011. Mapping of a substitutional A1-xBx alloy\ninto an Ising-like lattice model \\[39-40\\].\n\nAs a result, approximations must be made to the first-principles\ncalculations. At present, there are mainly three approaches to\ncalculating the disordered solution phases: the coherent potential\napproximation (CPA) \\[41\\], the cluster expansion (CE) \\[42\\], and the\nspecial quasirandom structures (SQS\u2019s) \\[43\\] approach.\n\nCPA \\[41\\] treats random alloys by considering the *average* occupations\nof lattice sites in solving the Kohn-Sham equation. Since a mean-field\napproach is employed, dependence of properties on the local environments\nsurrounding an atom is not treated explicitly in CPA. In a random\nsolution, there exists a distribution of local environments (e.g., in\nbcc alloys, A or B surrounded by the various\nA*X*B8-*X* coordination shells with *X* between 0\nand 8), resulting in local environmentally-dependent quantities such as\ncharge transfer and local displacements of atoms from their ideal\nlattice positions. Even in random A1-*x*B*x* solid\nsolutions, the average A-A, A-B and B-B bond lengths are generally\ndifferent. These effects can be considered by the CE and SQS approaches,\nwhich are the focus of the next two subsections. In following\nsubsections, unless specifically noted, the formulism for the binary\nsystem is discussed for the sake of simplicity.\n", + "source": [ + "## First-principles approaches to disordered alloys\n", + "\n", + "First-principles calcuations discussed so far strictly rely on the exact\n", + "atomic positions in the unit cells. A brutal-force approach for random\n", + "solution phase would be to directly construct a large supercell and\n", + "randomly decorate the host lattice with different types of atoms. Such\n", + "an approach would necessarily require very large supercells to\n", + "adequately mimic the statistics of the random solutions. Since\n", + "first-principles methods are computationally constrained by the number\n", + "of atoms that one can treat, this brute-force approach is\n", + "computationally prohibitive. Take a binary\n", + "A1-*x*B*x* substitutional alloy as illustrated in\n", + "as an example, for a system containing *N* atoms, there can be\n", + "2*N* possible number of configurations, which is an\n", + "astronomically large number when *N* is large. It is an impossible task\n", + "to explore such a huge configuration space with available computing\n", + "resources.\n", + "\n", + "Figure \u2011. Mapping of a substitutional A1-xBx alloy\n", + "into an Ising-like lattice model \\[39-40\\].\n", + "\n", + "As a result, approximations must be made to the first-principles\n", + "calculations. At present, there are mainly three approaches to\n", + "calculating the disordered solution phases: the coherent potential\n", + "approximation (CPA) \\[41\\], the cluster expansion (CE) \\[42\\], and the\n", + "special quasirandom structures (SQS\u2019s) \\[43\\] approach.\n", + "\n", + "CPA \\[41\\] treats random alloys by considering the *average* occupations\n", + "of lattice sites in solving the Kohn-Sham equation. Since a mean-field\n", + "approach is employed, dependence of properties on the local environments\n", + "surrounding an atom is not treated explicitly in CPA. In a random\n", + "solution, there exists a distribution of local environments (e.g., in\n", + "bcc alloys, A or B surrounded by the various\n", + "A*X*B8-*X* coordination shells with *X* between 0\n", + "and 8), resulting in local environmentally-dependent quantities such as\n", + "charge transfer and local displacements of atoms from their ideal\n", + "lattice positions. Even in random A1-*x*B*x* solid\n", + "solutions, the average A-A, A-B and B-B bond lengths are generally\n", + "different. These effects can be considered by the CE and SQS approaches,\n", + "which are the focus of the next two subsections. In following\n", + "subsections, unless specifically noted, the formulism for the binary\n", + "system is discussed for the sake of simplicity.\n" + ], "metadata": {} }, { - "id": "8789b01f", + "id": "e3730978", "cell_type": "markdown", - "source": "### Cluster Expansions\n\nMany properties of a solution phase such as energy are dependent on the\n*configurations* - the arrangements of atoms on the lattice sites. In\ncluster expansion \\[35, 42\\], the configuration dependence of properties\nis formulated efficiently by a \u201clattice algebra\u201d which maps a\nsubstitutional configuration into an Ising-like lattice model. Taking a\nbinary A1-*x*B*x* solution phase for instance, A\natoms are represented by the \u201cdown\u201d spins (*Si* = -1) and B\natoms are represented by the \u201cup\u201d spins (*Si* = +1) as\nillustrated in . Using the cluster expansion technique, for a system\ncontaining *N* atoms, the total energy of any alloy configuration *\u03c3* =\n(*S*1, *S*2, \u2026, *SN*) can be\nconveniently evaluated using the following Ising-like Hamiltonian:\n\n*Eq. 5\u2011147*\n\nwhere *J*\u2019s are the effective cluster interactions (ECI\u2019s); is a number\nrepresenting the atomic occupation at the lattice *i* under the\nconfiguration *\u03c3*, which takes the values -1 and 1 for binary and -1, 0,\nand 1 for ternary etc. In , the 2-site, 3-site, and 4-site correlations\nare written as follows,\n\nEq. 5\u2011148\n\nEq. 5\u2011149\n\nEq. 5\u2011150 .\n\nThe expansion in would be exact as long as *all* the *n*-site\ninteractions are included. For a binary system, this can be observed by\nthe combination law that where is the number of *n*-site interactions.\nHowever, in actual calculations, one never does an expansion to the\norder *N* (containing 2*N* terms for binary) since it is too\nlong to be practical. In fact, since the interactions between widely\nseparated atoms are expected to be weaker than the interactions between\nnearer atoms for most of the important properties, the expansion in is\nusually truncated at certain distance to include only a few short-ranged\npair (2-site), triple (3-site), and up to the most, the quadruple\n(4-site) interactions.\n\nOnce a configuration is assigned, the *S*\u2019s are just the geometrical\nfactors. The common practice in the cluster expansion is that i) perform\nfirst-principles calculations of a selected set of configurations\n(around 20-100); ii) evaluate the values of *J*\u2019s using with the\nenergies from i); iii) use the fitted *J*\u2019s to predict the energy for a\ndesired set of configurations; and iv) make the ensemble average at a\ngiven temperature for the energetics of the random alloys through Monte\nCarlo simulations.\n", + "source": [ + "### Cluster Expansions\n", + "\n", + "Many properties of a solution phase such as energy are dependent on the\n", + "*configurations* - the arrangements of atoms on the lattice sites. In\n", + "cluster expansion \\[35, 42\\], the configuration dependence of properties\n", + "is formulated efficiently by a \u201clattice algebra\u201d which maps a\n", + "substitutional configuration into an Ising-like lattice model. Taking a\n", + "binary A1-*x*B*x* solution phase for instance, A\n", + "atoms are represented by the \u201cdown\u201d spins (*Si* = -1) and B\n", + "atoms are represented by the \u201cup\u201d spins (*Si* = +1) as\n", + "illustrated in . Using the cluster expansion technique, for a system\n", + "containing *N* atoms, the total energy of any alloy configuration *\u03c3* =\n", + "(*S*1, *S*2, \u2026, *SN*) can be\n", + "conveniently evaluated using the following Ising-like Hamiltonian:\n", + "\n", + "*Eq. 5\u2011147*\n", + "\n", + "where *J*\u2019s are the effective cluster interactions (ECI\u2019s); is a number\n", + "representing the atomic occupation at the lattice *i* under the\n", + "configuration *\u03c3*, which takes the values -1 and 1 for binary and -1, 0,\n", + "and 1 for ternary etc. In , the 2-site, 3-site, and 4-site correlations\n", + "are written as follows,\n", + "\n", + "Eq. 5\u2011148\n", + "\n", + "Eq. 5\u2011149\n", + "\n", + "Eq. 5\u2011150 .\n", + "\n", + "The expansion in would be exact as long as *all* the *n*-site\n", + "interactions are included. For a binary system, this can be observed by\n", + "the combination law that where is the number of *n*-site interactions.\n", + "However, in actual calculations, one never does an expansion to the\n", + "order *N* (containing 2*N* terms for binary) since it is too\n", + "long to be practical. In fact, since the interactions between widely\n", + "separated atoms are expected to be weaker than the interactions between\n", + "nearer atoms for most of the important properties, the expansion in is\n", + "usually truncated at certain distance to include only a few short-ranged\n", + "pair (2-site), triple (3-site), and up to the most, the quadruple\n", + "(4-site) interactions.\n", + "\n", + "Once a configuration is assigned, the *S*\u2019s are just the geometrical\n", + "factors. The common practice in the cluster expansion is that i) perform\n", + "first-principles calculations of a selected set of configurations\n", + "(around 20-100); ii) evaluate the values of *J*\u2019s using with the\n", + "energies from i); iii) use the fitted *J*\u2019s to predict the energy for a\n", + "desired set of configurations; and iv) make the ensemble average at a\n", + "given temperature for the energetics of the random alloys through Monte\n", + "Carlo simulations.\n" + ], "metadata": {} }, { - "id": "ff30bb64", + "id": "fccf9ee9", "cell_type": "markdown", - "source": "### Special Quasirandom Structures\n\nSQS\u2019s \\[43-44\\] are specially designed *small-unit-cell* periodic\nstructures with minimal number of atoms per unit cell, which closely\nmimic the most relevant, near-neighbor pair and multisite correlation\nfunctions of random substitutional alloys. The correlation functions are\nclassified by their *n*-site component \u201cfigures\u201d *f* = (*n,m*), where\nthe index *n* is called vortex for pair, triple, and quadruple\ncorrelations (*n* = 2, 3, and 4); *m* measures the correlation distance.\n\nIn the SQS approach, a distribution of distinct local environments is\nmaintained and their average corresponds to the random alloy. Thus, a\nsingle DFT calculation of an SQS can give many important alloy\nproperties (e.g. equilibrium bond lengths, charge transfer, formation\nenthalpies, etc.) that depend on the existence of those distinct local\nenvironments. The SQS approach has been used extensively to study the\nformation enthalpies, bond length distributions, density of states, band\ngaps and optical properties in semiconductor alloys. It is noted that\nthe CE approach can treat short-range ordering efficiently, while it is\nnot clear how the SQS approach can be used to represent short-range\nordering.\n\nThe key quantities in the SQS approach are the *n*-site correlation\nfunctions. Specifically, the 2-site correlation function corresponding\nto the 2-site component \u201cfigures\u201d (2,*m*) is\n\nEq. 5\u2011151\n\nwhere represents the total number of possible pairs with correlation\ndistance (neighboring distance) *Rij* being equal to *m*. The\n3-site correlation function corresponding to the 3-site component\n\u201cfigures\u201d (3,*m*) is\n\nEq. 5\u2011152\n\nwhere represents the total number of all possible 3-site \u201cfigures\u201d with\nthe correlation distance (size and shape) *Rijk* being equal\nto *m*. The 4-site correlation function corresponding to the 4-site\ncomponent \u201cfigures\u201d (4,*m*) is\n\nEq. 5\u2011153 ,\n\nwhere represents the total number of all possible 4-site \u201cfigures\u201d with\ncorrelation distance (size and shape) *Rijkl* being equal to\n*m*.\n\nWith a given supercell size *N*, the essential task of the SQS approach\nis to search through all configurations that approach as close as\npossible to the correlation functions of the perfectly random (*R*)\nstructure and for the binary system it is\n\nEq. 5\u2011154 .\n\nDescribing random alloys by small unit-cell periodical structures surely\nintroduces erroneous correlations beyond a certain distance. However,\nsince interactions between nearest neighbors are generally more\nimportant than interactions between more distant neighbors, SQS\u2019s can be\nconstructed in such a way that they exactly reproduce the correlation\nfunctions of a random alloy between the first few nearest neighbors,\ndeferring errors due to periodicity to more distant neighbors. The\npractical procedure could be that to find the structures that match the\n2-site correlation functions up to a given neighboring distance, and\nthen add the conditions matching the high order correlation functions up\nto certain correlation distance.\n\nAppendix B is a collection of the SQS\u2019s with a variety of compositions\nfor binary fcc, bcc, hcp, and L12 structures, for ternary\nfcc, bcc, and B2 structures, and Perovskite in the cubic ABO3\nstructure. The used format is that of VASP.\n", + "source": [ + "### Special Quasirandom Structures\n", + "\n", + "SQS\u2019s \\[43-44\\] are specially designed *small-unit-cell* periodic\n", + "structures with minimal number of atoms per unit cell, which closely\n", + "mimic the most relevant, near-neighbor pair and multisite correlation\n", + "functions of random substitutional alloys. The correlation functions are\n", + "classified by their *n*-site component \u201cfigures\u201d *f* = (*n,m*), where\n", + "the index *n* is called vortex for pair, triple, and quadruple\n", + "correlations (*n* = 2, 3, and 4); *m* measures the correlation distance.\n", + "\n", + "In the SQS approach, a distribution of distinct local environments is\n", + "maintained and their average corresponds to the random alloy. Thus, a\n", + "single DFT calculation of an SQS can give many important alloy\n", + "properties (e.g. equilibrium bond lengths, charge transfer, formation\n", + "enthalpies, etc.) that depend on the existence of those distinct local\n", + "environments. The SQS approach has been used extensively to study the\n", + "formation enthalpies, bond length distributions, density of states, band\n", + "gaps and optical properties in semiconductor alloys. It is noted that\n", + "the CE approach can treat short-range ordering efficiently, while it is\n", + "not clear how the SQS approach can be used to represent short-range\n", + "ordering.\n", + "\n", + "The key quantities in the SQS approach are the *n*-site correlation\n", + "functions. Specifically, the 2-site correlation function corresponding\n", + "to the 2-site component \u201cfigures\u201d (2,*m*) is\n", + "\n", + "Eq. 5\u2011151\n", + "\n", + "where represents the total number of possible pairs with correlation\n", + "distance (neighboring distance) *Rij* being equal to *m*. The\n", + "3-site correlation function corresponding to the 3-site component\n", + "\u201cfigures\u201d (3,*m*) is\n", + "\n", + "Eq. 5\u2011152\n", + "\n", + "where represents the total number of all possible 3-site \u201cfigures\u201d with\n", + "the correlation distance (size and shape) *Rijk* being equal\n", + "to *m*. The 4-site correlation function corresponding to the 4-site\n", + "component \u201cfigures\u201d (4,*m*) is\n", + "\n", + "Eq. 5\u2011153 ,\n", + "\n", + "where represents the total number of all possible 4-site \u201cfigures\u201d with\n", + "correlation distance (size and shape) *Rijkl* being equal to\n", + "*m*.\n", + "\n", + "With a given supercell size *N*, the essential task of the SQS approach\n", + "is to search through all configurations that approach as close as\n", + "possible to the correlation functions of the perfectly random (*R*)\n", + "structure and for the binary system it is\n", + "\n", + "Eq. 5\u2011154 .\n", + "\n", + "Describing random alloys by small unit-cell periodical structures surely\n", + "introduces erroneous correlations beyond a certain distance. However,\n", + "since interactions between nearest neighbors are generally more\n", + "important than interactions between more distant neighbors, SQS\u2019s can be\n", + "constructed in such a way that they exactly reproduce the correlation\n", + "functions of a random alloy between the first few nearest neighbors,\n", + "deferring errors due to periodicity to more distant neighbors. The\n", + "practical procedure could be that to find the structures that match the\n", + "2-site correlation functions up to a given neighboring distance, and\n", + "then add the conditions matching the high order correlation functions up\n", + "to certain correlation distance.\n", + "\n", + "Appendix B is a collection of the SQS\u2019s with a variety of compositions\n", + "for binary fcc, bcc, hcp, and L12 structures, for ternary\n", + "fcc, bcc, and B2 structures, and Perovskite in the cubic ABO3\n", + "structure. The used format is that of VASP.\n" + ], "metadata": {} }, { - "id": "4cd57d86", + "id": "b1eee4e6", "cell_type": "markdown", - "source": "### Phonon calculations for SQS\n\nA somewhat more theoretically demanding application of the SQS approach\nis the calculation of the phonon dispersions of a random alloy.\nConsidering the fact that the size of an SQS cell in general is around\n8-32, phonon calculations based on SQS is doable, either using the SQS\ncell or its supercell, e.g., of it. However, one notes that while the\nphonon density of states can be calculated straightforwardly, the\ncalculations of the phonon dispersions run into a problem. That is,\nsince phonon calculation treats the SQS as primitive unit cell made of\nmore atoms than the primitive unit cell of the ideal lattice, the number\nof phonon dispersions derived from a regular phonon calculations is a\nlot greater than that measured for the random alloy. Say, one uses an\nSQS containing 16 atoms for an fcc solid solution, the regular phonon\ncalculations would produce phonon dispersions in comparison to that just\nthree phonon dispersions from measurement. By averaging over the force\nconstants of a SQS, the dynamical matrix can be calculated with respect\nto the wave vector space of the ideal lattice of the random alloys.\n\nOne consideration that must be taken into account is that the phonon\ndispersions measured from the inelastic neutron scattering experiments\nonly represent the averaged vibrations of the ideal lattice. For random\nalloys or phases with minor geometry distortion, it is suggested to\ncalculate the dynamical matrix by instead of *Eq. 5\u2011125* as (see \\[45\\])\n\n*Eq. 5\u2011155* ,\n\nwhere in the case of random alloy, represents the averaged atomic mass\nat the *j*th lattice site. The purpose of the summation over *Q* is to\naverage the effects of local distortions, making it possible of\ncomparing the calculated dispersions to the measured dispersions\nrepresenting the averaged vibrations of the ideal lattice. As a result,\nthe dimension of the SQS supercell dynamical matrix is thus reduced to\nmatch that of the primitive unit cell of the ideal lattice for the\ncalculation of the phonon frequencies. The calculational procedure is as\nfollows:\n\n1. Make an SQS supercell based on the primitive unit cell of ideal\n lattice to mimic the correlation functions of the random solution;\n\n2. Relax the SQS supercell with respect to the internal atomic\n positions while keeping the cell shape and volume fixed;\n\n3. Make the phonon supercell by further enlarging the SQS supercell and\n calculate the force constants; and\n\n4. Calculate the dynamical matrix , with the wave vector, **q**, being\n defined from the primitive unit cell of the ideal lattice, through\n the following Fourier transformation.\n\nThe calculated phonon dispersions, along the directions (00\u03be), (0\u03be\u03be),\nand (\u03be\u03be\u03be), are compared with the inelastic neutron scattering data in\nfor Cu3Au.\n\nFigure 5\u201113. Phonon dispersions for random Cu3Au. The solid\n(black) lines represent the present calculation and the open circles\nrepresent the inelastic neutron scattering data with details in Ref.\n\\[45\\]. The dashed (blue) lines represent the calculated results using\nthe ab initio transferable force-constant model by Dutta et al. \\[46\\].\n", + "source": [ + "### Phonon calculations for SQS\n", + "\n", + "A somewhat more theoretically demanding application of the SQS approach\n", + "is the calculation of the phonon dispersions of a random alloy.\n", + "Considering the fact that the size of an SQS cell in general is around\n", + "8-32, phonon calculations based on SQS is doable, either using the SQS\n", + "cell or its supercell, e.g., of it. However, one notes that while the\n", + "phonon density of states can be calculated straightforwardly, the\n", + "calculations of the phonon dispersions run into a problem. That is,\n", + "since phonon calculation treats the SQS as primitive unit cell made of\n", + "more atoms than the primitive unit cell of the ideal lattice, the number\n", + "of phonon dispersions derived from a regular phonon calculations is a\n", + "lot greater than that measured for the random alloy. Say, one uses an\n", + "SQS containing 16 atoms for an fcc solid solution, the regular phonon\n", + "calculations would produce phonon dispersions in comparison to that just\n", + "three phonon dispersions from measurement. By averaging over the force\n", + "constants of a SQS, the dynamical matrix can be calculated with respect\n", + "to the wave vector space of the ideal lattice of the random alloys.\n", + "\n", + "One consideration that must be taken into account is that the phonon\n", + "dispersions measured from the inelastic neutron scattering experiments\n", + "only represent the averaged vibrations of the ideal lattice. For random\n", + "alloys or phases with minor geometry distortion, it is suggested to\n", + "calculate the dynamical matrix by instead of *Eq. 5\u2011125* as (see \\[45\\])\n", + "\n", + "*Eq. 5\u2011155* ,\n", + "\n", + "where in the case of random alloy, represents the averaged atomic mass\n", + "at the *j*th lattice site. The purpose of the summation over *Q* is to\n", + "average the effects of local distortions, making it possible of\n", + "comparing the calculated dispersions to the measured dispersions\n", + "representing the averaged vibrations of the ideal lattice. As a result,\n", + "the dimension of the SQS supercell dynamical matrix is thus reduced to\n", + "match that of the primitive unit cell of the ideal lattice for the\n", + "calculation of the phonon frequencies. The calculational procedure is as\n", + "follows:\n", + "\n", + "1. Make an SQS supercell based on the primitive unit cell of ideal\n", + " lattice to mimic the correlation functions of the random solution;\n", + "\n", + "2. Relax the SQS supercell with respect to the internal atomic\n", + " positions while keeping the cell shape and volume fixed;\n", + "\n", + "3. Make the phonon supercell by further enlarging the SQS supercell and\n", + " calculate the force constants; and\n", + "\n", + "4. Calculate the dynamical matrix , with the wave vector, **q**, being\n", + " defined from the primitive unit cell of the ideal lattice, through\n", + " the following Fourier transformation.\n", + "\n", + "The calculated phonon dispersions, along the directions (00\u03be), (0\u03be\u03be),\n", + "and (\u03be\u03be\u03be), are compared with the inelastic neutron scattering data in\n", + "for Cu3Au.\n", + "\n", + "Figure 5\u201113. Phonon dispersions for random Cu3Au. The solid\n", + "(black) lines represent the present calculation and the open circles\n", + "represent the inelastic neutron scattering data with details in Ref.\n", + "\\[45\\]. The dashed (blue) lines represent the calculated results using\n", + "the ab initio transferable force-constant model by Dutta et al. \\[46\\].\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/firstprinciples_calculations_and_theory/firstprinciples_formulation_of_thermodynamics.ipynb b/src/psu410/src/firstprinciples_calculations_and_theory/firstprinciples_formulation_of_thermodynamics.ipynb index aeefb19..4029dc5 100644 --- a/src/psu410/src/firstprinciples_calculations_and_theory/firstprinciples_formulation_of_thermodynamics.ipynb +++ b/src/psu410/src/firstprinciples_calculations_and_theory/firstprinciples_formulation_of_thermodynamics.ipynb @@ -4,39 +4,311 @@ "metadata": {}, "cells": [ { - "id": "fc30e7f5", + "id": "ffde7f8b", "cell_type": "markdown", - "source": "## First-principles formulation of thermodynamics\n", + "source": [ + "## First-principles formulation of thermodynamics\n" + ], "metadata": {} }, { - "id": "a7e89071", + "id": "4990a52d", "cell_type": "markdown", - "source": "### Helmholtz energy\n\nIn this chapter so far, all our discussions are limited to the case of a\nsystem which is made from a single microstate (microscopic state). Here,\nthe terminology microstate refers to the microscopic structure that is\ndistinguished by crystal structure, atom distributions in the lattice\nsites, and the arrangements of the local atomic spin and electronic\nangular momentum distributions among the lattice sites. From this\nsection on, the index \u03c3 is employed to label the microstate. For a solid\nmaterial at finite temperatures, a phase can be formed by a single\nmicrostate or a mixture of multiple microstates.\n\nLet us consider a canonical system made of *N* atoms with an average\natomic volume *V*. The study is limited to the motions of atomic nuclei\nand electrons. For such a system, one can use to index the energy\neigenvalues of the corresponding microscopic Hamiltonian associated with\nmicrostate *\u03c3*. The subscript **g** symbolically labels the different\nvibrational states for the motions of atomic nuclei and the subscript\n**n** symbolically labels the electronic states distinguished by the\ndifferent distributions of the electrons between the electronic valence\nand conduction bands. Neglecting electron-phonon coupling, one can\nassume that the contributions to between the vibrational and electronic\nstates are additive, so that\n\n*Eq. 5\u201116* ,\n\nwhere *Ec* is the static total energy of the microstate *\u03c3*.\nNote that in and represent the energetics of the vibrational state and\nthe electronic state respectively.\n\nOne then can formulate the canonical partition function of the\nmicrostate *\u03c3* at the given temperature *T* and volume *V* as\n\n*Eq. 5\u201117* ,\n\nwhere *\u03b2* = 1/*kBT* with *kB* being the\nBoltzmann\u2019s constant. As a result, with, the Helmholtz energy *F* per\natom for the microstate *\u03c3* is derived as follows:\n\n*Eq. 5\u201118* ,\n\nwhere the variable *N* has been abbreviated using\n\n*Eq. 5\u201119* ,\n\n*Eq. 5\u201120* ,\n\n*Eq. 5\u201121* .\n\nThe calculation of *Ec* is straightforward in most of the\nexisting first-principles codes as discussed earlier.\n", + "source": [ + "### Helmholtz energy\n", + "\n", + "In this chapter so far, all our discussions are limited to the case of a\n", + "system which is made from a single microstate (microscopic state). Here,\n", + "the terminology microstate refers to the microscopic structure that is\n", + "distinguished by crystal structure, atom distributions in the lattice\n", + "sites, and the arrangements of the local atomic spin and electronic\n", + "angular momentum distributions among the lattice sites. From this\n", + "section on, the index \u03c3 is employed to label the microstate. For a solid\n", + "material at finite temperatures, a phase can be formed by a single\n", + "microstate or a mixture of multiple microstates.\n", + "\n", + "Let us consider a canonical system made of *N* atoms with an average\n", + "atomic volume *V*. The study is limited to the motions of atomic nuclei\n", + "and electrons. For such a system, one can use to index the energy\n", + "eigenvalues of the corresponding microscopic Hamiltonian associated with\n", + "microstate *\u03c3*. The subscript **g** symbolically labels the different\n", + "vibrational states for the motions of atomic nuclei and the subscript\n", + "**n** symbolically labels the electronic states distinguished by the\n", + "different distributions of the electrons between the electronic valence\n", + "and conduction bands. Neglecting electron-phonon coupling, one can\n", + "assume that the contributions to between the vibrational and electronic\n", + "states are additive, so that\n", + "\n", + "*Eq. 5\u201116* ,\n", + "\n", + "where *Ec* is the static total energy of the microstate *\u03c3*.\n", + "Note that in and represent the energetics of the vibrational state and\n", + "the electronic state respectively.\n", + "\n", + "One then can formulate the canonical partition function of the\n", + "microstate *\u03c3* at the given temperature *T* and volume *V* as\n", + "\n", + "*Eq. 5\u201117* ,\n", + "\n", + "where *\u03b2* = 1/*kBT* with *kB* being the\n", + "Boltzmann\u2019s constant. As a result, with, the Helmholtz energy *F* per\n", + "atom for the microstate *\u03c3* is derived as follows:\n", + "\n", + "*Eq. 5\u201118* ,\n", + "\n", + "where the variable *N* has been abbreviated using\n", + "\n", + "*Eq. 5\u201119* ,\n", + "\n", + "*Eq. 5\u201120* ,\n", + "\n", + "*Eq. 5\u201121* .\n", + "\n", + "The calculation of *Ec* is straightforward in most of the\n", + "existing first-principles codes as discussed earlier.\n" + ], "metadata": {} }, { - "id": "3902983f", + "id": "e74adeb3", "cell_type": "markdown", - "source": "### Mermin statistics to the thermal electronic contribution \n\nFor the calculation of *Fel* in , the most computationally\nflexible approach is to use the Mermin statistics \\[28\\] by which\n\n*Eq. 5\u201122*\n\nwhere the bare electronic entropy *Sel* takes the form, after\nreplacing the summation in over the electronic states with integration\n\n*Eq. 5\u201123* ,\n\nby means of utilizing , the electronic density of states *n*. *f* in is\nthe Fermi distribution that takes the form\n\n*Eq. 5\u201124* .\n\nNote that in is the electronic chemical potential that is strongly\ntemperature dependent. At each given *T*, should be carefully calculated\nkeeping the number of electrons unchanged in solving the following\nequation:\n\n*Eq. 5\u201125* ,\n\nnoting that is the Fermi energy calculated at 0 K. With respect to , the\nthermal electronic energy *Eel*, due to the thermal electron\nexcitations, can be calculated through\n\n*Eq. 5\u201126* .\n\nAt low temperature, , , and are reduced to\n\n*Eq. 5\u201127*\n\nwhere *\u03bb* is the so-called electronic specific heat coefficient\ncalculated as\n\n*Eq. 5\u201128* ,\n\nwhere is the electronic density of states at the Fermi level, and\n\n*Eq. 5\u201129*\n\n*Eq. 5\u201130*\n\nFrom , one can easily derive the electronic contribution to the specific\nheat at low temperature as\n\n*Eq. 5\u201131* .\n\nUsually, the dependence of on *V* is weak. Therefore for a normal\nconductor (except for the heavy Fermion metal or superconductor related\nmaterials), at low temperatures, the electronic contribution to the heat\ncapacity is linear against *T*. also indicates that for insulators, by\nmeans of , the electronic contribution to the heat capacity is zero\nsince for insulators .\n", + "source": [ + "### Mermin statistics to the thermal electronic contribution \n", + "\n", + "For the calculation of *Fel* in , the most computationally\n", + "flexible approach is to use the Mermin statistics \\[28\\] by which\n", + "\n", + "*Eq. 5\u201122*\n", + "\n", + "where the bare electronic entropy *Sel* takes the form, after\n", + "replacing the summation in over the electronic states with integration\n", + "\n", + "*Eq. 5\u201123* ,\n", + "\n", + "by means of utilizing , the electronic density of states *n*. *f* in is\n", + "the Fermi distribution that takes the form\n", + "\n", + "*Eq. 5\u201124* .\n", + "\n", + "Note that in is the electronic chemical potential that is strongly\n", + "temperature dependent. At each given *T*, should be carefully calculated\n", + "keeping the number of electrons unchanged in solving the following\n", + "equation:\n", + "\n", + "*Eq. 5\u201125* ,\n", + "\n", + "noting that is the Fermi energy calculated at 0 K. With respect to , the\n", + "thermal electronic energy *Eel*, due to the thermal electron\n", + "excitations, can be calculated through\n", + "\n", + "*Eq. 5\u201126* .\n", + "\n", + "At low temperature, , , and are reduced to\n", + "\n", + "*Eq. 5\u201127*\n", + "\n", + "where *\u03bb* is the so-called electronic specific heat coefficient\n", + "calculated as\n", + "\n", + "*Eq. 5\u201128* ,\n", + "\n", + "where is the electronic density of states at the Fermi level, and\n", + "\n", + "*Eq. 5\u201129*\n", + "\n", + "*Eq. 5\u201130*\n", + "\n", + "From , one can easily derive the electronic contribution to the specific\n", + "heat at low temperature as\n", + "\n", + "*Eq. 5\u201131* .\n", + "\n", + "Usually, the dependence of on *V* is weak. Therefore for a normal\n", + "conductor (except for the heavy Fermion metal or superconductor related\n", + "materials), at low temperatures, the electronic contribution to the heat\n", + "capacity is linear against *T*. also indicates that for insulators, by\n", + "means of , the electronic contribution to the heat capacity is zero\n", + "since for insulators .\n" + ], "metadata": {} }, { - "id": "927b0b2a", + "id": "2ca3897c", "cell_type": "markdown", - "source": "### Vibrational contribution by phonon theory\n\nUnder the harmonic/quasiharmonic approximation, the lattice dynamics or\nphonon theory is currently the most established method. It truncates the\ninteraction potential up to the second order. In such a case, in can be\nexpressed in terms of phonon frequency as follows\n\n*Eq. 5\u201132* ,\n\nwhere the label **g** has the meanings of (*g1*,\n*g2*, \u2026, *g3N*) that *gj* can take any\ninteger values from zero to infinite.\n\nAs a result, is reduced to\n\n*Eq. 5\u201133* ,\n\nor equivalently,\n\n*Eq. 5\u201134* ,\n\nwhere an integration has been used to replace the summation in by means\nof introducing a function, , named the phonon density of states (PDOS)\nwhose integration over *\u03c9* equals to three per atom.\n\nAccordingly, the formulation to calculate the entropy becomes\n\n*Eq. 5\u201135* ,\n\nthe formulation to calculate the internal energy becomes\n\n*Eq. 5\u201136* ,\n\nand the formulation to calculate the heat capacity at constant volume\nbecomes\n\n*Eq. 5\u201137* .\n", + "source": [ + "### Vibrational contribution by phonon theory\n", + "\n", + "Under the harmonic/quasiharmonic approximation, the lattice dynamics or\n", + "phonon theory is currently the most established method. It truncates the\n", + "interaction potential up to the second order. In such a case, in can be\n", + "expressed in terms of phonon frequency as follows\n", + "\n", + "*Eq. 5\u201132* ,\n", + "\n", + "where the label **g** has the meanings of (*g1*,\n", + "*g2*, \u2026, *g3N*) that *gj* can take any\n", + "integer values from zero to infinite.\n", + "\n", + "As a result, is reduced to\n", + "\n", + "*Eq. 5\u201133* ,\n", + "\n", + "or equivalently,\n", + "\n", + "*Eq. 5\u201134* ,\n", + "\n", + "where an integration has been used to replace the summation in by means\n", + "of introducing a function, , named the phonon density of states (PDOS)\n", + "whose integration over *\u03c9* equals to three per atom.\n", + "\n", + "Accordingly, the formulation to calculate the entropy becomes\n", + "\n", + "*Eq. 5\u201135* ,\n", + "\n", + "the formulation to calculate the internal energy becomes\n", + "\n", + "*Eq. 5\u201136* ,\n", + "\n", + "and the formulation to calculate the heat capacity at constant volume\n", + "becomes\n", + "\n", + "*Eq. 5\u201137* .\n" + ], "metadata": {} }, { - "id": "81df2df2", + "id": "a62c437f", "cell_type": "markdown", - "source": "### Debye-Gr\u00fcneisen approximation to the vibrational contribution \n\nStrictly speaking, the Debye theory is only accurate at very low\ntemperatures. It assumes a parabolic type of dependence of PDOS\u00a0on the\nphonon frequency. This assumption is only correct at the scale of 10\u2019s K\nbecause at low temperatures,\u00a0only the low frequency acoustic phonons are\nactivated which play the major role for the parabolic type of PDOS for\nthe low frequency range as shown in . That is why\u00a0there are two kinds of\nDebye temperature: low-temperature Debye temperature and\nhigh-temperature\u00a0Debye temperature. The low-temperature Debye\ntemperature can be strictly derived by fitting low temperature heat\ncapacity data. The high-temperature\u00a0Debye temperature is mostly a\nphenomenological fitting parameter.\n\nThe Debye model approximates the PDOS in by\n\n*Eq. 5\u201138* ,\n\nwhere is the so-called Debye cutoff frequency related to the Debye\ntemperature as\n\n*Eq. 5\u201139* .\n\nAs the result, the vibrational contribution to Helmholtz energy under\nthe Debye approximation becomes,\n\n*Eq. 5\u201140*\n\nwhere *D*(*\u0398D/T*) is the Debye function given by .\n\nUnder the Debye approximation, the formulation to calculate the entropy\nbecomes\n\n*Eq. 5\u201141* ,\n\nthe formulation to calculate the internal energy becomes\n\n*Eq. 5\u201142* ,\n\nwhere\n\n*Eq. 5\u201143* ,\n\nand the formulation to calculate the heat capacity at constant volume\nbecomes\n\n*Eq. 5\u201144* .\n\nHere it is noted that is volume dependent which is often written in\nterms of Gr\u00fcneisen constant:\n\n*Eq. 5\u201145* .\n\nIt has been found that the dependence of on *V* usually is weak and\nhence is often approximated as a constant. With , the formulation to\ncalculate the pressure becomes\n\n*Eq. 5\u201146* .\n\nAn important result of the Debye approximation is that when , together\nwith , the heat capacity in is reduced to\n\n*Eq. 5\u201147* .\n\nThis gives the Debye T3 law when the thermal electron\ncontribution is neglected. In the analysis of superconductor and heavy\nFermion materials, is often rewritten as\n\n*Eq. 5\u201148* ,\n\nwhich is more convenient for examining the heat capacity measured at\nextremely low temperatures.\n", + "source": [ + "### Debye-Gr\u00fcneisen approximation to the vibrational contribution \n", + "\n", + "Strictly speaking, the Debye theory is only accurate at very low\n", + "temperatures. It assumes a parabolic type of dependence of PDOS\u00a0on the\n", + "phonon frequency. This assumption is only correct at the scale of 10\u2019s K\n", + "because at low temperatures,\u00a0only the low frequency acoustic phonons are\n", + "activated which play the major role for the parabolic type of PDOS for\n", + "the low frequency range as shown in . That is why\u00a0there are two kinds of\n", + "Debye temperature: low-temperature Debye temperature and\n", + "high-temperature\u00a0Debye temperature. The low-temperature Debye\n", + "temperature can be strictly derived by fitting low temperature heat\n", + "capacity data. The high-temperature\u00a0Debye temperature is mostly a\n", + "phenomenological fitting parameter.\n", + "\n", + "The Debye model approximates the PDOS in by\n", + "\n", + "*Eq. 5\u201138* ,\n", + "\n", + "where is the so-called Debye cutoff frequency related to the Debye\n", + "temperature as\n", + "\n", + "*Eq. 5\u201139* .\n", + "\n", + "As the result, the vibrational contribution to Helmholtz energy under\n", + "the Debye approximation becomes,\n", + "\n", + "*Eq. 5\u201140*\n", + "\n", + "where *D*(*\u0398D/T*) is the Debye function given by .\n", + "\n", + "Under the Debye approximation, the formulation to calculate the entropy\n", + "becomes\n", + "\n", + "*Eq. 5\u201141* ,\n", + "\n", + "the formulation to calculate the internal energy becomes\n", + "\n", + "*Eq. 5\u201142* ,\n", + "\n", + "where\n", + "\n", + "*Eq. 5\u201143* ,\n", + "\n", + "and the formulation to calculate the heat capacity at constant volume\n", + "becomes\n", + "\n", + "*Eq. 5\u201144* .\n", + "\n", + "Here it is noted that is volume dependent which is often written in\n", + "terms of Gr\u00fcneisen constant:\n", + "\n", + "*Eq. 5\u201145* .\n", + "\n", + "It has been found that the dependence of on *V* usually is weak and\n", + "hence is often approximated as a constant. With , the formulation to\n", + "calculate the pressure becomes\n", + "\n", + "*Eq. 5\u201146* .\n", + "\n", + "An important result of the Debye approximation is that when , together\n", + "with , the heat capacity in is reduced to\n", + "\n", + "*Eq. 5\u201147* .\n", + "\n", + "This gives the Debye T3 law when the thermal electron\n", + "contribution is neglected. In the analysis of superconductor and heavy\n", + "Fermion materials, is often rewritten as\n", + "\n", + "*Eq. 5\u201148* ,\n", + "\n", + "which is more convenient for examining the heat capacity measured at\n", + "extremely low temperatures.\n" + ], "metadata": {} }, { - "id": "ea721412", + "id": "85af7136", "cell_type": "markdown", - "source": "### System with multiple microstates (MMS model)\n\nFor a system made of multiple microstates, the total partition function\nis the summation over the partition functions of all microstates, in ,\nas\n\n*Eq. 5\u201149* ,\n\nwhere is the multiplicity of the microstate *\u03c3.* It is immediately\napparent that is the thermal population of the microstate *\u03c3*.\nFurthermore, with , one obtains\n\n*Eq. 5\u201150* .\n\nrelates the total Helmholtz energy, *F(N,V,T)*, of a system with mixing\namong multiple microstates and the Helmholtz energy, , of individual\nmicrostates. An important result of is the configurational entropy due\nto the mixing among multiple microstates, named as microstate\nconfigurational entropy (MCE) in this book,\n\n*Eq. 5\u201151* ,\n\nwhich makes the entropy of a system with mixing among multiple\nmicrostates as\n\n*Eq. 5\u201152* .\n\nSimilarly, one can get the heat capacity at constant volume of a system\nwith mixing among multiple microstates as\n\n*Eq. 5\u201153* ,\n\nwhere the configurational contributions to the heat capacity due to the\nmixing among multiple microstates is\n\n*Eq. 5\u201154* .\n\nMoreover, the isothermal bulk modulus of a system with mixing among\nmultiple microstates can be also computed similarly\n\n*Eq. 5\u201155* ,\n\nWith\n\n*Eq. 5\u201156*\n\nwith being the partial pressure of the microstate *\u03c3*. This multiple\nmicrostate model (MMS model) is used in Chapter to quantitatively\npredict thermal expansion anomalies.\n", + "source": [ + "### System with multiple microstates (MMS model)\n", + "\n", + "For a system made of multiple microstates, the total partition function\n", + "is the summation over the partition functions of all microstates, in ,\n", + "as\n", + "\n", + "*Eq. 5\u201149* ,\n", + "\n", + "where is the multiplicity of the microstate *\u03c3.* It is immediately\n", + "apparent that is the thermal population of the microstate *\u03c3*.\n", + "Furthermore, with , one obtains\n", + "\n", + "*Eq. 5\u201150* .\n", + "\n", + "relates the total Helmholtz energy, *F(N,V,T)*, of a system with mixing\n", + "among multiple microstates and the Helmholtz energy, , of individual\n", + "microstates. An important result of is the configurational entropy due\n", + "to the mixing among multiple microstates, named as microstate\n", + "configurational entropy (MCE) in this book,\n", + "\n", + "*Eq. 5\u201151* ,\n", + "\n", + "which makes the entropy of a system with mixing among multiple\n", + "microstates as\n", + "\n", + "*Eq. 5\u201152* .\n", + "\n", + "Similarly, one can get the heat capacity at constant volume of a system\n", + "with mixing among multiple microstates as\n", + "\n", + "*Eq. 5\u201153* ,\n", + "\n", + "where the configurational contributions to the heat capacity due to the\n", + "mixing among multiple microstates is\n", + "\n", + "*Eq. 5\u201154* .\n", + "\n", + "Moreover, the isothermal bulk modulus of a system with mixing among\n", + "multiple microstates can be also computed similarly\n", + "\n", + "*Eq. 5\u201155* ,\n", + "\n", + "With\n", + "\n", + "*Eq. 5\u201156*\n", + "\n", + "with being the partial pressure of the microstate *\u03c3*. This multiple\n", + "microstate model (MMS model) is used in Chapter to quantitatively\n", + "predict thermal expansion anomalies.\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/firstprinciples_calculations_and_theory/index.ipynb b/src/psu410/src/firstprinciples_calculations_and_theory/index.ipynb index 4b99ed7..54dbaaa 100644 --- a/src/psu410/src/firstprinciples_calculations_and_theory/index.ipynb +++ b/src/psu410/src/firstprinciples_calculations_and_theory/index.ipynb @@ -4,9 +4,56 @@ "metadata": {}, "cells": [ { - "id": "cdf1b9b7", + "id": "7ae54358", "cell_type": "markdown", - "source": "# First-principles calculations and theory\n\nIn the previous chapter, the experimental techniques in obtaining the\nthermochemical and phase equilibrium data that are the inputs for the\nthermodynamic modeling of a system were summarized. However,\nexperimental data are not always available. This is due to the fact that\ni) the experiments are expensive, especially true in developing new\nmaterials; and ii) the experiments cannot reliably access the non-stable\nphases in most cases. The alternative approach is to predict the\nthermochemical data by first-principles calculations. The prediction of\nthe material properties, without using phenomenological parameters, is\nthe basic spirit of first-principles calculations. In particular, the\nsteady increase of both computer power and the efficiency of\ncomputational methods have made the first-principles predictions of most\nthermodynamic properties possible, including both enthalpy and entropy\nas a function of temperature, volume, and/or pressure.\n\nBy its definition, the term of \u201cfirst-principles\u201d represents a\nphilosophy that the prediction is to be based on a basic, fundamental\nproposition or assumption that cannot be deduced from any other\nproposition or assumption. This implies that the computational\nformulations are based on the most fundamental theory of quantum\nmechanics - Schr\u00f6dinger equation or density functional theory and the\ninputs to the calculations must be based on well-defined physical\nconstants \u2013 the nuclear and electronic charges. In another word, once\nthe atomic species of an assigned material are known, the theory should\npredict the energy of all possible crystalline structures, without\ninvoking any phenomenological fitting parameters.\n\nThis chapter organized in the sequence from thermodynamic calculations\nto fundamental theory to help those readers who are more interested in\nrealistic calculations using existing computer codes. The detailed\ntheoretical sections are presented to follow the section of\nthermodynamic calculations for those readers who are also interested in\nthe derivation of the formulations used in the thermodynamic\ncalculations. The following subsections are arranged accordingly in the\norder: (i) examples the commonly adopted calculation procedures for\nthermodynamic properties using the elemental metal nickel as the main\nprototype; (ii) derivation of the Helmholtz energy expression under the\nfirst-principles framework; (iii) introduction of the solution to the\nelectronic Schr\u00f6dinger equation within the two well developed frameworks\n\u2013 the quantum chemistry approach and the density functional theory; (iv)\ndetailed description of the procedure on how to solve the Schr\u00f6dinger\nequation for the motions of the atomic nuclei by means of lattice\ndynamics. The relation between the Helmholtz energy and Gibbs energy is\nshown by .\n", + "source": [ + "# First-principles calculations and theory\n", + "\n", + "In the previous chapter, the experimental techniques in obtaining the\n", + "thermochemical and phase equilibrium data that are the inputs for the\n", + "thermodynamic modeling of a system were summarized. However,\n", + "experimental data are not always available. This is due to the fact that\n", + "i) the experiments are expensive, especially true in developing new\n", + "materials; and ii) the experiments cannot reliably access the non-stable\n", + "phases in most cases. The alternative approach is to predict the\n", + "thermochemical data by first-principles calculations. The prediction of\n", + "the material properties, without using phenomenological parameters, is\n", + "the basic spirit of first-principles calculations. In particular, the\n", + "steady increase of both computer power and the efficiency of\n", + "computational methods have made the first-principles predictions of most\n", + "thermodynamic properties possible, including both enthalpy and entropy\n", + "as a function of temperature, volume, and/or pressure.\n", + "\n", + "By its definition, the term of \u201cfirst-principles\u201d represents a\n", + "philosophy that the prediction is to be based on a basic, fundamental\n", + "proposition or assumption that cannot be deduced from any other\n", + "proposition or assumption. This implies that the computational\n", + "formulations are based on the most fundamental theory of quantum\n", + "mechanics - Schr\u00f6dinger equation or density functional theory and the\n", + "inputs to the calculations must be based on well-defined physical\n", + "constants \u2013 the nuclear and electronic charges. In another word, once\n", + "the atomic species of an assigned material are known, the theory should\n", + "predict the energy of all possible crystalline structures, without\n", + "invoking any phenomenological fitting parameters.\n", + "\n", + "This chapter organized in the sequence from thermodynamic calculations\n", + "to fundamental theory to help those readers who are more interested in\n", + "realistic calculations using existing computer codes. The detailed\n", + "theoretical sections are presented to follow the section of\n", + "thermodynamic calculations for those readers who are also interested in\n", + "the derivation of the formulations used in the thermodynamic\n", + "calculations. The following subsections are arranged accordingly in the\n", + "order: (i) examples the commonly adopted calculation procedures for\n", + "thermodynamic properties using the elemental metal nickel as the main\n", + "prototype; (ii) derivation of the Helmholtz energy expression under the\n", + "first-principles framework; (iii) introduction of the solution to the\n", + "electronic Schr\u00f6dinger equation within the two well developed frameworks\n", + "\u2013 the quantum chemistry approach and the density functional theory; (iv)\n", + "detailed description of the procedure on how to solve the Schr\u00f6dinger\n", + "equation for the motions of the atomic nuclei by means of lattice\n", + "dynamics. The relation between the Helmholtz energy and Gibbs energy is\n", + "shown by .\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/firstprinciples_calculations_and_theory/lattice_dynamics.ipynb b/src/psu410/src/firstprinciples_calculations_and_theory/lattice_dynamics.ipynb index 8e333aa..058323d 100644 --- a/src/psu410/src/firstprinciples_calculations_and_theory/lattice_dynamics.ipynb +++ b/src/psu410/src/firstprinciples_calculations_and_theory/lattice_dynamics.ipynb @@ -4,33 +4,386 @@ "metadata": {}, "cells": [ { - "id": "f1788709", + "id": "da9f8821", "cell_type": "markdown", - "source": "## Lattice Dynamics\n", + "source": [ + "## Lattice Dynamics\n" + ], "metadata": {} }, { - "id": "0e6d239c", + "id": "c8223522", "cell_type": "markdown", - "source": "### Quantum theory for motion of atomic nuclei\n\nFor the convenience of discussion, the following convention of notations\nare used: *\u03b1* and *\u03b2* to label the Cartesian axes which is either *x*,\n*y*, or *z*, *j* and *k* to label atoms in the primitive unit cell,\n*mj* the atomic mass of the *j*th atom in the\nprimitive unit cell, **r**(*j*) the position of the *j*th\natom in the primitive unit cell, *P* and *Q* the index of the primitive\nunit cell in the system, **R**(*P*) the position of the *P*th\nprimitive unit cell in the system, and *V* the averaged volume of the\nprimitive unit cell.\n\nThe quantum theory for motion of atomic nuclei replicates closely the\nquantum theory for motion of electrons. That is to solve the wave\nfunction for the motions of the atomic nuclei for a Schr\u00f6dinger\u2019s\nequation with the potential being the total electronic energy derived\nfrom . The symbol is replaced by to represent the static total\nelectronic energy with **R** representing the static equilibrium\npositions of the atomic nuclei, **u** the displacements of atomic nuclei\naway from their static equilibrium positions, and *\u03c3* the additional\ndegree of freedom such as electronic states. The Schr\u00f6dinger\u2019s equation\nfor the motion of atomic nuclei is then\n\n*Eq. 5\u2011103* ,\n\nwhere\n\n*Eq. 5\u2011104* ,\n\nwith representing the kinetic energy operator\n\n*Eq. 5\u2011105* ,\n\nwhere *Nc* is the numbers of primitive unit cells in the\nsystem, *Na* is the numbers of atoms in the primitive unit\ncell, is the Plack constant, represents the *\u03b1*th Cartesian\ncomponent of **u** for the atom at *j*th lattice site in the\nprimitive unit cell and the *P*th primitive unit cell in the\nsystem.\n\nThe harmonic approximation \\[23, 33\\] truncates the into the second\norder in its Taylor\u2019s series\n\n*Eq. 5\u2011106* ,\n\nwhere is the real-space interatomic force constant. With the\napproximation of , it can be demonstrated that finding the solution of\nis equivalent to finding the vibrational frequencies of a classical\nsystem with *NcNa* particles for small mechanical\noscillations.\n\nLet us rewrite as\n\n*Eq. 5\u2011107* ,\n\nwhere\n\n*Eq. 5\u2011108* , and\n\n*Eq. 5\u2011109* .\n\nAccordingly, the kinetic energy operator in becomes\n\n*Eq. 5\u2011110* .\n", + "source": [ + "### Quantum theory for motion of atomic nuclei\n", + "\n", + "For the convenience of discussion, the following convention of notations\n", + "are used: *\u03b1* and *\u03b2* to label the Cartesian axes which is either *x*,\n", + "*y*, or *z*, *j* and *k* to label atoms in the primitive unit cell,\n", + "*mj* the atomic mass of the *j*th atom in the\n", + "primitive unit cell, **r**(*j*) the position of the *j*th\n", + "atom in the primitive unit cell, *P* and *Q* the index of the primitive\n", + "unit cell in the system, **R**(*P*) the position of the *P*th\n", + "primitive unit cell in the system, and *V* the averaged volume of the\n", + "primitive unit cell.\n", + "\n", + "The quantum theory for motion of atomic nuclei replicates closely the\n", + "quantum theory for motion of electrons. That is to solve the wave\n", + "function for the motions of the atomic nuclei for a Schr\u00f6dinger\u2019s\n", + "equation with the potential being the total electronic energy derived\n", + "from . The symbol is replaced by to represent the static total\n", + "electronic energy with **R** representing the static equilibrium\n", + "positions of the atomic nuclei, **u** the displacements of atomic nuclei\n", + "away from their static equilibrium positions, and *\u03c3* the additional\n", + "degree of freedom such as electronic states. The Schr\u00f6dinger\u2019s equation\n", + "for the motion of atomic nuclei is then\n", + "\n", + "*Eq. 5\u2011103* ,\n", + "\n", + "where\n", + "\n", + "*Eq. 5\u2011104* ,\n", + "\n", + "with representing the kinetic energy operator\n", + "\n", + "*Eq. 5\u2011105* ,\n", + "\n", + "where *Nc* is the numbers of primitive unit cells in the\n", + "system, *Na* is the numbers of atoms in the primitive unit\n", + "cell, is the Plack constant, represents the *\u03b1*th Cartesian\n", + "component of **u** for the atom at *j*th lattice site in the\n", + "primitive unit cell and the *P*th primitive unit cell in the\n", + "system.\n", + "\n", + "The harmonic approximation \\[23, 33\\] truncates the into the second\n", + "order in its Taylor\u2019s series\n", + "\n", + "*Eq. 5\u2011106* ,\n", + "\n", + "where is the real-space interatomic force constant. With the\n", + "approximation of , it can be demonstrated that finding the solution of\n", + "is equivalent to finding the vibrational frequencies of a classical\n", + "system with *NcNa* particles for small mechanical\n", + "oscillations.\n", + "\n", + "Let us rewrite as\n", + "\n", + "*Eq. 5\u2011107* ,\n", + "\n", + "where\n", + "\n", + "*Eq. 5\u2011108* , and\n", + "\n", + "*Eq. 5\u2011109* .\n", + "\n", + "Accordingly, the kinetic energy operator in becomes\n", + "\n", + "*Eq. 5\u2011110* .\n" + ], "metadata": {} }, { - "id": "668ad0ec", + "id": "5afe2868", "cell_type": "markdown", - "source": "### Normal coordinates, eigenenergetics, and phonons\n\nOne way to simplify the solution to the Schr\u00f6dinger\u2019s equation for\nmotion of atomic nuclei is to follow the study of the vibrations of the\natoms in a crystal \u2013 lattice dynamics. The essential step in lattice\ndynamics is to transform the problem of the correlated motions of 3*N*\nparticle into the problem of 3*N* independent harmonics. For this\npurpose, one can introduce a set of new coordinates ( *i*=1, 2, \u2026,\n3*N*), known as normal coordinates, by the transformation\n\n*Eq. 5\u2011111* ,\n\nwhere is the transformation coefficient that can be determined by\nsolving 3*N* simultaneous equations\n\n*Eq. 5\u2011112* ,\n\nwhere is to be determined so that one can find 3*N* , which are, not all\nzero. The equations are linear and homogeneous. Follow the basic theorem\nin linear algebra that, to find the nonzero solutions of the equations,\nthe determinant formed by must equal zero\n\n*Eq. 5\u2011113* ,\n\nwhere is the Kronecker delta symbol. Since is an equation with 3*N*\ndegrees, one can always find 3*N* values of (*i* = 1, \u2026, 3*N*). Each of\nyields a set of which can be chosen such that\n\n*Eq. 5\u2011114* ,\n\nwhere represents the Kronecker delta symbol and\n\n*Eq. 5\u2011115*\n\nThen with the normal coordinates defined in and utilizing , defined in\nis obtained as\n\n*Eq. 5\u2011116* .\n\nWith this equation, is simplified by the following process\n\n*Eq. 5\u2011117* ,\n\nnoting that in the above process and have been utilized.\n\nFurthermore, using , the kinetic energy operator in can be transformed\nas follows\n\n*Eq. 5\u2011118* ,\n\nnoting that in the above process and are utilized again.\n\nAs a result, under the harmonic approximation, the Hamiltonian in is\nsimplified as\n\n*Eq. 5\u2011119*\n\nwhich represents a quantum system containing 3*N* independent harmonics.\nCorresponding to the each of the *\u03c9i*, the quantum theory\ntells that the eigenenergy of a harmonics has the form\n\n*Eq. 5\u2011120* ,\n\nwith *gi* = 0, 1, \u2026, \u221e. Such a type of energetics behaves\nlike a Boson particle with energy and forms the concept of phonon.\n\nFurthermore, a state of the whole system is to be specified by the set\nof 3*N* independent quantum numbers **g** = (*g1*,\n*g1*, \u2026, *g3N*,). Finally, the energetics of a\nstate of the system formed by 3*N* independent harmonics, introduced by\n, is obtained with the summation of the energies of 3*N* independent\nharmonics as\n\n*Eq. 5\u2011121* .\n\nThis concludes the rationale by which how is derived.\n", + "source": [ + "### Normal coordinates, eigenenergetics, and phonons\n", + "\n", + "One way to simplify the solution to the Schr\u00f6dinger\u2019s equation for\n", + "motion of atomic nuclei is to follow the study of the vibrations of the\n", + "atoms in a crystal \u2013 lattice dynamics. The essential step in lattice\n", + "dynamics is to transform the problem of the correlated motions of 3*N*\n", + "particle into the problem of 3*N* independent harmonics. For this\n", + "purpose, one can introduce a set of new coordinates ( *i*=1, 2, \u2026,\n", + "3*N*), known as normal coordinates, by the transformation\n", + "\n", + "*Eq. 5\u2011111* ,\n", + "\n", + "where is the transformation coefficient that can be determined by\n", + "solving 3*N* simultaneous equations\n", + "\n", + "*Eq. 5\u2011112* ,\n", + "\n", + "where is to be determined so that one can find 3*N* , which are, not all\n", + "zero. The equations are linear and homogeneous. Follow the basic theorem\n", + "in linear algebra that, to find the nonzero solutions of the equations,\n", + "the determinant formed by must equal zero\n", + "\n", + "*Eq. 5\u2011113* ,\n", + "\n", + "where is the Kronecker delta symbol. Since is an equation with 3*N*\n", + "degrees, one can always find 3*N* values of (*i* = 1, \u2026, 3*N*). Each of\n", + "yields a set of which can be chosen such that\n", + "\n", + "*Eq. 5\u2011114* ,\n", + "\n", + "where represents the Kronecker delta symbol and\n", + "\n", + "*Eq. 5\u2011115*\n", + "\n", + "Then with the normal coordinates defined in and utilizing , defined in\n", + "is obtained as\n", + "\n", + "*Eq. 5\u2011116* .\n", + "\n", + "With this equation, is simplified by the following process\n", + "\n", + "*Eq. 5\u2011117* ,\n", + "\n", + "noting that in the above process and have been utilized.\n", + "\n", + "Furthermore, using , the kinetic energy operator in can be transformed\n", + "as follows\n", + "\n", + "*Eq. 5\u2011118* ,\n", + "\n", + "noting that in the above process and are utilized again.\n", + "\n", + "As a result, under the harmonic approximation, the Hamiltonian in is\n", + "simplified as\n", + "\n", + "*Eq. 5\u2011119*\n", + "\n", + "which represents a quantum system containing 3*N* independent harmonics.\n", + "Corresponding to the each of the *\u03c9i*, the quantum theory\n", + "tells that the eigenenergy of a harmonics has the form\n", + "\n", + "*Eq. 5\u2011120* ,\n", + "\n", + "with *gi* = 0, 1, \u2026, \u221e. Such a type of energetics behaves\n", + "like a Boson particle with energy and forms the concept of phonon.\n", + "\n", + "Furthermore, a state of the whole system is to be specified by the set\n", + "of 3*N* independent quantum numbers **g** = (*g1*,\n", + "*g1*, \u2026, *g3N*,). Finally, the energetics of a\n", + "state of the system formed by 3*N* independent harmonics, introduced by\n", + ", is obtained with the summation of the energies of 3*N* independent\n", + "harmonics as\n", + "\n", + "*Eq. 5\u2011121* .\n", + "\n", + "This concludes the rationale by which how is derived.\n" + ], "metadata": {} }, { - "id": "a7415cda", + "id": "80204d58", "cell_type": "markdown", - "source": "### Dynamical matrix and phonon mode\n\nBecause of the periodicity of a crystal, one can make an initial guess\nthat the solutions of are elastic plane waves made of a collective\natomic vibrations \\[23, 33\\], from the harmonic approximation of ,\n\n*Eq. 5\u2011122* ,\n\nwhere *\u03c9* represents the frequency of the plane wave, and is a wave\nvector designating the wave number and direction along which the plane\nwave propagates. It should be pointed out that in is now independent of\nthe index *P.* That is, except for a phase factor, the atoms that are\nequivalent by the translational symmetry among different primitive unit\ncells will experience the same type of atomic motion, independent of the\npositions of the primitive unit cell in the system. This is equivalent\nto applying the periodic condition so that in obeys\n\n*Eq. 5\u2011123* .\n\nNote that is now independent of the index *P*.\n\nFurthermore, one wants to limit **q***t* in and to those\nknown as the exact wave vectors which represent a special set of points\nin the reciprocal space that satisfy the condition\n\n*Eq. 5\u2011124* ,\n\nwhere is the Kronecker delta function. In fact, the number of equals to\nthe number of primitive unit cells contained in the system.\n\nUtilizing the translational invariance by which in \\[or in \\] depends on\n*P* and *Q* only through the difference , the following Fourier\ntransformation can be employed to simplify\n\n*Eq. 5\u2011125* ,\n\nand one obtains\n\n*Eq. 5\u2011126* .\n\nThe counterpart of with respect to is\n\n*Eq. 5\u2011127*\n\nis now an equation with 3*Na* degrees of freedom. At each ,\none can always find 3*Na* eigenvalues of (*i* = 1, \u2026,\n3*Na*). The 3*Na* vibrations are often known as\nphonon modes, noting again that *Na* is the number of atoms\nin the primitive unit cell. Each of yields a set of which can be chosen\nsuch that\n\n*Eq. 5\u2011128* ,\n\nwhere represents the complex conjugate of , and\n\n*Eq. 5\u2011129*\n\nFinally, for a solid, the summation over *Q* in can be abbreviated out,\nresulting in, after transforming back to together with using\n\n*Eq. 5\u2011130* ,\n\nbecause of translational invariance by which depends on *P* and *Q* only\nthrough the difference .\n\nIn the case that a system is built by a parallelepiped multiplication of\nthe primitive unit cell with lattice vectors of **a***\u03b1*\n(\u03b1=*x*, *y*, and *z*) in the form , in and takes the form\n\n*Eq. 5\u2011131* ,\n\nwhere = 0, 1, \u2026, , and is the primitive lattice vector in the reciprocal\nspace as\n\n*Eq. 5\u2011132* ,\n\nwith *Va* representing the volume of the primitive unit cell\ngiven as\n\n*Eq. 5\u2011133* .\n\nCombine and , it is easy to demonstrate\n\n*Eq. 5\u2011134* ,\n\nwhere is the Kronecker delta symbol.\n\nIt can be shown that the **q** points defined in represent the exact\nwave points. First, can be written as\n\n*Eq. 5\u2011135* ,\n\nwhere = 0, 1, \u2026, . Then, the left hand side of for the definition of\nexact wave vector becomes\n\n*Eq. 5\u2011136* .\n\nKnowing the fact that\n\n*Eq. 5\u2011137* .\n\nHence, it is proved the **q** points defined in are the exact wave\npoints.\n", + "source": [ + "### Dynamical matrix and phonon mode\n", + "\n", + "Because of the periodicity of a crystal, one can make an initial guess\n", + "that the solutions of are elastic plane waves made of a collective\n", + "atomic vibrations \\[23, 33\\], from the harmonic approximation of ,\n", + "\n", + "*Eq. 5\u2011122* ,\n", + "\n", + "where *\u03c9* represents the frequency of the plane wave, and is a wave\n", + "vector designating the wave number and direction along which the plane\n", + "wave propagates. It should be pointed out that in is now independent of\n", + "the index *P.* That is, except for a phase factor, the atoms that are\n", + "equivalent by the translational symmetry among different primitive unit\n", + "cells will experience the same type of atomic motion, independent of the\n", + "positions of the primitive unit cell in the system. This is equivalent\n", + "to applying the periodic condition so that in obeys\n", + "\n", + "*Eq. 5\u2011123* .\n", + "\n", + "Note that is now independent of the index *P*.\n", + "\n", + "Furthermore, one wants to limit **q***t* in and to those\n", + "known as the exact wave vectors which represent a special set of points\n", + "in the reciprocal space that satisfy the condition\n", + "\n", + "*Eq. 5\u2011124* ,\n", + "\n", + "where is the Kronecker delta function. In fact, the number of equals to\n", + "the number of primitive unit cells contained in the system.\n", + "\n", + "Utilizing the translational invariance by which in \\[or in \\] depends on\n", + "*P* and *Q* only through the difference , the following Fourier\n", + "transformation can be employed to simplify\n", + "\n", + "*Eq. 5\u2011125* ,\n", + "\n", + "and one obtains\n", + "\n", + "*Eq. 5\u2011126* .\n", + "\n", + "The counterpart of with respect to is\n", + "\n", + "*Eq. 5\u2011127*\n", + "\n", + "is now an equation with 3*Na* degrees of freedom. At each ,\n", + "one can always find 3*Na* eigenvalues of (*i* = 1, \u2026,\n", + "3*Na*). The 3*Na* vibrations are often known as\n", + "phonon modes, noting again that *Na* is the number of atoms\n", + "in the primitive unit cell. Each of yields a set of which can be chosen\n", + "such that\n", + "\n", + "*Eq. 5\u2011128* ,\n", + "\n", + "where represents the complex conjugate of , and\n", + "\n", + "*Eq. 5\u2011129*\n", + "\n", + "Finally, for a solid, the summation over *Q* in can be abbreviated out,\n", + "resulting in, after transforming back to together with using\n", + "\n", + "*Eq. 5\u2011130* ,\n", + "\n", + "because of translational invariance by which depends on *P* and *Q* only\n", + "through the difference .\n", + "\n", + "In the case that a system is built by a parallelepiped multiplication of\n", + "the primitive unit cell with lattice vectors of **a***\u03b1*\n", + "(\u03b1=*x*, *y*, and *z*) in the form , in and takes the form\n", + "\n", + "*Eq. 5\u2011131* ,\n", + "\n", + "where = 0, 1, \u2026, , and is the primitive lattice vector in the reciprocal\n", + "space as\n", + "\n", + "*Eq. 5\u2011132* ,\n", + "\n", + "with *Va* representing the volume of the primitive unit cell\n", + "given as\n", + "\n", + "*Eq. 5\u2011133* .\n", + "\n", + "Combine and , it is easy to demonstrate\n", + "\n", + "*Eq. 5\u2011134* ,\n", + "\n", + "where is the Kronecker delta symbol.\n", + "\n", + "It can be shown that the **q** points defined in represent the exact\n", + "wave points. First, can be written as\n", + "\n", + "*Eq. 5\u2011135* ,\n", + "\n", + "where = 0, 1, \u2026, . Then, the left hand side of for the definition of\n", + "exact wave vector becomes\n", + "\n", + "*Eq. 5\u2011136* .\n", + "\n", + "Knowing the fact that\n", + "\n", + "*Eq. 5\u2011137* .\n", + "\n", + "Hence, it is proved the **q** points defined in are the exact wave\n", + "points.\n" + ], "metadata": {} }, { - "id": "c3c81656", + "id": "d89bda80", "cell_type": "markdown", - "source": "### Linear-response method vs supercell method\n\nThe problem of lattice vibration for a solid is now transformed into\ncomputing the dynamical matrix in . The first-principles solution of the\nproblem is currently divided into two categories: the linear-response\nmethod \\[34\\] and the supercell method \\[35\\]. In the linear-response\nmethod, utilizing the electronic linear response upon the undistorted\ncrystals \\[36\\], the evaluations of the dynamical matrix can be\nperformed through the density-functional perturbation theory \\[34\\]\nwithout the approximation of the cutoff in neighboring interaction.\n\nCompared with the linear-response method, the supercell method is\nconceptually simple and is easy to implement computationally. The\nsupercell method adopts the frozen phonon approximation through which\nthe changes in total energy or forces are calculated in the direct space\nby displacing the atoms from their equilibrium positions. The advantage\nof the supercell method is that the phonon frequencies at the exact wave\nvectors, which are commensurable with the supercell, are calculated\nexactly with no further approximation \\[37\\]. The shortcoming of the\nsupercell method is that it is often limited by the size of the\nsupercell that can be handled with current computing resources.\n\nIn supercell approach, inaccuracies are thought to arise from the\ntruncation of the force constants \\[34-35\\]. This is only partially\ntrue. In the supercell method, the calculated represents the cumulative\ncontributions of the atom indexed by *k* and *P* in the supercell and\nall its images by translational transformation of the supercell in the\nwhole space. Let **L**\u03b1 represent the lattice vectors of the\nsupercell, then\n\n*Eq. 5\u2011138* .\n\nFor the exact wave vectors in , one has\n\n*Eq. 5\u2011139* **,**\n\nwhere is an integer. Replacing in with in , one obtains\n\nEq. 5\u2011140 .\n\nTherefore, the phonon frequencies calculated at the exact wave vectors\nby the cumulative force constants approach are exact, and the supercell\nsize will not lead to errors in the calculated phonon frequencies\n\\[37\\].\n\nGenerally speaking, if a supercell contains *Nc* primitive\nunit cells, one can always find *Nc* corresponding exact wave\nvectors in the **q** space. In most linear response calculations, the\ncommon choice of a 4\u00d74\u00d74 **q**-mesh is exactly equivalent to the 4\u00d74\u00d74\nsupercell in the real-space. Furthermore, since the supercell approach\ndoes not impose any approximation on the electronic response to the\ndistortion of the lattice geometry, the effects of electron-phonon\ninteractions can be accounted for by the supercell method.\n\nIn the supercell method, due to the impose of periodic condition, the\ncalculated force constant in the real-space cannot account for the\neffects of the vibration-induced electric field for the polar materials.\nIt has been demonstrated that such an effect adds an additional term to\ndynamical matrix in the reciprocal-space in the form,\n\n*Eq. 5\u2011141* ,\n\nwhere ***Z**\\* (j)* represents the Born effective charge\ntensor of the *j*th atom in the primitive unit cell and\n**\u03b5\u221e** the high frequency static dielectric tensor, i.e., the\ncontribution to the dielectric permittivity tensor from the electronic\npolarization. As a result for the polar materials, the matrix element at\nwave vector **q**\u21920 of the dynamical matrix in , by means of , should\nhave the form\n\n*Eq. 5\u2011142* .\n\nIt can be demonstrated \\[38\\] that this is equivalent to replacing the\nreal space force constant in by\n\n*Eq. 5\u2011143* .\n\nThe present implementation of the first-principles method in calculating\nthe phonon frequencies are mostly limited by the supercell size when\nusing or the number of exact wave vector points when using . A supercell\ncell built on the primitive unit cell or exact wave vector mesh is\nusually the common limit. If only the phonon frequencies derived from or\nare used, can be rather unsmooth which will result in inaccurate\nthermodynamic properties when it is used with . The mixed-space approach\n\\[38\\] can circumvent this bottleneck by using the Fourier interpolation\nto calculate the dynamical matrix for an arbitrary wave vector **q** as\n\n*Eq. 5\u2011144* ,\n\nwith the help of , , and for a polar materials. Note that the term is\nfor polar materials only.\n\nThe can be calculated as\n\n*Eq. 5\u2011145* ,\n\nwhere the function is usually taken as the Gaussian type\n\n*Eq. 5\u2011146* ,\n\nwhere is an adjustable damping (broadening) parameter whose role is to\nsmear the curve. in is the number of used **q** points. Empirically, a\n**q** mesh of is accurate enough for most purposes and doable with the\nYPHON code \\[15\\] efficiently, which is discussed in the Appendix A.\n", + "source": [ + "### Linear-response method vs supercell method\n", + "\n", + "The problem of lattice vibration for a solid is now transformed into\n", + "computing the dynamical matrix in . The first-principles solution of the\n", + "problem is currently divided into two categories: the linear-response\n", + "method \\[34\\] and the supercell method \\[35\\]. In the linear-response\n", + "method, utilizing the electronic linear response upon the undistorted\n", + "crystals \\[36\\], the evaluations of the dynamical matrix can be\n", + "performed through the density-functional perturbation theory \\[34\\]\n", + "without the approximation of the cutoff in neighboring interaction.\n", + "\n", + "Compared with the linear-response method, the supercell method is\n", + "conceptually simple and is easy to implement computationally. The\n", + "supercell method adopts the frozen phonon approximation through which\n", + "the changes in total energy or forces are calculated in the direct space\n", + "by displacing the atoms from their equilibrium positions. The advantage\n", + "of the supercell method is that the phonon frequencies at the exact wave\n", + "vectors, which are commensurable with the supercell, are calculated\n", + "exactly with no further approximation \\[37\\]. The shortcoming of the\n", + "supercell method is that it is often limited by the size of the\n", + "supercell that can be handled with current computing resources.\n", + "\n", + "In supercell approach, inaccuracies are thought to arise from the\n", + "truncation of the force constants \\[34-35\\]. This is only partially\n", + "true. In the supercell method, the calculated represents the cumulative\n", + "contributions of the atom indexed by *k* and *P* in the supercell and\n", + "all its images by translational transformation of the supercell in the\n", + "whole space. Let **L**\u03b1 represent the lattice vectors of the\n", + "supercell, then\n", + "\n", + "*Eq. 5\u2011138* .\n", + "\n", + "For the exact wave vectors in , one has\n", + "\n", + "*Eq. 5\u2011139* **,**\n", + "\n", + "where is an integer. Replacing in with in , one obtains\n", + "\n", + "Eq. 5\u2011140 .\n", + "\n", + "Therefore, the phonon frequencies calculated at the exact wave vectors\n", + "by the cumulative force constants approach are exact, and the supercell\n", + "size will not lead to errors in the calculated phonon frequencies\n", + "\\[37\\].\n", + "\n", + "Generally speaking, if a supercell contains *Nc* primitive\n", + "unit cells, one can always find *Nc* corresponding exact wave\n", + "vectors in the **q** space. In most linear response calculations, the\n", + "common choice of a 4\u00d74\u00d74 **q**-mesh is exactly equivalent to the 4\u00d74\u00d74\n", + "supercell in the real-space. Furthermore, since the supercell approach\n", + "does not impose any approximation on the electronic response to the\n", + "distortion of the lattice geometry, the effects of electron-phonon\n", + "interactions can be accounted for by the supercell method.\n", + "\n", + "In the supercell method, due to the impose of periodic condition, the\n", + "calculated force constant in the real-space cannot account for the\n", + "effects of the vibration-induced electric field for the polar materials.\n", + "It has been demonstrated that such an effect adds an additional term to\n", + "dynamical matrix in the reciprocal-space in the form,\n", + "\n", + "*Eq. 5\u2011141* ,\n", + "\n", + "where ***Z**\\* (j)* represents the Born effective charge\n", + "tensor of the *j*th atom in the primitive unit cell and\n", + "**\u03b5\u221e** the high frequency static dielectric tensor, i.e., the\n", + "contribution to the dielectric permittivity tensor from the electronic\n", + "polarization. As a result for the polar materials, the matrix element at\n", + "wave vector **q**\u21920 of the dynamical matrix in , by means of , should\n", + "have the form\n", + "\n", + "*Eq. 5\u2011142* .\n", + "\n", + "It can be demonstrated \\[38\\] that this is equivalent to replacing the\n", + "real space force constant in by\n", + "\n", + "*Eq. 5\u2011143* .\n", + "\n", + "The present implementation of the first-principles method in calculating\n", + "the phonon frequencies are mostly limited by the supercell size when\n", + "using or the number of exact wave vector points when using . A supercell\n", + "cell built on the primitive unit cell or exact wave vector mesh is\n", + "usually the common limit. If only the phonon frequencies derived from or\n", + "are used, can be rather unsmooth which will result in inaccurate\n", + "thermodynamic properties when it is used with . The mixed-space approach\n", + "\\[38\\] can circumvent this bottleneck by using the Fourier interpolation\n", + "to calculate the dynamical matrix for an arbitrary wave vector **q** as\n", + "\n", + "*Eq. 5\u2011144* ,\n", + "\n", + "with the help of , , and for a polar materials. Note that the term is\n", + "for polar materials only.\n", + "\n", + "The can be calculated as\n", + "\n", + "*Eq. 5\u2011145* ,\n", + "\n", + "where the function is usually taken as the Gaussian type\n", + "\n", + "*Eq. 5\u2011146* ,\n", + "\n", + "where is an adjustable damping (broadening) parameter whose role is to\n", + "smear the curve. in is the number of used **q** points. Empirically, a\n", + "**q** mesh of is accurate enough for most purposes and doable with the\n", + "YPHON code \\[15\\] efficiently, which is discussed in the Appendix A.\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/firstprinciples_calculations_and_theory/nickel_as_the_prototype.ipynb b/src/psu410/src/firstprinciples_calculations_and_theory/nickel_as_the_prototype.ipynb index 4951f56..97af25f 100644 --- a/src/psu410/src/firstprinciples_calculations_and_theory/nickel_as_the_prototype.ipynb +++ b/src/psu410/src/firstprinciples_calculations_and_theory/nickel_as_the_prototype.ipynb @@ -4,27 +4,336 @@ "metadata": {}, "cells": [ { - "id": "a54c87ec", + "id": "373ce861", "cell_type": "markdown", - "source": "## Nickel as the prototype\n\nThis section exemplifies the step-by-step procedure in calculating the\nthermal properties within the framework of first-principles phonon\napproach, using the elemental metal Ni as the prototype. The calculation\nof the formation enthalpy of Ni3Al is given at the end. The\ncalculation in this section is limited to the case for the ferromagnetic\nphase, i.e. single microstate, implying that no configurational mixtures\nor magnetic phase transitions is considered which is discussed in\nChapter 5.2.5 and Chapter 6.\n\nThe Vienna Ab-initio Simulation Package (VASP) \\[13-14\\] has been\nemployed for electronic calculations, and the YPHON code \\[15\\] has been\nemployed for phonon calculations. VASP is a code based on the\npseudopotential approach to the density functional theory using plane\nwave function as the basis set, by which only the valence electrons are\nhandled explicitly and the core electrons are approximated by an\neffective pseudopotential. The same energy cutoff values, which\ndetermine the number of plane waves in the expansion of electronic wave\nfunction, have been used for Ni, Al, and Ni3Al. The rationale\nfor the derivations of the formulations used in this section is to be\ngiven in Chapter for readers who want to have an in-depth understanding\nof the physics behind the used formulations.\n", + "source": [ + "## Nickel as the prototype\n", + "\n", + "This section exemplifies the step-by-step procedure in calculating the\n", + "thermal properties within the framework of first-principles phonon\n", + "approach, using the elemental metal Ni as the prototype. The calculation\n", + "of the formation enthalpy of Ni3Al is given at the end. The\n", + "calculation in this section is limited to the case for the ferromagnetic\n", + "phase, i.e. single microstate, implying that no configurational mixtures\n", + "or magnetic phase transitions is considered which is discussed in\n", + "Chapter 5.2.5 and Chapter 6.\n", + "\n", + "The Vienna Ab-initio Simulation Package (VASP) \\[13-14\\] has been\n", + "employed for electronic calculations, and the YPHON code \\[15\\] has been\n", + "employed for phonon calculations. VASP is a code based on the\n", + "pseudopotential approach to the density functional theory using plane\n", + "wave function as the basis set, by which only the valence electrons are\n", + "handled explicitly and the core electrons are approximated by an\n", + "effective pseudopotential. The same energy cutoff values, which\n", + "determine the number of plane waves in the expansion of electronic wave\n", + "function, have been used for Ni, Al, and Ni3Al. The rationale\n", + "for the derivations of the formulations used in this section is to be\n", + "given in Chapter for readers who want to have an in-depth understanding\n", + "of the physics behind the used formulations.\n" + ], "metadata": {} }, { - "id": "16d7283b", + "id": "cc102013", "cell_type": "markdown", - "source": "### Helmholtz energy and quasiharmonic approximation\n\nAt present, the most rigorous method to predict the thermodynamic\nproperties of a material at finite temperatures is the phonon approach.\nIn such an approach, the microscopic Hamiltonian is expanded up to the\nsecond order. All the thermodynamic quantities are calculated using\nformulations derived from the statistical physics without further\napproximation. The great importance of the phonon theory is that all the\ninput parameters can be obtained by means of first-principles\ncalculations without using any phenomenological parameters.\n\nLet us consider a system with an averaged atomic volume *V*. Neglecting\nthe electron phonon coupling, it is a well demonstrated procedure \\[16\\]\nto decompose the Helmholtz energy *F*(*V*,*T*) of the system at\ntemperature *T* into three additive contributions as follows\n\n*Eq. 5\u20111* ,\n\nwhere *Ec* is the static total energy which is the total\nenergy of the system at 0 K without any atomic vibrations,\n*Fvib* is the vibrational contribution due to the lattice\nions, and *Fel* is the electronic contribution due to the\nthermal electronic excitation at finite temperature which can become\nimportant for metals at high temperature.\n\nThe terminology of \u201cquasiharmonic approximation\u201d arises from the fact\nthat for a given volume, *Fvib*(*V*,*T*) is calculated under\nthe harmonic approximation and the anharmonic effects are carried out\nsolely through volume dependence of the phonon frequency. The easiest\ncomputational implementation of is to first independently calculate the\nHelmholtz energy at several selected volumes near the equilibrium volume\nand then use the numerical interpolation to find the Helmholtz energy at\nan arbitrary volume. The volume interval is usually at the scale of 3~5%\nof the equilibrium volume. Too small volume interval can result in\nnumerical instability due to the numerical uncertainties in the static\ntotal energy calculation, in particular, when one numerically computes\nthe first- and especially the second-order derivatives of the Helmholtz\nenergy in deriving the thermodynamic quantities. It should be noted that\nwhenever available, analytic formulas should be used instead of the\nnumerical second-order derivative to avoid numerical errors. For\ninstance, when the phonon approach is employed, the constant volume heat\ncapacity has the analytic expression in terms of phonon density of\nstates.\n\nNickel metal adopts the fcc structure at ambient conditions and the\nprimitive unit cell contains one atom. Almost all the existing\nfirst-principles codes have the function to calculate the static total\nenergy. The static total energy *Ec* in should be calculated\nusing the primitive unit cell. As the Helmholtz energy is to be\ncalculated at several volumes, a good practice is to plot the calculated\nstatic total energy points together with the interpolated energy curve\nto examine the convergence of the static total energy calculation. Since\nthe first-principles method often employs the self-consistent technique,\nit could occur that calculations at certain volumes may not convergent,\nwhich should be fixed by trying the various algebraic schemes provided\nin most of the existing codes. Furthermore, since certain calculations\ninvolve the second order derivative of the Helmholtz energy, a minor\nuncertainty along the static total energy curve can result in large\ndeviation for the calculated properties such as thermal expansion\ncoefficient and bulk modulus. In that case, a reasonable solution is to\nsmoothen the static total energy using the modified Birch-Murnaghan\nequation of states (EOS) \\[17-18\\]\n\n*Eq. 5\u20112* .\n\nPlotted in is the calculated static total energy of the elemental metal\nNi with the circles representing the calculated values and the curve\nrepresenting that by EOS fitting.\n\nFigure \u2011. Static total energy of nickel.\n\nThe vibrational contribution to the Helmholtz energy by phonon theory\ncan be computed by \\[19\\]\n\n*Eq. 5\u20113* ,\n\nwhere is the Boltzmann\u2019s constant, *\u03c9* represents the phonon frequency,\nand is the phonon density of states. It is recommended that is\ncalculated at the same volume set at which the static total static\nenergies are calculated.\n\nFor the present prototype of Ni, the supercell method for the\ncalculation of has been employed. The procedure is follows:\n\n1. Make supercell by enlarging the primitive unit cell according to the\n > defined neighbor interaction distance; Employ the first-principles\n > code (VASP, \\[13-14\\] in this work) to calculate the interatomic\n > force constants.\n\n2. Assign the mesh in the wave vector (**q**) space; Make the dynamical\n > matrix at each **q** point; Diagonalize the dynamical matrix to\n > find out the phonon frequencies at each **q** point; And finally\n > collect all the phonon frequencies for all **q** points. The\n > detailed formulation for phonon calculations is given in Chapter .\n\nFor the phonon calculations, one can use the open source code YPHON\n\\[15\\] by the present authors. Other choices can be the free ATAT code\n\\[20\\] or the free PHON code \\[21\\]. For the calculation of the phonon\ndensity of states, we have made a supercell containing 64 atoms which is\na 4\u00d74\u00d74 supercell of the primitive unit cell. Plotted in is the\ncalculated phonon density of states using YPHON code at the calculated\nstatic equilibrium volume compared with the measured data at 10 K \\[22\\]\n(symbols).\n\nFigure \u2011. Phonon density of states of nickel.\n\nFor a first-principles thermodynamic calculation, an important step to\navoid possible calculation errors is to examine the phonon dispersions\nfirst. Phonon dispersion \\[23\\] depicts the evolution of phonon\nfrequencies along the designated direction for a crystal. Phonon\ndispersion can be measured rather accurately by inelastic neutron\nscattering \\[24-26\\] or inelastic x-ray scattering \\[27\\] experiment.\nPlotted in are the calculated phonon dispersions (curves) along the\n\\[00\u03b6\\], \\[0\u03b61\\], \\[0\u03b6\u03b6\\], and \\[\u03b6\u03b6\u03b6\\] directions of Ni using YPHON code\ncompared with the neutron scattering data at 296 K (symbols) with\ndetails in Ref. \\[16\\].\n\nFigure \u2011. Phonon dispersions of nickel. The solid lines represent\nresults calculated using a supercell containing 256 atoms which is 4\u00d74\u00d74\nsupercell of the conventional cubic unit cell. The dot-dashed lines\nrepresent results calculated using a supercell containing 64 atoms which\nis 4\u00d74\u00d74 supercell of the primitive unit cell.\n\nFor the calculation of *Fel* in , the most computationally\nconvenient approach is to use the Mermin statistics as follows\n\n*Eq. 5\u20114* ,\n\nwhere is the thermal electronic energy, and *Sel* is the bare\nelectronic entropy. Both the calculations of and *Sel* need\nthe electronic density of states (EDOS) as input. The electronic density\nof states can be obtained during the step of the static total energy\ncalculation. The detailed formulations for and *Sel* are\ngiven in Chapter . Since Ni is magnetic, the EDOS of Ni can be\npartitioned into those of spin up and spin down due to the spin freedom\nof electron. The calculated EDOS for Ni is shown in where the solid,\ndot-dashed, and dashed lines represent the total, spin up, and spin down\nEDOS with the Fermi energy set to zero.\n\nFigure \u2011. Electronic density of states of nickel. That due to spin up is\nplotted as positive value and that due to spin down is plotted as\nnegative value purely for the clarity of the figure. The \u201ctotal\u201d is the\nsum of the absolute values of those of spin up and spin down.\n\nThe calculated temperature evolution of Helmholtz energy as a function\nof volume for Ni are illustrated in . The circles represent the\ncalculated static total energies. The solid curves represent the\nHelmholtz energy curves from 0 to 1600 K at a temperature increment of\n100 K as displayed from top to bottom in . The dashed line marks the\nevolution of the equilibrium volume at *P*=0 with increasing\ntemperature. It is noted that Helmholtz energy always decreases with\nincreasing temperature due to the entropy term of \u2013*TS*. Note that the\nat 0 K the Helmholtz energy is higher than the static total energy due\nto the zero point vibrational energy as can be seen when *T* \u21920 which\nreduces to\n\n*Eq. 5\u20115* ,\n\nwhich is positive.\n\nFigure \u2011. Temperature evolution of the Helmholtz energy for nickel.\n", + "source": [ + "### Helmholtz energy and quasiharmonic approximation\n", + "\n", + "At present, the most rigorous method to predict the thermodynamic\n", + "properties of a material at finite temperatures is the phonon approach.\n", + "In such an approach, the microscopic Hamiltonian is expanded up to the\n", + "second order. All the thermodynamic quantities are calculated using\n", + "formulations derived from the statistical physics without further\n", + "approximation. The great importance of the phonon theory is that all the\n", + "input parameters can be obtained by means of first-principles\n", + "calculations without using any phenomenological parameters.\n", + "\n", + "Let us consider a system with an averaged atomic volume *V*. Neglecting\n", + "the electron phonon coupling, it is a well demonstrated procedure \\[16\\]\n", + "to decompose the Helmholtz energy *F*(*V*,*T*) of the system at\n", + "temperature *T* into three additive contributions as follows\n", + "\n", + "*Eq. 5\u20111* ,\n", + "\n", + "where *Ec* is the static total energy which is the total\n", + "energy of the system at 0 K without any atomic vibrations,\n", + "*Fvib* is the vibrational contribution due to the lattice\n", + "ions, and *Fel* is the electronic contribution due to the\n", + "thermal electronic excitation at finite temperature which can become\n", + "important for metals at high temperature.\n", + "\n", + "The terminology of \u201cquasiharmonic approximation\u201d arises from the fact\n", + "that for a given volume, *Fvib*(*V*,*T*) is calculated under\n", + "the harmonic approximation and the anharmonic effects are carried out\n", + "solely through volume dependence of the phonon frequency. The easiest\n", + "computational implementation of is to first independently calculate the\n", + "Helmholtz energy at several selected volumes near the equilibrium volume\n", + "and then use the numerical interpolation to find the Helmholtz energy at\n", + "an arbitrary volume. The volume interval is usually at the scale of 3~5%\n", + "of the equilibrium volume. Too small volume interval can result in\n", + "numerical instability due to the numerical uncertainties in the static\n", + "total energy calculation, in particular, when one numerically computes\n", + "the first- and especially the second-order derivatives of the Helmholtz\n", + "energy in deriving the thermodynamic quantities. It should be noted that\n", + "whenever available, analytic formulas should be used instead of the\n", + "numerical second-order derivative to avoid numerical errors. For\n", + "instance, when the phonon approach is employed, the constant volume heat\n", + "capacity has the analytic expression in terms of phonon density of\n", + "states.\n", + "\n", + "Nickel metal adopts the fcc structure at ambient conditions and the\n", + "primitive unit cell contains one atom. Almost all the existing\n", + "first-principles codes have the function to calculate the static total\n", + "energy. The static total energy *Ec* in should be calculated\n", + "using the primitive unit cell. As the Helmholtz energy is to be\n", + "calculated at several volumes, a good practice is to plot the calculated\n", + "static total energy points together with the interpolated energy curve\n", + "to examine the convergence of the static total energy calculation. Since\n", + "the first-principles method often employs the self-consistent technique,\n", + "it could occur that calculations at certain volumes may not convergent,\n", + "which should be fixed by trying the various algebraic schemes provided\n", + "in most of the existing codes. Furthermore, since certain calculations\n", + "involve the second order derivative of the Helmholtz energy, a minor\n", + "uncertainty along the static total energy curve can result in large\n", + "deviation for the calculated properties such as thermal expansion\n", + "coefficient and bulk modulus. In that case, a reasonable solution is to\n", + "smoothen the static total energy using the modified Birch-Murnaghan\n", + "equation of states (EOS) \\[17-18\\]\n", + "\n", + "*Eq. 5\u20112* .\n", + "\n", + "Plotted in is the calculated static total energy of the elemental metal\n", + "Ni with the circles representing the calculated values and the curve\n", + "representing that by EOS fitting.\n", + "\n", + "Figure \u2011. Static total energy of nickel.\n", + "\n", + "The vibrational contribution to the Helmholtz energy by phonon theory\n", + "can be computed by \\[19\\]\n", + "\n", + "*Eq. 5\u20113* ,\n", + "\n", + "where is the Boltzmann\u2019s constant, *\u03c9* represents the phonon frequency,\n", + "and is the phonon density of states. It is recommended that is\n", + "calculated at the same volume set at which the static total static\n", + "energies are calculated.\n", + "\n", + "For the present prototype of Ni, the supercell method for the\n", + "calculation of has been employed. The procedure is follows:\n", + "\n", + "1. Make supercell by enlarging the primitive unit cell according to the\n", + " > defined neighbor interaction distance; Employ the first-principles\n", + " > code (VASP, \\[13-14\\] in this work) to calculate the interatomic\n", + " > force constants.\n", + "\n", + "2. Assign the mesh in the wave vector (**q**) space; Make the dynamical\n", + " > matrix at each **q** point; Diagonalize the dynamical matrix to\n", + " > find out the phonon frequencies at each **q** point; And finally\n", + " > collect all the phonon frequencies for all **q** points. The\n", + " > detailed formulation for phonon calculations is given in Chapter .\n", + "\n", + "For the phonon calculations, one can use the open source code YPHON\n", + "\\[15\\] by the present authors. Other choices can be the free ATAT code\n", + "\\[20\\] or the free PHON code \\[21\\]. For the calculation of the phonon\n", + "density of states, we have made a supercell containing 64 atoms which is\n", + "a 4\u00d74\u00d74 supercell of the primitive unit cell. Plotted in is the\n", + "calculated phonon density of states using YPHON code at the calculated\n", + "static equilibrium volume compared with the measured data at 10 K \\[22\\]\n", + "(symbols).\n", + "\n", + "Figure \u2011. Phonon density of states of nickel.\n", + "\n", + "For a first-principles thermodynamic calculation, an important step to\n", + "avoid possible calculation errors is to examine the phonon dispersions\n", + "first. Phonon dispersion \\[23\\] depicts the evolution of phonon\n", + "frequencies along the designated direction for a crystal. Phonon\n", + "dispersion can be measured rather accurately by inelastic neutron\n", + "scattering \\[24-26\\] or inelastic x-ray scattering \\[27\\] experiment.\n", + "Plotted in are the calculated phonon dispersions (curves) along the\n", + "\\[00\u03b6\\], \\[0\u03b61\\], \\[0\u03b6\u03b6\\], and \\[\u03b6\u03b6\u03b6\\] directions of Ni using YPHON code\n", + "compared with the neutron scattering data at 296 K (symbols) with\n", + "details in Ref. \\[16\\].\n", + "\n", + "Figure \u2011. Phonon dispersions of nickel. The solid lines represent\n", + "results calculated using a supercell containing 256 atoms which is 4\u00d74\u00d74\n", + "supercell of the conventional cubic unit cell. The dot-dashed lines\n", + "represent results calculated using a supercell containing 64 atoms which\n", + "is 4\u00d74\u00d74 supercell of the primitive unit cell.\n", + "\n", + "For the calculation of *Fel* in , the most computationally\n", + "convenient approach is to use the Mermin statistics as follows\n", + "\n", + "*Eq. 5\u20114* ,\n", + "\n", + "where is the thermal electronic energy, and *Sel* is the bare\n", + "electronic entropy. Both the calculations of and *Sel* need\n", + "the electronic density of states (EDOS) as input. The electronic density\n", + "of states can be obtained during the step of the static total energy\n", + "calculation. The detailed formulations for and *Sel* are\n", + "given in Chapter . Since Ni is magnetic, the EDOS of Ni can be\n", + "partitioned into those of spin up and spin down due to the spin freedom\n", + "of electron. The calculated EDOS for Ni is shown in where the solid,\n", + "dot-dashed, and dashed lines represent the total, spin up, and spin down\n", + "EDOS with the Fermi energy set to zero.\n", + "\n", + "Figure \u2011. Electronic density of states of nickel. That due to spin up is\n", + "plotted as positive value and that due to spin down is plotted as\n", + "negative value purely for the clarity of the figure. The \u201ctotal\u201d is the\n", + "sum of the absolute values of those of spin up and spin down.\n", + "\n", + "The calculated temperature evolution of Helmholtz energy as a function\n", + "of volume for Ni are illustrated in . The circles represent the\n", + "calculated static total energies. The solid curves represent the\n", + "Helmholtz energy curves from 0 to 1600 K at a temperature increment of\n", + "100 K as displayed from top to bottom in . The dashed line marks the\n", + "evolution of the equilibrium volume at *P*=0 with increasing\n", + "temperature. It is noted that Helmholtz energy always decreases with\n", + "increasing temperature due to the entropy term of \u2013*TS*. Note that the\n", + "at 0 K the Helmholtz energy is higher than the static total energy due\n", + "to the zero point vibrational energy as can be seen when *T* \u21920 which\n", + "reduces to\n", + "\n", + "*Eq. 5\u20115* ,\n", + "\n", + "which is positive.\n", + "\n", + "Figure \u2011. Temperature evolution of the Helmholtz energy for nickel.\n" + ], "metadata": {} }, { - "id": "60b7be55", + "id": "4c1240fb", "cell_type": "markdown", - "source": "### Volume, entropy, enthalpy, thermal expansion, bulk modulus, and heat capacity \n\nThe equilibrium volume *Veq* (*P*,*T*) at given *T* and *P*\ncan be obtained by finding the root of the following equation\n\n*Eq. 5\u20116* .\n\nThe dashed line in illustrates *Veq* (*P*,*T*) as a function\nof *T* from 0 to 1600 K at *P* = 0 for Ni.\n\nThe entropy can be calculated through *F* by\n\n*Eq. 5\u20117* .\n\nPlotted in is the calculated entropy (curve) of Ni as a function of\ntemperature from 0 to 1600 K at *P* = 0 compared with the recommended\ndata (symbols) with details in Ref. \\[16\\].\n\nFigure \u2011. Entropy of nickel as a function of temperature.\n\nBased on *F* and *S*, the enthalpy at given *P* and *T* can be computed\nas\n\n*Eq. 5\u20118* .\n\nPlotted in is the calculated enthalpy (curve) of Ni as function of\ntemperature from 0 to 1600 K at *P* = 0 comparing with the recommended\ndata (open circles) with details in Ref. \\[16\\]\n\nFigure \u2011. Enthalpy of nickel as a function of temperature.\n\nWith the equilibrium volume *Veq* (*P*,*T*) calculated by ,\nthe volume thermal expansion coefficient defined by can be calculated by\n\n*Eq. 5\u20119* .\n\nPlotted in is the calculated thermal expansion coefficient (curve) of\nnickel as function of temperature from 0 to 1600 K at *P* = 0 comparing\nwith experimental data (symbols) with details in Ref. \\[16\\]\n\nFigure \u2011. Volume thermal expansion coefficient of nickel as a function\nof temperature.\n\nThe bulk modulus of a material represents the substance's resistance to\nuniform compression. Depending on how the temperature varies during\ncompression, a distinction should be made between the isothermal bulk\nmodulus (constant temperature) and adiabatic bulk modulus (constant\nentropy or no heat transfer). As a matter of fact, most of the\nexperimental data are adiabatic whereas most of the published\ntheoretical data are isothermal.\n\nThe isothermal bulk modulus, as defined in terms of Gibbs energy shown\nby , can be calculated by\n\n*Eq. 5\u201110*\n\nBased on the isothermal bulk modulus, the adiabatic bulk modulus can be\ncalculated by\n\n*Eq. 5\u201111* ,\n\nwhere *CP* and *CV* represent the constant\npressure heat capacity and constant volume heat capacity respectively.\nPlotted in is the calculated bulk moduli (curves) of Ni as function of\ntemperature from 0 to 1600 K at *P* = 0. The experimental data are from\nultrasonic measurements (symbols, see Ref. \\[28\\] for more details) that\nare therefore adiabatic bulk moduli calculated based on the measured\nadiabatic elastic constants using the relation\n\n*Eq. 5\u201112* .\n\nFigure \u2011. Bulk moduli of nickel as a function of temperature. Solid\nline: adiabatic; Dashed line: isothermal.\n\nThe heat capacity at constant volume, as defined in terms of Gibbs\nenergy shown by Eq. 2\u201128, can be calculated by\n\n*Eq. 5\u201113* ,\n\nwhere represents the internal energy. The heat capacity at constant\npressure (see ) can then be calculated as\n\n*Eq. 5\u201114* ,\n\nutilizing the calculated thermal expansion coefficient in and bulk\nmodulus in .\n\nIt can be advocated that thermal expansion makes the difference between\nthe heat capacity at constant volume and the heat capacity at constant\npressure. The calculated contributions to the heat capacity of Ni as\nfunction of temperature from 0 to 1600 K at *P* = 0 are illustrated in\nwhere the lattice vibration and the thermal electron contributions have\nbeen separated out. From , it is observed a large difference between the\ncalcualted *CP* (solid line) and the experimental data\n(symbols, see Ref. \\[18\\] for more details) at 600 K due to the magnetic\nphase transition which has not been considered in the calculation. It\nshould be pointed out that for Ni the thermal electronic contribution to\nthe heat capacity (dashed line in ) is substantial at high temperatures.\n\nFigure \u2011. Heat capacity of nickel as a function of temperature.\nrepresents the calculated lattice vibration contribution; represents the\ncalculated thermal electronic contribution.\n", + "source": [ + "### Volume, entropy, enthalpy, thermal expansion, bulk modulus, and heat capacity \n", + "\n", + "The equilibrium volume *Veq* (*P*,*T*) at given *T* and *P*\n", + "can be obtained by finding the root of the following equation\n", + "\n", + "*Eq. 5\u20116* .\n", + "\n", + "The dashed line in illustrates *Veq* (*P*,*T*) as a function\n", + "of *T* from 0 to 1600 K at *P* = 0 for Ni.\n", + "\n", + "The entropy can be calculated through *F* by\n", + "\n", + "*Eq. 5\u20117* .\n", + "\n", + "Plotted in is the calculated entropy (curve) of Ni as a function of\n", + "temperature from 0 to 1600 K at *P* = 0 compared with the recommended\n", + "data (symbols) with details in Ref. \\[16\\].\n", + "\n", + "Figure \u2011. Entropy of nickel as a function of temperature.\n", + "\n", + "Based on *F* and *S*, the enthalpy at given *P* and *T* can be computed\n", + "as\n", + "\n", + "*Eq. 5\u20118* .\n", + "\n", + "Plotted in is the calculated enthalpy (curve) of Ni as function of\n", + "temperature from 0 to 1600 K at *P* = 0 comparing with the recommended\n", + "data (open circles) with details in Ref. \\[16\\]\n", + "\n", + "Figure \u2011. Enthalpy of nickel as a function of temperature.\n", + "\n", + "With the equilibrium volume *Veq* (*P*,*T*) calculated by ,\n", + "the volume thermal expansion coefficient defined by can be calculated by\n", + "\n", + "*Eq. 5\u20119* .\n", + "\n", + "Plotted in is the calculated thermal expansion coefficient (curve) of\n", + "nickel as function of temperature from 0 to 1600 K at *P* = 0 comparing\n", + "with experimental data (symbols) with details in Ref. \\[16\\]\n", + "\n", + "Figure \u2011. Volume thermal expansion coefficient of nickel as a function\n", + "of temperature.\n", + "\n", + "The bulk modulus of a material represents the substance's resistance to\n", + "uniform compression. Depending on how the temperature varies during\n", + "compression, a distinction should be made between the isothermal bulk\n", + "modulus (constant temperature) and adiabatic bulk modulus (constant\n", + "entropy or no heat transfer). As a matter of fact, most of the\n", + "experimental data are adiabatic whereas most of the published\n", + "theoretical data are isothermal.\n", + "\n", + "The isothermal bulk modulus, as defined in terms of Gibbs energy shown\n", + "by , can be calculated by\n", + "\n", + "*Eq. 5\u201110*\n", + "\n", + "Based on the isothermal bulk modulus, the adiabatic bulk modulus can be\n", + "calculated by\n", + "\n", + "*Eq. 5\u201111* ,\n", + "\n", + "where *CP* and *CV* represent the constant\n", + "pressure heat capacity and constant volume heat capacity respectively.\n", + "Plotted in is the calculated bulk moduli (curves) of Ni as function of\n", + "temperature from 0 to 1600 K at *P* = 0. The experimental data are from\n", + "ultrasonic measurements (symbols, see Ref. \\[28\\] for more details) that\n", + "are therefore adiabatic bulk moduli calculated based on the measured\n", + "adiabatic elastic constants using the relation\n", + "\n", + "*Eq. 5\u201112* .\n", + "\n", + "Figure \u2011. Bulk moduli of nickel as a function of temperature. Solid\n", + "line: adiabatic; Dashed line: isothermal.\n", + "\n", + "The heat capacity at constant volume, as defined in terms of Gibbs\n", + "energy shown by Eq. 2\u201128, can be calculated by\n", + "\n", + "*Eq. 5\u201113* ,\n", + "\n", + "where represents the internal energy. The heat capacity at constant\n", + "pressure (see ) can then be calculated as\n", + "\n", + "*Eq. 5\u201114* ,\n", + "\n", + "utilizing the calculated thermal expansion coefficient in and bulk\n", + "modulus in .\n", + "\n", + "It can be advocated that thermal expansion makes the difference between\n", + "the heat capacity at constant volume and the heat capacity at constant\n", + "pressure. The calculated contributions to the heat capacity of Ni as\n", + "function of temperature from 0 to 1600 K at *P* = 0 are illustrated in\n", + "where the lattice vibration and the thermal electron contributions have\n", + "been separated out. From , it is observed a large difference between the\n", + "calcualted *CP* (solid line) and the experimental data\n", + "(symbols, see Ref. \\[18\\] for more details) at 600 K due to the magnetic\n", + "phase transition which has not been considered in the calculation. It\n", + "should be pointed out that for Ni the thermal electronic contribution to\n", + "the heat capacity (dashed line in ) is substantial at high temperatures.\n", + "\n", + "Figure \u2011. Heat capacity of nickel as a function of temperature.\n", + "represents the calculated lattice vibration contribution; represents the\n", + "calculated thermal electronic contribution.\n" + ], "metadata": {} }, { - "id": "6b534057", + "id": "d94f5392", "cell_type": "markdown", - "source": "### Formation enthalpy of Ni3Al\n\nOne can do the similar calculations for the elemental metal Al and the\ncompound Ni3Al which has the L12 structure,\nfollowing the same steps in the calculations of Ni. The formation\nenthalpy in the unit of per mole atom can be calculated as\n\n*Eq. 5\u201115* ,\n\nwhere , , and represent the enthalpies of Ni3Al, Ni, and Al\nin the energy unit of per mole atom, respectively. Plotted in is the\ncalculated formation enthalpy of Ni3Al (curve) as a function\nof temperature from 0 to 1600 K at *P* = 0 compared with experimental\ndata (symbols) with details in Ref. \\[16\\].\n\n.\n\nFigure \u2011. Formation enthalpy of L12-Ni3Al with\nrespect to pure fcc Ni and Al.\n", + "source": [ + "### Formation enthalpy of Ni3Al\n", + "\n", + "One can do the similar calculations for the elemental metal Al and the\n", + "compound Ni3Al which has the L12 structure,\n", + "following the same steps in the calculations of Ni. The formation\n", + "enthalpy in the unit of per mole atom can be calculated as\n", + "\n", + "*Eq. 5\u201115* ,\n", + "\n", + "where , , and represent the enthalpies of Ni3Al, Ni, and Al\n", + "in the energy unit of per mole atom, respectively. Plotted in is the\n", + "calculated formation enthalpy of Ni3Al (curve) as a function\n", + "of temperature from 0 to 1600 K at *P* = 0 compared with experimental\n", + "data (symbols) with details in Ref. \\[16\\].\n", + "\n", + ".\n", + "\n", + "Figure \u2011. Formation enthalpy of L12-Ni3Al with\n", + "respect to pure fcc Ni and Al.\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/firstprinciples_calculations_and_theory/quantum_theory_for_the_motion_of_electrons.ipynb b/src/psu410/src/firstprinciples_calculations_and_theory/quantum_theory_for_the_motion_of_electrons.ipynb index 8b75f2a..5efbe02 100644 --- a/src/psu410/src/firstprinciples_calculations_and_theory/quantum_theory_for_the_motion_of_electrons.ipynb +++ b/src/psu410/src/firstprinciples_calculations_and_theory/quantum_theory_for_the_motion_of_electrons.ipynb @@ -4,39 +4,341 @@ "metadata": {}, "cells": [ { - "id": "2c82690a", + "id": "05af4bfa", "cell_type": "markdown", - "source": "## Quantum theory for the motion of electrons\n", + "source": [ + "## Quantum theory for the motion of electrons\n" + ], "metadata": {} }, { - "id": "6f383f95", + "id": "06b16305", "cell_type": "markdown", - "source": "### Schr\u00f6dinger equation\n\nThe Schr\u00f6dinger equation is typically written as follows\n\n*Eq. 5\u201157* ,\n\nwhere is the reduced Planck constant, **X** an abbreviation of the space\ncoordinates and spin states of the multiple particle system, the energy\noperator called Hamiltonian. When is independent of time *t*, one can\nseparate the coordinate **X** from the time *t* in finding the solution\nof by writing\n\n*Eq. 5\u201158* ,\n\nby which the stationary solutions of can be expressed through letting\n\n*Eq. 5\u201159* ,\n\nresulting in\n\n*Eq. 5\u201160*\n\n*Eq. 5\u201161*\n\nNote that is the frequency of de Broglie matter wave.\n\nFor any trial function \u039b(**X**) (in the Hilbert space) for \u03a8(**X**), the\nvariational principle tells us that the energy of the system always has\na lower bound through a ground state with energy *E0*, as\n\n*Eq. 5\u201162* ,\n\nwhere represents the complex conjugate of \u039b(**X**), resulting in\n\n*Eq. 5\u201163*\n\nwhich is known as the Rayleigh-Ritz variational principle,\n", + "source": [ + "### Schr\u00f6dinger equation\n", + "\n", + "The Schr\u00f6dinger equation is typically written as follows\n", + "\n", + "*Eq. 5\u201157* ,\n", + "\n", + "where is the reduced Planck constant, **X** an abbreviation of the space\n", + "coordinates and spin states of the multiple particle system, the energy\n", + "operator called Hamiltonian. When is independent of time *t*, one can\n", + "separate the coordinate **X** from the time *t* in finding the solution\n", + "of by writing\n", + "\n", + "*Eq. 5\u201158* ,\n", + "\n", + "by which the stationary solutions of can be expressed through letting\n", + "\n", + "*Eq. 5\u201159* ,\n", + "\n", + "resulting in\n", + "\n", + "*Eq. 5\u201160*\n", + "\n", + "*Eq. 5\u201161*\n", + "\n", + "Note that is the frequency of de Broglie matter wave.\n", + "\n", + "For any trial function \u039b(**X**) (in the Hilbert space) for \u03a8(**X**), the\n", + "variational principle tells us that the energy of the system always has\n", + "a lower bound through a ground state with energy *E0*, as\n", + "\n", + "*Eq. 5\u201162* ,\n", + "\n", + "where represents the complex conjugate of \u039b(**X**), resulting in\n", + "\n", + "*Eq. 5\u201163*\n", + "\n", + "which is known as the Rayleigh-Ritz variational principle,\n" + ], "metadata": {} }, { - "id": "23677eac", + "id": "a68fbf8b", "cell_type": "markdown", - "source": "### Born-Oppenheimer Approximation\n\nFor a time independent atomic system, it is often accurate enough to\nwrite in or in terms of the electron coordinate **r** and nuclei\ncoordinate **R**,\n\n*Eq. 5\u201164* ,\n\nwhere *e* represents the electron charge and *i* and *j* label the\nelectrons, *I* and *J* the atomic nuclei, *ZI* the atomic\nnuclear charge number of atom *I*, *e* the electron charge,\n*me* the electron mass, *MI* the mass of atomic\nnuclei *I*, the Laplace operator for electron *i*, and the Laplace\noperator for atomic nuclei *I,* noting\n\n*Eq. 5\u201165* ,\n\nwith respect to the Cartesian axis *x*, *y*, and *z*.\n\nConsider the fact that the electron mass is thousand times smaller than\nthe mass of the atomic nuclei, implying that the motions of the\nelectrons are much faster than the atomic nuclei, Born and Oppenheimer\nproposed that the wave function of the whole system can be simply\napproximated as the product of the electron wave function and the atomic\nnuclei wave function as\n\n*Eq. 5\u201166* .\n\nWith the auxiliary approximation of neglecting the dynamic coupling\nbetween the motions of electrons and atomic nuclei, the Schr\u00f6dinger\nequation for the motion of electrons becomes\n\n*Eq. 5\u201167* ,\n\nwhere\n\n*Eq. 5\u201168* ,\n\nand the Schr\u00f6dinger equation for the motion of atomic nuclei becomes\n\n*Eq. 5\u201169* ,\n\nwhere\n\n*Eq. 5\u201170* ,\n\nwith\n\n*Eq. 5\u201171* ,\n\nwhere represents the complex conjugate of .\n", + "source": [ + "### Born-Oppenheimer Approximation\n", + "\n", + "For a time independent atomic system, it is often accurate enough to\n", + "write in or in terms of the electron coordinate **r** and nuclei\n", + "coordinate **R**,\n", + "\n", + "*Eq. 5\u201164* ,\n", + "\n", + "where *e* represents the electron charge and *i* and *j* label the\n", + "electrons, *I* and *J* the atomic nuclei, *ZI* the atomic\n", + "nuclear charge number of atom *I*, *e* the electron charge,\n", + "*me* the electron mass, *MI* the mass of atomic\n", + "nuclei *I*, the Laplace operator for electron *i*, and the Laplace\n", + "operator for atomic nuclei *I,* noting\n", + "\n", + "*Eq. 5\u201165* ,\n", + "\n", + "with respect to the Cartesian axis *x*, *y*, and *z*.\n", + "\n", + "Consider the fact that the electron mass is thousand times smaller than\n", + "the mass of the atomic nuclei, implying that the motions of the\n", + "electrons are much faster than the atomic nuclei, Born and Oppenheimer\n", + "proposed that the wave function of the whole system can be simply\n", + "approximated as the product of the electron wave function and the atomic\n", + "nuclei wave function as\n", + "\n", + "*Eq. 5\u201166* .\n", + "\n", + "With the auxiliary approximation of neglecting the dynamic coupling\n", + "between the motions of electrons and atomic nuclei, the Schr\u00f6dinger\n", + "equation for the motion of electrons becomes\n", + "\n", + "*Eq. 5\u201167* ,\n", + "\n", + "where\n", + "\n", + "*Eq. 5\u201168* ,\n", + "\n", + "and the Schr\u00f6dinger equation for the motion of atomic nuclei becomes\n", + "\n", + "*Eq. 5\u201169* ,\n", + "\n", + "where\n", + "\n", + "*Eq. 5\u201170* ,\n", + "\n", + "with\n", + "\n", + "*Eq. 5\u201171* ,\n", + "\n", + "where represents the complex conjugate of .\n" + ], "metadata": {} }, { - "id": "f30dcaab", + "id": "8cd5cd6c", "cell_type": "markdown", - "source": "### Hartree-Fock approximation to solve Schr\u00f6dinger equation\n\nIt was Hartree who first assumed that the electron wave function in can\nbe expressed as a product of a collection of *N* independent\none-electron wave functions, where *i* = 1, 2, \u2026, *N* with *N* being the\nnumber of electrons in a system, in terms of the its space coordinate\n**r** and spin state **s**. After that, Fock modified the Hartree\napproximation by considering the fact that the wave function of a\nmulti-fermionic system should satisfy anti-symmetry requirements and\nsubsequently the Pauli exclusion principle that the total wave function\nchanges sign upon the exchange of fermions. Accordingly, the wave\nfunction of *N* electrons system under the Hartree-Fock approximation is\nexpressed as the Slater determinant \\[29\\]\n\n*Eq. 5\u201172*\n\nFor brevity, one can use the atomic unit that treats in , so that\n\n*Eq. 5\u201173*\n\nwhere , and\n\n*Eq. 5\u201174*\n\nAccordingly, the total energy of the system is expressed as\n\n*Eq. 5\u201175*\n\nwhere *Jij* is called as Coulomb/Hartree term\n\n*Eq. 5\u201176*\n\nwhere *Kij* is called as exchange term\n\n*Eq. 5\u201177* ,\n\nwhere = 1 if spin **s***i* and **s***j* points to\nthe same direction and = 0 if spin **s***i* and\n**s***j* points to the opposite direction.\n\nBy utilizing the variational condition of , one gets\n\n*Eq. 5\u201178* ,\n\nwhere is called one-electron energy, and\n\n*Eq. 5\u201179* ,\n\nwith being the electronic charge density whose expression is\n\n*Eq. 5\u201180* ,\n\n*Eq. 5\u201181* .\n\nIt should be especially noted here that to solve the Hartree-Fock\nequation , the most time consuming part is due to the nonlocal exchange\nterm, knowing the fact that the being evaluated one-electron wave\nfunction is also contained in the expression in the left hand side of by\nmeans of .\n\nThe configurational interaction method is the generalization of the\nHartree-Fock approximation. In such a case, *Y*, the number of\none-electron wave functions can be larger than the number of electrons,\n*N*, in the system. Accordingly, from the number of one-electron wave\nfunctions, , *y* = 1, 2, \u2026, *Y*, one can build the number of Stater\ndeterminants, *M*, by the combinatorial mathematics that the maximum of\n*M* can be\n\n*Eq. 5\u201182* .\n\nAs a result, the wave function of a collection of *N* electron system\nbecomes the recombination of the *M* Stater determinants as\n\n*Eq. 5\u201183* ,\n\nwhere the coefficients is to be found from the multiple linear equation\n\n*Eq. 5\u201184* .\n\nThe matrix element in *Eq. 5\u201184* is determined by the integral\n\n*Eq. 5\u201185* .\n", + "source": [ + "### Hartree-Fock approximation to solve Schr\u00f6dinger equation\n", + "\n", + "It was Hartree who first assumed that the electron wave function in can\n", + "be expressed as a product of a collection of *N* independent\n", + "one-electron wave functions, where *i* = 1, 2, \u2026, *N* with *N* being the\n", + "number of electrons in a system, in terms of the its space coordinate\n", + "**r** and spin state **s**. After that, Fock modified the Hartree\n", + "approximation by considering the fact that the wave function of a\n", + "multi-fermionic system should satisfy anti-symmetry requirements and\n", + "subsequently the Pauli exclusion principle that the total wave function\n", + "changes sign upon the exchange of fermions. Accordingly, the wave\n", + "function of *N* electrons system under the Hartree-Fock approximation is\n", + "expressed as the Slater determinant \\[29\\]\n", + "\n", + "*Eq. 5\u201172*\n", + "\n", + "For brevity, one can use the atomic unit that treats in , so that\n", + "\n", + "*Eq. 5\u201173*\n", + "\n", + "where , and\n", + "\n", + "*Eq. 5\u201174*\n", + "\n", + "Accordingly, the total energy of the system is expressed as\n", + "\n", + "*Eq. 5\u201175*\n", + "\n", + "where *Jij* is called as Coulomb/Hartree term\n", + "\n", + "*Eq. 5\u201176*\n", + "\n", + "where *Kij* is called as exchange term\n", + "\n", + "*Eq. 5\u201177* ,\n", + "\n", + "where = 1 if spin **s***i* and **s***j* points to\n", + "the same direction and = 0 if spin **s***i* and\n", + "**s***j* points to the opposite direction.\n", + "\n", + "By utilizing the variational condition of , one gets\n", + "\n", + "*Eq. 5\u201178* ,\n", + "\n", + "where is called one-electron energy, and\n", + "\n", + "*Eq. 5\u201179* ,\n", + "\n", + "with being the electronic charge density whose expression is\n", + "\n", + "*Eq. 5\u201180* ,\n", + "\n", + "*Eq. 5\u201181* .\n", + "\n", + "It should be especially noted here that to solve the Hartree-Fock\n", + "equation , the most time consuming part is due to the nonlocal exchange\n", + "term, knowing the fact that the being evaluated one-electron wave\n", + "function is also contained in the expression in the left hand side of by\n", + "means of .\n", + "\n", + "The configurational interaction method is the generalization of the\n", + "Hartree-Fock approximation. In such a case, *Y*, the number of\n", + "one-electron wave functions can be larger than the number of electrons,\n", + "*N*, in the system. Accordingly, from the number of one-electron wave\n", + "functions, , *y* = 1, 2, \u2026, *Y*, one can build the number of Stater\n", + "determinants, *M*, by the combinatorial mathematics that the maximum of\n", + "*M* can be\n", + "\n", + "*Eq. 5\u201182* .\n", + "\n", + "As a result, the wave function of a collection of *N* electron system\n", + "becomes the recombination of the *M* Stater determinants as\n", + "\n", + "*Eq. 5\u201183* ,\n", + "\n", + "where the coefficients is to be found from the multiple linear equation\n", + "\n", + "*Eq. 5\u201184* .\n", + "\n", + "The matrix element in *Eq. 5\u201184* is determined by the integral\n", + "\n", + "*Eq. 5\u201185* .\n" + ], "metadata": {} }, { - "id": "6be3ff7e", + "id": "40ade0dc", "cell_type": "markdown", - "source": "### Density functional theory (DFT) and 0 K Kohn-Sham equations\n\nThe density functional theory advocates that the properties of a matter\nare solely dictated by its electronic density distribution (or equally\nsay charge density), , in the real space. This is to say, that for an\narbitrary the total energy of the system, *E*, is always larger or equal\nto a value, *E0*, called as the ground state energy:\n\n*Eq. 5\u201186* .\n\nIn terms of variational principle, is equivalent to\n\n*Eq. 5\u201187* .\n\nKohn and Sham \\[8\\] proposed to write the total energy as\n\n*Eq. 5\u201188* ,\n\nwhere represents the kinetic energy of the system, is the external\npotential acting on the system, is the Hartree energy, and is the\nso-called exchange-correlation energy with where and represent the\ncharge density of electrons with spin down and spin up, respectively .\nUsing\n\n*Eq. 5\u201189* ,\n\ntogether with the variational principle of , one can get the\none-electron Schr\u00f6dinger equation\n\n*Eq. 5\u201190*\n\nwhere *me* represents the mass of an electron and\n\n*Eq. 5\u201191*\n\nso that the total energy is obtained as\n\n*Eq. 5\u201192*\n\nThe major challenge within DFT is that the accurate formulation of the\nexchange-correlation energy is unknown. Except for the uniform electron\ngas, no exact analytical form for the exchange-correlation energy has\nyet been obtained. Therefore approximations must be made for the\nexchange-correlation energy in calculating a realistic system. Until\nnow, the two most popular approximations are the local density\napproximation (LDA) \\[30\\] and the generalized gradient approximation\n(GGA) \\[31-32\\].\n\nThe local density approximation (LDA) states that the\nexchange-correlation energy is the same as that for a locally uniform\nelectron gas. In this case one can write *Vxc* as\n\n*Eq. 5\u201193* .\n\nAlthough this approximation is extremely simple, it works reasonably\nwell for many systems. The only remaining problem is to find an\napproximate solution to . One of most employed parameterized expression\nfor is that by Perdew and Zunger \\[30\\].\n\nMany modern DFT codes use the more advanced generalized gradient\napproximation (GGA) \\[31-32\\] to the exchange-correlation energy to\nimprove accuracy for certain physical properties. As the LDA\napproximates the energy of the true density by the energy of a local\nconstant density, it fails in situations where the density undergoes\nrapid changes such as in molecules. An improvement to this can be made\nby considering the gradient of the electron density. Symbolically, this\ncan be written as\n\n*Eq. 5\u201194*\n\nThe commonly used GGA is those due to Perdew et al. \\[31-32\\].\n", + "source": [ + "### Density functional theory (DFT) and 0 K Kohn-Sham equations\n", + "\n", + "The density functional theory advocates that the properties of a matter\n", + "are solely dictated by its electronic density distribution (or equally\n", + "say charge density), , in the real space. This is to say, that for an\n", + "arbitrary the total energy of the system, *E*, is always larger or equal\n", + "to a value, *E0*, called as the ground state energy:\n", + "\n", + "*Eq. 5\u201186* .\n", + "\n", + "In terms of variational principle, is equivalent to\n", + "\n", + "*Eq. 5\u201187* .\n", + "\n", + "Kohn and Sham \\[8\\] proposed to write the total energy as\n", + "\n", + "*Eq. 5\u201188* ,\n", + "\n", + "where represents the kinetic energy of the system, is the external\n", + "potential acting on the system, is the Hartree energy, and is the\n", + "so-called exchange-correlation energy with where and represent the\n", + "charge density of electrons with spin down and spin up, respectively .\n", + "Using\n", + "\n", + "*Eq. 5\u201189* ,\n", + "\n", + "together with the variational principle of , one can get the\n", + "one-electron Schr\u00f6dinger equation\n", + "\n", + "*Eq. 5\u201190*\n", + "\n", + "where *me* represents the mass of an electron and\n", + "\n", + "*Eq. 5\u201191*\n", + "\n", + "so that the total energy is obtained as\n", + "\n", + "*Eq. 5\u201192*\n", + "\n", + "The major challenge within DFT is that the accurate formulation of the\n", + "exchange-correlation energy is unknown. Except for the uniform electron\n", + "gas, no exact analytical form for the exchange-correlation energy has\n", + "yet been obtained. Therefore approximations must be made for the\n", + "exchange-correlation energy in calculating a realistic system. Until\n", + "now, the two most popular approximations are the local density\n", + "approximation (LDA) \\[30\\] and the generalized gradient approximation\n", + "(GGA) \\[31-32\\].\n", + "\n", + "The local density approximation (LDA) states that the\n", + "exchange-correlation energy is the same as that for a locally uniform\n", + "electron gas. In this case one can write *Vxc* as\n", + "\n", + "*Eq. 5\u201193* .\n", + "\n", + "Although this approximation is extremely simple, it works reasonably\n", + "well for many systems. The only remaining problem is to find an\n", + "approximate solution to . One of most employed parameterized expression\n", + "for is that by Perdew and Zunger \\[30\\].\n", + "\n", + "Many modern DFT codes use the more advanced generalized gradient\n", + "approximation (GGA) \\[31-32\\] to the exchange-correlation energy to\n", + "improve accuracy for certain physical properties. As the LDA\n", + "approximates the energy of the true density by the energy of a local\n", + "constant density, it fails in situations where the density undergoes\n", + "rapid changes such as in molecules. An improvement to this can be made\n", + "by considering the gradient of the electron density. Symbolically, this\n", + "can be written as\n", + "\n", + "*Eq. 5\u201194*\n", + "\n", + "The commonly used GGA is those due to Perdew et al. \\[31-32\\].\n" + ], "metadata": {} }, { - "id": "9d69a63e", + "id": "20b4c1b1", "cell_type": "markdown", - "source": "#### Solving the Kohn-Sham Equations for a Solid\n\nFor a solid, is still a mathematical challenge with infinite number of\none-electron wave functions to be solved and therefore cannot be solved\ndirectly in the real space. To reduce the dimension of the problem, one\ncan choose to solve the equation at a specific point in the reciprocal\nspace. According to Bloch\u2019s theorem, the wave function for a solid can\nbe written as the product of a wavelike part, , and a cell periodic\npart,\n\n*Eq. 5\u201195*\n\nwhere can be expressed as a sum of a finite number of plane waves whose\nwave vectors are reciprocal lattice vectors of the crystal\n\n*Eq. 5\u201196* .\n\nso that\n\n*Eq. 5\u201197*\n\nwhere the band index *j* is used to number the eigenenergy and the\neigenvector at a given . The number of plane waves is determined by the\nfollowing equation\n\n*Eq. 5\u201198*\n\nwhere is called energy cutoff.\n\nUtilize the obtained wave functions, the charge density can be\ncalculated with the Brillouin zone integration\n\n*Eq. 5\u201199*\n\nWhere\n\n*Eq. 5\u2011100*\n\nwhere the parameter is to make the integration over the charge density\nwithin the primitive unit cell equal to the number of electrons, *N,* in\nthe primitive unit cell. Numerically, the integration can be\napproximated by summation over a set of discrete *k*-mesh as\n\n*Eq. 5\u2011101*\n\nwhere *N*BZ represents the number of points in the first\nBrillouin zone in the *k*-mesh. When a solid possesses symmetry, the\nsummation in the above equation can be further reduced to the summation\nover the irreducible Brillouin zone (IBZ).\n\n*Eq. 5\u2011102*\n\nwhere is a weight factor that represents the number of points that are\nequivalent to by space group symmetry.\n", + "source": [ + "#### Solving the Kohn-Sham Equations for a Solid\n", + "\n", + "For a solid, is still a mathematical challenge with infinite number of\n", + "one-electron wave functions to be solved and therefore cannot be solved\n", + "directly in the real space. To reduce the dimension of the problem, one\n", + "can choose to solve the equation at a specific point in the reciprocal\n", + "space. According to Bloch\u2019s theorem, the wave function for a solid can\n", + "be written as the product of a wavelike part, , and a cell periodic\n", + "part,\n", + "\n", + "*Eq. 5\u201195*\n", + "\n", + "where can be expressed as a sum of a finite number of plane waves whose\n", + "wave vectors are reciprocal lattice vectors of the crystal\n", + "\n", + "*Eq. 5\u201196* .\n", + "\n", + "so that\n", + "\n", + "*Eq. 5\u201197*\n", + "\n", + "where the band index *j* is used to number the eigenenergy and the\n", + "eigenvector at a given . The number of plane waves is determined by the\n", + "following equation\n", + "\n", + "*Eq. 5\u201198*\n", + "\n", + "where is called energy cutoff.\n", + "\n", + "Utilize the obtained wave functions, the charge density can be\n", + "calculated with the Brillouin zone integration\n", + "\n", + "*Eq. 5\u201199*\n", + "\n", + "Where\n", + "\n", + "*Eq. 5\u2011100*\n", + "\n", + "where the parameter is to make the integration over the charge density\n", + "within the primitive unit cell equal to the number of electrons, *N,* in\n", + "the primitive unit cell. Numerically, the integration can be\n", + "approximated by summation over a set of discrete *k*-mesh as\n", + "\n", + "*Eq. 5\u2011101*\n", + "\n", + "where *N*BZ represents the number of points in the first\n", + "Brillouin zone in the *k*-mesh. When a solid possesses symmetry, the\n", + "summation in the above equation can be further reduced to the summation\n", + "over the irreducible Brillouin zone (IBZ).\n", + "\n", + "*Eq. 5\u2011102*\n", + "\n", + "where is a weight factor that represents the number of points that are\n", + "equivalent to by space group symmetry.\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/gibbs_energy_function/elastic_magnetic_and_electric_contributions_to_free_energy.ipynb b/src/psu410/src/gibbs_energy_function/elastic_magnetic_and_electric_contributions_to_free_energy.ipynb index 31c4e42..02730ea 100644 --- a/src/psu410/src/gibbs_energy_function/elastic_magnetic_and_electric_contributions_to_free_energy.ipynb +++ b/src/psu410/src/gibbs_energy_function/elastic_magnetic_and_electric_contributions_to_free_energy.ipynb @@ -4,9 +4,127 @@ "metadata": {}, "cells": [ { - "id": "5bcbb7fe", + "id": "3a9a396a", "cell_type": "markdown", - "source": "## Elastic, magnetic, and electric contributions to free energy\n\nChapters and focus on the thermal and hydrostatical pressure\ncontributions to Gibbs energy, which are the two prime variables\naffecting phase stability in typical experimental environments. However,\nthere are other internal and external contributions, which are\nparticularly important for crystalline phases. Two important internal\ncontributions are from magnetic and electric polarizations of materials\nwith the corresponding external contributions due to the magnetic and\nelectric fields. Furthermore, for non-hydrostatic pressing of solid\nphases, the $PV$ term in the combined law is to be replaced by elastic\nenergy calculated from elastic stress and elastic strain. The\ncorresponding works done to a system are as follows \\[3\\]\n\nEq. \u2011\n$dW_{elastic} = - V\\sum_{i,j,k,l}^{}{\\sigma_{ij}{d\\varepsilon}_{kl}}$\n\nEq. \u2011 $dW_{magnetic} = - V\\sum_{i}^{}{H_{i}{dB}_{i}}$\n\nEq. \u2011 $dW_{electric} = - V\\sum_{i}^{}{E_{i}{dD}_{i}}$\n\nwhere $i,j,\\ k,\\ l = 1,2,3$, $\\sigma_{ij}$ and $\\varepsilon_{kl}$ are\nthe components of stress and strain; $H_{i}$ and $B_{i}$ are the\ncomponents of magnetic field and magnetic induction; $E_{i}$ and $D_{i}$\nare the components of electric field and electric displacement; and $V$\nis the volume of the crystal. The negative sign in front of the equation\nis due to the fact that the system does work to the surroundings when it\nexpands its volume due to the strain, magnetic induction, and electric\ndisplacement.\n\nUsing the combined law of thermodynamics, can thus be re-written as\nfollows\n\nEq. \u2011\n$dU = TdS - V\\left( \\sum_{i,j,k,l}^{}{\\sigma_{ij}{d\\varepsilon}_{kl}} + \\sum_{i}^{}{H_{i}{dB}_{i}} + \\sum_{i}^{}{E_{i}{dD}_{i}} \\right) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$\n\nThe Legendre transformation similar to Helmholtz energy and Gibbs\nenergy, and , can be made to obtain the following characteristic free\nenergy functions\n\nEq. \u2011 $dF = d(U - TS)$\n\n$$= - SdT - V\\left( \\sum_{i,j,k,l}^{}{\\sigma_{ij}{d\\varepsilon}_{kl}} + \\sum_{i}^{}{H_{i}{dB}_{i}} + \\sum_{i}^{}{E_{i}{dD}_{i}} \\right) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$$\n\nEq. \u2011 $dF_{H} = d\\left( U - TS + \\sum_{i}^{}{H_{i}B_{i}} \\right)$\n\n$$= - SdT - V\\left( \\sum_{i,j,k,l}^{}{\\sigma_{ij}{d\\varepsilon}_{kl}} - \\sum_{i}^{}{B_{i}dH_{i}} + \\sum_{i}^{}{E_{i}{dD}_{i}} \\right) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$$\n\nEq. \u2011 $dF_{E} = d\\left( U - TS + \\sum_{i}^{}{E_{i}D_{i}} \\right)$\n\n$$= - SdT - V\\left( \\sum_{i,j,k,l}^{}{\\sigma_{ij}{d\\varepsilon}_{kl}} + \\sum_{i}^{}{H_{i}{dB}_{i}} - \\sum_{i}^{}{D_{i}{dE}_{i}} \\right) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$$\n\nEq. \u2011\n$dF_{EH} = d\\left( U - TS + \\sum_{i}^{}{H_{i}B_{i}} + \\sum_{i}^{}{E_{i}D_{i}} \\right)$\n\n$$= - SdT - V\\left( \\sum_{i,j,k,l}^{}{\\sigma_{ij}{d\\varepsilon}_{kl}} - \\sum_{i}^{}{B_{i}{dH}_{i}} - \\sum_{i}^{}{D_{i}{dE}_{i}} \\right) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$$\n\nEq. \u2011\n$dG = d\\left( U - TS + \\sum_{i,j,k,l}^{}{\\sigma_{ij}\\varepsilon_{kl}} + \\sum_{i}^{}{H_{i}B_{i}} + \\sum_{i}^{}{E_{i}D_{i}} \\right)$\n\n$$= - SdT + V\\left( \\sum_{i,j,k,l}^{}{\\varepsilon_{ij}d\\sigma}_{kl} + \\sum_{i}^{}{B_{i}dH_{i}} + \\sum_{i}^{}{D_{i}{dE}_{i}} \\right) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$$\n\nFrom above equations, it can be seen that the natural variables of\nvarious free energies are\n$F\\left( T,\\ \\varepsilon_{ij},\\ B_{i},\\ D_{i},\\ N_{i},\\ \\xi \\right)$,\n$F_{H}\\left( T,\\ \\varepsilon_{ij},\\ H_{i},\\ D_{i},\\ N_{i},\\ \\xi \\right)$,\n$F_{E}\\left( T,\\ \\varepsilon_{ij},\\ B_{i},\\ E_{i},\\ N_{i},\\ \\xi \\right)$,\n$F_{EH}\\left( T,\\ \\varepsilon_{ij},\\ H_{i},\\ E_{i},\\ N_{i},\\ \\xi \\right)$,\nand $G\\left( T,\\ \\sigma_{ij},\\ H_{i},\\ E_{i},\\ N_{i},\\ \\xi \\right)$.\nClearly, there can be more combinations when the components of\n$\\varepsilon_{ij}$, $D_{i}$, and $B_{i}$ are partially replaced by their\nconjugate potentials. The free energies listed above are useful\ndepending on how the system is constrained by the surroundings. For\npractical applications, the elastic, magnetic, and electric properties\nare usually considered for phases with fixed compositions, and at\nequilibrium can then be written as\n\nEq. \u2011\n$dG = - SdT + V\\left( \\sum_{i,j,k.l}^{}{\\varepsilon_{ij}d\\sigma}_{kl} + \\sum_{i}^{}{B_{i}dH_{i}} + \\sum_{i}^{}{D_{i}{dE}_{i}} \\right) + \\mu dN$\n\nThe corresponding Gibbs-Duhem equation follows as below\n\nEq. \u2011\n$0 = - SdT + V\\left( \\sum_{i,j,k,l}^{}{\\varepsilon_{ij}d\\sigma}_{kl} + \\sum_{i}^{}{B_{i}dH_{i}} + \\sum_{i}^{}{D_{i}{dE}_{i}} \\right) - Nd\\mu$\n\nThe general differential form of a molar quantity can be extended from\nas follows as a one-component system\n\nEq. \u2011\n\nEq. \u2011\n\nEq. \u2011\n\nEq. \u2011\n\nThe first derivatives in to are the second directives of Gibbs energy\nwith respect to its natural variables, i.e. potentials, and have their\nrespective nomenclatures as shown in . The limit of stability follows\nand can be re-written as\n\nEq. \u2011\n$\\left( \\frac{\\partial X_{i}}{\\partial Y_{i}} \\right)_{N,Y_{j}} = \\infty$\n\nThis means that the derivatives in to , i.e. the quantities in , diverge\nat the limit of stability.\n\nTable \u2011: Physical quantities related to the first derivatives in to .\nThe table is symmetric due to the Maxwell relations related to the\nsecond directives of Gibbs energy with respect to its natural variables.\n\n| | T | $$\\sigma_{kl}$$ | $$E_{k}$$ | $$H_{k}$$ |\n|----------------------|------------------------------------|------------------------------------|---------------------------------------|---------------------------------------|\n| S | C/T, heat capacity | $\\alpha_{kl}$, piezocaloric effect | $p_{k}$, electrocaloric effect | $m_{k}$, magnetocaloric effect |\n| $$\\varepsilon_{ij}$$ | $\\alpha_{ij}$, thermal expansion | $s_{ijkl}$, elastic compliance | $d_{ijk}$, converse piezoelectricity | $q_{ijk}$, piezomagnetic moduli |\n| $$D_{i}$$ | $p_{i}$, pyroelectric coefficients | $d_{ikl}$, piezoelectric moduli | $k_{ik}$, permittivities | $a_{ik}$, magnetoelectric coefficient |\n| $$B_{i}$$ | $m_{i}$, pyromagnetic coefficient | $q_{ikl}$, piezomagnetic moduli | $a_{ik}$, magnetoelectric coefficient | $\\mu_{ik}$, permeability |\n", + "source": [ + "## Elastic, magnetic, and electric contributions to free energy\n", + "\n", + "Chapters and focus on the thermal and hydrostatical pressure\n", + "contributions to Gibbs energy, which are the two prime variables\n", + "affecting phase stability in typical experimental environments. However,\n", + "there are other internal and external contributions, which are\n", + "particularly important for crystalline phases. Two important internal\n", + "contributions are from magnetic and electric polarizations of materials\n", + "with the corresponding external contributions due to the magnetic and\n", + "electric fields. Furthermore, for non-hydrostatic pressing of solid\n", + "phases, the $PV$ term in the combined law is to be replaced by elastic\n", + "energy calculated from elastic stress and elastic strain. The\n", + "corresponding works done to a system are as follows \\[3\\]\n", + "\n", + "Eq. \u2011\n", + "$dW_{elastic} = - V\\sum_{i,j,k,l}^{}{\\sigma_{ij}{d\\varepsilon}_{kl}}$\n", + "\n", + "Eq. \u2011 $dW_{magnetic} = - V\\sum_{i}^{}{H_{i}{dB}_{i}}$\n", + "\n", + "Eq. \u2011 $dW_{electric} = - V\\sum_{i}^{}{E_{i}{dD}_{i}}$\n", + "\n", + "where $i,j,\\ k,\\ l = 1,2,3$, $\\sigma_{ij}$ and $\\varepsilon_{kl}$ are\n", + "the components of stress and strain; $H_{i}$ and $B_{i}$ are the\n", + "components of magnetic field and magnetic induction; $E_{i}$ and $D_{i}$\n", + "are the components of electric field and electric displacement; and $V$\n", + "is the volume of the crystal. The negative sign in front of the equation\n", + "is due to the fact that the system does work to the surroundings when it\n", + "expands its volume due to the strain, magnetic induction, and electric\n", + "displacement.\n", + "\n", + "Using the combined law of thermodynamics, can thus be re-written as\n", + "follows\n", + "\n", + "Eq. \u2011\n", + "$dU = TdS - V\\left( \\sum_{i,j,k,l}^{}{\\sigma_{ij}{d\\varepsilon}_{kl}} + \\sum_{i}^{}{H_{i}{dB}_{i}} + \\sum_{i}^{}{E_{i}{dD}_{i}} \\right) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$\n", + "\n", + "The Legendre transformation similar to Helmholtz energy and Gibbs\n", + "energy, and , can be made to obtain the following characteristic free\n", + "energy functions\n", + "\n", + "Eq. \u2011 $dF = d(U - TS)$\n", + "\n", + "$$= - SdT - V\\left( \\sum_{i,j,k,l}^{}{\\sigma_{ij}{d\\varepsilon}_{kl}} + \\sum_{i}^{}{H_{i}{dB}_{i}} + \\sum_{i}^{}{E_{i}{dD}_{i}} \\right) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$$\n", + "\n", + "Eq. \u2011 $dF_{H} = d\\left( U - TS + \\sum_{i}^{}{H_{i}B_{i}} \\right)$\n", + "\n", + "$$= - SdT - V\\left( \\sum_{i,j,k,l}^{}{\\sigma_{ij}{d\\varepsilon}_{kl}} - \\sum_{i}^{}{B_{i}dH_{i}} + \\sum_{i}^{}{E_{i}{dD}_{i}} \\right) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$$\n", + "\n", + "Eq. \u2011 $dF_{E} = d\\left( U - TS + \\sum_{i}^{}{E_{i}D_{i}} \\right)$\n", + "\n", + "$$= - SdT - V\\left( \\sum_{i,j,k,l}^{}{\\sigma_{ij}{d\\varepsilon}_{kl}} + \\sum_{i}^{}{H_{i}{dB}_{i}} - \\sum_{i}^{}{D_{i}{dE}_{i}} \\right) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$$\n", + "\n", + "Eq. \u2011\n", + "$dF_{EH} = d\\left( U - TS + \\sum_{i}^{}{H_{i}B_{i}} + \\sum_{i}^{}{E_{i}D_{i}} \\right)$\n", + "\n", + "$$= - SdT - V\\left( \\sum_{i,j,k,l}^{}{\\sigma_{ij}{d\\varepsilon}_{kl}} - \\sum_{i}^{}{B_{i}{dH}_{i}} - \\sum_{i}^{}{D_{i}{dE}_{i}} \\right) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$$\n", + "\n", + "Eq. \u2011\n", + "$dG = d\\left( U - TS + \\sum_{i,j,k,l}^{}{\\sigma_{ij}\\varepsilon_{kl}} + \\sum_{i}^{}{H_{i}B_{i}} + \\sum_{i}^{}{E_{i}D_{i}} \\right)$\n", + "\n", + "$$= - SdT + V\\left( \\sum_{i,j,k,l}^{}{\\varepsilon_{ij}d\\sigma}_{kl} + \\sum_{i}^{}{B_{i}dH_{i}} + \\sum_{i}^{}{D_{i}{dE}_{i}} \\right) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$$\n", + "\n", + "From above equations, it can be seen that the natural variables of\n", + "various free energies are\n", + "$F\\left( T,\\ \\varepsilon_{ij},\\ B_{i},\\ D_{i},\\ N_{i},\\ \\xi \\right)$,\n", + "$F_{H}\\left( T,\\ \\varepsilon_{ij},\\ H_{i},\\ D_{i},\\ N_{i},\\ \\xi \\right)$,\n", + "$F_{E}\\left( T,\\ \\varepsilon_{ij},\\ B_{i},\\ E_{i},\\ N_{i},\\ \\xi \\right)$,\n", + "$F_{EH}\\left( T,\\ \\varepsilon_{ij},\\ H_{i},\\ E_{i},\\ N_{i},\\ \\xi \\right)$,\n", + "and $G\\left( T,\\ \\sigma_{ij},\\ H_{i},\\ E_{i},\\ N_{i},\\ \\xi \\right)$.\n", + "Clearly, there can be more combinations when the components of\n", + "$\\varepsilon_{ij}$, $D_{i}$, and $B_{i}$ are partially replaced by their\n", + "conjugate potentials. The free energies listed above are useful\n", + "depending on how the system is constrained by the surroundings. For\n", + "practical applications, the elastic, magnetic, and electric properties\n", + "are usually considered for phases with fixed compositions, and at\n", + "equilibrium can then be written as\n", + "\n", + "Eq. \u2011\n", + "$dG = - SdT + V\\left( \\sum_{i,j,k.l}^{}{\\varepsilon_{ij}d\\sigma}_{kl} + \\sum_{i}^{}{B_{i}dH_{i}} + \\sum_{i}^{}{D_{i}{dE}_{i}} \\right) + \\mu dN$\n", + "\n", + "The corresponding Gibbs-Duhem equation follows as below\n", + "\n", + "Eq. \u2011\n", + "$0 = - SdT + V\\left( \\sum_{i,j,k,l}^{}{\\varepsilon_{ij}d\\sigma}_{kl} + \\sum_{i}^{}{B_{i}dH_{i}} + \\sum_{i}^{}{D_{i}{dE}_{i}} \\right) - Nd\\mu$\n", + "\n", + "The general differential form of a molar quantity can be extended from\n", + "as follows as a one-component system\n", + "\n", + "Eq. \u2011\n", + "\n", + "Eq. \u2011\n", + "\n", + "Eq. \u2011\n", + "\n", + "Eq. \u2011\n", + "\n", + "The first derivatives in to are the second directives of Gibbs energy\n", + "with respect to its natural variables, i.e. potentials, and have their\n", + "respective nomenclatures as shown in . The limit of stability follows\n", + "and can be re-written as\n", + "\n", + "Eq. \u2011\n", + "$\\left( \\frac{\\partial X_{i}}{\\partial Y_{i}} \\right)_{N,Y_{j}} = \\infty$\n", + "\n", + "This means that the derivatives in to , i.e. the quantities in , diverge\n", + "at the limit of stability.\n", + "\n", + "Table \u2011: Physical quantities related to the first derivatives in to .\n", + "The table is symmetric due to the Maxwell relations related to the\n", + "second directives of Gibbs energy with respect to its natural variables.\n", + "\n", + "| | T | $$\\sigma_{kl}$$ | $$E_{k}$$ | $$H_{k}$$ |\n", + "|----------------------|------------------------------------|------------------------------------|---------------------------------------|---------------------------------------|\n", + "| S | C/T, heat capacity | $\\alpha_{kl}$, piezocaloric effect | $p_{k}$, electrocaloric effect | $m_{k}$, magnetocaloric effect |\n", + "| $$\\varepsilon_{ij}$$ | $\\alpha_{ij}$, thermal expansion | $s_{ijkl}$, elastic compliance | $d_{ijk}$, converse piezoelectricity | $q_{ijk}$, piezomagnetic moduli |\n", + "| $$D_{i}$$ | $p_{i}$, pyroelectric coefficients | $d_{ikl}$, piezoelectric moduli | $k_{ik}$, permittivities | $a_{ik}$, magnetoelectric coefficient |\n", + "| $$B_{i}$$ | $m_{i}$, pyromagnetic coefficient | $q_{ikl}$, piezomagnetic moduli | $a_{ik}$, magnetoelectric coefficient | $\\mu_{ik}$, permeability |\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/gibbs_energy_function/index.ipynb b/src/psu410/src/gibbs_energy_function/index.ipynb index 377709e..b422a98 100644 --- a/src/psu410/src/gibbs_energy_function/index.ipynb +++ b/src/psu410/src/gibbs_energy_function/index.ipynb @@ -4,9 +4,173 @@ "metadata": {}, "cells": [ { - "id": "dbdfb491", + "id": "447fe0e1", "cell_type": "markdown", - "source": "# Gibbs energy function\n\nAs shown in through , all functions have $N_{i}$ and $\\xi$ as natural\nvariables while they differ in other two natural variables. In typical\nmaterials-related experiments, temperature and pressure are the two\nvariables controlled. They are also the natural variables of Gibbs\nenergy. Consequently, Gibbs energy is the most widely used function in\nthermodynamics of materials science. The rest of this book focuses on\nGibbs energy for this reason. In this chapter, the mathematical formulas\nfor Gibbs energy of phases with fixed and variable compositions are\ndiscussed which are needed for quantitative calculations of Gibbs energy\nunder given values of its natural variables.\n\nFrom , the molar Gibbs energy can be defined as\n\nEq. \u2011\n$G_{m}\\left( T,P,x_{i},\\xi \\right) = \\frac{G}{N} = \\sum_{}^{}\\mu_{i}x_{i}$\n\nThe molar entropy, molar volume, chemical potential, and the driving\nforce can be obtained from as\n\nEq. \u2011\n$S_{m} = \\frac{S}{N} = - \\frac{1}{N}\\left( \\frac{\\partial G}{\\partial T} \\right)_{P,\\ N_{i},\\ \\xi} = {- \\left( \\frac{\\partial G_{m}}{\\partial T} \\right)}_{P,\\ x_{i},\\ \\xi}$\n\nEq. \u2011\n$V_{m} = \\frac{V}{N} = \\frac{1}{N}\\left( \\frac{\\partial G}{\\partial P} \\right)_{T,\\ N_{i},\\ \\xi} = \\left( \\frac{\\partial G_{m}}{\\partial P} \\right)_{T,\\ x_{i},\\ \\xi}$\n\nEq. \u2011\n$\\mu_{i} = \\left( \\frac{\\partial G}{\\partial N_{i}} \\right)_{T,P,N_{j \\neq i},\\ \\xi}$\n\nEq. \u2011\n$- D = \\left( \\frac{\\partial G}{\\partial\\xi} \\right)_{T,P,N_{i}\\ }$\n\nBased on , the molar enthalpy is written as\n\nEq. \u2011 $H_{m} = G_{m} + TS_{m}$\n\nOther physical properties of the system can also be represented by the\npartial derivatives of Gibbs energy such as heat capacity, $C_{P}$,\nvolume thermal expansivity, $\\alpha_{V}$, isothermal compressibility,\n$\\kappa_{T}$, as follows under constant pressure or temperature\n\nEq. \u2011\n$C_{P} = \\left( \\frac{\\partial Q}{\\partial T} \\right)_{P} = \\left( \\frac{\\partial H}{\\partial T} \\right)_{P} = T\\left( \\frac{\\partial(G + TS)}{\\partial T} \\right)_{P} = T\\left( \\frac{\\partial S}{\\partial T} \\right)_{P} = - T\\left( \\frac{\\partial^{2}G}{\\partial T^{2}} \\right)_{P}$\n\nEq. \u2011\n$\\alpha_{V} = \\frac{\\left( \\frac{\\partial V}{\\partial T} \\right)_{P}}{V} = \\frac{\\left( \\frac{\\left( \\partial G/\\partial( - P) \\right)_{T}}{\\partial T} \\right)_{P}}{\\left( \\partial G/\\partial( - P) \\right)_{T}} = \\frac{\\frac{\\partial^{2}G}{\\partial T\\partial( - P)}}{\\left( \\partial G/\\partial( - P) \\right)_{T}}$\n\nEq. \u2011\n$\\kappa_{T} = \\frac{\\left( \\frac{\\partial V}{\\partial( - P)} \\right)_{T}}{V} = \\frac{\\left( \\frac{\\left( \\partial G/\\partial( - P) \\right)_{T}}{\\partial( - P)} \\right)_{T}}{\\left( \\partial G/\\partial( - P) \\right)_{T}} = \\frac{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}}{\\left( \\partial G/\\partial( - P) \\right)_{T}} = \\frac{1}{B}$\n\nwhere the $N_{i}$ and $\\xi$ are kept constant for all partial\nderivatives, and $B$ is the bulk modulus.\n\nIn , $G$ cannot be directly replaced by $G_{m}$ because *N* also depends\non *Ni*. The thermodynamic quantities under such conditions,\ni.e. varying the amount of a component at constant temperature and\npressure, are called partial quantities which are introduced in Eq. 1\u20118\nfor partial entropy and for partial enthalpy. This definition can be\nextended to all molar quantities such as partial volume and partial\nGibbs energy. Partial quantities of a molar quantity, $A$, can thus be\ndefined in general as\n\nEq. \u2011\n$A_{i} = \\left( \\frac{\\partial A}{\\partial N_{i}} \\right)_{T,P,N_{j \\neq i},\\ \\xi}$\n\nThe general differential form of a molar quantity for a system at\nequilibrium can be represented by its partial quantities as\n\nEq. \u2011\n$dA = \\left( \\frac{\\partial A}{\\partial T} \\right)dT + \\left( \\frac{\\partial A}{\\partial P} \\right)dP + \\sum_{}^{}\\left( \\frac{\\partial A}{\\partial N_{i}} \\right){dN}_{i}$\n\nwhere the subscripts representing variables kept constant, i.e. the\nremaining natural variables of Gibbs energy not in the denominator, are\nomitted for simplicity. This will be done throughout the book unless\nspecified otherwise.\n\nUsing the following relations: $A = NA_{m}$, $N = \\sum_{}^{}N_{j}$,\n$x_{i} = N_{i}/N$,\n$\\frac{{\\partial x}_{i}}{{\\partial N}_{i}} = \\left( 1 - x_{i} \\right)/N$,\nand $\\frac{{\\partial x}_{k}}{{\\partial N}_{i}} = {- x}_{k}/N$, can be\nderived as, under constant T and P,\n\nEq. \u2011\n$A_{i} = A_{m} + N\\sum_{j = 1}^{c}{\\frac{\\partial A_{m}}{\\partial x_{j}}\\frac{\\partial x_{j}}{\\partial N_{i}}} = A_{m} + \\frac{\\partial A_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial A_{m}}{\\partial x_{j}}$\n\nwhere the summation is for all *c* components and the partial\nderivatives are taken with other mole fractions kept constant. However,\nmole fractions are not independent, but follow the relation\n$\\sum_{}^{}x_{i} = 1$. Taking $x_{1} = 1 - \\sum_{j = 2}^{c}x_{j}$ as the\ndependent mole fraction, can be rewritten as\n\nEq. \u2011\n$A_{i} = A_{m} + \\left( \\frac{\\partial A_{m}}{\\partial x_{i}} - \\frac{\\partial A_{m}}{\\partial x_{1}} \\right) - \\sum_{j = 2}^{c}x_{j}\\left( \\frac{\\partial A_{m}}{\\partial x_{j}} - \\frac{\\partial A_{m}}{\\partial x_{1}} \\right)$\n\nThe difference of the partial derivatives in the parenthesis in\nrepresents the partial derivative of $A_{m}$ with respect to the mole\nfraction of one component when the first component is selected as the\ndependent component. Applying and to Gibbs energy, the partial Gibbs\nenergy or chemical potential of component $i$ is obtained as\n\nEq. \u2011\n$\\mu_{i} = G_{i} = G_{m} + \\frac{\\partial G_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial G_{m}}{\\partial x_{j}} = G_{m} + \\left( \\frac{\\partial G_{m}}{\\partial x_{i}} - \\frac{\\partial G_{m}}{\\partial x_{1}} \\right) - \\sum_{j = 2}^{c}x_{j}\\left( \\frac{\\partial G_{m}}{\\partial x_{j}} - \\frac{\\partial G_{m}}{\\partial x_{1}} \\right)$\n\nThe derivatives in the stability equation, , are defined with the molar\nquantities kept constant. On the other hand, Gibbs energy has two\npotentials, temperature and pressure, as natural variables instead. One\nwould thus need to compare the stability conditions when a variable kept\nfixed is changed from a molar quantity to its conjugate potential. This\ncan be carried out through the use of Jacobians to change the\nindependent variables\n\nEq. \u2011\n$\\frac{\\partial\\left( Y_{i},Y_{j} \\right)}{\\partial\\left( X_{i},X_{j} \\right)} = \\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{Y_{j}}\\left( \\frac{\\partial Y_{j}}{\\partial X_{j}} \\right)_{X_{i}} = \\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}}\\left( \\frac{\\partial Y_{j}}{\\partial X_{j}} \\right)_{X_{i}} - \\left( \\frac{\\partial Y_{i}}{\\partial X_{j}} \\right)_{X_{i}}\\left( \\frac{\\partial Y_{j}}{\\partial X_{i}} \\right)_{X_{j}}$\n\nFor a stable system, both\n$\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}}$ and\n$\\left( \\frac{\\partial Y_{j}}{\\partial X_{j}} \\right)_{X_{i}}$ are\npositive based on . Using the Maxwell relation shown by , one thus\nobtains\n\nEq. \u2011\n$\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}} - \\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{Y_{j}} = \\left( \\frac{\\partial Y_{i}}{\\partial X_{j}} \\right)_{X_{i}}\\left( \\frac{\\partial Y_{j}}{\\partial X_{i}} \\right)_{X_{j}}/\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}} \\geq 0$\n\nThis means that\n$\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{Y_{j}}$ will go\nto zero before\n$\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}}$ does. It\nindicates that the stability condition becomes more restrictive when\npotentials are kept constant in place of their conjugate molar\nquantities. Based on the Gibbs-Duhem equation of , the maximum number of\nindependent potentials is *c+1*, and the last potential is dependent,\ni.e.\n\nEq. \u2011\n$\\left( \\frac{\\partial Y_{c + 2}}{\\partial X_{c + 2}} \\right)_{Y_{j \\leq c + 1}} = 0$\n\nTherefore, the limit of stability is determined when the derivative\nbecomes zero with one molar quantity kept constant, e.g.\n\nEq. \u2011\n$\\left( \\frac{\\partial Y_{c + 1}}{\\partial X_{c + 1}} \\right)_{Y_{j < c + 1},X_{c + 2}} = 0$\n\nThis is because this derivative reaches zero faster than any other\nderivatives with more molar quantities kept constant. shows that all\nmolar quantities diverge at the limit of stability. The consolute point\nis obtained with $c$ additional conditions as follows based on\n\nEq. \u2011\n$\\left( \\frac{\\partial^{2}Y_{i}}{\\partial\\left( X_{i} \\right)^{2}} \\right)_{Y_{j \\leq c + 1, \\neq i},X_{c + 2}} = 0$\n\nTogether with , all $c + 1$ independent potentials at the consolute\npoint can be determined. It is evident that the consolute point is a\nzero-dimensional point in a two-dimensional space of independent\npotentials in a one-component system. With the addition of a second\ncomponent to form a binary system, this consolute point in the\none-component system extends into a one-dimensional line. This line\nrepresents the limit of stability of the binary system, and a consolute\npoint is located at the end of this line. It is thus evident that in a\nsystem with $c$ independent components, the limit of stability is a\n*c-1*-dimensional hypersurface in a space of $c + 1$ independent\npotentials, while the consolute point is a zero-dimensional point in all\nsystems, which may be called the invariant critical point.\n", + "source": [ + "# Gibbs energy function\n", + "\n", + "As shown in through , all functions have $N_{i}$ and $\\xi$ as natural\n", + "variables while they differ in other two natural variables. In typical\n", + "materials-related experiments, temperature and pressure are the two\n", + "variables controlled. They are also the natural variables of Gibbs\n", + "energy. Consequently, Gibbs energy is the most widely used function in\n", + "thermodynamics of materials science. The rest of this book focuses on\n", + "Gibbs energy for this reason. In this chapter, the mathematical formulas\n", + "for Gibbs energy of phases with fixed and variable compositions are\n", + "discussed which are needed for quantitative calculations of Gibbs energy\n", + "under given values of its natural variables.\n", + "\n", + "From , the molar Gibbs energy can be defined as\n", + "\n", + "Eq. \u2011\n", + "$G_{m}\\left( T,P,x_{i},\\xi \\right) = \\frac{G}{N} = \\sum_{}^{}\\mu_{i}x_{i}$\n", + "\n", + "The molar entropy, molar volume, chemical potential, and the driving\n", + "force can be obtained from as\n", + "\n", + "Eq. \u2011\n", + "$S_{m} = \\frac{S}{N} = - \\frac{1}{N}\\left( \\frac{\\partial G}{\\partial T} \\right)_{P,\\ N_{i},\\ \\xi} = {- \\left( \\frac{\\partial G_{m}}{\\partial T} \\right)}_{P,\\ x_{i},\\ \\xi}$\n", + "\n", + "Eq. \u2011\n", + "$V_{m} = \\frac{V}{N} = \\frac{1}{N}\\left( \\frac{\\partial G}{\\partial P} \\right)_{T,\\ N_{i},\\ \\xi} = \\left( \\frac{\\partial G_{m}}{\\partial P} \\right)_{T,\\ x_{i},\\ \\xi}$\n", + "\n", + "Eq. \u2011\n", + "$\\mu_{i} = \\left( \\frac{\\partial G}{\\partial N_{i}} \\right)_{T,P,N_{j \\neq i},\\ \\xi}$\n", + "\n", + "Eq. \u2011\n", + "$- D = \\left( \\frac{\\partial G}{\\partial\\xi} \\right)_{T,P,N_{i}\\ }$\n", + "\n", + "Based on , the molar enthalpy is written as\n", + "\n", + "Eq. \u2011 $H_{m} = G_{m} + TS_{m}$\n", + "\n", + "Other physical properties of the system can also be represented by the\n", + "partial derivatives of Gibbs energy such as heat capacity, $C_{P}$,\n", + "volume thermal expansivity, $\\alpha_{V}$, isothermal compressibility,\n", + "$\\kappa_{T}$, as follows under constant pressure or temperature\n", + "\n", + "Eq. \u2011\n", + "$C_{P} = \\left( \\frac{\\partial Q}{\\partial T} \\right)_{P} = \\left( \\frac{\\partial H}{\\partial T} \\right)_{P} = T\\left( \\frac{\\partial(G + TS)}{\\partial T} \\right)_{P} = T\\left( \\frac{\\partial S}{\\partial T} \\right)_{P} = - T\\left( \\frac{\\partial^{2}G}{\\partial T^{2}} \\right)_{P}$\n", + "\n", + "Eq. \u2011\n", + "$\\alpha_{V} = \\frac{\\left( \\frac{\\partial V}{\\partial T} \\right)_{P}}{V} = \\frac{\\left( \\frac{\\left( \\partial G/\\partial( - P) \\right)_{T}}{\\partial T} \\right)_{P}}{\\left( \\partial G/\\partial( - P) \\right)_{T}} = \\frac{\\frac{\\partial^{2}G}{\\partial T\\partial( - P)}}{\\left( \\partial G/\\partial( - P) \\right)_{T}}$\n", + "\n", + "Eq. \u2011\n", + "$\\kappa_{T} = \\frac{\\left( \\frac{\\partial V}{\\partial( - P)} \\right)_{T}}{V} = \\frac{\\left( \\frac{\\left( \\partial G/\\partial( - P) \\right)_{T}}{\\partial( - P)} \\right)_{T}}{\\left( \\partial G/\\partial( - P) \\right)_{T}} = \\frac{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}}{\\left( \\partial G/\\partial( - P) \\right)_{T}} = \\frac{1}{B}$\n", + "\n", + "where the $N_{i}$ and $\\xi$ are kept constant for all partial\n", + "derivatives, and $B$ is the bulk modulus.\n", + "\n", + "In , $G$ cannot be directly replaced by $G_{m}$ because *N* also depends\n", + "on *Ni*. The thermodynamic quantities under such conditions,\n", + "i.e. varying the amount of a component at constant temperature and\n", + "pressure, are called partial quantities which are introduced in Eq. 1\u20118\n", + "for partial entropy and for partial enthalpy. This definition can be\n", + "extended to all molar quantities such as partial volume and partial\n", + "Gibbs energy. Partial quantities of a molar quantity, $A$, can thus be\n", + "defined in general as\n", + "\n", + "Eq. \u2011\n", + "$A_{i} = \\left( \\frac{\\partial A}{\\partial N_{i}} \\right)_{T,P,N_{j \\neq i},\\ \\xi}$\n", + "\n", + "The general differential form of a molar quantity for a system at\n", + "equilibrium can be represented by its partial quantities as\n", + "\n", + "Eq. \u2011\n", + "$dA = \\left( \\frac{\\partial A}{\\partial T} \\right)dT + \\left( \\frac{\\partial A}{\\partial P} \\right)dP + \\sum_{}^{}\\left( \\frac{\\partial A}{\\partial N_{i}} \\right){dN}_{i}$\n", + "\n", + "where the subscripts representing variables kept constant, i.e. the\n", + "remaining natural variables of Gibbs energy not in the denominator, are\n", + "omitted for simplicity. This will be done throughout the book unless\n", + "specified otherwise.\n", + "\n", + "Using the following relations: $A = NA_{m}$, $N = \\sum_{}^{}N_{j}$,\n", + "$x_{i} = N_{i}/N$,\n", + "$\\frac{{\\partial x}_{i}}{{\\partial N}_{i}} = \\left( 1 - x_{i} \\right)/N$,\n", + "and $\\frac{{\\partial x}_{k}}{{\\partial N}_{i}} = {- x}_{k}/N$, can be\n", + "derived as, under constant T and P,\n", + "\n", + "Eq. \u2011\n", + "$A_{i} = A_{m} + N\\sum_{j = 1}^{c}{\\frac{\\partial A_{m}}{\\partial x_{j}}\\frac{\\partial x_{j}}{\\partial N_{i}}} = A_{m} + \\frac{\\partial A_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial A_{m}}{\\partial x_{j}}$\n", + "\n", + "where the summation is for all *c* components and the partial\n", + "derivatives are taken with other mole fractions kept constant. However,\n", + "mole fractions are not independent, but follow the relation\n", + "$\\sum_{}^{}x_{i} = 1$. Taking $x_{1} = 1 - \\sum_{j = 2}^{c}x_{j}$ as the\n", + "dependent mole fraction, can be rewritten as\n", + "\n", + "Eq. \u2011\n", + "$A_{i} = A_{m} + \\left( \\frac{\\partial A_{m}}{\\partial x_{i}} - \\frac{\\partial A_{m}}{\\partial x_{1}} \\right) - \\sum_{j = 2}^{c}x_{j}\\left( \\frac{\\partial A_{m}}{\\partial x_{j}} - \\frac{\\partial A_{m}}{\\partial x_{1}} \\right)$\n", + "\n", + "The difference of the partial derivatives in the parenthesis in\n", + "represents the partial derivative of $A_{m}$ with respect to the mole\n", + "fraction of one component when the first component is selected as the\n", + "dependent component. Applying and to Gibbs energy, the partial Gibbs\n", + "energy or chemical potential of component $i$ is obtained as\n", + "\n", + "Eq. \u2011\n", + "$\\mu_{i} = G_{i} = G_{m} + \\frac{\\partial G_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial G_{m}}{\\partial x_{j}} = G_{m} + \\left( \\frac{\\partial G_{m}}{\\partial x_{i}} - \\frac{\\partial G_{m}}{\\partial x_{1}} \\right) - \\sum_{j = 2}^{c}x_{j}\\left( \\frac{\\partial G_{m}}{\\partial x_{j}} - \\frac{\\partial G_{m}}{\\partial x_{1}} \\right)$\n", + "\n", + "The derivatives in the stability equation, , are defined with the molar\n", + "quantities kept constant. On the other hand, Gibbs energy has two\n", + "potentials, temperature and pressure, as natural variables instead. One\n", + "would thus need to compare the stability conditions when a variable kept\n", + "fixed is changed from a molar quantity to its conjugate potential. This\n", + "can be carried out through the use of Jacobians to change the\n", + "independent variables\n", + "\n", + "Eq. \u2011\n", + "$\\frac{\\partial\\left( Y_{i},Y_{j} \\right)}{\\partial\\left( X_{i},X_{j} \\right)} = \\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{Y_{j}}\\left( \\frac{\\partial Y_{j}}{\\partial X_{j}} \\right)_{X_{i}} = \\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}}\\left( \\frac{\\partial Y_{j}}{\\partial X_{j}} \\right)_{X_{i}} - \\left( \\frac{\\partial Y_{i}}{\\partial X_{j}} \\right)_{X_{i}}\\left( \\frac{\\partial Y_{j}}{\\partial X_{i}} \\right)_{X_{j}}$\n", + "\n", + "For a stable system, both\n", + "$\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}}$ and\n", + "$\\left( \\frac{\\partial Y_{j}}{\\partial X_{j}} \\right)_{X_{i}}$ are\n", + "positive based on . Using the Maxwell relation shown by , one thus\n", + "obtains\n", + "\n", + "Eq. \u2011\n", + "$\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}} - \\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{Y_{j}} = \\left( \\frac{\\partial Y_{i}}{\\partial X_{j}} \\right)_{X_{i}}\\left( \\frac{\\partial Y_{j}}{\\partial X_{i}} \\right)_{X_{j}}/\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}} \\geq 0$\n", + "\n", + "This means that\n", + "$\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{Y_{j}}$ will go\n", + "to zero before\n", + "$\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}}$ does. It\n", + "indicates that the stability condition becomes more restrictive when\n", + "potentials are kept constant in place of their conjugate molar\n", + "quantities. Based on the Gibbs-Duhem equation of , the maximum number of\n", + "independent potentials is *c+1*, and the last potential is dependent,\n", + "i.e.\n", + "\n", + "Eq. \u2011\n", + "$\\left( \\frac{\\partial Y_{c + 2}}{\\partial X_{c + 2}} \\right)_{Y_{j \\leq c + 1}} = 0$\n", + "\n", + "Therefore, the limit of stability is determined when the derivative\n", + "becomes zero with one molar quantity kept constant, e.g.\n", + "\n", + "Eq. \u2011\n", + "$\\left( \\frac{\\partial Y_{c + 1}}{\\partial X_{c + 1}} \\right)_{Y_{j < c + 1},X_{c + 2}} = 0$\n", + "\n", + "This is because this derivative reaches zero faster than any other\n", + "derivatives with more molar quantities kept constant. shows that all\n", + "molar quantities diverge at the limit of stability. The consolute point\n", + "is obtained with $c$ additional conditions as follows based on\n", + "\n", + "Eq. \u2011\n", + "$\\left( \\frac{\\partial^{2}Y_{i}}{\\partial\\left( X_{i} \\right)^{2}} \\right)_{Y_{j \\leq c + 1, \\neq i},X_{c + 2}} = 0$\n", + "\n", + "Together with , all $c + 1$ independent potentials at the consolute\n", + "point can be determined. It is evident that the consolute point is a\n", + "zero-dimensional point in a two-dimensional space of independent\n", + "potentials in a one-component system. With the addition of a second\n", + "component to form a binary system, this consolute point in the\n", + "one-component system extends into a one-dimensional line. This line\n", + "represents the limit of stability of the binary system, and a consolute\n", + "point is located at the end of this line. It is thus evident that in a\n", + "system with $c$ independent components, the limit of stability is a\n", + "*c-1*-dimensional hypersurface in a space of $c + 1$ independent\n", + "potentials, while the consolute point is a zero-dimensional point in all\n", + "systems, which may be called the invariant critical point.\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/gibbs_energy_function/phases_with_fixed_compositions.ipynb b/src/psu410/src/gibbs_energy_function/phases_with_fixed_compositions.ipynb index b3c1ab7..cf032aa 100644 --- a/src/psu410/src/gibbs_energy_function/phases_with_fixed_compositions.ipynb +++ b/src/psu410/src/gibbs_energy_function/phases_with_fixed_compositions.ipynb @@ -4,9 +4,275 @@ "metadata": {}, "cells": [ { - "id": "cdda5cf8", + "id": "011f5401", "cell_type": "markdown", - "source": "## Phases with fixed compositions\n\nThe homogeneous system discussed so far means that there is only one\nphase in the system, i.e. a single-phase system. A phase with a fixed\ncomposition can be a pure element or a stoichiometric compound. There is\nthus only one independent component in the system. A stoichiometric\ncompound contains more than one element, but the relative amounts of\neach element are fixed by the stoichiometry and cannot vary\nindependently, i.e., $dN_{i} = x_{i}dN$. The combined law of\nthermodynamics becomes\n\nEq. \u2011\n$dG = - SdT - Vd( - P) + \\left( \\sum_{}^{}{x_{i}\\mu_{i}} \\right)dN - Dd\\xi = - SdT - Vd( - P) + G_{m}dN - Dd\\xi$\n\n$G_{m}$ is the molar Gibbs energy of the stoichiometric compound and can\nbe regarded as the chemical potential of the stoichiometric\nphase,$\\ \\alpha$,\n\nEq. \u2011 $G_{m} = \\mu^{\\alpha} = \\sum_{}^{}{x_{i}\\mu_{i}}$\n\nThe chemical potential of individual components in the phase cannot be\ndefined because the amount of each component cannot be varied\nindependently. For a stoichiometric phase of $N$ moles of atoms at\nequilibrium with $dG$=$Nd\\mu^{\\alpha} + \\mu^{\\alpha}dN$, reduces to\n\nEq. \u2011 $\\ 0 = - SdT - Vd( - P) - Nd\\mu^{\\alpha}$\n\nwhich is the Gibbs-Duhem equation, , applied to a stoichiometric phase.\nIt can be represented graphically by a surface in a three-dimensional\nspace composed of $\\ \\mu^{\\alpha}$, *T* and *\u2013P*. The direction of the\nsurface is represented by the three partial directives between any two\nof $\\ \\mu^{\\alpha}$, *T* and *\u2013P* with the third one kept constant, i.e.\n\nEq. \u2011\n$\\ \\left( \\frac{\\partial\\mu^{\\alpha}}{\\partial T} \\right)_{P} = - \\frac{S}{N} = - S_{m}$\n\nEq. \u2011\n$\\ \\left( \\frac{\\partial\\mu^{\\alpha}}{\\partial( - P)} \\right)_{T} = - \\frac{V}{N} = - V_{m}$\n\nEq. \u2011\n$\\ \\left( \\frac{\\partial( - P)}{\\partial T} \\right)_{\\mu^{\\alpha}} = - \\frac{S}{V} = - \\frac{S_{m}}{V_{m}}$\n\nBased on Nernst\u2019s heat theorem, the entropy difference between two\ncrystals approaches zero when the temperature approaches absolute zero.\nIt is thus a common practice to put $S = 0$ for a crystal at 0 K. This\nis usually referred as the third law of thermodynamics. From the\ndefinition of entropy change by , $S$ or $S_{m}$ is always positive at\nfinite temperatures as the system or the crystal absorbs heat from the\nsurroundings to increase its temperature. $V$ or $V_{m}$ of a phase is a\nwell-defined physical quantity, and its absolute value can be given and\nis always positive. The above three equations can be written in a\ngeneral form as\n\nEq. \u2011\n$\\ \\left( \\frac{\\partial Y_{i}}{\\partial Y_{j}} \\right)_{Y_{k}} = - \\frac{X_{j}}{X_{i}} < 0$\n\nThe surface thus has negative slopes in all its directions. The\ncurvature of the surface can be derived from\n\nEq. \u2011\n$\\ \\left( \\frac{\\partial^{2}Y_{i}}{\\partial\\left( Y_{j} \\right)^{2}} \\right)_{Y_{k}} = - \\left( \\frac{\\partial\\left( \\frac{X_{j}}{X_{i}} \\right)}{\\partial Y_{j}} \\right)_{Y_{k}} = - \\frac{1}{X_{i}}\\left( \\frac{\\partial X_{j}}{\\partial Y_{j}} \\right)_{Y_{k}} + \\frac{X_{j}}{\\left( X_{i} \\right)^{2}}\\left( \\frac{\\partial X_{i}}{\\partial Y_{j}} \\right)_{Y_{k}} = - \\frac{1}{X_{i}}\\left\\lbrack \\left( \\frac{\\partial X_{j}}{\\partial Y_{j}} \\right)_{Y_{k}} - \\frac{X_{j}}{X_{i}}{\\left( \\frac{\\partial X_{i}}{\\partial Y_{i}} \\right)\\left( \\frac{\\partial Y_{i}}{\\partial Y_{j}} \\right)}_{Y_{k}} \\right\\rbrack = - \\frac{1}{X_{i}}\\left\\lbrack \\left( \\frac{\\partial X_{j}}{\\partial Y_{j}} \\right)_{Y_{k}} + \\left( \\frac{X_{j}}{X_{i}} \\right)^{2}\\left( \\frac{\\partial X_{i}}{\\partial Y_{i}} \\right)_{Y_{k}} \\right\\rbrack < 0$\n\nBoth terms inside the last bracket are positive for a system in a state\nof stable internal equilibrium, and the surface thus has a negative\ncurvature and is convex everywhere as shown in .\n\nFigure \u2011: Gibbs energy of a one-component phase as a function of\ntemperature and negative pressure, showing the convex shape\n\nFrom experimental observations, it is known that\n$S_{m}^{vapor} \\gg S_{m}^{liquid} > S_{m}^{solid}$. The curves of\n$G_{m}$ or $\\mu^{\\alpha}$ plotted with respect to $T$ at constant $P$\nwould thus have the most negative slope for a vapour phase followed by\nits liquid and solid phases. As an example, shows Gibbs energy of Zn in\nits solid, liquid, and vapour forms as a function of $T$ at constant\n$P = 1$ atmospheric pressure.\n\nFigure \u2011: Molar Gibbs energy of Zn as a function of T at constant P\n\nSimilarly it is common that\n$V_{m}^{vapor} \\gg V_{m}^{liquid} > V_{m}^{solid}$, and the curves of\n$G_{m}$ or $\\mu^{\\alpha}$ plotted with respect to $P$ at constant $T$\nwould thus have the most positive slope for a vapour phase followed by\nits liquid and solid phases, though there are cases that\n$V_{m}^{liquid} < V_{m}^{solid}$ such as those of water and ice. As an\nexample, shows Gibbs energy of Fe in its two solid (fcc and bcc),\nliquid, and vapour forms as a function of $P$ at constant $T = 1273K$.\n\nFigure \u2011: Molar Gibbs energy of Zn as a function of P at constant T\n\nThe quantities measurable by experiments typically include temperature,\npressure, volume, composition, and amount of heat flow in the combined\nlaw of thermodynamics discussed so far. By measuring the heat needed to\nincrease the temperature of a phase, the heat capacity of the phase is\nobtained as shown by Eq. 2\u20117. A typical heat capacity curve as a\nfunction of temperature is shown in for fcc-Al, hcp-Mg, and an\nintermetallic phase Al12Mg17.\n\nFigure \u2011: Heat capacity of fcc-Al, hcp-Mg, and\nAl12Mg17 as a function of temperature\n\nThere are various theoretical models for the heat capacity under\nconstant volume to be discussed in Chapter 5 of this book, which is\ndefined as\n\nEq. \u2011\n$C_{V} = \\left( \\frac{\\partial U}{\\partial T} \\right)_{V} = T\\left( \\frac{\\partial(F + TS)}{\\partial T} \\right)_{V} = T\\left( \\frac{\\partial S}{\\partial T} \\right)_{V} = - T\\left( \\frac{\\partial^{2}F}{\\partial T^{2}} \\right)_{V}$\n\nTo establish the relationship between $C_{P}$ defined by and $C_{V}$,\n$U$ needs to be represented as a function of $T$ and $V$ in terms of $G$\nand its derivatives with respect to Gibbs energy\u2019s natural variables of\n$T$ and $P$. It can be done as follows\n\nEq. \u2011\n$dV = \\frac{\\partial V}{\\partial T}dT + \\frac{\\partial V}{\\partial( - P)}d( - P) = - \\frac{\\partial^{2}G}{\\partial T( - P)}dT - \\frac{\\partial^{2}G}{\\partial( - P)^{2}}d( - P)$\n\nEq. \u2011\n$dU = \\frac{\\partial(G + TS - PV)}{\\partial T}dT + \\frac{\\partial(G + TS - PV)}{\\partial( - P)}d( - P) = - \\left( T\\frac{\\partial^{2}G}{\\partial T^{2}} - P\\frac{\\partial^{2}G}{\\partial T( - P)} \\right)dT\u2014\\left( T\\frac{\\partial^{2}G}{\\partial T( - P)} + P\\frac{\\partial^{2}G}{\\partial( - P)^{2}} \\right)\\left( - \\frac{1}{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}}dV + \\frac{\\frac{\\partial^{2}G}{\\partial T( - P)}}{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}}dT \\right) = - \\left\\lbrack T\\frac{\\partial^{2}G}{\\partial T^{2}} - T\\frac{\\left( \\frac{\\partial^{2}G}{\\partial T( - P)} \\right)^{2}}{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}} \\right\\rbrack dT + \\left( - T\\frac{\\frac{\\partial^{2}G}{\\partial T( - P)}}{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}} + P \\right)dV$\n\nEq. \u2011\n$C_{V} = C_{P} + T\\frac{\\left( \\frac{\\partial^{2}G}{\\partial T( - P)} \\right)^{2}}{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}} = C_{P} - \\frac{\\alpha_{V}^{2}VT}{\\kappa_{T}} = C_{P} - \\alpha_{V}^{2}BVT$\n\nwhere the thermal expansion, $\\alpha_{V}$, and the compressibility or\nbulk modulus, $\\kappa_{T}$ or $B$, are defined by and , respectively.\nFrom the heat capacity, the enthalpy and entropy can be obtained by\nintegration of at a constant pressure\n\nEq. \u2011\n$S = S_{0} + \\int_{0}^{T}\\frac{C_{P}}{T}dT = S_{0} + \\int_{0}^{298.15}\\frac{C_{P}}{T}dT + \\int_{298.15}^{T}\\frac{C_{P}}{T}dT = S_{298.15} + \\int_{298.15}^{T}\\frac{C_{P}}{T}dT$\n\nEq. \u2011\n$H = H_{0} + \\int_{0}^{T}C_{P}dT = H_{0} + \\int_{0}^{298.15}C_{P}dT + \\int_{298.15}^{T}C_{P}dT = H_{298.15} + \\int_{298.15}^{T}C_{P}dT$\n\nIn the above equations, two temperature ranges of integration are chosen\nfor practical applications as most processing procedures in the field of\nmaterials science and engineering take place at temperatures above the\nroom temperature. Based on the third-law of thermodynamics, $S_{0} = 0$,\n$S_{298.15}$ can be obtained by integration. On the other hand for\n$H_{0} = U_{0} + PV$, one does not know the absolute value of the\ninternal energy and thus have to select a reference state for $H$. In\nprinciple, the reference state can be arbitrarily chosen. A widely used\nreference state in the thermodynamic modeling practice is to set\n$H_{298.15}^{SER} = 0$ at ambient pressure for pure elements at their\nrespective stable structures at room temperature, called stable element\nreference (SER) state with\n\nEq. \u2011\n$G_{298.15}^{SER} = H_{298.15}^{SER} - TS_{298.15}^{SER} = - TS_{298.15}^{SER}$\n\nIt is further noted that after defining $S_{298.15}$ and $H_{298.15}$,\none only needs the heat capacity at higher temperatures. This makes the\nmathematical representation of heat capacity simpler due to a relatively\nsimple temperature dependence of heat capacity at higher temperatures in\ncomparison with the variation at lower temperatures. One common\nexpression for heat capacity at high temperatures and ambient pressure\nis as follows\n\nEq. \u2011 $C_{P} = c + dT + \\frac{e}{T^{2}} + fT^{2}$\n\nwhere c, d, e, and f are parameters fitted to experimental or theoretic\ndata and compiled in various handbooks.\n\nThe corresponding $S$, $H$, and $G$ are obtained as\n\nEq. \u2011 $S = b^{'} + clnT + dT - \\frac{e}{{2T}^{2}} + \\frac{f}{2}T^{2}$\n\nEq. \u2011 $H = a + cT + \\frac{d}{2}T^{2} - \\frac{e}{T} + \\frac{f}{3}T^{3}$\n\nEq. \u2011\n$G = H - TS = a - bT - cTlnT - \\frac{d}{2}T^{2} - \\frac{e}{2T} - \\frac{f}{6}T^{3}$\n\nwith $b = b^{'} - c$. The integration constants $b^{'}$ and $a$ are\nevaluated from $S_{298.15}$ and $H_{298.15}$. As an example, the\nenthalpy and entropy of Zn in solid (hcp), liquid, and gas forms are\nplotted in and , respectively. The distances between any two curves in\nand represent the enthalpy or entropy differences between the two\nphases. It can be seen that the gas has much higher enthalpy and entropy\nthan the solid and liquid.\n\nFigure \u2011: Enthalpy of Zn as a function of temperature at one atmospheric\npressure\n\nFigure \u2011: Entropy of Zn as a function of temperature at one atmospheric\npressure\n\nSimilarly, one can add the pressure dependence into the Gibbs energy\nfunction such as\n\nEq. \u2011\n$G = a - bT - cTlnT - \\frac{d}{2}T^{2} - \\frac{e}{2T} - \\frac{f}{6}T^{3} + gP + hTP + mP^{2}$\n\nwhere g, h, and m are parameters fitted to experimental or theoretic\ndata and compiled in various handbooks.\n\nThe expression for $V$ can be derived as\n\nEq. \u2011 $V = g + hT + 2mP$\n\nThe Helmholtz energy can be expressed as a function of its natural\nvariables by solving $P\\ $ from\n\nEq. \u2011\n$F = G - PV = a - bT - cTlnT - \\frac{d}{2}T^{2} - \\frac{e}{2T} - \\frac{f}{6}T^{3} - \\frac{(g + hT - V)^{2}}{4m}$\n\nIn the literature there are many models to represent the temperature and\npressure dependences of thermodynamic properties. The Gibbs energy\ndifference between a stoichiometric compound and the components at their\nreference states of which the compound is composed,\n${_{\\ }^{0}G}_{i}^{ref}$, is termed as Gibbs energy of formation, i.e.\n\nEq. \u2011 $\\mathrm{\\Delta}_{f}G = G - \\sum_{}^{}N_{i}{_{\\ }^{0}G}_{i}^{ref}$\n\nwith $N_{i}$ being the stoichiometry of the compound. Similarly,\nenthalpy of formation, entropy of formation, and heat capacity of\nformation with respect to components at their reference states,\n$_{\\ }^{0}H_{i}^{ref}$, $_{\\ }^{0}S_{i}^{ref}$, and\n$_{\\ }^{0}{C_{P}}_{i}^{ref}$, can be defined as\n\nEq. \u2011 $\\mathrm{\\Delta}_{f}H = H - \\sum_{}^{}{N_{i}_{\\ }^{0}H_{i}^{ref}}$\n\nEq. \u2011 $\\mathrm{\\Delta}_{f}S = S - \\sum_{}^{}{N_{i}_{\\ }^{0}S_{i}^{ref}}$\n\nEq. \u2011\n$\\mathrm{\\Delta}_{f}C_{P} = C_{P} - \\sum_{}^{}{N_{i}_{\\ }^{0}{C_{P}}_{i}^{ref}}$\n\nIt should be mentioned that one mole of a compound usually refers to one\nmole of formula of stoichiometry of the compound. With a formula like\n$A_{a}B_{b}C_{c}$, the compound is composed of total $(a + b + c)$ moles\nof components. One should thus be very careful when dealing with\nnumerical values to be sure whether the data is in terms of per mole of\nformula or per mole of components. At the same time the reference states\nmust be clearly defined. When the SER state defined in is selected as\nthe reference state, the above formation quantities are called standard\nformation quantities such as standard enthalpy of formation.\n\nSince there are only two independent potentials in a one-component\nsystem, its limit of stability can be evaluated with one potential kept\nconstant, i.e. either $T$ or $P$. Consequently, either Helmholtz energy\nor enthalpy is to be used in deriving the limit of stability of a\nhomogeneous system. For the practical usefulness, let us use Helmholtz\nenergy because its natural variables of $T$ and *V* are measurable\nquantities in typical experiments, while one of the natural variables of\nenthalpy, entropy, is not. From and , the limit of stability for a\none-component system at constant temperature can be written as\n\nEq. \u2011\n$\\left( \\frac{\\partial( - P)}{\\partial V} \\right)_{T,N} = F_{VV} = \\frac{1}{V\\kappa_{T}} = \\frac{B}{V} = 0$\n\nwhere the isothermal compressibility and bulk modulus, $\\kappa_{T}$ and\n$B$, are defined in . The limit of stability is thus determined when the\nisothermal compressibility diverges or the bulk modulus becomes zero\nbecause $V$ has finite values at any temperatures. It is evident that\nHelmholtz energy must have higher order dependence on volume than in for\na system with instability because $F_{VV}$ from is constant.\n\nFrom , the consolute point is defined by\n\nEq. \u2011\n$F_{VVV} = \\left( \\frac{\\partial^{2}( - P)}{\\partial V^{2}} \\right)_{T,N} = \\frac{\\partial\\left( \\frac{1}{V\\kappa_{T}} \\right)}{\\partial V} = - \\frac{1 + \\frac{V}{\\kappa_{T}}\\frac{\\partial\\kappa_{T}}{\\partial V}}{\\kappa_{T}V^{2}} = 0$\n\nSince $\\kappa_{T}$ becomes infinite at the limit of stability,\n$\\frac{\\partial\\kappa_{T}}{\\partial V}$ approaches negative infinite\nwhen the critical/consolute point is approached so that\n$\\frac{V}{\\kappa_{T}}\\frac{\\partial\\kappa_{T}}{\\partial V} = - 1$ and\n$F_{VVV} = 0$.\n", + "source": [ + "## Phases with fixed compositions\n", + "\n", + "The homogeneous system discussed so far means that there is only one\n", + "phase in the system, i.e. a single-phase system. A phase with a fixed\n", + "composition can be a pure element or a stoichiometric compound. There is\n", + "thus only one independent component in the system. A stoichiometric\n", + "compound contains more than one element, but the relative amounts of\n", + "each element are fixed by the stoichiometry and cannot vary\n", + "independently, i.e., $dN_{i} = x_{i}dN$. The combined law of\n", + "thermodynamics becomes\n", + "\n", + "Eq. \u2011\n", + "$dG = - SdT - Vd( - P) + \\left( \\sum_{}^{}{x_{i}\\mu_{i}} \\right)dN - Dd\\xi = - SdT - Vd( - P) + G_{m}dN - Dd\\xi$\n", + "\n", + "$G_{m}$ is the molar Gibbs energy of the stoichiometric compound and can\n", + "be regarded as the chemical potential of the stoichiometric\n", + "phase,$\\ \\alpha$,\n", + "\n", + "Eq. \u2011 $G_{m} = \\mu^{\\alpha} = \\sum_{}^{}{x_{i}\\mu_{i}}$\n", + "\n", + "The chemical potential of individual components in the phase cannot be\n", + "defined because the amount of each component cannot be varied\n", + "independently. For a stoichiometric phase of $N$ moles of atoms at\n", + "equilibrium with $dG$=$Nd\\mu^{\\alpha} + \\mu^{\\alpha}dN$, reduces to\n", + "\n", + "Eq. \u2011 $\\ 0 = - SdT - Vd( - P) - Nd\\mu^{\\alpha}$\n", + "\n", + "which is the Gibbs-Duhem equation, , applied to a stoichiometric phase.\n", + "It can be represented graphically by a surface in a three-dimensional\n", + "space composed of $\\ \\mu^{\\alpha}$, *T* and *\u2013P*. The direction of the\n", + "surface is represented by the three partial directives between any two\n", + "of $\\ \\mu^{\\alpha}$, *T* and *\u2013P* with the third one kept constant, i.e.\n", + "\n", + "Eq. \u2011\n", + "$\\ \\left( \\frac{\\partial\\mu^{\\alpha}}{\\partial T} \\right)_{P} = - \\frac{S}{N} = - S_{m}$\n", + "\n", + "Eq. \u2011\n", + "$\\ \\left( \\frac{\\partial\\mu^{\\alpha}}{\\partial( - P)} \\right)_{T} = - \\frac{V}{N} = - V_{m}$\n", + "\n", + "Eq. \u2011\n", + "$\\ \\left( \\frac{\\partial( - P)}{\\partial T} \\right)_{\\mu^{\\alpha}} = - \\frac{S}{V} = - \\frac{S_{m}}{V_{m}}$\n", + "\n", + "Based on Nernst\u2019s heat theorem, the entropy difference between two\n", + "crystals approaches zero when the temperature approaches absolute zero.\n", + "It is thus a common practice to put $S = 0$ for a crystal at 0 K. This\n", + "is usually referred as the third law of thermodynamics. From the\n", + "definition of entropy change by , $S$ or $S_{m}$ is always positive at\n", + "finite temperatures as the system or the crystal absorbs heat from the\n", + "surroundings to increase its temperature. $V$ or $V_{m}$ of a phase is a\n", + "well-defined physical quantity, and its absolute value can be given and\n", + "is always positive. The above three equations can be written in a\n", + "general form as\n", + "\n", + "Eq. \u2011\n", + "$\\ \\left( \\frac{\\partial Y_{i}}{\\partial Y_{j}} \\right)_{Y_{k}} = - \\frac{X_{j}}{X_{i}} < 0$\n", + "\n", + "The surface thus has negative slopes in all its directions. The\n", + "curvature of the surface can be derived from\n", + "\n", + "Eq. \u2011\n", + "$\\ \\left( \\frac{\\partial^{2}Y_{i}}{\\partial\\left( Y_{j} \\right)^{2}} \\right)_{Y_{k}} = - \\left( \\frac{\\partial\\left( \\frac{X_{j}}{X_{i}} \\right)}{\\partial Y_{j}} \\right)_{Y_{k}} = - \\frac{1}{X_{i}}\\left( \\frac{\\partial X_{j}}{\\partial Y_{j}} \\right)_{Y_{k}} + \\frac{X_{j}}{\\left( X_{i} \\right)^{2}}\\left( \\frac{\\partial X_{i}}{\\partial Y_{j}} \\right)_{Y_{k}} = - \\frac{1}{X_{i}}\\left\\lbrack \\left( \\frac{\\partial X_{j}}{\\partial Y_{j}} \\right)_{Y_{k}} - \\frac{X_{j}}{X_{i}}{\\left( \\frac{\\partial X_{i}}{\\partial Y_{i}} \\right)\\left( \\frac{\\partial Y_{i}}{\\partial Y_{j}} \\right)}_{Y_{k}} \\right\\rbrack = - \\frac{1}{X_{i}}\\left\\lbrack \\left( \\frac{\\partial X_{j}}{\\partial Y_{j}} \\right)_{Y_{k}} + \\left( \\frac{X_{j}}{X_{i}} \\right)^{2}\\left( \\frac{\\partial X_{i}}{\\partial Y_{i}} \\right)_{Y_{k}} \\right\\rbrack < 0$\n", + "\n", + "Both terms inside the last bracket are positive for a system in a state\n", + "of stable internal equilibrium, and the surface thus has a negative\n", + "curvature and is convex everywhere as shown in .\n", + "\n", + "Figure \u2011: Gibbs energy of a one-component phase as a function of\n", + "temperature and negative pressure, showing the convex shape\n", + "\n", + "From experimental observations, it is known that\n", + "$S_{m}^{vapor} \\gg S_{m}^{liquid} > S_{m}^{solid}$. The curves of\n", + "$G_{m}$ or $\\mu^{\\alpha}$ plotted with respect to $T$ at constant $P$\n", + "would thus have the most negative slope for a vapour phase followed by\n", + "its liquid and solid phases. As an example, shows Gibbs energy of Zn in\n", + "its solid, liquid, and vapour forms as a function of $T$ at constant\n", + "$P = 1$ atmospheric pressure.\n", + "\n", + "Figure \u2011: Molar Gibbs energy of Zn as a function of T at constant P\n", + "\n", + "Similarly it is common that\n", + "$V_{m}^{vapor} \\gg V_{m}^{liquid} > V_{m}^{solid}$, and the curves of\n", + "$G_{m}$ or $\\mu^{\\alpha}$ plotted with respect to $P$ at constant $T$\n", + "would thus have the most positive slope for a vapour phase followed by\n", + "its liquid and solid phases, though there are cases that\n", + "$V_{m}^{liquid} < V_{m}^{solid}$ such as those of water and ice. As an\n", + "example, shows Gibbs energy of Fe in its two solid (fcc and bcc),\n", + "liquid, and vapour forms as a function of $P$ at constant $T = 1273K$.\n", + "\n", + "Figure \u2011: Molar Gibbs energy of Zn as a function of P at constant T\n", + "\n", + "The quantities measurable by experiments typically include temperature,\n", + "pressure, volume, composition, and amount of heat flow in the combined\n", + "law of thermodynamics discussed so far. By measuring the heat needed to\n", + "increase the temperature of a phase, the heat capacity of the phase is\n", + "obtained as shown by Eq. 2\u20117. A typical heat capacity curve as a\n", + "function of temperature is shown in for fcc-Al, hcp-Mg, and an\n", + "intermetallic phase Al12Mg17.\n", + "\n", + "Figure \u2011: Heat capacity of fcc-Al, hcp-Mg, and\n", + "Al12Mg17 as a function of temperature\n", + "\n", + "There are various theoretical models for the heat capacity under\n", + "constant volume to be discussed in Chapter 5 of this book, which is\n", + "defined as\n", + "\n", + "Eq. \u2011\n", + "$C_{V} = \\left( \\frac{\\partial U}{\\partial T} \\right)_{V} = T\\left( \\frac{\\partial(F + TS)}{\\partial T} \\right)_{V} = T\\left( \\frac{\\partial S}{\\partial T} \\right)_{V} = - T\\left( \\frac{\\partial^{2}F}{\\partial T^{2}} \\right)_{V}$\n", + "\n", + "To establish the relationship between $C_{P}$ defined by and $C_{V}$,\n", + "$U$ needs to be represented as a function of $T$ and $V$ in terms of $G$\n", + "and its derivatives with respect to Gibbs energy\u2019s natural variables of\n", + "$T$ and $P$. It can be done as follows\n", + "\n", + "Eq. \u2011\n", + "$dV = \\frac{\\partial V}{\\partial T}dT + \\frac{\\partial V}{\\partial( - P)}d( - P) = - \\frac{\\partial^{2}G}{\\partial T( - P)}dT - \\frac{\\partial^{2}G}{\\partial( - P)^{2}}d( - P)$\n", + "\n", + "Eq. \u2011\n", + "$dU = \\frac{\\partial(G + TS - PV)}{\\partial T}dT + \\frac{\\partial(G + TS - PV)}{\\partial( - P)}d( - P) = - \\left( T\\frac{\\partial^{2}G}{\\partial T^{2}} - P\\frac{\\partial^{2}G}{\\partial T( - P)} \\right)dT\u2014\\left( T\\frac{\\partial^{2}G}{\\partial T( - P)} + P\\frac{\\partial^{2}G}{\\partial( - P)^{2}} \\right)\\left( - \\frac{1}{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}}dV + \\frac{\\frac{\\partial^{2}G}{\\partial T( - P)}}{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}}dT \\right) = - \\left\\lbrack T\\frac{\\partial^{2}G}{\\partial T^{2}} - T\\frac{\\left( \\frac{\\partial^{2}G}{\\partial T( - P)} \\right)^{2}}{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}} \\right\\rbrack dT + \\left( - T\\frac{\\frac{\\partial^{2}G}{\\partial T( - P)}}{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}} + P \\right)dV$\n", + "\n", + "Eq. \u2011\n", + "$C_{V} = C_{P} + T\\frac{\\left( \\frac{\\partial^{2}G}{\\partial T( - P)} \\right)^{2}}{\\frac{\\partial^{2}G}{\\partial( - P)^{2}}} = C_{P} - \\frac{\\alpha_{V}^{2}VT}{\\kappa_{T}} = C_{P} - \\alpha_{V}^{2}BVT$\n", + "\n", + "where the thermal expansion, $\\alpha_{V}$, and the compressibility or\n", + "bulk modulus, $\\kappa_{T}$ or $B$, are defined by and , respectively.\n", + "From the heat capacity, the enthalpy and entropy can be obtained by\n", + "integration of at a constant pressure\n", + "\n", + "Eq. \u2011\n", + "$S = S_{0} + \\int_{0}^{T}\\frac{C_{P}}{T}dT = S_{0} + \\int_{0}^{298.15}\\frac{C_{P}}{T}dT + \\int_{298.15}^{T}\\frac{C_{P}}{T}dT = S_{298.15} + \\int_{298.15}^{T}\\frac{C_{P}}{T}dT$\n", + "\n", + "Eq. \u2011\n", + "$H = H_{0} + \\int_{0}^{T}C_{P}dT = H_{0} + \\int_{0}^{298.15}C_{P}dT + \\int_{298.15}^{T}C_{P}dT = H_{298.15} + \\int_{298.15}^{T}C_{P}dT$\n", + "\n", + "In the above equations, two temperature ranges of integration are chosen\n", + "for practical applications as most processing procedures in the field of\n", + "materials science and engineering take place at temperatures above the\n", + "room temperature. Based on the third-law of thermodynamics, $S_{0} = 0$,\n", + "$S_{298.15}$ can be obtained by integration. On the other hand for\n", + "$H_{0} = U_{0} + PV$, one does not know the absolute value of the\n", + "internal energy and thus have to select a reference state for $H$. In\n", + "principle, the reference state can be arbitrarily chosen. A widely used\n", + "reference state in the thermodynamic modeling practice is to set\n", + "$H_{298.15}^{SER} = 0$ at ambient pressure for pure elements at their\n", + "respective stable structures at room temperature, called stable element\n", + "reference (SER) state with\n", + "\n", + "Eq. \u2011\n", + "$G_{298.15}^{SER} = H_{298.15}^{SER} - TS_{298.15}^{SER} = - TS_{298.15}^{SER}$\n", + "\n", + "It is further noted that after defining $S_{298.15}$ and $H_{298.15}$,\n", + "one only needs the heat capacity at higher temperatures. This makes the\n", + "mathematical representation of heat capacity simpler due to a relatively\n", + "simple temperature dependence of heat capacity at higher temperatures in\n", + "comparison with the variation at lower temperatures. One common\n", + "expression for heat capacity at high temperatures and ambient pressure\n", + "is as follows\n", + "\n", + "Eq. \u2011 $C_{P} = c + dT + \\frac{e}{T^{2}} + fT^{2}$\n", + "\n", + "where c, d, e, and f are parameters fitted to experimental or theoretic\n", + "data and compiled in various handbooks.\n", + "\n", + "The corresponding $S$, $H$, and $G$ are obtained as\n", + "\n", + "Eq. \u2011 $S = b^{'} + clnT + dT - \\frac{e}{{2T}^{2}} + \\frac{f}{2}T^{2}$\n", + "\n", + "Eq. \u2011 $H = a + cT + \\frac{d}{2}T^{2} - \\frac{e}{T} + \\frac{f}{3}T^{3}$\n", + "\n", + "Eq. \u2011\n", + "$G = H - TS = a - bT - cTlnT - \\frac{d}{2}T^{2} - \\frac{e}{2T} - \\frac{f}{6}T^{3}$\n", + "\n", + "with $b = b^{'} - c$. The integration constants $b^{'}$ and $a$ are\n", + "evaluated from $S_{298.15}$ and $H_{298.15}$. As an example, the\n", + "enthalpy and entropy of Zn in solid (hcp), liquid, and gas forms are\n", + "plotted in and , respectively. The distances between any two curves in\n", + "and represent the enthalpy or entropy differences between the two\n", + "phases. It can be seen that the gas has much higher enthalpy and entropy\n", + "than the solid and liquid.\n", + "\n", + "Figure \u2011: Enthalpy of Zn as a function of temperature at one atmospheric\n", + "pressure\n", + "\n", + "Figure \u2011: Entropy of Zn as a function of temperature at one atmospheric\n", + "pressure\n", + "\n", + "Similarly, one can add the pressure dependence into the Gibbs energy\n", + "function such as\n", + "\n", + "Eq. \u2011\n", + "$G = a - bT - cTlnT - \\frac{d}{2}T^{2} - \\frac{e}{2T} - \\frac{f}{6}T^{3} + gP + hTP + mP^{2}$\n", + "\n", + "where g, h, and m are parameters fitted to experimental or theoretic\n", + "data and compiled in various handbooks.\n", + "\n", + "The expression for $V$ can be derived as\n", + "\n", + "Eq. \u2011 $V = g + hT + 2mP$\n", + "\n", + "The Helmholtz energy can be expressed as a function of its natural\n", + "variables by solving $P\\ $ from\n", + "\n", + "Eq. \u2011\n", + "$F = G - PV = a - bT - cTlnT - \\frac{d}{2}T^{2} - \\frac{e}{2T} - \\frac{f}{6}T^{3} - \\frac{(g + hT - V)^{2}}{4m}$\n", + "\n", + "In the literature there are many models to represent the temperature and\n", + "pressure dependences of thermodynamic properties. The Gibbs energy\n", + "difference between a stoichiometric compound and the components at their\n", + "reference states of which the compound is composed,\n", + "${_{\\ }^{0}G}_{i}^{ref}$, is termed as Gibbs energy of formation, i.e.\n", + "\n", + "Eq. \u2011 $\\mathrm{\\Delta}_{f}G = G - \\sum_{}^{}N_{i}{_{\\ }^{0}G}_{i}^{ref}$\n", + "\n", + "with $N_{i}$ being the stoichiometry of the compound. Similarly,\n", + "enthalpy of formation, entropy of formation, and heat capacity of\n", + "formation with respect to components at their reference states,\n", + "$_{\\ }^{0}H_{i}^{ref}$, $_{\\ }^{0}S_{i}^{ref}$, and\n", + "$_{\\ }^{0}{C_{P}}_{i}^{ref}$, can be defined as\n", + "\n", + "Eq. \u2011 $\\mathrm{\\Delta}_{f}H = H - \\sum_{}^{}{N_{i}_{\\ }^{0}H_{i}^{ref}}$\n", + "\n", + "Eq. \u2011 $\\mathrm{\\Delta}_{f}S = S - \\sum_{}^{}{N_{i}_{\\ }^{0}S_{i}^{ref}}$\n", + "\n", + "Eq. \u2011\n", + "$\\mathrm{\\Delta}_{f}C_{P} = C_{P} - \\sum_{}^{}{N_{i}_{\\ }^{0}{C_{P}}_{i}^{ref}}$\n", + "\n", + "It should be mentioned that one mole of a compound usually refers to one\n", + "mole of formula of stoichiometry of the compound. With a formula like\n", + "$A_{a}B_{b}C_{c}$, the compound is composed of total $(a + b + c)$ moles\n", + "of components. One should thus be very careful when dealing with\n", + "numerical values to be sure whether the data is in terms of per mole of\n", + "formula or per mole of components. At the same time the reference states\n", + "must be clearly defined. When the SER state defined in is selected as\n", + "the reference state, the above formation quantities are called standard\n", + "formation quantities such as standard enthalpy of formation.\n", + "\n", + "Since there are only two independent potentials in a one-component\n", + "system, its limit of stability can be evaluated with one potential kept\n", + "constant, i.e. either $T$ or $P$. Consequently, either Helmholtz energy\n", + "or enthalpy is to be used in deriving the limit of stability of a\n", + "homogeneous system. For the practical usefulness, let us use Helmholtz\n", + "energy because its natural variables of $T$ and *V* are measurable\n", + "quantities in typical experiments, while one of the natural variables of\n", + "enthalpy, entropy, is not. From and , the limit of stability for a\n", + "one-component system at constant temperature can be written as\n", + "\n", + "Eq. \u2011\n", + "$\\left( \\frac{\\partial( - P)}{\\partial V} \\right)_{T,N} = F_{VV} = \\frac{1}{V\\kappa_{T}} = \\frac{B}{V} = 0$\n", + "\n", + "where the isothermal compressibility and bulk modulus, $\\kappa_{T}$ and\n", + "$B$, are defined in . The limit of stability is thus determined when the\n", + "isothermal compressibility diverges or the bulk modulus becomes zero\n", + "because $V$ has finite values at any temperatures. It is evident that\n", + "Helmholtz energy must have higher order dependence on volume than in for\n", + "a system with instability because $F_{VV}$ from is constant.\n", + "\n", + "From , the consolute point is defined by\n", + "\n", + "Eq. \u2011\n", + "$F_{VVV} = \\left( \\frac{\\partial^{2}( - P)}{\\partial V^{2}} \\right)_{T,N} = \\frac{\\partial\\left( \\frac{1}{V\\kappa_{T}} \\right)}{\\partial V} = - \\frac{1 + \\frac{V}{\\kappa_{T}}\\frac{\\partial\\kappa_{T}}{\\partial V}}{\\kappa_{T}V^{2}} = 0$\n", + "\n", + "Since $\\kappa_{T}$ becomes infinite at the limit of stability,\n", + "$\\frac{\\partial\\kappa_{T}}{\\partial V}$ approaches negative infinite\n", + "when the critical/consolute point is approached so that\n", + "$\\frac{V}{\\kappa_{T}}\\frac{\\partial\\kappa_{T}}{\\partial V} = - 1$ and\n", + "$F_{VVV} = 0$.\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/gibbs_energy_function/phases_with_variable_compositions_random_solutions.ipynb b/src/psu410/src/gibbs_energy_function/phases_with_variable_compositions_random_solutions.ipynb index eabca3e..4285468 100644 --- a/src/psu410/src/gibbs_energy_function/phases_with_variable_compositions_random_solutions.ipynb +++ b/src/psu410/src/gibbs_energy_function/phases_with_variable_compositions_random_solutions.ipynb @@ -4,33 +4,540 @@ "metadata": {}, "cells": [ { - "id": "469ee6b7", + "id": "55b690ab", "cell_type": "markdown", - "source": "## Phases with variable compositions: Random solutions\n\nThe combined law of thermodynamics and the Gibbs-Duhem equation of a\nsolution phase with variable compositions are shown by and ,\nrespectively. A phase can be represented by a *c+1*-dimensional surface\nin a *c+2*-dimensional space of potentials based on the Gibbs-Duhem\nequation. The directions and curvature of the surface are represented by\nthe partial derivatives shown by and secondary derivatives shown by ,\nboth being negative for a stable phase. To develop a mathematical\nformula for Gibbs energy of a phase with variable compositions, one can\nconsider a phase as a mixture of independent components that the phase\nis made of. Its Gibbs energy function can be postulated as the summation\nof Gibbs energy of the independent components of the same structure of\nthe solution, $_{\\ }^{0}G_{i}$, plus the contribution due to the mixing,\n$_{\\ }^{mixing}G$ or $_{\\ }^{M}G$\n\nEq. \u2011 $G = \\sum_{}^{}{N_{i}_{\\ }^{0}G_{i}} +_{\\ }^{M}G$\n\nSince the system size usually is not important in thermodynamics, its\nproperties are typically normalized to one mole with its composition\nrepresented by mole fractions of components. The molar Gibbs energy is\nobtained as shown below with the molar Gibbs energy of mixing separated\ninto two parts: ideal Gibbs energy of mixing assuming no chemical\ninteraction among components, $_{\\ }^{ideal}G_{m}$ or $_{\\ }^{I}G_{m}$,\nand excess Gibbs energy of mixing due to chemical reaction among\ncomponents, $_{\\ }^{excess}G_{m}$ or $_{\\ }^{E}G_{m}$\n\nEq. \u2011\n$G_{m} = \\sum_{}^{}{x_{i}_{\\ }^{0}G_{i}} +_{\\ }^{M}G_{m} = \\sum_{}^{}{x_{i}_{\\ }^{0}G_{i}} +_{\\ }^{I}G_{m} +_{\\ }^{E}G_{m}$\n\nFrom , the chemical potential of a component is thus\n\nEq. \u2011\n$\\mu_{i} =_{\\ }^{0}G_{i} +_{\\ }^{I}{G_{m} +}_{\\ }^{E}G_{m} + \\frac{\\partial\\left(_{\\ }^{I}{G_{m} +}_{\\ }^{E}G_{m} \\right)}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial\\left(_{\\ }^{I}{G_{m} +}_{\\ }^{E}G_{m} \\right)}{\\partial x_{j}}$\n\nOne may define the chemical activity of component *i*, $a_{i}^{\\ }$, as\nfollows\n\nEq. \u2011\n$RTlna_{i}^{\\ } = \\mu_{i} -_{\\ }^{0}G_{i}^{\\ } =_{\\ }^{I}{G_{m} +}\\frac{\\partial_{\\ }^{I}G_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial_{\\ }^{I}G_{m}}{\\partial x_{j}} +_{\\ }^{E}G_{m} + \\frac{\\partial_{\\ }^{E}G_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial_{\\ }^{E}G_{m}}{\\partial x_{j}}$\n\nIn this definition, the chemical activity or simply activity is\ncalculated with respect to the pure elements in the structure of the\nsolution for practical reasons as one would like to understand the\nchemical potential difference of components in the solution and by\nitself with the same structure. It should be noted that this reference\nstate for chemical activity is usually different from the SER reference\nstate defined in as the solution may have a different structure than\nthose of pure components in their SER states. On the other hand, the\nactivity under the SER reference state can be easily obtained by\nreplacing $_{\\ }^{0}G_{i}$ with ${_{\\ }^{0}G}_{i}^{SER}$ from . In\nprinciple, one may choose any structure as the reference state for\nactivity to be useful for practical applications, i.e.\n\nEq. \u2011 $RTlna_{i}^{ref} = \\mu_{i} -_{\\ }^{0}G_{i}^{ref}$\n\nFor example, the activity of a component in a liquid solution is defined\nwith respect to the pure component in its liquid form from , but can\nalso be referred to its SER state which is solid using . The following\nsections will discuss in more details how components mix when they are\nbrought together including concepts such as random mixing, short-range\nordering, and long-range ordering.\n\nThe limit of stability of a solution with respect to composition\nfluctuation under constant *T*, *P*, and *N1* can be derived\nas follows from and\n\nEq. \u2011\n$\\left( \\frac{\\partial\\mu_{i}}{\\partial N_{i}} \\right)_{T,P,N_{j \\neq i},i > 1} > \\left( \\frac{\\partial\\mu_{i}}{\\partial N_{i}} \\right)_{T,P,N_{1},{\\mu_{2},N}_{j \\neq i},i,j > 2}\\ldots... > \\left( \\frac{\\partial\\mu_{c}}{\\partial N_{c}} \\right)_{T,P,N_{1},\\mu_{2}..\\mu_{c - 1}} = 0$\n\nThe first term can be derived from as follows\n\nEq. \u2011\n$\\left( \\frac{\\partial\\mu_{i}}{\\partial N_{i}} \\right)_{T,P,N_{j \\neq i},i > 1} = \\sum_{j = 1}^{c}\\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{i}\\partial x_{j}}\\frac{\\partial x_{j}}{\\partial N_{i}} - \\sum_{j = 1}^{c}{x_{j}\\sum_{k = 1}^{c}{\\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{j}\\partial x_{k}}\\frac{\\partial x_{k}}{\\partial N_{i}}}} = \\frac{1}{N}\\left( \\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{i}^{2}} - \\sum_{j = 1}^{c}{x_{j}\\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{j}^{2}}} - \\sum_{j = 1}^{c}{x_{j}\\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{i}\\partial x_{j}}} + \\sum_{j = 1}^{c}{\\sum_{k = 1}^{c}{{x_{j}x}_{k}\\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{j}\\partial x_{k}}}} \\right)$\n\nDenoting\n$G_{ij} = \\left( \\frac{\\partial\\mu_{i}}{\\partial N_{j}} \\right)_{T,P,N_{k \\neq j}}$and\nusing to change the variables kept constant from molar quantities to\npotentials one-by-one (see \\[1\\]), the limit of stability can be further\nobtained as\n\nEq. \u2011\n$\\left( \\frac{\\partial\\mu_{c}}{\\partial N_{c}} \\right)_{T,P,N_{1},\\mu_{2}..\\mu_{c - 1}} = \\frac{\\det\\left( G_{ij}:2 \\leq i,j \\leq c \\right)}{\\det\\left( G_{ij}:2 \\leq i,j \\leq c - 1 \\right)} = 0$\n\nwhere $\\det$ stands for determinant. indicates that\n$\\det\\left( G_{ij}:2 \\leq i,j \\leq c \\right) = 0$ at the limit of\nstability. Considering $x_{1} = 1 - \\sum_{j \\neq 1}^{}x_{j}$, let us\nintroduce\n\nEq. \u2011\n$g_{i} = \\mu_{i} - \\mu_{1} = \\left( \\frac{\\partial G_{m}}{\\partial x_{i}} \\right)_{x_{k \\neq i}} - \\left( \\frac{\\partial G_{m}}{\\partial x_{1}} \\right)_{x_{k \\neq 1}}$\n\nand\n\nEq. \u2011\n$g_{ij} = \\frac{\\partial g_{i}}{\\partial x_{j}} = \\frac{\\partial\\left( \\mu_{i} - \\mu_{1} \\right)}{\\partial x_{j}} = \\frac{\\partial^{2}G_{m}}{\\partial x_{i}\\partial x_{j}} - \\frac{\\partial^{2}G_{m}}{\\partial x_{1}\\partial x_{j}} - \\frac{\\partial^{2}G_{m}}{\\partial x_{i}\\partial x_{1}} + \\frac{\\partial^{2}G_{m}}{\\partial\\left( x_{1} \\right)^{2}}$\n\nThe limit of stability can be re-written as\n\nEq. \u2011\n$\\left( \\frac{\\partial\\left( \\mu_{c} - \\mu_{1} \\right)}{\\partial x_{c}} \\right)_{T,P,N,\\mu_{2} - \\mu_{1},\\ ...\\mu_{c - 1} - \\mu_{1}} = \\frac{\\det\\left( g_{ij}:2 \\leq i,j \\leq c \\right)}{\\det\\left( g_{ij}:2 \\leq i,j \\leq c - 1 \\right)} = 0$\n\ni.e. $\\det\\left( g_{ij}:2 \\leq i,j \\leq c \\right) = 0$. The consolute\npoint can be defined following\n\nEq. \u2011\n$\\left( \\frac{\\partial^{2}\\mu_{c}}{\\partial\\left( N_{c} \\right)^{2}} \\right)_{T,P,N_{1},\\mu_{2}..\\mu_{c - 1}} = \\left( \\frac{\\partial^{2}\\left( \\mu_{c} - \\mu_{1} \\right)}{\\partial\\left( x_{c} \\right)^{2}} \\right)_{T,P,N,\\mu_{2} - \\mu_{1},\\ ...\\mu_{c - 1} - \\mu_{1}} = 0$\n\nNo closed mathematic form has been published in the literature.\n", + "source": [ + "## Phases with variable compositions: Random solutions\n", + "\n", + "The combined law of thermodynamics and the Gibbs-Duhem equation of a\n", + "solution phase with variable compositions are shown by and ,\n", + "respectively. A phase can be represented by a *c+1*-dimensional surface\n", + "in a *c+2*-dimensional space of potentials based on the Gibbs-Duhem\n", + "equation. The directions and curvature of the surface are represented by\n", + "the partial derivatives shown by and secondary derivatives shown by ,\n", + "both being negative for a stable phase. To develop a mathematical\n", + "formula for Gibbs energy of a phase with variable compositions, one can\n", + "consider a phase as a mixture of independent components that the phase\n", + "is made of. Its Gibbs energy function can be postulated as the summation\n", + "of Gibbs energy of the independent components of the same structure of\n", + "the solution, $_{\\ }^{0}G_{i}$, plus the contribution due to the mixing,\n", + "$_{\\ }^{mixing}G$ or $_{\\ }^{M}G$\n", + "\n", + "Eq. \u2011 $G = \\sum_{}^{}{N_{i}_{\\ }^{0}G_{i}} +_{\\ }^{M}G$\n", + "\n", + "Since the system size usually is not important in thermodynamics, its\n", + "properties are typically normalized to one mole with its composition\n", + "represented by mole fractions of components. The molar Gibbs energy is\n", + "obtained as shown below with the molar Gibbs energy of mixing separated\n", + "into two parts: ideal Gibbs energy of mixing assuming no chemical\n", + "interaction among components, $_{\\ }^{ideal}G_{m}$ or $_{\\ }^{I}G_{m}$,\n", + "and excess Gibbs energy of mixing due to chemical reaction among\n", + "components, $_{\\ }^{excess}G_{m}$ or $_{\\ }^{E}G_{m}$\n", + "\n", + "Eq. \u2011\n", + "$G_{m} = \\sum_{}^{}{x_{i}_{\\ }^{0}G_{i}} +_{\\ }^{M}G_{m} = \\sum_{}^{}{x_{i}_{\\ }^{0}G_{i}} +_{\\ }^{I}G_{m} +_{\\ }^{E}G_{m}$\n", + "\n", + "From , the chemical potential of a component is thus\n", + "\n", + "Eq. \u2011\n", + "$\\mu_{i} =_{\\ }^{0}G_{i} +_{\\ }^{I}{G_{m} +}_{\\ }^{E}G_{m} + \\frac{\\partial\\left(_{\\ }^{I}{G_{m} +}_{\\ }^{E}G_{m} \\right)}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial\\left(_{\\ }^{I}{G_{m} +}_{\\ }^{E}G_{m} \\right)}{\\partial x_{j}}$\n", + "\n", + "One may define the chemical activity of component *i*, $a_{i}^{\\ }$, as\n", + "follows\n", + "\n", + "Eq. \u2011\n", + "$RTlna_{i}^{\\ } = \\mu_{i} -_{\\ }^{0}G_{i}^{\\ } =_{\\ }^{I}{G_{m} +}\\frac{\\partial_{\\ }^{I}G_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial_{\\ }^{I}G_{m}}{\\partial x_{j}} +_{\\ }^{E}G_{m} + \\frac{\\partial_{\\ }^{E}G_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial_{\\ }^{E}G_{m}}{\\partial x_{j}}$\n", + "\n", + "In this definition, the chemical activity or simply activity is\n", + "calculated with respect to the pure elements in the structure of the\n", + "solution for practical reasons as one would like to understand the\n", + "chemical potential difference of components in the solution and by\n", + "itself with the same structure. It should be noted that this reference\n", + "state for chemical activity is usually different from the SER reference\n", + "state defined in as the solution may have a different structure than\n", + "those of pure components in their SER states. On the other hand, the\n", + "activity under the SER reference state can be easily obtained by\n", + "replacing $_{\\ }^{0}G_{i}$ with ${_{\\ }^{0}G}_{i}^{SER}$ from . In\n", + "principle, one may choose any structure as the reference state for\n", + "activity to be useful for practical applications, i.e.\n", + "\n", + "Eq. \u2011 $RTlna_{i}^{ref} = \\mu_{i} -_{\\ }^{0}G_{i}^{ref}$\n", + "\n", + "For example, the activity of a component in a liquid solution is defined\n", + "with respect to the pure component in its liquid form from , but can\n", + "also be referred to its SER state which is solid using . The following\n", + "sections will discuss in more details how components mix when they are\n", + "brought together including concepts such as random mixing, short-range\n", + "ordering, and long-range ordering.\n", + "\n", + "The limit of stability of a solution with respect to composition\n", + "fluctuation under constant *T*, *P*, and *N1* can be derived\n", + "as follows from and\n", + "\n", + "Eq. \u2011\n", + "$\\left( \\frac{\\partial\\mu_{i}}{\\partial N_{i}} \\right)_{T,P,N_{j \\neq i},i > 1} > \\left( \\frac{\\partial\\mu_{i}}{\\partial N_{i}} \\right)_{T,P,N_{1},{\\mu_{2},N}_{j \\neq i},i,j > 2}\\ldots... > \\left( \\frac{\\partial\\mu_{c}}{\\partial N_{c}} \\right)_{T,P,N_{1},\\mu_{2}..\\mu_{c - 1}} = 0$\n", + "\n", + "The first term can be derived from as follows\n", + "\n", + "Eq. \u2011\n", + "$\\left( \\frac{\\partial\\mu_{i}}{\\partial N_{i}} \\right)_{T,P,N_{j \\neq i},i > 1} = \\sum_{j = 1}^{c}\\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{i}\\partial x_{j}}\\frac{\\partial x_{j}}{\\partial N_{i}} - \\sum_{j = 1}^{c}{x_{j}\\sum_{k = 1}^{c}{\\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{j}\\partial x_{k}}\\frac{\\partial x_{k}}{\\partial N_{i}}}} = \\frac{1}{N}\\left( \\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{i}^{2}} - \\sum_{j = 1}^{c}{x_{j}\\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{j}^{2}}} - \\sum_{j = 1}^{c}{x_{j}\\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{i}\\partial x_{j}}} + \\sum_{j = 1}^{c}{\\sum_{k = 1}^{c}{{x_{j}x}_{k}\\frac{\\partial_{\\ }^{2}G_{m}}{\\partial x_{j}\\partial x_{k}}}} \\right)$\n", + "\n", + "Denoting\n", + "$G_{ij} = \\left( \\frac{\\partial\\mu_{i}}{\\partial N_{j}} \\right)_{T,P,N_{k \\neq j}}$and\n", + "using to change the variables kept constant from molar quantities to\n", + "potentials one-by-one (see \\[1\\]), the limit of stability can be further\n", + "obtained as\n", + "\n", + "Eq. \u2011\n", + "$\\left( \\frac{\\partial\\mu_{c}}{\\partial N_{c}} \\right)_{T,P,N_{1},\\mu_{2}..\\mu_{c - 1}} = \\frac{\\det\\left( G_{ij}:2 \\leq i,j \\leq c \\right)}{\\det\\left( G_{ij}:2 \\leq i,j \\leq c - 1 \\right)} = 0$\n", + "\n", + "where $\\det$ stands for determinant. indicates that\n", + "$\\det\\left( G_{ij}:2 \\leq i,j \\leq c \\right) = 0$ at the limit of\n", + "stability. Considering $x_{1} = 1 - \\sum_{j \\neq 1}^{}x_{j}$, let us\n", + "introduce\n", + "\n", + "Eq. \u2011\n", + "$g_{i} = \\mu_{i} - \\mu_{1} = \\left( \\frac{\\partial G_{m}}{\\partial x_{i}} \\right)_{x_{k \\neq i}} - \\left( \\frac{\\partial G_{m}}{\\partial x_{1}} \\right)_{x_{k \\neq 1}}$\n", + "\n", + "and\n", + "\n", + "Eq. \u2011\n", + "$g_{ij} = \\frac{\\partial g_{i}}{\\partial x_{j}} = \\frac{\\partial\\left( \\mu_{i} - \\mu_{1} \\right)}{\\partial x_{j}} = \\frac{\\partial^{2}G_{m}}{\\partial x_{i}\\partial x_{j}} - \\frac{\\partial^{2}G_{m}}{\\partial x_{1}\\partial x_{j}} - \\frac{\\partial^{2}G_{m}}{\\partial x_{i}\\partial x_{1}} + \\frac{\\partial^{2}G_{m}}{\\partial\\left( x_{1} \\right)^{2}}$\n", + "\n", + "The limit of stability can be re-written as\n", + "\n", + "Eq. \u2011\n", + "$\\left( \\frac{\\partial\\left( \\mu_{c} - \\mu_{1} \\right)}{\\partial x_{c}} \\right)_{T,P,N,\\mu_{2} - \\mu_{1},\\ ...\\mu_{c - 1} - \\mu_{1}} = \\frac{\\det\\left( g_{ij}:2 \\leq i,j \\leq c \\right)}{\\det\\left( g_{ij}:2 \\leq i,j \\leq c - 1 \\right)} = 0$\n", + "\n", + "i.e. $\\det\\left( g_{ij}:2 \\leq i,j \\leq c \\right) = 0$. The consolute\n", + "point can be defined following\n", + "\n", + "Eq. \u2011\n", + "$\\left( \\frac{\\partial^{2}\\mu_{c}}{\\partial\\left( N_{c} \\right)^{2}} \\right)_{T,P,N_{1},\\mu_{2}..\\mu_{c - 1}} = \\left( \\frac{\\partial^{2}\\left( \\mu_{c} - \\mu_{1} \\right)}{\\partial\\left( x_{c} \\right)^{2}} \\right)_{T,P,N,\\mu_{2} - \\mu_{1},\\ ...\\mu_{c - 1} - \\mu_{1}} = 0$\n", + "\n", + "No closed mathematic form has been published in the literature.\n" + ], "metadata": {} }, { - "id": "051f94e5", + "id": "3e83c37f", "cell_type": "markdown", - "source": "### Random solutions\n\nThe ideal Gibbs energy of mixing represents an ideal solution in which\nall sites are equivalent and the distributions of components on the\nsites are completely random. The number of different configurations to\narrange all components is\n\nEq. \u2011 $w = \\frac{N!}{\\prod_{}^{}\\left( N_{i}! \\right)}$\n\nBased on Boltzmann\u2019s relation from statistic thermodynamics when all\nconfigurations have the same probability to be observed, the ideal\nconfigurational molar entropy of mixing due to the distribution is\n\nEq. \u2011\n$_{\\ }^{ideal}S_{m} =_{\\ }^{I}S_{m} = \\frac{Rlnw}{N} = R\\frac{lnN! - \\sum_{}^{}\\ln\\left( N_{i}! \\right)}{N} \\cong R\\frac{NlnN - \\sum_{}^{}{N_{i}l{nN}_{i}}}{N} = - R\\sum_{}^{}{x_{i}l{nx}_{i}}$\n\nwhere $R$ is the gas constant. represents the entropy difference between\nthe ideal solution and the states when individual components are by\nthemselves, i.e. the mechanical mixing of components. As $x_{i}$ is\nsmaller than unity, the entropy production to form an ideal solution\nfrom pure components is thus positive, indicating that it is a\nspontaneous process. In such an ideal solution, it is assumed that there\nare no interactions between components, and the enthalpy of mixing is\nthus zero as the internal energy and the volume of the system do not\nchange. The ideal Gibbs energy of mixing is written as\n\nEq. \u2011 $_{\\ }^{I}G = - T_{\\ }^{I}S_{m} = RT\\sum_{}^{}{x_{i}l{nx}_{i}}$\n\nThe Gibbs energy of real solutions, i.e. , becomes\n\nEq. \u2011\n$G_{m} = \\sum_{}^{}{x_{i}_{\\ }^{0}G_{i}} + RT\\sum_{}^{}{x_{i}l{nx}_{i}} +_{\\ }^{E}G_{m}$\n\nFrom , the chemical potential is obtained as\n\nEq. \u2011\n$\\mu_{i} = G_{i} =_{\\ }^{0}G_{i} + RTlnx_{i} +_{\\ }^{E}G_{m} + \\frac{\\partial_{\\ }^{E}G_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial_{\\ }^{E}G_{m}}{\\partial x_{j}}$\n\nFrom the chemical activity of , the activity coefficient, $\\gamma_{i}$,\ncan be defined as follows\n\nEq. \u2011\n$\\gamma_{i} = \\frac{a_{i}}{x_{i}} = \\frac{1}{x_{i}}\\exp\\frac{G_{i} -_{\\ }^{0}G_{i}\\ }{RT}$\n\nThe solution is an ideal solution with $\\gamma_{i} = 1$, and is said to\nbe positively or negatively deviating from an ideal solution with\n$\\gamma_{i} > 1$ or $\\gamma_{i} < 1$, respectively. The chemical\npotential is related to the activity and activity coefficient by the\nfollowing equation\n\nEq. \u2011\n$\\mu_{i} =_{\\ }^{0}G_{i} + RTlna_{i} =_{\\ }^{0}G_{i} + RTln\\gamma_{i}x_{i} =_{\\ }^{0}G_{i} + RTlnx_{i} + RTln\\gamma_{i}$\n\nLet us further exam in more details in order to better understand the\nrelation between $G_{m}$ and $\\mu_{i}$. The partial derivatives in\nrepresent the directions of molar Gibbs energy in the composition space,\ni.e. the tangents of molar Gibbs energy with respect to mole fractions\nof independent components. Collectively, they define the\nmulti-dimensional tangent plane of molar Gibbs energy at the given\ncomposition, $x_{i}^{0}$. The mathematical representation of this\ntangent plane, $z_{G_{m}}$, is defined by its directional derivatives\nand the distance from the point where the derivatives are taken,\n\nEq. \u2011\n$z_{G_{m}} = G_{m}\\left( x_{i}^{0} \\right) + \\sum_{i = 1}^{c}{\\left( \\frac{\\partial G_{m}}{\\partial x_{i}} \\right)_{x_{i}^{0}}\\left( x_{i} - x_{i}^{0} \\right)}$\n\nThe intercept of this tangent plane at each pure component axis, i.e.\n$x_{i} = 1$ and $x_{j \\neq i} = 0$, is obtained as\n\nEq. \u2011\n$z_{G_{m},x_{i} = 1} = G_{m}\\left( x_{i}^{0} \\right) + \\left( \\frac{\\partial G_{m}}{\\partial x_{i}} \\right)_{x_{i}^{\\ } = x_{i}^{0}} - \\sum_{j = 1}^{c}{x_{j}^{0}\\left( \\frac{\\partial G_{m}}{\\partial x_{j}} \\right)_{x_{i}^{\\ } = x_{i}^{0}}}$\n\nThis is identical to at the point $x_{i}^{0}$. It is thus shown that the\nchemical potential of a component in a solution is represented by the\nintercept of the tangent plane of Gibbs energy of the solution on the\n$G_{m}$ axis of the component. The distance between the intercept and\nthe Gibbs energy of the pure component in the same structure of the\nsolution is related to the chemical activity of the component as defined\nby . On the other hand, it is evident that one can choose any other\nstructure of the pure element to define the chemical activity to compare\nchemical potential of the component as shown by .\n\nThe stability of a solution is evaluated following , and the derivatives\nof chemical potential with respect to its moles, i.e. the elements in\nthe determinant, are obtained as follows from and ,\n\nEq. \u2011\n$\\frac{N}{RT}\\frac{\\partial\\mu_{i}}{\\partial N_{i}} = \\frac{N}{RT}G_{ii} = \\frac{1 - x_{i}}{x_{i}} + \\frac{1}{\\gamma_{i}}\\left( \\frac{\\partial\\gamma_{i}}{\\partial x_{i}} - \\sum_{j = 1}^{c}{x_{j}\\frac{\\partial\\gamma_{i}}{\\partial x_{j}}} \\right)$\n\nEq. \u2011\n$\\frac{N}{RT}\\frac{\\partial\\mu_{i}}{\\partial N_{k}} = \\frac{N}{RT}G_{ik} = - \\frac{x_{k}}{x_{i}} + \\frac{1}{\\gamma_{i}}\\left( 1 - \\sum_{j = 1}^{c}{x_{j}\\frac{\\partial\\gamma_{i}}{\\partial x_{j}}} \\right)$\n\nTo further study Gibbs energy of solution phases, let us discuss the\ndetails on the excess Gibbs energy of mixing. At this point, one can\nstart with lower-order systems with fewer components, i.e. two component\nand three-component systems, noting that the Gibbs energy of phases with\none component is already presented in Chapter .\n", + "source": [ + "### Random solutions\n", + "\n", + "The ideal Gibbs energy of mixing represents an ideal solution in which\n", + "all sites are equivalent and the distributions of components on the\n", + "sites are completely random. The number of different configurations to\n", + "arrange all components is\n", + "\n", + "Eq. \u2011 $w = \\frac{N!}{\\prod_{}^{}\\left( N_{i}! \\right)}$\n", + "\n", + "Based on Boltzmann\u2019s relation from statistic thermodynamics when all\n", + "configurations have the same probability to be observed, the ideal\n", + "configurational molar entropy of mixing due to the distribution is\n", + "\n", + "Eq. \u2011\n", + "$_{\\ }^{ideal}S_{m} =_{\\ }^{I}S_{m} = \\frac{Rlnw}{N} = R\\frac{lnN! - \\sum_{}^{}\\ln\\left( N_{i}! \\right)}{N} \\cong R\\frac{NlnN - \\sum_{}^{}{N_{i}l{nN}_{i}}}{N} = - R\\sum_{}^{}{x_{i}l{nx}_{i}}$\n", + "\n", + "where $R$ is the gas constant. represents the entropy difference between\n", + "the ideal solution and the states when individual components are by\n", + "themselves, i.e. the mechanical mixing of components. As $x_{i}$ is\n", + "smaller than unity, the entropy production to form an ideal solution\n", + "from pure components is thus positive, indicating that it is a\n", + "spontaneous process. In such an ideal solution, it is assumed that there\n", + "are no interactions between components, and the enthalpy of mixing is\n", + "thus zero as the internal energy and the volume of the system do not\n", + "change. The ideal Gibbs energy of mixing is written as\n", + "\n", + "Eq. \u2011 $_{\\ }^{I}G = - T_{\\ }^{I}S_{m} = RT\\sum_{}^{}{x_{i}l{nx}_{i}}$\n", + "\n", + "The Gibbs energy of real solutions, i.e. , becomes\n", + "\n", + "Eq. \u2011\n", + "$G_{m} = \\sum_{}^{}{x_{i}_{\\ }^{0}G_{i}} + RT\\sum_{}^{}{x_{i}l{nx}_{i}} +_{\\ }^{E}G_{m}$\n", + "\n", + "From , the chemical potential is obtained as\n", + "\n", + "Eq. \u2011\n", + "$\\mu_{i} = G_{i} =_{\\ }^{0}G_{i} + RTlnx_{i} +_{\\ }^{E}G_{m} + \\frac{\\partial_{\\ }^{E}G_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial_{\\ }^{E}G_{m}}{\\partial x_{j}}$\n", + "\n", + "From the chemical activity of , the activity coefficient, $\\gamma_{i}$,\n", + "can be defined as follows\n", + "\n", + "Eq. \u2011\n", + "$\\gamma_{i} = \\frac{a_{i}}{x_{i}} = \\frac{1}{x_{i}}\\exp\\frac{G_{i} -_{\\ }^{0}G_{i}\\ }{RT}$\n", + "\n", + "The solution is an ideal solution with $\\gamma_{i} = 1$, and is said to\n", + "be positively or negatively deviating from an ideal solution with\n", + "$\\gamma_{i} > 1$ or $\\gamma_{i} < 1$, respectively. The chemical\n", + "potential is related to the activity and activity coefficient by the\n", + "following equation\n", + "\n", + "Eq. \u2011\n", + "$\\mu_{i} =_{\\ }^{0}G_{i} + RTlna_{i} =_{\\ }^{0}G_{i} + RTln\\gamma_{i}x_{i} =_{\\ }^{0}G_{i} + RTlnx_{i} + RTln\\gamma_{i}$\n", + "\n", + "Let us further exam in more details in order to better understand the\n", + "relation between $G_{m}$ and $\\mu_{i}$. The partial derivatives in\n", + "represent the directions of molar Gibbs energy in the composition space,\n", + "i.e. the tangents of molar Gibbs energy with respect to mole fractions\n", + "of independent components. Collectively, they define the\n", + "multi-dimensional tangent plane of molar Gibbs energy at the given\n", + "composition, $x_{i}^{0}$. The mathematical representation of this\n", + "tangent plane, $z_{G_{m}}$, is defined by its directional derivatives\n", + "and the distance from the point where the derivatives are taken,\n", + "\n", + "Eq. \u2011\n", + "$z_{G_{m}} = G_{m}\\left( x_{i}^{0} \\right) + \\sum_{i = 1}^{c}{\\left( \\frac{\\partial G_{m}}{\\partial x_{i}} \\right)_{x_{i}^{0}}\\left( x_{i} - x_{i}^{0} \\right)}$\n", + "\n", + "The intercept of this tangent plane at each pure component axis, i.e.\n", + "$x_{i} = 1$ and $x_{j \\neq i} = 0$, is obtained as\n", + "\n", + "Eq. \u2011\n", + "$z_{G_{m},x_{i} = 1} = G_{m}\\left( x_{i}^{0} \\right) + \\left( \\frac{\\partial G_{m}}{\\partial x_{i}} \\right)_{x_{i}^{\\ } = x_{i}^{0}} - \\sum_{j = 1}^{c}{x_{j}^{0}\\left( \\frac{\\partial G_{m}}{\\partial x_{j}} \\right)_{x_{i}^{\\ } = x_{i}^{0}}}$\n", + "\n", + "This is identical to at the point $x_{i}^{0}$. It is thus shown that the\n", + "chemical potential of a component in a solution is represented by the\n", + "intercept of the tangent plane of Gibbs energy of the solution on the\n", + "$G_{m}$ axis of the component. The distance between the intercept and\n", + "the Gibbs energy of the pure component in the same structure of the\n", + "solution is related to the chemical activity of the component as defined\n", + "by . On the other hand, it is evident that one can choose any other\n", + "structure of the pure element to define the chemical activity to compare\n", + "chemical potential of the component as shown by .\n", + "\n", + "The stability of a solution is evaluated following , and the derivatives\n", + "of chemical potential with respect to its moles, i.e. the elements in\n", + "the determinant, are obtained as follows from and ,\n", + "\n", + "Eq. \u2011\n", + "$\\frac{N}{RT}\\frac{\\partial\\mu_{i}}{\\partial N_{i}} = \\frac{N}{RT}G_{ii} = \\frac{1 - x_{i}}{x_{i}} + \\frac{1}{\\gamma_{i}}\\left( \\frac{\\partial\\gamma_{i}}{\\partial x_{i}} - \\sum_{j = 1}^{c}{x_{j}\\frac{\\partial\\gamma_{i}}{\\partial x_{j}}} \\right)$\n", + "\n", + "Eq. \u2011\n", + "$\\frac{N}{RT}\\frac{\\partial\\mu_{i}}{\\partial N_{k}} = \\frac{N}{RT}G_{ik} = - \\frac{x_{k}}{x_{i}} + \\frac{1}{\\gamma_{i}}\\left( 1 - \\sum_{j = 1}^{c}{x_{j}\\frac{\\partial\\gamma_{i}}{\\partial x_{j}}} \\right)$\n", + "\n", + "To further study Gibbs energy of solution phases, let us discuss the\n", + "details on the excess Gibbs energy of mixing. At this point, one can\n", + "start with lower-order systems with fewer components, i.e. two component\n", + "and three-component systems, noting that the Gibbs energy of phases with\n", + "one component is already presented in Chapter .\n" + ], "metadata": {} }, { - "id": "b159479c", + "id": "6fd8d94b", "cell_type": "markdown", - "source": "### Binary random solutions\n\nFrom , the Gibbs-Duhem equation of a binary system consisting of\ncomponents $A$ and $B$ is written as\n\nEq. \u2011 $0 = - SdT - Vd( - P) - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B}$\n\nThis equation represents a four-dimensional surface. It is self-evident\nthat both and hold for stable binary solutions too, i.e. the directions\nand the curvature of the surface are all negative. To visualize the\nfour-dimensional surface in the three-dimensional space, one needs to\nfix one of the four potentials. As $T$ and $P$ are the natural variables\nof Gibbs energy, they are usually chosen to be kept constant. One can\ntypically investigate behaviors of systems consisting of condensed\nphases by varying the temperature at constant pressure. at constant\npressure thus becomes\n\nEq. \u2011 $0 = - SdT - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B}$\n\nSimilar to and , the property of a phase can be represented by a\ntwo-dimensional surface in the three-dimensional space composed of $T$,\n$\\mu_{A}$, and $\\mu_{B}$ under constant $P$, keeping in mind the\nfollowing\n\nEq. \u2011\n$G_{m} = x_{A}\\mu_{A} + x_{B}\\mu_{B} = x_{A}_{\\ }^{0}G_{A} + x_{B}_{\\ }^{0}G_{B} + RT\\left( x_{A}l{nx}_{A} + x_{B}l{nx}_{B} \\right) +_{\\ }^{E}G_{m}$\n\nSince $_{\\ }^{E}G_{m}$ must be zero for pure components $A$ and $B$, it\nneeds to be in the following form\n\nEq. \u2011 $_{\\ }^{E}G_{m} = x_{A}x_{B}L_{AB}$\n\nwith $L_{AB}$ being a parameter denoting the interaction between\ncomponents $A$ and $B$, called interaction parameter. When $L_{AB} = 0$,\nthe solution is an ideal solution. When $L_{AB}$ is a non-zero constant\nindependent of temperature and composition, the solution is called a\nregular solution. Its excess entropy and excess enthalpy of mixing are\nobtained as\n\nEq. \u2011 $_{\\ }^{E}S_{m} = \\frac{\\partial_{\\ }^{E}G_{m}}{\\partial T} = 0$\n\nEq. \u2011\n$_{\\ }^{E}H_{m} =_{\\ }^{E}G_{m} - T_{\\ }^{E}S_{m} = x_{A}x_{B}L_{AB}$\n\nThe chemical potential of component $A$ or $B$ in a binary regular\nsolution can be derived as\n\nEq. \u2011\n$\\mu_{i} =_{\\ }^{0}G_{i} + RTlnx_{i} + \\left( 1 - x_{i} \\right)^{2}L_{AB}$\n\nIn a dilute solution with $x_{i} \\rightarrow 0$, one can have\n\nEq. \u2011\n$RTln\\gamma_{i} = \\left( 1 - x_{i} \\right)^{2}L_{AB} \\approx L_{AB}$\n\nEq. \u2011 $\\gamma_{i} = e^{\\frac{L_{AB}}{RT}}$\n\nThe activity is thus proportional to its mole fraction, which is called\nHenry\u2019s law. By the same token, for the solvent, i.e.\n$x_{i} \\rightarrow 1$,\n\nEq. \u2011 $RTln\\gamma_{i} = \\left( 1 - x_{i} \\right)^{2}L_{AB} \\approx 0$\n\nwhich gives $\\gamma_{i} \\approx 1$, and its activity approaches its mole\nfraction. This is called Raoult\u2019s law.\n\nThe stability of a binary solution is derived from as\n\nEq. \u2011\n$\\left( \\frac{\\partial\\mu_{A}}{\\partial N_{A}} \\right)_{T,P,N_{B}} = \\left\\lbrack \\frac{RT}{x_{A}} - 2\\left( 1 - x_{A} \\right)L_{AB} \\right\\rbrack\\frac{1 - x_{A}}{N}$\n\nEq. \u2011\n$\\left( \\frac{\\partial\\mu_{B}}{\\partial N_{B}} \\right)_{T,P,N_{A}} = \\left\\lbrack \\frac{RT}{x_{B}} - 2\\left( 1 - x_{B} \\right)L_{AB} \\right\\rbrack\\frac{1 - x_{B}}{N}$\n\nIt should be noted that the two chemical potentials in a binary system\nat constant temperature and pressure are dependent on each other due to\nthe Gibbs-Duhem equation shown in , i.e.\n\nEq. \u2011 $0 = - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B}$\n\nand the two chemical potentials depend on each other by the following\nrelation\n\nEq. \u2011\n$\\left( \\frac{\\partial\\mu_{A}}{\\partial\\mu_{B}} \\right)_{T,P} = - \\frac{N_{B}}{N_{A}}$\n\nTherefore, at the limit of stability, both and go to zero at the same\ntime, which is obtained when\n\nEq. \u2011 $RT = 2{x_{A}x_{B}L}_{AB}$\n\nAs the absolute temperature cannot be negative, has no solution for a\nsolution phase with $L_{AB} < 0$, i.e. the solution phase is stable with\nrespect to the composition fluctuation. For a solution with\n$L_{AB} > 0$, its limit of stability is represented .\n\nA schematic molar Gibbs energy of a solution with $L_{AB} < 0$ at\nconstant temperature and pressure is shown in along with the ideal and\nexcess Gibbs energy of mixing. A tangent line on the molar Gibbs energy\nof the solution is drawn, and its two intercepts at $x_{B} = 0$ and\n$x_{B} = 1$ give the chemical potentials of components $A$ and $B$,\n$\\mu_{A}$ and $\\mu_{B}$ by , respectively. It is evident that $\\mu_{A}$\nand $\\mu_{B}$ are not independent on each other as they are two points\non the same straight line. This is a graphic representation of the\nGibbs-Duhem equation of . The chemical activity of component $B$ is also\ndepicted with the reference state being the pure B with the same\nstructure. As shown in , other structures of pure B can be selected as\nthe reference states of the chemical activity of component B, resulting\nin the different distances to its chemical potential in the solution,\nthus different values of its chemical activities. It is clear that this\nchange of reference state for chemical activity does not affect the\nchemical potential of the component in the solution.\n\nFigure \u2011: Schematic molar Gibbs energy diagram with $L_{AB} < 0$\n\nWhen $L_{AB} > 0$, represents a parabola in the $T - x_{i}$\ntwo-dimensional coordinate, symmetric with respect to $x_{A}$ and\n$x_{B}$, shown in , i.e. the spinodal of the solution. The consolute\npoint is obtained by applying to and letting equal to zero at the\nconsolute point\n\nEq. \u2011\n$\\left( \\frac{\\partial^{2}\\mu_{A}}{\\partial N_{A}^{2}} \\right)_{T,P,N_{B}} = \\left\\lbrack - \\frac{RT}{x_{A}^{2}} + 2L_{AB} \\right\\rbrack\\left( \\frac{1 - x_{A}}{N} \\right)^{2} = 0$\n\nwhich gives\n\nEq. \u2011 $T_{cons} = 2x_{A}^{2}L_{AB}$\n\nSolving and , one obtains $x_{A} = x_{B} = 0.5$ and\n\nEq. \u2011 $T_{cons} = \\frac{L_{AB}}{2R}$\n\nFigure \u2011: A Spinodal curve with $L_{AB} > 0$\n\nA schematic molar Gibbs energy diagram at temperatures below the\nconsolute point is shown in . It can be seen that part of the molar\nGibbs energy has negative curvature, and the solution becomes unstable.\nThe chemical potential thus does not change monotonically with respect\nto composition and its derivative changes sign at the inflexion point.\n\nFigure \u2011: Schematic molar Gibbs energy diagram with $L_{AB} > 0$\n\nFor more complex solutions, $L_{AB}$ can be a function of temperature,\npressure, and compositions. In principle, the temperature and pressure\ndependences can be treated by means of formula similar to . There are\nvarious approaches in the literature to consider the composition\ndependence of $L_{AB}$. The empirical Redlich-Kister polynomial stands\nout as the one most widely used because it can be extrapolated to\nternary and multi-component systems consistently, which will be\ndiscussed in Chapter .\n", + "source": [ + "### Binary random solutions\n", + "\n", + "From , the Gibbs-Duhem equation of a binary system consisting of\n", + "components $A$ and $B$ is written as\n", + "\n", + "Eq. \u2011 $0 = - SdT - Vd( - P) - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B}$\n", + "\n", + "This equation represents a four-dimensional surface. It is self-evident\n", + "that both and hold for stable binary solutions too, i.e. the directions\n", + "and the curvature of the surface are all negative. To visualize the\n", + "four-dimensional surface in the three-dimensional space, one needs to\n", + "fix one of the four potentials. As $T$ and $P$ are the natural variables\n", + "of Gibbs energy, they are usually chosen to be kept constant. One can\n", + "typically investigate behaviors of systems consisting of condensed\n", + "phases by varying the temperature at constant pressure. at constant\n", + "pressure thus becomes\n", + "\n", + "Eq. \u2011 $0 = - SdT - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B}$\n", + "\n", + "Similar to and , the property of a phase can be represented by a\n", + "two-dimensional surface in the three-dimensional space composed of $T$,\n", + "$\\mu_{A}$, and $\\mu_{B}$ under constant $P$, keeping in mind the\n", + "following\n", + "\n", + "Eq. \u2011\n", + "$G_{m} = x_{A}\\mu_{A} + x_{B}\\mu_{B} = x_{A}_{\\ }^{0}G_{A} + x_{B}_{\\ }^{0}G_{B} + RT\\left( x_{A}l{nx}_{A} + x_{B}l{nx}_{B} \\right) +_{\\ }^{E}G_{m}$\n", + "\n", + "Since $_{\\ }^{E}G_{m}$ must be zero for pure components $A$ and $B$, it\n", + "needs to be in the following form\n", + "\n", + "Eq. \u2011 $_{\\ }^{E}G_{m} = x_{A}x_{B}L_{AB}$\n", + "\n", + "with $L_{AB}$ being a parameter denoting the interaction between\n", + "components $A$ and $B$, called interaction parameter. When $L_{AB} = 0$,\n", + "the solution is an ideal solution. When $L_{AB}$ is a non-zero constant\n", + "independent of temperature and composition, the solution is called a\n", + "regular solution. Its excess entropy and excess enthalpy of mixing are\n", + "obtained as\n", + "\n", + "Eq. \u2011 $_{\\ }^{E}S_{m} = \\frac{\\partial_{\\ }^{E}G_{m}}{\\partial T} = 0$\n", + "\n", + "Eq. \u2011\n", + "$_{\\ }^{E}H_{m} =_{\\ }^{E}G_{m} - T_{\\ }^{E}S_{m} = x_{A}x_{B}L_{AB}$\n", + "\n", + "The chemical potential of component $A$ or $B$ in a binary regular\n", + "solution can be derived as\n", + "\n", + "Eq. \u2011\n", + "$\\mu_{i} =_{\\ }^{0}G_{i} + RTlnx_{i} + \\left( 1 - x_{i} \\right)^{2}L_{AB}$\n", + "\n", + "In a dilute solution with $x_{i} \\rightarrow 0$, one can have\n", + "\n", + "Eq. \u2011\n", + "$RTln\\gamma_{i} = \\left( 1 - x_{i} \\right)^{2}L_{AB} \\approx L_{AB}$\n", + "\n", + "Eq. \u2011 $\\gamma_{i} = e^{\\frac{L_{AB}}{RT}}$\n", + "\n", + "The activity is thus proportional to its mole fraction, which is called\n", + "Henry\u2019s law. By the same token, for the solvent, i.e.\n", + "$x_{i} \\rightarrow 1$,\n", + "\n", + "Eq. \u2011 $RTln\\gamma_{i} = \\left( 1 - x_{i} \\right)^{2}L_{AB} \\approx 0$\n", + "\n", + "which gives $\\gamma_{i} \\approx 1$, and its activity approaches its mole\n", + "fraction. This is called Raoult\u2019s law.\n", + "\n", + "The stability of a binary solution is derived from as\n", + "\n", + "Eq. \u2011\n", + "$\\left( \\frac{\\partial\\mu_{A}}{\\partial N_{A}} \\right)_{T,P,N_{B}} = \\left\\lbrack \\frac{RT}{x_{A}} - 2\\left( 1 - x_{A} \\right)L_{AB} \\right\\rbrack\\frac{1 - x_{A}}{N}$\n", + "\n", + "Eq. \u2011\n", + "$\\left( \\frac{\\partial\\mu_{B}}{\\partial N_{B}} \\right)_{T,P,N_{A}} = \\left\\lbrack \\frac{RT}{x_{B}} - 2\\left( 1 - x_{B} \\right)L_{AB} \\right\\rbrack\\frac{1 - x_{B}}{N}$\n", + "\n", + "It should be noted that the two chemical potentials in a binary system\n", + "at constant temperature and pressure are dependent on each other due to\n", + "the Gibbs-Duhem equation shown in , i.e.\n", + "\n", + "Eq. \u2011 $0 = - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B}$\n", + "\n", + "and the two chemical potentials depend on each other by the following\n", + "relation\n", + "\n", + "Eq. \u2011\n", + "$\\left( \\frac{\\partial\\mu_{A}}{\\partial\\mu_{B}} \\right)_{T,P} = - \\frac{N_{B}}{N_{A}}$\n", + "\n", + "Therefore, at the limit of stability, both and go to zero at the same\n", + "time, which is obtained when\n", + "\n", + "Eq. \u2011 $RT = 2{x_{A}x_{B}L}_{AB}$\n", + "\n", + "As the absolute temperature cannot be negative, has no solution for a\n", + "solution phase with $L_{AB} < 0$, i.e. the solution phase is stable with\n", + "respect to the composition fluctuation. For a solution with\n", + "$L_{AB} > 0$, its limit of stability is represented .\n", + "\n", + "A schematic molar Gibbs energy of a solution with $L_{AB} < 0$ at\n", + "constant temperature and pressure is shown in along with the ideal and\n", + "excess Gibbs energy of mixing. A tangent line on the molar Gibbs energy\n", + "of the solution is drawn, and its two intercepts at $x_{B} = 0$ and\n", + "$x_{B} = 1$ give the chemical potentials of components $A$ and $B$,\n", + "$\\mu_{A}$ and $\\mu_{B}$ by , respectively. It is evident that $\\mu_{A}$\n", + "and $\\mu_{B}$ are not independent on each other as they are two points\n", + "on the same straight line. This is a graphic representation of the\n", + "Gibbs-Duhem equation of . The chemical activity of component $B$ is also\n", + "depicted with the reference state being the pure B with the same\n", + "structure. As shown in , other structures of pure B can be selected as\n", + "the reference states of the chemical activity of component B, resulting\n", + "in the different distances to its chemical potential in the solution,\n", + "thus different values of its chemical activities. It is clear that this\n", + "change of reference state for chemical activity does not affect the\n", + "chemical potential of the component in the solution.\n", + "\n", + "Figure \u2011: Schematic molar Gibbs energy diagram with $L_{AB} < 0$\n", + "\n", + "When $L_{AB} > 0$, represents a parabola in the $T - x_{i}$\n", + "two-dimensional coordinate, symmetric with respect to $x_{A}$ and\n", + "$x_{B}$, shown in , i.e. the spinodal of the solution. The consolute\n", + "point is obtained by applying to and letting equal to zero at the\n", + "consolute point\n", + "\n", + "Eq. \u2011\n", + "$\\left( \\frac{\\partial^{2}\\mu_{A}}{\\partial N_{A}^{2}} \\right)_{T,P,N_{B}} = \\left\\lbrack - \\frac{RT}{x_{A}^{2}} + 2L_{AB} \\right\\rbrack\\left( \\frac{1 - x_{A}}{N} \\right)^{2} = 0$\n", + "\n", + "which gives\n", + "\n", + "Eq. \u2011 $T_{cons} = 2x_{A}^{2}L_{AB}$\n", + "\n", + "Solving and , one obtains $x_{A} = x_{B} = 0.5$ and\n", + "\n", + "Eq. \u2011 $T_{cons} = \\frac{L_{AB}}{2R}$\n", + "\n", + "Figure \u2011: A Spinodal curve with $L_{AB} > 0$\n", + "\n", + "A schematic molar Gibbs energy diagram at temperatures below the\n", + "consolute point is shown in . It can be seen that part of the molar\n", + "Gibbs energy has negative curvature, and the solution becomes unstable.\n", + "The chemical potential thus does not change monotonically with respect\n", + "to composition and its derivative changes sign at the inflexion point.\n", + "\n", + "Figure \u2011: Schematic molar Gibbs energy diagram with $L_{AB} > 0$\n", + "\n", + "For more complex solutions, $L_{AB}$ can be a function of temperature,\n", + "pressure, and compositions. In principle, the temperature and pressure\n", + "dependences can be treated by means of formula similar to . There are\n", + "various approaches in the literature to consider the composition\n", + "dependence of $L_{AB}$. The empirical Redlich-Kister polynomial stands\n", + "out as the one most widely used because it can be extrapolated to\n", + "ternary and multi-component systems consistently, which will be\n", + "discussed in Chapter .\n" + ], "metadata": {} }, { - "id": "4a5ace2b", + "id": "7e5249cd", "cell_type": "markdown", - "source": "### Ternary random solutions\n\nFrom , the Gibbs-Duhem equation of a ternary system consisting of\ncomponents $A$, $B$ and $C$ is written as\n\nEq. \u2011\n$0 = - SdT - Vd( - P) - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B} - {N_{C}d\\mu}_{C}$\n\nThis equation represents a five-dimensional surface. It can be\nvisualized in a three-dimensional space with two of the five potentials\nfixed. Usually $T$ and $P$ are kept constant as they are the natural\nvariables of $G$, and reduces to\n\nEq. \u2011 $0 = - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B} - {N_{C}d\\mu}_{C}$\n\nA phase can thus be represented by a surface in the three-dimensional\nspace of $\\mu_{A}$, $\\mu_{B}$, and $\\mu_{C}$ at constant $T$ and $P$\nwith similar geometric appearance as .\n\nFrom , Gibbs energy of a ternary solution is written as\n\nEq. \u2011\n$G_{m} = x_{A}_{\\ }^{0}G_{A} + x_{B}_{\\ }^{0}G_{B} + x_{C}_{\\ }^{0}G_{C} + RT\\left( x_{A}l{nx}_{A} + x_{B}l{nx}_{B} + x_{C}l{nx}_{C} \\right) +_{\\ }^{E}G_{m}$\n\nWhen the mole fraction of one component approaches zero,\n$_{\\ }^{E}G_{m}$ reduces to the excess Gibbs energy of mixing of the\nbinary systems of the remaining two components, represented by .\nHowever, for a given composition of a ternary solution, there is no\nunique way to assign the contributions from $_{\\ }^{E}G_{m}$ of each\nbinary to $_{\\ }^{E}G_{m}$ of the ternary solution because\n$_{\\ }^{E}G_{m}$ of the ternary solution contains information of both\nbinary and ternary interactions. A variety of models is available in the\nliterature (see \\[1\\]). One intuitive approach would be to use the same\nformula as that in the binary system, i.e. , with the mole fractions\nsubstituted by the values in the ternary system, and $_{\\ }^{E}G_{m}$ of\na ternary solution may thus be defined as the following by including the\nternary interaction involving all three components,\n\nEq. \u2011\n$_{\\ }^{E}G_{m} = x_{A}x_{B}L_{AB} + x_{A}x_{C}L_{AC} + x_{B}x_{C}L_{BC} + x_{A}x_{B}x_{C}L_{ABC}$\n\nThe chemical potential of a component is represented by . When all\ninteraction parameters in are constant, i.e. a ternary regular solution,\nthe chemical potential of component $A$ can be derived as\n\nEq. \u2011\n$\\mu_{A} = G_{A} =_{\\ }^{0}G_{A} + RTlnx_{A} + x_{B}L_{AB} + x_{C}L_{AC} -_{\\ }^{E}G_{m} =_{\\ }^{0}G_{A} + RTlnx_{A} + x_{B}{\\left( 1 - x_{A} \\right)L}_{AB} + x_{C}\\left( 1 - x_{A} \\right)L_{AC} - x_{B}x_{C}L_{BC} + x_{B}x_{C}\\left( 1 - 2x_{A} \\right)L_{ABC} =_{\\ }^{0}G_{A} + RTlnx_{B} + x_{B}^{2}L_{AB} + x_{C}^{2}L_{AC} + x_{B}x_{C}\\left( L_{AB} + L_{AC} - L_{BC} \\right) + x_{B}x_{C}\\left( 1 - 2x_{A} \\right)L_{ABC}$\n\nSimilar equations can be derived for component $B$ and C with\n$L_{AB} = L_{BA}$, $L_{AC} = L_{CA}$, and $L_{BC} = L_{CB}$. A schematic\nmolar Gibbs energy diagram at constant temperature and pressure is shown\nin with all three binary systems having $L_{ij} < 0$ of similar values.\n\nFigure \u2011: Schematic ternary molar Gibbs energy diagram as a function of\ncompositions for given temperature and pressure\n\nTo evaluate the stability of a ternary solution, one needs to calculate\nthe elements in the determinant shown in . Using the mole of component\n$C$ as the independent molar quantity, the limit of stability is\nexpresses as\n\nEq. \u2011 $G_{AA}G_{BB} - G_{AB}G_{BA} = 0$\n\nAs an example, $G_{AA}$ is shown in the following equation, which must\nbe positive for the solution to be stable\n\nEq. \u2011\n${N\\left( \\frac{\\partial\\mu_{A}}{\\partial N_{A}} \\right)}_{T,P,N_{B},N_{C}} = NG_{AA} = \\frac{RT\\left( 1 - x_{A} \\right)}{x_{A}} - 2x_{B}^{2}L_{AB} - 2x_{C}^{2}L_{AC} - 2x_{B}x_{C}\\left( L_{AB} + L_{AC} - L_{BC} \\right) - 2x_{B}x_{C}\\left( 2 - 3x_{A} \\right)L_{ABC}$\n\nIt is evident that any instability in binary systems with positive\ninteraction parameters extends into the ternary system. It can also be\nseen that even if all binary interaction parameters are negative, i.e.\nno instability in the binary systems, it is possible that becomes\nnegative for some combinations of the binary interaction parameters such\nthat $\\mathrm{\\Delta}L = L_{AB} + L_{AC} - L_{BC}$ becomes very positive\nand overshadows the contributions due to $L_{AB}$ and $L_{AC}$, i.e.\n$L_{BC}$ is more negative than $L_{AB}$ and $L_{AC}$ combined. In an\nextreme case with $L_{AB} = L_{AC} = L_{ABC} = 0$ and $L_{BC} < 0$, i.e.\nideal solutions for the $A - B$ and $A - C$ binary systems, stable\nsolution in the $B - C$ binary system, and no additional ternary\ninteraction, reduces to\n\nEq. \u2011\n$N\\left( \\frac{\\partial\\mu_{A}}{\\partial N_{A}} \\right)_{T,P,N_{B},N_{C}} = \\frac{RT\\left( 1 - x_{A} \\right)}{x_{A}} + 2x_{B}x_{C}L_{BC}$\n\nSetting\n$\\left( \\frac{\\partial\\mu_{A}}{\\partial N_{A}} \\right)_{T,P,N_{B},N_{C}} = 0$,\none obtains\n\nEq. \u2011\n$- \\frac{RT}{2L_{BC}} = \\frac{x_{A}x_{B}x_{C}}{1 - x_{A}} = \\frac{{\\left( 1 - x_{B} - x_{C} \\right)x}_{B}x_{C}}{x_{B} + x_{C}}$\n\nWith $- \\frac{RT}{2L_{BC}}$ being positive due to $L_{BC} < 0$, there is\na parabola-shaped composition area in which the solution is unstable at\nconstant temperature and pressure. This is reasonable because the system\ntends to maximize the number of B-C bonds due to its lower energy, which\ncompetes with the entropy of mixing among the three elements and results\nin segregation of B-C bonds, thus miscibility gap at low temperatures.\n\nTo evaluate the ternary consolute point, the second derivatives for\ncomponent A and B are obtained as\n\nEq. \u2011\n${N\\left( \\frac{\\partial_{\\ }^{2}\\mu_{A}}{\\partial N_{A}^{2}} \\right)}_{T,P,N_{B},N_{C}} = \\frac{RT\\left( 1 - x_{A} \\right)}{x_{A}^{2}} + 4x_{B}^{2}L_{AB} + 4x_{C}^{2}L_{AC} + 4x_{B}x_{C}\\left( L_{AB} + L_{AC} - L_{BC} \\right) + 2x_{B}x_{C}\\left( 7 - 9x_{A} \\right)L_{ABC} = 0$\n\nEq. \u2011\n${N\\left( \\frac{\\partial_{\\ }^{2}\\mu_{B}}{\\partial N_{B}^{2}} \\right)}_{T,P,N_{A},N_{C}} = \\frac{RT\\left( 1 - x_{B} \\right)}{x_{B}^{2}} + 4x_{A}^{2}L_{AB} + 4x_{C}^{2}L_{BC} + 4x_{A}x_{C}\\left( L_{AB} + L_{BC} - L_{AC} \\right) + 2x_{A}x_{C}\\left( 7 - 9x_{B} \\right)L_{ABC} = 0$\n\nThe consolute point can then be obtained using , and .\n\nIt is observed in that $\\left( 1 - 2x_{A} \\right)L_{ABC} = 0$ at\n$x_{A} = 0.5$, i.e. the ternary interaction parameter does not\ncontribute to the chemical potential of $A$. It is also observed in that\nthe contribution from the ternary interaction parameter changes sign at\n$x_{i} = 2/3$ due to $\\left( 2 - 3x_{A} \\right)L_{ABC} = 0$.\n", + "source": [ + "### Ternary random solutions\n", + "\n", + "From , the Gibbs-Duhem equation of a ternary system consisting of\n", + "components $A$, $B$ and $C$ is written as\n", + "\n", + "Eq. \u2011\n", + "$0 = - SdT - Vd( - P) - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B} - {N_{C}d\\mu}_{C}$\n", + "\n", + "This equation represents a five-dimensional surface. It can be\n", + "visualized in a three-dimensional space with two of the five potentials\n", + "fixed. Usually $T$ and $P$ are kept constant as they are the natural\n", + "variables of $G$, and reduces to\n", + "\n", + "Eq. \u2011 $0 = - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B} - {N_{C}d\\mu}_{C}$\n", + "\n", + "A phase can thus be represented by a surface in the three-dimensional\n", + "space of $\\mu_{A}$, $\\mu_{B}$, and $\\mu_{C}$ at constant $T$ and $P$\n", + "with similar geometric appearance as .\n", + "\n", + "From , Gibbs energy of a ternary solution is written as\n", + "\n", + "Eq. \u2011\n", + "$G_{m} = x_{A}_{\\ }^{0}G_{A} + x_{B}_{\\ }^{0}G_{B} + x_{C}_{\\ }^{0}G_{C} + RT\\left( x_{A}l{nx}_{A} + x_{B}l{nx}_{B} + x_{C}l{nx}_{C} \\right) +_{\\ }^{E}G_{m}$\n", + "\n", + "When the mole fraction of one component approaches zero,\n", + "$_{\\ }^{E}G_{m}$ reduces to the excess Gibbs energy of mixing of the\n", + "binary systems of the remaining two components, represented by .\n", + "However, for a given composition of a ternary solution, there is no\n", + "unique way to assign the contributions from $_{\\ }^{E}G_{m}$ of each\n", + "binary to $_{\\ }^{E}G_{m}$ of the ternary solution because\n", + "$_{\\ }^{E}G_{m}$ of the ternary solution contains information of both\n", + "binary and ternary interactions. A variety of models is available in the\n", + "literature (see \\[1\\]). One intuitive approach would be to use the same\n", + "formula as that in the binary system, i.e. , with the mole fractions\n", + "substituted by the values in the ternary system, and $_{\\ }^{E}G_{m}$ of\n", + "a ternary solution may thus be defined as the following by including the\n", + "ternary interaction involving all three components,\n", + "\n", + "Eq. \u2011\n", + "$_{\\ }^{E}G_{m} = x_{A}x_{B}L_{AB} + x_{A}x_{C}L_{AC} + x_{B}x_{C}L_{BC} + x_{A}x_{B}x_{C}L_{ABC}$\n", + "\n", + "The chemical potential of a component is represented by . When all\n", + "interaction parameters in are constant, i.e. a ternary regular solution,\n", + "the chemical potential of component $A$ can be derived as\n", + "\n", + "Eq. \u2011\n", + "$\\mu_{A} = G_{A} =_{\\ }^{0}G_{A} + RTlnx_{A} + x_{B}L_{AB} + x_{C}L_{AC} -_{\\ }^{E}G_{m} =_{\\ }^{0}G_{A} + RTlnx_{A} + x_{B}{\\left( 1 - x_{A} \\right)L}_{AB} + x_{C}\\left( 1 - x_{A} \\right)L_{AC} - x_{B}x_{C}L_{BC} + x_{B}x_{C}\\left( 1 - 2x_{A} \\right)L_{ABC} =_{\\ }^{0}G_{A} + RTlnx_{B} + x_{B}^{2}L_{AB} + x_{C}^{2}L_{AC} + x_{B}x_{C}\\left( L_{AB} + L_{AC} - L_{BC} \\right) + x_{B}x_{C}\\left( 1 - 2x_{A} \\right)L_{ABC}$\n", + "\n", + "Similar equations can be derived for component $B$ and C with\n", + "$L_{AB} = L_{BA}$, $L_{AC} = L_{CA}$, and $L_{BC} = L_{CB}$. A schematic\n", + "molar Gibbs energy diagram at constant temperature and pressure is shown\n", + "in with all three binary systems having $L_{ij} < 0$ of similar values.\n", + "\n", + "Figure \u2011: Schematic ternary molar Gibbs energy diagram as a function of\n", + "compositions for given temperature and pressure\n", + "\n", + "To evaluate the stability of a ternary solution, one needs to calculate\n", + "the elements in the determinant shown in . Using the mole of component\n", + "$C$ as the independent molar quantity, the limit of stability is\n", + "expresses as\n", + "\n", + "Eq. \u2011 $G_{AA}G_{BB} - G_{AB}G_{BA} = 0$\n", + "\n", + "As an example, $G_{AA}$ is shown in the following equation, which must\n", + "be positive for the solution to be stable\n", + "\n", + "Eq. \u2011\n", + "${N\\left( \\frac{\\partial\\mu_{A}}{\\partial N_{A}} \\right)}_{T,P,N_{B},N_{C}} = NG_{AA} = \\frac{RT\\left( 1 - x_{A} \\right)}{x_{A}} - 2x_{B}^{2}L_{AB} - 2x_{C}^{2}L_{AC} - 2x_{B}x_{C}\\left( L_{AB} + L_{AC} - L_{BC} \\right) - 2x_{B}x_{C}\\left( 2 - 3x_{A} \\right)L_{ABC}$\n", + "\n", + "It is evident that any instability in binary systems with positive\n", + "interaction parameters extends into the ternary system. It can also be\n", + "seen that even if all binary interaction parameters are negative, i.e.\n", + "no instability in the binary systems, it is possible that becomes\n", + "negative for some combinations of the binary interaction parameters such\n", + "that $\\mathrm{\\Delta}L = L_{AB} + L_{AC} - L_{BC}$ becomes very positive\n", + "and overshadows the contributions due to $L_{AB}$ and $L_{AC}$, i.e.\n", + "$L_{BC}$ is more negative than $L_{AB}$ and $L_{AC}$ combined. In an\n", + "extreme case with $L_{AB} = L_{AC} = L_{ABC} = 0$ and $L_{BC} < 0$, i.e.\n", + "ideal solutions for the $A - B$ and $A - C$ binary systems, stable\n", + "solution in the $B - C$ binary system, and no additional ternary\n", + "interaction, reduces to\n", + "\n", + "Eq. \u2011\n", + "$N\\left( \\frac{\\partial\\mu_{A}}{\\partial N_{A}} \\right)_{T,P,N_{B},N_{C}} = \\frac{RT\\left( 1 - x_{A} \\right)}{x_{A}} + 2x_{B}x_{C}L_{BC}$\n", + "\n", + "Setting\n", + "$\\left( \\frac{\\partial\\mu_{A}}{\\partial N_{A}} \\right)_{T,P,N_{B},N_{C}} = 0$,\n", + "one obtains\n", + "\n", + "Eq. \u2011\n", + "$- \\frac{RT}{2L_{BC}} = \\frac{x_{A}x_{B}x_{C}}{1 - x_{A}} = \\frac{{\\left( 1 - x_{B} - x_{C} \\right)x}_{B}x_{C}}{x_{B} + x_{C}}$\n", + "\n", + "With $- \\frac{RT}{2L_{BC}}$ being positive due to $L_{BC} < 0$, there is\n", + "a parabola-shaped composition area in which the solution is unstable at\n", + "constant temperature and pressure. This is reasonable because the system\n", + "tends to maximize the number of B-C bonds due to its lower energy, which\n", + "competes with the entropy of mixing among the three elements and results\n", + "in segregation of B-C bonds, thus miscibility gap at low temperatures.\n", + "\n", + "To evaluate the ternary consolute point, the second derivatives for\n", + "component A and B are obtained as\n", + "\n", + "Eq. \u2011\n", + "${N\\left( \\frac{\\partial_{\\ }^{2}\\mu_{A}}{\\partial N_{A}^{2}} \\right)}_{T,P,N_{B},N_{C}} = \\frac{RT\\left( 1 - x_{A} \\right)}{x_{A}^{2}} + 4x_{B}^{2}L_{AB} + 4x_{C}^{2}L_{AC} + 4x_{B}x_{C}\\left( L_{AB} + L_{AC} - L_{BC} \\right) + 2x_{B}x_{C}\\left( 7 - 9x_{A} \\right)L_{ABC} = 0$\n", + "\n", + "Eq. \u2011\n", + "${N\\left( \\frac{\\partial_{\\ }^{2}\\mu_{B}}{\\partial N_{B}^{2}} \\right)}_{T,P,N_{A},N_{C}} = \\frac{RT\\left( 1 - x_{B} \\right)}{x_{B}^{2}} + 4x_{A}^{2}L_{AB} + 4x_{C}^{2}L_{BC} + 4x_{A}x_{C}\\left( L_{AB} + L_{BC} - L_{AC} \\right) + 2x_{A}x_{C}\\left( 7 - 9x_{B} \\right)L_{ABC} = 0$\n", + "\n", + "The consolute point can then be obtained using , and .\n", + "\n", + "It is observed in that $\\left( 1 - 2x_{A} \\right)L_{ABC} = 0$ at\n", + "$x_{A} = 0.5$, i.e. the ternary interaction parameter does not\n", + "contribute to the chemical potential of $A$. It is also observed in that\n", + "the contribution from the ternary interaction parameter changes sign at\n", + "$x_{i} = 2/3$ due to $\\left( 2 - 3x_{A} \\right)L_{ABC} = 0$.\n" + ], "metadata": {} }, { - "id": "fd9ad1e0", + "id": "ae59d19d", "cell_type": "markdown", - "source": "### Multi-component random solutions\n\nSimilar to a ternary solution, the excess Gibbs energy of mixing of a\nmulti-component solution can be written as\n\nEq. \u2011\n$_{\\ }^{E}G_{m} = \\sum_{i}^{}{\\sum_{j}^{}{x_{i}x_{j}L_{ij}}} + \\sum_{i}^{}{\\sum_{j}^{}{\\sum_{k}^{}{x_{i}x_{j}x_{k}L_{ijk}}}}$\n\nIn principle, one can add interaction parameters for quaternary and\nhigher order systems, but their contributions to Gibbs energy are\nrelatively minor because the major contributions have already been taken\ninto account by the binary and ternary interactions. It is anticipated\nthat not only the interaction parameters of four or more components are\nsmall, but also the multiplication of mole fractions in front the\ninteraction parameters diminishes their contribution to the Gibbs energy\neven further.\n\nUnder the condition that all interaction parameters are constant, the\nchemical potential of a component in a multi-component system with\nbinary and ternary interaction parameters can be extended from as\n\nEq. \u2011\n$\\mu_{i} =_{\\ }^{0}G_{i} + RTlnx_{i} + \\sum_{j \\neq i}^{}{x_{j}^{2}L_{ij}} + \\sum_{k > j}^{}{\\sum_{j \\neq i}^{}{x_{j}x_{k}}}\\left\\lbrack L_{ij} + L_{ik} - L_{jk} + \\left( 1 - 2x_{i} \\right)L_{ijk} \\right\\rbrack$\n\nThe stability of the solution can also be extended from as\n\nEq. \u2011\n${N\\left( \\frac{\\partial\\mu_{i}}{\\partial N_{i}} \\right)}_{T,P,N_{j \\neq i}} = NG_{ii} = \\frac{RT\\left( 1 - x_{i} \\right)}{x_{i}} - 2\\sum_{j \\neq i}^{}{x_{j}^{2}L_{ij}} - 2\\sum_{k > j}^{}{\\sum_{j \\neq i}^{}{x_{j}x_{k}}}\\left\\lbrack L_{ij} + L_{ik} - L_{jk} + \\left( 2 - 3x_{i} \\right)L_{ijk} \\right\\rbrack$\n\nThe limit of stability of a multi-component random solution can be\nrepresented by or .\n", + "source": [ + "### Multi-component random solutions\n", + "\n", + "Similar to a ternary solution, the excess Gibbs energy of mixing of a\n", + "multi-component solution can be written as\n", + "\n", + "Eq. \u2011\n", + "$_{\\ }^{E}G_{m} = \\sum_{i}^{}{\\sum_{j}^{}{x_{i}x_{j}L_{ij}}} + \\sum_{i}^{}{\\sum_{j}^{}{\\sum_{k}^{}{x_{i}x_{j}x_{k}L_{ijk}}}}$\n", + "\n", + "In principle, one can add interaction parameters for quaternary and\n", + "higher order systems, but their contributions to Gibbs energy are\n", + "relatively minor because the major contributions have already been taken\n", + "into account by the binary and ternary interactions. It is anticipated\n", + "that not only the interaction parameters of four or more components are\n", + "small, but also the multiplication of mole fractions in front the\n", + "interaction parameters diminishes their contribution to the Gibbs energy\n", + "even further.\n", + "\n", + "Under the condition that all interaction parameters are constant, the\n", + "chemical potential of a component in a multi-component system with\n", + "binary and ternary interaction parameters can be extended from as\n", + "\n", + "Eq. \u2011\n", + "$\\mu_{i} =_{\\ }^{0}G_{i} + RTlnx_{i} + \\sum_{j \\neq i}^{}{x_{j}^{2}L_{ij}} + \\sum_{k > j}^{}{\\sum_{j \\neq i}^{}{x_{j}x_{k}}}\\left\\lbrack L_{ij} + L_{ik} - L_{jk} + \\left( 1 - 2x_{i} \\right)L_{ijk} \\right\\rbrack$\n", + "\n", + "The stability of the solution can also be extended from as\n", + "\n", + "Eq. \u2011\n", + "${N\\left( \\frac{\\partial\\mu_{i}}{\\partial N_{i}} \\right)}_{T,P,N_{j \\neq i}} = NG_{ii} = \\frac{RT\\left( 1 - x_{i} \\right)}{x_{i}} - 2\\sum_{j \\neq i}^{}{x_{j}^{2}L_{ij}} - 2\\sum_{k > j}^{}{\\sum_{j \\neq i}^{}{x_{j}x_{k}}}\\left\\lbrack L_{ij} + L_{ik} - L_{jk} + \\left( 2 - 3x_{i} \\right)L_{ijk} \\right\\rbrack$\n", + "\n", + "The limit of stability of a multi-component random solution can be\n", + "represented by or .\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/gibbs_energy_function/phases_with_variable_compositions_solutions_with_ordering.ipynb b/src/psu410/src/gibbs_energy_function/phases_with_variable_compositions_solutions_with_ordering.ipynb index c4936f0..31e39f5 100644 --- a/src/psu410/src/gibbs_energy_function/phases_with_variable_compositions_solutions_with_ordering.ipynb +++ b/src/psu410/src/gibbs_energy_function/phases_with_variable_compositions_solutions_with_ordering.ipynb @@ -4,33 +4,416 @@ "metadata": {}, "cells": [ { - "id": "30fe313a", + "id": "b876aec3", "cell_type": "markdown", - "source": "## Phases with variable compositions: Solutions with ordering\n", + "source": [ + "## Phases with variable compositions: Solutions with ordering\n" + ], "metadata": {} }, { - "id": "ff7e47a3", + "id": "8ac0a3e4", "cell_type": "markdown", - "source": "### Solutions with short-range ordering\n\nThe order in a system can be measured by correlation functions which\ndescribes how various components are correlated in space. For\nsimplicity, let us consider only the pairs between nearest neighbors\nwith the correlation function represented by the pair probability of\nnearest neighbor bonds between two components. In a random solution, the\nprobability to find nearest neighbor bonds between two components $i$\nand $j$ is\n\nEq. \u2011 $p_{ij} = x_{i}x_{j}$\n\nWhen $p_{ij} \\neq x_{i}x_{j}$, the nearest neighbors of component $i$\nare not occupied randomly by component $j$, rather certain components\nare favored, resulting in short-range ordering or local clustering in\nthe solution. When short-range ordering develops throughout the\nsolution, long-range ordering takes place, and each component has its\nown primary sites in the solution, to be discussed in Chapter . There\nare relations between bond probabilities and mole fractions of\ncomponents due to the mass balance as follows, with the assumption of\n$p_{ij} = p_{ji}$\n\nEq. \u2011 $\\sum_{i}^{}{\\sum_{j}^{}p_{ij}} = 1$\n\nEq. \u2011 $x_{i} = \\sum_{j}^{}p_{ij}$\n\nFor small deviations from a random solution, one can consider the\nformation of $i - j$ bonds from $i - i\\ $ and $j - j$ bonds and the\nideal mixing of three types of bonds, similar to a typical Ising model.\nThe bond reaction can be written as\n\nEq. \u2011 $(i - i\\ )bonds + (j - j)\\ bonds = 2\\ (i - j)\\ bonds$\n\nwith the Gibbs energy of reaction being\n\nEq. \u2011\n${\\mathrm{\\Delta}G}_{ij} = 2G_{ij} - \\left( G_{ii} + G_{jj} \\right)\\ $\n\nGibbs energy of the solution per mole of atom is thus represented by the\nbond energies and the ideal mixing of bonds plus non-ideal interactions\nbetween pairs,\n\nEq. \u2011\n$G_{m} = \\sum_{i}^{}{\\sum_{j}^{}{p_{ij}G_{ij}}} + \\frac{Z}{2}RT\\sum_{i}^{}{\\sum_{j}^{}{p_{ij}\\ln p_{ij}}} +_{\\ }^{E}G_{m}$\n\nwith $G_{ij}$ being the molar bond energy between components $i$ and\n$j$, $Z$ the number of bonds per atom which is divided by two in the\nequation due to two atoms needed to form one bond, and\n$_{\\ }^{E}G_{m} = \\sum_{}^{}{p_{ij}p_{kl}I_{ijkl}}$ the excess Gibbs\nenergy of mixing between bonds. This approach proposed by Guggenheim\n\\[2\\] is called quasi-chemical method as it is based on the chemical\nreaction shown by .\n\nHowever, the entropy of mixing in does not reduce to ideal entropy of\nmixing for a solution without short-range ordering as defined by . An\napproximated correction may be added for small degree of short-range\nordering as follows\n\nEq. \u2011\n$G_{m} = \\sum_{i}^{}{\\sum_{j}^{}{p_{ij}G_{ij}}} + \\frac{Z}{2}RT\\sum_{i}^{}{\\sum_{j}^{}{p_{ij}\\ln\\frac{p_{ij}}{x_{i}x_{j}}}} + RT\\sum_{}^{}{x_{i}l{nx}_{i}} +_{\\ }^{E}G_{m}$\n\nFor a random solution defined by , becomes\n\nEq. \u2011\n$G_{m} = \\sum_{}^{}{x_{i}_{\\ }^{0}G_{i}} + RT\\sum_{}^{}{x_{i}l{nx}_{i}} + \\sum_{}^{}{x_{i}x_{j}{\\mathrm{\\Delta}G}_{ij}} +_{\\ }^{E}G_{m}$\n\nwith $_{\\ }^{0}G_{i} = G_{ii}$, ${\\mathrm{\\Delta}G}_{ij}$ from Eq. 2\u2011107\nrepresenting the interaction parameter between components $i$ and $j$,\nand $_{\\ }^{E}G_{m} = \\sum_{}^{}{x_{i}{x_{j}x_{k}x}_{l}I_{ijkl}}$\ndenoting the higher order interactions, in comparison with .\n\nWhen short-range ordering exists in a solution, one typically uses the\nlaw of mass reaction for the chemical reaction represented by to define\nthe equilibrium among all bonds, i.e.\n\nEq. \u2011\n$\\frac{\\left( p_{ij} \\right)^{2}}{p_{ii}p_{jj}} = e^{- \\frac{{\\mathrm{\\Delta}G}_{ij}}{kT}}$\n\nHowever, this is under the assumption that the chemical activities of\nall bonds can be represented by their respective probabilities, which is\nonly true for an ideal solution even excluding dilute solutions due to\nthe Henry\u2019s law shown by . Preferrably, the bond probabilities can be\nobtained by calculating the driving force for the fluctuation of bond\nprobabilities under constant temperature, pressure, and amount of each\ncomponent along with the constraints defined by and and equating the\ndriving force to zero, i.e.\n\nEq. \u2011\n${\\frac{1}{N}\\left( \\frac{\\partial G}{\\partial\\xi} \\right)}_{T,P,N_{n}} = \\left( \\frac{\\partial G_{m}}{\\partial p_{ij}} \\right)_{T,P,N_{n}} = \\left( \\frac{\\partial G_{m}}{\\partial p_{ij}} \\right)_{T,P,x_{n},p_{kl \\neq ij}} - \\sum_{kl \\neq ij}^{}\\left( \\frac{\\partial G_{m}}{\\partial p_{kl}} \\right)_{T,P,x_{n},p_{op \\neq kl}} + \\left( \\frac{\\partial G_{m}}{\\partial x_{i}} \\right)_{T,P,x_{q \\neq i},p_{kl}} + \\left( \\frac{\\partial G_{m}}{\\partial x_{j}} \\right)_{T,P,x_{q \\neq j},p_{kl}} = 0$\n\nwhere $\\frac{\\partial p_{kl}}{\\partial p_{ij}} = - 1$ and\n$\\frac{\\partial x_{i}}{\\partial p_{ij}} = \\frac{\\partial x_{j}}{\\partial p_{ij}} = 1$\nare used from and . Numerical values of $p_{ij}$ can be obtained by\nminimization of Gibbs energy under constraints given by and .\n\nThe chemical potential of independent component $i$ is defined as in and\ncan be represented by the following equation\n\nEq. \u2011\n$\\mu_{i} = G_{m} + \\frac{\\partial G_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial G_{m}}{\\partial x_{j}} + 2\\sum_{j = 1}^{c}\\frac{\\partial G_{m}}{\\partial p_{ij}} - \\frac{\\partial G_{m}}{\\partial p_{ii}} - 2\\sum_{j = 1}^{c}x_{j}\\sum_{k = 1}^{c}\\frac{\\partial G_{m}}{\\partial p_{jk}} + \\sum_{j = 1}^{c}x_{j}\\frac{\\partial G_{m}}{\\partial p_{jj}}$\n\nThe stability of the solution can be derived similar to .\n\nWhen the bonding between components becomes very strong, distinctive new\ncomponents may form. They are not independent components and often\ncalled associates. Both the independent and dependent components are\ncollectively called species. The formation of an associate\n$i_{a_{i}}j_{b_{j}}$ consisting of $a_{i}$ mole of $i$ and $a_{j}$ mole\n$j$ can be written as\n\nEq. \u2011 $a_{i}i + a_{j}j = i_{a_{i}}j_{a_{j}}$\n\nGibbs energy of the associate follows the same format of that of a\nstoichiometric phase as shown by ,\n\nEq. \u2011\n${_{\\ }^{0}G}_{i_{a_{i}}j_{a_{j}}} = \\sum_{}^{}a_{i}{_{\\ }^{0}G}_{i}^{SER} + \\mathrm{\\Delta}_{f}G_{i_{a_{i}}j_{a_{j}}}$\n\nGibbs energy of the solution is obtained by extending to all species\n\nEq. \u2011\n$G_{m} = \\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}_{\\ }^{0}G_{i_{a_{i}}j_{a_{j}}}} + RT\\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}\\ln y_{i_{a_{i}}j_{a_{j}}}} +_{\\ }^{E}G_{m}$\n\nwhere $y_{i_{a_{i}}j_{a_{j}}}$ is the mole fraction of species\n$i_{a_{i}}j_{a_{j}}$ in the solution with $a_{i} = 1$ and $a_{j} = 0$\nfor component $i$ and $a_{i} = 0$ and $a_{j} = 1$ for component $j$. The\nequilibrium amount of each associate $i_{a_{i}}j_{a_{j}}$ is obtained in\ncombination of mass balance and the zero driving force for the variation\nof the amount of the associate similar to , i.e.\n\nEq. \u2011 $\\sum_{i}^{}{\\sum_{j}^{}y_{i_{a_{i}}j_{a_{j}}}} = 1$\n\nEq. \u2011 $x_{i} = \\sum_{}^{}{a_{i}y_{i_{a_{i}}j_{a_{j}}}}$\n\nEq. \u2011\n$\\left( \\frac{{\\partial G}_{m}}{\\partial y_{i_{a_{i}}j_{a_{j}}}} \\right)_{x_{i}} = 0$\n\nAssociates are particularly plentiful in a gas phase, and their amounts\nare significantly affected by pressure. For an ideal gas phase with\n$_{\\ }^{E}G_{m} = 0$ and $PV_{m} = RT$, the effect of pressure is added\nas follows\n\nEq. \u2011\n$G_{m} = \\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}_{\\ }^{0}G_{i_{a_{i}}j_{a_{j}}}} + RT\\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}\\ln y_{i_{a_{i}}j_{a_{j}}}} + \\int_{P_{0}}^{P}{V_{m}dP} = \\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}_{\\ }^{0}G_{i_{a_{i}}j_{a_{j}}}} + RT\\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}\\ln y_{i_{a_{i}}j_{a_{j}}}} + RT\\ln\\frac{P}{P_{0}}$\n\nwhere $P$ is the total pressure, and $P_{0}$ the reference pressure at\nwhich $_{\\ }^{0}G_{i_{a_{i}}j_{a_{j}}}$ is defined, usually chosen to be\none atmospheric pressure. thus becomes\n\nEq. \u2011\n$G_{m} = \\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}_{\\ }^{0}{G_{i_{a_{i}}j_{a_{j}}}(P = 1atm)}} + RT\\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}\\ln y_{i_{a_{i}}j_{a_{j}}}} + RTlnP$\n\nwith the unit of the total pressure $P$ being in atmospheric pressure\n(atm). The chemical potential of species $i_{a_{i}}j_{a_{j}}$ is equal\nto\n\nEq. \u2011\n$\\mu_{i_{a_{i}}j_{a_{j}}} =_{\\ }^{0}{G_{i_{a_{i}}j_{a_{j}}}(P = 1atm)} + RTlny_{i_{a_{i}}j_{a_{j}}}P =_{\\ }^{0}{G_{i_{a_{i}}j_{a_{j}}}(P = 1atm)} + RTlnP_{i_{a_{i}}j_{a_{j}}}$\n\nwhere $P_{i_{a_{i}}j_{a_{j}}}$ is the partial pressure of species\n$i_{a_{i}}j_{a_{j}}$ defined as\n\nEq. \u2011 $P_{i_{a_{i}}j_{a_{j}}} = y_{i_{a_{i}}j_{a_{j}}}P$\n\nCombining and with , the relation between chemical potentials of an\nassociate and its constituents is expressed as\n\nEq. \u2011\n$\\mu_{i_{a_{i}}j_{a_{j}}} = a_{i}\\mu_{i} + {a_{j}\\mu}_{j} = a_{i}_{\\ }^{0}G_{i} + a_{j}_{\\ }^{0}G_{j} + RTln\\left( {P_{i}}^{a_{i}}{P_{j}}^{a_{j}} \\right)$\n\nThe equilibrium condition for the chemical reaction of an associate\nforming from its constituents in an ideal gas phase is obtained as\n\nEq. \u2011\n$\\mathrm{\\Delta}_{f}G_{i_{a_{i}}j_{a_{j}}} + RTln\\frac{{P_{i}}^{a_{i}}{P_{j}}^{a_{j}}}{P_{i_{a_{i}}j_{a_{j}}}} = 0$\n\nFor non-ideal phases, the mole fractions of various associates can be\ncalculated numerically through the minimization of Gibbs energy under\nthe constraints of and .\n", + "source": [ + "### Solutions with short-range ordering\n", + "\n", + "The order in a system can be measured by correlation functions which\n", + "describes how various components are correlated in space. For\n", + "simplicity, let us consider only the pairs between nearest neighbors\n", + "with the correlation function represented by the pair probability of\n", + "nearest neighbor bonds between two components. In a random solution, the\n", + "probability to find nearest neighbor bonds between two components $i$\n", + "and $j$ is\n", + "\n", + "Eq. \u2011 $p_{ij} = x_{i}x_{j}$\n", + "\n", + "When $p_{ij} \\neq x_{i}x_{j}$, the nearest neighbors of component $i$\n", + "are not occupied randomly by component $j$, rather certain components\n", + "are favored, resulting in short-range ordering or local clustering in\n", + "the solution. When short-range ordering develops throughout the\n", + "solution, long-range ordering takes place, and each component has its\n", + "own primary sites in the solution, to be discussed in Chapter . There\n", + "are relations between bond probabilities and mole fractions of\n", + "components due to the mass balance as follows, with the assumption of\n", + "$p_{ij} = p_{ji}$\n", + "\n", + "Eq. \u2011 $\\sum_{i}^{}{\\sum_{j}^{}p_{ij}} = 1$\n", + "\n", + "Eq. \u2011 $x_{i} = \\sum_{j}^{}p_{ij}$\n", + "\n", + "For small deviations from a random solution, one can consider the\n", + "formation of $i - j$ bonds from $i - i\\ $ and $j - j$ bonds and the\n", + "ideal mixing of three types of bonds, similar to a typical Ising model.\n", + "The bond reaction can be written as\n", + "\n", + "Eq. \u2011 $(i - i\\ )bonds + (j - j)\\ bonds = 2\\ (i - j)\\ bonds$\n", + "\n", + "with the Gibbs energy of reaction being\n", + "\n", + "Eq. \u2011\n", + "${\\mathrm{\\Delta}G}_{ij} = 2G_{ij} - \\left( G_{ii} + G_{jj} \\right)\\ $\n", + "\n", + "Gibbs energy of the solution per mole of atom is thus represented by the\n", + "bond energies and the ideal mixing of bonds plus non-ideal interactions\n", + "between pairs,\n", + "\n", + "Eq. \u2011\n", + "$G_{m} = \\sum_{i}^{}{\\sum_{j}^{}{p_{ij}G_{ij}}} + \\frac{Z}{2}RT\\sum_{i}^{}{\\sum_{j}^{}{p_{ij}\\ln p_{ij}}} +_{\\ }^{E}G_{m}$\n", + "\n", + "with $G_{ij}$ being the molar bond energy between components $i$ and\n", + "$j$, $Z$ the number of bonds per atom which is divided by two in the\n", + "equation due to two atoms needed to form one bond, and\n", + "$_{\\ }^{E}G_{m} = \\sum_{}^{}{p_{ij}p_{kl}I_{ijkl}}$ the excess Gibbs\n", + "energy of mixing between bonds. This approach proposed by Guggenheim\n", + "\\[2\\] is called quasi-chemical method as it is based on the chemical\n", + "reaction shown by .\n", + "\n", + "However, the entropy of mixing in does not reduce to ideal entropy of\n", + "mixing for a solution without short-range ordering as defined by . An\n", + "approximated correction may be added for small degree of short-range\n", + "ordering as follows\n", + "\n", + "Eq. \u2011\n", + "$G_{m} = \\sum_{i}^{}{\\sum_{j}^{}{p_{ij}G_{ij}}} + \\frac{Z}{2}RT\\sum_{i}^{}{\\sum_{j}^{}{p_{ij}\\ln\\frac{p_{ij}}{x_{i}x_{j}}}} + RT\\sum_{}^{}{x_{i}l{nx}_{i}} +_{\\ }^{E}G_{m}$\n", + "\n", + "For a random solution defined by , becomes\n", + "\n", + "Eq. \u2011\n", + "$G_{m} = \\sum_{}^{}{x_{i}_{\\ }^{0}G_{i}} + RT\\sum_{}^{}{x_{i}l{nx}_{i}} + \\sum_{}^{}{x_{i}x_{j}{\\mathrm{\\Delta}G}_{ij}} +_{\\ }^{E}G_{m}$\n", + "\n", + "with $_{\\ }^{0}G_{i} = G_{ii}$, ${\\mathrm{\\Delta}G}_{ij}$ from Eq. 2\u2011107\n", + "representing the interaction parameter between components $i$ and $j$,\n", + "and $_{\\ }^{E}G_{m} = \\sum_{}^{}{x_{i}{x_{j}x_{k}x}_{l}I_{ijkl}}$\n", + "denoting the higher order interactions, in comparison with .\n", + "\n", + "When short-range ordering exists in a solution, one typically uses the\n", + "law of mass reaction for the chemical reaction represented by to define\n", + "the equilibrium among all bonds, i.e.\n", + "\n", + "Eq. \u2011\n", + "$\\frac{\\left( p_{ij} \\right)^{2}}{p_{ii}p_{jj}} = e^{- \\frac{{\\mathrm{\\Delta}G}_{ij}}{kT}}$\n", + "\n", + "However, this is under the assumption that the chemical activities of\n", + "all bonds can be represented by their respective probabilities, which is\n", + "only true for an ideal solution even excluding dilute solutions due to\n", + "the Henry\u2019s law shown by . Preferrably, the bond probabilities can be\n", + "obtained by calculating the driving force for the fluctuation of bond\n", + "probabilities under constant temperature, pressure, and amount of each\n", + "component along with the constraints defined by and and equating the\n", + "driving force to zero, i.e.\n", + "\n", + "Eq. \u2011\n", + "${\\frac{1}{N}\\left( \\frac{\\partial G}{\\partial\\xi} \\right)}_{T,P,N_{n}} = \\left( \\frac{\\partial G_{m}}{\\partial p_{ij}} \\right)_{T,P,N_{n}} = \\left( \\frac{\\partial G_{m}}{\\partial p_{ij}} \\right)_{T,P,x_{n},p_{kl \\neq ij}} - \\sum_{kl \\neq ij}^{}\\left( \\frac{\\partial G_{m}}{\\partial p_{kl}} \\right)_{T,P,x_{n},p_{op \\neq kl}} + \\left( \\frac{\\partial G_{m}}{\\partial x_{i}} \\right)_{T,P,x_{q \\neq i},p_{kl}} + \\left( \\frac{\\partial G_{m}}{\\partial x_{j}} \\right)_{T,P,x_{q \\neq j},p_{kl}} = 0$\n", + "\n", + "where $\\frac{\\partial p_{kl}}{\\partial p_{ij}} = - 1$ and\n", + "$\\frac{\\partial x_{i}}{\\partial p_{ij}} = \\frac{\\partial x_{j}}{\\partial p_{ij}} = 1$\n", + "are used from and . Numerical values of $p_{ij}$ can be obtained by\n", + "minimization of Gibbs energy under constraints given by and .\n", + "\n", + "The chemical potential of independent component $i$ is defined as in and\n", + "can be represented by the following equation\n", + "\n", + "Eq. \u2011\n", + "$\\mu_{i} = G_{m} + \\frac{\\partial G_{m}}{\\partial x_{i}} - \\sum_{j = 1}^{c}x_{j}\\frac{\\partial G_{m}}{\\partial x_{j}} + 2\\sum_{j = 1}^{c}\\frac{\\partial G_{m}}{\\partial p_{ij}} - \\frac{\\partial G_{m}}{\\partial p_{ii}} - 2\\sum_{j = 1}^{c}x_{j}\\sum_{k = 1}^{c}\\frac{\\partial G_{m}}{\\partial p_{jk}} + \\sum_{j = 1}^{c}x_{j}\\frac{\\partial G_{m}}{\\partial p_{jj}}$\n", + "\n", + "The stability of the solution can be derived similar to .\n", + "\n", + "When the bonding between components becomes very strong, distinctive new\n", + "components may form. They are not independent components and often\n", + "called associates. Both the independent and dependent components are\n", + "collectively called species. The formation of an associate\n", + "$i_{a_{i}}j_{b_{j}}$ consisting of $a_{i}$ mole of $i$ and $a_{j}$ mole\n", + "$j$ can be written as\n", + "\n", + "Eq. \u2011 $a_{i}i + a_{j}j = i_{a_{i}}j_{a_{j}}$\n", + "\n", + "Gibbs energy of the associate follows the same format of that of a\n", + "stoichiometric phase as shown by ,\n", + "\n", + "Eq. \u2011\n", + "${_{\\ }^{0}G}_{i_{a_{i}}j_{a_{j}}} = \\sum_{}^{}a_{i}{_{\\ }^{0}G}_{i}^{SER} + \\mathrm{\\Delta}_{f}G_{i_{a_{i}}j_{a_{j}}}$\n", + "\n", + "Gibbs energy of the solution is obtained by extending to all species\n", + "\n", + "Eq. \u2011\n", + "$G_{m} = \\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}_{\\ }^{0}G_{i_{a_{i}}j_{a_{j}}}} + RT\\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}\\ln y_{i_{a_{i}}j_{a_{j}}}} +_{\\ }^{E}G_{m}$\n", + "\n", + "where $y_{i_{a_{i}}j_{a_{j}}}$ is the mole fraction of species\n", + "$i_{a_{i}}j_{a_{j}}$ in the solution with $a_{i} = 1$ and $a_{j} = 0$\n", + "for component $i$ and $a_{i} = 0$ and $a_{j} = 1$ for component $j$. The\n", + "equilibrium amount of each associate $i_{a_{i}}j_{a_{j}}$ is obtained in\n", + "combination of mass balance and the zero driving force for the variation\n", + "of the amount of the associate similar to , i.e.\n", + "\n", + "Eq. \u2011 $\\sum_{i}^{}{\\sum_{j}^{}y_{i_{a_{i}}j_{a_{j}}}} = 1$\n", + "\n", + "Eq. \u2011 $x_{i} = \\sum_{}^{}{a_{i}y_{i_{a_{i}}j_{a_{j}}}}$\n", + "\n", + "Eq. \u2011\n", + "$\\left( \\frac{{\\partial G}_{m}}{\\partial y_{i_{a_{i}}j_{a_{j}}}} \\right)_{x_{i}} = 0$\n", + "\n", + "Associates are particularly plentiful in a gas phase, and their amounts\n", + "are significantly affected by pressure. For an ideal gas phase with\n", + "$_{\\ }^{E}G_{m} = 0$ and $PV_{m} = RT$, the effect of pressure is added\n", + "as follows\n", + "\n", + "Eq. \u2011\n", + "$G_{m} = \\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}_{\\ }^{0}G_{i_{a_{i}}j_{a_{j}}}} + RT\\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}\\ln y_{i_{a_{i}}j_{a_{j}}}} + \\int_{P_{0}}^{P}{V_{m}dP} = \\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}_{\\ }^{0}G_{i_{a_{i}}j_{a_{j}}}} + RT\\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}\\ln y_{i_{a_{i}}j_{a_{j}}}} + RT\\ln\\frac{P}{P_{0}}$\n", + "\n", + "where $P$ is the total pressure, and $P_{0}$ the reference pressure at\n", + "which $_{\\ }^{0}G_{i_{a_{i}}j_{a_{j}}}$ is defined, usually chosen to be\n", + "one atmospheric pressure. thus becomes\n", + "\n", + "Eq. \u2011\n", + "$G_{m} = \\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}_{\\ }^{0}{G_{i_{a_{i}}j_{a_{j}}}(P = 1atm)}} + RT\\sum_{}^{}{y_{i_{a_{i}}j_{a_{j}}}\\ln y_{i_{a_{i}}j_{a_{j}}}} + RTlnP$\n", + "\n", + "with the unit of the total pressure $P$ being in atmospheric pressure\n", + "(atm). The chemical potential of species $i_{a_{i}}j_{a_{j}}$ is equal\n", + "to\n", + "\n", + "Eq. \u2011\n", + "$\\mu_{i_{a_{i}}j_{a_{j}}} =_{\\ }^{0}{G_{i_{a_{i}}j_{a_{j}}}(P = 1atm)} + RTlny_{i_{a_{i}}j_{a_{j}}}P =_{\\ }^{0}{G_{i_{a_{i}}j_{a_{j}}}(P = 1atm)} + RTlnP_{i_{a_{i}}j_{a_{j}}}$\n", + "\n", + "where $P_{i_{a_{i}}j_{a_{j}}}$ is the partial pressure of species\n", + "$i_{a_{i}}j_{a_{j}}$ defined as\n", + "\n", + "Eq. \u2011 $P_{i_{a_{i}}j_{a_{j}}} = y_{i_{a_{i}}j_{a_{j}}}P$\n", + "\n", + "Combining and with , the relation between chemical potentials of an\n", + "associate and its constituents is expressed as\n", + "\n", + "Eq. \u2011\n", + "$\\mu_{i_{a_{i}}j_{a_{j}}} = a_{i}\\mu_{i} + {a_{j}\\mu}_{j} = a_{i}_{\\ }^{0}G_{i} + a_{j}_{\\ }^{0}G_{j} + RTln\\left( {P_{i}}^{a_{i}}{P_{j}}^{a_{j}} \\right)$\n", + "\n", + "The equilibrium condition for the chemical reaction of an associate\n", + "forming from its constituents in an ideal gas phase is obtained as\n", + "\n", + "Eq. \u2011\n", + "$\\mathrm{\\Delta}_{f}G_{i_{a_{i}}j_{a_{j}}} + RTln\\frac{{P_{i}}^{a_{i}}{P_{j}}^{a_{j}}}{P_{i_{a_{i}}j_{a_{j}}}} = 0$\n", + "\n", + "For non-ideal phases, the mole fractions of various associates can be\n", + "calculated numerically through the minimization of Gibbs energy under\n", + "the constraints of and .\n" + ], "metadata": {} }, { - "id": "b63ee9dd", + "id": "2148b250", "cell_type": "markdown", - "source": "### Solutions with long-range ordering\n\nSo far, solutions in which a component can occupy any site in a phase\nare discussed. In many phases, this is not the case. For example, in the\nfcc solid solution of Fe and C, Fe atoms take the fcc lattice sites, and\nC atoms occupy the interstitial sites between the fcc lattice sites.\nTherefore, Fe atoms do not mix with C atoms on the fcc lattice sites,\nand they rather develop long-rang ordering by occupying their own\ndistinct sites in the phase. Long-range ordering can also develop when\nshort-range ordering extends to the whole lattice. New formula for the\nGibbs energy of mixing is needed by considering the details how\ncomponents are distributed and mixed in various sites in a phase.\n\nOne way to group various sites in a phase is based on equivalent\ncrystallographic positions in a phase, i.e. Wyckoff positions. Various\nsets of equivalent positions divide the lattice into sub-sets of\nlattices. Each set of equivalent positions forms a sublattice. The\ndistributions of components on each sublattice can be represented by\nmole fractions of components in the sublattice, commonly referred to as\nsite fractions and defined as\n\nEq. \u2011 $y_{i}^{t} = \\frac{N_{i}^{t}}{\\sum_{j}^{}N_{j}^{t}}$\n\nEq. \u2011 $\\sum_{i}^{}y_{i}^{t} = 1$\n\nwhere the superscript $t$ denotes the sublattice in which the component\nresides, and the summation is for all species in sublattice $t$\nincluding vacancy. Site fractions and mole fractions are related through\nthe mass balance as follows\n\nEq. \u2011\n$x_{i} = \\frac{\\sum_{}^{}{a^{t}y_{i}^{t}}}{\\sum_{}^{}{a^{t}\\left( 1 - y_{va}^{t} \\right)}}$\n\nwhere $a^{t}$ and $y_{va}^{t}$ are the number of sites and the site\nfraction of vacancy in the sublattice *t*.\n\nRandom solutions form when each component enters all sublattices\nequally. Mole fractions and site fractions thus become identical.\nSolutions with both substitutional and interstitial component, like the\nfcc Fe-C solution mentioned above, can be represented by two\nsublattices. Stoichiometric compounds have its site fractions equal to\nunity in each sublattice. When site fractions in a compound deviate from\nunity, the compound is no longer stoichiometric and develops a\ncomposition range of homogeneity. When the composition range is small,\nthe deviations are often referred to as defects. Since many properties\nof a compound are determined by defects, a distinct field of defect\nchemistry exists, predominantly for charged species. As will be\ndemonstrated in Chapter and the rest of the book, defects can be treated\nas an integral part of the thermodynamics of a phase with more than one\nsublattices.\n\nLet us consider a case where there is only one component in each\nsublattice, which represents one possible stoichiometric composition of\nthe phase and is often called an end-member of the phase. The Gibbs\nenergy of an end-member is the same as that of a phase with a fixed\ncomposition shown by or or . By re-arranging , Gibbs energy of an\nend-member, $_{\\ }^{0}G_{em}$, is obtained as\n\nEq. \u2011\n$_{\\ }^{0}G_{em} = \\sum_{t}^{}{a^{t}{_{\\ }^{0}G}_{i}^{t,ref}} + \\mathrm{\\Delta}_{f}G_{em}$\n\nwhere ${_{\\ }^{0}G}_{i}^{t,ref}$ represents the Gibbs energy of\ncomponent $i$ in a given reference state which occupies sublattice $t$\nin the end-member. For vacancy, $_{\\ }^{0}G_{Va} = 0$ is defined. The\ncontribution of each end-member to the Gibbs energy of the phase is the\nproduct of site fraction of each component in their respective\nsublattices and the Gibbs energy of the end-member per mole of formula\nunit (*mf*), i.e.\n\nEq. \u2011\n$_{\\ }^{0}G_{mf} = \\sum_{em}^{}\\left( \\prod_{t}^{}y_{i}^{t}_{\\ }^{0}G_{em} \\right)$\n\nThe ideal mixing in each sublattice is similar to that in a random\nsolution with mole fractions substituted by site fractions. The excess\nGibbs energy of mixing consists of two contributions: (i) the mixing in\none sublattice with all other sublattices containing only one component\nin each sublattice and (ii) the mixing simultaneously in more than one\nsublattices. The Gibbs energy of a solution phase with multi-sublattices\ncan thus be written in terms of per mole of formula unit as\n\nEq. \u2011\n$G_{mf} =_{\\ }^{0}G_{mf} + RT\\sum_{t}^{}{a^{t}\\sum_{i}^{}{y_{i}^{t}\\ln y_{i}^{t}}} +_{\\ }^{E}G_{mf}$\n\nwith $_{\\ }^{E}G_{mf}$ given by\n\nEq. \u2011\n$_{\\ }^{E}G_{mf} = \\sum_{t}^{}{\\prod_{s \\neq t}^{}y_{l}^{s}\\sum_{i > j}^{}{\\sum_{j}^{}{y_{i}^{t}y_{j}^{t}L_{i,j:l}^{t}}}} + \\sum_{t}^{}{\\prod_{s \\neq t}^{}y_{l}^{s}\\sum_{i > j}^{}{\\sum_{j > k}^{}{\\sum_{k}^{}{y_{i}^{t}y_{j}^{t}y_{k}^{t}L_{i,j,k:l}^{t}}}}} + \\sum_{t}^{}{\\prod_{s \\neq t,u}^{}y_{l}^{s}\\sum_{i > j}^{}{\\sum_{j > k}^{}{\\sum_{k}^{}{y_{i}^{t}y_{j}^{t}y_{m}^{u}y_{n}^{u}L_{i,j:m,n:l}^{t}}}}}$\n\nThe first term in represents the binary interaction between component\n$i$ and $j$ in sublattice $t$ with sublattice $s$ occupied by component\n$l$ with comma separating interacting components in one sublattice and\ncolon separating sublattices. The product,\n$\\prod_{s \\neq t}^{}y_{l}^{s}$, runs through all other sublattices with\none component in each sublattice, except sublattice $t$ in which the\ninteraction is considered. The second term denotes the ternary\ninteraction among $i$, $j$, and $k$ in sublattice $t$ with sublattice\n$s$ occupied by component $l$. The third term depicts the interactions\nsimultaneously in both sublattice $t$ and $u$, and the product runs\nthrough all other sublattices with one component in each sublattice,\nexcept sublattice $t$ and $u$ in which the interactions are considered.\nThe third term thus partially reflects the short-range ordering among\ncomponents between two sublattices. In principles, high-order\ninteraction parameters such as quaternary, quinary, and multiple\nsublattice interaction parameters could be added, but their\ncontributions to $_{\\ }^{E}G_{mf}$ are small due to the physical\ninsignificance of co-location of four or five components indicated by\nthe product of their site fractions in front of the interaction\nparameters.\n\nIn , the chemical potential of a stoichiometric compound was defined in\nterms of a summation of chemical potential of individual components in\nthe compound because the relative amounts of components are constrained\nby the stoichiometry of the compound and chemical potentials of\nindividual components can not vary independently. By the same token, the\nchemical potential of an end-member in a solution can be written as\n\nEq. \u2011 $\\mu_{em} = G_{em} = \\sum_{t}^{}a^{t}\\mu_{i}^{t}$\n\nwhere $\\mu_{i}^{t}$ is the chemical potential of component $i$ that\noccupies the sublattice $t$ in the end-member, and can be derived using\n\nEq. \u2011\n$\\mu_{i}^{t} = a^{t}_{\\ }^{0}G_{i}^{t,ref} + a^{t}RTlny_{i}^{t} +_{\\ }^{E}G_{mf}^{\\ } + \\frac{\\partial_{\\ }^{E}G_{mf}^{\\ }}{\\partial y_{i}^{t}} - \\sum_{j}^{\\ }y_{j}^{t}\\frac{\\partial_{\\ }^{E}G_{mf}^{\\ }}{\\partial y_{j}^{t}}$\n\nFor constant interaction parameters in , for chemical potential reduces\nto the following expression from\n\nEq. \u2011\n$\\mu_{i}^{t} = a^{t}_{\\ }^{0}G_{i}^{t} + a^{t}RTlny_{i}^{t} + \\sum_{j \\neq i}^{}{\\left( y_{i}^{t} \\right)^{2}L}_{i,j}^{t} + \\sum_{k > j}^{}{\\sum_{j \\neq i}^{}{y_{j}^{t}y_{k}^{t}\\left\\lbrack L_{i,j}^{t} + L_{i,k}^{t} - L_{j,k}^{t} + \\left( 1 - 2y_{i}^{t} \\right)L_{i,j,k}^{t} \\right\\rbrack}}$\n\nThe stability of the solution is defined by\n$\\frac{\\partial\\mu_{em}}{\\partial N_{em}}$ with $N_{em}$ being the moles\nof the end-member in the solution given by\n$N_{em} = N\\prod_{u}^{}y_{j}^{u}$ and\n\nEq. \u2011\n$\\frac{\\partial N_{em}}{\\partial y_{j}^{u}} = \\frac{N\\prod_{t}^{}y_{i}^{t}}{y_{j}^{u}\\left( 1 - \\prod_{t}^{}y_{i}^{t} \\right)}$\n\nFollowing , one obtains\n\nEq. \u2011\n$\\frac{\\partial\\mu_{em}}{\\partial N_{em}} = \\sum_{u}^{}{\\frac{\\partial\\left( \\sum_{t}^{}a^{t}\\mu_{i}^{t} \\right)}{\\partial y_{j}^{u}}\\frac{\\partial y_{j}^{u}}{\\partial N_{em}}} = \\frac{\\left( 1 - \\prod_{t}^{}y_{j}^{t} \\right)}{N\\prod_{t}^{}y_{j}^{t}}\\sum_{u}^{}{a^{u}y_{i}^{u}\\frac{\\partial\\mu_{i}^{u}}{\\partial y_{i}^{u}}} = \\frac{\\left( 1 - \\prod_{t}^{}y_{j}^{t} \\right)}{N\\prod_{t}^{}y_{j}^{t}}\\sum_{u}^{}{a^{u}y_{i}^{u}\\left\\{ \\frac{RT\\left( 1 - y_{i}^{u} \\right)}{y_{i}^{u}} - 2\\sum_{j \\neq i}^{}\\left( y_{j}^{u} \\right)^{2}L_{i,j}^{u} - 2\\sum_{k > j}^{}{\\sum_{j \\neq i}^{}{y_{j}^{u}y_{k}^{u}\\left\\lbrack L_{i,j}^{u} + L_{i,k}^{u} - L_{j,k}^{u} + \\left( 2 - 3y_{i}^{u} \\right)L_{i,j,k}^{u} \\right\\rbrack}} \\right\\}}$\n\nIt is self-evident from that a site fraction is only uniquely defined\nfrom the mole fraction of the component when the component enters into\none sublattice only and does not form any associates. Therefore, in\ngeneral, the distribution of components on sublattices and different\nkinds of molecules can only be obtained by equilibrium calculations, and\nthe thermodynamic properties for such a phase can thus not be\nrepresented in a closed form using mole fractions of independent\ncomponents. This was demonstrated in Chapter when short-range ordering\nexists in solution phases, where the energy minimization procedure was\nused to obtain the distribution of components on different kinds of\nbonds and the amounts of individual associates.\n", + "source": [ + "### Solutions with long-range ordering\n", + "\n", + "So far, solutions in which a component can occupy any site in a phase\n", + "are discussed. In many phases, this is not the case. For example, in the\n", + "fcc solid solution of Fe and C, Fe atoms take the fcc lattice sites, and\n", + "C atoms occupy the interstitial sites between the fcc lattice sites.\n", + "Therefore, Fe atoms do not mix with C atoms on the fcc lattice sites,\n", + "and they rather develop long-rang ordering by occupying their own\n", + "distinct sites in the phase. Long-range ordering can also develop when\n", + "short-range ordering extends to the whole lattice. New formula for the\n", + "Gibbs energy of mixing is needed by considering the details how\n", + "components are distributed and mixed in various sites in a phase.\n", + "\n", + "One way to group various sites in a phase is based on equivalent\n", + "crystallographic positions in a phase, i.e. Wyckoff positions. Various\n", + "sets of equivalent positions divide the lattice into sub-sets of\n", + "lattices. Each set of equivalent positions forms a sublattice. The\n", + "distributions of components on each sublattice can be represented by\n", + "mole fractions of components in the sublattice, commonly referred to as\n", + "site fractions and defined as\n", + "\n", + "Eq. \u2011 $y_{i}^{t} = \\frac{N_{i}^{t}}{\\sum_{j}^{}N_{j}^{t}}$\n", + "\n", + "Eq. \u2011 $\\sum_{i}^{}y_{i}^{t} = 1$\n", + "\n", + "where the superscript $t$ denotes the sublattice in which the component\n", + "resides, and the summation is for all species in sublattice $t$\n", + "including vacancy. Site fractions and mole fractions are related through\n", + "the mass balance as follows\n", + "\n", + "Eq. \u2011\n", + "$x_{i} = \\frac{\\sum_{}^{}{a^{t}y_{i}^{t}}}{\\sum_{}^{}{a^{t}\\left( 1 - y_{va}^{t} \\right)}}$\n", + "\n", + "where $a^{t}$ and $y_{va}^{t}$ are the number of sites and the site\n", + "fraction of vacancy in the sublattice *t*.\n", + "\n", + "Random solutions form when each component enters all sublattices\n", + "equally. Mole fractions and site fractions thus become identical.\n", + "Solutions with both substitutional and interstitial component, like the\n", + "fcc Fe-C solution mentioned above, can be represented by two\n", + "sublattices. Stoichiometric compounds have its site fractions equal to\n", + "unity in each sublattice. When site fractions in a compound deviate from\n", + "unity, the compound is no longer stoichiometric and develops a\n", + "composition range of homogeneity. When the composition range is small,\n", + "the deviations are often referred to as defects. Since many properties\n", + "of a compound are determined by defects, a distinct field of defect\n", + "chemistry exists, predominantly for charged species. As will be\n", + "demonstrated in Chapter and the rest of the book, defects can be treated\n", + "as an integral part of the thermodynamics of a phase with more than one\n", + "sublattices.\n", + "\n", + "Let us consider a case where there is only one component in each\n", + "sublattice, which represents one possible stoichiometric composition of\n", + "the phase and is often called an end-member of the phase. The Gibbs\n", + "energy of an end-member is the same as that of a phase with a fixed\n", + "composition shown by or or . By re-arranging , Gibbs energy of an\n", + "end-member, $_{\\ }^{0}G_{em}$, is obtained as\n", + "\n", + "Eq. \u2011\n", + "$_{\\ }^{0}G_{em} = \\sum_{t}^{}{a^{t}{_{\\ }^{0}G}_{i}^{t,ref}} + \\mathrm{\\Delta}_{f}G_{em}$\n", + "\n", + "where ${_{\\ }^{0}G}_{i}^{t,ref}$ represents the Gibbs energy of\n", + "component $i$ in a given reference state which occupies sublattice $t$\n", + "in the end-member. For vacancy, $_{\\ }^{0}G_{Va} = 0$ is defined. The\n", + "contribution of each end-member to the Gibbs energy of the phase is the\n", + "product of site fraction of each component in their respective\n", + "sublattices and the Gibbs energy of the end-member per mole of formula\n", + "unit (*mf*), i.e.\n", + "\n", + "Eq. \u2011\n", + "$_{\\ }^{0}G_{mf} = \\sum_{em}^{}\\left( \\prod_{t}^{}y_{i}^{t}_{\\ }^{0}G_{em} \\right)$\n", + "\n", + "The ideal mixing in each sublattice is similar to that in a random\n", + "solution with mole fractions substituted by site fractions. The excess\n", + "Gibbs energy of mixing consists of two contributions: (i) the mixing in\n", + "one sublattice with all other sublattices containing only one component\n", + "in each sublattice and (ii) the mixing simultaneously in more than one\n", + "sublattices. The Gibbs energy of a solution phase with multi-sublattices\n", + "can thus be written in terms of per mole of formula unit as\n", + "\n", + "Eq. \u2011\n", + "$G_{mf} =_{\\ }^{0}G_{mf} + RT\\sum_{t}^{}{a^{t}\\sum_{i}^{}{y_{i}^{t}\\ln y_{i}^{t}}} +_{\\ }^{E}G_{mf}$\n", + "\n", + "with $_{\\ }^{E}G_{mf}$ given by\n", + "\n", + "Eq. \u2011\n", + "$_{\\ }^{E}G_{mf} = \\sum_{t}^{}{\\prod_{s \\neq t}^{}y_{l}^{s}\\sum_{i > j}^{}{\\sum_{j}^{}{y_{i}^{t}y_{j}^{t}L_{i,j:l}^{t}}}} + \\sum_{t}^{}{\\prod_{s \\neq t}^{}y_{l}^{s}\\sum_{i > j}^{}{\\sum_{j > k}^{}{\\sum_{k}^{}{y_{i}^{t}y_{j}^{t}y_{k}^{t}L_{i,j,k:l}^{t}}}}} + \\sum_{t}^{}{\\prod_{s \\neq t,u}^{}y_{l}^{s}\\sum_{i > j}^{}{\\sum_{j > k}^{}{\\sum_{k}^{}{y_{i}^{t}y_{j}^{t}y_{m}^{u}y_{n}^{u}L_{i,j:m,n:l}^{t}}}}}$\n", + "\n", + "The first term in represents the binary interaction between component\n", + "$i$ and $j$ in sublattice $t$ with sublattice $s$ occupied by component\n", + "$l$ with comma separating interacting components in one sublattice and\n", + "colon separating sublattices. The product,\n", + "$\\prod_{s \\neq t}^{}y_{l}^{s}$, runs through all other sublattices with\n", + "one component in each sublattice, except sublattice $t$ in which the\n", + "interaction is considered. The second term denotes the ternary\n", + "interaction among $i$, $j$, and $k$ in sublattice $t$ with sublattice\n", + "$s$ occupied by component $l$. The third term depicts the interactions\n", + "simultaneously in both sublattice $t$ and $u$, and the product runs\n", + "through all other sublattices with one component in each sublattice,\n", + "except sublattice $t$ and $u$ in which the interactions are considered.\n", + "The third term thus partially reflects the short-range ordering among\n", + "components between two sublattices. In principles, high-order\n", + "interaction parameters such as quaternary, quinary, and multiple\n", + "sublattice interaction parameters could be added, but their\n", + "contributions to $_{\\ }^{E}G_{mf}$ are small due to the physical\n", + "insignificance of co-location of four or five components indicated by\n", + "the product of their site fractions in front of the interaction\n", + "parameters.\n", + "\n", + "In , the chemical potential of a stoichiometric compound was defined in\n", + "terms of a summation of chemical potential of individual components in\n", + "the compound because the relative amounts of components are constrained\n", + "by the stoichiometry of the compound and chemical potentials of\n", + "individual components can not vary independently. By the same token, the\n", + "chemical potential of an end-member in a solution can be written as\n", + "\n", + "Eq. \u2011 $\\mu_{em} = G_{em} = \\sum_{t}^{}a^{t}\\mu_{i}^{t}$\n", + "\n", + "where $\\mu_{i}^{t}$ is the chemical potential of component $i$ that\n", + "occupies the sublattice $t$ in the end-member, and can be derived using\n", + "\n", + "Eq. \u2011\n", + "$\\mu_{i}^{t} = a^{t}_{\\ }^{0}G_{i}^{t,ref} + a^{t}RTlny_{i}^{t} +_{\\ }^{E}G_{mf}^{\\ } + \\frac{\\partial_{\\ }^{E}G_{mf}^{\\ }}{\\partial y_{i}^{t}} - \\sum_{j}^{\\ }y_{j}^{t}\\frac{\\partial_{\\ }^{E}G_{mf}^{\\ }}{\\partial y_{j}^{t}}$\n", + "\n", + "For constant interaction parameters in , for chemical potential reduces\n", + "to the following expression from\n", + "\n", + "Eq. \u2011\n", + "$\\mu_{i}^{t} = a^{t}_{\\ }^{0}G_{i}^{t} + a^{t}RTlny_{i}^{t} + \\sum_{j \\neq i}^{}{\\left( y_{i}^{t} \\right)^{2}L}_{i,j}^{t} + \\sum_{k > j}^{}{\\sum_{j \\neq i}^{}{y_{j}^{t}y_{k}^{t}\\left\\lbrack L_{i,j}^{t} + L_{i,k}^{t} - L_{j,k}^{t} + \\left( 1 - 2y_{i}^{t} \\right)L_{i,j,k}^{t} \\right\\rbrack}}$\n", + "\n", + "The stability of the solution is defined by\n", + "$\\frac{\\partial\\mu_{em}}{\\partial N_{em}}$ with $N_{em}$ being the moles\n", + "of the end-member in the solution given by\n", + "$N_{em} = N\\prod_{u}^{}y_{j}^{u}$ and\n", + "\n", + "Eq. \u2011\n", + "$\\frac{\\partial N_{em}}{\\partial y_{j}^{u}} = \\frac{N\\prod_{t}^{}y_{i}^{t}}{y_{j}^{u}\\left( 1 - \\prod_{t}^{}y_{i}^{t} \\right)}$\n", + "\n", + "Following , one obtains\n", + "\n", + "Eq. \u2011\n", + "$\\frac{\\partial\\mu_{em}}{\\partial N_{em}} = \\sum_{u}^{}{\\frac{\\partial\\left( \\sum_{t}^{}a^{t}\\mu_{i}^{t} \\right)}{\\partial y_{j}^{u}}\\frac{\\partial y_{j}^{u}}{\\partial N_{em}}} = \\frac{\\left( 1 - \\prod_{t}^{}y_{j}^{t} \\right)}{N\\prod_{t}^{}y_{j}^{t}}\\sum_{u}^{}{a^{u}y_{i}^{u}\\frac{\\partial\\mu_{i}^{u}}{\\partial y_{i}^{u}}} = \\frac{\\left( 1 - \\prod_{t}^{}y_{j}^{t} \\right)}{N\\prod_{t}^{}y_{j}^{t}}\\sum_{u}^{}{a^{u}y_{i}^{u}\\left\\{ \\frac{RT\\left( 1 - y_{i}^{u} \\right)}{y_{i}^{u}} - 2\\sum_{j \\neq i}^{}\\left( y_{j}^{u} \\right)^{2}L_{i,j}^{u} - 2\\sum_{k > j}^{}{\\sum_{j \\neq i}^{}{y_{j}^{u}y_{k}^{u}\\left\\lbrack L_{i,j}^{u} + L_{i,k}^{u} - L_{j,k}^{u} + \\left( 2 - 3y_{i}^{u} \\right)L_{i,j,k}^{u} \\right\\rbrack}} \\right\\}}$\n", + "\n", + "It is self-evident from that a site fraction is only uniquely defined\n", + "from the mole fraction of the component when the component enters into\n", + "one sublattice only and does not form any associates. Therefore, in\n", + "general, the distribution of components on sublattices and different\n", + "kinds of molecules can only be obtained by equilibrium calculations, and\n", + "the thermodynamic properties for such a phase can thus not be\n", + "represented in a closed form using mole fractions of independent\n", + "components. This was demonstrated in Chapter when short-range ordering\n", + "exists in solution phases, where the energy minimization procedure was\n", + "used to obtain the distribution of components on different kinds of\n", + "bonds and the amounts of individual associates.\n" + ], "metadata": {} }, { - "id": "578e73db", + "id": "527f6d64", "cell_type": "markdown", - "source": "### Solutions with both short-range and long-range ordering\n\nThe short-range ordering in a solution with long-rang ordering can take\nplace in each sublattice or between two sublattices. The short-range\nordering in one sublattice can be treated similarly as in Chapter 2.3.1\nwith mole fractions substituted by site fractions. In case that\nassociates form, the relation between mole fractions and site fractions\nbecomes more complicated as follows\n\nEq. \u2011\n$x_{i} = \\frac{\\sum_{}^{}{a^{t}\\sum_{k}^{}{i_{k}y}_{k}^{t}}}{\\sum_{}^{}{a^{t}\\left( 1 - y_{va}^{t} \\right)}}$\n\nwhere the summation for $k$ goes over all associates in sublattice $t$\ncontaining component $i$.\n\nThe short-range ordering between two-sublattices indicates that a\ncomponent in one sublattice has different interactions with different\ncomponents in another sublattice. This results in local ordering of one\ncomponent around another component in two neighboring sublattices. Such\nlocal ordering involves interactions between two sublattices shown as\nthe third term in .\n", + "source": [ + "### Solutions with both short-range and long-range ordering\n", + "\n", + "The short-range ordering in a solution with long-rang ordering can take\n", + "place in each sublattice or between two sublattices. The short-range\n", + "ordering in one sublattice can be treated similarly as in Chapter 2.3.1\n", + "with mole fractions substituted by site fractions. In case that\n", + "associates form, the relation between mole fractions and site fractions\n", + "becomes more complicated as follows\n", + "\n", + "Eq. \u2011\n", + "$x_{i} = \\frac{\\sum_{}^{}{a^{t}\\sum_{k}^{}{i_{k}y}_{k}^{t}}}{\\sum_{}^{}{a^{t}\\left( 1 - y_{va}^{t} \\right)}}$\n", + "\n", + "where the summation for $k$ goes over all associates in sublattice $t$\n", + "containing component $i$.\n", + "\n", + "The short-range ordering between two-sublattices indicates that a\n", + "component in one sublattice has different interactions with different\n", + "components in another sublattice. This results in local ordering of one\n", + "component around another component in two neighboring sublattices. Such\n", + "local ordering involves interactions between two sublattices shown as\n", + "the third term in .\n" + ], "metadata": {} }, { - "id": "180919dd", + "id": "f54abe2b", "cell_type": "markdown", - "source": "### Solutions with charged species\n\nOne special type of solutions with both short-range and long-range\nordering is solutions with charged species, i.e. ionic solutions, plus\nelectrons and holes. There is an additional constraint on species\nconcentrations to maintain the charge neutrality of such solutions, i.e.\n\nEq. \u2011 $0 = \\sum_{t}^{}{\\sum_{i}^{}{a^{t}y_{i}^{t}v_{i}^{t}}}$\n\nwhere $v_{i}^{t}$ is the valance of species $i$ in sublattice $t$\nincluding its sign which is positive for cations, negative for anions,\nand zero for neutral species. The conventional defect chemistry theory\nis typically based on the ideal mass action laws and applicable to a\nsingle set of defects and at very low defect concentrations, i.e. in the\nlimit of ideal solutions. For interacting defects, their concentrations\nshould be replaced by their activities, which can be obtained from\nthermodynamic principles similarly as discussed in previous sections. It\nshould be emphasized that in addition to formation of many more charged\nspecies, one component may have different valences, particularly the\ntransition metals. Consequently, there can be many more species in an\nionic phase than the number of independent components in the system, and\ntheir concentrations can be found by equilibrium calculations as\ndiscussed in Chapter .\n", + "source": [ + "### Solutions with charged species\n", + "\n", + "One special type of solutions with both short-range and long-range\n", + "ordering is solutions with charged species, i.e. ionic solutions, plus\n", + "electrons and holes. There is an additional constraint on species\n", + "concentrations to maintain the charge neutrality of such solutions, i.e.\n", + "\n", + "Eq. \u2011 $0 = \\sum_{t}^{}{\\sum_{i}^{}{a^{t}y_{i}^{t}v_{i}^{t}}}$\n", + "\n", + "where $v_{i}^{t}$ is the valance of species $i$ in sublattice $t$\n", + "including its sign which is positive for cations, negative for anions,\n", + "and zero for neutral species. The conventional defect chemistry theory\n", + "is typically based on the ideal mass action laws and applicable to a\n", + "single set of defects and at very low defect concentrations, i.e. in the\n", + "limit of ideal solutions. For interacting defects, their concentrations\n", + "should be replaced by their activities, which can be obtained from\n", + "thermodynamic principles similarly as discussed in previous sections. It\n", + "should be emphasized that in addition to formation of many more charged\n", + "species, one component may have different valences, particularly the\n", + "transition metals. Consequently, there can be many more species in an\n", + "ionic phase than the number of independent components in the system, and\n", + "their concentrations can be found by equilibrium calculations as\n", + "discussed in Chapter .\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/gibbs_energy_function/polymer_solutions_and_polymer_blends.ipynb b/src/psu410/src/gibbs_energy_function/polymer_solutions_and_polymer_blends.ipynb index 3ebed0f..ccaa83d 100644 --- a/src/psu410/src/gibbs_energy_function/polymer_solutions_and_polymer_blends.ipynb +++ b/src/psu410/src/gibbs_energy_function/polymer_solutions_and_polymer_blends.ipynb @@ -4,9 +4,78 @@ "metadata": {}, "cells": [ { - "id": "0aedfb02", + "id": "7f7d018e", "cell_type": "markdown", - "source": "## Polymer solutions and polymer blends\n\nA polymer solution is a mixture between polymer molecules and solvents,\nwhile a polymer blend is a mixture between different polymer molecules.\nA polymer molecule consists of the same repeating units of one or more\nmonomers, which can be an atom or a small molecule. The number of\nrepeating units is called the degree of polymerization and can be as\nlarge as 104\u2013105. It defines the molecular mass,\ni.e. the mass of one polymer molecule. There are three typical\narchitectures of polymerization: a linear chain, a branched chain, and a\ncross-linked polymer. Nearly all polymers are mixtures of molecules with\na different degree of polymerization with a molecular mass distribution,\ncomplicating the modeling of their thermodynamic properties because of\nthe dependence of properties on molecular mass.\n\nGibbs energy functions of polymers with a single molecular mass can be\ntreated similarly as in Chapter . For a polymer solution, the ideal\nentropy of mixing is quite different from that of atomically random\nsolutions discussed in Chapter because the monomers in a polymer\nmolecule are connected to each other and cannot move freely. One common\napproach to calculate the ideal entropy of mixing is to evoke a lattice\nmodel and assume that one monomer occupies a lattice site with a fixed\nvolume. The number of translational states of a single molecule is equal\nto the number of lattice sites available. In a homogeneous solution, the\ntotal number of lattice sites available is\n\nEq. \u2011 $n = \\sum_{i}^{}{m_{i}n_{i}}$\n\nwhere $n_{i}$ and $m_{i}$ are the number of molecule $i$ and the number\nof lattice sites per molecule $i$, respectively. While in its pure\nstate, i.e. before mixing, the number of states of molecule $i$ in terms\nof the number of lattice sites is\n\nEq. \u2011 $w_{i} = m_{i}n_{i} = n\\varphi_{i}$\n\nwhere $\\varphi_{i}$ is the volume fraction of molecule $i$ in the\nsolution. The entropy change per molecule $i$ is thus\n\nEq. \u2011\n$S_{i} = kln(n) - klnw_{i} = kln\\frac{1}{\\varphi_{i}} = - kln\\varphi_{i}$\n\nThe total entropy of mixing is the summation for all molecules,\nnormalized to per mole of lattice site\n\nEq. \u2011\n$_{}^{I}S_{m} = \\frac{N_{a}}{n}\\sum_{i}^{}{n_{i}S_{i}} = - R\\sum_{i}^{}{\\frac{\\varphi_{i}}{m_{i}}\\ln\\varphi_{i}}$\n\nwhere $N_{a}$ is the Avogadro number. When $m_{i} = 1$ for all\nmolecules, Eq. 2\u2011143 reduces to . Since $m_{i}$ values are typically\nvery large numbers for polymers, the entropy of mixing in polymer\nsolutions and blends is thus significantly lower than those in\nnon-polymer solutions as shown schematically in for binary systems with\nvarious $m_{i}$ values.\n\nFigure \u2011: Schematic entropy of mixing in solutions with the numbers of\nlattice sites per molecule shown.\n\nSimilar to , Gibbs energy of a multi-component random polymer\nsolution/blend can be written as\n\nEq. \u2011\n$G_{m} = \\sum_{}^{}\\frac{\\varphi_{i}}{m_{i}}_{\\ }^{0}G_{im} + RT\\left( \\sum_{}^{}{\\frac{\\varphi_{i}}{m_{i}}\\ln\\varphi_{i}} + \\sum_{}^{}{\\varphi_{i}\\varphi_{j}\\chi_{ij}} \\right)$\n\nwhere $_{\\ }^{0}G_{im}$ is the Gibbs energy of molecular $i$ per mole of\nlattice site, and $\\chi_{ij}$ the unitless interaction parameter between\nmolecule $i$ and $j$. Other equations shown in Chapter can be derived\nsimilarly too. It is to be noted that is very similar to the\nFlory\u2013Huggins solution equation widely used in the polymer community.\n", + "source": [ + "## Polymer solutions and polymer blends\n", + "\n", + "A polymer solution is a mixture between polymer molecules and solvents,\n", + "while a polymer blend is a mixture between different polymer molecules.\n", + "A polymer molecule consists of the same repeating units of one or more\n", + "monomers, which can be an atom or a small molecule. The number of\n", + "repeating units is called the degree of polymerization and can be as\n", + "large as 104\u2013105. It defines the molecular mass,\n", + "i.e. the mass of one polymer molecule. There are three typical\n", + "architectures of polymerization: a linear chain, a branched chain, and a\n", + "cross-linked polymer. Nearly all polymers are mixtures of molecules with\n", + "a different degree of polymerization with a molecular mass distribution,\n", + "complicating the modeling of their thermodynamic properties because of\n", + "the dependence of properties on molecular mass.\n", + "\n", + "Gibbs energy functions of polymers with a single molecular mass can be\n", + "treated similarly as in Chapter . For a polymer solution, the ideal\n", + "entropy of mixing is quite different from that of atomically random\n", + "solutions discussed in Chapter because the monomers in a polymer\n", + "molecule are connected to each other and cannot move freely. One common\n", + "approach to calculate the ideal entropy of mixing is to evoke a lattice\n", + "model and assume that one monomer occupies a lattice site with a fixed\n", + "volume. The number of translational states of a single molecule is equal\n", + "to the number of lattice sites available. In a homogeneous solution, the\n", + "total number of lattice sites available is\n", + "\n", + "Eq. \u2011 $n = \\sum_{i}^{}{m_{i}n_{i}}$\n", + "\n", + "where $n_{i}$ and $m_{i}$ are the number of molecule $i$ and the number\n", + "of lattice sites per molecule $i$, respectively. While in its pure\n", + "state, i.e. before mixing, the number of states of molecule $i$ in terms\n", + "of the number of lattice sites is\n", + "\n", + "Eq. \u2011 $w_{i} = m_{i}n_{i} = n\\varphi_{i}$\n", + "\n", + "where $\\varphi_{i}$ is the volume fraction of molecule $i$ in the\n", + "solution. The entropy change per molecule $i$ is thus\n", + "\n", + "Eq. \u2011\n", + "$S_{i} = kln(n) - klnw_{i} = kln\\frac{1}{\\varphi_{i}} = - kln\\varphi_{i}$\n", + "\n", + "The total entropy of mixing is the summation for all molecules,\n", + "normalized to per mole of lattice site\n", + "\n", + "Eq. \u2011\n", + "$_{}^{I}S_{m} = \\frac{N_{a}}{n}\\sum_{i}^{}{n_{i}S_{i}} = - R\\sum_{i}^{}{\\frac{\\varphi_{i}}{m_{i}}\\ln\\varphi_{i}}$\n", + "\n", + "where $N_{a}$ is the Avogadro number. When $m_{i} = 1$ for all\n", + "molecules, Eq. 2\u2011143 reduces to . Since $m_{i}$ values are typically\n", + "very large numbers for polymers, the entropy of mixing in polymer\n", + "solutions and blends is thus significantly lower than those in\n", + "non-polymer solutions as shown schematically in for binary systems with\n", + "various $m_{i}$ values.\n", + "\n", + "Figure \u2011: Schematic entropy of mixing in solutions with the numbers of\n", + "lattice sites per molecule shown.\n", + "\n", + "Similar to , Gibbs energy of a multi-component random polymer\n", + "solution/blend can be written as\n", + "\n", + "Eq. \u2011\n", + "$G_{m} = \\sum_{}^{}\\frac{\\varphi_{i}}{m_{i}}_{\\ }^{0}G_{im} + RT\\left( \\sum_{}^{}{\\frac{\\varphi_{i}}{m_{i}}\\ln\\varphi_{i}} + \\sum_{}^{}{\\varphi_{i}\\varphi_{j}\\chi_{ij}} \\right)$\n", + "\n", + "where $_{\\ }^{0}G_{im}$ is the Gibbs energy of molecular $i$ per mole of\n", + "lattice site, and $\\chi_{ij}$ the unitless interaction parameter between\n", + "molecule $i$ and $j$. Other equations shown in Chapter can be derived\n", + "similarly too. It is to be noted that is very similar to the\n", + "Flory\u2013Huggins solution equation widely used in the polymer community.\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/laws_of_thermodynamics/combined_law_of_thermodynamics_and_equilibrium_conditions.ipynb b/src/psu410/src/laws_of_thermodynamics/combined_law_of_thermodynamics_and_equilibrium_conditions.ipynb index 7b460eb..c25c002 100644 --- a/src/psu410/src/laws_of_thermodynamics/combined_law_of_thermodynamics_and_equilibrium_conditions.ipynb +++ b/src/psu410/src/laws_of_thermodynamics/combined_law_of_thermodynamics_and_equilibrium_conditions.ipynb @@ -4,9 +4,225 @@ "metadata": {}, "cells": [ { - "id": "b6dde096", + "id": "eb09e8ad", "cell_type": "markdown", - "source": "## Combined law of thermodynamics and equilibrium conditions\n\nFor a system with an irreversible internal process taking place, the\nentropy change in the system thus consists of three parts: the heat\nexchange with the surrounding defined by , the entropy production due to\nthe internal process represented by , and the entropy of mass exchange\nwith the surrounding. The total entropy change of the system can thus be\nwritten as follows\n\nEq. \u2011 $dS = \\frac{dQ}{T} + d_{ip}S + \\sum_{}^{}{S_{i}dN_{i}}$\n\nwhere $S_{i}$ is the unit entropy of component *i* in the surroundings,\noften called partial entropy of component *i*, and will be further\ndiscussed in Chapter\n\nCombining and and re-arranging, one obtains\n\nEq. \u2011 $dQ = TdS - Dd\\xi - \\sum_{}^{}{TS_{i}dN_{i}}$\n\nInserting and into yields the combined law of thermodynamics from the\nfirst and second laws of thermodynamics,\n\nEq. \u2011\n$dU = TdS - PdV + \\sum_{}^{}\\left( H_{i} - TS_{i} \\right){dN}_{i} - Dd\\xi$\n\nThe internal energy of the system is thus a function of *S*, *V*,\n*Ni* and *\u03be* of the system, which are called natural\nvariables of the internal energy, i.e. *U*(*S*,*V*,*Ni*,*\u03be*).\nThe other variables are dependent variables and can be represented by\npartial derivatives of the internal energy with respect to their\nrespective natural variables with other natural variables kept constant\nas shown below\n\nEq. \u2011\n$T = \\left( \\frac{\\partial U}{\\partial S} \\right)_{V,\\ N_{i},\\ \\xi}$\n\nEq. \u2011\n$- P = \\left( \\frac{\\partial U}{\\partial V} \\right)_{S,\\ N_{i},\\ \\xi}$\n\nEq. \u2011\n$\\mu_{i} = H_{i} - TS_{i} = \\left( \\frac{\\partial U}{\\partial N_{i}} \\right)_{S,\\ V,N_{j \\neq i},\\ \\xi} = U_{i}$\n\nEq. \u2011\n$- D = \\left( \\frac{\\partial U}{\\partial\\xi} \\right)_{S,\\ V,N_{i}\\ }$\n\nIn , a new variable, $\\mu_{i}$, is introduced. It is called chemical\npotential and defined as the internal energy change with respect of the\naddition of the component *i* when the entropy, volume and the amount of\nother components of the system are kept constant. It may be worth\npointing out that for a system at equilibrium, i.e. $d_{ip}S = 0$, and\nwith constant entropy, $dS = 0$, if it exchanges mass with the\nsurroundings, $dN_{i} \\neq 0$, the system must also exchange heat with\nthe surroundings at the same time in order to keep the entropy invariant\nas demonstrated by .\n\nThe pairs of the natural variables and their corresponding partial\nderivatives are called conjugate variables, i.e. *S* and *T*, *V* and\n*\u2013P*, *Ni* and $\\mu_{i}$, and *\u03be* and *\u2013D*. There are minus\nsign in front of *P* and *D* as the increase of volume and the progress\nof the internal process decrease the internal energy of the system. The\nimportance of this conjugate relation will be evident when various forms\nof combined thermodynamic laws and various types of phase diagrams are\nintroduced in the book.\n\nThe last pair of conjugate variables, *\u03be* and *\u2013D*, is worthy of further\ndiscussion. Based on the second law of thermodynamics, i.e. , no\ninternal processes take place spontaneously if there is no entropy\nproductions, i.e. D\u22640 or *d\u03be*=0 and *D*\\>0. With D\u22640, there is no\ndriving for any internal processes, and the system is at a full\nequilibrium state. The last term in drops off, and *\u03be* becomes a\ndependent variable of the system and can be calculated from the\nequilibrium conditions. With *d\u03be*=0 and *D*\\>0, the system is under a\nconstrained equilibrium or freezing-in condition when the internal\nprocess is constrained not to take place, and *\u03be* remains to be an\nindependent variable of the system.\n\nThese two cases represent the two branches of thermodynamics:\nequilibrium, reversible thermodynamics and irreversible thermodynamics.\nIt is clear from the above discussions that these two branches are\nidentical if the internal energy is not only a function of *S*, *V*, and\n*Ni* , but also any internal process variable *\u03be*. This means\nthat one should be able to evaluate the internal energy of a system for\nany freezing-in equilibrium conditions in addition to the full\nequilibrium condition. In the rest of the book, the freezing-in\nequilibrium and full equilibrium are not differentiated unless\nspecified.\n\nAs the mechanical work under hydrostatic pressure is very important in\nexperiments, let us define a new quantity called enthalpy as follows\n\nEq. \u2011 $H = U + PV$\n\nIts differential form can be obtained from as\n\nEq. \u2011 $dH = dU + d(PV) = dQ + VdP + \\sum_{}^{}H_{i}{dN}_{i}$\n\nThere are two significant consequences of the above equation. First, for\na close system under constant pressure, i.e. ${dN}_{i} = dP = 0$, one\nhas $dH = dQ$. This implies that the enthalpy change in a system is\nequal to the heat exchange between the system and the surrounding of the\nsystem, which is why enthalpy and heat are often exchangeable in the\nliterature. Second, for an adiabatic system under constant pressure,\ni.e. $dQ = dP = 0$, can be re-arranged to the following equation\n\nEq. \u2011\n$H_{i} = \\left( \\frac{\\partial H}{\\partial N_{i}} \\right)_{N_{j \\neq i,\\ \\ dQ = dP = 0}}$\n\n$H_{i}$ is thus the partial enthalpy of component *i* and will be\nfurther discussed in Chapter . The chemical potential of component *i*\ndefined in is thus related to the partial enthalpy and partial entropy\nof the component.\n\nTo further define equilibrium conditions of a system, consider a\nhomogeneous system in a state of internal equilibrium, i.e. no\nspontaneous internal processes are possible with $Dd\\xi = 0$, and\nbecomes\n\nEq. \u2011\n$dU = TdS - PdV + \\sum_{}^{}\\mu_{i}{dN}_{i} = \\sum_{}^{}{Y_{i}dX_{i}}$\n\nwhere *X* represents *S*, *V*, *Ni*, and *Y* their conjugate\nvariables *T*, *-P*, $\\mu_{i}$. The state of the system with *c*\nindependent components is completely determined by the *c+2* variables,\ni.e. *S*, *V*, and *Ni* with *i* from 1 to *c*.\n\nTo simplify the situation, let us limit the discussion to an isolated\nequilibrium system, i.e. $dU = 0$, and conduct a virtual internal\nexperiment inside the system by moving an infinitesimal amount of\n$X_{i}$, ${dX}_{i}$, with other $X_{j}$ kept constant, from one region\nof the system to another region of the system as schematically shown in\n.\n\nFigure \u2011: Virtual experiment for a system at equilibrium\n\nAs the system is homogeneous and at equilibrium,\n$- dX_{i}^{'} = dX_{i}^{\"} = dX_{i}$. The total change of the internal\nenergy of this internal process is the combination of the changes in the\ntwo regions, i.e.\n\nEq. \u2011\n$dU = dU^{'} + dU^{\"} = Y_{i}^{'}dX_{i}^{'} + Y_{i}^{\"}dX_{i}^{\"} = \\left( - Y_{i}^{'} + Y_{i}^{\"} \\right)dX_{i} = 0$\n\nTherefore, $Y_{i}^{'} = Y_{i}^{\"}$ for *T*, *-P*, and $\\mu_{i}$,\nindicating that *T*, *-P*, and $\\mu_{i}$ are homogeneous in the system,\nrespectively, and are thus named as potentials of the system.\nFurthermore these potentials are independent of the size of the system\nand are often referred to as intensive variables in the literature. On\nthe other hand, all *X:s*, i.e. *S*, *V*, and *Ni*, are\nproportional to the size of the system and may be normalized with\nrespect to the size of the system, usually in terms of total moles,\n\nEq. \u2011 $N = \\sum_{}^{}N_{i}$\n\nThey are thus called molar quantities and often referred to as extensive\nvariables, and the respective normalized variables are molar entropy,\nmolar volume, and mole fractions, defined as follows\n\nEq. \u2011 $S_{m} = \\frac{S}{N}$\n\nEq. \u2011 $V_{m} = \\frac{V}{N}$\n\nEq. \u2011 $x_{i} = \\frac{N_{i}}{N}$\n\nConsider a small subsystem in this homogeneous system at equilibrium and\nlet the subsystem grow in size. The entropy, volume, and mass enclosed\nin the subsystem increase as follows\n\nEq. \u2011 ${dS = S}_{m}dN$\n\nEq. \u2011 $dV = V_{m}dN$\n\nEq. \u2011 $dN_{i} = x_{i}dN$\n\nThe corresponding change in the internal energy of the subsystem becomes\n\nEq. \u2011\n$dU = TdS - PdV + \\sum_{}^{}\\mu_{i}{dN}_{i} = \\left( TS_{m} - PV_{m} + \\sum_{}^{}\\mu_{i}x_{i} \\right)dN = U_{m}dN$\n\nBy integration one obtains the integral form of the internal energy as\n\nEq. \u2011\n$U = TS - PV + \\sum_{}^{}\\mu_{i}N_{i} = \\left( TS_{m} - PV_{m} + \\sum_{}^{}\\mu_{i}x_{i} \\right)N = U_{m}N$\n\nSimilarly, the molar enthalpy can be defined as follows\n\nEq. \u2011\n$H = U + PV = U_{m}N + PV_{m}N = \\left( U_{m} + PV_{m} \\right)N = H_{m}N$\n\nIn case a potential is not homogeneous in a system, the system will not\nbe in a state of equilibrium. Let us consider the same virtual\nexperiment as shown in for an isolated system that is not in\nequilibrium, i.e. by moving an infinitesimal amount of $X_{i}$,\n${dX}_{i}$, with other $X_{j}$ kept constant, from one region of the\nsystem to another region of the system with the two regions having\ndifferent potentials. The total internal energy change equals to zero as\nthe virtual experiment has $dU = 0$. Similarly, each region can be\nconsidered to be homogeneous by itself, and one has\n$- dX_{i}^{'} = dX_{i}^{\"} = dX_{i}$. The total internal energy change\nin the system is thus the sum of these two regions plus the entropy\nproduction due to the internal process with $d\\xi = dX_{i}$, i.e.\n\nEq. \u2011\n$dU = dU^{'} + dU^{\"} + Dd\\xi = Y_{i}^{'}dX_{i}^{'} + Y_{i}^{\"}dX_{i}^{\"} + Dd\\xi = \\left( - Y_{i}^{'} + Y_{i}^{\"} \\right)dX_{i} + Dd\\xi = 0$\n\nConsequently, one obtains the following\n\n*Eq. 1\u201131* $D = Y_{i}^{'} - Y_{i}^{\"}$\n\nThe driving force thus represents the difference of the potential at the\ntwo regions, and the internal process is to eliminate inhomogeneity of\nthe potential with the heat transfer from high temperature regions to\nlow temperature regions, volume shrink of low pressure regions (high\n$\u2013P$) and volume expansion of high pressure regions (low $\u2013P$), and\ntransport of components from high chemical potential regions to low\nchemical potential regions.\n", + "source": [ + "## Combined law of thermodynamics and equilibrium conditions\n", + "\n", + "For a system with an irreversible internal process taking place, the\n", + "entropy change in the system thus consists of three parts: the heat\n", + "exchange with the surrounding defined by , the entropy production due to\n", + "the internal process represented by , and the entropy of mass exchange\n", + "with the surrounding. The total entropy change of the system can thus be\n", + "written as follows\n", + "\n", + "Eq. \u2011 $dS = \\frac{dQ}{T} + d_{ip}S + \\sum_{}^{}{S_{i}dN_{i}}$\n", + "\n", + "where $S_{i}$ is the unit entropy of component *i* in the surroundings,\n", + "often called partial entropy of component *i*, and will be further\n", + "discussed in Chapter\n", + "\n", + "Combining and and re-arranging, one obtains\n", + "\n", + "Eq. \u2011 $dQ = TdS - Dd\\xi - \\sum_{}^{}{TS_{i}dN_{i}}$\n", + "\n", + "Inserting and into yields the combined law of thermodynamics from the\n", + "first and second laws of thermodynamics,\n", + "\n", + "Eq. \u2011\n", + "$dU = TdS - PdV + \\sum_{}^{}\\left( H_{i} - TS_{i} \\right){dN}_{i} - Dd\\xi$\n", + "\n", + "The internal energy of the system is thus a function of *S*, *V*,\n", + "*Ni* and *\u03be* of the system, which are called natural\n", + "variables of the internal energy, i.e. *U*(*S*,*V*,*Ni*,*\u03be*).\n", + "The other variables are dependent variables and can be represented by\n", + "partial derivatives of the internal energy with respect to their\n", + "respective natural variables with other natural variables kept constant\n", + "as shown below\n", + "\n", + "Eq. \u2011\n", + "$T = \\left( \\frac{\\partial U}{\\partial S} \\right)_{V,\\ N_{i},\\ \\xi}$\n", + "\n", + "Eq. \u2011\n", + "$- P = \\left( \\frac{\\partial U}{\\partial V} \\right)_{S,\\ N_{i},\\ \\xi}$\n", + "\n", + "Eq. \u2011\n", + "$\\mu_{i} = H_{i} - TS_{i} = \\left( \\frac{\\partial U}{\\partial N_{i}} \\right)_{S,\\ V,N_{j \\neq i},\\ \\xi} = U_{i}$\n", + "\n", + "Eq. \u2011\n", + "$- D = \\left( \\frac{\\partial U}{\\partial\\xi} \\right)_{S,\\ V,N_{i}\\ }$\n", + "\n", + "In , a new variable, $\\mu_{i}$, is introduced. It is called chemical\n", + "potential and defined as the internal energy change with respect of the\n", + "addition of the component *i* when the entropy, volume and the amount of\n", + "other components of the system are kept constant. It may be worth\n", + "pointing out that for a system at equilibrium, i.e. $d_{ip}S = 0$, and\n", + "with constant entropy, $dS = 0$, if it exchanges mass with the\n", + "surroundings, $dN_{i} \\neq 0$, the system must also exchange heat with\n", + "the surroundings at the same time in order to keep the entropy invariant\n", + "as demonstrated by .\n", + "\n", + "The pairs of the natural variables and their corresponding partial\n", + "derivatives are called conjugate variables, i.e. *S* and *T*, *V* and\n", + "*\u2013P*, *Ni* and $\\mu_{i}$, and *\u03be* and *\u2013D*. There are minus\n", + "sign in front of *P* and *D* as the increase of volume and the progress\n", + "of the internal process decrease the internal energy of the system. The\n", + "importance of this conjugate relation will be evident when various forms\n", + "of combined thermodynamic laws and various types of phase diagrams are\n", + "introduced in the book.\n", + "\n", + "The last pair of conjugate variables, *\u03be* and *\u2013D*, is worthy of further\n", + "discussion. Based on the second law of thermodynamics, i.e. , no\n", + "internal processes take place spontaneously if there is no entropy\n", + "productions, i.e. D\u22640 or *d\u03be*=0 and *D*\\>0. With D\u22640, there is no\n", + "driving for any internal processes, and the system is at a full\n", + "equilibrium state. The last term in drops off, and *\u03be* becomes a\n", + "dependent variable of the system and can be calculated from the\n", + "equilibrium conditions. With *d\u03be*=0 and *D*\\>0, the system is under a\n", + "constrained equilibrium or freezing-in condition when the internal\n", + "process is constrained not to take place, and *\u03be* remains to be an\n", + "independent variable of the system.\n", + "\n", + "These two cases represent the two branches of thermodynamics:\n", + "equilibrium, reversible thermodynamics and irreversible thermodynamics.\n", + "It is clear from the above discussions that these two branches are\n", + "identical if the internal energy is not only a function of *S*, *V*, and\n", + "*Ni* , but also any internal process variable *\u03be*. This means\n", + "that one should be able to evaluate the internal energy of a system for\n", + "any freezing-in equilibrium conditions in addition to the full\n", + "equilibrium condition. In the rest of the book, the freezing-in\n", + "equilibrium and full equilibrium are not differentiated unless\n", + "specified.\n", + "\n", + "As the mechanical work under hydrostatic pressure is very important in\n", + "experiments, let us define a new quantity called enthalpy as follows\n", + "\n", + "Eq. \u2011 $H = U + PV$\n", + "\n", + "Its differential form can be obtained from as\n", + "\n", + "Eq. \u2011 $dH = dU + d(PV) = dQ + VdP + \\sum_{}^{}H_{i}{dN}_{i}$\n", + "\n", + "There are two significant consequences of the above equation. First, for\n", + "a close system under constant pressure, i.e. ${dN}_{i} = dP = 0$, one\n", + "has $dH = dQ$. This implies that the enthalpy change in a system is\n", + "equal to the heat exchange between the system and the surrounding of the\n", + "system, which is why enthalpy and heat are often exchangeable in the\n", + "literature. Second, for an adiabatic system under constant pressure,\n", + "i.e. $dQ = dP = 0$, can be re-arranged to the following equation\n", + "\n", + "Eq. \u2011\n", + "$H_{i} = \\left( \\frac{\\partial H}{\\partial N_{i}} \\right)_{N_{j \\neq i,\\ \\ dQ = dP = 0}}$\n", + "\n", + "$H_{i}$ is thus the partial enthalpy of component *i* and will be\n", + "further discussed in Chapter . The chemical potential of component *i*\n", + "defined in is thus related to the partial enthalpy and partial entropy\n", + "of the component.\n", + "\n", + "To further define equilibrium conditions of a system, consider a\n", + "homogeneous system in a state of internal equilibrium, i.e. no\n", + "spontaneous internal processes are possible with $Dd\\xi = 0$, and\n", + "becomes\n", + "\n", + "Eq. \u2011\n", + "$dU = TdS - PdV + \\sum_{}^{}\\mu_{i}{dN}_{i} = \\sum_{}^{}{Y_{i}dX_{i}}$\n", + "\n", + "where *X* represents *S*, *V*, *Ni*, and *Y* their conjugate\n", + "variables *T*, *-P*, $\\mu_{i}$. The state of the system with *c*\n", + "independent components is completely determined by the *c+2* variables,\n", + "i.e. *S*, *V*, and *Ni* with *i* from 1 to *c*.\n", + "\n", + "To simplify the situation, let us limit the discussion to an isolated\n", + "equilibrium system, i.e. $dU = 0$, and conduct a virtual internal\n", + "experiment inside the system by moving an infinitesimal amount of\n", + "$X_{i}$, ${dX}_{i}$, with other $X_{j}$ kept constant, from one region\n", + "of the system to another region of the system as schematically shown in\n", + ".\n", + "\n", + "Figure \u2011: Virtual experiment for a system at equilibrium\n", + "\n", + "As the system is homogeneous and at equilibrium,\n", + "$- dX_{i}^{'} = dX_{i}^{\"} = dX_{i}$. The total change of the internal\n", + "energy of this internal process is the combination of the changes in the\n", + "two regions, i.e.\n", + "\n", + "Eq. \u2011\n", + "$dU = dU^{'} + dU^{\"} = Y_{i}^{'}dX_{i}^{'} + Y_{i}^{\"}dX_{i}^{\"} = \\left( - Y_{i}^{'} + Y_{i}^{\"} \\right)dX_{i} = 0$\n", + "\n", + "Therefore, $Y_{i}^{'} = Y_{i}^{\"}$ for *T*, *-P*, and $\\mu_{i}$,\n", + "indicating that *T*, *-P*, and $\\mu_{i}$ are homogeneous in the system,\n", + "respectively, and are thus named as potentials of the system.\n", + "Furthermore these potentials are independent of the size of the system\n", + "and are often referred to as intensive variables in the literature. On\n", + "the other hand, all *X:s*, i.e. *S*, *V*, and *Ni*, are\n", + "proportional to the size of the system and may be normalized with\n", + "respect to the size of the system, usually in terms of total moles,\n", + "\n", + "Eq. \u2011 $N = \\sum_{}^{}N_{i}$\n", + "\n", + "They are thus called molar quantities and often referred to as extensive\n", + "variables, and the respective normalized variables are molar entropy,\n", + "molar volume, and mole fractions, defined as follows\n", + "\n", + "Eq. \u2011 $S_{m} = \\frac{S}{N}$\n", + "\n", + "Eq. \u2011 $V_{m} = \\frac{V}{N}$\n", + "\n", + "Eq. \u2011 $x_{i} = \\frac{N_{i}}{N}$\n", + "\n", + "Consider a small subsystem in this homogeneous system at equilibrium and\n", + "let the subsystem grow in size. The entropy, volume, and mass enclosed\n", + "in the subsystem increase as follows\n", + "\n", + "Eq. \u2011 ${dS = S}_{m}dN$\n", + "\n", + "Eq. \u2011 $dV = V_{m}dN$\n", + "\n", + "Eq. \u2011 $dN_{i} = x_{i}dN$\n", + "\n", + "The corresponding change in the internal energy of the subsystem becomes\n", + "\n", + "Eq. \u2011\n", + "$dU = TdS - PdV + \\sum_{}^{}\\mu_{i}{dN}_{i} = \\left( TS_{m} - PV_{m} + \\sum_{}^{}\\mu_{i}x_{i} \\right)dN = U_{m}dN$\n", + "\n", + "By integration one obtains the integral form of the internal energy as\n", + "\n", + "Eq. \u2011\n", + "$U = TS - PV + \\sum_{}^{}\\mu_{i}N_{i} = \\left( TS_{m} - PV_{m} + \\sum_{}^{}\\mu_{i}x_{i} \\right)N = U_{m}N$\n", + "\n", + "Similarly, the molar enthalpy can be defined as follows\n", + "\n", + "Eq. \u2011\n", + "$H = U + PV = U_{m}N + PV_{m}N = \\left( U_{m} + PV_{m} \\right)N = H_{m}N$\n", + "\n", + "In case a potential is not homogeneous in a system, the system will not\n", + "be in a state of equilibrium. Let us consider the same virtual\n", + "experiment as shown in for an isolated system that is not in\n", + "equilibrium, i.e. by moving an infinitesimal amount of $X_{i}$,\n", + "${dX}_{i}$, with other $X_{j}$ kept constant, from one region of the\n", + "system to another region of the system with the two regions having\n", + "different potentials. The total internal energy change equals to zero as\n", + "the virtual experiment has $dU = 0$. Similarly, each region can be\n", + "considered to be homogeneous by itself, and one has\n", + "$- dX_{i}^{'} = dX_{i}^{\"} = dX_{i}$. The total internal energy change\n", + "in the system is thus the sum of these two regions plus the entropy\n", + "production due to the internal process with $d\\xi = dX_{i}$, i.e.\n", + "\n", + "Eq. \u2011\n", + "$dU = dU^{'} + dU^{\"} + Dd\\xi = Y_{i}^{'}dX_{i}^{'} + Y_{i}^{\"}dX_{i}^{\"} + Dd\\xi = \\left( - Y_{i}^{'} + Y_{i}^{\"} \\right)dX_{i} + Dd\\xi = 0$\n", + "\n", + "Consequently, one obtains the following\n", + "\n", + "*Eq. 1\u201131* $D = Y_{i}^{'} - Y_{i}^{\"}$\n", + "\n", + "The driving force thus represents the difference of the potential at the\n", + "two regions, and the internal process is to eliminate inhomogeneity of\n", + "the potential with the heat transfer from high temperature regions to\n", + "low temperature regions, volume shrink of low pressure regions (high\n", + "$\u2013P$) and volume expansion of high pressure regions (low $\u2013P$), and\n", + "transport of components from high chemical potential regions to low\n", + "chemical potential regions.\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/laws_of_thermodynamics/first_and_second_laws_of_thermodynamics.ipynb b/src/psu410/src/laws_of_thermodynamics/first_and_second_laws_of_thermodynamics.ipynb index 18870a4..1c75f73 100644 --- a/src/psu410/src/laws_of_thermodynamics/first_and_second_laws_of_thermodynamics.ipynb +++ b/src/psu410/src/laws_of_thermodynamics/first_and_second_laws_of_thermodynamics.ipynb @@ -4,9 +4,124 @@ "metadata": {}, "cells": [ { - "id": "57a35cb6", + "id": "f8e20f3c", "cell_type": "markdown", - "source": "## First and second laws of thermodynamics\n\nA system typically consists of many chemical components. The first law\nof thermodynamics states that the exchanges of heat, work, and\nindividual components with its surroundings must obey the law of\nconversation of energy. In the domain of materials science and\nengineering, the energy of interest is at the atomic and molecular\nlevels. The energy at the higher and lower levels such as nuclear energy\nand kinetic and potential energies of a rigid body are usually excluded\nfrom the discussion of thermodynamics of materials.\n\nLet us consider a system receiving an amount of heat, *dQ*, and an\namount of work, *dW*, and an amount of each independent component *i*,\n*dNi*, from the surroundings. Such a system is called an open\nsystem in contrast to a closed system when *dNi*=0 for all\ncomponents, i.e. no exchange of mass between the system and the\nsurrounding. Other types of systems commonly defined in thermodynamics\ninclude adiabatic systems without exchange of heat, i.e. *dQ*=0, and\nisolated systems without exchange of any, i.e. *dQ*= *dW*=\n*dNi*=0.\n\nThe corresponding change of energy in the system, i.e. the internal\nenergy change, *dU*, is formulated in terms of the first law of\nthermodynamics as follows,\n\nEq. \u2011 $dU = dQ + dW + \\sum_{}^{}{H_{i}dN_{i}}$\n\nwhere $H_{i}$ is the unit energy of component *i* in the surroundings,\nand the summation is for all components in the system, which can be\ncontrolled independently from the surroundings, i.e. the independent\ncomponents of the system.\n\nIt is self-evident that the left-hand side of refers to the change\ninside the system, while its right-hand side is for the contributions\nfrom the surroundings to the system. In principle, no matter how the\nheat and mass are added, or how the work is done to the system, as far\nas their summation is the same, the change of the internal energy will\nbe the same based on the first law of thermodynamics, indicating that\nthe system reaches the same state for a closed system. The internal\nenergy is thus a state function in a close system as it does not depend\non how the state is reached.\n\nOn the other hand, for the purpose of easy mathematical treatment, a\nreversible process can be considered for a closed system, in which the\ninitial state of the system can be restored reversibly without any\nchanges left to the surroundings. Therefore, the heat transferred and\nthe work done to the system are identical to the heat and work lost by\nthe surroundings and vice versa. The classic example of reversible\nprocesses is the Carnot\u2019s cycle, which is shown in . It consists of four\nreversible processes for a closed system. The four reversible processes\nare compression at constant temperature *T1* (isothermal,),\ncompression without heat exchange (adiabatic) ending at *T2*,\nisothermal expansion at *T2*, and adiabatic expansion ending\nat *T1*.\n\nFigure \u2011: Schematics of the Carnot\u2019s cycle\n\nThe Carnot\u2019s cycle involves a simple type of mechanical work, either\nhydrostatic expansion or compression, with the work that the surrounding\ndoes to the system represented by\n\nEq. \u2011 $dW = - PdV$\n\nwith *P* being the external pressure that the surrounding exerts on the\nsystem and *V* the volume of the system. It is now necessary to\ndifferentiate the external and internal variables for further discussion\nwith the former representing variables in the surroundings and the\nlatter representing variables in the system. For the isothermal\nprocesses in the Carnot\u2019s cycle, the entropy change of the system, *dS*,\ncan be defined as the heat exchange divided by temperature\n\nEq. \u2011 $dS = \\frac{dQ}{T}$\n\nIn addition to processes involving heat, work, and mass exchanges\nbetween the system and the surroundings, there could be internal\nprocesses taking place inside the system. As the system cannot do work\nto itself, the criterion for whether an internal process can occur\nspontaneously must be related to the heat exchange, which is related to\nthe entropy change as shown by .\n\nIt is a known fact that heat can spontaneously transfer from a higher\ntemperature (*T2*) region to a lower temperature\n(*T1*) region inside a system if the heat conduction is\nallowed, and this process is irreversible because heat cannot be\nconducted from a low temperature region to a high temperature region by\nitself. indicates that for the same amount of heat change, the entropy\nchange at *T1* is higher than that at *T2*, and\nthe heat conduction thus results in a positive entropy change in the\nsystem, i.e.\n\nEq. \u2011\n$\\mathrm{\\Delta}S = - \\frac{dQ}{T_{2}} + \\frac{dQ}{T_{1}} = \\frac{dQ}{{T_{2}T}_{1}}\\left( {T_{2} - T}_{1} \\right) > 0$\n\nConsequently, the second law of thermodynamics is obtained, which states\nthat for an internal process to take place spontaneously or\nirreversibly, this internal process (*ip*) must have a positive entropy\nproduction, which can be written in a differential form as follows\n\nEq. \u2011 $d_{ip}S > 0$\n\nFrom the definition of entropy change shown by , the amount of heat\nproduced by this irreversible internal process can be calculated as\nfollows\n\nEq. \u2011 $d_{ip}Q = Td_{ip}S$\n\nLet us represent this internal process by *d\u03be* and define the driving\nforce for this internal process by *D*. The work done by this internal\nprocess is thus *Dd\u03be*, which is released as heat, i.e.\n\nEq. \u2011 $Dd\\xi = d_{ip}Q = Td_{ip}S$\n\nAn irreversible process thus must have a positive driving force in order\nfor it to take place spontaneously.\n", + "source": [ + "## First and second laws of thermodynamics\n", + "\n", + "A system typically consists of many chemical components. The first law\n", + "of thermodynamics states that the exchanges of heat, work, and\n", + "individual components with its surroundings must obey the law of\n", + "conversation of energy. In the domain of materials science and\n", + "engineering, the energy of interest is at the atomic and molecular\n", + "levels. The energy at the higher and lower levels such as nuclear energy\n", + "and kinetic and potential energies of a rigid body are usually excluded\n", + "from the discussion of thermodynamics of materials.\n", + "\n", + "Let us consider a system receiving an amount of heat, *dQ*, and an\n", + "amount of work, *dW*, and an amount of each independent component *i*,\n", + "*dNi*, from the surroundings. Such a system is called an open\n", + "system in contrast to a closed system when *dNi*=0 for all\n", + "components, i.e. no exchange of mass between the system and the\n", + "surrounding. Other types of systems commonly defined in thermodynamics\n", + "include adiabatic systems without exchange of heat, i.e. *dQ*=0, and\n", + "isolated systems without exchange of any, i.e. *dQ*= *dW*=\n", + "*dNi*=0.\n", + "\n", + "The corresponding change of energy in the system, i.e. the internal\n", + "energy change, *dU*, is formulated in terms of the first law of\n", + "thermodynamics as follows,\n", + "\n", + "Eq. \u2011 $dU = dQ + dW + \\sum_{}^{}{H_{i}dN_{i}}$\n", + "\n", + "where $H_{i}$ is the unit energy of component *i* in the surroundings,\n", + "and the summation is for all components in the system, which can be\n", + "controlled independently from the surroundings, i.e. the independent\n", + "components of the system.\n", + "\n", + "It is self-evident that the left-hand side of refers to the change\n", + "inside the system, while its right-hand side is for the contributions\n", + "from the surroundings to the system. In principle, no matter how the\n", + "heat and mass are added, or how the work is done to the system, as far\n", + "as their summation is the same, the change of the internal energy will\n", + "be the same based on the first law of thermodynamics, indicating that\n", + "the system reaches the same state for a closed system. The internal\n", + "energy is thus a state function in a close system as it does not depend\n", + "on how the state is reached.\n", + "\n", + "On the other hand, for the purpose of easy mathematical treatment, a\n", + "reversible process can be considered for a closed system, in which the\n", + "initial state of the system can be restored reversibly without any\n", + "changes left to the surroundings. Therefore, the heat transferred and\n", + "the work done to the system are identical to the heat and work lost by\n", + "the surroundings and vice versa. The classic example of reversible\n", + "processes is the Carnot\u2019s cycle, which is shown in . It consists of four\n", + "reversible processes for a closed system. The four reversible processes\n", + "are compression at constant temperature *T1* (isothermal,),\n", + "compression without heat exchange (adiabatic) ending at *T2*,\n", + "isothermal expansion at *T2*, and adiabatic expansion ending\n", + "at *T1*.\n", + "\n", + "Figure \u2011: Schematics of the Carnot\u2019s cycle\n", + "\n", + "The Carnot\u2019s cycle involves a simple type of mechanical work, either\n", + "hydrostatic expansion or compression, with the work that the surrounding\n", + "does to the system represented by\n", + "\n", + "Eq. \u2011 $dW = - PdV$\n", + "\n", + "with *P* being the external pressure that the surrounding exerts on the\n", + "system and *V* the volume of the system. It is now necessary to\n", + "differentiate the external and internal variables for further discussion\n", + "with the former representing variables in the surroundings and the\n", + "latter representing variables in the system. For the isothermal\n", + "processes in the Carnot\u2019s cycle, the entropy change of the system, *dS*,\n", + "can be defined as the heat exchange divided by temperature\n", + "\n", + "Eq. \u2011 $dS = \\frac{dQ}{T}$\n", + "\n", + "In addition to processes involving heat, work, and mass exchanges\n", + "between the system and the surroundings, there could be internal\n", + "processes taking place inside the system. As the system cannot do work\n", + "to itself, the criterion for whether an internal process can occur\n", + "spontaneously must be related to the heat exchange, which is related to\n", + "the entropy change as shown by .\n", + "\n", + "It is a known fact that heat can spontaneously transfer from a higher\n", + "temperature (*T2*) region to a lower temperature\n", + "(*T1*) region inside a system if the heat conduction is\n", + "allowed, and this process is irreversible because heat cannot be\n", + "conducted from a low temperature region to a high temperature region by\n", + "itself. indicates that for the same amount of heat change, the entropy\n", + "change at *T1* is higher than that at *T2*, and\n", + "the heat conduction thus results in a positive entropy change in the\n", + "system, i.e.\n", + "\n", + "Eq. \u2011\n", + "$\\mathrm{\\Delta}S = - \\frac{dQ}{T_{2}} + \\frac{dQ}{T_{1}} = \\frac{dQ}{{T_{2}T}_{1}}\\left( {T_{2} - T}_{1} \\right) > 0$\n", + "\n", + "Consequently, the second law of thermodynamics is obtained, which states\n", + "that for an internal process to take place spontaneously or\n", + "irreversibly, this internal process (*ip*) must have a positive entropy\n", + "production, which can be written in a differential form as follows\n", + "\n", + "Eq. \u2011 $d_{ip}S > 0$\n", + "\n", + "From the definition of entropy change shown by , the amount of heat\n", + "produced by this irreversible internal process can be calculated as\n", + "follows\n", + "\n", + "Eq. \u2011 $d_{ip}Q = Td_{ip}S$\n", + "\n", + "Let us represent this internal process by *d\u03be* and define the driving\n", + "force for this internal process by *D*. The work done by this internal\n", + "process is thus *Dd\u03be*, which is released as heat, i.e.\n", + "\n", + "Eq. \u2011 $Dd\\xi = d_{ip}Q = Td_{ip}S$\n", + "\n", + "An irreversible process thus must have a positive driving force in order\n", + "for it to take place spontaneously.\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/laws_of_thermodynamics/gibbsduhem_equation.ipynb b/src/psu410/src/laws_of_thermodynamics/gibbsduhem_equation.ipynb index 24d1f86..7858d83 100644 --- a/src/psu410/src/laws_of_thermodynamics/gibbsduhem_equation.ipynb +++ b/src/psu410/src/laws_of_thermodynamics/gibbsduhem_equation.ipynb @@ -4,9 +4,87 @@ "metadata": {}, "cells": [ { - "id": "34832f69", + "id": "e63729bb", "cell_type": "markdown", - "source": "## Gibbs-Duhem equation\n\nIn experiments, it is difficult to control *S* and *V* of a system in\ncomparison with their conjugate variables *T* and *-P*. It is thus\ndesirable to construct new functions to represent the system with *T*\nand *-P* as natural variables of the functions. One of them is enthalpy\ndefined in , and other two can be defined as follows\n\nEq. \u2011 $F = U - TS$\n\nEq. \u2011 $G = U - TS + PV = \\sum_{}^{}\\mu_{i}N_{i} = H - TS = F + PV$\n\nwith *F* and *G* called Helmholtz energy and Gibbs energy, respectively.\nThe middle part of is obtained using $U$ from . The corresponding\ncombined law of thermodynamics in terms of *H*, *F*, and *G* can be\nobtained through the Legendre transformation of as\n\nEq. \u2011 $dH = TdS - Vd( - P) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$\n\nEq. \u2011 $dF = - SdT - PdV + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$\n\nEq. \u2011 $dG = - SdT - Vd( - P) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$\n\nThe independent variables in each of the above form are regarded as the\nnatural variables to the corresponding function. The integral forms of\nall the functions can thus be written as the following with their\nnatural variables listed in the parenthesis\n\nEq. \u2011 $U = U\\left( S,V,N_{i},\\xi \\right)$\n\nEq. \u2011 $H = H\\left( S, - P,N_{i},\\xi \\right)$\n\nEq. \u2011 $F = F\\left( T,V,N_{i},\\xi \\right)$\n\nEq. \u2011 $G = G\\left( T, - P,N_{i},\\xi \\right)$\n\nBy differentiating , one obtains\n\nEq. \u2011\n$dG = \\sum_{}^{}\\mu_{i}{dN}_{i} + \\sum_{}^{}{N_{i}d\\mu}_{i} = - SdT - Vd( - P) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$\n\nFor a system at equilibrium, $Dd\\xi = 0$, re-arranging gives the\nGibbs-Duhem equation\n\nEq. \u2011 $0 = - SdT - Vd( - P) - \\sum_{}^{}{N_{i}d\\mu}_{i}$\n\nThis equation indicates that for a homogeneous system with *c*\nindependent components at equilibrium, there is a direct relation among\nall the *c+2* potentials, and they are $c$ chemical potentials\n($\\mu_{i}$), temperature, and pressure. Consequently, only *c+1*\npotentials can change independently, and the remaining potential is\ndependent on the other potentials. As discussed in connection with ,\nthere are $c + 2$ independent variables for an equilibrium system with\n*c* independent components, where all of them are molar quantities.\n\nWith the relationships between potentials and molar quantities defined\nby to , one can switch between potentials and molar quantities as\nnatural variables of the system. For example, one can define a new free\nenergy function when the chemical potential of one component is\ncontrolled from the surroundings instead of its content and obtain the\nfollowing combined first and second law of thermodynamics\n\nEq. \u2011 $\\Phi = G - \\mu_{1}N_{1} = \\sum_{i = 2}^{c}{\\mu_{i}N_{i}}$\n\nEq. \u2011\n$d\\Phi = - SdT - Vd( - P) - N_{1}{d\\mu}_{1} + \\sum_{i = 2}^{c}{\\mu_{i}{dN}_{i}} - Dd\\xi$\n\nHowever, even though the $c + 2$ molar quantities are independent of\neach other, indicates that not all the $c + 2$ potentials are\nindependent, i.e., if chemical potentials of all components are changed\nto natural variables, one would obtain . Therefore, among the *c+2*\nindependent variables used to define the system, the maximum number of\nindependent potential is *c+1*, and at least one of the *c+2*\nindependent variables must be a molar quantity. This variable is usually\nchosen to be the size of the system or the major element in the system.\nThe Gibbs-Duhem equation is used to derive Gibbs phase rule in\nheterogeneous systems, which is discussed in Chapter of the book.\n", + "source": [ + "## Gibbs-Duhem equation\n", + "\n", + "In experiments, it is difficult to control *S* and *V* of a system in\n", + "comparison with their conjugate variables *T* and *-P*. It is thus\n", + "desirable to construct new functions to represent the system with *T*\n", + "and *-P* as natural variables of the functions. One of them is enthalpy\n", + "defined in , and other two can be defined as follows\n", + "\n", + "Eq. \u2011 $F = U - TS$\n", + "\n", + "Eq. \u2011 $G = U - TS + PV = \\sum_{}^{}\\mu_{i}N_{i} = H - TS = F + PV$\n", + "\n", + "with *F* and *G* called Helmholtz energy and Gibbs energy, respectively.\n", + "The middle part of is obtained using $U$ from . The corresponding\n", + "combined law of thermodynamics in terms of *H*, *F*, and *G* can be\n", + "obtained through the Legendre transformation of as\n", + "\n", + "Eq. \u2011 $dH = TdS - Vd( - P) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$\n", + "\n", + "Eq. \u2011 $dF = - SdT - PdV + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$\n", + "\n", + "Eq. \u2011 $dG = - SdT - Vd( - P) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$\n", + "\n", + "The independent variables in each of the above form are regarded as the\n", + "natural variables to the corresponding function. The integral forms of\n", + "all the functions can thus be written as the following with their\n", + "natural variables listed in the parenthesis\n", + "\n", + "Eq. \u2011 $U = U\\left( S,V,N_{i},\\xi \\right)$\n", + "\n", + "Eq. \u2011 $H = H\\left( S, - P,N_{i},\\xi \\right)$\n", + "\n", + "Eq. \u2011 $F = F\\left( T,V,N_{i},\\xi \\right)$\n", + "\n", + "Eq. \u2011 $G = G\\left( T, - P,N_{i},\\xi \\right)$\n", + "\n", + "By differentiating , one obtains\n", + "\n", + "Eq. \u2011\n", + "$dG = \\sum_{}^{}\\mu_{i}{dN}_{i} + \\sum_{}^{}{N_{i}d\\mu}_{i} = - SdT - Vd( - P) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi$\n", + "\n", + "For a system at equilibrium, $Dd\\xi = 0$, re-arranging gives the\n", + "Gibbs-Duhem equation\n", + "\n", + "Eq. \u2011 $0 = - SdT - Vd( - P) - \\sum_{}^{}{N_{i}d\\mu}_{i}$\n", + "\n", + "This equation indicates that for a homogeneous system with *c*\n", + "independent components at equilibrium, there is a direct relation among\n", + "all the *c+2* potentials, and they are $c$ chemical potentials\n", + "($\\mu_{i}$), temperature, and pressure. Consequently, only *c+1*\n", + "potentials can change independently, and the remaining potential is\n", + "dependent on the other potentials. As discussed in connection with ,\n", + "there are $c + 2$ independent variables for an equilibrium system with\n", + "*c* independent components, where all of them are molar quantities.\n", + "\n", + "With the relationships between potentials and molar quantities defined\n", + "by to , one can switch between potentials and molar quantities as\n", + "natural variables of the system. For example, one can define a new free\n", + "energy function when the chemical potential of one component is\n", + "controlled from the surroundings instead of its content and obtain the\n", + "following combined first and second law of thermodynamics\n", + "\n", + "Eq. \u2011 $\\Phi = G - \\mu_{1}N_{1} = \\sum_{i = 2}^{c}{\\mu_{i}N_{i}}$\n", + "\n", + "Eq. \u2011\n", + "$d\\Phi = - SdT - Vd( - P) - N_{1}{d\\mu}_{1} + \\sum_{i = 2}^{c}{\\mu_{i}{dN}_{i}} - Dd\\xi$\n", + "\n", + "However, even though the $c + 2$ molar quantities are independent of\n", + "each other, indicates that not all the $c + 2$ potentials are\n", + "independent, i.e., if chemical potentials of all components are changed\n", + "to natural variables, one would obtain . Therefore, among the *c+2*\n", + "independent variables used to define the system, the maximum number of\n", + "independent potential is *c+1*, and at least one of the *c+2*\n", + "independent variables must be a molar quantity. This variable is usually\n", + "chosen to be the size of the system or the major element in the system.\n", + "The Gibbs-Duhem equation is used to derive Gibbs phase rule in\n", + "heterogeneous systems, which is discussed in Chapter of the book.\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/laws_of_thermodynamics/index.ipynb b/src/psu410/src/laws_of_thermodynamics/index.ipynb index 9d399cd..eaf8566 100644 --- a/src/psu410/src/laws_of_thermodynamics/index.ipynb +++ b/src/psu410/src/laws_of_thermodynamics/index.ipynb @@ -4,9 +4,11 @@ "metadata": {}, "cells": [ { - "id": "2d5be24d", + "id": "3e5734cb", "cell_type": "markdown", - "source": "# Laws of thermodynamics\n", + "source": [ + "# Laws of thermodynamics\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/laws_of_thermodynamics/stability_at_equilibrium_and_property_anomaly.ipynb b/src/psu410/src/laws_of_thermodynamics/stability_at_equilibrium_and_property_anomaly.ipynb index e9c049e..7d5ff7b 100644 --- a/src/psu410/src/laws_of_thermodynamics/stability_at_equilibrium_and_property_anomaly.ipynb +++ b/src/psu410/src/laws_of_thermodynamics/stability_at_equilibrium_and_property_anomaly.ipynb @@ -4,9 +4,188 @@ "metadata": {}, "cells": [ { - "id": "2aeff033", + "id": "9010bd96", "cell_type": "markdown", - "source": "## Stability at equilibrium and property anomaly\n\nAs shown by , potentials are homogenous for a homogeneous system in a\nstate of internal equilibrium. To study the stability of the equilibrium\nstate, one considers the entropy production due to a fluctuation of a\nmolar quantity as an internal process. Based on the second law of\nthermodynamics, the driving force, as the first derivative of the\nentropy production with respect to the internal process, is zero for\nsuch a fluctuation at equilibrium, i.e. *D*=0, and the entropy of\nproduction thus depends on the second derivative. It can be written as\nfollows\n\nEq. \u2011\n$Td_{ip}S = \\frac{\\partial_{ip}S}{\\partial\\xi}d\\xi + \\frac{1}{2}{\\frac{\\partial_{ip}^{2}S}{\\partial\\xi^{2}}(d\\xi)}^{2} = Dd\\xi - \\frac{1}{2}D_{2}(d\\xi)^{2}$\n\nwith $D_{2} = - \\frac{\\partial_{ip}^{2}S}{\\partial\\xi^{2}}$. When\n$\\frac{\\partial_{ip}^{2}S}{\\partial\\xi^{2}} < 0$ or $D_{2} > 0$ along\nwith $D = 0$, the fluctuation does not produce positive entropy of\nproduction and can thus not develop further. The equilibrium state of\nthe system is therefore stable against the fluctuation. On the other\nhand, when $\\frac{\\partial_{ip}^{2}S}{\\partial\\xi^{2}} > 0$ or\n$D_{2} < 0$ along with $D = 0$, the fluctuation creates positive entropy\nof production and can continue to grow. The equilibrium state of the\nsystem is therefore unstable against the fluctuation. In connection with\n, one can realize that for a system at stable equilibrium without heat\nand mass exchanges with the surroundings, its entropy is at its maximum,\nand there are no other internal processes, which could produce any more\nentropy. This is schematically shown in .\n\nFigure \u2011: Schematic diagram showing maximum entropy\n\nUsing , , and , the combined law of thermodynamics can be written as\n\n*Eq. 1\u201133*\n$dU = \\sum_{}^{}{Y_{i}dX_{i}} - Dd\\xi + \\frac{1}{2}D_{2}(d\\xi)^{2}$\n\nLet us carry out the same virtual internal experiment shown in Chapter ,\ni.e. moving an infinitesimal amount of *Xi* in a homogenous\nsystem with other $X_{j}$ kept constant in an isolated system, i.e.\n$dU = 0$ and $D = 0$. The internal energy change due to this internal\nprocess is\n\nEq. \u2011\n$dU = \\frac{1}{2}D_{2}\\left\\{ \\left( dX_{i}^{'} \\right)^{2} + \\left( dX_{i}^{\"} \\right)^{2} \\right\\}$\n\nFor a homogeneous system in a state of stable equilibrium with\n$\\left( dX_{i}^{'} \\right)^{2} = \\left( dX_{i}^{\"} \\right)^{2} = \\left( dX_{i} \\right)^{2}$,\nthis internal process must result in an increase of internal energy,\n$dU > 0$, and thus gives\n\nEq. \u2011\n$D_{2} = 2\\left( \\frac{\\partial^{2}U}{\\partial\\left( X_{i} \\right)^{2}} \\right)_{X_{j}} = 2\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}} > 0$\n\nshows that for a system to be stable, any pair of conjugate variables\nmust change in the same direction when other independent molar\nquantities are kept constant. For the conjugate variables discussed so\nfar, it means that for a stable system, the addition of entropy\nincreases temperature with $\\frac{\\partial T}{\\partial S}$\\>0, the\nvolume decreases with pressure or increases with the negative of\npressure with $\\frac{\\partial( - P)}{\\partial V} > 0$, and the chemical\npotential of a component increases with its amount, i.e.\n$\\frac{\\partial\\mu_{i}}{\\partial N_{i}} > 0$, where the derivatives are\ntaken with all other molar quantities kept constant. The limit of\nstability is reached when becomes zero, i.e.\n\nEq. \u2011\n$D_{2} = 2\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}} = 0$\n\nshows schematically the energy as a function of configurations including\nthree states: unstable, stable, and metastable. Both the stable and\nmetastable states have positive curvatures due to $D_{2} > 0$, while the\nunstable state has a negative curvature due to $D_{2} < 0$. There is an\ninflection point of $D_{2} = 0$ for a state between a stable or\nmetastable state with $D_{2} > 0$ and an unstable state with\n$D_{2} < 0$. These two inflection points, called spinodal, represent the\nlimit of stability. The states between the two inflection points are\nunstable, and other states are either stable or metastable. The two\ninflection points can move apart from or close to each other depending\non the change of external conditions, i.e. the natural variables. One\nextreme situation is when these two inflection points merge into one\npoint, and the instability occurs only at this particular point. It is\nevident that all three states, stable, metastable, and unstable, also\nmerge into one point. This point is called critical or consolute point,\nbeyond which the instability no longer exists.\n\nFigure \u2011: Schematic diagram showing the stable and unstable equilibrium\nstates\n\nTo mathematically define the consolute point, the third derivative needs\nto be added to because both $D$ and $D_{2}$ vanish at this point, i.e.\n\nEq. \u2011\n$Td_{ip}S = \\frac{\\partial_{ip}S}{\\partial\\xi}d\\xi + \\frac{1}{2}{\\frac{\\partial_{ip}^{2}S}{\\partial\\xi^{2}}(d\\xi)}^{2} + \\frac{1}{6}{\\frac{\\partial_{ip}^{3}S}{\\partial\\xi^{3}}(d\\xi)}^{3} = Dd\\xi - \\frac{1}{2}D_{2}(d\\xi)^{2} + \\frac{1}{6}{D_{3}(d\\xi)}^{3}$\n\nEq. \u2011\n$dU = \\sum_{}^{}{Y_{i}dX_{i}} - Dd\\xi + \\frac{1}{2}D_{2}(d\\xi)^{2} - \\frac{1}{6}{D_{3}(d\\xi)}^{3}$\n\nAt the consolute point, the third derivative also becomes zero, i.e.\n\nEq. \u2011 $D_{3} = \\frac{\\partial_{ip}^{3}S}{\\partial\\xi^{3}}^{3} = 0$\n\nLet us further discuss the properties of the system in relation to the\ncritical point. By taking the inverse of the equation of the limit of\nstability, , one obtains\n\nEq. \u2011\n$\\left( \\frac{\\partial X_{i}}{\\partial Y_{i}} \\right)_{X_{j}} = + \\ \\infty$\n\ni.e. all $X_{i}$ quantities diverge at the critical point. Therefore,\nwhen a system approaches the critical point from its stable region, the\nchange of a molar quantity with respect to its conjugate potential\nvaries dramatically and becomes infinite at the critical point,\nresulting in property anomalies in the system. In the unstable region,\nthe system would thus separate into stable subsystems and becomes\nheterogeneous, and $X_{i}$:s change discontinuously between subsystems.\nWhile in the stable region, the change of a molar quantity with respect\nto its conjugate potential decreases as the system moves away from the\ncritical point and remains positive due to the stability criteria\ndenoted by .\n\nHowever, it is not clear how a molar quantity changes with respect to a\nnon-conjugate potential at the critical point. From the Maxwell\nrelation, one has\n\nEq. \u2011\n$\\left( \\frac{\\partial Y_{i}}{\\partial X_{j}} \\right)_{X_{k \\neq j}} = \\frac{\\partial^{2}U}{\\partial X_{i}\\partial X_{j}} = \\left( \\frac{\\partial Y_{j}}{\\partial X_{i}} \\right)_{X_{k \\neq i}}$\n\nEq. \u2011\n$\\left( \\frac{\\partial X_{j}}{\\partial Y_{i}} \\right)_{X_{k \\neq j}} = \\left( \\frac{\\partial X_{i}}{\\partial Y_{j}} \\right)_{X_{k \\neq i}}$\n\nSince all $X_{i}$:s diverge at the critical point, both derivatives in\nshould also go to infinite at the critical point. To investigate their\nsigns, let us carry out a virtual experiment similar to the one in\nderiving the stability condition ( and ). In this case, two internal\nprocesses are needed for moving two molar quantities simultaneously in\nan isolated system, i.e.\n\n*Eq. 1\u201143*\n$dU = - D_{\\xi_{1}}d\\xi_{1} - D_{\\xi_{2}}d\\xi_{2} + D_{\\xi_{1}\\xi_{2}}d\\xi_{1}d\\xi_{2} + \\frac{1}{2}D_{2\\xi_{1}}\\left( d\\xi_{1} \\right)^{2} + \\frac{1}{2}D_{2\\xi_{2}}\\left( d\\xi_{2} \\right)^{2}$\n\nBased on the above discussions, in a stable system at equilibrium with\n$D_{\\xi_{1}} = D_{\\xi_{2}} = 0$, $D_{2\\xi_{1}} > 0$ and\n$D_{2\\xi_{2}} > 0$, the sign of $D_{\\xi_{1}\\xi_{2}}$ cannot be\nunambiguously determined in keeping the change of internal energy\npositive, i.e. $dU > 0$. This indicates that the quantities in can be\neither positive or negative in the stable region and become zero at the\ncritical point. By the same token, the quantities in can be either\npositive or negative and become positive or negative infinite at the\ncritical point.\n\nA profound conclusion from this analysis is that in a stable system even\nthough a molar quantity always changes in the same direction as its\nconjugate potential, the same molar quantity may change in the opposite\ndirection of a non-conjugate potential, resulting in additional\nanomalies represented by Eq. 1\u201140. One example of is the thermal\nexpansion in a closed system, i.e. $dN_{i} = 0$, as follows\n\nEq. \u2011\n$\\left( \\frac{\\partial V}{\\partial T} \\right)_{S} = \\left( \\frac{\\partial S}{\\partial( - P)} \\right)_{V}$\n\nThe left-hand side of can be understood as follows: with the increase of\ntemperature, the system regulates its pressure in order to keep the\nentropy from increasing, which results in the volume change of the\nsystem. The behavior of the system depends on whether the pressure\ndecreases or increases in order to maintain the entropy of the system\nconstant. If the pressure decreases to maintain the entropy of the\nsystem constant, the volume would increase with the increase of\ntemperature, i.e. the left-hand side of the equation has a positive\nsign, which is also shown by the right-hand side of the equation as the\nchanges of $S$ and $\u2013P$ have the same sign. That the volume increases\nwith temperature is the normal scenario. On the other hand, if the\npressure increases to maintain the entropy of the system constant, the\nvolume would decrease with the increase of temperature, resulting in a\nnegative sign for the left-hand side of the equation. This decrease of\nvolume with the increase of temperature is usually considered to be\nanomalous, originated from the increase of entropy by the decrease of\n$\u2013P$ or the increase of pressure. More discussions on entropy will\nfollow in Chapter 5.2.5 and Chapter 9.\n", + "source": [ + "## Stability at equilibrium and property anomaly\n", + "\n", + "As shown by , potentials are homogenous for a homogeneous system in a\n", + "state of internal equilibrium. To study the stability of the equilibrium\n", + "state, one considers the entropy production due to a fluctuation of a\n", + "molar quantity as an internal process. Based on the second law of\n", + "thermodynamics, the driving force, as the first derivative of the\n", + "entropy production with respect to the internal process, is zero for\n", + "such a fluctuation at equilibrium, i.e. *D*=0, and the entropy of\n", + "production thus depends on the second derivative. It can be written as\n", + "follows\n", + "\n", + "Eq. \u2011\n", + "$Td_{ip}S = \\frac{\\partial_{ip}S}{\\partial\\xi}d\\xi + \\frac{1}{2}{\\frac{\\partial_{ip}^{2}S}{\\partial\\xi^{2}}(d\\xi)}^{2} = Dd\\xi - \\frac{1}{2}D_{2}(d\\xi)^{2}$\n", + "\n", + "with $D_{2} = - \\frac{\\partial_{ip}^{2}S}{\\partial\\xi^{2}}$. When\n", + "$\\frac{\\partial_{ip}^{2}S}{\\partial\\xi^{2}} < 0$ or $D_{2} > 0$ along\n", + "with $D = 0$, the fluctuation does not produce positive entropy of\n", + "production and can thus not develop further. The equilibrium state of\n", + "the system is therefore stable against the fluctuation. On the other\n", + "hand, when $\\frac{\\partial_{ip}^{2}S}{\\partial\\xi^{2}} > 0$ or\n", + "$D_{2} < 0$ along with $D = 0$, the fluctuation creates positive entropy\n", + "of production and can continue to grow. The equilibrium state of the\n", + "system is therefore unstable against the fluctuation. In connection with\n", + ", one can realize that for a system at stable equilibrium without heat\n", + "and mass exchanges with the surroundings, its entropy is at its maximum,\n", + "and there are no other internal processes, which could produce any more\n", + "entropy. This is schematically shown in .\n", + "\n", + "Figure \u2011: Schematic diagram showing maximum entropy\n", + "\n", + "Using , , and , the combined law of thermodynamics can be written as\n", + "\n", + "*Eq. 1\u201133*\n", + "$dU = \\sum_{}^{}{Y_{i}dX_{i}} - Dd\\xi + \\frac{1}{2}D_{2}(d\\xi)^{2}$\n", + "\n", + "Let us carry out the same virtual internal experiment shown in Chapter ,\n", + "i.e. moving an infinitesimal amount of *Xi* in a homogenous\n", + "system with other $X_{j}$ kept constant in an isolated system, i.e.\n", + "$dU = 0$ and $D = 0$. The internal energy change due to this internal\n", + "process is\n", + "\n", + "Eq. \u2011\n", + "$dU = \\frac{1}{2}D_{2}\\left\\{ \\left( dX_{i}^{'} \\right)^{2} + \\left( dX_{i}^{\"} \\right)^{2} \\right\\}$\n", + "\n", + "For a homogeneous system in a state of stable equilibrium with\n", + "$\\left( dX_{i}^{'} \\right)^{2} = \\left( dX_{i}^{\"} \\right)^{2} = \\left( dX_{i} \\right)^{2}$,\n", + "this internal process must result in an increase of internal energy,\n", + "$dU > 0$, and thus gives\n", + "\n", + "Eq. \u2011\n", + "$D_{2} = 2\\left( \\frac{\\partial^{2}U}{\\partial\\left( X_{i} \\right)^{2}} \\right)_{X_{j}} = 2\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}} > 0$\n", + "\n", + "shows that for a system to be stable, any pair of conjugate variables\n", + "must change in the same direction when other independent molar\n", + "quantities are kept constant. For the conjugate variables discussed so\n", + "far, it means that for a stable system, the addition of entropy\n", + "increases temperature with $\\frac{\\partial T}{\\partial S}$\\>0, the\n", + "volume decreases with pressure or increases with the negative of\n", + "pressure with $\\frac{\\partial( - P)}{\\partial V} > 0$, and the chemical\n", + "potential of a component increases with its amount, i.e.\n", + "$\\frac{\\partial\\mu_{i}}{\\partial N_{i}} > 0$, where the derivatives are\n", + "taken with all other molar quantities kept constant. The limit of\n", + "stability is reached when becomes zero, i.e.\n", + "\n", + "Eq. \u2011\n", + "$D_{2} = 2\\left( \\frac{\\partial Y_{i}}{\\partial X_{i}} \\right)_{X_{j}} = 0$\n", + "\n", + "shows schematically the energy as a function of configurations including\n", + "three states: unstable, stable, and metastable. Both the stable and\n", + "metastable states have positive curvatures due to $D_{2} > 0$, while the\n", + "unstable state has a negative curvature due to $D_{2} < 0$. There is an\n", + "inflection point of $D_{2} = 0$ for a state between a stable or\n", + "metastable state with $D_{2} > 0$ and an unstable state with\n", + "$D_{2} < 0$. These two inflection points, called spinodal, represent the\n", + "limit of stability. The states between the two inflection points are\n", + "unstable, and other states are either stable or metastable. The two\n", + "inflection points can move apart from or close to each other depending\n", + "on the change of external conditions, i.e. the natural variables. One\n", + "extreme situation is when these two inflection points merge into one\n", + "point, and the instability occurs only at this particular point. It is\n", + "evident that all three states, stable, metastable, and unstable, also\n", + "merge into one point. This point is called critical or consolute point,\n", + "beyond which the instability no longer exists.\n", + "\n", + "Figure \u2011: Schematic diagram showing the stable and unstable equilibrium\n", + "states\n", + "\n", + "To mathematically define the consolute point, the third derivative needs\n", + "to be added to because both $D$ and $D_{2}$ vanish at this point, i.e.\n", + "\n", + "Eq. \u2011\n", + "$Td_{ip}S = \\frac{\\partial_{ip}S}{\\partial\\xi}d\\xi + \\frac{1}{2}{\\frac{\\partial_{ip}^{2}S}{\\partial\\xi^{2}}(d\\xi)}^{2} + \\frac{1}{6}{\\frac{\\partial_{ip}^{3}S}{\\partial\\xi^{3}}(d\\xi)}^{3} = Dd\\xi - \\frac{1}{2}D_{2}(d\\xi)^{2} + \\frac{1}{6}{D_{3}(d\\xi)}^{3}$\n", + "\n", + "Eq. \u2011\n", + "$dU = \\sum_{}^{}{Y_{i}dX_{i}} - Dd\\xi + \\frac{1}{2}D_{2}(d\\xi)^{2} - \\frac{1}{6}{D_{3}(d\\xi)}^{3}$\n", + "\n", + "At the consolute point, the third derivative also becomes zero, i.e.\n", + "\n", + "Eq. \u2011 $D_{3} = \\frac{\\partial_{ip}^{3}S}{\\partial\\xi^{3}}^{3} = 0$\n", + "\n", + "Let us further discuss the properties of the system in relation to the\n", + "critical point. By taking the inverse of the equation of the limit of\n", + "stability, , one obtains\n", + "\n", + "Eq. \u2011\n", + "$\\left( \\frac{\\partial X_{i}}{\\partial Y_{i}} \\right)_{X_{j}} = + \\ \\infty$\n", + "\n", + "i.e. all $X_{i}$ quantities diverge at the critical point. Therefore,\n", + "when a system approaches the critical point from its stable region, the\n", + "change of a molar quantity with respect to its conjugate potential\n", + "varies dramatically and becomes infinite at the critical point,\n", + "resulting in property anomalies in the system. In the unstable region,\n", + "the system would thus separate into stable subsystems and becomes\n", + "heterogeneous, and $X_{i}$:s change discontinuously between subsystems.\n", + "While in the stable region, the change of a molar quantity with respect\n", + "to its conjugate potential decreases as the system moves away from the\n", + "critical point and remains positive due to the stability criteria\n", + "denoted by .\n", + "\n", + "However, it is not clear how a molar quantity changes with respect to a\n", + "non-conjugate potential at the critical point. From the Maxwell\n", + "relation, one has\n", + "\n", + "Eq. \u2011\n", + "$\\left( \\frac{\\partial Y_{i}}{\\partial X_{j}} \\right)_{X_{k \\neq j}} = \\frac{\\partial^{2}U}{\\partial X_{i}\\partial X_{j}} = \\left( \\frac{\\partial Y_{j}}{\\partial X_{i}} \\right)_{X_{k \\neq i}}$\n", + "\n", + "Eq. \u2011\n", + "$\\left( \\frac{\\partial X_{j}}{\\partial Y_{i}} \\right)_{X_{k \\neq j}} = \\left( \\frac{\\partial X_{i}}{\\partial Y_{j}} \\right)_{X_{k \\neq i}}$\n", + "\n", + "Since all $X_{i}$:s diverge at the critical point, both derivatives in\n", + "should also go to infinite at the critical point. To investigate their\n", + "signs, let us carry out a virtual experiment similar to the one in\n", + "deriving the stability condition ( and ). In this case, two internal\n", + "processes are needed for moving two molar quantities simultaneously in\n", + "an isolated system, i.e.\n", + "\n", + "*Eq. 1\u201143*\n", + "$dU = - D_{\\xi_{1}}d\\xi_{1} - D_{\\xi_{2}}d\\xi_{2} + D_{\\xi_{1}\\xi_{2}}d\\xi_{1}d\\xi_{2} + \\frac{1}{2}D_{2\\xi_{1}}\\left( d\\xi_{1} \\right)^{2} + \\frac{1}{2}D_{2\\xi_{2}}\\left( d\\xi_{2} \\right)^{2}$\n", + "\n", + "Based on the above discussions, in a stable system at equilibrium with\n", + "$D_{\\xi_{1}} = D_{\\xi_{2}} = 0$, $D_{2\\xi_{1}} > 0$ and\n", + "$D_{2\\xi_{2}} > 0$, the sign of $D_{\\xi_{1}\\xi_{2}}$ cannot be\n", + "unambiguously determined in keeping the change of internal energy\n", + "positive, i.e. $dU > 0$. This indicates that the quantities in can be\n", + "either positive or negative in the stable region and become zero at the\n", + "critical point. By the same token, the quantities in can be either\n", + "positive or negative and become positive or negative infinite at the\n", + "critical point.\n", + "\n", + "A profound conclusion from this analysis is that in a stable system even\n", + "though a molar quantity always changes in the same direction as its\n", + "conjugate potential, the same molar quantity may change in the opposite\n", + "direction of a non-conjugate potential, resulting in additional\n", + "anomalies represented by Eq. 1\u201140. One example of is the thermal\n", + "expansion in a closed system, i.e. $dN_{i} = 0$, as follows\n", + "\n", + "Eq. \u2011\n", + "$\\left( \\frac{\\partial V}{\\partial T} \\right)_{S} = \\left( \\frac{\\partial S}{\\partial( - P)} \\right)_{V}$\n", + "\n", + "The left-hand side of can be understood as follows: with the increase of\n", + "temperature, the system regulates its pressure in order to keep the\n", + "entropy from increasing, which results in the volume change of the\n", + "system. The behavior of the system depends on whether the pressure\n", + "decreases or increases in order to maintain the entropy of the system\n", + "constant. If the pressure decreases to maintain the entropy of the\n", + "system constant, the volume would increase with the increase of\n", + "temperature, i.e. the left-hand side of the equation has a positive\n", + "sign, which is also shown by the right-hand side of the equation as the\n", + "changes of $S$ and $\u2013P$ have the same sign. That the volume increases\n", + "with temperature is the normal scenario. On the other hand, if the\n", + "pressure increases to maintain the entropy of the system constant, the\n", + "volume would decrease with the increase of temperature, resulting in a\n", + "negative sign for the left-hand side of the equation. This decrease of\n", + "volume with the increase of temperature is usually considered to be\n", + "anomalous, originated from the increase of entropy by the decrease of\n", + "$\u2013P$ or the increase of pressure. More discussions on entropy will\n", + "follow in Chapter 5.2.5 and Chapter 9.\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/phase_equilibria_in_heterogeneous_systems/general_condition_of_equilibrium.ipynb b/src/psu410/src/phase_equilibria_in_heterogeneous_systems/general_condition_of_equilibrium.ipynb index 62ed602..1291561 100644 --- a/src/psu410/src/phase_equilibria_in_heterogeneous_systems/general_condition_of_equilibrium.ipynb +++ b/src/psu410/src/phase_equilibria_in_heterogeneous_systems/general_condition_of_equilibrium.ipynb @@ -4,9 +4,119 @@ "metadata": {}, "cells": [ { - "id": "448bcda8", + "id": "fdd1ab93", "cell_type": "markdown", - "source": "## General condition of equilibrium\n\nA system is heterogeneous when some properties have different values at\ndifferent portions of the system when the system is at equilibrium. Two\nscenarios may exist where the variations of properties can be either\ncontinuous or discontinuous. In the scenario of continuous variations,\nthe gradients of those variations must be coupled so that the system\nremains at equilibrium. The number of independent variables is thus\nreduced. These gradients must also be constrained along the boundaries\nbetween the system and the surroundings. This type of constrained\nequilibrium is not discussed in the book as it involves heterogeneous\nboundary conditions between the system and the surroundings and depends\non the morphology of the system.\n\nIn the second scenario with discontinuous variations, those properties\nhave different values in different portions of the system, but remain\nhomogenous within each portion. The system is in equilibrium as each\nportion is in equilibrium with all other portions of the system. Those\nhomogeneous portions represent different phases in the system with the\nproperties in each phase being homogeneous at equilibrium. In the\nprevious chapter, it has been shown that all potentials are homogeneous\nin a homogeneous system.\n\nFor a heterogeneous system, the same conclusion can be obtained. If the\ntemperature is inhomogeneous, heat can be conducted from high\ntemperature locations to low temperature locations, and this process is\nirreversible based on the second law of thermodynamics because it\nincreases the internal entropy of the system. If the pressure is\ninhomogeneous, the amounts of lower molar volume phases will increase to\nreduce the internal energy of the system. If the chemical potential of a\ncomponent is inhomogeneous, the chemical potential difference of the\ncomponent will drive that component to the locations with a lower\nchemical potential so the internal energy of the system can be\ndecreased. Therefore, it can be concluded that all potentials are\nhomogeneous in a heterogeneous system at equilibrium, and the variables\nthat are not homogeneous are thus their conjugate molar quantities.\nUnder certain special circumstances to be discussed in later part of\nthis book, some molar quantities may also have the same values in\ndifference phases.\n\nIn a system at equilibrium with $c$ independent components, there are\n$c + 2$ pairs of conjugate variables based on or though more can be\nadded as shown by depending on experimental conditions. For simplicity,\nmost discussions in this book are limited to systems with $c + 2$ pairs\nof conjugate variables unless otherwise specified, with number \u201c2\u201d\nrepresenting conjugate variables of $T - S$ and $( - P) - V$.\n\nFor a system under constant temperature, pressure, and moles of each\nindependent component, the equilibrium condition derives from as\n\nEq. \u2011\n$dG = - SdT - Vd( - P) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi = - Dd\\xi = 0$\n\nConsequently, the equilibrium state is defined by the minimization of\nGibbs energy of the system at constant $T$, $P$ and $N_{i}$ because the\nsecond derivatives need to be positive for the equilibrium system to be\nstable as stipulated by . For heterogeneous systems with two or more\nphases, Gibbs energy of the system is the weighted summation of Gibbs\nenergies of individual phases, i.e.\n\nEq. \u2011 $\\frac{G}{N} = G_{m} = \\sum_{\\beta}^{}{f^{\\beta}G_{m}^{\\beta}}$\n\nwhere $f^{\\beta}$ and $G_{m}^{\\beta}$ are the mole fraction and molar\nGibbs energy of the phase $\\beta$, respectively, and the summation goes\nover all phases in the system. $f^{\\beta}$ is equal to zero for phases\nnot present in the equilibrium state.\n\nThe minimization of Gibbs energy of the system is carried out under the\nfollowing mass balance conditions\n\nEq. \u2011\n$x_{i} = \\sum_{\\beta}^{}{f^{\\beta}x_{i}^{\\beta}} = \\sum_{\\beta}^{}{f^{\\beta}\\frac{\\sum_{\\beta - t}^{}{a^{\\beta - t}\\sum_{k}^{}{i_{k}^{\\beta - t}y_{k}^{\\beta - t}}}}{\\sum_{\\beta - t}^{}{a^{\\beta - t}\\left( 1 - y_{va}^{\\beta - t} \\right)}}}$\n\nEq. \u2011 $\\sum_{i}^{}x_{i} = 1$\n\nEq. \u2011 $\\sum_{k}^{}y_{k}^{\\beta - t} = 1$\n\nwhere $a^{\\beta - t}$ and $y_{k}^{\\beta - t}$ are the number of site in\nsublattice $t$ in the $\\beta$ phase and the corresponding site fraction\nof species $k$ in the sublattice, respectively, and $i_{k}^{\\beta - t}$\nis the stoichiometry of the component $i$ in the species $k$, as used in\n. The summation in runs over species for each sublattice. For phases\ncontaining ionic species, electroneutrality also needs to be maintained,\ni.e. is applied to each phase. This minimization problem of Gibbs energy\nunder the constraint of mass conservation can be solved by means of a\nrange of algorithms. It should be noted that the mole fractions of\nphases and site fractions of species are bounded between 0 and 1.\n\nThis minimization procedure must result in that potentials are\nhomogeneous in the system as discussed above. Since the present book\ndeals with thermodynamics of materials, the chemical potential of each\ncomponent is of particular interest and must be homogenous in all phases\nof the system at equilibrium, i.e.\n\nEq. \u2011\n$\\mu_{i}^{\\alpha} = \\mu_{i}^{\\beta} = \\mu_{i}^{\\gamma}\\ldots\\ldots$\n\nFor phases in which the chemical potentials of individual components\ncannot be evaluated due to stoichiometry, the combined chemical\npotentials can be used to relate individual potentials as shown by and .\nAs proved in Chapter shown by , the chemical potential of a component in\na solution is represented by the intercept on the Gibbs energy axis by\nthe multi-dimensional tangent surface of Gibbs energy of the solution\nplotted with respect to mole fractions of independent components. The\nGibbs energy functions of all phases in equilibrium must thus share the\nsame tangent surface. This is usually referred to as the common tangent\nconstruction for phases at equilibrium. Any phase with its Gibbs energy\ncurve above the tangent surface is not stable under given compositions\nof the system.\n", + "source": [ + "## General condition of equilibrium\n", + "\n", + "A system is heterogeneous when some properties have different values at\n", + "different portions of the system when the system is at equilibrium. Two\n", + "scenarios may exist where the variations of properties can be either\n", + "continuous or discontinuous. In the scenario of continuous variations,\n", + "the gradients of those variations must be coupled so that the system\n", + "remains at equilibrium. The number of independent variables is thus\n", + "reduced. These gradients must also be constrained along the boundaries\n", + "between the system and the surroundings. This type of constrained\n", + "equilibrium is not discussed in the book as it involves heterogeneous\n", + "boundary conditions between the system and the surroundings and depends\n", + "on the morphology of the system.\n", + "\n", + "In the second scenario with discontinuous variations, those properties\n", + "have different values in different portions of the system, but remain\n", + "homogenous within each portion. The system is in equilibrium as each\n", + "portion is in equilibrium with all other portions of the system. Those\n", + "homogeneous portions represent different phases in the system with the\n", + "properties in each phase being homogeneous at equilibrium. In the\n", + "previous chapter, it has been shown that all potentials are homogeneous\n", + "in a homogeneous system.\n", + "\n", + "For a heterogeneous system, the same conclusion can be obtained. If the\n", + "temperature is inhomogeneous, heat can be conducted from high\n", + "temperature locations to low temperature locations, and this process is\n", + "irreversible based on the second law of thermodynamics because it\n", + "increases the internal entropy of the system. If the pressure is\n", + "inhomogeneous, the amounts of lower molar volume phases will increase to\n", + "reduce the internal energy of the system. If the chemical potential of a\n", + "component is inhomogeneous, the chemical potential difference of the\n", + "component will drive that component to the locations with a lower\n", + "chemical potential so the internal energy of the system can be\n", + "decreased. Therefore, it can be concluded that all potentials are\n", + "homogeneous in a heterogeneous system at equilibrium, and the variables\n", + "that are not homogeneous are thus their conjugate molar quantities.\n", + "Under certain special circumstances to be discussed in later part of\n", + "this book, some molar quantities may also have the same values in\n", + "difference phases.\n", + "\n", + "In a system at equilibrium with $c$ independent components, there are\n", + "$c + 2$ pairs of conjugate variables based on or though more can be\n", + "added as shown by depending on experimental conditions. For simplicity,\n", + "most discussions in this book are limited to systems with $c + 2$ pairs\n", + "of conjugate variables unless otherwise specified, with number \u201c2\u201d\n", + "representing conjugate variables of $T - S$ and $( - P) - V$.\n", + "\n", + "For a system under constant temperature, pressure, and moles of each\n", + "independent component, the equilibrium condition derives from as\n", + "\n", + "Eq. \u2011\n", + "$dG = - SdT - Vd( - P) + \\sum_{}^{}\\mu_{i}{dN}_{i} - Dd\\xi = - Dd\\xi = 0$\n", + "\n", + "Consequently, the equilibrium state is defined by the minimization of\n", + "Gibbs energy of the system at constant $T$, $P$ and $N_{i}$ because the\n", + "second derivatives need to be positive for the equilibrium system to be\n", + "stable as stipulated by . For heterogeneous systems with two or more\n", + "phases, Gibbs energy of the system is the weighted summation of Gibbs\n", + "energies of individual phases, i.e.\n", + "\n", + "Eq. \u2011 $\\frac{G}{N} = G_{m} = \\sum_{\\beta}^{}{f^{\\beta}G_{m}^{\\beta}}$\n", + "\n", + "where $f^{\\beta}$ and $G_{m}^{\\beta}$ are the mole fraction and molar\n", + "Gibbs energy of the phase $\\beta$, respectively, and the summation goes\n", + "over all phases in the system. $f^{\\beta}$ is equal to zero for phases\n", + "not present in the equilibrium state.\n", + "\n", + "The minimization of Gibbs energy of the system is carried out under the\n", + "following mass balance conditions\n", + "\n", + "Eq. \u2011\n", + "$x_{i} = \\sum_{\\beta}^{}{f^{\\beta}x_{i}^{\\beta}} = \\sum_{\\beta}^{}{f^{\\beta}\\frac{\\sum_{\\beta - t}^{}{a^{\\beta - t}\\sum_{k}^{}{i_{k}^{\\beta - t}y_{k}^{\\beta - t}}}}{\\sum_{\\beta - t}^{}{a^{\\beta - t}\\left( 1 - y_{va}^{\\beta - t} \\right)}}}$\n", + "\n", + "Eq. \u2011 $\\sum_{i}^{}x_{i} = 1$\n", + "\n", + "Eq. \u2011 $\\sum_{k}^{}y_{k}^{\\beta - t} = 1$\n", + "\n", + "where $a^{\\beta - t}$ and $y_{k}^{\\beta - t}$ are the number of site in\n", + "sublattice $t$ in the $\\beta$ phase and the corresponding site fraction\n", + "of species $k$ in the sublattice, respectively, and $i_{k}^{\\beta - t}$\n", + "is the stoichiometry of the component $i$ in the species $k$, as used in\n", + ". The summation in runs over species for each sublattice. For phases\n", + "containing ionic species, electroneutrality also needs to be maintained,\n", + "i.e. is applied to each phase. This minimization problem of Gibbs energy\n", + "under the constraint of mass conservation can be solved by means of a\n", + "range of algorithms. It should be noted that the mole fractions of\n", + "phases and site fractions of species are bounded between 0 and 1.\n", + "\n", + "This minimization procedure must result in that potentials are\n", + "homogeneous in the system as discussed above. Since the present book\n", + "deals with thermodynamics of materials, the chemical potential of each\n", + "component is of particular interest and must be homogenous in all phases\n", + "of the system at equilibrium, i.e.\n", + "\n", + "Eq. \u2011\n", + "$\\mu_{i}^{\\alpha} = \\mu_{i}^{\\beta} = \\mu_{i}^{\\gamma}\\ldots\\ldots$\n", + "\n", + "For phases in which the chemical potentials of individual components\n", + "cannot be evaluated due to stoichiometry, the combined chemical\n", + "potentials can be used to relate individual potentials as shown by and .\n", + "As proved in Chapter shown by , the chemical potential of a component in\n", + "a solution is represented by the intercept on the Gibbs energy axis by\n", + "the multi-dimensional tangent surface of Gibbs energy of the solution\n", + "plotted with respect to mole fractions of independent components. The\n", + "Gibbs energy functions of all phases in equilibrium must thus share the\n", + "same tangent surface. This is usually referred to as the common tangent\n", + "construction for phases at equilibrium. Any phase with its Gibbs energy\n", + "curve above the tangent surface is not stable under given compositions\n", + "of the system.\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/phase_equilibria_in_heterogeneous_systems/gibbs_phase_rule.ipynb b/src/psu410/src/phase_equilibria_in_heterogeneous_systems/gibbs_phase_rule.ipynb index 89e0c82..2fcd012 100644 --- a/src/psu410/src/phase_equilibria_in_heterogeneous_systems/gibbs_phase_rule.ipynb +++ b/src/psu410/src/phase_equilibria_in_heterogeneous_systems/gibbs_phase_rule.ipynb @@ -4,9 +4,60 @@ "metadata": {}, "cells": [ { - "id": "bb986421", + "id": "a89214fd", "cell_type": "markdown", - "source": "## Gibbs phase rule\n\nThe Gibbs-Duhem equation, i.e. , states that only $c + 1$ potentials are\nindependent in a homogeneous system with $c$ independent components and\nthe additional two variables of temperature and pressure. In a\nheterogeneous system at equilibrium, this equation can be applied to\nindividual phases as each phase is homogeneous. Noting that each\npotential has the same value in all phases at equilibrium, can be\nwritten as follows for each individual phase, $\\ \\beta$, in the system\nat equilibrium\n\nEq. \u2011\n$0 = - S^{\\beta}dT - V^{\\beta}d( - P) - \\sum_{}^{}{N_{i}^{\\beta}d\\mu}_{i}$\n\nFor a system with $p$ phases at equilibrium, there are $p$ such\nequations relating the potentials in the system. The number of\nindependent potentials thus becomes\n\nEq. \u2011 $\\upsilon = c + 2 - p$\n\nis called Gibbs phase rule. It dictates the number of potentials that\ncan change independently for a given number of phases co-existing at\nequilibrium, commonly called degree of freedom of the system at\nequilibrium. It stipulates that the maximum number of phases can\nco-exist in a system at equilibrium is obtained by setting\n$\\upsilon = 0$, called an invariant equilibrium due to the zero degree\nof freedom,\n\nEq. \u2011 $p_{\\max} = c + 2$\n\nThere are thus maximum three phases in a one-component system, four\nphases in a binary system, five phases in a ternary system, and so on,\nthat can co-exist simultaneously at equilibrium with all potentials in\nthe system at fixed values. This should not be confused with the total\nnumber of phases that could exist, but not co-exist in a system, which\nof course are not limited by Gibbs phase rule.\n\nIt should be emphasized that the degree of freedom, $\\upsilon$, is\nreferred to the number of potentials only, not to molar quantities of\nthe system because molar quantities are generally not homogeneous in a\nheterogeneous system. For example, in a system at equilibrium with\n$\\upsilon = 0$, the amount of each component can be varied, while\nkeeping the number of phases at $p_{\\max} = c + 2$. This can be done\nthrough changing the amount of each phase in the system through the mass\nbalance equation, , without altering the composition of each phase and\nthus the chemical potentials in the system. As mentioned at the\nbeginning of Chapter , the number of independent variables in a system\nat equilibrium, i.e. the sum of independent potentials and independent\nmolar quantities, is $c + 2$ with the maximum number of independent\npotentials determined by Gibbs phase rule, .\n", + "source": [ + "## Gibbs phase rule\n", + "\n", + "The Gibbs-Duhem equation, i.e. , states that only $c + 1$ potentials are\n", + "independent in a homogeneous system with $c$ independent components and\n", + "the additional two variables of temperature and pressure. In a\n", + "heterogeneous system at equilibrium, this equation can be applied to\n", + "individual phases as each phase is homogeneous. Noting that each\n", + "potential has the same value in all phases at equilibrium, can be\n", + "written as follows for each individual phase, $\\ \\beta$, in the system\n", + "at equilibrium\n", + "\n", + "Eq. \u2011\n", + "$0 = - S^{\\beta}dT - V^{\\beta}d( - P) - \\sum_{}^{}{N_{i}^{\\beta}d\\mu}_{i}$\n", + "\n", + "For a system with $p$ phases at equilibrium, there are $p$ such\n", + "equations relating the potentials in the system. The number of\n", + "independent potentials thus becomes\n", + "\n", + "Eq. \u2011 $\\upsilon = c + 2 - p$\n", + "\n", + "is called Gibbs phase rule. It dictates the number of potentials that\n", + "can change independently for a given number of phases co-existing at\n", + "equilibrium, commonly called degree of freedom of the system at\n", + "equilibrium. It stipulates that the maximum number of phases can\n", + "co-exist in a system at equilibrium is obtained by setting\n", + "$\\upsilon = 0$, called an invariant equilibrium due to the zero degree\n", + "of freedom,\n", + "\n", + "Eq. \u2011 $p_{\\max} = c + 2$\n", + "\n", + "There are thus maximum three phases in a one-component system, four\n", + "phases in a binary system, five phases in a ternary system, and so on,\n", + "that can co-exist simultaneously at equilibrium with all potentials in\n", + "the system at fixed values. This should not be confused with the total\n", + "number of phases that could exist, but not co-exist in a system, which\n", + "of course are not limited by Gibbs phase rule.\n", + "\n", + "It should be emphasized that the degree of freedom, $\\upsilon$, is\n", + "referred to the number of potentials only, not to molar quantities of\n", + "the system because molar quantities are generally not homogeneous in a\n", + "heterogeneous system. For example, in a system at equilibrium with\n", + "$\\upsilon = 0$, the amount of each component can be varied, while\n", + "keeping the number of phases at $p_{\\max} = c + 2$. This can be done\n", + "through changing the amount of each phase in the system through the mass\n", + "balance equation, , without altering the composition of each phase and\n", + "thus the chemical potentials in the system. As mentioned at the\n", + "beginning of Chapter , the number of independent variables in a system\n", + "at equilibrium, i.e. the sum of independent potentials and independent\n", + "molar quantities, is $c + 2$ with the maximum number of independent\n", + "potentials determined by Gibbs phase rule, .\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/phase_equilibria_in_heterogeneous_systems/index.ipynb b/src/psu410/src/phase_equilibria_in_heterogeneous_systems/index.ipynb index dc56fcd..c4fddd5 100644 --- a/src/psu410/src/phase_equilibria_in_heterogeneous_systems/index.ipynb +++ b/src/psu410/src/phase_equilibria_in_heterogeneous_systems/index.ipynb @@ -4,9 +4,11 @@ "metadata": {}, "cells": [ { - "id": "84575b0c", + "id": "f853d6af", "cell_type": "markdown", - "source": "# Phase equilibria in heterogeneous systems\n", + "source": [ + "# Phase equilibria in heterogeneous systems\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/phase_equilibria_in_heterogeneous_systems/molar_phase_diagrams.ipynb b/src/psu410/src/phase_equilibria_in_heterogeneous_systems/molar_phase_diagrams.ipynb index 112a8ec..4f73268 100644 --- a/src/psu410/src/phase_equilibria_in_heterogeneous_systems/molar_phase_diagrams.ipynb +++ b/src/psu410/src/phase_equilibria_in_heterogeneous_systems/molar_phase_diagrams.ipynb @@ -4,33 +4,634 @@ "metadata": {}, "cells": [ { - "id": "40a62bf1", + "id": "c05b952a", "cell_type": "markdown", - "source": "## Molar phase diagrams\n\nThe potential phase diagrams discussed in Chapter present information on\nwhich phases are in equilibrium under given values of potentials, but do\nnot have any information on the properties of phases in equilibrium. On\nthe other hand, there are direct relations between potentials and their\nconjugate molar quantities for each phase at equilibrium depicted by to\n. One may thus substitute the potentials by their conjugate molar\nquantities in the potential phase diagrams as molar quantities provide\nmore information on the properties of phases and the system. This is\nparticularly true when chemical potentials are replaced by compositions\nas the compositions of the system are often the variables controlled and\nmeasured in experiments instead of chemical potentials.\n", + "source": [ + "## Molar phase diagrams\n", + "\n", + "The potential phase diagrams discussed in Chapter present information on\n", + "which phases are in equilibrium under given values of potentials, but do\n", + "not have any information on the properties of phases in equilibrium. On\n", + "the other hand, there are direct relations between potentials and their\n", + "conjugate molar quantities for each phase at equilibrium depicted by to\n", + ". One may thus substitute the potentials by their conjugate molar\n", + "quantities in the potential phase diagrams as molar quantities provide\n", + "more information on the properties of phases and the system. This is\n", + "particularly true when chemical potentials are replaced by compositions\n", + "as the compositions of the system are often the variables controlled and\n", + "measured in experiments instead of chemical potentials.\n" + ], "metadata": {} }, { - "id": "f8421ad2", + "id": "0f9a32f5", "cell_type": "markdown", - "source": "### Tie-lines and lever rule\n\nIt is self-evident from to Eq. 2\u20114 that while potentials are homogeneous\nin all phases in a heterogeneous system at equilibrium, the molar\nquantities usually have different values in individual phases. This is\nalso stipulated in various Clausius-Clapeyron equations such as to and\nto . The difference in molar quantities thus increases the\ndimensionality of the phase region by the number of potentials replaced\nby their conjugate molar quantities. The maximum dimensionality of a\nphase region is the dimensionality of the phase diagram under\nconsideration. This thus creates a finite space between phases in\nequilibrium in the phase diagram with some axes being molar quantities.\n\nFor an equilibrium system under constant $T$, $P$ and $N_{i}$, the\npotentials in the system and their conjugate molar quantities in each\nphase are all uniquely defined. In a phase diagram with one or more\npotentials replaced by their conjugate molar quantities, two phases in\nequilibrium in a system with $c$ independent components are connected by\na $c$ dimensional line in a *c*+2 dimensional space or\nits$\\ $*c+1*-dimensional projection as discussed in Chapter . These\nlines are called tie-lines and collectively represent a two-phase\nequilibrium region. For a $k$-phase equilibrium, there are total\n$C_{k}^{2} = \\frac{k(k - 1)}{2}$ tie-lines connecting every two phases\nwith $k - 1\\ $of them independent because the number of independent\ntie-lines increases by one with each new phase added. For the invariant\nequilibrium with zero degree of freedom, the number of phases in\nequilibrium is $c + 2$ shown by containing\n$C_{c + 2}^{2} = \\frac{(c + 2)(c + 1)}{2}$ tie-lines with $c + 1$ of\nthem being independent.\n\nInside the space encapsulated by the tie-lines, the axis variables of\nthe phase diagram (a mixture of potentials and molar quantities) can be\nchanged independently without changing the phases in equilibrium and\ntheir properties. Only the relative amounts of individual phases are\nadjusted accordingly to maintain the conservation of the molar\nquantities in the system specified by the molar quantity axes of the\nphase diagram. The geometric feature circumscribing the space\nencapsulated by the tie-lines no longer represents any phase regions,\nbut a boundary between the neighbouring phase regions. Its\ncharacteristics will be discussed in more details in the next few\nsections. As properties in each phase are homogeneous, the values of\nmolar quantities of a system are simply the sum of individual phases and\ncan be represented by the following equation\n\nEq. \u2011 $A_{m} = \\sum_{\\alpha}^{}{f^{\\alpha}A_{m}^{\\alpha}}$\n\nwhere $A_{m}$ and $A_{m}^{\\alpha}$ represent the values of a molar\nquantity of the system and the $\\alpha$ phase, respectively,\n$f^{\\alpha}$ the mole fraction of the $\\alpha$ phase, and the summation\ngoes over all phases in equilibrium with each other. With\n$\\sum_{\\alpha}^{}f^{\\alpha} = 1$, can be re-arranged into the following\nequation\n\nEq. \u2011\n$\\sum_{\\alpha}^{}{f^{\\alpha}\\left( {A_{m} - A}_{m}^{\\alpha} \\right)} = 0$\n\nis commonly referred as lever rule. For a two-phase equilibrium of\n$\\alpha$ and $\\beta$, it becomes\n\nEq. \u2011\n$f^{\\alpha} = \\frac{A_{m}^{\\beta} - A_{m}}{A_{m}^{\\beta} - A_{m}^{\\alpha}}$\n\nEq. \u2011\n$f^{\\beta} = \\frac{A_{m}^{\\alpha} - A_{m}}{A_{m}^{\\alpha} - A_{m}^{\\beta}}$\n\nFor a phase diagram with the number of axes being\n$n = \\left( c - n_{s} \\right) + 1$, the number of possible axes being\nmolar quantities is thus $k \\leq n$. There are thus $k$ equations\nsimilar to with one for each molar quantity, $A_{mi}$, resulting in the\nfollowing $k + 1$ equations\n\nEq. \u2011\n$\\sum_{\\alpha}^{}{f^{\\alpha}\\left( {A_{mi} - A}_{mi}^{\\alpha} \\right)} = 0$\n\nEq. \u2011 $1 - \\sum_{\\alpha}^{}f^{\\alpha} = 0$\n\nThe summations in and go over the phases in equilibrium, and the amount\nof each phase is obtained by solving these $k + 1$ equations\nsimultaneously along with the equilibrium conditions.\n", + "source": [ + "### Tie-lines and lever rule\n", + "\n", + "It is self-evident from to Eq. 2\u20114 that while potentials are homogeneous\n", + "in all phases in a heterogeneous system at equilibrium, the molar\n", + "quantities usually have different values in individual phases. This is\n", + "also stipulated in various Clausius-Clapeyron equations such as to and\n", + "to . The difference in molar quantities thus increases the\n", + "dimensionality of the phase region by the number of potentials replaced\n", + "by their conjugate molar quantities. The maximum dimensionality of a\n", + "phase region is the dimensionality of the phase diagram under\n", + "consideration. This thus creates a finite space between phases in\n", + "equilibrium in the phase diagram with some axes being molar quantities.\n", + "\n", + "For an equilibrium system under constant $T$, $P$ and $N_{i}$, the\n", + "potentials in the system and their conjugate molar quantities in each\n", + "phase are all uniquely defined. In a phase diagram with one or more\n", + "potentials replaced by their conjugate molar quantities, two phases in\n", + "equilibrium in a system with $c$ independent components are connected by\n", + "a $c$ dimensional line in a *c*+2 dimensional space or\n", + "its$\\ $*c+1*-dimensional projection as discussed in Chapter . These\n", + "lines are called tie-lines and collectively represent a two-phase\n", + "equilibrium region. For a $k$-phase equilibrium, there are total\n", + "$C_{k}^{2} = \\frac{k(k - 1)}{2}$ tie-lines connecting every two phases\n", + "with $k - 1\\ $of them independent because the number of independent\n", + "tie-lines increases by one with each new phase added. For the invariant\n", + "equilibrium with zero degree of freedom, the number of phases in\n", + "equilibrium is $c + 2$ shown by containing\n", + "$C_{c + 2}^{2} = \\frac{(c + 2)(c + 1)}{2}$ tie-lines with $c + 1$ of\n", + "them being independent.\n", + "\n", + "Inside the space encapsulated by the tie-lines, the axis variables of\n", + "the phase diagram (a mixture of potentials and molar quantities) can be\n", + "changed independently without changing the phases in equilibrium and\n", + "their properties. Only the relative amounts of individual phases are\n", + "adjusted accordingly to maintain the conservation of the molar\n", + "quantities in the system specified by the molar quantity axes of the\n", + "phase diagram. The geometric feature circumscribing the space\n", + "encapsulated by the tie-lines no longer represents any phase regions,\n", + "but a boundary between the neighbouring phase regions. Its\n", + "characteristics will be discussed in more details in the next few\n", + "sections. As properties in each phase are homogeneous, the values of\n", + "molar quantities of a system are simply the sum of individual phases and\n", + "can be represented by the following equation\n", + "\n", + "Eq. \u2011 $A_{m} = \\sum_{\\alpha}^{}{f^{\\alpha}A_{m}^{\\alpha}}$\n", + "\n", + "where $A_{m}$ and $A_{m}^{\\alpha}$ represent the values of a molar\n", + "quantity of the system and the $\\alpha$ phase, respectively,\n", + "$f^{\\alpha}$ the mole fraction of the $\\alpha$ phase, and the summation\n", + "goes over all phases in equilibrium with each other. With\n", + "$\\sum_{\\alpha}^{}f^{\\alpha} = 1$, can be re-arranged into the following\n", + "equation\n", + "\n", + "Eq. \u2011\n", + "$\\sum_{\\alpha}^{}{f^{\\alpha}\\left( {A_{m} - A}_{m}^{\\alpha} \\right)} = 0$\n", + "\n", + "is commonly referred as lever rule. For a two-phase equilibrium of\n", + "$\\alpha$ and $\\beta$, it becomes\n", + "\n", + "Eq. \u2011\n", + "$f^{\\alpha} = \\frac{A_{m}^{\\beta} - A_{m}}{A_{m}^{\\beta} - A_{m}^{\\alpha}}$\n", + "\n", + "Eq. \u2011\n", + "$f^{\\beta} = \\frac{A_{m}^{\\alpha} - A_{m}}{A_{m}^{\\alpha} - A_{m}^{\\beta}}$\n", + "\n", + "For a phase diagram with the number of axes being\n", + "$n = \\left( c - n_{s} \\right) + 1$, the number of possible axes being\n", + "molar quantities is thus $k \\leq n$. There are thus $k$ equations\n", + "similar to with one for each molar quantity, $A_{mi}$, resulting in the\n", + "following $k + 1$ equations\n", + "\n", + "Eq. \u2011\n", + "$\\sum_{\\alpha}^{}{f^{\\alpha}\\left( {A_{mi} - A}_{mi}^{\\alpha} \\right)} = 0$\n", + "\n", + "Eq. \u2011 $1 - \\sum_{\\alpha}^{}f^{\\alpha} = 0$\n", + "\n", + "The summations in and go over the phases in equilibrium, and the amount\n", + "of each phase is obtained by solving these $k + 1$ equations\n", + "simultaneously along with the equilibrium conditions.\n" + ], "metadata": {} }, { - "id": "c5827dc1", + "id": "faa2022e", "cell_type": "markdown", - "source": "### Phase diagrams with both potentials and molar quantities\n\nBased on Gibbs phase rule discussed in Chapters and , the\ndimensionalities of phase regions in a potential phase diagram are given\nby or . With every potential substituted by its conjugate molar\nquantity, the dimensionalities of phase regions increase by one until\nthe phase region reaches the dimensionality of the phase diagram. The\naxes of this phase diagram now consist of both potentials and molar\nquantities. The dimensionality of a phase region can thus be represented\nby the following equation based on\n\nEq. \u2011\n$\\upsilon_{m} = \\left( c - n_{s} \\right) + 2 - p + n_{m} \\leq \\left( c - n_{s} \\right) + 1$\n\nwhere $n_{m}$ is the number of molar axes. This equation is applicable\nto phase regions with more than $n_{m} + 1$ phases. For phase regions\nwith $n_{m} + 1$ phases or fewer, the dimensionalities are the same as\nthe phase diagram, i.e. $\\upsilon_{m} = \\left( c - n_{s} \\right) + 1$,\nand no longer vary with the number of molar quantity axes. When all\n$\\left( c - n_{s} \\right) + 1$ potentials are substituted by their\nconjugate molar quantities, one obtains a complete molar phase diagrams\nto be discussed in Chapter , and all phase regions have the\nsame-dimensionality of $\\left( c - n_{s} \\right) + 1$.\n\nFor the sake of graphic visualization, let us exam a two-dimensional\nphase diagram of a one-component system. Topologically, it is equivalent\nto a multi-component system with $n_{s} = c - 1$. In , three\ntwo-dimensional phase diagrams are shown for pure Fe. In principle, one\ncan use any one of them to illustrate mixed potential and molar phase\ndiagrams. For the purpose of practical usefulness, one selects the\n$T - ( - P)$ potential diagram as temperature and pressure are the two\ntypical variables controlled in experiments of one-component systems.\nThe conjugate molar quantities of $- P$ and $T$ are molar volume and\nmolar entropy, respectively. For stable phases, any pair of conjugate\nvariables changes in the same direction as illustrated by and , i.e. the\nphase stable at higher $T$ has higher molar entropy, and the phase\nstable at higher $\u2013P$, i.e. lower pressure, has higher molar volume. Let\nus first substitute $\u2013P$ by $V_{m}$ as shown in a. The dimensionality of\na sing-phase region remains unchanged because of\n$p = 1 < 2 = n_{m} + 1$. The dimensionality of two-phase regions is\nchanged from 1 to 2 due to $\\upsilon_{m} = 3 - 2 + 1 = 2$ from .\n\nFigure \u2011: T-Vm and Sm-(-P) phase diagrams of Fe\n\nAs both phases in a two-phase equilibrium have the same temperature, all\ntie-lines, depicted by dotted lines, are perpendicular to the\ntemperature axis in a. When the molar volume of the system changes from\none end of a tie-line to another end at a constant temperature, the mole\nfraction of one phase increases from 0 to 1, and the mole fraction of\nanother phase decreases from 1 to 0. The tine-lines at various\ntemperatures combine together to form a two-dimensional two-phase\nregion. The two curves at the two ends of tie-lines represent the\nboundaries between the single-phase and two-phase regions and are no\nlonger phase regions themselves. They are thus called phase boundaries.\n\nBy the same token, by changing the temperature at a constant molar\nvolume of the system, the system will locate on different tie-lines with\nthe amounts of the two phases determined by the lever rule. It is thus\nclear that the system maintains the two-phase equilibrium state with\nboth $T$ and $V_{m}$ changing independently inside the two-dimensional\ntwo-phase region. This seems in contradiction to Gibbs phase rule being\n$\\upsilon = 3 - p = 1$ from , but it is not because Gibbs phase rule\napplies strictly to potential phase diagrams only, while the $T - V_{m}$\nphase diagram has one of its axis being a molar quantity. As an\nalternative, one may consider as a modified Gibbs phase rule in\ndescribing the dimensionality of a phase region in a mixed potential and\nmolar phase diagram with $p \\geq n_{m} + 1$.\n\nThe two three-phase equilibria in pure Fe are also represented by\ntie-lines connecting all three phases. The dimensionality of three-phase\nregions is $\\upsilon_{m} = 3 - 3 + 1 = 1$ from , and the three two-phase\ntie-lines for a three-phase equilibrium thus overlap each other with\ntheir three molar volumes on the same tie-line.\n\nLet us examine the two three-phase equilibria in more detail. In the\n\u03b3/\u03b1/\u03b5 three-phase equilibrium at $T_{E} = 756.6\\ K$ and\n$\\left( - P_{E} \\right) = - 1.046 \\bullet 10^{10}\\ Pa$, the molar\nvolumes of \u03b1, \u03b3, and \u03b5 are 6.837, 6.677, 6.582 $10^{- 6}\\ m^{3}/mol$,\nrespectively. There are two two-phase regions at higher temperatures and\none two-phase region at lower temperatures. This is also shown in the\npotential phase diagram of a with two two-phase curves entering into and\none two-phase curve leaving from the three-phase equilibrium point with\ndecreasing temperature. Considering the system with the fixed molar\nvolume equal to that of the $\\gamma$ phase, i.e.\n$V_{m}^{\\ }\\left( T_{E} \\right) = 6.677 \\bullet 10^{- 6}\\ m^{3}/mol$. At\n$T > T_{E}$, the system is in the single \u03b3 phase region. With decrease\nin temperature across $T_{E}$, it enters into the $\\alpha + \\varepsilon$\ntwo-phase region. This transformation can be written as follows and is\ncalled a eutectoid reaction\n\nEq. \u2011 $\\gamma \\rightarrow \\alpha + \\varepsilon$\n\nThis type of transformations is named as eutectic reaction if the high\ntemperature phase is a liquid phase. If the system molar volume is\nbetween $V_{m}^{\\gamma}$ and $V_{m}^{\\alpha}$ at $756.6K$, with decrease\nof temperature, the system first moves from the single \u03b3 phase region to\nthe $\\alpha + \\gamma$ two-phase region when the\n$\\gamma/(\\alpha + \\gamma)$ phase boundary is crossed. When the\ntemperature reaches $T_{E}$, the eutectoid transformation takes place in\nthe remaining \u03b3 phase. The $\\alpha$ formed prior to the eutectoid\ntransformation is called proeutectoid $\\alpha$. By the same token, when\nthe system molar volume is between $V_{m}^{\\gamma}$ and\n$V_{m}^{\\varepsilon}$ at $756.6K$, proeutectoid $\\varepsilon$ would form\nfollowed by the eutectoid transformation with decrease in temperature.\n\nOn the other hand, the $L/\\delta/\\gamma$ three-phase equilibrium at\n$T_{P} = 1977.9\\ K$ and\n$\\left( - P_{P} \\right) = - 5.111 \\bullet \\ 10^{9}\\ Pa$ has different\ncharacteristics, where the subscript P will be defined shortly. There\nare one two-phase equilibrium above and two two-phase equilibria below\nthe invariant temperature, shown in the potential phase diagram of a\nwith one two-phase curve entering into and two two-phase curves leaving\nfrom the three-phase equilibrium point with decreasing temperature. The\nmolar volumes of $L$, $\\delta$, and $\\gamma$ at $T_{P}$ are 7.735,\n7.542, and 7.498 $10^{- 6}\\ m^{3}/mol$, respectively. For $T > T_{P}$,\nthe two-phase region is $L + \\gamma$. If the system molar volume is\nbetween 7.735 and 7.498 $10^{- 6}\\ m^{3}/mol$, i.e.\n$V_{m}^{L}\\left( T_{P} \\right)$ and\n$V_{m}^{\\gamma}\\left( T_{P} \\right)$, when the temperature reaches\n$T_{P}$, $L$ and $\\gamma$ are combined to form $\\delta$ with the\ntransformation written as\n\nEq. \u2011 $L + \\gamma \\rightarrow \\delta$\n\nThis type of reactions is called as peritectic reaction or peritectoid\nreaction when the liquid phase is replaced by a solid phase, denoted by\nthe subscript P. At $T < T_{P}$, one or both high temperature phases may\nno longer be present in equilibrium depending on the value of the system\nmolar volume. For\n$V_{m}^{\\ } = V_{m}^{\\delta}\\left( T_{P} \\right) = 7.542 \\bullet 10^{- 6}\\ m^{3}/mol$,\nthe peritectic reaction, , can come to completion with no $L$ and\n$\\gamma$ left upon decreasing temperature. For\n$V_{m}^{\\gamma}\\left( T_{P} \\right) = 7.498 < V_{m}^{\\ } < V_{m}^{\\delta}\\left( T_{P} \\right) = 7.542\\ \\left( 10^{- 6}\\ m^{3}/mol \\right)$,\nthe liquid phase is consumed, and the system enters the\n$\\gamma + \\delta$ two-phase region. On the other hand, for\n$V_{m}^{\\delta}\\left( T_{P} \\right) = 7.542 < V_{m}^{\\ } < V_{m}^{\\ L}\\left( T_{P} \\right) = 7.735\\ \\ \\left( 10^{- 6}\\ m^{3}/mol \\right)$,\nthe $\\gamma$ phase is consumed instead, and the system enters into the\n$L + \\delta$ two-phase equilibrium region upon cooling.\n\nLet us now replace $T$ by $S_{m}$ to obtain the $( - P) - S_{m}$ phase\ndiagram shown in b. The morphology of this phase diagram is identical to\nthe $T - V_{m}$ phase diagram just discussed with all tie-lines\nperpendicular to the pressure axis. The transformations at the two\nthree-phase equilibria with $( - P)$ decreasing or $P$ increasing are as\nfollows\n\nEq. \u2011 $\\gamma + \\alpha \\rightarrow \\varepsilon$\n\nEq. \u2011 $\\delta \\rightarrow L + \\gamma$\n\nTo visualize two-dimensional phase diagrams of binary systems, one\nusually keeps the pressure constant. One type of commonly used binary\nphase diagram is the temperature-composition ($T - x$) phase diagram. As\nan example, let us re-plot the $T - \\mu_{C}$ potential diagram shown in\ninto a $T - x_{C}$ mixed potential and molar phase diagram by replacing\nthe chemical potential of $C$ by its mole fraction. The $T - x_{C}$\nphase diagram thus obtained is shown in . In this phase diagram there\nare one peritectic reaction and two eutectic reactions as follows\n\nEq. \u2011 $L + \\delta \\rightarrow \\gamma$\n\nEq. \u2011 $L \\rightarrow \\gamma + C$\n\nEq. 3\u201161 $\\gamma \\rightarrow \\alpha + C$\n\nFigure \u2011: $T - x_{C}\\ $ phase diagram of the Fe-C binary system\n\nIn in Chapter , it was discussed the formation of miscibility gaps due\nto repulsive interactions between components. One example is shown in\nfor the Al-Zn binary system in terms of both $T - \\mu_{Zn}$ potential\nphase diagram and $\\ T - x_{Zn}\\ $ mixed potential and molar phase\ndiagram.\n\nFigure \u2011: $T - \\mu_{Zn}$ potential phase diagram (a) and\n$\\ T - x_{Zn}\\ $ mixed potential and molar phase diagram (b) of the\nAl-Zn binary system\n\nIn ,there are one eutectic reaction and one eutectoid reaction as\nfollows\n\nEq. \u2011 $L \\rightarrow fcc + hcp$\n\nEq. \u2011 $fcc\\# 1 \\rightarrow fcc\\# 2 + hcp$\n\nThe eutectoid reaction, , is also termed as monotectoid reaction because\nthe fcc phase appears on both sides of the reaction with different\ncompositions, $fcc\\# 1$ and $fcc\\# 2$, due to the miscibility gap. The\nhighest temperature of the miscibility gap is called the consolute point\nas discussed in Chapter , which can be clearly seen in the\n$T - \\mu_{Zn}$ potential phase diagram shown in . This is a critical\npoint, marking the limit of instability as shown in .\n\nWhen there is only one phase on either side of the reaction, i.e. both\nphases have the same composition, the reaction is called a congruent\nreaction. One example is shown in for the $T - x_{SiO_{2}}\\ $ mixed\npotential and molar phase diagram of the CaO-SiO2\npseudo-binary system with two congruent reactions as follows\n\nEq. 3\u201164 $L \\rightarrow CaSiO_{3}$\n\nEq. 3\u201165 $L \\rightarrow {Ca}_{2}SiO_{4}$\n\nThey are not invariant reactions based on the Gibbs phase rule. In , it\nis noted that there are one miscibility gap in the liquid phase close to\nthe $SiO_{2}$ side, four eutectic reactions with one being monotectic\ninvolving two liquid phases due to the miscibility gap, and three\nperitectic reactions.\n\nFigure \u2011: $T - x_{SiO_{2}}\\ $ mixed potential and molar phase diagram of\nthe CaO-SiO2 pseudo-binary system.\n\nLet us generalize the above discussion to phase diagrams with\n$\\left( c - n_{s} \\right) + 1$ axes. In such a phase diagram, the\nmaximum number of phases is given by as\n$p_{\\max} = \\left( c - n_{s} \\right) + 2$. The number of phases on the\neither side of an invariant reaction can vary from one phase to\n$p_{\\max} - 1 = \\left( c - n_{s} \\right) + 1$ phases with the remaining\nphases on the other side of the reaction, typically with the potential\ndecreasing from left to right. The invariant reaction with one phase on\nthe left of the reaction is named as eutectic, or eutectoid reaction\ndepending on if the phase on the left of the reaction is liquid or\nsolid. The rest invariant reactions are named as peritectic or\nperitectoid reactions with or without a liquid phase.\n", + "source": [ + "### Phase diagrams with both potentials and molar quantities\n", + "\n", + "Based on Gibbs phase rule discussed in Chapters and , the\n", + "dimensionalities of phase regions in a potential phase diagram are given\n", + "by or . With every potential substituted by its conjugate molar\n", + "quantity, the dimensionalities of phase regions increase by one until\n", + "the phase region reaches the dimensionality of the phase diagram. The\n", + "axes of this phase diagram now consist of both potentials and molar\n", + "quantities. The dimensionality of a phase region can thus be represented\n", + "by the following equation based on\n", + "\n", + "Eq. \u2011\n", + "$\\upsilon_{m} = \\left( c - n_{s} \\right) + 2 - p + n_{m} \\leq \\left( c - n_{s} \\right) + 1$\n", + "\n", + "where $n_{m}$ is the number of molar axes. This equation is applicable\n", + "to phase regions with more than $n_{m} + 1$ phases. For phase regions\n", + "with $n_{m} + 1$ phases or fewer, the dimensionalities are the same as\n", + "the phase diagram, i.e. $\\upsilon_{m} = \\left( c - n_{s} \\right) + 1$,\n", + "and no longer vary with the number of molar quantity axes. When all\n", + "$\\left( c - n_{s} \\right) + 1$ potentials are substituted by their\n", + "conjugate molar quantities, one obtains a complete molar phase diagrams\n", + "to be discussed in Chapter , and all phase regions have the\n", + "same-dimensionality of $\\left( c - n_{s} \\right) + 1$.\n", + "\n", + "For the sake of graphic visualization, let us exam a two-dimensional\n", + "phase diagram of a one-component system. Topologically, it is equivalent\n", + "to a multi-component system with $n_{s} = c - 1$. In , three\n", + "two-dimensional phase diagrams are shown for pure Fe. In principle, one\n", + "can use any one of them to illustrate mixed potential and molar phase\n", + "diagrams. For the purpose of practical usefulness, one selects the\n", + "$T - ( - P)$ potential diagram as temperature and pressure are the two\n", + "typical variables controlled in experiments of one-component systems.\n", + "The conjugate molar quantities of $- P$ and $T$ are molar volume and\n", + "molar entropy, respectively. For stable phases, any pair of conjugate\n", + "variables changes in the same direction as illustrated by and , i.e. the\n", + "phase stable at higher $T$ has higher molar entropy, and the phase\n", + "stable at higher $\u2013P$, i.e. lower pressure, has higher molar volume. Let\n", + "us first substitute $\u2013P$ by $V_{m}$ as shown in a. The dimensionality of\n", + "a sing-phase region remains unchanged because of\n", + "$p = 1 < 2 = n_{m} + 1$. The dimensionality of two-phase regions is\n", + "changed from 1 to 2 due to $\\upsilon_{m} = 3 - 2 + 1 = 2$ from .\n", + "\n", + "Figure \u2011: T-Vm and Sm-(-P) phase diagrams of Fe\n", + "\n", + "As both phases in a two-phase equilibrium have the same temperature, all\n", + "tie-lines, depicted by dotted lines, are perpendicular to the\n", + "temperature axis in a. When the molar volume of the system changes from\n", + "one end of a tie-line to another end at a constant temperature, the mole\n", + "fraction of one phase increases from 0 to 1, and the mole fraction of\n", + "another phase decreases from 1 to 0. The tine-lines at various\n", + "temperatures combine together to form a two-dimensional two-phase\n", + "region. The two curves at the two ends of tie-lines represent the\n", + "boundaries between the single-phase and two-phase regions and are no\n", + "longer phase regions themselves. They are thus called phase boundaries.\n", + "\n", + "By the same token, by changing the temperature at a constant molar\n", + "volume of the system, the system will locate on different tie-lines with\n", + "the amounts of the two phases determined by the lever rule. It is thus\n", + "clear that the system maintains the two-phase equilibrium state with\n", + "both $T$ and $V_{m}$ changing independently inside the two-dimensional\n", + "two-phase region. This seems in contradiction to Gibbs phase rule being\n", + "$\\upsilon = 3 - p = 1$ from , but it is not because Gibbs phase rule\n", + "applies strictly to potential phase diagrams only, while the $T - V_{m}$\n", + "phase diagram has one of its axis being a molar quantity. As an\n", + "alternative, one may consider as a modified Gibbs phase rule in\n", + "describing the dimensionality of a phase region in a mixed potential and\n", + "molar phase diagram with $p \\geq n_{m} + 1$.\n", + "\n", + "The two three-phase equilibria in pure Fe are also represented by\n", + "tie-lines connecting all three phases. The dimensionality of three-phase\n", + "regions is $\\upsilon_{m} = 3 - 3 + 1 = 1$ from , and the three two-phase\n", + "tie-lines for a three-phase equilibrium thus overlap each other with\n", + "their three molar volumes on the same tie-line.\n", + "\n", + "Let us examine the two three-phase equilibria in more detail. In the\n", + "\u03b3/\u03b1/\u03b5 three-phase equilibrium at $T_{E} = 756.6\\ K$ and\n", + "$\\left( - P_{E} \\right) = - 1.046 \\bullet 10^{10}\\ Pa$, the molar\n", + "volumes of \u03b1, \u03b3, and \u03b5 are 6.837, 6.677, 6.582 $10^{- 6}\\ m^{3}/mol$,\n", + "respectively. There are two two-phase regions at higher temperatures and\n", + "one two-phase region at lower temperatures. This is also shown in the\n", + "potential phase diagram of a with two two-phase curves entering into and\n", + "one two-phase curve leaving from the three-phase equilibrium point with\n", + "decreasing temperature. Considering the system with the fixed molar\n", + "volume equal to that of the $\\gamma$ phase, i.e.\n", + "$V_{m}^{\\ }\\left( T_{E} \\right) = 6.677 \\bullet 10^{- 6}\\ m^{3}/mol$. At\n", + "$T > T_{E}$, the system is in the single \u03b3 phase region. With decrease\n", + "in temperature across $T_{E}$, it enters into the $\\alpha + \\varepsilon$\n", + "two-phase region. This transformation can be written as follows and is\n", + "called a eutectoid reaction\n", + "\n", + "Eq. \u2011 $\\gamma \\rightarrow \\alpha + \\varepsilon$\n", + "\n", + "This type of transformations is named as eutectic reaction if the high\n", + "temperature phase is a liquid phase. If the system molar volume is\n", + "between $V_{m}^{\\gamma}$ and $V_{m}^{\\alpha}$ at $756.6K$, with decrease\n", + "of temperature, the system first moves from the single \u03b3 phase region to\n", + "the $\\alpha + \\gamma$ two-phase region when the\n", + "$\\gamma/(\\alpha + \\gamma)$ phase boundary is crossed. When the\n", + "temperature reaches $T_{E}$, the eutectoid transformation takes place in\n", + "the remaining \u03b3 phase. The $\\alpha$ formed prior to the eutectoid\n", + "transformation is called proeutectoid $\\alpha$. By the same token, when\n", + "the system molar volume is between $V_{m}^{\\gamma}$ and\n", + "$V_{m}^{\\varepsilon}$ at $756.6K$, proeutectoid $\\varepsilon$ would form\n", + "followed by the eutectoid transformation with decrease in temperature.\n", + "\n", + "On the other hand, the $L/\\delta/\\gamma$ three-phase equilibrium at\n", + "$T_{P} = 1977.9\\ K$ and\n", + "$\\left( - P_{P} \\right) = - 5.111 \\bullet \\ 10^{9}\\ Pa$ has different\n", + "characteristics, where the subscript P will be defined shortly. There\n", + "are one two-phase equilibrium above and two two-phase equilibria below\n", + "the invariant temperature, shown in the potential phase diagram of a\n", + "with one two-phase curve entering into and two two-phase curves leaving\n", + "from the three-phase equilibrium point with decreasing temperature. The\n", + "molar volumes of $L$, $\\delta$, and $\\gamma$ at $T_{P}$ are 7.735,\n", + "7.542, and 7.498 $10^{- 6}\\ m^{3}/mol$, respectively. For $T > T_{P}$,\n", + "the two-phase region is $L + \\gamma$. If the system molar volume is\n", + "between 7.735 and 7.498 $10^{- 6}\\ m^{3}/mol$, i.e.\n", + "$V_{m}^{L}\\left( T_{P} \\right)$ and\n", + "$V_{m}^{\\gamma}\\left( T_{P} \\right)$, when the temperature reaches\n", + "$T_{P}$, $L$ and $\\gamma$ are combined to form $\\delta$ with the\n", + "transformation written as\n", + "\n", + "Eq. \u2011 $L + \\gamma \\rightarrow \\delta$\n", + "\n", + "This type of reactions is called as peritectic reaction or peritectoid\n", + "reaction when the liquid phase is replaced by a solid phase, denoted by\n", + "the subscript P. At $T < T_{P}$, one or both high temperature phases may\n", + "no longer be present in equilibrium depending on the value of the system\n", + "molar volume. For\n", + "$V_{m}^{\\ } = V_{m}^{\\delta}\\left( T_{P} \\right) = 7.542 \\bullet 10^{- 6}\\ m^{3}/mol$,\n", + "the peritectic reaction, , can come to completion with no $L$ and\n", + "$\\gamma$ left upon decreasing temperature. For\n", + "$V_{m}^{\\gamma}\\left( T_{P} \\right) = 7.498 < V_{m}^{\\ } < V_{m}^{\\delta}\\left( T_{P} \\right) = 7.542\\ \\left( 10^{- 6}\\ m^{3}/mol \\right)$,\n", + "the liquid phase is consumed, and the system enters the\n", + "$\\gamma + \\delta$ two-phase region. On the other hand, for\n", + "$V_{m}^{\\delta}\\left( T_{P} \\right) = 7.542 < V_{m}^{\\ } < V_{m}^{\\ L}\\left( T_{P} \\right) = 7.735\\ \\ \\left( 10^{- 6}\\ m^{3}/mol \\right)$,\n", + "the $\\gamma$ phase is consumed instead, and the system enters into the\n", + "$L + \\delta$ two-phase equilibrium region upon cooling.\n", + "\n", + "Let us now replace $T$ by $S_{m}$ to obtain the $( - P) - S_{m}$ phase\n", + "diagram shown in b. The morphology of this phase diagram is identical to\n", + "the $T - V_{m}$ phase diagram just discussed with all tie-lines\n", + "perpendicular to the pressure axis. The transformations at the two\n", + "three-phase equilibria with $( - P)$ decreasing or $P$ increasing are as\n", + "follows\n", + "\n", + "Eq. \u2011 $\\gamma + \\alpha \\rightarrow \\varepsilon$\n", + "\n", + "Eq. \u2011 $\\delta \\rightarrow L + \\gamma$\n", + "\n", + "To visualize two-dimensional phase diagrams of binary systems, one\n", + "usually keeps the pressure constant. One type of commonly used binary\n", + "phase diagram is the temperature-composition ($T - x$) phase diagram. As\n", + "an example, let us re-plot the $T - \\mu_{C}$ potential diagram shown in\n", + "into a $T - x_{C}$ mixed potential and molar phase diagram by replacing\n", + "the chemical potential of $C$ by its mole fraction. The $T - x_{C}$\n", + "phase diagram thus obtained is shown in . In this phase diagram there\n", + "are one peritectic reaction and two eutectic reactions as follows\n", + "\n", + "Eq. \u2011 $L + \\delta \\rightarrow \\gamma$\n", + "\n", + "Eq. \u2011 $L \\rightarrow \\gamma + C$\n", + "\n", + "Eq. 3\u201161 $\\gamma \\rightarrow \\alpha + C$\n", + "\n", + "Figure \u2011: $T - x_{C}\\ $ phase diagram of the Fe-C binary system\n", + "\n", + "In in Chapter , it was discussed the formation of miscibility gaps due\n", + "to repulsive interactions between components. One example is shown in\n", + "for the Al-Zn binary system in terms of both $T - \\mu_{Zn}$ potential\n", + "phase diagram and $\\ T - x_{Zn}\\ $ mixed potential and molar phase\n", + "diagram.\n", + "\n", + "Figure \u2011: $T - \\mu_{Zn}$ potential phase diagram (a) and\n", + "$\\ T - x_{Zn}\\ $ mixed potential and molar phase diagram (b) of the\n", + "Al-Zn binary system\n", + "\n", + "In ,there are one eutectic reaction and one eutectoid reaction as\n", + "follows\n", + "\n", + "Eq. \u2011 $L \\rightarrow fcc + hcp$\n", + "\n", + "Eq. \u2011 $fcc\\# 1 \\rightarrow fcc\\# 2 + hcp$\n", + "\n", + "The eutectoid reaction, , is also termed as monotectoid reaction because\n", + "the fcc phase appears on both sides of the reaction with different\n", + "compositions, $fcc\\# 1$ and $fcc\\# 2$, due to the miscibility gap. The\n", + "highest temperature of the miscibility gap is called the consolute point\n", + "as discussed in Chapter , which can be clearly seen in the\n", + "$T - \\mu_{Zn}$ potential phase diagram shown in . This is a critical\n", + "point, marking the limit of instability as shown in .\n", + "\n", + "When there is only one phase on either side of the reaction, i.e. both\n", + "phases have the same composition, the reaction is called a congruent\n", + "reaction. One example is shown in for the $T - x_{SiO_{2}}\\ $ mixed\n", + "potential and molar phase diagram of the CaO-SiO2\n", + "pseudo-binary system with two congruent reactions as follows\n", + "\n", + "Eq. 3\u201164 $L \\rightarrow CaSiO_{3}$\n", + "\n", + "Eq. 3\u201165 $L \\rightarrow {Ca}_{2}SiO_{4}$\n", + "\n", + "They are not invariant reactions based on the Gibbs phase rule. In , it\n", + "is noted that there are one miscibility gap in the liquid phase close to\n", + "the $SiO_{2}$ side, four eutectic reactions with one being monotectic\n", + "involving two liquid phases due to the miscibility gap, and three\n", + "peritectic reactions.\n", + "\n", + "Figure \u2011: $T - x_{SiO_{2}}\\ $ mixed potential and molar phase diagram of\n", + "the CaO-SiO2 pseudo-binary system.\n", + "\n", + "Let us generalize the above discussion to phase diagrams with\n", + "$\\left( c - n_{s} \\right) + 1$ axes. In such a phase diagram, the\n", + "maximum number of phases is given by as\n", + "$p_{\\max} = \\left( c - n_{s} \\right) + 2$. The number of phases on the\n", + "either side of an invariant reaction can vary from one phase to\n", + "$p_{\\max} - 1 = \\left( c - n_{s} \\right) + 1$ phases with the remaining\n", + "phases on the other side of the reaction, typically with the potential\n", + "decreasing from left to right. The invariant reaction with one phase on\n", + "the left of the reaction is named as eutectic, or eutectoid reaction\n", + "depending on if the phase on the left of the reaction is liquid or\n", + "solid. The rest invariant reactions are named as peritectic or\n", + "peritectoid reactions with or without a liquid phase.\n" + ], "metadata": {} }, { - "id": "21db7899", + "id": "b03f070d", "cell_type": "markdown", - "source": "### Phase diagrams with only molar quantities\n\nWhen all $\\left( c - n_{s} \\right) + 1$ potentials in a potential phase\ndiagrams are replaced by their conjugate molar quantities, one obtains a\nmolar phase diagram with molar quantities on all axes of the phase\ndiagram. For regions with the number of phases\n$p \\leq n_{m} + 1 = \\left( c - n_{s} \\right) + 2 = p_{\\max}$ (see ), the\nphase regions have the same-dimensionality as that of the phase diagram,\ni.e. all phase regions have the same-dimensionality of\n$\\left( c - n_{s} \\right) + 1$, and any geometric feature with\nlower-dimensionalities, i.e. from 0 to $\\ \\left( c - n_{s} \\right)$, is\nnot phase regions, but phase boundaries between neighbouring phase\nregions. For the sake of graphic visualization, the molar phase diagram\nof pure Fe is shown in obtained by combining the two mixed phase\ndiagrams in .\n\nFigure \u2011: Molar phase diagram of Fe\n\nIn this molar phase diagram, all one-, two-, and three-phase regions are\ntwo-dimensional, the same as the dimensionality of the phase diagram. A\ntwo-phase region is made up by tie-lines connecting the two phases in\nequilibrium, while a three phase-region is surrounded by three two-phase\ntie-lines, i.e. a tie-triangle. The amount of each phase in the\ntie-triangle can be obtained using the lever rule represented by and .\nAs can be seen, phase boundaries between a one-phase region and a\ntwo-phase region are one-dimensional. When the system crosses such a\nphase boundary, the number of phases changes by one from two to one or\nvice versa. Phase boundaries between two- and three-phase regions are\nrepresented by two-phase tie-lines. When the system crosses such a phase\nboundary, the number of phases also changes by one. The\nlowest-dimensional phase boundaries are the points between one- and\nthree-phase regions that are zero-dimensional and the intercept of four\none-dimensional phase boundaries. When the system crosses such a phase\nboundary, the number of phases changes by two.\n\nFor multi-component systems, the phase relations cannot be directly\nvisualized. By representing the system equations in terms of equilibrium\nconditions and level rules on a phase boundary using phases separately\nfrom the two adjacent phase regions, Palatnik and Landau \\[4\\]\npostulated that the difference between the number of unknowns and\nequations gives the dimensionality of the phase boundary and derived the\nfollowing relationships\n\nEq. \u2011 $D^{+} + D^{-} = r - b = c - n_{s} - b$\n\nwhere $D^{+}$ and $D^{-}$ are the numbers of phases added and removed\nwhen the phase boundary is crossed, and $r$ and $b$ are the\ndimensionalities of the phase diagram and the phase boundary,\nrespectively. They termed the as the contact rule, which is named as the\nMPL boundary rule by Hillert \\[1\\].\n\nBy the same token, is applicable to any phase boundary where the two\nadjacent phase regions have the same-dimensionality of the phase\ndiagram, even in phase diagrams with a mixture of potentials and molar\nquantities as the diagram axes. This can be understood because the\npotentials are homogeneous in all phases in equilibrium on the phase\nboundary. The phase boundary is thus equivalent to those in a complete\nmolar phase diagram with its number of components equal to the number of\nmolar axes in a mixed potential and molar phase diagrams minus one, i.e.\n\nEq. \u2011 $c^{'} = n_{m} - 1 = c - n_{s}$\n\nThe last part of stems from the discussion related to when all\n$c^{'} + 1$ potentials are replaced by their conjugate molar quantities,\nwhich is analogous to a molar phase diagram with $\\ c$ independent\ncomponents and $n_{s}$ potentials fixed.\n\nFor a two-dimensional phase diagram with $r = 2$, the phase boundary can\nbe either zero- or one--dimensional. As shown in , the basic element of\na molar phase diagram is a joint of four one-dimensional phase boundary\nlines. When a phase boundary line is crossed, the number of phases is\neither increased or decreased by one. The joint of four one-dimensional\nphase boundary lines is zero-dimensional. The number of phases differs\nby two between the phase regions across the zero-dimensional joint. Two\nscenarios are possible\n\nTwo phases are added or removed, i.e. $D^{+} = 2$ and $D^{-} = 0$ or\n$D^{+} = 0$ and $D^{-} = 2$, and the number of phases in the two phase\nregions differs by two;\n\nOne phase is added, one phase is removed, i.e. $D^{+} = D^{-} = 1$, and\nthe phase regions have the same number of phases.\n\nBy combining the contact rules for both zero- and one-dimensional phase\nboundaries, it is evident that the above two scenarios co-exist in a\njoint of four-phase regions with two-phase regions following the first\nscenario and other two-phase regions following the second scenario.\nBased on the Schreinemakers\u2019 rule generalized by Hillert \\[5\\], each of\nthe two-phase regions with the same number of phases contains one\nextrapolation of the phase boundaries, while the other two-phase regions\ncontain either zero or two extrapolations of the phase boundaries. This\ncan be observed for all the zero-dimensional phase boundaries in and is\nfurther schematically illustrated in for general cases.\n\nFigure \u2011: Schematic molar phase diagram, demonstrating the\nSchreinemakers\u2019 rule\n", + "source": [ + "### Phase diagrams with only molar quantities\n", + "\n", + "When all $\\left( c - n_{s} \\right) + 1$ potentials in a potential phase\n", + "diagrams are replaced by their conjugate molar quantities, one obtains a\n", + "molar phase diagram with molar quantities on all axes of the phase\n", + "diagram. For regions with the number of phases\n", + "$p \\leq n_{m} + 1 = \\left( c - n_{s} \\right) + 2 = p_{\\max}$ (see ), the\n", + "phase regions have the same-dimensionality as that of the phase diagram,\n", + "i.e. all phase regions have the same-dimensionality of\n", + "$\\left( c - n_{s} \\right) + 1$, and any geometric feature with\n", + "lower-dimensionalities, i.e. from 0 to $\\ \\left( c - n_{s} \\right)$, is\n", + "not phase regions, but phase boundaries between neighbouring phase\n", + "regions. For the sake of graphic visualization, the molar phase diagram\n", + "of pure Fe is shown in obtained by combining the two mixed phase\n", + "diagrams in .\n", + "\n", + "Figure \u2011: Molar phase diagram of Fe\n", + "\n", + "In this molar phase diagram, all one-, two-, and three-phase regions are\n", + "two-dimensional, the same as the dimensionality of the phase diagram. A\n", + "two-phase region is made up by tie-lines connecting the two phases in\n", + "equilibrium, while a three phase-region is surrounded by three two-phase\n", + "tie-lines, i.e. a tie-triangle. The amount of each phase in the\n", + "tie-triangle can be obtained using the lever rule represented by and .\n", + "As can be seen, phase boundaries between a one-phase region and a\n", + "two-phase region are one-dimensional. When the system crosses such a\n", + "phase boundary, the number of phases changes by one from two to one or\n", + "vice versa. Phase boundaries between two- and three-phase regions are\n", + "represented by two-phase tie-lines. When the system crosses such a phase\n", + "boundary, the number of phases also changes by one. The\n", + "lowest-dimensional phase boundaries are the points between one- and\n", + "three-phase regions that are zero-dimensional and the intercept of four\n", + "one-dimensional phase boundaries. When the system crosses such a phase\n", + "boundary, the number of phases changes by two.\n", + "\n", + "For multi-component systems, the phase relations cannot be directly\n", + "visualized. By representing the system equations in terms of equilibrium\n", + "conditions and level rules on a phase boundary using phases separately\n", + "from the two adjacent phase regions, Palatnik and Landau \\[4\\]\n", + "postulated that the difference between the number of unknowns and\n", + "equations gives the dimensionality of the phase boundary and derived the\n", + "following relationships\n", + "\n", + "Eq. \u2011 $D^{+} + D^{-} = r - b = c - n_{s} - b$\n", + "\n", + "where $D^{+}$ and $D^{-}$ are the numbers of phases added and removed\n", + "when the phase boundary is crossed, and $r$ and $b$ are the\n", + "dimensionalities of the phase diagram and the phase boundary,\n", + "respectively. They termed the as the contact rule, which is named as the\n", + "MPL boundary rule by Hillert \\[1\\].\n", + "\n", + "By the same token, is applicable to any phase boundary where the two\n", + "adjacent phase regions have the same-dimensionality of the phase\n", + "diagram, even in phase diagrams with a mixture of potentials and molar\n", + "quantities as the diagram axes. This can be understood because the\n", + "potentials are homogeneous in all phases in equilibrium on the phase\n", + "boundary. The phase boundary is thus equivalent to those in a complete\n", + "molar phase diagram with its number of components equal to the number of\n", + "molar axes in a mixed potential and molar phase diagrams minus one, i.e.\n", + "\n", + "Eq. \u2011 $c^{'} = n_{m} - 1 = c - n_{s}$\n", + "\n", + "The last part of stems from the discussion related to when all\n", + "$c^{'} + 1$ potentials are replaced by their conjugate molar quantities,\n", + "which is analogous to a molar phase diagram with $\\ c$ independent\n", + "components and $n_{s}$ potentials fixed.\n", + "\n", + "For a two-dimensional phase diagram with $r = 2$, the phase boundary can\n", + "be either zero- or one--dimensional. As shown in , the basic element of\n", + "a molar phase diagram is a joint of four one-dimensional phase boundary\n", + "lines. When a phase boundary line is crossed, the number of phases is\n", + "either increased or decreased by one. The joint of four one-dimensional\n", + "phase boundary lines is zero-dimensional. The number of phases differs\n", + "by two between the phase regions across the zero-dimensional joint. Two\n", + "scenarios are possible\n", + "\n", + "Two phases are added or removed, i.e. $D^{+} = 2$ and $D^{-} = 0$ or\n", + "$D^{+} = 0$ and $D^{-} = 2$, and the number of phases in the two phase\n", + "regions differs by two;\n", + "\n", + "One phase is added, one phase is removed, i.e. $D^{+} = D^{-} = 1$, and\n", + "the phase regions have the same number of phases.\n", + "\n", + "By combining the contact rules for both zero- and one-dimensional phase\n", + "boundaries, it is evident that the above two scenarios co-exist in a\n", + "joint of four-phase regions with two-phase regions following the first\n", + "scenario and other two-phase regions following the second scenario.\n", + "Based on the Schreinemakers\u2019 rule generalized by Hillert \\[5\\], each of\n", + "the two-phase regions with the same number of phases contains one\n", + "extrapolation of the phase boundaries, while the other two-phase regions\n", + "contain either zero or two extrapolations of the phase boundaries. This\n", + "can be observed for all the zero-dimensional phase boundaries in and is\n", + "further schematically illustrated in for general cases.\n", + "\n", + "Figure \u2011: Schematic molar phase diagram, demonstrating the\n", + "Schreinemakers\u2019 rule\n" + ], "metadata": {} }, { - "id": "a5607756", + "id": "2a1c4452", "cell_type": "markdown", - "source": "### Projection and section of phase diagrams with potential and molar quantities\n\nAs discussed in Chapter , projections of high-dimensional phase diagrams\nusually cannot keep all information. However, there is one type of\nwidely used projection in the literature, i.e. the liquidus surface in\nternary systems under constant pressure with temperature and mole\nfractions of two components as its axes. The projection along the\ntemperature axis reveals the composition regions for primary phases that\nsolidify from liquid upon cooling. These regions are separated by\nunivariant lines of three-phase equilibria. The projections along one of\nthe two mole fractions show the temperature as a function of composition\non the univariant three-phase equilibrium lines and also depict whether\na four-phase equilibrium is peritectic or eutectic. There are four\nscenarios for the three univariant three-phase equilibrium lines to meet\nat the four-phase equilibrium as depicted in and discussed individually\nbelow.\n\nFigure \u2011: Schematic four options for three univariant three-phase\nequilibrium lines to meet at the invariant four-phase equilibrium\n\nThe first scenario is that with decreasing temperature, all three\nunivariant lines merge into the four-phase equilibrium. It indicates\nthat the liquid phase does not exist at temperatures below the\nfour-phase invariant reaction. This invariant reaction is thus a ternary\neutectic reaction with liquid completely transformed to three solid\nphases upon cooling, i.e.\n\nEq. \u2011 $L \\rightarrow \\alpha + \\beta + \\gamma$\n\nIn the second scenario, two univariant lines merge into and one leaves\nfrom the four-phase equilibrium with decreasing temperature. This means\nthat one solid phase at higher temperature is no longer stable at lower\ntemperature, and it must react with the liquid phase to form the\nremaining two solid phases. The four-phase invariant reaction is thus\nperitectic. The solid phase common to both univariant lines at high\ntemperatures reacts with the liquid phase. Assuming that this phase is\n$\\alpha$, the four-phase invariant reaction becomes\n\nEq. \u2011 $L + \\alpha \\rightarrow \\beta + \\gamma$\n\nIn the third scenario, one univariant line points to and two leave from\nthe four-phase equilibrium with decreasing temperature. A new phase\nforms at low temperatures from the three high temperature phases, e.g.\nliquid, $\\alpha$, and $\\beta$, with the four-phase invariant reaction as\n\nEq. \u2011 $L + \\alpha + \\beta \\rightarrow \\gamma$\n\nThe fourth scenario is the inverse of the first scenario, indicating the\nformation of liquid from solid phases upon cooling, i.e.\n\nEq. \u2011 $\\alpha + \\beta + \\gamma \\rightarrow L$\n\nThis case has not been observed in reality.\n\nAs an example, the liquidus projections of the Al-Fe-Si ternary system\nare shown in in two formats\\[6\\], i.e. (a) three-dimensional liquidus\nsurface with the isotherms showing the liquidus contours; (b)\nconventional projection to the composition axis with the temperature\ndecrease shown by arrows; (c) projection to the temperature and weight\nfraction of Si. The first to third scenarios of invariant reactions\ndiscussed above can clearly be identified and listed in . It is evident\nthat c provides the easiest route to visualize the type of invariant\nreactions as show by .\n\nFigure \u2011: Liquidus of the Al-Fe-Si ternary system\\[6\\], (a)\nthree-dimensional presentation of the liquidus; (b) projection to the\ncomposition triangle with isotherms (dotted lines) superimposed and\ntheir temperatures indicated close to the horizontal axis.\n\nTable \u2011: Invariant liquidus reactions of the Al-Fe-Si ternary system\nwith the composition of the liquid phase \\[6\\]\n\n| Reaction | T, \u00b0C | wFe, % | wSi, % |\n|----------|-------|-------------------|-------------------|\n| | 1178 | 77.9 | 21.0 |\n| | 1155 | 49.0 | 0.16 |\n| | 1127 | 52.7 | 2.81 |\n| -H-L | 1076 | 41.6 | 44.0 |\n| | 1073 | 51.0 | 7.19 |\n| | 1050 | 53.8 | 18.4 |\n| -H-L | 1019 | 34.8 | 44.4 |\n| | 1004 | 49.0 | 12.6 |\n| | 1000 | 46.7 | 13.2 |\n| -L | 940 | 37.9 | 32.0 |\n| | 921 | 33.7 | 20.1 |\n| | 899 | 33.8 | 32.2 |\n| | 884 | 30.8 | 26.0 |\n| -L | 877 | 29.5 | 35.2 |\n| | 851 | 23.3 | 21.6 |\n| | 834 | 22.2 | 31.7 |\n| | 825 | 22.1 | 25.7 |\n| | 823 | 21.8 | 25.4 |\n| | 715 | 6.64 | 10.8 |\n| | 694 | 6.11 | 17.1 |\n| | 680 | 4.68 | 11.6 |\n| | 630 | 2.11 | 4.10 |\n| | 616 | 1.76 | 6.56 |\n| | 598 | 1.22 | 14.3 |\n| | 575 | 0.73 | 12.7 |\n\nIn contrast to projections, sectioning is used more often to understand\nphase relations in multi-component systems. Sectioning of a potential\nphase diagrams is relatively simple as the resulted phase diagram\nbehaves like a system with one component less. The same is true if\npotentials are sectioned in phase diagrams with both potential and molar\nquantities as the section is along the tie-lines of the fixed\npotentials. As an example, shows the ternary Al-Fe-Si potential and\nmolar phase diagrams sectioned at T=1273K and P=1atm, commonly referred\nto as isothermal section. It is evident that the geometric features of\nboth phase diagrams are identical to those of pure Fe shown in and ,\nrespectively, with one-, two-, and three-phase regions and corresponding\nphase boundaries.\n\nFigure \u2011: Ternary isothermal section of the Al-Fe-Si ternary system at\nT=1273K and P=1atm\n\nOn the other hand, when the phase diagram is sectioned along a molar\nquantity, it would usually not follow a tie-line because phases in\nequilibrium usually have different values for the same molar quantity.\nConsequently, there are no tie-lines inside such phase diagrams in\ngeneral, and any phase regions only show which phases are in equilibrium\nwith each other without any information on the values of molar\nquantities of individual phases.\n\nThis type of sectioning reduces both the dimensionalities of the phase\ndiagram and phase boundary by the same number, but does not alter the\nnumber of phases in the adjacent phase regions. The contact rule, i.e. ,\nthus remains valid and is applicable to phase regions with the\nsame-dimensionality as that of the sectioned phase diagram. Similarly,\nthe Schreinemakers\u2019 rule shown in is valid under the same conditions.\n\nFor example, the two-dimensional phase diagram of the Mg-Al-Zn ternary\nsystem sectioned with one atmospheric pressure and the weight fraction\nof Zn fixed at 0.01 is shown in plotted with temperature and mole\nfraction of Al \\[7\\]. This phase diagram is commonly called isopleth and\nis generated by fixing one potential, the pressure, $P$, changing the\nchemical potentials of Al and Zn to their conjugate molar quantities\nrepresented by weight fractions of Al and Zn, and sectioning at\n$w_{Zn} = 0.01$. From the discussions in Chapter , the phase regions\nwith the number of phases equal to three or fewer, i.e.\n$p \\leq n_{m} + 1 = 3$, have the same-dimensionality as the phase\ndiagram, i.e. two-dimensional in the present case, and the phase\nboundary rule is applicable. The maximum number of phases co-existing at\nequilibrium is given by as following for the present case\n\nEq. \u2011 $p_{\\max} = \\left( c - n_{s} \\right) + 2 = 3 - 1 + 2 = 4$\n\nThis is because introducing molar quantities only increases the\ndimensionality of phase regions and does not change the maximum number\nof co-existing phases.\n\nThe dimensionality of a four-phase region is calculated from as\n\nEq. \u2011\n$\\upsilon_{m} = \\left( c - n_{s} \\right) + 2 - p + n_{m} - n_{ms} = 3 - 1 + 2 - 4 + 2 - 1 = 1$\n\nwhere $n_{ms}$ is the number of sectioned molar quantities. Since the\ndimensionality of a four-phase region is lower than that of the phase\ndiagram, the phase boundary rule cannot be applied directly. Such a\nfour-phase region, liquid+Mg+\u03b3+\u03c6, is shown in between three three-phase\nregions of liquid+Mg+\u03b3, liquid+Mg+\u03c6, and Mg+\u03b3+\u03c6.\n\nFigure \u2011: Isopleth with the weight fraction of Zn fixed at 0.01 of the\nMg-Al-Zn ternary system.\n\nalso displays information on what phases are in equilibrium for a given\nalloy at various temperatures. One example is shown by the dotted\nvertical line marking the weight fraction of Al being 0.09, a widely\nused Mg alloy called AZ91. Various phases are present at different\ntemperature ranges, but the equilibrium phase fractions and phase\ncompositions are not shown in the figure as the tie-lines are not in the\nplane of the phase diagram and have to be calculated at each temperature\nindividually. shows the amount of each phase of the AZ91 alloy as a\nfunction of temperature with the dotted lines depicting the values under\nthe equilibrium condition and the solid lines depicting the values under\nthe so-called Scheil condition assuming no diffusion in solid phases and\ninfinitelyfast diffusion in liquid. Similarly, the composition of each\nphase can also be plotted as shown in .\n\nFigure \u2011: Mole fraction of individual phases under equilibrium (dotted\ncurves) and Scheil (solid curves) conditions in the AZ91 alloy\n\nFigure \u2011: Mass fraction of Al and Zn in the Mg solid solution phase\nunder equilibrium (dotted curves) and Scheil (solid curves) conditions\nin AZ91\n", + "source": [ + "### Projection and section of phase diagrams with potential and molar quantities\n", + "\n", + "As discussed in Chapter , projections of high-dimensional phase diagrams\n", + "usually cannot keep all information. However, there is one type of\n", + "widely used projection in the literature, i.e. the liquidus surface in\n", + "ternary systems under constant pressure with temperature and mole\n", + "fractions of two components as its axes. The projection along the\n", + "temperature axis reveals the composition regions for primary phases that\n", + "solidify from liquid upon cooling. These regions are separated by\n", + "univariant lines of three-phase equilibria. The projections along one of\n", + "the two mole fractions show the temperature as a function of composition\n", + "on the univariant three-phase equilibrium lines and also depict whether\n", + "a four-phase equilibrium is peritectic or eutectic. There are four\n", + "scenarios for the three univariant three-phase equilibrium lines to meet\n", + "at the four-phase equilibrium as depicted in and discussed individually\n", + "below.\n", + "\n", + "Figure \u2011: Schematic four options for three univariant three-phase\n", + "equilibrium lines to meet at the invariant four-phase equilibrium\n", + "\n", + "The first scenario is that with decreasing temperature, all three\n", + "univariant lines merge into the four-phase equilibrium. It indicates\n", + "that the liquid phase does not exist at temperatures below the\n", + "four-phase invariant reaction. This invariant reaction is thus a ternary\n", + "eutectic reaction with liquid completely transformed to three solid\n", + "phases upon cooling, i.e.\n", + "\n", + "Eq. \u2011 $L \\rightarrow \\alpha + \\beta + \\gamma$\n", + "\n", + "In the second scenario, two univariant lines merge into and one leaves\n", + "from the four-phase equilibrium with decreasing temperature. This means\n", + "that one solid phase at higher temperature is no longer stable at lower\n", + "temperature, and it must react with the liquid phase to form the\n", + "remaining two solid phases. The four-phase invariant reaction is thus\n", + "peritectic. The solid phase common to both univariant lines at high\n", + "temperatures reacts with the liquid phase. Assuming that this phase is\n", + "$\\alpha$, the four-phase invariant reaction becomes\n", + "\n", + "Eq. \u2011 $L + \\alpha \\rightarrow \\beta + \\gamma$\n", + "\n", + "In the third scenario, one univariant line points to and two leave from\n", + "the four-phase equilibrium with decreasing temperature. A new phase\n", + "forms at low temperatures from the three high temperature phases, e.g.\n", + "liquid, $\\alpha$, and $\\beta$, with the four-phase invariant reaction as\n", + "\n", + "Eq. \u2011 $L + \\alpha + \\beta \\rightarrow \\gamma$\n", + "\n", + "The fourth scenario is the inverse of the first scenario, indicating the\n", + "formation of liquid from solid phases upon cooling, i.e.\n", + "\n", + "Eq. \u2011 $\\alpha + \\beta + \\gamma \\rightarrow L$\n", + "\n", + "This case has not been observed in reality.\n", + "\n", + "As an example, the liquidus projections of the Al-Fe-Si ternary system\n", + "are shown in in two formats\\[6\\], i.e. (a) three-dimensional liquidus\n", + "surface with the isotherms showing the liquidus contours; (b)\n", + "conventional projection to the composition axis with the temperature\n", + "decrease shown by arrows; (c) projection to the temperature and weight\n", + "fraction of Si. The first to third scenarios of invariant reactions\n", + "discussed above can clearly be identified and listed in . It is evident\n", + "that c provides the easiest route to visualize the type of invariant\n", + "reactions as show by .\n", + "\n", + "Figure \u2011: Liquidus of the Al-Fe-Si ternary system\\[6\\], (a)\n", + "three-dimensional presentation of the liquidus; (b) projection to the\n", + "composition triangle with isotherms (dotted lines) superimposed and\n", + "their temperatures indicated close to the horizontal axis.\n", + "\n", + "Table \u2011: Invariant liquidus reactions of the Al-Fe-Si ternary system\n", + "with the composition of the liquid phase \\[6\\]\n", + "\n", + "| Reaction | T, \u00b0C | wFe, % | wSi, % |\n", + "|----------|-------|-------------------|-------------------|\n", + "| | 1178 | 77.9 | 21.0 |\n", + "| | 1155 | 49.0 | 0.16 |\n", + "| | 1127 | 52.7 | 2.81 |\n", + "| -H-L | 1076 | 41.6 | 44.0 |\n", + "| | 1073 | 51.0 | 7.19 |\n", + "| | 1050 | 53.8 | 18.4 |\n", + "| -H-L | 1019 | 34.8 | 44.4 |\n", + "| | 1004 | 49.0 | 12.6 |\n", + "| | 1000 | 46.7 | 13.2 |\n", + "| -L | 940 | 37.9 | 32.0 |\n", + "| | 921 | 33.7 | 20.1 |\n", + "| | 899 | 33.8 | 32.2 |\n", + "| | 884 | 30.8 | 26.0 |\n", + "| -L | 877 | 29.5 | 35.2 |\n", + "| | 851 | 23.3 | 21.6 |\n", + "| | 834 | 22.2 | 31.7 |\n", + "| | 825 | 22.1 | 25.7 |\n", + "| | 823 | 21.8 | 25.4 |\n", + "| | 715 | 6.64 | 10.8 |\n", + "| | 694 | 6.11 | 17.1 |\n", + "| | 680 | 4.68 | 11.6 |\n", + "| | 630 | 2.11 | 4.10 |\n", + "| | 616 | 1.76 | 6.56 |\n", + "| | 598 | 1.22 | 14.3 |\n", + "| | 575 | 0.73 | 12.7 |\n", + "\n", + "In contrast to projections, sectioning is used more often to understand\n", + "phase relations in multi-component systems. Sectioning of a potential\n", + "phase diagrams is relatively simple as the resulted phase diagram\n", + "behaves like a system with one component less. The same is true if\n", + "potentials are sectioned in phase diagrams with both potential and molar\n", + "quantities as the section is along the tie-lines of the fixed\n", + "potentials. As an example, shows the ternary Al-Fe-Si potential and\n", + "molar phase diagrams sectioned at T=1273K and P=1atm, commonly referred\n", + "to as isothermal section. It is evident that the geometric features of\n", + "both phase diagrams are identical to those of pure Fe shown in and ,\n", + "respectively, with one-, two-, and three-phase regions and corresponding\n", + "phase boundaries.\n", + "\n", + "Figure \u2011: Ternary isothermal section of the Al-Fe-Si ternary system at\n", + "T=1273K and P=1atm\n", + "\n", + "On the other hand, when the phase diagram is sectioned along a molar\n", + "quantity, it would usually not follow a tie-line because phases in\n", + "equilibrium usually have different values for the same molar quantity.\n", + "Consequently, there are no tie-lines inside such phase diagrams in\n", + "general, and any phase regions only show which phases are in equilibrium\n", + "with each other without any information on the values of molar\n", + "quantities of individual phases.\n", + "\n", + "This type of sectioning reduces both the dimensionalities of the phase\n", + "diagram and phase boundary by the same number, but does not alter the\n", + "number of phases in the adjacent phase regions. The contact rule, i.e. ,\n", + "thus remains valid and is applicable to phase regions with the\n", + "same-dimensionality as that of the sectioned phase diagram. Similarly,\n", + "the Schreinemakers\u2019 rule shown in is valid under the same conditions.\n", + "\n", + "For example, the two-dimensional phase diagram of the Mg-Al-Zn ternary\n", + "system sectioned with one atmospheric pressure and the weight fraction\n", + "of Zn fixed at 0.01 is shown in plotted with temperature and mole\n", + "fraction of Al \\[7\\]. This phase diagram is commonly called isopleth and\n", + "is generated by fixing one potential, the pressure, $P$, changing the\n", + "chemical potentials of Al and Zn to their conjugate molar quantities\n", + "represented by weight fractions of Al and Zn, and sectioning at\n", + "$w_{Zn} = 0.01$. From the discussions in Chapter , the phase regions\n", + "with the number of phases equal to three or fewer, i.e.\n", + "$p \\leq n_{m} + 1 = 3$, have the same-dimensionality as the phase\n", + "diagram, i.e. two-dimensional in the present case, and the phase\n", + "boundary rule is applicable. The maximum number of phases co-existing at\n", + "equilibrium is given by as following for the present case\n", + "\n", + "Eq. \u2011 $p_{\\max} = \\left( c - n_{s} \\right) + 2 = 3 - 1 + 2 = 4$\n", + "\n", + "This is because introducing molar quantities only increases the\n", + "dimensionality of phase regions and does not change the maximum number\n", + "of co-existing phases.\n", + "\n", + "The dimensionality of a four-phase region is calculated from as\n", + "\n", + "Eq. \u2011\n", + "$\\upsilon_{m} = \\left( c - n_{s} \\right) + 2 - p + n_{m} - n_{ms} = 3 - 1 + 2 - 4 + 2 - 1 = 1$\n", + "\n", + "where $n_{ms}$ is the number of sectioned molar quantities. Since the\n", + "dimensionality of a four-phase region is lower than that of the phase\n", + "diagram, the phase boundary rule cannot be applied directly. Such a\n", + "four-phase region, liquid+Mg+\u03b3+\u03c6, is shown in between three three-phase\n", + "regions of liquid+Mg+\u03b3, liquid+Mg+\u03c6, and Mg+\u03b3+\u03c6.\n", + "\n", + "Figure \u2011: Isopleth with the weight fraction of Zn fixed at 0.01 of the\n", + "Mg-Al-Zn ternary system.\n", + "\n", + "also displays information on what phases are in equilibrium for a given\n", + "alloy at various temperatures. One example is shown by the dotted\n", + "vertical line marking the weight fraction of Al being 0.09, a widely\n", + "used Mg alloy called AZ91. Various phases are present at different\n", + "temperature ranges, but the equilibrium phase fractions and phase\n", + "compositions are not shown in the figure as the tie-lines are not in the\n", + "plane of the phase diagram and have to be calculated at each temperature\n", + "individually. shows the amount of each phase of the AZ91 alloy as a\n", + "function of temperature with the dotted lines depicting the values under\n", + "the equilibrium condition and the solid lines depicting the values under\n", + "the so-called Scheil condition assuming no diffusion in solid phases and\n", + "infinitelyfast diffusion in liquid. Similarly, the composition of each\n", + "phase can also be plotted as shown in .\n", + "\n", + "Figure \u2011: Mole fraction of individual phases under equilibrium (dotted\n", + "curves) and Scheil (solid curves) conditions in the AZ91 alloy\n", + "\n", + "Figure \u2011: Mass fraction of Al and Zn in the Mg solid solution phase\n", + "under equilibrium (dotted curves) and Scheil (solid curves) conditions\n", + "in AZ91\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/phase_equilibria_in_heterogeneous_systems/potential_phase_diagrams.ipynb b/src/psu410/src/phase_equilibria_in_heterogeneous_systems/potential_phase_diagrams.ipynb index 17228c2..ceb371c 100644 --- a/src/psu410/src/phase_equilibria_in_heterogeneous_systems/potential_phase_diagrams.ipynb +++ b/src/psu410/src/phase_equilibria_in_heterogeneous_systems/potential_phase_diagrams.ipynb @@ -4,27 +4,396 @@ "metadata": {}, "cells": [ { - "id": "167e045a", + "id": "cb9fb312", "cell_type": "markdown", - "source": "## Potential phase diagrams\n\nGibbs phase rule can be further understood through the\n$c + 2$-dimensional space of potentials consisting of $T$, $- P$, and\n$\\mu_{i}$ with $i$ from 1 to $c$. Each phase is a $c + 1$-dimensional\nfeature in this $c + 2$-dimensional space characterized by . The\ndirections of this $c + 1$-dimensional feature are represented by their\nmolar quantities as shown by following equations,\n\nEq. \u2011\n$\\ \\left( \\frac{\\partial\\mu_{i}}{\\partial T} \\right)_{P,\\mu_{j \\neq i}} = - \\frac{S^{\\beta}}{N_{i}^{\\beta}} = - \\frac{S_{m}^{\\beta}}{x_{i}^{\\beta}}$\n\nEq. \u2011\n$\\ \\left( \\frac{\\partial\\mu_{i}}{\\partial( - P)} \\right)_{T,\\mu_{j \\neq i}} = - \\frac{V^{\\beta}}{N_{i}^{\\beta}} = - \\frac{V_{m}^{\\beta}}{x_{i}^{\\beta}}$\n\nEq. \u2011\n$\\ \\left( \\frac{\\partial( - P)}{\\partial T} \\right)_{\\mu_{i}} = - \\frac{S^{\\beta}}{V^{\\beta}} = - \\frac{S_{m}^{\\beta}}{V_{m}^{\\beta}}$\n\nAs can be seen, all the direction derivatives are negative, indicating\nthat the $c + 1$-dimensional feature is convex. The intercept of any two\n$c + 1$-dimensional features is thus a $c$-dimensional feature. On this\n$c$-dimensional feature, these two phases are in equilibrium with each\nother because each potential has the same value in both phases. This\nfeature thus represents a two-phase equilibrium. By the same token, the\nintercept of any three $c + 1$-dimensional features is a\n$c - 1$-dimensional feature in the $c + 2$-dimensional space of\npotentials and represents a three-phase equilibrium. This continues\nuntil the number of phases reaches $c + 2$ with all $c + 2$ potentials\ncompletely determined, and the dimension of their intercepts becomes\nzero.\n\nThose $c + 1$ to zero-dimensional geometrical features in the\n$c + 2$-dimensional space of potentials thus denote one-phase,\ntwo-phase, three-phase to $(c + 2)$-phase equilibria of the system with\nthe dimensionality of the feature and the number of phases in\nequilibrium related by , i.e. Gibbs phase rule. Their arrangements in\nthe $c + 2$-dimensional space of potentials thus depict the phase\nrelations in the system and are commonly called phase diagrams. Since\nall the diagram axes in the phase diagram discussed above are\npotentials, the diagram is called potential phase diagram in order to\ndifferentiate it from phase diagrams with some or all diagram axes being\ntheir conjugate molar quantities. Both potential and molar phase\ndiagrams are discussed in this chapter.\n", + "source": [ + "## Potential phase diagrams\n", + "\n", + "Gibbs phase rule can be further understood through the\n", + "$c + 2$-dimensional space of potentials consisting of $T$, $- P$, and\n", + "$\\mu_{i}$ with $i$ from 1 to $c$. Each phase is a $c + 1$-dimensional\n", + "feature in this $c + 2$-dimensional space characterized by . The\n", + "directions of this $c + 1$-dimensional feature are represented by their\n", + "molar quantities as shown by following equations,\n", + "\n", + "Eq. \u2011\n", + "$\\ \\left( \\frac{\\partial\\mu_{i}}{\\partial T} \\right)_{P,\\mu_{j \\neq i}} = - \\frac{S^{\\beta}}{N_{i}^{\\beta}} = - \\frac{S_{m}^{\\beta}}{x_{i}^{\\beta}}$\n", + "\n", + "Eq. \u2011\n", + "$\\ \\left( \\frac{\\partial\\mu_{i}}{\\partial( - P)} \\right)_{T,\\mu_{j \\neq i}} = - \\frac{V^{\\beta}}{N_{i}^{\\beta}} = - \\frac{V_{m}^{\\beta}}{x_{i}^{\\beta}}$\n", + "\n", + "Eq. \u2011\n", + "$\\ \\left( \\frac{\\partial( - P)}{\\partial T} \\right)_{\\mu_{i}} = - \\frac{S^{\\beta}}{V^{\\beta}} = - \\frac{S_{m}^{\\beta}}{V_{m}^{\\beta}}$\n", + "\n", + "As can be seen, all the direction derivatives are negative, indicating\n", + "that the $c + 1$-dimensional feature is convex. The intercept of any two\n", + "$c + 1$-dimensional features is thus a $c$-dimensional feature. On this\n", + "$c$-dimensional feature, these two phases are in equilibrium with each\n", + "other because each potential has the same value in both phases. This\n", + "feature thus represents a two-phase equilibrium. By the same token, the\n", + "intercept of any three $c + 1$-dimensional features is a\n", + "$c - 1$-dimensional feature in the $c + 2$-dimensional space of\n", + "potentials and represents a three-phase equilibrium. This continues\n", + "until the number of phases reaches $c + 2$ with all $c + 2$ potentials\n", + "completely determined, and the dimension of their intercepts becomes\n", + "zero.\n", + "\n", + "Those $c + 1$ to zero-dimensional geometrical features in the\n", + "$c + 2$-dimensional space of potentials thus denote one-phase,\n", + "two-phase, three-phase to $(c + 2)$-phase equilibria of the system with\n", + "the dimensionality of the feature and the number of phases in\n", + "equilibrium related by , i.e. Gibbs phase rule. Their arrangements in\n", + "the $c + 2$-dimensional space of potentials thus depict the phase\n", + "relations in the system and are commonly called phase diagrams. Since\n", + "all the diagram axes in the phase diagram discussed above are\n", + "potentials, the diagram is called potential phase diagram in order to\n", + "differentiate it from phase diagrams with some or all diagram axes being\n", + "their conjugate molar quantities. Both potential and molar phase\n", + "diagrams are discussed in this chapter.\n" + ], "metadata": {} }, { - "id": "6b91a7a1", + "id": "279783f1", "cell_type": "markdown", - "source": "### Potential phase diagrams of one-component systems\n\nAs physical vision of human being is limited to three-dimensions, only\none-component system can be completely visualized as shown in for one\nphase where any two of the three potentials can change independently.\nFrom Gibbs phase rule, when two phases are in equilibrium, only one\npotential can vary freely if the two-phase equilibrium is to be\nmaintained. While three phases are in equilibrium, the degree of freedom\nis zero, and all three potentials are fixed.\n\nFor a two-phase equilibrium, two surfaces intersect each other as\ndepicted by the dashed line in . This two-phase equilibrium line is\nobtained by applying to both phases in a one-component system. Since one\nof the potentials is dependent on the other two, one can eliminate it by\ndividing the equation by its conjugate molar quantity and subtracting\nthe two equations, resulting in the following three equations\n\nEq. \u2011\n$0 = \\left( \\frac{S^{\\alpha}}{N_{A}^{\\alpha}} - \\frac{S^{\\beta}}{N_{A}^{\\beta}} \\right)dT + \\left( \\frac{V^{\\alpha}}{N_{A}^{\\alpha}} - \\frac{V^{\\beta}}{N_{A}^{\\beta}} \\right)d( - P) = \\mathrm{\\Delta}S_{m}^{\\alpha\\beta}dT + \\mathrm{\\Delta}V_{m}^{\\alpha\\beta}d( - P)$\n\nEq. \u2011\n$0 = \\left( \\frac{S^{\\alpha}}{V^{\\alpha}} - \\frac{S^{\\beta}}{V^{\\beta}} \\right)dT + \\left( \\frac{N_{A}^{\\alpha}}{V^{\\alpha}} - \\frac{N_{A}^{\\beta}}{V^{\\beta}} \\right){d\\mu}_{A} = \\mathrm{\\Delta}\\left( \\frac{S_{m}}{V_{m}} \\right)^{\\alpha\\beta}dT + \\mathrm{\\Delta}\\left( \\frac{1}{V_{m}} \\right)^{\\alpha\\beta}{d\\mu}_{A}$\n\nEq. \u2011\n$0 = \\left( \\frac{V^{\\alpha}}{S^{\\alpha}} - \\frac{V^{\\beta}}{S^{\\beta}} \\right)d( - P) + \\left( \\frac{N_{A}^{\\alpha}}{S^{\\alpha}} - \\frac{N_{A}^{\\beta}}{S^{\\beta}} \\right){d\\mu}_{A} = \\mathrm{\\Delta}\\left( \\frac{V_{m}}{S_{m}} \\right)^{\\alpha\\beta}d( - P) + \\mathrm{\\Delta}\\left( \\frac{1}{S_{m}} \\right)^{\\alpha\\beta}{d\\mu}_{A}$\n\nFigure \u2011: Gibbs energy surfaces of two phases and their intersection,\nrepresenting the two-phase equilibrium.\n\nThe directions of the two-phase equilibrium line can thus be obtained as\n\nEq. \u2011\n$\\frac{d( - P)}{dT} = - \\frac{\\mathrm{\\Delta}S_{m}^{\\alpha\\beta}}{\\mathrm{\\Delta}V_{m}^{\\alpha\\beta}}$\n\nEq. \u2011\n$\\frac{dT}{{d\\mu}_{A}} = - \\frac{\\mathrm{\\Delta}\\left( \\frac{1}{V_{m}} \\right)^{\\alpha\\beta}}{\\mathrm{\\Delta}\\left( \\frac{S_{m}}{V_{m}} \\right)^{\\alpha\\beta}}$\n\nEq. \u2011\n$\\frac{d( - P)}{{d\\mu}_{A}} = - \\frac{\\mathrm{\\Delta}\\left( \\frac{1}{S_{m}} \\right)^{\\alpha\\beta}}{\\mathrm{\\Delta}\\left( \\frac{V_{m}}{S_{m}} \\right)^{\\alpha\\beta}}$\n\nThese three equations define the mathematical forms of the two-phase\nequilibrium line in the two-dimensional spaces of $T - ( - P)$,\n$\\mu_{A} - T$ and $\\mu_{A} - ( - P)$, respectively, and can thus be\nplotted as two-dimensional diagrams. is commonly called\nClausius-Clapeyron equation in the literature. One may thus call all\nthree equations above as generalized Clausius-Clapeyron equations. At\nequilibrium, the chemical potentials of the component in both phases are\nequal to each other, so are their Gibbs energies. One thus has\n\nEq. \u2011\n$G_{m}^{\\alpha} - G_{m}^{\\beta} = 0 = \\mathrm{\\Delta}G_{m} = \\mathrm{\\Delta}H_{m}^{\\alpha\\beta} - T\\mathrm{\\Delta}S_{m}^{\\alpha\\beta}$\n\nThe Clausius-Clapeyron equation, , can be re-written as\n\nEq. \u2011\n$\\frac{d( - P)}{dT} = - \\frac{\\mathrm{\\Delta}H_{m}^{\\alpha\\beta}}{T\\mathrm{\\Delta}V_{m}^{\\alpha\\beta}}$\n\nAs an example, three potential phase diagrams of pure Fe are shown in .\nThere are four phases in the system, bcc, fcc, hcp, and liquid. In the\nliterature, the high temperature and low temperature bcc phases are\nusually denoted by \u03b4 (high temperature) and \u03b1 (low temperature), the fcc\nand hcp phases by \u03b3 and \u03b5, and the liquid phase by L, respectively. In\nthese figures, the two-dimensional areas are single-phase regions where\ntwo potentials can change independently with the system remaining as\nsingle-phase. The lines denote two-phase equilibrium regions where only\none potential can vary independently if the two-phase equilibrium is to\nbe maintained. The points where three two-phase equilibrium lines meet\nrepresent the invariant three-phase equilibria with three potentials\nfixed.\n\nFigure \u2011 $T - ( - P)$, $\\mu_{A} - T$ and $\\mu_{A} - ( - P)$ phase\ndiagrams of pure Fe\n\nBased on the discussions in Chapter , enthalpy and entropy of a phase\nincrease monotonically with temperature, and phases stable at higher\ntemperatures have higher enthalpy and entropy than phases stable at\nlower temperatures. Consequently, the two-phase equilibrium lines in a\n$T - ( - P)$ potential phase diagram have negative slopes if the phase\nstable at higher temperatures also has larger molar volume than the\nphase stable at lower temperatures (note that if $P$ is plotted instead\nof $- P$, the slope is positive). This is the case for the two-phase\nequilibrium lines of \u03b4/L, \u03b3/L, and \u03b3/\u03b4 at high temperatures, and \u03b5/\u03b3\nshown in a. On the other hand, the two-phase equilibrium lines of \u03b1/\u03b5\nand \u03b1/\u03b3 at low temperatures have positive slopes, indicating that \u03b5 and\n\u03b3 have smaller molar volume than \u03b1 as \u03b5 and \u03b3 are more stable at higher\npressures than \u03b1 at constant temperatures. It is thus evident that the\nphase stable at higher pressure can have either higher or lower entropy\nthan the phase stable at lower pressure, and the phase stable at higher\ntemperature can have either higher or lower volume than the phase stable\nat lower temperature. This is the property anomaly discussed in Chapter\n.\n\nAnother useful example of potential phase diagram is the\npressure-temperature phase diagram of $H_{2}O$ shown in with three\nphases: ice, water, and vapour. It is known that the solid ice has many\npolymorphic structures at high pressures, which are not included in this\ndiagram. As in the pure Fe potential phase diagram discussed above, the\nsingle-phase regions of ice, water, and vapour are represented by the\ntwo-dimensional areas with two-degree of freedom based on the Gibbs\nphase rule, the lines are for the two-phase regions of ice-water,\nice-vapour, and water-vapour, and the three-phase equilibrium has\nzero-degree of freedom represented by a point at $273.16\\ K$ and\n$611.73\\ Pa$.\n\nThere are two features in which are different from those of Fe shown in\na. The first feature is that the slope of the liquid-solid two-phase\nequilibrium line in has the opposite sign of that in a. This is because\nsolid ice has larger molar volume than liquid water, while the molar\nvolume of liquid Fe is larger than those of fcc-Fe and bcc-Fe. The\nsecond feature is that the two-phase equilibrium line of water-vapour\nends at $647\\ K$ and $22.064 \\cdot 10^{6}\\ Pa$. Beyond this point, the\ndifference between vapour and water disappears when the pressure and\ntemperature are changed, i.e. it behaves like one phase. This point is a\ncritical point, as discussed in Chapter . However, it should be pointed\nout that it does not represent an invariant reaction as the degree of\nfreedom based on the Gibbs phase rule is equal to one and not zero. On\nthe other hand, both the temperature and pressure of the critical point\nare invariant due to the two constraints introduced by the limit of\nstability of a single phase, i.e. the second and third derivatives of\ntemperature to entropy or pressure to volume are zero.\n\nFigure \u2011: P-T phase diagram of $H_{2}O$\n", + "source": [ + "### Potential phase diagrams of one-component systems\n", + "\n", + "As physical vision of human being is limited to three-dimensions, only\n", + "one-component system can be completely visualized as shown in for one\n", + "phase where any two of the three potentials can change independently.\n", + "From Gibbs phase rule, when two phases are in equilibrium, only one\n", + "potential can vary freely if the two-phase equilibrium is to be\n", + "maintained. While three phases are in equilibrium, the degree of freedom\n", + "is zero, and all three potentials are fixed.\n", + "\n", + "For a two-phase equilibrium, two surfaces intersect each other as\n", + "depicted by the dashed line in . This two-phase equilibrium line is\n", + "obtained by applying to both phases in a one-component system. Since one\n", + "of the potentials is dependent on the other two, one can eliminate it by\n", + "dividing the equation by its conjugate molar quantity and subtracting\n", + "the two equations, resulting in the following three equations\n", + "\n", + "Eq. \u2011\n", + "$0 = \\left( \\frac{S^{\\alpha}}{N_{A}^{\\alpha}} - \\frac{S^{\\beta}}{N_{A}^{\\beta}} \\right)dT + \\left( \\frac{V^{\\alpha}}{N_{A}^{\\alpha}} - \\frac{V^{\\beta}}{N_{A}^{\\beta}} \\right)d( - P) = \\mathrm{\\Delta}S_{m}^{\\alpha\\beta}dT + \\mathrm{\\Delta}V_{m}^{\\alpha\\beta}d( - P)$\n", + "\n", + "Eq. \u2011\n", + "$0 = \\left( \\frac{S^{\\alpha}}{V^{\\alpha}} - \\frac{S^{\\beta}}{V^{\\beta}} \\right)dT + \\left( \\frac{N_{A}^{\\alpha}}{V^{\\alpha}} - \\frac{N_{A}^{\\beta}}{V^{\\beta}} \\right){d\\mu}_{A} = \\mathrm{\\Delta}\\left( \\frac{S_{m}}{V_{m}} \\right)^{\\alpha\\beta}dT + \\mathrm{\\Delta}\\left( \\frac{1}{V_{m}} \\right)^{\\alpha\\beta}{d\\mu}_{A}$\n", + "\n", + "Eq. \u2011\n", + "$0 = \\left( \\frac{V^{\\alpha}}{S^{\\alpha}} - \\frac{V^{\\beta}}{S^{\\beta}} \\right)d( - P) + \\left( \\frac{N_{A}^{\\alpha}}{S^{\\alpha}} - \\frac{N_{A}^{\\beta}}{S^{\\beta}} \\right){d\\mu}_{A} = \\mathrm{\\Delta}\\left( \\frac{V_{m}}{S_{m}} \\right)^{\\alpha\\beta}d( - P) + \\mathrm{\\Delta}\\left( \\frac{1}{S_{m}} \\right)^{\\alpha\\beta}{d\\mu}_{A}$\n", + "\n", + "Figure \u2011: Gibbs energy surfaces of two phases and their intersection,\n", + "representing the two-phase equilibrium.\n", + "\n", + "The directions of the two-phase equilibrium line can thus be obtained as\n", + "\n", + "Eq. \u2011\n", + "$\\frac{d( - P)}{dT} = - \\frac{\\mathrm{\\Delta}S_{m}^{\\alpha\\beta}}{\\mathrm{\\Delta}V_{m}^{\\alpha\\beta}}$\n", + "\n", + "Eq. \u2011\n", + "$\\frac{dT}{{d\\mu}_{A}} = - \\frac{\\mathrm{\\Delta}\\left( \\frac{1}{V_{m}} \\right)^{\\alpha\\beta}}{\\mathrm{\\Delta}\\left( \\frac{S_{m}}{V_{m}} \\right)^{\\alpha\\beta}}$\n", + "\n", + "Eq. \u2011\n", + "$\\frac{d( - P)}{{d\\mu}_{A}} = - \\frac{\\mathrm{\\Delta}\\left( \\frac{1}{S_{m}} \\right)^{\\alpha\\beta}}{\\mathrm{\\Delta}\\left( \\frac{V_{m}}{S_{m}} \\right)^{\\alpha\\beta}}$\n", + "\n", + "These three equations define the mathematical forms of the two-phase\n", + "equilibrium line in the two-dimensional spaces of $T - ( - P)$,\n", + "$\\mu_{A} - T$ and $\\mu_{A} - ( - P)$, respectively, and can thus be\n", + "plotted as two-dimensional diagrams. is commonly called\n", + "Clausius-Clapeyron equation in the literature. One may thus call all\n", + "three equations above as generalized Clausius-Clapeyron equations. At\n", + "equilibrium, the chemical potentials of the component in both phases are\n", + "equal to each other, so are their Gibbs energies. One thus has\n", + "\n", + "Eq. \u2011\n", + "$G_{m}^{\\alpha} - G_{m}^{\\beta} = 0 = \\mathrm{\\Delta}G_{m} = \\mathrm{\\Delta}H_{m}^{\\alpha\\beta} - T\\mathrm{\\Delta}S_{m}^{\\alpha\\beta}$\n", + "\n", + "The Clausius-Clapeyron equation, , can be re-written as\n", + "\n", + "Eq. \u2011\n", + "$\\frac{d( - P)}{dT} = - \\frac{\\mathrm{\\Delta}H_{m}^{\\alpha\\beta}}{T\\mathrm{\\Delta}V_{m}^{\\alpha\\beta}}$\n", + "\n", + "As an example, three potential phase diagrams of pure Fe are shown in .\n", + "There are four phases in the system, bcc, fcc, hcp, and liquid. In the\n", + "literature, the high temperature and low temperature bcc phases are\n", + "usually denoted by \u03b4 (high temperature) and \u03b1 (low temperature), the fcc\n", + "and hcp phases by \u03b3 and \u03b5, and the liquid phase by L, respectively. In\n", + "these figures, the two-dimensional areas are single-phase regions where\n", + "two potentials can change independently with the system remaining as\n", + "single-phase. The lines denote two-phase equilibrium regions where only\n", + "one potential can vary independently if the two-phase equilibrium is to\n", + "be maintained. The points where three two-phase equilibrium lines meet\n", + "represent the invariant three-phase equilibria with three potentials\n", + "fixed.\n", + "\n", + "Figure \u2011 $T - ( - P)$, $\\mu_{A} - T$ and $\\mu_{A} - ( - P)$ phase\n", + "diagrams of pure Fe\n", + "\n", + "Based on the discussions in Chapter , enthalpy and entropy of a phase\n", + "increase monotonically with temperature, and phases stable at higher\n", + "temperatures have higher enthalpy and entropy than phases stable at\n", + "lower temperatures. Consequently, the two-phase equilibrium lines in a\n", + "$T - ( - P)$ potential phase diagram have negative slopes if the phase\n", + "stable at higher temperatures also has larger molar volume than the\n", + "phase stable at lower temperatures (note that if $P$ is plotted instead\n", + "of $- P$, the slope is positive). This is the case for the two-phase\n", + "equilibrium lines of \u03b4/L, \u03b3/L, and \u03b3/\u03b4 at high temperatures, and \u03b5/\u03b3\n", + "shown in a. On the other hand, the two-phase equilibrium lines of \u03b1/\u03b5\n", + "and \u03b1/\u03b3 at low temperatures have positive slopes, indicating that \u03b5 and\n", + "\u03b3 have smaller molar volume than \u03b1 as \u03b5 and \u03b3 are more stable at higher\n", + "pressures than \u03b1 at constant temperatures. It is thus evident that the\n", + "phase stable at higher pressure can have either higher or lower entropy\n", + "than the phase stable at lower pressure, and the phase stable at higher\n", + "temperature can have either higher or lower volume than the phase stable\n", + "at lower temperature. This is the property anomaly discussed in Chapter\n", + ".\n", + "\n", + "Another useful example of potential phase diagram is the\n", + "pressure-temperature phase diagram of $H_{2}O$ shown in with three\n", + "phases: ice, water, and vapour. It is known that the solid ice has many\n", + "polymorphic structures at high pressures, which are not included in this\n", + "diagram. As in the pure Fe potential phase diagram discussed above, the\n", + "single-phase regions of ice, water, and vapour are represented by the\n", + "two-dimensional areas with two-degree of freedom based on the Gibbs\n", + "phase rule, the lines are for the two-phase regions of ice-water,\n", + "ice-vapour, and water-vapour, and the three-phase equilibrium has\n", + "zero-degree of freedom represented by a point at $273.16\\ K$ and\n", + "$611.73\\ Pa$.\n", + "\n", + "There are two features in which are different from those of Fe shown in\n", + "a. The first feature is that the slope of the liquid-solid two-phase\n", + "equilibrium line in has the opposite sign of that in a. This is because\n", + "solid ice has larger molar volume than liquid water, while the molar\n", + "volume of liquid Fe is larger than those of fcc-Fe and bcc-Fe. The\n", + "second feature is that the two-phase equilibrium line of water-vapour\n", + "ends at $647\\ K$ and $22.064 \\cdot 10^{6}\\ Pa$. Beyond this point, the\n", + "difference between vapour and water disappears when the pressure and\n", + "temperature are changed, i.e. it behaves like one phase. This point is a\n", + "critical point, as discussed in Chapter . However, it should be pointed\n", + "out that it does not represent an invariant reaction as the degree of\n", + "freedom based on the Gibbs phase rule is equal to one and not zero. On\n", + "the other hand, both the temperature and pressure of the critical point\n", + "are invariant due to the two constraints introduced by the limit of\n", + "stability of a single phase, i.e. the second and third derivatives of\n", + "temperature to entropy or pressure to volume are zero.\n", + "\n", + "Figure \u2011: P-T phase diagram of $H_{2}O$\n" + ], "metadata": {} }, { - "id": "9f0752e8", + "id": "5c6b46cf", "cell_type": "markdown", - "source": "### Potential phase diagrams of two-component systems\n\nFrom Gibbs-Duhem equation (see ), a single phase in a two-component\nsystem has three independent potentials, out of the four potentials of\n$T$, $- P$, $\\mu_{A}$, and $\\mu_{B}$, and is a three-dimensional\ngeometric feature in a four-dimensional space. Alternatively, it can be\nrepresented by a three-dimensional space constructed by the three\nindependent potentials. A two-phase equilibrium is thus a\ntwo-dimensional surface in this three-dimensional space created by the\nintercept of two three-dimensional spaces, and a three-phase equilibrium\nis a one-dimensional line, and a four-phase equilibrium is a\nzero-dimensional point. This is shown in for the Fe-C binary system\ninvolving four phases: fcc, bcc, Fe3C and graphite. Since any\none of the four potentials can be chosen as the dependent one, four\nthree-dimensional potential phase diagrams are depicted in .\n\nFigure \u2011: Projected potential phase diagram of Fe-C system with fcc,\nbcc, Fe3C, and graphite\n\nThe two-phase equilibrium surfaces are obtained by choosing any one of\nthe four potentials as the dependent one and solving the Gibbs-Duhem\nequations for both phases, resulting in following four equations\n\nEq. \u2011\n$0 = \\left( \\frac{S^{\\alpha}}{N_{A}^{\\alpha}} - \\frac{S^{\\beta}}{N_{A}^{\\beta}} \\right)dT + \\left( \\frac{V^{\\alpha}}{N_{A}^{\\alpha}} - \\frac{V^{\\beta}}{N_{A}^{\\beta}} \\right)d( - P) + \\left( \\frac{N_{B}^{\\alpha}}{N_{A}^{\\alpha}} - \\frac{N_{B}^{\\beta}}{N_{A}^{\\beta}} \\right){d\\mu}_{B} = \\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}dT + \\mathrm{\\Delta}V_{mA}^{\\alpha\\beta}d( - P) + \\mathrm{\\Delta}z_{B}^{\\alpha\\beta}{d\\mu}_{B}$\n\nEq. \u2011\n$0 = \\left( \\frac{S^{\\alpha}}{N_{B}^{\\alpha}} - \\frac{S^{\\beta}}{N_{B}^{\\beta}} \\right)dT + \\left( \\frac{V^{\\alpha}}{N_{B}^{\\alpha}} - \\frac{V^{\\beta}}{N_{B}^{\\beta}} \\right)d( - P) + \\left( \\frac{N_{B}^{\\alpha}}{N_{B}^{\\alpha}} - \\frac{N_{B}^{\\beta}}{N_{B}^{\\beta}} \\right){d\\mu}_{B} = \\mathrm{\\Delta}S_{mB}^{\\alpha\\beta}dT + \\mathrm{\\Delta}V_{mB}^{\\alpha\\beta}d( - P) + \\mathrm{\\Delta}z_{A}^{\\alpha\\beta}{d\\mu}_{A}$\n\nEq. \u2011\n$0 = \\left( \\frac{S^{\\alpha}}{V^{\\alpha}} - \\frac{S^{\\beta}}{V^{\\beta}} \\right)dT + \\left( \\frac{N_{A}^{\\alpha}}{V^{\\alpha}} - \\frac{N_{A}^{\\beta}}{V^{\\beta}} \\right){d\\mu}_{A} + \\left( \\frac{N_{B}^{\\alpha}}{V^{\\alpha}} - \\frac{N_{B}^{\\beta}}{V^{\\beta}} \\right){d\\mu}_{B} = \\mathrm{\\Delta}\\left( \\frac{S_{m}}{V_{m}} \\right)^{\\alpha\\beta}dT + \\mathrm{\\Delta}\\left( \\frac{1}{V_{mA}} \\right)^{\\alpha\\beta}{d\\mu}_{A} + \\mathrm{\\Delta}\\left( \\frac{1}{V_{mB}} \\right)^{\\alpha\\beta}{d\\mu}_{B}$\n\nEq. \u2011\n$0 = \\left( \\frac{V^{\\alpha}}{S^{\\alpha}} - \\frac{V^{\\beta}}{S^{\\beta}} \\right)d( - P) + \\left( \\frac{N_{A}^{\\alpha}}{S^{\\alpha}} - \\frac{N_{A}^{\\beta}}{S^{\\beta}} \\right){d\\mu}_{A} + \\left( \\frac{N_{B}^{\\alpha}}{S^{\\alpha}} - \\frac{N_{B}^{\\beta}}{S^{\\beta}} \\right){d\\mu}_{B} = \\mathrm{\\Delta}\\left( \\frac{V_{m}}{S_{m}} \\right)^{\\alpha\\beta}d( - P) + \\mathrm{\\Delta}\\left( \\frac{1}{S_{mA}} \\right)^{\\alpha\\beta}{d\\mu}_{A} + \\mathrm{\\Delta}\\left( \\frac{1}{S_{mB}} \\right)^{\\alpha\\beta}{d\\mu}_{B}$\n\nA three-phase equilibrium line is represented by the intercept of two\ntwo-phase surfaces by applying any one of the above four equations to\ntwo two-phase equilibria. Let us use as an example\n\nEq. \u2011\n$0 = \\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}dT + \\mathrm{\\Delta}V_{mA}^{\\alpha\\beta}d( - P) + {\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}{d\\mu}_{B}$\n\nEq. \u2011\n$0 = \\mathrm{\\Delta}S_{mA}^{\\alpha\\gamma}dT + \\mathrm{\\Delta}V_{mA}^{\\alpha\\gamma}d( - P) + \\mathrm{\\Delta}{z_{B}^{\\alpha\\gamma}d\\mu}_{B}$\n\nIt is self-evident that the two-phase equilibrium surface between\n$\\beta$ and $\\gamma$ is not independent and can be obtained by\nsubtraction of and\n\nEq. \u2011\n$0 = \\left( \\mathrm{\\Delta}S_{mA}^{\\alpha\\beta} - \\mathrm{\\Delta}S_{mA}^{\\alpha\\gamma} \\right)dT + \\left( \\mathrm{\\Delta}V_{mA}^{\\alpha\\beta} - \\mathrm{\\Delta}V_{mA}^{\\alpha\\gamma} \\right)d( - P) + \\left( {\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta} - \\mathrm{\\Delta}z_{B}^{\\alpha\\gamma} \\right){d\\mu}_{B} = \\mathrm{\\Delta}S_{mA}^{\\gamma\\beta}dT + \\mathrm{\\Delta}V_{mA}^{\\gamma\\beta}d( - P) + {\\mathrm{\\Delta}z}_{B}^{\\gamma\\beta}{d\\mu}_{B}$\n\nEliminating one of three potentials in and , one can obtain three\nequations for the three-phase equilibrium line\n\nEq. \u2011\n$\\frac{d( - P)}{dT} = - \\frac{\\frac{\\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}}{{\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}S_{mA}^{\\alpha\\gamma}}{\\mathrm{\\Delta}z_{B}^{\\alpha\\gamma}}}{\\frac{\\mathrm{\\Delta}V_{mA}^{\\alpha\\beta}}{{\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}V_{mA}^{\\alpha\\gamma}}{\\mathrm{\\Delta}z_{B}^{\\alpha\\gamma}}}$\n\nEq. \u2011\n$\\frac{dT}{{d\\mu}_{B}} = - \\frac{\\frac{{\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}}{\\mathrm{\\Delta}V_{mA}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}z_{B}^{\\alpha\\gamma}}{\\mathrm{\\Delta}V_{mA}^{\\alpha\\gamma}}}{\\frac{\\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}}{\\mathrm{\\Delta}V_{mA}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}S_{mA}^{\\alpha\\gamma}}{\\mathrm{\\Delta}V_{mA}^{\\alpha\\gamma}}}$\n\nEq. \u2011\n$\\frac{d( - P)}{{d\\mu}_{B}} = - \\frac{\\frac{{\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}}{\\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}z_{B}^{\\alpha\\gamma}}{\\mathrm{\\Delta}S_{mA}^{\\alpha\\gamma}}}{\\frac{\\mathrm{\\Delta}V_{mA}^{\\alpha\\beta}}{\\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}V_{mA}^{\\alpha\\gamma}}{\\mathrm{\\Delta}S_{mA}^{\\alpha\\gamma}}}$\n\nto can be referred as generalized Clausius-Clapeyron equations for\nbinary systems. Similar equations can be derived for\n$T - ( - P) - \\mu_{A}$, $T - \\mu_{A} - \\mu_{B}$, and\n$( - P) - \\mu_{A} - \\mu_{B}$ potential phase diagrams from to , and are\nlisted below\n\n- Generalized Clausius-Clapeyron equations for a three-phase equilibrium\n in $T - ( - P) - \\mu_{A}$ potential phase diagrams\n\nEq. \u2011\n$\\frac{d( - P)}{dT} = - \\frac{\\frac{\\mathrm{\\Delta}S_{mB}^{\\alpha\\beta}}{{\\mathrm{\\Delta}z}_{A}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}S_{mB}^{\\alpha\\gamma}}{\\mathrm{\\Delta}z_{A}^{\\alpha\\gamma}}}{\\frac{\\mathrm{\\Delta}V_{mB}^{\\alpha\\beta}}{{\\mathrm{\\Delta}z}_{A}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}V_{mB}^{\\alpha\\gamma}}{\\mathrm{\\Delta}z_{A}^{\\alpha\\gamma}}}$\n\nEq. \u2011\n$\\frac{dT}{{d\\mu}_{A}} = - \\frac{\\frac{{\\mathrm{\\Delta}z}_{A}^{\\alpha\\beta}}{\\mathrm{\\Delta}V_{mB}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}z_{A}^{\\alpha\\gamma}}{\\mathrm{\\Delta}V_{mB}^{\\alpha\\gamma}}}{\\frac{\\mathrm{\\Delta}S_{mB}^{\\alpha\\beta}}{\\mathrm{\\Delta}V_{mB}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}S_{mB}^{\\alpha\\gamma}}{\\mathrm{\\Delta}V_{mB}^{\\alpha\\gamma}}}$\n\nEq. \u2011\n$\\frac{d( - P)}{{d\\mu}_{A}} = - \\frac{\\frac{{\\mathrm{\\Delta}z}_{A}^{\\alpha\\beta}}{\\mathrm{\\Delta}S_{mB}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}z_{A}^{\\alpha\\gamma}}{\\mathrm{\\Delta}S_{mB}^{\\alpha\\gamma}}}{\\frac{\\mathrm{\\Delta}V_{mB}^{\\alpha\\beta}}{\\mathrm{\\Delta}S_{mB}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}V_{mB}^{\\alpha\\gamma}}{\\mathrm{\\Delta}S_{mB}^{\\alpha\\gamma}}}$\n\n- Generalized Clausius-Clapeyron equations for a three-phase equilibrium\n in $\\ T - \\mu_{A} - \\mu_{B}$ potential phase diagrams\n\nEq. \u2011\n$\\frac{dT}{{d\\mu}_{A}} = - \\frac{\\frac{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}}{\\frac{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}}$\n\nEq. \u2011\n$\\frac{dT}{{d\\mu}_{B}} = - \\frac{\\frac{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}}{\\frac{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}}$\n\nEq. \u2011\n$\\frac{{d\\mu}_{A}}{{d\\mu}_{B}} = - \\frac{\\frac{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\gamma}}}{\\frac{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\gamma}}}$\n\n- Generalized Clausius-Clapeyron equations for a three-phase equilibrium\n in $( - P) - \\mu_{A} - \\mu_{B}$ potential phase diagrams\n\nEq. \u2011\n$\\frac{d( - P)}{{d\\mu}_{A}} = - \\frac{\\frac{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}}{\\frac{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}}$\n\nEq. \u2011\n$\\frac{d( - P)}{{d\\mu}_{B}} = - \\frac{\\frac{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}}{\\frac{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}}$\n\nEq. \u2011\n$\\frac{{d\\mu}_{A}}{{d\\mu}_{B}} = - \\frac{\\frac{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\gamma}}}{\\frac{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\gamma}}}$\n", + "source": [ + "### Potential phase diagrams of two-component systems\n", + "\n", + "From Gibbs-Duhem equation (see ), a single phase in a two-component\n", + "system has three independent potentials, out of the four potentials of\n", + "$T$, $- P$, $\\mu_{A}$, and $\\mu_{B}$, and is a three-dimensional\n", + "geometric feature in a four-dimensional space. Alternatively, it can be\n", + "represented by a three-dimensional space constructed by the three\n", + "independent potentials. A two-phase equilibrium is thus a\n", + "two-dimensional surface in this three-dimensional space created by the\n", + "intercept of two three-dimensional spaces, and a three-phase equilibrium\n", + "is a one-dimensional line, and a four-phase equilibrium is a\n", + "zero-dimensional point. This is shown in for the Fe-C binary system\n", + "involving four phases: fcc, bcc, Fe3C and graphite. Since any\n", + "one of the four potentials can be chosen as the dependent one, four\n", + "three-dimensional potential phase diagrams are depicted in .\n", + "\n", + "Figure \u2011: Projected potential phase diagram of Fe-C system with fcc,\n", + "bcc, Fe3C, and graphite\n", + "\n", + "The two-phase equilibrium surfaces are obtained by choosing any one of\n", + "the four potentials as the dependent one and solving the Gibbs-Duhem\n", + "equations for both phases, resulting in following four equations\n", + "\n", + "Eq. \u2011\n", + "$0 = \\left( \\frac{S^{\\alpha}}{N_{A}^{\\alpha}} - \\frac{S^{\\beta}}{N_{A}^{\\beta}} \\right)dT + \\left( \\frac{V^{\\alpha}}{N_{A}^{\\alpha}} - \\frac{V^{\\beta}}{N_{A}^{\\beta}} \\right)d( - P) + \\left( \\frac{N_{B}^{\\alpha}}{N_{A}^{\\alpha}} - \\frac{N_{B}^{\\beta}}{N_{A}^{\\beta}} \\right){d\\mu}_{B} = \\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}dT + \\mathrm{\\Delta}V_{mA}^{\\alpha\\beta}d( - P) + \\mathrm{\\Delta}z_{B}^{\\alpha\\beta}{d\\mu}_{B}$\n", + "\n", + "Eq. \u2011\n", + "$0 = \\left( \\frac{S^{\\alpha}}{N_{B}^{\\alpha}} - \\frac{S^{\\beta}}{N_{B}^{\\beta}} \\right)dT + \\left( \\frac{V^{\\alpha}}{N_{B}^{\\alpha}} - \\frac{V^{\\beta}}{N_{B}^{\\beta}} \\right)d( - P) + \\left( \\frac{N_{B}^{\\alpha}}{N_{B}^{\\alpha}} - \\frac{N_{B}^{\\beta}}{N_{B}^{\\beta}} \\right){d\\mu}_{B} = \\mathrm{\\Delta}S_{mB}^{\\alpha\\beta}dT + \\mathrm{\\Delta}V_{mB}^{\\alpha\\beta}d( - P) + \\mathrm{\\Delta}z_{A}^{\\alpha\\beta}{d\\mu}_{A}$\n", + "\n", + "Eq. \u2011\n", + "$0 = \\left( \\frac{S^{\\alpha}}{V^{\\alpha}} - \\frac{S^{\\beta}}{V^{\\beta}} \\right)dT + \\left( \\frac{N_{A}^{\\alpha}}{V^{\\alpha}} - \\frac{N_{A}^{\\beta}}{V^{\\beta}} \\right){d\\mu}_{A} + \\left( \\frac{N_{B}^{\\alpha}}{V^{\\alpha}} - \\frac{N_{B}^{\\beta}}{V^{\\beta}} \\right){d\\mu}_{B} = \\mathrm{\\Delta}\\left( \\frac{S_{m}}{V_{m}} \\right)^{\\alpha\\beta}dT + \\mathrm{\\Delta}\\left( \\frac{1}{V_{mA}} \\right)^{\\alpha\\beta}{d\\mu}_{A} + \\mathrm{\\Delta}\\left( \\frac{1}{V_{mB}} \\right)^{\\alpha\\beta}{d\\mu}_{B}$\n", + "\n", + "Eq. \u2011\n", + "$0 = \\left( \\frac{V^{\\alpha}}{S^{\\alpha}} - \\frac{V^{\\beta}}{S^{\\beta}} \\right)d( - P) + \\left( \\frac{N_{A}^{\\alpha}}{S^{\\alpha}} - \\frac{N_{A}^{\\beta}}{S^{\\beta}} \\right){d\\mu}_{A} + \\left( \\frac{N_{B}^{\\alpha}}{S^{\\alpha}} - \\frac{N_{B}^{\\beta}}{S^{\\beta}} \\right){d\\mu}_{B} = \\mathrm{\\Delta}\\left( \\frac{V_{m}}{S_{m}} \\right)^{\\alpha\\beta}d( - P) + \\mathrm{\\Delta}\\left( \\frac{1}{S_{mA}} \\right)^{\\alpha\\beta}{d\\mu}_{A} + \\mathrm{\\Delta}\\left( \\frac{1}{S_{mB}} \\right)^{\\alpha\\beta}{d\\mu}_{B}$\n", + "\n", + "A three-phase equilibrium line is represented by the intercept of two\n", + "two-phase surfaces by applying any one of the above four equations to\n", + "two two-phase equilibria. Let us use as an example\n", + "\n", + "Eq. \u2011\n", + "$0 = \\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}dT + \\mathrm{\\Delta}V_{mA}^{\\alpha\\beta}d( - P) + {\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}{d\\mu}_{B}$\n", + "\n", + "Eq. \u2011\n", + "$0 = \\mathrm{\\Delta}S_{mA}^{\\alpha\\gamma}dT + \\mathrm{\\Delta}V_{mA}^{\\alpha\\gamma}d( - P) + \\mathrm{\\Delta}{z_{B}^{\\alpha\\gamma}d\\mu}_{B}$\n", + "\n", + "It is self-evident that the two-phase equilibrium surface between\n", + "$\\beta$ and $\\gamma$ is not independent and can be obtained by\n", + "subtraction of and\n", + "\n", + "Eq. \u2011\n", + "$0 = \\left( \\mathrm{\\Delta}S_{mA}^{\\alpha\\beta} - \\mathrm{\\Delta}S_{mA}^{\\alpha\\gamma} \\right)dT + \\left( \\mathrm{\\Delta}V_{mA}^{\\alpha\\beta} - \\mathrm{\\Delta}V_{mA}^{\\alpha\\gamma} \\right)d( - P) + \\left( {\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta} - \\mathrm{\\Delta}z_{B}^{\\alpha\\gamma} \\right){d\\mu}_{B} = \\mathrm{\\Delta}S_{mA}^{\\gamma\\beta}dT + \\mathrm{\\Delta}V_{mA}^{\\gamma\\beta}d( - P) + {\\mathrm{\\Delta}z}_{B}^{\\gamma\\beta}{d\\mu}_{B}$\n", + "\n", + "Eliminating one of three potentials in and , one can obtain three\n", + "equations for the three-phase equilibrium line\n", + "\n", + "Eq. \u2011\n", + "$\\frac{d( - P)}{dT} = - \\frac{\\frac{\\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}}{{\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}S_{mA}^{\\alpha\\gamma}}{\\mathrm{\\Delta}z_{B}^{\\alpha\\gamma}}}{\\frac{\\mathrm{\\Delta}V_{mA}^{\\alpha\\beta}}{{\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}V_{mA}^{\\alpha\\gamma}}{\\mathrm{\\Delta}z_{B}^{\\alpha\\gamma}}}$\n", + "\n", + "Eq. \u2011\n", + "$\\frac{dT}{{d\\mu}_{B}} = - \\frac{\\frac{{\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}}{\\mathrm{\\Delta}V_{mA}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}z_{B}^{\\alpha\\gamma}}{\\mathrm{\\Delta}V_{mA}^{\\alpha\\gamma}}}{\\frac{\\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}}{\\mathrm{\\Delta}V_{mA}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}S_{mA}^{\\alpha\\gamma}}{\\mathrm{\\Delta}V_{mA}^{\\alpha\\gamma}}}$\n", + "\n", + "Eq. \u2011\n", + "$\\frac{d( - P)}{{d\\mu}_{B}} = - \\frac{\\frac{{\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}}{\\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}z_{B}^{\\alpha\\gamma}}{\\mathrm{\\Delta}S_{mA}^{\\alpha\\gamma}}}{\\frac{\\mathrm{\\Delta}V_{mA}^{\\alpha\\beta}}{\\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}V_{mA}^{\\alpha\\gamma}}{\\mathrm{\\Delta}S_{mA}^{\\alpha\\gamma}}}$\n", + "\n", + "to can be referred as generalized Clausius-Clapeyron equations for\n", + "binary systems. Similar equations can be derived for\n", + "$T - ( - P) - \\mu_{A}$, $T - \\mu_{A} - \\mu_{B}$, and\n", + "$( - P) - \\mu_{A} - \\mu_{B}$ potential phase diagrams from to , and are\n", + "listed below\n", + "\n", + "- Generalized Clausius-Clapeyron equations for a three-phase equilibrium\n", + " in $T - ( - P) - \\mu_{A}$ potential phase diagrams\n", + "\n", + "Eq. \u2011\n", + "$\\frac{d( - P)}{dT} = - \\frac{\\frac{\\mathrm{\\Delta}S_{mB}^{\\alpha\\beta}}{{\\mathrm{\\Delta}z}_{A}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}S_{mB}^{\\alpha\\gamma}}{\\mathrm{\\Delta}z_{A}^{\\alpha\\gamma}}}{\\frac{\\mathrm{\\Delta}V_{mB}^{\\alpha\\beta}}{{\\mathrm{\\Delta}z}_{A}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}V_{mB}^{\\alpha\\gamma}}{\\mathrm{\\Delta}z_{A}^{\\alpha\\gamma}}}$\n", + "\n", + "Eq. \u2011\n", + "$\\frac{dT}{{d\\mu}_{A}} = - \\frac{\\frac{{\\mathrm{\\Delta}z}_{A}^{\\alpha\\beta}}{\\mathrm{\\Delta}V_{mB}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}z_{A}^{\\alpha\\gamma}}{\\mathrm{\\Delta}V_{mB}^{\\alpha\\gamma}}}{\\frac{\\mathrm{\\Delta}S_{mB}^{\\alpha\\beta}}{\\mathrm{\\Delta}V_{mB}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}S_{mB}^{\\alpha\\gamma}}{\\mathrm{\\Delta}V_{mB}^{\\alpha\\gamma}}}$\n", + "\n", + "Eq. \u2011\n", + "$\\frac{d( - P)}{{d\\mu}_{A}} = - \\frac{\\frac{{\\mathrm{\\Delta}z}_{A}^{\\alpha\\beta}}{\\mathrm{\\Delta}S_{mB}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}z_{A}^{\\alpha\\gamma}}{\\mathrm{\\Delta}S_{mB}^{\\alpha\\gamma}}}{\\frac{\\mathrm{\\Delta}V_{mB}^{\\alpha\\beta}}{\\mathrm{\\Delta}S_{mB}^{\\alpha\\beta}} - \\frac{\\mathrm{\\Delta}V_{mB}^{\\alpha\\gamma}}{\\mathrm{\\Delta}S_{mB}^{\\alpha\\gamma}}}$\n", + "\n", + "- Generalized Clausius-Clapeyron equations for a three-phase equilibrium\n", + " in $\\ T - \\mu_{A} - \\mu_{B}$ potential phase diagrams\n", + "\n", + "Eq. \u2011\n", + "$\\frac{dT}{{d\\mu}_{A}} = - \\frac{\\frac{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}}{\\frac{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}}$\n", + "\n", + "Eq. \u2011\n", + "$\\frac{dT}{{d\\mu}_{B}} = - \\frac{\\frac{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}}{\\frac{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}}$\n", + "\n", + "Eq. \u2011\n", + "$\\frac{{d\\mu}_{A}}{{d\\mu}_{B}} = - \\frac{\\frac{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/V_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\gamma}}}{\\frac{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/V_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( S_{m}/V_{m} \\right)}_{\\ }^{\\alpha\\gamma}}}$\n", + "\n", + "- Generalized Clausius-Clapeyron equations for a three-phase equilibrium\n", + " in $( - P) - \\mu_{A} - \\mu_{B}$ potential phase diagrams\n", + "\n", + "Eq. \u2011\n", + "$\\frac{d( - P)}{{d\\mu}_{A}} = - \\frac{\\frac{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}}{\\frac{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}}$\n", + "\n", + "Eq. \u2011\n", + "$\\frac{d( - P)}{{d\\mu}_{B}} = - \\frac{\\frac{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}}{\\frac{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}}$\n", + "\n", + "Eq. \u2011\n", + "$\\frac{{d\\mu}_{A}}{{d\\mu}_{B}} = - \\frac{\\frac{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/S_{mB} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\gamma}}}{\\frac{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\beta}}{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\beta}} - \\frac{{\\mathrm{\\Delta}\\left( 1/S_{mA} \\right)}_{\\ }^{\\alpha\\gamma}}{{\\mathrm{\\Delta}\\left( V_{m}/S_{m} \\right)}_{\\ }^{\\alpha\\gamma}}}$\n" + ], "metadata": {} }, { - "id": "0b41a83f", + "id": "320479b8", "cell_type": "markdown", - "source": "### Section of potential phase diagrams\n\nBased on the Gibbs-Duhem equation (see ), a single-phase equilibrium in\na system with more than two independent components has more than three\nindependent potentials. There is no problem in representing them using\nthe mathematical formulas discussed so far, but it is not possible for\nus to visualize graphically the full potential phase diagrams in\nmulti-component systems with more than two independent components. In\nprinciples, there are two options. One option is to project the\nmulti-dimensional potential phase diagram into a two- or three-dimension\ndiagram, and another option is to section the multi-dimensional\npotential phase diagram by fixing the values of some potentials.\n\nThe projection approach is used for one-component systems in Chapter .\nSince a two-phase equilibrium in a one-component potential phase diagram\nis one-dimensional, the projection does not lose any information, and\nthe same is true for a three-phase equilibrium in a one-component\npotential phase diagram. In a binary system, the projections of three-\nand four-phase equilibria do not lose any information, while the\nprojections of two-phase equilibria become two-dimensional and cannot\nretain all the information as the original-dimensionality of these\ntwo-phase equilibria is three. Consequently, sectioning at fixed values\nof some potentials is necessary in order to visualize the phase\nrelations in systems with two or more components. Gibbs phase rule shown\nin and are thus modified to\n\nEq. \u2011 $\\upsilon = c + 2 - p - n_{s} = \\left( c - n_{s} \\right) + 2 - p$\n\nEq. \u2011 $p_{\\max} = \\left( c - n_{s} \\right) + 2$\n\nwhere $n_{s}$ is the number of potentials fixed in sectioning. As can be\nseen in the last part of , the number of sectioning is equivalent to the\nreduction of the effective number of independent components. Therefore,\nany multi-component systems with $n_{s} = c - i$ behave like an\n$i$-component system. The equations presented in Chapter and are thus\ndirectly applicable to multi-component systems with $n_{s} = c - 1$ and\n$n_{s} = c - 2$, respectively.\n\nA common practice in experiments is to fix pressure, temperature or\nchemical potentials of volatile components as they are usually the\nvariables controlled experimentally. In a binary system, the potential\nphase diagram at constant pressure can be represented by any two of the\nthree potentials, i.e. two chemical potentials and temperature, with the\nremaining potential being dependent, and has the identical morphology as\na one-component system. The Gibbs-Duhem equation under such conditions\nbecomes\n\nEq. \u2011 $0 = - SdT - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B}$\n\nThe corresponding two-phase Clausius-Clapeyron equations are written as\n\nEq. \u2011\n$\\frac{dT}{{d\\mu}_{A}} = - \\frac{{\\mathrm{\\Delta}z}_{A}^{\\alpha\\beta}}{\\mathrm{\\Delta}S_{mB}^{\\alpha\\beta}}$\n\nEq. \u2011\n$\\frac{dT}{{d\\mu}_{B}} = - \\frac{{\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}}{\\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}}$\n\nEq. \u2011\n$\\frac{{d\\mu}_{A}}{{d\\mu}_{B}} = - \\frac{\\mathrm{\\Delta}\\left( \\frac{1}{S_{mB}^{}} \\right)^{\\alpha\\beta}}{\\mathrm{\\Delta}\\left( \\frac{1}{S_{mA}^{}} \\right)^{\\alpha\\beta}}$\n\nAs an example, the $T$-$\\mu_{C}$ potential phase diagram for the Fe-C\nbinary system at one atmospheric pressure is shown in .\n\nFigure \u2011: $T$-$\\mu_{C}$ potential phase diagram for the Fe-C binary\nsystem at $P = 1atm$\n\nIn a ternary system, two potentials need to be fixed in order to obtain\ntwo-dimensional potential phase diagrams. When the pressure and the\nchemical potential of one species are fixed, the system behaves like a\nbinary system discussed above. When the system temperature and pressure\nare fixed, the Gibbs-Duhem equation is written as\n\nEq. \u2011 $0 = - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B} - {N_{C}d\\mu}_{C}$\n\nTaking the component A as the dependent element, the two-phase\nClausius-Clapeyron equation is simplified as\n\nEq. \u2011\n$\\frac{{d\\mu}_{B}}{{d\\mu}_{C}} = \\frac{d\\left( \\ln a_{B} \\right)}{d\\left( \\ln a_{C} \\right)} = - \\frac{\\mathrm{\\Delta}z_{C}^{\\alpha\\beta}}{{\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}}$\n\nWhen the two phases in equilibrium are stoichiometric phases, the\ntwo-phase equilibrium is thus a straight line. For example, the Ti-O-Cl\npotential phase diagram at 600\u00b0C and one atmospheric pressure is shown\nin . Since both O and Cl are volatile components, their activities are\nusually represented by their partial pressures with the pure\nO2 and Cl2 gas as their respective reference\nstates at the given temperature and pressure.\n\nFigure \u2011: Ti-O-Cl potential phase diagram at 600\u00b0C and one atmospheric\npressure\n\nFor systems with more than three components, the chemical potentials of\none or more components must be fixed in order to obtain a\ntwo-dimensional potential phase diagram similar to the potential phase\ndiagrams discussed above.\n", + "source": [ + "### Section of potential phase diagrams\n", + "\n", + "Based on the Gibbs-Duhem equation (see ), a single-phase equilibrium in\n", + "a system with more than two independent components has more than three\n", + "independent potentials. There is no problem in representing them using\n", + "the mathematical formulas discussed so far, but it is not possible for\n", + "us to visualize graphically the full potential phase diagrams in\n", + "multi-component systems with more than two independent components. In\n", + "principles, there are two options. One option is to project the\n", + "multi-dimensional potential phase diagram into a two- or three-dimension\n", + "diagram, and another option is to section the multi-dimensional\n", + "potential phase diagram by fixing the values of some potentials.\n", + "\n", + "The projection approach is used for one-component systems in Chapter .\n", + "Since a two-phase equilibrium in a one-component potential phase diagram\n", + "is one-dimensional, the projection does not lose any information, and\n", + "the same is true for a three-phase equilibrium in a one-component\n", + "potential phase diagram. In a binary system, the projections of three-\n", + "and four-phase equilibria do not lose any information, while the\n", + "projections of two-phase equilibria become two-dimensional and cannot\n", + "retain all the information as the original-dimensionality of these\n", + "two-phase equilibria is three. Consequently, sectioning at fixed values\n", + "of some potentials is necessary in order to visualize the phase\n", + "relations in systems with two or more components. Gibbs phase rule shown\n", + "in and are thus modified to\n", + "\n", + "Eq. \u2011 $\\upsilon = c + 2 - p - n_{s} = \\left( c - n_{s} \\right) + 2 - p$\n", + "\n", + "Eq. \u2011 $p_{\\max} = \\left( c - n_{s} \\right) + 2$\n", + "\n", + "where $n_{s}$ is the number of potentials fixed in sectioning. As can be\n", + "seen in the last part of , the number of sectioning is equivalent to the\n", + "reduction of the effective number of independent components. Therefore,\n", + "any multi-component systems with $n_{s} = c - i$ behave like an\n", + "$i$-component system. The equations presented in Chapter and are thus\n", + "directly applicable to multi-component systems with $n_{s} = c - 1$ and\n", + "$n_{s} = c - 2$, respectively.\n", + "\n", + "A common practice in experiments is to fix pressure, temperature or\n", + "chemical potentials of volatile components as they are usually the\n", + "variables controlled experimentally. In a binary system, the potential\n", + "phase diagram at constant pressure can be represented by any two of the\n", + "three potentials, i.e. two chemical potentials and temperature, with the\n", + "remaining potential being dependent, and has the identical morphology as\n", + "a one-component system. The Gibbs-Duhem equation under such conditions\n", + "becomes\n", + "\n", + "Eq. \u2011 $0 = - SdT - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B}$\n", + "\n", + "The corresponding two-phase Clausius-Clapeyron equations are written as\n", + "\n", + "Eq. \u2011\n", + "$\\frac{dT}{{d\\mu}_{A}} = - \\frac{{\\mathrm{\\Delta}z}_{A}^{\\alpha\\beta}}{\\mathrm{\\Delta}S_{mB}^{\\alpha\\beta}}$\n", + "\n", + "Eq. \u2011\n", + "$\\frac{dT}{{d\\mu}_{B}} = - \\frac{{\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}}{\\mathrm{\\Delta}S_{mA}^{\\alpha\\beta}}$\n", + "\n", + "Eq. \u2011\n", + "$\\frac{{d\\mu}_{A}}{{d\\mu}_{B}} = - \\frac{\\mathrm{\\Delta}\\left( \\frac{1}{S_{mB}^{}} \\right)^{\\alpha\\beta}}{\\mathrm{\\Delta}\\left( \\frac{1}{S_{mA}^{}} \\right)^{\\alpha\\beta}}$\n", + "\n", + "As an example, the $T$-$\\mu_{C}$ potential phase diagram for the Fe-C\n", + "binary system at one atmospheric pressure is shown in .\n", + "\n", + "Figure \u2011: $T$-$\\mu_{C}$ potential phase diagram for the Fe-C binary\n", + "system at $P = 1atm$\n", + "\n", + "In a ternary system, two potentials need to be fixed in order to obtain\n", + "two-dimensional potential phase diagrams. When the pressure and the\n", + "chemical potential of one species are fixed, the system behaves like a\n", + "binary system discussed above. When the system temperature and pressure\n", + "are fixed, the Gibbs-Duhem equation is written as\n", + "\n", + "Eq. \u2011 $0 = - {N_{A}d\\mu}_{A} - {N_{B}d\\mu}_{B} - {N_{C}d\\mu}_{C}$\n", + "\n", + "Taking the component A as the dependent element, the two-phase\n", + "Clausius-Clapeyron equation is simplified as\n", + "\n", + "Eq. \u2011\n", + "$\\frac{{d\\mu}_{B}}{{d\\mu}_{C}} = \\frac{d\\left( \\ln a_{B} \\right)}{d\\left( \\ln a_{C} \\right)} = - \\frac{\\mathrm{\\Delta}z_{C}^{\\alpha\\beta}}{{\\mathrm{\\Delta}z}_{B}^{\\alpha\\beta}}$\n", + "\n", + "When the two phases in equilibrium are stoichiometric phases, the\n", + "two-phase equilibrium is thus a straight line. For example, the Ti-O-Cl\n", + "potential phase diagram at 600\u00b0C and one atmospheric pressure is shown\n", + "in . Since both O and Cl are volatile components, their activities are\n", + "usually represented by their partial pressures with the pure\n", + "O2 and Cl2 gas as their respective reference\n", + "states at the given temperature and pressure.\n", + "\n", + "Figure \u2011: Ti-O-Cl potential phase diagram at 600\u00b0C and one atmospheric\n", + "pressure\n", + "\n", + "For systems with more than three components, the chemical potentials of\n", + "one or more components must be fixed in order to obtain a\n", + "two-dimensional potential phase diagram similar to the potential phase\n", + "diagrams discussed above.\n" + ], "metadata": {} } ] diff --git a/src/psu410/src/references/index.ipynb b/src/psu410/src/references/index.ipynb index 8760a2d..8bc0e0d 100644 --- a/src/psu410/src/references/index.ipynb +++ b/src/psu410/src/references/index.ipynb @@ -4,9 +4,270 @@ "metadata": {}, "cells": [ { - "id": "96a5d429", + "id": "fdeb5814", "cell_type": "markdown", - "source": "# References\n\n1\\. M. 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Fries, \"An\nassessment of the entire Al-Fe system including D0(3) ordering\", *Acta\nMater.* **57** (2009) 2896-2908.\n\n57\\. A. Kusoffsky, N. Dupin and B. Sundman, \"On the compound energy\nformalism applied to fcc ordering\", *Calphad-Comput. Coupling Ph.\nDiagrams Thermochem.* **25** (2001) 549-565.\n\n58\\. T. Abe and B. Sundman, \"A description of the effect of short range\nordering in the compound energy formalism\", *Calphad-Comput. Coupling\nPh. Diagrams Thermochem.* **27** (2003) 403-408.\n\n59\\. Scientific Group Thermodata Europe (SGTE), *Thermodynamic\nProperties of Inorganic Materials*. Lehrstuhl f\u00fcr Theoretische\nH\u00fcttenkunde, Ed. Landolt-Boernstein New Series, Group IV, Springer,\nVerlag Berlin Heidelberg, 1999, vol. 19.\n\n60\\. J. O. Andersson, T. Helander, L. H. Hoglund, P. F. Shi and B.\nSundman, \"THERMO-CALC & DICTRA, computational tools for materials\nscience\", **26** (2002) 273-312.\n\n61\\. D. D. 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Shang, \"Thermal Expansion Anomaly\nRegulated by Entropy\", **4** (2014).\n\n67\\. Y. Wang, S. L. Shang, H. Zhang, L. Q. Chen and Z. K. Liu,\n\"Thermodynamic fluctuations in magnetic states: Fe3Pt as a prototype\",\n*Phil. Mag. Lett.* **90** (2010) 851-859.\n\n68\\. National Science and Technology Council, \u201cMaterials Genome\nInitiative for Global Competitiveness\u201d,\n*[http://www.whitehouse.gov/sites/default/files/microsites/ostp/materials_genome_initiative-final.pdf]()\nOffice of Science and Technology Policy, Washington DC*, June, 2011.\n\n69\\. L. Kaufman and J. Agren, \"CALPHAD, first and second generation -\nBirth of the materials genome\", **70** (2014) 3-6.\n\n70\\. Z. K. Liu, \"Perspective on Materials Genome\u00ae\", *Chin. Sci. Bull.*\n**59** (2014) 1619-1623.\n", + "source": [ + "# References\n", + "\n", + "1\\. M. Hillert, *Phase Equilibria, Phase Diagrams and Phase\n", + "Transformations,* Cambridge University Press, Cambridge, ed. 2nd, 2007.\n", + "\n", + "2\\. 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Bull.*\n", + "**59** (2014) 1619-1623.\n" + ], "metadata": {} } ] From f8e770dd2c31754d8d8b38c60c3a0318f59828d4 Mon Sep 17 00:00:00 2001 From: tonyfast Date: Fri, 12 Jan 2024 10:51:24 -0800 Subject: [PATCH 8/9] add dataframe utils ive written before and extend to this problem --- src/nobook/bootstrap.ipynb | 2355 ++++++++++++++---------------------- src/nobook/mime.types | 6 + src/nobook/utils.py | 582 +++++++++ 3 files changed, 1493 insertions(+), 1450 deletions(-) create mode 100644 src/nobook/mime.types create mode 100644 src/nobook/utils.py diff --git a/src/nobook/bootstrap.ipynb b/src/nobook/bootstrap.ipynb index c4f1f8b..7487658 100644 --- a/src/nobook/bootstrap.ipynb +++ b/src/nobook/bootstrap.ipynb @@ -13,13 +13,6 @@ "execution_count": 1, "id": "c4601326-6456-47e8-99c9-ae8c8f59e32b", "metadata": { - "execution": { - "iopub.execute_input": "2024-01-10T20:16:08.137247Z", - "iopub.status.busy": "2024-01-10T20:16:08.137068Z", - "iopub.status.idle": "2024-01-10T20:16:08.437332Z", - "shell.execute_reply": "2024-01-10T20:16:08.436939Z", - "shell.execute_reply.started": "2024-01-10T20:16:08.137237Z" - }, "tags": [] }, "outputs": [], @@ -31,6 +24,7 @@ " INTERACTIVE = MAIN and not FILE\n", " singleton = functools.lru_cache(1)\n", " from pathlib import Path\n", + " from utils import apply\n", " HERE = Path(__file__).parent if FILE else Path(subprocess.check_output([\"pwd\"]).decode().strip())\n", " PSU = HERE.parent / \"psu410\" / \"src\" / \"psu410\"\n", "\n", @@ -53,72 +47,19 @@ "`get_docx` loads in the original document using `python-docx` data structures." ] }, - { - "cell_type": "code", - "execution_count": 2, - "id": "2f23bf91-3d5d-4bd5-8619-e05ac0df6c53", - "metadata": { - "execution": { - "iopub.execute_input": "2024-01-10T20:16:08.438433Z", - "iopub.status.busy": "2024-01-10T20:16:08.438095Z", - "iopub.status.idle": "2024-01-10T20:16:08.563425Z", - "shell.execute_reply": "2024-01-10T20:16:08.563084Z", - "shell.execute_reply.started": "2024-01-10T20:16:08.438421Z" - }, - "tags": [] - }, - "outputs": [ - { - "data": { - "text/plain": [ - "" - ] - }, - "execution_count": 2, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - " def get_docx() -> \"docx.document.Document\":\n", - " return __import__(\"docx\").Document(\"2015-07-15-Textbook-Cambridge-U-P.docx\")\n", - " INTERACTIVE and get_docx()" - ] - }, { "cell_type": "markdown", - "id": "4ea5f5c8-52de-4e9c-8eb9-db264a61a5d0", - "metadata": { - "execution": { - "iopub.execute_input": "2024-01-06T04:44:46.234364Z", - "iopub.status.busy": "2024-01-06T04:44:46.234086Z", - "iopub.status.idle": "2024-01-06T04:44:46.238225Z", - "shell.execute_reply": "2024-01-06T04:44:46.237652Z", - "shell.execute_reply.started": "2024-01-06T04:44:46.234339Z" - }, - "tags": [] - }, + "id": "c8cfea5a-26ac-4e53-a9ba-62db108e866d", + "metadata": {}, "source": [ - "`get_paragraphs` represents the document a collections of `docx.Paragraph` like object.\n", - "we extract features about each row to use further in our translation of docx to notebooks.\n", - "\n", - "the snippet discovers the style id's of each paragraph. this means we could remediate the the headings here and make them formal headings" + "loads the docx into a dataframe and extracts features for organizing the contents into cells." ] }, { "cell_type": "code", - "execution_count": 3, - "id": "bd674c8e-4a31-4544-b6b9-0aa2573385c8", - "metadata": { - "execution": { - "iopub.execute_input": "2024-01-10T20:16:08.564092Z", - "iopub.status.busy": "2024-01-10T20:16:08.563929Z", - "iopub.status.idle": "2024-01-10T20:16:09.378948Z", - "shell.execute_reply": "2024-01-10T20:16:09.378644Z", - "shell.execute_reply.started": "2024-01-10T20:16:08.564076Z" - }, - "tags": [] - }, + "execution_count": 2, + "id": "0e4d1725-66d2-49f1-9d3f-d363c3f019d1", + "metadata": {}, "outputs": [ { "data": { @@ -133,51 +74,63 @@ " vertical-align: top;\n", " }\n", "\n", - " .dataframe thead tr th {\n", - " text-align: left;\n", + " .dataframe thead th {\n", + " text-align: right;\n", " }\n", "\n", "\n", " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", + " \n", + " \n", " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", " \n", " \n", " \n", " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", " \n", " \n", " \n", @@ -203,30 +156,78 @@ " \n", " \n", " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", " \n", " \n", - " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", " \n", " \n", " \n", @@ -250,7 +251,7 @@ " \n", " \n", " \n", - " \n", + " \n", " \n", " \n", " \n", @@ -273,1200 +274,417 @@ " \n", " \n", " \n", - " \n", - "
package0...10
doc0...48
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mdComputational Thermodynamics of Materials\\n\\nZ...# Laws of thermodynamics\\n## First and second laws of thermodynamics\\n\\n...## Combined law of thermodynamics and equilibr...## Stability at equilibrium and property anoma...## Gibbs-Duhem equation\\n\\nIn experiments, it ...# Gibbs energy function\\n\\nAs shown in through...## Phases with fixed compositions\\n\\nThe homog...## Phases with variable compositions: Random s...### Random solutions\\n\\nThe ideal Gibbs energy...0000000000...### Galvanic protection\\n\\nA galvanic reaction...### Fuel cells\\n\\nFuel cells are devices to co...### Ion transport membranes\\n\\nIon transport m...### Electrical batteries\\n\\nBatteries utilize ...# Critical phenomena, thermal expansion, and M...## MMS model applied to thermal expansion\\n\\nA...## Application to cerium\\n\\nCerium (Ce) displa...## Application to Fe<sub>3</sub>Pt\\n\\nInvar wa...## Concept of Materials Genome®\\n\\n“A genome i...# References\\n\\n1\\. M. Hillert, *Phase Equilib...
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Hillert, *Phase Equilib... \n", - "\n", - "[2 rows x 97 columns]" - ] - }, - "execution_count": 4, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - " async def get_documents(p):\n", - " compact = p.groupby([\"package\", \"doc\", \"cell\"]).apply(\n", - " lambda df: pandas.Series([\n", - " df.sort_values(\"original\").p.pipe(get_new_docx)\n", - " ], pandas.Index([df.original.min()], name=\"original\"))\n", - " ).to_frame(\"docx\")\n", - " return compact.assign(md=await asyncio.gather(*map(get_docx_markdown, compact.docx)))\n", - "\n", - " def get_new_docx(elements: list[\"docx.Type\"]) -> \"docx.document.Document\":\n", - " \"creates a new `docx` data structure using only the specified elements.\"\n", - " try: return (new := docx.Document())\n", - " finally: new._body._element.extend(x._element for x in elements) \n", - " \n", - " async def get_docx_markdown(document) -> str:\n", - " \"\"\"save docx to disc and return a stringified markdown version. this is async because it shells out everytime\"\"\"\n", - " document.save((tmp := tempfile.NamedTemporaryFile(suffix=\".docx\")).name)\n", - " return subprocess.check_output([PANDOC, \"--to\", PANDOC_TO, \"--from\", PANDOC_FROM, tmp.name]).decode()\n", - " \n", - " INTERACTIVE and (documents := await get_documents(p)).T" - ] - }, - { - "cell_type": "markdown", - "id": "44b336b4-a228-47e4-ab5d-aac8f776f3e4", - "metadata": {}, - "source": [ - "## composing the target files for the documents\n", - "\n", - "from the information in our dataframes we need to compose the target locations for the collected documents." - ] - }, - { - "cell_type": "markdown", - "id": "211b1dd8-278c-4496-a1fa-90cf81e8c705", - "metadata": {}, - "source": [ - "`get_contents` reduces our data to just the h1 and h2 headings in the document.\n", - "we use this to compose our virtual file system layout." - ] - }, - { - "cell_type": "code", - "execution_count": 5, - "id": "58d15767-aa89-4bd3-b6c5-66acc848ee7a", - "metadata": { - "execution": { - "iopub.execute_input": "2024-01-10T20:16:22.446764Z", - "iopub.status.busy": "2024-01-10T20:16:22.446651Z", - "iopub.status.idle": "2024-01-10T20:16:22.470426Z", - "shell.execute_reply": "2024-01-10T20:16:22.470113Z", - "shell.execute_reply.started": "2024-01-10T20:16:22.446753Z" - }, - "tags": [] - }, - "outputs": [ - { - "data": { - "text/html": [ - "
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mdComputational Thermodynamics of Materials\\n\\nZ...# Laws of thermodynamics\\n## First and second laws of thermodynamics\\n\\n...## Combined law of thermodynamics and equilibr...## Stability at equilibrium and property anoma...## Gibbs-Duhem equation\\n\\nIn experiments, it ...# Gibbs energy function\\n\\nAs shown in through...## Phases with fixed compositions\\n\\nThe homog...## Phases with variable compositions: Random s...### Random solutions\\n\\nThe ideal Gibbs energy......### Galvanic protection\\n\\nA galvanic reaction...### Fuel cells\\n\\nFuel cells are devices to co...### Ion transport membranes\\n\\nIon transport m...### Electrical batteries\\n\\nBatteries utilize ...# Critical phenomena, thermal expansion, and M...## MMS model applied to thermal expansion\\n\\nA...## Application to cerium\\n\\nCerium (Ce) displa...## Application to Fe<sub>3</sub>Pt\\n\\nInvar wa...## Concept of Materials Genome®\\n\\n“A genome i...# References\\n\\n1\\. M. Hillert, *Phase Equilib...
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M. Hillert, *Phase Equilib... \n", + "\n", + "[2 rows x 97 columns]" + ] + }, + "execution_count": 4, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + " async def get_documents(p):\n", + " documents = p.groupby([\"package\", \"doc\", \"cell_id\"]).p.agg(get_new_docx).to_frame(\"docx\")\n", + " # shell out to pandoc to get the markdown form\n", + " documents = documents.assign(md=await documents.docx.apply(get_docx_markdown).gather())\n", + " return documents\n", + "\n", + " def get_new_docx(elements: list[\"docx.Type\"]) -> \"docx.document.Document\":\n", + " \"creates a new `docx` data structure using only the specified elements.\"\n", + " try: return (new := docx.Document())\n", + " finally: new._body._element.extend(x._element for x in elements) \n", + " \n", + "\n", + " async def get_docx_markdown(document) -> str:\n", + " \"\"\"save docx to disc and return a stringified markdown version. this is async because it shells out everytime\"\"\"\n", + " document.save((tmp := tempfile.NamedTemporaryFile(suffix=\".docx\")).name)\n", + " return subprocess.check_output([PANDOC, \"--to\", PANDOC_TO, \"--from\", PANDOC_FROM, tmp.name]).decode()\n", + "\n", + " INTERACTIVE and (documents := await get_documents(p)).T" + ] + }, + { + "cell_type": "markdown", + "id": "c6b2f8c8-3a02-4f2b-a6f8-4d943ed819f4", + "metadata": {}, + "source": [ + "`get_notebooks` synthesizes the blocks of markdown into notebook files" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "id": "7ea8fb81-5b2f-44de-b178-93e32f2f2774", + "metadata": {}, "outputs": [ { "data": { @@ -1658,27 +1180,27 @@ " \n", " \n", " target\n", - " /home/tbone/Documents/nobook/src/psu410/src/laws_of_thermodynamics/index.ipynb\n", - " /home/tbone/Documents/nobook/src/psu410/src/laws_of_thermodynamics/first_and_second_laws_of_thermodynamics.ipynb\n", - " /home/tbone/Documents/nobook/src/psu410/src/laws_of_thermodynamics/combined_law_of_thermodynamics_and_equilibrium_conditions.ipynb\n", - " /home/tbone/Documents/nobook/src/psu410/src/laws_of_thermodynamics/stability_at_equilibrium_and_property_anomaly.ipynb\n", - " /home/tbone/Documents/nobook/src/psu410/src/laws_of_thermodynamics/gibbsduhem_equation.ipynb\n", - " /home/tbone/Documents/nobook/src/psu410/src/gibbs_energy_function/index.ipynb\n", - " /home/tbone/Documents/nobook/src/psu410/src/gibbs_energy_function/phases_with_fixed_compositions.ipynb\n", - " /home/tbone/Documents/nobook/src/psu410/src/gibbs_energy_function/phases_with_variable_compositions_random_solutions.ipynb\n", - " /home/tbone/Documents/nobook/src/psu410/src/gibbs_energy_function/phases_with_variable_compositions_solutions_with_ordering.ipynb\n", - " /home/tbone/Documents/nobook/src/psu410/src/gibbs_energy_function/polymer_solutions_and_polymer_blends.ipynb\n", + " /home/tbone/Documents/nobook/src/psu410/src/psu410/applications_to_chemical_reactions/ellingham_diagram_and_buffered_systems.ipynb\n", + " /home/tbone/Documents/nobook/src/psu410/src/psu410/applications_to_chemical_reactions/index.ipynb\n", + " /home/tbone/Documents/nobook/src/psu410/src/psu410/applications_to_chemical_reactions/internal_process_and_differential_and_integrated_driving_forces.ipynb\n", + " /home/tbone/Documents/nobook/src/psu410/src/psu410/applications_to_chemical_reactions/maximum_reaction_rate_and_chemical_transport_reactions_.ipynb\n", + " /home/tbone/Documents/nobook/src/psu410/src/psu410/applications_to_chemical_reactions/trends_of_entropies_of_reactions.ipynb\n", + " /home/tbone/Documents/nobook/src/psu410/src/psu410/applications_to_electrochemical_systems/application_examples.ipynb\n", + " /home/tbone/Documents/nobook/src/psu410/src/psu410/applications_to_electrochemical_systems/aqueous_solution_and_pourbaix_diagram.ipynb\n", + " /home/tbone/Documents/nobook/src/psu410/src/psu410/applications_to_electrochemical_systems/concentrations_activities_and_reference_states_of_electrolyte_species.ipynb\n", + " /home/tbone/Documents/nobook/src/psu410/src/psu410/applications_to_electrochemical_systems/electrochemical_cells_and_half_cell_potentials.ipynb\n", + " /home/tbone/Documents/nobook/src/psu410/src/psu410/applications_to_electrochemical_systems/electrolyte_reactions_and_electrochemical_reactions.ipynb\n", " ...\n", - " /home/tbone/Documents/nobook/src/psu410/src/applications_to_electrochemical_systems/concentrations_activities_and_reference_states_of_electrolyte_species.ipynb\n", - " /home/tbone/Documents/nobook/src/psu410/src/applications_to_electrochemical_systems/electrochemical_cells_and_half_cell_potentials.ipynb\n", - " /home/tbone/Documents/nobook/src/psu410/src/applications_to_electrochemical_systems/aqueous_solution_and_pourbaix_diagram.ipynb\n", - " /home/tbone/Documents/nobook/src/psu410/src/applications_to_electrochemical_systems/application_examples.ipynb\n", - " /home/tbone/Documents/nobook/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/index.ipynb\n", - " /home/tbone/Documents/nobook/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/mms_model_applied_to_thermal_expansion.ipynb\n", - " /home/tbone/Documents/nobook/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_cerium.ipynb\n", - " /home/tbone/Documents/nobook/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_fept.ipynb\n", - " /home/tbone/Documents/nobook/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/concept_of_materials_genome.ipynb\n", - " /home/tbone/Documents/nobook/src/psu410/src/references/index.ipynb\n", + " /home/tbone/Documents/nobook/src/psu410/src/psu410/laws_of_thermodynamics/first_and_second_laws_of_thermodynamics.ipynb\n", + " /home/tbone/Documents/nobook/src/psu410/src/psu410/laws_of_thermodynamics/gibbsduhem_equation.ipynb\n", + " /home/tbone/Documents/nobook/src/psu410/src/psu410/laws_of_thermodynamics/index.ipynb\n", + " /home/tbone/Documents/nobook/src/psu410/src/psu410/laws_of_thermodynamics/stability_at_equilibrium_and_property_anomaly.ipynb\n", + " /home/tbone/Documents/nobook/src/psu410/src/psu410/phase_equilibria_in_heterogeneous_systems/general_condition_of_equilibrium.ipynb\n", + " /home/tbone/Documents/nobook/src/psu410/src/psu410/phase_equilibria_in_heterogeneous_systems/gibbs_phase_rule.ipynb\n", + " /home/tbone/Documents/nobook/src/psu410/src/psu410/phase_equilibria_in_heterogeneous_systems/index.ipynb\n", + " /home/tbone/Documents/nobook/src/psu410/src/psu410/phase_equilibria_in_heterogeneous_systems/molar_phase_diagrams.ipynb\n", + " /home/tbone/Documents/nobook/src/psu410/src/psu410/phase_equilibria_in_heterogeneous_systems/potential_phase_diagrams.ipynb\n", + " /home/tbone/Documents/nobook/src/psu410/src/psu410/references/index.ipynb\n", " \n", " \n", " \n", @@ -1712,83 +1234,87 @@ "" ], "text/plain": [ - "target /home/tbone/Documents/nobook/src/psu410/src/laws_of_thermodynamics/index.ipynb \\\n", - "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", + "target /home/tbone/Documents/nobook/src/psu410/src/psu410/applications_to_chemical_reactions/ellingham_diagram_and_buffered_systems.ipynb \\\n", + "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", "\n", - "target /home/tbone/Documents/nobook/src/psu410/src/laws_of_thermodynamics/first_and_second_laws_of_thermodynamics.ipynb \\\n", - "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", + "target /home/tbone/Documents/nobook/src/psu410/src/psu410/applications_to_chemical_reactions/index.ipynb \\\n", + "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", "\n", - "target /home/tbone/Documents/nobook/src/psu410/src/laws_of_thermodynamics/combined_law_of_thermodynamics_and_equilibrium_conditions.ipynb \\\n", - "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", + "target /home/tbone/Documents/nobook/src/psu410/src/psu410/applications_to_chemical_reactions/internal_process_and_differential_and_integrated_driving_forces.ipynb \\\n", + "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", "\n", - "target /home/tbone/Documents/nobook/src/psu410/src/laws_of_thermodynamics/stability_at_equilibrium_and_property_anomaly.ipynb \\\n", - "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", + "target /home/tbone/Documents/nobook/src/psu410/src/psu410/applications_to_chemical_reactions/maximum_reaction_rate_and_chemical_transport_reactions_.ipynb \\\n", + "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", "\n", - "target /home/tbone/Documents/nobook/src/psu410/src/laws_of_thermodynamics/gibbsduhem_equation.ipynb \\\n", - "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", + "target /home/tbone/Documents/nobook/src/psu410/src/psu410/applications_to_chemical_reactions/trends_of_entropies_of_reactions.ipynb \\\n", + "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", "\n", - "target /home/tbone/Documents/nobook/src/psu410/src/gibbs_energy_function/index.ipynb \\\n", - "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", + "target /home/tbone/Documents/nobook/src/psu410/src/psu410/applications_to_electrochemical_systems/application_examples.ipynb \\\n", + "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", "\n", - "target /home/tbone/Documents/nobook/src/psu410/src/gibbs_energy_function/phases_with_fixed_compositions.ipynb \\\n", - "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", + "target /home/tbone/Documents/nobook/src/psu410/src/psu410/applications_to_electrochemical_systems/aqueous_solution_and_pourbaix_diagram.ipynb \\\n", + "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", "\n", - "target /home/tbone/Documents/nobook/src/psu410/src/gibbs_energy_function/phases_with_variable_compositions_random_solutions.ipynb \\\n", - "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", + "target /home/tbone/Documents/nobook/src/psu410/src/psu410/applications_to_electrochemical_systems/concentrations_activities_and_reference_states_of_electrolyte_species.ipynb \\\n", + "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", "\n", - "target /home/tbone/Documents/nobook/src/psu410/src/gibbs_energy_function/phases_with_variable_compositions_solutions_with_ordering.ipynb \\\n", - "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", + "target /home/tbone/Documents/nobook/src/psu410/src/psu410/applications_to_electrochemical_systems/electrochemical_cells_and_half_cell_potentials.ipynb \\\n", + "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", "\n", - "target /home/tbone/Documents/nobook/src/psu410/src/gibbs_energy_function/polymer_solutions_and_polymer_blends.ipynb \\\n", - "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", + "target /home/tbone/Documents/nobook/src/psu410/src/psu410/applications_to_electrochemical_systems/electrolyte_reactions_and_electrochemical_reactions.ipynb \\\n", + "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", "\n", "target ... \\\n", "nb ... \n", "\n", - "target /home/tbone/Documents/nobook/src/psu410/src/applications_to_electrochemical_systems/concentrations_activities_and_reference_states_of_electrolyte_species.ipynb \\\n", - "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", + "target /home/tbone/Documents/nobook/src/psu410/src/psu410/laws_of_thermodynamics/first_and_second_laws_of_thermodynamics.ipynb \\\n", + "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", "\n", - "target /home/tbone/Documents/nobook/src/psu410/src/applications_to_electrochemical_systems/electrochemical_cells_and_half_cell_potentials.ipynb \\\n", - "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", + "target /home/tbone/Documents/nobook/src/psu410/src/psu410/laws_of_thermodynamics/gibbsduhem_equation.ipynb \\\n", + "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", "\n", - "target /home/tbone/Documents/nobook/src/psu410/src/applications_to_electrochemical_systems/aqueous_solution_and_pourbaix_diagram.ipynb \\\n", - "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", + "target /home/tbone/Documents/nobook/src/psu410/src/psu410/laws_of_thermodynamics/index.ipynb \\\n", + "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", "\n", - "target /home/tbone/Documents/nobook/src/psu410/src/applications_to_electrochemical_systems/application_examples.ipynb \\\n", - "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", + "target /home/tbone/Documents/nobook/src/psu410/src/psu410/laws_of_thermodynamics/stability_at_equilibrium_and_property_anomaly.ipynb \\\n", + "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", "\n", - "target /home/tbone/Documents/nobook/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/index.ipynb \\\n", - "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", + "target /home/tbone/Documents/nobook/src/psu410/src/psu410/phase_equilibria_in_heterogeneous_systems/general_condition_of_equilibrium.ipynb \\\n", + "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", "\n", - "target /home/tbone/Documents/nobook/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/mms_model_applied_to_thermal_expansion.ipynb \\\n", - "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", + "target /home/tbone/Documents/nobook/src/psu410/src/psu410/phase_equilibria_in_heterogeneous_systems/gibbs_phase_rule.ipynb \\\n", + "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", "\n", - "target /home/tbone/Documents/nobook/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_cerium.ipynb \\\n", - "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", + "target /home/tbone/Documents/nobook/src/psu410/src/psu410/phase_equilibria_in_heterogeneous_systems/index.ipynb \\\n", + "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", "\n", - "target /home/tbone/Documents/nobook/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_fept.ipynb \\\n", - "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", + "target /home/tbone/Documents/nobook/src/psu410/src/psu410/phase_equilibria_in_heterogeneous_systems/molar_phase_diagrams.ipynb \\\n", + "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", "\n", - "target /home/tbone/Documents/nobook/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/concept_of_materials_genome.ipynb \\\n", - "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", + "target /home/tbone/Documents/nobook/src/psu410/src/psu410/phase_equilibria_in_heterogeneous_systems/potential_phase_diagrams.ipynb \\\n", + "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", "\n", - "target /home/tbone/Documents/nobook/src/psu410/src/references/index.ipynb \n", - "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", + "target /home/tbone/Documents/nobook/src/psu410/src/psu410/references/index.ipynb \n", + "nb {'nbformat': 4, 'nbformat_minor': 5, 'metadata... \n", "\n", "[1 rows x 48 columns]" ] }, - "execution_count": 7, + "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ - " def get_notebooks(documents, contents):\n", - " notebooks = documents.groupby([\"package\", \"doc\"]).apply(lambda df: v4.new_notebook(cells=[v4.new_markdown_cell(x.splitlines(True)) for x in df.md]))\n", - " return contents.drop_duplicates(subset=\"target\").join(notebooks.rename(\"nb\"))[[\"target\", \"nb\"]].set_index(\"target\")\n", + " def get_notebooks(documents, headings):\n", + " notebooks = documents.join(headings.target).groupby([\"target\"]).apply(get_new_notebook).to_frame(\"nb\")\n", + " return notebooks\n", + " \n", + " def get_new_notebook(df):\n", + " \"\"\"create a new notebook format from markdown paragraphs\"\"\"\n", + " return v4.new_notebook(cells=[v4.new_markdown_cell(x.splitlines(True)) for x in df.sort_index().md])\n", "\n", - " INTERACTIVE and (notebooks := get_notebooks(documents, contents)).T" + " INTERACTIVE and (notebooks := get_notebooks(documents, headings)).T" ] }, { @@ -1801,16 +1327,9 @@ }, { "cell_type": "code", - "execution_count": 8, + "execution_count": 6, "id": "86f04370-5233-4f0f-b920-06f5af9b92cc", "metadata": { - "execution": { - "iopub.execute_input": "2024-01-10T20:16:22.580339Z", - "iopub.status.busy": "2024-01-10T20:16:22.580231Z", - "iopub.status.idle": "2024-01-10T20:16:37.433327Z", - "shell.execute_reply": "2024-01-10T20:16:37.432938Z", - "shell.execute_reply.started": "2024-01-10T20:16:22.580328Z" - }, "tags": [] }, "outputs": [ @@ -1822,7 +1341,7 @@ "the table of contents, 9 primary chapters, and the references are captured ... ok\n", "\n", "----------------------------------------------------------------------\n", - "Ran 1 test in 14.847s\n", + "Ran 1 test in 14.030s\n", "\n", "OK\n" ] @@ -1830,10 +1349,10 @@ { "data": { "text/plain": [ - "" + "" ] }, - "execution_count": 8, + "execution_count": 6, "metadata": {}, "output_type": "execute_result" } @@ -1848,18 +1367,19 @@ " except RuntimeError: return __import__(\"nest_asyncio\").apply() or cls.arun(co)\n", " \n", " def setUp(self):\n", - " self.paragraphs = get_paragraphs()\n", - " self.contents = get_contents(self.paragraphs)\n", + " self.paragraphs = get_docx_features()\n", + " self.headings = get_headings(self.paragraphs)\n", " self.documents = self.arun(get_documents(self.paragraphs))\n", + " self.notebooks = get_notebooks(self.documents, self.headings)\n", " INTERACTIVE and globals().update(self=self)\n", " \n", " def test_number_of_sections(self):\n", " \"\"\"the table of contents, 9 primary chapters, and the references are captured\"\"\"\n", " # all of the main headings are captured as Heading1s\n", - " assert self.paragraphs.style_id.eq(\"Heading1\").sum() == 10\n", + " assert self.paragraphs.styleId.eq(\"Heading1\").sum() == 10\n", " assert len(self.documents.index.get_level_values(\"package\").unique()) == 11\n", " \n", - " INTERACTIVE and unittest.main(argv=[\"discover\"], verbosity=2, exit=False)" + " INTERACTIVE and (test_results := unittest.main(argv=[\"discover\"], verbosity=2, exit=False))" ] }, { @@ -1872,73 +1392,19 @@ }, { "cell_type": "code", - "execution_count": 9, + "execution_count": 7, "id": "bda27d6b-3be8-47d6-bb37-1c19c7e0153d", "metadata": { - "execution": { - "iopub.execute_input": "2024-01-10T20:16:37.434099Z", - "iopub.status.busy": "2024-01-10T20:16:37.433930Z", - "iopub.status.idle": "2024-01-10T20:16:37.462315Z", - "shell.execute_reply": "2024-01-10T20:16:37.462022Z", - "shell.execute_reply.started": "2024-01-10T20:16:37.434084Z" - }, "tags": [] }, "outputs": [ { "data": { "text/plain": [ - "[230,\n", - " 7327,\n", - " 12929,\n", - " 12381,\n", - " 4717,\n", - " 11916,\n", - " 18271,\n", - " 34412,\n", - " 26361,\n", - " 4538,\n", - " 8221,\n", - " 249,\n", - " 7680,\n", - " 3372,\n", - " 31853,\n", - " 40796,\n", - " 1668,\n", - " 15971,\n", - " 8319,\n", - " 3663,\n", - " 19250,\n", - " 12878,\n", - " 14173,\n", - " 18091,\n", - " 14994,\n", - " 3254,\n", - " 7739,\n", - " 3906,\n", - " 2040,\n", - " 3562,\n", - " 11518,\n", - " 1193,\n", - " 6567,\n", - " 10326,\n", - " 3959,\n", - " 10446,\n", - " 1649,\n", - " 5052,\n", - " 4257,\n", - " 16935,\n", - " 15674,\n", - " 22812,\n", - " 1465,\n", - " 6998,\n", - " 14044,\n", - " 8518,\n", - " 4168,\n", - " 14905]" + "'[10326, 1193, 6567, 10446, 3959, 22812, 15674, 4257, 16935, 5052, 1649, 7739, 3254, 3906, 3562, 11518, 2040, 14044, 8518, 4168, 1465, 6998, 1668, 15971, 8319, 14994, 12878, 3663, 18091, 19250, 14173, 8221, 11916, 18271, 34412, 26361, 4538, 12929, 7327, 4717, 230, 12381, 7680, 3372, 249, 40796, 31853, 14905]'" ] }, - "execution_count": 9, + "execution_count": 7, "metadata": {}, "output_type": "execute_result" } @@ -1952,10 +1418,9 @@ " async def write_nb(notebooks):\n", " return await asyncio.gather(*(\n", " write_json(target, nb) for target, nb in notebooks.nb.items()\n", - " )) \n", - "\n", + " ))\n", "\n", - " 10 and INTERACTIVE and await write_nb(notebooks)" + " 10 and INTERACTIVE and str(await write_nb(notebooks))" ] }, { @@ -1968,18 +1433,9 @@ }, { "cell_type": "code", - "execution_count": 10, - "id": "ae617c09-6440-48a7-a015-2c60896fb1a1", - "metadata": { - "execution": { - "iopub.execute_input": "2024-01-10T20:16:37.462885Z", - "iopub.status.busy": "2024-01-10T20:16:37.462699Z", - "iopub.status.idle": "2024-01-10T20:16:37.471213Z", - "shell.execute_reply": "2024-01-10T20:16:37.470863Z", - "shell.execute_reply.started": "2024-01-10T20:16:37.462876Z" - }, - "tags": [] - }, + "execution_count": 8, + "id": "41c2db45-1d82-42a7-9773-8c9dece52bd8", + "metadata": {}, "outputs": [ { "data": { @@ -3576,8 +3032,7 @@ } ], "source": [ - " if INTERACTIVE:\n", - " display(*documents.md[documents.md.str.contains(\"\\$\")].iloc[:10].apply(Markdown))" + " INTERACTIVE and display(*documents.md[documents.md.str.contains(\"\\$\")].iloc[:10].apply(Markdown))" ] }, { diff --git a/src/nobook/mime.types b/src/nobook/mime.types new file mode 100644 index 0000000..76ac134 --- /dev/null +++ b/src/nobook/mime.types @@ -0,0 +1,6 @@ +application/x-ipynb+json ipynb +text/markdown md markdown +text/toml toml +application/x-yaml yaml yml +text/ini ini cfg +application/vnd.openxmlformats-officedocument.wordprocessingml.document docx \ No newline at end of file diff --git a/src/nobook/utils.py b/src/nobook/utils.py new file mode 100644 index 0000000..57429c7 --- /dev/null +++ b/src/nobook/utils.py @@ -0,0 +1,582 @@ +"""reusable pandas and jinja components for dacs""" + +import asyncio +from configparser import ConfigParser +import enum +from functools import lru_cache, partial, partialmethod, singledispatch +from mimetypes import MimeTypes +from operator import attrgetter +from re import S +from textwrap import dedent, indent +from typing import AsyncGenerator, AsyncIterable, AsyncIterator, Coroutine +from numpy import isin, nan +from pandas import DataFrame, Index, MultiIndex, Series +import pandas +import anyio +import jinja2.ext +from pathlib import Path +from enum import auto, IntFlag + +import slugify + +HERE = Path(__file__).parent +TEMPLATES = HERE / "templates" + +MIME = MimeTypes((HERE / "mime.types",)) + + +class Kind(IntFlag): + INDEX, SERIES, DATAFRAME = auto(), auto(), auto() + + +INDEX, SERIES, FRAME = Kind.INDEX, Kind.SERIES, Kind.DATAFRAME + + +@singledispatch +def apply(x, f, *args, **kwargs): + """a generalized, async-aware apply method for pandas collections""" + return apply_index(Index(x), f, *args, **kwargs) + + +@apply.register(Index) +def apply_index(x, f, *args, **kwargs): + return apply_series(x.to_series(), f, *args, **kwargs) + + +@apply.register(Series) +def apply_series(x, f, *args, _name=None, **kwargs): + if _name is None: + _name = getattr(f, "__name__", None) + return x.apply(f, args=args, **kwargs).rename(_name).pipe(_sync_or_async) + + +@apply.register(DataFrame) +def apply_frame(x, f, *args, **kwargs): + return x.apply(f, args=args, axis=1, **kwargs).pipe(_sync_or_async) + + +@apply.register(pandas.core.groupby.DataFrameGroupBy) +def apply_group(x, f, *args, **kwargs): + return Series(dict((k, f(y, *args, **kwargs)) for k, y in x)).pipe(_sync_or_async) + + +async def _asyncgen(x): + return [x async for x in x] + + +async def _update(s): + from asyncio import gather + + if not len(s): + return () + if isinstance(s[0], (AsyncGenerator, AsyncIterable, AsyncIterator)): + s = s.apply(_asyncgen) + if isinstance(s[0], Coroutine): + y = await gather(*s) + if isinstance(s, Series): + s.update(Series(y, s.index)) + elif isinstance(s, DataFrame): + s = s.combine_first(pandas.DataFrame(y, s.index)) + return s + + +def _sync_or_async(s): + return _update(s) if _isasync(s) else s + + +def _isasync(x): + if len(x): + return isinstance( + x.iloc[0], (AsyncGenerator, AsyncIterable, AsyncIterator, Coroutine) + ) + return False + + +class Accessor: + """a base class for accessor extensions""" + + def __init__(self, object): + self.object = object + + def __init_subclass__(cls, method=None, types=INDEX | SERIES | FRAME, name=None): + cls.method = method + + for t in (INDEX, SERIES, FRAME): + if t in types: + getattr(pandas.api.extensions, f"register_{t.name.lower()}_accessor")( + name or cls.__name__.lower() + )(cls) + + def apply(self, f, *args, **kwargs): + return apply(self.object, f, *args, **kwargs) + + ACC_METHODS = {Index: "index", Series: "series", DataFrame: "dataframe"} + + +class Method(Accessor): + """the method accessor base class places ALL the public methods and of the `method` provided.""" + + def __init_subclass__(cls, method=None, types=INDEX | SERIES | FRAME, name=None): + super().__init_subclass__(method=method, types=types, name=name) + cls._register_methods(method) + + @classmethod + def _register_methods(cls, method, properties=True, methods=True): + for k in dir(method): + if hasattr(cls, k): + continue # break when the attribute exists + v = getattr(method, k) + if methods and callable(v) and not isinstance(v, classmethod): + cls._register_method(k, v) + elif properties and isinstance(v, property): + cls._register_property(k, v) + + @classmethod + def _register_method(cls, k, v): + setattr(cls, k, partialmethod(cls.apply, v)) + + @classmethod + def _register_property(cls, k, v): + setattr(cls, k, property(lambda x: x.apply(attrgetter(k)))) + + +class _attrgetter(Accessor, types=INDEX | SERIES, name="attrgetter"): + def __call__(self, *args, **kwargs): + return apply(self.object, attrgetter(*args, **kwargs)) + + +class _methodcaller(Accessor, types=INDEX | SERIES, name="methodcaller"): + def __call__(self, *args, **kwargs): + from operator import methodcaller + + return apply(self.object, methodcaller(*args, **kwargs)) + + +class _itemgetter(Accessor, types=INDEX | SERIES, name="itemgetter"): + def __call__(self, *args, **kwargs): + from operator import itemgetter + + return apply(self.object, itemgetter(*args, **kwargs)) + + +class explode_index(Accessor, types=SERIES, name="explode_index"): + def __call__(self, *args, **kwargs): + if isinstance(self.object, pandas.Series): + return pandas.concat({k: pandas.Series(v) for k, v in self.object.items()}) + elif isin(self.object, pandas.DataFrame): + return + raise NotImplemented() + + +class _aseries(Accessor, types=SERIES, name="aseries"): + def __call__(self, *args, **kwargs): + return self.object.apply(pandas.Series, *args, **kwargs) + + +class _gather(Accessor, types=SERIES, name="gather"): + async def __call__(self, *args, **kwargs): + return pandas.Series( + await asyncio.gather(*self.object), self.object.index, *args, **kwargs + ) + + +async def run(command, **kwargs): + import anyio + + result = await anyio.run_process(command, **kwargs) + return pandas.Series([result.stdout, result.stderr], ["stdout", "stderr"]) + + +class Shell(Accessor, types=INDEX | SERIES, name="sh"): + async def run(self, **kwargs): + return (await self.apply(run, **kwargs)).apply(pandas.Series) + + +class Bytes(Method, types=INDEX | SERIES, method=bytes): + pass + + +def get_mimetype(object, mime=MIME): + """infer the mime type of our object""" + if isinstance(object, str): + if object.startswith("."): + object = "x" + object + return mime.guess_type(object)[0] + + +@lru_cache +def get_markdown(): + import midgy.tangle + + md = midgy.tangle.Tangle().parser + md.options.update(highlight=highlight) + return md + + +def markdown(md, **kwargs): + return get_markdown().render(md, **kwargs) + + +def markdown_parse(md, **kwargs): + return get_markdown().render(md, **kwargs) + + +def highlight(code, lang="python", attrs=None): + import pygments, html + + try: + return pygments.highlight( + code, + pygments.lexers.get_lexer_by_name(lang or "python"), + pygments.formatters.get_formatter_by_name( + "html", debug_token_types=True, title=f"{lang} code" + ), + ) + except: + return f"""
{html.escape(code)}
""" + + +HERE = Path(__file__).parent +TEMPLATES = HERE / "templates" + + +@lru_cache +def get_environment(): + import jinja2, builtins + from html import escape + from slugify import slugify + + env = jinja2.Environment( + enable_async=True, + loader=jinja2.ChoiceLoader( + [jinja2.FileSystemLoader(TEMPLATES), jinja2.DictLoader({})] + ), + extensions=["jinja2.ext.loopcontrols", "jinja2.ext.with_", "jinja2.ext.do"], + ) + env.globals.update(vars(builtins), markdown=get_markdown().render) + env.filters.update( + markdown=get_markdown().render, + highlight=highlight, + escape=escape, + slug=slugify, + dedent=dedent, + indent=indent, + ) + return env + + +class _Jinja2(Accessor, types=SERIES | FRAME | INDEX, name="template"): + template_name = None # need a default template + + def __init__(self, object): + super().__init__(object) + self.environment = get_environment() + + def render_one(self, row, template=None, **kwargs): + if not isinstance(template, jinja2.Template): + template = self.environment.get_template( + template or row.get("template_name") + ) + data = row.to_dict() + data.update(kwargs) + + data.update(row=row) + if self.environment.is_async: + return template.render_async(row=row, **{**kwargs, **row.to_dict()}) + return template.render(row=row, **{**kwargs, **row.to_dict()}) + + def render_template(self, template=None, **kwargs): + object = self.object + if isinstance(object, MultiIndex): + object = object.to_frame() + elif isinstance(object, Index): + object = object.to_series() + object = object.replace({nan: None}) + if isinstance(object, Series): + object = object.to_frame() + else: + try: + object = object.reset_index().set_index(object.index) + except ValueError: + pass # dont reset on duplicate names + return apply_frame(object, self.render_one, template=template, **kwargs) + + def render_string(self, template, **kwargs): + return self.render_template(self.environment.from_string(template), **kwargs) + + +async def get__file_index( + path=None, include="", exclude=None, recursive=False +) -> pandas.DataFrame: + if not isinstance(recursive, bool): + recursive -= 1 + return [ + path + async for path in get__file_index_iter( + anyio.Path(path), include, exclude, recursive + ) + ] + + +async def get__file_index_iter( + path=None, include=[".ipynb", ".py", ".md"], exclude=None, recursive=False +) -> AsyncGenerator[anyio.Path, None]: + if isinstance(exclude, str): + import pathspec + + exclude = pathspec.PathSpec.from_lines( + pathspec.patterns.GitWildMatchPattern, exclude.splitlines() + ) + if await path.is_dir(): + async for path in path.iterdir(): + if await path.is_file(): + if path.suffix in include: + if (not exclude) or not exclude.match_file(path): + yield path + elif recursive: + async for path in get__file_index_iter( + path, include, exclude, recursive + ): + yield path + + +def get_mimetype(object): + if isinstance(object, str): + if object.startswith("."): + object = "xxxxx" + object + return MIME.guess_type(object)[0] + + +def loads(x, mime=None): + try: + f = LOADERS[mime] + except KeyError: + return x + return f(x) + + +LOADERS = dict() + + +@partial(setattr, loads, "register") +def loads_register(mime, callable=None): + if callable is None: + return partial(loads.register, mime) + LOADERS[mime] = callable + return callable + + +@loads.register("text/toml") +def loads_toml(x): + return __import__("tomli").loads(x.read_text()) + + +@loads.register("application/x-ipynb+json") +@loads.register("application/json") +def loads_json(x): + data = __import__("json").loads(x.read_text()) + for i, x in enumerate(data["cells"]): + x.update(source="".join(x["source"]), count=i) + return data + + +@loads.register("application/x-yaml") +def loads_yaml(x): + try: + return __import__("ruamel.yaml").yaml.safe_load(x.read_text()) + except ModuleNotFoundError: + return __import__("yaml").safe_load(x.read_text()) + + +@loads.register("text/ini") +def loads_ini(x): + parser = ConfigParser() + parser.read_string(x.read_text()) + return parser._sections + + +@loads.register( + "application/vnd.openxmlformats-officedocument.wordprocessingml.document" +) +def loads_docx(x): + import docx + + return docx.Document(x) + + +def get_cell(x, cell_type="markdown", **kwargs): + return dict(cell_type=cell_type, source=x) + + +@singledispatch +def get_notebook(x, **kwargs): + return get_cell(x, kwargs.pop("cell_type", "raw"), **kwargs) + + +@get_notebook.register +def get_notebook_dict(x: dict, **kwargs): + if "cells" in x and "metadata" in x: + return x + return dict(metadata=x) + + +@get_notebook.register +def get_notebook_str(x: str, cell_type="markdown", **kwargs): + return dict(cells=[get_cell(x, cell_type, **kwargs)]) + + +def get_parent_glob(path, pattern): + path = path.absolute() + while path.parent is not path: + x = list(path.glob(pattern)) + if x: + return x + path = path.parent + + +class Path(type(Path())): + def load(self): + return loads(self.read_text(), Path.mime(self)) + + def notebook(self, **kwargs): + return get_notebook(Path.load(self), **kwargs) + + def mime(self): + return get_mimetype(self) + + iglob = get_parent_glob + + +async def aget_parent_glob(path, pattern): + path = await path.absolute() + paths = list() + while path.parent is not path: + async for x in path.glob(pattern): + paths.append(x) + if paths: + return paths + path = path.parent + return [] + + +class APath(anyio.Path): + async def load(self): + return loads(await self.read_text(), APath.mime(self)) + + async def notebook(self, **kwargs): + return get_notebook(await APath.load(self), **kwargs) + + def mime(self): + return get_mimetype(self) + + iglob = aget_parent_glob + + +class _Path(Method, method=Path, types=INDEX | SERIES, name="path"): + def __call__(self, *args, **kwargs): + if isinstance(self.object, Index): + return self.object.map(self.method) + return self.object.apply(self.method) + + def find(self, *args, **kwargs): + x = self.apply(get__file_index, *args, **kwargs) + return x.explode().pipe(pandas.Index, name="path") + + def glob(self, *patterns, method="glob"): + return pandas.concat( + self.apply(getattr(self.method, method), x).apply(list) for x in patterns + ).explode() + + rglob = partialmethod(glob, method="rglob") + iglob = partialmethod(glob, method="iglob") + + async def notebook(self, **kwargs): + return self.apply(self.method.notebook).apply(pandas.Series) + + +class _APath(Method, method=APath, types=INDEX | SERIES, name="apath"): + def __call__(self, *args, **kwargs): + if isinstance(self.object, Index): + return self.object.map(self.method) + return self.object.apply(self.method) + + async def find(self, *args, **kwargs): + x = await self.apply(get__file_index, *args, **kwargs) + return x.explode().pipe(pandas.Index, name="path") + + async def glob(self, *patterns, method="glob"): + return pandas.concat( + await asyncio.gather( + *(self.apply(getattr(self.method, method), x) for x in patterns) + ) + ).explode() + + rglob = partialmethod(glob, method="rglob") + iglob = partialmethod(glob, method="iglob") + + async def notebook(self, **kwargs): + return (await self.apply(self.method.notebook)).apply(pandas.Series) + + +@singledispatch +def get_soup(x, *args, **kwargs): + return x + + +@get_soup.register(str) +def get_soup_str(x, *args, **kwargs): + import bs4 + + kwargs.setdefault("features", "html.parser") + return bs4.BeautifulSoup(x, *args, **kwargs) + + +class _html(Accessor, types=SERIES | INDEX, name="html"): + import bs4 + from html import escape + + escape = partialmethod(Method.apply, escape) + soup = partialmethod(Method.apply, get_soup) + select_one = partialmethod(Method.apply, bs4.BeautifulSoup.select_one) + + def select(self, *args, **kwargs): + import bs4 + + return ( + self.object.html.soup().method.apply( + bs4.BeautifulSoup.select, *args, **kwargs + ) + ).explode() + + def select_one(self, *args, **kwargs): + import bs4 + + return self.object.html.soup().method.apply( + bs4.BeautifulSoup.select_one, *args, **kwargs + ) + + +class _git(Accessor, types=SERIES | INDEX, name="git"): + async def authors(self): + return ( + ( + await ( + await self.object.rename("path").template.render_string( + "cd {{path.parent}} && git log --format='%an<%ae>' -- {{path.name}} | sort | uniq" + ) + ).sh.run() + ) + .stdout.bytes.decode() + .apply(str.splitlines) + .pipe(lambda x: x[x.apply(bool)]) + .explode() + .str.rstrip(">") + .str.rpartition("<", expand=True)[[0, 2]] + .rename(columns={0: "name", 2: "email"}) + .groupby(["path", "email"]) + .name.agg(list) + .reset_index("email") + .groupby("path") + .apply(lambda x: x.to_dict(orient="records")) + .rename("authors") + ) From f7492b8f17c413ac15eb575c391127f9dba83453 Mon Sep 17 00:00:00 2001 From: tonyfast Date: Fri, 12 Jan 2024 10:53:05 -0800 Subject: [PATCH 9/9] use src layout for chapter --- .../ellingham_diagram_and_buffered_systems.ipynb | 2 +- .../applications_to_chemical_reactions/index.ipynb | 2 +- ..._differential_and_integrated_driving_forces.ipynb | 2 +- ...tion_rate_and_chemical_transport_reactions_.ipynb | 2 +- .../trends_of_entropies_of_reactions.ipynb | 2 +- .../application_examples.ipynb | 12 ++++++------ .../aqueous_solution_and_pourbaix_diagram.ipynb | 2 +- ...and_reference_states_of_electrolyte_species.ipynb | 2 +- ...ctrochemical_cells_and_half_cell_potentials.ipynb | 6 +++--- ...yte_reactions_and_electrochemical_reactions.ipynb | 2 +- .../index.ipynb | 2 +- .../importance_of_lattice_stability.ipynb | 2 +- .../calpahd_modeling_of_thermodynamics/index.ipynb | 2 +- .../modeling_of_pure_elements.ipynb | 2 +- .../modeling_of_random_solution_phases.ipynb | 2 +- ..._of_solution_phases_with_longrange_ordering.ipynb | 2 +- .../modeling_of_stoichiometric_phases.ipynb | 2 +- .../application_to_cerium.ipynb | 2 +- .../application_to_fept.ipynb | 2 +- .../concept_of_materials_genome.ipynb | 2 +- .../index.ipynb | 2 +- .../mms_model_applied_to_thermal_expansion.ipynb | 2 +- .../index.ipynb | 2 +- .../phase_equilibrium_data_from_experiments.ipynb | 8 ++++---- .../thermodynamic_data_from_experiments.ipynb | 8 ++++---- ...tprinciples_approaches_to_disordered_alloys.ipynb | 8 ++++---- ...rstprinciples_formulation_of_thermodynamics.ipynb | 12 ++++++------ .../index.ipynb | 2 +- .../lattice_dynamics.ipynb | 10 +++++----- .../nickel_as_the_prototype.ipynb | 8 ++++---- .../quantum_theory_for_the_motion_of_electrons.ipynb | 12 ++++++------ ...c_and_electric_contributions_to_free_energy.ipynb | 2 +- .../{ => psu410}/gibbs_energy_function/index.ipynb | 2 +- .../phases_with_fixed_compositions.ipynb | 2 +- ...with_variable_compositions_random_solutions.ipynb | 10 +++++----- ...riable_compositions_solutions_with_ordering.ipynb | 10 +++++----- .../polymer_solutions_and_polymer_blends.ipynb | 2 +- ...f_thermodynamics_and_equilibrium_conditions.ipynb | 2 +- .../first_and_second_laws_of_thermodynamics.ipynb | 2 +- .../laws_of_thermodynamics/gibbsduhem_equation.ipynb | 2 +- .../{ => psu410}/laws_of_thermodynamics/index.ipynb | 2 +- ...ability_at_equilibrium_and_property_anomaly.ipynb | 2 +- .../general_condition_of_equilibrium.ipynb | 2 +- .../gibbs_phase_rule.ipynb | 2 +- .../index.ipynb | 2 +- .../molar_phase_diagrams.ipynb | 10 +++++----- .../potential_phase_diagrams.ipynb | 8 ++++---- src/psu410/src/{ => psu410}/references/index.ipynb | 2 +- 48 files changed, 96 insertions(+), 96 deletions(-) rename src/psu410/src/{ => psu410}/applications_to_chemical_reactions/ellingham_diagram_and_buffered_systems.ipynb (99%) rename src/psu410/src/{ => psu410}/applications_to_chemical_reactions/index.ipynb (97%) rename src/psu410/src/{ => psu410}/applications_to_chemical_reactions/internal_process_and_differential_and_integrated_driving_forces.ipynb (99%) rename src/psu410/src/{ => psu410}/applications_to_chemical_reactions/maximum_reaction_rate_and_chemical_transport_reactions_.ipynb (99%) rename src/psu410/src/{ => psu410}/applications_to_chemical_reactions/trends_of_entropies_of_reactions.ipynb (99%) rename src/psu410/src/{ => psu410}/applications_to_electrochemical_systems/application_examples.ipynb (99%) rename src/psu410/src/{ => psu410}/applications_to_electrochemical_systems/aqueous_solution_and_pourbaix_diagram.ipynb (99%) rename src/psu410/src/{ => psu410}/applications_to_electrochemical_systems/concentrations_activities_and_reference_states_of_electrolyte_species.ipynb (99%) rename src/psu410/src/{ => psu410}/applications_to_electrochemical_systems/electrochemical_cells_and_half_cell_potentials.ipynb (99%) rename src/psu410/src/{ => psu410}/applications_to_electrochemical_systems/electrolyte_reactions_and_electrochemical_reactions.ipynb (99%) rename src/psu410/src/{ => psu410}/applications_to_electrochemical_systems/index.ipynb (98%) rename src/psu410/src/{ => psu410}/calpahd_modeling_of_thermodynamics/importance_of_lattice_stability.ipynb (99%) rename src/psu410/src/{ => psu410}/calpahd_modeling_of_thermodynamics/index.ipynb (99%) rename src/psu410/src/{ => psu410}/calpahd_modeling_of_thermodynamics/modeling_of_pure_elements.ipynb (99%) rename src/psu410/src/{ => psu410}/calpahd_modeling_of_thermodynamics/modeling_of_random_solution_phases.ipynb (99%) rename src/psu410/src/{ => psu410}/calpahd_modeling_of_thermodynamics/modeling_of_solution_phases_with_longrange_ordering.ipynb (99%) rename src/psu410/src/{ => psu410}/calpahd_modeling_of_thermodynamics/modeling_of_stoichiometric_phases.ipynb (98%) rename src/psu410/src/{ => psu410}/critical_phenomena_thermal_expansion_and_materials_genome/application_to_cerium.ipynb (99%) rename src/psu410/src/{ => psu410}/critical_phenomena_thermal_expansion_and_materials_genome/application_to_fept.ipynb (99%) rename src/psu410/src/{ => psu410}/critical_phenomena_thermal_expansion_and_materials_genome/concept_of_materials_genome.ipynb (99%) rename src/psu410/src/{ => psu410}/critical_phenomena_thermal_expansion_and_materials_genome/index.ipynb (98%) rename src/psu410/src/{ => psu410}/critical_phenomena_thermal_expansion_and_materials_genome/mms_model_applied_to_thermal_expansion.ipynb (99%) rename src/psu410/src/{ => psu410}/experimental_data_for_thermodynamic_modeling/index.ipynb (98%) rename src/psu410/src/{ => psu410}/experimental_data_for_thermodynamic_modeling/phase_equilibrium_data_from_experiments.ipynb (99%) rename src/psu410/src/{ => psu410}/experimental_data_for_thermodynamic_modeling/thermodynamic_data_from_experiments.ipynb (98%) rename src/psu410/src/{ => psu410}/firstprinciples_calculations_and_theory/firstprinciples_approaches_to_disordered_alloys.ipynb (99%) rename src/psu410/src/{ => psu410}/firstprinciples_calculations_and_theory/firstprinciples_formulation_of_thermodynamics.ipynb (98%) rename src/psu410/src/{ => psu410}/firstprinciples_calculations_and_theory/index.ipynb (99%) rename src/psu410/src/{ => psu410}/firstprinciples_calculations_and_theory/lattice_dynamics.ipynb (99%) rename src/psu410/src/{ => psu410}/firstprinciples_calculations_and_theory/nickel_as_the_prototype.ipynb (99%) rename src/psu410/src/{ => psu410}/firstprinciples_calculations_and_theory/quantum_theory_for_the_motion_of_electrons.ipynb (98%) rename src/psu410/src/{ => psu410}/gibbs_energy_function/elastic_magnetic_and_electric_contributions_to_free_energy.ipynb (99%) rename src/psu410/src/{ => psu410}/gibbs_energy_function/index.ipynb (99%) rename src/psu410/src/{ => psu410}/gibbs_energy_function/phases_with_fixed_compositions.ipynb (99%) rename src/psu410/src/{ => psu410}/gibbs_energy_function/phases_with_variable_compositions_random_solutions.ipynb (99%) rename src/psu410/src/{ => psu410}/gibbs_energy_function/phases_with_variable_compositions_solutions_with_ordering.ipynb (99%) rename src/psu410/src/{ => psu410}/gibbs_energy_function/polymer_solutions_and_polymer_blends.ipynb (99%) rename src/psu410/src/{ => psu410}/laws_of_thermodynamics/combined_law_of_thermodynamics_and_equilibrium_conditions.ipynb (99%) rename src/psu410/src/{ => psu410}/laws_of_thermodynamics/first_and_second_laws_of_thermodynamics.ipynb (99%) rename src/psu410/src/{ => psu410}/laws_of_thermodynamics/gibbsduhem_equation.ipynb (99%) rename src/psu410/src/{ => psu410}/laws_of_thermodynamics/index.ipynb (89%) rename src/psu410/src/{ => psu410}/laws_of_thermodynamics/stability_at_equilibrium_and_property_anomaly.ipynb (99%) rename src/psu410/src/{ => psu410}/phase_equilibria_in_heterogeneous_systems/general_condition_of_equilibrium.ipynb (99%) rename src/psu410/src/{ => psu410}/phase_equilibria_in_heterogeneous_systems/gibbs_phase_rule.ipynb (99%) rename src/psu410/src/{ => psu410}/phase_equilibria_in_heterogeneous_systems/index.ipynb (90%) rename src/psu410/src/{ => psu410}/phase_equilibria_in_heterogeneous_systems/molar_phase_diagrams.ipynb (99%) rename src/psu410/src/{ => psu410}/phase_equilibria_in_heterogeneous_systems/potential_phase_diagrams.ipynb (99%) rename src/psu410/src/{ => psu410}/references/index.ipynb (99%) diff --git a/src/psu410/src/applications_to_chemical_reactions/ellingham_diagram_and_buffered_systems.ipynb b/src/psu410/src/psu410/applications_to_chemical_reactions/ellingham_diagram_and_buffered_systems.ipynb similarity index 99% rename from src/psu410/src/applications_to_chemical_reactions/ellingham_diagram_and_buffered_systems.ipynb rename to src/psu410/src/psu410/applications_to_chemical_reactions/ellingham_diagram_and_buffered_systems.ipynb index 0ed9998..dd897f9 100644 --- a/src/psu410/src/applications_to_chemical_reactions/ellingham_diagram_and_buffered_systems.ipynb +++ b/src/psu410/src/psu410/applications_to_chemical_reactions/ellingham_diagram_and_buffered_systems.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "e079ac39", + "id": "86ba3e23", "cell_type": "markdown", "source": [ "## Ellingham diagram and buffered systems\n", diff --git a/src/psu410/src/applications_to_chemical_reactions/index.ipynb b/src/psu410/src/psu410/applications_to_chemical_reactions/index.ipynb similarity index 97% rename from src/psu410/src/applications_to_chemical_reactions/index.ipynb rename to src/psu410/src/psu410/applications_to_chemical_reactions/index.ipynb index 978853a..5cba631 100644 --- a/src/psu410/src/applications_to_chemical_reactions/index.ipynb +++ b/src/psu410/src/psu410/applications_to_chemical_reactions/index.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "3d15fa98", + "id": "9ccb7396", "cell_type": "markdown", "source": [ "# Applications to chemical reactions\n", diff --git a/src/psu410/src/applications_to_chemical_reactions/internal_process_and_differential_and_integrated_driving_forces.ipynb b/src/psu410/src/psu410/applications_to_chemical_reactions/internal_process_and_differential_and_integrated_driving_forces.ipynb similarity index 99% rename from src/psu410/src/applications_to_chemical_reactions/internal_process_and_differential_and_integrated_driving_forces.ipynb rename to src/psu410/src/psu410/applications_to_chemical_reactions/internal_process_and_differential_and_integrated_driving_forces.ipynb index 1ec0188..13630bf 100644 --- a/src/psu410/src/applications_to_chemical_reactions/internal_process_and_differential_and_integrated_driving_forces.ipynb +++ b/src/psu410/src/psu410/applications_to_chemical_reactions/internal_process_and_differential_and_integrated_driving_forces.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "15a6786f", + "id": "1a5ed192", "cell_type": "markdown", "source": [ "## Internal process and differential and integrated driving forces\n", diff --git a/src/psu410/src/applications_to_chemical_reactions/maximum_reaction_rate_and_chemical_transport_reactions_.ipynb b/src/psu410/src/psu410/applications_to_chemical_reactions/maximum_reaction_rate_and_chemical_transport_reactions_.ipynb similarity index 99% rename from src/psu410/src/applications_to_chemical_reactions/maximum_reaction_rate_and_chemical_transport_reactions_.ipynb rename to src/psu410/src/psu410/applications_to_chemical_reactions/maximum_reaction_rate_and_chemical_transport_reactions_.ipynb index e745a0a..fc91abf 100644 --- a/src/psu410/src/applications_to_chemical_reactions/maximum_reaction_rate_and_chemical_transport_reactions_.ipynb +++ b/src/psu410/src/psu410/applications_to_chemical_reactions/maximum_reaction_rate_and_chemical_transport_reactions_.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "2abbd0a0", + "id": "e2171527", "cell_type": "markdown", "source": [ "## Maximum reaction rate and chemical transport reactions \n", diff --git a/src/psu410/src/applications_to_chemical_reactions/trends_of_entropies_of_reactions.ipynb b/src/psu410/src/psu410/applications_to_chemical_reactions/trends_of_entropies_of_reactions.ipynb similarity index 99% rename from src/psu410/src/applications_to_chemical_reactions/trends_of_entropies_of_reactions.ipynb rename to src/psu410/src/psu410/applications_to_chemical_reactions/trends_of_entropies_of_reactions.ipynb index e01968a..3521e82 100644 --- a/src/psu410/src/applications_to_chemical_reactions/trends_of_entropies_of_reactions.ipynb +++ b/src/psu410/src/psu410/applications_to_chemical_reactions/trends_of_entropies_of_reactions.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "c11d5e72", + "id": "fb0726b5", "cell_type": "markdown", "source": [ "## Trends of entropies of reactions\n", diff --git a/src/psu410/src/applications_to_electrochemical_systems/application_examples.ipynb b/src/psu410/src/psu410/applications_to_electrochemical_systems/application_examples.ipynb similarity index 99% rename from src/psu410/src/applications_to_electrochemical_systems/application_examples.ipynb rename to src/psu410/src/psu410/applications_to_electrochemical_systems/application_examples.ipynb index 0b5c281..f780138 100644 --- a/src/psu410/src/applications_to_electrochemical_systems/application_examples.ipynb +++ b/src/psu410/src/psu410/applications_to_electrochemical_systems/application_examples.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "bb41f564", + "id": "58e74cef", "cell_type": "markdown", "source": [ "## Application examples\n", @@ -15,7 +15,7 @@ "metadata": {} }, { - "id": "bc3b8e1c", + "id": "153afcb4", "cell_type": "markdown", "source": [ "### Metastability and passivation\n", @@ -73,7 +73,7 @@ "metadata": {} }, { - "id": "c4f37fc5", + "id": "83ea110b", "cell_type": "markdown", "source": [ "### Galvanic protection\n", @@ -128,7 +128,7 @@ "metadata": {} }, { - "id": "80535ac5", + "id": "8f8d4045", "cell_type": "markdown", "source": [ "### Fuel cells\n", @@ -181,7 +181,7 @@ "metadata": {} }, { - "id": "8fcd81c6", + "id": "3014d564", "cell_type": "markdown", "source": [ "### Ion transport membranes\n", @@ -233,7 +233,7 @@ "metadata": {} }, { - "id": "b28ad146", + "id": "863f2b4a", "cell_type": "markdown", "source": [ "### Electrical batteries\n", diff --git a/src/psu410/src/applications_to_electrochemical_systems/aqueous_solution_and_pourbaix_diagram.ipynb b/src/psu410/src/psu410/applications_to_electrochemical_systems/aqueous_solution_and_pourbaix_diagram.ipynb similarity index 99% rename from src/psu410/src/applications_to_electrochemical_systems/aqueous_solution_and_pourbaix_diagram.ipynb rename to src/psu410/src/psu410/applications_to_electrochemical_systems/aqueous_solution_and_pourbaix_diagram.ipynb index 711457e..f542ee5 100644 --- a/src/psu410/src/applications_to_electrochemical_systems/aqueous_solution_and_pourbaix_diagram.ipynb +++ b/src/psu410/src/psu410/applications_to_electrochemical_systems/aqueous_solution_and_pourbaix_diagram.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "fc524093", + "id": "54708f15", "cell_type": "markdown", "source": [ "## Aqueous solution and Pourbaix diagram\n", diff --git a/src/psu410/src/applications_to_electrochemical_systems/concentrations_activities_and_reference_states_of_electrolyte_species.ipynb b/src/psu410/src/psu410/applications_to_electrochemical_systems/concentrations_activities_and_reference_states_of_electrolyte_species.ipynb similarity index 99% rename from src/psu410/src/applications_to_electrochemical_systems/concentrations_activities_and_reference_states_of_electrolyte_species.ipynb rename to src/psu410/src/psu410/applications_to_electrochemical_systems/concentrations_activities_and_reference_states_of_electrolyte_species.ipynb index fec8d3f..c173033 100644 --- a/src/psu410/src/applications_to_electrochemical_systems/concentrations_activities_and_reference_states_of_electrolyte_species.ipynb +++ b/src/psu410/src/psu410/applications_to_electrochemical_systems/concentrations_activities_and_reference_states_of_electrolyte_species.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "0564f0db", + "id": "0dbe4f22", "cell_type": "markdown", "source": [ "## Concentrations, activities, and reference states of electrolyte species\n", diff --git a/src/psu410/src/applications_to_electrochemical_systems/electrochemical_cells_and_half_cell_potentials.ipynb b/src/psu410/src/psu410/applications_to_electrochemical_systems/electrochemical_cells_and_half_cell_potentials.ipynb similarity index 99% rename from src/psu410/src/applications_to_electrochemical_systems/electrochemical_cells_and_half_cell_potentials.ipynb rename to src/psu410/src/psu410/applications_to_electrochemical_systems/electrochemical_cells_and_half_cell_potentials.ipynb index e04a471..707ab02 100644 --- a/src/psu410/src/applications_to_electrochemical_systems/electrochemical_cells_and_half_cell_potentials.ipynb +++ b/src/psu410/src/psu410/applications_to_electrochemical_systems/electrochemical_cells_and_half_cell_potentials.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "e3027b69", + "id": "0fbd6eac", "cell_type": "markdown", "source": [ "## Electrochemical cells and half cell potentials\n", @@ -33,7 +33,7 @@ "metadata": {} }, { - "id": "bc5e76b6", + "id": "6e9264ef", "cell_type": "markdown", "source": [ "### Electrochemical cells\n", @@ -146,7 +146,7 @@ "metadata": {} }, { - "id": "d4493883", + "id": "e194b4c0", "cell_type": "markdown", "source": [ "### Half cell potentials\n", diff --git a/src/psu410/src/applications_to_electrochemical_systems/electrolyte_reactions_and_electrochemical_reactions.ipynb b/src/psu410/src/psu410/applications_to_electrochemical_systems/electrolyte_reactions_and_electrochemical_reactions.ipynb similarity index 99% rename from src/psu410/src/applications_to_electrochemical_systems/electrolyte_reactions_and_electrochemical_reactions.ipynb rename to src/psu410/src/psu410/applications_to_electrochemical_systems/electrolyte_reactions_and_electrochemical_reactions.ipynb index bf7faa3..cb1a075 100644 --- a/src/psu410/src/applications_to_electrochemical_systems/electrolyte_reactions_and_electrochemical_reactions.ipynb +++ b/src/psu410/src/psu410/applications_to_electrochemical_systems/electrolyte_reactions_and_electrochemical_reactions.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "d68abe36", + "id": "db31d1be", "cell_type": "markdown", "source": [ "## Electrolyte reactions and electrochemical reactions\n", diff --git a/src/psu410/src/applications_to_electrochemical_systems/index.ipynb b/src/psu410/src/psu410/applications_to_electrochemical_systems/index.ipynb similarity index 98% rename from src/psu410/src/applications_to_electrochemical_systems/index.ipynb rename to src/psu410/src/psu410/applications_to_electrochemical_systems/index.ipynb index edbd9ee..c7d54fe 100644 --- a/src/psu410/src/applications_to_electrochemical_systems/index.ipynb +++ b/src/psu410/src/psu410/applications_to_electrochemical_systems/index.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "1f40160b", + "id": "3bb06c52", "cell_type": "markdown", "source": [ "# Applications to electrochemical systems\n", diff --git a/src/psu410/src/calpahd_modeling_of_thermodynamics/importance_of_lattice_stability.ipynb b/src/psu410/src/psu410/calpahd_modeling_of_thermodynamics/importance_of_lattice_stability.ipynb similarity index 99% rename from src/psu410/src/calpahd_modeling_of_thermodynamics/importance_of_lattice_stability.ipynb rename to src/psu410/src/psu410/calpahd_modeling_of_thermodynamics/importance_of_lattice_stability.ipynb index 2f091d2..d559428 100644 --- a/src/psu410/src/calpahd_modeling_of_thermodynamics/importance_of_lattice_stability.ipynb +++ b/src/psu410/src/psu410/calpahd_modeling_of_thermodynamics/importance_of_lattice_stability.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "4d2c637c", + "id": "46bb1c6e", "cell_type": "markdown", "source": [ "## Importance of lattice stability\n", diff --git a/src/psu410/src/calpahd_modeling_of_thermodynamics/index.ipynb b/src/psu410/src/psu410/calpahd_modeling_of_thermodynamics/index.ipynb similarity index 99% rename from src/psu410/src/calpahd_modeling_of_thermodynamics/index.ipynb rename to src/psu410/src/psu410/calpahd_modeling_of_thermodynamics/index.ipynb index 3a4e34d..c5d2f92 100644 --- a/src/psu410/src/calpahd_modeling_of_thermodynamics/index.ipynb +++ b/src/psu410/src/psu410/calpahd_modeling_of_thermodynamics/index.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "ca65cc0e", + "id": "e744c055", "cell_type": "markdown", "source": [ "# CALPAHD modeling of thermodynamics\n", diff --git a/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_pure_elements.ipynb b/src/psu410/src/psu410/calpahd_modeling_of_thermodynamics/modeling_of_pure_elements.ipynb similarity index 99% rename from src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_pure_elements.ipynb rename to src/psu410/src/psu410/calpahd_modeling_of_thermodynamics/modeling_of_pure_elements.ipynb index 87c5f77..6dff134 100644 --- a/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_pure_elements.ipynb +++ b/src/psu410/src/psu410/calpahd_modeling_of_thermodynamics/modeling_of_pure_elements.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "ba726078", + "id": "9020233c", "cell_type": "markdown", "source": [ "## Modeling of pure elements\n", diff --git a/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_random_solution_phases.ipynb b/src/psu410/src/psu410/calpahd_modeling_of_thermodynamics/modeling_of_random_solution_phases.ipynb similarity index 99% rename from src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_random_solution_phases.ipynb rename to src/psu410/src/psu410/calpahd_modeling_of_thermodynamics/modeling_of_random_solution_phases.ipynb index c493ede..3fae5e1 100644 --- a/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_random_solution_phases.ipynb +++ b/src/psu410/src/psu410/calpahd_modeling_of_thermodynamics/modeling_of_random_solution_phases.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "1375de02", + "id": "9c4cc802", "cell_type": "markdown", "source": [ "## Modeling of random solution phases\n", diff --git a/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_solution_phases_with_longrange_ordering.ipynb b/src/psu410/src/psu410/calpahd_modeling_of_thermodynamics/modeling_of_solution_phases_with_longrange_ordering.ipynb similarity index 99% rename from src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_solution_phases_with_longrange_ordering.ipynb rename to src/psu410/src/psu410/calpahd_modeling_of_thermodynamics/modeling_of_solution_phases_with_longrange_ordering.ipynb index cd2e24e..2605ebb 100644 --- a/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_solution_phases_with_longrange_ordering.ipynb +++ b/src/psu410/src/psu410/calpahd_modeling_of_thermodynamics/modeling_of_solution_phases_with_longrange_ordering.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "7036eab8", + "id": "4737c640", "cell_type": "markdown", "source": [ "## Modeling of solution phases with long-range ordering\n", diff --git a/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_stoichiometric_phases.ipynb b/src/psu410/src/psu410/calpahd_modeling_of_thermodynamics/modeling_of_stoichiometric_phases.ipynb similarity index 98% rename from src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_stoichiometric_phases.ipynb rename to src/psu410/src/psu410/calpahd_modeling_of_thermodynamics/modeling_of_stoichiometric_phases.ipynb index 038e9c9..bfc2de9 100644 --- a/src/psu410/src/calpahd_modeling_of_thermodynamics/modeling_of_stoichiometric_phases.ipynb +++ b/src/psu410/src/psu410/calpahd_modeling_of_thermodynamics/modeling_of_stoichiometric_phases.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "a6ae79da", + "id": "51b3fec2", "cell_type": "markdown", "source": [ "## Modeling of stoichiometric phases\n", diff --git a/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_cerium.ipynb b/src/psu410/src/psu410/critical_phenomena_thermal_expansion_and_materials_genome/application_to_cerium.ipynb similarity index 99% rename from src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_cerium.ipynb rename to src/psu410/src/psu410/critical_phenomena_thermal_expansion_and_materials_genome/application_to_cerium.ipynb index 45ebd7e..2259d94 100644 --- a/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_cerium.ipynb +++ b/src/psu410/src/psu410/critical_phenomena_thermal_expansion_and_materials_genome/application_to_cerium.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "e6bcdb88", + "id": "2b36eca7", "cell_type": "markdown", "source": [ "## Application to cerium\n", diff --git a/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_fept.ipynb b/src/psu410/src/psu410/critical_phenomena_thermal_expansion_and_materials_genome/application_to_fept.ipynb similarity index 99% rename from src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_fept.ipynb rename to src/psu410/src/psu410/critical_phenomena_thermal_expansion_and_materials_genome/application_to_fept.ipynb index 7686fa4..dc7dbe1 100644 --- a/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/application_to_fept.ipynb +++ b/src/psu410/src/psu410/critical_phenomena_thermal_expansion_and_materials_genome/application_to_fept.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "8f719255", + "id": "a65d7622", "cell_type": "markdown", "source": [ "## Application to Fe3Pt\n", diff --git a/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/concept_of_materials_genome.ipynb b/src/psu410/src/psu410/critical_phenomena_thermal_expansion_and_materials_genome/concept_of_materials_genome.ipynb similarity index 99% rename from src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/concept_of_materials_genome.ipynb rename to src/psu410/src/psu410/critical_phenomena_thermal_expansion_and_materials_genome/concept_of_materials_genome.ipynb index 29bd85b..4d85473 100644 --- a/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/concept_of_materials_genome.ipynb +++ b/src/psu410/src/psu410/critical_phenomena_thermal_expansion_and_materials_genome/concept_of_materials_genome.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "2e83b1bc", + "id": "51d02db3", "cell_type": "markdown", "source": [ "## Concept of Materials Genome\u00ae\n", diff --git a/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/index.ipynb b/src/psu410/src/psu410/critical_phenomena_thermal_expansion_and_materials_genome/index.ipynb similarity index 98% rename from src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/index.ipynb rename to src/psu410/src/psu410/critical_phenomena_thermal_expansion_and_materials_genome/index.ipynb index 3eb3c04..b9574b4 100644 --- a/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/index.ipynb +++ b/src/psu410/src/psu410/critical_phenomena_thermal_expansion_and_materials_genome/index.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "bb30430b", + "id": "12bb1dc5", "cell_type": "markdown", "source": [ "# Critical phenomena, thermal expansion, and Materials Genome\u00ae\n", diff --git a/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/mms_model_applied_to_thermal_expansion.ipynb b/src/psu410/src/psu410/critical_phenomena_thermal_expansion_and_materials_genome/mms_model_applied_to_thermal_expansion.ipynb similarity index 99% rename from src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/mms_model_applied_to_thermal_expansion.ipynb rename to src/psu410/src/psu410/critical_phenomena_thermal_expansion_and_materials_genome/mms_model_applied_to_thermal_expansion.ipynb index 2928154..849809b 100644 --- a/src/psu410/src/critical_phenomena_thermal_expansion_and_materials_genome/mms_model_applied_to_thermal_expansion.ipynb +++ b/src/psu410/src/psu410/critical_phenomena_thermal_expansion_and_materials_genome/mms_model_applied_to_thermal_expansion.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "840ab7ad", + "id": "5554609d", "cell_type": "markdown", "source": [ "## MMS model applied to thermal expansion\n", diff --git a/src/psu410/src/experimental_data_for_thermodynamic_modeling/index.ipynb b/src/psu410/src/psu410/experimental_data_for_thermodynamic_modeling/index.ipynb similarity index 98% rename from src/psu410/src/experimental_data_for_thermodynamic_modeling/index.ipynb rename to src/psu410/src/psu410/experimental_data_for_thermodynamic_modeling/index.ipynb index 9d04047..1c57a2f 100644 --- a/src/psu410/src/experimental_data_for_thermodynamic_modeling/index.ipynb +++ b/src/psu410/src/psu410/experimental_data_for_thermodynamic_modeling/index.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "14788efe", + "id": "f5d00303", "cell_type": "markdown", "source": [ "# Experimental data for thermodynamic modeling\n", diff --git a/src/psu410/src/experimental_data_for_thermodynamic_modeling/phase_equilibrium_data_from_experiments.ipynb b/src/psu410/src/psu410/experimental_data_for_thermodynamic_modeling/phase_equilibrium_data_from_experiments.ipynb similarity index 99% rename from src/psu410/src/experimental_data_for_thermodynamic_modeling/phase_equilibrium_data_from_experiments.ipynb rename to src/psu410/src/psu410/experimental_data_for_thermodynamic_modeling/phase_equilibrium_data_from_experiments.ipynb index d0e1bcb..c6acb65 100644 --- a/src/psu410/src/experimental_data_for_thermodynamic_modeling/phase_equilibrium_data_from_experiments.ipynb +++ b/src/psu410/src/psu410/experimental_data_for_thermodynamic_modeling/phase_equilibrium_data_from_experiments.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "fbbb8610", + "id": "23a7020d", "cell_type": "markdown", "source": [ "## Phase equilibrium data from experiments\n" @@ -12,7 +12,7 @@ "metadata": {} }, { - "id": "cae88515", + "id": "7334a053", "cell_type": "markdown", "source": [ "### Equilibrated materials\n", @@ -99,7 +99,7 @@ "metadata": {} }, { - "id": "dc62bc68", + "id": "a64fc988", "cell_type": "markdown", "source": [ "### Diffusion couples/multiples\n", @@ -188,7 +188,7 @@ "metadata": {} }, { - "id": "1da46570", + "id": "e4019523", "cell_type": "markdown", "source": [ "### Additional methods\n", diff --git a/src/psu410/src/experimental_data_for_thermodynamic_modeling/thermodynamic_data_from_experiments.ipynb b/src/psu410/src/psu410/experimental_data_for_thermodynamic_modeling/thermodynamic_data_from_experiments.ipynb similarity index 98% rename from src/psu410/src/experimental_data_for_thermodynamic_modeling/thermodynamic_data_from_experiments.ipynb rename to src/psu410/src/psu410/experimental_data_for_thermodynamic_modeling/thermodynamic_data_from_experiments.ipynb index 3a98bce..c5245a9 100644 --- a/src/psu410/src/experimental_data_for_thermodynamic_modeling/thermodynamic_data_from_experiments.ipynb +++ b/src/psu410/src/psu410/experimental_data_for_thermodynamic_modeling/thermodynamic_data_from_experiments.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "b922c877", + "id": "50fa3767", "cell_type": "markdown", "source": [ "## Thermodynamic data from experiments\n", @@ -25,7 +25,7 @@ "metadata": {} }, { - "id": "6407a501", + "id": "28799218", "cell_type": "markdown", "source": [ "### Solution calorimetry\n", @@ -72,7 +72,7 @@ "metadata": {} }, { - "id": "37c416cc", + "id": "a3827671", "cell_type": "markdown", "source": [ "### Combustion, direct reaction, and heat capacity calorimetry\n", @@ -107,7 +107,7 @@ "metadata": {} }, { - "id": "6205849e", + "id": "f2cb6d03", "cell_type": "markdown", "source": [ "### Vapor pressure method\n", diff --git a/src/psu410/src/firstprinciples_calculations_and_theory/firstprinciples_approaches_to_disordered_alloys.ipynb b/src/psu410/src/psu410/firstprinciples_calculations_and_theory/firstprinciples_approaches_to_disordered_alloys.ipynb similarity index 99% rename from src/psu410/src/firstprinciples_calculations_and_theory/firstprinciples_approaches_to_disordered_alloys.ipynb rename to src/psu410/src/psu410/firstprinciples_calculations_and_theory/firstprinciples_approaches_to_disordered_alloys.ipynb index 1da27c5..87467a4 100644 --- a/src/psu410/src/firstprinciples_calculations_and_theory/firstprinciples_approaches_to_disordered_alloys.ipynb +++ b/src/psu410/src/psu410/firstprinciples_calculations_and_theory/firstprinciples_approaches_to_disordered_alloys.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "0a55a5c4", + "id": "f6f98fcb", "cell_type": "markdown", "source": [ "## First-principles approaches to disordered alloys\n", @@ -53,7 +53,7 @@ "metadata": {} }, { - "id": "e3730978", + "id": "e63b45c0", "cell_type": "markdown", "source": [ "### Cluster Expansions\n", @@ -109,7 +109,7 @@ "metadata": {} }, { - "id": "fccf9ee9", + "id": "a3f07054", "cell_type": "markdown", "source": [ "### Special Quasirandom Structures\n", @@ -185,7 +185,7 @@ "metadata": {} }, { - "id": "b1eee4e6", + "id": "fe975054", "cell_type": "markdown", "source": [ "### Phonon calculations for SQS\n", diff --git a/src/psu410/src/firstprinciples_calculations_and_theory/firstprinciples_formulation_of_thermodynamics.ipynb b/src/psu410/src/psu410/firstprinciples_calculations_and_theory/firstprinciples_formulation_of_thermodynamics.ipynb similarity index 98% rename from src/psu410/src/firstprinciples_calculations_and_theory/firstprinciples_formulation_of_thermodynamics.ipynb rename to src/psu410/src/psu410/firstprinciples_calculations_and_theory/firstprinciples_formulation_of_thermodynamics.ipynb index 4029dc5..90be934 100644 --- a/src/psu410/src/firstprinciples_calculations_and_theory/firstprinciples_formulation_of_thermodynamics.ipynb +++ b/src/psu410/src/psu410/firstprinciples_calculations_and_theory/firstprinciples_formulation_of_thermodynamics.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "ffde7f8b", + "id": "577c34be", "cell_type": "markdown", "source": [ "## First-principles formulation of thermodynamics\n" @@ -12,7 +12,7 @@ "metadata": {} }, { - "id": "4990a52d", + "id": "1ba93fb4", "cell_type": "markdown", "source": [ "### Helmholtz energy\n", @@ -70,7 +70,7 @@ "metadata": {} }, { - "id": "e74adeb3", + "id": "96ad1634", "cell_type": "markdown", "source": [ "### Mermin statistics to the thermal electronic contribution \n", @@ -133,7 +133,7 @@ "metadata": {} }, { - "id": "2ca3897c", + "id": "b0eff4bb", "cell_type": "markdown", "source": [ "### Vibrational contribution by phonon theory\n", @@ -177,7 +177,7 @@ "metadata": {} }, { - "id": "a62c437f", + "id": "0349caf1", "cell_type": "markdown", "source": [ "### Debye-Gr\u00fcneisen approximation to the vibrational contribution \n", @@ -256,7 +256,7 @@ "metadata": {} }, { - "id": "85af7136", + "id": "48b64474", "cell_type": "markdown", "source": [ "### System with multiple microstates (MMS model)\n", diff --git a/src/psu410/src/firstprinciples_calculations_and_theory/index.ipynb b/src/psu410/src/psu410/firstprinciples_calculations_and_theory/index.ipynb similarity index 99% rename from src/psu410/src/firstprinciples_calculations_and_theory/index.ipynb rename to src/psu410/src/psu410/firstprinciples_calculations_and_theory/index.ipynb index 54dbaaa..248f881 100644 --- a/src/psu410/src/firstprinciples_calculations_and_theory/index.ipynb +++ b/src/psu410/src/psu410/firstprinciples_calculations_and_theory/index.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "7ae54358", + "id": "097789bc", "cell_type": "markdown", "source": [ "# First-principles calculations and theory\n", diff --git a/src/psu410/src/firstprinciples_calculations_and_theory/lattice_dynamics.ipynb b/src/psu410/src/psu410/firstprinciples_calculations_and_theory/lattice_dynamics.ipynb similarity index 99% rename from src/psu410/src/firstprinciples_calculations_and_theory/lattice_dynamics.ipynb rename to src/psu410/src/psu410/firstprinciples_calculations_and_theory/lattice_dynamics.ipynb index 058323d..de9b4bd 100644 --- a/src/psu410/src/firstprinciples_calculations_and_theory/lattice_dynamics.ipynb +++ b/src/psu410/src/psu410/firstprinciples_calculations_and_theory/lattice_dynamics.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "da9f8821", + "id": "bcb06510", "cell_type": "markdown", "source": [ "## Lattice Dynamics\n" @@ -12,7 +12,7 @@ "metadata": {} }, { - "id": "c8223522", + "id": "48a67c32", "cell_type": "markdown", "source": [ "### Quantum theory for motion of atomic nuclei\n", @@ -83,7 +83,7 @@ "metadata": {} }, { - "id": "5afe2868", + "id": "f78bd8f1", "cell_type": "markdown", "source": [ "### Normal coordinates, eigenenergetics, and phonons\n", @@ -166,7 +166,7 @@ "metadata": {} }, { - "id": "80204d58", + "id": "0a0a3f25", "cell_type": "markdown", "source": [ "### Dynamical matrix and phonon mode\n", @@ -276,7 +276,7 @@ "metadata": {} }, { - "id": "d89bda80", + "id": "a7d6dd73", "cell_type": "markdown", "source": [ "### Linear-response method vs supercell method\n", diff --git a/src/psu410/src/firstprinciples_calculations_and_theory/nickel_as_the_prototype.ipynb b/src/psu410/src/psu410/firstprinciples_calculations_and_theory/nickel_as_the_prototype.ipynb similarity index 99% rename from src/psu410/src/firstprinciples_calculations_and_theory/nickel_as_the_prototype.ipynb rename to src/psu410/src/psu410/firstprinciples_calculations_and_theory/nickel_as_the_prototype.ipynb index 97af25f..416ecd4 100644 --- a/src/psu410/src/firstprinciples_calculations_and_theory/nickel_as_the_prototype.ipynb +++ b/src/psu410/src/psu410/firstprinciples_calculations_and_theory/nickel_as_the_prototype.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "373ce861", + "id": "907eac6b", "cell_type": "markdown", "source": [ "## Nickel as the prototype\n", @@ -34,7 +34,7 @@ "metadata": {} }, { - "id": "cc102013", + "id": "548a5194", "cell_type": "markdown", "source": [ "### Helmholtz energy and quasiharmonic approximation\n", @@ -202,7 +202,7 @@ "metadata": {} }, { - "id": "4c1240fb", + "id": "102628bc", "cell_type": "markdown", "source": [ "### Volume, entropy, enthalpy, thermal expansion, bulk modulus, and heat capacity \n", @@ -311,7 +311,7 @@ "metadata": {} }, { - "id": "d94f5392", + "id": "20203042", "cell_type": "markdown", "source": [ "### Formation enthalpy of Ni3Al\n", diff --git a/src/psu410/src/firstprinciples_calculations_and_theory/quantum_theory_for_the_motion_of_electrons.ipynb b/src/psu410/src/psu410/firstprinciples_calculations_and_theory/quantum_theory_for_the_motion_of_electrons.ipynb similarity index 98% rename from src/psu410/src/firstprinciples_calculations_and_theory/quantum_theory_for_the_motion_of_electrons.ipynb rename to src/psu410/src/psu410/firstprinciples_calculations_and_theory/quantum_theory_for_the_motion_of_electrons.ipynb index 5efbe02..3dd25aa 100644 --- a/src/psu410/src/firstprinciples_calculations_and_theory/quantum_theory_for_the_motion_of_electrons.ipynb +++ b/src/psu410/src/psu410/firstprinciples_calculations_and_theory/quantum_theory_for_the_motion_of_electrons.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "05af4bfa", + "id": "4098f321", "cell_type": "markdown", "source": [ "## Quantum theory for the motion of electrons\n" @@ -12,7 +12,7 @@ "metadata": {} }, { - "id": "06b16305", + "id": "f7876e4d", "cell_type": "markdown", "source": [ "### Schr\u00f6dinger equation\n", @@ -56,7 +56,7 @@ "metadata": {} }, { - "id": "a68fbf8b", + "id": "4ded1d86", "cell_type": "markdown", "source": [ "### Born-Oppenheimer Approximation\n", @@ -114,7 +114,7 @@ "metadata": {} }, { - "id": "8cd5cd6c", + "id": "cdad6ca5", "cell_type": "markdown", "source": [ "### Hartree-Fock approximation to solve Schr\u00f6dinger equation\n", @@ -203,7 +203,7 @@ "metadata": {} }, { - "id": "40ade0dc", + "id": "ed57d05a", "cell_type": "markdown", "source": [ "### Density functional theory (DFT) and 0 K Kohn-Sham equations\n", @@ -281,7 +281,7 @@ "metadata": {} }, { - "id": "20b4c1b1", + "id": "3fc99f7b", "cell_type": "markdown", "source": [ "#### Solving the Kohn-Sham Equations for a Solid\n", diff --git a/src/psu410/src/gibbs_energy_function/elastic_magnetic_and_electric_contributions_to_free_energy.ipynb b/src/psu410/src/psu410/gibbs_energy_function/elastic_magnetic_and_electric_contributions_to_free_energy.ipynb similarity index 99% rename from src/psu410/src/gibbs_energy_function/elastic_magnetic_and_electric_contributions_to_free_energy.ipynb rename to src/psu410/src/psu410/gibbs_energy_function/elastic_magnetic_and_electric_contributions_to_free_energy.ipynb index 02730ea..c7aa78c 100644 --- a/src/psu410/src/gibbs_energy_function/elastic_magnetic_and_electric_contributions_to_free_energy.ipynb +++ b/src/psu410/src/psu410/gibbs_energy_function/elastic_magnetic_and_electric_contributions_to_free_energy.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "3a9a396a", + "id": "87365766", "cell_type": "markdown", "source": [ "## Elastic, magnetic, and electric contributions to free energy\n", diff --git a/src/psu410/src/gibbs_energy_function/index.ipynb b/src/psu410/src/psu410/gibbs_energy_function/index.ipynb similarity index 99% rename from src/psu410/src/gibbs_energy_function/index.ipynb rename to src/psu410/src/psu410/gibbs_energy_function/index.ipynb index b422a98..dca2b89 100644 --- a/src/psu410/src/gibbs_energy_function/index.ipynb +++ b/src/psu410/src/psu410/gibbs_energy_function/index.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "447fe0e1", + "id": "efe43b42", "cell_type": "markdown", "source": [ "# Gibbs energy function\n", diff --git a/src/psu410/src/gibbs_energy_function/phases_with_fixed_compositions.ipynb b/src/psu410/src/psu410/gibbs_energy_function/phases_with_fixed_compositions.ipynb similarity index 99% rename from src/psu410/src/gibbs_energy_function/phases_with_fixed_compositions.ipynb rename to src/psu410/src/psu410/gibbs_energy_function/phases_with_fixed_compositions.ipynb index cf032aa..3ad796d 100644 --- a/src/psu410/src/gibbs_energy_function/phases_with_fixed_compositions.ipynb +++ b/src/psu410/src/psu410/gibbs_energy_function/phases_with_fixed_compositions.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "011f5401", + "id": "40c1c423", "cell_type": "markdown", "source": [ "## Phases with fixed compositions\n", diff --git a/src/psu410/src/gibbs_energy_function/phases_with_variable_compositions_random_solutions.ipynb b/src/psu410/src/psu410/gibbs_energy_function/phases_with_variable_compositions_random_solutions.ipynb similarity index 99% rename from src/psu410/src/gibbs_energy_function/phases_with_variable_compositions_random_solutions.ipynb rename to src/psu410/src/psu410/gibbs_energy_function/phases_with_variable_compositions_random_solutions.ipynb index 4285468..be2d299 100644 --- a/src/psu410/src/gibbs_energy_function/phases_with_variable_compositions_random_solutions.ipynb +++ b/src/psu410/src/psu410/gibbs_energy_function/phases_with_variable_compositions_random_solutions.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "55b690ab", + "id": "62204a33", "cell_type": "markdown", "source": [ "## Phases with variable compositions: Random solutions\n", @@ -120,7 +120,7 @@ "metadata": {} }, { - "id": "3e83c37f", + "id": "6f077042", "cell_type": "markdown", "source": [ "### Random solutions\n", @@ -224,7 +224,7 @@ "metadata": {} }, { - "id": "6fd8d94b", + "id": "9cc6b18b", "cell_type": "markdown", "source": [ "### Binary random solutions\n", @@ -381,7 +381,7 @@ "metadata": {} }, { - "id": "7e5249cd", + "id": "891d7281", "cell_type": "markdown", "source": [ "### Ternary random solutions\n", @@ -503,7 +503,7 @@ "metadata": {} }, { - "id": "ae59d19d", + "id": "2eab4d60", "cell_type": "markdown", "source": [ "### Multi-component random solutions\n", diff --git a/src/psu410/src/gibbs_energy_function/phases_with_variable_compositions_solutions_with_ordering.ipynb b/src/psu410/src/psu410/gibbs_energy_function/phases_with_variable_compositions_solutions_with_ordering.ipynb similarity index 99% rename from src/psu410/src/gibbs_energy_function/phases_with_variable_compositions_solutions_with_ordering.ipynb rename to src/psu410/src/psu410/gibbs_energy_function/phases_with_variable_compositions_solutions_with_ordering.ipynb index 31e39f5..236f8ac 100644 --- a/src/psu410/src/gibbs_energy_function/phases_with_variable_compositions_solutions_with_ordering.ipynb +++ b/src/psu410/src/psu410/gibbs_energy_function/phases_with_variable_compositions_solutions_with_ordering.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "b876aec3", + "id": "ef80df3d", "cell_type": "markdown", "source": [ "## Phases with variable compositions: Solutions with ordering\n" @@ -12,7 +12,7 @@ "metadata": {} }, { - "id": "8ac0a3e4", + "id": "7908011c", "cell_type": "markdown", "source": [ "### Solutions with short-range ordering\n", @@ -198,7 +198,7 @@ "metadata": {} }, { - "id": "2148b250", + "id": "dc74ea53", "cell_type": "markdown", "source": [ "### Solutions with long-range ordering\n", @@ -359,7 +359,7 @@ "metadata": {} }, { - "id": "527f6d64", + "id": "31d4168e", "cell_type": "markdown", "source": [ "### Solutions with both short-range and long-range ordering\n", @@ -387,7 +387,7 @@ "metadata": {} }, { - "id": "f54abe2b", + "id": "bb96bbcc", "cell_type": "markdown", "source": [ "### Solutions with charged species\n", diff --git a/src/psu410/src/gibbs_energy_function/polymer_solutions_and_polymer_blends.ipynb b/src/psu410/src/psu410/gibbs_energy_function/polymer_solutions_and_polymer_blends.ipynb similarity index 99% rename from src/psu410/src/gibbs_energy_function/polymer_solutions_and_polymer_blends.ipynb rename to src/psu410/src/psu410/gibbs_energy_function/polymer_solutions_and_polymer_blends.ipynb index ccaa83d..450ad26 100644 --- a/src/psu410/src/gibbs_energy_function/polymer_solutions_and_polymer_blends.ipynb +++ b/src/psu410/src/psu410/gibbs_energy_function/polymer_solutions_and_polymer_blends.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "7f7d018e", + "id": "5dc6e2f1", "cell_type": "markdown", "source": [ "## Polymer solutions and polymer blends\n", diff --git a/src/psu410/src/laws_of_thermodynamics/combined_law_of_thermodynamics_and_equilibrium_conditions.ipynb b/src/psu410/src/psu410/laws_of_thermodynamics/combined_law_of_thermodynamics_and_equilibrium_conditions.ipynb similarity index 99% rename from src/psu410/src/laws_of_thermodynamics/combined_law_of_thermodynamics_and_equilibrium_conditions.ipynb rename to src/psu410/src/psu410/laws_of_thermodynamics/combined_law_of_thermodynamics_and_equilibrium_conditions.ipynb index c25c002..411cc3d 100644 --- a/src/psu410/src/laws_of_thermodynamics/combined_law_of_thermodynamics_and_equilibrium_conditions.ipynb +++ b/src/psu410/src/psu410/laws_of_thermodynamics/combined_law_of_thermodynamics_and_equilibrium_conditions.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "eb09e8ad", + "id": "a75b942c", "cell_type": "markdown", "source": [ "## Combined law of thermodynamics and equilibrium conditions\n", diff --git a/src/psu410/src/laws_of_thermodynamics/first_and_second_laws_of_thermodynamics.ipynb b/src/psu410/src/psu410/laws_of_thermodynamics/first_and_second_laws_of_thermodynamics.ipynb similarity index 99% rename from src/psu410/src/laws_of_thermodynamics/first_and_second_laws_of_thermodynamics.ipynb rename to src/psu410/src/psu410/laws_of_thermodynamics/first_and_second_laws_of_thermodynamics.ipynb index 1c75f73..436bf7f 100644 --- a/src/psu410/src/laws_of_thermodynamics/first_and_second_laws_of_thermodynamics.ipynb +++ b/src/psu410/src/psu410/laws_of_thermodynamics/first_and_second_laws_of_thermodynamics.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "f8e20f3c", + "id": "b09c8a86", "cell_type": "markdown", "source": [ "## First and second laws of thermodynamics\n", diff --git a/src/psu410/src/laws_of_thermodynamics/gibbsduhem_equation.ipynb b/src/psu410/src/psu410/laws_of_thermodynamics/gibbsduhem_equation.ipynb similarity index 99% rename from src/psu410/src/laws_of_thermodynamics/gibbsduhem_equation.ipynb rename to src/psu410/src/psu410/laws_of_thermodynamics/gibbsduhem_equation.ipynb index 7858d83..923eaac 100644 --- a/src/psu410/src/laws_of_thermodynamics/gibbsduhem_equation.ipynb +++ b/src/psu410/src/psu410/laws_of_thermodynamics/gibbsduhem_equation.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "e63729bb", + "id": "aceb69c3", "cell_type": "markdown", "source": [ "## Gibbs-Duhem equation\n", diff --git a/src/psu410/src/laws_of_thermodynamics/index.ipynb b/src/psu410/src/psu410/laws_of_thermodynamics/index.ipynb similarity index 89% rename from src/psu410/src/laws_of_thermodynamics/index.ipynb rename to src/psu410/src/psu410/laws_of_thermodynamics/index.ipynb index eaf8566..8068dc8 100644 --- a/src/psu410/src/laws_of_thermodynamics/index.ipynb +++ b/src/psu410/src/psu410/laws_of_thermodynamics/index.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "3e5734cb", + "id": "2bb2daa7", "cell_type": "markdown", "source": [ "# Laws of thermodynamics\n" diff --git a/src/psu410/src/laws_of_thermodynamics/stability_at_equilibrium_and_property_anomaly.ipynb b/src/psu410/src/psu410/laws_of_thermodynamics/stability_at_equilibrium_and_property_anomaly.ipynb similarity index 99% rename from src/psu410/src/laws_of_thermodynamics/stability_at_equilibrium_and_property_anomaly.ipynb rename to src/psu410/src/psu410/laws_of_thermodynamics/stability_at_equilibrium_and_property_anomaly.ipynb index 7d5ff7b..d35cf1a 100644 --- a/src/psu410/src/laws_of_thermodynamics/stability_at_equilibrium_and_property_anomaly.ipynb +++ b/src/psu410/src/psu410/laws_of_thermodynamics/stability_at_equilibrium_and_property_anomaly.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "9010bd96", + "id": "9ccd12f0", "cell_type": "markdown", "source": [ "## Stability at equilibrium and property anomaly\n", diff --git a/src/psu410/src/phase_equilibria_in_heterogeneous_systems/general_condition_of_equilibrium.ipynb b/src/psu410/src/psu410/phase_equilibria_in_heterogeneous_systems/general_condition_of_equilibrium.ipynb similarity index 99% rename from src/psu410/src/phase_equilibria_in_heterogeneous_systems/general_condition_of_equilibrium.ipynb rename to src/psu410/src/psu410/phase_equilibria_in_heterogeneous_systems/general_condition_of_equilibrium.ipynb index 1291561..4aae221 100644 --- a/src/psu410/src/phase_equilibria_in_heterogeneous_systems/general_condition_of_equilibrium.ipynb +++ b/src/psu410/src/psu410/phase_equilibria_in_heterogeneous_systems/general_condition_of_equilibrium.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "fdd1ab93", + "id": "7fc1ab97", "cell_type": "markdown", "source": [ "## General condition of equilibrium\n", diff --git a/src/psu410/src/phase_equilibria_in_heterogeneous_systems/gibbs_phase_rule.ipynb b/src/psu410/src/psu410/phase_equilibria_in_heterogeneous_systems/gibbs_phase_rule.ipynb similarity index 99% rename from src/psu410/src/phase_equilibria_in_heterogeneous_systems/gibbs_phase_rule.ipynb rename to src/psu410/src/psu410/phase_equilibria_in_heterogeneous_systems/gibbs_phase_rule.ipynb index 2fcd012..22182f2 100644 --- a/src/psu410/src/phase_equilibria_in_heterogeneous_systems/gibbs_phase_rule.ipynb +++ b/src/psu410/src/psu410/phase_equilibria_in_heterogeneous_systems/gibbs_phase_rule.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "a89214fd", + "id": "9550cd18", "cell_type": "markdown", "source": [ "## Gibbs phase rule\n", diff --git a/src/psu410/src/phase_equilibria_in_heterogeneous_systems/index.ipynb b/src/psu410/src/psu410/phase_equilibria_in_heterogeneous_systems/index.ipynb similarity index 90% rename from src/psu410/src/phase_equilibria_in_heterogeneous_systems/index.ipynb rename to src/psu410/src/psu410/phase_equilibria_in_heterogeneous_systems/index.ipynb index c4fddd5..7752e80 100644 --- a/src/psu410/src/phase_equilibria_in_heterogeneous_systems/index.ipynb +++ b/src/psu410/src/psu410/phase_equilibria_in_heterogeneous_systems/index.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "f853d6af", + "id": "45d835d9", "cell_type": "markdown", "source": [ "# Phase equilibria in heterogeneous systems\n" diff --git a/src/psu410/src/phase_equilibria_in_heterogeneous_systems/molar_phase_diagrams.ipynb b/src/psu410/src/psu410/phase_equilibria_in_heterogeneous_systems/molar_phase_diagrams.ipynb similarity index 99% rename from src/psu410/src/phase_equilibria_in_heterogeneous_systems/molar_phase_diagrams.ipynb rename to src/psu410/src/psu410/phase_equilibria_in_heterogeneous_systems/molar_phase_diagrams.ipynb index 4f73268..5d44860 100644 --- a/src/psu410/src/phase_equilibria_in_heterogeneous_systems/molar_phase_diagrams.ipynb +++ b/src/psu410/src/psu410/phase_equilibria_in_heterogeneous_systems/molar_phase_diagrams.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "c05b952a", + "id": "cf1f5e65", "cell_type": "markdown", "source": [ "## Molar phase diagrams\n", @@ -24,7 +24,7 @@ "metadata": {} }, { - "id": "0f9a32f5", + "id": "8ea13828", "cell_type": "markdown", "source": [ "### Tie-lines and lever rule\n", @@ -110,7 +110,7 @@ "metadata": {} }, { - "id": "faa2022e", + "id": "5fd889c2", "cell_type": "markdown", "source": [ "### Phase diagrams with both potentials and molar quantities\n", @@ -340,7 +340,7 @@ "metadata": {} }, { - "id": "b03f070d", + "id": "86435606", "cell_type": "markdown", "source": [ "### Phase diagrams with only molar quantities\n", @@ -443,7 +443,7 @@ "metadata": {} }, { - "id": "2a1c4452", + "id": "b0c34261", "cell_type": "markdown", "source": [ "### Projection and section of phase diagrams with potential and molar quantities\n", diff --git a/src/psu410/src/phase_equilibria_in_heterogeneous_systems/potential_phase_diagrams.ipynb b/src/psu410/src/psu410/phase_equilibria_in_heterogeneous_systems/potential_phase_diagrams.ipynb similarity index 99% rename from src/psu410/src/phase_equilibria_in_heterogeneous_systems/potential_phase_diagrams.ipynb rename to src/psu410/src/psu410/phase_equilibria_in_heterogeneous_systems/potential_phase_diagrams.ipynb index ceb371c..021ee87 100644 --- a/src/psu410/src/phase_equilibria_in_heterogeneous_systems/potential_phase_diagrams.ipynb +++ b/src/psu410/src/psu410/phase_equilibria_in_heterogeneous_systems/potential_phase_diagrams.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "cb9fb312", + "id": "f2770bd3", "cell_type": "markdown", "source": [ "## Potential phase diagrams\n", @@ -54,7 +54,7 @@ "metadata": {} }, { - "id": "279783f1", + "id": "c4d45002", "cell_type": "markdown", "source": [ "### Potential phase diagrams of one-component systems\n", @@ -183,7 +183,7 @@ "metadata": {} }, { - "id": "5c6b46cf", + "id": "a261a7d1", "cell_type": "markdown", "source": [ "### Potential phase diagrams of two-component systems\n", @@ -295,7 +295,7 @@ "metadata": {} }, { - "id": "320479b8", + "id": "a8f22c51", "cell_type": "markdown", "source": [ "### Section of potential phase diagrams\n", diff --git a/src/psu410/src/references/index.ipynb b/src/psu410/src/psu410/references/index.ipynb similarity index 99% rename from src/psu410/src/references/index.ipynb rename to src/psu410/src/psu410/references/index.ipynb index 8bc0e0d..251b924 100644 --- a/src/psu410/src/references/index.ipynb +++ b/src/psu410/src/psu410/references/index.ipynb @@ -4,7 +4,7 @@ "metadata": {}, "cells": [ { - "id": "fdeb5814", + "id": "f5ad9e97", "cell_type": "markdown", "source": [ "# References\n",