DCModeling

Using the Z-Z-Z binary model to model diffusion coefficients in binary systems.

Table of Contents

• Equations
  • End Member Diffusion Coefficients
  • Gibbs Energy
  • Thermodynamic Factor
  • Tracer Diffusion Coefficients
  • Intrinsic Diffusion Coefficients
  • Inter-diffusion Coefficient (Darken Equation)
  • Objective Function
• Results
• Installation

Equations

For a binary system with A and B elements.

End Member Diffusion Coefficients

Arrhenius equation (single-term):

$$D = D_0 \exp\left(-\frac{Q}{RT}\right)$$

Two-term form:

$$D = D_0 \exp\left(-\frac{Q_0}{RT}\right) + D_1 \exp\left(-\frac{Q_1}{RT}\right)$$

Brown-Ashby correlation (structure-dependent estimation):

$$D = D_0 \exp\left(-K \frac{T_m}{T}\right)$$

where $D_0$ and $K$ are constants that depend on crystal structure (FCC, BCC, HCP, etc.) and $T_m$ is the melting temperature.

Gibbs Energy

$$G = G^{\text{mech}} + G^{\text{ideal}} + G^{\text{excess}} + G^{\text{mag}}$$

Mechanical mixing:

$$G^{\text{mech}} = x_A G_A + x_B G_B$$

Ideal mixing:

$$G^{\text{ideal}} = RT\left(x_A \ln x_A + x_B \ln x_B\right)$$

Excess (Redlich-Kister):

$$G^{\text{excess}} = x_A x_B \sum_k L_k \left(x_A - x_B\right)^k$$

Magnetic contribution:

$$G^{\text{mag}} = RT \ln(\beta + 1) f(\tau)$$

where $\tau = T / T_c$, $T_c$ is the Curie temperature, $\beta$ is the Bohr magneton number, and $f(\tau)$ is the Hillert-Jarl function with structure-dependent parameter $p$ (0.28 for FCC/HCP, 0.4 for BCC).

Thermodynamic Factor

$$\Psi = \frac{x_A x_B}{RT} \frac{\partial^2 G}{\partial x^2}$$

Tracer Diffusion Coefficients

$$D^{*}_{A} = \exp\left(x_A \ln D_{AA} + x_B \ln D_{AB} + \frac{\Phi_A x_A x_B}{RT}\right)$$

$$D^{*}_{B} = \exp\left(x_A \ln D_{BA} + x_B \ln D_{BB} + \frac{\Phi_B x_A x_B}{RT}\right)$$

where $D_{AA}$, $D_{AB}$, $D_{BA}$, $D_{BB}$ are end member diffusion coefficients (self- and impurity diffusion), and $\Phi$ is the interaction parameter. Supported model variants:

Model	Parameters
1-para	$\Phi_A = \Phi_B = \Phi$
2-para	$\Phi_A$, $\Phi_B$ independent
4-para	$\Phi_A = a_0 + a_1 T$, $\Phi_B = b_0 + b_1 T$

Intrinsic Diffusion Coefficients

$$D^{I}_{A} = \Psi \cdot D^{*}_{A}, \qquad D^{I}_{B} = \Psi \cdot D^{*}_{B}$$

where $\Psi$ is the thermodynamic factor.

Inter-diffusion Coefficient (Darken Equation)

$$\tilde{D} = x_B D^{I}_{A} + x_A D^{I}_{B}$$

Objective Function

Weighted mean squared error:

$$\text{MSE} = \frac{1}{N} \sum_{i=1}^{N} \left[w_i \ln\left(\frac{D_i^{\text{pred}}}{D_i^{\text{exp}}}\right)\right]^2$$

Results

Thermodynamic factor calculation

Installation

Install required packages

pip install -r requirements.txt

Add support for ThermoCalc

• Install ThermoCalc (with valid license connection)
  • Install tc-python into preferred python environment
    • For further instructions, see the help file at C:/Program Files/Thermo-Calc/<version>/HTML5/content/installation/sdks/tc-python-install-advanced.htm
    • In summary, start the ThermoCalc GUI once (with valid license), and a Python wheel will be created at a path such as C:\Users\<YourUser>\Documents\Thermo-Calc\<version>\SDK\TC-Python\ (on Linux, this will be at /home/YourUser/Thermo‑Calc/<version>/SDK/TC-Python)
    • Install this wheel package into your environment using a command like the following:

      pip install C:\Users\<YourUser>\Documents\Thermo-Calc\<version>\SDK\TC-Python\TC_Python-<version>-py3-none-any.whl