Mobility

The growth rate of precipitates is dependent on the diffusivity of each component in the system. kawin supports the usage of Calphad-based mobility models to determine both the mobility and diffusivity. This is handled by an extension of pycalphad’s Model class. The following equation shows how the mobility is determined by two terms: MF and MQ, where both are treated as Redlich-Kister polynomials. These two terms can be stored in a thermodynamic database (.tdb) file as MF and MQ.

$$M = \frac{M_F}{RT} \exp{\left(\frac{M_Q}{RT}\right)} = \frac{1}{RT} \exp{\left(M_F + \frac{M_Q}{RT}\right)}$$

In addition, diffusivity can be directly stored in .tdb files as DF and DQ.

$$D = D_F \exp{⁡\left(\frac{D_Q}{RT}\right)} = \exp{⁡\left(D_F + \frac{D_Q}{RT}\right)}$$

If using mobility terms, the diffusivities are determined from the chemical potential gradient. Using Onsager’s relationships and chain rule derivatives, the diffusivities can be expressed as a function of the chemical potential gradient with respect to the composition.

$$J_k = -\sum_{i}{L_{ki} \frac{\partial \mu_i}{\partial z}} = -\sum_{i}{L_{ki} \sum_{j}{\frac{\partial \mu_i}{\partial x_j} \frac{\partial x_j}{\partial z}}}$$

$$J_k = -\sum_{j}{D_{kj} \frac{\partial x_j}{\partial z}}$$

$$D_{kj} = \sum_{i}{L_{ki} \frac{\partial \mu_i}{\partial x_j}}$$

$$L_{ki} = \left(\delta_{ik} - x_k\right) x_i M_i$$

Due to the summation constraint, there are only n-1 independent components. Thus, a reference element can be defined and the interdiffusivities can be determined. Only n-1 components need to be tracked if interdiffusivities are used.

$$\sum_{j}{\frac{\partial x_j}{\partial z}} = 0$$

$$D_{kj}^n = D_{kj} - D_{kn} = \sum_{i}{L_{ki} \left(\frac{\partial \mu_i}{\partial x_j} - \frac{\partial \mu_i}{\partial x_n}\right)} \text{ if k is substituional}$$

$$D_{kj}^n = D_{kj} \text{ if k is interstitial}$$

$$J_k = -\sum_{i}^{n-1}{D_{kj}^n \frac{\partial x_j}{\partial z}}$$

References #

1. A. Borgenstam, A. Engstrom, L. Hoglund and J. Agren, “DICTRA, a tool for simulation of diffusional transformations in alloys” Journal of Phase Equilibria 21 (2000) p. 269

2. U. R. Kattner and C. E. Campbell, “Modelling of thermodynamics and diffusion in multicomponent systems” Materials Science and Technology 25 (2009) p. 443