Single phase diffusion
Diffusion within a single phase can modeled using interdiffusivities. Flux of a solute species depends on the mobility and chemical potential gradient or the diffusivity and the concentration gradient.
$$ J_k = -\sum_j{L_{kj} \frac{\partial \mu_j}{\partial z}} = -\sum_j{D_{kj} \frac{\partial x_j}{\partial z}} $$
In a volume fixed frame, $\sum_{k \epsilon S}{J_k^v} = 0$. The constraint results in only n-1 components being independent. By using interdiffusivities, only n-1 components need to be tracked, reducing the number of computations needed.
$$ D_{kj}^n = D_{kj} - D_{kn} $$
$$ J_k^v = -\sum_i^{n-1}{D_{kj}^n \frac{\partial x_j}{\partial z}} $$
More information regarding the relationship between mobility, diffusivities and interdiffusivities can be found in the mobility documentation.
The built-in kawin meshes uses the finite volume method to solve Fick’s second law, where the domain is split up into small cells each holding a concentration at the center. Fluxes between the cells are calculated and the concentration is updated by the flux of solute entering and leaving the cell. Since the fluxes are calculated at the edges of the cells, the diffusivity has to be calculated there as well. In kawin, the diffusivity is calculated at the cell centers, then interpolating at the edges.
$$ x_k^{x,t+1} = x_k^{x,t} - \left(\frac{J_k^{x+1,t} - J_k^{x,t}}{\Delta z}\right) \Delta t $$
The finite volume method is a conservative method in that the total amount of solute in the system will be conserved. However, this does not guarantee numerical stability. Since the diffusivities are already calculated when determining the fluxes, an adequate time step can be determined by the von Neumann condition. In kawin, the $\frac{1}{2}$ is replaced with a slightly smaller value of 0.4.
$$ \Delta t \leq \frac{1}{2} \frac{\Delta x^2}{\text{max}\left( D_{jk}^n \right)} $$
References
- 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