Model of dissolution

Mathematical model of dissolution

The mathematical model of dissolution is composed of two coupled PDEs. The first one for interface-tracking on \(\phi\) and the second one on the composition \(c\). The interface movement is based on the Allen-Cahn equation

(102)\[\frac{\partial\phi}{\partial t}=M_{\phi}\boldsymbol{\nabla}^{2}\phi-\frac{M_{\phi}}{W^{2}}2\phi(1-\phi)(1-2\phi)+\frac{\lambda M_{\phi}}{W^{2}}\mathscr{S}_{\phi}(\phi,\,\overline{\mu})\]

where \(W\) is the interface width, \(M_{\phi}\) is the interface mobility, and \(M_{\lambda}\) is the coupling coefficient which is related to the interface surface tension. The source term \(\mathscr{S}_{\phi}(\phi,\,\overline{\mu})\) is defined by

(103)\[\mathscr{S}_{\phi}(\phi,\,\overline{\mu})=6\phi(1-\phi)(c_{s}^{eq}-c_{l}^{eq})(\overline{\mu}-\overline{\mu}^{eq})\]

where \(c_{s}^{eq}\) and \(c_{l}^{eq}\) are equilibrium compositions in solid and liquid respectively. They are both scalar values. \(\overline{\mu}^{eq}\) is the equilibrium chemical potentiel which is also a scalar value. \(\overline{\mu}\) is the chemical potential which is involved in the diffusion equation below:

(104)\[\frac{\partial c}{\partial t}=\boldsymbol{\nabla}\cdot\left[D_{l}\phi\boldsymbol{\nabla}\overline{\mu}-\boldsymbol{j}_{at}(\phi,\,\overline{\mu})\right]\]

where \(c\) is the composition, \(D_l\) is the diffusion coefficient in the liquid phase. The flux is given by the gradient of chemical potential. It is related to the composition by:

(105)\[\overline{\mu}=\overline{\mu}^{eq}+c(\phi,\,\overline{\mu})-\left[c_{l}^{eq}\phi+c_{s}^{eq}\left(1-\phi\right)\right]\]

Finally in Eq. (104), \(\boldsymbol{j}_{at}\) is the anti-trapping current which is defined by:

(106)\[\boldsymbol{j}_{at}=\frac{1}{4}W(c_{s}^{eq}-c_{l}^{eq})\frac{\partial\phi}{\partial t}\boldsymbol{n}\]

Closure relationships

Input parameters in .ini file