Model of crystal growth

Mathematical model

(95)\[\tau(\mathbf{n})\frac{\partial\phi}{\partial t}=W_{0}^{2}\pmb{\nabla}\cdot(a_{s}^{2}(\mathbf{n})\pmb{\nabla}\phi)+W_{0}^{2}\pmb{\nabla}\cdot\pmb{\mathcal{N}}+(\phi-\phi^{3})-\lambda u(1-\phi^{2})^{2}\]

(96)\[\frac{\partial u}{\partial t}=\kappa\pmb{\nabla}^{2}u+\frac{1}{2}\frac{\partial\phi}{\partial t}\]

where \(\pmb{\mathcal{N}}\equiv\pmb{\mathcal{N}}(\mathbf{x},\,t)\) is defined by:

\[\pmb{\mathcal{N}}(\mathbf{x},\,t)=\bigl|\pmb{\nabla}\phi\bigr|^{2}a_{s}(\mathbf{n})\left(\frac{\partial a_{s}(\mathbf{n})}{\partial(\partial_{x}\phi)},\,\frac{\partial a_{s}(\mathbf{n})}{\partial(\partial_{y}\phi)},\,\frac{\partial a_{s}(\mathbf{n})}{\partial(\partial_{z}\phi)}\right)^{T}.\label{eq:TermesAnisotropes}\]

The anisotropy function \(a_{s}(\mathbf{n})\) (dimensionless) for a growing direction \(\left\langle 100\right\rangle\) is:

\[a_{s}(\mathbf{n})=1-3\varepsilon_{s}+4\varepsilon_{s}\sum_{\alpha=x,y,z}n_{\alpha}^{4}.\label{eq:As_Function_Classical}\]

In Eq. ([eq:PhaseField_KarmaRappel]), \(\phi\) is the phase-field, \(W_{0}\) is the interface thickness, \(\lambda\) is the coupling coefficient with the normalized temperature \(u(\mathbf{x},\,t)\). The normal vector \(\mathbf{n}\equiv\mathbf{n}(\mathbf{x},\,t)\) is defined such as:

\[\mathbf{n}(\mathbf{x},\,t)=-\frac{\pmb{\nabla}\phi}{\bigl|\pmb{\nabla}\phi\bigr|},\label{eq:Def_Normal_Vector}\]

directed from the solid to the liquid. The coefficient \(\tau(\mathbf{n})\) is the kinetic coefficient of the interface, it is defined as \(\tau(\mathbf{n})=\tau_{0}a_{s}^{2}(\mathbf{n})\) where \(\tau_{0}\) is the kinetic characteristic time. Let us notice that each term of Eq. ([eq:PhaseField_KarmaRappel]) is dimensionless. The physical dimensions of \(W_{0}\), \(\pmb{\mathcal{N}}\), \(\tau_{0}\) and \(\lambda\) are respectively \([W_{0}]\equiv[\mathscr{L}]\), \([\pmb{\mathcal{N}}]\equiv[\mathscr{L}]^{-1}\), \([\tau_{0}]\equiv[\mathcal{T}]\) and \([\lambda]\equiv[-]\), where \([\mathscr{L}]\) indicates the length dimension and \([\mathcal{T}]\) indicates the time dimension. In Eq. ([eq:Temp_Ramirez]), the physical dimension of \(\kappa\) is \([\mathscr{L}]^{2}/[\mathcal{T}]\).

Model of dissolution

Mathematical model of dissolution

The mathematical model of dissolution is derived in [1].

Mathematical model

The model 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

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

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

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

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:

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

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:

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

Finally in Eq. (99), \(\boldsymbol{j}_{at}\) is the anti-trapping current which is defined by:

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

Closure relationships

Input parameters in .ini file