Model of crystal growth

\[\begin{split}\begin{aligned} \tau(\mathbf{n})\frac{\partial\phi}{\partial t} & =W_{0}^{2}\pmb{\nabla}\cdot(a_{s}^{2}(\mathbf{n})\pmb{\nabla}\phi)+W_{0}^{2}\pmb{\nabla}\cdot\pmb{\mathcal{N}}+(\phi-\phi^{3})-\lambda u(1-\phi^{2})^{2},\label{eq:PhaseField_KarmaRappel}\\ \frac{\partial u}{\partial t} & =\kappa\pmb{\nabla}^{2}u+\frac{1}{2}\frac{\partial\phi}{\partial t},\label{eq:Temp_Ramirez}\end{aligned}\end{split}\]

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 \([\mathscr{L}]^{2}/[\mathcal{T}]\).