Table 1 Classical models of particle growth by mass diffusion in a binary system. Only one species is tracked in the systema
Stefan type models | Mullins-Sekerka models | ||
|---|---|---|---|
Classical (equilibrium) model | Model with kinetic drag | Classical (equilibrium) model | Model with kinetic drag |
\( {\displaystyle \begin{array}{l}\mathrm{for}\ x\in {\Omega}_{\pm}\\ {}{\partial}_tc=-\nabla \cdot \mathbf{J}\\ {}\mathbf{J}=-M\;\nabla \mu (c)\\ {}\\ {}\mathrm{for}\ x\in \Gamma \\ {}{\mu}^{\alpha }={\mu}^{\beta }=\mu \\ {}\mathrm{v}{\left[c\right]}_{\alpha}^{\beta }={\left[\mathbf{J}\right]}_{\alpha}^{\beta}\cdot \mathbf{n}\\ {}\mu =0\end{array}} \) | \( {\displaystyle \begin{array}{l}\mathrm{for}\ x\in {\Omega}_{\pm}\\ {}{\partial}_tc=-\nabla \cdot \mathbf{J}\\ {}\mathbf{J}=-M\;\nabla \mu (c)\\ {}\\ {}\mathrm{for}\ x\in \Gamma \\ {}{\mu}^{\alpha }={\mu}^{\beta }=\mu \\ {}\mathrm{v}{\left[c\right]}_{\alpha}^{\beta }={\left[\mathbf{J}\right]}_{\alpha}^{\beta}\cdot \mathbf{n}\\ {}-\lambda \mathrm{v}=\gamma \kappa +{\left[f\right]}_{\alpha}^{\beta }-\mu {\left[c\right]}_{\alpha}^{\beta}\end{array}} \) | \( {\displaystyle \begin{array}{l}\mathrm{for}\ x\in {\Omega}_{\pm}\\ {}\nabla \cdot \mathbf{J}=0\\ {}\mathbf{J}=-M\;\nabla \mu (c)\\ {}\\ {}\mathrm{for}\ x\in \Gamma \\ {}{\mu}^{\alpha }={\mu}^{\beta }=\mu \\ {}\mathrm{v}{\left[c\right]}_{\alpha}^{\beta }={\left[\mathbf{J}\right]}_{\alpha}^{\beta}\cdot \mathbf{n}\\ {}\mu {\left[c\right]}_{\alpha}^{\beta }=\gamma \kappa \end{array}} \) | \( {\displaystyle \begin{array}{l}\mathrm{for}\ x\in {\Omega}_{\pm}\\ {}\nabla \cdot \mathbf{J}=0\\ {}\mathbf{J}=-M\;\nabla \mu (c)\\ {}\\ {}\mathrm{for}\ x\in \Gamma \\ {}{\mu}^{\alpha }={\mu}^{\beta }=\mu \\ {}\mathrm{v}{\left[c\right]}_{\alpha}^{\beta }={\left[\mathbf{J}\right]}_{\alpha}^{\beta}\cdot \mathbf{n}\\ {}\mu\;{\left[c\right]}_{\alpha}^{\beta }=\gamma \kappa +\lambda \mathrm{v}\end{array}} \) |