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Table 1 Classical models of particle growth by mass diffusion in a binary system. Only one species is tracked in the systema

From: An analysis of two classes of phase field models for void growth and coarsening in irradiated crystalline solids

Stefan type models Mullins-Sekerka models
Classical (equilibrium) model Model with kinetic drag Classical (equilibrium) model Model with kinetic drag
\( {\displaystyle \begin{array}{l}\mathrm{for}\ x\in {\Omega}_{\pm}\\ {}{\partial}_tc=-\nabla \cdot \mathbf{J}\\ {}\mathbf{J}=-M\;\nabla \mu (c)\\ {}\\ {}\mathrm{for}\ x\in \Gamma \\ {}{\mu}^{\alpha }={\mu}^{\beta }=\mu \\ {}\mathrm{v}{\left[c\right]}_{\alpha}^{\beta }={\left[\mathbf{J}\right]}_{\alpha}^{\beta}\cdot \mathbf{n}\\ {}\mu =0\end{array}} \) \( {\displaystyle \begin{array}{l}\mathrm{for}\ x\in {\Omega}_{\pm}\\ {}{\partial}_tc=-\nabla \cdot \mathbf{J}\\ {}\mathbf{J}=-M\;\nabla \mu (c)\\ {}\\ {}\mathrm{for}\ x\in \Gamma \\ {}{\mu}^{\alpha }={\mu}^{\beta }=\mu \\ {}\mathrm{v}{\left[c\right]}_{\alpha}^{\beta }={\left[\mathbf{J}\right]}_{\alpha}^{\beta}\cdot \mathbf{n}\\ {}-\lambda \mathrm{v}=\gamma \kappa +{\left[f\right]}_{\alpha}^{\beta }-\mu {\left[c\right]}_{\alpha}^{\beta}\end{array}} \) \( {\displaystyle \begin{array}{l}\mathrm{for}\ x\in {\Omega}_{\pm}\\ {}\nabla \cdot \mathbf{J}=0\\ {}\mathbf{J}=-M\;\nabla \mu (c)\\ {}\\ {}\mathrm{for}\ x\in \Gamma \\ {}{\mu}^{\alpha }={\mu}^{\beta }=\mu \\ {}\mathrm{v}{\left[c\right]}_{\alpha}^{\beta }={\left[\mathbf{J}\right]}_{\alpha}^{\beta}\cdot \mathbf{n}\\ {}\mu {\left[c\right]}_{\alpha}^{\beta }=\gamma \kappa \end{array}} \) \( {\displaystyle \begin{array}{l}\mathrm{for}\ x\in {\Omega}_{\pm}\\ {}\nabla \cdot \mathbf{J}=0\\ {}\mathbf{J}=-M\;\nabla \mu (c)\\ {}\\ {}\mathrm{for}\ x\in \Gamma \\ {}{\mu}^{\alpha }={\mu}^{\beta }=\mu \\ {}\mathrm{v}{\left[c\right]}_{\alpha}^{\beta }={\left[\mathbf{J}\right]}_{\alpha}^{\beta}\cdot \mathbf{n}\\ {}\mu\;{\left[c\right]}_{\alpha}^{\beta }=\gamma \kappa +\lambda \mathrm{v}\end{array}} \)
  1. aThe equilibrium versions of the models assume the interface to be in local equilibrium and hence the growth is diffusion-controlled. The kinetic drag accounts for interface attachment kinetics. The Mullins-Sekerka models are the quasistatic limits of the Stefan problems valid for the case of low supersaturation. The Mullins-Sekerka models are usually used as the starting point for deriving the classical mean-field Lifshitz-Slyozov-Wagner (LSW) theories of Ostwald ripening (Lifshitz & Slyozov, 1961; Rahaman, 2003; Mullins & Sekerka, 1963; Niethammer, 2000; Dai et al., 2010)