Phase-Field Modeling and Simulation of the Zone Melting Purification Process

Zone melting is an efficient purification method that is widely used in the manufacture of semiconductors. Recently, it has been used in the purification of industrial phosphorus [Z. Zhang et al., Separation Purification Tech., 98 (2012), pp. 249--254]. It is a process consisting of both the melting-solidification process and the solute diffusion process. There is a great demand for a quantitative understanding of the underlying mechanism. In this paper, we start from a two-component, two-phase physical system and propose a variational phase-field model to model the zone melting process. The model consisting of the Allen--Cahn equation, the heat equation, and the solute diffusion equation is solved numerically. To overcome the stiffness caused by the nonlinear term in the Allen--Cahn equation, we propose a semi-implicit scheme based on a convex splitting of the free energy. By a proper splitting of the free energy functional, we show that our semi-implicit scheme is unconditionally energy stable. Numerica...

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