Homogenization of a phase field model for binary mixtures

The aim of this article is the derivation of a two-scale model that describes the evolution of equiaxed dendritic microstructure in liquid-solid phase transitions of binary mixtures. The approach is based on a phase field model proposed by Caginalp and Xie [Arch. Rational Mech. Anal., 142 (1998), pp. 293-329]. Assuming a periodic initial distribution of solid kernels with a period of scale epsilon > 0 and scaling certain physical parameters in the dependence of E, a formal asymptotic expansion is carried out. The result is a two-scale model that consists of a global homogenized heat equation, and, at each point of the macroscopic domain, local cell problems for the evolution of single solid crystals. In order to justify the asymptotic expansion, an estimate for the difference of the solution of the two-scale model and the solution of the original model of scale E is derived. This estimate shows an error of order epsilon(1/2). Finally, the two-scale model is illustrated by the results of numerical solutions of the two-scale model for some simple examples.