A two‐scale model for liquid‐phase epitaxy
C. F. Eck, Heike Emmerich · Mathematical Methods in the Applied Sciences · 2008
Abstract We study a model for liquid‐phase epitaxy that is based on a continuum description of the transport processes in the liquid and a Burton–Cabrera–Frank (BCF) model for the growth of the solid by epitaxy. In order to develop a model that is capable to incorporate structures of a very small scale in the solid phase within a computation for a technically relevant macroscopic length scale, we apply homogenization methods. The result of the homogenization procedure is a two‐scale model that consists of macroscopic equations for fluid flow and solute diffusion in the fluid volume, coupled to microscopic BCF models for the evolution of the microstructure in the solid phase. The obtained two‐scale model is justified by an estimate for the model error that is valid under appropriate assumptions on the regularity of the solutions. This estimate is proved for a phase field approximation of the BCF model. Copyright © 2008 John Wiley & Sons, Ltd.