A diffuse-interface approximation for step flow in epitaxial growth

We consider a step-flow model for epitaxial growth, as proposed by Burton et al. This type of model is discrete in the growth direction but continuous in the lateral directions. The effect of the Ehrlich-Schwoebel barrier, which limits the attachment rate of adatoms to a step from an upper terrace, is included. Mathematically, this model is a 2+1-dimensional dynamic free boundary problem for the steps. In this paper, we propose a diffuse-interface approximation which reproduces an arbitrary Ehrlich-Schwoebel barrier. This is achieved by introducing a degenerate mobility into the so-called viscous Cahn-Hilliard equation. We relate this modified Cahn-Hilliard equation to the sharp interface model via formal matched asymptotic expansion.