Front migration in the nonlinear Cahn-Hilliard equation

Abstract

The method of matched asymptotic expansions is used to describe solutions of the nonlinear Cahn-Hilliard equation for phase separation in N > 1 space dimensions. The expansion is formally valid when the thickness of internal transition layers is small compared with the distance separating layers and with their radii of curvature. On the dominant (slowest) timescale the interface velocity is determined by the mean curvature of the interface, by a non-local relation which is identical to that in a well-known quasi-static model of solidification, which exhibits a shape instability discovered by Mullins & Sekerka (J. appl. Phys. 34, 323-329 (1963)). On a faster timescale, the Cahn-Hilliard equation regularizes a classic two-phase Stefan problem. Similarity solutions of the two-phase Stefan problem should describe boundary layers. Existence and uniqueness of such similarity solutions which admit metastable states is proved rigorously in an appendix.