Metastable internal layer dynamics for the viscous Cahn–Hilliard equation

Abstract

A formal asymptotic method is used to derive a differential-algebraic system of equations characterizing the metastable motion of a pattern of n (n ≥ 2) internal layers for the one-dimensional viscous Cahn-Hilliard modeling slow phase separation. Similar slow motion results are obtained for the Cahn-Hilliard equation and the constrained Allen-Cahn equation by introducing a homotopy parameter into the viscous Cahn-Hilliard equation and letting this parameter take on limiting values. For each of these phase separation models, the asymptotic results for the slow internal layer motion associated with two-layer metastable patterns are found to compare very favorably over very long time intervals with corresponding full numerical results computed using a finite-difference scheme. Finally, an example is given that clearly illustrates the very sensitive effect of boundary conditions on metastable internal layer dynamics.