The Dynamics and Coarsening of Interfaces for the ViscousCahn - Hilliard Equation in One Spatial

In one spatial dimension, the metastable dynamics and coarsening process of an n-layer pattern of internal layers is studied for the Cahn-Hilliard equation, the viscous Cahn-Hilliard equation, and the constrained Allen-Cahn equation. These models from the continuum theory of phase transitions provide a caricature of the physical process of the phase separation of a binary alloy. A homotopy parameter is used to encapsulate these three phase separation models into one parameter-dependent model. By studying a diierential-algebraic system of ordinary diierential equations describing the locations of the internal layers for a metastable pattern for this parameter-dependent model, we are able to provide detailed comparisons between the internal layer dynamics for the three models. Layer collapse events are studied in detail and the analytical theory is supplemented by numerical results showing the diierent behaviors for the diierent models. Finally, an asymptotic-numerical algorithm, based on our asymptotic information of layer collapse events and the conservation of mass condition, is devised to characterize the entire coarsening process for each of these models. Numerical realizations of this algorithm are shown.