Dynamic transitions and pattern formations for a cahn-hilliard model with long-range repulsive interactions

Abstract

The main objective of this article is to study the order-disorder phase transition and pattern formation for systems with long-range repulsive interactions. The main focus is on a Cahn-Hilliard model with a nonlocal term in the corresponding energy functional, representing certain long-range repulsive interaction. We show that as soon as the trivial steady state loses its linear stability, the system always undergoes a dynamic transition of one of three types - continuous, catastrophic and random - forming different patterns/structures, such as lamellae, hexagonally packed cylinders, rectangles, and spheres. The types of transitions are dictated by a non-dimensional parameter, measuring the interactions between the long-range repulsive term and the quadratic and cubic nonlinearities in the model. In particular, the hexagonal pattern is unique to this long-range interaction, and it is captured by the corresponding two-dimensional reduced equations on the center manifold, which involve (degenerate) quadratic terms and non-degenerate cubic terms. Explicit information on the metastability and basins of attraction of different ordered states, corresponding to different patterns, are derived as well.