Dynamic Transitions and Pattern Formations for Cahn-Hilliard Model with Long-Range Repulsive Interactions

TL;DR

This paper investigates phase transitions and pattern formations in a nonlocal Cahn-Hilliard model with long-range repulsive interactions, revealing how stability loss leads to various pattern types and a unique hexagonal pattern due to interaction competition.

Contribution

It introduces a new explicit parameter for phase diagram analysis and uncovers a novel pattern selection mechanism specific to long-range interactions.

Findings

System undergoes dynamic transition upon stability loss

Hexagonal pattern is unique to long-range interactions

Explicit metastability and basin of attraction information

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 the Cahn-Hilliard model with a nonlocal term in the corresponding energy functional, representing the long-range repulsive interaction. First, we show that as soon as the linear problem loses stability, the system always undergoes a dynamic transition to one of the three types, forming different patterns/structures. The types of transition are then dictated by a nondimensional parameter, measuring the interactions between the long-range repulsive term and the quadratic and cubic nonlinearities in the model. The derived explicit form of this parameter offers precise information for the phase diagrams. Second, we obtain a novel and explicit pattern selection mechanism associated with the competition between the long-range repulsive interaction and the short-range attractive interactions. In particular, the hexagonal pattern is unique to the long-range interaction, and is associated with a novel two-dimensional reduced transition equations on the center manifold generated by the unstable modes, consisting of (degenerate) quadratic terms and non-degenerate cubic terms. Finally, explicit information on the metastability and basin of attraction of different disordered/ordered states and patterns are derived as well.