Cahn-Hilliard Equations on Lattices: Dynamic Transitions and Pattern Formations

TL;DR

This paper investigates how lattice geometry and physical parameters influence pattern formation and phase transitions in binary systems modeled by the Cahn-Hilliard equation, revealing geometry-dependent patterns like hexagons, rolls, and squares.

Contribution

It provides a detailed analysis of dynamic transitions and pattern formations on two-dimensional lattices, highlighting the impact of domain geometry and parameters on stability and pattern types.

Findings

Emergence of hexagonally-packed circles, rolls, and squares depending on domain geometry.

Transition characterized by domain geometry and physical parameters.

Long-range interactions produce similar pattern formation results.

Abstract

This article examines the dynamic phase transitions and pattern formations attributed to binary systems modeled by the Cahn-Hilliard equation. In particular, we consider a two-dimensional lattice structure and determine how different choices of the spanning vectors influence the resulting dynamical tramsitions and pattern formations. As the basic steady-state loses its linear stability, the binary system undergoes a dynamic transition which is shown to be characterized by both the geometry of the domain and the choice of physical parameters of the model. Unlike rectangular domains, we are able to observe the emergence of hexagonally-packed circles, as well as the familiar rolls and square structures. We begin with the decomposition of our function space into a stable and unstable eigenspace before calculating the center manifold that maps the former to the later. In analyzing the resulting reduced equations, we consider the different multiplicities that the critical eigenvalue can have, which is shown to be geometry-dependent. We briefly consider the long-range interaction model and determine that it produces similar results to the original model.