Complex pattern formation governed by a Cahn-Hilliard-Swift-Hohenberg system: Analysis and numerical simulations

TL;DR

This paper analyzes a complex mathematical model combining Cahn-Hilliard and Swift-Hohenberg equations, establishing solution existence, and demonstrating its ability to simulate intricate pattern formations in materials science.

Contribution

It introduces a novel extended model with logarithmic potentials, proving solution existence and developing a stable finite element method for simulating complex patterns.

Findings

Model captures phase separation and pattern formation

Numerical simulations reproduce natural pigment and material patterns

Enhanced physical accuracy with logarithmic potentials

Abstract

This paper investigates a Cahn-Hilliard-Swift-Hohenberg system, focusing on a three-species chemical mixture subject to physical constraints on volume fractions. The resulting system leads to complex patterns involving a separation into phases as typical of the Cahn-Hilliard equation and small scale stripes and dots as seen in the Swift-Hohenberg equation. We introduce singular potentials of logarithmic type to enhance the model's accuracy in adhering to essential physical constraints. The paper establishes the existence and uniqueness of weak solutions within this extended framework. The insights gained contribute to a deeper understanding of phase separation in complex systems, with potential applications in materials science and related fields. We introduce a stable finite element approximation based on an obstacle formulation. Subsequent numerical simulations demonstrate that the model allows for complex structures as seen in pigment patterns of animals and in porous polymeric materials.