On the Thermodynamics of the Swift-Hohenberg Theory

TL;DR

This paper develops a thermodynamic framework for the Swift-Hohenberg equation, introducing microforces and microstresses, and demonstrates their interactions through detailed numerical simulations.

Contribution

It formulates a consistent thermodynamic theory for the Swift-Hohenberg equation, including microforces and microstresses, and extends the theory via a new constitutive process.

Findings

Explicit microstress structure derived for second-order free-energy functional

Thermodynamic consistency achieved with free-energy imbalance

Numerical simulation illustrates microstress interactions in evolution

Abstract

We present the microbalance including the microforces, the first- and second-order microstresses for the Swift--Hohenberg equation concomitantly with their constitutive equations, which are consistent with the free-energy imbalance. We provide an explicit form for the microstress structure for a free-energy functional endowed with second-order spatial derivatives. Additionally, we generalize the Swift--Hohenberg theory via a proper constitutive process. Finally, we present one highly-resolved three-dimensional numerical simulation to demonstrate the particular form of the resulting microstresses and their interactions in the evolution of the Swift--Hohenberg equation.