A homogenization result in the gradient theory of phase transitions

TL;DR

This paper investigates how small-scale heterogeneities influence phase transitions in fluids, showing that as heterogeneity scale diminishes, the resulting interfacial energy becomes anisotropic, blending homogenization with phase transition theory.

Contribution

It provides a homogenization result in the gradient theory of phase transitions, specifically addressing the case where heterogeneity scale matches the phase transition scale.

Findings

Limit of the interfacial energy becomes anisotropic as heterogeneity scale approaches zero.

Homogenization interacts with phase transition processes to produce anisotropic effects.

The model extends understanding of phase transitions in heterogeneous media.

Abstract

A variational model in the context of the gradient theory for fluid-fluid phase transitions with small scale heterogeneities is studied. In particular, the case where the scale $\varepsilon$ of the small homogeneities is of the same order of the scale governing the phase transition is considered. The interaction between homogenization and the phase transitions process will lead, in the limit as $\varepsilon\to0$, to an anisotropic interfacial energy.