Phase separation in multiply periodic materials with fine microstructures

TL;DR

This paper analyzes phase separation in complex composite materials with multiple microstructural scales, proving that the energy converges to a perimeter-based limit using multi-scale homogenization techniques.

Contribution

It introduces a multi-scale homogenization approach for the Cahn-Hilliard model in materials with microstructures smaller than the phase separation scale.

Findings

Gamma-limit of energy is a multiple of the perimeter

Unfolding operator effectively handles multiple scales

Limit process can be performed sequentially from smaller to larger scales

Abstract

We study a Cahn-Hilliard model for phase separation in composite materials with multiple periodic microstructures. These are modeled by considering a highly oscillating potential. The focus of this paper is in the case where the scales of the microstructures are smaller than that of phase separation. We provide a compactness result and prove that the {\Gamma}-limit of the energy is a multiple of the perimeter. In particular, using the recently introduced unfolding operator for multiple scales, we show that the taking the limit of all of the scales together is equivalent to taking one limit at the time, starting from the smaller scale and keeping the larger fixed.