Domain formation in magnetic polymer composites: an approach via stochastic homogenization

TL;DR

This paper analyzes the magnetic energy behavior in polymer composites with randomly distributed magnetic particles, using stochastic homogenization and $ ext{Gamma}$-convergence to derive a deterministic continuum limit.

Contribution

It introduces a stochastic lattice model for magnetic particles and proves the $ ext{Gamma}$-convergence of the energy to a deterministic homogenized limit under ergodic assumptions.

Findings

The $ ext{Gamma}$-limit is finite on magnetic domain partitions.

The limit energy has a local integral structure.

The homogenized energy is deterministic under ergodicity.

Abstract

We study the magnetic energy of magnetic polymer composite materials as the average distance between magnetic particles vanishes. We model the position of these particles in the polymeric matrix as a stochastic lattice scaled by a small parameter $\varepsilon$ and the magnets as classical $\pm 1$ spin variables interacting via an Ising type energy. Under surface scaling of the energy we prove, in terms of $\Gamma$-convergence that, up to subsequences, the (continuum) $\Gamma$-limit of these energies is finite on the set of Caccioppoli partitions representing the magnetic Weiss domains where it has a local integral structure. Assuming stationarity of the stochastic lattice, we can make use of ergodic theory to further show that the $\Gamma$-limit exists and that the integrand is given by an asymptotic homogenization formula which becomes deterministic if the lattice is ergodic.