Sharp interface limit of an energy modelling nanoparticle-polymer blends

TL;DR

This paper derives the limiting behavior of an energy model for nanoparticle-polymer blends as particle number increases and sizes decrease, revealing the structure of phase boundaries and nanoparticle distribution effects.

Contribution

It identifies the Gamma-limit of the energy model, incorporating interface perimeter and nanoparticle density effects, and analyzes local minimizers and pattern formations.

Findings

Limiting energy includes interface perimeter and nanoparticle density penalization.

Local minimizers have regular phase boundaries.

Patterns vary from classical isoperimetric minimizers.

Abstract

We identify the $\Gamma$-limit of a nanoparticle-polymer model as the number of particles goes to infinity and as the size of the particles and the phase transition thickness of the polymer phases approach zero. The limiting energy consists of two terms: the perimeter of the interface separating the phases and a penalization term related to the density distribution of the infinitely many small nanoparticles. We prove that local minimizers of the limiting energy admit regular phase boundaries and derive necessary conditions of local minimality via the first variation. Finally we discuss possible critical and minimizing patterns in two dimensions and how these patterns vary from global minimizers of the purely local isoperimetric problem.