Pattern formation for a local/nonlocal interaction functional arising in colloidal systems

TL;DR

This paper analyzes a mathematical model combining local and nonlocal interactions to explain pattern formation in colloidal systems, proving that minimizers form periodic stripes across dimensions.

Contribution

It establishes that minimizers of a specific local/nonlocal interaction functional are periodic stripes, extending previous methods to a new physical model.

Findings

Minimizers are periodic stripes in any dimension.

The model is a Gamma-limit of a double Yukawa interaction.

Pattern formation is explained through energy minimization.

Abstract

In this paper we study pattern formation for a physical local/nonlocal interaction functional where the local attractive term is given by the $1$-perimeter and the nonlocal repulsive term is the Yukawa (or screened Coulomb) potential. This model is physically interesting as it is the $\Gamma$-limit of a double Yukawa model used to explain and simulate pattern formation in colloidal systems \cite{BBCH,CCA,IR,GCLW}. Following a strategy introduced in~\cite{DR} we prove that in a suitable regime minimizers are periodic stripes, in any space dimension.