Hexagonal patterns in a simplified model for block copolymers

TL;DR

This paper introduces a simplified, analyzable model for two-dimensional block copolymer patterns that reproduces hexagonal structures, with numerical and theoretical evidence showing particles arrange on a triangular lattice.

Contribution

The paper derives a new nonlocal energy model from a continuum model using Gamma-convergence, capable of predicting hexagonal patterns in block copolymers.

Findings

Particles tend to form a triangular lattice as their number increases.

Numerical minimization confirms hexagonal packing patterns.

The model's predictions align with experimental observations.

Abstract

In this paper we study a new model for patterns in two dimensions, inspired by diblock copolymer melts with a dominant phase. The model is simple enough to be amenable not only to numerics but also to analysis, yet sophisticated enough to reproduce hexagonally packed structures that resemble the cylinder patterns observed in block copolymer experiments. Starting from a sharp-interface continuum model, a nonlocal energy functional involving a Wasserstein cost, we derive the new model using Gamma-convergence in a limit where the volume fraction of one phase tends to zero. The limit energy is defined on atomic measures; in three dimensions the atoms represent small spherical blobs of the minority phase, in two dimensions they represent thin cylinders of the minority phase. We then study minimisers of the limit energy. Numerical minimisation is performed in two dimensions by recasting the problem as a computational geometry problem involving power diagrams. The numerical results suggest that the small particles of the minority phase tend to arrange themselves on a triangular lattice as the number of particles goes to infinity. This is proved in the companion paper Bourne, Peletier & Theil and agrees with patterns observed in block copolymer experiments. This is a rare example of a nonlocal energy-driven pattern formation problem in two dimensions where it can be proved that the optimal pattern is periodic.