Optimal periodic structures with general space group symmetries in the Ohta-Kawasaki problem

TL;DR

This paper develops a rigorous computational method to find and analyze periodic minimizers of the Ohta-Kawasaki energy, revealing qualitative differences from theoretical predictions in certain regimes.

Contribution

It introduces a method to rigorously prove existence and bounds of solutions for the Ohta-Kawasaki problem with general symmetries, including numerical exploration of phase space.

Findings

Existence of periodic minimizers with prescribed symmetries established.

Quantitative bounds on approximation errors and energy differences provided.

Qualitative differences observed between phase diagrams and self-consistent field theory away from weak segregation.

Abstract

We consider the problem of rigorously computing periodic minimizers to the Ohta-Kawasaki energy. We develop a method to prove existence of solutions and determine rigorous bounds on the distance between our numerical approximations and the true infinite dimensional solution and also on the energy. We use a method with prescribed symmetries to explore the phase space, computing candidate minimizers both with and without experimentally observed symmetries. We find qualitative differences between the phase diagram of the Ohta-Kawasaki energy and self consistent field theory when well away form the weak segregation limit.