On the Ternary Ohta-Kawasaki Free Energy and Its One-dimensional Global Minimizers

TL;DR

This paper investigates the one-dimensional global minimizers of the ternary Ohta-Kawasaki free energy, revealing that noncyclic patterns can be energetically favored over cyclic ones depending on parameters, thus challenging previous conjectures.

Contribution

It demonstrates that the conjecture of cyclic patterns as minimizers does not always hold and introduces a generalized charge interpretation to unify the model's parameters.

Findings

Noncyclic patterns can be global minimizers for certain parameters.

The cyclic pattern conjecture does not universally apply.

A reformulation of the long-range term clarifies conditions for physically relevant structures.

Abstract

We study the ternary Ohta-Kawasaki free energy that has been used to model triblock copolymer systems. Its one-dimensional global minimizers are conjectured to have cyclic patterns. However, some physical experiments and computer simulations found triblock copolymers forming noncyclic lamellar patterns. In this work, by comparing the free energies of the cyclic pattern and some noncyclic candidates, we show that the conjecture does not hold for some choices of parameters. Our results suggest that even in one dimension, the global minimizers may take on very different patterns in different parameter regimes. To unify the existing choices of the long range coefficient matrix, we present a reformulation of the long range term using a generalized charge interpretation, and thereby propose conditions on the matrix in order for the global minimizers to reproduce physically relevant nanostructures of block copolymers.