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

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.

35 pages; 12 figures; appeared at AMS 2021 Fall Southeastern Sectional Meeting; minor typos corrected; several references added

References in corpus (5)

• Pattern formation in systems with competing interactions
• Lamellar phase solutions for diblock copolymers with nonlocal diffusions
• Nonlocal Effect on a Generalized Ohta-Kawasaki Model
• Striped patterns for generalized antiferromagnetic functionals with power law kernels of exponent smaller than $d+2$
• Periodic Minimizers of a Ternary Non-Local Isoperimetric Problem

Cited by in corpus (2)

• Ohta-Kawasaki energy for amphiphiles: asymptotics and phase-field simulations
• Bifurcation and fission in the liquid drop model: a phase-field approach