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.

arXiv comment35 pages; 12 figures; appeared at AMS 2021 Fall Southeastern Sectional Meeting; minor typos corrected; several references added

References in corpus (17)

• Theory of domain patterns in systems with long-range interactions of Coulomb type
• Droplet phases in non-local Ginzburg-Landau models with Coulomb repulsion in two dimensions
• The $\mathbf{\Gamma }$-limit of the two-dimensional Ohta-Kawasaki energy. I. Droplet density
• Low density phases in a uniformly charged liquid
• Dissipative Particle Dynamics for Directed Self-Assembly of Block Copolymers
• Exact periodic stripes for a minimizers of a local/non-local interaction functional in general dimension
• Periodic striped ground states in Ising models with competing interactions
• A nonlocal isoperimetric problem with dipolar repulsion
• The non-centrosymmetric lamellar phase in blends of ABC triblock and ac diblock copolymers
• Pattern formation for a local/nonlocal interaction functional arising in colloidal systems
• Pattern formation in systems with competing interactions
• Lamellar phase solutions for diblock copolymers with nonlocal diffusions
• Non-spherical equilibrium shapes in the liquid drop model
• Nonlocal Effect on a Generalized Ohta-Kawasaki Model
• One-dimensionality of the minimizers for a diffuse interface generalized antiferromagnetic model in general dimension
• 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