Delaunay-type interface in a screened model of diblock copolymer melts

Abstract

A diblock copolymer is a soft-matter composed of two chemically distinct block of repeating monomers covalently bonded together at an end-to-end junction to form a single polymer chain. In this paper, we establish the existence of infinitely many smooth periodic unbounded domain patterns of Delaunay-type in $\mathbb{R}^3$ that optimize the energy distribution in diblock copolymer melts. We emphasize that pattern domains at the equilibrium correspond to stationary sets of the screened Ohta--Kawasaki free energy functional \begin{align*} \mathcal{P}_γ(Ω) := |\partialΩ| + γ\int_Ω\int_Ω G_κ(|x-y|) \,\mathrm{d}x\mathrm{d}y, \end{align*} where $γ>0$, $κ>0$ and $G_κ(r)=\frac{1}{r} e^{-κr}$ is the repulisive Yukawa potential. Equivalently, these equilibria satisfy the corresponding Euler--Lagrange equation \begin{align*} \mathcal{H}_Ω(x):= H_{\partialΩ}(x) + γ\int_Ω G_κ(|x-y|) \mathrm{d}y = \textrm{Const} \quad \text{on } \partialΩ, \end{align*} where $H_{\partialΩ}$ denotes the mean curvature of the surface $\partialΩ$. By analyzing the linearization of $Ω\mapsto \mathcal{H}_Ω$ around flat cylinders and applying the Crandall--Rabinowitz bifurcation theorem, for any $κ> 0$ and sufficiently small $γ> 0$, we prove the existence of non-trivial, $2π$-periodic Delaunay-type equilibrium cylinder interfaces with shapes close to a Delaunay unduloid surface of constant mean curvature.