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}_\gamma(\Omega) := |\partial\Omega| + \gamma \int_{\Omega}\int_{\Omega} G_{\kappa}(|x-y|) \,\mathrm{d}x\mathrm{d}y, \end{align*} where $\gamma>0$, $\kappa>0$ and $G_\kappa(r)=\frac{1}{r} e^{-\kappa r}$ is the repulisive Yukawa potential. Equivalently, these equilibria satisfy the corresponding Euler--Lagrange equation \begin{align*} \mathcal{H}_\Omega (x):= H_{\partial\Omega}(x) + \gamma \int_{\Omega} G_{\kappa}(|x-y|) \mathrm{d}y = \textrm{Const} \quad \text{on } \partial\Omega, \end{align*} where $H_{\partial\Omega}$ denotes the mean curvature of the surface $\partial\Omega$. By analyzing the linearization of $\Omega \mapsto \mathcal{H}_\Omega$ around flat cylinders and applying the Crandall--Rabinowitz bifurcation theorem, for any $\kappa > 0$ and sufficiently small $\gamma > 0$, we prove the existence of non-trivial, $2\pi$-periodic Delaunay-type equilibrium cylinder interfaces with shapes close to a Delaunay unduloid surface of constant mean curvature.