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 $\gamma >0$, $\kappa >0$ 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}_Ω(x):= H_{\partialΩ}(x) + γ\int_Ω G_κ(|x-y|) \mathrm{d}y = \textrm{Const} \quad \text{on } \partialΩ, \end{align*} where $H_{\partial \Omega }$ 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.

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