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

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 = $\text{Const}$ \quad $\text{on }$ \partialΩ, \end{align*} where $H_{\mathrm{\partial }\mathrm{\Omega }}$ denotes the mean curvature of the surface $\mathrm{\partial }\mathrm{\Omega }$. By analyzing the linearization of $\mathrm{\Omega }\mapsto {\mathcal{H}}_{\mathrm{\Omega }}$ 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.