Spherical Solutions to a Nonlocal Free Boundary Problem from Diblock Copolymer Morphology

Abstract

The $\Gamma$-limit of the Ohta–Kawasaki density functional theory of diblock copolymers is a nonlocal free boundary problem. For some values of block composition and the nonlocal interaction, an equilibrium pattern of many spheres exists in a three-dimensional domain. A subrange of the parameters is found where the multiple sphere pattern is stable. This stable pattern models the spherical phase in the diblock copolymer morphology. The spheres are approximately round. They satisfy an equation that involves their mean curvature and a quantity that depends nonlocally on the whole pattern. The locations of the spheres are determined via a Green's function of the domain.