Single Droplet Pattern in the Cylindrical Phase of Diblock Copolymer Morphology

The Ohta-Kawasaki density functional theory of diblock copolymers gives rise to a nonlocal free boundary problem. Under a proper condition between the block composition fraction and the nonlocal interaction parameter, a pattern of a single droplet is proved to exist in a general planar domain. A smaller parameter range is identified where the droplet solution is stable. The droplet is a set which is close to a round disc. The boundary of the droplet satisfies an equation that involves the curvature of the boundary and a quantity that depends nonlocally on the whole pattern. The location of the droplet is determined by the regular part of a Green’s function of the domain. This droplet pattern describes one cylinder in space in the cylindrical phase of diblock copolymer morphology.