Many 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. In a proper range of the block composition parameter and the nonlocal interaction parameter, an equilibrium pattern of many droplets exists in a general planar domain. A sub-range of the parameters is identified where the multiple droplet pattern is stable. This stable droplet pattern models the cylindrical phase in the diblock copolymer morphology. Each droplet is close to a round disc. The boundaries of the droplets satisfy an equation that involves the curvature of the boundaries and a quantity that depends nonlocally on the whole pattern. The locations of the droplets are determined via a Green’s function of the domain. In constructing the droplet pattern we overcome three obstacles: interface oscillation, droplet coarsening, and droplet translation.