Numerical investigations of pattern formation in binary systems with inhibitory long-range interaction

We investigate pattern formation in a two-dimensional manifold using the Otha-Kawasaki model for micro-phase separation of diblock copolymers. In this model, the total energy includes a short-range and a long-range term. The short-range term is a Landau-type free energy that is common in phase separation problems and favors large domains with minimum perimeter. The inhibitory long-range interaction term is the Otha-Kawasaki functional derived from the theory of diblock copolymers and favors small domains. The balance of these terms leads to equilibrium states that exhibit a variety of patterns, including disk-like droplets, droplet assemblies, elongated droplets, dog-bone shaped droplets, stripes, annular rings, wriggled stripes and combinations thereof. For problems where analytical results are known, we compare our numerical results and find good agreement. Where analytical results are not available, our numerical methods allow us to explore the solution space revealing new stable patterns. We focus on the triaxial ellipsoid, but our methods are general and can be applied to higher genus surfaces and surfaces with boundaries.