Numerical Investigations of Pattern Formation in Binary Systems Open Access
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In this paper, we investigate pattern formation in a two-phase system on a two-dimensional manifold by numerically computing the minimizers of a Cahn-Hilliard-like model for micro-phase separation of diblock copolymers. The total energy of the system includes a short-range term - a Landau free energy and a long-range term - the Otha-Kawasaki functional. The short range term favors large domains with minimum perimeter and the long-range inhibitory term favors small domains. The balance of these terms leads to minimizers with a variety of patterns, including single droplets, droplet assemblies, stripes, wriggled stripes and combinations thereof. We compare the results of our numerical simulations with known analytical results and discuss the stability of the computed solutions and the role of key parameters in pattern formation. For demonstration purposes, we focus on the triaxial ellipsoid, but our methods are general and can be applied to higher genus surfaces and surfaces with boundaries.