Numerical investigations of the role of curvature in strong segregation problems on a given surface

Below the transition temperature in a multi-phase lipid vesicle membrane, phases separate into non-connecting domains that coarsen into larger areas. The free energy of phase properties determines the length of the boundaries separating the regions. Mathematical theory predicts that in a two-phase lipid vesicle, a small geodesic disk of the minority lipids forms at a point of the membrane where the Gauss curvature attains a maximum. The lipid bilayer enables crucial signaling and clustering activities in cell membranes and so it is critical to understand the interplay between membrane curvature and lipid separation. We use numerical simulation with finite elements to probe the connection between curvature and phase. Our numerical solutions affirm the assertion regarding patch formation of the minority lipid and suggest the analytical results are applicable to patches that are not necessarily small. To demonstrate these results, we focus on an ellipsoid-shaped vesicle and determine the phase distribution on this domain by directly minimizing a Landau-type free energy subject to a constraint that describes the proportion of each phase. We investigate the sensitivity of the solution process on the grid size h and the relation between h, the diffusion coefficient α, the conservation constant m and the initial phase configuration.