Shapes of crystalline domains on spherical fluid vesicles

The energies of crystalline domains of different shapes on a spherical surface are calculated using continuum elasticity theory and simulations. Additivity of the stretching energy and the line energy is assumed, and a phase diagram is constructed considering discs/caps, rings, and ribbons of the defect-free crystal as well as discs/caps with a 5-fold disclination at their center. The shapes and the number of the crystalline domains are found to depend only on the relative area covered by the domains and the ratio of the reduced line tension and the reduced Young modulus. The results are expected to be relevant for systems like two-dimensional colloidal crystals on emulsion droplets or lipid-bilayer vesicles in the two-phase region.