Ordered states on spheres require a minimum number of topological defects. The difficulty of constructing ordered states was recognized by J. J. Thomson, who discovered the electron and then attempted regular tilings of the sphere in an ill-fated attempt to explain the periodic table. One set of solutions to this "Thomson problem" requires that regular triangular lattices be interrupted by an array of at least 12 five-fold disclination defects, typically sitting at the vertices of an icosahedron. For R>>a, where R is the sphere radius and a is the particle spacing, the energy associated with these defects is very large. This energy can be lowered, however, either by buckling, as appears to be the case for large viruses, or by introducing unusual finite length grain boundary scars. The latter have been observed recently for colloidal particles adsorbed onto water droplets in oil. Related problems involving crystalline and liquid crystal patterns on other curved surfaces will be discussed as well.

Host: Bob Ecke, CNLS