Crystallography on Curved Surfaces: Virus Buckling and Grain Boundary
Scars

David R. Nelson
Lyman Laboratory of Physics
Harvard University

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.