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.