Interacting topological defects on frozen topographies

We propose and analyze an effective free energy describing the physics of disclination defects in particle arrays constrained to move on an arbitrary two-dimensional surface. At finite temperature the physics of interacting disclinations is mapped to a Laplacian sine-Gordon Hamiltonian suitable for numerical simulations. We discuss general features of the ground state and thereafter specialize to the spherical case. The ground state is analyzed as a function of the ratio of the defect core energy to the Young’s modulus. We argue that the core energy contribution becomes less and less important in the limit $R\gg a,$ where R is the radius of the sphere and a is the particle spacing. For large core energies there are 12 disclinations forming an icosahedron. For intermediate core energies unusual finite-length grain boundaries are preferred. The complicated regime of small core energies, appropriate to the limit $R/\overrightarrow{a}\infty ,$ is also addressed. Finally we discuss the application of our results to the classic Thomson problem of finding the ground state of electrons distributed on a two sphere.