Surface energy anisotropy for dipolar lattices

Lattices of parallel dipoles, which appear in, e.g., electrorheological fluids, have strongly anisotropic surface energies. We compute these energies for a number of lattices as a function of polar angle θ and azimuthal angle φ with respect to dipole direction. Logarithmic cusps appear at low order lattice planes, so that the surface energy near a lattice plane oriented at a polar angle θ̄ is given by σ(θ)=σ(θ̄)+K±‖θ−θ̄‖log‖θ−θ̄‖, where K− or K+ are chosen depending on whether θ is less or greater then θ̄. K± for any cusp are exactly calculable with a continuum method.