Steps and facets at the surface of soft crystals

Abstract

We consider the shape of crystals which are soft in the sense that their elastic modulus $μ$ is small compared to their surface tension $γ$, more precisely $ μa < γ$ where $a$ is the lattice spacing. We show that their surface steps penetrate inside the crystal as edge dislocations. As a consequence, these steps are broad with a small energy which we calculate. We also calculate the elastic interaction between steps a distance $d$ apart, which is a $1/d^2$ repulsion. We finally calculate the roughening temperatures of successive facets in order to compare with the remarkable shapes of lyotropic crystals recently observed by P. Pieranski et al. Good agreement is found.