Continuum Descriptions of Crystal Surface Evolution

Abstract

It is well known that liquid surfaces evolve in shape due to the effect of surface tension, which drives configurations towards lower energy. The break-up of an initially cylindrical fluid thread into spherical droplets, first quantified experimentally by Plateau and analyzed by Rayleigh, is a popular and illustrative example. Solid surfaces, in particular surfaces of crystals, also evolve according to the analogous principle of minimizing their surface energy. The evolution in this case, however, is more complicated to describe physically and mathematically than the analogous phenomena for fluid interfaces, because there is a richer variety of competing mechanisms that are available for the solid to change its shape. In addition, a solid supports strain, which leads to the surface energy depending on the slope of the crystal surface. In this article we summarize the basic physical ideas that underlie crystal surface evolution, introduce continuum descriptions in terms of continuum thermodynamics and partial differential equations (PDEs), and provide solutions to some analytically tractable prototypical problems.