Phase-field theory of edges in an anisotropic crystal

In the presence of sufficiently strong surface energy anisotropy, the equilibrium shape of an isothermal crystal may include corners or edges. Models of edges have, to date, involved the regularization of the corresponding free-boundary problem resulting in equilibrium shapes with smoothed out edges. In this paper, we take a new approach and consider how a phase-field model, which provides a diffuse description of an interface, can be extended to the consideration of edges by an appropriate regularization of the underlying mathematical model. Using the method of matched asymptotic expansions, we develop an approximate solution which corresponds to a smoothed out edge from which we are able to determine the associated edge energy.