Corners in phase-field theory

Phase-field models for strongly anisotropic surface energy need to be regularized to remove the ill posedness of the dynamic equations. Regularization introduces a new length scale, the corner size, also called the bending length. For large corner size, with respect to interface thickness, the phase-field method is known to converge asymptotically toward the sharp-interface theory when the appropriate approximation of the Willmore energy is used. In this work we study the opposite limit, i.e., corner size smaller than the interface width, and show that the shape of corners, at equilibrium, differs from the sharp-interface picture. However, we find that the phase transition at the interface is preserved and presents the same properties as the classical problem.

Automated summary

Regularization is necessary for phase-field models with strongly anisotropic surface energy to remove the ill posedness of dynamic equations. A new length scale, the corner size, is introduced, and for large corner sizes relative to interface thickness, the phase-field method converges asymptotically towards the sharp-interface theory. When the corner size is smaller than the interface width, the shape of corners differs from the sharp-interface picture, but the phase transition at the interface retains the same properties as the classical problem.