Phase field models for step flow.
The relation between phase field and discontinuous models for crystal steps is analyzed. Different formulations of the kinetic boundary conditions of the discontinuous model are first presented. We show that (i) step transparency, usually interpreted as the possibility for adatoms to jump through steps, may be seen as a modification of the equilibrium concentration engendered by step motion. (ii) The interface definition (i.e., the position of the dividing line) intervenes in the expression of the kinetic coefficients only in the case of fast attachment kinetics. (iii) We also identify the thermodynamically consistent reference state for kinetic boundary conditions. Asymptotic expansions of the phase field models in the limit where the interface width is small, lead to various discontinuous models. (1) A phase field model with one global concentration field and variable mobility is shown to lead to a discontinuous model with fast step kinetics. (2) A phase field model with one concentration field per terrace allows one to recover arbitrary step kinetics (i.e., arbitrarily strong Ehrlich-Schwoebel effect and step transparency). Quantitative agreement is found, in both the linear and nonlinear regimes, between the numerical solution of the phase field models and the analytical solution of the discontinuous model.