Crystal growth at long times: Critical behavior at the crossover from diffusion to kinetics-limited regimes

We simulate one-dimensional crystal growth from an undercooled melt using a phase-field model and find interesting behavior when the liquid is undercooled by L/${\mathit{c}}_{\mathit{p}}$ degrees (‘‘unit undercooling’’). L is the latent heat and ${\mathit{c}}_{\mathit{p}}$ the specific heat. For smaller undercoolings, the diffusion of latent heat limits growth and the velocity of the solid-liquid interface decays with time as ${\mathit{t}}^{\mathrm{-}1/2}$. For larger undercoolings, non- equilibrium interface kinetics limits growth, and the interface velocity is constant. At unit undercooling, there are two scenarios, depending on the ratio of order parameter to thermal diffusivity (p). If p is small, the front-decay velocity is very well described by a power law ${\mathit{t}}^{\mathrm{-}\nu }$, with ν≊0.3. If p is large, the velocity at unit undercooling is finite. The branch of steady-state solutions then extends to smaller undercoolings, where the solid created is superheated. At the end of the branch, the solution jumps to a ${\mathit{t}}^{\mathrm{-}1/2}$ velocity-decay law. Although pure materials have small p’s, impure materials can have large p’s, so that the two scenarios at unit undercooling should be observable experimentally. Although our simulations apply strictly only to one-dimensional fronts, similar behavior is expected in two and three dimensions. The presence of the Mullins-Sekerka instability is unlikely to change our conclusions.