Pattern formation in directional solidification

We simulate directional solidification using the phase field method solved with adaptive mesh refinement. We examine length scale selection for two cases. For small surface tension anisotropy directed at forty five degrees relative to the pulling direction, we observe a transition from a seaweed to dendrite morphology as the thermal gradient is lowered, consistent with recent experimental findings. We show that the morphology of crystal structures can be unambiguously characterized through the local interface velocity distribution. We derive semi-empirically a phase diagram for the transition from seaweed to dendrites as a function of thermal gradient and pulling speed. As surface tension anisotropy is increased and aligned with the pulling direction we observe cellular and dendritic arrays directed in the pulling direction. We characterize wavelength selection and obtain a new universal scaling of the wavelength that differs from previous theories.