Finite element simulation of planar instabilities during solidification of an undercooled melt

Abstract

Two-dimensional finite element solutions for planar solidification from an undercooled melt are presented. These simulations are based on the transient heat equation in both solid and liquid and show the onset and propagation of both stable and unstable numerical waveforms which reproduce those predicted in the continuum analysis with fidelity. The inherent instabilities associated with the freezing process dictate a more comprehensive treatment of the interfacial temperature than that specified in stable Stefan-type problems. Herein, we apply radiation-type boundary conditions on the interface that incorporate temperature effects associated with curvature and interfacial kinetics. The interfacial temperature depression due to curvature is the primary restraining factor during dendritic growth. Its numerical representation requires special care to avoid fatal discretization error; additionally, curvature must be treated implicitly within the thermal iteration and within the time step to overcome otherwise severe time-step constraints. The numeric simulations of anisotropic ice show similar waveform patterns at the onset of the instability to those of isotropic cases. However, as the amplitude of the waveform increases significant lengths of interface become orientated along the C axis where interfacial kinetics inhibit growth. This alters the interface shape by elongating the dendrite finger.