Oscillatory instabilities in rapid directional solidification: bifurcation theory

Abstract

Merchant and Davis performed a linear stability analysis on a model for the rapid directional solidification a dilute binary alloy. The model has a velocity-dependent distribution coefficient and liquidus slope, and a linear form for attachment kinetics. The analysis revealed that in addition to the Mullins and Sekerka cellular mode there is a new oscillatory instability; the zero wavenumber pulsatile mode is most dangerous in the linear theory. In the present paper, a weakly nonlinear analysis is performed to describe the nonlinear behavior of this oscillatory mode. The analysis leads to a complex Ginzburg-Landau equation governing the evolution of the solid-liquid interface shape. Parametric regions of supercritical (subcritical) bifurcation giving smooth (jump) transitions to the pulsatile mode are demarked. Further analysis is made to determine how non-zero wavenumber disturbances compete with the zero wavenumber pulsation. In the low speed limit, these disturbances have a relatively broad range of wavenumbers in which they are stable. In the high speed limit, only very long waves are selected. The very long waves correspond approximately to solute bands, in which the solute distribution in the solid is layered in the direction of growth with no lateral segregation. In the three-dimensional case, spiral waves are solutions to the Ginzburg-Landau equation and these may be related to the spiral waves that have been observed by Thoma et al. in melt spinning.