Cellular instability in rapid directional solidification: Bifurcation theory

Merchant and Davis performed a linear stability analysis on a model for the directional solification of a dilute binary alloy valid for all speeds. The analysis revealed that non-equilibrium segregation effects modify the Mullins and Sekerka cellular mode, whereas attachment kinetics has no effect on these cells. In this paper, the nonlinear stability of the steady cellular mode is analyzed. A Landau equation is obtained that determines the amplitude of the cells. The Landau coefficient here depends on both non-equilibrium segregation effects and attachment kinetics. This equation gives the ranges of parameters for subcritical bifurcation (jump transition) or supercritical bifurcation (smooth transition) to cells.