Three-dimensional cellular instabilities in directional solidification considering interfacial kinetics

In this work, three-dimensional cellular instabilities in directional solidification are investigated. Local thermodynamic nonequilibrium at the solid-liquid interface is taken into account by applying a model of velocity-dependent segregation coefficient and attachment kinetics developed by Boettinger, Aziz, and Jackson et al. The infinitely one-sided model and the frozen-temperature approximation are adopted in the analysis. An evolution equation was first derived by an integral technique. Then the Segel-Stuart method was used to determine the solvability constants of the amplitude equations. The equilibrium solutions associated with different morphologies were evaluated and the stability analysis of them to three-dimensional disturbances was studied. Finally, five stability basins of bifurcation were addressed. The result shows that the interface morphologies depend on which basin the chosen critical condition is located in and how far the operating point is exceeded from the critical condition. The presence of disequilibrium can stabilize the steady cellular mode, cause the wave number to be smaller, and affect the location of the chosen critical condition in the basin of bifurcation.