Phase-Field Study of the Cellular Bifurcation in Dilute Binary Alloys

Phase-field simulations in both two and three dimensions are used to investigate the microstructures that form closely above the threshold of the Mullins-Sekerka instability in the directional solidification of dilute binary alloys. It is found that in this regime of shallow cells the simulation results strongly depend on the thickness of the diffuse interfaces even for model parameters that yield quantitative results for deep cells. For the material parameters of a dilute Sn-Bi alloy, the bifurcation is found to be supercritical, whereas weakly nonlinear amplitude expansions predict a subcritical bifurcation. Furthermore, an oscillatory instability of the cell grooves is found, which leads to the pinch-off of liquid inclusions even for relatively shallow cells. Finally, in three dimensions, three different morphologies are found, in agreement with experiments and previous numerical studies: regular hexagons, elongated cells ("stripes"), and inverted hexagons ("node" or "pox" structure, a hexagonal array of local depressions of the solidification front). Nodes and stripes are stable steady-state solutions only very close to the bifurcation.