Directional solidification with interface dissipation

The front dynamics during directional solidification of a dilute binary mixture with noninstantaneous interface kinetic attachment is investigated in the weakly and highly nonlinear regimes. In the linear regime we find that, even for small enough kinetic coefficient, the most unstable mode is appreciably modified. With the help of multiscale analysis, we derive the amplitude equation that governs the front dynamics close to criticality. It is found that, for the case of a constant miscibility gap, studied here, the supercritical nature of the bifurcation remains unaltered. However, the computation of the Landau coefficient shows that the interface excursion is reduced by kinetics. Moreover, the analysis of the Mullins-Sekerka spectrum close to the instability threshold indicates that higher harmonics should be activated by interface kinetics. This is confirmed numerically. We study the transition from cells to dendrites by adopting a recent code to the present situation. We find scaling laws that confirm those obtained in the free-growth situation [Brener and Mel’nikov (unpublished)]. The interface structure is ‘‘angular,’’ and side branches are much less pronounced, even when the Péclet number exceeds a few unities, than those obtained in the free-kinetics case [Saito, Misbah, and Müller-Krumbhaar, Phys. Rev. Lett. 63, 2377 (1989)]. Our findings agree well with experiments [P. Kurowski, C. Guthmann, and S. de Cheveigné (unpublished)] on impure ${\mathrm{CBr}}_4$. We speculate that dendrites in this system are selected by kinetic anisotropy rather than by surface-tension anisotropy.