Steps, kinetic anisotropy, and long-wavelength instabilities in directional solidification

We consider the effect of anisotropic interface kinetics on long-wavelength instabilities during the directional solidification of a binary alloy having a vicinal interface. Linear theory predicts that a planar solidification front is stabilized under the effect of anisotropy as long as the segregation coefficient is small enough, whereas a novel instability appears at high rates of solidification. Furthermore, the neutral stability curve, indicating the values of the principal control parameter (here the morphological number) for which the growth rate of a sinusoidal perturbation of a given wavelength changes its sign, is shown to have up to three branches, two of them combining to form an isola for certain values of the control parameters. We identify conditions for which linear stability theory predicts the instability of the planar interface to long-wavelength traveling waves. A number of distinguished limits provide evolution equations that describe the resulting dynamical behavior of the crystal-melt interface and generalize previous work by Sivashinsky, Brattkus, and Davis and Riley and Davis. Bifurcation analysis and numerical computations for the derived evolution equations show that the anisotropy is able to promote the tendency to supercritical bifurcation, and also leads to the development of strongly preferred interface orientations for finite-amplitude deformations.