Nonlinear stability analysis of self-similar crystal growth: control of the Mullins-Sekerka instability

In this paper, we perform a stability analysis of 2D, noncircular self-similar crystals with isotropic surface tension growing in a supercooled melt. The existence of such self-similarly growing crystals was demonstrated recently in our previous work (J. Crystal Growth 267 (2004) 703). Here, we characterize the nonlinear morphological stability of the self-similar crystals, using a new spectrally accurate 2D boundary integral method in which a novel time and space rescaling is implemented (J. Crystal Growth 266 (2004) 552). This enables us to accurately simulate the long-time, nonlinear dynamics of evolving crystals. Our analysis and simulations reveal that self-similar shapes are stable to perturbations of the critical flux for self-similar growth. This suggests that in experiments, small oscillations in the critical flux will not change the main features of self-similar growth. Shape perturbations may either grow or decay. However, at long times there is nonlinear stabilization even though unstable growth may be significant at early times. Interestingly, this stabilization leads to the existence of universal limiting shapes. In particular, we find that the morphologies of the nonlinearly evolving crystals tend to limiting shapes that evolve self-similarly and depend on the flux. A number of limiting shapes exist for each flux (the number of possible shapes actually depends on the flux), but only one is universal in the sense that a crystal with an arbitrary initial shape will evolve to this universal shape. The universal shape can actually be retrograde. By performing a series of simulations, we construct a phase diagram that reveals the relationship between the applied flux and the achievable symmetries of the limiting shapes. Finally, we use the phase diagram to design a nonlinear protocol that might be used in a physical experiment to control the nonlinear morphological evolution of a growing crystal. Because our analysis shows that interactions among the perturbation modes are similar in both 2D and 3D, our results apply qualitatively to 3D.