Needle crystals with nonlinear diffusion

Abstract
We study the free-boundary problem of the steady growth of a solid into its undercooled melt, allowing the thermal diffusion coefficient to have an arbitrary temperature dependence. By developing a novel approach to the Ivantsov method, we show that needle-crystal solutions can be found in two and three dimensions. The calculation establishes that the only steadily advancing, shape-preserving solidification fronts the method can produce, for linear or nonlinear diffusion, are parabolas in two dimensions and elliptic (or circular) paraboloids in three dimensions. We discuss the limitations on the Ivantsov method, pointing out that it is only capable of finding families of solutions of the problem, whose members are related by a rescaling of space and time. We show explicitly that including a heat-loss term in the equations causes the method to fail, and argue that any term which introduces a length scale into the problem will, in general, do likewise.