Analytic solution for a non-axisymmetric isothermal dendrite

Abstract

The Ivantsov solution for an isothermal paraboloid of revolution growing into a pure, supercooled melt provides a relation between the bulk supercooling and a dimensionless product (the Peclet number P) of the growth velocity and tip radius of a dendrite. Horvay and Cahn generalized this axisymmetric analytical solution to a paraboloid with elliptical cross-section. They found that as the deviation of the dendrite cross-section from a circle increases, the two-fold symmetry of the interface shape causes a systematic deviation from the supercooling/Peclet number relation of the Ivantsov solution. To model dendritic growth in cubic materials, we find approximate solutions for paraboloids having perturbations with four-fold axial asymmetry. These solutions are valid through second order in the perturbation amplitude, and provide self-consistent corrections through this order to the supercooling/Peclet number relation of the Ivantsov solution. Glicksman and colleagues have measured the shape and the supercooling/Peclet number relation for growth of succinonitrile dendrites in microgravity. For a Peclet number of P≈0.004 and the experimentally observed shape, we calculate a correction corresponding to a 9% increase in the supercooling, in general agreement with the experimental results.