A free boundary model for shape preserving dendritic growth at high undercooling

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Abstract
Within the widely quoted Lipton, Kurz, and Trivedi model for the calculation of dendritic growth velocities [Acta Metall. 35, 957 (1987)], the kinetic and Gibbs–Thomson undercoolings evaluated at the dendrite tip are assumed to apply equally over the whole dendrite surface, approximating the nonisothermal dendrite as an isothermal dendrite with a reduced interface temperature. In a previous article [J. Appl. Phys. 78, 4137 (1995)] we described a finite difference model to calculate the growth velocity of a parabolic, nonisothermal dendrite growing into an undercooled melt, and showed that proper consideration of the nonisothermal interface reduced the growth velocity by ≊35%, relative to that predicted by the analytical model. We report an improved computational scheme which solves the free boundary problem for the shape preserving needle dendrite. At all undercoolings we find that the shape preserving needle dendrite is broadened with respect to the Ivantsov paraboloid with the same tip radius. Moreover, the extent of the broadening increases with undercooling. Thus, unlike the Ivantsov solutions, the form of the dimensionless, shape preserving needle dendrite is not invariant with undercooling. Growth velocities for the shape preserving solutions are found to be within 2% of those of our previous nonisothermal model.