The solution of the dendrite problem for the edgewise growth of a platelet

Abstract

The dendritic steady-state growth of a platelet into its pure undercooled melt has been solved using a self-consistent method. The method requires solution of the shape of the dendrite that simultaneously satisfies the required heat flow and temperature variations over the dendrite surface. To find this shape, a Taylor expansion in terms of geometric constants is carried out around the dendrite tip for a given assumed tip radius and velocity. The melt undercooling is a derived quantity. It is necessary to carry out expansions of the heat flow and thermal boundary conditions to fourth order in order to specify the first geometric shape factor which shows departure from a parabola. The geometric constants are found to be multi-valued, and are interpreted to mean that the original assumption of steady-state is probably generally invalid. This then may be the fundamental origin of dendritic branching. The range of validity of these conclusions is briefly discussed.