Relationship between dendrite tip characteristics and dendrite spacings in alloy directional solidification

We summarize and discuss a new theoretical model for the directional solidification of alloys in the form of dendrite arrays. Slender body theory is used to obtain an integral equation for the dendrite shape in the asymptotic limit of the dendrite tip radius being much smaller than the solute diffusion length. This equation has a solvability condition that selects the shape and tip undercooling for prescribed solidification conditions and array spacings. A consequence of our results is that we obtain a unique solution to the well-known indeterminacy for the single-dendrite [2] similarity solution by considering the interaction between individual members of an array of dendrites. Further, the dependence of the solutions on the dendrite spacing gives a family of 'array solutions,' in which the tip radius is related directly to the dendrite spacing. These solutions agree well with experiments in parameter ranges where the slender dendrite theory is expected to be valid. Finally, we discuss how these array solutions, together with surface energy and stability considerations, can describe the selection of dendrite spacings during directional solidification.