On the solidification of dendritic arrays: Selection of the tip characteristics of slender needle crystals by array interactions

We obtain a unique solution to the well known indeterminacy for Ivantsov dendrites [Dokl. Akad. Nauk. SSSR, 58, 567 (1947)] by considering the directional solidification of a binary alloy as an array of interacting needle crystal dendrites. From the results of an asymptotic theory for the steady-state solidification of slender needle crystal arrays, the shape of the dendrite can be obtained from the solution of a non-linear integral equation. Here we solve this integral equation numerically to determine the characteristics of the solutions and compare the results to dendrite morphologies observed in experiments. The integral equation has a solvability condition that selects a distinct tip radius and tip undercooling for a given set of experimental conditions and dendrite spacings. This selection criteria is fundamentally different from traditional tip selection theories based on surface energy, and is linked to the interactions of dendrites in the array. Predictions of the tip radius are in good agreement with experimental measurements for conditions where the asymptotic theory is expected to be valid. Our results suggest that the tip radius increases with array spacing in the experimentally relevant parameter range. This relationship is consistent with the existence of a range of stable array spacings during directional solidification: the lower bound on array spacings is set by the stability criteria for array overgrowth, and an upper bound may be determined by the condition for tip splitting. In the asymptotic limit of small dendrite spacings, our integral equation interestingly has a degenerate set of solutions, indicating a transition from selection to degeneracy in the limit of small spacings. The explanation of this transition is beyond the scope of our theory and remains to be addressed.