Bj. Spencer et He. Huppert, ON THE SOLIDIFICATION OF DENDRITIC ARRAYS - SELECTION OF THE TIP CHARACTERISTICS OF SLENDER NEEDLE CRYSTALS BY ARRAY INTERACTIONS, Acta materialia, 46(8), 1998, pp. 2645-2662
Citations number
22
Categorie Soggetti
Material Science","Metallurgy & Metallurigical Engineering
We obtain a unique solution to the well known indeterminacy for Ivants
ov dendrites [Dokl. Akad Nauk. SSSR, 58, 567 (1947)] by considering th
e directional solidification of a binary alloy as an array of interact
ing 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 numeric
ally to determine the characteristics of the solutions and compare the
results to dendrite morphologies observed in experiments. The integra
l equation has a solvability condition that selects a distinct lip rad
ius and tip undercooling for a given set of experimental conditions an
d dendrite spacings. This selection criteria is fundamentally differen
t 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. O
ur results suggest that the tip radius increases with array spacing in
the experimentally relevant parameter range. This relationship is con
sistent with the existence of a range of stable array spacings during
directional solidification: the lower bound on array spacings is set b
y the stability criteria for array overgrowth, and an upper bound may
be determined by the condition for tip splitting. In the asymptotic li
mit of small dendrite spacings, our integral equation interestingly ha
s a degenerate set of solutions, indicating a transition from selectio
n to degeneracy in the limit of small spacings. The explanation of thi
s transition is beyond the scope of our theory and remains to be addre
ssed. (C) 1998 Acta Metallurgica Inc.