Cellular and dendritic growth in a binary alloy melt: A marginal stability approach

Abstract

A simple model describing “constrained” growth of an “array” of cells or dendrites in a binary alloy melt, in the presence of an imposed positive temperature gradient in the liquid, has been presented. The dendrite or cell tip radius is calculated using the marginal stability criterion proposed by Langer and Müller-Krumbhaar. The results obtained are compared with the predictions deriving from two other approaches: the first approach adopts the “ad hoc” assumption of “minimum undercooling” at the cell or dendrite tips, the second approach is based on a closely similar stability criterion proposed by Trivedi. It has been shown that all three approaches predict tip radii, differing by no more than 30%. All three approaches yield the following simple relationship between the tip radius, rt, and the growth conditions: σc = A(1 − s) where σc = 2lcDL/Rr2t and s = DLGL/RΔT0 are dimensionless parameters, and A is a constant whose value depends on the criterion used to determine the tip radius. Recent experimental data in succinonitrile-acetone system have been shown to be in good agreement with these predictions. These comparisons also indicate that during “constrained” dendritic growth in an alloy, in the presence of a positive temperature gradient in the liquid, σc = 1/(2π2), exactly twice the value of σ∗ = 1/(4π2) = 0.0253 obtained by Langer and Müller-Krumbhaar for “free” dendritic growth in a supercooled pure melt. It is shown that the increase in the value of the tip stability parameter, σc, may be related to a transition in morphology from a cellular structure, with just a few side branches (for which s = 0.50), to a more “fully developed” dendritic structure (for which s → 0 or GL/R → 0).