Analysis of stability of a planar solid-liquid interface in a dilute binary alloy
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The question of stability of a planar solid-liquid interface in undercooled pure and alloy melts has been reconsidered without the restrictive assumption of no heat flow in the solid made in earlier works. The modified analysis indicates that provided the thermal gradient on the solid side of the interface, Gs, is positive, stability can be achieved in an undercooled alloy melt for growth rates R > Ra, whereas a recent analysis by Trivedi and Kurz, which assumes Gs = 0, suggests that stability is possible only if R > Ra + Rat. Here Ra is the familiar absolute stability limit of Mullins and Sekerka and Ra, is the absolute stability limit in an undercooled pure melt, as identified by Trivedi and Kurz. The absolute stability criterion for steady-state planar growth in an undercooled alloy melt is thus the same as derived earlier by Mullins and Sekerka for directional solidification. Relaxing the restrictive assumption of Gs = 0 also reveals that there is a regime of stability for low growth rates and low supercoolings. Stability is possible under these conditions if Gs > 0, and the bath undercooling ATb < AT0 + ATh/2, where AT0 is the freezing range of the alloy and ATh is the hypercooling limit for the pure melt. For large supercoolings, Gs < 0, and the interface will be unstable with respect to large wavelength perturbations, even if R > Ra + Rat.
I. INTRODUCTION
The question of stability of a planar solid-liquid interface in a dilute binary alloy melt has been considered by Mullins and Sekerka,1 Coriell and Sekerka,2 and most recently by Trivedi and Kurz.3 Table I summarizes the principal findings and the important assumptions made by these authors. It is interesting to note that for the special case of solidification in an undercooled pure melt, Coriell and Sekerka conclude that the situation is probably unstable whereas Trivedi and Kurz (T-K) conclude that the interface will be absolutely stable if the growth rate R > Rat- Here Ral = aL/d0 is the thermal version of the absolute stability limit of Mullins and Sekerka (M-S). aL is the thermal diffusivity of the liquid phase and d0 is the thermal capillary length.4 The M-S absolute stability limit may be expressed as Ra = DL/I'C where DL is the diffusivity of the solute in the liquid phase and l'c is the chemical capillary length; see nomenclature for a more complete definition of d0 and l'c. In an undercooled alloy, T-K suggest an "additive" stability criterion; viz., the interface will be absolutely stable if R > Ra + Rat- In most materials Ra < RatHence the T-K "additive" stability criterion implies that the absolute stability limit in an undercooled alloy is virtually indistinguishable from the absolute stability limit in an undercooled pure melt. Considering high purity succinonitrile as an example, Ral = 43 m/s whereas Ra = 127 microns/s for an impurity content Co = 5 x 10"5 mole %.5~7 In metallic alloys Rat is of the order of J. Mater. Res., Vol. 5, No. 1, Jan 1990
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several tens to a few hundred km/s,
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