Dendrite characteristics in directionally solidified Pb-8 Pct Au and Pb-3 Pct Pd alloys
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INTRODUCTION
MECHANICAL properties of cast alloys are highly dependent on their solidification structure. It is therefore important to understand evolution of the usual dendritic microstructure. Several theories have been proposed to explain the cellular/dendritic solidification behavior of binary alloy melts in positive temperature gradients. 1-12 The aim of these models is to predict the radius of curvature (r~), temperature (T~), and liquid composition (C,) at the cell/dendrite tip and for some models6'8-1~the primary arm spacing (h i) as a function of the variables, alloy composition (Co), imposed thermal gradient at the liquid-solid interface (G), and alloy growth velocity (R). These models can be classified into three groups. The first, consisting of models due to Bower, Brody, and Flemings, 1 Burden and Hunt, 2 and Laxmanan, 3'4 consider steady-state behavior of a dendrite array and assume that the dendrite grows with minimum tip undercooling, i.e., maximum temperature at the tips. The second group, consisting of the Trivedi5 and Kurz and Fisher 6 models, assume that marginal stability concepts determine the dendrite tip characteristics at the operating point of the dendrite. The third group, the models of Jin and Purdy 7 and Kirkaldy,8 assume that the observed cell shape minimizes the rate of entropy production for the liquid to solid transformation. The Burden and Hunt model, an improvement over the Bower, Brody, and Flemings model, incorporated for the first time the effect of the dendrite tip curvature. Tip undercooling measurements showed a good agreement with the model. 2 However, it has been criticized for not predicting the proper dependence of r, on G. 10,12It also fails to predict S.N. TEWARI is with the Chemical Engineering Department, Cleveland State University, Cleveland, OH 44115. Manuscript submitted January 6, 1986. METALLURGICAL TRANSACTIONS A
the proper undercooling at the absolute stability limit of Mullins and Sekerka. 12.13Laxmanan3.4 has recently proposed a further improved model, which while incorporating the "minimum tip undercooling" assumption, predicts r, dependence on G and satisfactorily explains the limiting behavior, both in the small growth velocity regime (Chalmers constitutional stability limit14), and at rapid growth rates (absolute stability limit13). In another approach Laxmanan has examined the dendrite growth problem by incorporating the "stability" requirements obtained by Trivedi5 into the solute transport equations and predicted the corresponding r,, T,, and Ct .3 The Trivedi model 5 based on "marginal stability" concepts has been shown to predict accurately the observed growth rate dependence of i", in the succinonitrile-acetone model system. 1~ Trivedi 1~has incorporated Hunt's approach 9 in his model to predict the primary arm spacing. Trivedi and his colleagues have directionally solidified several metallic alloys to examine the dependence of AI on R and G.16'17'18It has been shown that the primary arm spacings observed in Pb-Au, Pb-Pd, and Pb-Sn alloys qualita
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