Growth-speed dependence of primary arm spacings in directionally solidified Pb-10 Wt Pct Sn
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I.
INTRODUCTION
T H E microstructure at the liquid-solid interface during directional solidification of binary alloys in a positive thermal gradient is planar, cellular, or dendritic, depending upon the gradient of constitutional supercooling (1 - S*), where S* = [DiGk/mzRCo(k - 1)]. [1'2] Here, D l is the solute diffusivity in the melt, G is the effective thermal gradient at the liquid-solid interface, k is the solute partition coefficient, mt is the slope of the liquidus line from the phase diagram, R is the growth speed, and Co is the solute content of the melt. Figure 1 schematically represents the literature reported I3-2~ dependence of the primary arm spacings (A1) on growth speed [or (1 S*)]. In region A, starting from Rp, the growth speed for the initial breakdown of a planar liquid-solid interface at (1 - S*) = 0, A1, decreases with increasing R. A minimum in A~ occurs at Rmi,. The spacings increase with increasing R in region B, until a maximum is reached at R m a x , beyond which they decrease in region C. Region C represents the most extensively investigated growth regime, t3-~1] Behavior shown near R,~x has also been reported in several a l l o y s . [4-6'12-16] The primary arm spacing minimum, near Rm~,, has been reported in A1-4 wt pct Cu t13] and succinonitrile-acetone, t17] However, contradictory behaviors are reported for A1-Cu alloys: observations in agreement with Figure 1t13] and those where region A was not seen. I14'19j The decreasing arm spacing behavior near Rp has been reported in Pb-T1 [~s] and succinonitrile-acetone. [17] Several reasons have been presented in the literature to explain the observed primary arm spacing maximum in the spacings vs growth speed plot. It has been attributed to the thermosolutal convection in the melt. [6,~~ It has been explained as an artifact, introduced in the primary arm size measurement technique, because of the different spatial distributions of dendrites and cells, t~~ The primary arm spacing maximum has been suggested to correspond with the cell-to-dendrite transition. I~6] However, data on aluminum-copper alloys t13,a4] show that
M.A. CHOPRA, formerly Research Associate with Chemical Engineering Department, Cleveland State University and the N A S A Lewis Research Center, Cleveland, OH, 44135, is with IBM, Austin, TX 78758. S.N. TEWARI, Professor, is with the Chemical Engineering Department, Cleveland State University, Cleveland, OH 44115. Manuscript submitted September 11, 1990. METALLURGICAL TRANSACTIONS A
the transition occurs at R < R , ~ in the region B (Figure 1). It has also been suggested that the contradictory behaviors reported in the literature may be because of the differing nature of the planar-to-cellular/ dendritic transformation, i.e., whether the bifurcation is subcritical or supercritical. [17] The subcritical bifurcation would result in a growth-speed dependance of primary arm spacings, which contains the region A shown in Figure 1. For supercritical bifurcation, only the B and C regions (Figure 1) would be observed. By assuming
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