Macrosegregation during steady-state arrayed growth of dendrites in directionally solidified Pb-Sn alloys
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I.
INTRODUCTION
THE temperature and composition profiles in the melt, both in the interdendritic mushy region and in the melt column ahead, are schematically shown in Figure 1 for a steady-state arrayed growth of dendrites obtained during directional solidification of binary alloys in a positive thermal gradient. This figure also shows the leadrich portion of the Pb-Sn phase diagram. The solute content of the interdendritic melt decreases from CE (eutectic composition) at the base of the array to C, at the dendrite tip. The composition in the bulk melt, ahead of the dendritic array, decreases from C, to the bulk melt composition, Co, over some distance, which, for the diffusive mass transport, is approximately equal to Dt/R, where Dt is the solutal diffusivity in the melt and R the growth speed. The solutal buildup at the tips (C, - Co) has been shown 11,2,31 to depend on the gradient of constitutional supercooling (1 - S), where S = DtGt/ mtRCo(k - 1). Here, Gl is the thermal gradient in the melt at the liquid-solid interface, mt is the liquidus slope, and k is the solute partition coefficient. Ignoring the rapid solidification regime, (Ct - Co) is negligible for S -~ 0, where the dendritic microstructures are observed. Larger (C, - Co) are expected for growth at small gradients of constitutional supercoolings, where the array morphology changes to cellular. For the growth conditions depicted in Figure 1, melt on the top and the solid below, with gravity pointing down, the temperature profile alone is expected to be stabilizing against natural convection. However, the solutal profile will be destabilizing for those alloys where rejection of the solute into the melt during solidification results in reduced melt density, as is expected for Pb-Sn alloys with Co < CE (Figure 1). Depending upon the temperature and composition dependence of the melt S.N. TEWARI, Professor, and RAJESH SHAH, Graduate Student, are with the Chemical Engineering Department, Cleveland State University, Cleveland, OH 44115. Manuscript submitted December 16, 1991. METALLURGICAL TRANSACTIONS A
density (schematically shown in Figure 1), thermosolutal convection t41 can start in the melt, both within the interdendritic region and ahead of the dendrite array. Besides the thermophysical properties of the alloy and the growth parameters (Gl, Co, R), several other factors, such as the primary arm spacings (A 1), the liquid volume fraction (f~), and the length of the mushy zone (H), are expected to control the onset and extent of interdendritic thermosolutal convection. The extent of solute enrichment (C, - Co) and the thermal gradient Gt will determine the onset and intensity of the convection in the melt ahead. Because of the negligibly small (C, - Co), the composition profile in the bulk melt ahead of the array is not expected to be a major factor in thermosolutal convection for dendritic arrays, and the interdendritic convection will dominate. Thermosolutal convecion in the bulk melt will, however, be very important for cellular arrays because o
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