Microsegregation in cellular solidification

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I.

INTRODUCTION

MICROSEGREGATION, i.e., the presence of concentration gradients and nonequilibrium second phases in solidified alloys, is found when the rate of solute diffusion in the solid alloy is too low to homogenize this phase while it grows. In practice, solid diffusion rates, microstructural dimensional scales, and alloy solidification times are such that some diffusion takes place in the solid during solidification. This, in turn, reduces microsegregation compared to the upper bound predicted by analytical equations that assume no solid state diffusion (such as the Scheil equation for dendritic solidification). Predicting microsegregation with limited solid diffusion is generally difficult, because solidifying microstructures are geometrically complex and because the diffusion problem features a moving boundary. Given the practical importance of the problem, a relatively large number of approaches have been adopted to predict solidification microsegregation in cellular or dendritic alloys. These range in complexity from simple but approximate analytical solutions to numerical models based on the finite difference or the finite element methods. Modeling of microsegregation has been the subject of a few recent and comprehensive reviews, tl,2,31 It is found that the more sophisticated numerical models, which treat the liquid/solid interface as a moving boundary and are based on realistic solidification geometries, generally agree with experimental data gathered predominantly on alloys solidified in the dendritic growth regime. At both extremes of very slow cellular or very rapid dendritic or cellular solidification, prediction of microsegregation is further complicated because the cell or dendrite tips are significantly undercooled. The initial solid composition then deviates from kCo, where k is the equilibrium partition ratio and Co is the average alloy composition. Recent publications have addressed this NANCY F. DEAN, formerly Research Assistant, Department of Materials Science and Engineering, Massachusetts Institute of Technology, is Metallurgical Engineer, Precision Castparts Corp. Portland, OR. ANDREAS MORTENSEN, Associate Professor, and MERTON C. FLEMINGS, Head of Department, are with the Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139. Manuscript submitted August 16, 1993. METALLURGICALAND MATERIALS TRANSACTIONSA

problem for rapid solidification, a case in which solute concentration gradients in the liquid must also be taken into account, t4,51In slow cellular solidification, the composition at the cell tip differs significantly from kCo due to the comparatively high average rate of axial solute diffusion in the intercellular liquid. Its comparatively simple geometry has made steady state cellular solidification an attractive problem for theoretical modeling, being addressed by a relatively large number of authors using various methods and assumptions. The effects of finite solid state diffusion during cellular solidification have in par