An analytical model for solute redistribution during solidification of planar, columnar, or equiaxed morphology
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BACKGROUND
A S S E S S M E N T of microsegregation occurring in solidifying alloys is important, since it influences mechanical properties. Also, a comprehensive theoretical treatment of dendritic growth requires an accurate eva/uation of the solutal field (microsegregation) during solidification. Models for calculation of microsegregation differ by the problem that they tackle as well as by the approach. Most models are restricted to the case of planar (plate) solidification (Figure l(a)), or columnar solidification (Figure l(b)), with the volume element over which the calculation is performed being selected between the primary or the secondary dendrite arms. One-dimensional (l-D) Cartesian or cylindrical coordinates solutions have been proposed. When equiaxed solidification is considered (Figure l(c)) a threedimensional (3-D) problem (or at least 1-D spherical coordinates) must be considered. The earliest description of solute redistribution during solidification by Scheil I~t involves several assumptions, such as negligible undercooling during solidification, complete solute diffusion in liquid, no diffusion in solid, no mass flow into or out of the volume element, constant physical properties, and fixed volume element (no dendrite arm coarsening). However, the diffusion of solute into the solid phase can affect microsegregation significantly, especially toward the end of solidification. For example, calculations by Brooks et al. 12] showed that little solid-state diffusion occurs during the solidification and cooling of primary austenite solidified welds of FeNi-Cr ternary alloys, whereas structures that solidify as ferrite may become almost completely homogenized as a result of diffusion. Although Brody and Flemings t3j (BF) have proposed a model that assumed complete diffusion in liquid and incomplete back diffusion, they have L. N A S T A C , G r a d u a t e R e s e a r c h A s s i s l a n t , a n d D . M . STEFANESCU, University Research Professor and Director of the Solidification Laboratory, are with the Department of Metallurgical and Materials Engineering, The University of Alabama, Tuscaloosa, AL 35487. Manuscript submitted October 19, 1992.
METALLURGICAL TRANSACTIONS A
not solved the "Fickian" diffusion equation. When significant solid-state diffusion occurs, mass balance is violated. Consequently, the application of their result is limited to slow diffusion. Clyne and Kurz 141 (CK) have used the BF model and added a spline fit to match predictions by the Scheil equation and the equilibrium equation for infinitesimal and infinite diffusion coefficients, respectively. This relation has no physical basis. All these models are i-D Cartesian and describe "plate" dendrite solidification. The BF and CK analyses were used to explain microsegregation in A1-Cu and A1-Si alloys at cooling rates of up to 200 K / s by Sarreal and Abbaschian. 151For higher cooling rates, a new equation based on the BF model that includes the effects of dendrite tip undercooling and eutectic temperature depression was develo
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