A unified model of microsegregation and coarsening
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I. INTRODUCTION
DURING the solidification of a dendritic alloy, the solute rejected at the solid/liquid interface is redistributed, at the local scale of the secondary dendrite arm spacings (10 to 100 mm), by mass diffusion or convection. This process, referred to as microsegregation, controls the composition of the microstructure and the fraction of eutectic or other phases that form. The closed-form, limiting models are the lever rule (complete mixing in the solid and liquid phases) and the Gulliver–Scheil equation (complete mixing in the liquid, no diffusion in the solid.[1] When the microstructure can be characterized by a fixed length scale (usually a secondary arm spacing), modifications of the Gulliver–Scheil equation that account for finite-rate diffusion in the solid phase (so called back-diffusion) have been presented in the literature. Specific examples can be found in References 2 through 5 and a complete review in Kraft and Cheng.[6] Typically, these models consist of an explicit expression for the solute concentration at the solid-liquid interface in terms of the local solid fraction. Most models are semianalytical in nature, based on an approximation for the diffusion behavior in the solid. An exception is the work of Kobayashi,[7] who develops an exact solution for the solid-state diffusion under the assumption of a parabolic growth rate for the solid. In all models, the solid-state diffusion is characterized by the Fourier number, as follows: a5
DtF X2
[1]
where D is the mass diffusivity in the solid, X is a length scale usually taken to be equal to half the secondary arm spacing, and tF is the local solidification time. Microsegregation models are useful components in largescale models of alloy solidification systems (refer to Reference 8 for a review). In this context, a main role of a microsegregation model is to provide an estimate of the dilution (assuming a partition coefficient of less than unity) of the liquid phase due to the back-diffusion into the solid. In binary eutectic alloys, the dilution will affect the amount of eutectic phase that forms. If the back-diffusion is low (small
V.R. VOLLER, Professor, is with the Saint Anthony Falls Laboratory, Department of Civil Engineering, University of Minnesota, Minneapolis, MN 55455. C. BECKERMANN, Professor, is with the Department of Mechanical Engineering, The University of Iowa, Iowa City, IA 52242. Manuscript submitted August 18, 1998. METALLURGICAL AND MATERIALS TRANSACTIONS A
dilution), the amount of eutectic that forms will be larger than that formed with a large back-diffusion (large dilution). There are, however, other micro scale mechanisms besides back-diffusion that control the dilution of the liquid phase, in particular the coarsening of the microstructure. In general, it is not correct to assume that the characteristic length scale of the microstructure is constant. A more complete model would account for the coarsening of the microstructure, measured by the secondary arm spacings, that typically follows a time t1/3 dependen
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