A Unified Model of Microsegregation and Coarsening

than that formed with a large back-diffusion (large dilution). DURING the solidification of a dendritic alloy, the solThere are, however, other micro scale mechanisms besides ute rejected at the solid/liquid interface is redistributed, at back-diffusion that control the dilution of the liquid phase, the local scale of the secondary dendrite arm spacings (10 in particular the coarsening of the microstructure. In general, to 100 mm), by mass diffusion or convection. This process, it is not correct to assume that the characteristic length scale referred to as microsegregation, controls the composition of of the microstructure is constant. A more complete model the microstructure and the fraction of eutectic or other phases would account for the coarsening of the microstructure, meathat form. The closed-form, limiting models are the lever sured by the secondary arm spacings, that typically follows rule (complete mixing in the solid and liquid phases) and a time t1/3 dependence.[9] the Gulliver–Scheil equation (complete mixing in the liquid, To understand how coarsening dilutes the liquid phase, no diffusion in the solid.[1] When the microstructure can be consider a representative elemental volume (REV) in the characterized by a fixed length scale (usually a secondary two-phase mushy region of a solidifying alloy (Figure 1). arm spacing), modifications of the Gulliver–Scheil equation As the microstructure coarsens, new liquid is introduced that account for finite-rate diffusion in the solid phase (so into the REV. If there is no macroscopic transport of solute, called back-diffusion) have been presented in the literature. in order to maintain a constant average composition in the Specific examples can be found in References 2 through 5 REV, this incoming liquid is at the average composition and and a complete review in Kraft and Cheng.[6] Typically, dilutes the liquid phase in the REV, which in a binary eutectic these models consist of an explicit expression for the solute alloy reduces the amount of eutectic formed. Note that this concentration at the solid-liquid interface in terms of the dilution occurs even in cases where there is no back-diffusion local solid fraction. Most models are semianalytical in in the solid. nature, based on an approximation for the diffusion behavior In light of this, it is reasonable to expect a more complete in the solid. An exception is the work of Kobayashi,[7] who model of microsegregation in alloy solidification to include develops an exact solution for the solid-state diffusion under the effects of both back-diffusion and coarsening. To date, the assumption of a parabolic growth rate for the solid. In in terms of analytical models, the effect of coarsening has all models, the solid-state diffusion is characterized by the only been accounted for in the limiting case of no backFourier number, as follows: diffusion, by Mortensen.[10] Most microsegregation models