Multiphase micro-macroscopic model of solute diffusion in dendritic alloy solidification

A new model, aimed at predicting microstructure formation in metal castings, is proposed for solute diffusion during dendritic solidification of alloys. The model accounts for the different length scales existing in a dendritic structure. This is accomplished by utilizing a multiphase approach, in which not only the various physical phases but also phases associated with different length scales are considered separately. The macroscopic conservation equations are derived for each phase using the volume averaging technique, with constitutive relations developed for the interfacial transfer terms. It is shown that the multiphase model can rigorously incorporate a kinetic law for dendrite tips growing into an undercooled melt, as well as finite-rate solute diffusion in the solid. In addition, the distinction of different length scales enables the inclusion of realistic descriptions of the dendrite topology and relations for key metallurgical parameters. Another novel aspect of the model is that a single set of equations for solute diffusion is developed for both equiaxed and columnar dendritic solidification. Finally, several simplified versions of the general model are discussed and compared to previous studies.