Segregation during solidification with spongy deformation of the mushy zone

Mathematical expressions of segregation rates occurring during solidification with spongy deformation of an alloyed mushy zone are derived from classical mass balance analyses. The boundaries of the domain considered for the mass balances deform with the same velocity as the solid phase: the reference frame is attached to the solid phase. The three variables needed to fully describe segregation within this “Lagrangian” approach are the mass of solid in the domain, the volume of the domain and the relative velocity of the liquid with respect to the solid. The main result is a mathematical expression for the local macrosegregation rate. The analysis is further extended to produce an expression for the local microsegregation rate that can be compared with the one proposed by Flemings and Nereo (Trans. AIME 239 (1967) 1449–1461). An additional term thus enters into our expression. This term is proportional to the local strain rate multiplied by the ratio of the volume fraction of solid over the volume fraction of liquid. It is expected to become dominant when the mushy zone is fully developed (large value of the ratio) and when the strain can be transmitted through a coherent solid skeleton that deforms like a sponge. Still, the origin of the deformation remains arbitrary and the derived expressions could be used to quantify segregation due to the thermal contraction of the solid and liquid phases, the shrinkage associated with the phase transformations, and/or the mechanical deformation of the domain. Finally, the expression for the local microsegregation rate is combined with the first law of thermodynamics. The rate of progress of solidification is thus related to the rate of heat extraction, the transport of heat due to the relative liquid flow with respect to the solid, and the enthalpy of dissolution of the solid in the liquid.