Modelling mushy zones in binary alloys

Two outstanding problems in the formulation of models for the evolution of dendritic or granular mushy zones in solidifying binary alloys are the self-selection of the micro-scale which controls the flow permeability, and the prescription of an appropriate boundary condition for the solid fraction at the mush liquid interface. We suggest that the microscale is selected to be that where diffusive removal of solute from the dendrite surfaces becomes rate-limiting, and the interface regains its stability. In freely growing mushes (but not necessarily in directional solidification), the microscale is then determined by the (large) value of the Lewis number. With the microscale selected in this way, we find that if the averaged model equations are derived by the method of homogenisation, the appropriate boundary condition for the solid fraction at the mush-liquid interface (that it equal zero) can be formally derived through consideration of solute conservation at the interface. A particular consequence of our study is the derivation of an explicit microscale model describing the evolution of the dendritic interface. In principle this model should be able to describe the formation of secondary dendrites, and we show that a uniform solution corresponding to a cellular-dendritic array is always unstable to the formation of secondary dendrites, although the linear response is ultimately damped. We use this assessment of the microscale to comment on possible pore spacings in dendritic solidification of the Earth's inner core.