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Revisiting Jackson-Hunt calculations: Unified theoretical analysis for generic multi-phase growth in a multi-component system

A straight-forward extension of the Jackson-Hunt theory for directionally solidifying multi-phase growth where the number of components exceeds the number of solid phases becomes difficult on account of the absence of the required number of equations to determine the boundary layer compositions ahead of the interface. In this paper, we therefore revisit the Jackson-Hunt(JH) type calculations for any given situation of multi-phase growth in a multi-component system and self-consistently derive the variations of the compositions of the solid phases as well as their volume fractions, which grow such that the composite solid-liquid interface is isothermal. This allows us to unify the (JH) calculation schemes for both in-variant as well as multi-variant eutectic reactions. The derived analytical expressions are then utilized to study the effect of dissimilar solute diffusivities and interfacial energies on the undercoolings and the solidified fractions. We also perform phase field simulations to confirm our theoretical predictions and find a good agreement between our analytical calculations and model predictions for model symmetric alloys as well as for a particular Ni-Al-Zr alloy.