Second order sharp-interface and thin-interface asymptotic analyses and error minimization for phase-field descriptions of two-sided dilute binary alloy solidification

Sharp-interface and thin-interface asymptotic analyses are presented for a generalization of the Beckermann–Karma phase-field model for solidification of a dilute binary alloy when the interface curvature is macroscopic. The ratio of diffusivities, Rm≡Ds′/Dm′, in the solid and melt is arbitrary with 0⩽Rm⩽10⩽Rm⩽1. Discrepancies between this diffuse-interface model and the classical, two sided solutal model (TSM) description are quantified up to second order in the small parameter εε that controls the interface thickness. We uncover extra terms in the interface species flux balance and in the Gibbs–Thomson equilibrium condition introduced by the finite width of the interface. Asymptotic results in the limit of rapid-interfacial kinetics are presented for both finite phase-field mobility and a quasi-steady state approximation for the phase-field wherein the phase-field responds passively to the concentration field. The possibility of adding additional terms to the phase-field version of the species conservation equation is explored as a means of achieving O(ε2)O(ε2) consistency with the classical model. Our results naturally lead to a generalization of the anti-trapping solutal flux suggested by Karma [Phase-field-formulation for quantitative modeling of alloy solidification, Phys. Rev. Lett. 87(11) (2001) 115701] for the limit Rm=0Rm=0. Achieving second order accuracy for arbitrary RmRm requires judicious choices for the interpolating functions; these are calculated a posteriori using the functional forms of the error terms as a guide.