Solidification fronts and solute trapping in a binary alloy

A phase-field model with order parameter, concentration, and temperature as field variables is used to study the properties of solidification fronts in a binary alloy. As in previous papers, the model includes dependence of the free energy density not only on these field variables but also on the gradients of the order parameter and concentration. Terms with these gradients represent surface free energy associated with the phase interface and with the jump in concentration. We treat them as conceptually and physically different; in particular, the thicknesses of the two interfaces will generally be different. Based on the smallness of the coefficients of these gradient terms, and the largeness of the ratio of solute diffusivity in the liquid to that of the solid, asymptotic analyses in various parameter regimes are performed which reveal information on such things as the dependence of the discontinuity of concentration at the front on its velocity and on the above-mentioned parameters. More broadly, we investigate the spatial structure of the concentration jump interface in various parameter ranges. Formulations are given to the problem of directional solidification and to free boundary problems for the free motion of a solidification front. Corrections due to curvature of the interface are found.