Sharp interface limit of a two-time scale phase field model of a binary mixture

Due to its analytical flexibility and thermodynamic consistency, the phase field methodology is widely used in the analysis of equilibrium states and transformation between phases. The present review is devoted to a class of hyperbolic phase field models, which applies to slow and fast phase transformations. Focusing on the example of solidification of metastable liquid, an analysis is presented for the important procedure of reducing the diffuse interface to the sharp interface. An asymptotic analysis is discussed for application to solidifying binary mixture with diffuse phase interface under arbitrary concentration of species and isothermal and isobaric conditions. The analysis reveals that the hyperbolic phase field model can be mapped onto the known hyperbolic Stefan problem within the sharp interface limit. This result, together with the common tangent construction, allows us to analyze (i) nonequilibrium effects in the form of solute trapping and (ii) the complete transition from the diffusion-limited to the diffusionless (chemically partitionless) solidification at finite interface velocity. A comparison with other theoretical models is summarized and a discussion, which is attributed to experimental results, is given.