Hodograph-equation for rapid solidification of Si-0.1 at.% As alloy melt

Hyperbolic-type equations of both phase field and concentration arising from a phase-field model for fast phase transformations in binary dilute systems yield in the one-dimension moving frame of reference to the concentration-and phase field governing equations, respectively. These equations have been solved numerically and applied to the case of Si-0.1 at.% As binary alloy [P.K. Galenko et al., Phys. Rev. E 84 , 041143 (2011)]. In this paper, the coupling of the hodograph equation for the interface with the solute diffusion equation leads to an exact analytical solution of the one-point Cauchy problem of an ordinary differential equation in a parametric form. Application of this solution to the case of Si-0.1 at.% As gives (i) the same tendency of concentration variation along dimensionless spatial coordinate (ii) the same values of interface velocity with a very slight difference in the value of concentration for a given undercooling at the interface. Based on the results obtained, the established hodograph-equation confirms again its usefulness to predict, for instance, certain aspects of rapid solidification processes for binary alloys.