Eur. Phys. J. Special Topics 229, 439-445 (2020)
https://doi.org/10.1140/epjst/e2019-900122-4

Regular Article

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

¹ University of Hassan II Casablanca, Faculty of Sciences Ben M’Sik, Department of Physics, Laboratory of Condensed Matter Physics (LPMC), BP 7955, Casablanca, Morocco
² Ural Federal University, Department of Theoretical and Mathematical Physics, Laboratory of Multi-Scale Mathematical Modeling, 620000 Ekaterinburg, Russia
³ Friedrich-Schiller-Universität-Jena, Faculty of Physics and Astronomy, Otto Schott Institute of Materials Research, 07743 Jena, Germany

^a e-mail: ahmedsalhoumi@gmail.com

Received: 25 June 2019
Published online: 11 February 2020

Abstract

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.