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Germany; Russia Izhevsk; Jena; Karlsruhe
Section Computer science
Title Computer simulation of the rapid solidification for diluted melt Si-As
Author(-s) Lebedev V.G.a, Sysoeva A.A.a, Knyazheva I.S.a, Danilov D.A.b, Galenko P.K.c
Affiliations Udmurt State Universitya, Karlsruhe Institute of Technologyb, University of Jenac
Abstract We consider a locally nonequilibrium process of solidification for a supercooled binary melt. For sake of simplicity, it is assumed, that the solidifying binary system is at constant temperature and pressure. Also there are two phases corresponding to the solid and the liquid states. The mathematical description of the solidification process is based on the phase-field model that generalizes the approach of Plapp (M. Plapp, Phys. Rev. E 84, 031601 (2011)) to the case of locally nonequilibrium processes. We use the method of extended irreversible thermodynamics to derive thermodynamically consistent equations of the model, in contrast to the phenomenological approach of Plapp. A concentration as a dynamic variable (and not the chemical potential of the impurity) is another difference from Plapp's model. The equivalence of describing the process of solidification through the concentration field and through the chemical potential of the system is shown in the framework of the resulting model. In view of the smallness of the relaxation times, the present model is reduced to the singular-perturbed system of partial differential parabolic equations describing the dynamics of concentration and phase fields. In the paper, it is assumed that the description of the thermodynamic equilibrium states on the basis of the experimentally obtained Gibbs potentials is given. To verify the model, the numerical simulation of the one-dimensional problem of solidification of the melt was performed in the approximation of the diluted melt Si-As, which had been repeatedly investigated experimentally. In this paper, we propose a gradient-stable explicit method of integrating equations of the second order of accuracy in time in order to solve the system of singularly-perturbed equations numerically. We reduced an infinite space interval to a finite interval by the method of “periodic translation”. The estimation of stability was performed using numerical experiments. The concentration profile, the phase-field profile and the distribution coefficient of the impurity at the front of solidification depending upon the value of supercooling were obtained from the numerical simulation of the solidification process for diluted melt Si-As. An analytical expression for the distribution coefficient as a function of supercooling that follows from the locally nonequilibrium model with a sharp interface was used to test the adequacy of the results of numerical experiments. The effect of the model parameters on the solidification process and behavior of the numerical solutions near the diffuse boundary were investigated.
Keywords diluted solution, rapid solidification, phase field, grand potential, modeling
UDC 538.911
MSC 74N20
DOI 10.20537/vm140111
Received 7 October 2013
Language Russian
Citation Lebedev V.G., Sysoeva A.A., Knyazheva I.S., Danilov D.A., Galenko P.K. Computer simulation of the rapid solidification for diluted melt Si-As, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2014, issue 1, pp. 123-140.
  1. Wheeler A.A., Boettinger W.J., McFadden G.B. Phase-field model of solute trapping during solidification, Phys. Rev. E, 1993, vol. 47, no. 3, pp. 1893-1909.
  2. Ahmad N.A., Wheeler A.A., Boettinger W.J., McFadden G.B. Solute trapping and solute drag in a phase-field model of rapid solidification, Phys. Rev. E, 1998, vol. 58, no. 3, pp. 3436-3450.
  3. Echebarria B., Folch R., Karma A., Plapp M. Quantitative phase field model of alloy solidification, Phys. Rev. E, 2004, vol. 70, no. 6, pp. 061604-061625.
  4. Singer-Loginova I., Singer H.M. The phase field technique for modeling multiphase materials, Rep. Prog. Phys., 2008, vol. 71, no. 10, pp. 106501-106532.
  5. Emmerich H. Advances of and by phase-field modelling in condensed-matter physics, Adv. Phys., 2008, vol. 57, no. 1, pp. 1-87.
  6. Provatas N., Elder K. Phase-field methods in materials science and engineering, Weinheim: Wiley-VCH Verlag, 2010.
  7. Steinbach I. Phase-field model for microstructure evolution at the mesoscopic scale, Annu. Rev. Mater. Res., 2013, vol. 43, pp. 89-107.
  8. Plapp M. Unified derivation of phase-field models for alloy solidification from a grand-potential functional, Phys. Rev. E, 2011, vol. 84, no. 3, pp. 031601-031614.
  9. (19.07.2013)
  10. Hillert M. Phase equilibria, phase diagrams and phase transformations, 2nd Ed., Cambridge: Univ. Press, 2007.
  11. Jou D., Casas-Vazquez J., Lebon G. Extended irreversible thermodynamics, 4th Ed., Berlin: Springer, 2010.
  12. Galenko P., Jou D. Diffuse-interface model for rapid phase transformations in nonequilibrium systems, Phys. Rev. E, 2005, vol. 71, no. 4, pp. 046125-046138.
  13. Wang H., Liu F., Zhai H., Wang K. Application of the maximal entropy production principle to rapid solidification: A sharp interface model, Acta Mater., 2012, vol. 60, no. 4, pp. 1444-1454.
  14. Galenko P.K., Abramova E.V., Jou D., Danilov D.A., Lebedev V.G., Herlach D.M. Solute trapping in rapid solidification of a binary dilute system: a phase-field study, Phys. Rev. E, 2011, vol. 84, no. 4, pp. 041143-041159.
  15. Galenko P. Solute trapping and diffusionless solidification in a binary system, Phys. Rev. E, 2007, vol. 76, no. 3, pp. 031606-031615.
  16. Cahn J.W., Hilliard J.E. Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 1958, vol. 28, no. 2, pp. 258-267.
  17. Kessler D. Sharp interface limits of a thermodynamically consistent solutal phase field model, J. of Crys. Growth, 2001, vol. 224, no. 1-2, pp. 175-186.
  18. Lebedev V., Sysoeva A., Galenko P. Unconditionally gradient-stable computational schemes in problems of fast phase transitions, Phys. Rev. E, 2011. vol. 83, no. 2, pp. 026705-026715.
  19. Kittl J.A., Aziz M.J., Brunco D.P., Thompson M.O. Nonequilibrium partitioning during rapid solidification of Si-As alloys, J. Cryst. Growth, 1995, vol. 148, no. 1-2, pp. 172-182.
  20. Kittl J.A., Sanders P.G., Aziz M.J., Brunco D.P., Thompson M.O. Complete experimental test of kinetic models for rapid alloy solidification, Acta Mater., 2000, vol. 48, no. 20, pp. 4797-4811.
  21. (12.07.2013).
  22. Galenko P.K., Gomez H., Kropotin N.V., Elder K.R. Unconditionally stable method and numerical solution of the hyperbolic phase-field crystal equation, Phys. Rev. E, 2013, vol. 88, no. 1, pp. 013310-013320.
  23. Galenko P.K., Krivilyov M.D. Model for isothermal pattern formation of growing crystals in undercooled binary alloys, Model. Simul. Mater. Sci. Eng., 2000, vol. 8, pp. 67-79.
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