Skip to main content

A mathematical model of solidification dynamics of binary alloys


This paper proposes a mathematical model of solidification dynamics of binary alloys. We express the state of an alloy by the phase parameter and the concentration, and describe the dynamics as a free energy minimizing process. The most advantageous feature of the model is that the interaction energy is directly given to each pair of atoms according to types of alloys. Thus, we can easily know the conditions to let the model correspond to all basic kind of alloys including eutectic, peritectic alloys, for example. We also perform some numerical simulations and reproduce typical structures observed in real alloys.

This is a preview of subscription content, access via your institution.


  1. G. Caginalp, An analysis of a phase field model of a free boundary. Arch. Rat. Mech. Anal.,92 (1986), 205–245.

    MATH  Article  MathSciNet  Google Scholar 

  2. G. Caginalp and W. Xie, Phase-field and sharp-interface models. Phys. Rev. E,48 (1993), 1897–1909.

    Article  MathSciNet  Google Scholar 

  3. G. Caginalp and W. Xie, Mathematical models of phase boundaries in alloys: Phase field and Sharp interface. Motion by Mean Curvature and Related Topics, Proceeding of the International Conference held at Trento, 1992 (eds. G. Buttazzo and A. Visintin), Walter de Gruyter, Berlin, 1994.

    Google Scholar 

  4. J.W. Cahn and J.E. Hilliard, Free boundary of a nonuniform system. I. Interfacial free energy. J. Chem. Phys.,28 (1958), 258–267.

    Article  Google Scholar 

  5. N.E. Cusack, The Physics of Structurally Disordered Matter: An Introduction. IOP Publishing, 1987.

  6. R.H. Doremus, Rates of Phase Transformations. Academic Press, 1985.

  7. C. Godrèche (ed.), Solids Far from Equilibrium. Cambridge Univ. Press, 1992.

  8. S.R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics. Dover, New York, 1984.

    Google Scholar 

  9. M. Doi and A. Onuki, Koubunshi butsuri · Souten’i dainamikusu (in Japanese). Iwanami kouza gendai no butsurigaku 19, Iwanami Shoten, Tokyo, 1992.

    Google Scholar 

  10. K.R. Elder, F. Drolet, J.M. Kosterlitz and M. Grant, Stochastic eutectic growth. Phys. Rev. Lett.,72 (1994), 677–680.

    Article  Google Scholar 

  11. M. Hiraoka and M. Tanaka, Shinban Idougensyouron (in Japanese). Asakura Shoten, Tokyo, 1994.

    Google Scholar 

  12. T.S. Hutchison and D.C. Baird, The Physics of Engineering Solids (2nd edition). John Wiley & Sons, 1968.

  13. A. Karma, Phase-field model of eutectic growth. Phys. Rev. E,49 (1994), 2245–2249.

    Article  Google Scholar 

  14. R. Kobayashi, Modeling and numerical simulation of dendritic crystal growth. Physica D,63 (1993), 410–423.

    MATH  Article  Google Scholar 

  15. R. Kobayashi, A numerical approach to three-dimensional dendritic solidification. Experimental Math.,3 (1994), 59–81.

    MATH  Google Scholar 

  16. H. Komiyama, Sokudoron (in Japanese). Asakura Shoten, Tokyo, 1990.

    Google Scholar 

  17. T. McLeish (ed.), Theoretical Challenges in the Dynamics of Complex Fluids. Kluwer Academic Publishers, 1997.

  18. C. Misbah and D.E. Temkin, Model for eutectic organization: The purely kinetic regime. Phys. Rev. E,49 (1994), 3159–3165.

    Article  Google Scholar 

  19. W.W. Mullins and R.F. Sekerka, Stability of a planar interface during solidification of a dilute binary alloy. J. Appl. Phys.,35 (1964), 444–451.

    Article  Google Scholar 

  20. F. Nakano and H. Kimura, Souten’i no toukei-netsurikigaku (in Japanese). Asakura Shoten, Tokyo, 1988.

    Google Scholar 

  21. A. Oono, Kinzoku no gyouko (in Japanese). Chijin Shokan, Tokyo, 1984.

    Google Scholar 

  22. Y. Oono and S. Puri, Study of phase-separation dynamics by use of cell dynamical systems. I. Modeling. Phys. Rev. A,38 (1988), 434–453.

    Google Scholar 

  23. K. Osamura et al, Zairyousoshikigaku (in Japanese). Asakura Shoten, Tokyo, 1991.

    Google Scholar 

  24. O. Penrose and P.C. Fife, Thermodynamically consistent models of phase-field type for the kinetics of phase transitions. Physica D,43 (1990), 44–62.

    MATH  Article  MathSciNet  Google Scholar 

  25. A. Prince, Alloy Phase Equilibria. Elsevier, 1966.

  26. J.S. Rowlinson, Translation of J. D. van der Waals’ “The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density”. J. Stat. Phys.,20 (1979), 197–244.

    Article  MathSciNet  Google Scholar 

  27. K. Sakai, A mathematical model of casting process of binary alloys and its application to numerical simulation of structure formation in eutectic alloys (in Japanese). Submitted to Trans. JSIAM.

  28. J. Strain, Spectral methods for nonlinear parabolic systems. J. Comp. Phys.,122 (1995), 1–12.

    MATH  Article  MathSciNet  Google Scholar 

  29. A.A. Wheeler, A numerical scheme to model the evolution of the morphological instability of a freezing binary alloy. Q. J. Mech. Appl. Math.,39 (1986), 381–401.

    Article  Google Scholar 

  30. A.A. Wheeler, W.J. Boettinger and G.B. McFadden, Phase-field model for isothermal phase transitions in binary alloys. Phys. Rev. A,45 (1992), 7424–7439.

    Article  Google Scholar 

  31. A.A. Wheeler, W.J. Boettinger and G.B. McFadden, Phase-field model of solute trapping during solidification. Phys. Rev. E,47 (1993), 1893–1909.

    Article  Google Scholar 

  32. A.A. Wheeler, W.J. Boettinger and G.B. McFadden, Phase-field model for solidification of a eutectic alloy. Proc. R. Soc. Lond. A,452 (1996), 495–525.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Kazushige Sakai.

About this article

Cite this article

Sakai, K. A mathematical model of solidification dynamics of binary alloys. Japan J. Indust. Appl. Math. 17, 43 (2000).

Download citation

  • Received:

  • Revised:

  • DOI:

Key words

  • binary alloys
  • isothermal solidification dynamics
  • mathematical model using phase parameter