Abstract
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.
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References
G. Caginalp, An analysis of a phase field model of a free boundary. Arch. Rat. Mech. Anal.,92 (1986), 205–245.
G. Caginalp and W. Xie, Phase-field and sharp-interface models. Phys. Rev. E,48 (1993), 1897–1909.
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.
J.W. Cahn and J.E. Hilliard, Free boundary of a nonuniform system. I. Interfacial free energy. J. Chem. Phys.,28 (1958), 258–267.
N.E. Cusack, The Physics of Structurally Disordered Matter: An Introduction. IOP Publishing, 1987.
R.H. Doremus, Rates of Phase Transformations. Academic Press, 1985.
C. Godrèche (ed.), Solids Far from Equilibrium. Cambridge Univ. Press, 1992.
S.R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics. Dover, New York, 1984.
M. Doi and A. Onuki, Koubunshi butsuri · Souten’i dainamikusu (in Japanese). Iwanami kouza gendai no butsurigaku 19, Iwanami Shoten, Tokyo, 1992.
K.R. Elder, F. Drolet, J.M. Kosterlitz and M. Grant, Stochastic eutectic growth. Phys. Rev. Lett.,72 (1994), 677–680.
M. Hiraoka and M. Tanaka, Shinban Idougensyouron (in Japanese). Asakura Shoten, Tokyo, 1994.
T.S. Hutchison and D.C. Baird, The Physics of Engineering Solids (2nd edition). John Wiley & Sons, 1968.
A. Karma, Phase-field model of eutectic growth. Phys. Rev. E,49 (1994), 2245–2249.
R. Kobayashi, Modeling and numerical simulation of dendritic crystal growth. Physica D,63 (1993), 410–423.
R. Kobayashi, A numerical approach to three-dimensional dendritic solidification. Experimental Math.,3 (1994), 59–81.
H. Komiyama, Sokudoron (in Japanese). Asakura Shoten, Tokyo, 1990.
T. McLeish (ed.), Theoretical Challenges in the Dynamics of Complex Fluids. Kluwer Academic Publishers, 1997.
C. Misbah and D.E. Temkin, Model for eutectic organization: The purely kinetic regime. Phys. Rev. E,49 (1994), 3159–3165.
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.
F. Nakano and H. Kimura, Souten’i no toukei-netsurikigaku (in Japanese). Asakura Shoten, Tokyo, 1988.
A. Oono, Kinzoku no gyouko (in Japanese). Chijin Shokan, Tokyo, 1984.
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.
K. Osamura et al, Zairyousoshikigaku (in Japanese). Asakura Shoten, Tokyo, 1991.
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.
A. Prince, Alloy Phase Equilibria. Elsevier, 1966.
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.
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.
J. Strain, Spectral methods for nonlinear parabolic systems. J. Comp. Phys.,122 (1995), 1–12.
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.
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.
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.
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.
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Sakai, K. A mathematical model of solidification dynamics of binary alloys. Japan J. Indust. Appl. Math. 17, 43–58 (2000). https://doi.org/10.1007/BF03167335
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DOI: https://doi.org/10.1007/BF03167335