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On the theory of spinodal decomposition in solid and liquid binary mixtures

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Zeitschrift für Physik B Condensed Matter

Abstract

The kinetics of phase separation is discussed with emphasis on the transition between spinodal decomposition and nucleation. A reanalysis of the theory of Langer, Baron and Miller shows that it exhibits a spinodal line somewhat closer to the coexistence curve than the meanfield spinodal. There the same (as we think unphysical) critical singularities occur as in Cahn-Hilliard theory. The precise location of this spinodal line depends on the cell size of the coarse graining. For concentrations less than the spinodal one the structure factorS(k, t) converges then towards the structure factor of the metastable onephase state, implying an infinite lifetime of the latter.

In order to include the effects of nucleation and growth we hence present an alternative treatment, extending our previous work on cluster dynamics. From a simple approximation for the radial concentration distribution function of clustersS(k, t) is computed numerically. Even at rather low concentrations the time evolution ofS(k, t) is then similar to what Langer et al. find at high concentrations, implying a very gradual transition from nucleation and growth to spinodal decomposition, at least for parameter values appropriate to the Ising model. This treatment, which is consistent with Lifshitz-Slyozov's coarsening law at late times, is extended to the early stages of phase separation in liquid mixtures.

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References

  1. See e.g. Rundman, K.B., Hilliard, J.E.: Acta Met.15, 1025 (1967) Butler, E.P., Thomas, G.: Acta Met.18, 347 (1970) Bouchard, M., Thomas, G.: Acta Met.23, 1485 (1975) Naudon, A., Allain, J.: Scripta Met.8, 1105 (1974) Laughlin, D.E.: Acta Met.24, 53 (1976) and Ref. 34

    Google Scholar 

  2. Laslaz, G., Guyot, P.: Acta Met.25, 277 (1977) Laslaz, G., Guyot, P., Kostorz, G.: preprint

    Google Scholar 

  3. Goldburg, W.S., Huang, J.S.: In: Fluctuations, Instabilities and Phase Transitions (T. Riste, ed.) New York: Plenum Press 1975 Goldburg, W.I., Shaw, C., Huang, J.S., Pilant, M.S.: J. Chem. Phys.68, 484 (1978) Morrison, G., Knobler, C.M.: J. Chem. Phys.65, 5507 (1976) Wong, N., Knobler, C.M.: J. Chem. Phys.66, 4707 (1977), and preprint Kim, M.W., Schwartz, A.J., Goldburg, W.I.: preprint

    Google Scholar 

  4. Cahn, J.W., Hilliard, J.E.: J. Chem. Phys.28, 258 (1958);31, 688 (1959) Cahn, J.W.: Trans. Metall. Soc. AIME242, 166 (1968), and references therein

    Google Scholar 

  5. Langer, J.S.: Ann. Phys. (N.Y.)65, 53 (1971) Langer, J.S., Baron, M.: ibid.78, 421 (1973) Langer, J.S.: Acta Met.21, 1649 (1973)

    Google Scholar 

  6. Langer, J.S., Baron, M., Miller, H.D.: Phys. Rev. A11, 1417 (1975) Langer, J.S.: J. de phys.12c-7, 395 (1977)

    Google Scholar 

  7. Cook, H.E.: Acta Met.18, 297 (1970)

    Google Scholar 

  8. Lifshitz, I.M., Slyozov, V.V.: J. Phys. Chem. Solids19, 35 (1961)

    Google Scholar 

  9. Binder, K.: Z. Physik267, 213 (1974)

    Google Scholar 

  10. Janssen, H.K.: Z. Physik B23, 245 (1976)

    Google Scholar 

  11. Binder, K., Stauffer, D.: Phys. Rev. Lett.33, 1006 (1974)

    Google Scholar 

  12. Binder, K.: Phys. Rev. B15, 4425 (1977)

    Google Scholar 

  13. Mirold, P., Binder, K.: Acta Met.25, 1435 (1977)

    Google Scholar 

  14. Kawasaki, K.: Progr. Theor. Phys.57, 826 (1977) Kawasaki, K., Ohta, T.: preprint

    Google Scholar 

  15. Tomita, H.: Progr. Theor. Phys.57, 1547 (1977), and preprints

    Google Scholar 

  16. Furukawa, H.: Phys. Lett.62A, 377 (1977); Prog. Theor. Phys.56, 464 (1976);58, 1127 (1977); and preprints

    Google Scholar 

  17. Abraham, F.F.: to be published

  18. Flynn, P.A.: J. Statist. Phys.10, 89 (1974)

    Google Scholar 

  19. Bortz, A.B., Kalos, M.H., Lebowitz, J.L., Zendejas, M.A.: Phys. Rev. B10, 535 (1974) Marro, J., Bortz, A.B., Kalos, M.H., Lebowitz, J.L.: Phys. Rev. B12, 2000 (1975) Lebowitz, J.L., Kalos, M.H.: Scripta Met.10, 9 (1976) Rao, M., Kalos, M.H., Lebowitz, J.L., Marro, J.: Phys. Rev. B13, 7325 (1976) Sur, A., Lebowitz, J.L., Marro, J., Kalos, M.H.: Phys. Rev. B15, 535 (1977) Marro, J., Lebowitz, J.L., Kalos, M.H.: preprint Kalos, M.H., Lebowitz, J.L., Penrose, O., Sur, A.: preprint Penrose, O., Lebowitz, J.L., Marro, J., Kalos, M.H., Sur, A.: preprint

    Google Scholar 

  20. Pundarika, E.S., Hanson, K., Morris, J.W.: to be published

  21. Saito, Y.: J. Phys. Soc. Japan41, 1129 (1976)

    Google Scholar 

  22. Cahn, J.W.: Acta Met.10, 907 (1962)

    Google Scholar 

  23. For recent review of nucleation theory, see Binder, K., Stauffer, D.: Adv. Phys.25, 343 (1976) Binder, K.: In: Proceedings of the IUPAP Conference on Statistical Physics (Budapest, 1975), p. 219

    Google Scholar 

  24. Zettlemoyer, A.C. (ed.): Nucleation. New York: M. Dekker 1969; Nucleation Phenomena. New York: Elsevier 1977; Nucleation III (to be published) Abraham, F.F., Homogeneous Nucleation Theory. New York: Academic 1975

    Google Scholar 

  25. Langer, J.S., Turski, A.: Phys. Rev. A8, 3230 (1973) Sarkies, K.W., Frankel, N.E.: Phys. Rev. A11, 1724 (1975)

    Google Scholar 

  26. Kiang, C.S., Stauffer, D., Walker, G.H., Puri, O.P., Wise, J.D., Jr., Patterson, E.M.: J. Atm. Sci.28, 1112 (1971) Stauffer, D., Kiang, C.S.: Adv. Coll. Interface Sci.7, 103 (1977)

    Google Scholar 

  27. Fisher, M.E.: Physics3, 255 (1967)

    Google Scholar 

  28. Chu, B., Schoenes, F.J., Fisher, M.E.: Phys. Rev.185, 219 (1969)

    Google Scholar 

  29. Andreev, A.F.: Sov. Phys. JETP18, 1415 (1964)

    Google Scholar 

  30. Binder, K.: Phys. Rev. B8, 3423 (1973)

    Google Scholar 

  31. Binder, K., Stoll, E.: Phys. Rev. Lett.31, 47 (1973) Binder, K., Müller-Krumbhaar, H.: Phys. Rev. B9, 2328 (1974) Binder, K.: Ann. Phys.98, 390 (1976)

    Google Scholar 

  32. Langer, J.S.: Ann. Phys. (N.Y.)41, 108 (1967); physica73, 61 (1974)

    Google Scholar 

  33. Penrose, O., Lebowitz, J.L.: J. Statist. Phys.3, 211 (1971)

    Google Scholar 

  34. Grilhé, J.: to be published

  35. Compagner, A.: physica72, 115 (1974) Saito, Y.: Progr. Theor. Phys.59, 375 (1978)

    Google Scholar 

  36. For a review, see Halperin, B.I., Hohenberg, P.C.: Revs. Mod. Phys.49, 435 (1977)

    Google Scholar 

  37. Billotet, C.: Dissertation, Universität des Saarlandes (1978, unpublished)

  38. Billotet, C., Binder, K.: to be published

  39. Kawasaki, K.: Phys. Rev.145, 224 (1966);148, 375 (1966);150, 285 (1966)

    Google Scholar 

  40. Kikuchi, R.: J. Chem. Phys.47, 1664 (1967)

    Google Scholar 

  41. Binder, K., Stauffer, D., Müller-Krumbhaar, H.: Phys. Rev. B12, 5261 (1975)

    Google Scholar 

  42. Ardell, A.J.: In: The Mechanics of Phase Transitions in Crystalline Solids, Vol. 33 (Institute of Metals Monograph Series, London, 1969) p. 111 Greenwood, G.W.: ibid. p. 103 Fischmeister, H., Grimvall, G.: In: Sintering and Related Phenomena (edited by Kuczynski, G.C.). New York: Plenum 1973

    Google Scholar 

  43. Müller-Krumbhaar, H.: Phys. Lett.48 A, 459 (1974)

    Google Scholar 

  44. Binder, K., Stauffer, D.: Adv. Phys.25, 343 (1976)

    Google Scholar 

  45. Cahn, J.W.: private communication

  46. Knobler, C.M.: private communication; Goldburg, W.I.: private communication

  47. Kirkpatrick, S.: Rev. Mod. Phys.45, 574 (1973)

    Google Scholar 

  48. Müller-Krumbhaar, H.: Phys. Lett. A50, 27 (1974)

    Google Scholar 

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Authors and Affiliations

  1. Institut für Festkörperforschung, Kernforschungsanlage Jülich GmbH, Postfach 1913, D-5170, Jülich, Germany

    K. Binder

  2. Theoretische Physik, Universität des Saarlandes, D-6600, Saarbrücken, Germany

    C. Billotet & P. Mirold

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  1. K. Binder
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  2. C. Billotet
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  3. P. Mirold
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Binder, K., Billotet, C. & Mirold, P. On the theory of spinodal decomposition in solid and liquid binary mixtures. Z Physik B 30, 183–195 (1978). https://doi.org/10.1007/BF01320985

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