Growth of Domains and Scaling in the Late Stages of Phase Separation and Diffusion-Controlled Ordering Phenomena

  • K. Binder
  • D. W. Heermann


These lectures consider the kinetics of phase changes, induced by a sudden change of external thermodynamic parameters. E.g., we treat a system with a second-order transition at a critical temperature Tc (Fig. 1, left part). For T0 > Tc the system is disordered, while for T < Tc there is an order parameter ± ψ (implying one-component orderings, e.g., an Ising model; later we discuss generalizations). We consider a “quenching experiment”: The system is brought from an initially disordered state at T0 to a state at T where in equilibrium the system should be orderedl. Since no sign of ψ is preferred, the system starts forming locally ordered regions of either sign, separated by domain walls. Due to the unfavorable interface free energy cost, this situation is not thermodynamically stable — there is a driving force to reduce this free energy. Thus the random motion of walls, induced by statistical fluctuations, leads to a growth of a characteristic length L(t) of the ordered regions with the time t after the sudden quench performed at t = 0. Typically, one expects L(t)ta for large t, and the excess internal energy ΔE(t)∝t-a a’ : a, a’ are the characteristic exponents of interest here. Sometimes even slower growth [L(t)∝lnt] might occur, see below.


Ising Model Critical Phenomenon Cluster Dynamic Coexistence Curve Cluster Size Distribution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    For an extensive recent review of this area, see J. D. Gunton, M. San Miguel and P. S. Sahni,in “Phase Transitions and Critical Phenomena”, Vol. 8, C. Domb and J. L. Lebowitz, Eds., Academic Press, New York (1983) p. 267.Google Scholar
  2. 2.
    For more detailed discussions of this point and earlier literature, see e.g. K. Binder, Phys. Rev A29:341 (1984) and in “Condensed Matter Research Using Neutrons”, S. W. Lovesey and R. Scherm, Eds., Plenum, New York (1984) p. I.Google Scholar
  3. 3.
    E.g., this has been seen in Monte-Carlo simulations (K. ‘Binder, Solid State Comm 34:191 (1980)) and experiments on phase-separating fluid mixtures (W. I. Goldburg, private communication).Google Scholar
  4. 4.
    D. W. Heermann, Z.Phys . B55: 309 (1984).CrossRefGoogle Scholar
  5. 5.
    K. Binder, Ann. Phys 98:390 (1976); R. Kretschmer, K. Binder and D. Stauffer, J..Stat. Phys 15:267 (1976). See also A. D. Bruce and D. J. Wallace, J. Phys A16:1721 (1983) and A. D. Bruce, preprints.Google Scholar
  6. 6.
    A. Coniglio and W. Klein, J. Phys A13: 2775 (1980).MathSciNetGoogle Scholar
  7. 7.
    C.-K. Hu, Phys. Rev B29: 5103 (1984).CrossRefGoogle Scholar
  8. 8.
    K. Binder and D. Stauffer, Phys. Rev. Lett 33:1006 (1974).Google Scholar
  9. 9.
    K. Binder, Phys. Rev B15:4425 (1977).Google Scholar
  10. 10.
    K. Binder, C. Billotet and P. Mirold, Z. Phys B30:183 (1978); P. Mirold and K. Binder, Acta met 25:1435 (1977).Google Scholar
  11. 11.
    C. Billotet and K. Binder, Z. Phys B32: 195 (1979).Google Scholar
  12. 12.
    K. Binder, in “Stochastic Nonlinear Systems in Physics, Chemistry and Biology”, L. Arnold and R. Lefever, eds., Springer, Berlin (1981) p. 62.Google Scholar
  13. 13.
    H. Furukawa, Progr. Theor 59:1072 (1978); Phys. Lett 66A:60 (1978); 97A:346 (1983).Google Scholar
  14. 14.
    H. Furukawa, Phys. Rev. Lett 43:136 (1979); Phys. Rev A23:1535 (1981); A28:1717 (1983).Google Scholar
  15. 15.
    P. C. Hohenberg and B. I. Halperin, Rev. Mod. Phys 49:435 (1977).Google Scholar
  16. 16.
    A. Sadiq and K. Binder, J. Statis Phys. 35:617 (1984); Phys. Rev. Lect 51:674 (1983).Google Scholar
  17. 17.
    G. S. Grest and P. S. Sahni, Phys. Rev B30: 226 (1984).CrossRefGoogle Scholar
  18. 18.
    F. G. Mazenko and 0. T. Valls, Phys. Rev. Lett 51:2044 (1983).Google Scholar
  19. 19.
    G. F. Mazenko, 0. T. Valls and F. C. Zhang, to be published.Google Scholar
  20. 20.
    J. D. Gunton, J. Stat. Phys 34: 1019 (1984).MathSciNetCrossRefGoogle Scholar
  21. 21.
    K. Kawasaki and T. Ohta, Progr. Theor. Phys 59:361 (1978).Google Scholar
  22. 22.
    E. Siggia, Phys. Rev A20:595 (1979).Google Scholar
  23. 23.
    G. F. Mazenko and M. Zannetti, Phys. Rev. Lett 53:21Q6 (1984).Google Scholar
  24. 14.
    J. K. Bhattacharjee, P. Meakin and D. J. Scalapino, Phys. Rev A30:1026 (1984).Google Scholar
  25. 25.
    G. S. Grest, D. J. Srolovitz and M. P. Anderson, Phys. Rev. Lett 52:1321 (1984).Google Scholar
  26. 26.
    G. Mouritsen, Phys. Rev. B (1985); B28:3150 (1983) and preprint.Google Scholar
  27. 27.
    P. S. Sahni, G. S. Grest, M. P. Anderson and D. J. Srolovitz, Phys. Rev. Lett 50:263 (1983); Phys Rev. B28:2705 (1983); K. Kaska J. Nierinen and J. D. Gunton, preprint; see also D. J. Srolovitz, M. P. Anderson, P. S. Sahni and G. S. Grest, Acta met 32: 783 (1984).Google Scholar
  28. 28.
    F. Y. Wu, Rev. Mod. Phys 54:235 (1982).Google Scholar
  29. 29.
    I. M. Lifshitz, Sov. Phys. JETP 15:939 (1962).Google Scholar
  30. 30.
    S. A. Safran, Phys. Rev. Lett 46:1581 (1981).Google Scholar
  31. 31.
    For the Ising model of a binary mixture at the critical concentration ccrit = 1/2, such frozen-in structures were first studied by P. PB Meakin and S. Reich, Phys. Lett. A92, 247 (1982) and by A. Levy, S. Reich and P. Meakin, Phys. Lett. 78A (1982).Google Scholar
  32. 32.
    J. Villain, Phys. Rev. Lett 52:1543 (1984); G. Grinstein and J. F. Fernandex, Phys. Rev B29:389 (1984).Google Scholar
  33. 33.
    D. Stauffer, C. Hartzstein, K. Binder and A. Aharony, Z. Phys B55:325 (1984). However, E. Pytte and J. F. Fernandez (Phys. Rev B31:616 (1985)) do find L(t) = Int.Google Scholar
  34. 34.
    C. Rottman and M. Wortis, Phys. Rev B24:6274 (1981); R. K. P. Zia and J. E. Avron, Phys. Rev 1352:2042 (1982); J. E. Avron, L. S. Schulman, H. van Beijeren and R. K. P. Zia, J. Phys A15:L81 (1982); R. K. P. Zia, preprint.Google Scholar
  35. 35.
    J. R. Simon, P. Guyot and A. Fhilarducci de Salva, Phil. Mag A49:151 (1984).Google Scholar
  36. 36.
    P. A. Rikvold and J. D. Gunton, Phys. Rev. Lett 49 (1982).Google Scholar
  37. 37.
    K. Tokuyama and K. Kawasaki, Physica 123A:386 (1984); T. Ohta, preprint.Google Scholar
  38. 38.
    P. W. Vvorhees and M. E. Glicksman, Acta met. (1984); J. A. Marqusee and J. Ross, J. Chem Phys. 80: 536 1984 ).Google Scholar
  39. 39.
    S. W. Allen and J. W. Cahn, Acta met. 27:1017 (1979); 27: 1085 (1979).Google Scholar
  40. 40.
    K. Kawasaki, M. C. Yalabik and J. D. Gunton, Phys. Rev A17:455 (1978).Google Scholar
  41. 41.
    T. Ohta, D. Gamow and K. Kawasaki, Phys. Rev. Lett 49:1223 (1982).Google Scholar
  42. 42.
    I. Ohta, ref. 37; M. Grant and J. D. Gunton, Phys. Rev B38: 5496 (1983); K. Kawasaki and T. Ohta, Progr. Theor. Phys 67:142 (1982).Google Scholar
  43. 43.
    G. F. Mazenko, Phys. Rev B26:5103 (1982); G. F. Mazenko and 0. T. Valls, Phys. Rev B27:6811 (1983); G. F. Mazenko and 0. T. Valls, Phys:. Rev. B30:6732 (1984); F. C. Chang, 0. T. Valls and G. F. Mazenko, preprInt.Google Scholar
  44. 44.
    Note that this statement refers to domain growth in a system with a second-order transition. If one quenches to a metastable regime underneath a first-order transition, one rather finds L(t) a t (S. K. Chan, J. Chem. Phys 67:5755 (1977)).Google Scholar
  45. 45.
    M. K. Phani, J. L. Lebowitz, M. H. Kalos and 0. Penrose, Phys. Rev. Lett 45:366 (1980).Google Scholar
  46. 46.
    P. S. Sahní, G. Dee, J. D. Gunton, M. K. Phaní, J. L. Lebowitz and M. H. Kalos, Phys. Rev B24: 410 (1981).CrossRefGoogle Scholar
  47. 47.
    K. Kaski, M. D. Yalabík, J. D. Gunton and P. S. Sahni, Phys Rev. B28:5263 (1983); E. T. Gawlinski, M. Grant, J. D. Gunton and K. Kaski, Phys. Rev B31:281 (1985).Google Scholar
  48. 48.
    Y. C. Chou and W. I. Goldburg, Phys Rev. A23: 858 (1981).CrossRefGoogle Scholar
  49. 49.
    J. Marro, J. L. Lebowitz and M. H. Kalos, Phys. Rev Lett. 43: 282 (1979).CrossRefGoogle Scholar
  50. 50.
    J. L. Lebowitz, J. Marro and M. H. Kalos, Acta met. 30: 297 (1982).CrossRefGoogle Scholar
  51. 51.
    P. Fratzl, J. L. Lebowitz, J. Marro and M.s, Acta met 31: 1849 (1983).CrossRefGoogle Scholar
  52. 52.
    F. F. Abraham, S. W. Koch and R. C. Desai, Phys. Rev. Lett 49:923 (1982); S. W. Koch, R. C. Desai and F. F. Abraham, Phys. Rev A27:2152 (1983); S. W. Koch and R. Liebmann, J. Statist. Phys 33:31 (1983).Google Scholar
  53. 53.
    P. S. Salmi and J. D. Gunton, Phys. Rev. Lett 45:368 (1980); K. Kaski and J. D. Gunton, Phys. Rev. B28:5371 (1983); K. Kaski, M. Grant and J. D Gunton, preprint-`; K. Kaski, S. Kumar, J. D. Gunton and P. A. Rikvold, Phys. Rev B29:4420 (1984); K. Kaski, T. Ala-Nissila and J. D. Gunton, Phys. Rev B31:310 (1985).Google Scholar
  54. 54.
    J. S. Langer and A. J. Schwartz, Phys. Rev A21:948 (1980).Google Scholar
  55. 55.
    H. L. Snyder and P. A. Meakín, J. Chem Phys. 79: 5588 (1983).CrossRefGoogle Scholar
  56. 56.
    H. Furukawa, Phys. Rev A30:1052 (1984); A29:2160 (1984) and preprints.Google Scholar
  57. 57.
    K. Binder and M. H. Kalos, J. Stat. Phys 22:363 (1980).Google Scholar
  58. 58.
    S. Katano and M. Iízumi, Phys. Rev Lett. 52:835 (1984).Google Scholar
  59. 59.
    A. B. Bortz, M. H. Kalos, J. L. Lebowitz and M. A. Zendejas, Phys. Rev B10:535 (1974).Google Scholar
  60. 60.
    J. Marro, A. B. Bortz, M. H. Kalos and J. L. Lebowitz, Phys. Rev B12:2000 (1975).Google Scholar
  61. 61.
    M. Rao, M. H. Kalos, J. L. Lebowitz and J. Marro, Phys Rev. B13: 4328 (1976).CrossRefGoogle Scholar
  62. 62.
    A. Sur, J. L. Lebowitz, J. Marro and M. H. Kalos, Phys. Rev B15: 3014 (1978).CrossRefGoogle Scholar
  63. 63.
    K. Kawasaki, in “Phase Transitions and Critical Phenomena”, Vol. 2, C. Domb and M. S. Green, Eds., Academic Press, London (1972) p. 443.Google Scholar
  64. 64.
    C. M. Knobler and N. C. Wong, J. Phys. Chem 85:1972 (1981).Google Scholar
  65. 65.
    M. Hennion, P. Guyot and D. Ronzaud, Acta met 30:599 (1982); P. Guyot, preprint.Google Scholar
  66. 66.
    S. Komura, K. Osamura, H. Fujií and T. Takeda, Phys. Rev B30:2944 (1984).Google Scholar
  67. 67.
    A. Craivich and J. M. Sanchez, Phys. Rev. Lett 47:1308 (1981).Google Scholar
  68. 68.
    Blaschko, G. Ernst, P. Fratzl, M. Bernole and P. Auger, Acta met 30:547 (1982).Google Scholar
  69. 69.
    P. Guyot and J.P. Simon, in “Solid-solid phase transformations”, H. I. Aaronson, D. E. Laughlus, R. F. Sekeka and C. M. Wayman, Eds., American Institute of Metals (1982) p. 325.Google Scholar
  70. 70.
    S. Komura, K. Osamura, H. Fujii and T. Takeda, Physica 120B:397 (1983); G. Kosterz, Physica 120B:387 (1983); S. Katano and M. Iizumí, J. Phys. Soc. Japan 51:347 (1982); Physica 120B:302 (1983).Google Scholar
  71. 71.
    D. N. Sinha and J.K. Hoffer, Physica 107B: 155 (1981).Google Scholar
  72. 72.
    I. M. Lifshitz and V. V. Slyozov, J. Phys Chem. Solids 19:35 (1961); see also W. Wagner, Z. Elektro Chem 65:581 (1961).Google Scholar
  73. 73.
    E. Stoll, K. Binder and T. Schneider, Phys. Rev B6:2777 (1972).Google Scholar
  74. 74.
    K. Binder and E. Stoll, Phys Rev. Lett. 31:47 (1973); K. Binder and H. Müller-Krumbhaar, Phys. Rev B9: 3228 (1974).Google Scholar
  75. 75.
    H. Müller-Krumbhaar and E. Stoll, J. Chem. Phys. 65: 4294 (1976).Google Scholar
  76. 76.
    D. Stauffer, A. Coniglio and D. W. Heermann, Phys. Rev. Lett. 49: 1299 (1982).CrossRefGoogle Scholar
  77. 77.
    R. Dickman and W. C. Schieve, Physica 112A: 51 (1982).MathSciNetCrossRefGoogle Scholar
  78. 78.
    J. Marro and R. Toral, Physíca 122A: 563 (1983).Google Scholar
  79. 79.
    G. Jacuccí, A. Perini and G. Martin, J. Phys. A16: 369 (1983).Google Scholar
  80. 80.
    Penrose, J. L. Lebowitz, J. Marro, M. H. Kalos and A. Sur, J. Stat. Phys. 19:243 (1978); M. H. Kalos, J. L. Lebowitz, 0. Penrose and A. Sur, J. Stat. Phys 18:39 (1978).Google Scholar
  81. 81.
    H. Müller-Krumbhaar, Phys. Lett. A50: 27 (1974).CrossRefGoogle Scholar
  82. 82.
    In ref. 7, active and inaacctiive bonds are also introduced ím the other component and then there is need to consider clusters of both types, nt and nil. By this approach the symmetry of the disordered phase at cB = cgrit above Tc is ensured.Google Scholar
  83. 83.
    D. W. Heermann, A. Coniglio, W. Klein and D. Stauffer, J. Stat. Phys 36:447 (1984).Google Scholar
  84. 84.
    D. W. Heermann and W. Klein, Phys. Rev. Lett 50:1962 (1983); Phys. Rev. B27:1732 (1983).Google Scholar
  85. 85.
    D. W. Heermann, to be published.Google Scholar
  86. 86.
    M. von Smoluchowski, Phys. Z 17:593 (1916); S. Chandrasekhar, Rev. Mod. Phys 15:1 (1943).Google Scholar
  87. 87.
    R. L. Drake, in “Topics in Current Aerosol Research”, G. M. Hidy and G. R. Brock, Eds., Vol. 3, Pergamon, New York (1972) p. 201.Google Scholar
  88. 88.
    S. K. Friedlander, J. Meteorology 17:479 (1960); 753 (1961); S. K. Friedlander and C. S. Wang, J. Coll. Interface Sci 22:126 (1966); S. K. Friedlander, Phys. Fluids 3:693 (1970); J. Pich, S. K. Friedlander and F. S. Lai, Aerosol Sci 1:115 (1970); C. S. Wang and S. K. Friedlander, J. Coll. Interface Sc. 24:170 (1967); S. K. Friedlander, “Smoke, Dust and Haze”, Wiley, New York (1977).Google Scholar
  89. 89.
    G. M. Hidy and J. R. Brock, “The Dynamics of Aerocolloidal Systems”, Pergamon Press, New York (1970).Google Scholar
  90. 90.
    F. Leyvraz, Phys. Rev A29:854 (1984); F. Leyvraz and H. R.’Tschudi., J. Phys A14:3389 (1981); 15=.1951 (1982); F. Leyvraz, J. Phys A16:1861 (1983).Google Scholar
  91. 91.
    R. M. Ziff, E. M. Hendriks and M. H. Ernst, Phys. Rev. Lett 49:593 (1982); E. M. Hendríks, M. H. Ernst and R. M. Ziff, J. Stat. Phys. 31:519 (1983); R. M. Ziff, J. Stat. Phys 23:241 (1980).Google Scholar
  92. 92.
    T. Vicsek and F. Family, in “Kinetics of Aggregation and Gelation, F. Family and D.P. Landau, Eds., North-Holland, Amsterdam (1984) p. 101; P. Meakin, T. Vicsek and F. Family, Phys. Rev B31:564 (1985); T. Vicsek and F. Family, Phys Rev. Lett. 52:1661 (1984); K. Kang and S. Redner, Phys. Rev A30:2833 (1984).Google Scholar
  93. 93.
    R. Jullien, M. Kolb and R. Botet, in “Kinetics of Aggregation and Gelation”, F. Family and D. P. Landau, Eds., North-Holland, Amsterdam (1984); M. Kelle, R. Botet and R. Jullien, Phys. Rev. Lett 51:1123 (1983); R. Botet, R. Jullien and M. Kolb, J. Phys A17:175 (1984); M. Kolb, preprint.Google Scholar
  94. 94.
    A. C. Zettlemoyer, ed., “Nucleation”, Marcel Dekker, New York (1969).Google Scholar
  95. 95.
    K. Binder and D. Stauffer, Adv. Phys 25:343 (1976).Google Scholar
  96. 96.
    D. Stauffer, Phys. Repts 54 (1979).Google Scholar
  97. 97.
    M. E. Fisher, Physics 3: 255 (1967).Google Scholar
  98. 98.
    For reviews of older experimental data, see ref. 1 and W. I. Goldburg, in “Scattering Techniques Applied to Supramolecular and Nonequilibrium Systems”, S. H. Chen, B. Chu and R. Nossal, Eds., Plenum, New York (1981) p. 383.Google Scholar
  99. 99.
    J. S. Langer, M. Baron and H. D. Miller, Phys. Rev. A11: 1417 (1975).CrossRefGoogle Scholar
  100. 100.
    B. B. Mandelbrot, “The Fractal Geometry off Nom, Freeman, San Francisco (1982).Google Scholar
  101. 101.
    D. W. Heermann and K. Binder, to be published.Google Scholar
  102. 102.
    M. Kahlweit, Adv. Coll. Interface Sci 5:1 (1975).Google Scholar
  103. 103.
    Another effect relevant for late stages is the possible loss of coherence of precipitated clusters with the host lattice, due to building of strong elastic distortions and resulting grain-boundary formation.Google Scholar

Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • K. Binder
    • 1
  • D. W. Heermann
    • 1
  1. 1.Institut für PhysikUniversität MainzMainzWest Germany

Personalised recommendations