Advertisement

Journal of Statistical Physics

, Volume 110, Issue 3–6, pp 1411–1514 | Cite as

Monte Carlo Studies of Wetting, Interface Localization and Capillary Condensation

  • Kurt Binder
  • David Landau
  • Marcus Müller
Article

Abstract

We present a brief review of Monte Carlo simulations of ferromagnetic Ising lattices in a film geometry with surface magnetic fields. The seminal work of Nakanishi and Fisher [Phys. Rev. Lett.49:1565 (1982)] showed how phase transitions in such models are related to wetting in systems with short range forces; and we will show how theoretical concepts about critical and tricritical wetting, interface localization-delocalization, and capillary condensation can be tested in this and similar models. After reviewing the qualitative, phenomenological description of these phenomena on a mean field level, we will summarize predictions of scaling theories. Comments will be made about the models studied and simulation techniques as well as the specific problems that occur in the relevant finite size scaling analysis. The resulting simulational data have prompted considerable new theoretical efforts, but there are still unsolved problems with respect to critical wetting. We will also present results for interface localization-delocalization transitions in both Ising models and lattice polymer mixtures in a thin film geometry and show that theory can account for many, but not all, aspects of the simulations. In systems with asymmetric boundary fields rather complex phase diagrams can result, and these should be relevant for corresponding experiments. The simulational evidence is fully compatible with the scaling predictions of Fisher and Nakanishi [J. Chem. Phys.75:5875 (1981)] on capillary condensation. To conclude we shall summarize the major unanswered theoretical questions in this rich field of inquiry.

Ising models Monte Carlo simulation wetting transition capillary waves finite size scaling 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

REFERENCES

  1. 1.
    T. Young, Philos. Trans. Roy. Soc. London 95:65(1805)Google Scholar
  2. 2.
    J. W. Cahn, J. Chem. Phys. 66:3667(1977).Google Scholar
  3. 3.
    On an atomic scale, the density profile of a fluid against a hard wall shows well-known density oscillations (i.e., a "layered structure"). We assume here that we are so close to the bulk critical point, that the correlation length of density fluctuations is much larger than interatomic distances and can be used as a linear dimension of coarse-graining cells. Using the average of the density over such coarse graining cells as an order parameter in Fig. 2 rather than the density itself, we average the layered structure out.Google Scholar
  4. 4.
    D. E. Sullivan and M. M. Telo da Gama, in Fluid Interfacial Phenomena, C. A. Croxton, ed. (Wiley, New York 1986), p. 45.Google Scholar
  5. 5.
    S. Dietrich, in Phase Transitions and Critical Phenomena, Vol. XII, C. Domb and J. L. Lebowitz, eds. (Academic, New York, 1988), p. 1.Google Scholar
  6. 6.
    M. Schick, in Liquids at Interfaces, J. Charvolin, J. F. Joanny, and J. Zinn-Justin, eds. (North-Holland, Amsterdam, 1990), p. 415.Google Scholar
  7. 7.
    K. Binder and P. C. Hohenberg, Phys. Rev. B 6:3461(1972)Google Scholar
  8. 8.
    K. Binder and P. C. Hohenberg, Phys. Rev. B ibid B9:2194(1974).Google Scholar
  9. 8.
    K. Binder, in Phase Transitions and Critical Phenomena, Vol. VIII, C. Domb and J. L. Lebowitz, eds. (Academic, New York, 1983), p. 1.Google Scholar
  10. 9.
    A. J. Liu and M. E. Fisher, Phys. Rev. A 40:7202(1989), and references therein.Google Scholar
  11. 10.
    H. W. Diehl and M. Smock, Phys. Rev. B 47:5841(1993)Google Scholar
  12. 12.
    H. W. Diehl and M. Smock, Phys. Rev. B 48, 6740(E)(1993).Google Scholar
  13. 13.
    A. J. Bray and M. A. Moore, J. Phys. A 10:1927(1977).Google Scholar
  14. 14.
    K. Ohno and Y. Okabe, Phys. Rev. B 39:9764(1989).Google Scholar
  15. 15.
    H. Nakanishi and M. E. Fisher, Phys. Rev. Lett. 49:1565(1982)Google Scholar
  16. 16.
    We disregard here temperatures far below the bulk critical temperature Tcb and thus shall not discuss the fact that near the roughening transition temperature TR complete wetting is replaced by a sequence of layering transitions, as discussed by R. Pandit, M. Schick, and M. Wortis, Phys. Rev. B 25:5112 (1982) and by K. Binder and D. P. Landau, Phys. Rev. B 46:4844 (1992).Google Scholar
  17. 17.
    K. Binder, R. Evans, D. P. Landau, and A. M. Ferrenberg, Phys. Rev. E 53:5023(1996).Google Scholar
  18. 18.
    A. O. Parry and R. Evans, Phys. Rev. Lett. 64:439(1990).Google Scholar
  19. 19.
    M. Müller, K. Binder, and E. V. Albano, Physica A 279:188(2000), M. Müller, E. V. Albano, and K. Binder, Phys. Rev. E 62:5281(2000).Google Scholar
  20. 20.
    K. Binder, M. Müller, and E. V. Albano, Phys. Chem. Chem. Phys. 3:1160(2001).Google Scholar
  21. 21.
    J. Rogiers and J. O. Indekeu, Europhys. Lett. 24:21(1993).Google Scholar
  22. 22.
    E. Carlon and A. Drzewinski, Phys. Rev. Lett. 79:1591(1997).Google Scholar
  23. 23.
    M. Müller, K. Binder, and E. V. Albano, Europhys. Lett. 50:724(2000).Google Scholar
  24. 24.
    S. J. Gregg and K. S. W. Sing, Adsorption, Surface Area and Porosity (Academic, New York, 1982).Google Scholar
  25. 25.
    W. T. Thomson (Lord Kelvin), Philos. Mag. 42:448(1871).Google Scholar
  26. 26.
    M. E. Fisher and H. Nakanishi, J. Chem. Phys. 75:5857(1981).Google Scholar
  27. 27.
    H. Nakanishi and M. E. Fisher, J. Chem. Phys. 78:3279(1983).Google Scholar
  28. 28.
    R. Evans and P. Taranzona, Phys. Rev. Lett. 52:557(1984).Google Scholar
  29. 29.
    R. Evans, J. Phys. Condens. Matter 2:8989(1990).Google Scholar
  30. 30.
    L. D. Gelb, K. E. Gubbins, R. Radhakrishnan, and M. Sliwinska-Bartkoviak, Rep. Progr. Phys. 62:1573(1999).Google Scholar
  31. 31.
    D. Nicolaides and R. Evans, Phys. Rev. B 39:9336(1989).Google Scholar
  32. 32.
    A. Milchev and K. Binder, J. Chem. Phys. 114:8610(2001).Google Scholar
  33. 33.
    A. Milchev and A. Milchev, Europhys. Lett. 56:695(2001).Google Scholar
  34. 34.
    L. G. MacDowell, M. Müller, and K. Binder, Colloids Surf. A 206:277(2002).Google Scholar
  35. 35.
    J. W. Gibbs. The Collected Works of J. Willard Gibbs (Yale University Press, London, 1957), p. 288.Google Scholar
  36. 36.
    P. G. deGennes, Rev. Mod. Phys. 57:827(1985).Google Scholar
  37. 37.
    J. O. Indekeu, Internat. J. Modern Phys. 138:309(1994), and references therein.Google Scholar
  38. 38.
    T. Getta and S. Dietrich, Phys. Rev. E 57:655(1998), and references therein.Google Scholar
  39. 39.
    J. Drelich, Colloids Surf. A 116:43(1996).Google Scholar
  40. 40.
    C. Bauer and S. Dietrich, Euro. Phys. J. B 10:767(1999).Google Scholar
  41. 41.
    M. P. Gelfand and R. Lipowsky, Phys. Rev. B 36:8725(1987).Google Scholar
  42. 42.
    T. Bieker and S. Dietrich, Phys. A 252:85(1998) andGoogle Scholar
  43. 43.
    T. Bieker and S. Dietrich Phys. A 259:466(1998), and refereces therein.Google Scholar
  44. 44.
    A. Milchev and K. Binder, J. Chem. Phys. 117:6852(2002).Google Scholar
  45. 45.
    A. J. Liu and G. S. Grest, Phys. Rev. A 44:R7894(1991).Google Scholar
  46. 46.
    D. B. Abraham, in Phase Transitions and Critical Phenomena, Vol. 10, C. Domb and J. L. Lebowitz, eds. (Academic Press, New York, 1986).Google Scholar
  47. 47.
    F. Igloi and J. O. Indekeu, Phys. Rev. B 41:6836(1990), and references therein.Google Scholar
  48. 48.
    J. O. Indekeu, Phys. A 177:428(1991).Google Scholar
  49. 49.
    I. Schmidt and K. Binder, Z. Phys. B 67:369(1987).Google Scholar
  50. 50.
    S. Puri and K. Binder, Z. Phys. B 86:263(1992).Google Scholar
  51. 51.
    K. Binder and H. L. Frisch, Z. Phys. B 84:403(1991).Google Scholar
  52. 52.
    R. Pandit and M. Wortis, Phys. Rev. B 25:3226(1982).Google Scholar
  53. 53.
    J. D. Weeks, in Ordering in Strongly Fluctuating Condensed Matter Systems, T. Riste, ed. (Plenum Press, New York, 1980), p. 293.Google Scholar
  54. 54.
    H. Van Beijeren and I. Nolden, in Structure and Dynamics of Surfaces II, W. Schommers and P. Blankenhagen, eds. (Springer, Berlin, 1987), p. 259.Google Scholar
  55. 55.
    B. Widom, in Phase Transitions and Critical Phenomena, Vol. 2, C. Domb and M. S. Green, ed. (Academic Press, London 1972), p. 79.Google Scholar
  56. 56.
    J. S. Rowlinson and B. Widom, Molecular Theory of Capillary (Clarendon, Oxford, 1982).Google Scholar
  57. 57.
    F. P. Buff, R. A. Lovett, and F. H. Stillinger, Phys. Rev. Lett 15:621(1965).Google Scholar
  58. 58.
    J. D. Weeks, J. Chem. Phys. 67:3106(1977).Google Scholar
  59. 59.
    J. D. Weeks, Phys. Rev. Lett. 52:2160(1984).Google Scholar
  60. 60.
    H. W. Diehl, in Phase Transitions and Critical Phenomena, Vol. 10, C. Domb and J. L. Lebowitz, eds. (Academic Press, London, 1986), p. 75.Google Scholar
  61. 61.
    M. E. Fisher, Rev. Mod. Phys. 46:597(1974).Google Scholar
  62. 62.
    A. O. Parry and R. Evans, Phys. A 181:250(1992).Google Scholar
  63. 63.
    M. R. Swift, A. L. Owczarek, and J. D. Indekeu, Europhys. Lett. 14:475(1991).Google Scholar
  64. 64.
    K. Binder, M. Müller, F. Schmid, and A. Werner, J. Statist. Phys. 95:1045(1999).Google Scholar
  65. 65.
    V. Privman, Internat. J. Modern Phys. C 3:857(1992).Google Scholar
  66. 66.
    M. E. Fisher and J. A. Jin, Phys. Rev. Lett. 69:792(1992); A. J. Jin and M. E. Fisher, Phys. Rev. B 47:7365(1993).Google Scholar
  67. 67.
    M. E. Fisher, A. J. Jin, and A. O. Parry, Ber. Bunsenges. Phys. Chem. 98:357(1994).Google Scholar
  68. 68.
    M. Müller and K. Binder, Phys. Rev. E 63:021602(2001).Google Scholar
  69. 69.
    A. M. Ferrenberg, D. P. Landau, and K. Binder, Phys. Rev. E 58:3353(1998).Google Scholar
  70. 70.
    E. Brezin, B. I. Halperin, and S. Leibler, Phys. Rev. Lett. 50:1387(1983).Google Scholar
  71. 71.
    E. Brezin, B. I. Halperin, and S. Leibler, J. Phys. (Paris) 44:775(1983).Google Scholar
  72. 72.
    R. Lipowsky, D. M. Kroll, and R. K. P. Zia, Phys. Rev. B 27:4499(1983).Google Scholar
  73. 73.
    D. S. Fisher and D. A. Huse, Phys. Rev. B 32:247(1985).Google Scholar
  74. 74.
    E. H. Hauge and K. Olaussen, Phys. Rev. B 32:4766(1985).Google Scholar
  75. 75.
    R. Lipowsky and M. E. Fisher, Phys. Rev. B 36:2126(1987).Google Scholar
  76. 76.
    E. Brézin and T. Halpin-Healey, J. Phys. (Paris) 48:747. (1987).Google Scholar
  77. 77.
    E. Brézin and T. Halpin-Healey, Phys. Rev. Lett. 58:1220. (1987).Google Scholar
  78. 78.
    T. Halpin-Healey, Phys. Rev. B 40:772(1989).Google Scholar
  79. 79.
    A. O. Parry and J. C. Boulter, Phys. Rev. E 53:6577(1996).Google Scholar
  80. 80.
    M. Napiorkowski and S. Dietrich, Z. Phys. B 89:263(1992)Google Scholar
  81. 81.
    M. Napiorkowski and S. DietrichPhys. Rev. E 47:1836(1993).Google Scholar
  82. 82.
    A. O. Parry and C. J. Boulter, J. Phys. Condens. Matter 6:7199(1994).Google Scholar
  83. 83.
    K. Mecke and S. Dietrich, Phys. Rev. E 59:6766(1999).Google Scholar
  84. 84.
    A. Werner, F. Schmid, M. Müller, and K. Binder, Phys. Rev. E 59:728(1999).Google Scholar
  85. 85.
    M. Müller and L. G. MacDowell, Macromolecules 33:3902(2000).Google Scholar
  86. 86.
    K. Binder and M. Müller, Internat. J. Modern Phys. C 11:1093(2000).Google Scholar
  87. 87.
    V. L. Ginzburg, Fiz. Tverd. Tela (Leningrad) 2:2031(1960)Google Scholar
  88. 88.
    V. L. Ginzburg[Sov. Phys. Solid State 2:1824(1960)].Google Scholar
  89. 89.
    J. Als-Nielsen and R. J. Birgeneau, Amer. J. Phys. 45:554(1970).Google Scholar
  90. 90.
    M. A. Anisimov, S. B. Kiselev, J. V. Sengers, and S. Tang, Phys. A 188:487(1992).Google Scholar
  91. 91.
    C. J. Boulter and A. O. Parry, Phys. Rev. Lett. 74:3403(1995).Google Scholar
  92. 92.
    A. O. Parry and C. J. Boulter, Phys. A 218:77(1995).Google Scholar
  93. 93.
    C. J. Boulter and A. O. Parry, Phys. A 218:109(1995).Google Scholar
  94. 94.
    A. O. Parry, C. J. Boulter, and P. S. Swain, Phys. Rev. E 52:R5768(1995).Google Scholar
  95. 95.
    P. S. Swain and A. O. Parry, Europhys. Lett. 37:207(1997).Google Scholar
  96. 96.
    A. O. Parry, J. Phys.: Condens. Matter 8:10761(1996).Google Scholar
  97. 97.
    C. J. Boulter, Mod. Phys. Lett. 15:993(2001).Google Scholar
  98. 98.
    J. C. Le Guillou and J. Zinn-Justin, Phys. Rev. B 21:3976(1980).Google Scholar
  99. 99.
    K. Binder and E. Luijten, Phys. Rep. 344:179. (2001).Google Scholar
  100. 100.
    M. E. Fisher, in Critical Phenomena, Proc. 1970 E. Fermi Int. School of Physics, M. S. Green, ed. (Academic, London, 1971), p.1.Google Scholar
  101. 101.
    M. E. Fisher and M. N. Barber, Phys. Rev. Lett. 28:1516(1972).Google Scholar
  102. 102.
    M. N. Barber, in Phase Transitions and Critical Phenomena, Vol. 8, C. Domb and J. L. Lebowitz, eds. (Academic, New York, 1983), p. 145.Google Scholar
  103. 103.
    V. Privman, ed., Finite Size Scaling and the Numerical Simulation of Statistical Systems (Singapore, World Scientific, 1990).Google Scholar
  104. 104.
    K. Binder, in Computational Methods in Field Theory, H. Gausterer and C. B. Lang, eds. (Springer, Berlin, 1992), p. 59.Google Scholar
  105. 105.
    K. Binder and D. P. Landau, Phys. Rev. Lett. 52:318(1984).Google Scholar
  106. 106.
    D. P. Landau and K. Binder, Phys. Rev. B 41:4633(1990).Google Scholar
  107. 107.
    C. Ruge, S. Dunkelmann, F. Wagner, and J. Wulf, J. Statist. Phys. 73:293(1993).Google Scholar
  108. 108.
    C. Ruge and F. Wagner, Phys. Rev. B 52:4209(1995).Google Scholar
  109. 109.
    H. W. Diehl and M. Shpot, Nucl. Phys. B 528:595(1998).Google Scholar
  110. 110.
    H. W. Diehl, Internat. J. Modern Phys. B 11:3503(1997).Google Scholar
  111. 118.
    A. D. Bruce and N. B. Wilding, Phys. Rev. Lett. 86:193(1992).Google Scholar
  112. 119.
    K. Binder, Thin Solid Films 20:367(1974).Google Scholar
  113. 120.
    T. W. Capehart and M. E. Fisher, Phys. Rev. B 13:6021(1976).Google Scholar
  114. 121.
    M. E. Fisher and H. Au-Yang, Phys. A 101:255(1980)Google Scholar
  115. 115.
    H. Au-Yang and M. E. Fisher, Phys. Rev. B 21:3956(1980).Google Scholar
  116. 110.
    F. Freire, D. O'Connor, and C. R. Stephens, J. Statist. Phys. 74:219(1994)Google Scholar
  117. 117.
    D. O'Connor and R. C. Stephens, Phys. Rev. Lett. 72:506(1994).Google Scholar
  118. 118.
    Y. Rouault, J. Baschnagel, and K. Binder, J. Statist. Phys. 80:1009(1995).Google Scholar
  119. 119.
    D. M. Kroll, R. Lipowsky, and R. K. P. Zia, Phys. Rev. B 32:1862(1985).Google Scholar
  120. 120.
    M. E. Fisher and H. Wen, Phys. Rev. Lett. 68:3654(1992).Google Scholar
  121. 121.
    C. J. Boulter and F. Clarysse, Eur. Phys. J. E 5:465(2001).Google Scholar
  122. 122.
    E. Riedel and F. J. Wegner, Z. Phys. 225:195(1969).Google Scholar
  123. 123.
    M. Hasenbusch, Internat. J. Modern Phys. C 12:91(2001).Google Scholar
  124. 124.
    D. P. Landau and K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics (Cambridge University Press, Cambridge, 2000).Google Scholar
  125. 125.
    R. H. Swendson and J. S. Wang, Phys. Rev. Lett. 58:86(1987).Google Scholar
  126. 126.
    U. Wolff, Phys. Rev. Lett. 62:361(1989).Google Scholar
  127. 127.
    R. H. Swendsen, J. S. Wang, and A. M. Ferrenberg, The Monte Carlo Method in Condensed Matter Physics (Springer, Berlin, 1992), p. 75.Google Scholar
  128. 128.
    J. S. Wang, Phys. A 161:249(1989).Google Scholar
  129. 129.
    P. G. Lauwers and V. Rittenberg, Phys. Lett. B 233:197(1989).Google Scholar
  130. 130.
    V. S. Dotsenko, W. Selke, and A. L. Talapov, Phys. A 170, 278(1990).Google Scholar
  131. 131.
    O. Redner, J. Machta, and L. F. Chayes, Phys. Rev. E 58:2749(1998)Google Scholar
  132. 132.
    L. Chayes, J. Machta, and O. Redner, J. Statist. Phys. 93:17(1998)Google Scholar
  133. 133.
    J. Machta, Internat. J. Modern Phys. C 10:1427(1999).Google Scholar
  134. 134.
    O. Dillmann, W. Janke, M. Müller, and K. Binder, J. Chem. Phys. 114:5853(2001).Google Scholar
  135. 135.
    K. Binder and D. P. Landau, J. Appl. Phys. 57:3306(1985).Google Scholar
  136. 136.
    K. Binder, D. P. Landau, and D. M. Kroll, Phys. Rev. Lett. 56, 2276(1986).Google Scholar
  137. 137.
    K. Binder and D. P. Landau, Phys. Rev. B 37:1745(1988).Google Scholar
  138. 138.
    K. Binder, D. P. Landau, and S. Wansleben, Phys. Rev. B 40:6971(1989).Google Scholar
  139. 139.
    K. Binder and D. P. Landau, Phys. Rev. B 46:4844(1992).Google Scholar
  140. 140.
    K. Binder and D. P. Landau, J. Chem. Phys. 96:1444(1992).Google Scholar
  141. 141.
    K. Binder, A. M. Ferrenberg, and D. P. Landau, Ber. Bunsenges. Phys. Chem. 98:340(1994).Google Scholar
  142. 142.
    K. Binder, D. P. Landau, and A. M. Ferrenberg, Phys. Rev. Lett. 74:298(1985).Google Scholar
  143. 143.
    K. Binder, D. P. Landau, and A. M. Ferrenberg, Phys. Rev. E 51:2823(1995).Google Scholar
  144. 144.
    D. P. Landau and K. Binder, in STATPHYS 19, B. Hao, ed. (World Scientific, Singapore, 1996), p. 446.Google Scholar
  145. 145.
    N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, J. Chem. Phys. 21:1087(1953)Google Scholar
  146. 146.
    M. Hasenbusch and S. Meyer, Phys. Rev. Lett. 66:530(1991).Google Scholar
  147. 147.
    B. J. Schulz, K. Binder, and M. Müller, Int. J. Mod. Phys. C 13:477(2002).Google Scholar
  148. 148.
    F. Wang and D. P. Landau, Phys. Rev. Lett. 86:2050(2001).Google Scholar
  149. 149.
    A. B. Bortz, M. M. Kalos, and J. L. Lebowitz, J. Comput. Phys. 17:10(1975).Google Scholar
  150. 150.
    A. M. Ferrenberg, D. P. Landau, and Y. J. Wong, Phys. Rev. Lett. 69:3382(1992)Google Scholar
  151. 151.
    W. Janke, in Computational Physics: Selected Methods, Simple Exercises, Serious Applications, K. H. Hoffmann and M. Schreiber, eds. (Springer, Berlin, 1996), p. 10.Google Scholar
  152. 152.
    S. Kirkpatrick and E. Stoll, J. Comput. Phys. 40:517(1981).Google Scholar
  153. 153.
    A. M. Ferrenberg and R. H. Swendsen, Phys. Rev. Lett. 63:1195(1989)Google Scholar
  154. 154.
    K. Binder and D. P. Landau, Phys. Rev. B 30:147(1984).Google Scholar
  155. 155.
    C. Borgs and R. Kotecky, J. Statist. Phys. 61:79(1990)Google Scholar
  156. 156.
    C. Borgs, R. Kotecky, and S. Miracle-Sole, J. Statist. Phys. 62:529(1991).Google Scholar
  157. 157.
    C. Borgs and W. Janke, Phys. Rev. Lett. 68:1738(1992)Google Scholar
  158. 158.
    W. Janke, Phys. Rev. B 47:1453(1993).Google Scholar
  159. 159.
    H. W. Blöte, E. Luijten, and J. R. Heringa, J. Phys. A 28:6289(1995).Google Scholar
  160. 160.
    K. Binder, Phys. Rev. Lett. 47:693(1981)Google Scholar
  161. 161.
    K. BinderZ. Phys. B 43:119(1981).Google Scholar
  162. 162.
    D. P. Landau, Phys. A 205:41(1994).Google Scholar
  163. 163.
    A. Werner, F. Schmid, M. Müller, and K. Binder, J. Chem. Phys. 107:8175(1997).Google Scholar
  164. 164.
    D. Ross, D. Bonn, and J. Meunier, Nature 400:737(1999)Google Scholar
  165. 165.
    D. Ross, D. Bonn, and J. MeunierJ. Chem. Phys. 114:2784(2001).Google Scholar
  166. 166.
    A. J. Liu and M. E. Fisher, Phys. A 156:35(1989).Google Scholar
  167. 167.
    M. Hasenbusch and K. Pinn, Phys. A 192:342(1993).Google Scholar
  168. 168.
    M. Müller and K. Binder, Macromolecules 31:8323(1998).Google Scholar
  169. 169.
    T. Kerle, J. Klein, and K. Binder, Phys. Rev. Lett. 77:1318(1996).Google Scholar
  170. 170.
    T. Kerle, J. Klein, and K. Binder, Eur. Phys. J. B 7:401(1999).Google Scholar
  171. 171.
    A. O. Parry, R. Evans, J. Phys. A 25:275(1992).Google Scholar
  172. 172.
    E. Montevecchi and J. O. Indekeu, Phys. Rev. B 62:14359(2000).Google Scholar
  173. 173.
    I. Carmesin and K. Kremer, Macromolecules 21:2819(1988).Google Scholar
  174. 174.
    H.-P. Deutsch and K. Binder, J. Chem. Phys. 95:2294(1991).Google Scholar
  175. 175.
    M. Müller, Macromol. Theory Simul. 8:343(1999).Google Scholar
  176. 176.
    M. Müller and F. Schmid, in Annual Reviews of Computational Physics VI, D. Stauffer, ed. (World Scientific, Singapore, 1999), p. 59.Google Scholar
  177. 177.
    A. Sariban and K. Binder, Macromolecules 21:711(1988).Google Scholar
  178. 178.
    H.-P. Deutsch and K. Binder, Macromolecules 25:6214(1992).Google Scholar
  179. 179.
    B. A. Berg, U. Hansmann, and T. Neuhaus, Z. Phys. B 90:229(1993).Google Scholar
  180. 180.
    R. Evans and U. Marini Bettolo Marconi, Phys. Rev. A 32:3817(1985).Google Scholar
  181. 181.
    N. B. Wilding and P. Nielaba, Phys. Rev. E 53:926(1996).Google Scholar
  182. 182.
    D. B. Abraham, Phys. Rev. Lett. 44:1165(1980)Google Scholar
  183. 183.
    T. M. Burkhardt, J. Phys. A: Math. Gen. 14, L63(1981)Google Scholar
  184. 184.
    J. T. Chalker, ibidJ. Phys. A: Math. Gen. 14:2431(1981)Google Scholar
  185. 185.
    S. T. Chui and J. D. Weeks, Phys. Rev. B 23:2438(1981)Google Scholar
  186. 186.
    H. Hilhorst and J. M. J. van Leeuwen, Phys. A 107:319(1981)Google Scholar
  187. 187.
    D. M. Kroll, Z. Phys. B 41:345(1981)Google Scholar
  188. 188.
    M. Vallade and J. Lajczerowicz, J. Phys. (Paris) 42:1505(1981).Google Scholar
  189. 189.
    D. B. Abraham and E. R. Smith, Phys. Rev. B 26:1480(1982)Google Scholar
  190. 190.
    D. B. Abraham and E. R. SmithJ. Statist. Phys. 43:621(1986).Google Scholar
  191. 191.
    M. E. Fisher, J. Statist. Phys. 34:667(1984).Google Scholar
  192. 192.
    E. V. Albano, K. Binder, D. W. Heermann, and W. Paul, J. Statist. Phys. 61:161(1990).Google Scholar
  193. 193.
    A. Maciolek, J. Phys. A: Math. Gen. 29:3837(1996).Google Scholar
  194. 194.
    E. V. Albano, K. Binder, and W. Paul, J. Phys.: Condens. Matter 12:2701(2000), and references therein.Google Scholar

Copyright information

© Plenum Publishing Corporation 2003

Authors and Affiliations

  • Kurt Binder
    • 1
  • David Landau
    • 2
  • Marcus Müller
    • 1
  1. 1.Institut für PhysikJohannes Gutenberg-UniversitätMainzGermany
  2. 2.Center for Simulational PhysicsThe University of GeorgiaAthens

Personalised recommendations