Finite Size Effects in Thin Film Simulations

  • K. Binder
Part of the NATO Science Series book series (NAII, volume 114)


Phase transitions in thin films are discussed, with an emphasis on Ising-type systems (liquid-gas transition in slit-like pores, unmixing transition in thin films, orderdisorder transitions on thin magnetic films, etc.) The typical simulation geometry then is a L xL x D system, where at the low confining L x L surfaces appropriate boundary “fields” are applied, while in the lateral directions periodic boundary conditions are used. In the z-direction normal to the film, the order parameter always is inhomogeneous, due to the boundary “fields” at the confining surfaces. When one varies the temperature T from the region of the bulk disordered phase to a temperature below the critical temperature of the bulk (T cb ), one may encounter the onset of a stratified structure, i.e. domains of appropriate sign of the order parameter form, separated by a domain wall parallel to the confining surfaces. If the boundary fields favor the same phase, a structure with two parallel interfaces may form, if the boundary fields favor opposite phases, formation of a single interface results. In both cases, this vertical phase separation is a rounded transition due to the finite size of the linear dimension D, not a sharp phase transition as in the bulk. However, for L → ∞ phase transitions at temperatures T C (D) can occur, which can be interpreted as a symmetry breaking of a quasi-two dimensional character, leading to phase separation in lateral directions if one keeps the value of the total order parameter in the thin film fixed. Phenomenological theories of these phenomena (“capillary condensation”, “interface localization transition”) will be discussed, and it will be shown that there can exist a very large correlation length in parallel direction, leading to strong finite size effects if L is not extremely large. The general considerations will be exemplified with selected Monte Carlo results.


Coexistence Curve Thin Magnetic Film Dimensional Crossover Lateral Phase Separation Ising Universality Class 
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  1. 1.
    M. E. Fisher, in Critical Phenomena, M. S. Green, ed. (Academic, London, 1971), p. 1Google Scholar
  2. 2.
    V. Privman (ed.) Finite Size Scaling and the Numerical Simulation of Statistical Systems (Singapore, World Scientific, 1990)Google Scholar
  3. 3.
    K. Binder, in Computational Methods in Field Theory, H. Gausterer and C. B. Lang, eds. (Springer, Berlin, 1992) p. 59CrossRefGoogle Scholar
  4. 4.
    K. Binder, in Phase Transitions and Critical Phenomena, Vol. 8, C. Domb and J. L. Lebowitz, eds. (Academic, New York, 1983) p. 45Google Scholar
  5. 5.
    D. E. Sullivan and M. M. Telo da Gama, in Fluid Interfacial Phenomena, C. A. Croxton, ed. (Wiley, New York, 1986) p. 45Google Scholar
  6. 6.
    S. Dietrich, in Phase Transitions and Critical Phenomena, Vol. 12, C Domb and J. L. Lebowitz, eds. (Academic, New York, 1988) p. 1Google Scholar
  7. 7.
    Y. Rouault, J. Baschnagel, and K. Binder, J. Stat. Phys. 80, 1009 (1995)ADSMATHCrossRefGoogle Scholar
  8. 8.
    K. Binder, Adv. Polymer Sci. 138, 1 (1999)MathSciNetCrossRefGoogle Scholar
  9. 9.
    M. E. Fisher and N. Nakamishi, J. Chem. Phys. 75, 5857 (1981); H. Nakanishi and M. E. Fisher, J. Chem. Phys. 78, 3279 (1983)ADSCrossRefGoogle Scholar
  10. 10.
    R. Evans, J. Phys-Condens. Matter 2, 8989 (1990)ADSCrossRefGoogle Scholar
  11. 11.
    K. Binder and D. P. Landau, J. Chem. Phys. 96, 1444 (1992)ADSCrossRefGoogle Scholar
  12. 12.
    L. D. Gelb, K. E. Gubbins, R. Radhakrishnan, and M. Sliwinska-Bartkoviak, Rep. Progr. Phys. 62, 1573 (1999)ADSCrossRefGoogle Scholar
  13. 13.
    O. Dillmann, W. Janke, M. Müller, and K. Binder, J. Chem. Phys. 114, 5853 (2001)ADSCrossRefGoogle Scholar
  14. 14.
    M. Müller and K. Binder, Macromolecules 31, 8323 (1998)ADSCrossRefGoogle Scholar
  15. 15.
    K. Binder, D. P. Landau, and A. M. Ferrenberg, Phys. Rev. E51, 2823 (1995)ADSGoogle Scholar
  16. 16.
    K. Binder, R. Evans, D. P. Landau, and A. M. Ferrenberg, Phys. Rev. E53, 5023 (1996ADSGoogle Scholar
  17. 17.
    A. M. Ferrenberg, D. P. Landau, and K. Binder, Phys. Rev. E58, 3353 (1998)ADSGoogle Scholar
  18. 18.
    A. Werner, F. Schmid, M. Müller, and K. Binder, J. Chem. Phys. 107, 8175 (1997)ADSCrossRefGoogle Scholar
  19. 19.
    M. Müller, K. Binder, and E. V. Albano, Physica A279, 188 (2000); Europhys. Lett. 49, 724 (2000)ADSGoogle Scholar
  20. 20.
    M. Müller and K. Binder, Phys. Rev. E63, 021602 (2001)ADSGoogle Scholar
  21. 21.
    A. O. Parry and R. Evans, Physica A181, 250 (1992)Google Scholar
  22. 22.
    T. Kerle, J. Klein and K. Binder, Eur. Phys. J. B7, 401 (1999)ADSGoogle Scholar
  23. 23.
    K. Binder, D. P. Landau, and M. Müller, J. Stat. Phys. (2002, in press)Google Scholar
  24. 24.
    I. Carmesin and K. Kremer, Macromolecules 21, 2819 (1988); H.-P. Deutsch and K. Binder, J. Chem. Phys. 95, 2294 (1991)ADSCrossRefGoogle Scholar
  25. 25.
    H. P. Deutsch and K. Binder, Macromolecules 25, 6214 (1992)ADSCrossRefGoogle Scholar
  26. 26.
    K. Binder (ed.) Monte Carlo and Molecular Dynamics Simulations in Polymer Science, (Oxford University Press, Oxford, 1995)Google Scholar
  27. 27.
    J. C. Le Guillou and J. Zinn-Justin, Phys. Rev. B21, 3976 (1980)ADSGoogle Scholar
  28. 28.
    M. R. Swift, A. L. Owczarek, and J. D. Indekeu, Europhys. Lett. 14, 475 (1991)ADSCrossRefGoogle Scholar
  29. 29.
    N. B. Wilding and P. Nielaba, Phys. Rev. E53, 926 (1996)ADSGoogle Scholar
  30. 30.
    A. O. Parry and C. J. Boulter, Physica A218, 77 (1995; A. O. Parry, J. Phys.: Condens. Matter 8, 10761 (1996)ADSGoogle Scholar
  31. 31.
    M. E. Fisher and H. Wen, Phys. Rev. Lett. 68, 3654 (1992)ADSCrossRefGoogle Scholar
  32. 32.
    A. Werner, M. Müller, F. Schmid, and K. Binder, J. Chem. Phys. (1999)Google Scholar
  33. 33.
    A. Werner, F. Schmid, M. Müller, and K. Binder, Phys. Rev. E59, 728 (1999)ADSGoogle Scholar
  34. 34.
    K. Binder and M. Müller, Int. J. Mod. Phys. C11, 1093 (2000)ADSGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2003

Authors and Affiliations

  • K. Binder
    • 1
  1. 1.Institut für PhysikJohannes Gutenberg-Universität MainzMainzGermany

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