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Interfacial Roughening in Field Theory

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Abstract

In the rough phase, the width of interfaces separating different phases of statistical systems increases logarithmically with the system size. This phenomenon is commonly described in terms of the capillary wave model, which deals with fluctuating, infinitely thin membranes, requiring ad hoc cut-offs in momentum space. We investigate the interface roughening in a unified approach, which does not rely on joining different models, namely in the framework of the Landau-Ginzburg model, that is renormalized field theory, in the one-loop approximation. The interface profile and width are calculated analytically, resulting in finite expressions with definite coefficients. They are valid in the scaling region and depend on the known renormalized coupling constant.

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References

  1. Rowlinson, J., Widom, B.: Molecular Theory of Capillarity. Clarendon Press, Oxford (1982)

    Google Scholar 

  2. Huang, J.S., Webb, W.W.: Viscous damping of thermal excitations on the interface of critical fluid mixtures. Phys. Rev. Lett. 23, 160–163 (1969)

    Article  ADS  Google Scholar 

  3. Langevin, D. (ed.): Light Scattering by Liquid Surfaces and Complementary Techniques. Dekker, New York (1992)

    Google Scholar 

  4. McClain, B.R., Yoon, M., Litster, J.D., Mochrie, S.G.J.: Interfacial roughness in a near-critical binary fluid mixture: X-ray reflectivity and near-specular diffuse scattering. Eur. Phys. J. B 10, 45–52 (1999)

    Article  ADS  Google Scholar 

  5. Thomas, R.K.: Neutron reflection from liquid interfaces. Annu. Rev. Phys. Chem. 55, 391–426 (2004)

    Article  Google Scholar 

  6. Werner, A., Schmid, F., Müller, M., Binder, K.: Anomalous size-dependence of interfacial profiles between coexisting phases of polymer mixtures in thin film geometry: a Monte-Carlo simulation. J. Chem. Phys. 107, 8175–8188 (1997)

    Article  ADS  Google Scholar 

  7. Werner, A., Schmid, F., Müller, M., Binder, K.: “Intrinsic” profiles and capillary waves at homopolymer interfaces: a Monte Carlo study. Phys. Rev. E 59, 728–738 (1999)

    Article  ADS  Google Scholar 

  8. Dünweg, B., Landau, D.P., Milchev, A. (eds.): Computer Simulations of Surfaces and Interfaces. Kluwer Acad. Publ., Dordrecht (2003)

    MATH  Google Scholar 

  9. Buff, F., Lovett, R., Stillinger, F.: Interfacial density profile for fluids in the critical region. Phys. Rev. Lett. 15, 621–623 (1965)

    Article  ADS  Google Scholar 

  10. Jasnow, D.: Critical phenomena at interfaces. Rep. Prog. Phys. 47, 1059–1132 (1984)

    Article  ADS  Google Scholar 

  11. Widom, B.: In: Domb, C., Green, M. (eds.) Phase Transitions and Critical Phenomena, vol. 2. Academic Press, New York (1972)

    Google Scholar 

  12. Ohta, T., Kawasaki, K.: Renormalization group approach to the interfacial order parameter profile near the critical point. Prog. Theor. Phys. 58, 467–481 (1977)

    Article  ADS  Google Scholar 

  13. Rudnick, J., Jasnow, D.: ε expansion for the interfacial profile. Phys. Rev. B 17, 1351–1354 (1978)

    Article  ADS  Google Scholar 

  14. Sikkenk, J.H., van Leeuwen, J.M.J.: ε-expansion for the interfacial profile in an external field. Physica A 137, 156–177 (1986)

    Article  ADS  Google Scholar 

  15. Parisi, G.: Field-theoretic approach to second-order phase transition in two-and three-dimensional systems. J. Stat. Phys. 23, 49–82 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  16. Le Guillou, J.C., Zinn-Justin, J.: Critical exponents from field theory. Phys. Rev. B 21, 3976–3998 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  17. Münster, G.: Interface tension in three-dimensional systems from field theory. Nucl. Phys. B 340, 559–567 (1990)

    Article  ADS  Google Scholar 

  18. Münster, G., Heitger, J.: Field-theoretic calculation of the universal amplitude ratio of correlation lengths in 3D Ising systems. Nucl. Phys. B 424, 582–594 (1994)

    Article  ADS  Google Scholar 

  19. Gutsfeld, C., Küster, J., Münster, G.: Calculation of universal amplitude ratios in three-loop order. Nucl. Phys. B 479, 654–662 (1996)

    Article  ADS  Google Scholar 

  20. Brézin, E., Zinn-Justin, J.: Finite size effects in phase transitions. Nucl. Phys. B 257, 867–893 (1985)

    Article  ADS  Google Scholar 

  21. Zinn-Justin, J.: Quantum Field Theory and Critical Phenomena. Cambridge University Press, Cambridge (2002)

    Google Scholar 

  22. Jasnow, D., Rudnick, J.: Interfacial profile in three dimensions. Phys. Rev. Lett. 41, 698–701 (1978)

    Article  ADS  Google Scholar 

  23. Le Bellac, M.: Quantum and Statistical Field Theory. Clarendon Press, Oxford (1991)

    Google Scholar 

  24. van der Waals, J.: Thermodynamische theorie der capillariteit in de onderstelling van continue dichtheidsverandering. Verhandel. Konink. Akad. Weten. Amsterdam 1 (1893) (in Dutch). English translation: Rowlinson, J.: The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density. J. Stat. Phys. 20, 197–244 (1979)

    Google Scholar 

  25. Cahn, J., Hilliard, J.: Free energy of a nonuniform system. J. Chem. Phys. 28, 258–267 (1958)

    Article  ADS  Google Scholar 

  26. Rajamaran, R.: Non-perturbative semi-classical methods in quantum field theory (a pedagogical review). Phys. Rep. 21C, 227–313 (1975)

    Article  ADS  Google Scholar 

  27. Gervais, J.L., Sakita, B.: Extended particles in quantum field theories. Phys. Rev. D 11, 2943–2945 (1975)

    ADS  Google Scholar 

  28. Küster, J., Münster, G.: The interfacial profile in two-loop order. J. Stat. Phys. 129, 441–451 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. Lüscher, M., Weisz, P.: Scaling laws and triviality bounds in the lattice φ 4-theory; one-component model in the phase with spontaneous broken symmetry. Nucl. Phys. B 295, 65–92 (1987)

    Article  Google Scholar 

  30. Köpf, M.H.: Rauhigkeit von Grenzflächen in der Ising-Universalitätsklasse. Diploma thesis, Univ. of Münster, 2007

  31. Fisk, S., Widom, B.: Structure and free energy of the interface between fluid phases in equilibrium near the critical point. J. Chem. Phys. 50, 3219–3227 (1960)

    Article  ADS  Google Scholar 

  32. Hoppe, P., Münster, G.: The interface tension of the three-dimensional Ising model in two-loop order. Phys. Lett. A 238, 265–269 (1998)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. Caselle, M., Hasenbusch, M.: Universal amplitude ratios in the 3D Ising model. J. Phys. A 30, 4963–4982 (1997)

    ADS  MATH  Google Scholar 

  34. Mon, K., Landau, D., Stauffer, D.: Interface roughening in the three-dimensional Ising model. Phys. Rev. B 42, 545–547 (1990)

    Article  ADS  Google Scholar 

  35. Bürkner, E., Stauffer, D.: Monte Carlo study of surface roughening in the three-dimensional Ising model. Z. Phys. B 53, 241–243 (1983)

    ADS  Google Scholar 

  36. Hasenbusch, M., Pinn, K.: Surface tension, surface stiffness, and surface width of the 3-dimensional Ising model on a cubic lattice. Physica A 192, 342–374 (1992)

    Article  ADS  Google Scholar 

  37. Müller, M., Münster, G.: Profile and width of rough interfaces. J. Stat. Phys. 118, 669–686 (2005)

    Article  ADS  MATH  Google Scholar 

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Authors and Affiliations

  1. Institut für Theoretische Physik, Universität Münster, Wilhelm-Klemm-Str. 9, 48149, Münster, Germany

    Michael H. Köpf & Gernot Münster

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  1. Michael H. Köpf
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  2. Gernot Münster
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Correspondence to Gernot Münster.

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Köpf, M.H., Münster, G. Interfacial Roughening in Field Theory. J Stat Phys 132, 417–430 (2008). https://doi.org/10.1007/s10955-008-9572-5

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  • DOI: https://doi.org/10.1007/s10955-008-9572-5

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