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Relaxation rate of the shape fluctuation of a fluid membrane immersed in a near-critical binary fluid mixture

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Abstract.

We consider the two-time correlation of the shape fluctuation of a fluid membrane immersed in a near-critical binary fluid mixture. Usually one component of the mixture is preferably attracted by the membrane. Adsorption layers, where the preferred component is more concentrated, are generated on both sides of the membrane significantly because of the near-criticality. The resultant gradient of the local mass-density difference between the two components generates additional stress, including the osmotic pressure, to influence the membrane motion. Assuming the mixture to be in the homogeneous phase near, but not too close to, the demixing critical point, we use the Gaussian free-energy functional to calculate the relaxation rate for a wavelength much longer than the correlation length of the mixture. Our calculation supposes weak preferential attraction and weak dependence of the mixture viscosity on the mass-density difference, and is performed within the linear approximation with respect to the undulation amplitude. It is shown for small wave number that the additional stress makes the relaxation more rapid independently of whether the preferred component is more viscous or not and that the relaxation rate can be regarded as proportional to the wave number even for a tensionless membrane. This linear dependence comes from the balance between the frictional force due to the mixture viscosity and the restoring force of the adsorption layer.

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References

  1. P.B. Canham, J. Theor. Biol. 26, 61 (1970)

    Article  Google Scholar 

  2. W. Helfrich, Z. Naturforsch. 28c, 693 (1973)

    Google Scholar 

  3. W. Helfrich, in Liquids at Interfaces, edited by J. Charvolin, J.F. Joanny, J. Zinn-Justin (North-Holland, Amsterdam, 1990)

  4. S.J. Singer, G.L. Nicolson, Science 175, 720 (1972)

    Article  ADS  Google Scholar 

  5. B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts, P. Walter, Molecular Biology of the Cell (Garland, New York, 2008) Chapt. 10

  6. D. Roux, A.M. Bellocq, in Physics of Amphiphiles, edited by V. Degiorgio, M. Corti (North Holland, Amsterdam, 1985)

  7. O. Diat, D. Roux, F. Nallet, J. Phys. II 3, 1427 (1993)

    Google Scholar 

  8. D. Roux, C.R. Safinya, J. Phys. (Paris) 49, 307 (1988)

    Article  Google Scholar 

  9. N. Lei, C.R. Safinya, R.F. Bruinsma, J. Phys. II 5, 1155 (1995)

    Google Scholar 

  10. W. Cai, T.C. Lubensky, P. Nelson, T. Powers, J. Phys. II 4, 931 (1994)

    Google Scholar 

  11. A. Imparato, J. Chem. Phys. 124, 154714 (2006)

    Article  ADS  Google Scholar 

  12. J.-B. Fournier, C. Barbetta, Phys. Rev. Lett. 100, 078103 (2008)

    Article  ADS  Google Scholar 

  13. F. Schmid, EPL 95, 28008 (2011)

    Article  ADS  Google Scholar 

  14. H. Shiba, H. Noguchi, Phys. Rev. E 84, 031926 (2011)

    Article  ADS  Google Scholar 

  15. O. Farago, Phys. Rev. E 84, 051914 (2011)

    Article  ADS  Google Scholar 

  16. J.-B. Fournier, Europhys. Lett. 97, 18001 (2012)

    Article  ADS  Google Scholar 

  17. H. Shiba, H. Noguchi, J.-B. Fournier, Soft Matter 12, 2373 (2016)

    Article  Google Scholar 

  18. F. Brochard, J.F. Lennon, J. Phys. (Paris) 11, 1035 (1975)

    Article  Google Scholar 

  19. F. Brochard, P.G. de Gennes, P. Pfeuty, J. Phys. (Paris) 37, 1099 (1976)

    Article  Google Scholar 

  20. P.G. de Gennes, C. Taupin, J. Phys. Chem. 86, 2294 (1982)

    Article  Google Scholar 

  21. P.G. de Gennes, F. Brochard-Wyart, D. Quéré, Capillarity and Wetting Phenomena (Springer, New York, 2004) Chapt. 8.2

  22. E. Evans, A. Yeung, Chem. Phys. Liquids 73, 39 (1994)

    Article  Google Scholar 

  23. U. Seifert, S.A. Langer, Europhys. Lett. 23, 71 (1993)

    Article  ADS  Google Scholar 

  24. W. Pfeiffer, S. König, J.F. Legrand, T. Bayerl, D. Richter, E. Sackmann, Europhys. Lett. 23, 457 (1993)

    Article  ADS  Google Scholar 

  25. T. Pott, P. Méléard, Europhys. Lett. 59, 87 (2002)

    Article  ADS  Google Scholar 

  26. S.A. Shkulipa, W.K. den Otter, W.J. Briels, Phys. Rev. Lett. 96, 178302 (2006)

    Article  ADS  Google Scholar 

  27. S.A. Shkulipa, W.K. den Otter, W.J. Briels, J. Chem. Phys. 125, 234905 (2006)

    Article  ADS  Google Scholar 

  28. J.-B. Fournier, Int. J. Non-Linear Mech. 75, 67 (2015)

    Article  ADS  Google Scholar 

  29. M.B. Schneider, J.T. Jenkins, W.W. Webb, J. Phys. (Paris) 45, 1457 (1984)

    Article  Google Scholar 

  30. S.T. Milner, S.A. Safran, Phys. Rev. A 36, 4371 (1987)

    Article  ADS  Google Scholar 

  31. A. Onuki, Europhys. Lett. 24, 151 (1993)

    Article  ADS  Google Scholar 

  32. Y. Fujitani, Physica A 203, 214 (1994)

    Google Scholar 

  33. V. Lisy, B. Brutovsky, Phys. Rev. E 61, 4045 (2000)

    Article  ADS  Google Scholar 

  34. V. Lisy, B. Brutovsky, A.V. Zatovsky, A.V. Zvelindovsky, J. Mol. Liquids 93, 113 (2001)

    Article  Google Scholar 

  35. R. Dimova, R. Lipowsky, Soft Matter 8, 6409 (2012)

    Article  Google Scholar 

  36. J.W. Cahn, J. Chem. Phys. 66, 3667 (1977)

    Article  ADS  Google Scholar 

  37. M.N. Binder, in Phase Transition and Critical Phenomena VIII, edited by C. Domb, J.L. Lebowitz (Academic, London, 1983)

  38. A.J. Bray, M.A. Moore, J. Phys. A: Math. Gen. 10, 1927 (1977)

    Article  ADS  MathSciNet  Google Scholar 

  39. H.W. Diehl, in Phase Transition and Critical Phenomena X, edited by C. Domb, J.L. Lebowitz (Academic, London, 1986)

  40. T.W. Burkhardt, H.W. Diehl, Phys. Rev. B 50, 3894 (1994)

    Article  ADS  Google Scholar 

  41. H.W. Diehl, Int. J. Mod. Phys. B 11, 3503 (1997)

    Article  ADS  Google Scholar 

  42. R. Okamoto, A. Onuki, J. Chem. Phys. 136, 114704 (2012)

    Article  ADS  Google Scholar 

  43. L. Onsager, Phys. Rev. 38, 2265 (1931)

    Article  ADS  Google Scholar 

  44. P.C. Hohenberg, B.I. Halperin, Rev. Mod. Phys. 49, 435 (1977)

    Article  ADS  Google Scholar 

  45. A. Onuki, Phase Transition Dynamics (Cambridge University Press, Cambridge, 2002) Chapt. 4 and 6

  46. R. Okamoto, Y. Fujitani, S. Komura, J. Phys. Soc. Jpn. 82, 084003 (2013)

    Article  ADS  Google Scholar 

  47. A. Furukawa, A. Gambassi, S. Dietrich, H. Tanaka, Phys. Rev. Lett. 111, 055701 (2013)

    Article  ADS  Google Scholar 

  48. Y. Fujitani, J. Phys. Soc. Jpn. 82, 124601 (2013)

    Article  ADS  Google Scholar 

  49. Y. Fujitani, J. Phys. Soc. Jpn. 83, 024401 (2014)

    Article  ADS  Google Scholar 

  50. Y. Fujitani, J. Phys. Soc. Jpn. 83, 084401 (2014)

    Article  ADS  Google Scholar 

  51. Y. Fujitani, Phys. Rev. E 91, 042402 (2015)

    Article  ADS  Google Scholar 

  52. S. Yabunaka, R. Okamoto, A. Onuki, Soft Matter 11, 5738 (2015)

    Article  ADS  Google Scholar 

  53. Y. Fujitani, arXiv:1510.02870 [condmat.soft]

  54. Y. Fujitani, J. Phys. Soc. Jpn. 85, 044401 (2016)

    Article  ADS  Google Scholar 

  55. K. Kawasaki, Ann. Phys. (N.Y.) 61, 1 (1970)

    Article  ADS  Google Scholar 

  56. T. Ohta, K. Kawasaki, Prog. Theor. Phys. 55, 1384 (1976)

    Article  ADS  Google Scholar 

  57. R. Kubo, M. Toda, N. Hashitsume, Statistical Physics II (Springer, Berlin, 1991) Chapt. 1.6

  58. M. Grimm, T. Franosch, S. Jeney, Phys. Rev. E 86, 021912 (2012)

    Article  ADS  Google Scholar 

  59. H. Lamb, Hydrodynamics, 6th edition (Dover, New York, 1932) Sect. 92

  60. H.W. Diehl, H.K. Janssen, Phys. Rev. A 45, 7145 (1992)

    Article  ADS  Google Scholar 

  61. Ou-Yang Zhong-can, W. Helfrich, Phys. Rev. Lett. 59, 2486 (1987)

    Article  ADS  Google Scholar 

  62. Ou-Yang Zhong-can, W. Helfrich, Phys. Rev. A 39, 5280 (1989)

    Article  ADS  Google Scholar 

  63. D. Nelson, in Statistical Mechanics of Membranes and Surfaces, Vol. 5, edited by D. Nelson, T. Piran, S. Weinberg (World Scientific, Singapore, 1989)

  64. M. Deserno, Chem. Phys. Lipids 185, 11 (2015)

    Article  Google Scholar 

  65. T.R. Powers, Rev. Mod. Phys. 82, 1607 (2010)

    Article  ADS  Google Scholar 

  66. Y. Fujitani, J. Chem. Phys. 116, 7787 (2002)

    Article  ADS  Google Scholar 

  67. N. Lei, X. Lei, Langmuir 14, 2155 (1998)

    Article  Google Scholar 

  68. H.P. Duwe, J. Kaes, E. Sackmann, J. Phys. (Paris) 51, 945 (1990)

    Article  Google Scholar 

  69. M. Mutz, W. Helfrich, J. Phys. (Paris) 51, 991 (1990)

    Article  Google Scholar 

  70. G.W. Brady, J. Chem. Phys. 32, 45 (1960)

    Article  ADS  Google Scholar 

  71. S.Z. Mizraev, I. Iwanovski, U. Kaatze, J. Phys. D: Appl. Phys. 40, 3248 (2007)

    Article  ADS  Google Scholar 

  72. A.J. Liu, M.E. Fisher, Phys. Rev. A 40, 7202 (1989)

    Article  ADS  Google Scholar 

  73. B.S. Carley, L.E. Scriven, H.T. Davis, AIChE J. 26, 705 (1980)

    Article  Google Scholar 

  74. H. Kahl, S. Enders, Phys. Chem. Chem. Phys. 4, 931 (2002)

    Article  Google Scholar 

  75. P.M.W. Cornelisse, C.J. Peters, J. de Swaan Arons, Fluid Phase Equilib. 117, 312 (1996)

    Article  Google Scholar 

  76. C.A. Grattoni, R.A. Dawe, C.Y. Seah, J.D. Gray, J. Chem. Eng. Data 38, 516 (1993)

    Article  Google Scholar 

  77. H.W. Diehl, M. Smock, Phys. Rev. B 47, 5841 (1993)

    Article  ADS  Google Scholar 

  78. H.W. Diehl, Ber. Bunsenges, Phys. Chem. 98, 466 (1994)

    Article  Google Scholar 

  79. G. Flöter, S. Dietrich, Z. Phys. B 97, 213 (1995)

    Article  ADS  Google Scholar 

  80. A. Hanke, S. Dietrich, Phys. Rev. E 59, 5081 (1999)

    Article  ADS  Google Scholar 

Download references

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  1. School of Fundamental Science and Technology, Keio University, 223-8522, Yokohama, Japan

    Youhei Fujitani

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Fujitani, Y. Relaxation rate of the shape fluctuation of a fluid membrane immersed in a near-critical binary fluid mixture. Eur. Phys. J. E 39, 31 (2016). https://doi.org/10.1140/epje/i2016-16031-8

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