Numerical solutions of thin-film equations for polymer flows

  • Thomas Salez
  • Joshua D. McGraw
  • Sara L. Cormier
  • Oliver Bäumchen
  • Kari Dalnoki-Veress
  • Elie Raphaël
Regular Article

Abstract

We report on the numerical implementation of thin-film equations that describe the capillary-driven evolution of viscous films, in two-dimensional configurations. After recalling the general forms and features of these equations, we focus on two particular cases inspired by experiments: the leveling of a step at the free surface of a polymer film, and the leveling of a polymer droplet over an identical film. In each case, we first discuss the long-term self-similar regime reached by the numerical solution before comparing it to the experimental profile. The agreement between theory and experiment is excellent, thus providing a versatile probe for nanorheology of viscous liquids in thin-film geometries.

Graphical abstract

Keywords

Flowing Matter: Liquids and Complex Fluids 

References

  1. 1.
    F. Brochard Wyart, P.-G. de Gennes, Eur. Phys. J. E 1, 93 (2000)CrossRefGoogle Scholar
  2. 2.
    H. Bodiguel, C. Fretigny, Phys. Rev. Lett. 97, 266105 (2006)ADSCrossRefGoogle Scholar
  3. 3.
    Z. Fakhraai, J.A. Forrest, Science 319, 600 (2008)CrossRefGoogle Scholar
  4. 4.
    L. Si, M.V. Massa, K. Dalnoki-Veress, H.R. Brown, R.A.L. Jones, Phys. Rev. Lett. 94, 127801 (2005)ADSCrossRefGoogle Scholar
  5. 5.
    K. Shin, S. Obukhov, J.-T. Chen, J. Huh, Y. Hwang, S. Mok, P. Dobriyal, P. Thiyagarajan, T.P. Russell, Nat. Mater. 6, 961 (2007)ADSCrossRefGoogle Scholar
  6. 6.
    G. Reiter, P.-G. de Gennes, Eur. Phys. J. E 6, 25 (2001)CrossRefGoogle Scholar
  7. 7.
    D.R. Barbero, U. Steiner, Phys. Rev. Lett. 102, 248303 (2009)ADSCrossRefGoogle Scholar
  8. 8.
    A. Raegen, M. Chowdhury, C. Calers, A. Schmatulla, U. Steiner, G. Reiter, Phys. Rev. Lett. 105, 227801 (2010)ADSCrossRefGoogle Scholar
  9. 9.
    P.-G. de Gennes, C. R. Acad. Sci. (Paris) 288, 219 (1979)Google Scholar
  10. 10.
    O. Bäumchen, R. Fetzer, K. Jacobs, Phys. Rev. Lett. 103, 247801 (2009)ADSCrossRefGoogle Scholar
  11. 11.
    A. Münch, B. Wagner, T.P. Witelski, J. Eng. Math. 53, 359 (2005)CrossRefMATHGoogle Scholar
  12. 12.
    L. H. Tanner, J. Phys. D 12, 1473 (1979)ADSCrossRefGoogle Scholar
  13. 13.
    P.-G. de Gennes, Rev. Mod. Phys. 57, 827 (1985)ADSCrossRefGoogle Scholar
  14. 14.
    P.-G. de Gennes, F. Brochard-Wyart, D. Quéré, Capillarity and wetting phenomena: drops, bubbles, pearls, waves (Springer, 2003)Google Scholar
  15. 15.
    D. Bonn, J. Eggers, J. Indekeu, J. Meunier, E. Rolley, Rev. Mod. Phys. 81, 739 (2009)ADSCrossRefGoogle Scholar
  16. 16.
    A. Aradian, E. Raphaël, P.-G. de Gennes, Eur. Phys. J. E 2, 367 (2000)CrossRefGoogle Scholar
  17. 17.
    F. Pierce, D. Perahia, G.S. Grest, Europhys. Lett. 86, 64004 (2009)ADSCrossRefGoogle Scholar
  18. 18.
    S.L. Cormier, J.D. McGraw, T. Salez, E. Raphaël, K. Dalnoki-Veress, Phys. Rev. Lett. 109, 154501 (2012)ADSCrossRefGoogle Scholar
  19. 19.
    L.D. Landau, E.M. Lifshitz, Fluid Mechanics (Pergamon Press, 1987)Google Scholar
  20. 20.
    A. Oron, S.H. Davis, S.G. Bankoff, Rev. Mod. Phys. 69, 931 (1997)ADSCrossRefGoogle Scholar
  21. 21.
    R.V. Craster, O.K. Matar, Rev. Mod. Phys. 81, 1131 (2009)ADSCrossRefGoogle Scholar
  22. 22.
    R. Blossey, Thin liquid films (Springer, 2012)Google Scholar
  23. 23.
    M. Bowen, T.P. Witelski, SIAM J. Appl. Math. 66, 1727 (2006)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    T. Salez, J.D. McGraw, O. Bäumchen, K. Dalnoki-Veress, E. Raphaël, Phys. Fluids 24, 102111 (2012)ADSCrossRefGoogle Scholar
  25. 25.
    F. Bernis, A. Friedman, J. Diff. Eq. 83, 179 (1990)MathSciNetCrossRefADSMATHGoogle Scholar
  26. 26.
    T.G. Myers, SIAM Rev. 40, 441 (1998)MathSciNetADSCrossRefMATHGoogle Scholar
  27. 27.
    L. Kondic, SIAM Rev. 45, 95 (2003)MathSciNetADSCrossRefMATHGoogle Scholar
  28. 28.
    A. Bertozzi, Notices AMS 45, 689 (1998)MathSciNetMATHGoogle Scholar
  29. 29.
    L. Zhornitskaya, A. Bertozzi, SIAM J. Numer. Anal. 37, 523 (2000)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    J. D. McGraw, N.M. Jago, K. Dalnoki-Veress, Soft Matter 7, 7832 (2011)ADSCrossRefGoogle Scholar
  31. 31.
    J.D. McGraw, T. Salez, O. Bäumchen, E. Raphaël, K. Dalnoki-Veress, Phys. Rev. Lett. 109, 128303 (2012)ADSCrossRefGoogle Scholar
  32. 32.
    G. I. Barenblatt, Scaling, self-similarity, and intermediate asymptotics (Cambridge University Press, 1996)Google Scholar
  33. 33.
    S. Wu, J. Phys. Chem. 74, 632 (1970)CrossRefGoogle Scholar
  34. 34.
    M. Rubinstein, R.H. Colby, Polymer physics (Oxford University Press, 2003)Google Scholar
  35. 35.
    J. Brandrup, E.H. Immergut, E.A. Grulke, A. Abe, D.R. Bloch, Polymer handbook (John Wiley and Sons, 2005)Google Scholar
  36. 36.
    H. Huppert, J. Fluid Mech. 121, 43 (1982)ADSCrossRefGoogle Scholar
  37. 37.
    R. Seemann, S. Herminghaus, K. Jacobs, Phys. Rev. Lett. 86, 5534 (2001)ADSCrossRefGoogle Scholar
  38. 38.
    L.E. Stillwagon, R.G. Larson, J. Appl. Phys. 63, 5251 (1988)ADSCrossRefGoogle Scholar
  39. 39.
    L.E. Stillwagon, R.G. Larson, Phys. Fluids A: Fluid Dyn. 2, 1937 (1990)ADSCrossRefGoogle Scholar
  40. 40.
    W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in Fortran 90 (Cambridge University Press, 1996)Google Scholar
  41. 41.
    A. Aradian, E. Raphaël, P.-G. de Gennes, Europhys. Lett. 55, 834 (2001)ADSCrossRefGoogle Scholar
  42. 42.
    I.C. Christov, H.A. Stone, Proc. Natl. Acad. Sci. U.S.A. 109, 16012 (2012)ADSCrossRefGoogle Scholar
  43. 43.
    A. Bach, K. Almdal, H.K. Rasmussen, O. Hassager, Macromolecules 36, 5174 (2003)ADSCrossRefGoogle Scholar
  44. 44.
    M.L. Williams, R.F. Landel, J.D. Ferry, J. Am. Chem. Soc. 77, 3701 (1955)CrossRefGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Thomas Salez
    • 1
  • Joshua D. McGraw
    • 2
  • Sara L. Cormier
    • 2
  • Oliver Bäumchen
    • 2
  • Kari Dalnoki-Veress
    • 2
  • Elie Raphaël
    • 1
  1. 1.Laboratoire de Physico-Chimie Théorique, UMR CNRS Gulliver 7083, ESPCIParisFrance
  2. 2.Department of Physics & Astronomy and the Brockhouse Institute for Materials ResearchMcMaster UniversityHamiltonCanada

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