The European Physical Journal E

, Volume 33, Issue 4, pp 307–311 | Cite as

Polymer-brush lubrication in the limit of strong compression

  • L. Spirin
  • A. Galuschko
  • T. Kreer
  • A. Johner
  • J. Baschnagel
  • K. Binder
Regular Article


By means of molecular dynamics simulations we demonstrate power laws for macroscopic transport properties of strongly compressed polymer-brush bilayers to stationary shear motion beyond the Newtonian response. The corresponding exponents are derived from a recently developed scaling theory, where the interpenetration between the brushes is taken as the relevant length scale. This allows to predict the dependence of the critical shear rate, which separates linear and non-linear behavior, on compression and molecular parameters of the bilayer. We present scaling plots for chain extension (R , viscosity (\( \eta\) , and shear force (F over a wide range of Weissenberg numbers, W . In agreement with our theory, the simulation reveals simple power laws, RW 0.53 , \( \eta\)W -0.46 , and FW 0.54 , for the non-Newtonian regime.


Shear Rate Chain Extension Dissipative Particle Dynamic Polymer Brush Weissenberg Number 


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  1. 1.
    J. Klein, D. Perahia, S. Warburg, Nature 352, 143 (1991)CrossRefADSGoogle Scholar
  2. 2.
    J. Klein, Annu. Rev. Mater. Sci. 26, 581 (1996)CrossRefADSGoogle Scholar
  3. 3.
    P.A. Schorr, T.C.B. Kwan, S.M. Kilbey II, E.S.G. Shaqfey, M. Tirrell, Macromolecules 36, 389 (2003)CrossRefADSGoogle Scholar
  4. 4.
    J. Klein, Proc. IMechE J. 220, 691 (2006)CrossRefGoogle Scholar
  5. 5.
    J. Klein, Science 323, 47 (2009)CrossRefGoogle Scholar
  6. 6.
    M. Murat, G.S. Grest, Phys. Rev. Lett. 63, 1074 (1989)CrossRefADSGoogle Scholar
  7. 7.
    P.-Y. Lai, K. Binder, J. Chem. Phys. 98, 2366 (1993)CrossRefADSGoogle Scholar
  8. 8.
    P.S. Doyle, E.S.G. Shaqfeh, A.P. Gast, Phys. Rev. Lett. 78, 1182 (1997)CrossRefADSGoogle Scholar
  9. 9.
    P.S. Doyle, E.S.G. Shaqfeh, A.P. Gast, Macromolecules 31, 5474 (1998)CrossRefADSGoogle Scholar
  10. 10.
    T. Kreer, K. Binder, M.H. Müser, Langmuir 19, 7551 (2003)CrossRefGoogle Scholar
  11. 11.
    F. Goujon, P. Malfreyt, D.J. Tildesley, Chem. Phys. Chem. 5, 457 (2004)Google Scholar
  12. 12.
    F. Goujon, P. Malfreyt, D.J. Tildesley, Mol. Phys. 103, 2675 (2005)CrossRefADSGoogle Scholar
  13. 13.
    F. Goujon, Dissertation (Clermont-Ferrand, 2003)Google Scholar
  14. 14.
    C. Pastorino, T. Kreer, M. Müller, K. Binder, Phys. Rev. E 76, 026706 (2007)CrossRefADSGoogle Scholar
  15. 15.
    A. Galuschko, L. Spirin, T. Kreer, A. Johner, C. Pastorino, J. Wittmer, J. Baschnagel, Langmuir 26, 6418 (2010)CrossRefGoogle Scholar
  16. 16.
    J.-F. Joanny, Langmuir 8, 989 (1992)CrossRefGoogle Scholar
  17. 17.
    F. Clement, T. Charitat, A. Johner, J.-F. Joanny, Europhys. Lett. 54, 65 (2001)CrossRefADSGoogle Scholar
  18. 18.
    M. Rubinstein, S.P. Obukhov, Macromolecules 26, 1740 (1993)CrossRefADSGoogle Scholar
  19. 19.
    T. Moro, Y. Takatori, K. Ishihara, T. Konno, Y. Takigawa, T. Matsushita, U. Chung, K. Nakamura, H. Kawaguchi, Nat. Mater. 3, 829 (2004)CrossRefADSGoogle Scholar
  20. 20.
    R.C. Advincula, W.J. Brittain, K.C. Caster, J. Rühe (Editors), Polymer Brushes (Wiley, 2004)Google Scholar
  21. 21.
    R. Everaers, S.K. Sukumaran, G.S. Grest, C. Svaneborg, A. Sivasubramanian, K. Kremer, Science 303, 823 (2004)CrossRefADSGoogle Scholar
  22. 22.
    We anticipate that the entanglement length for directed chains in a brush is expected to be larger than for bulk systemsGoogle Scholar
  23. 23.
    K. Kremer, G.S. Grest, I. Carmesin, Phys. Rev. Lett. 61, 566 (1988)CrossRefADSGoogle Scholar
  24. 24.
    P.J. Hoogerbrugge, J.M.V.A. Koelman, Europhys. Lett. 19, 155 (1992)CrossRefADSGoogle Scholar
  25. 25.
    P. Espanol, P. Warren, Europhys. Lett. 30, 191 (1995)CrossRefADSGoogle Scholar
  26. 26.
    A detailed discussion about the performance of the DPD thermostat in non-equilibrium MD simulations of polymer brushes can be found in ref. claudio and in P. Virnau, K. Binder, H. Heinz, T. Kreer, M. Müller, Encyclopedia of polymer blends Vol. I: Foundamentals, edited by A.I. Isayev (Wiley-VCH, Weinheim, 2010)Google Scholar
  27. 27.
    All lengths are measured in Lennard-Jones unitsGoogle Scholar
  28. 28.
    T. Kreer, S. Metzger, M. Müller, K. Binder, J. Baschnagel, J. Chem. Phys. 120, 4012 (2004)CrossRefADSGoogle Scholar
  29. 29.
    T.A. Witten, L. Leibler, P.A. Pincus, Macromolecules 23, 824 (1990)CrossRefADSGoogle Scholar
  30. 30.
    P.G. de Gennes, Scaling Concepts in Polymer Physics (Cornell University Press, 1979)Google Scholar
  31. 31.
    In this context, the limit of ``strong'' compression does not imply melt conditions. Instead, we refer to a semidilute bilayer with a uniform monomer density profileGoogle Scholar
  32. 32.
    In fact, previous investigations kreer, where hydrodynamic interactions were strongly screened due to the application of a different (Langevin) thermostat, report a somewhat larger exponent, $R^2/R^2_0 \sim W^{0.6}$. This can be understood from our approach by reformulating eq. (Fwet.eq) for ``dry'' bilayers. Without hydrodynamic interactions, the force per area in linear response is proportional to the number of monomers in the interpenetration zone, $F/A \sim cL\dot{\gamma} D$. Repeating our analysis we obtain $R^2/R^2_0 \sim W^{0.65}$, $F/F(W = 1)$ $\sim W^{0.73}$, $\eta/\eta_0 \sim W^{-0.27}$ for the non-Newtonian response of semidilute, dry bilayers. Note that these exponents clearly differ from the present approach giving rise to the assertion that hydrodynamic interactions are represented in our simulations for all solvent models used. Dry bilayers, as modeled in ref. kreer, are physically much less relevant, of courseGoogle Scholar
  33. 33.
    Note that experimental shear rates are usually much smaller than in the simulation. On the other hand, simulations typically work with much smaller chain lengths. Equation (tau.eq) suggests that both effects partially cancel, such that the related Weissenberg numbers become comparableGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • L. Spirin
    • 1
  • A. Galuschko
    • 2
  • T. Kreer
    • 1
    • 2
  • A. Johner
    • 2
  • J. Baschnagel
    • 2
  • K. Binder
    • 1
  1. 1.Institute of PhysicsJohannes Gutenberg-UniversityMainzGermany
  2. 2.Institut Charles SadronStrasbourg Cedex 2France

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