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The European Physical Journal E

, Volume 13, Issue 4, pp 353–358 | Cite as

Exact effective force between star-polymers in a \(\Theta\)-solvent

  • M. BenhamouEmail author
  • M. Himmi
  • F. Benzouine
  • A. Bettachy
  • A. Derouiche
Article

Abstract.

We re-examine here the computation of the effective force between two star-polymers of respective numbers of branches f 1 and f 2, immersed in a common \(\Theta\)-solvent. Such a force originates essentially from the repulsive three-body interactions. To achieve this, we take advantage of some established results using renormalization theory for three-dimensional star-polymers, or conformal invariance for two-dimensional ones. We first show that, in dimension d = 3, the force, \(F\left(r\right) \), decreases with the center-to-center distance r as \(F\left(r\right) /k_{\rm B}T\simeq A_{f_1f_2} \cdot \left[ r\ln \left(R^2/r^2\right) \right]^{-1}\) \( \left(r < R\right) \), with the exact universal amplitude \( A_{f_1f_2} = f_1f_2\left(f_1 + f_2-2\right) /22\). Second, in dimension d = 2, we find that the force decays more slowly as \(F\left(r\right) /k_{\rm B}T\simeq B_{f_1f_2} \cdot r^{-1}\) \( \left(r < R\right) \), with the exact universal amplitude \(B_{f_1f_2} = \left(2 + 4f_1f_2\right) /21\). For high distances compared to the gyration radius, \(R\thicksim a\sqrt{N}\), of a single polymer chain at the \(\Theta\)-point, an exponential decay of the force is expected.

Keywords

Polymer Polymer Chain Exponential Decay Conformal Invariance Respective Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    G.S. Grest, L.J. Fetters, J.H. Huang, D. Richter, Adv. Chem. Phys. XCIV, 6 (1996).Google Scholar
  2. 2.
    M. Daoud, J.P. Cotton, J. Phys. (Paris) 43, 531 (1982).Google Scholar
  3. 3.
    T.M. Birshtein, E.B. Zhulina, Polymer 25, 1453 (1984); T.M. Birshtein, E.B. Zhulina, O.V. Borisov, Polymer 27, 1078 (1986).CrossRefGoogle Scholar
  4. 4.
    C.H. Vlahos, K. Kosmas, Polymer 25, 1607 (1984).CrossRefGoogle Scholar
  5. 5.
    A. Miyake, K.F. Freed, Macromolecules 16, 1228 (1983).Google Scholar
  6. 6.
    K.F. Freed, J. Chem. Phys. 79, 6357 (1983).CrossRefGoogle Scholar
  7. 7.
    B. Duplantier, Renormalization and Conformal Invariance for Polymers, in Fundamental Problems in Statistical Mechanics VII, edited by H. van Beijeren (Elsevier Science Publishers B.V, 1990).Google Scholar
  8. 8.
    B. Duplantier, J. Stat. Phys. 54, 581 (1988); D.J. Wallace, R.K.P. Zia, J. Phys. C 8, 839 (1975); U. Lehr, L. Schäfer, C. von Ferber, B. Duplantier, Nucl. Phys. 374, 473 (1990).Google Scholar
  9. 9.
    A. Rey, J.J. Freire, J.G. de la Torre, Macromolecules 20, 342 (1987).Google Scholar
  10. 10.
    J. Batoulis, K. Kremer, Macromolecules 22, 531 (1989).Google Scholar
  11. 11.
    G. Grest, K. Kremer, T.A. Witten, Macromolecules 20, 1376 (1987).Google Scholar
  12. 12.
    D. Richter, B. Farago, L.J. Fetters, J.S. Huang, B. Ewen, Macromolecules 23, 1845 (1990).Google Scholar
  13. 13.
    C. von Ferber, A. Jusfi, C.N. Likos, H. Löwen, M. Watzlawek, Eur. Phys. J. E 2, 311 (2000).CrossRefGoogle Scholar
  14. 14.
    M. Badia, M. Benhamou, A. Derouiche, J.-L. Bretonnet, Colloid Polym. Sci. 279, 763 (2001).CrossRefGoogle Scholar
  15. 15.
    L. Millner et al. , Macromolecules 27, 3821 (1994).Google Scholar
  16. 16.
    C.N. Likos et al. , Phys. Rev. Lett. 80, 4450 (1998).CrossRefGoogle Scholar
  17. 17.
    C.M. Marques, D. Izzo, T. Chariat, E. Mendes, Eur. Phys. J. B 3, 353 (1998).CrossRefGoogle Scholar
  18. 18.
    T.A. Witten, P.A. Pincus, Macromolecules 19, 2509 (1986).Google Scholar
  19. 19.
    M. Benhamou, F. Benzouine, Eur. Phys. J. E 5, 275 (2001).CrossRefGoogle Scholar
  20. 20.
    F. Benzouine, M. Benhamou, M. Himmi, this issue, p. 345.Google Scholar
  21. 21.
    B. Duplantier, Europhys. Lett. 7, 677 (1988).Google Scholar
  22. 22.
    B. Duplantier, H. Saleur, Phys. Rev. Lett. 59, 541 (1987).CrossRefGoogle Scholar
  23. 23.
    J. des Cloizeaux, J. Phys. (Paris) 42, 653 (1981).Google Scholar
  24. 24.
    J. des Cloizeaux, G. Jannink, Polymers in Solution (Oxford University Press, Oxford, 1990).Google Scholar
  25. 25.
    C.N. Likos et al. , Phys. Rev. E 58, 6299 (1998).CrossRefGoogle Scholar
  26. 26.
    J. Mewis et al. , AIChE J. 35, 418 (1989).Google Scholar
  27. 27.
    B.V. Derjaguin, Kolloid Z. 69, 155 (1934).Google Scholar
  28. 28.
    S.T. Milner, T.A. Witten, M.E. Cates, Macromolecules 21, 2610 (1988); S.T. Milner, Science 251, 905 (1991).Google Scholar
  29. 29.
    M. Watzlawek, H. Löwen, C.N. Likos, J. Phys. Condens. Matter 10, 8189 (1998), and references therein.CrossRefGoogle Scholar
  30. 30.
    P.-G. de Gennes, Phys. Lett. A 38, 339 (1972).Google Scholar
  31. 31.
    P.-G. de Gennes, J. Phys. (Paris) Lett. 36, L-55 (1975).Google Scholar
  32. 32.
    P.J. Flory, J. Chem. Phys. 17, 303 (1949).Google Scholar
  33. 33.
    B. Duplantier, J. Phys. (Paris) 43, 991 (1982).Google Scholar
  34. 34.
    B. Duplantier, Europhys. Lett. 1, 491 (1986).Google Scholar
  35. 35.
    S.F. Edwards, Proc. Phys. Soc. London 85, 613 (1965).CrossRefzbMATHGoogle Scholar
  36. 36.
    B. Duplantier, J. Chem. Phys. 86, 4233 (1987).CrossRefMathSciNetGoogle Scholar
  37. 37.
    M. Benhamou, G. Mahoux, J. Phys. (Paris) 47, 559 (1986).Google Scholar
  38. 38.
    M. Benhamou, Thesis, Paris XI-University (1987).Google Scholar
  39. 39.
    B. Duplantier, J. Phys. (Paris) 47, 569 (1986).Google Scholar
  40. 40.
    P.-G. de Gennes, Scaling Concepts in Polymer Physics (Cornell University Press, 1979).Google Scholar
  41. 41.
    B. Duplantier, J. des Cloizeaux, G. Jannink, Phys. Rev. Lett. 56, 2080 (1986).CrossRefGoogle Scholar
  42. 42.
    M. Himmi, M. Benhamou, A. Bettachy, to be published in J. Mol. Liq. (2004).Google Scholar

Copyright information

© Springer-Verlag Berlin/Heidelberg 2004

Authors and Affiliations

  • M. Benhamou
    • 1
    Email author
  • M. Himmi
    • 1
  • F. Benzouine
    • 1
  • A. Bettachy
    • 1
  • A. Derouiche
    • 1
  1. 1.Faculté des Sciences Ben M’sikLaboratoire de Physique des Polyméres et Phénoménes CritiquesCasablancaMorocco

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