The European Physical Journal E

, Volume 27, Issue 1, pp 57–62 | Cite as

Field-theoretical Renormalization-Group approach to critical dynamics of crosslinked polymer blends

Article

Abstract

We consider a crosslinked polymer blend that may undergo a microphase separation. When the temperature is changed from an initial value towards a final one very close to the spinodal point, the mixture is out equilibrium. The aim is the study of dynamics at a given time t, before the system reaches its final equilibrium state. The dynamics is investigated through the structure factor, S(q, t), which is a function of the wave vector q, temperature T, time t, and reticulation dose D. To determine the phase behavior of this dynamic structure factor, we start from a generalized Langevin equation (model C) solved by the time composition fluctuation. Beside the standard de Gennes Hamiltonian, this equation incorporates a Gaussian local noise, ζ. First, by averaging over ζ, we get an effective Hamiltonian. Second, we renormalize this dynamic field theory and write a Renormalization-Group equation for the dynamic structure factor. Third, solving this equation yields the behavior of S(q, t), in space of relevant parameters. As result, S(q, t) depends on three kinds of lengths, which are the wavelength q−1, a time length scale R(t) ∼ t1/z, and the mesh size ξ*. The scale R(t) is interpreted as the size of growing microdomains at time t. When R(t) becomes of the order of ξ*, the dynamics is stopped. The final time, t*, then scales as t*ξ*z, with the dynamic exponent z = 6−η. Here, η is the usual Ising critical exponent. Since the final size of microdomains ξ* is very small (few nanometers), the dynamics is of short time. Finally, all these results we obtained from renormalization theory are compared to those we stated in some recent work using a scaling argument.

PACS

64.75.-g Phase equilibria 64.60.Ht Dynamic critical phenomena 83.80.Tc Polymer blends 

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References

  1. 1.
    P.-G. de Gennes, J. Phys. Lett. 40, 69 (1979).CrossRefGoogle Scholar
  2. 2.
    A. Bettachy, A. Derouiche, M. Benhamou, M. Daoud, J. Phys. II 1, 153 (1991).CrossRefGoogle Scholar
  3. 3.
    A. Derouiche, A. Bettachy, M. Benhamou, M. Daoud, Macromolecules 25, 7188 (1992).CrossRefGoogle Scholar
  4. 4.
    T.A. Vilgis, M. Benmouna, M. Daoud, M. Benhamou, A. Bettachy, A. Derouiche, Polym. Network Blends 3, 59 (1993).Google Scholar
  5. 5.
    M. Benmouna, T.A. Vilgis, M. Daoud, M. Benhamou, Macromolecules 27, 1172 (1994).CrossRefGoogle Scholar
  6. 6.
    M. Benmouna, T.A. Vilgis, M. Benhamou, A. Babaoui, M. Daoud, Macromol: Theory Simul. 3, 557 (1994).CrossRefGoogle Scholar
  7. 7.
    A. Bettachy, A. Derouiche, M. Benhamou, M. Benmouna, T.A. Vilgis, M. Daoud, Macromol: Theory Simul. 4, 67 (1995).CrossRefGoogle Scholar
  8. 8.
    M. Benhamou, J. Chem. Phys. 102, 5854 (1995).CrossRefADSGoogle Scholar
  9. 9.
    D.J. Read, M.G. Brereton, T.C.B. McLeish, J. Phys. II 5, 1679 (1995).CrossRefGoogle Scholar
  10. 10.
    A. Bettachy, Thesis, Hassan II-Mohammedia University, 1995.Google Scholar
  11. 11.
    A. Derouiche, Thesis, Hassan II-Mohammedia University, 1995.Google Scholar
  12. 12.
    M. Riva, V.G. Benza, J. Phys. II 7, 285 (1997).CrossRefGoogle Scholar
  13. 13.
    M. Benhamou, A. Derouiche, A. Bettachy, J. Chem. Phys. 106, 2513 (1997).CrossRefADSGoogle Scholar
  14. 14.
    A. Derouiche, M. Benhamou, A. Bettachy, Eur. Phys. J. E 13, 353 (2005).CrossRefGoogle Scholar
  15. 15.
    R.M. Briber, B.J. Bauer, Macromolecules 21, 3296 (1988).CrossRefGoogle Scholar
  16. 16.
    M. Benhamou, M. Chahid, Physica A 373, 153 (2007).CrossRefADSGoogle Scholar
  17. 17.
    J. Zinn-Justin, Quantum Field Theory and Critical Phenomena (Clarendon Press, Oxford, 1989).Google Scholar
  18. 18.
    C. Itzykson, J.-M. Drouffe, Statistical Field Theory: 1 and 2 (Cambridge University Press, 1989).Google Scholar
  19. 19.
    L. van Hove, Phys. Rev. 93, 249 (1954); 95, 1374 (1954).CrossRefADSGoogle Scholar
  20. 20.
    M. Benhamou, Int. J. Mod. Phys. A 8, 2581 (1993).CrossRefADSGoogle Scholar
  21. 21.
    P.J. Flory, Principles of Polymer Chemistry (Cornell University Press, Ithaca, NY, 1953).Google Scholar
  22. 22.
    P.-G. de Gennes, Scaling Concept in Polymer Physics (Cornell University Press, 1979).Google Scholar
  23. 23.
    G. ’t Hooft, M. Veltman, Nucl. Phys. B 44, 189 (1972).CrossRefADSMathSciNetGoogle Scholar
  24. 24.
    G. ’t Hooft, Nucl. Phys. B 61, 455 (1973).CrossRefADSMathSciNetGoogle Scholar
  25. 25.
    J.C. Collins, Nucl. Phys. B 80, 341 (1974).CrossRefADSGoogle Scholar
  26. 26.
    H.K. Jansen, Z. Phys. B 23, 377 (1976); R. Bausch, H.K. Jansen, H. Wagner, Z. Phys. B 24, 113 (1976).CrossRefADSMathSciNetGoogle Scholar
  27. 27.
    F. Langouche, D. Roekaerts, E. Tirapegui, Physica A 95, 252 (1979).CrossRefADSMathSciNetGoogle Scholar
  28. 28.
    D. Amit, Field Theory, the Renormalization Group and Critical Phenomena (McGraw-Hill, New-York, 1978).Google Scholar
  29. 29.
    P.-G. de Gennes, J. Phys. Lett. (Paris) 38, L–441 (1977); J.-F. Joanny, J. Phys. A 11, L-177 (1978); K. Binder, J. Chem. Phys. 79, 6387 (1983).Google Scholar
  30. 30.
    F.S. Battes et al., Phys. Rev. Lett. 65, 1893 (1990).CrossRefADSGoogle Scholar

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© Springer 2008

Authors and Affiliations

  1. 1.Laboratoire de Physique des Polymères et Phénomènes CritiquesFaculté des Sciences Ben M’sikCasablancaMorocco

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