Cross-linked polymer chains: Scaling and exact results

  • Thomas A. Vilgis
  • Michael P. Solf
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 492)


The paper discusses the size of a randomly cross-linked polymer chain. The calculations are based on an exact theorem for the characteristic function of a polydisperse phantom network that allows for treating the cross-links between pairs of randomly selected monomers as quenched variables without resorting to replica methods. By variation of the cross-linking potential from infinity (hard constraints) to zero (free chain), we have studied the cross-over of the radius of gyration from the branched polymer regime where R gO(1) to the extended regime R gO(√N). In the cross-over regime the network size R g is found to be proportional to (N/M)1/4, where M is the total number of cross-links and N the number of monomers in the system. Our exact results can also be understood in terms of simple scaling arguments.


Free Chain Ideal Network Debye Function Replica Trick Kratky Plot 
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  1. Edwards S.F. (1982): Ann. N.Y. Soc. 371, 210.ADSGoogle Scholar
  2. S. F. Edwards in Polymer Networks, eds. A. J. Chompff and S. Newman (Plenum Press, New York, 1971).Google Scholar
  3. R. T. Deam and S. F. Edwards, Proc. Trans. R. Soc. London A 280 (1976) 317.ADSMathSciNetCrossRefGoogle Scholar
  4. Goldbart P. and Goldenfeld N., (1989): Phys. Rev. Lett. 58 (1987) 2676; Phys. Rev. A 39 1402; ibid 1412.CrossRefADSGoogle Scholar
  5. Vilgis T. A. and Solf M. P. (1995: J. Phys. I France 5 1241.CrossRefGoogle Scholar
  6. Panyukow S. V. and Rabin Y. (1996): Phys. Rep. 269 no. 1 & 2.Google Scholar
  7. Higgs P. G. and Ball R. C. (1988): J. Phys. France 49 1785.CrossRefGoogle Scholar
  8. Warner M. and Edwards S.F. (1978): J. Phys. A 11 1649.CrossRefADSGoogle Scholar
  9. Edwards S.F. and Vilgis T.A. (1988): Rep. Prog. Phys. 51 243.CrossRefADSMathSciNetGoogle Scholar
  10. Solf M.P. and Vilgis T. A. (1995): J. Phys. A: Math. Gen. 28 6655.zbMATHCrossRefADSMathSciNetGoogle Scholar
  11. Gutin A. M. and Shakhnovich E. I. (1993): J. Chem. Phys. 100 5290.CrossRefADSGoogle Scholar
  12. Bryngelson J. D. and Thirumalai D. (1996): Phys. Rev. Lett. 76 542.CrossRefADSGoogle Scholar
  13. Kantor Y. and Kardar M. (1996): preprint.Google Scholar
  14. Lancaster P. and Tismenetsky M. (1985): The Theory of Matrices (Academic Press, New York, 2nd Ed.).zbMATHGoogle Scholar
  15. Doi M. and Edwards S. F. (1986): The Theory of Polymer Dynamics (Clarendon Press, Oxford).Google Scholar
  16. Press W. H., Teukolsky S. A., Vetterling W. T., Flannery B. P. (1992): Numerical Recipes (University Press, Cambridge).Google Scholar
  17. de Gennes P. G. (1979): Scaling Concepts in Polymer Physics (Cornell University Press, Ithaca).Google Scholar

Copyright information

© Springer-Verlag 1997

Authors and Affiliations

  • Thomas A. Vilgis
    • 1
  • Michael P. Solf
    • 1
  1. 1.Max-Planck-Institut für PolymerforschungMainzF.R.G.

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