Cross-linked polymer chains: Scaling and exact results
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 g⋍O(1) to the extended regime R g⋍O(√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.
KeywordsFree Chain Ideal Network Debye Function Replica Trick Kratky Plot
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- S. F. Edwards in Polymer Networks, eds. A. J. Chompff and S. Newman (Plenum Press, New York, 1971).Google Scholar
- Panyukow S. V. and Rabin Y. (1996): Phys. Rep. 269 no. 1 & 2.Google Scholar
- Kantor Y. and Kardar M. (1996): preprint.Google Scholar
- Doi M. and Edwards S. F. (1986): The Theory of Polymer Dynamics (Clarendon Press, Oxford).Google Scholar
- Press W. H., Teukolsky S. A., Vetterling W. T., Flannery B. P. (1992): Numerical Recipes (University Press, Cambridge).Google Scholar
- de Gennes P. G. (1979): Scaling Concepts in Polymer Physics (Cornell University Press, Ithaca).Google Scholar