J. Phys. I France
Volume 6, Number 11, November 1996
Page(s) 1451 - 1460
DOI: 10.1051/jp1:1996157
J. Phys. I France 6 (1996) 1451-1460

Size and Scaling in Ideal Polymer Networks. Exact Results

Michael P. Solf and Thomas A. Vilgis

Max-Planck-Institut für Polymerforschung, Postfach 3148, 55021 Mainz, Germany

(Received 29 May 1996, received in final form 12 July 1996, accepted 26 July 1996)

The scattering function and radius of gyration of an ideal polymer network are calculated depending on the strength of the bonds that form the crosslinks. Our calculations are based on an exact theorem for the characteristic function of a polydisperse phantom network that allows for treating the crosslinks between pairs of randomly selected monomers as quenched variables without resorting to replica methods. From this new approach it is found that the scattering function of an ideal network obeys a master curve which depends on one single parameter x=(ak)2 N/M, where ak is the product of the persistence length times the scattering wavevector, N the total number of monomers and M the crosslinks in the system. By varying the crosslinking potential from infinity (hard $\delta$-constraints) to zero (free chain), we have also studied the crossover of the radius of gyration from the collapsed regime where $R_{\rm g}\simeq \mathcal{O}(1)$ to the extended regime $R_{\rm g}\simeq \mathcal{O}(\sqrt{N})$. In the crossover regime the network size $R_{\rm g}$ is found to be proportional to (N/M)1/4. The latter result can be understood in terms of a simple Flory argument.

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