Size and Scaling in Ideal Polymer Networks. Exact Results

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