Size and scaling in ideal polymer networks

TL;DR

This paper derives an exact theoretical framework for analyzing the size and scattering properties of ideal polymer networks, revealing a universal master curve and the crossover behavior of the radius of gyration based on crosslink strength.

Contribution

It introduces an exact theorem for the characteristic function of polydisperse phantom networks, enabling analysis without replica methods and uncovering universal scaling laws.

Findings

Scattering function obeys a master curve depending on a single parameter

Radius of gyration exhibits a crossover from collapsed to extended regime

Network size scales as (N/M)^{1/4} in the crossover regime

Abstract

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 {\it 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 $R_{\mbox{\tiny g}}\simeq {\cal O}(1)$ to the extended regime $R_{\mbox{\tiny g}}\simeq {\cal O}(\sqrt{N})$. In the crossover regime the network size $R_{\mbox{\tiny g}}$ is found to be proportional to $(N/M)^{1/4}$.