Statistical mechanics of macromolecular networks without replicas

TL;DR

This paper introduces an exact analytical approach to the statistical mechanics of macromolecular networks without using replicas, deriving key properties like the partition function, scattering function, and radius of gyration for randomly crosslinked Gaussian networks.

Contribution

It presents a novel, exact solution for the Deam-Edwards model of polymer networks without replicas, including derivations of the partition function and scattering function.

Findings

The scattering function $S_0$ is self-averaging and depends only on crosslink concentration.

The radius of gyration follows a universal form $R_g^2 = (0.26 imes a^2 N)/M$.

The approach can be extended to crosslinked polymer blends using mean field theory.

Abstract

We report on a novel approach to the Deam-Edwards model for interacting polymeric networks without using replicas. Our approach utilizes the fact that a network modelled from a single non-interacting Gaussian chain of macroscopic size can be solved exactly, even for randomly distributed crosslinking junctions. We derive an {\it exact} expression for the partition function of such a generalized Gaussian structure in the presence of random external fields and for its scattering function $S_0$. We show that $S_0$ of a randomly crosslinked Gaussian network (RCGN) is a self-averaging quantity and depends only on crosslink concentration $M/N$, where $M$ and $N$ are the total numbers of crosslinks and monomers. From our derivation we find that the radius of gyration $R_{\mbox{\tiny g}}$ of a RCGN is of the universal form $R_{\mbox{\tiny g}}^2=(0.26 \pm 0.01)a^2 N/M$, with $a$ being the Kuhn length. To treat the excluded volume effect in a systematic, perturbative manner, we expand the Deam-Edwards partition function in terms of density fluctuations analogous to the theory of linear polymers. For a highly crosslinked interacting network we derive an expression for the free energy of the system in terms of $S_0$ which has the same role in our model as the Debye function for linear polymers. Our ideas are easily generalized to crosslinked polymer blends which are treated within a modified version of Leibler's mean field theory for block copolymers.