Statistical mechanics of macromolecular networks without replicas

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Abstract
We report on a novel approach to the Deam-Edwards model (1976) 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 exact expression for the partition function of such a generalized Gaussian structure in the presence of random external fields and for its scattering function S0. We show that S0 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 Rg of a RCGN is of the universal form Rg2=(0.26+or-0.01)a2N/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 S0 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 modifed version of Leibler`s mean-field theory (1980) for block copolymers.
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