Elementary statistical theory for idealized networks

Abstract

As was mentioned in Chapter 1, the elementary molecular theory of polymer networks rests on the postulate that the elastic free energy of a network is equal to the sum of the elastic free energies of the individual chains. The elastic free energy for the single chain was discussed in Chapter 3. Intermolecular contributions to the total elastic free energy are assumed to be insignificant and are entirely neglected in the elementary theory (Flory, 1953; Treloar, 1975). Understanding of the theory thus requires a precise description of the statistical behavior of the individual chains, given in Chapter 3, and of the relationship between their dimensions and the macroscopic strain that will be discussed in this chapter.

The chains in a network formed in the amorphous bulk state exhibit unperturbed dimensions identical to that of a single chain in Θ-solvents (Flory, 1953; Flory, 1976). This results from the fact that the distribution of the end-to-end vectors r for the chains in the bulk state is unchanged upon formation of network junctions, i.e., extended chains in the bulk state are equally susceptible to the interlinking or cross-linking reaction as others. The distribution of the end-to-end vector r of the chains in the network may therefore be identified with that of the single free chain. In rubber networks, chains that join two cross links typically have 100 to 700 bonds (Flory, 1976).