Statistical thermodynamics of random networks

Abstract Polymeric networks exhibiting high elasticity consist, typically, of linear chains several hundred bonds in length joined at their ends to ϕ-functional junctions (ϕ 2). For random unconstrained chains of this length, the density distribution of chain vectors r is Gaussian in satisfactory approximation. The chains interpenetrate copiously in the network; the domain described by the set of ϕ junctions that are topological neighbours of a given junction encompasses many (20–100) spatial neighbours. A phantom network is expressly defined as a hypothetical one whose chains may move freely through one another; the chains act exclusively by introducing a force that is proportional to the distance between each pair of junctions so connected. The following results of James r2≻0, a result obtained previously by Graessley. It follows from (1) and (3) that the transformation of chain vectors r of the phantom network is not affine in λ, and hence that junctions exchange neighbours with strain. In real networks, the mutual interpenetration of chains pendent at a given junction must obstruct this process of local rearrangement of junctions; the transformation of chain vectors may therefore be more nearly affine in λ, especially when the network is undiluted. The elastic free energy derived for a phantom network of any functionality and degree of imperfection reduces to ∆Ae1 ═ ½ξkT(I1 – 3), where I1 ═ trace (λTλ) is the first invariant of the strain and ξ is the cycle rank of the network. If the fluctuations of junctions in a real network are suppressed for the reasons stated, the elastic free energy is ∆A*e1 = ξ(1 – 2ϕ)–1 ½kT(I1 – 3) – (2ξ/ϕ) (1 – 2/ϕ)–1kT In (V/V0), where V and V0 are the actual and reference volumes, respectively. The expected trend from ∆A*e1 to ∆Ae1 with dilution may account, qualitatively at least, for the effect of dilution on the stress–strain relation. A similar trend with extension may explain the familiar departure of the observed tension–elongation relation from theory.