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 & Guth are rederived for a Gaussian phantom network using a simplified version of their procedure: (1) the mean values r̄ of the individual chain vectors are linear functions of the tensor λ of the principal extension ratios specifying the macroscopic strain, (2) fluctuations ∆ r ═ r – r̄ about these mean values are Gaussian, and (3) the mean-square fluctuations depend only on the structure of the network and not on the strain. Additionally, we show (4) that the distribution of the average vectors r̄ is Gaussian, and (5) that ≺(∆ r ) 2 ≻ ═ (2/ ϕ ) < r 2 ≻ 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 ∆ A e1 ═ ½ξ kT ( I 1 – 3), where I 1 ═ trace ( λ T λ ) 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 – 2 ϕ ) –1 ½ kT ( I 1 – 3) – (2ξ/ ϕ ) (1 – 2/ϕ) –1 kT In ( V / V 0 ), where V and V 0 are the actual and reference volumes, respectively. The expected trend from ∆ A* e1 to ∆ A e1 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.