## 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.

## Footnotes

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- Received March 25, 1976.

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