## Abstract

Polymeric networks exhibiting high elasticity consist, typically, of linear chains several hundred bonds in length joined at their ends to $\phi $-functional junctions $(\phi >2)$. 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 $\phi $ 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 $\overline{r}$ of the individual chain vectors are linear functions of the tensor $\lambda $ of the principal extension ratios specifying the macroscopic strain, (2) fluctuations $\Delta r=r-\overline{r}$ 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 $\overline{r}$ is Gaussian, and (5) that $\langle (\Delta r)^{2}\rangle =(2/\phi)\langle r^{2}\rangle _{0}$, 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 $\lambda $, 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 $\lambda $, especially when the network is undiluted. The elastic free energy derived for a phantom network of any functionality and degree of imperfection reduces to $\Delta A_{\text{e1}}=\frac{1}{2}\xi kT(I_{1}-3)$, where $I_{1}$ = trace $(\lambda ^{T}\lambda)$ is the first invariant of the strain and $\xi $ 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 $\Delta A_{\text{e1}}^{\ast}=\xi (1-2/\phi)^{-1}\frac{1}{2}kT(I_{1}-3)-(2\xi /\phi)^{-1}kT$ln $(V/V^{0})$, where $V$ and $V^{0}$ are the actual and reference volumes, respectively. The expected trend from $\Delta A_{\text{e1}}^{\ast}$ to $\Delta A_{\text{e1}}$ 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.

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