Defining Ideal Phantom Polymer Networks

Elastomers are modeled as networks with $\phi$-functional junctions containing $N$ ideal, $n$-segment, freely jointed chains (FJCs) per unit volume (p.u.v.). Our compact model of the exact FJC length probability density (Treloar, 1975), accurately yields their exact distribution moments (Flory, 1969). The governing geometry of fluctuations of $N_X=2N\mathrm{/}\phi$ junctions p.u.v., parametrically maps their $\lambda$(elongation ratio)-dependent distribution to an equivalent FJC consisting of $n_{f\phi }=(n\mathrm{/}\phi )(1-\mathrm{\Lambda })$ segments, where $\mathrm{\Lambda }=(1\mathrm{/}3n)({\lambda }^2+2\mathrm{/}\lambda ))$. The resulting elastic pre-factor, $N_{\text{eff}}kT=(N-\eta N_X)kT$, with junction effectiveness $\eta =\phi (1-\mathrm{\Lambda })\mathrm{/}(\phi -\mathrm{\Lambda })$, defines ideal phantom networks