We show that the front factor appearing in the shear modulus of a phantom network, $G_{\mathrm{ph}}=(1-2/f)\left(\rho k_{B}T\right)/N_s$, also controls the ratio of the strand length, $N_s$, and the number of monomers per Kuhn length of the primitive paths, $N_{\mathrm{ph}}^{\mathrm{PPKuhn}}$, characterizing the average network conformation. In particular, $N_{\mathrm{ph}}^{\mathrm{PPKuhn}}=N_s/(1-2/f)$ and $G_{\mathrm{ph}}=\left(\rho k_{B}T\right)/N_{\mathrm{ph}}^{\mathrm{PPKuhn}}$. Neglecting the difference between cross-links and slip-links, these results can be transferred to entangled systems and the interpretation of primitive path analysis data. In agreement with the tube model, the analogy to phantom networks suggest that the rheological entanglement length, $N_e^{\mathrm{rheo}}=\left(\rho k_{B}T\right)/G_e$, should equal $N_e^{\mathrm{PPKuhn}}$. Assuming binary entanglements with $f=4$ functional junctions, we expect that $N_e^{\mathrm{rheo}}$ should be twice as large as the topological entanglement length, $N_e^{\mathrm{topo}}$. These results are in good agreement with reported primitive path analysis results for model systems and a wide range of polymeric materials. Implications for tube and slip-link models are discussed.

• Received 23 November 2011

DOI:https://doi.org/10.1103/PhysRevE.86.022801