Topological versus rheological entanglement length in primitive-path analysis protocols, tube models, and slip-link models

We show that the front factor appearing in the shear modulus of a phantom network, Gₚₕ=(1-2/f)0.16em0ex(ρ0.16em0exkBT)/Nₛ, also controls the ratio of the strand length, Nₛ, and the number of monomers per Kuhn length of the primitive paths, NₚₕPPKuhn, characterizing the average network conformation. In particular, NₚₕPPKuhn=Nₛ/(1-2/f) and Gₚₕ=(ρ0.16em0exkBT)/NₚₕPPKuhn. 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, Nₑʳʰᵉᵒ=(ρ0.16em0exkBT)/Gₑ, should equal NₑPPKuhn. Assuming binary entanglements with $f=4$ functional junctions, we expect that Nₑʳʰᵉᵒ should be twice as large as the topological entanglement length, Nₑᵗᵒᵖᵒ. 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.