Unified Analytic Expressions for the Entanglement Length, Tube Diameter, and Plateau Modulus of Polymer Melts.

By combining molecular dynamics simulations and topological analyses with scaling arguments, we obtain analytic expressions that quantitatively predict the entanglement length N_{e}, the plateau modulus G, and the tube diameter a in melts that span the entire range of chain stiffnesses for which systems remain isotropic. Our expressions resolve conflicts between previous scaling predictions for the loosely entangled [Lin-Noolandi, Gℓ_{K}^{3}/k_{B}T∼(ℓ_{K}/p)^{3}], semiflexible [Edwards-de Gennes: Gℓ_{K}^{3}/k_{B}T∼(ℓ_{K}/p)^{2}], and tightly entangled [Morse, Gℓ_{K}^{3}/k_{B}T∼(ℓ_{K}/p)^{1+ϵ}] regimes, where ℓ_{K} and p are, respectively, the Kuhn and packing lengths. We also find that maximal entanglement (minimal N_{e}) coincides with the onset of local nematic order.