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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: $G\ell_K^3/k_\textrm{B}T \sim (\ell_K/p)^3$], semiflexible [Edwards/de Gennes: $G\ell_K^3/k_\textrm{B}T \sim (\ell_K/p)^2$], and tightly-entangled [Morse: $G\ell_K^3/k_\textrm{B}T \sim (\ell_K/p)^{1+ε}$] regimes, where $\ell_K$ 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.