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

Figure 1

Relation of entanglement to local structure. Panel (a): $N_e$ vs $C_{\infty }={\ell }_K/{\ell }_0$ for our Kremer-Grest melts [37]. $Z1$ data (blue and green symbols) and the analytic expression given in Eq. (7) (red and orange curves). The dashed and dotted vertical lines indicate $C_{\infty }^{*}(\rho )$ for $\rho =0.85{\sigma }^{-3}$ and $\rho =0.7{\sigma }^{-3}$ systems. Panel (b): $F_{bb}(\mathrm{\Delta })$ for selected $k_{\text{bend}}$ for the $\rho =0.85{\sigma }^{-3}$ systems.

Figure 2

$L_e/{\ell }_K$, $G{\ell }_K^3/k_{B}T$, and $a/{\ell }_K$ vs ${\ell }_K/p$ for our Kremer-Grest melts. Blue and green symbols in panels (a)–(c) respectively show results for $L_e/{\ell }_K\equiv N_e^{M\text{-kink}}/C_{\infty }$, $G{\ell }_K^3/k_{B}T\equiv \rho {\ell }_0^3{C}_{\infty }^3/N_e^{M\text{-kink}}$, and $a/{\ell }_K\equiv \sqrt{N_e^{M\text{-kink}}/C_{\infty }}$ vs ${\ell }_K/p\equiv \rho {\ell }_0^3{C}_{\infty }^2$, where $C_{\infty }$ is calculated from systems’ chain statistics [Eq. (1)]. Orange curves show Eqs. (9)–(11) with the $\{c_i\}$ given below and $ϵ=1/3$. The inset to panel (b) shows the local scaling exponent ${\mu }^{*}$ calculated from Eq. (10) using Eq. (12).