Stress Relaxation in Monodisperse Entangled Polymer Melts: Correlation Between Viscoelastic Response and Single-Chain Relaxation via Molecular Dynamics Simulations

Abstract

We study stress relaxation in several types of entangled monodisperse linear polymer melts by comparing the shear stress relaxation modulus, $G(t)$, with the end-to-end vector autocorrelation function, $P(t)$. The study includes three Kremer-Grest bead-spring models with varying chain stiffness, as well as a chemistry-specific coarse-grained model of \emph{cis}-1,4-polybutadiene. For each model, multiple chain lengths were simulated, spanning a range of $N/N_e = 5$-$50$ entanglements per chain. We observe that in all cases the behavior of $G(t)$, beyond the short-time Rouse regime, is accurately described by $G^0_{\mathrm{N}}[P(t)]^2$, where the chain-length-independent prefactor $G^0_{\mathrm{N}}$ denotes the plateau modulus. This correlation is consistent with both double reptation and dynamic tube dilation models of polymer relaxation, although the two models are based on different physical pictures. The double reptation model represents the melt as a transient network in which stress relaxation is governed by the survival probability of pairwise entanglements. The dynamic tube dilation model, however, assumes that the tube of constraints surrounding a polymer chain progressively enlarges as relaxation proceeds. The relation $G(t) = G^0_\mathrm{N}[P(t)]^2$ can serve as a basis for determining the plateau modulus and the corresponding entanglement length. It also simplifies the modeling of $G(t)$, since an accurate analytical expression for $P(t)$ is sufficient to describe the long-time behavior of $G(t)$. We further compare the simulation data for $P(t)$ and $G(t)$ with theoretical predictions.

Figures (5)

• Figure 1: Mean-squared internal distances for the simulated Kremer-Grest chains. $R^2(n)$ denotes the average squared distance between two monomers separated by $n$ consecutive bonds along one chain.
• Figure 2: The $G(t)$ curves for the bead–spring chains with a bending stiffness of $k_{\theta} = 1.5$. The $G(t)$ curves are compared to $G^0_\text{N}[P(t)]$ (dashed magenta), $G^0_\text{N}[P(t)]^2$ (solid red), and $G^0_\text{N,ppa}[P(t)]^2$ (dashed green), with $G^0_\text{N} = 0.0324\,\varepsilon/\sigma^3$ (corresponding to $N_\text{e} = 21$ beads) and $G^0_\text{N,ppa} = 0.0243,\varepsilon/\sigma^3$ (corresponding to $N_\text{e,ppa} = 28$ as calculated from primitive path analysis). Panels correspond to chain lengths $N = 100$ (a), $200$ (b), $400$ (c), and $1000$ (d). The dashed orange lines in panel (a) also show the short-time prediction of the Rouse model with monomeric time $\tau_0 = 2.1\,\tau$. Additionally, panel (c) shows an estimate of the entanglement time, obtained from the condition $G(\tau_\text{e}) = \rho k_\text{B} T / N_\text{e} = 5 G^0_\text{N}/4$.
• Figure 3: Comparison between $G(t)$ and $G^0_\text{N}[P(t)]^2$ for three different polymer models. Panel (a) shows the results for the bead–spring model with bending stiffness $k_\theta = 2.5$, where $G^0_\text{N} = 0.052\,\varepsilon/\sigma^3$, corresponding to $N_\text{e} = 13$. Panel (b) presents the results for the bead–spring model with $k_\theta = -1$, for which $G^0_\text{N} = 0.0085\,\varepsilon/\sigma^3$, corresponding to $N_\text{e} = 80$. In this panel, an estimate of the entanglement time is also shown, obtained from the condition $G(\tau_\text{e}) = \rho k_\text{B} T / N_\text{e} = 5 G^0_\text{N}/4$. Panel (c) shows the relaxation functions for a chemistry-specific coarse-grained model of cis-PB at $413$ K, where $G^0_\text{N} = 15.5\,\text{bar}$ corresponding to $N_\text{e} = 28$ monomers.
• Figure 4: (a) The $P(t)$ data for various chain lengths of the semi-flexible bead–spring model with $k_\theta = 1.5$ (symbols), shown together with the $\mu(t)$ in the Likhtman–McLeish model, \ref{['Eq:LM']} (solid lines), using $N_\text{e} = 21$ and $\tau_\text{e} = \tau_0 N_\text{e}^2 = 2.1 N_\text{e}^2$. The model inputs were determined independently and were not treated as free fitting parameters. (b) The same data presented in the format adopted by Likhtman and McLeish likhtman2002quantitative, where $\widehat{P}(t) = -4Z\tau_\text{e}^{1/4} t^{3/4} \partial P/\partial t$.
• Figure 5: The $G(t)$ curves for the polymer melts studied (symbols), shown together with the predictions of \ref{['Eq:Gt-LM']} (dashed lines). Panels (a), (b), and (c) show the results for the bead-spring models with $k_{\theta} = 1.5$, $k_{\theta} = 2.5$, and $k_{\theta} = -1$, respectively. Panel (d) shows the results for cis-PB at 413 K. \ref{['Eq:Gt-LM']} uses $N_\text{e}$ and $\tau_\text{0}$ as explicit inputs. The values of $N_\text{e}$ are taken from the double reptation analysis reported in \ref{['Tab:Ne']}, and the monomeric times $\tau_0$ are obtained from the Rouse regimes of the $G(t)$ curves, as reported in \ref{['Tab:tau_0']}.