Stress Relaxation in Entangled Polymer Melts

We present an extensive set of simulation results for the stress relaxation in equilibrium and step-strained bead-spring polymer melts. The data allow us to explore the chain dynamics and the shear relaxation modulus, $G(t)$, into the plateau regime for chains with $Z=40$ entanglements and into the terminal relaxation regime for $Z=10$. Using the known (Rouse) mobility of unentangled chains and the melt entanglement length determined via the primitive path analysis of the microscopic topological state of our systems, we have performed parameter-free tests of several different tube models. We find excellent agreement for the Likhtman-McLeish theory using the double reptation approximation for constraint release, if we remove the contribution of high-frequency modes to contour length fluctuations of the primitive chain.

Figure 1

(a) Normal tensions, $\sigma (\lambda ,t)$ for step-strained melts $N=50$ (black), 100 (blue), 175 (green), 350 (red), 700 (cyan), 1000 (violet), and 3500 (orange) and elongation $\lambda =2$ (★), 3 (◇), and 4 (△). (b)–(d) Green-Kubo shear relaxation moduli, $G(t)$, (this work: large solid ○ with solid line; Ref. 23: no symbol, solid line) compared to extrapolations, $G(\lambda ,t)=\sigma (\lambda ,t)/h(\lambda ,t)$, from the nonlinear response using the (b) classical, (c) Doi-Edwards, (d) slip-tube damping function [same symbols as in (a)]. Colored ticks indicate the Rouse time of the corresponding systems.

Figure 2

Comparison of the measured relaxation moduli to the predictions of various theories. Symbols and colors for the simulation data as in Fig. 1 with Green-Kubo data shown as large solid ○ and □. Theoretical predictions are shown as thick lines using the same color code. Thin lines indicate the uncertainty in the theoretical predictions due to the uncertainty of the PPA entanglement length $N_e=85\pm 7$.

Figure 3

Comparison of the measured relaxation moduli to Eq. (1) using (a),(b) the independently measured autocorrelation function of the primitive chain end-to-end vectors to estimate of the tube memory function, $\mu (t)$; (c),(d) our proposition Eq. (2) for removing high-frequency modes from the Likhtman-McLeish theory [27] of contour length fluctuations with $\alpha =1.7$. Symbols and colors as in Fig. 2.