Scaling Behavior of Anisotropy Relaxation in Deformed Polymers

Drawing an analogy to the paradigm of quasielastic neutron scattering, we present a general approach for quantitatively investigating the spatiotemporal dependence of structural anisotropy relaxation in deformed polymers by using small-angle neutron scattering. Experiments and nonequilibrium molecular dynamics simulations on polymer melts over a wide range of molecular weights reveal that their conformational relaxation at relatively high momentum transfer $Q$ and short time can be described by a simple scaling law, with the relaxation rate proportional to $Q$. This peculiar scaling behavior, which cannot be derived from the classical Rouse and tube models, is indicative of a surprisingly weak direct influence of entanglement on the microscopic mechanism of single-chain anisotropy relaxation.

Figure 1

(a) Dynamic mechanical spectra of PS500K, PS100K, and PS30K at $130°\mathrm{C}$. (b)–(d): Relaxation of engineering stress ${\sigma }_{\mathrm{eng}}$ after a large step uniaxial extension to a stretching ratio $\lambda =1.8$. The initial Rouse Weissenberg numbers ${\mathrm{Wi}}_{R,i}=v{\tau }_R/l_0$ of the step deformation are 5.6, 5.0, and 40, for PS30K, PS100K, and PS500K, respectively.

Figure 2

Relaxation of the leading anisotropic expansion coefficient $S_2^0(Q)$ after a step deformation. (a) PS30K at 0, 0.1, 0.2, 0.4, 0.6, and $1{\tau }_R$. (b) PS100K at 0, 0.5, 1, 2, 3, 6, and $10{\tau }_R$. (c) PS500K at 0, 0.5, 1, 3, 10, and $20{\tau }_R$. (Solid lines) Guide to eye. The SANS spectra at the beginning and the end of the relaxation experiments are shown on the right, with different color schemes for each sample.

Figure 3

Anisotropic structural relaxation function ${\varphi }_2^0(Q;t)$ as a function of the normalized relaxation time $t/{\tau }_R$ (with ${\tau }_R$ being the Rouse relaxation time) at different $Q$’s. (a)–(c) Comparison of experiments and theories. (d)–(f) Comparison of simulations and theories. To put these results in perspective, the momentum transfer $Q$ is normalized by the equilibrium radius of gyration $R_g$ of the polymer. The initial Rouse Weissenberg numbers in the MD simulations are 5.56, 6.08, and 41.8 for $N=120$, $N=500$, and $N=2000$, respectively. The same rates are used in the corresponding theoretical calculations. The GLaMM calculations include the “local fluctuation” effect [39].

Figure 4

$\ln [{\varphi }_2^0(Q;t)]$ as a function of the scaling variable $(R_{g}Q{)}^{1/2}(t/{\tau }_R{)}^{1/2}$. (a) Scaling analysis for the experimental data. (Inset) Shows $\ln [{\varphi }_2^0(Q;t)]$ as a function of $(R_{g}Q{)}^{1/2}(t/\tau {)}^{1/2}$, where $\tau$ is the longest viscoelastic relaxation time of the sample. (b)–(d) Scaling analysis of the theoretical and simulation results.