Power Law Stretching of Associating Polymers in Steady-State Extensional Flow

We present a tube model for the Brownian dynamics of associating polymers in extensional flow. In linear response, the model confirms the analytical predictions for the sticky diffusivity by Leibler-Rubinstein-Colby theory. Although a single-mode Doi-Edwards-Marrucci-Grizzuti approximation accurately describes the transient stretching of the polymers above a “sticky” Weissenberg number (product of the strain rate with the sticky-Rouse time), the preaveraged model fails to capture a remarkable development of a power law distribution of stretch in steady-state extensional flow: while the mean stretch is finite, the fluctuations in stretch may diverge. We present an analytical model that shows how strong stochastic forcing drives the long tail of the distribution, gives rise to rare events of reaching a threshold stretch, and constitutes a framework within which nucleation rates of flow-induced crystallization may be understood in systems of associating polymers under flow. The model also exemplifies a wide class of driven systems possessing strong, and scaling, fluctuations.

Figure 1

Comparison between the stretch ratio $\lambda$ of a sticky polymer ($Z_e=Z_s=10$, ${\tau }_s={10}^4{\tau }_e$, $p=0.95$, $Z_s=10$) against time $t$ in units of the sticky Rouse time ${\tau }_{\mathrm{SR}}$ at a range of flow rates from $\dot{ϵ}=0.056{\tau }_{\mathrm{SR}}^{-1}$ to $22.3{\tau }_{\mathrm{SR}}^{-1}$ in logarithmic steps. The sticky Rouse time is ${\tau }_{\mathrm{SR}}=[D_R/D_{\mathrm{SR}}]{\tau }_R$ with $D_R$ the bare Rouse diffusivity, ${\tau }_R={\tau }_e{Z}_e^2$ the bare Rouse time, and $D_{\mathrm{SR}}$ the sticky diffusivity (see inset). In the main panel, the symbols are obtained by averaging over five Brownian dynamics simulations with different random number seeds; the lines represent the single-mode model in Eq. (5). The inset shows consistence of the simulated sticky-Rouse diffusivity (symbols; averaged over 25 random number seeds) with the sticky-reptation model (lines) of Leibler et al. [19].

Figure 2

The steady-state probability distribution $P(\lambda )$ is plotted against the stretch ratio $\lambda$. The symbols are obtained from the steady-state simulations of Fig. 1 at the flow rates ($\dot{ϵ}{\tau }_{\mathrm{SR}}=0.446$, 0.668, and 0.780); the curves are Gaussian fits. For an increasing flow rate, the high-stretch tail is no longer Gaussian but becomes a power law $P(\lambda )\propto {\lambda }^{-\nu }$. The inset shows the stretch ratio against time for $\dot{ϵ}{\tau }_{\mathrm{SR}}=0.780$ and visualizes how this distribution includes “rare events” of enormous chain stretch. For a sufficiently large flow rate, $\nu$ decreases. If $\nu >2$, the mean value of $\lambda$ is finite (as it should in steady state); however, if also $\nu \le 3$, the fluctuations in stretch, characterized by the expectation value of ${\lambda }^2$, diverge.

Figure 3

The power law stretch distribution, $P(\lambda )\propto {\lambda }^{-\nu }$ for large $\lambda$, observed in Fig. 2 is replicated analytically in a sticky dumbbell model for a sticky polymer ($Z_e=10$, $p=0.9$, ${\tau }_s=1000{\tau }_e$), which has two stickers near the end of the chain that are simultaneously either open or closed (lines). The dashed curve is the Gaussian stretch distribution under quiescent conditions. In linear steps, the flow rate is increased up to $\dot{ϵ}{\tau }_R=0.05$. The symbols are obtained in simulations with 2, 6, 12, and 36 beads (from red to light blue). For small flow rates, where $\nu <3$, the simulated power law tails of $P(\lambda )$ (symbols) are in agreement with Eq. (8). The inset shows the transient behavior of the simulation with $\dot{ϵ}{\tau }_R=0.05$.

Figure 4

State diagram of a sticky dumbbell. For a short sticker lifetime, polymer stretching takes place if the Weissenberg number $(1-p)\dot{ϵ}{\tau }_R$ is larger than unity. $p$ is the time-averaged fraction of closed stickers and ${\tau }_R$ is the bare Rouse time. For a finite sticker lifetime, the mean and the variance of the stress $\sigma$ and the stretch $\lambda$ diverge in different regimes. The curves are given by Eq. (8) for $\nu =2$, 3, 4, 6 as discussed in the main text.