Accurate Statistics of a Flexible Polymer Chain in Shear Flow

We present exact and analytically accurate results for the problem of a flexible polymer chain in shear flow. Under such a flow the polymer tumbles, and the probability distribution of the tumbling times $\tau$ of the polymer decays exponentially as $\sim \exp (-\alpha \tau /{\tau }_0)$ (where ${\tau }_0$ is the longest relaxation time). We show that for a Rouse chain this nontrivial constant $\alpha$ can be calculated in the limit of a large Weissenberg number (high shear rate) and is in excellent agreement with our simulation result of $\alpha \simeq 0.324$. We also derive exactly the distribution functions for the length and the orientational angles of the end-to-end vector $\mathit{R}$ of the polymer.

Figure 1

A polymer configuration with shear flow along the $x$ direction.

Figure 2

Linear-log plot of $P(T)$ versus $T$: the fitted $\alpha$’s for the four curves with $\mathrm{Wi}=0$, 2.0, 6.1, and 20.3 are 1.20, 0.57, 0.375, and 0.326, respectively. The two data sets with symbols ⊡ and ▴ in (c) are obtained by switching off the thermal noise along the $x$ direction (${\eta }_1=0$) in Eq. (1) and $\dot{\gamma }=0.2$ and 0.6 respectively—both fit well with the analytical $\alpha =0.324$ line. All data are for $N=10$. Inset: Typical $R_x(t)$ versus $t$ corresponding to the cases (a) both $\dot{\gamma }\ne 0$ and ${\eta }_1\ne 0$, (b) $\dot{\gamma }=0$ and ${\eta }_1\ne 0$, and (c) $\dot{\gamma }\ne 0$ and ${\eta }_1=0$.