Cyclic Motion of a Grafted Polymer under Shear Flow

The long-time dynamics of a single end-tethered chain under shear flow are studied using molecular and Brownian dynamics simulations of a flexible polymer. As observed in previous experiments with tethered DNA [Phys. Rev. Lett. 84, 4769 (2000)], under a flow sheared at constant rate $\dot{\gamma }$ the chain performs a cyclic motion. But, contrary to what has been previously suggested, a well-defined characteristic period exists and it is clearly revealed in the cross spectra of the chain extension along flow and gradient directions. The main cycling time scales like the time needed to stretch the polymer by convection, being about 10 times the relaxation time of the chain in flow. This coherent recursive motion introduces long memory in the fluid and suggests resonance effects under periodic external forcing.

Figure 1

Tethered polymer under shear flow from a MD simulation with $\mathrm{Wi}=2$. Each figure shows 25 successive superimposed snapshots of the chain, covering a time interval of $[t,t+9.375]$. The sequence (a)–(e) illustrates the “cyclic dynamics” observed at time scales longer than its relaxation time, which for this case is $1250\tau$. Time is given in LJ units ($\tau$).

Figure 2

(a) The normalized autocorrelation of the chain extension along flow direction $X$. (b) The cross correlation between $X$ and the chain extension in gradient direction $Y$. Results are obtained from MD simulations of one chain at $\mathrm{Wi}=2$ and during a sampling time of ${10}^5\tau$; error bars come from the standard deviation of $\delta X(t)\delta Y(0)$.

Figure 3

CPSD of the chain extension in flow and gradient directions, $X$ and $Y$. Frequencies are scaled with the relaxation time in flow ${\tau }_0(\mathrm{Wi})$ (values indicated in the figure) and the CPSD are normalized with the maximum value (some curves has been displaced for clarity). Diamonds corresponds to a chain with $N=30$ beads, while the rest to $N=60$.

Figure 4

Reciprocal of the peak frequency ${\omega }_{\mathrm{peak}}^{-1}$ of the cross power spectral density (circles) and the chain longest relaxation time in flow ${\tau }_0(\mathrm{Wi})$ (triangles) versus the Weissenberg number. Times are scaled with ${\tau }_0(0)=2000\tau$ (MD results) or ${\tau }_0(0)=4000\tau$ (BD results). Solid lines (best fits to BD results) are $0.9{\mathrm{Wi}}^{-0.78}$ and $12{\mathrm{Wi}}^{-0.78}$ and dashed lines $1.6{\mathrm{Wi}}^{0.70}$ and $19{\mathrm{Wi}}^{0.70}$.