Characteristic Periodic Motion of Polymers in Shear Flow

The motion of both free and tethered polymer molecules as well as rigid Brownian rods in unbound shear flow is found to be characterized by a clear periodicity or tumbling frequency. Periodicity is shown using a combination of single molecule DNA experiments and computer simulations. In all cases, we develop scaling laws for this behavior and demonstrate that the frequency of characteristic periodic motion scales sublinearly with flow rate.

Figure 1

Dynamics of DNA in steady, unbound shear flow. (a) Time sequence of images of fluorescent DNA ($L=80\mu \mathrm{m}$) at $\mathrm{Wi}=109$ in shear flow clearly illustrating end-over-end tumbling motion. Time between images is $\approx 10\mathrm{s}$. (b) Trajectories of fractional polymer extension $x/L$, gradient-direction polymer thickness ${\delta }_2$, and polymer orientation angle $\theta$ for lambda DNA at $\mathrm{Wi}=12$. Accumulated fluid strain ($\gamma$) is defined as observation time $t_{\mathrm{obs}}$ nondimensionalized with shear rate ($\gamma =t_{\mathrm{obs}}\dot{\gamma }$). (c) Descriptive cycle of periodic polymer motion in shear flow.

Figure 2

Periodic dynamical behavior of polymers in steady shear flow. (a) Power spectral density (PSD) of polymer orientation angle for DNA in steady shear flow for experiment and simulation at $\mathrm{Wi}=49$. PSDs and frequencies are made dimensionless with polymer relaxation times, $\tau$. (b) Scaling of peak frequencies (${\nu }^{\mathrm{peak}}$) from PSDs of $\theta$ from experiment (□) and simulation (●). Tumbling frequencies (${\nu }^{\mathrm{tumb}}$) from experimental observations of $22\mu \mathrm{m}$ (▽) and $84\mu \mathrm{m}$ (◇) DNA motion agree well with PSD peaks. Also shown is frequency of negative to positive $\theta$ transitions reaching ${\theta }^{\mathrm{average}}$ as a function of $\mathrm{Wi}$ (▴).

Figure 3

Power spectral densities of orientation angle $\theta$ for free DNA in unbounded shear (●), tethered DNA in shear flow (△), and infinite aspect ratio Brownian rods (□) in steady shear flow. (Inset) Scaling of peak frequencies from PSDs of $\theta$ for tethered polymers and Brownian rods in shear flow.