Dynamics of a semiflexible polymer or polymer ring in shear flow

Polymers exposed to shear flow exhibit a remarkably rich tumbling dynamics. While rigid rods rotate on Jeffery orbits, a flexible polymer stretches and coils up during tumbling. Theoretical results show that in both of these asymptotic regimes the corresponding tumbling frequency $f_c$ in a linear shear flow of strength $\gamma$ scales as a power law ${\mathrm{Wi}}^{2/3}$ in the Weissenberg number $\mathrm{Wi}=\gamma \tau$, where $\tau$ is a characteristic time of the polymer's relaxational dynamics. For a flexible polymer these theoretical results are well confirmed by a large body of experimental single molecule studies. However, for the intermediate semiflexible regime, especially relevant for cytoskeletal filaments like F-actin and microtubules, the situation is less clear. While recent experiments on single F-actin filaments are still interpreted within the classical ${\mathrm{Wi}}^{2/3}$ scaling law, theoretical results indicated deviations from it. Here we perform extensive Brownian dynamics simulations to explore the tumbling dynamics of semiflexible polymers over a broad range of shear strength and the polymer's persistence length $l_p$. We find that the Weissenberg number alone does not suffice to fully characterize the tumbling dynamics, and the classical scaling law breaks down. Instead, both the polymer's stiffness and the shear rate are relevant control parameters. Based on our Brownian dynamics simulations we postulate that in the parameter range most relevant for cytoskeletal filaments there is a distinct scaling behavior with $f_c{\tau }^{*}={\mathrm{Wi}}^{3/4}{\widehat{f}}_c\left(x\right)$ with $\mathrm{Wi}=\gamma {\tau }^{*}$ and the scaling variable $x=(l_p/L){\left(\mathrm{Wi}\right)}^{-1/3}$; here ${\tau }^{*}$ is the time the polymer's center of mass requires to diffuse its own contour length $L$. Comparing these results with experimental data on F-actin we find that the ${\mathrm{Wi}}^{3/4}$ scaling law agrees quantitatively significantly better with the data than the classical ${\mathrm{Wi}}^{2/3}$ law. Finally, we extend our results to single ring polymers in shear flow, and find similar results as for linear polymers with slightly different power laws.

Figure 1

Time evolution of the mean-square displacement of the diameter of cyclic polymers, $\delta {\mathbf{D}}^2$, for different polymer stiffness $l_p/L$ indicated in the graph. For better visibility, the curves with $l_p/L=5$ and $l_p/L=2$ are shifted by a factor of 10 and 100, respectively. Symbols give the numerical data; standard error is below symbol size. Solid lines give the corresponding analytic prediction as obtained from Eqs. (11) and (13).

Figure 2

Mean-square displacement of the ring diameter, $\delta {\mathbf{D}}^2$, as a function of time for two ring polymers with stiffness $l_p/L=5$ and $l_p/L=50$, respectively. As predicted from the calculations, the stiffness does not affect the rotational relaxation time nor the saturation value. For comparison the mean-square displacement of the end-to-end vector of a linear polymer is shown, and we adjusted the contour length of the linear chain $L_l$ to be identical to the diameter of the ring polymer, i.e., $L_l=L/\pi$. For this linear polymer $l_p/L_l=5$. Statistical errors of the Brownian dynamics simulations are of the order of the symbol size.

Figure 3

Schematic representation of the flow geometry for a linear polymer in linear shear flow $\mathbf{v}=\gamma y{\widehat{\mathbf{e}}}_x$ [dashed-dotted red (dark gray) arrows]. The inclination of the polymer's end-to-end distance $\mathbf{R}$ [dotted yellow (light gray) line] with respect to the neutral plane ($xz$ plane) is measured in terms of the angle $\varphi$ between the projection of $\mathbf{R}$ onto the shear plane ($xy$ plane) (dashed line) and the $x$ axis.

Figure 4

Typical time evolution of the angle $\varphi$ for a linear polymer with $l_p/L=2$ in shear flow with $\gamma {\tau }^{*}=1.35\times {10}^4$.

Figure 5

Typical power spectra $E\left(f\right)$ for the angle $\varphi$ at a shear rate $\gamma {\tau }^{*}=5\times {10}^6$ and for two different values of the polymer stiffness as indicated in the graph. The dashed lines show Gaussian fits to the central region of the peak at the tumbling frequency $f_c{\tau }^{*}$.

Figure 6

Scaling plot showing $f_c{\tau }^{*}/{\left(\gamma {\tau }^{*}\right)}^{2/3}$ as a function of $l_p/L$ for a set of values for the shear rate $\gamma {\tau }^{*}$ as indicated in the graph. The dashed line in the stiff limit is the tumbling frequency of a rigid rod derived from Jeffery's equation [49, 50]. This Jeffery plateau is reached the later the stronger the shear flow.

Figure 7

Typical examples for the different types of tumbling, depending on which type of Euler buckling occurs. For all examples shown here $\gamma {\tau }^{*}=8\times {10}^4$. (a) For a flexible polymer with $l_p/L=0.025$ the polymer coils up during the tumbling. (b) A polymer with an intermediate stiffness of $l_p/L=2$ exhibits localized bends (hairpin configurations) during tumbling. (c) Finally, for stiff polymers (here $l_p/L=80$) the polymer configurations are weakly curved over their full length.

Figure 8

Modified scaling plot for the tumbling frequency to achieve data collapse for the stiff to rigid regime $0.01{\left(\gamma {\tau }^{*}\right)}^{2/3}\lesssim l_p/L\lesssim 1{\left(\gamma {\tau }^{*}\right)}^{2/3}$. The rescaled tumbling frequency $f_c{\tau }^{*}{\left(\gamma {\tau }^{*}\right)}^{-2/3}$ is shown as a function of $(l_p/L){\left(\gamma {\tau }^{*}\right)}^{-2/3}$ for a series of shear rates $\gamma {\tau }^{*}$ indicated in the graph.

Figure 9

Modified scaling plot for the tumbling frequency to achieve data collapse for the intermediate stiffness regime $0.001{\left(\gamma {\tau }^{*}\right)}^{1/3}\lesssim l_p/L\lesssim 1{\left(\gamma {\tau }^{*}\right)}^{1/3}$. The rescaled tumbling frequency $f_c{\tau }^{*}{\left(\gamma {\tau }^{*}\right)}^{-3/4}$ is shown as a function of $(l_p/L){\left(\gamma {\tau }^{*}\right)}^{-1/3}$ for a series of shear rates $\gamma {\tau }^{*}$ indicated in the graph.

Figure 10

For different values of $l_p/L$ the dependence of the characteristic tumbling frequency on the shear rate $\gamma {\tau }^{*}$ is determined. In (a) we show $f_c{\tau }^{*}{\left(\gamma {\tau }^{*}\right)}^{-2/3}$ for linear polymer of weak to intermediate stiffness. With increasing stiffness a characteristic deviation of about $0.1$ occurs for small, but increasing values of $\gamma {\tau }^{*}$. For $l_p/L=2$ the deviation persists throughout the whole range of $\gamma {\tau }^{*}$ used here, which should also cover the experimentally accessible regime. In (b) we show directly the data for $f_c{\tau }^{*}$ for stiff polymers. The scaling exponent changes in the other direction than for the flexible to intermediate filament, i.e., the new scaling exponent occurs at high values of $\gamma {\tau }^{*}$, which increase with the stiffness of the polymer.

Figure 11

Comparison of our simulational results, exhibiting an increase of the frequency with ${\left(\gamma {\tau }^{*}\right)}^{3/4}$, to experimental data (squares in the plot) for F-actin [37]. The experimental data agree well with the Brownian dynamics data and follow a ${\left(\gamma {\tau }^{*}\right)}^{3/4}$ scaling law. The data as well as the simulations are not consistent with the classical $2/3$ scaling law (solid line).

Figure 12

Schematic representation of the flow geometry for a ring polymer [yellow (light gray)] in linear shear flow $\mathbf{v}=\gamma y{\widehat{\mathbf{e}}}_x$ [dashed-dotted red (dark gray) arrows]. The ring plane is characterized by its normal (dashed line). To fully characterize the ring's orientation relative to the shear we use the angle $\varphi$ between the normal and the $xz$ plane, and the angle $\theta$ between the normal and the $z$ axis.

Figure 13

(a) Typical time evolution of the angle $\varphi$ and $\theta$ for a ring with $l_p/L=2$ under a shear flow with $\gamma {\tau }^{*}=62500$. (b) Power spectrum of the same ring polymer as in (a) for the time trace of $\varphi$.

Figure 14

Rescaled tumbling frequency $f_c$ as a function of $l_p/L$ for a ring polymer. As shown in (a), rescaling only the dimensionless frequency with ${\left(\gamma {\tau }^{*}\right)}^{2/3}$ again results in overlapping curves in the flexible and stiff limit. In (b) we show that using an exponent of $0.72$ when rescaling the frequencies and shifting the rigidity by a factor ${\left(\gamma {\tau }^{*}\right)}^{0.5}$ gives good agreement from the flexible regime to the peak. For higher values of $\left(\gamma {\tau }^{*}\right)$ there seem to exist small deviations in the fall to the stiff regime.