Kayaking and Wagging of Rods in Shear Flow

For the first time, we have simulated the periodic collective orientational motions performed by rigid liquid-crystalline polymers with large aspect ratio in the nematic state in shear flow. In order to be able to do so, we developed a new, event-driven Brownian dynamics technique. We present the results of simulations of rods with aspect ratios $L/d$ ranging from 20 to 60 at volume fractions $\phi$ given by $L\phi /d=3.5$ and 4.5. By studying the path of the director, i.e., the average direction of the rods, we observe kayaking, wagging, flow aligning, and log-rolling type of orbits, depending on the parameters of the simulation and the initial orientation. We find that the tumbling periods depend on $L\phi /d$ and the shear rate but not on the type of motion. Our simulation results qualitatively confirm theoretical predictions and are in good agreement with the experimental measurements of tumbling times of fd viruses.

Figure 1

Snapshots of a simulation box with rods collectively changing orientation by 180°. The four snapshots are taken at the times indicated as (a), (b), (c), and (d), respectively, in the upper part of Fig. 2. The box contains 1750 rods of which only 500 are shown in each frame, so the density is 3.5 times as high as it appears from the pictures.

Figure 2

Order parameter $P_2$ (thick solid line) and director components ${\widehat{n}}_x$ (solid line), ${\widehat{n}}_y$ (dashed line), and ${\widehat{n}}_z$ (dotted line) as a function of time. The upper and the lower figures illustrate the kayaking and wagging motions, respectively. In this figure are also indicated the times at which the snapshots in the upper part of Fig. 1 were taken.

Figure 3

The path on the unit sphere traced out by the director of a kayaking (a) and wagging (b) box of rodlike particles with $L/d=25$ and $L\phi /d=3.5$. The arrows attached to the $y$ axis represent the flow field.

Figure 4

Simulated and experimental tumbling times $T$ as a function of the shear rate $\dot{\gamma }$ at a volume fraction $\phi$ determined by $L\phi /d=4.5$. Experiments were done [24] on the fd virus for which $L/d=60$ and $d=0.0148\mu \mathrm{m}$ like in our simulations.