^{1}, W. K. den Otter

^{1}and W. J. Briels

^{1,a)}

### Abstract

The collective periodic motions of liquid-crystalline polymers in a nematic phase in shear flow have, for the first time, been simulated at the particle level by Brownian dynamics simulations. A wide range of parameter space has been scanned by varying the aspect ratio between 10 and 60 at three different scaled volume fractions and an extensive series of shear rates. The influence of the start configuration of the box on the final motion has also been studied. Depending on these parameters, the motion of the director is either characterized as tumbling, kayaking, log-rolling, wagging, or flow-aligning. The periods of kayaking and wagging motions are given by for high aspect ratios. Our simulation results are in agreement with theoretical predictions and recent shear experiments on viruses in solution. These calculations of elongated rigid rods have become feasible with a newly developed event-driven Brownian dynamics algorithm.

This work is part of the SoftLink research program of the Stichting voor Fundamenteel Onderzoek der Materie (FOM), which is financially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO). We thank Professor Jan K. G. Dhont for many useful discussions.

I. INTRODUCTION

II. SIMULATION METHOD

III. RESULTS AT LOW AND INTERMEDIATE SHEAR RATES

A. Tumbling and kayaking

B. Shear rate and volume fraction dependence of the tumbling and kayaking period

C. Finite size analysis

D. Comparison with theoretical results

E. Comparison with experiments

IV. RESULTS AT HIGH SHEAR RATE

A. Kayaking to wagging transition

B. The flow-aligned state

V. INFLUENCE OF THE INITIAL DIRECTOR

VI. CONCLUSION

### Key Topics

- Shear flows
- 15.0
- Polymer liquid crystals
- 10.0
- Viruses
- 10.0
- Vortex dynamics
- 10.0
- Brownian dynamics
- 8.0

## Figures

Snapshots of the normalized end-to-end vectors of rigid rods in shear flow, plotted as dots on the surface of a unit sphere. The flow direction, which is also the initial orientation of every rod, runs from the left pole to the right pole. Drawn lines represent meridians and parallels. A tumbling/kayaking motion is represented by the snapshots A to F, which are not evenly spaced in time. The simulation box contains 1750 rods with aspect ratio , at a scaled volume fraction of 3.5 and an applied shear rate of . Note that gradually a small secondary cluster develops, diametrically opposed to the main cluster, of rods that have individually flipped from parallel to antiparallel to the director.

Snapshots of the normalized end-to-end vectors of rigid rods in shear flow, plotted as dots on the surface of a unit sphere. The flow direction, which is also the initial orientation of every rod, runs from the left pole to the right pole. Drawn lines represent meridians and parallels. A tumbling/kayaking motion is represented by the snapshots A to F, which are not evenly spaced in time. The simulation box contains 1750 rods with aspect ratio , at a scaled volume fraction of 3.5 and an applied shear rate of . Note that gradually a small secondary cluster develops, diametrically opposed to the main cluster, of rods that have individually flipped from parallel to antiparallel to the director.

The scalar order parameter (thick solid line) as well as the three components (solid), (dashed), and (dotted) of the director as functions of time. The component reveals how the motion of an initially flow-aligned system evolves from in-plane tumbling to out-of-plane kayaking.

The scalar order parameter (thick solid line) as well as the three components (solid), (dashed), and (dotted) of the director as functions of time. The component reveals how the motion of an initially flow-aligned system evolves from in-plane tumbling to out-of-plane kayaking.

The path traced out by the director of Fig. 2 on the surface of a unit sphere as seen along the vorticity (a) and flow (b) direction. The in-plane tumbling motion, characterized by a circle in (a) and a vertical line in (b), turns into an out-of-plane kayaking motion, illustrated by an ellipse in (a) and a >-shaped path in (b).

The path traced out by the director of Fig. 2 on the surface of a unit sphere as seen along the vorticity (a) and flow (b) direction. The in-plane tumbling motion, characterized by a circle in (a) and a vertical line in (b), turns into an out-of-plane kayaking motion, illustrated by an ellipse in (a) and a >-shaped path in (b).

The scalar order parameter (solid lines) and the (dashed) and (dotted) components of the director as functions of elapsed strain. From left to right the scaled volume fractions are 3.5, 4.0, and 4.5 at fixed , while from bottom to top the shear rate doubles.

The scalar order parameter (solid lines) and the (dashed) and (dotted) components of the director as functions of elapsed strain. From left to right the scaled volume fractions are 3.5, 4.0, and 4.5 at fixed , while from bottom to top the shear rate doubles.

The dimensionless kayaking periods for as functions of shear rate. Horizontal lines are averages over the simulations corresponding to , 4.0, and 4.5, respectively.

The dimensionless kayaking periods for as functions of shear rate. Horizontal lines are averages over the simulations corresponding to , 4.0, and 4.5, respectively.

The dependence of the kayaking period on the aspect ratio, the scaled volume fraction, and the applied shear rate. Solid, gray, and open symbols represent , 4.0, and 4.5, respectively.

The dependence of the kayaking period on the aspect ratio, the scaled volume fraction, and the applied shear rate. Solid, gray, and open symbols represent , 4.0, and 4.5, respectively.

The dimensionless kayaking period as a function of , for various aspect ratios in a range from 10 to 60. The simulation results are seen to converge with increasing to the master curve of Eq. (4), shown as a solid line.

The dimensionless kayaking period as a function of , for various aspect ratios in a range from 10 to 60. The simulation results are seen to converge with increasing to the master curve of Eq. (4), shown as a solid line.

The correlation between rods’ end-to-end vector at time 0 and at time , in cubic simulation boxes with volumes twice (b), four times (c), and eight times (d) as large as the standard box size (a), respectively. In all four systems, , , and . For comparison reasons, the origins of time were shifted to make the first occasions of coincide (dashed line).

The correlation between rods’ end-to-end vector at time 0 and at time , in cubic simulation boxes with volumes twice (b), four times (c), and eight times (d) as large as the standard box size (a), respectively. In all four systems, , , and . For comparison reasons, the origins of time were shifted to make the first occasions of coincide (dashed line).

The dimensionless kayaking period as a function of the scalar order parameter . Solid squares denote the direct measurements in this work. Open circles represent calculated periods using the theory of Larson (Ref. 12), Eqs. (8)–(11), based on and from equilibrium simulations. The dashed curve shows a calculation of on the basis of alone, proposed by Hinch and Leal (Ref. 53). Experimental measurements by Lettinga *et al.* (Ref. 38) are expressed as stars.

The dimensionless kayaking period as a function of the scalar order parameter . Solid squares denote the direct measurements in this work. Open circles represent calculated periods using the theory of Larson (Ref. 12), Eqs. (8)–(11), based on and from equilibrium simulations. The dashed curve shows a calculation of on the basis of alone, proposed by Hinch and Leal (Ref. 53). Experimental measurements by Lettinga *et al.* (Ref. 38) are expressed as stars.

Comparison of the experimental kayaking periods of the virus (Ref. 38) (stars, ) and simulation results (open symbols, ).

Comparison of the experimental kayaking periods of the virus (Ref. 38) (stars, ) and simulation results (open symbols, ).

The paths traced out by the directors on unit spheres (viewed along the vorticity direction) at shear rates of 75 (a), 250 (b), and (c). and .

The paths traced out by the directors on unit spheres (viewed along the vorticity direction) at shear rates of 75 (a), 250 (b), and (c). and .

The critical rotational Peclet number of the kayaking to wagging transition as a function of aspect ratio for three scaled volume fractions. The experimental result for the virus (Ref. 38), with , is shown as a star. See the main text for a discussion of this measurement.

The critical rotational Peclet number of the kayaking to wagging transition as a function of aspect ratio for three scaled volume fractions. The experimental result for the virus (Ref. 38), with , is shown as a star. See the main text for a discussion of this measurement.

The scalar order parameter and the three components of as a function of strain for wagging (left) and flow-aligning (right) motions. The paths of these directors are depicted by Figs. 11(b) and 11(c).

The scalar order parameter and the three components of as a function of strain for wagging (left) and flow-aligning (right) motions. The paths of these directors are depicted by Figs. 11(b) and 11(c).

The shear-dependent average angle between the arrested director and the flow direction in the flow-aligned system, as a function of .

The shear-dependent average angle between the arrested director and the flow direction in the flow-aligned system, as a function of .

The maximum averaged flow-aligned angle as a function of , for three scaled volume fractions.

The maximum averaged flow-aligned angle as a function of , for three scaled volume fractions.

The scalar order parameter (thick solid line) as well as the three components (solid), (dashed), and (dotted) of the director as functions of time. Nematic simulation boxes initially aligned along the flow (a) and gradient (b) direction, as well as an initially isotropic (c) box, quickly evolve a kayaking motion. The nematic box aligned along the vorticity direction (d), however, shows a log-rolling motion. In all plots, , , and .

The scalar order parameter (thick solid line) as well as the three components (solid), (dashed), and (dotted) of the director as functions of time. Nematic simulation boxes initially aligned along the flow (a) and gradient (b) direction, as well as an initially isotropic (c) box, quickly evolve a kayaking motion. The nematic box aligned along the vorticity direction (d), however, shows a log-rolling motion. In all plots, , , and .

At a low shear rate of , a system initially aligned along the vorticity direction shows a log-rolling motion (a), which is stable against occasional fluctuations. At an intermediate shear rate of , the system performs a log-rolling motion at the beginning, which gives way to a wagging (b) motion, while the same system immediately progresses to a flow-aligning (c) state at the high shear rate of . , and .

At a low shear rate of , a system initially aligned along the vorticity direction shows a log-rolling motion (a), which is stable against occasional fluctuations. At an intermediate shear rate of , the system performs a log-rolling motion at the beginning, which gives way to a wagging (b) motion, while the same system immediately progresses to a flow-aligning (c) state at the high shear rate of . , and .

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