Hydrodynamic rotlet dipole driven by spinning chiral liquid crystal droplets

Chirality is an essential evolutionary-conserved physical aspect of swimming microorganisms. However, the role of chirality on the hydrodynamics of such microswimmers is still being elucidated. Hydrodynamic theories have so far predicted that, under a torque-free condition satisfied in the system of microswimmers, a rotlet dipole generating a twisting flow is the leading-order singularity of the chiral flow field. Nevertheless, such a chiral flow field has never been experimentally detected. Here we explore a hydrodynamic field generated in a system of a chiral microswimmer, where a droplet of a cholesteric liquid crystal (CLC) exhibits helical and spinning motions in surfactant solutions due to a chiral nonequilibrium cross coupling between the rotation and the Marangoni flow. Combining measurement of the flow field around the spinning CLC droplets and a computational flow modeling, we revealed that the CLC droplets generate a flow field of a rotlet dipole. Remarkably, we found that the chiral component of the flow field decays with distance $r$ as $r^{-3}$, which is consistent with the theoretical prediction for the flow field produced by a point singularity of a rotlet dipole. Our findings will promote the understanding of roles of chirality on the hydrodynamics in active matter as well as liquid crystals.

Figure 1

Rotating behavior of a CLC droplet and the experimental setup. (a) A rotating spiral texture of a CLC droplet. A curved arrow indicates the rotational direction (CW). The scale bar is $10\mu \mathrm{m}$. (b) The rotating behavior of a CLC droplet in a surfactant solution was observed after the droplet sank to the bottom of the chamber. (c) A schematic of a cross-sectional view of the director field inside a CLC droplet with a spiral texture. A solid line represents a disclination line. (d) Confocal observation of a cross-sectional view of a CLC droplet with a spiral texture. The top and bottom figures show the cross-sectional images in the $xy$ plane (transmitted light image) and $xz$ plane (fluorescent confocal image). The dashed line in the $xy$ plane indicates the position of the $xz$ plane. The position of the cover glass detected by a fluorescent bead attached to the surface is also shown. The scale bar is $10\mu \mathrm{m}$.

Figure 2

Angular velocity of rotating CLC droplets, and texture of CLC droplets with opposite handedness and a NLC droplet. (a) Angular velocity $\omega$ of the rotating spiral texture of the CLC droplets ($D\approx 16\mu \mathrm{m}$) vs density difference $\mathrm{\Delta }\rho$ between the surfactant solution and the CLC. The blue solid line is a linear fit to the data. (b) The texture of the CLC droplets doped with R811 when $\mathrm{\Delta }\rho <0$. The droplet floats at the top glass of the chamber. The curved arrows indicate the rotational direction. (c) The texture of the CLC droplets doped with chiral dopants which have opposite chirality [(c-1) R811 and (c-2) S811], respectively. (d) A nonrotating texture of a NLC droplet observed under crossed polarizers. The direction of the crossed polarizers are shown as a pair of the double-headed arrows. A cross-sectional view of the radial director field inside the NLC droplet is also schematically drawn. The scale bars in panels (b), (c), and (d) are $10\mu \mathrm{m}$.

Figure 3

Flow fields around a NLC and a CLC droplets at each height. (a)(b)The white arrows indicate the velocity vector in the $xy$ plane. The color represents the amplitudes of $v_{\parallel }$ and $v_{\perp }$. The definition of $v_{\parallel }$ and $v_{\perp }$ is also schematically drawn in panel (a). The dark circular region at the center of each panel is the area masked to avoid the texture of the LC droplets in the PIV analysis (see Appendix pp3). The scale bars are $20\mu \mathrm{m}$.

Figure 4

The radial profiles of $v_{\parallel }$ and $v_{\perp }$ obtained from the experiments with (a) the NLC and (b, c) the CLC droplets. (a, b) The squares with the error bars indicate the experimental data. The solid curves are the numerical results. The different colors correspond to the different heights of the $xy$ plane cross sections (blue: $z=6\mu \mathrm{m}$, red: $z=12\mu \mathrm{m}$, yellow: $z=18\mu \mathrm{m}$, purple: $z=24\mu \mathrm{m}$, green: $z=30\mu \mathrm{m}$). The dotted lines indicate the radius of the droplets. (c) The log-log plots of $|v_{\parallel }|$ and $|v_{\perp }|$ are shown for the CLC droplets. A solid line in the plot for $|v_{\perp }|$ indicates a power law with an exponent $-3$.

Figure 5

3D streamlines around (a) the NLC and (b) the CLC droplets. The streamlines were drawn for the velocity fields calculated in the numerical simulation applied in Fig. 4. The yellow spheres represent the LC droplets. The black arrows indicate the 3D velocity fields, and the size is proportional to the absolute value of the velocity field. The color of the streamlines indicates the angular component $v_{\perp }$ of the velocity field. The color map is also shown in each figure.

Figure 6

Rotational behavior of the CLC droplets with different size and chirality. (a) A phase diagram of the types of the CLC droplets with the different sizes and concentration of R811. The different symbols correspond to the different types of the CLC droplets (blue circle: spiral droplet, red diamond: disclination droplet, brown upward triangle: concentric droplet, orange square: winding droplet, green downward triangle: others). The dotted curves are guides for eyes. (b) A rotating texture and the schematic cross-sectional view of the director field of a disclination droplet are shown. Arrows show the position of the disclination line. A curved arrow indicates the rotational direction. The scale bar is $10\mu \mathrm{m}$. (c) A texture of a concentric droplet observed is shown. The right panel is an image obtained using cross polarizers. The direction of the polarizers is shown as the double-headed arrows. The scale bar is $10\mu \mathrm{m}$. (d) A time evolution of a texture of a winding droplet is shown. The scale bar is $20\mu \mathrm{m}$.