Curling Liquid Crystal Microswimmers: A Cascade of Spontaneous Symmetry Breaking

We report curling self-propulsion in aqueous emulsions of common mesogenic compounds. Nematic liquid crystal droplets self-propel in a surfactant solution with concentrations above the critical micelle concentration while undergoing micellar solubilization [Herminghaus et al., Soft Matter 10, 7008 (2014)]. We analyzed trajectories both in a Hele-Shaw geometry and in a 3D setup at variable buoyancy. The coupling between the nematic director field and the convective flow inside the droplet leads to a second symmetry breaking which gives rise to curling motion in 2D. This is demonstrated through a reversible transition to nonhelical persistent swimming by heating to the isotropic phase. Furthermore, autochemotaxis can spontaneously break the inversion symmetry, leading to helical trajectories in 3D.

Figure 1

(a) Example trajectories of $50\text{−}\mu \mathrm{m}$ droplets at $c_{\mathrm{TTAB}}=7.5\mathrm{wt}\%$ over the course of 90 s at $34°\mathrm{C}$ (nematic, solid line) and $37°\mathrm{C}$ (isotropic, dashed). Above $T_{\mathrm{NI}}$, the curling is suppressed. Inset: trajectories of 100-s duration with a temperature ramp across the nematic-to-isotropic transition. The black arrows emphasize the transition. (b) Comparison of angular autocorrelation $C(\tau )$ in the rescaled time frame $\tau$. The isotropic $C(\tau )$ fits $\exp (-\tau /{\tau }_d)$ with a decay time ${\tau }_d=22.3$ (dash-dotted, black). Inset: Comparison of the mean-squared displacement (MSD) $⟨\mathrm{\Delta }{r}^2(\tau )⟩$.

Figure 2

(a) MSD and (b) angular autocorrelation $C(\tau )$ for experiments in the nematic state with $c_{\mathrm{TTAB}}=7.5\mathrm{wt}\%$ at $22°\mathrm{C}$ (solid), $30°\mathrm{C}$ (dashed), and $34°\mathrm{C}$ (dotted) in the rescaled time frame $\tau$. We see data collapse by rescaling for MSD and $C(\tau )$ up to one rotation period $T^{\prime }$.

Figure 3

Self-propelling nematic droplet. (a) Schematic flow field (arrows) inside and outside of the droplet and distorted nematic director field (black lines). The Marangoni flow (arrows on the interface) induced by the inhomogeneous surfactant coverage leads to internal convection and self-propelled motion. (b) A 5CB droplet ($D\approx 50\mu \mathrm{m}$), moving from left to right in a straight capillary, imaged between crossed polarizers with an added $\lambda$ retardation plate. Schematic arrows denote polarizer and $\lambda$ plate orientations.

Figure 4

(a) Curling trajectory of a $50\text{−}\mu \mathrm{m}$ droplet with defect vectors (arrows) and swimmer trajectory (solid line). (b) Averaged phase shift ${\theta }_s$ between velocity and defect vs droplet speed for two droplet sizes. (c) Tangential velocity $\approx 4\mu \mathrm{m}$ away from the droplet interface. The flow is increased around the defect location.

Figure 5

(a) Light sheet microscope: Fluorescently dyed droplets are excited by a wide laser sheet scanning the sample in $z$. A video microscope focused on the sheet records time and $xy$ positions. (b) Reconstructed helical 3D trajectory of a buoyant $50\text{−}\mu \mathrm{m}$ swimmer at $c_{\mathrm{TTAB}}=7.5\mathrm{wt}\%$. (c) Azimuthal angle $\varphi$, height $h$ along axis, and curvature $r$ vs time for the same trajectory.

Figure 6

(left) Trajectories of 11 droplets in a buoyancy-matched aqueous TTAB solution of 5 wt%, at room temperature, over a time of 5 min, with dots marking the initial position. The trajectories show no global bias of orientation or chirality (5 right handed, 6 left handed). (right) Conceptual sketch of trail avoidance strategies after the initial undisturbed orbit (1). In 2D (2), the resulting autochemotactic force (black arrows) is in plane and leads to a trajectory crossing; in 3D (3), it is parallel or antiparallel to the orbit normal.