Self-propelled motion switching in nematic liquid crystal droplets in aqueous surfactant solutions

The self-propelled motions of micron-sized nematic liquid crystal droplets in an aqueous surfactant solution have been studied by tracking individual droplets over long time periods. Switching between self-propelled modes is observed as the droplet size decreases at a nearly constant dissolution rate: from random to helical and then straight motion. The velocity of the droplet decreases with its size for straight and helical motions but is independent of size for random motion. The switching between helical and straight motions is found to be governed by the self-propelled velocity, and is confirmed by experiments at various surfactant concentrations. The helical motion appears along with a shifting of a point defect from the self-propelled direction of the droplet. The critical velocity for this shift of the defect position is found to be related with the Ericksen number, which is defined by the ratio of the viscous and elastic stresses. In a thin cell whose thickness is smaller than that of the initial droplet size, the droplets show more complex trajectories, including “figure-8s” and zigzags. The appearance of those characteristic motions is attributed to autochemotaxis of the droplet.

Figure 1

Experimental setup: (a) Schematic of the preparation steps for the sample cell. A droplet of 5CB of the appropriate size is injected into a surfactant solution using a microinjector. The thickness of the cell is controlled using the silicone rubber ring. The solution is enclosed with a cover slip after the droplets are injected. (b) A bright-field microscope image of a nematic droplet dispersed in an aqueous mixture solution of TTAB and ${\mathrm{D}}_2\mathrm{O}$ at 25 ${}^{\circ }\mathrm{C}$.

Figure 2

Change in self-propelled motion of a nematic liquid crystal droplet in the 3D system. (a) Change in diameter over time for two droplets with different initial sizes. Different symbols represent different modes of motion. (b) Characteristic trajectories of a droplet over 30 s. By decreasing the droplet size or waiting for time to elapse, the motion was observed to change random $\to$ helical $\to$ straight. The scale bars represent 50 $\mu \mathrm{m}$.

Figure 3

Mean square displacement (MSD) and angular autocorrelation function $C\left(t\right)$ for random, helical, straight, and Brownian-like motions. (a) MSD calculated from a single droplet trajectory. (b) $C\left(t\right)$ calculated from the instantaneous velocities.

Figure 4

Dependence of the averaged instantaneous 2D projected velocity on the diameter of the droplet in 10 wt% TTAB solution. Different symbols represent different modes of the self-propelled motion. (a) Nematic phase at 25 ${}^{\circ }\mathrm{C}$. (b) Isotropic phase at 40 ${}^{\circ }\mathrm{C}$.

Figure 5

Characteristic parameters of helical motion: (a) Dependence of rotational period $T$ on the diameter of the droplet. (b) Dependence of the pitch $p$ of the helical trajectory on the diameter of the droplet. (c) Dependence of radius $r$ of the helical trajectory on the velocity of the droplet. The extrapolated best-fit line to the data crosses the zero-radius axis at 19.4 $\mu \mathrm{m}/\mathrm{s}$. The definitions of $p$ and $r$ are shown in the inset of (b).

Figure 6

Dependence of the self-propelled velocity on the diameter of droplet for various TTAB concentrations: 3.75, 7.5, and 10 wt%. For all concentrations, the droplets show random (R), helical (H), or straight (S) motion as their sizes decreased. The boundary between the R and H motion appears to be governed by size, which is marked with a thick vertical line. The crossover between H and S motion was governed by velocity, as marked with a thick horizontal line.

Figure 7

The motion of a defect during helical motion. (a) Time series of bright-field microscope images. The contrast was controlled to make the defect more easily identifiable. The black point indicated by a triangle is a point defect. (b) The dependence of the polar angle $\theta$ on the velocity of a droplet. The solid line represents the best-fit approximate curve obtained by dimensional analysis. The schematic shows the definition of polar angle $\theta$. $v_c$ =18.3 $\mu \mathrm{m}/\mathrm{s}$ is the critical velocity for which the defect begins to shift its position away from the velocity vector.

Figure 8

Characteristic motions of a self-propelled 5CB droplet in the 2D system. The concentration of the TTAB aqueous solution is 10 wt%. (a) Typical trajectories of droplets of various sizes: “figure-8”, random, zigzag, and straight. The scale bars represent 50 $\mu \mathrm{m}$. (b) Temporal change of droplet diameter. The data marked with empty symbols represent the diameters of spheres with equivalent volumes. (c) Dependence of the averaged instantaneous 2D projected velocities on the diameter of the droplets. Different symbols represent different types of motion.

Figure 9

Temporal change of MSD and $C\left(t\right)$ for figure-8 and zigzag motions. (a) MSD for figure-8 and zigzag motions. (b) $C\left(t\right)$ for figure-8 motion. (c) $C\left(t\right)$ for zigzag motion.