Anomalous Diffusion of Motile Colloids Dispersed in Liquid Crystals

We study the superdiffusion of driven colloidal particles dispersed in a nematic liquid crystal. While motion is ballistic in the driving direction, our experiments show that transversal fluctuations become superdiffusive depending on the topological defect pattern around the inclusions. The phenomenon can be reproduced with different driving methods and propulsion speeds, while it is strongly dependent on particle size and temperature. We propose a mechanism based on the geometry of the liquid crystal backflow around the inclusions to justify the persistence of thermal fluctuations and to explain the observed temperature and particle size dependence of the superdiffusive behavior based on material and geometrical parameters.

Figure 1

Sedimentation of a $5\mu \mathrm{m}$ particle with dipolar LC configuration. (a) Sketch of the experimental geometry and LC director surrounding the inclusion. (b) Particle trajectory in the $x/y$ plane at $T-T_{NI}=-27.5°\mathrm{C}$. (c) MSD in the transversal direction vs time window. The solid line is a power law fit that yields an exponent $\nu =1.41\pm 0.04$. In the inset, the mean parallel displacement is plotted vs time, showing a ballistic trend. The solid line has a slope 1.

Figure 2

(a) Temperature dependence of the superdiffusive exponent in the transversal MSD of nematic colloids featuring a dipolar configuration. Particles diameters are $10\mu \mathrm{m}$ (blue, solid symbols) and $5\mu \mathrm{m}$ (red, empty symbols), and driving mechanisms are sedimentation (circle) and electrokinetic (rectangle). (b) Exponents ($\nu$) vs speed at $T-T_{NI}=-20°\mathrm{C}$ for $5\mu \mathrm{m}$ electrokinetically-driven particles featuring a dipolar defect. Different speeds are achieved by tuning the electric field amplitude with a constant frequency of $10\mathrm{Hz}$. Error bars indicate the standard deviation over different experimental realizations.

Figure 3

Sedimentation of a $5\mu \mathrm{m}$ particle with quadrupolar LC configuration. (a) Sketch of the LC director field around the inclusion. (b) Particle trajectory in the $x/y$ plane at $T-T_{NI}=-17.5°\mathrm{C}$. (c) Mean square displacement in the transversal direction vs time window. The solid line is a power law fit that yields an exponent $\nu =1.0\pm 0.1$. In the inset, the mean parallel displacement is plotted vs time, showing a ballistic trend. (d) Temperature dependence of the nearly diffusive exponent in the transversal MSD. Error bars indicate the standard deviation over different experimental realizations.

Figure 4

(a) Position, $d$, and relative orientation, $\theta$, of the hedgehog defect for a particle of diameter $20\mu \mathrm{m}$, observed between crossed polarizers, in a cell of thickness $25\mu \mathrm{m}$. Scale, $10\mu \mathrm{m}$. (b) Defect orientation with respect to the horizontal LC far field as a function of time for a particle of diameter $10\mu \mathrm{m}$ under electrokinetic drive at $T-T_{NI}=-15°\mathrm{C}$. (c) Autocorrelation of the driving torque, $f\propto \sin (2\theta )$, for the data in (b). The solid line is a power law fit to $⟨f(t)f(t+\tau )⟩\sim t^{-\alpha }$, with $\alpha =0.22$, for times larger than the time needed for the particle to travel its own size. (d) Exponent of the mean square displacement, $\nu$ (open rectangle), and force autocorrelation, $\alpha$ (filled circle), as a function of temperature, for $10\mu \mathrm{m}$ particles under electrokinetic drive.