Entangled Nematic Colloidal Dimers and Wires

It has been predicted, but never confirmed, that colloidal particles in a nematic liquid crystal could be self-assembled by delocalized topological defects and entangled disclinations. We show experimentally and theoretically that colloidal dimers and 1D structures bound by entangled topological defect loops can indeed be created by locally thermally quenching a thin layer of the nematic liquid crystal around selected colloidal particles. The topological entanglement provides a strong stringlike binding, which is ten thousand times stronger compared to water-based colloids. This unique binding mechanism could be used to assemble resonator optical waveguides and robust chiral and achiral structures of topologically entangled colloids that we call colloidal wires.

Figure 1

Assembling entangled colloidal pairs by thermal quench using light. The diameter of the particles is $19\mu \mathrm{m}$, the cell thickness is $21\mu \mathrm{m}$. (a) Figure of eight entangled state. The disclination loops are visible under a nonpolarizing optical microscope due to the scattering of light. (b) Numerical simulation of the time evolution of entanglement, measured in the number of iteration steps. Defects are represented by drawing the surfaces of constant order parameter $S$ (colored red, corresponding to $S=0.5$). (c) Evolution of the figure of omega entangled state. (d) The figure of omega was unstable and transformed into the entangled hyperbolic defect. (e) The calculated figure of eight structure. (f) Close-up look at the director field in the gap between the microspheres for the figure of eight. (g) Calculated figure of omega structure. (h) Calculated entangled hyperbolic defect structure.

Figure 2

Stretching and releasing the topologically bound colloidal pair using light. (a) Using focused light of the laser tweezers (two red crosses on left and right), the figure of eight colloidal pair is first stretched and then released by switching-off the light. The diameter of the particles is $4.7\mu \mathrm{m}$, the thickness of the cell is $6\mu \mathrm{m}$. The force and the binding energy are calculated from the video frames (a), and shown in (c). (b) The same is done for entangled hyperbolic defects, shown in (d). (e) The pair binding energy, calculated for the figure of eight, as a function of particle separation $x$, normalized to the particle diameter $d$. (f) The pair binding energy, calculated for the entangled point defect as a function of separation $x$, normalized to the particle diameter $d$.

Figure 3

Colloidal wires assembled by entanglement. (a) Right-handed figure of eight colloidal wire assembled from $19\mu \mathrm{m}$ glass spheres. (b) Figure of eight colloidal wire made of $4.7\mu \mathrm{m}$ microspheres, stretched by light of the laser tweezers (red crosses). In the absence of light, the average equilibrium surface-surface separation between neighboring spheres is 160 nm. (c) Entangled hyperbolic defect colloidal wire, stretched by laser tweezers. Without light, the average equilibrium surface-surface separation between neighboring spheres is 1100 nm. (d) Calculated structure of stretched figure of eight colloidal wire. (e) Calculated structure of stretched entangled hyperbolic defect colloidal wire.