2D Interactions and Binary Crystals of Dipolar and Quadrupolar Nematic Colloids

In this Letter, we demonstrate that the symmetry of the elastic interaction between the dipolar and quadrupolar colloidal particles in the nematic liquid crystal leads to a novel variety of 2D nematic “binary” colloidal crystals, which have not been observed in any colloidal system. The dipolar-quadrupolar interaction is highly anisotropic and shows a power-law dependence when the particles approach each other along the director field with a pair-binding energy of the order of several thousands of $k_{B}T$ for $4\mu \mathrm{m}$ diameter colloids.

Figure 1

Microscopic images of (a) dipolar and (b) quadrupolar $4\mu \mathrm{m}$ colloidal particles. Dipolar colloids are accompanied by a hyperbolic hedgehog defect (dark spot); quadrupolar colloids have the Saturn ring around them (two dark spots are projections of the ring) [2, 5]. (c) A pair of dipole-quadrupole particles, attracted along the LC director and (d) schematics of the director field. (e) Dipole-quadrupole pair, attracted in a direction perpendicular to the bulk director and (f) the corresponding director.

Figure 2

(a) Trajectories of the dipolar colloidal particle after it was released from the optical trap near the quadrupolar colloidal particle, which is placed at the origin. Blue lines (2–4, 6–8, 10–12) show the attractive trajectories, red lines (1, 5, 9) show repulsive trajectories, and black dashed lines are calculated by using an electrostatic ansatz [6]. Open circles and black dots indicate the starting positions. The Saturn ring orientation corresponds to trajectories 5–9. It is tilted like in Fig. 1f for trajectories 10–12 and in the opposite direction for trajectories 2–4. The thickness of the cell is $h=7\mu \mathrm{m}$. (b) The interaction potential as a function of separation between a pair of a dipolar and a quadrupolar colloidal particle for the starting position 7. Inset: log-log plot of the interaction force with the best fit to a power-law exponent of $-5.1\pm 0.25$. Vertical dashed lines indicate boundaries of the fitting region; their values are indicated at the top.

Figure 3

(a),(c) Microscopic images of two different, mixed dipolar-quadrupolar 2D lattices of $4\mu \mathrm{m}$ particles in a $6.5\mu \mathrm{m}$ nematic liquid crystal cell, together with (b),(d) schematic presentations.

Figure 4

Landau–de Gennes simulation of the stability of mixed dipolar-quadrupolar 2D colloidal crystals: (a) the most dense mixed 2D colloidal crystal, (b) close-up of the crystallite showing strongly deformed defect rings, and (c) binding potential of the 2D crystal as a function of the unit cell lattice constants $X$ and $Y$. Particle radius is $R=0.5\mu \mathrm{m}$, the cell thickness is $h=2\mu \mathrm{m}$, and the Landau–de Gennes parameters are identical to those used in Refs. 20, 29.