Interactions of quadrupolar nematic colloids

We present experimental and theoretical study of colloidal interactions in quadrupolar nematic liquid crystal colloids, confined to a thin planar nematic cell. Using the laser tweezers, the particles have been positioned in the vicinity of other colloidal particles and their interactions have been determined using particle tracking video microscopy. Several types of interactions have been analyzed: (i) quadrupolar pair interaction, (ii) the interaction of an isolated quadrupole with a quadrupolar chain, and (iii) the interaction of an isolated quadrupolar colloidal particle with a two-dimensional (2D) quadrupolar crystallite. In all cases, the interactions are of the order of several $100k_{B}T$ for $2\mu \text{m}$ particles, which gives rise to relatively stable 2D colloidal crystals. The experimental results are compared to the predictions of Landau–de Gennes theory and we find a relatively good qualitative agreement.

Figure 1

(a) A Saturn ring is clearly visible when larger colloidal particles with homeotropic surface anchoring are observed in a planar nematic cell under an optical microscope. (b) With smaller microspheres, the ring is barely observed and appears in a form of two dark spots. (c) Schematic drawing of the director field around Saturn-like colloidal particle.

Figure 2

(a) Schematics: in thinner parts of a planar wedge-type nematic cell, the particles with homeotropic surface anchoring are of quadrupolar symmetry. In thicker parts of the cell, dipolar colloids are observed. (b) At a critical cell thickness, both types of particles are observed. Dipolar colloids form linear chains, directed along the rubbing direction, whereas quadrupolar colloids form kinked chains, perpendicular to the rubbing direction. (c) Histogram presenting the number of dipolar and quadrupolar colloidal particles at different cell thickness. Note rather broad region of the coexistence of both colloidal types.

Figure 3

When the cell thickness is close to the critical thickness $h_c$ for the dipolar-to-quadrupolar transition, quenching a small isotropic island around a selected colloidal particle can result either in (a) quadrupolar colloidal structure or (b) dipolar colloidal structure. In this particular case the cell thickness was $7\mu \text{m}$ and the diameter of the silica colloidal particles with DMOAP surface coating was $4.7\mu \text{m}$.

Figure 4

A majority of self-assembled quadrupolar colloidal structures are just kinked chains, shown in (a). However, in some parts of the sample, spontaneously ordered 2D quadrupolar crystallites are observed, such as the one in the encircled region in (b).

Figure 5

(Color online) Quadrupolar colloidal chains can grow in a form of kinked (a) or straight chains (b). (c) The quadrupole-quadrupole interaction, as a function of separation between the particle and the closest particle in the chain. The squares correspond to $a$, the circles correspond to $b$.

Figure 6

(a) An isolated quadrupolar colloidal particle is attracted laterally to the already formed quadrupolar chain. Note that the chain is slightly distorted after inclusion of an extra particle, as the “kink” angle increases from $17°$ to $29°$. (b) The lateral quadrupolar interaction is weaker and amounts only to $\sim 1/3$ of the head-to-head quadrupolar interaction. The binding of quadrupoles in 2D quadrupolar colloidal crystals is therefore highly anisotropic.

Figure 7

(a) Large scale quadrupolar nematic colloidal crystals could be assembled using laser tweezers. (b) The unit cell of the quadrupolar nematic colloidal crystal is oblique. The lattice vectors for $2.32\mu \text{m}$ colloidal particles in a $2.7\mu \text{m}$ planar cell are $a=\left(2.69\pm 0.04\right)\mu \text{m}$, $b=\left(3.01\pm 0.05\right)\mu \text{m}$, and $\gamma =\left(56\pm 1\right)°$.

Figure 8

Unlike the dipolar interaction [22], the quadrupolar interaction, especially lateral, is not sufficient to resist the external perturbations, such as the change of temperature and the flow of the liquid crystal. This results to gradual (over a month period) decomposition of the order.

Figure 9

(Color online) Numerical study of a dipolar to quadrupolar defect transition. The upper four frames show the initial configurations of the director field, which are different in the position and size of the defect ring. Initial configurations with $q\gtrsim 1.68$ relax into the final dipolar configuration, shown in lower left image, and the starting configurations with $q\lesssim 1.04$ relax into the quadrupolar state, shown in the lower right image. The defects are visualized in black, corresponding to isosurfaces of $S=0.48$ (compared to $S_{\text{bulk}}=0.533$. The numerical parameters: the colloidal particle diameter $2R=2.0\mu \text{m}$ and the cell thickness $h=2.5\mu \text{m}$.

Figure 10

(Color online) (a) A kinked quadrupolar colloidal trimer in the equilibrium configuration. Defect rings in black are visualized as isosurfaces of the scalar order parameter $S=0.48$ $\left(S_{\text{bulk}}=0.533\right)$; streamlines show the corresponding director field profile. (b) Binding potential for stretching of the kinked trimer. Numerical parameters: colloidal particle diameter $2R=1.0\mu \text{m}$ and cell thickness $h=1.4\mu \text{m}$.

Figure 11

(Color online) (a) A quadrupolar colloidal rhomb: defect rings are colored black, corresponding to isosurfaces of $S=0.48$, whereas the streamlines visualize the director field. (b) The binding potential for stretching of the rhomb particle configuration. Note that the binding energy is $\sim 1000k_{B}T$ per particle. The numerical parameters: the colloidal particle diameter $2R=1.0\mu \text{m}$, the cell thickness $h=1.4\mu \text{m}$.

Figure 12

(Color online) A numerically simulated stable configuration of a two-dimensional quadrupolar colloidal crystal. The defect rings are visualized in black, corresponding to isosurfaces of $S=0.48$. The director field in two perpendicular planes is shown. The numerical parameters are: the colloidal particle diameter is $2R=1.0\mu \text{m}$, and the cell thickness is $h=2.0\mu \text{m}$.