Two-dimensional dipolar nematic colloidal crystals

We study the interactions and directed assembly of dipolar nematic colloidal particles in planar nematic cells using laser tweezers. The binding energies for two stable configurations of a colloidal pair with homeotropic surface alignment are determined. It is shown that the orientation of the dipolar colloidal particle can efficiently be controlled and changed by locally quenching the nematic liquid crystal from the laser-induced isotropic phase. The interaction of a single colloidal particle with a single colloidal chain is determined and the interactions between pairs of colloidal chains are studied. We demonstrate that dipolar colloidal chains self-assemble into the two-dimensional (2D) dipolar nematic colloidal crystals. An odd-even effect is observed with increasing number of colloidal chains forming the 2D colloidal crystal.

Figure 1

Dipolar nematic colloids and their aggregation. (a) Unpolarized optical micrograph of a single silica microsphere of dipolar symmetry in 5CB. The hyperbolic hedgehog point defect is visible as a dark spot due to the strong scattering of light. (b) Schematic presentation of the nematic director field around a dipolar colloidal particle. “A” denotes the real hyperbolic hedgehog point defect, “B” denotes the virtual radial hedgehog, located inside the colloidal particle. They form a topological dipole. (c) Dipolar colloidal chains are spontaneously formed in nematic colloids. (d) Spontaneous formation of 2D crystalline colloidal clusters, and various types of “nematic bonds.”

Figure 2

(Color online) The interaction between two dipolar colloidal particles. (a) Two dipolar colloidal particles with parallel topological dipoles were positioned using the laser tweezers at the initial separation of $\sim 10\mu \text{m}$ and released. The particles were attracted along the straight line and in the equilibrium position both dipoles were aligned “head to tail.” (b) The direction of the upper topological dipole was reversed and the particles were released from the laser traps. Curved trajectories of both particles are clearly revealed, indicating strongly anisotropic attraction. In the equilibrium, both dipoles were antiparallel, with their centers positioned on a line, tilted at $\alpha =36°$ with respect to the rubbing direction. (c) The potential between the two dipolar colloidal particles, as measured along their trajectories, for parallel (circles) and antiparallel (squares) orientation of the two dipoles. The inset shows the dipolar force for parallel orientation and head-to-tail attraction in the log-log plot. The solid line is the $1∕r^4$ law.

Figure 3

(Color online) The orientation of the dipolar colloidal particle can be reversed by melting locally the nematic liquid crystal around the particle and “dragging” the molten crystal in a preferred direction for the new orientation of the dipole.

Figure 4

(Color online) The interaction between a dipolar colloidal chain and an isolated dipolar colloidal particle. The arrows indicate the directions of the topological dipoles. (a) For parallel orientation of the topological dipoles and side position, the colloidal particle is repelled from the chain. (b) If the direction of the topological dipole is reversed, the colloidal particle is strongly attracted to the chain. (c) Using the particle tracking software, the energies of repulsion (squares) and attraction (circles) of a single colloidal particle from the dipolar chain are calculated. Note the energy scale and the strength of interaction.

Figure 5

Attraction between a pair of dipolar nematic colloidal chains. (a) Two dipolar colloidal chains with reversed directions of their topological dipoles were assembled, positioned close to each other, and released. Note the tilting of the assembly, as it approaches the equilibrium position. (b) The interaction energy between a pair of dipolar colloidal chains.

Figure 6

(a)–(d) Odd-even effect in the tilt of 2D dipolar nematic colloidal assemblies. (e) The tilt angle of the dipolar colloidal assembly as a function of the number of colloidal chains. Even numbers only are shown.

Figure 7

Origin of the odd-even effect in the tilt of the colloidal chains. (a) For an even number of dipolar nematic colloidal chains, regions of strong elastic deformation are present, when the assembly is oriented along the rubbing direction. This strain can be released by tilting the 2D assembly. (b) The 2D nematic colloidal crystal, composed of an odd number of dipolar colloidal chains, is symmetrically distorted. Any rotation does not decrease the energy of deformation.

Figure 8

(Color online) (a) An example of a 2D dipolar nematic colloidal crystal, assembled by the laser tweezer's manipulation. The crystal consists of 280 silica spheres and is stable for more than a year at laboratory conditions. (b) Schematic drawing of the direction of the topological dipoles of each dipolar chain. Each chain can be considered as a “ferroelectric” object, which are packed in the “antiferroelectric,” up-down manner, thus filling the 2D space. (c) The lattice vectors and the unit cell of the dipolar colloidal crystal shown in (a). (d) The result of the LdG modeling of the stability of 2D dipolar nematic colloidal crystal in a thin nematic cell. The lines denote the orientation of the director field. Thick dark circles between the colloidal particles are small topological rings that have developed from a hedgehog point defect due to strong confinement.