Spherical colloidal particles with normal and tangential surface director alignment in a nematic liquid crystal induce elastic quadrupoles of opposite signs that attract one another along and perpendicular to the director. We utilize this unique angular profile of the mixed quadrupolar interaction to build 2D crystals with square lattices by laser tweezers.

We formulate a numerical method for predicting the tensorial linear response of a rigid, asymmetrically charged body to an applied electric field. This prediction requires calculating the response of the fluid to the Stokes drag forces on the moving body and on the countercharges near its surface. To determine the fluid's motion, we represent both the body and the countercharges using many point sources of drag known as stokeslets. Finding the correct flow field amounts to finding the set of drag forces on the stokeslets that is consistent with the relative velocities experienced by each stokeslet. The method rigorously satisfies the condition that the object moves with no transfer of momentum to the fluid. We demonstrate that a sphere represented by 1999 well-separated stokeslets on its surface produces flow and drag force like a solid sphere to one-percent accuracy. We show that a uniformly-charged sphere with 3998 body and countercharge stokeslets obeys the Smoluchowski prediction [1] for electrophoretic mobility when the countercharges lie close to the sphere. Spheres with dipolar and quadrupolar charge distributions rotate and translate as predicted analytically to four percent accuracy or better. We describe how the method can treat general asymmetric shapes and charge distributions. This method offers promise as a way to characterize and manipulate asymmetrically charged colloid-scale objects from biology (eg. viruses) and technology (eg. self-assembled clusters).

The interaction between particle-like sources of the nematic director distortions (e.g., colloids, point defects, macromolecules in nematic emulsions) allows for a useful analogy with the electrostatic multipole interaction between charged bodies. In this paper we develop this analogy to the level corresponding to the charge density and consider the general status of the pairwise approach to the nematic emulsions with finite-size colloids. It is shown that the elastic analog of the surface electric charge density is represented by the two transverse director components on the surface imposing the director distortions. The elastic multipoles of a particle are expressed as integrals over the charge density distribution on this surface. Because of the difference between the scalar electrostatics and vector nematostatics, the number of elastic multipoles of each order is doubled compared to that in the electrostatics: there are two elastic charges, two vectors of dipole moments, two quadrupolar tensors, and so on. The two-component elastic charge is expressed via the vector of external mechanical torque applied on the particle. As a result, the elastic Coulomb-like coupling between two particles is found to be proportional to the scalar product of the two external torques and does not directly depend on the particles' form and anchoring. The real-space Green function method is used to develop the pairwise approach to nematic emulsions and determine its form and restrictions. The pairwise potentials are obtained in the familiar form, but, in contrast to the electrostatics, they describe the interaction between pairs (dyads) of the elastic multipole moments. The multipole moments are shown to be uniquely determined by the single-particle director field, unperturbed by other particles. The pairwise approximation is applicable only in the leading order in the small ratio particle size-to-interparticle distance as the next order contains irreducible three-body terms.

An external mechanical torque on colloids immersed in a nematic liquid crystal can induce a Coulomb-like 1/r interaction between them [Lev and Tomchuk, Phys. Rev. E 59, 591 (1999); Lev, ibid. 65, 021709 (2002)]. In this paper we show that the director-mediated Coulomb-like interaction of two colloids is determined by the vectors Gamma perpendicular (1) and Gamma perpendicular (2) of the transverse external torques exerted upon these colloids. We derive the 1/r potential in which the scalar product -(Gamma perpendicular (1) x Gamma perpendicular (2)) of the two torques plays the role of the product of two electrostatic charges. The 1/r interaction is attractive for (Gamma perpendicular (1) x Gamma perpendicular (2))>0 and repulsive for (Gamma perpendicular (1) x Gamma perpendicular (2))<0 ("parallel torques" attract whereas "antiparallel torques" repel each other). The vector of transverse torque determines the two-component "elastic charge" (dyad), which is illustrated by the 1/r2 and 1/r3 terms in the elastic energy (the elastic analogs of the monopole-dipole and dipole-dipole interactions). The general status of the pairwise approach to nematic emulsions is considered in terms of the elastic charge density.

We give a review of the theory of large distance colloidal interaction via the nematic director field. The new area of nematic colloidal systems (or nematic emulsions) has been guided by the analogy between the colloidal nematostatics and electrostatics. The elastic charge density representation of the colloidal nematostatics [V.M. Pergamenshchik, V.O. Uzunova, Eur. Phys. J. E, 2007, 23, 161; Phys. Rev. E, 2007, 76, 011707] develops this analogy at the level of charge density and Coulomb interaction. The analogy is shown to lie in common mathematics based on the solutions of Laplace equation. However, the 3d colloidal nematostatics substantially differs from electrostatics both in its mathematical structure and physical implications. The elastic charge is a vector fully determined by the torque exerted upon colloid, the role of Gauss' theorem is played by conservation of the torque components. Elastic multipoles consist of two tensors (dyads). Formulas for the elastic multipoles, the Coulomb-like, dipole-dipole, and quadrupole-quadrupole pair interaction potentials are derived and illustrated by particular examples. Based on the tensorial structure, we list possible types of elastic dipoles and quadrupoles. An elastic dipole is characterized by its isotropic strength, anisotropy, chirality, and its longitudinal component. An elastic quadrupole can be uniaxial and biaxial. Relation between the multipole type and its symmetry is discussed, sketches of some types of multipoles are given. Using the mirror image method of electrostatics as a guiding idea, we develop the mirror image method in nematostatics for arbitrary director tilt at the wall. The method is applied to the charge-wall and dipole-wall interaction. : 61.30.Dk, 61.30.Jf, 82.70Dd, 01.55.+b 1. The intrigue of nematic colloids: similarity to the electrostatics, its origin, and how it is developed in this article. Key words: nematic colloid, electrostatic analogy, elastic charge and multipoles PACSIsotropic liquids doped with colloidal particles or just colloids are the classical many-particle systems which, over many decades , have been studied using the methods of molecular and statistical physics, electrolyte theory, physical chemistry, and so on [1,2]. Colloidal particles can be of different physical and chemical origin, but in all cases their interaction via isotropic liquid is of a short range [3], even if the colloids are charged since their electric field is rapidly screened by a numerous couterions. The nematic colloids, that have become widely known just over the last 15 years, are fundamentally different: interaction of colloids mediated by distortions of the nematic director is of a long range. That is why nematic colloids are usually compared not with standard isotropic colloids, but with a system of electric charges. The similarity with the electrostatics has always been a powerful, if not dominating, factor of the development of the field of nematic colloids. But what is the origin of this similarity?The interaction of particles in a ne...