Publisher's Note: Colloidal interactions in a homeotropic nematic cell with different elastic constants [Phys. Rev. E 92, 042505 (2015)].

This corrects the article DOI: 10.1103/PhysRevE.92.042505.

Colloidal interactions in a homeotropic nematic cell with different elastic constants.

We propose a theoretical description of the interaction mediated by a nematic-liquid-crystal host with different Frank elastic constants. A general expression for the energy of such an interaction between colloidal particles of arbitrary size and shape suspended in a homeotropic cell is obtained. In the cells of large thickness, the presented potential converges to that found previously for small particles in the nematic bulk. In general, our results confirm the validity of the one-constant approximation for weakly elastically anisotropic nematic liquid crystals. For nematics with a high splay-to-bend ratio we predict a larger range of the interaction. Using the dependence of this range on the elastic constants, we show that there exists a qualitative similarity between the interactions in a nematic and in a smectic-A phase. It manifests itself, in particular, in a decrease of the angle between a chain of quadrupole particles and the uniform far-field director across a nematic-smectic-A phase transition. We also demonstrate that the anisotropy of the elastic constants can lead to the formation of thermodynamically stable linear superstructures of asymmetric particles (elastic monopoles) with large, compared to usual dipole chains, interparticle distances.

Surface-induced structures in nematic liquid crystal colloids.

We predict theoretically the existence of a class of colloidal structures in nematic liquid crystal (NLC) cells, which are induced by surface patterns on the plates of the cell (like cells with UV-irradiated polyamide surfaces using micron sized masks in front of the cell). These bulk structures arise from nonuniform boundary conditions for the director distortions at the confining surfaces. In particular, we demonstrate that quadrupole spherical particles (like spheres with boojums or Saturn-ring director configurations) form a square lattice inside a planar NLC cell, which has checkerboard patterns on both its plates.

Elastic octopoles and colloidal structures in nematic liquid crystals.

We propose a simple theoretical model which explains the formation of dipolar two- (2D) and three-dimensional (3D) colloidal structures in nematic liquid crystals. The colloidal particles are treated as effective hard spheres interacting via their elastic dipole, quadrupole, and octopole moments. It is shown that the octopole moment plays an important role in the formation of 2D and 3D nematic colloidal crystals. We generalize this assumption to the case of an external electric field and theoretically explain a giant electrostriction effect in 3D crystals observed recently.

Interaction of small spherical particles in confined cholesteric liquid crystals.

The theory of the elastic interaction of spherical colloidal particles immersed into a confined cholesteric liquid crystal is proposed. The case of weak anchoring on the particle surfaces is considered. We derive a general expression for the energy of the interaction between small spherical particles (with diameter much smaller than the cholesteric pitch) suspended in a cholesteric confined by two parallel planes. The resulting form of the interaction energy has a more complex spatial pattern and energy versus distance dependence than that in nematic colloids. The absence of translational symmetry related to helical periodicity and local nematic ordering in cholesteric liquid crystals manifest themselves in the complex nature of the interaction maps.

High-order elastic terms, boojums and general paradigm of the elastic interaction between colloidal particles in the nematic liquid crystals.

The theoretical description of the elastic interaction between colloidal particles in NLC with incorporation of higher-order elastic terms beyond the limit of dipole and qudrupole interactions was proposed. The expression for the elastic interaction potential between axially symmetric colloidal particles, taking into account the high-order elastic terms, was obtained. The general paradigm of elastic interaction between colloidal particles in NLC was proposed so that every particle with strong anchoring and radius a has three zones surrounding itself. The first zone for a < r ⪅ 1.3a is the zone of topological defects; the second zone at the approximate distance range 1.3a ⪅ r ⪅ 4a is the zone where crossover from topological defects to the main multipole moment takes place. The higher-order elastic terms are essential here (from 10% to 60% of the total deformation). The third zone is the zone of the main multipole moment, where higher-order terms make a contribution of less than 10%. This zone extends to distances where r ⪆ 4a = 2D . The case of spherical particles with planar anchoring conditions and boojums at the poles was considered as an example. It was found that boojums can be described analitically via multipole expansion with accuracy up to 1/r(7) and the whole spherical particle can be effectively considered as the multipole of the order 6 where multipolarity equal 2(6) = 64. The corresponding elastic interaction with higher-order elastic terms gives the angle θ(min) = 34.5° of minimum energy between two contact beads which is close to the experimental value of θ(min) = 30° . In addition, high-order elastic terms make the effective power of the repulsive potential to be non-integer at the range 4.5 < γ(eff) < 5 for different distances. The incorporation of the high-order elastic terms in the confined NLC also produce results that agree with experimental data.

Theory of elastic interaction between colloidal particles in a nematic cell in the presence of an external electric or magnetic field.

The Green's function method developed previously [S. B. Chernyshuk and B. I. Lev, Phys. Rev. E 81, 041701 (2010)] is used to describe elastic interactions between axially symmetric colloidal particles in a nematic cell in the presence of an external electric or magnetic field. Formulas for dipole-dipole, dipole-quadrupole, and quadrupole-quadrupole interactions in the homeotropic and planar nematic cells with parallel and perpendicular field orientations are obtained. A set of predictions has been made: (1) The deconfinement effect for dipole particles in the homeotropic nematic cell when an electric field is approaching its Freedericksz threshold value E⇒E(t). This means cancellation of the confinement effect found in [M. Vilfan et al., Phys. Rev. Lett. 101, 237801 (2008)] near the Freedericksz transition. In the planar nematic cell this deconfinement effect exists for both dipole and quadrupole particles and depends on the field orientation as well as on the sign of dielectric anisotropy Δε. (2) The effect of tunable stabilization of the particles is predicted. The equilibrium distance between two particles, which are attracted along the electric field parallel to the planes of a homeotropic nematic cell with Δε<0, depends on the strength of the field. (3) Attraction and repulsion zones for all elastic interactions are changed dramatically under the action of the external field.

Theory of elastic interaction between arbitrary colloidal particles in confined nematic liquid crystals.

We develop the method proposed by Chernyshuk and Lev [Phys. Rev. E 81, 041701 (2010)] for theoretical investigation of elastic interactions between colloidal particles of arbitrary shape and chirality (polar as well as azimuthal anchoring) in the confined nematic liquid crystal (NLC). General expressions for six different types of multipole elastic interactions are obtained in the confined NLC: monopole-monopole (Coulomb type), monopole-dipole, monopole-quadrupole, dipole-dipole, dipole-quadrupole, and quadrupole-quadrupole interactions. The obtained formulas remain valid in the presence of the external electric or magnetic fields. The exact equations are found for all multipole coefficients for the weak anchoring case. For the strong anchoring coupling, the connection between the symmetry of the shape or director and multipole coefficients is obtained, which enables us to predict which multipole coefficients vanish and which remain nonzero. The particles with azimuthal helicoid anchoring are considered as an example. Dipole-dipole interactions between helicoid cylinders and cones are found in the confined NLC. In addition, the banana-shaped particles in homeotropic and planar nematic cells are considered. It is found that the dipole-dipole interaction between banana-shaped particles differs greatly from the dipole-dipole interaction between the axially symmetrical particles in the nematic cell. There is a crossover from attraction to repulsion between banana particles along some directions in nematic cells. It is shown that monopoles do not "feel" the type of nematic cell: monopole-monopole interaction turns out to be the same in homeotropic and planar nematic cells and converges to the Coulomb law as thickness increases, L→∞.

Theory of elastic interaction of colloidal particles in nematic liquid crystals near one wall and in the nematic cell.

We apply the method developed [Chernyshuk and Lev, Phys. Rev. E 81, 041701 (2010)] for theoretical investigation of colloidal elastic interactions between axially symmetric particles in the confined nematic liquid crystal near one wall and in the nematic cell with thickness L. Both cases of homeotropic and planar director orientations are considered. Particularly, dipole-dipole, dipole-quadrupole, and quadrupole-quadrupole interactions of the one particle with the wall and within the nematic cell are found as well as corresponding two particle elastic interactions. A set of results has been predicted: The effective power of repulsion between two dipole particles at height h near the homeotropic wall is reduced gradually from inverse 3 to 5 with an increase of dimensionless distance r / h; near the planar wall, the effect of dipole-dipole isotropic attraction is predicted for large distances r > r(dd) = 4.76 h; maps of attraction and repulsion zones are crucially changed for all interactions near the planar wall and in the planar cell; and one dipole particle in the homeotropic nematic cell was found to be shifted by the distance δ(eq) from the center of the cell. The proposed theory fits very well with experimental data for the confinement effect of elastic interaction between spheres in the homeotropic cell [Vilfan et al., Phys. Rev. Lett. 101, 237801 (2008)] in the range 1-1000 kT. The influence of the K(24) and K(13) terms as well as connection with other theoretical approaches are discussed.

Elastic interaction between colloidal particles in confined nematic liquid crystals.

The theory of elastic interaction of micrometer-sized axially symmetric colloidal particles immersed into confined nematic liquid crystal has been proposed. General formulas are obtained for the self-energy of one colloidal particle and interaction energy between two particles in arbitrary confined nematic liquid crystals with strong anchoring condition on the bounding surfaces. Particular cases of dipole-dipole interaction in the homeotropic and planar nematic cell with thickness L are considered and found to be exponentially screened on far distances with decay length lambdadd=L/pi. It is predicted that bounding surfaces in the planar cell crucially change the attraction and repulsion zones of usual dipole-dipole interaction. As well it is predicted that the decay length in quadrupolar interaction is two times smaller than for the dipolar case in the homeotropic cell.

Photochemical switching between colloidal photonic crystals at the nematic-air interface.

A direct observation of the photochemical switching between colloidal crystals with different lattice constants in a liquid-crystal (LC) emulsion is reported. Glycerol droplets introduced in a nematic liquid crystal form two-dimensional hexagonal colloidal crystal at the nematic-air interface with a lattice constant depending on the surface tension sigmaLCA. We dope an azobenzene derivative into the LC emulsion to modulate the colloidal structures by using cis-trans photoisomerization of the doped dye. The photoisomerization changes sigmaLCA and the lattice constant of colloidal crystals with a relaxation time T approximately 10s . A simple theoretical description, which qualitatively agrees with experimental results, has been proposed. Possible applications in infrared range photonic crystals are discussed. The number of different kinds of dyes N set the number of different possible photonic crystals as 2N, which may be transformed one into another by corresponding 2N light beams inducing cis-trans photoisomerization of the doped dyes.

Anisotropic laser trapping in nematic colloidal dispersion.

The interaction between a colloidal particle and a focused laser beam in a nematic liquid crystal reveals an unusual anisotropic Coulomb-like character. Experiments demonstrate two opposite directions in which the particle is attracted to and repelled from the nematic region deformed by the light-induced director reorientation. In this work we present analytical analysis of such behavior and derive the energy of interaction between colloidal particle and deformed director field. The analytical solution is in good agreement with recent results obtained by computer simulation.

Paranematic interaction between nanoparticles of ordinary shape.

We propose a general approach to the description of the long-ranged interaction between nanoparticles (1-10 nm) of ordinary shape in the paranematic phase, i.e., nematic liquid crystal in the isotropic phase. In general case interaction potential is attractive of Yukawa form with derivatives. But it can be anisotropic despite the isotropy of the paranematic phase. The origin of such anisotropy is the shape of nanoparticles. Particular potentials for spherical and cylindrical particles are considered. For the case of nanocylinders anisotropic part of the interaction potential can lead to the orientational ordering of them in the isotropic phase of nematic liquid crystals.

Symmetry breaking and interaction of colloidal particles in nematic liquid crystals.

We propose a general approach to the description of the long-ranged elastic interaction in the nematic colloids, based on the symmetry breaking of the director field. The type of the far-field interaction between particles immersed in a nematic host is determined by the way the symmetry is broken in the near-field region around the colloidal particle. This is caused both by the particle's shape and the anchoring at the surface. If the director field near the particle has a set of three symmetry planes, the far-field interaction falls off as d(-5) with d being the distance between particles. If one symmetry plane is absent, a dipolar moment perpendicular to it is allowed and yields dipole-dipole interactions, which decays as d(-3). If both the horizontal and vertical mirror symmetries are broken (it is equivalent to the case when the nonzero torque moment is applied to the particle by the nematic liquid crystal), the particles are shown to attract each other following the Coulomb law. We propose a simple method for the experimental observation of this Coulomb attraction. The behavior of colloid particles in curved director fields is analyzed. Quadrupolar particles with planar anchoring are shown to be attracted toward the regions with high splay deformations, while quadrupoles with homeotropic anchoring are depleted from such regions. When there are many colloidal particles in the nematic solvent, the distortions of the director from all of them are overlapped and lead to the exponential screening in the elastic pair interaction potential. This is a many-body interaction effect. This screening is essential in the real dense colloid systems, such as ferronematics--suspensions of magnetic cylindrical grains in the nematic liquid crystal. External magnetic field induces an elastic Yukawa attraction between them. We apply this attraction to the explanation of the cellular texture in magnetically doped liquid crystals.

Full energy expression of a uniaxial nematic phase with spatially dependent density and order parameters: From microscopic to macroscopic theory.

We present a microscopic derivation of the full macroscopic energy expression of a spatially bounded uniaxial nematic phase. The surface is described by spatial variations of the density and scalar order parameters of all even orders. The method developed in the paper allowed us to unambiguously separate the surface elastic K24 and K13 terms and isotropic and anisotropic surface tension (anchoring). The full energy expression incorporating variations of the director, scalar order parameters, and density is obtained. The macroscopic coefficients are derived in terms of the isotropic and anisotropic fractions of the microscopic intermolecular interaction. An important physical consequence of the obtained formulas, in particular, is that the observed considerable difference K33-K11 between the bend and splay elastic constants unambiguously indicates that (i) the intermolecular interaction has a large anisotropic fraction, and thus, the effective constant K13 and intrinsic anchoring are considerable; (ii) at least some scalar order parameters of order four and higher are essentially nonzero. Relation of the developed theory with the Nehring-Saupe theory and Landau-de Gennes approach is considered.