Interaction between two spherical particles in a nematic liquid crystal

Design and Preparation of Nematic Colloidal Particles

Colloidal systems are abundant in technology, in biomedical settings, and in our daily life. The so-called "colloidal atoms" paradigm exploits interparticle interactions to self-assemble colloidal analogs of atomic and molecular crystals, liquid crystal glasses, and other types of condensed matter from nanometer- or micrometer-sized colloidal building blocks. Nematic colloids, which comprise colloidal particles dispersed within an anisotropic nematic fluid host medium, provide a particularly rich variety of physical behaviors at the mesoscale, not only matching but even exceeding the diversity of structural and phase behavior in conventional atomic and molecular systems. This feature article, using primarily examples of works from our own group, highlights recent developments in the design, fabrication, and self-assembly of nematic colloidal particles, including the capabilities of preprogramming their behavior by controlling the particle's surface boundary conditions for liquid crystal molecules at the colloidal surfaces as well as by defining the shape and topology of the colloidal particles. Recent progress in defining particle-induced defects, elastic multipoles, self-assembly, and dynamics is discussed along with open issues and challenges within this research field.

Magnetic hybrid materials in liquid crystals

The integration of nanoparticles with magnetic, ferroelectric or semiconducting properties into liquid crystals (LCs) has attracted great interest both for fundamental investigations and for technological applications. Here, an overview of hybrid materials based on magnetic nanoparticles (MNPs) and thermotropic LCs is given. After a general introduction to thermotropic LCs and LC-MNP hybrid materials, various preparation methods established by us are presented. The synthesis of shape-(an)isotropic MNPs, their functionalization by tailored (pro)mesogenic ligands with linear or dendritic structures and their integration into LC hosts are discussed. The characterization of the MNPs, (pro)mesogenic ligands and resulting MNP-LC hybrid materials is described to show the influence of MNP functionalization on the MNP-LC interactions including aspects such as colloidal stability and structuring in the LC host. Overall, we show that the physical properties of the hybrid material are significantly influenced not only by the MNPs (i.e., their size, shape and composition) but also by their surface properties (i.e., the structure of the (pro)mesogenic ligands).

Chaining of hard disks in nematic needles: particle-based simulation of colloidal interactions in liquid crystals

Colloidal particles suspended in liquid crystals can exhibit various effective anisotropic interactions that can be tuned and utilized in self-assembly processes. We simulate a two-dimensional system of hard disks suspended in a solution of dense hard needles as a model system for colloids suspended in a nematic lyotropic liquid crystal. The novel event-chain Monte Carlo technique enables us to directly measure colloidal interactions in a microscopic simulation with explicit liquid crystal particles in the dense nematic phase. We find a directional short-range attraction for disks along the director, which triggers chaining parallel to the director and seemingly contradicts the standard liquid crystal field theory result of a quadrupolar attraction with a preferred [Formula: see text] angle. Our results can be explained by a short-range density-dependent depletion interaction, which has been neglected so far. Directionality and strength of the depletion interaction are caused by the weak planar anchoring of hard rods. The depletion attraction robustly dominates over the quadrupolar elastic attraction if disks come close. Self-assembly of many disks proceeds via intermediate chaining, which demonstrates that in lyotropic liquid crystal colloids depletion interactions play an important role in structure formation processes.

Modifying Nobili-Durand surface energy for conically degenerate anchorings at the interface of liquid crystal colloids

We propose a surface energy for conically degenerate anchorings of uniaxial liquid crystal mesogens by modifying tensorial Nobili-Durand surface energy that is usually employed for fixed anchoring orientations with preferred polar angles. By minimizing Landau-de Gennes free energy and the proposed surface energy, we obtain the equilibrium director configuration around a spherical colloid in the uniform nematic liquid crystal. Our calculations show that the proposed surface energy can cause boojum or/and Saturn-ring defect textures depending on the equilibrium conic angle. We also study the interactions between two spherical colloids with the equilibrium conic angle 45^{∘}, where the surface energy provides both boojum and Saturn-ring defects on the surface of particles. We compare the calculated anisotropic colloidal interactions with experimental observations [B. Senyuk et al., Nat. Commun. 7, 10659 (2016)2041-172310.1038/ncomms10659]. In agreement with experiment, our results show two stable angular assemblies in the close particle-particle separations. Also, the long-range elastic interactions are almost consistent with the hexadecapolar elastic distortion.

Self-assembly of fractal liquid crystal colloids

Nematic liquid crystals are anisotropic fluids that self-assemble into vector fields, which are governed by geometrical and topological laws. Consequently, particulate or droplet inclusions self-assemble in nematic domains through a balance of topological defects. Here, we use double emulsions of water droplets inside radial nematic liquid crystal droplets to form various structures, ranging from linear chains to three-dimensional fractal structures. The system is modeled as a formation of satellite droplets, distributed around a larger, central core droplet and we extend the problem to explain the formation of fractal structures. We show that a distribution of droplet sizes plays a key role in determining the symmetry properties of the resulting geometric structures. The results are relevant to a variety of inclusions, ranging from colloids suspensions to multi-emulsion systems. Such systems have potential applications for novel switchable photonic structures as well as providing wider insights into the packing of self-assembled structures.

Electrokinetic effects in nematic suspensions: Single-particle electro-osmosis and interparticle interactions

Electrokinetic phenomena in a nematic suspension are considered when one or more dielectric particles are suspended in a liquid crystal matrix in its nematic phase. The long-range orientational order of the nematic constitutes a fluid with anisotropic properties. This anisotropy enables charge separation in the bulk under an applied electric field, and leads to streaming flows even when the applied field is oscillatory. In the cases considered, charge separation is seen to result from director field distortions in the matrix that are created by the suspended particles. We use a recently introduced electrokinetic model to study the motion of a single-particle hyperbolic hedgehog pair. We find this motion to be parallel to the defect-particle center axis, independent of field orientation. For a two-particle configuration, we find that the relative force of electrokinetic origin is attractive in the case of particles with perpendicular director anchoring, and repulsive for particles with tangential director anchoring. The study reveals large scale flow properties that are respectively derived from the topology of the configuration alone and from short scale hydrodynamics phenomena in the vicinity of the particle and defect.

Formation of three-dimensional colloidal crystals in a nematic liquid crystal

We investigate the possible structures of three-dimensional colloidal crystals formed when these spherical particles are dispersed in a liquid crystal. The case of a strong homeotropic boundary condition is considered here. Their corresponding defect structures in the space-filler nematic liquid crystal are induced by the presence of the spherical surface of the colloids and produce an attraction between colloidal particles. Here, a standard Landau-de Gennes free energy model for a spatially inhomogeneous liquid crystal is numerically minimized to yield an optimal configuration of both spherical colloids and the orientational field. The stable and metastable structures obtained in this work are described and analyzed according to the type of periodic liquid-crystal defect lines that couple the colloidal spheres together. A large range of the spherical size is covered in this study, corresponding to a 5CB-liquid-crystal comparison for assembling micron- to nano-sized colloidal spheres. Multiple configurations are found for each given particle size and the most stable state is determined by a comparison of the free energies. From large to small colloidal particles, a sequence of structures, which range from quasi-one-dimensional (columnar), to quasi-two-dimensional (planar), and to truly three-dimensional, are found to exist.

Nematic Liquid-Crystal Colloids

This article provides a concise review of a new state of colloidal matter called nematic liquid-crystal colloids. These colloids are obtained by dispersing microparticles of different shapes in a nematic liquid crystal that acts as a solvent for the dispersed particles. The microparticles induce a local deformation of the liquid crystal, which then generates topological defects and long-range forces between the neighboring particles. The colloidal forces in nematic colloids are much stronger than the forces in ordinary colloids in isotropic solvents, exceeding thousands of kBT per micrometer-sized particle. Of special interest are the topological defects in nematic colloids, which appear in many fascinating forms, such as singular points, closed loops, multitudes of interlinked and knotted loops or soliton-like structures. The richness of the topological phenomena and the possibility to design and control topological defects with laser tweezers make colloids in nematic liquid crystals an excellent playground for testing the basic theorems of topology.

Structuralization of magnetic nanoparticles in 5CB liquid crystals

This work is devoted to the study of highly stable composite systems of the liquid crystal 4-n-pentyl-4'-cyanobiphenyl (5CB) doped with CoFe₂O₄ magnetic nanoparticles. Ferronematic samples were prepared with two different weight concentrations: sample A 0.085 wt% and sample B 0.062 wt%. The interaction of CoFe₂O₄ nanoparticles with the liquid crystal was investigated by small-angle X-ray-scattering and magnetization measurements. The obtained results reveal aggregates formed by magnetic nanoparticles that are oriented in the nematic phase. Moreover, the prepared samples show unexpected behaviour of a sudden change in magnetization, which is unusual for such ferronematics.

Topological defects in an unconfined nematic fluid induced by single and double spherical colloidal particles

We present numerical solutions to the Landau-de Gennes free-energy model under the one-constant approximation for systems of single and double spherical colloidal particles immersed in an otherwise uniformly aligned nematic liquid crystal. A perfect homeotropic surface anchoring of liquid-crystal molecules on the spherical surface is considered. A large parameter space is carefully examined, including those in the free-energy model and those describing the dimer configurations and the background liquid-crystal orientation. The stability of the resulting liquid-crystal defects appearing in the neighborhood of the colloidal dimer pair is analyzed in light of the numerical results for their free energies. A number of scenarios are considered: a free dimer pair in a nematic fluid where the free-energy ground states are described in terms of a phase diagram, and a constrained dimer pair where the interparticle distance and the relative orientation of the distance vector to the nematic director can be manipulated. We pay particular attention to the nonsymmetric solutions, which yield several metastable defect states that can be observed in real systems. The high-precision numerical calculations are based on a spectral method, which is an enabling factor that allows us to compare the subtle difference in the free energies of different defect structures.

Analytical description of the Saturn-ring defect in nematic colloids

We derive an analytical formula for the Saturn-ring configuration around a small colloidal particle suspended in nematic liquid crystal. In particular we obtain an explicit expression for the ring radius and its dependence on the anchoring energy. We work within Landau-de Gennes theory: Nematic alignment is described by a tensorial order parameter. For nematic colloids this model had previously been used exclusively to perform numerical computations. Our method demonstrates that the tensorial theory can also be used to obtain analytical results, suggesting a different approach to the understanding of nematic colloidal interactions.

Thermo-optical properties and nonlinear optical response of smectic liquid crystals containing gold nanoparticles

The present work is devoted to the study of the thermo-optical and nonlinear optical properties of smectic samples containing gold nanoparticles with different shapes. By using the time-resolved Z-scan technique, we determine the effects of nanoparticle addition on the critical behavior of the thermal diffusivity and thermo-optical coefficient at the vicinity of the smectic-A-nematic phase transition. Our results reveal that introduction of gold nanoparticles affects the temperature dependence of thermo-optical parameters, due to the local distortions in the orientational order and heat generation provided by guest particles during the laser exposure. Further, we show that a nonlinear optical response may take place at temperatures where the smectic order is well established. We provide a detailed discussion of the effects associated with the introduction gold nanoparticles on the mechanisms behind the thermal transport and optical nonlinearity in liquid-crystal samples.

Three-dimensional positioning and control of colloidal objects utilizing engineered liquid crystalline defect networks

Topological defects in liquid crystals not only affect the optical and rheological properties of the host, but can also act as scaffolds in which to trap nano or micro-sized colloidal objects. The creation of complex defect shapes, however, often involves confining the liquid crystals in curved geometries or adds complex-shaped colloidal objects, which are unsuitable for device applications. Using topologically patterned substrates, here we demonstrate the controlled generation of three-dimensional defect lines with non-trivial shapes and even chirality, in a flat slab of nematic liquid crystal. By using the defect lines as templates and the electric response of the liquid crystals, colloidal superstructures are constructed, which can be reversibly reconfigured at a voltage as low as 1.3 V. Three-dimensional engineering of the defect shapes in liquid crystals is potentially useful in the fabrication of self-healing composites and in stabilizing artificial frustrated phases.

Topological defects can be used not only to modify the properties of liquid crystals but also as scaffolds to build new structures by trapping particles. Here, Yoshida et al. construct three-dimensional colloidal superstructures in a nematic host, which are reconfigurable in an electric field.

Topological defect transformation and structural transition of two-dimensional colloidal crystals across the nematic to smectic-A phase transition

We observe that topological defects in nematic colloids are strongly influenced by the elasticity and onset of smectic layering across the nematic (N) to smectic-A (SmA) phase transition. When approaching the SmA phase from above, the nematic hyperbolic hedgehog defect that accompanies a spherical colloidal inclusion is transformed into a focal conic line in the SmA phase. This phase transformation has a strong influence on the pairwise colloidal interaction and is responsible for a structural transition of two-dimensional colloidal crystals. The pretransitional behavior of the point defect is supported by Landau-de Gennes Q-tensor modeling accounting for the increasing elastic anisotropy.

Equilibrium state of a cylindrical particle with flat ends in nematic liquid crystals

A continuum theory is employed to numerically study the equilibrium orientation and defect structures of a circular cylindrical particle with flat ends under a homeotropic anchoring condition in a uniform nematic medium. Different aspect ratios of this colloidal geometry from thin discotic to long rodlike shapes and several colloidal length scales ranging from mesoscale to nanoscale are investigated. We show that the equilibrium state of this colloidal geometry is sensitive to the two geometrical parameters: aspect ratio and length scale of the particle. For a large enough mesoscopic particle, there is a specific asymptotic equilibrium angle associated to each aspect ratio. Upon reducing the particle size to nanoscale, the equilibrium angle follows a descending or ascending trend in such a way that the equilibrium angle of a particle with the aspect ratio bigger than 1:1 (a discotic particle) goes to a parallel alignment with respect to the far-field nematic, whereas the equilibrium angle for a particle with the aspect ratio 1:1 and smaller (a rodlike particle) tends toward a perpendicular alignment to the uniform nematic direction. The discrepancy between the equilibrium angles of the mesoscopic and nanoscopic particles originates from the significant differences between their defect structures. The possible defect structures related to mesoscopic and nanoscopic colloidal particles of this geometry are also introduced.

Elastic multipoles in the field of the nematic director distortions

Theory of the interaction between all types of elastic dipoles and quadrupoles and distortions of the nematic director is presented. If a particle is small relative to the characteristic distortion length, the interaction is determined by the director derivatives at the particle location. We consider a spherical particle since, even under the standard assumptions of the multipole theory (weak deformations, one constant approximation), the problem can be solved analytically only in this case. Different dipoles interact with different distortion modes (e.g., isotropic dipole interacts with the splay, chiral dipole with the twist, and so on). In the main order, the interaction of a dipole is linear in the director derivatives, and the interaction of a quadrupole is linear in the second-order director derivatives. The theory goes beyond the main-order terms and predicts an effective distortion-induced dipolar component on a particle. This effect is described by the free energy term quadratic in the director derivatives and has contributions both of a bulk and surface origin. The bulk effect takes place even if the director at the particle surface is fixed, whereas the surface effect appears if the surface director is perturbed by the distortions due to a weak surface anchoring. The theory is illustrated by simple examples of the interaction of elastic dipoles with a disclination line, with cholesteric spiral, and with the director distortions in a hybrid cell.

Twist transition of nematic hyperbolic hedgehogs

Stability of an idealized hyperbolic hedgehog in a nematic liquid crystal against a twist transition is investigated by extending the methodology of Rüdinger and Stark [Liq. Cryst. 26, 753 (1999)], where the hedgehog is confined between two concentric spheres. In the ideal hyperbolic-hedgehog the molecular orientation is assumed to rotate proportionally with respect to the inclination angle, θ (and in the opposite sense). However, when splay, k11, and bend, k33, moduli differ this proportionality is lost and the liquid crystal deforms relative to the ideal with bend and splay. Although slight, these deformations are shown to significantly shift the transition if k11/k33 is small. By increasing the degree of confinement the twist transition can be inhibited, a characteristic both hyperbolic and radial hedgehogs have in common. The twist transition of a hyperbolic defect that accompanies a particle is found to be well predicted by the earlier stability analysis of a thick shell.

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.

Hydrodynamic effects in the measurement of interparticle forces in nematic colloids

We propose improved measurement methods of interparticle force between nematic colloids. Although various methods have been utilized for the force measurement, the comparison between the forces obtained by different methods has not been reported. In the frequently used method called the "free-release" method, the hydrodynamic interaction between moving particles has a serious influence on the measurement. In this study we modified those measurement methods by taking the long-ranged hydrodynamic interaction into account. The evaluated forces have been compared with that obtained by the dual beam "optical trap" method, which is free from the hydrodynamic effect. The agreement between them is quantitatively fairly good.

Confined nematic liquid crystal between two spherical boundaries with planar anchoring

Nematic shells of liquid crystals have been provided in microscales. Defect structures in the shells are very essential in the electro-optical applications of such colloidal objects. We have numerically minimized the free energy of symmetric and asymmetric spherical shells of the nematic liquid crystal. Considering degenerate planar anchoring on the surfaces and isotropic nematic elasticity, a variety of defect structures are observed by controlling or varying the thicknesses of the shell and its degree of asymmetry. In symmetric shells, our calculations show that boojums (bipolar) defects appear in thick shells and tetrahedral (baseball) defects in thin shells. In asymmetric shells, while we are in the bipolar regime, the boojums defects transform to trigonal configurations. Free energy landscape shows that in this regime the inner droplet is not stable in the center and it is trapped in an off-center minimum energy position. For the case of thin shells, there are two degenerate director textures with similar tetrahedral configuration of the disclination lines. The levels are split in asymmetric shells. The stability of the inner droplet in the center position depends on director texture. It is stable for one texture and unstable for the other one. For an unstable pattern there is no minimum energy position for the inner droplet and it moves until it touches the outer boundary.

Effect of anchoring energy and elastic anisotropy on spherical inclusions in a nematic liquid crystal

This paper explores how pairs of spherical particles with homeotropic (normal) surface anchoring cluster when immersed in nematic liquid crystal. By means of the Landau-de Gennes continuum theory we calculate how the equilibrium separation of a particle pair depends on the anchoring energy at the particle surface and the elastic anisotropy of the liquid crystal. We find that, for modest to strong anchoring strengths, the particle separation depends linearly on the elastic anisotropy and the inverse of the anchoring strength. Thus, the anchoring strength can be estimated by measuring the particle-pair separation.

Interparticle force between different types of nematic colloids

We have studied the interparticle force between colloidal particles with three different types of defects in nematic liquid crystal by dual-beam optical tweezers. The force between a dipole (D)- and a Saturn-ring (S)-type particle at large interparticle distance R is proportional to R(-4.95±0.05). The force between a D- and a planar (P)-type particle and that between an S- and a P-type particle are, respectively, proportional to R(-5.04±0.08) and R(-5.78±0.13). The observed dependence of the interparticle force on R at large R is in agreement with that predicted by electrostatic analogy. The topological quadrupole moments for S and P particles are evaluated from experimental data. We have also studied the force curves in oblique arrangement against the far-field director for respective pairs. The experimental force curves at large R quantitatively agree with those predicted by electrostatic analogy, but they always become attractive at small R due to the reorientation and deformation of defects. The force profiles for the S-P pair are also compared with those obtained by the recent numerical simulation.

Elastic mediated force between nanoparticles adsorbed on smectic films under an external field

Within the harmonic approximation, we analytically determine the elastic-mediated interaction between colloidal nanoparticles adsorbed on the surface of smectic films under the influence of an external field. Both cases of free-standing films and films deposited over a solid substrate are considered. We show that the asymptotic decay (1/R in free-standing and exponential in deposited films) is not altered by the external field. However, the external field plays distinct roles according to the film configuration, the interparticle distance, the film thickness, and the surface tension at the film-gas interface. We provide a detailed discussion under the light of the distinct mechanisms controlling the undulations of the surface layer.

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 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.

Interparticle force in nematic colloids: comparison between experiment and theory

We have studied the interparticle force between two colloidal particles in a nematic liquid crystal experimentally and theoretically. The force F was directly measured using dual-beam optical tweezers and was numerically calculated from the equilibrium tensor field around the particles. The dependence of F on the center-to-center distance R between the particles was studied not only for equal-sized particles but also for different-sized ones in various kinds of configurations and arrangements. The magnitude of F between different-sized particles in the dipole configuration depends on their relative arrangement. Both experimental and theoretical force curves are found to be in good agreement with each other. At large R, they also make agreement with those predicted by an electrostatic analogy of nematic field.

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.

Light-driven oscillations of entangled nematic colloidal chains

Laser tweezers have been used to drive the oscillations of a chain of entangled colloidal particles in the nematic liquid crystal 5CB. The amplitude and phase of light-driven oscillations have been determined for the motion of individual colloidal particles. The collective motion of 4.8μm silica particles is highly damped for a driving frequency above 0.5Hz. The results were compared to an effective bead-spring model, where the motion of elastically coupled particles is hindered by viscous damping and hydrodynamic coupling. Qualitative agreement between theory and experiment was obtained.

Dependence of interparticle force on temperature and cell thickness in nematic colloids

We have experimentally studied the interparticle force between two particles accompanied by hyperbolic hedgehog defects in a nematic liquid crystal. The force F was measured with dual-beam optical tweezers at various temperatures and in cells with various thicknesses. In a thick cell, the dependence of F on the interparticle distance R obtained at different temperatures can be scaled to a universal curve of F∝R^(-4) for R>3a , where a is the radius of a particle. The effective elastic constant evaluated from F is found to be in good agreement with splay constant of the nematic liquid crystal. In a thin cell, the magnitude of F decreases and the dependence of F on R becomes short-ranged as the thickness of a cell, L , decreases. The reduced force curves, FL(4) against R/L , at different L are found to be scaled to a single theoretical curve which has been proposed recently.

Small surface pretilt strikingly affects the director profile during Poiseuille flow of a nematic liquid crystal

Conventional nematic liquid crystal cells are fabricated with small surface pretilt of the director induced by rubbed polymer alignment. Depending on the orientation of the bounding surfaces, this may lead to two slightly different untwisted director configurations, splay and parallel. This small difference leads to remarkably different director profiles during pressure-driven flow, observed here using optical conoscopy. Data show excellent agreement with numerical modeling from Leslie-Ericksen-Parodi theory.

Arrangement dependence of interparticle force in nematic colloids

We have experimentally and theoretically studied the interparticle force between two colloidal particles with different sizes accompanied by hyperbolic hedgehog defects in a nematic liquid crystal. The force f was directly measured using dual-beam optical tweezers and calculated theoretically from the equilibrium tensor field around the particles. The dependence of f on the center-to-center distance between particles of different sizes R is different from that for particles with the same size. The magnitude of f depends on the relative arrangement of the particles: f is larger when a defect between the particles belongs to the larger particle. From the theoretical calculation, the difference in f between the two arrangements, deltaf, monotonically increases with increasing size difference. The difference deltaf was experimentally and theoretically found to be proportional to R(-4.6) at large R. The obtained exponent is comparable to the exponent of -5 predicted by electrostatic analogy.

Long-range elastic-mediated interaction between nanoparticles adsorbed on free-standing smectic films

We determine the elastic-mediated interaction between colloidal nanoparticles adsorbed on the surface of free-standing smectic films. In contrast with the short-range character of the elastic-mediated force between particles adsorbed on smectic films supported by a solid substrate, the effective force acquires a long-range character in free-standing films, decaying with the particles distance R as slow as 1/R . We also discuss the dependence of the effective interaction potential on the surface tension gamma and film thickness. We show that it decays as 1/gamma in the regime of large surface tensions and becomes independent of the film thickness at a characteristic surface tension.

Quadrupolar particles in a nematic liquid crystal: effects of particle size and shape

We investigate the effects of particle size and shape on the quadrupolar (Saturn-ring-like) defect structures formed by a nematic liquid crystal around nm-sized and mum -sized particles with spherical and spherocylindrical shapes. We also report results for the potentials of mean force in our systems, calculated using a mesoscale theory for the tensor order parameter Q of the nematic. Our results indicate that for pairs of nm-sized particles in close proximity, the nematic forms "entangled hyperbolic" defect structures regardless of the shape of the nanoparticles. In our calculations with nanoparticles we did not observe any other entangled or unentangled defect structures, in contrast to what was reported for pairs of mum -sized spherical particles. Such a finding suggests that the "entangled hyperbolic" defect structures are the most stable for pairs of nanoparticles in close proximity. For pairs of mum -sized particles, our results indicate that the nematic forms entangled "figure-of-eight" defect structures around pairs of spheres and spherocylinders. Our results suggest that the transition between "entangled hyperbolic" and figure-of-eight defect structures takes place when the diameter of the particle is between D=100 nm and 1 microm . We have also calculated the torques that develop when pairs of spherocylindrical nanoparticles in a nematic approach each other. Our calculations suggest that the nematic-mediated interactions between the nm-sized particles are fairly strong, up to 5700 k{B}T for the case of pairs of spherocylindrical nanoparticles arranged with their long axis parallel to each other. Furthermore, these interactions can make the particles to bind together at specific locations, and thus could be used to assemble the particles into ordered structures with different morphologies.

Defect-mediated emulsification in two dimensions

We consider two-dimensional dispersions of droplets of isotropic phase in a liquid with an XY -like order parameter, tilt, nematic, and hexatic symmetries being included. Strong anchoring boundary conditions are assumed. Textures for a single droplet and a pair of droplets are calculated and a universal droplet-droplet pair potential is obtained. The interaction of dispersed droplets via the ordered phase is attractive at large distances and repulsive at short distances, which results in a well defined preferred separation for two droplets and topological stabilization of the emulsion. This interaction also drives self-assembly into chains. Preferred separations and energy barriers to coalescence are calculated, and the effects of thermal fluctuations and film thickness are discussed.

Dynamic simulation of droplet interaction and self-assembly in a nematic liquid crystal

We use dynamic simulations to explore the pairwise interaction and multiparticle assembly of droplets suspended in a nematic liquid crystal. The computation is based on a regularized Leslie-Ericksen theory that allows orientational defects. The homeotropic anchoring on the drop surface is of sufficient strength as to produce a satellite point defect near the droplet. Based on the position of the defects relative to the host droplet and the far-field molecular orientation, we have identified five types of pairwise attractive and repulsive forces. In particular, long-range attraction between two droplets with their line of centers along the far-field orientation decays as R-4, with R being the center-to-center separation. This agrees with prior static calculations and a phenomenological model that treats the attraction as that between two dipoles. For interaction in shorter ranges, our simulations agree qualitatively with experimental measurements and static calculations. However, there is considerable quantitative discrepancy among the few existing studies and our simulation. We suggest that this is partly due to the dynamic nature of the process, which has never been taken into account in prior calculations. Multidrop simulations show the formation of linear chains through pairwise interactions between nearby droplets. Parallel chains repel or attract each other depending on the relative orientation of the drop-to-defect vector. These are consistent with experimental observations of chain formation and two-dimensional self-assembly in bulk nematics and smectic-C films.

Inclusions in free standing smectic liquid crystal films

Colloidal inclusions in thin free standing liquid crystal films are ideal model systems for 2D anisotropic dispersions. Different types of self-organization in chain and lattice structures have been observed. The orientational elasticity of the anisotropic matrix and capillary forces are the dominating interaction mechanisms between solid or liquid inclusions, the director field, and dislocations of the films. We give an overview of the progress in this field, focussing on different inclusion types and their interactions in thermotropic smectic films.

Force between colloidal particles in a nematic liquid crystal studied by optical tweezers

We measure the dependence of the interparticle force F on the distance R between two colloidal particles with hyperbolic hedgehog defects in a nematic liquid crystal using optical tweezers. The particle-defect pair can be regarded as an elastic "dipole" in the electrostatic analogy. In a parallel configuration, where the dipole vectors are parallel with each other, F is attractive and proportional to R(-4). However, F becomes repulsive at small R due to the existence of a defect between the particles. In an antiparallel configuration, where the particles directly face each other, F is repulsive over the whole range of R and proportional to R(-3.6). In another antiparallel configuration, where two hyperbolic hedgehog defects directly face each other, F is proportional to R(-3.6) and F at small R turns out to be attractive upon tilting the dipoles. Furthermore, we yield the force between particles connected by a stringlike defect called a bubblegum defect.

Different mechanisms of nucleation and self-organization of droplets in ferroelectric smectic membranes

New mechanisms of droplet nucleation and self-organization in ferroelectric membranes are described. The droplets may be accompanied by different number of topological defects (zero, one, two) whose location may be on the droplet boundary or in the membrane. Nucleation and self-organization of droplets with total topological charge S = 0 , S = + 1 and S = - 1 were investigated. We found that an S = - 1 topological defect may be the center of both droplet nucleation and chain formation. This mechanism of chaining drastically differs from the droplet self-organization described earlier which is realized by attraction of droplet-defect pairs. Our observations demonstrate new possibilities for manipulating the inclusions and their self-organization in smectic membranes.

Nanoparticles in nematic liquid crystals: Interactions with nanochannels

A mesoscale theory for the tensor order parameter Q is used to investigate the structures that arise when spherical nanoparticles are suspended in confined nematic liquid crystals (NLCs). The NLC is "sandwiched" between a wall and a small channel. The potential of mean force is determined between particles and the bottom of the channels or between several particles. Our results suggest that strong NLC-mediated interactions between the particles and the sidewalls of the channels, on the order of hundreds of k(B)T, arise when the colloids are inside the channels. The magnitude of the channel-particle interactions is dictated by a combination of two factors, namely, the type of defect structures that develop when a nanoparticle is inside a channel, and the degree of ordering of the nematic in the region between the colloid and the nanochannel. The channel-particle interactions become stronger as the nanoparticle diameter becomes commensurate with the nanochannel width. Nanochannel geometry also affects the channel-particle interactions. Among the different geometries considered, a cylindrical channel seems to provide the strongest interactions. Our calculations suggest that small variations in geometry, such as removing the sharp edges of the channels, can lead to important reductions in channel-particle interactions. Our calculations for systems of several nanoparticles indicate that linear arrays of colloids with Saturn ring defects, which for some physical conditions are not stable in a bulk system, can be stabilized inside the nanochannels. These results suggest that nanochannels and NLCs could be used to direct the assembly of nanoparticles into ordered arrays with unusual morphologies.

Configuration of a chiral smectic-C film with a circular inclusion

It was shown experimentally (P.V. Dolganov et al., Europhys. Lett. 76, 250 (2006)) and by numerical calculations (C. Bohley, R. Stannarius, Eur. Phys. J. E 23, 25 (2007)) that the c -director profile of a two-dimensional chiral smectic-C (SmC) film around a circular inclusion adopts dipolar rather than quadrupolar configuration observed in achiral SmC films. We give an analytical argument on how spontaneous bend inherent in chiral SmC liquid crystals influences the configuration of a SmC liquid crystal film around a circular inclusion imposing tangential anchoring. We find how the angle alpha between two surface defects seen from the center of the inclusion depends on the radius of the inclusion R and the strength of the spontaneous bend q . We show, however, that the contribution of the spontaneous bend to the free energy suffers from mathematical ambiguity; it depends on the mathematical treatment of the outer boundary even when it is at infinity. This might indicate that the shape as well as the treatment of the outer boundary of the film can significantly influence the equilibrium configuration of the c -director and the position of the surface defects.

Colloidal inclusions in smectic films with spontaneous bend

Inclusions in free-standing smectic films are simple model systems for two-dimensional anisotropic dispersions. From theory and experiment, different topologies of elastic distortions of the embedding liquid crystal are known. Quadrupolar and different dipolar defect configurations in the vicinity of the inclusion are possible, and these configurations determine the type of interactions between the inclusions. The quadrupolar configuration is often energetically preferred. We show, however, that dipolar director configurations around inclusions can be energetically favourable over quadrupolar arrangements in chiral smectics, as a consequence of a spontaneous-bend term in the elastic-energy formulation. As the inclusion size influences the selection of the deformation types, the corresponding spontaneous-bend constant can be estimated for the strong anchoring limit if the c -director fields around inclusions of different diameters are taken into account.

Air tube formation at the freezing transition in nematic liquid crystals

A phenomenon is presented, which changes the shape of gas bubbles in liquid crystals and also creates long gas tubes. The system consists of air bubbles which are injected into a nematic liquid crystal host. The shape of these air bubbles changes from spherical to ellipsoidal by initiating freezing of the sample. Furthermore, long gas tubes are formed from the air which was formerly dissolved in the liquid crystal. The gas tubes are created by the progression of the crystalline-liquid interface. Their length can reach up to 40 times their diameter. The diameter of the tubes depends on the pressure applied to the system, as well as on the interface velocity.

Director-configurational transitions around microbubbles of hydrostatically regulated size in liquid crystals

A high-pressure technique is introduced which allows a continuous variation of the inclusion size in liquid crystal colloids. We use a nematic liquid crystal host into which micrometer-sized gas bubbles are injected. By applying hydrostatic pressures, the diameter of these gas bubbles can be continuously decreased via compression and absorption of gas into the host liquid crystal, so that the director configurations around a single bubble can be investigated as a function of the bubble size. The theoretically predicted transition from a hyperbolic hedgehog to a Saturn-ring configuration, on reduction of the particle size below a certain threshold, is confirmed to occur at the radius of a few micrometers.

Colloidal aggregation in a nematic liquid crystal: topological arrest of particles by a single-stroke disclination line

We numerically study many-body interactions among colloidal particles suspended in a nematic liquid crystal, using a fluid particle dynamics method, which properly incorporates dynamical coupling among particles, nematic orientation, and flow field. Based on simulation results, we propose a new type of interparticle interaction in addition to well-known quadrupolar interaction for particles accompanying Saturn-ring defects. This interaction is mediated by the defect of the nematic phase: upon nematic ordering, a closed disclination loop binds more than two particles to form a sheetlike dynamically arrested structure. The interaction depends upon the topology of a disclination loop binding particles, which is determined by aggregation history.

Anisotropic nanoparticles immersed in a nematic liquid crystal: defect structures and potentials of mean force

We report results for the potential of mean force (PMF) and the defect structures that arise when spherocylindrical nanoparticles are immersed in a nematic liquid crystal. Using a dynamic field theory for the tensor order parameter Q of the liquid crystal, we analyzed configurations, including one, two, and three elongated particles, with strong homeotropic anchoring at their surfaces. For systems with one nanoparticle, the most stable configuration is achieved when the spherocylinder is placed with its long axis perpendicular to the far-field director, for which the defect structure consists of an elongated Saturn ring. For systems with two or three nanoparticles with their long axes placed perpendicular to the far-field director, at small separations the defect structures consist of incomplete Saturn rings fused with new disclination rings orthogonal to the original ones, in analogy to results previously observed for spherical nanoparticles. The shape of these orthogonal rings depends on the nanoparticles' configuration, i.e., triangular, linear, or parallel with respect to their long axis. A comparison of the PMFs indicates that the latter configuration is the most stable. The stability of the different arrays depends on whether orthogonal disclination rings form or not, their size, and the curvature effects in the interparticle regions. Our results suggest that the one-elastic-constant approximation is valid for the considered systems; similar results were obtained when a three-constant expression is used to represent the elastic free energy. The attractive interactions between the elongated particles were compared to those observed for spheres of similar diameters. Similar interparticle energies were observed for linear arrays; in contrast, parallel and triangular arrays of spherocylinders yielded interactions that were up to 3.4 times stronger than those observed for spherical particles.

Energetics of 2D colloids in free-standing smectic-C films

The formation of regular colloid patterns in free-standing smectic films at the transition from the smectic-C to the isotropic or nematic phase is well known experimentally. The self-organization of isotropic or nematic droplets is caused by their mutual interaction, mediated by elastic distortions of the local director in the surrounding liquid crystal. These distortions are related to the anchoring conditions of the director at the droplet border. We describe analytically the energetics of the liquid crystal environment of a single droplet in one-constant approximation. A method of complex analysis, Conformal Mapping, is employed. Following a suggestion of Dolganov et al. (Phys. Rev. E. 73, 041706 (2006)), energetics of chain and grid patterns built from the colloids are investigated numerically in order to explain experimentally observed formations and their director fields.

Measurement of elastic forces between iron colloidal particles in a nematic liquid crystal

The forces that arise between two iron particles in a nematic liquid crystal with a strong homeotropic anchoring were studied. For the first time, the short range repulsive force resulting from the presence of a hedgehog defect between two particles was precisely determined thanks to application of a small magnetic field and observation of the equilibrium position resulting from the balance between the elastic and magnetic forces. Above a given threshold force, the particles stuck together whereas the hedgehog defect was expelled and transformed into a Saturn ring located between the particles. The attractive part of the interparticle force was determined with the same method on the entire range of separation distances; we found that the equilibrium distance between two particles was r = 1.19 +/- 0.05 <d> (<d> was the average diameter of the pair of particles).