Confinement effect on interparticle potential in nematic colloids

Out-of-equilibrium interactions and collective locomotion of colloidal spheres with squirming of nematoelastic multipoles

Many living and artificial systems show similar emergent behavior and collective motions on different scales, starting from swarms of bacteria to synthetic active particles, herds of mammals, and crowds of people. What all these systems often have in common is that new collective properties like flocking emerge from interactions between individual self-propelled or driven units. Such systems are naturally out-of-equilibrium and propel at the expense of consumed energy. Mimicking nature by making self-propelled or externally driven particles and studying their individual and collective motility may allow for deeper understanding of physical underpinnings behind collective motion of large groups of interacting objects or beings. Here, using a soft matter system of colloids immersed into a liquid crystal, we show that resulting so-called nematoelastic multipoles can be set into a bidirectional locomotion by external oscillating electric fields. Out-of-equilibrium elastic interactions between such colloidal objects lead to collective flock-like behaviors emerging from time-varying elasticity-mediated interactions between externally driven propelling particles. Repulsive elastic interactions in the equilibrium state can be turned into attractive interactions in the out-of-equilibrium state under applied external electric fields. We probe this behavior at different number densities of colloidal particles and show that particles in dense dispersions collectively select the same direction of a coherent motion due to elastic interactions between near neighbors. In our experimentally implemented design, their motion is highly ordered and without clustering or jamming often present in other colloidal transport systems, which is promising for technological and fundamental-science applications, like nano-cargo transport, out-of-equilibrium assembly, and microrobotics.

Self-Timed and Spatially Targeted Delivery of Chemical Cargo by Motile Liquid Crystal

A key challenge underlying the design of miniature machines is encoding materials with time- and space-specific functional behaviors that require little human intervention. Dissipative processes that drive materials beyond equilibrium and evolve continuously with time and location represent one promising strategy to achieve such complex functions. This work reports how internal nonequilibrium states of liquid crystal (LC) emulsion droplets undergoing chemotaxis can be used to time the delivery of a chemical agent to a targeted location. During ballistic motion, hydrodynamic shear forces dominate LC elastic interactions, dispersing microdroplet inclusions (microcargo) within double emulsion droplets. Scale-dependent colloidal forces then hinder the escape of dispersed microcargo from the propelling droplet. Upon arrival at the targeted location, a circulatory flow of diminished strength allows the microcargo to cluster within the LC elastic environment such that hydrodynamic forces grow to exceed colloidal forces and thus trigger the escape of the microcargo. This work illustrates the utility of the approach by using microcargo that initiate polymerization upon release through the outer interface of the carrier droplet. These findings provide a platform that utilizes nonequilibrium strategies to design autonomous spatial and temporal functions into active materials.

Effect of confinement and topology: 2-TIPS vs. MIPS

2-TIPS (two temperature induced phase separation) refers to the phase separation phenomenon observed in mixtures of active and passive particles which are modelled using scalar activity. The active particles are connected to a thermostat at high temperature while the passive particles are connected to the thermostat at low temperature and the relative temperature difference between "hot" and "cold" particles is taken as the measure of the activity χ of the non-equilibrium system. The study of such binary mixtures of hot and cold particles under various kinds of confinement is an important problem in many physical and biological processes. The nature and extent of phase separation are heavily influenced by the geometry of confinement, activity, and density of the non-equilibrium binary mixture. Investigating such 3D binary mixtures confined by parallel walls, we observe that the active and passive particles phase separate, but the extent of phase separation is reduced compared to bulk phase separation at high densities and enhanced at low densities. However, when the binary mixture of active and passive particles is confined inside a spherical cavity, the phase separation is radial for small radii of the confining sphere and the extent of phase separation is higher compared to their bulk counterparts. Confinement leads to interesting properties in the passive (cold) region like enhanced layering and high compression in the direction parallel to the confining wall. In 2D, both the bulk and confined systems of the binary mixture show a significant decrement in the extent of phase separation at higher densities. This observation is attributed to the trapping of active particles inside the passive cluster, which increases with density. Thus the 2D systems show structures more akin to dense-dilute phase co-existence, which is observed in motility induced phase separation in 2D active systems. The binary mixture constrained on the spherical surface also shows similar phase co-existence. Our analyses reveal that the coexistent densities observed in 2-TIPS on the spherical surface agree with the findings of previous studies on MIPS in active systems on a sphere.

Thin pyramidal cones in nematic liquid crystals

The present study investigates the arrangement of hollow pyramidal cone shells and their interactions with degenerate planar anchoring on the inner and outer surfaces of particles within the nematic host. The shell thickness is in order of the nematic coherence length. The numerical behavior of colloids is determined by minimizing the Landau-de Gennes free energy in the presence of the Fournier surface energy and using the finite element method. Colloidal pyramidal cones can orient parallel and perpendicular with the far director orientation. In the parallel alignment, we found the splay director distortion into the pyramid with two boojum defects at the inner and outer tips. The director shows bending distortion without defect patterns when the pyramid is aligned perpendicularly. They induce long-range dipolar interaction and can form nested structures in close contact.

Field-triggered vertical positional transition of a microparticle suspended in a nematic liquid crystal cell

In this paper, based on the numerical calculation of total energy utilizing the Green's function method, we investigate how a field-triggered vertical positional transition of a microparticle suspended in a nematic liquid crystal cell is influenced by the direction of the applied field, surface anchoring feature, and nematic's dielectric properties. The new equilibrium position of the translational movement is decided via a competition between the buoyant force and the effective force built on the microparticle by the elastic energy gradient along the vertical direction. The threshold value of external field depends on thickness L and Frank elastic constant K and slightly on the microparticle size and density, in a Fréedericksz-like manner, but by a factor. For a nematic liquid crystal cell with planar surface alignment, a bistable equilibrium structure for the transition is found when the direction of the applied electric field is (a) perpendicular to the two plates of the cell with positive molecular dielectric anisotropy or (b) parallel to the two plates and the anchoring direction of the cell with negative molecular dielectric anisotropy. When the electric field applied is parallel to both plates and perpendicular to the anchoring direction, the microparticle suspended in the nematic liquid crystal tends to be trapped in the midplane, regardless of the sign of the molecular dielectric anisotropy. Such a phenomenon also occurs for negative molecular dielectric anisotropy if the external field is applied perpendicular to the two plates. Explicit formulas proposed for the critical electric field agree extremely well with the numerical calculation.

Perspectives in Liquid-Crystal-Aided Nanotechnology and Nanoscience

The research field of liquid crystals and their applications is recently changing from being largely focused on display applications and optical shutter elements in various fields, to quite novel and diverse applications in the area of nanotechnology and nanoscience. Functional nanoparticles have recently been used to a significant extent to modify the physical properties of liquid crystals by the addition of ferroelectric and magnetic particles of different shapes, such as arbitrary and spherical, rods, wires and discs. Also, particles influencing optical properties are increasingly popular, such as quantum dots, plasmonic, semiconductors and metamaterials. The self-organization of liquid crystals is exploited to order templates and orient nanoparticles. Similarly, nanoparticles such as rods, nanotubes and graphene oxide are shown to form lyotropic liquid crystal phases in the presence of isotropic host solvents. These effects lead to a wealth of novel applications, many of which will be reviewed in this publication.

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.

Tunable colloid trajectories in nematic liquid crystals near wavy walls

The ability to dictate the motion of microscopic objects is an important challenge in fields ranging from materials science to biology. Field-directed assembly drives microparticles along paths defined by energy gradients. Nematic liquid crystals, consisting of rod-like molecules, provide new opportunities in this domain. Deviations of nematic liquid crystal molecules from uniform orientation cost elastic energy, and such deviations can be molded by bounding vessel shape. Here, by placing a wavy wall in a nematic liquid crystal, we impose alternating splay and bend distortions, and define a smoothly varying elastic energy field. A microparticle in this field displays a rich set of behaviors, as this system has multiple stable states, repulsive and attractive loci, and interaction strengths that can be tuned to allow reconfigurable states. Microparticles can transition between defect configurations, move along distinct paths, and select sites for preferred docking. Such tailored landscapes have promise in reconfigurable systems and in microrobotics applications.

Nematic liquid crystals have a rich energy landscape which can define elastic fields to guide colloidal assembly. Here the authors show controllable trapping of colloidal particles by placing them in a system with wavy walls which are exploited to obtain stable, metastable and unstable equilibria.

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.

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.

Motion of a colloidal particle in a nonuniform director field of a nematic liquid crystal

We investigate the dynamics of a single spherical particle immersed in a nematic liquid crystal. A nonuniform director field is imposed on the substrate by a stripe alignment pattern with splay deformation. The particle of homeotropic anchoring at the surface is accompanied by hyperbolic hedgehog or Saturn-ring defects. The particle motion is dependent on the defect structure. We study the two types of motions theoretically and confirm the obtained results experimentally. The particle accompanied by a hyperbolic hedgehog defect is pulled to a deformed region to relax the elastic deformation energy. The motion occurs in the direction heading the hyperbolic hedgehog defect of a particle in a twist region. The position exhibits a weak S-shaped change as a function of time. The particle accompanied by a Saturn-ring defect shows insignificant motion due to its relatively small deformation energy.

Colloidal spirals in nematic liquid crystals

One of the central experimental efforts in nematic colloids research aims to explore how the interplay between the geometry of particles along with the accompanying nematic director deformations and defects around them can provide a means of guiding particle self-assembly and controlling the structure of particle-induced defects. In this work, we design, fabricate, and disperse low-symmetry colloidal particles with shapes of spirals, double spirals, and triple spirals in a nematic fluid. These spiral-shaped particles, which are controlled by varying their surface functionalization to provide tangential or perpendicular boundary conditions of the nematic molecular alignment, are found inducing director distortions and defect configurations with non-chiral or chiral symmetry. Colloidal particles also exhibit both stable and metastable multiple orientational states in the nematic host, with a large number of director configurations featuring both singular and solitonic nonsingular topological defects accompanying them, which can result in unusual forms of colloidal self-assembly. Our findings directly demonstrate how the symmetry of particle-generated director configurations can be further lowered, or not, as compared to the low point group symmetry of solid micro-inclusions, depending on the nature of induced defects while satisfying topological constraints. We show that achiral colloidal particles can cause chiral symmetry breaking of elastic distortions, which is driven by complex three-dimensional winding of induced topological line defects and solitons.

Spherical microparticles with Saturn ring defects and their self-assembly across the nematic to smectic-A phase transition

We report experimental studies on the Saturn ring defect associated with a spherical microparticle across the nematic (N) to smectic-A (SmA) phase transition. We observe that the director distortion around the microparticle changes rapidly with temperature. The equilibrium interparticle separation and the angle between two quadrupolar particles in the N phase are larger than those of the SmA phase. They are almost independent of the temperature in both phases, except for a discontinuous jump at the transition. We assembled a few particles using a laser tweezer to form a two-dimensional colloidal crystal in the N phase. The lattice structure of the crystal dissolves irreversibly across the N-SmA phase transition. The results on the pretransitional behavior of the defect are supported by the Landau-de Gennes Q-tensor modeling.

Orientation, interaction and laser assisted self-assembly of organic single-crystal micro-sheets in a nematic liquid crystal

Colloidal self-assembly has been one of the major driving themes in material science to obtain functional and advanced optical materials with complex architecture. Most of the nematic colloids reported so far are based on the optically isotropic spherical microparticles. We study organic single crystal micro-sheets and investigate their orientation, interaction and directed assembly in a nematic liquid crystal. The micro-sheets induce planar surface anchoring of the liquid crystal. The elasticity mediated pair interaction of micro-sheets shows quadrupolar characteristics. The average orientation angle of the micro-sheets in a planar cell and the angle between two micro-sheets in a homeotropic cell are supported by the Landau-de Gennes Q-tensor modeling. The self-assembly of the micro-sheets is assisted by a laser tweezer to form larger two-dimensional structures which have the potential for application of colloids in photonics.

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.

Effects of confinement, surface-induced orientations and strain on dynamical behaviors of bacteria in thin liquid crystalline films

We report on the organization and dynamics of bacteria (Proteus mirabilis) dispersed within lyotropic liquid crystal (LC) films confined by pairs of surfaces that induce homeotropic (perpendicular) or hybrid (homeotropic and parallel orientations at each surface) anchoring of the LC. By using motile vegetative bacteria (3 µm in length) and homeotropically aligned LC films with thicknesses that exceed the length of the rod-shaped cells, a key finding reported in this paper is that elastic torques generated by the LC are sufficiently large to overcome wall-induced hydrodynamic torques acting on the cells, thus leading to LC-guided bacterial motion near surfaces that orient LCs. This result extends to bacteria within LC films with hybrid anchoring, and leads to the observation that asymmetric strain within a hybrid aligned LC rectifies motions of motile cells. In contrast, when the LC film thickness is sufficiently small that confinement prevents alignment of the bacteria cells along a homeotropically aligned LC director (achieved using swarm cells of length 10-60 µm), the bacterial cells propel in directions orthogonal to the director, generating transient distortions in the LC that have striking "comet-like" optical signatures. In this limit, for hybrid LC films, we find LC elastic stresses deform the bodies of swarm cells into bent configurations that follow the LC director, thus unmasking a coupling between bacterial shape and LC strain. Overall, these results provide new insight into the influence of surface-oriented LCs on dynamical bacterial behaviors and hint at novel ways to manipulate bacteria using confined LC phases that are not possible in isotropic solutions.

Geometry-guided colloidal interactions and self-tiling of elastic dipoles formed by truncated pyramid particles in liquid crystals

The progress of realizing colloidal structures mimicking natural forms of organization in condensed matter is inherently limited by the availability of suitable colloidal building blocks. To enable new forms of crystalline and quasicrystalline self-organization of colloids, we develop truncated pyramidal particles that form nematic elastic dipoles with long-range electrostaticlike and geometry-guided low-symmetry short-range interactions. Using a combination of nonlinear optical imaging, laser tweezers, and video microscopy, we characterize colloidal pair interactions and demonstrate unusual forms of self-tiling of these particles into crystalline, quasicrystalline, and other arrays. Our findings are explained using an electrostatics analogy along with liquid crystal elasticity and symmetry breaking considerations, potentially expanding photonic and electro-optic applications of colloids.

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.

Particle selection through topographic templates in nematic colloids

Liquid crystal colloids have been proposed as suitable candidates for responsive photonic crystals. Large scale growth of such colloidal systems is, however, a challenge and recently template-assisted assembly has been proposed to guide the growth of colloidal crystals, with controlled symmetries, in nematic liquid crystals. Known for their long-range anisotropic interactions, these colloidal systems are stabilized typically at the center of the cells due to strong particle-wall repulsion from the confining substrates. This behaviour is dramatically changed in the presence of topographic patterning. Here we propose the use of topographic modulation of surfaces to select and localize particles in nematic colloids. By considering convex and concave deformations of one of the confining surfaces we show that the colloid-flat surface repulsion may be enhanced or switched into an attraction. In particular, we find that when the colloidal particles have the same anchoring conditions as the patterned surfaces, they are strongly attracted to concave dimples, while if they exhibit different anchoring conditions they are pinned at the top of convex protrusions. Although dominated by elastic interactions the first mechanism is reminiscent of the depletion induced attraction or of the key-lock mechanism, while the second is specific to liquid crystal colloids. These long-ranged, highly tunable, surface-colloid interactions contribute to the development of template-assisted assembly of large colloidal crystals, with well defined symmetries, as required for applications.

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.

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.

Bonded boojum-colloids in nematic liquid crystals

We investigate bonded boojum-colloids in nematic liquid crystals, configurations where two colloids with planar degenerate anchoring are double-bonded through line defects connecting their surfaces. This bonded structure promotes the formation of linear chains aligned with the nematic director. We show that the bonded configuration is the global minimum in systems that favor twist deformations. In addition, we investigate the influence of confinement on the stability of bonded boojum-colloids. Although the unbonded colloid configuration, where the colloids bundle at oblique angles, is favored by confinement, the bonded configuration is again the global minimum for liquid crystals with sufficiently small twist elastic constants.

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.

Transport of particles by a thermally induced gradient of the order parameter in nematic liquid crystals

We demonstrate manipulation and transport of microparticles and even fluorescent molecules by the thermally induced gradient of the order parameter in the nematic liquid crystal. We use IR light absorption of the tightly focused beam of laser tweezers to heat locally a thin layer of the nematic liquid crystal by several degrees. This creates a spatial gradient of temperature of the nematic liquid crystal over separations of several tens of micrometers. We show that a dipolar colloidal particle is attracted into the hot spot of the laser tweezers. The depth of the trapping potential scales linearly with particle radius, indicating that the trapping mechanism is due to elastic self-energy of the distorted nematic liquid crystal around the particle and softening of the elasticity with increased temperature of the liquid crystal. We also demonstrate that this thermal trapping mechanism is efficient down to the nanoscale, as fluorescent molecules are also transported into hotter regions of the liquid crystal. This effect is absent in the isotropic phase, which calls into question particle transport due to the Soret effect.

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.

Nematic colloids, topology and photonics

We review and discuss recent progress in the field of nematic colloids, with an emphasis on possible future applications in photonics. The role of the topology is described, based on experimental manipulations of the topological defects in nematic colloids. The topology of the ordering field in nematics provides the forces between colloidal particles that are unique to these materials. We also discuss recent progress in the new field of active microphotonic devices based on liquid crystals (LCs), where chiral nematic microlasers and tuneable nematic microresonators are just two of the recently discovered examples. We conclude that the combination of topology and microphotonic devices based on LCs provides an interesting platform for future progress in the field of LCs.

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.

Chiral dipole induced by azimuthal anchoring on the surface of a planar elastic quadrupole

A spherical colloid with the tangential surface nematic director, aligned along the surface meridians, is known as a planar elastic quadrupole. The azimuthal anchoring, however, can induce a deviation of the planar director from the meridional lines. We show that a helical component of the planar surface director at the spherical surface of a planar quadrupole removes all the reflection symmetry planes and gives rise to a chiral elastic dipolar component. Using an ansatz approach, we consider the interplay between the quadrupole and anchoring-induced chiral dipole components. The chirality is enhanced by the bend-twist anisotropy. The interaction of the chiral components changes the attraction directions of two such colloids. In particular, a point appears at which the quadrupolar repulsion is balanced by the dipolar attraction.

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.

Square colloidal lattices and pair interaction in a binary system of quadrupolar nematic colloids

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.

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.

Mesoscopic modelling of colloids in chiral nematics

We present numerical modelling of colloidal particles in chiral nematics with cubic symmetry (blue phases) within the framework of the Landau-de Gennes free energy. The interaction potential of a single, nano-sized colloidal particle with a -1/2 disclination line is calculated as a generic trapping mechanism for particles within the cholesteric blue phases. The interaction potential is shown to be highly anisotropic and have threefold rotational symmetry. We discuss the equilibration of the colloidal texture with respect to particle positions and the unit cell size of the blue phase. We also describe how preservation of the liquid crystal volume and the number of particles allows blue phase colloidal structures with different unit cell sizes and configurations to be compared numerically.

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.

Self-assembled artificial cilia

Due to their small dimensions, microfluidic devices operate in the low Reynolds number regime. In this case, the hydrodynamics is governed by the viscosity rather than inertia and special elements have to be introduced into the system for mixing and pumping of fluids. Here we report on the realization of an effective pumping device that mimics a ciliated surface and imitates its motion to generate fluid flow. The artificial biomimetic cilia are constructed as long chains of spherical superparamagnetic particles, which self-assemble in an external magnetic field. Magnetic field is also used to actuate the cilia in a simple nonreciprocal manner, resulting in a fluid flow. We prove the concept by measuring the velocity of a cilia-pumped fluid as a function of height above the ciliated surface and investigate the influence of the beating asymmetry on the pumping performance. A numerical simulation was carried out that successfully reproduced the experimentally obtained data.

Wall-colloid interaction in nematic solvents: external field effects

We propose a molecular theory of colloid-wall interactions in nematic media that predicts a new effective force acting on colloidal particles in the presence of an external field. In contrast to the so-called 'image' interaction that is always repulsive at long distances, the force identified here can be attractive or repulsive, depending on the type of anchoring at the wall and colloidal surfaces. The effective force on a colloidal particle decreases with distance s from the wall as exp(-s/ξ), where ξ is a magnetic (electric) coherence length. At weak fields the force is proportional to (Σ/ξ)(3) for 'quadrupolar' colloids and to (Σ/ξ)(2) for 'dipoles', where Σ is the colloidal diameter. A brief discussion of recent experiments in the light of our findings is presented.

Confinement effect on the interaction between colloidal particles in a nematic liquid crystal: an analytical study

Motivated by a recent experimental study on the interaction between colloidal particles in a confined nematic liquid crystal [M. Vilfan, N. Osterman, M. Copic, M. Ravnik, S. Zumer, J. Kotar, D. Babic, and I. Poberaj, Phys. Rev. Lett. 101, 237801 (2008)], we discuss in an analytical manner how the interaction potential U between spherical colloidal particles in a confined nematic cell behaves as a function of the interparticle distance r. We show that the short-range potential follows a power law U(r) approximately r(-5) as expected from the quadrupolar nature of the interaction, while the long-range potential is dominated by an exponential function U(r) approximately sqrt[d/r] exp(-2pir/d), where d is the cell thickness. These two regimes are interchanged at r/d approximately 0.8. This behavior of U(r) is in a good semiquantitative agreement with the experimental finding.