Colloid-wall interaction in a nematic liquid crystal: the mirror-image method of colloidal nematostatics

Entangled nematic disclinations using multi-particle collision dynamics

Colloids dispersed in nematic liquid crystals form topological composites in which colloid-associated defects mediate interactions while adhering to fundamental topological constraints. Better realising the promise of such materials requires numerical methods that model nematic inclusions in dynamic and complex scenarios. We employ a mesoscale approach for simulating colloids as mobile surfaces embedded in a fluctuating nematohydrodynamic medium to study the kinetics of colloidal entanglement. In addition to reproducing far-field interactions, topological properties of disclination loops are resolved to reveal their metastable states and topological transitions during relaxation towards ground state. The intrinsic hydrodynamic fluctuations distinguish formerly unexplored far-from-equilibrium disclination states, including configurations with localised positive winding profiles. The adaptability and precision of this numerical approach offers promising avenues for studying the dynamics of colloids and topological defects in designed and out-of-equilibrium situations.

Brownian Dynamics of Particles "Dressed" by Chiral Director Configurations in Lyotropic Chromonic Liquid Crystals

We study Brownian dynamics of colloidal spheres, with planar anchoring conditions, suspended in the nematic phase of the lyotropic chromonic liquid crystal disodium chromoglycate (DSCG). Unlike typical liquid crystals, the unusually small twist elastic modulus of DSCG permits two energetically distinct helical distortions (twisted tails) of the nematic director to "dress" the suspended spheres. Video microscopy is used to characterize the helical distortions versus particle size and to measure particle mean-square displacements. Diffusion coefficients parallel and perpendicular to the far-field director, and their anisotropy ratio, are different for the two twisted tail configurations. Moreover, the crossover from subdiffusive to diffusive behavior is anomalously slow for motion perpendicular to the director (>100  s). Simple arguments using Miesowicz viscosities and ideas about twist relaxation are suggested to understand the mean-square displacement observations.

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.

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.

Periodic dynamics, localization metastability, and elastic interaction of colloidal particles with confining surfaces and helicoidal structure of cholesteric liquid crystals

Nematic and cholesteric liquid crystals are three-dimensional fluids that possess long-range orientational ordering and can support both topological defects and chiral superstructures. Implications of this ordering remain unexplored even for simple dynamic processes such as the ones found in so-called "fall experiments," or motion of a spherical inclusion under the effects of gravity. Here we show that elastic and surface anchoring interactions prompt periodic dynamics of colloidal microparticles in confined cholesterics when gravity acts along the helical axis. We explore elastic interactions between colloidal microparticles and confining surfaces as well as with an aligned ground-state helical structure of cholesterics for different sizes of spheres relative to the cholesteric pitch, demonstrating unexpected departures from Stokes-like behavior at very low Reynolds numbers. We characterize metastable localization of microspheres under the effects of elastic and surface anchoring periodic potential landscapes seen by moving spheres, demonstrating the important roles played by anchoring memory, confinement, and topological defect transformation. These experimental findings are consistent with the results of numerical modeling performed through minimizing the total free energy due to colloidal inclusions at different locations along the helical axis and with respect to the confining substrates. A potential application emerging from this work is colloidal sorting based on particle shapes and sizes.

Magnetically responsive gourd-shaped colloidal particles in cholesteric liquid crystals

Particle shape and medium chirality are two key features recently used to control anisotropic colloidal self-assembly and dynamics in liquid crystals. Here, we study magnetically responsive gourd-shaped colloidal particles dispersed in cholesteric liquid crystals with periodicity comparable or smaller than the particle's dimensions. Using magnetic manipulation and optical tweezers, which allow one to position colloids near the confining walls, we measured the elastic repulsive interactions of these particles with confining surfaces and found that separation-dependent particle-wall interaction force is a non-monotonic function of separation and shows oscillatory behavior. We show that gourd-shaped particles in cholesterics reside not on a single sedimentation level, but on multiple long-lived metastable levels separated by a distance comparable to cholesteric periodicity. Finally, we demonstrate three-dimensional laser tweezers assisted assembly of gourd-shaped particles taking advantage of both orientational order and twist periodicity of cholesterics, potentially allowing new forms of orientationally and positionally ordered colloidal organization in these media.

The interaction of colloidal particles with weak homeotropic anchoring energy in homogeneous nematic liquid crystal cells

We have investigated interactions of colloidal particles with weak homeotropic anchoring energy in homogeneous nematic liquid crystal cells. Particle-wall and inter-particle interactions were observed experimentally and analyzed using typical dipole-dipole and quadrupole-quadrupole interactions, including substrate effects as the image charges. Both experimental results matched well with the calculated results for the effective particle radius reflecting the weak anchoring. The effective radius is reduced by the amount of extrapolation length than the actual particle radius. The effective radii of polyethylene micro-particles were reduced to a coefficient ζ (0.78 ≥ ζ ≥ 0.52) times the actual radius with anchoring coefficients in the range of 3.8 × 10(-6) to 1.4 × 10(-6) J m(-2). The anchoring energy of the particles is, therefore, a key component for explaining liquid crystal colloidal systems.

Transport of particles in liquid crystals

Colloidal particles in a liquid crystal (LC) behave very differently from their counterparts in isotropic fluids. Elastic nature of the orientational order and surface anchoring of the director cause long-range anisotropic interactions and lead to the phenomenon of levitation. The LC environment enables new mechanisms of particle transport that are reviewed in this work. Among them the motion of particles caused by gradients of the director, and effects in the electric field: backflow powered by director reorientations, dielectrophoresis in LC with varying dielectric permittivity and LC-enabled nonlinear electrophoresis with velocity that depends on the square of the applied electric field and can be directed differently from the field direction.

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→∞.

Unconventional structure-assisted optical manipulation of high-index nanowires in liquid crystals

Stable optical trapping and manipulation of high-index particles in low-index host media is often impossible due to the dominance of scattering forces over gradient forces. Here we explore optical manipulation in liquid crystalline structured hosts and show that robust optical manipulation of high-index particles, such as GaN nanowires, is enabled by laser-induced distortions in long-range molecular alignment, via coupling of translational and rotational motions due to helicoidal molecular arrangement, or due to elastic repulsive interactions with confining substrates. Anisotropy of the viscoelastic liquid crystal medium and particle shape give rise to a number of robust unconventional trapping capabilities, which we use to characterize defect structures and study rheological properties of various thermotropic liquid crystals.

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.

Large-area optoelastic manipulation of colloidal particles in liquid crystals using photoresponsive molecular surface monolayers

Noncontact optical trapping and manipulation of micrometer- and nanometer-sized particles are typically achieved by use of forces and torques exerted by tightly focused high-intensity laser beams. Although they were instrumental for many scientific breakthroughs, these approaches find few technological applications mainly because of the small-area manipulation capabilities, the need for using high laser powers, limited application to anisotropic fluids and low-refractive-index particles, as well as complexity of implementation. To overcome these limitations, recent research efforts have been directed toward extending the scope of noncontact optical control through the use of optically-guided electrokinetic forces, vortex laser beams, plasmonics, and optofluidics. Here we demonstrate manipulation of colloidal particles and self-assembled structures in nematic liquid crystals by means of single-molecule-thick, light-controlled surface monolayers. Using polarized light of intensity from 1,000 to 100,000 times smaller than that in conventional optical tweezers, we rotate, translate, localize, and assemble spherical and complex-shaped particles of various sizes and compositions. By controlling boundary conditions through the monolayer, we manipulate the liquid crystal director field and the landscape of ensuing elastic forces exerted on colloids by the host medium. This permits the centimeter-scale, massively parallel manipulation of particles and complex colloidal structures that can be dynamically controlled by changing illumination or assembled into stationary stable configurations dictated by the "memorized" optoelastic potential landscape due to the last illumination pattern. We characterize the strength of optically guided elastic forces and discuss the potential uses of this noncontact manipulation in fabrication of novel optically- and electrically-tunable composites from liquid crystals and colloids.

Direct and inverted nematic dispersions for soft matter photonics

General properties and recent developments in the field of nematic colloids and emulsions are discussed. The origin and nature of pair colloidal interactions in the nematic colloids are explained and an overview of the stable colloidal 2D crystalline structures and superstructures discovered so far is given. The nature and role of topological defects in the nematic colloids is discussed, with an emphasis on recently discovered entangled colloidal structures. Applications of inverted nematic emulsions and binding force mechanisms in nematic colloids for soft matter photonic devices are discussed.

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.

Spatially modulated structures in nematic colloids: Statistical thermodynamics and kinetics

We examine the spatial distribution of rigid-sphere-like particles in a nematic host. Using a continuum model we analyse the conditions necessary for the appearance of a modulated lamellar structure. There is a long-range effective interaction between the particles, which can lead to the formation of superstructures. In general, this interaction includes several contributions: van der Waals-type direct interaction and indirect interaction via the director field distortions. The latter depends on the temperature of the sample, the coupling energy between a colloidal particle and a nematic host, and the particle concentration. This effective interaction controls the spatial structure and the kinetic properties of the system. We obtained the analytical expression for the temperature when the system loses the stability with respect to the modulated structure formation. Typical contours of the diffuse light scattering are presented.

Dipolar colloids in nematostatics: tensorial structure, symmetry, different types, and their interaction

In spite of the analogy to the electrostatics, the three-dimensional colloidal nematostatics is substantially different in both its mathematical structure and its physical implications. The general tensorial structure of elastic multipoles derived in V. M. Pergamenshchik and V. O. Uzunova [Eur. Phys. J. E 23, 161 (2007); Phys. Rev. E 76, 011707 (2007)] allows for a classification of different types of colloids in the nematostatics. In comparison to their electrostatic counterparts, the elastic multipoles have one extra tensorial index. Based on this structure, we identify possible types of elastic dipoles. An elastic dipole is characterized by three coefficients--isotropic strength, anisotropy, and chirality--and a two-component vector along the unperturbed director. The relationship between the dipole type and symmetry groups is established and sketches of various representative types of dipolar colloids are given. Instead of a single electric dipole, in the nematostatics there are four different pure types (dipolar singlets) and eight mixed types of elastic dipoles (one quintet, one quartet, two triplets, and four doublets). It is shown that the full symmetry of the colloid-induced director field and the colloid's shape (body) symmetry determine different dipole components. For instance, a helicoidal component of the anchoring easy axes can make a chiral elastic dipole of a colloid with the quadrupolar shape symmetry. The interaction potentials for different singlet and doublet dipoles are derived and illustrated in terms of the dipolar dyads and elastic Coulomb law. We argue that multipole parameters must be found by pure numerical means, as from ansatz director distributions one can find only orders of their magnitudes.

Design of 2D binary colloidal crystals in a nematic liquid crystal

In this paper, we examine directed self-assembly in a 2D binary system of dipolar and quadrupolar colloidal particles with normal surface boundary conditions, dispersed in the nematic liquid crystal. Using the laser tweezers, we assembled a large variety of stable 2D colloidal crystal structures. In all analyzed structures, the particles, their surface treatment and the cell conditions were the same, which gives us the ability to systematically follow the evolution of colloidal assembly when many particles are present. We present an analogy between molecular self-assembly and organization of colloidal microspheres in liquid crystalline medium to extend the strategy for designing colloidal crystalline structures of different level of complexity.

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.