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.     

Director configuration and dynamics of a nematic liquid crystal around a two-dimensional spherical particle: Numerical analysis using adaptive grids

We study numerically the director and orientational order parameter configurations in a nematic liquid crystal around a two-dimensional spherical particle on the basis of the tensor order parameter formalism. To properly account for the large length scale difference between the particle and the accompanying orientational defect, we devise an adaptive grid scheme in which the lattice spacing is automatically and locally adjusted in response to the spatial gradient of the orientational order parameter. This adaptive grid scheme is useful in studying dynamical as well as static orientational structures. We present a simulation result which shows how a hedgehog defect of topological charge -1 becomes unstable in two dimensions, and splits into a defect pair of topological charge -1/2, located symmetrically around the particle. Received 14 September 2000 and Received in final form 27 December 2000     

Anisotropic laser trapping in nematic colloidal dispersion

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

Screened electrostatic interactions between clay platelets

An effective pair potential for systems of uniformly charged lamellar colloids in the presence of an electrolytic solution of microscopic co- and counterions is derived. The charge distribution on the discs is expressed as a collection of multipole moments, and the tensors which determine the interactions between these multipoles are derived from a screened Coulomb potential. Unlike previous theoretical studies of such systems, the interaction energy may now be expressed for discs at arbitrary mutual orientation. The potential is shown to be exactly equivalent to the use of linearized Poisson–Boltzmann theory.     

Colloidal interactions in two-dimensional nematics

The interaction between two disks immersed in a 2D nematic is investigated i) analytically using the tensor order parameter formalism for the nematic configuration around isolated disks and ii) numerically using finite-element methods with adaptive meshing to minimize the corresponding Landau-de Gennes free energy. For strong homeotropic anchoring, each disk generates a pair of defects with one-half topological charge responsible for the 2D quadrupolar interaction between the disks at large distances. At short distance, the position of the defects may change, leading to unexpected complex interactions with the quadrupolar repulsive interactions becoming attractive. This short-range attraction in all directions is still anisotropic. As the distance between the disks decreases, their preferred relative orientation with respect to the far-field nematic director changes from oblique to perpendicular. Received 1 October 2002 and Received in final form 12 November 2002 RID="a" ID="a"e-mail: miko@cii.fc.ul.pt     

Elastic properties of nematoid arrangements formed by amoeboid cells

In culture migrating and interacting amoeboid cells can form nematoid arrangements in analogy to a nematic liquid crystal phase. A nematoid arrangement is formed if the interaction has an apolar symmetry. Different cell types like human melanocytes (= pigment cells of the skin), human fibroblasts (= connective tissue cells), human osteoblasts (= bone cells), human adipocytes (= fat cells) etc., form a nematoid structure. Our hypothesis is that elastic properties of these nematoid structures can be described in analogy to that of classical nematic liquid crystals. The orientational elastic energy is derived and the orientational defects (disclination) of nematoid arrangements are investigated. The existence of half-numbered disclinations shows that the nematoid structure has an apolar symmetry. The density- and order parameter dependence of the orientational elastic constants and their absolute values are estimated. From the defect structure, one finds that the splay elastic constant is smaller than the bend elastic constant (melanocytes). The core of a disclination is either a cell free space or occupied by non oriented cells (isotropic state), by a cell with a different symmetry, or by another cell type. Received 3 May 1999 and Received in final form 29 September 1999     

Soliton-like molecular deformations in a nematic liquid crystal film

We study the nature of molecular deformations in a nematic liquid crystal film with elastic energy under homeotropic boundary conditions. The deformation in terms of splay, twist and bend fields of the director axis is found to be governed by the completely integrable Davey-Stewartson-I (DS-I) equation in (2+1) dimensions. Using the line soliton and breather solutions of the DS-I equation, the director axis is constructed, the components of which exhibit damped spatial oscillations. However, the splay and bend fields of the director axis exhibit localized structures of deformation.     

Critical exponents for Fréedericskz transition in nematics between concentric cylinders

The equilibrium tilt angle profile in a cell limited by two concentric cylinders filled with nematic liquid crystals is determined for strong homeotropic anchoring at the surfaces. The anchoring condition is such that the nematic director is perpendicular to the cylinder axes and a radial nonuniform electric field is applied to investigate a Fréedericksz transition. The distortions induced by the field remain in the plane perpendicular to the cylinder axes, and a threshold field is analytically determined indicating a transition from a pure splay to a splay-bend conformation of the director. It is shown that this transition can be induced by the thickness of the region between the two cylinders, and can be detected even in the absence of an external field. If the maximum value of the tilt angle is assumed as an order parameter, its behavior near to the transition can be used to obtain the critical exponent, which is the same as the one obtained in the mean field approximation. These results are indications that nontrivial consequences may occur when complex fluids are subject to non-planar geometries.     

Elastic constants of a nematic liquid crystal: Quadrupole-quadrupole interaction in the ellipsoidal approximation

The elastic constants of a nematic liquid crystal are calculated by means of a pseudo-molecular method that considers the quadrupolar intermolecular interaction with a screening length governing the range of the forces. The bulk as well as the surface-like elastic constants are determined as functions of the eccentricity of the interaction volume of ellipsoidal shape. It is shown that the elastic constants become negative for some values of the eccentricity, and, therefore, that the screened quadrupolar interaction could be the source of instabilities in the nematic phase and should be taken into account to interpret threshold phenomena in these systems.     

Nonlinear molecular deformations and solitons in a nematic liquid crystal

We studied nonlinear molecular deformations in a nematic liquid crystal with homeotropically aligned molecules and hard boundaries. As the basic dynamical equation for the director axis of the liquid crystal resembles the Landau-Lifshitz equation representing spin dynamics in a one dimensional classical continuum isotropic Heisenberg ferromagnetic spin chain, we invoke here the space curve formalism and the stereographic projection technique used in the case of the Heisenberg spin chain. Under space curve mapping, the director dynamics with elastic deformation is found to be governed by a perturbed nonlinear Schrödinger equation. A multiple-scale perturbation analysis brings out perturbed solitons to represent molecular deformations in the nematic liquid crystal. However, when a constant electric field is applied, the director dynamics is expressed under stereographic projection and the molecular deformations are found to be governed by periodic and localized static planar director configurations. A linear stability analysis on the static planar configurations shows that the system exhibits stable deformations.     

Elastic energies for nematic elastomers

We discuss several elastic energies for nematic elastomers and their small strain expansions both in the regime of large director rotations, and in the case that director changes are small. We propose two fully non-linear model anisotropic energies and compare the behavior they predict with the currently available experimental evidence.     

Orientation-dependent depletion interaction in rodlike colloid-polymer mixtures

Depletion interaction in a suspension of rodlike colloids with added non-adsorbing polymer coils is theoretically studied. We calculate an overlap volume of depletion zone between two rodlike colloids, based on the second virial approximation. We examine nematic-isotropic phase transition (NIT) and two-phase coexistence between an isotropic and a nematic phase at low polymer concentrations. We find that the depletion interaction is dependent on the orientational order parameter of rodlike colloids and leads to a decrease in the NIT concentration on the addition of polymer. The coexistence curves have a leaning-chimney shape and are shifted to lower rod concentrations on increasing the polymer concentration. Received 23 May 2001 and Received in final form 18 July 2001     

The two-body multipole problem of electrodynamics

A 2-body system composed of two objects having arbitrary distributions of charge and current is discussed. An expression for the velocity dependent potential between these two objects has been obtained in the non-relativistic approximation. This potential consists of two parts viz. a velocity independent scalar potential Φ_eff and another part which is linearly dependent on the relative velocity between the objects. The second part naturally suggests a vector potential A_eff. The potentials have been expanded into multipole terms. It has been found that Φ_eff is a sum of two components viz. Φ_EE and Φ_MM such that each multipole term in Φ_EE represents an interaction between the electric multipoles of the two systems, each term in Φ_MM represents an interaction between their magnetic multipoles whereas each term in A_eff represents an interaction between an electric multipole of one and a magnetic multipole of the other. The results have been applied to the interaction between an electric dipole and a magnetic dipole. The symmetry among the multipole terms in A_eff suggests vanishing vector potential between two identical objects. A corollary of this appears to be absence of spin orbit interaction between two identical particles in the same spin state.     

Elastic model of protein-protein interaction

Summary The elastic distortions of a model membrane are discussed, by taking into account those due to incorporated proteins. Two types of distortions exist: i) membrane compression and ii) displacement at constant thickness. The first mode decays on a very-short-length scale (≈nm). The second one is important since the characteristic lengh is of the order of the μm. This last distortion is characterized by a multipole expansion, in a way similar to electrostatics. The interaction of different elastic distortions is described in the context of their multipole expansion: for example, i) the monopole-monopole interaction is attractive (repulsive) in the case of opposite (equal) signs of the monopoles. The interaction energy behaves asW _AB≈Inr _AB (r _AB distance between the monopoles A and B) and may be even at large distance much higher thankT. ii) The dipole-dipole interaction is always attractive if the elastic dipole can rotate (fluid membrane). iii) The interaction energy varies asW _AB≈r _AB ⁻² (r _AB distance between the dipoles) and so on. This multipole concept may be used for the understanding of the recognition of different proteins. In addition, the particles which creaste the distortion may be polarizable. This means that shape changes of the particle could be induced by an external elastic field. We expect for a large particle a large polarizability. As a consequence, two static monopoles which reped each other may attract if they become polarizable. This feature is reminiscent of the Van der Waals interaction in electrostatics. Paper presented at the “Meeting on Lyotropics and Related Fields?, held in Rende, Cosenza, September 13–18, 1982.     

Orientational phase transition and the solvation force in a nematic liquid crystal confined between inhomogeneous substrates

A nematic liquid crystal confined between two identical flat solid substrates, with an alternating stripe pattern of planar and homeotropic anchoring, is studied in the framework of the Frank-Oseen theory. By means of numerical minimization of the free energy functional we study the effect of the sample thickness D on the location of the phase transition between a uniform alignment, either planar or homeotropic, and a distorted nematic texture. The solvation force f due to distortions of the nematic director is also studied. It is found that f is always attractive, and for D small compared to the periodicity of the surface structure it exhibits two distinct asymptotic behaviors: f ∼ - D ^-1/2 or f ∼ - D ^-1, depending on the relation between D and the extrapolation lengths. Received 12 November 2002 Published online: 16 April 2003     

Langmuir-Blodgett film and nematic interface

The orientation induced by a Langmuir-Blodgett film on a Nematic Liquid Crystal (NLC) is theoretically analyzed. We show that the effective surface energy is due to different contributions connected with steric and van der Waals interactions between the nematic and the solid substrate. The analysis shows that the Langmuir-Blodgett film orientation depends on the surface density of the molecules. The initial homeotropic orientation may become unstable giving rise to a tilted film. The average orientation of the nematic molecules is also analyzed. We show that, in the event in which the steric interaction Nematic-Langmuir-Blodgett film is very large with respect to the dispersion interaction Nematic-Substrate, the nematic orientation coincides with the one of the film. On the contrary, when the two interactions are comparable, the orientation of the two media may differ. In particular, we analyze how the stable orientation depends on the surface molecular density of the film.     

On spatial variations of nematic ordering

We present a theory of orientational order in nematic liquid crystals which interpolates between several distinct approaches based on the director field (Oseen and Frank), order parameter tensor (Landau and de Gennes), and orientation probability density function (Onsager). As in density-functional theories, the suggested free energy is a functional of spatially-dependent orientation distribution, however, the nonlocal effects are taken into account via phenomenological elastic terms rather than by means of a direct pair-correlation function. In illustration of this approach we consider a simplified model with orientation parameter on a circle and reveal its relation to the complex Ginzburg-Landau theory.     

Spherical magnetic multipole moments system of currents

Concepts of spherical magnetic multipoles that represent distributions of electric currents over a spherical surface are introduced. Vector potentials of magnetic multipoles meet solenoidal- and harmonic-field conditions outside of the spherical surface and are continuous on it. Within the sphere, the vector potential of currents flowing outside of it is represented by the sum of vector potentials of basis magnetic multipoles with coefficients expressed by spherical multipole moments of system of currents. This expansion of the vector potential is in many respects analogous to the multipole expansion known from electrostatics. The first three terms of the expansion represent components of the well-known magnetic moment, the next five terms represent components of the magnetic quadrupole moment, etc. Possible applications of the magnetic spherical multipole technique are discussed. Krasnoyarsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 66–72, October, 1999.     

Electrostatics of the point dipole and higher multipoles

It is shown that a distribution theory form for the electrostatics of point multipoles can be constructed, which reduces to the usual theory away from the position of the multipole, but in which the total electromagnetic self-energy is zero and the self-force density is zero.     

Line defect dynamics around a colloidal particle

We study experimentally the dynamics of a topological defect located around a colloidal particle suspended in a thermotropic nematic liquid crystal. The considered defect consists of a disclination loop encircling the particle at the equator. Under specific conditions, it is shown that this disclination continuously shrinks to a hedgehog defect located in the immediate vicinity of the particle. This phenomenon corresponds to a transition between an elastic quadrupolar configuration and an elastic dipolar configuration. We performed a basic numerical calculation to get an estimate of the dissipated energy during the transition; we compare the results with theoretical predictions that describe the elastic energy of particles surrounded by defects. Received 21 December 2001