Anchoring-induced nonmonotonic velocity versus temperature dependence of motile bacteria in a lyotropic nematic liquid crystal.

The elastic and viscous properties of lyotropic chromonic liquid crystals have a very sharp, often exponential temperature dependence. Self-propelled bacteria swimming in this viscoelastic medium induce director deformations which can strongly influence their velocity, and we study the temperature behavior of their motility in the whole range of the nematic phase. We observe experimentally that, with increasing temperature, while the viscosity drops exponentially and the frequency of the flagellum rotation grows linearly, the swimmers' speed first conventionally increases but then, above some crossover temperature, slows down and at the same time bacteria-induced director distortions become visible. It is shown that the physics behind this temperature-driven effect is in a sharp rise in the ability of the bacterium's flagellum to induce director deformations. As temperature increases, the splay and bend elastic constants sharply decrease and the anchoring extrapolation length of the flagellum surface gets shorter and shorter. At the crossover temperature the resulting effective anchoring effect dominates the fast dropping viscosity and the distortion strengthens. As a result, a fraction of the torque the flagellum applies for the propulsion is spent for the elastic degrees of freedom, which results in a bacterium slowdown. To find the director distortions, the flagellum is presented as a collection of anchoring-induced elastic monopoles, and the bacterium velocity is found from the balance of the energy spent for the propulsion and the viscous drag and nematodynamic dissipation.

Analytical canonical partition function of a quasi-one-dimensional system of hard disks.

The exact canonical partition function of a hard disk system in a narrow quasi-one-dimensional pore of given length and width is derived analytically in the thermodynamic limit. As a result, the many body problem is reduced to solving the single transcendental equation. The pressures along and across the pore, distributions of contact distances along the pore, and disks' transverse coordinates are found analytically and presented in the whole density range for three different pore widths. The transition from the solidlike zigzag to the liquidlike state is found to be quite sharp in the density scale but shows no genuine singularity. This transition is quantitatively described by the distribution of zigzag's windows through which disks exchange their positions across the pore. The windowlike defects vanish only in the densely packed zigzag, which is in line with a continuous Kosterlitz-Thouless transition.

Interaction of supramolecular aggregates and the enhanced optical torque on the director in a dye doped nematic liquid crystal.

There has been strong experimental evidence that molecules of some dyes in an anisotropic solvent, nematic liquid crystal, form aggregates. We present a detailed experimental analysis of the light-induced director reorientation (DR) in a dye-doped nematic liquid crystal (known as the Jánossy effect) and a theoretical model of its strong enhancement based on the aggregates' interaction. The DR transition is found to be very different from the Frederiks effect. If the light polarization is normal to the director, the transition is jump-like first order. Moreover, light polarization along the director also induces a DR which is a smooth second order transition with a very low threshold intensity. The theoretical model which explains these effects is based on the idea that dye molecules form rodlike supramolecular aggregates. The aggregates interact via the director distortions and their effective diameter gets certain field-dependence. As a result, the related entropy depletion depends on the light intensity and polarization and can be decreased by a certain DR along with the aggregate subsystem. This entropy gain is proportional to the square of light intensity which is a two-photon effect: the first resonance photon excites the dye molecule and the second photon polarizes the aggregate. This is in line with the experimental dependence of the critical intensity on the sample thickness. A special experiment shows that the effect is not connected with a possible heat-induced isotropic phase and hydrodynamic motion.

Statistical model of a flexible inextensible polymer chain: The effect of kinetic energy.

Because of the holonomic constraints, the kinetic energy contribution in the partition function of an inextensible polymer chain is difficult to find, and it has been systematically ignored. We present the first thermodynamic calculation incorporating the kinetic energy of an inextensible polymer chain with the bending energy. To explore the effect of the translation-rotation degrees of freedom, we propose and solve a statistical model of a fully flexible chain of N+1 linked beads which, in the limit of smooth bending, is equivalent to the well-known wormlike chain model. The partition function with the kinetic and bending energies and correlations between orientations of any pair of links and velocities of any pair of beads are found. This solution is precise in the limits of small and large rigidity-to-temperature ratio b/T. The last exact solution is essential as even very "harmless" approximation results in loss of the important effects when the chain is very rigid. For very high b/T, the orientations of different links become fully correlated. Nevertheless, the chain does not go over into a hard rod even in the limit b/T→∞: While the velocity correlation length diverges, the correlations themselves remain weak and tend to the value ∝T/(N+1). The N dependence of the partition function is essentially determined by the kinetic energy contribution. We demonstrate that to obtain the correct energy and entropy in a constrained system, the T derivative of the partition function has to be applied before integration over the constraint-setting variable.

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.

Stability and minimum size of colloidal clusters on a liquid-air interface.

A vertical force applied to each of two colloids, trapped at a liquid-air interface, induces their logarithmic pairwise attraction. I recently showed [Phys. Rev. E 79, 011407 (2009)] that in clusters of size R much larger than the capillary length λ, the attraction changes to that of a power law and is much stronger due to a many-body effect, and I derived two equations that describe the equilibrium coarse-grained meniscus profile and colloid density in such clusters. In this paper, this theory is shown also to describe small clusters with R≪ λ provided the number N of colloids therein is sufficiently large. An analytical solution for a small circular cluster with an arbitrary short-range power-law pairwise repulsion is found. The energy of a cluster is obtained as a function of its radius R and colloid number N. As in large clusters, the attraction force and energy universally scale with the distance L between colloids as L(-3) and L(-2), respectively, for any repulsion forces. The states of an equilibrium cluster, predicted by the theory, are shown to be stable with respect to small perturbations of the meniscus profile and colloid density. The minimum number of colloids in a circular cluster, which sustains the thermal motion, is estimated. For standard parameters, it can be very modest, e.g., in the range 20-200, which is in line with experimental findings on reversible clusterization on a liquid-air interface.

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.

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.

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

The new area of nematic colloidal systems (or nematic emulsions) has been greatly guided by the fruitful analogy between the colloidal nematostatics and electrostatics. The elastic charge density representation of the colloidal nematostatics [V. M. Pergamenshchik and V. O. Uzunova, Eur. Phys. J. E 23, 161 (2007); Phys. Rev. E 76, 011707 (2007)] develops this analogy at the level of charge density and Coulomb interaction. It shows, however, that the colloidal nematostatics in three dimensions substantially differs from the electrostatics both in its mathematical structure and physical implications: the elastic charge and multipoles are dyads; similar charges attract while opposite charges repel each other, and so on. In this paper we consider the interaction between an elastic charge and elastic dipole with a nematic surface (wall) at which the director alignment is fixed. Using the mirror image method of electrostatics as a guiding idea, we develop the mirror image method in the nematostatics for arbitrary director tilt at the wall. A wall is shown to induce a repulsive 1R{4} force on the elastic dipole which, in general, is accompanied by its reorientation. External torque on the colloid induces an elastic charge therein and triggers switching to the 1R{2} repulsion. The dyadic nature of an elastic dipole is shown to be essential: a particle-wall interaction potential cannot be obtained in phenomenological theories with a single component dipole. In the introductory sections we discuss connection between the director-mediated interaction in two and three dimensions and the electrostatic interaction and consider different symmetries of elastic dipoles. Conservation of the torque components exerted upon colloids is shown to play the role of Gauss' theorem and determines the elastic charge dyad.

Strong collective attraction in colloidal clusters on a liquid-air interface.

It is shown that in a cluster of many colloids, trapped at a liquid-air interface, the well-known vertical-force-induced pairwise logarithmic attraction changes to a strongly enhanced power-law attraction. In large two-dimensional clusters, the attraction energy scales as the inverse square of the distance between colloids. The enhancement is given by the ratio eta = (square of the capillary length) / (interface surface area per colloid) and can be as large as 10;{5} . This explains why a very small vertical force on colloids, which is too weak to bring two of them together, can stabilize many-body structures on a liquid-air interface. The profile of a cluster is shown to consist of a large slow collective envelope modulated by a fast low-amplitude perturbation due to individual colloids. A closed equation for the slow envelope, which incorporates an arbitrary power-law repulsion between colloids, is derived. For example, this equation is solved for a large circular cluster with the hard-core colloid repulsion. It is suggested that the predicted effect is responsible for mysterious stabilization of colloidal structures observed in experiments on a surface of isotropic liquid and nematic liquid crystal.

Stripe domains in a nearly homeotropic nematic liquid crystal: a bend escaped state at a nematic-smectic-A transition.

We report an observation and mechanism of spontaneous periodic modulations of the nematic director close to the temperature T(NA) of a nematic-to-smectic-A phase transition if the surface alignment slightly differs from a pure homeotropic one. Stripe domains appear in the nematic phase about one degree above T(NA) and persist into the Sm A phase. The instability of the homogeneous state with respect to stripe domains is shown to be related to a very large bend constant which is much larger than the twist and splay elastic constants. The instability mechanism consists of reduction of the highly energetic bend deformation, induced by small surface director tilts, at the expense of a spontaneous periodic splay-twist modulation. Using smallness of the twist-to-bend and splay-to-bend elastic constant ratios, the critical condition of the instability and the modulation period are found analytically. Both the experimentally obtained and theoretically predicted domain period scales very closely to a square root of the cell thickness.

Coulomb-like interaction in nematic emulsions induced by external torques exerted on the colloids.

An external mechanical torque on colloids immersed in a nematic liquid crystal can induce a Coulomb-like 1/r interaction between them [Lev and Tomchuk, Phys. Rev. E 59, 591 (1999); Lev, ibid. 65, 021709 (2002)]. In this paper we show that the director-mediated Coulomb-like interaction of two colloids is determined by the vectors Gamma perpendicular (1) and Gamma perpendicular (2) of the transverse external torques exerted upon these colloids. We derive the 1/r potential in which the scalar product -(Gamma perpendicular (1) x Gamma perpendicular (2)) of the two torques plays the role of the product of two electrostatic charges. The 1/r interaction is attractive for (Gamma perpendicular (1) x Gamma perpendicular (2))>0 and repulsive for (Gamma perpendicular (1) x Gamma perpendicular (2))<0 ("parallel torques" attract whereas "antiparallel torques" repel each other). The vector of transverse torque determines the two-component "elastic charge" (dyad), which is illustrated by the 1/r2 and 1/r3 terms in the elastic energy (the elastic analogs of the monopole-dipole and dipole-dipole interactions). The general status of the pairwise approach to nematic emulsions is considered in terms of the elastic charge density.

Elastic charge density representation of the interaction via the nematic director field.

The interaction between particle-like sources of the nematic director distortions (e.g., colloids, point defects, macromolecules in nematic emulsions) allows for a useful analogy with the electrostatic multipole interaction between charged bodies. In this paper we develop this analogy to the level corresponding to the charge density and consider the general status of the pairwise approach to the nematic emulsions with finite-size colloids. It is shown that the elastic analog of the surface electric charge density is represented by the two transverse director components on the surface imposing the director distortions. The elastic multipoles of a particle are expressed as integrals over the charge density distribution on this surface. Because of the difference between the scalar electrostatics and vector nematostatics, the number of elastic multipoles of each order is doubled compared to that in the electrostatics: there are two elastic charges, two vectors of dipole moments, two quadrupolar tensors, and so on. The two-component elastic charge is expressed via the vector of external mechanical torque applied on the particle. As a result, the elastic Coulomb-like coupling between two particles is found to be proportional to the scalar product of the two external torques and does not directly depend on the particles' form and anchoring. The real-space Green function method is used to develop the pairwise approach to nematic emulsions and determine its form and restrictions. The pairwise potentials are obtained in the familiar form, but, in contrast to the electrostatics, they describe the interaction between pairs (dyads) of the elastic multipole moments. The multipole moments are shown to be uniquely determined by the single-particle director field, unperturbed by other particles. The pairwise approximation is applicable only in the leading order in the small ratio particle size-to-interparticle distance as the next order contains irreducible three-body terms.

Coexistence of two colloidal crystals at the nematic-liquid-crystal-air interface.

Glycerol droplets at a nematic-liquid-crystal-air interface form two different lattices--hexagonal and dense quasihexagonal--which are separated by the energy barrier and can coexist. Director distortions around each droplet form an elastic dipole. The first order transition between the two lattices is driven by a reduction of the dipole-dipole repulsion through reorientation of these dipoles. The elastic-capillary attraction is essential for the both lattices. The effect has a many-body origin.

Intermediate periodic "saddle-splay" nematic phase in the vicinity of a nematic-smectic-A transition.

We consider possible spontaneous modulations of the nematic director induced by the elastic saddle-splay K24 term when the value of the elastic constant K24 does not satisfy the Ericksen stability condition for the homogeneous ground state. According to the standard formula expressing K24 in terms of the twist elastic constant K22, this can be expected close to the nematic-smectic-A transition where K22 becomes very large. It is predicted that in a planar nematic layer (or, more generally, if the surface director alignment is sufficiently close to a planar one), a modulated phase with observable long wavelength period can occur in samples considerably thicker than the anchoring extrapolation length. The modulated nematic phase is expected to persist into the smectic phase so that its temperature of the transition to smectic phase has to be lower than that for the homogeneous nematic liquid crystal. Low amplitude short wavelength modulations are predicted for any thickness if the surface director is sufficiently far from a pure homeotropic alignment. At the expense of this mode the temperature of a nematic-smectic-A transition in a planar cell with isotropic surfaces has to be lower than that for a homeotropic cell even if the periodic structure is not accessible for the direct observation.

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

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

Non-Debye screening of a surface charge and a bulk-ion-controlled anchoring transition in a nematic liquid crystal.

We study the anchoring mechanism due to substrate-adsorbed ions by examining a related anchoring transition. An analytical solution to the Poisson equation shows that, as their number suffices for a non-negligible anchoring contribution, the surface field is screened over some characteristic microscopic distance. It is shown both theoretically and experimentally that the critical temperature of the transition can be controlled by bulk ion density through its relation to the density of adsorbed ions.