Theory of nine elastic constants of biaxial nematics

In this paper, a rotational invariant of interaction energy between two biaxial-shaped molecules is assumed and in the mean field approximation, nine elastic constants for simple distortion patterns in biaxial nematics are derived in terms of the thermal average （Dmn^（l）） （Dm＇n＇^（l＇））, where Dmn^（l） is the Wigner rotation matrix. In the lowest order terms, the elastic constants depend on coefficients Γ,Γ＇, λ, order parameters Q0 = Q0（D00^（2）） ＋Q2（D02^（2）＋D0-2^（2）） and Q2 = Q0（D20^（2）） ＋ Q2（D22^（2）＋D2-2^（2））. Here Γ and Γ＇ depend on the function form of molecular interaction energy vj′j″j （τ12） and probability function fk′k″k （τ12）, where r12 is the distance between two molecules, and λ is proportional to temperature. Q0 and Q2 are parameters related to multiple moments of molecules. Comparing these results with those obtained from Landau-de Gennes theory, we have obtained relationships between coefficients, order parameters used in both theories. In the special case of uniaxial nematics, both results are reduced to a degenerate case where K11=K33.     

The molecular-statistical calculation of the elastic constants of uniaxial nematics

Using the molecular-statistical theory of the thermodynamic properties of nematics and perturbation theory, the expressions for the elastic constants of uniaxial nematics are derived in terms of the mutual interactions between their constituent molecules. Next the elastic constants of the Maier-Saupe model are calculated and compared with the original result. The appearing difference is discussed briefly. Finally, in view of existing discrepancies, the elastic constants of the Onsager model are recalculated. For that purpose the hard rod interaction is conceived as the limit of a non-singular interaction. The elastic constants of hard core models are shown to increase with increasing temperature at constant density. In view of this behaviour the usefulness of hard core models for an understanding of the nematic state seems rather limited.     

Theory of quantum Hall nematics

Transport measurements on two-dimensional electron systems in moderate magnetic fields suggest the existence of a spontaneously orientationally ordered, compressible liquid state. We develop and analyze a microscopic theory of such a "quantum Hall nematic" (QHN) phase, predict the existence of a novel, highly anisotropic q(3) density-director mode, find that the T = 0 long-range orientational order is unstable to weak disorder, and compute the tunneling into such a strongly correlated state. This microscopic approach is supported and complemented by a hydrodynamic model of the QHN, which, in the dissipationless limit, reproduces the modes of the microscopic model.     

Disclinations in nematics in the presence of flexoelectricity

The stability problems of wedge disclinations (with pure splay and bend distortions) in nematic liquid crystals (NLC) are considered in the presence of flexoelectricity. We write NLC director equations in the cylindrical coordinate system, taking into account flexoelectricity. These equations allow us to solve many interesting problems on the influence of flexoelectricity on the orientational structures of NLC. In particular, it is shown that the line of disclination with radial distribution of director can be stabilized in the presence of flexoelectricity even far from nematicsmectic phase transition point. For the disclination with azimuthal distribution of director for all physical sizes the line is not stable and escapes in the third dimension.     

On the theory of the dynamical properties of nematics

The Leslie–Ericksen coefficients of a thermotropic nematic are determined by using an approximate solution of the Fokker–Planck equation for the one-particle distribution function over orientations of the nematic molecules. The results show that the well-known Doi–Edwards theory of the dynamical properties of nematics leads to a qualitatively wrong result for the Leslie angle. The “isotropic medium - nematic” (I–N$I–N$) transition induced by the shear flow is considered. When the temperature decreases, the I–N$I–N$ transition in the shear flowing system takes place at the temperature _T1$T_1$ higher than the temperature _Tc$T_c$ of the equilibrium transition in the motionless system. The interface boundary in this case is parallel to the plane formed by the flow velocity and its gradient. When the shear flowing nematic phase is heated, the N–I$N–I$ transition occurs at another temperature _T2$T_2$, and the following inequalities _T1>_T2>_Tc$T_1>T_2>T_c$ hold. In this case the boundary between the isotropic and nematic phases is perpendicular to the flow velocity. Thus, unlike the equilibrium phase transition, a temperature hysteresis of the phase transition is expected.     

Theory of rate constants: Master equation approach

Starting from the master equations we derive the kinetic equations for the concentrations of chemical species. Both adiabatic and nonadiabatic rate processes are analyzed. In limiting cases, the results of the work conform to those of Widom⁽⁹⁾ and Gibbs, Fleming, and co-workers.^(4,5)     

Representation of pair correlations in nematics

A complete expansion of equilibrium pair correlation functions for a uniaxial nematic phase composed of axially symmetric, non-polar and chiral molecules is proposed. Full advantage is taken of the symmetry of the nematic state and of the molecules. The explicit analysis and classification of the terms involving spherical harmonics with indices not exceeding 4 is given and illustrated using computer simulations for the nematic phase of a Gay-Berne mesogen. The theory is contrasted with the commonly used approach employing invariants that describe orientational correlations in molecular fluids. The role of the new representation in obtaining a correct understanding of a variety of observables, like the elastic constants, is demonstrated. In particular, the long-standing puzzle concerning the splay-bend degeneracy is resolved. Received 11 April 2000 and Received in final form 19 July 2000     

Some elastic properties of solid nematics

The general approach to study the properties of the mechanical deformations of solid nematics, which are the macroscopic homogeneous elastic media having the rotational symmetry of the nematic liquid crystals is proposed. The stress tensor, the Young modulus and the Poisson ratios for the parallel and perpendicular homogeneous orientations of nematic molecules relative to the axis of external forces influence are obtained by the varying of the free energy of mechanical deformation. It is shown that these constants have the anisotropic character and the experiments for the direct measurement of five elasticity coefficients entering the free energy expression are proposed.     

On the hard core equation of state for nematics

An equation of state is proposed for a system of rods interacting only through hard core repulsions. A quantitative comparison is made with recently proposed equations of state based upon the scaled-particle and Percus-Yevick theory.     

Excitable patterns in active nematics

We analyze a model of mutually propelled filaments suspended in a two-dimensional solvent. The system undergoes a mean-field isotropic-nematic transition for large enough filament concentrations, and the nematic order parameter is allowed to vary in space and time. We show that the interplay between nonuniform nematic order, activity, and flow results in spatially modulated relaxation oscillations, similar to those seen in excitable media. In this regime the dynamics consists of nearly stationary periods separated by "bursts" of activity in which the system is elastically distorted and solvent is pumped throughout. At even higher activity, the dynamics becomes chaotic.     

Topological defects in spherical nematics

We study the organization of topological defects in a system of nematogens confined to the two-dimensional sphere (S2). We first perform Monte Carlo simulations of a fluid system of hard rods (spherocylinders) living in the tangent plane of S2. The sphere is adiabatically compressed until we reach a jammed nematic state with maximum packing density. The nematic state exhibits four +1/2 disclinations arrayed on a great circle. This arises from the high elastic anisotropy of the system in which splay (K1) is far softer than bending (K3). We also introduce and study a lattice nematic model on S2 with tunable elastic constants and map out the preferred defect locations as a function of elastic anisotropy. We find a one-parameter family of degenerate ground states in the extreme splay-dominated limit K_{3}/K_{1}-->infinity. Thus the global defect geometry is controllable by tuning the relative splay to bend modulus.     

The effect of magnetic fields and boundary conditions on the shear flow of nematics

The flow of a nematic liquid crystal between plane parallel plates, with one plate moving with uniform velocity relative to the other, is discussed. The apparent viscosity, orientation and velocity profiles are computed forp-azoxyanisole as functions of shear rate and magnetic field for symmetric and asymmetric molecular alignment at the plates. For symmetric homeotropic boundary condition, a magnetic field applied along the flow direction exhibits a threshold reminiscent of a Freedericksz transition in the hydrostatic case. In general the apparent viscosity for the asymmetric boundary condition is less than that for the symmetric case.     

The elastic constants of the alkali-metals

Experimental results of the elastic constants c₁₁, c₄₄ and c are compared with theoretical predictions. It is shown that the alkali-metals obey a law of corresponding states rather closely, if lithium is excluded to some extent. The absolute magnitudes of the reduced elastic constants and their temperature dependence are finally discussed in the light of recent more sophisticated theories and other experimental evidence.     

The elastic constants of the alkali-metals

Experimental results of the elastic constants c₁₁, c₄₄ and c are compared with theoretical predictions. It is shown that the alkali-metals obey a law of corresponding states rather closely, if lithium is excluded to some extent. The absolute magnitudes of the reduced elastic constants and their temperature dependence are finally discussed in the light of recent more sophisticated theories and other experimental evidence.     

Dispersion and absorption of sound in nematics

The unified hydrodynamic theory for systems with a liquid crystalline phase is generalized to include a frequency dependence of the elastic and dissipative parameters of the system. Application is made to nematics for which the frequency dependent anisotropies of the velocity and absorption of sound are calculated. It is shown that the relaxation approximation for the dispersion leads to reasonable results.     

Light transmission at the uniaxial-biaxial lyotropic nematics phase transitions

Results are reported on light transmission versus the temperature near the phase transitions between lyotropic nematics. Strong pretransitional effects are observed when the phase transitions occur between biaxial nematic and uniaxial phases. The light is strongly depolarized both at the calamitic-biaxial and at biaxial-discotic phase transitions. The results are qualitatively explained.