Theory Of Nematic Liquid Crystals Research Articles

Phenomenological approach to the problem of the K13 surfacelike elastic term in the free energy of a nematic liquid crystal.

We consider the mathematical foundations of continuum theories of nematic liquid crystals of the Frank-Oseen form which include, in addition, the surfacelike ${\mathit{K}}_{13}$ term. Such theories present problems because (i) the free-energy functional ${\mathit{F}}_{2}$, quadratic in the director derivatives, is unbounded from below and hence possesses no minima unless ${\mathit{K}}_{13}$ is strictly zero; and (ii) microscopic theories indicate that in the general case ${\mathit{K}}_{13}$ does not vanish. The continuum theory presupposes the existence of weak director deformations. This is not consistent with the idea, proposed by Oldano and Barbero, that there should be strong subsurface director deformations, which are shown in the present paper to be a formal consequence of (i). Instead we propose a resolution of the ${\mathit{K}}_{13}$ problem which is consistent with weak director distortions alone. The resolution involves a formal consideration of all the terms in the total free energy containing high-order derivatives, the infinite sum of which, ${\mathit{R}}_{\mathrm{\ensuremath{\infty}}}$, bounds the total free energy ${\mathit{F}}_{2}$+${\mathit{R}}_{\mathrm{\ensuremath{\infty}}}$ from below. A consequence of this resolution is that the Euler-Lagrange equations which follow from a naive consideration of the Oseen-Frank free-energy functional ${\mathit{F}}_{2}$, and which appear to give rise to a nonminimal family thereof, in fact give rise to a minimal family of director distributions of the total free energy ${\mathit{F}}_{2}$+${\mathit{R}}_{\mathrm{\ensuremath{\infty}}}$. Moreover, no specific information on higher-order elastic terms enters the theory. The theory further allows consideration of the derivative-dependent terms in the anchoring energy. Each such derivative is shown to be proportional to a small parameter. As a result, all derivative-dependent anchoring terms are much smaller than the usual Rapini-Popoular term.

Numerical Simulation of Flows of Nematic Liquid Crystals. Inlet Flow between Parallel Plates.

Two-dimensional flow problems of the Leslie-Ericksen theory for nematic liquid crystals are considered. As a concrete example, inlet flows between parallel plates are calculated using a finite difference method. The calculations are performed at low and intermediate Ericksen numbers for three patterns of surface alignment, and the effects of elasticity and viscosity in the nematic liquid crystals on the development in director orientation and velocity are examined. The validity of the numerical calculation is confirmed by making a comparison between the director orientation in the fully developed region in the present calculation and that obtained from the calculation in unidirectional flow. The director orientation in the inlet flow region depends on the Ericksen number and the surface alignment. Furthermore the velocity profile develops in a different manner from that for Newtonian fluids because of the effect of director orientation.

Hydrodynamic theory of micellar solutions: suppression of micellar growth by interactions

A hydrodynamic theory of rigid-rod micellar solutions is presented. The previously omitted intermicelle interactions are incorporated by generalizing the Onsager theory for nematic liquid crystals to account for polydispersity and self-assembly in the presence of flow. The micellar size distribution is derived as a function of shear rate in the form of a highly nonlinear integral equation. It is found that intermicellar repulsion postpones the micellar growth in flow to higher shear rates.

Continuum theory for nematic liquid crystals

This paper presents a formulation of continuum theory for nematic liquid crystals based upon the balance laws for linear and angular momentum, that derives directly expressions for stress and couple stress in these transversely isotropic liquids. This approach therefore avoids the introduction of generalised forces or torques associated with the director describing the axis of transverse isotropy.

On a generalization of the Rapini-Papoular expression of the surface free energy for nematic liquid crystals

The effect of the K 13 and K 1 elastic constants on the equilibrium orientation of nematic liquid crystals is considered. Our analysis shows that these terms can be ignored if the surface energy is modified in a particular way. In this manner the difficulties in the elastic theory of nematic liquid crystals connected to the proper minimization of the free energy can be solved. Several suggestions have been made in the past to overcome this lacuna in the elastic theory of nematics. In our paper we suggest a different approach leading to a new surface free energy.The surface variation of the nematic average orientation induced by K 13 and K 1 is also estimated.

Numerical simulation of rodlike polymers in a uniaxial extensional flow: The distribution-function version versus the order-parameter version of Doi’s theory

The molecular theory of nematic liquid crystals, proposed by Doi in 1981, exists in two versions: the original kinetic equation written in terms of the orientational distribution function and an averaged approximation of the original kinetic equation written in terms of the order parameter tensor. In this study, the two versions, as well as a third (hybrid) approach by Edwards and Beris (1989), are compared. Due to the complexity of the original kinetic equation, only uniaxial extensional flow in a homogeneous monodomain is considered. The solution of the original kinetic equation without simplifications is obtained for the first time—using a finite difference scheme with an iterative procedure for an integral equation. The solution of the kinetic equation written in terms of the order-parameter tensor (a nonlinear algebraic equation) is obtained by Newton’s procedure. For the ordered phase, the order-parameter version provides acceptable approximation only in the low-concentration region, the extent of which decreases with increasing velocity gradient. The hybrid approach of Edwards and Beris (1989) provides no improvement over the order-parameter version; in the presence of a strong flow, it actually gives worse results at low concentrations.

A van der Waals theory of nematic liquid crystals: a ‘convex peg in a round hole’ potential

The convex peg potential has been incorporated into a van der Waals theory of the (uniaxial) nematic to isotropic liquid crystal transition. In the convex peg model, molecules are envisioned to have a hard (here biaxial) core embedded in a spherically symmetric square well. Anisotropies in the potential are derived from both its repulsive and attractive regions. In accord with generalized van der Waals theories, the repulsive interactions are treated to all orders (albeit approximately using a resummation technique) and the attractive interactions are incorporated to lowest order. Global phase diagrams are determined and these predict the formation of one uniaxial nematic and two isotropic phases (vapour and liquid). From the phase diagram, one observes the coexistence of the nematic-vapour phases, the nematic-isotropic liquid phases, the vapour-liquid phases as well as a nematic-vapour-isotropic liquid triple point. By virtue of the effective anisotropic attractive forces, the first order character of the nematic to isotropic transition is significantly enhanced from that of a hard uniaxial body. However, the influence of biaxiality more than compensates for this enhancement and the resulting isotropic to nematic transition becomes weakly first order. The temperature dependence of the order parameters at fixed density (and also at fixed pressure) compare favourably with the experimental values of PAA.

Some Considerations on the Elastic Theory for Nematic Liquid Crystals

Abstract The elastic theory for nematic liquid crystals is critically analysed. After a review on the variational calculus formalism, the range of applicability of the Lagrangian method for the solution of practical problems is discussed. It is underlined that only a limited number of problems can be solved by means of a variational approach. The role at the Jacobi equation is also discussed. The importance of the non linear character of the K13-problem is analyzed in the framework of a simple molecular model. Finally, the principle of virtual work is applied to the elastic theory of nematic liquid crystals. Our analysis shows that the K13 elastic problem is an ill-posed one, since this problem can only be solved by means of a variational, or virtual work, approach by modifying the bulk elastic free energy and taking into account new terms quadratic in the second order deviatives. However it is necessary to remember that, in the proximity of a surface, a spatial variation of the density and of the scalar ...

Role of dispersion potential anisotropy in the theory of nematic liquid crystals. I. The mean-field potential

We reexamine the effect of the anisotropy in the intermolecular attractive (dispersion) interaction on the attractive mean-field potential used in the generalized van der Waals theory of the nematic–isotropic transition. Contrary to results in the literature, we show (1) that the anisotropic part of the dispersion potential reduces the anisotropy of the mean field contributed by the isotropic part of the dispersion potential; (2) that the form of anisotropy in the attractive intermolecular interaction usually chosen for simplicity in thermodynamic calculations corresponds to a preaveraged form of the dispersion interaction. The anisotropy of both the dispersion and of the mean included angle potential (which corresponds to the mean-field potential in a perfectly ordered nematic) are shown graphically, and the expansion coefficients for the latter in terms of even-ordered Legendre polynomials are tabulated.

Role of anisotropic dispersion potential in the theory of nematic liquid crystals. II. Thermodynamic calculations

We reexamine the effect of the anisotropy in the intermolecular attractive (dispersion) interaction on the thermodynamic quantities characterizing the nematic–isotropic phase transition, as estimated from generalized van der Waals theory. Contrary to statements in the literature, this anisotropy significantly affects the transition temperature and the slope of the coexistence curve. The results obtained here, as in generalized van der Waals calculations in the literature, in general do not well reproduce the experimental data. Sources of these discrepancies are discussed briefly.

More about the relations between the Ericksen-Leslie-Parodi and Eringen-Lee theories of nematic liquid crystals

This paper is a further consideration of the relation between two main phenomenological theories of nematic liquid crystals: Ericksen-Leslie-Parodi (ELP) and Eringen-Lee (EL). The aim of the study is to establish the generality of the conclusion which claims that the ELP theory is a particular case of the EL theory. According to the analysis presented in the paper this conclusion may be treated as true but only after modification of the constitutive equation for Em ij in the EL theory by one term arising from splay deformation. The results of the study are formulated in four conclusions given at the end of the paper.

Theory of nematic liquid crystals

Abstract Theories of nematic liquid crystals are briefly reviewed, and the generalized van der Waals (GVDW) theory is analyzed in some detail. In the latter approach, the short-range repulsion is approximated by hard volume exclusion, and the long-range attraction by dispersion interaction, mean-field averaged. It is shown that if the full London dispersion potential is used the anisotropy in the attractive dispersion potential substantially modifies the mean-field potential anisotropy caused by volume exclusion and affects the nematic-isotropic phase transition temperature and the slope of the coexistence curve, statements can be found in the literature to the contrary. Overall the agreement between theory and experiment, however, is poor. A rigorous formulation is presented of the configurational free energy functions in terms of one-particle distribution functions and correlation functions, applicable to any ordered system at arbitrary density, and practical means for improving the GVDW approach to nematic liquid theory are suggested.

Attenuation and Dispersion of Acoustic Waves in Reacting Nematic Liquid Crystal Mixtures

Abstract The Ericksen-Leslie theory of nematic liquid crystals is extended to non-diffusive reacting binary mixtures. This mixture theory allows for the reaction rate to depend on both the rate-of-deformation tensor and the anisotropic axis. Thermodynamic arguments result in reactive terms appearing in the expression for the stress tensor. The theory is applied to the propagation of acoustic waves. The equations are linearized assuming a forced infinitesimal harmonic plane wave, and a biquadratic polynomial is obtained for the complex wave number. In a low frequency expansion the attenuation is the sum of three parts: (1) the classical reaction contribution, (2) the thermoviscous contribution, and (3) the viscoreactive contribution arising from the dependence of the reaction rate on the alignment of the anisotropic axis and rate-of-deformation tensor. Additionally, the sound speed is a function of orientation of the anisotropic axis and frequency. The observed velocity anisotropy of acoustic waves in N-p-...

Perturbation theory for nematic liquid crystals of axially symmetric molecules: Numerical results at high pressures including second- and fourth rank orientational order parameters

Abstract A statistical mechanical perturbation theory is applied to study the thermo-dynamic properties of nematic liquid crystals at the nematic-isotropic (N-I) phase transition under pressure, retaining second- and fourth-rank long-range orientational order parameters. We report calculations for a hard ellipsoidal system superposed with an attractive interaction and subjected to different external pressures. The repulsive interaction is represented by a repulsion between hard ellipsoids characterized by a length-to-width ratio and the interaction arising from the dispersion interaction between two asymmetric molecules represents the attractive interaction. The inclusion of the fourth-rank order parameter in the effective one-body potential ψ(Ω) does not lead to a significant overall improvement. The influence of pressure on the stability, ordering and thermodynamic functions at the N-I transition is analysed. We find that our theoretical predictions are in accordance with experimental observations.

SHORT-RANGE CORRELATIONS IN MOLECULAR THEORIES OF NEMATIC LIQUID CRYSTALS

A cell theory of nematic liquid crystals is presented, which provides a calculable theoretical model for investigation of molecular short-range correlations. Numerical calculations are made for PAA The theoretical values are improved in compared with the other theories'. The effects of molecular correlations on calculated results and relations between the cell theory and the other theories are discussed.

Computer simulation studies of anisotropic systems. XI. Second- and fourth-rank molecular interactions

Abstract The Maier-Saupe theory for nematic liquid crystals provides a reasonable account of their orientational order and its temperature dependence. The theory is based on second-rank anisotropic interactions and its predictions can be improved by the introduction of higher-rank terms as in the Humphries-James-Luckhurst theory. However comparison with the properties of real nematogens does not allow an unambiguous test of the theory because the form of the pair potential is unknown. This is not the case for computer simulations where the intermolecular potential is defined. We have therefore undertaken a Monte Carlo study of the influence fourth-rank interactions on nematic behaviour and report the results of our simulations here. The model nematogen used as a reference is that developed by Lebwohl and Lasher in which the particles are confined to the sites of a simple cubic lattice and interact via a second-rank anisotropic potential. The simulation gives the internal energy, the heat capacity at constant volume and the second-rank order parameter as a function of temperature, as well as the nematic-isotropic transition temperature. These results are used to provide the first unambiguous test of the Humphries-James-Luckhurst theory. We also discuss the changes in the transition temperature which are caused by the introduction of the fourth-rank term in the pair potential using thermodynamic perturbation theory for the Helmoltz free energy.

Dynamics of orientation fluctuations. Order parameter of a nematic liquid crystal

This paper develops a dynamical approach to the theory of nematic liquid crystals on the basis of the Maier-Saupe model. The relative mean square orientation displacements of the particles from the equilibrium position (the local fluctuations) are calculated as functions of the temperature up to the instability point of the system. It is shown that at sufficiently low temperatures there is a linear growth of the fluctuations, this acquiring a strongly anomalous nature in the neighborhood of the phase transition from the nematic liquid crystal to an isotropic liquid. The temperature dependences of the specific heat and the order parameter are discussed. The results are compared with experimental data and theoretical calculations.

Perturbation Theory for Nematic Liquid Crystals of Axially Symmetric Molecules: Extension of Calculations to High Pressures

Abstract We have applied a statistical mechanical perturbation theory to study the thermodynamic properties of nematic liquid crystals under pressure. In this theory the reference potential function is nonspherical and consists of the short range rapidly varying repulsive part of the pair potential. We report calculations on the nematic-isotropic (NI) transition properties for a hard spherocylindrical system superposed with an attractive potential, which is a function of only the centre of mass distance and the relative orientation between the two molecules, and subjected to different external pressures. The interaction arising from dispersion interaction between two asymmetric molecules represents the attractive interaction. The influence of pressure on the stability, ordering and thermodynamic functions for the NI transition is analysed. It is found that the theoretical predictions are in accordance with the experimental observations.

Some topics in continuum theory of nematics

This paper presents a concise formulation of continuum theory for nematic liquid crystals, both static and dynamic theory being discussed in turn. The emphasis is on the various assumptions behind the theory, and the experimental evidence supporting them. The resulting theory contains a fair number of material parameters, and progress concerning their measurement is assessed. Also, whenever possible, mention is made of problems that remain unresolved.

Weak Boundary Storage Effect in Homogeneous Liquid Crystal Cells

Based on the Oseen and Frank elastic continuum theory of nematic liquid crystals and on a generalized torque balance equation at the liquid crystal-to-substrate interface, static deformations of a semi-infinite liquid crystal (LC) medium and a homogeneous LC cell have been investigated using the anisotropic LC-to-substrate anchoring energy C as a parameter. Both the elastic and dielectric anisotropies are considered in obtaining the solutions. Storage effects are predicted for sufficiently small anchoring anisotropy. Numerical ranges of C, within which the weak boundary storage effect may occur, have been calculated for twenty commercially-avaialble nematic LC mixtures in the homo-geneous-cell configuration. An equation for the storage time as a function of C has also been derived. The treatment has also been extended to include the most general from of a LC-to-substrate interfacial potential. Such a weak boundary storage effect could provide a means for realizing low power, non-refreshed, matrix-addressed flat panel displays.