Orientation instability of the layer of a nematic liquid crystal

The origin of periodic structures in a layer of a lyotropic nematic liquid crystal observed in the director (vector, describing the anisotropic properties of the medium) reorientation experiment is studied. Such perturbations with the wavevector perpendicular to the initial orientation can develop in a liquid crystal layer in the unstable equilibrium state when the director is parallel to the walls under the condition that its orthogonality to the boundary corresponds to the minimum anchoring energy. It is shown that the linear dependence of the domain period on the layer thickness observed experimentally can be theoretically described when the Frank orientation elasticity energy is considered in the most general form taking the divergence terms into account and the anchoring energy of orientation is small as compared with the bulk energy. A relation between the coefficient of the divergence terms (saddlesplay elastic constant) and two other coefficients in the Frank energy is obtained.     

Orientational instability of a lyotropic nematic liquid crystal layer

The problem of periodic domain initiation in a thin lyotropic nematic liquid crystal layer is studied. This layer has a planar director initial orientation, but the anchoring energy is minimized by the homeotropic one. The periodic structures whose wave vector is perpendicular to the director exist during the director reorientation process from the planar orientation to the homeotropic one when the reorientation wave front appears. It is shown that the divergent terms of the Prank orientation elasticity energy plays an important role in this effect. The saddle-splay Prank constant and the anisotropic anchoring energy coefficient are estimated.     

Refinement of boundary conditions for nematic liquid crystals in the one-constant approximation

The boundary conditions are studied for nematic liquid crystals in the case of weak anchoring. The cases of the general expression and one-constant approximation are considered for the Frank energy of elastic distortions of the director field. It is shown that the one-constant approximation is correct for one-dimensional problems only and, for two- and three-dimensional problems, this model significantly simplifies the boundary conditions and changes their type.     

Magnetic field-induced changes in molecular order in nematic liquid crystals

d there are two inherent characteristic lengths, the nematic correlation length and the magnetic coherence length . As the magnetic field increases the magnetic coherence length decreases and the relative ordering of the three length scales determines the director and scalar order parameter configuration through the cell. We use asymptotic expansions in regions defined by these length scales to analytically determine the molecular configuration in terms of these variables. Specifically, we investigate the boundary layer between the cell substrate and the bulk nematic material when strong anchoring forces the nematic director in a different direction to that of the applied field. We find that at low field strengths the classical picture of liquid crystal/magnetic field interaction occurs, that is, the director orientation is governed by the surface alignment until a transition occurs as the magnetic coherence length becomes comparable to the cell thickness and the director changes orientation so as to align with the magnetic field. At high field strengths, we find that a field-induced reduction of the molecular order occurs in a region close to the cell boundary. We are able to analytically determine the director and scalar order parameter configurations for the majority of field strengths and where analytical solutions are not found a numerical solution is presented. It is hoped that further work will extend this basis of analytical solutions to include a solution for all field strengths and for different cell configurations. Received July 31, 2001 / Published online May 21, 2002     

Thermomechanical modeling of the thermo-order-mechanical coupling behaviors in liquid crystal elastomers

Liquid crystal elastomer is a kind of anisotropic polymeric material, with complicated micro-structures and thermo-order-mechanical coupling behaviors. In this paper, we propose a method to systematically model these coupling behaviors. We derive the constitutive model in full tensor structure according to the Clausius-Duhem inequality. Two of the constitutive equations represent the mechanical equilibrium and the other two represent the phase equilibrium. Choosing the total free energy as the combination of the neo-classical free energy and the Landau-de Gennes nematic free energy, we obtain the Cauchy stress-deformation gradient relation and the order-mechanical coupling equations. We find the analytical homogeneous solutions of the deformation for the typical mechanical loadings, such as uniaxial stretch, and simple shear in any directions. We also compare the compression behavior of prolate liquid crystal elastomers with the stretch behavior of oblate liquid crystal elastomers. As a result, the stress, strain, temperature, order parameter, biaxiality and the direction of the director of liquid crystal elastomers couple with each other. When the prolate liquid crystal elastomer sample is stretched in the direction parallel to its director, the deviatoric stress makes the mesogens more order and increase the transition temperature. When the sample is sheared or stretched in the direction non-parallel to the director, the director of the liquid crystal elastomer will rotate, and the biaxiality will be induced. Because of the order-mechanical coupling, under infinitesimal deformation, liquid crystal elastomer has anisotropic Young’s modulus and zero shear modulus in the direction parallel or perpendicular to the director. While for the oblate liquid crystal elastomers, the stretch parallel to the director will cause the rotation of the director and induce the biaxiality.     

Continuum Electromechanical Theory for Nematic Continua with Application to Freedericksz Instability

Of interest in this work are nematic continua that exhibit electromechanical coupling. The first part of this paper presents a novel variational formulation with a potential energy depending on four independent variables (the displacement, director, specific polarization and electric displacement perturbation). Variations of the potential energy with respect to each one of these variables lead to the governing mechanical equilibrium and constitutive relations plus Maxwell’s equations.The proposed variational formulation is next applied to the study of bifurcation of an infinite layer of a nematic liquid crystal confined between two parallel plates and subjected to a uniform electric field perpendicular to these plates under full anchoring boundary conditions. As the electric field exceeds a critical value, the nematic directors which are initially parallel to the plates, rotate and tend to align with the electric field orientation. This phenomenon, termed in the literature as Freedericksz transition, is treated here as a bifurcation problem using a fully 2D formulation. It is shown that the solution corresponding to the lowest applied electric field, also termed the critical load, is uniform in the direction parallel to the plates and that the corresponding bifurcated path is stable near this critical load. This result holds for arbitrary positive constants of the Frank-Oseen energy (and values of electric susceptibility constants that allow bifurcation) and justifies the 1D treatment of the Freedericksz transition in 2D settings that is widely adopted in the liquid crystal literature. An asymptotic analysis of the supercritical, stable bifurcated equilibrium path about the critical load is also presented and compared with the exact bifurcated solution.     

Dynamic rheology of the immiscible blends of liquid crystalline polymers and flexible chain polymers

Immiscible blends containing liquid crystalline polymers (LCP) as dispersed phases show different dynamic rheological properties than those composed of flexible polymers. The widely used Palierne’s model was shown by many authors to be insufficient to describe the frequency dependence of dynamic modulus of such blends. A new model was presented to describe the dynamic rheology of the immiscible blend containing LCP as a dispersed phase. The flexible chain polymer matrix was assumed to be a linear viscoelastic material under small amplitude oscillatory shear flow, and the LCP was assumed to be an Ericksen’s transversely isotropic fluid. The Rapini-Papoular equation of anisotropic interfacial energy was used to account for the effect of nematic orientation on the interfacial tension. It was found that the orientation of the director and the anchoring energy greatly influenced the storage modulus at the “shoulder” regime. The overall dynamic modulus of the blend can be well described by the model with suitable choice of the orientation of the director and anchoring energy of LCP.     

Curvature effects on the surface viscosity of nematic liquid crystals

We analyse the flow of a nematic liquid crystal close to a cylindrical surface. As a first result, we show that it is not possible to decouple the motion equations for the director and the macroscopic velocity. Nevertheless, we derive an expression for an effective viscosity, which can be used in the dynamic boundary condition linking the time derivative and the gradient of the director angle on the surface. The effective viscosity so obtained may be greater or smaller than its planar counterpart, depending on both the concavity of the surface and the sign of the Leslie coefficients α₂ and α₃. In particular, in materials that align in shear, the correction to the surface viscosity changes sign depending on whether the angle between the director and the surface normal exceeds or not a critical value.     

Director reorientation and order reconstruction: competing mechanisms in a nematic cell

We propose a model to explore the competition between two mechanisms possibly at work in a nematic liquid crystal confined within a flat cell with strong uniaxial planar conditions on the bounding plates and subject to an external field. To obtain an electric field perpendicular to the plates, a voltage is imposed across the cell; no further assumption is made on the electric potential within the cell, which is therefore calculated together with the nematic texture. The Landau-de Gennes theory of liquid crystals is used to derive the equilibrium nematic order tensor Q. When the voltage applied is low enough, the equilibrium texture is nearly homogeneous. Above a critical voltage, there exist two different possibilities for adjusting the order tensor to the applied field within the cell: plain director reorientation, i.e., the classical Freedericksz transition, and order reconstruction. The former mechanism entails the rotation of the eigenvectors of Q and can be described essentially by the orientation of the ordinary uniaxial nematic director, whilst the latter mechanism implies a significant variation of the eigenvalues of Q within the cell, virtually without any rotation of its eigenvectors, but with the intervention of a wealth of biaxial states. Either mechanism can actually occur, which yields different nematic textures, depending on material parameters, temperature, cell thickness and the applied potential. The equilibrium phase diagram illustrating the prevailing mechanism is constructed for a significant set of parameters.      

Orientation instability of shear flow of a nematic liquid crystal

The instability of shear flow of a nematic liquid crystal layer is studied. The case when the orientation vector and the flow velocity vector are parallel is considered. It is shown that the orientation instability of this flow is possible if the anchoring boundary condition is weak and if the splay-bend constants in the Frank energy are taken into account. For this type of instability, periodic structures are possible to appear. Their wave vector belongs to the plane of flow and is perpendicular to the velocity vector. The medium parameters are estimated on the basis of the existence condition for this instability. The period of the appearing periodic structures is evaluated.     

Poiseuille flow of Leslie–Ericksen discotic liquid crystals: solution multiplicity, multistability, and non-Newtonian rheology

Computational modeling of the steady capillary Poiseuille flow of flow-aligning discotic nematic liquid crystals (DNLCs) using the Leslie–Ericksen (LE) equations predicts solution multiplicity and multistability. The phenomena are independent of boundary conditions. The steady state solutions are classified into: (a) primary, (b) secondary, and (c) hybrid. Primary solutions exist for all orientation boundary conditions and all flow rates, and are characterized by a flow-alignment angle that is closest to the anchoring angle at the bounding surface. Secondary solutions exist for all orientation boundary conditions and flow rates above a certain critical value. The secondary solutions are characterized by a flow-alignment angle which can be either the nearest neighbor below the primary solution or any multiple of π above. Hybrid solutions interpolate between the primary and the nearest secondary solutions, and hence exhibit two alignment angles. All solutions are stable to planar finite amplitude perturbations. Hybrid solutions are unstable to front propagation and lead to primary or secondary solutions. The non-Newtonian rheology of the primary and secondary solutions is characterized by non-classical shear thinning and thickening apparent viscosity behavior. Well-aligned monodomains can lead to shear thickening, thinning, or a sequence of both. The degree of rheological uncertainty is present for planar and homeotropic anchoring conditions. The non-Newtonian rheology of non-aligned samples leads to shear thinning and lack the uncertainty of well-aligned samples, since the apparent viscosity becomes insensitive to orientation.     

Shear response of monodomains of flow-aligning nematic liquid crystals: TIF model comparisons and effect of pre-tilt

This work is the continuation of a previous study [Van Horn BL, Winter HH (2000) Dynamics of shear aligning nematic liquid crystal monodomains. Rheol Acta 39:294–300] on the shear dynamics of monodomains of shear aligning nematic liquid crystals [NLC]. The strain dependence of director orientation has been experimentally investigated for monodomains of a NLC with various initial orientations. Comparison of experimental results to predictions using Ericksen's transversly isotropic fluid [TIF] model supports the validity of the TIF model for systems of low molecular weight NLCs. The TIF model has been used to examine the effect of pre-tilt on the dynamics of flow-aligning NLC monodomains. It is shown that small deviations from planar alignment (no pre-tilt) have a large effect on orientation dynamics.     

Influence of surface tension on the stability of heterogeneous equilibrium of barotropic liquid phases

On the basis of the results of earlier work of the author [1] a study is made of the equilibrium and stability of a two-phase single-component heterogeneous liquid system with respect to perturbations of arbitrary shape. Allowance is made for the influence of surface tension, which plays a critical part in the formation of nucleating centers of a new phase [2]. Conditions of equilibrium are derived, and also a criterion of radial stability of a nucleating center of a new phase bounded by a closed spherical boundary. It is shown that radial stability of spherical nucleating centers also guarantees stability with respect to perturbations of arbitrary shape. The part played by the finite size of the system and the boundary conditions is elucidated. For this, two different cases are studied: a) a system under a constant external pressure, b) a system with fixed volume. In the first case, all equilibrium states are unstable. In the second, there are both unstable and stable configurations (depending on the corresponding values of two dimensionless parameters). The equation of the hyperbola of neutral stability is derived. The limits of a very small coefficient of surface tension and a very large size of the container are considered. The first situation corresponds to stable configurations, the second to unstable. For simplicity, the considered systems are assumed to be isothermal, and the equilibrium and stability are analyzed on the basis of the mechanical analog of Gibbs's principle, namely, the principle of a minimum of the mechanical potential energy of the barotropic heterogeneous liquid system. The case of nonisothermal perturbations leads to similar results, but the expressions for the corresponding dimensionless parameters are more cumbersome and less physically perspicuous.     

Orienting properties of discotic nematic liquid crystals in Jeffrey-Hamel flows

The orienting properties of incompressible discotic nematic liquid crystals for creeping flows between converging and diverging planar walls (Jeffrey-Hamel) are analyzed using the Leslie-Ericksen theory. The dependence of director orientation on the reactive parameter and the flow kinematics is presented. Closed form stationary solutions for the director orientation are found when elastic effects are neglected. Stationary numerical solutions for the velocity and director fields using the full Leslie-Ericksen theory are presented. The director field in converging flow is characterized by azimuthal (radial) centerline orientation, by being asymmetric with respect to the azimuthal (radial) direction, and by having an allowed orientation range that spans two half-quadrants (full quadrants). In the limiting case of perfectly flat disk ( –) the flow-induced director orientation in converging flow is the azimuthal direction, while in diverging flow the director rotates by a full n radians. By reducing the vertex angle between the walls to vanishingly small values, converging flow solutions properly reduce to those of flow between parallel plates, but diverging flows are expected to lead to a new instability.     

Stability of motion of the plane boundary separating two fluids

The effect of viscosity on the disintegration of liquid jets can be considered in two ways. First, viscous forces alter the basic flow: they form a boundary layer whose presence necessarily alters wave formation. Second, viscous forces can have a direct effect on the development of perturbations for a given velocity profile of the basic flow. In this case the study of stability must be based on the Navier-Stokes equations instead of the equations of an ideal fluid. This complicates analysis considerably. Available data [1] indicate that this influence is very minor in the case of moderately viscous fluids. It appears, therefore, that the principal role is played by changes in the velocity profile alone, and that the behavior of the perturbations is described by the equations of an ideal fluid.In the present study we investigated the stability of motion and wave formation at the boundary between two fluids in order to determine the effect of viscosity on the drop formation mechanism. The simplest case of oscillation of the boundary is chosen in order to keep the analysis as simple as possible.     

Computational analysis of liquid crystalline elastomer membranes: Changing Gaussian curvature without stretch energy

Liquid crystalline elastomers (LCEs) can undergo extremely large reversible shape changes when exposed to external stimuli, such as mechanical deformations, heating or illumination. The deformation of LCEs result from a combination of directional reorientation of the nematic director and entropic elasticity. In this paper, we study the energetics of initially flat, thin LCE membranes by stress driven reorientation of the nematic director. The energy functional used in the variational formulation includes contributions depending on the deformation gradient and the second gradient of the deformation. The deformation gradient models the in-plane stretching of the membrane. The second gradient regularises the non-convex membrane energy functional so that infinitely fine in-plane microstructures and infinitely fine out-of-plane membrane wrinkling are penalised. For a specific example, our computational results show that a non-developable surface can be generated from an initially flat sheet at cost of only energy terms resulting from the second gradients. That is, Gaussian curvature can be generated in LCE membranes without the cost of stretch energy in contrast to conventional materials.     

The stability of travelling waves induced by crossed electric and magnetic fields in nematic liquid crystals

A theoretical study is carried out into the stability of travelling wave solutions to an approximate dynamic equation for the problem in which a nematic liquid crystal is subjected to crossed electric and magnetic fields. The authors recently found three types of travelling wave solutions for this problem [2], each characterised by the control parameter which describes the relationship between the magnitudes of the fields and their crossed angle. Two types of stability are ex amined: the first considers perturbations which vanish outside some finite interval in the moving coordinate of the travelling wave, while the second considers quite general perturbations belonging to a weighted space, the weighting function being determined by the particular solution and the control parameter . When the first type of stability occurs, perturbations decay to zero as time increases. In the second type of stability perturbations may eith er decay to zero or induce a small phase shift to the original travelling wave. Both these versions of stability depend crucially on and on the type of travelling wave solution being considered. Received April 15, 1997     

Surface waves in nematic liquid crystals

The problem of small-amplitude harmonic wave propagation along the surface of an incompressible nematic liquid crystal is considered when the evolution of the orientation vector is specified by viscous stresses and the orientational elasticity can be ignored. An analytic solution and a dispersion relation are obtained in the case of the planar and homeotropic initial orientation of this vector.