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.     

The role of divergent terms in the Frank energy of nematic liquid crystals

The effect of divergent terms in the Frank orientation energy of nematic liquid crystals on the equilibrium state of the director field is studied. Such terms have no effect on the equations of motion or on the equilibrium of the medium under consideration; however, they should be taken into account in the derivation of boundary conditions. It is shown that, in the case of boundary perturbations or in the case of polar orientation angle perturbations, the divergent terms can be considered as a surface energy for the azimuth angle (this energy is similar to the Rapini-Papoular energy). In addition, these terms may cause a deviation of the director in the plane parallel to the boundary. The equilibrium problem for a nematic liquid crystal is considered as an example in the case of small periodic boundary perturbations.     

Shear cell rupture of nematic liquid crystal droplets in viscous fluids

We model the hydrodynamics of a shear cell experiment with an immiscible nematic liquid crystal droplet in a viscous fluid using an energetic variational approach and phase-field methods [86]. The model includes the coupled system for the flow field for each phase, a phase-field function for the diffuse interface and the orientational director field of the liquid crystal phase. An efficient numerical scheme is implemented for the two-dimensional evolution of the shear cell experiment for this initial data. The same model reduces to an immiscible viscous droplet in a viscous fluid, which we simulate first to compare with other numerical and experimental behavior. Then we simulate drop deformation by varying capillary number (independent of liquid crystal physics), liquid crystal interfacial anchoring energy and Oseen–Frank distortional elastic energy. We show the number of eventual droplets (one to several) and “beads on a string” behavior are tunable with these three physical parameters. All stable droplets possess signature quadrupolar shear and normal stress distributions. The liquid crystal droplets always possess a global surface defect structure, called a boojum, when tangential surface anchoring is imposed. Boojums [79], [32] consist of degree +1/2 and ?1/2 surface defects within a bipolar global orientational structure.     

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.     

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     

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.     

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.     

Transitions in Poiseuille flow of nematic liquid crystal

Recent experiments by Sengupta et al. (Phys. Rev. Lett. 2013) [9] revealed interesting transitions that can occur in flow of nematic liquid crystal under carefully controlled conditions within a long microfluidic channel of width much larger than height, and homeotropic anchoring at the walls. At low flow rates the director field of the nematic adopts a configuration that is dominated by the surface anchoring, being nearly parallel to the channel height direction over most of the cross-section; but at high flow rates there is a transition to a flow-dominated state, where the director configuration at the channel centerline is aligned with the flow (perpendicular to the channel height direction). We analyze simple channel-flow solutions to the Leslie–Ericksen model for nematics. We demonstrate that two solutions exist, at all flow rates, but that there is a transition between the elastic free energies of these solutions: the anchoring-dominated solution has the lowest energy at low flow rates, and the flow-dominated solution has lowest energy at high flow rates.     

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.     

Function Spaces for Liquid Crystals

We consider the relationship between three continuum liquid crystal theories: Oseen–Frank, Ericksen and Landau–de Gennes. It is known that the function space is an important part of the mathematical model and by considering various function space choices for the order parameters s, n, and Q, we establish connections between the variational formulations of these theories. We use these results to justify a version of the Oseen–Frank theory using special functions of bounded variation. This proposed model can describe both orientable and non-orientable defects. Finally we study a number of frustrated nematic and cholesteric liquid crystal systems and show that the model predicts the existence of point and surface discontinuities in the director.     

Conoscopic observations of shear-induced rotations in nematic liquid crystals

Orientational changes in monodomains of flow-aligning liquid crystals, 4-n-pentyl-4′-cyanobiphenyl and N-(4-methoxybenzylidene)-4-butylaniline, were studied during shear and recovery in a linear shearing device fitted to an optical microscope. Planar alignment (director in the shear plane) allows the study of twist effects and was generated by strong planar anchoring at the walls with orientations in a range of 0–90° with the shear direction. While being held back by the anchoring walls, shear caused the bulk director to rotate towards a steady-state alignment angle in the shear direction (Leslie angle θ_L). The transient director rotation was observed with conoscopy. It was found that increasing the initial alignment towards the vorticity direction increased the measured θ_L. Upon stopping the flow, the bulk director returned to its initial state. With initial alignment orientation changing from parallel to perpendicular to the flow direction, the rate of the twist-driven recovery process increases. This rate increase is not seen in the splay-driven recovery which is constant and consistently faster than twist-driven recovery at all orientations studied. Received: 10 December 1998/Accepted: 7 June 1999     

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.     

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.     

The electrical untwisting of a homeotropically aligned smectic C* liquid crystal

This article is concerned with the application of an electric field along the planar layers of a finite sample of ferroelectric smectic C^* liquid crystal with strong smectic anchoring at the boundaries. It will be seen by use of continuum theory that above a certain critical field strength it is energetically favourable for the helix to unwind such that the c-director becomes aligned perpendicular to the applied field throughout the sample. This problem is similar to that of untwisting the helical structure of a cholesteric liquid crystal [1], [2]; the differences lie in the symmetry and polarization of the smectic C^* phase. Comparison of this work with that detailed in [2] will show that these two factors lead to significant differences in the analysis. The effects of sample width on the minimum field strength required for this untwisting are also examined and an expression is gained for the critical field strength in a large sample. In addition, director reorientation due to a reversal of the applied field is discussed.     

Effects of strong anchoring on the dynamic moduli of heterogeneous nematic polymers II: oblique anchoring angles

We extend previous work on the linear viscoelastic moduli of heterogeneous nematic polymers in a small-amplitude oscillatory shear flow, focusing on the role of the orientational anchoring conditions at the plates. When tangential or normal anchoring conditions are applied, the Doi–Marrucci–Greco orientation tensor-flow model effectively reduces to the Leslie–Ericksen director-flow model, predicting that director distortions control the dynamic moduli with negligible contributions from tensor-order parameters. In this paper, we examine oblique anchoring angles. We use a combination of analysis and numerical simulation on the generalized tensor-flow system for arbitrary anchoring conditions to show that any oblique anchoring condition induces a nontrivial order parameter contribution to the dynamic moduli, which vanishes only in the limit of tangential or normal anchoring. Our approach reveals that the storage and loss moduli admit an approximate decomposition in terms of two reduced problems that are exactly solvable: the heterogeneous director–flow response plus the monodomain tensor response to an imposed shear. The importance of this result is that we gain scaling properties of the moduli with respect to material parameters and experimental conditions without having to compute and assimilate across the full parameter space. These results provide insight into the relative importance of the distortional vs bulk nematic elastic stress in determining the viscoelastic moduli, predicting that anchoring conditions tune the relative contributions.     

Ordering effects in electric splay Freedericksz transitions

We analyze the interaction between a nematic liquid crystal and an electric field, in a cell in which splay Freedericksz geometry is enforced. Equilibrium configurations are explored both close to the Freedericksz threshold and in the limit of strong applied voltages. We frame within de Gennes’ order-tensor theory, which allows us to detect the effects of a variable degree of orientation on critical fields and bifurcation shapes. The applied voltage induces nontrivial effects on the degree of orientation as well. Up to the Freedericksz transition, the degree of orientation decreases, whereas ordering is recovered when the applied voltage drop increases. We also stress the role played by the dielectric anisotropy. In particular, the limit in which the dielectric anisotropy approaches the dielectric permittivities deserves attention, since the order-tensor theory regularizes some of the critical phenomena exhibited by classical Frank solutions.      

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.     

Brownian dynamics simulations for suspension of ellipsoids in liquid crystalline phase under simple shear flows

Brownian dynamics simulations of shear flows are carried out for various suspensions of ellipsoids interacting via the Gay-Berne potential. In this simulation all the systems of the suspension are in a liquid crystalline phase at rest. In a shear flow they exhibit various motions of the director depending on the shear rate: the continuous rotation, the intermittent rotation, the wagging-like oscillation, and the aligning. The director is almost always out of the vorticity plane when it rotates, that is the kayaking. The number density of the system and the inter-particle potential intensity significantly affect the shear rate dependence of orientation. In particular, the continuous rotation of director is maintained to higher shear rates for the system with a stronger potential. Furthermore, the rheological properties are examined. The shear-thinning in viscosity is observed, but the negative first normal difference is not obtained.     

High‐Frequency Changes in the Orientation of Nematic Liquid Crystals in Radio Frequency Crossed Electric Fields

The orientational dynamics of the director of a nematic liquid crystal located in radio frequency crossed electric fields were studied by numerical calculations and experimentally. This system is shown to be a physical object of nonlinear dynamics. Depending on the parameters of the problem, the following types of states of the director were observed: stationary (an analog of the nonthreshold Freedericksz transition), periodic, quasiperiodic (multimode), and stochastic of the strange attractor type. In the calculations, all states were obtained by solving a deterministic system of two timedependent nonlinear differential equations of the first order with no electrohydrodynamic terms. All types of solutions obtained, including stochastic ones, were observed experimentally.