Numerical analysis for a nematic droplet in polymer dispersed liquid crystals

Starting from the Landau-de Gennes free energy expression, the author has numerically analysed the director pattern in a nematic droplet of polymer dispersed liquid crystals. The nematic director has been understood as the eigenvector, which corresponds to the largest eigenvalue of the tensor order parameter. To investigate the droplet structure influence, all equations have been treated on the curvilinear coordinate system which is generated along the droplet boundary. In the case of spherical and spheroidal droplets with normal strong anchoring, the director exhibits an axial configuration and a disclination ring. The ring radius and the capactiance of the system change without hysteresis with the applied voltage.     

Confinement of a liquid crystal to small droplets and its effect on nuclear magnetic relaxation

Abstract

The spatial dependence of the orientation of the molecular director and of the nematic order parameter is obtained by minimization of the Landau–de Gennes free energy of the nematic liquid crystal confined in a spherical droplet. Special attention is given to the vicinity of the nematic–isotropic transition. The influence of the resulting nematic structure, large liquid crystal–polymer interface and restricted molecular diffusion on the nuclear magnetic relaxation is analysed. The translationally-induced molecular reorientation and the liquid crystal–polymer cross relaxation are discussed in particular. The possibility of an indirect study of the molecular anchoring on the polymer surface is demonstrated.     

Modelling of director equilibrium states in a nematic cell with relief surface

We study two-dimensional equilibrium configurations of nematic liquid crystal (NLC) director in a cell bounded by two parallel surfaces. One surface is planar and the other one is spatially modulated. The relief of the modulated surface is described by a smooth periodic sine-like function. The orientation of NLC director easy axis is assumed to be homeotropic at one cell surface and planar at the other one. The NLC director anchoring with cell surfaces is assumed to be strong. We consider the case where disclination lines occur in the bulk of NLC above the extrema of the modulated surface. These disclination lines run along the crests and troughs of the surface relief. If the orientation of director at both bounding surfaces is of the same type, then NLC director field is continuous. For both configurations mentioned above (with defects and without defects), we obtain analytical expressions for director distribution in the bulk of NLC in the approximation of planar director deformations. Equilibrium distances from disclination lines to the spatially modulated surface are calculated when the defects occur. The dependences of these equilibrium distances on the period and depth of surface relief and the cell thickness are investigated in detail.     

A thermodynamic approach to the anchoring phenomenon in the nematic liquid crystal-substrate system

The phenomenon of anchoring in the nematic liquid crystal-amorphous substrate system is considered and model independent definitions of the surface nematic director, the surface tension and the anchoring energy coefficient are proposed. Then the Landau-de Gennes model of the system is studied for a specific choice of the surface parameters, which leads to a continuous homeotropic-conical anchoring transition. The free energy as a function of the director at a distance l from the surface is found. It is shown that its form is different in two regions of the temperature-distance plane separated by the line of a Freedericksz transition. The asymptotic behaviour of the free energy for large l and for infinitesimal deviations of the director from the anchoring direction is studied. It is found that the asymptotic formula holds also in the vicinity of the anchoring transition. Finally, the results of numerical studies of the Landau-de Gennes model are compared with the predictions of a simple phenomenological model.     

Nematic director configurations in high pretilt, parallel aligned layers

We describe the director configurations associated with a negative dielectric anisotropy, high pretilt (typically ~ 85°) nematic cell. A number of distinct director configurations are observed, with both a transition to a spontaneously twisted state and a nucleated transition mediated by disclination line movement. We discuss the role of surface pretilt and elastic constant anisotropy on behaviour and give results on the relative energetic stability of the director states.     

Nematic director configurations in high pretilt,parallel aligned layers

We describe the director configurations associated with a negative dielectric anisotropy, high pretilt (typically ~ 85°) nematic cell. A number of distinct director configurations are observed, with both a transition to a spontaneously twisted state and a nucleated transition mediated by disclination line movement. We discuss the role of surface pretilt and elastic constant anisotropy on behaviour and give results on the relative energetic stability of the director states.     

Influence of anchoring at a nematic cell surface on threshold spatially periodic reorientation of a director

We have analysed the influence of surface director anchoring in a planar flexoelectric nematic cell on the threshold spatially periodic reorientation of the director in an external dc electric field. By minimizing the free energy of the nematic cell we obtained the equations for a director and numerically solved them in the one elastic constant approximation. The dependences of the threshold electric field and the spatial period of director structure on the azimuthal and polar anchoring energy, as well as the flexoelectric parameters, are determined. It is shown that the domain of the flexoelectric parameter values, at which the spatially periodic reorientation of a director takes place, increases with decreasing azimuthal anchoring energy and increasing polar anchoring energy.     

Interaction of electromagnetic waves in planar waveguide with a nematic layer

This work is a theoretical study of energy exchange between two coupled TE-wave modes on director diffraction grating in a planar waveguide containing a layer of nematic liquid crystal. The diffraction grating is produced by an external electric field in the nematic layer with spatial periodic anchoring energy between director and waveguide surface. The intensity of a signal mode at the output of the nematic layer has been calculated in dependence of anchoring energy amplitude and modulation period, the size of nematic layer and electrical field value. The cases of co-propagating and oppositely propagating modes have been analysed. The analytical expressions that describe the maximum values of signal mode intensity have been derived. The maximum intensity value output from the nematic has been shown to depend monotonously on the anchoring energy parameters in the case of oppositely propagating wave modes and non-monotonously in the case of co-propagating wave modes. In both cases, the maximum value of signal mode intensity grows with the increase in electric field.     

Numerical analysis of nematic liquid crystal alignment on asymmetric surface grating structures

The influence of an asymmetric periodic grooved cell surface on the 2D static director configuration of a nematic liquid crystal has been investigated. The minimum in the Frank-Oseen free energy was solved numerically with the Rapini-Papoular form of the surface anchoring energy at the nematic-grating interface. Results are presented for the variation of pretilt angle in the tilted bulk director field as a function of the surface groove depth, pitch and asymmetry and the bulk parameters. The simulations demonstrate the existence of two energetically degenerate high and low pretilted bulk alignment configurations. The pretilt values in these two regimes and also for the low tilt regime with finite surface anchoring are consistent with experimental results. An effective increase in the resolution of the model is obtained by using an irregular grid to describe the surface profile.     

Numerical analysis of nematic liquid crystal alignment on asymmetric surface grating structures

The influence of an asymmetric periodic grooved cell surface on the 2D static director configuration of a nematic liquid crystal has been investigated. The minimum in the Frank-Oseen free energy was solved numerically with the Rapini-Papoular form of the surface anchoring energy at the nematic-grating interface. Results are presented for the variation of pretilt angle in the tilted bulk director field as a function of the surface groove depth, pitch and asymmetry and the bulk parameters. The simulations demonstrate the existence of two energetically degenerate high and low pretilted bulk alignment configurations. The pretilt values in these two regimes and also for the low tilt regime with finite surface anchoring are consistent with experimental results. An effective increase in the resolution of the model is obtained by using an irregular grid to describe the surface profile.     

Two-dimensional director configurations in a nematic-filled cylindrical capillary with the hybrid director alignment on its surface

ABSTRACT

We analytically calculate two-dimensional equilibrium configurations of the nematic liquid crystal (NLC) director in a right cylindrical capillary with the hybrid director alignment on its inner surface. A part of this surface is treated to impose the strong homeotropic alignment, while on the remaining part the strong circular alignment, perpendicular to the capillary axis, is imposed. Owing to the system homogeneity along the capillary axis, a disclination line emerges in the bulk of the NLC, which runs parallel to the axis. It is shown that there exist either one or two equilibrium locations of the disclination line depending on the ratio of capillary surface parts with homeotropic and circular alignment. The technique that allows to obtain an analytical expression for the system free energy as a function of a disclination line location is presented. It considerably simplifies calculations and can be used while solving a variety of problems in which a defect location is sought.     

Second order elasticity in nematics: A new anchoring source

By considering, in the expression of the nematic free energy density, an additional term in the square of the director second derivatives, an unexpected anchoring source results, due only to surface and bulk elastic constants. As an example, the case of a planar homogeneous and of a homeotropic nematic cell, equally anchored on both walls, is discussed. In both situations the new anchoring source has a destabilizing effect.     

A new configurational transition in inhomogeneous nematics

At the wall in a hybrid cell with strong anchoring, the nematic director is parallel to one wall and perpendicular to the other. Usually, the free energy is minimized by a configuration where the director orientation changes continuously with position across the cell. The boundary conditions can also be satisfied, however, by a biaxial configuration without such rotation. Under certain conditions, such as under increased curvature strains, a transition can take place between these configurations. The transition typically occurs when the wavelength of the deformation becomes comparable to the coherence length of the material. The hybrid cell considered is a simple illustrative example; in real systems, such a transition may be expected in highly strained thermotropics, or in strained lyotropics which are easily made biaxial.     

Disclinations dynamics in confined nematic liquid crystals: strong anchoring

We have investigated the dynamics of pair annihilation of disclination lines in strong anchoring. This work is based on the Frank free energy. The director angle, φ(x, y, z), is obtained by the continuous theory. We show that the form of the viscous force in a confined nematic liquid crystal and in strong anchoring is a function of the initial distance between the two disclination lines. The asymptotic velocity, v _asy, is also a function of the initial distance. Our theoretical result on the asymptotic velocity is in good agreement with previous experimental results.     

Electric field studies of liquid crystal droplet suspensions

The responses of freely-suspended micron-sized liquid crystal droplets subjected to an alternating electric field are presented. By examining droplets of isotropic, nematic bipolar, and nematic radial configurations, we test the effect of anchoring on the droplet response. Specifically, using birefringence and scattering dichroism we measure the relaxation of electric field-induced orientation following a field pulse. Results indicate that bipolar and radial droplets in suspension orient in the field through very different mechanisms. Bipolar droplets are observed to rotate their defect axes in the field while radial droplets orient through a nematic distortion. By varying the field pulse, we observe that droplets also respond differently to the field depending on their relative sizes. In radial droplet suspensions we quantitatively measure time scales associated with the reorientation and restructuring of the defect region.     

Surface viscosity and reorientation process in an asymmetric nematic cell

The influence of surface viscosity and anchoring energy on the reorientation process of a nematic liquid crystal cell is theoretically investigated. The cell is a slab of thickness, d, whose limiting surfaces are characterised by different anchoring strengths and present easy directions parallel to the bounding surfaces, changing with time due to some external action. The exact space-time profile of the director angle is obtained by means of integral transform techniques and a Green function approach. From this formalism, the time dependence of the optical path difference is exactly determined and its behaviour is analysed in connection with the presence of surface viscosity and different anchoring energies. The problem is also exactly solved in the presence of a constant electric field. It is shown that the compatibility problem between the time derivative of the director field on the surface and in the bulk can be avoided.     

Dynamic behaviour at a nematic liquid crystal-rubbednylon interface using evanescent wave photon correlation spectroscopy

Photon correlation spectroscopy of light scattered by director fluctuations from an evanescent optical wave propagating in the nematic liquid crystal 5CB is used to study the interfacial dynamic behaviour of the liquid crystal. The intensity correlation function of light scattered by interfacial orientation fluctuations is measured by illuminating to give a short optical penetration depth within the nematic. These surface scattering correlation functions strongly differ from the bulk correlation function and are interpreted in terms of a nematic surface orientation mode arising from the coupling between the director field and the fluid velocity. It is shown that the analysis of the surface mode gives a method for measuring anchoring energies in liquid crystals. The anchoring energy obtained for rotation of the director away from the rubbing direction about an axis normal to the surface for 5CB at a rubbed nylon surface is 7.14±0.7 × 10-² ergcm-².     

A new configurational transition in inhomogeneous nematics

Abstract

At the wall in a hybrid cell with strong anchoring, the nematic director is parallel to one wall and perpendicular to the other. Usually, the free energy is minimized by a configuration where the director orientation changes continuously with position across the cell. The boundary conditions can also be satisfied, however, by a biaxial configuration without such rotation. Under certain conditions, such as under increased curvature strains, a transition can take place between these configurations. The transition typically occurs when the wavelength of the deformation becomes comparable to the coherence length of the material. The hybrid cell considered is a simple illustrative example; in real systems, such a transition may be expected in highly strained thermotropics, or in strained lyotropics which are easily made biaxial.     

Effect of an electric field on defects in a nematic liquid crystal with variable surface anchoring

We report experimental studies on defects in a nematic liquid crystal with negative dielectric anisotropy mounted in a cell with perfluoropolymer-coated surfaces. The sample exhibits a discontinuous anchoring transition from planar to homeotropic on cooling at zero or a small electric field, and above a cross-over voltage a continuous ‘inverse Freedericksz transition’, at which the director starts tilting in opposite directions at the two surfaces. Defects of strength ±1/2 are either annihilated or expelled when the director tilts. On the other hand, disclination lines of ±1 which end in partial point defects (boojums) at the surfaces in the planar alignment regime acquire point defects of strength ±1 at the midplane of the cell when the director tilts. At a low enough temperature, the homeotropic anchoring becomes strong, and an electric field above the Freedericksz threshold generates the usual umbilic defects, which follow the dynamic scaling laws found in earlier studies.