Solution of the mixed Dirichlet–Neumann problem for molecular orientation in liquid crystals

The orientation of the nematic director field under the action of an external time‐dependent field is theoretically investigated as a mixed Dirichlet–Neumann boundary‐value problem. This mathematical problem represents the situation in which a nematic liquid crystal sample is limited by two inhomogeneous flat surfaces, separated by a distance d, on which the anchoring is weak. By considering the one‐constant approximation and a parabolic approximation for the surface energy, the initial conditions and boundary‐value problem for the profile of the tilt angle can be analytically solved even in the case in which the surfaces are not identical, which represents the more general situation. The results are valid for small deviations from the homeotropic orientation and for θ?Θ?1, where θ is the actual tilt angle and Θ characterizes the easy direction imposed by the surface, and can be relevant to investigation of the molecular orientation in a nematic cell submitted to a small external voltage.     

Simulation of reorientation dynamics in bipolar nematic droplets

A two-dimensional model composed of a synthesis of the Leslie-Ericksen continuumtheory of nematics and the Euler-Lagrange equation for surface director motion is used to study the magnetic-induced director reorientation dynamics confined in spherical bipolar droplets with viscoelastic surfaces. The magnetic field is restricted to the droplet axis of symmetry direction. The numerical results indicate that the surface viscosity and anchoring strength must be taken into account to describe accurately director reorientation dynamics in droplets. In addition, the numerical results replicate frequently reported experimental observations on the performance of polymer dispersed liquid crystal films. These observations include the familiar exponential increase followed by saturation in light transmittance as the external applied field increases, and the exponential increase (decrease) followed by saturation as time increases in the on (off) state. Furthermore, this model is able to predict precisely the relationships between the rise and decay times and the external applied field strength, and the fact that the switching field strength is inversely proportional to droplet size.     

Exact tilt angle profiles for splay–bend deformations in nematic liquid crystals

The exact tilt angle profiles for splay–bend deformations, in nematic liquid crystal samples limited by inhomogeneous surfaces, are determined in the one‐constant approximation. The boundary value problem concerning the situation of strong anchoring at the surfaces of a sample of slab shape of thickness d (Dirichlet's problem) is analytically solved in the presence of an external uniform field. The boundary value problem concerning the weak anchoring situation (mixed problem) is also exactly solved in the absence of an external field. The results are used to obtain the thickness dependence of the optical path difference between the ordinary and extraordinary rays, from which the physical properties of the sample can be deduced.     

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.     

Photo-reversible liquid crystal alignment using azobenzene-based self-assembled monolayers: comparison of the bare monolayer and liquid crystal reorientation dynamics

Photosensitive surfaces treated to have in-plane structural anisotropy by illumination with polarized light can be used to orient liquid crystals (LCs). Here we report a detailed study of the dynamic behavior of this process at both short and long times, comparing the ordering induced in the bare active surface with that of the LC in contact with the surface using a high-sensitivity polarimeter that enables detailed characterization of the anisotropy of the active surface. The experiments were carried out using self-assembled monolayers (SAMs) made from dimethylaminoazobenzene covalently bonded to a glass surface through a triethoxysilane terminus. This surface gives planar alignment of the liquid crystal director with an azimuthal orientation that can be controlled by the polarization of actinic light. We find a remarkable long-term collective interaction between the orientationally ordered SAM and the director field of the LC: while an azobenzene based SAM in contact with an isotropic gas or liquid relaxes to an azimuthally isotropic state in the absence of light due to thermal fluctuations, an orientationally written SAM in contact with LC in the absence of light can maintain the LC director twist permanently, that is, the SAM is capable of providing azimuthal anchoring to the LC even in the presence of a torque about the surface normal. We find that the short-time, transient LC reorientation is limited by the weak azimuthal anchoring strength of the SAM and by the LC viscosity.     

Oscillating Poiseuille flow in photo‐aligned liquid crystal cells

We present an experimental study of thin liquid crystal (LC) layers under the action of a harmonically varied pressure gradient. Optical measurements were performed to register the linear oscillations of a nematic director related to homeotropic and homeoplanar (hybrid) initial states. In the latter case one of the inner surfaces of the rectangular channels was treated by ultraviolet light to provide a relatively weak planar anchoring. The optical response of hybrid and homeotropic LC cells under an oscillating pressure gradient was investigated in relation to on the amplitude and frequency of the pressure gradient. A hydrodynamic model is developed taking into account the LC polar anchoring strength and the surface viscosity responsible for a fast LC surface dynamics. Our estimates show that the thickness of the boundary layer corresponding to the surface viscosity does not exceed 10^?6 m, and further experiments are needed with thinner LC cells and higher frequency oscillations to achieve a more precise value. An oscillating Poiseuille flow in the hybrid cell was found to be useful for characterizing elastic and viscous properties of a weakly anchoring LC surface layer in a fast surface dynamic process.     

Oscillating Poiseuille flow in photo-aligned liquid crystal cells

We present an experimental study of thin liquid crystal (LC) layers under the action of a harmonically varied pressure gradient. Optical measurements were performed to register the linear oscillations of a nematic director related to homeotropic and homeoplanar (hybrid) initial states. In the latter case one of the inner surfaces of the rectangular channels was treated by ultraviolet light to provide a relatively weak planar anchoring. The optical response of hybrid and homeotropic LC cells under an oscillating pressure gradient was investigated in relation to on the amplitude and frequency of the pressure gradient. A hydrodynamic model is developed taking into account the LC polar anchoring strength and the surface viscosity responsible for a fast LC surface dynamics. Our estimates show that the thickness of the boundary layer corresponding to the surface viscosity does not exceed 10^-6 m, and further experiments are needed with thinner LC cells and higher frequency oscillations to achieve a more precise value. An oscillating Poiseuille flow in the hybrid cell was found to be useful for characterizing elastic and viscous properties of a weakly anchoring LC surface layer in a fast surface dynamic process.     

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.     

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.     

Exact tilt angle profiles for splay-bend deformations in nematic liquid crystals

The exact tilt angle profiles for splay-bend deformations, in nematic liquid crystal samples limited by inhomogeneous surfaces, are determined in the one-constant approximation. The boundary value problem concerning the situation of strong anchoring at the surfaces of a sample of slab shape of thickness d (Dirichlet's problem) is analytically solved in the presence of an external uniform field. The boundary value problem concerning the weak anchoring situation (mixed problem) is also exactly solved in the absence of an external field. The results are used to obtain the thickness dependence of the optical path difference between the ordinary and extraordinary rays, from which the physical properties of the sample can be deduced.     

Anchoring of a nematic liquid crystal on an anisotropic substrate

A nematic liquid crystal in contact with a flat solid substrate is studied by means of a mesoscopic Landau-de Gennes theory. It is assumed that the substrate is anisotropic, i.e. the directions x and y in the surface of the substrate are not equivalent, and the only symmetry is the mirror symmetry y ? - y. Assuming the simplest form of the bare surface free energy, where only the linear terms in the nematic order parameter are taken into account, we study anchoring directions induced by the interaction of the liquid crystal with the substrate. A phase diagram in terms of the surface fields and the temperature is obtained. Depending on the values of the surface fields we find four types of anchoring: the symmetric planar anchoring, with the director along x, the symmetric tilted anchoring, with the director in the xz plane, the antisymmetric planar anchoring, with the director along y, and the asymmetric tilted anchoring, with the director tilted with respect to all three axes.     

Thresholdless switching induced by polar anchoring in antiferroelectric liquid crystals

We present a novel thresholdless switching mode in an antiferroelectric liquid crystal cell which is stabilized by the presence of polar anchoring at the cell surfaces and the antiferroelectric nature of the material. We also suggest other possible configurations which are induced by strong polar anchoring and possess quite different director structures and optical characteristics.     

Thresholdless switching induced by polar anchoring in antiferroelectric liquid crystals

We present a novel thresholdless switching mode in an antiferroelectric liquid crystal cell which is stabilized by the presence of polar anchoring at the cell surfaces and the antiferroelectric nature of the material. We also suggest other possible configurations which are induced by strong polar anchoring and possess quite different director structures and optical characteristics.     

Threshold flexoelectric deformations in homeotropic nematic layers

Elastic deformations of nematic liquid crystal layers subjected to a d.c. electric field were studied numerically. The flexoelectric properties of the nematic material and the presence of ionic space charge were taken into account. Homeotropic alignment with finite surface anchoring strength was assumed. The director orientation and the electric potential distribution were calculated; the space charge density was also determined. It was found that the threshold voltage strongly depended on the parameters of the system. In particular, a threshold as low as a few tenths of a volt occurred under suitable circumstances. In the case of a negative dielectric anisotropy, Δ ε, such low values of the threshold voltage existed when the ion concentration was sufficiently high, and given sufficiently large magnitudes of the flexoelectric coefficients and a sufficiently small anchoring energy. If the ion concentration was low or if the flexoelectric coefficients were small or if the surface anchoring was strong, the threshold was equal to several volts. In the case of positive dielectric anisotropy, the threshold amounted to several tenths of a volt for a weakly anisotropic and highly conductive material. If the dielectric anisotropy was sufficiently high or if the ion concentration was sufficiently low, the threshold voltage increased with Δ ε and reached tens of volts. These results can be explained as the effect of the inhomogeneous electric field arising in the vicinity of the surfaces, due to the ionic space charge redistributed by the external voltage. They are qualitatively consistent with earlier experiments which show the effect of the ion concentration on the elastic deformations in flexoelectric nematics. They correspond also with theoretical results concerning the effect of the electric field produced by the surface polarization or by the adsorption of ions.     

The elastic distortion and stability of biaxial nematic liquid crystal on the surface grooves

Fukuda et al. reexamined the Berreman's model which attributes the surface anchoring to the elastic distortion of the uniaxial nematic liquid crystal induced by the grooves of a surface. They showed that at the variance with the assumption made in the original approach of Berreman, the azimuthal distortion of the director cannot be considered as negligibly small. Now this method is generalized to the biaxial nematic liquid crystals, with some approximations for the elastic constants. We obtain an additional term in the elastic distortion energy per unit area which depends on the second power of the cosine of the angle made between the main director n at infinity and the direction of the surface grooves. This additional term describes the distortion energy of the minor director m induced by the surface grooves when the n director is anchored exactly along the grooves. We have studied the stability of the n director around the grooves, and in one-constant model for each director the stability condition is that the elastic constant of the n director is the maximum.     

Orientational transition in the cholesteric layer induced by electrically controlled ionic modification of the surface anchoring

A reorientation of cholesteric liquid crystal with a large helix pitch induced by the electrically controlled ionic modification of the surface anchoring has been studied. In initial state, the cholesteric helix is untwisted completely owing to the normal surface anchoring specified by the cations adsorbed at the substrates. As a result, the homeotropic director configuration is observed within the cell. Under the action of DC electric field, one of the substrates becomes free from the layer of surface active cations, therefore, setting the planar surface anchoring. The latter, in turn, leads to the formation of the hybrid chiral structure. The threshold value and dynamic parameters have been estimated for this process as well as the range of control voltages, which do not allow the electrohydrodynamic instabilities. The twisted hybrid director configuration observed in the experiment has been analysed by means of the simulation of polarisation change of light propagating through the cholesteric layer with asymmetric (planar and homeotropic) surface anchoring on the cell substrates.     

On the non-symmetric planar aligned NLC cell

The planar aligned nematic liquid crystal cell with different anchoring for the two substrates (i.e. a non-symmetric NLC cell) is investigated by an analytical method. We deduce the basic equations and the boundary conditions of the tilt angle θ of the LC director. Expressions for threshold and saturation magnetic field are obtained, and numerical results of these two quantities with variation in anchoring parameters of the two substrates are given. A symmetry breaking parameter Δ is introduced and the relations between Δ and applied field, as well as the two sets of anchoring parameters are discussed in detail. A feasible experimental plan for measurement of anchoring strengths of a series of different substrates is proposed.     

Determination of the surface anchoring potential of a nematic in contact with a substrate

We have developed a method for determining the surface anchoring potential for nematics in contact with a substrate that provides director alignment. Our main result is that the surface torque and hence the anchoring potential may be determined from either dielectric or optical phase response of a nematic undergoing a Freedericksz transition. The method is based on the Frank-Oseen continuum theory, and makes no assumptions about the functional form of the potential. We have measured the surface anchoring potential of two types of substrate in contact with the nematic liquid crystal 4-n-pentyl-4′-cyanobiphenyl. The surfaces were ITO-coated float glass, coated either with obliquely evaporated SiO or a buffed polymer film. Comparison of the results obtained from capacitance and optical measurements provides an estimate of the goodness of the method.     

Optical response of a nematic liquid crystal cell at the splay – bend transition: a model and dynamic simulation

We study the dynamical optical response of a nematic liquid crystal cell that undergoes the splay–bend transition after applying a voltage across the cell. We formulate a simplified model that takes into account both the flexoelectric coupling and the surface rotational viscosity. The dynamic equations of the model are solved numerically to calculate the temporal evolution of the director profile and of the transmittance. We evaluate the response time as a function of a number of parameters, such as dielectric and elastic anisotropies, asymmetry of the surface pretilt angles, anchoring energy, surface rotational viscosity and flexoelectricity.