Search

We describe an experimental situation with a looped line defect in nematic liquid crystals observed by polarizing optical microscopy. We measured the critical size of the loop below which it spontaneously shrinks and transforms into a point defect. The experiment was done with 5CB which gives rise to twist disclinations as do most of the usual nematics. For this kind of disclination an in-plane force due to the boundary conditions acts on the line and influences the critical radius. W e have constructed a model which is in good agreement with experimental measurements and deduced the line tension of the disclination.

~Received 19 March 1998! The response of a homeotropically aligned nematic liquid crystal layer to oscillatory rectilinear shear ~Couette flow ! was investigated for frequencies f between 0.01 and 200 Hz and layer thickness d between 10 and 130 mm. Below the onset of instability the cell was placed between crossed polars and light transmission was studied using a parallel light beam. The experimental results for the transmitted light intensity agree quantitatively with numerical solutions of the nematodynamic equations for different cell thicknesses, oscillation frequencies, and amplitudes. For frequencies between 25 and 150 Hz the critical oscillation amplitude for the onset of a spatial pattern, observed in polarized white light, could be reached. The pattern consisted of stationary rolls perpendicular to the direction of the oscillation. The experimentally obtained frequency dependence of the critical shear amplitude for the roll instability for different cell thicknesses is compared with an existing theory and the results of numerical calculations. @S1063-651X~98!09411-2# Nematic liquid crystals ~NLCs! exhibit interesting flow phenomena due to the coupling between the local molecular orientation ~director n ˆ! and the velocity field v. The flow properties of NLCs are characterized by Leslie viscosity coefficients a 1 ,...,a6 , two of which a 2(,0) and a 3 are important in describing the coupling between flow and director orientation @1‐4#. In the case of steady, plane, parallel shear flow, e.g., along the x axis with a velocity field v x(z )( v y 5 v z 50) and in the absence of other torques, the director will tend to align in the flow plane ( x-z plane! at a fixed angle u fl 56arctan(a3 /a2) 1/2 with the x axis if a 3,0 ~the 6

The morphologies of a homeotropic smectic-$B$ germ growing into an undercooled homeotropic or planar nematic melt were studied. The two sets of growth shapes observed in the experiment were reproduced by computer simulations using a phase-field model. From the comparison of the experiment and numerical simulations we give an estimate for the anisotropy of the surface tension and the kinetic coefficient for the case of the homeotropic nematic melt. In the case of the planarly aligned melt the twofold anisotropy of the nematic superimposes onto the hexagonal symmetry of the smectic $B.$ An explanation of the phenomenon of the ``inverted growth'' is given.

An anisotropic heat di⁄usion coeƒcient is introduced in order to study some interfacial growth phenomena. This anisotropy has been incorporated in a phase field model which has been studied numerically to reproduce some fundamental solidification situations (needle crystal growth) as well as the dynamics of a nematic—smectic-B interface. As a general result, we find that dendrites grow faster in the lower heat di⁄usion direction. Simulation results are compared with experiments with remarkable qualitative agreement. ( 1998 Elsevier Science B.V. All rights reserved.

We report on experiments in nematics with homeotropic alignment where the electrohydrodynamic instability occurs after a bend Freedericksz transition. The threshold of convection, the wave number at onset, and the threshold for the transition between order and disorder in the presence of a stabilizing magnetic field agree with theory. Whereas in the oblique-roll range the behavior of the correlation time of the disordered state exhibits near onset the expected scaling behavior, the disordered state in the normal-roll range at zero magnetic field shows no dynamics below a well-defined value epsilon s of the reduced control parameter, in contrast to theoretical predictions. At epsilon s one has reproducibly a continuous and reversible transition to a dynamic state. We attribute these features to the appearance of chevronlike structures.

Pattern forming instabilities induced by ultralow frequency sinusoidal voltages were studied in a rodlike nematic liquid crystal by microscopic observations and simultaneous electric current measurements. Two pattern morphologies, electroconvection (EC) and flexodomains (FD), were distinguished, both appearing as time separated flashes within each half period of driving. A correlation was found between the time instants of the EC flashes and those of the nonlinear current response. The voltage dependence of the pattern contrast $C(U)$ for EC has a different character than that for the FD. The flattening of $C(U)$ at reducing the frequency was described in terms of an imperfect bifurcation model. Analyzing the threshold characteristics of FD, the temperature dependence of the difference $|{e}_{1}\ensuremath{-}{e}_{3}|$ of the flexoelectric coefficients was also determined by considering elastic anisotropy.

This book intends to give a state-of-the-art overview of flexoelectricity, a linear physical coupling between mechanical (orientational) deformations and electric polarization, which is specific to systems with orientational order, such as liquid crystals. Chapters written by experts in the field shed light on theoretical as well as experimental aspects of research carried out since the discovery of flexoelectricity. Besides a common macroscopic (continuum) description the microscopic theory of flexoelectricity is also addressed. Electro-optic effects due to or modified by flexoelectricity as well as various (direct and indirect) measurement methods are discussed. Special emphasis is given to the role of flexoelectricity in pattern-forming instabilities. While the main focus of the book lies in flexoelectricity in nematic liquid crystals, peculiarities of other mesophases (bent-core systems, cholesterics, and smectics) are also reviewed. Flexoelectricity has relevance to biological (living) systems and can also offer possibilities for technical applications. The basics of these two interdisciplinary fields are also summarized.

We present numerical simulations directed at the description of smectic-B germs growing into the supercooled nematic phase for two different liquid crystalline substances. The simulations are done by means of a phase-field model appropriate to study strong anisotropy and also faceted interfaces. The most important ingredient is the angle-dependent surface energy, but kinetic effects are also relevant. The simulations reproduce qualitatively a rich variety of morphologies observed in the experiments for different value of undercooling, extending from the faceted equilibrium shape to fully developed dendrites.

Remarkable differences in the shape of the nematic--smectic-B interface in a quasi-two-dimensional geometry have been experimentally observed in three liquid crystals of very similar molecular structure, i.e., neighboring members of a homologous series. In the thermal equilibrium of the two mesophases a faceted rectanglelike shape was observed with considerably different shape anisotropies for the three homologs. Various morphologies such as dendritic, dendriticlike, and faceted shapes of the rapidly growing smectic-B germ were also observed for the three substances. Experimental results were compared with computer simulations based on the phase field model. The pattern forming behavior of a binary mixture of two homologs was also studied. \textcopyright{} 1996 The American Physical Society.

The equilibrium morphology of the nematic (N) - smectic B (SmB) interface, and its growth properties have been studied in quasi-twodimensional geometry. After thermal equilibration at the N - SmB phase transition temperature TNS two different equilibrium shapes of the SmB phase were found, depending on its orientation. The surface tension anisotropy has been determined from the shape anisotropy of the smectic body. An interesting “inverse process” has been detected in melting of the rapidly grown smectic phase. The stable nematic phase nucleated substancially below TNS . Most of these nematic germs had uniform oval shape, totally different from that of the equilibrium smectic seed. The morphology of the growing smectic B phase in a supercooled nematic also has been studied. Different morphologies were observed, depending on the substance, undercooling ΔT = TNS - T, and alignment of the growing SmB germ. In order to compare the growth dynamics, the area of the patterns was measured and the square ...

The temporal evolution of patterns within the driving period of the ac voltage was studied in the 10-mHz-250-Hz frequency range. It was shown that the stationary electroconvection pattern of the conductive regime transforms into a flashing one at ultralow frequencies, existing only in narrow time windows within the period. Furthermore a transition between electroconvection and flexoelectric domains was detected which is repeating in each half period. The two patterns are well separated in time and in Fourier space. Simultaneous current measurements uncovered that the electric properties of the polyimide orienting layers influence the redistribution of the applied voltage. The experimental findings are in good qualitative agreement with the theoretical predictions based on an extended standard model including flexoelectricity.

A nematic liquid crystal with high, positive dielectric anisotropy (5CB) has been studied under the influence of the combined action of a dc and an ac electric field. Broad frequency, voltage, and cell thickness ranges were considered. Pattern morphologies were identified; the thresholds and critical wave numbers were measured and analyzed as a function of frequency, dc-to-ac voltage ratio, and thickness. The current-voltage characteristics were simultaneously detected.

Two optical methods were applied to obtain information about the director configuration in electrically induced convective instabilities. The intensity of the transmitted light and the contrast of the patterns were measured. Normal, oblique and ‘abnormal’ rolls were studied and compared for initially planar and homeotropic samples.

Equilibrium shapes of the nematic--smectic-B liquid-crystal interface have been investigated. Three types of smectic-B germs with different orientation in a quasi-two-dimensional geometry have been found at small undercooling. Two of these have a similar, rectanglelike shape with two long faceted sides; the third is circular with a small hexagonal modulation. By heating the rapidly grown smectic phase the nematic phase is nucleated below the phase transition. Nematic monodomains were detected for different alignments having a nonfaceted, oval form. From the shape of the germs the angle dependence of the surface tension was determined.

It is demonstrated that the key findings of the paper by Y.G. Marinov and H.P. Hinov, Eur. Phys. J. E 31, 179-189 (2010), are in direct conflict with the general physical background of flexoelectric domains. This is caused by a methodological error in the theoretical analysis of the paper.

The temporal evolution of the spatially periodic electroconvection (EC) patterns has been studied within the period of the driving ac voltage by monitoring the light intensity diffracted from the pattern. Measurements have been carried out on a variety of nematic systems, including those with negative dielectric and positive conductivity anisotropy, exhibiting "standard EC" (s-EC), those with both anisotropies negative exhibiting "non-standard EC" (ns-EC), as well as those with the two anisotropies positive. Theoretical predictions have been confirmed for stationary s-EC and ns-EC patterns. Transitions with Hopf bifurcation have also been studied. While traveling had no effect on the temporal evolution of dielectric s-EC, traveling conductive s-EC and ns-EC patterns exhibited a substantially altered temporal behavior with a dependence on the Hopf frequency. It has also been shown that in nematics with both anisotropies positive, the pattern develops and decays within an interval much shorter than the period, even at relatively large driving frequencies., Comment: 19 pages, 5 figures

Using a simple averaging method we show that under plane oscillatory shear a net torque acts on the director of a nematic liquid crystal leading to unexpected flow alignment. Under quite general conditions the usual flow-alignment orientations are unstable, and the ori- entation parallel to the shear (perpendicular to the velocity) becomes most stable within the shear plane. The effect drops out exactly under (unrealistic) linear shear unless the director has a nontrivial space dependence. An explicit application to plane Poiseuifle flow is presented. Pre- liminary experimental data are shown which demonstrate a transition under planar oscillatory shear from a homeotropically aligned nematic to a spatially homogeneously deformed state.

Two electroconvection (EC) pattern morphologies---a cellular and a subsequent roll pattern---have been detected in the same frequency range in a nematic with positive permittivity and conductivity anisotropies. The frequency dependences of the onset voltages and critical wave numbers have been determined both for homeotropic and planar alignments. It has been proven that both pattern morphologies have a dielectric time symmetry. We also discuss possible sources for the pattern formation in the frame of both the isotropic (Felici-Benard) mechanism, as well as the standard model of EC.

The effect of the electric-field inhomogeneity has been experimentally studied by polarizing microscopy in a homeotropic nematic liquid crystal in confined electrode geometries which may be relevant in display applications. Defects related to tilt inversion have been detected by monitoring the transmitted intensity profile as a function of the applied voltage. The position of the defects could be controlled by an additional magnetic field breaking the symmetry of the original arrangement. The phenomenon has been interpreted via numerical calculation of the director distribution using the continuum theory of nematics. The influence of oblique light incidence and of weak anchoring has also been analyzed. Simulations have provided good qualitative agreement with the observations. The method has turned out to be a sensitive tool to detect small misalignment angles between the magnetic field and the cell plane.

A nonlinear theory of transient patterns in the electrically induced splay Fr\'eedericksz transition in nematic liquid crystals is presented. Growth conditions for periodic perturbations with a wave vector perpendicular to the initial director alignment, superposed on the homogeneous realignment process, have ben investigated. The decisive effect leading to this type of pattern formation arises from the field distortion resulting from the electric anisotropies. The nonlinear analysis with rigid boundary conditions delivers the onset of the pattern-forming instability at a critical value of the control parameter ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{c}}$, which appears to be too large.

Dielectric spectroscopy measurements have been performed on a bent-core nematic liquid crystal and on its binary mixtures with a calamitic nematic. We have detected more dispersions in the bent-core compound than in the calamitic one, including one at an unusually low frequency of a few kilohertz. The dispersions detected in the mixtures have been identified and the spectra have been split into contributions of the constituents. In order to connect the dielectric increment with the molecular dipole moment we have applied a sophisticated conformational calculation not performed before for a large, flexible mesogen molecule with numerous polar groups.

Different morphological phases have been studied in several pattern forming processes in liquid crystals. Shape descriptors of the two-dimensional patterns were introduced and computational methods applied to identify and classify the different morphological phases.

Novel behavior in electroconvection (EC) has been detected in nematic liquid crystals (NLCs) under the condition of comparable timescales of the director relaxation and the period of the driving ac voltage. The studied NLCs exhibit standard EC (s-EC) at the onset of the instability, except one compound in which nonstandard EC (ns-EC) has been detected. In the relevant frequency region, the threshold voltage for conductive s-EC bends down considerably, while for dielectric s-EC it bends up strongly with the decrease of the driving frequency. We show that inclusion of the flexoelectric effect into the theoretical description of conductive s-EC leads to quantitative agreement, while for dielectric s-EC a qualitative agreement is achieved. The frequency dependence of the threshold voltage for ns-EC strongly resembles that of the dielectric s-EC., Comment: 13 pages, 14 figures, submitted

For many years it has been commonly accepted that electroconvection (EC) as primary instability in nematic liquid crystals for the "classical" planar geometry requires a positive anisotropy of the electric conductivity, sigma(a), and a slightly negative dielectric anisotropy, epsilon(a). This firm belief was supported by many experimental and theoretical studies. Recent experiments, which have surprisingly revealed EC patterns at negative conduction anisotropy as well, have motivated the theoretical studies in this paper. It will be demonstrated that extending the common hydrodynamic description of nematics by the usually neglected flexoelectric effect allows for a simple explanation of EC in the "nonstandard" case sigma(a)

Electric-field-driven pattern formation has been investigated in a nematic liquid crystal with negative dielectric and conductivity anisotropies. Despite the fact that the standard Carr-Helfrich theory predicts no hydrodynamic instability for such compound, experiments reveal convection patterns which we call nonstandard electroconvection (ns-EC). In this work, we characterize the ns-EC patterns by measuring the frequency, thickness, and temperature dependence of the threshold voltage, wave number, roll orientation, etc., and compare them with the standard-EC (s-EC) characteristics. For the first time, we report traveling rolls in ns-EC, and we give the dependence of the Hopf frequency on the driving frequency, temperature, and sample thickness. Finally, we discuss possible sources for the existence of these patterns.

We present experimental measurements near the onset of electroconvection (EC) of homeotropically aligned nematic liquid crystals Phase 5A and MBBA. A voltage of amplitude square root 2V0 and frequency f was applied. With increasing V0, EC occurred after the bend Freedericksz transition. We found supercritical bifurcations to EC that were either stationary bifurcations or Hopf bifurcations to traveling convection rolls, depending on the sample conductances. Results for the onset voltages Vc, the critical wave numbers kc, the obliqueness angles thetac, and the traveling-wave (Hopf) frequencies at onset omegac over a range of sample conductances and driving frequencies are presented and compared, to the extent possible, with theoretical predictions. For the most part good agreement was found. However, the experiment revealed some unusual results for the orientations of the convection rolls relative to the direction selected by the Freedericksz domain.

The decay of stripe patterns in planarly aligned nematic liquid crystals has been studied experimentally and theoretically. The initial patterns have been generated by the electrohydrodynamic instability and a light diffraction technique has been used to monitor their decay. In our experiments different decay rates have been observed as a function of the pattern wave number. According to our theoretical analysis they belong to a spectrum of decay modes and are individually selected in dependence on the initial conditions. Additional insight has emerged from a refined physical optical description of the diffraction intensity. The results compare well with experiments, which include also controlled modifications of the initial conditions to assess different decay modes.

A detailed theoretical and experimental analysis of the decay of electroconvection patterns is presented in a planarly aligned nematic liquid crystal. The relaxation time is measured as a function of the wave number of the pattern using a light diffraction technique. A theoretical analysis exhibits a rich structure of the dispersion curves for the decay rates. An interesting relation between the realistic case of no-slip boundary conditions and the simpler free-slip case is found. The experimentally determined relaxation rates for both ``conductive'' and ``dielectric'' initial patterns follow the theoretical solution with subsequent jumps between branches when the wave number is increased.

We present an experimental and theoretical investigation of a variant of electroconvection using an unusual nematic liquid crystal in an isotropic configuration (homeotropic alignment). The significance of the system is a direct transition to the convecting state due to the negative conductivity anisotropy and positive dielectric anisotropy. We observe at onset rolls or squares depending on the frequency and amplitude of the applied ac voltage with a strong signature of the zigzag instability. Good agreement with calculations based on the underlying hydrodynamic theory is found. We also construct an extended Swift–Hohenberg model which allows us to capture complex patterns like squares with a quasiperiodic modulation.

Experimental electroconvection (EC) investigations on four homologs with negative dielectric anisotropy ${ϵ}_{a}$ and conductivity anisotropies ${\ensuremath{\sigma}}_{a}$ of various signs (the relative ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}}_{a}={\ensuremath{\sigma}}_{a}∕{\ensuremath{\sigma}}_{\ensuremath{\perp}}$ ranges from $\ensuremath{-}0.6$ to $+0.3$) are reported. The homolog with positive conductivity anisotropy follows the predictions of the standard theoretical model, while the homolog with negative ${\ensuremath{\sigma}}_{a}$ shows a phenomenon, i.e., nonstandard electroconvection, not described by the Carr-Helfrich mechanism but manifesting in forms of stripes mainly parallel to the initial director, thus having a morphological similarity to the rolls of the standard EC. The remaining two homologs change the sign of their conductivity anisotropy with temperature that leads to a twofold character by crossing over from standard to nonstandard electroconvection. The patterns of both the standard and nonstandard EC have been characterized and compared.

We study a new variant of electroconvection using a homeotropically aligned nematic liquid crystal. The novelty of this system is a direct transition to roll- or square-type patterns controlled by the frequency of the applied voltage with a rich crossover scenario and strong influence of the zigzag instability even at onset. From the weakly nonlinear theory and simulations of an adapted Swift-Hohenberg model one can understand essential features of the phase diagram. In particular we find a quasiperiodic pattern with square symmetry.

Electroconvection in an unusual nematic compound with strongly positive dielectric anisotropy and negative anisotropy of the conductivity is investigated. For homeotropic alignment, where one has a direct transition to rolls or squares depending on the frequency of the applied voltage, we present a quantitative theory. From the comparison we infer values for some viscosities, which are rather unusual, but not unreasonable in view of the vicinity of the nematic-smectic transition. For planar alignment, electroconvection sets in above a splay Freedericksz transition with ``parallel rolls,'' which is also captured by the theory.

The influence of an external electric field (applied to the nematic liquid crystal layer) on the morphology of the nematic-liquid-crystal-air interface has been studied experimentally in radial Hele-Shaw geometry. The effective viscosity mu(eff) of the nematic has been tuned by the electric field E and by the flow. At low excess pressure p(e) (where the growth of the interface is controlled mainly by the surface tension sigma), the applied E has no significant influence on the morphology of the interface, but decreases its normal velocity due to the increase of mu(eff). At higher p(e) (where the growth is not only controlled by sigma, but also by the kinetic term that depends on the effective viscosity) a significant difference in the morphology has been observed as a function of E. Experiments have shown that the influence of the electric field on the pattern morphology increases with the driving force (pressure gradient).

We present a detailed analysis of the characteristics of electroconvection patterns in a homeotropic nematic liquid crystal under the influence of a variable magnetic field. An unambiguous observation of low frequency "reentrant" normal rolls and a nonmonotonic magnetic field dependence of the threshold voltages is reported. The effect of the magnetic field on the normal roll-abnormal roll transition is determined, which is in good agreement with theoretical predictions of the weakly nonlinear analysis.

Liquid crystals (LCs) are substances possessing one or more mesophases between their liquid and solid phase. The sequence of these mesophases represents a step by step ordering of the structure. The nematic (N) phase is characterized by an orientational order of the elongated molecules described by the director n, but the centers of mass of the molecules are arranged randomly. In the smectic (Sm) phases besides the orientational order the centers of mass of molecules form a layered structure. The smectic-A (SmA) phase has no positional order within the layers, while the smectic-B (SmB) phase is characterized by a long-range hexagonal order in each layer and by a weak correlation between the layers. The features of these and other LC phases are described in detail in the literature— see e.g. [1],[2],[3].

Free growth properties of the smectic B liquid crystalline phase into the supercooled nematic have been investigated in quasi-two-dimensional geometry. Different orientation combinations of the two phases have been achieved experimentally and the interfacial patterns have been studied and analysed as a function of undercooling. The angular dependence of the surface tension has been deduced from the shape of the interface in thermal equilibrium. The experimentally determined surface tension anisotropy has been incorporated into computer simulations based on the phase-field model. The simulations have reproduced qualitatively the rich variety of morphologies (extending from the faceted shape to fully developed dendrites) observed in the experiments for a given set of undercoolings in three geometries. Anisotropic heat diffusion on the nematic side, relevant to our experimental system has also been introduced. Both in the experiments and in the simulations we find that the growth is faster in the lower heat diffusion direction.

Elliptic shear applied to a homeotropically oriented nematic above the electric bend Freedericks transition (FT) generates slow precession of the director. The character of the accompanying nonlinear waves changes from diffusive phase waves to dispersive ones exhibiting spirals and spatio-temporal chaos as the FT is approached from above. An exact solution of the flow alignment equations captures the observed precession and predicts its reversal for non-flow aligning materials. The FT transforms into a Hopf bifurcation opening the way to understand the wave phenomena., Comment: 4 pages, 4 figures, submitted to PRL

We present the first quantitative verification of an amplitude description for systems with (nearly) spontaneously broken isotropy, in particular for the recently discovered abnormal-roll states. We also obtain a conclusive picture of the 3d director configuration in a spatial period doubling phenomenon involving disclination loops (CRAZY rolls). The first observation of two Lifshitz frequencies in electroconvection is reported., Comment: 4 pages; 4 figures

A homeotropic nematic liquid crystal was exposed to low frequency oscillatory shear (Couette flow). In the presence ofrectilinear shear a stationary roll pattern was found above a critical amplitude (Ar). In case of elliptical excitation a spatiallynonuniform slow rotation of the director was observed, which lead to the propagation of the so called phase waves in two different systems: in a two-phase (nematic-isotropic) system and in a homeotropic nematic layer in the presence of a destabilizing external electric field. Phase waves also have been observed in the ultrasonic frequency range at much smallershear amplitudes.Keywords: Nematic liquid crystal. shear flow. pattern formation. 1. INTRODUCTION The formation of spatial patterns in nematic liquid crystals (NLCs) due to oscillatory rectilinear or elliptical shear was wellstudied in the last 3 0 years (for an overview see Ref. 1). Applying rectilinear shear different types of stationary patterns werefound for homeotropic initial alignment. In the experimental results obtained in Ref. 2 a transition to stationary rolls wasobtained with roll axis perpendicular or parallel to the oscillatory shear flow depending on the applied frequency in therange of 1 <

We investigated travelling waves appearing as the primary pattern-forming instability in the nematic Phase 5 (Merck) in the planar geometry in order to test the recently developed weak electrolyte model of ac-driven electroconvection [M. Treiber and L. Kramer, Mol. Cryst. Liq. Cryst 261, 311 (1995)]. Travelling waves are observed over the full conductive range of applied frequencies for three cells of different layer thickness d. We also measured the elastic constants, the electric conductivity, and the dielectric constant. The other parameters of Phase 5 are known, apart from the (relatively unimportant) viscosity $\alpha_1$ and the two parameters of the weak electrolyte model that are proportional to the geometric mean of the mobilities, and the recombination rate, respectively. Assuming a sufficiently small recombination rate, the predicted dependence of the frequency of the travelling waves at onset (Hopf frequency) on d and on the applied frequency agreed quantitatively with the experiments, essentially without fit parameters. The absolute value of the Hopf frequency implies that the geometric mean of the mobilities amounts to $1.1 \times 10^{-10} m^2/(Vs)$., Comment: ReVTeX, 24 pages, 4 figures, to appear in Journal de Physique II

After a brief introduction into the subject a simple one-dimensional model is presented to illustrate typical features of pattern forming systems and to demonstrate the linear stability analysis. The special features of liquid crystals are outlined and two particular topics are discussed: transient patterns in the Freedericksz transition and patterns obtained in rotating magnetic and electric fields.

Different features of electroconvection patterns at threshold in a homeotropically aligned nematic sample are studied as a function of the driving frequency and the conductivity of the sample. The primary instability is the bend Fr\'eedericksz transition that spontaneously breaks the rotational symmetry. The director orientation is optically determined in the Fr\'eedericksz distorted state. Stationary normal and oblique convection rolls appear as a secondary instability. A complex time dependence of the pattern at low frequencies involving a reorientation of the rolls together with the underlying director configuration is observed as predicted by theory. For low conductivity samples a transition to traveling waves is detected and the Hopf frequency is measured as a function of the driving frequency for samples with different conductivities.

We reexamine the influence of the flexoelectricity on the electroconvection (EC), the effect which has previously been studied in the conductive EC regime only. Now we extend our studies to the dielectric EC, and to the parameter range for which the standard model of EC excludes the existence of the instability. First, we demonstrate that inclusion of flexoelectricity into the standard model of EC provides a finite EC threshold even for this “forbidden” material parameter range (in accordance with the experiments). Second, we show that flexoelectricity considerably decreases the threshold voltage for the dielectric regime. Finally, we present a novel frequency dependence of the threshold voltage for all EC patterns in the case when the period of the driving frequency becomes comparable with the director relaxation time.