Abnormal rolls and regular arrays of disclinations in homeotropic electroconvection

Quantitative definition of patterns in soft-mode turbulence suppressing the Nambu-Goldstone mode

Chaotic patterns in electroconvection of homeotropic nematics, soft-mode turbulence (SMT), and the related spatiotemporal chaos (STC) are discussed, and the quantitative definition of the patterns is considered. The order parameter S, obtained directly from the 2D spectra of the patterns, is introduced. The contribution of the Nambu-Goldstone mode and the increase in pattern regularity under the influence of an external magnetic field H are evaluated. We propose a schematic phase diagram of STC patterns based on the value of S.

Nanoparticles in liquid crystals and liquid crystalline nanoparticles

Combinations of liquid crystals and materials with unique features as well as properties at the nanoscale are reviewed. Particular attention is paid to recent developments, i.e., since 2007, in areas ranging from liquid crystal-nanoparticle dispersions to nanomaterials forming liquid crystalline phases after surface modification with mesogenic or promesogenic moieties. Experimental and synthetic approaches are summarized, design strategies compared, and potential as well as existing applications discussed. Finally, a critical outlook into the future of this fascinating field of liquid crystal research is provided.

Onset of electroconvection of homeotropically aligned nematic liquid crystals

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.

Spatiotemporal chaos in electroconvection of a homeotropically aligned nematic liquid crystal

We present patterns of electroconvection (EC) for the homeotropically aligned nematic liquid crystal MBBA. A voltage V = square root of 2V0 sin(2pift) was applied. With increasing V0, the bend Freedericksz transition at VF was followed by the onset of EC at Vc > VF. We found four distinct pattern types. First, a primary supercritical Hopf bifurcation to traveling waves (TW's) of convection rolls occurred. The structure factor S(k) of this state reflected the azimuthal anisotropy of the underlying Freedericksz state. For f < fL approximately 75 Hz there was a superposition of two oblique-roll modes (pattern I). These patterns were chaotic in space and time. For larger f the patterns consisted of chaotic TW normal rolls (pattern II). Here the chaos was attributable to the motion of dislocations and domain walls between left- and right-traveling waves. A secondary bifurcation yielded pattern III; it had no dominant TW frequency but had broadband chaotic dynamics dominated by the motion of dislocations. This pattern type had been referred to by others as a "chevron pattern;" its structure factor still revealed azimuthal anisotropy. Finally, at somewhat larger identical with epsilon = V2/Vc2 -1 a highly disordered pattern IV with defect dynamics was found. This state had been studied before by Kai and co-workers and was referred to by them as "phase turbulence." It had a structure factor that was (within our resolution) invariant under rotation. For patterns I, II, and III, S(k) contained crescent-shaped peaks. The peak shape was qualitatively different from the case of planar EC where the structure factor has an elliptical cross section. We present measurements of the widths 1/xik and 1/xitheta in the radial (k) and the azimuthal (theta) directions. For small epsilon (patterns I and II) we found that xik was consistent with the usual Ginzburg-Landau scaling xik approximately epsilon(-nuk) with nuk approximately 1/2. However, for xitheta we found xitheta approximately epsilon(-nutheta) with nutheta approximately 3/4. Presumably this anomalous scaling of xitheta is associated with the Goldstone mode of homeotropic EC. We also show data for the height S0 of the structure factor that are consistent with S0 approximately epsilonbeta with beta approximately -0.5, implying that S0 diverges at onset. This differs from the case of domain chaos in rotating Rayleigh-Bénard convection where experiment is consistent with beta = 1/2 and thus with a vanishing S0. The difference between the shape of the structure-factor cross section and between the exponents, for the present case, for planar EC, and for domain chaos suggests that there are different universality classes for spatiotemporal chaos.

Wavelength halving in a transition between standing waves and traveling waves

In the Belousov-Zhabotinsky reaction-diffusion system dispersed in a newly developed water-in-oil aerosol OT/Span-20 microemulsion, the transition between standing waves and traveling waves is accompanied by a halving of the wavelength.

Formation of a defect lattice in electroconvection of nematics

The formation of a defect lattice--i.e., a periodic orientational structure of numerous defect pairs--is experimentally investigated in the electroconvection of nematics. Specific twist structures of the director as a background field play an important role in the formation of a defect lattice. The observed formation sequence is as follows. With increasing applied voltage, normal rolls change into abnormal rolls due to the twisting deformation of the director. This process leads to belt-shaped domains along the abnormal rolls, in which the twist angle of the director alternates between positive and negative angles. The period of the defect lattice perpendicular to the rolls is equal to that of the domain structure in the abnormal rolls. Further increasing the applied voltage induces a secondary short-wavelength mode by the skewed varicose instability, which in turn induces defects. The periodicity of the defect lattice parallel to the rolls is due to the beating mode of the normal rolls and the secondary mode.

Modulated structures in electroconvection in nematic liquid crystals

Motivated by experiments in electroconvection in nematic liquid crystals with homeotropic alignment we study the coupled amplitude equations describing the formation of a stationary roll pattern in the presence of a weakly damped mode that breaks isotropy. The equations can be generalized to describe the planarly aligned case if the orienting effect of the boundaries is small, which can be achieved by a destabilizing magnetic field. The slow mode represents the in-plane director at the center of the cell. The simplest uniform states are normal rolls, which may undergo a pitchfork bifurcation to abnormal rolls with a misaligned in-plane director. We present a new class of defect-free solutions with spatial modulations perpendicular to the rolls. In a parameter range where the zigzag instability is not relevant these solutions are stable attractors, as observed in experiments. We also present two-dimensionally modulated states with and without defects which result from the destabilization of the one-dimensionally modulated structures. Finally, for no (or very small) damping, and away from the rotationally symmetric case, we find static chevrons made up of a periodic arrangement of defect chains (or bands of defects) separating homogeneous regions of oblique rolls with very small amplitude. These states may provide a model for a class of poorly understood stationary structures observed in various highly conducting materials ("prechevrons" or "broad domains").

Electroconvection in nematic liquid crystals with positive dielectric and negative conductivity anisotropy

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.

Magnetic field effect on the thresholds of a sequence of transitions in the electroconvection of a homeotropic nematic liquid crystal

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.

Three-dimensional pattern formation, multiple homogeneous soft modes, and nonlinear dielectric electroconvection

Patterns forming spontaneously in extended, three-dimensional, dissipative systems are likely to excite several homogeneous soft modes ( approximately hydrodynamic modes) of the underlying physical system, much more than quasi-one- (1D) and two-dimensional (2D) patterns are. The reason is the lack of damping boundaries. This paper compares two analytic techniques to derive the pattern dynamics from hydrodynamics, which are usually equivalent but lead to different results when applied to multiple homogeneous soft modes. Dielectric electroconvection in nematic liquid crystals is introduced as a model for 3D pattern formation. The 3D pattern dynamics including soft modes are derived. For slabs of large but finite thickness the description is reduced further to a 2D one. It is argued that the range of validity of 2D descriptions is limited to a very small region above threshold. The transition from 2D to 3D pattern dynamics is discussed. Experimentally testable predictions for the stable range of ideal patterns and the electric Nusselt numbers are made. For most results analytic approximations in terms of material parameters are given.

Zigzag structures and domain walls in electroconvection of nematic liquid crystal

To describe the secondary-bifurcation scenario in ac-driven electroconvection of a planarly aligned nematic liquid crystal layer, we have constructed a generic phase equation coupled to the cinsertion mark director. The equations are applicable in particular in the vicinity of the codimension-2 point, where the zigzag and the abnormal roll instabilities meet. This point is also the origin of a line of homoclinic bifurcations, which separates a region where one has stationary zigzag walls from one with spontaneously accelerated abnormal roll walls. The final velocity of the walls depends linearly on the distance from the bifurcation. We analyze the scenario analytically, test it numerically and propose an experimental check.