Recent results on flexodomain

We investigate flexodomains, which are observed in planar layers of certain nematic liquid crystals, when a dc voltage U above a critical value Uc is applied across the layer. They are characterized by stationary stripelike spatial variations of the director in the layer plane with a wave number p(U). Our experiments for different nematics demonstrate that p(U) varies almost linearly with U for U >Uc. That is confirmed by a numerical analysis of the full nonlinear equations for the director field and the induced electric potential. Beyond this numerical study, we demonstrate that the linearity of p(U) follows even analytically, when considering a special parameter set first used by Terent’ev and Pikin [Sov. Phys. JETP 56, 587 (1982)]. Their theoretical paper serves until now as the standard reference on the nonlinear analysis of flexodomains, since it has arrived at a linear variation of p(U) for large U [1] Uc. Unfortunately, the corresponding analysis suffers from mistakes, which in a combination led to that result.

W. Pesch, A. Krekhov, N. Éber and Á. Buka, Nonlinear analysis of flexodomains in nematic liquid crystals, Phys. Rev. E 98, 032702 (2018) [pdf]

The possibilities to induce regular, stationary stripe patterns with easily controllable wavenumber have been investigated with the aim to apply them in optical devices as gratings. Several, electric field induced phenomena have been studied on three members of a homologous series of a bent core nematic. Pattern morphologies, threshold voltages and wave numbers have been determined. Temporal evolution and switching dynamics have been analyzed.

N. Éber, Y. Xiang, and Á. Buka: Bent core nematics as optical gratings, J. Mol. Liq. http://dx.doi.org/10.1016/j.molliq.2017.09.025

Optical gratings have been created by flexoelectric domains in a bent-core nematic liquid crystal. A unique feature of this structure is that its wavelength can be controlled by the amplitude of the applied voltage, as demonstrated by polarizing microscopy and light diffraction techniques. In order to understand the reaction of the system to the voltage change, the dynamics of the switching process has been studied via digital processing of recorded image sequences. It has been shown that the characteristics and the switching mechanisms are different, if the lower voltage level is below or above the threshold of pattern onset. In both cases, the response to increasing voltage levels is much slower than the response to decreasing voltage levels.

Ying Xiang, Hong-Zhen Jing, Zhi-Dong Zhang, Wen-Jiang Ye, Ming-Ya Xu, Everett Wang, Péter Salamon, Nándor Éber, and Ágnes Buka: Tunable optical grating based on the flexoelectric effect in a bent-core nematic liquid crystal, Phys. Rev. Appl. 7, 064032 (2017). [pdf]

Electric field-induced patterns in liquid crystals have been observed and studied for about 50 years. During this time, a great variety of structures, detected under different conditions, have been described; theoretical descriptions were also developed parallel with the experiments and a huge number of papers have been published. The non-vanishing interest in the topic is due to several factors. First, most experimentalists working with new (or even well-known) liquid crystals apply sooner or later an electric field for different purposes and, as a response, often (maybe undesirably or unexpectedly) have to face with emergence of patterns. Second, understanding the complexity of the formation mechanism of regular patterns in a viscous, anisotropic fluid is an extremely challenging theoretical task. Third, specialists in display fabrication or in other applications are also interested in the results; either to make use of them or in order to avoid field-induced patterns.

In this review, we attempt to provide a systematic overview of the large amount of published results, focusing on recent achievements, about the three main types of electric field-induced patterns: transient patterns during the Freedericksz transition, flexoelectric domains and electroconvection. As a result of different instability mechanisms, a variety of pattern morphologies may arise. We address the physical background of the mechanisms, specify the conditions under which they may become effective, discuss the characteristics of the patterns, and summarize the possibilities of morphological transitions induced by frequency, voltage or temperature variations. Special emphasis is given to certain topics, which recently have gained enhanced interest from experimental as well as theoretical point of view, like driving with ultra-low frequencies or non-sinusoidal (superposed) waveforms, and the dynamics of defects and embedded colloidal particles. Assisting newcomers to the field, we also mention some, yet unresolved, problems, which may need further experimental and/or theoretical studies.

N. Éber, P. Salamon and Á. Buka: Electrically induced patterns in nematics and how to avoid them, Liquid Crystals Reviews 4 (2), 101-134 (2016). [pdf]

The effect of superposed dc and ac applied voltages on two types of spatially periodic instabilities in nematic liquid crystals, flexoelectric domains (FD), and electroconvection (EC) was studied. The onset characteristics, threshold voltages, and critical wave vectors were determined. We found that in general the superposition of driving with different time symmetries inhibits the pattern forming mechanisms for FD and EC as well. As a consequence, the onset extends to much higher voltages than the individual dc or ac thresholds. A dc-bias-induced reduction of the crossover frequency from the conductive to the dielectric EC regimes and a peculiar transition between two types of flexodomains with different wavelengths were detected. Direct measurements of the change of the electrical conductivity and its anisotropy, induced by the applied dc voltage component, showed that the dc bias substantially affects both parameters. Taking into account the experimentally detected variations of the conductivity in the linear stability analysis of the underlying nematohydrodynamic equations, a qualitative agreement with the experimental findings on the onset behavior of spatially periodic instabilities was obtained.

N. Éber, P. Salamon, B. A. Fekete, R. Karapinar, A. Krekhov, and Á. Buka: Suppression of spatially periodic patterns by dc voltage, Phys. Rev. E 93, 042701 (2016). [pdf]

A regular domain structure consisting of parallel stripes – flexodomains – have been induced by low frequency (subHz) electric voltage in a bent core nematic liquid crystal. The wavelength of the pattern is in the range of 1–10 micrometers and thus can conveniently be observed in a polarizing microscope. It also serves as an optical grating and produces a regular system of laser diffraction spots. The pattern was found to emerge and disappear consecutively in each half period of the driving, with the wavelength of the flexodomains changing periodically as the ac voltage oscillates. Analyzing the polarization characteristics of the diffracted light, the polarization of the first order spot was found perpendicular to that of the incident light, in accordance with a recent theoretical calculation.

Ming-Ya Xu, Meng-jie Zhou, Ying Xiang, Péter Salamon, Nándor Éber, and Ágnes Buka: Domain structures as optical gratings controlled by electric field in a bent-core nematic. Optics Express 23(12), 15224 (2015) [pdf]

The effect of superimposed ac and dc electric fields on the formation of electroconvection and flexoelectric patterns in nematic liquid crystals was studied. For selected ac frequencies, an extended standard model of the electrohydrodynamic instabilities was used to characterize the onset of pattern formation in the two-dimensional parameter space of the magnitudes of the ac and dc electric field components. Numerical as well as approximate analytical calculations demonstrate that depending on the type of patterns and on the ac frequency, the combined action of ac and dc fields may either enhance or suppress the formation of patterns. The theoretical predictions are qualitatively confirmed by experiments in most cases. Some discrepancies, however, seem to indicate the need to extend the theoretical description.

Alexei Krekhov, Werner Decker, Werner Pesch, Nándor Éber, Péter Salamon, Balázs Fekete, and Ágnes Buka: Patterns driven by combined ac and dc electric fields in nematic liquid crystals, Phys. Rev. E 89, 052507/1-9 (2014) [pdf]

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 |e1 − e3| of the flexoelectric coefficients was also determined by considering elastic anisotropy.

P. Salamon, N. Éber, A. Krekhov and Á. Buka: Flashing flexodomains and electroconvection rolls in a nematic liquid crystal. Phys. Rev. E. 87, 032505/1-10 (2013) [pdf]

In this chapter the influence of flexoelectricity on pattern formation induced by an electric field in nematics will be summarized. Two types of patterns will be discussed in the linear regime, the equilibrium structure of flexoelectric domains and the dissipative electroconvection (EC) rolls. In a separate section, recent experimental and theoretical results on the competition and crossover between the flexoelectric domains and EC patterns will be described.

Á. Buka, T. Tóth-Katona, N. Éber, A. Krekhov and W. Pesch, Chapter 4. The role of flexoelectricity in pattern formation. In eds. Á. Buka and N. Éber, Flexoelectricity in Liquid Crystals. Theory, Experiments and Applications, Imperial College Press, London, 2012. pp. 101–135

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.

[N. Éber,L.O. Palomares, P. Salamon, A. Krekhov and Á. Buka: Temporal evolution and alternation of mechanisms of electric-field-induced patterns at ultralow-frequency driving. Phys. Rev. E 86, 021702/1-9 (2012).] [pdf]

[N. Éber, L.O. Palomares, P. Salamon, A. Krekhov, Á. Buka: Competition between Electric Field Induced Equilibrium and Dissipative Patterns at Low Frequency Driving in Nematics. Invited talk at the 24th International Liquid Crystal Conference, Mainz, August 19th - 24th, 2012] [pdf]

The temporal evolution of electric field induced patterns within the driving period was studied in the nematic Phase 5 in a wide frequency range. The compound exhibits a transition from conductive to dielectric regime of electroconvection (EC) at high frequency. At low frequencies we found that the conductive EC rolls evolve and decay in each half period of driving. Following EC rolls another pattern, flexoelectric domains (FD), also appear as flashes in the same half period. This scenario thus represents a repetitive morphological transition between dissipative (EC) and equilibrium (FD) patterns.

[N. Éber, P. Salamon and Á. Buka: Competition between Electric Field Induced Equilibrium and Non-Equilibrium Patterns at Low Frequency Driving in Nematics. In Proceedings of the 13^th Small Triangle Meeting on Theoretical Physics, Stará Lesná, November 14-16, 2011, J. Busa, M. Hnatic and P. Kopcansky (eds.), IEP SAS, Kosice, 2012, pp. 56-63.]  [pdf]

We present in this paper a detailed analysis of the flexoelectric instability of a planar nematic layer in the presence of an alternating electric field (frequency ω), which leads to stripe patterns (flexodomains) in the plane of the layer. This equilibrium transition is governed by the free energy of the nematic, which describes the elasticity with respect to the orientational degrees of freedom supplemented by an electric part. Surprisingly the limit ω → 0 is highly singular. In distinct contrast to the dc case, where the patterns are stationary and time independent, they appear at finite, small ω periodically in time as sudden bursts. Flexodomains are in competition with the intensively studied electrohydrodynamic instability in nematics, which presents a nonequilibrium dissipative transition. It will be demonstrated that ω is a very convenient control parameter to tune between flexodomains and convection patterns, which are clearly distinguished by the orientation of their stripes.

[A. Krekhov, W. Pesch, Á. Buka: Flexoelectricity and pattern formation in nematic liquid crystals. Phys. Rev. E 83, 051706 (2011)] [pdf]

Periodic stripe patterns which form when an electric field is applied to a thin nematic liquid crystal layer with a very low conductivity are discussed. In this case the dielectric electroconvection mode persists down to very low frequencies of the driving voltage. A Lifschitz point, i.e., a transition from normal to oblique rolls is detected in the dielectric regime. A crossover from electroconvection to flexoelectric domains occurs for extremely low frequencies of about 0.1 Hz. The crossover scenario yields pattern morphologies characteristic for both mechanisms, i.e., electroconvection and flexoelectric domains which appear consecutively within one period of the driving voltage. A theoretical description of the onset characteristics of dielectric convection, which is based on an extended model including flexoelectricity, is also presented.

[M. May, W. Schöpf, I. Rehberg, A. Krekhov, A. Buka: Transition from longitudinal to transversal patterns in an anisotropic system. Phys. Rev. E 78, 046215 (2008)] [pdf]