Bent core nematics as optical gratings

    Bent core nematics as optical gratings ´ ´ N´andor Eber, Ying Xiang, Agnes Buka PII: DOI: Reference:

S0167-7322(17)33261-0 doi:10.1016/j.molliq.2017.09.025 MOLLIQ 7861

To appear in:

Journal of Molecular Liquids

Received date: Revised date: Accepted date:

20 July 2017 30 August 2017 7 September 2017

´ ´ Please cite this article as: N´andor Eber, Ying Xiang, Agnes Buka, Bent core nematics as optical gratings, Journal of Molecular Liquids (2017), doi:10.1016/j.molliq.2017.09.025

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Bent core nematics as optical gratings a,∗ ´ ´ N´ andor Eber , Ying Xiangb , Agnes Bukaa a Institute

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for Solid State Physics and Optics, Wigner Research Centre for Physics, Hungarian Academy of Sciences, H-1525 Budapest, P. O. Box 49, Hungary b School of Information Engineering, Guangdong University of Technology, Guangzhou 510006, Peoples Republic of China

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Abstract

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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 analysed.

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Keywords: liquid crystals, pattern formation, optical grating, flexoelectricity PACS: 61.30.Gd, 47.54.-r, 89.75.Kd 1. Introduction

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Diffraction gratings are devices to separate light of different wavelengths with high resolution. The monochromatic light beam diffracts in multiple orders when striking the grating. Gratings are very widely used optical components; they are usually made of solids with an array of closely spaced parallel, equidistant ridges or rulings on their surface. As a consequence, the grating spacing is a fixed parameter of the device. It is well known that liquid crystal (LC) cells can also create gratings. The set of equidistant parallel lines may be induced by different applied fields (such as shear, electric field, etc) [1] and as a consequence, the grating parameters are tunable by the control parameters. The most convenient way of inducing a regular set of parallel lines is to apply electric field to the liquid crystal. There is a great variety of pattern morphologies induced by a voltage depending on the frequency, field strength, temperature, surface alignment, etc. [2]. Even the basic phenomena induced by the field can be classified into several groups: (a) the transient patterns at the Freedericksz transition [3, 4], (b) the dissipative phenomenon of electroconvection (EC) [5–10], associated with electric current and fluid flow, and (c) the periodically deformed equilibrium state called flexodomains (FDs) [9–13]. In this article we will address groups (b) and (c). Pattern formation in calamitic nematics has intensely been studied in recent decades, both experimentally and theoretically [2, 5–8]. The basic phenomenon – standard electroconvection – has been fairly well explored and un∗ Corresponding

author ´ Email address: [email protected] (N´ andor Eber) ´ URL: www.szfki.hu/∼eber (N´ andor Eber)

Preprint submitted to Journal of Molecular Liquids

derstood on the basis of the Standard Model of electroconvection [14] for certain classes of nematics, the (- +) and (+ -) materials. Here the first sign stands for the dielectric anisotropy, while the second one is for the conductivity anisotropy of the material. Nematics not belonging to the above classes may also exhibit patterns – nonstandard electroconvection [8]; however, their formation mechanisms are only partially understood [15] and many details are still waiting for clarification. Bent-core nematics [16], a recently invented group of liquid crystals with unconventional, bent molecular structure belong to the latter category. They have been reported exhibiting unusual physical properties (e.g., high viscosities, small bend elastic constants, giant flexoelectricity [17], etc.) as well as novel electroconvection scenarios [18, 19], unprecedented among calamitic nematics; that makes bent-core nematics materials with wide perspectives for research as well as for applications. In view of these arguments, the investigations reported in the present paper have been carried out using bent-core nematic compounds. 2. Experimental A newly synthesised homologous series of an etherbridged bent core nematic 2,5-di4-[(4-alkylphenyl)-difluoromethoxy]-phenyl-1,3,4-oxadiazole (nP-CF2OODBP) has been used for studying the conditions for the formation of the patterns. The chemical structure of the molecule is shown in Fig. 1. Three members (n = 7, 8, 9) of the series have been investigated. Their phase sequence and phase transition temperatures are listed in Table 1. The materials have a conveniently broad nematic temperature range; the type of the smectic phase has not been identified yet. Though full characterization of all three compounds could not be performed, some important material paramSeptember 8, 2017

ACCEPTED MANUSCRIPT is called the shadowgraph technique; its theoretical basis, i.e., the relation between the periodical director distortions and the observed intensity modulation, has been provided in [21–23]). A digital camera with adjustable frame rate (Mikrotron EoSens MC 1362, max. 2000 frames/s) attached to the microscope allowed recording of snapshot images of 520×512 pixels or their sequences for later analysis. The recorded images were digitally processed for obtaining the contrast C, defined as the mean square deviation of the pixel intensities Ixy : C = h(Ixy − hIxy i)2 i. Here h· · ·i denotes averaging over all pixels. Additionally, a two-dimensional Fast Fourier Transformation (FFT) yielded the wavelength Λ and the wave vector q of the patterns, i.e., the wave number q = |q| = 2π/Λ and the obliqueness angle α = arccos(n0 q/q) enclosed by q and n0 . As a supplementary experimental technique, light diffraction on the gratings was also utilized. The sample was illuminated by a He-Ne laser beam (λ = 633 nm) and the resulting diffraction spots were detected on a far screen. They were either recorded by a camera or the intensity variation of certain diffraction spots was monitored using a fast photo-detector. The location of the diffraction spots is connected to the periodicity of the director distortion via Bragg’s law. The relation between the distortion amplitude and the intensity of the diffracted light of various order is by far not trivial. For the simplest liquid crystalline optical gratings (standard electroconvection patterns) both approximative analytical formulae and numerical results have been derived, using either a geometrical optical [24, 25] or a wave-optical approach [26–29]. We note, however, that the distortion profiles of the gratings reported below (nonstandard electroconvection and flexodomains) differ from that assumed in the above calculations. Recently a theoretical treatment applicable for general director distortions have been developed [23]; its application for specific grating structures is still a task for the future.

TN S [◦ C] 90.3 ◦ C 88 ◦ C 91.2 ◦ C

TIN [◦ C] 131.5 ◦ C 129 ◦ C 126.4 ◦ C

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TSCr 77 ◦ C 76 ◦ C 87.3 ◦ C

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n 7 8 9

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Figure 1: Chemical structure of the symmetrical bent core homologous series 2,5-di4-[(4-alkylphenyl)-difluoromethoxy]-phenyl-1,3,4oxadiazole (nP-CF2 OODBP).

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eters of the longest homolog 9P-CF2 OODBP (the elastic moduli K1 , K2 , K3 and the dielectric permittivities εk , ε⊥ , and εa = εk − ε⊥ ) have been measured at temperatures close to and far away from the isotropic-to-nematic phase transition temperature TIN [20]. These data are collected in Table 2. Due to their polar groups the compounds have a relatively high negative dielectric anisotropy, enabling us to study formation of patterns in planar alignment. As the electrical conductivity anisotropy was also found negative, the compounds belong to the (- -) family and as such, they are expected to exhibit nonstandard EC. Usual sandwich cells were used with a thickness of d = 6 µm. Planar alignment (the initial director n0 along x) was ensured by rubbed polyimide coatings covering the transparent electrodes. Cells were filled with the LC in the isotropic phase utilizing capillarity.

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Table 1: Temperatures of the isotropic-to-nematic (TIN ), nematicto-smectic (TN S ) and smectic-to-crystal (TSCr ) phase transitions of the studied members of the nP-CF2OODBP homologous series.

T − TIN K1 K2 K3 εk ε⊥ εa σa

-26 ◦ C 10.7 pN 10.6 pN 25.6 pN 5.2 9.5 -4.3 <0

-3 ◦ C 7.3 pN 6.9 pN 14.1 pN 5.7 8.5 -2.8 <0

3. Controlling the wave vector of the grating As mentioned in the Introduction, a disadvantage of the solid state gratings - the fixed wave vector - can be overcome by using electric field-induced stripe patterns of LCs. The morphology (viz. the wave vector q)) of such patterns depends, besides the cell geometry and the used nematic LC material, on suitable control parameters. In this chapter we demonstrate that the frequency and the amplitude of the applied voltage as well as temperature may serve as convenient control parameters, allowing adjustment of the wave vector - both its direction and magnitude. The pattern forming phenomenon with the largest morphological richness is electroconvection. Exploration of the phenomena requires, for example, answering the following questions: In what frequency and temperature ranges can one observe EC patterns? What is the threshold voltage

Table 2: Some material parameters of the compound 9PCF2OODBP measured at two temperatures [20].

For the measurements the cells were placed in a heating stage (Linkam LTS 350/TMS 94), which provided temperature controlled environment with a relative accuracy of ±0.1√◦ C. A dc voltage U or a sinusoidal ac voltage U (t) = 2U0 sin(2πf t) with rms value of U0 and frequency f was provided by function generators and a high voltage amplifier. Applying it to the electrodes, an electric field E along z was generated in the sample. The field-induced gratings were observed by a polarizing microscope (Leica DM RXP), either with crossed polarizer and analyzer, or with a single polarizer. This latter 2

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for the onset of the pattern? What is the pattern morphology at the onset? Are there morphological transitions upon applying higher voltages? Figure 2 exhibits snapshots of the three basic EC morphologies observed in our compounds: oblique stripes (OS), longitudinal stripes (LS) and prewavy stripes (PW). These morphologies trivially differ in the direction and the magnitude of the wave vector. The longitudinal stripes [8, 30] are parallel to n0 , thus q⊥n0 . In contrast to that, the prewavy stripes [18, 19, 31–33] are perpendicular to n0 , thus qkn0 . In the OS pattern, there are stripes running obliquely with respect to n0 in two, symmetrical directions forming a zigzag structure [8, 30]; consequently, q makes an angle ±α with n0 .

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Figure 3: Temperature dependence of the frequency range of existence for electroconvection patterns in 8P-CF2 OODBP. In the regions with lines hatched to the left (fOL (T ) < f < fLL (T )) and right (fOH (T ) < f < fLH (T )), oblique and longitudinal stripes, respectively, are observed. In the crosshatched region (fLL (T ) < f < fOH (T )), increasing the voltage induces morphological transitions.

Figure 2: Characteristic pattern morphologies of 8P-CF2 OODBP observed in a polarizing microscope at T −TIN = −5◦ C: (a) Oblique stripes (OS) at f = 125 Hz, U0 = 16.3 V (46 Vpp ); (b) longitudinal stripes (LS) at f = 500 Hz, U0 = 28.3 V (80 Vpp ); (c) prewavy stripes (PW) at f = 500 Hz, U0 = 56.6 V (160 Vpp . The length of the scale bar is 20 µm; the rubbing direction is horizontal.

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changing too (the wavelength of the PW stripes are much larger than that of the LS). The longer homologue, 9P-CF2 OODBP, was tested only at a low temperature (T = TIN − 29 ◦ C), however, in a considerably wider frequency range, from subhertz up to several ten kHz [20]. Figures 5a, 5b and 5c depict the frequency dependence of the threshold voltage Uc , the dimensionless wave number q ∗ = qd/π = 2d/Λ and the obliqueness angle α, respectively. Analysing these figures, three distinct frequency ranges can be distinguished. At intermediate frequencies (50 Hz < f < 3 kHz), the behaviour is qualitatively similar to that discussed above for 8P-CF2 OODBP; the primary instability results in a zigzag (OS) pattern, where the obliqueness angle increases from about α = ±60◦ to |α| ≥ 80◦ , approaching the LS state as the frequency is increased. The wavelength of the pattern is smaller than d (Λ ≈ 0.5–0.77d) and decreases with increasing f . At high frequencies (3 kHz < f < 80 kHz), one finds a different scenario; here the prewavy (PW) pattern is the primary instability instead of OS or LS. PW stripes are normal to n0 and have a much larger (Λ ≈ 3d) wavelength independent of f . At low f (f < 25 Hz), OS pattern with Λ ≈ 0.6– 0.9d and α ≈ 30–40◦ emerges at onset. Increasing the frequency reduces Λ, but has no significant influence on α. Increasing the voltage one finds a secondary instability yielding PW stripes with α ≈ 0 and Λ ≈ 2.1–2.5d. In this frequency range the patterns have an unusual dependence on the polarity of the applied field, which is discussed in detail in the next section.

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In order to reply to the questions raised above, compound 8P-CF2 OODBP was tested experimentally in the frequency range of 30 Hz ≤ f ≤ 1100 Hz. It was found that patterns do not occur at any frequency and temperature [34]. Instead, the frequency range fOL ≤ f ≤ fLH – where regular EC pattern emerges – exhibits a strong temperature dependence, as shown in Fig. 3. This frequency range is narrow in the lower half of the nematic temperature range, but it gradually extends as the sample is heated toward the clearing point. Both fOL (T ) and fLH (T ) have positive slope. As far as the morphology at the onset is concerned, at low f (fOL < f < fOH , left-hatched and crosshatched regions in Fig. 3) the primary instability yields oblique stripes [2(a)], while for fOH < f < fLH (right-hatched region in Fig. 3) it results in longitudinal rolls [2(b)] [34]. The frequency dependence of the threshold rms voltage Uc is illustrated in Figs. 4(a) and 4(b), which represent a different behaviour in the lower and higher halves of the nematic temperature range, respectively. At low temperature [e.g., at T = TIN − 24 ◦ C, Fig. 4(a)], fOH ≈ fLL and increasing the voltage above Uc , appearance and then motion of defects (dislocations) is induced and finally a chaotic state is reached. No regular stripes with another q appear. At high temperature [e.g., at T = TIN − 3 ◦ C, Fig. 4(b)], there is a frequency range fLL < f < fOH (crosshatched in Fig. 4), where the voltage increase exerts morphological transitions: first, from OS to LS and then, at a higher voltage, from LS to PW stripes [[2(c)]]. The latter transition is especially spectacular as it is accompanied with a 90◦ rotation of the wave vector, whose magnitude is 3

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Figure 4: Frequency dependence of threshold rms voltages of various electroconvection patterns in 8P-CF2 OODBP. (a) T = TIN − 24 ◦ C; (b) T = TIN − 3 ◦ C. Increasing the voltage induces morphological transitions.

4. Temporal evolution and the effect of the polarity of the electric field

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The temporal evolution of the pattern, especially the behaviour within a driving period t0 = 1/f of the applied voltage, provides important information on the mechanism and applicability of the LC grating. In polarizing microscopes, the maximal available frame rate of the attached digital camera yields a strong upper limit on the test frequency. In diffraction, however, this constraint is absent; there are photo-detectors fast enough to follow the variation of the diffracted intensities even at high f . The temporal behaviour of all morphologies exhibited in Fig. 2 have been studied by the diffraction technique, i.e., by monitoring the temporal evolution of the intensity I1 (t) of the first order diffraction spot, both for the polarization Pin of the illuminating laser beam being parallel with or perpendicular to n0 . Diffraction occurred for both polarizations. The polarization direction of the first order diffracted beam was found to be perpendicular to Pin [34]. For f ≫ 25 Hz, t0 is shorter than the growth or decay times of the pattern; therefore, stabilization of the grating requires several periods. After stabilization of the diffraction pattern, at high f , I1 (t) was found to be stationary with a small (< 10%) 2f modulation for all pattern types, as expected. At lower f , however, a huge 2f modulation was detected, following nearly the I1 (t) ∝ sin2 (2πf t) dependence.

Figure 5: Frequency dependence of (a) the threshold rms voltages Uc , (b) the dimensionless wave number q ∗ , and (C) the obliqueness angle α for various electroconvection patterns. 9P-CF2 OODBP, T = TIN − 29 ◦ C.)

The findings above indicate that (i) the director oscillates with the driving frequency at low f , but is stationary at high f ; (ii) the director involves out-of-plane tilt as well as in-plane twist deformations. For f < 25 Hz, the situation becomes more delicate, as the growth or decay times are comparable or shorter than t0 . As a result, the grating may emerge and decay within a driving period; i.e., it exists in the form of flashes [20]. Due to the low f , this scenario can already be monitored by polarizing microscopy, using a fast digital camera. It is seen in Fig. 6 that the pattern contrast C increases to a maximum, then returns to the background value (no pattern) and remains there for a while. This sequence is repeated in each half of the driving period. We note that this flashing behaviour (which becomes more pronounced at ultralow, f < 1 Hz, frequencies) seems to be quite general; it has 4

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Figure 6: Temporal dependence of the contrast of oblique roll pattern as well as of the applied voltage within one full period of the low frequency ac voltage. The thick arrows indicate where the representative snapshots were taken: near onset in the positive half period (OS+ ), around the zero crossing of the voltage and near onset in the negative half period (OS− ). All snapshots are of 70 × 70 µm size. The initial director n0 is horizontal, the crossed polarizers are at ±45◦ . 9P-CF2 OODBP, f = 12 Hz, U0 = 8.3 V, T = TIN − 20 ◦ C.

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been reported for standard EC and flexodomains [9, 35– 37] as well as for nonstandard EC [38] at a sufficiently low driving frequency.

Figure 7: Temporal dependence of the contrast of the prewavy pattern as well as of the applied voltage within one full period of the low frequency ac voltage. The thick arrows indicate where the representative snapshots were taken: PW in the positive half period, PW around the zero crossing of the voltage and PW in the negative half period. All snapshots are of 70 × 70 µm size. The initial director n0 is horizontal, the crossed polarizers are at ±45◦ . 9P-CF2 OODBP, f = 12 Hz, U0 = 10.0 V, T = TIN − 20 ◦ C.

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The appearance of EC patterns is typically independent of the polarity of the electric field. According to theoretical considerations, q should depend on U 2 only. Even if the director is oscillating with the field (in the dielectric regime of standard EC or due to the linear flexoelectric interaction), in the case of normal light incidence that is implemented in polarizing microscopes, optics is sensitive only to the magnitude of the director tilt, not to its sign. Therefore, the contrast spikes of both half periods are expected to have equal maxima. Though Fig. 6 shows a minor asymmetry of these maxima, it may be associated either to cell imperfection (e.g., a director pretilt) or to the presence of a small dc offset voltage of the amplifier. Despite of the expectations, astonishingly, compound 9P-CF2 OODBP exhibited at low f a polarity dependent pattern [20]. It is clear from the snapshots of Fig. 6 that the OS stripes in the positive half period (OS+ ,’zig’) are leaning to opposite direction compared to those in the negative half period (OS− , ’zag’); i.e., the direction of the wave vector is changing with the polarity of the applied voltage. This polarity sensitivity of the wave vector holds up to voltages not very far from the threshold. Increasing the voltage further, OS stripes of the opposite obliqueness angle either coexist forming zigzag domains or are superposed yielding a grid structure. Finally, above the threshold of the secondary prewavy instability (see Fig. 5, PW stripes emerge. In any case, the at those voltages the structure (its q) does not depend on the polarity of the voltage any more. The temporal evolution of the secondary PW pattern within a driving period is shown in Fig. 7. The pattern

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The alternation of OS+ and OS− gratings near the onset at low frequencies is an unexpected feature of 9PCF2 OODBP. Though such a phenomenon is not fully unprecedented, to our knowledge, all previous reports related to twisted cells, i.e. to cells where the initial directors on the two boundary plates do not coincide, but enclose an angle ψ > 0 (either ψ = 90◦ [39, 40] or ψ = 18◦ [41]. In those twisted cells the pattern is claimed to form always close to the cathode’s surface, thus q is determined by n0 on the cathode. Changing the polarity relocates the pattern to the other surface with a different n0 ; so q rotates by the twist angle of the cell. This mechanism cannot, however, explain our observations. First, we used an untwisted planar cell where n0 is the same on both electrodes. Therefore, whichever electrode is the cathode, q should be the same. Second, though we admit that EC patterns may be generated near the surface at some combination of surface treatments and nematic materials, the existence of OS pattern with α ± 40◦ clearly demonstrates that in our case the pattern is rather formed in the bulk (in the middle of the cell), similarly to the case of standard EC. Exploring the origin of this unusual, polarity sensitive wave vector selection thus requires further experimental and theoretical efforts.

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Figure 8: Temporal dependence of the contrast of the EC and FD patterns as well as of the applied voltage within one full period of the low frequency ac voltage. The symbols indicate where the representative snapshots in Fig. 9 were taken. 7P-CF2 OODBP, f = 0.1 Hz, U0 = 35 V, T = TIN − 7 ◦ C.

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In the previous chapters we showed that by adjusting a few control parameters one can drastically change the wave vector (both its magnitude and direction) by driving the system into different types of electroconvection. Now we concentrate only on the case, where voltage changes leave the direction of the wave vector unaltered, but allows a fine tuning of the wave number. The grating type which we can achieve this goal with is the flexodomain structure. Though the phenomenon of the voltage dependent wave number of FDs has been known for decades, detailed experiments are very scarce [12, 42–44]. Moreover, the early (and so far only) theory intending to interpret this nonlinear behaviour [13, 45] has recently proved to be not sufficiently rigorous [46]. Our experiments revealed new aspects especially about the temporal behaviour - building up and relaxing - of the flexodomain structure. We recall that for the existence of flexodomains the material parameter combination µ = ε0 εa K/(e1 −e3 )2 should fall into a specific interval, whose boundaries can be calculated numerically [9]. Here ε0 is the electric constant, εa is the dielectric anisotropy, K = (K1 + K2 )/2 is the averaged elastic constant, and (e1 − e3 ) is the difference of the flexoelectric coefficients. In case of equal elastic constants, the constraint above simplifies to |µ| < 1) [11, 13]. Additionally, for the observability of flexodomains their threshold voltage must be lower than that of electroconvection. Based on accumulated experience one can conclude that this latter condition can easier fulfil in thin cells, in materials with a lower electrical conductivity and at low frequency or dc excitation. Among the investigated compounds, the shortest homologue, 7P-CF2 OODBP, proved to be suitable for studying FDs, therefore experiments focussed on its subHz [47] and dc [48] behaviour. At f = 1 Hz EC is the primary instability; it appears in the form of flashes of OS in a limited part of each half period of the driving voltage. This behaviour is similar to that of 9P-CF2 OODBP, however, no polarity dependent features have been detected. Reducing f , however, a crossover to flexodomains occurs, as FD has lower threshold. At f = 0.1 Hz, EC still is observable at a high enough voltage. As an example, Fig. 8 shows the temporal evolution of the contrast and of the applied voltage within a single period. The contrast curve has two peaks in each half period. The symbols denote the time instants where the representative snapshots in Fig. 9 were taken. The first, small peak of the C(t) curve, closely following the zero crossing of the voltage, corresponds to EC [oblique stripes, Fig. 9(b)]. The second, large and wide peak belongs to FDs [Figs. 9(d) and 9(e)]. The two pattern types (EC and FD) are continuously alternating; they emerge and decay, existing in different time windows [47]. Between these windows there is no pattern at all [Figs. 9(a), 9(c) and 9(f)]. This behaviour is similar to that reported recently for calamitic nematics [35–37].

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Figure 9: Representative shadowgraph snapshots of 70 × 70 µm size taken within a driving period. The initial director n0 is horizontal, the polarizer is parallel with n0 . Time is counted from the beginning of the driving period. 7P-CF2 OODBP, f = 0.1 Hz, t0 = 10 s, U0 = 35 V, T = TIN − 7 ◦ C.

Comparing Figs. 9(d) and 9(e) it is seen that as time evolves and thus the instantaneous value of the sinusoidal voltage increases, the wavelength of FDs diminishes. This voltage sensitivity of the wavelength Λ (and thus of the wave number q) can also be clearly demonstrated by laser diffraction experiments. According to the Bragg criterion, the diffraction angle increases if Λ becomes smaller; at sinusoidal driving with voltages exceeding Uc , the first order diffracted beam sweeps back-and forth through a range of the diffraction angle φmin ≤ φ ≤ φmax [47]. This can be observed by bare eyes looking at a far screen, as well as by photodetectors positioned at various diffraction angles within the angle range φmin –φmax . The most evident proof for the voltage tunability of the FD wavelength can be obtained by observing the patterns (either with POM or with diffraction) at dc driving. Figures 10(a)-(f) show a set of snapshots taken at increasing dc driving voltages using a d = 6 µm thick cell of 7PCF2 OODBP. FDs appear at Uc ≈ 22 V; increasing U the wavelength becomes evidently much smaller [48]. 6

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Figure 11 plots the relative variation of the dimensionless wave number q ∗ /qc∗ in the function of the relative voltage U/Uc . Data points are well fitted (solid line) by the relation q ∗ /qc∗ = βU/Uc + γ with β = 1.46 ± 0.03, γ = −0.55 and qc∗ = 2.53. This linear relation allows for convenient tunability of the wavelength of the grating, opening wide perspectives for applications, e.g., in beam steering without moving parts.

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Figure 10: Shadowgraph snapshots of 125 × 125 µm size of flexodomains at different dc voltages. The initial director n0 is horizontal, the polarizer is parallel with n0 . 7P-CF2 OODBP, T = TIN − 7 ◦ C.

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cays to the homogeneous state. At growth, the wave number selection occurs mostly at the appearance of the pattern. The contrast first increases exponentially; then it saturates as approaching its equilibrium value, which depends on U2 . Stabilization (saturation) of the contrast takes more time if U2 is close to Uc . At decay, the contrast diminishes exponentially and no significant change in q could be detected before the pattern is fading away. For both the growth and the decay, the characteristic times depend on U2 : the larger U2 , the faster the transient. Surprisingly, the decay times were much shorter than the growth times. (ii) When Uc < U1 < U2 , the switching occurs between two FD gratings of different wavelengths. The mechanism of adjusting the wave numbers is different from that of pattern appearance. As usual in case of extended stripe patterns of any origin, change of q cannot occur continuously, but requires generation, motion and finally (partial) annihilation of defects (dislocations). This holds both for increasing or decreasing the voltage and thus q. The characteristic times of this procedure is significantly larger than those in case (i). Nevertheless the dependence on U2 and the slowness of the response on voltage increase compared to that on voltage decrease still holds in case (ii) too.

6. Conclusions

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The mechanisms mentioned in the Introduction, which can produce a system of regularly ordered parallel lines, can be rated from the point of view of application as follows: (a) Transient patterns are obviously out of the question, being not stationary and, moreover, usually being not regular enough (though having a dominant wave number). (b) Electroconvection could be used; the pattern is regular and stationary, but it has also disadvantages. Namely, the wave number is only slightly tunable by changing just one control parameter. A simultaneous adjustment of two parameters (frequency and voltage) is more efficient, but this procedure is too complicated for implementing it in a device. Summarising our experiences with this phenomenon there are a few issues that we want to note. First, the various EC morphologies exist in different, only partially overlapping frequency ranges. Therefore, there are frequencies, where (i) only one morphology is present (either OS or LS); (ii) only one morphological transition occurs (OS–LS or LS–Pw) upon increasing U ; (iii) a sequence of two transitions (OS– LS–PW) can be observed. Second, the OS threshold voltage increases with decreasing f at f < fm ≈ 50–100 Hz for 8P-CF2 OODBP

Figure 11: The dimensionless wavenumber q ∗ scaled with its onset value qc∗ in the function of the dc voltage U scaled by its threshold value Uc . 7P-CF2 OODBP, T = TIN − 7 ◦ C.

For application purposes it is important to know, how fast the system responds to the external excitation. In order to answer this question the response of the FD-based gratings on dc voltage jumps were investigated. The voltage jumps from U1 to U2 were actually realized by bipolar square waves of ultra-low frequency (f = 0.01 Hz) with an amplitude of (U2 − U1 )/2 biased with a dc voltage of (U2 + U1 )/2, where the length of the half period of this periodic signal (t0 /2 = 50 s ) was much longer than any characteristic times of the system. According to the measurements two regimes with different switching characteristics could be distinguished [48]: (i) When U1 < Uc < U2 , the grating emerges from or de7

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This work was supported by the National Natural Science Foundation of China (Grant No. 11374067, 11374087), the Guangdong Provincial Science and Technology Plan (Grant No. 2014A050503064 and 2016A050502055), the Hungarian Research Fund (OTKA NN110672), and the 2012 Inter-Governmental S&T Cooperation Proposal Between Hungary and China (6-8 Complex study of nonlinear ´ 12 CN-1structures in novel types of liquid crystals, TET 2012-0039). References

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[1] A. Buka and L. Kramer, editors, Pattern Formation in Liquid Crystals. Springer, New York, 1996. ´ ´ Buka, Electrically induced patterns [2] N. Eber, P. Salamon and A. in nematics and how to avoid them, Liquid Crystals Reviews 4 (2016) 101–134. [3] Sagues F, san Miguel M. Transient patterns in nematic liquid crystals: Domain-wall dynamics. Phys Rev A. 39 (1989) 6567– 6572. [4] Buka A, de la Torre Juarez M, Kramer L, Rehberg I. Transient structures in the Fr´ eedericksz transition. Phys Rev A. 40 (1989) 7427–7430. [5] Kramer L, Pesch W. Electrohydrodynamic instabilities in ne´ Kramer L, editors. Pattern matic liquid crystals. In: Buka A, formation in liquid crystals. New York: Springer-Verlag; 1996. p. 221–256. [6] Pesch W, Behn U. Electrohydrodynamic convection in nematics. In: Busse FH, M¨ uller SC, editors. Evolution of spontaneous structures in dissipative continuous systems. Berlin–Heidelberg: Springer; 1998. p. 335–383. (Lecture Notes in Physics, Vol. 55) [7] Kramer L, Pesch W. Electrohydrodynamics in nematics. In Dunmur DA, Fukuda A, Luckhurst GR, editors. Physical properties of nematic liquid crystals. London: Inspec; 2001. p. 441– 454. ´ Buka, N. Eber, ´ [8] A. W. Pesch, L. Kramer, Convective patterns in liquid crystals driven by electric field. In. Self-Assembly, Pattern Formation and Growth Phenomena in Nano-Systems. Eds. A. A. Golovin, A. A. Nepomnyashchy, NATO Science Series II, Mathematica, Physics and Chemistry, Vol. 218, Springer, Dordrecht, 2006, pp. 55-82. [9] Krekhov A, Pesch W, Buka A. Flexoelectricity and pattern formation in nematic liquid crystals. Phys Rev E. 83 (2011) 051706. ´ T´ ´ [10] Buka A, oth-Katona T, Eber N, Krekhov A, Pesch W. Chapter 4. The role of flexoelectricity in pattern formation. In: Buka ´ Eber ´ A, N, editors. Flexoelectricity in liquid crystals. Theory, experiments and applications. London: Imperial College Press; 2012. p. 101–135. [11] Bobylev YuP, Pikin SA. Threshold piezoelectric instability in a liquid crystal. Sov Phys JETP. 45 (1977) 195–198 [Zh Eksp Teor Phys. 72 (1977) 369–374]. [12] M. I. Barnik, L. M. Blinov, A. N. Trufanov, and B. A. Umanski, Flexo-electric domains in liquid crystals, J. Phys. (France). 39 (1978) 417–422. [13] Pikin SA. Structural transformations in liquid crystals. New York: Gordon and Breach Science Publishers; 1991. [14] Bodenschatz E, Zimmermann W, Kramer L. On electrically driven pattern-forming instabilities in planar nematics. J. Phys. (France). 49 (1988) 1875–1899. ´ ´ Buka, [15] A. Krekhov, W. Pesch, N. Eber, T. T´ oth-Katona, A. Nonstandard electroconvection and flexoelectricity in nematic liquid crystals. Phys. Rev. E. 77 (2008) 021705. [16] J´ akli A. Liquid crystals of the twenty-first century nematic phase of bent-core molecules. Liq. Cryst. Rev. 1 (2013) 65-82.

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[Fig. 4] and at f < fm ≈ 5 Hz for 9P-CF2 OODBP [Fig. 5(a)]. This may be an experimental artifact related to the cell construction. The polyimide coating on the electrodes, used as orienting layer, is actually insulating; therefore, its impedance ZP I forms an internal voltage attenuator with the impedance ZLC of the LC. Consequently, the voltage sensed by the LC is smaller than U , which is applied to the electrodes [36]. This attenuation, ζ = ZLC /(ZP I + ZLC ), is typically negligible at high f (ζ ≈ 1), but becomes important as the frequency is reduced; thus turning the theoretically monotonic Uc (f ) to a function with a minimum. The higher the frequency of this minimum, the lower ZLC , i.e., the higher the electrical conductivity σ of the nematic. fm ≈ 50–100 Hz implies a fairly large σ ∼ 10−7 –10−6 S/m for 8PCF2 OODBP, considerably larger than that of typical electroconvecting calamitic nematics (σ ∼ 10−9 – 10−8 S/m). This conclusion is in agreement with estimation of the conductivity from direct measurements. Third, all three EC pattern types remain observable up to the clearing point TIN and no divergence of the thresholds is experienced as T → TIN . Fourth, the studied compounds did not exhibit those banana-specific EC scenarios (diverging PW thresholds and negative slope of Uc (f ) at high frequencies), which were reported earlier for other bent-core nematics [18, 19]; their features rather resemble the behaviour of calamitic (–) nematics. (c) The best candidate for constructing a tunable grating is the flexodomain structure. It is regular, stationary, involves no flow and ideally neither electric current. Its most important property is the easy tunability of the wavelength with the applied voltage, up to rather high (4-5 times the threshold) values. At device construction one should consider that the internal attenuation mentioned above is present (actually is most pronounced) at dc driving. Our results on the dynamics of the growth and the decay of the pattern helps to select the parameter ranges convenient for applications. We found that, similarly to the behaviour of the homogeneous Fredericksz transition, decay is faster than the growth for low voltages (in the vicinity of the threshold). We have also found that, surprisingly, the decay time depends strongly on the voltage, which is not the case for the relaxation of a homogeneous (e.g., Freedericksz) state. For explanation, we recall our experiences collected on the EC patterns. There it was theoretically proved and experimentally justified that the characteristic times depend strongly (being roughly proportional to q −2 ) on the wavenumber [49]. The shorter the wavelength, the faster the decay. This holds for the flexodomains too.

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[39] Krishnamurthy KS, Kumar P, Vijay Kumar M. Polaritysensitive transient patterned state in a twisted nematic liquid crystal driven by very low frequency fields. Phys Rev E. 87 (2013) 022504. [40] Krishnamurthy KS. Spatiotemporal character of the BobylevPikin flexoelectric instability in a twisted nematic bent-core liquid crystal exposed to very low frequency fields. Phys Rev E. 89 (2014) 052508. [41] Omaima AE. Bent core nematics. Alignment and electro-optic effects [PhD dissertation]. Gothenburg (Sweden): University of Gothenburg; 2013. [42] W. Greubel, U. Wolff, Electrically Controllable Domains in Nematic Liquid Crystals. Appl. Phys. Lett. 19 (1971) 213–215. [43] P. Kumar and K. S. Krishnamurthy, Gradient flexoelectric switching response in a nematic phenyl benzoate, Liq. Cryst. 34 (2007) 257. ´ ´ Buka, T. Ostapenko, S. D¨ [44] P. Salamon, N. Eber, A. olle, and R. Stannarius, Magnetic control of flexoelectric domains in a nematic fluid, Soft Matter 10 (2014) 4487 . [45] Terent’ev EM, Pikin SA. Nonlinear effects in a real flexoelectric structure. Sov Phys JETP. 56 (1982) 587–590 [Zh Eksp Teor Fiz. 83 (1982) 1038–1044]. [46] W. Pesch, A. Krekhov, private communication. ´ [47] M.-Y. Xu, M.-j. Zhou, Y. Xiang, P. Salamon, N. Eber, and ´ Buka, Domain structures as optical gratings controlled by A. electric field in a bent-core nematic, Optics Express 23 (2015) 15224–15234. [48] Y. Xiang, H.-Z. Jing, Z.-D. Zhang, W.-J. Ye, M.-Y. Xu, E. ´ ´ Buka, Tunable optical gratWang, P. Salamon, N. Eber, and A. ing based on the flexoelectric effect in a bent-core nematic liquid crystal, Phys. Rev. Appl. 7 (2017) 064032. ´ ´ Buka, W. Pesch, L. [49] N. Eber, S. A. Rozanski, S. N´ emeth, A. Kramer, Decay of spatially periodic patterns in a nematic liquid crystal. Phys. Rev. E 70 (2004) 061706.

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´ [17] J. Hardeen, B. Mbanga, N. Eber, K. Fodor-Csorba, S. Sprunt, J.T. Gleeson, A. J´ akli, Giant Flexoelectricity Of Bent-Core Nematic Liquid Crystals. Phys. Rev. Lett. 97 (2006) 157802. ´ [18] D. B. Wiant, J. T. Gleeson, N. Eber, K. Fodor-Csorba, A. J´ akli, T. T´ oth-Katona, Non-Standard Electroconvection in a Bent Core Nematic. Phys. Rev. E 72 (2005) 041712. ´ [19] S. Tanaka, H. Takezoe, N. Eber, K. Fodor-Csorba, A. Vajda, ´ Buka, Electroconvection in nematic mixtures of bentand A. core and calamitic molecules. Phys. Rev. E 80 (2009) 021702. ´ [20] Y. Xiang, M.-j. Zhou, M.-Y. Xu, P. Salamon, N. Eber, and ´ Buka, Unusual polarity-dependent patterns in a bent-core A. nematic liquid crystal under low-frequency ac field, Phys. Rev. E 91 (2015) 042501 [21] Rasenat S, Hartung G, Winkler BL, Rehberg I. The shadowgraph method in convection experiments. Exp Fluids. 7 (1989) 412–420. [22] S. P. Trainoff, D. S. Cannell, Physical optics treatment of the shadowgraph. Phys. Fluids 14 (2002) 1340–1363. [23] W. Pesch, A. Krekhov, Optical analysis of spatially periodic patterns in nematic liquid crystals: Diffraction and shadowgraphy. Phys. Rev. E 87 (2013) 52504.2013 [24] J. A. Kosmopoulos, H. M. Zenginoglou, Geometrical optics approach to the nematic liquid crystal grating: numerical results. Appl. Opt. 26 (1987) 1714–1721. [25] H. M. Zenginoglou, J. A. Kosmopoulos, Geometrical optics approach to the nematic liquid crystal grating: leading term formulas. Appl. Opt. 28 (1989) 3516–3519. [26] H. M. Zenginoglou, J. A. Kosmopoulos, Linearized wave-optical approach to the grating effect of a periodically distorted nematic liquid crystal layer. J. Opt. Soc. Am. A 14 (1997) 669–675. [27] H. M. Zenginoglou, P. L. Papadopoulos, Analytical expression for the fringe intensities of a nematic grating excited electrohydrodynamically in the dielectric mode, J. Opt. Soc. Am. A 18 (2001) 573–576. [28] T. John, U. Behn, R. Stannarius, Laser diffraction by periodic dynamic patterns in anisotropic fluids. Eur. Phys. J. B 35 (2003) 267–278. [29] Ch. Bohley, J. Heuer, R. Stannarius, Optical properties of electrohydrodynamic convection patterns: rigorous and approximate methods. J. Opt. Soc. Am. A 22 (2005) 2817–2826. ´ ´ Buka: Non[30] T. T´ oth-Katona, A. Cauquil-Vergnes, N. Eber, A. standard electroconvection with Hopf-bifurcation in a nematic with negative electric anisotropies. Phys. Rev. E 75 (2007) 066210. [31] Petrescu P, Giurgea M. A new type of domain structure in nematic liquid crystals. Phys Lett A. 59 (1976) 41–42. ´ [32] Huh J-H, Hidaka Y, Yusuf Y, Eber N, T´ oth-Katona T, Buka ´ Kai S. Prewavy pattern: a director-modulation structure in A, nematic liquid crystals. Mol Cryst Liq Cryst. 364 (2001) 111– 122. [33] Huh J-H, Yusuf Y, Hidaka Y, Kai S. Prewavy instability of nematic liquid crystals in a high-frequency electric field. Phys Rev E. 66 (2002) 031705. ´ Buka, N. Eber, ´ [34] Y. Xiang, Y.-K. Liu, A. Z.-Y. Zhang, M.-Y. Xu, and E. Wang, Electric-field-induced patterns and their temperature dependence in a bent-core liquid crystal, Phys. Rev. E 89 (2014) 012502. ´ Transition [35] May M, Sch¨ opf W, Rehberg I, Krekhov A, Buka A. from longitudinal to transversal patterns in an anisotropic system. Phys Rev E. 78 (2008) 046215. ´ ´ [36] N. Eber, L.O. Palomares, P. Salamon, A. Krekhov and A. Buka, Temporal evolution and alternation of mechanisms of electric field induced patterns at ultra-low-frequency driving, Phys. Rev. E 86, (2012) 021702. ´ ´ Buka, Flashing flex[37] P. Salamon, N. Eber, A. Krekhov and A. odomains and electroconvection rolls in a nematic liquid crystal. Phys. Rev. E. 87 (2013) 032505. ´ ´ [38] Kumar P, Heuer J, T´ oth-Katona T, Eber N, Buka A. Convection-roll instability in spite of a large stabilizing torque. Phys Rev E. 81 (2010) 020702(R).

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Graphical abstract

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ACCEPTED MANUSCRIPT Highlights • Electric field induced gratings in a homologous series of bent-core nematics are explored. • Diversity of morphologies in electroconvection and flexodomains are demonstrated.

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• Threshold voltages and wave numbers are determined.

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• Flexodomains have voltage tunable wave numbers.

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• The dynamics of switching upon voltage increase or decrease is studied.

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