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We examined the flexodomains (FDs) in chiral bent-core nematics (BCNs), and demonstrated the morphology changed from parallel stripes in pure BCN to oblique ones in chiral BCNs. While the magnitude of obliqueness angle strongly depended on the concentration of chiral dopant, its sign was determined by the polarity of the driving voltage, thus FDs appeared alternately as symmetrical oblique stripes in the positive and negative half a.c. voltage cycles, respectively. Also the HTP value of chiral dopant in BCNs can be determined based on this phenomenon. The polarity-dependent behavior of FDs can be potentially exploited in photonic devices with a bistable function.
© 2016 Optical Society of America
1. Introduction
The remarkable electro-optic effects of nematic liquid crystals (LCs) arise from their super large optical anisotropy and high sensitivity to external fields. In addition to a wide range of applications in displays, LCs are widely used to manipulate light, including beam shaping or steering, switchable holograms, and others. These optical processings become increasingly important in the information age [1–3].
Among photonic devices, optical phase gratings are fundamental components whose performance will determine the capability of information processing. Gratings based on LCs have potential advantages over other holographic gratings due to large optical anisotropies and the controllable electro-optical switching ability. Usually, LC gratings are realized by a patterned electric- or optical-field, which changes the distribution of refractive index, thus modifies the corresponding spatial phase. However, LCs are mesophase media possessing orientation and fluidity, they self-assemble into long-range ordering micro-structures even under uniform external field, such as field-induced periodic instability in chiral nematics [4].
Among these self-assembled structures, flexodomains (FDs) are periodically deformed states arising from the flexoelectric effect [5–9], due to the linear coupling (–Pfl E) between the electric field E and the flexoelectric polarization Pfl = e11 n div n – e33 n × curl n (here e11, e33 are phenomenological flexocoefficients belonging to the splay and bend elastic deformations respectively, and n is the director of the LCs) [10]. When the applied voltage exceeds a threshold value Vc, FDs manifest themselves as longitudinal stripes running parallel to the initial director alignment n0 [11,12].
Besides the occurrence of static FDs under dc or quasi-static ac electric field, if the ac frequency is increased higher where electrical conductivity is non-negligible, dissipative effects will enhance significantly, thus LCs will undergo electroconvection (EC) effect arising from the couplings among the orientation of the LC molecules, the flow of both material and charge, and the induced electric field [13]. EC effects are often manifested in the appearance of various domains (such as Williams domain), whose characteristics (such as the threshold voltage, the critical wave vector, and the frequency range of existence) are closely related to the electric transport properties (such as dielectric and conductivity anisotropies) of LCs as well as the boundary conditions.
For traditional rodlike LC molecules, the flexocoefficients are quite small due to their cylindrical symmetry, which makes them hard to form FD structures [14]. Recently, bent-core nematic molecules (BCNs) generated considerable interest since their banana-shaped molecules lead to non-standard electro-convections [15–21], biaxial nematic phase [22], wide blue phase range [23–25], and large flexocoefficients [26–28]. Thus BCNs offer significant potential for exploring the formation mechanism and applications of FD gratings.
So far, most of the research efforts on FDs have focused on pure BCN systems without doping in planar cell [29–31], where FD stripes run parallel to the initial alignment n0, irrespective of the polarity of the driving voltage, or in twist cell [32]where FDs demonstrate as symmetrical oblique rolls in each voltage polarity. The behavior of FDs in chiral BCN system should be, but not yet explored. Here, we intend to provide a deeper insight into the characteristics of FDs in chiral BCNs, and compare them with that in pure BCNs. As one might expect, the inclusion of chiral dopants induce helical structure with a pitch P, which depends on dopant concentration C, modifying the morphology of FDs. Indeed, we observed experimentally that the FD morphologies changed from longitudinal stripes in pure BCNs to the oblique ones in chiral BCNs, while the magnitude of the obliqueness angle α strongly depended on P and the positive or negative signs of α is determined by the polarity of the driving voltage. As these polarity-dependent FDs can be potentially used as bistable optical gratings in optical processing, we here present a simple model to explain the underlying mechanism, where the dependences of α and Vc on P are obtained and in good agreement with experimental results.
2. Material and experimental set-up
The experiments were primarily performed on chiral BCNs, where the achiral host BCN compound is 2,5-bis (4-(difluoro (4-heptylphenyl) methoxy) phenyl)-1,3,4-oxadiazole (7P-CF2O-ODBP), provided from Wuhan Polytechnic University in P.R.China, whose structural formula is shown in Fig. 1(a). The corresponding phase sequence is: Crystal - 77°C - Smectic - 90.3°C - Nematic - 131.5°C - Isotropic. In order to demonstrate the peculiar behaviors of BCN, rodlike LC compound 1008 from East China University of Sciences and Technology was used for comparison. The molecular structure of 1008 is shown in Fig. 1(b) with phase sequence: Crystal - 53°C - Nematic - 77°C - Isotropic.
Fig. 1 .Structurals of the achiral BCN compound 7P-CF2O-ODBP (a) and rodlike compound 1008 (b), respectively.
The left-handed chiral dopant S811 was doped into the BCN host at concentrations of 1.2 wt%, 0.6wt% and 0.4wt%, which induced helical structures with various pitch P. Pure BCNs with P = ∞ was tested for comparison. So does the same for rodlike 1008.
The above two kinds of samples were filled into standard planar sandwich cells with thickness of d = 6 μm, where two ITO glass substrates were coated with a high temperature polyimide (KPI-200, Kelead photoelectric Materials Co., Ltd.), and then rubbed unidirectionally to obtain planar alignment. The surface alignment n0 is along the x-axis, while the electric field E is applied along the z-axis, i.e., perpendicular to the substrates and n0. The temperature of the samples was controlled by a precise hot-stage.
Driven by sinusoidal or square a.c. voltages with ultra-low frequency from a combination of a function generator and a high voltage amplifier, the induced FD patterns were observed and captured by a polarizing microscope (POM) equipped with a CCD camera. Besides the POM, a complementary laser (λ = 633nm) diffraction technique was also used to probe the diffraction spots of FD gratings, especially for visual investigation of the temporal evolution of FDs.
It should be noted that both kind of LC samples exhibit negative dielectric anisotropy [33–35], thus there will be no uniform deformation (Freedericksz transition) to take place in planar cell.
3. Experimental results
Using complementary experimental techniques, POM and laser diffraction, the dynamic and static behavior of FD gratings in BCNs were explored at the fixed temperature of T = 105°C.
3.1 Dynamic behavior
Under an ultra-low frequency (f = 0.02Hz) sinusoidal a.c. driving, whose amplitude Vp>Vc, FD stripes will evolve and decay according to the instantaneous value of the applied voltage within each half period. Figures 2(a) to 2(j) shows a sequence of snapshots of the diffraction spots from the FD stripes, taken at representative time instants within one period τ = 1/f. The off-axis diffraction spots with two distinct directions (OS+ and OS-) occured alternately, according to the polarity of the driving voltage. The off-axis angles α of the spots were almost symmetric with respect to n0. Inspection of the snapshots proved that OS+ always occurs in the positive, and OS- in the negative half period of the voltage, hence the FDs in chiral BCN system exhibit a pronounced polarity dependent behavior, appearing alternately in the 1st and 2nd half cycles of the driving field.
Fig. 2 (Color online) Temporal evolution of diffraction spots from FD stripes driven by sinusoidal ac voltages (f = 0.02Hz, τ = 50.0S) within one period; (a)-(j) are representative snapshots taken at different time instants (t = 0.1τ, 0.2τ, 0.25τ, 0.30τ, 0.40τ, 0.60τ, 0.70τ, 0.75τ, 0.80τ, 0.90τ) in a chiral BCNs (C = 0.6wt%, and Vp = 21V); (k)-(t) are representative snapshots taken at the same time instants in the pure BCNs (C = 0.0wt%, and Vp = 25V). The arrows indicate the initial director n0 of pure BCNs.
On the contrary, for pure BCNs, as shown in Figs. 2(k) to 2(t), the orientation of diffraction spots are always perpendicular to n0, irrespective of the polarity of the driving voltage, representing the degenerated OS+ and OS- directions.
In addition, during their time evolution, the wavelength Λ of FDs changes periodically as the ac voltage oscillates, resulting in a collinear swing of the diffraction spots. So, Λ of FDs is closely related to the instantaneous value of the ac voltage, as noticed in [36].
Apart from the above differences in the morphology of FDs in chiral and pure achiral BCNs, there are common behaviors in dynamics of these two kinds of FDs: ① the formation of FDs is a quasi-static process, taking several seconds to reach the equilibrium states; ② the growth of FDs are in phase with the applied voltage [37,38]. Figure 2 clearly demonstrates that with the time t elapsing, OS+ and OS- become strongest with maximal α at t→τ/4 or t→3τ/4, i.e., where the applied voltage reaches its maxima Vp, while they gradually decay as t→0, t→τ/2 or t→τ, when the voltage is approaching zero. The temporal behavior of FDs in the pure achiral BCNs is the same.
It should be noted that the dynamic behaviors of FDs strongly depend on the driving ac frequency f: ①when f<0.1Hz, only FDs present, whose growth is a rather slow process if Vp>Vc, demonstrating equilibrium attribute of the FDs; ②when 0.1Hz<f<1.0Hz, both FDs and EC domains take place, which differ in the textures; ③when frequency is increased to f>1.0Hz, only EC domains were detected, characterized by stripes perpendicular to the n0 with a larger periodicity (as shown in Fig. 5), whose occurrence were immediately after the application of ac driving, demonstrating the dynamic attribute of the ECs.
3.2 Static behavior
Apparently, the above dynamic behavior has already demonstrated that chirality modifies the structure of FDs. In order to quantitatively measure the pitch P dependence of FD parameters (such as α and Vc), a bipolar square wave driving voltage was used to ensure reaching the equilibrium states, as shown in Fig. 3.
Fig. 3 (Color online) Representative snapshots of FD stripes in BCNs under POM driven by square voltages (f = 0.02Hz, Vp = Vc). (a) (b) C = 1.2wt% and Vp = 14V; (c) (d) C = 0.6wt% and Vp = 19V; (e) (f) C = 0.4wt% and Vp = 21V; (g) (h) C = 0.0wt% and Vp = 22V. The arrows indicate the initial director n0 of pure BCNs. The sizes of all snapshots are 50μm × 50μm.
The pattern morphologies observed by POM indicated that two FD states alternately appeared in the chiral system (P≠∞) according to the different polarity of voltage, characterized by symmetrical obliqueness angles ± α. We found FD1 with obliqueness angle + α during the positive half-cycle of the square voltage turned into FD2 with -α during the negative half-cycle. However, in the pure achiral system (P = ∞), the orientations of FDs corresponding to different polarity were identical and parallel to n0, i.e., α = 0, independent of polarity. These observations once again demonstrate the effect of chirality.
However, such oblique FD stripes were absent in the case of doped rodlike LC system, in which only grids composed of two sets of left-tilted and right-tilted stripes were present, as shown in Fig. 4. The threshold voltage and obliqueness angles were independent of concentration of chiral dopant, thus the morphologies cannot be tuned and it is impossible to use them in bistable processing.
Fig. 4 (Color online) Representative snapshots of FD stripes in 1008 under POM driven by square voltages (f = 0.02Hz, Vp = Vc), where T = 60°C. (a) (b) C = 1.2wt% and Vp = 48V; (c) (d) C = 0.6wt% and Vp = 51V; (e) (f) C = 0.4wt% and Vp = 52V; (g) (h) C = 0.0wt% and Vp = 30V. The arrows indicate the initial director n0 of pure 1008. The sizes of all snapshots are 50μm × 50μm.
Fig. 5 Representative snapshots of EC stripes in chiral BCN of C = 1.2wt% under POM driven by square ac voltages. (a) f = 15.0Hz, Vp = 7.0 V; (b) f = 6.0Hz, Vp = 7.0V. The arrows indicate the initial director n0 of pure BCNs. The sizes of all snapshots are 50μm × 50μm.
4. Theoretical models
In order to clarify the role of chirality in the formation of oblique FDs in BCNs, we present here a concise derivation within the framework of linear model developed in Refs [39]. Our natural system of reference is (x,y,z), where the x-axis is along the initial orientation n0 and the z-axis (the direction of the electric field E) is normal to the substrates. In this frame the director is characterized by the θ tilt and the ϕ azimuthal angles. As in the chiral BCNs FD stripes are oblique (α≠0), it is more convenient to switch to another frame of axes, (ξ, ζ, z), rotated around the z-axis by α, so that the ξ-axis is parallel with the FDs orientation. Accordingly, the corresponding deformation director can be expressed as:
The total free energy density F of chiral system is composed of three terms:
where K is the averaged elastic modulus.
where q0 = 2π/P is the wave vector corresponding to the helical pitch P.
For simplicity, we neglect the dielectric contribution as the dielectric anisotropy is negative; also, E is spatial inhomogeneity due to the deformation of BCN molecules, which makes the results rather complicated, thus here E is assumed to be uniform.
We are interested in the behavior of FDs at onset. A trial solution of periodic modulation of the director is put forward in order to obtain a concise physical image (The cosine-like expressions with respect to z-coordinate satisfy the strong surface anchoring condition):
where θ0 and ϕ0 are small deformation angles at the threshold electric field Ec, q = 2π/Λ is the wave vector of the FD stripes, and -d/2≤z≤d/2.After minimize F under the constraint set by the Euler-Lagrange equations, yielding
The threshold Ec can be determined when these homogeneous linear equations have non-zero solution:In addition, after averaging F over one period of FDs and taking the derivative with respect to α, we obtainFinally, with the combination of Eqs. (8) and (9), we obtain concise relations between the helical pitch P, obliqueness angle α, and Ec of FDs in the chiral BCN system.The Eqs. (10) and (11) predict that the higher the concentration C (P is inversely proportional to C), the larger the α and the lower Vc, which are consistent with the experimental results in Fig. 3.
Finally, when the chiral system is degenerated to a pure achiral one (P→∞), the resulting and coincide with the known conclusions of pure system [40].
In addition, here we would briefly compare the director arrangements of FDs in the case of an achiral BCN and case of a chiral BCN: In ground state without E, due to strong surface planar alignment, the director fields in both cases are identical and uniform according to n0; However, when excited by E, the director fields in both cases will undergo periodic deformation with a tilted angle θ, azimuthal angle ϕ, and a wave vector q. The profiles of reoriented director fields in both cases are almost the same, except that the azimuthal angle ϕ was rotated from that of achiral BCN to the chiral BCN by angle α, whose magnitude is determined by the helical pitch P of chiral BCN.
5. Discussion and conclusion
A characteristic feature of FDs in chiral BCNs, distinguishing them from that in pure achiral BCNs, is that their orientation changes according to the polarity of the driving voltage, and the resulting α strongly depends on pitch P. This polarity sensitivity is easy to understand, as the flexoelectric torque is linear in the electric field E, resulting in symmetrical α due to the opposite sign of [41]. In Table 1 we summarize the results on the characteristics of FD stripes obtained by measurement and calculation, respectively.
Table 1. Chirality dependence of the properties of FDs in the positive half period.
First, we calculated P and Vc from Eqs. (10) and (11) according to the various α (depending on C) obtained in experiments, where the measured threshold Vc = 22Vof the achiral host system was used. The calculated values of Vc are in good agreement with the experiment results.
From the calculated pitch P and the corresponding concentration C of the chiral dopants, we can acquire the helical twisting power (HTP) of the dopant in the BCN system via the equation P = 1/(HTP.C). It has been known from supplier datasheet that HTP for a calamitic LC is 10.5-11.1 μm−1, but no data have been available so far for BCNs. The HTP calculated here from our experiments is about 5.35-5.70 μm−1, which is much weaker than that for the calamitic LCs, maybe due to the banana-shaped molecular structure.
In order to prove the validity of theoretical predictions in Eqs. (10) and (11), we also measured the pitches of chiral BCNs directly by Grandjean-Cano wedge cell method. The measured results P≈38μm (C = 0.4wt%), 25μm (C = 0.6wt%), 12μm (C = 1.2wt%) are roughly consistent with the theoretical results.
From microcosmic point of view, a strong flexoelectric effect in BCNs origins from the special V-shaped molecular structure and oxadiazole core, where a strong lateral dipole presents at the centre, together with electron donator and acceptor groups linked by conjugated electronic bridges and aromatic rings in the arms of molecules [42]. In contrast, the absence of oblique FD stripes in rodlike nematics can be attributed to a weak flexoelectric effect, which is due to the symmetry of molecules.
It should be noted that although from the appearance, EC domain and FD seem similar, they fundamentally differ in mechanism: the former is driven by electric current with fluid flow; whereas the latter is driven by electric field with static deformation.
In addition, another kind of 2D pattern has been reported in [43], which arises from the layer undulation under electric field. In planar cell, one of wave vector of this pattern is parallel to the rubbing direction n0 while the other one is perpendicular to n0. Besides the different morphology, our 2D grating in Fig. 4 differs from the above reported 2D grating in following aspects: ①Our grating occurs only under dc or ac with very low frequency, while the reported grating can present at frequency as high as f = 1kHz;②The appearance, depth of modulation of our grating depends on the ac polarity;③The grating constant Λ of our FD closely correlated with the amplitude of applied voltage, whereas the Λ of the reported grating is determined by pitch and thickness of cell, irrespective of electric polarity. These different behaviors are the results of different mechanisms: our grating origins from static flexoelectricity effect whereas the reported 2D grating origins from dynamic coupling between the elastic torque and dielectric torque.
In conclusion, by comparing the BCN systems with and without chiral dopants, we gained a deeper understanding on how the helical structures affect the morphology and dynamics of FDs. The tunability of the obliqueness angle by the concentration of the chiral dopant and by the polarity of the applied voltage is a specific feature of FDs in chiral BCNs, which can be potentially exploited to make certain photonic devices, such as optical bistable devices or optical storage devices. However, such effects are absent in the counterpart calamitic nematics, which severely limits their applications in novel photonic devices.
Furthermore, from the relation between the obliqueness angle α and concentration C of the chiral dopant, we obtained the HTP value, which is an essential parameter of the BCNs. So far only HTP values for calamitics are available.
Acknowledgments
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, 2016A050502055 and 2016A030313698).
References and links
1. I. C. Khoo, “Nonlinear optics, active plasmonics and metamaterials with liquid crystals,” Prog. Quantum Electron. 38(2), 77–117 (2014). [CrossRef]
2. W. Lee and S. T. Wu, “Focus issue introduction: Liquid crystal materials for photonic applications,” Opt. Mater. Express 1(8), 1585–1587 (2011). [CrossRef]
3. J. Kim, J. H. Suh, B. Y. Lee, S. U. Kim, and S. D. Lee, “Optically switchable grating based on dye-doped ferroelectric liquid crystal with high efficiency,” Opt. Express 23(10), 12619–12627 (2015). [CrossRef] [PubMed]
4. V. G. Chigrinov, V. V. Belyaev, S. V. Belyaev, and M. F. Grebenkin, “Instability of cholesteric liquid crystals in an electric field,” Sov. Phys. JETP 50(5), 994–999 (1979).
5. P. Tadapatri, K. S. Krishnamurthy, and W. Weissflog, “Patterned flexoelectric instability in a bent-core nematic liquid crystal,” Soft Matter 8(4), 1202–1214 (2012). [CrossRef]
6. Á. Buka and N. Éber, Flexoelectricity in Liquid Crystals: Theory, Experiments and Applications (Imperial College, London, 2012).
7. L. M. Blinov and V. G. Chigrinov, Electrooptic Effects in Liquid Crystal Materials (Springer-Verlag, 1994).
8. S. Kaur, V. P. Panov, C. Greco, A. Ferrarini, V. Görtz, J. W. Goodby, and H. F. Gleeson, “Flexoelectricity in an oxadiazole bent-core nematic liquid crystal,” Appl. Phys. Lett. 105(22), 223505 (2014). [CrossRef]
9. A. Krekhov, W. Pesch, and A. Buka, “Flexoelectricity and pattern formation in nematic liquid crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 83(5 Pt 1), 051706 (2011). [CrossRef] [PubMed]
10. R. B. Meyer, “Piezoelectric Effects in Liquid Crystals,” Phys. Rev. Lett. 22(18), 918–921 (1969). [CrossRef]
11. L. K. Vistin, “Electrostructural effect and optical properties of a certain class of liquid crystals and their binary mixtures,” Sov. Phys. Crystallogr.15(3), 514–515 (1970) (Kristallografiya 15(3), 594–595 (1970)).
12. P. Salamon, Ph.D dissertation, Institute for Solid State Physics and Optics, Hungarian Academy of Sciences.
13. W. Helfrich, “Conduction-Induced Alignment of Nematic Liquid Crystals: Basic Model and Stability Considerations,” J. Chem. Phys. 51(9), 4092 (1969). [CrossRef]
14. C. L. Trabi, C. V. Brown, A. A. T. Smith, and N. J. Mottram, “Interferometric method for determining the sum of the flexoelectric coefficients (e1+e3) in an ionic nematic material,” Appl. Phys. Lett. 92(22), 223509 (2008). [CrossRef]
15. S. Kumar and A. N. Gowda, “The chemistry of bent-core molecules forming nematic liquid crystals,” Liq. Cryst. Rev. 3(2), 99–145 (2015). [CrossRef]
16. S. Sreenilayam, Y. Panarin, and J. K. Vij, “Dielectric Study of Nematic LC Built with Bent-core Molecules,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 610(1), 63–67 (2015). [CrossRef]
17. Y. Xiang, M. J. Zhou, M. Y. Xu, P. Salamon, N. Éber, and Á. Buka, “Unusual polarity-dependent patterns in a bent-core nematic liquid crystal under low-frequency ac field,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 91(4), 042501 (2015). [CrossRef] [PubMed]
18. H. Takezoe and Y. Takanishi, “Bent-Core Liquid Crystals: Their Mysterious and Attractive World,” Jpn. J. Appl. Phys. 45(2A), 597–625 (2006). [CrossRef]
19. D. Wiant, J. T. Gleeson, N. Éber, K. Fodor-Csorba, A. Jákli, and T. Tóth-Katona, “Nonstandard electroconvection in a bent-core nematic liquid crystal,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(4), 041712 (2005). [CrossRef] [PubMed]
20. Y. Xiang, Y.-K. Liu, Á. Buka, N. Éber, 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 Stat. Nonlin. Soft Matter Phys. 89(1), 012502 (2014). [CrossRef] [PubMed]
21. R. Stannarius and J. Heuer, “Electroconvection in nematics above the splay Fréedericksz transition,” Eur Phys J E Soft Matter 24(1), 27–33 (2007). [CrossRef] [PubMed]
22. Y. Xiang, J. W. Goodby, V. Gortz, and H. F. Gleeson, “Revealing the uniaxial to biaxial nematic liquid crystal phase transition via distinctive electroconvection,” Appl. Phys. Lett. 94(19), 193507 (2009). [CrossRef]
23. Z. G. Zheng, D. Shen, and P. Huang, “Wide blue phase range of chiral nematic liquid crystal doped with bentshaped molecules,” New J. Phys. 12(11), 113018 (2010). [CrossRef]
24. G. Zhu, X. W. Lin, W. Hu, Z. G. Zheng, H. F. Wang, H. Q. Cui, D. Shen, and Y. Q. Lu, “Liquid crystal blue phase induced by bent-shaped molecules with allylic end groups,” Opt. Mater. Express 1(8), 1478–1483 (2011). [CrossRef]
25. Z. G. Zheng, W. Hu, G. Zhu, M. Sun, D. Shen, and Y. Q. Lu, “Brief review of recent research on blue phase liquid crystal materials and devices,” Chin. Opt. Lett. 11(1), 011601 (2013). [CrossRef]
26. S. P. Sreenilayam, Y. P. Panarin, J. K. Vij, S. I. Torgova, A. Lehmann, and C. Tschierske, “Flexoelectric polarization studies in bent-core nematic liquid crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 92(2), 022502 (2015). [CrossRef] [PubMed]
27. J. Harden, B. Mbanga, N. Éber, K. Fodor-Csorba, S. Sprunt, J. T. Gleeson, and A. Jákli, “Giant flexoelectricity of bent-core nematic liquid crystals,” Phys. Rev. Lett. 97(15), 157802 (2006). [CrossRef] [PubMed]
28. A. Jákli, “Liquid crystals of the twenty-first century – nematic phase of bent-core molecules,” Liq. Cryst. Rev. 1(1), 65–82 (2013). [CrossRef]
29. P. Salamon, N. Éber, Á. Buka, T. Ostapenko, S. Dölle, and R. Stannarius, “Magnetic control of flexoelectric domains in a nematic fluid,” Soft Matter 10(25), 4487–4497 (2014). [CrossRef] [PubMed]
30. P. Kumar and K. S. Krishnamurthy, “Gradient flexoelectric switching response in a nematic phenyl benzoate,” Liq. Cryst. 34(2), 257–266 (2007). [CrossRef]
31. A. Krekhov, W. Pesch, N. Éber, T. Tóth-Katona, and A. Buka, “Nonstandard electroconvection and flexoelectricity in nematic liquid crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 77(2), 021705 (2008). [CrossRef] [PubMed]
32. K. S. Krishnamurthy, “Spatiotemporal character of the Bobylev-Pikin flexoelectric instability in a twisted nematic bent-core liquid crystal exposed to very low frequency fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 89(5), 052508 (2014). [CrossRef] [PubMed]
33. P. Salamon, N. Eber, A. Krekhov, and A. Buka, “Flashing flexodomains and electroconvection rolls in a nematic liquid crystal,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 87(3), 032505 (2013). [CrossRef]
34. M. Y. Xu, Y. K. Liu, Y. Xiang, T. Xia, E. Wang, and Z.-D. Cheng, “A fast light-induced grating in bent-core nematic liquid crystals with in-plane switching,” Appl. Phys. Lett. 103(8), 083507 (2013). [CrossRef]
35. Y. Xiang, Y.-K. Liu, Z.-Y. Zhang, H.-J. You, T. Xia, E. Wang, and Z.-D. Cheng, “Observation of the photorefractive effects in bent-core liquid crystals,” Opt. Express 21(3), 3434–3444 (2013). [CrossRef] [PubMed]
36. M. Y. Xu, M. J. Zhou, Y. Xiang, P. Salamon, N. Éber, and Á. Buka, “Domain structures as optical gratings controlled by electric field in a bent-core nematic,” Opt. Express 23(12), 15224–15234 (2015). [CrossRef] [PubMed]
37. P. Salamon, N. Éber, A. Krekhov, and Á. Buka, “Flashing flexodomains and electroconvection rolls in a nematic liquid crystal,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 87(3), 032505 (2013). [CrossRef]
38. 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 Stat. Nonlin. Soft Matter Phys. 86(2), 021702 (2012). [CrossRef] [PubMed]
39. B. A. Umansky, V. G. Chigrinov, L. M. Blinov, and Yu. B. Podyachev, “Flexoelectric instability in twisted liquid crystal structures,” Fourth Liquid Crystal Conference of Socialist Countries, Tbilisi. Sov. Phys. JETP 81, 1307–1317 (1981).
40. Yu. P. Bobylev and S. A. Pikin, “Threshold piezoelectric instability in a liquid crystal,” Sov. Phys. JETP 45(1), 195–198 (1977).
41. Y. Xiang, Y. K. Liu, X. S. Xie, J. M. Li, J. H. Wang, and Z. G. Cai, “A kind of two-dimensional electroconvection at ultralow electric frequency,” Appl. Phys. Lett. 97(20), 203507 (2010). [CrossRef]
42. V. Yelamaggad, M. Mathews, S. A. Nagamani, D. S. Rao, S. K. Prasad, S. Findeisen, and W. Weissflog, “A novel family of salicylaldimine-based five-ring symmetric and non-symmetric banana-shaped mesogens derived from laterally substituted resorcinol: synthesis and characterization,” J. Mater. Chem. 17(3), 284–298 (2007). [CrossRef]
43. B. I. Senyuk, I. I. Smalyukh, and O. D. Lavrentovich, “Switchable two-dimensional gratings based on field-induced layer undulations in cholesteric liquid crystals,” Opt. Lett. 30(4), 349–351 (2005). [CrossRef] [PubMed]
