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Abstract
The structures of flexodomains, which are similar to optical gratings and can be controlled by the amplitude of applied voltage and temperature, were verified through polarizing microscopy and light diffraction techniques. The properties of the optical grating induced by a bent-core nematic liquid crystal in planar cells with varied cell gaps and pretilt angles were studied. The period of optical grating decreases with the increase in the amplitude of the applied voltage and pretilt angle. In addition, the period increases with the increase in cell gap and temperature. The period of optical grating has a linear relationship with temperature. The continuously adjustable period has the potential to become an important and extended application of optical grating.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Corrections
12 February 2018: Typographical corrections were made to the author listing and author affiliations.
1. Introduction
As one of the well-known types of functional materials, liquid crystal (LC) is widely applied in electrically switchable LC grating and has become increasingly important for future information processing due to its advantages such as low cost, easy preparation, and low power consumption [1–3]. In general, traditional electrically switchable phase gratings based on LC have two categories: one is with periodical electric field distribution induced by striped or interdigitated patterned electrodes on one or both substrates of the LC grating [4–11], and the other is with a periodic refractive index profile by defining two or more alignment regions or holographic recordings in polymer-dispersed liquid crystal (PDLC) [12–15]. Furthermore, LC grating can form by the finger print patterns in cholesteric LC cells [16,17]. However, the manufacture and control of these types of LC phase grating are complicated, and the diffraction angle is fixed or cannot be continuously changed.
Based on fingerprint textures of the bent-core nematic (BCN) LC, the diffraction angle could be adjusted by an external electric field and continuously changed [18,19]. In addition, this type of grating can form in planar cells without special treatment and shows significant potential applications for manipulating light such as diffraction optics, laser displays, beam shaping or steering, switchable holograms, and adaptive microdevices. Different from traditional rod-shaped LC compounds, BCN LC exhibits special V-shaped and electro-optical phenomena, usually along with obvious flexoelectricity, which is the cross-coupling between mechanical and electric properties [20].
Flexoelectricity is characterized by two coefficients, namely, e1 and e3, which denote coupling splay and bend elastic deformations, respectively. An induced electric polarization under an external applied electric is expressed as P = e1ndivn – e3n × curln, where n is the LC director pointing in the direction of the average molecular orientation. A linear relationship exists between electric field E and flexoelectric polarization P [21]. The flexoelectric instability, which is called flexodomains (FDs), can be observed in a planar cell filled with certain BCN LC. When the temperature is appropriate and the external applied voltage exceeds the threshold value, the FDs appear, which are examples of an electric field induced deformation solely due to flexoelectricity [18]. The FDs, which manifest themselves as a spatially periodic array of bright and dark stripes and runs parallel to the initial director alignment, was first observed by Vistin [22]. The emergence of FDs in BCN LC is related to the amplitude and frequency of external applied voltage and temperature [18]. In addition, the period of the pattern is on a micron scale and can be conveniently observed by a polarizing microscope (POM) [18–24]. The properties of the mixture of bent-core and rod-shaped LC molecules have been widely studied [25–31], such as elastic constants, viscosities, permittivity, conductivity, as well as optical properties such as light scattering [32] and electrically tunable color in the mixtures [33]. Understanding the properties of a pure BCN compound, such as LC mixtures, or improving its properties by doping other compounds or monomers, is important for future studies.
The properties or phenomena of a BCN compound in planar cells with varied cell conditions are different and have not been considered [18-19, 24]. Due to the patterns generated by a BCN LC in planar cells, which consist of FDs similar to optical grating, the periods of these optical gratings as the key characteristic of FDs were recorded and analyzed in this letter. The photomicrographs of FDs and their period with different voltages, temperatures, cell gaps, and pretilt angles were studied by a POM and light diffraction techniques. The period of the optical grating that changes linearly with temperature was found. We also detected the diffraction efficiency of this type of optical grating in which the diffraction efficiency could be increased by the increased cell gaps.
2. Material and liquid crystal cells
Most BCN LC materials are generally in a crystal state at room temperature. When the temperature increases at a certain phase transformation point, the BCN LC changes to nematic from crystal state. Unlike common rod-shaped nematic liquid crystal (NLC) compounds, BCN LC molecules have a special bending core. In this study, the BCN LC compound of 2,5-bis{4-[difluoro(4-heptylphenyl)methoxy]phenyl}-1,3,4-oxadiazole (7P-CF2O-ODBP) is used in our experiments. Figure 1 shows the BCN LC structural formula, whereas its phase transformation sequence is crystal, 77 °C smetic, 90.3 °C nematic, and 131.5 °C isotropic.
Fig. 1 Structural formula of BCN LC compound 2,5-bis{4-[difluoro(4-heptylphenyl)methoxy] phenyl}-1,3,4-oxadiazole (7P-CF2O-ODBP).
The BCN LC sample is filled into the planar cells with a simplified cross-sectional structure as shown in Fig. 2. The indium–tin-oxide (ITO) electrodes are covered with an insulated layer of SiO2 to prevent the injection of charge carriers through the electrode. The polyimide (PI) is used to provide planar alignment with rubbing direction along the x axis. The various types of PI, and rubbing direction and strength, cause the director of LC at the boundaries to have different orientations. A potential difference U may be applied on the bottom and top ITO electrodes. Thereby, the electric field direction is along the z direction.
Table 1 shows six planar cells with different cell gaps, friction torques, and pretilt angles. The six cells are numbered 1 to 6, respectively. The cell gap represents the distance between the upper and lower PIs or the thickness of the LC layer while the pretilt angle represents the prior orientation of the LC director at the boundaries. Friction torque represents the different rubbing strength on the upper and lower substrates.
Table 1. Planar cells with different cell gaps, friction torques, and pretilt angles at boundaries.
Most experiments were performed using an Olympus BX51 POM because the FDs of BCN LC can be obviously observed and manifests itself as a spatially periodic array of bright and dark stripes in the planar cell. The structure of FDs is similar to optical grating, which produces laser diffraction spots. The BCN LC sample was heated and temperature was kept within ± 0.01 °C by a Linkam LTS-350 precision hot stage regulated by a TMS 94 temperature controller. The applied voltage signals were produced by a precision signal generator and voltage amplifier. FDs of the BCN LC compound usually appear under DC voltage or ultra-low frequency AC voltage signal. Thus, the ultra-low frequency of square AC voltage is 0.01 Hz in the experiments.
3. Results and Discussion
3.1 Period of optical grating with different external applied voltages
Figures 3 and 4 show the photomicrograph of optical grating under the charge coupled device (CCD) on the eyepiece of the POM with two orthogonal polarizers at different voltages (U = 18, 24, 30, and 36 V) in the first and second cells, respectively. In addition, the liquid crystal is 15 °C below clearing temperature TNI (TNI = 130.2 °C, and was measured by precision hot stage which used in our experiment). The period of optical grating consists of the FDs, which are parallel to the initial director n, which obviously decreases with the increase in external applied voltage as shown in Figs. 3 and 4. Compared with the photomicrographs of optical grating in the first cell with those in the second cell at the same voltage, period Λ in the second cell is larger than that in the first.
Fig. 3 Photomicrograph of optical grating in the first cell changes with increased voltage U, where the temperature T-TNI = −15 °C; (A) and (P) refer to the analyzer and polarizer, respectively, and are orthogonal to each other. The initial director n indicates the rubbing direction of the PI. Sizes of all photomicrographs are 50 μm × 50 μm. (a) U = 18 V, Λ = 2.82 μm; (b) U = 24 V, Λ = 1.99 μm; (c) U = 30 V, Λ = 1.54 μm; and (d) U = 36 V, Λ = 1.24 μm.
Fig. 4 Photomicrograph of optical grating in the second cell changes with increased voltage U, where the temperature T-TNI = −15 °C; (A) and (P) refer to the analyzer and polarizer, respectively, and are orthogonal to each other; initial director n indicates the rubbing direction of the PI. Sizes of all photomicrographs are 50 μm × 50 μm; (a) U = 18 V, Λ = 5.41 μm; (b) U = 24 V, Λ = 4.13 μm; (c) U = 30 V, Λ = 3.05 μm; and (d) U = 36 V, Λ = 2.55 μm.
To disclose the relationship between period Λ and external applied voltage U, the period of optical grating Λ, which varies with external applied voltage U in the first and second cells, are presented in Fig. 5. Figure 5 shows that the temperature and cell gap can influence period Λ. These two factors are individually analyzed in a later section. In Fig. 5, if the temperature is certain, then period Λ decreases with the increase in external applied voltage, which has a reciprocal relationship with voltage. This relationship is maintained with the changed temperature and cell gap, which can be confirmed in Fig. 5. Note that this result is consistent with the number of FDs within a specified length, which has a linear function with voltage [19]. The difference is that the variation trend of period Λ and voltage U was verified to be invariable with the change in temperature and cell gap.
Fig. 5 Period of optical grating Λ varies with voltage U under different temperature T-TNI in first cell and second cell.
3.2 Period of optical grating with varying temperature
Figure 5 shows that period Λ increases with the increase in temperature under a certain external applied voltage. To further study the relationship between period Λ and temperature T-TNI, the morphology and period Λ, which vary with various temperature in the third cell, were recorded and measured using the POM. The morphology of the optical grating changes with different temperatures in the third cell, whose external applied voltage is 15 V, were recorded and the observations from the CCD are shown in Fig. 6. Period Λ increases with the increase in temperature T-TNI and the FDs run parallel to the initial director n.
Fig. 6 Photomicrograph of optical grating reflected in changed period Λ in the third cell changes with the increase in temperature T-TNI, where the external voltage is U = 15 V; (A) and (P) refer to the analyzer and polarizer, respectively, and are orthogonal to each other. The initial director n indicates the rubbing direction of PI. (a) T-TNI = −30 °C, Λ = 2.71 μm; (b) T-TNI = −24 °C, Λ = 2.78 μm; (c) T-TNI = −18 °C, Λ = 2.94 μm; (d) T-TNI = −12 °C, Λ = 3.13 μm.
To show the relationship between period and temperature, the period, which varies with the temperature under different external applied voltages was obtained. Figure 7 shows that period Λ linearly changes with various temperature T-TNI. Linear fittings were also conducted between period Λ and various temperature T-TNI. Figure 7 shows the fitted curves. Linear fitting provides
where A(θ, d) = (2.700 ± 0.001) × 10 − 2μm / °C indicates the slope of the Λ(T) curve, which may be related to the pretilt and cell gaps, whereas B(U) is a voltage-related variable which corresponds to critical period when the temperature is close to clearing temperature. This result explains the phenomenon shown in Fig. 5 in that period Λ decreases with the increase in voltage U. However, the Λ–U curves have no overlap within various temperature T-TNI .Fig. 7 Period of optical grating Λ in third cell varies with temperature T-TNI under different voltages U.
3.3 Period of optical grating with small difference in anchoring energy under strong anchoring
Friction torque on the upper and lower substrates of the planar cells from the first to the fourth has a small difference which is inevitable in quantity production, and all of them correspond to strong anchoring according to the experience in enterprise production. For the traditional rod-shaped NLC, the influence of the tiny difference on the optics through the LC cell is negligible. However, for the BCN LC with a special bending structure, whether the small difference in anchoring strength is reflected in the period of optical grating is important and needs to be experimentally confirmed. Thus, the curves of period Λ, which vary with voltage U, were compared between the first and fifth cells, and the second and sixth cells in different temperatures, as shown in Fig. 8. Although the difference in the friction torque on the two substrates in the first and fifth cells is relatively obvious, the curves of Λ–U almost overlap at temperature T-TNI = −25 °C. This situation does not change at temperature T-TNI = −10 °C, as shown in Fig. 8. This result is confirmed in the second and sixth cells at temperature T-TNI = −25 °C and −10 °C, respectively. Note that a certain difference exists in the curves of Λ–U given in the second and sixth cells at a relatively low voltage, which correspond to a small change in the period of optical grating. However, the difference between the two curves in different temperature is small and negligible. In other words, a small difference exists in friction torque on the upper and lower substrates of the planar cells from the first to the fourth. The effect of this difference on period Λ can be neglected because despite the difference in friction torque, all of these friction torques correspond to strong anchoring for LC molecules. Despite this condition, BCN LC molecules may be insensitive to such a difference in friction torque on two substrates because of the special bending structures in the structural formula of the BCN compound.
Fig. 8 Period of optical grating Λ in planar cell of numbers 1, 2, 5, and 6 varies with voltage U at different temperatures T-TNI.
3.4 Period of optical grating with different pretilt angles
The difference in pretilt angle at the boundaries of the planar cell can influence the alignment of director of BCN LC molecules at the substrates, as well as the rod-shaped NLC compound. The rod-shaped NLC molecules are evenly arranged on the substrate surface and alignment of director for different pretilt angles can be easily determined. For the BCN LC, the situation becomes more complicated. However, at the macro level, the influence of the pretilt angle can be reflected in the period of optical grating in the planar cells with the BCN compound. Thus, the first and third cells were selected to reveal this condition. In these two cells, the pretilt angle at the boundaries is different. The first and third cells correspond to the pretilt angles 1–2° and 5–7°, respectively. Figure 9 shows that the period Λ varies with voltage U with different pretilt angles at the boundaries. Period Λ in the third cell is larger than that in the first cell as shown in Fig. 9(a). This result does not change with temperature, which can be confirmed in Fig. 9(b). The pretilt angles at the boundaries in the third cell are larger than those in the first cell. Therefore, the larger the pretilt angle, the smaller the period of optical grating in the planar cell filled with the BCN compound.
Fig. 9 Period of optical grating Λ in the first and third cells varies with voltage U at different temperatures T-TNI: (a) T-TNI = −20 °C and (b) T-TNI = −10 °C.
To confirm the preceding conclusion, the same experiment was conducted on the second and fourth cells. Although the difference in period Λ between the second and fourth cells is small, period Λ in the second cell is larger than that in the fourth cell, as shown in Fig. 10. The small difference can be neglected probably because the difference in the period of optical grating in planar cells with different pretilt angles can be decreased by enlarging the cell gap or insensitivity of the BCN compound to the pretilt angle in the planar cell with a large cell gap.
Fig. 10 Period of optical grating in the second and fourth cells varies with voltage U at different temperatures T-TNI: (a) T-TNI = −20 °C and (b) T-TNI = −10 °C.
3.5 Period of optical grating with different cell gaps
Figure 5 show that period Λ and cell gap d under the same temperature and voltage satisfy the following expression:
where Λ1 and Λ2 are the period of optical grating in the first and second cells, respectively, d1 and d2 are the cell gap of the first and second cells, respectively; C21 may be a constant when these conditions of the planar cell, such as pretilt angle, and anchoring energy at the boundaries, are certain. The result of Λ2 / Λ1 = 2.080 ± 0.163 can be obtained from Fig. 5. Additionally, according to the data shown in Table 1 and Eq. (2), the constant C21 = 1.068 ± 0.084 is related to period Λ and cell gap d in the first and second cells.To verify the preceding result, period Λ was varied with voltage U at different temperatures and the ratio of the period in the third and fourth cells, whose pretilt angles are 5–7°, at the same temperature and voltage were calculated to obtain the constant C43 as shown in Fig. 11. Figures 11(a) and 11(b) show that period Λ decreased with the increase in voltage U in the third and fourth cells. The maximum and minimum of C21 are 1.152 and 0.984, respectively.
Fig. 11 Period Λ in the third and fourth cells and the constant C43, which reflects the relationship of the period Λ and cell gap d, varies with voltage U at different temperatures T-TNI: (a) T-TNI = −25 °C and (b) T-TNI = −15 °C.
According to Eq. (2), C43 reflects the relationship of period Λ and cell gap d in the third and fourth cells, and the values of C43 are between the maximum and minimum of C21 (i.e., 0.984 ≦ C43 ≦ 1.152). Equation (2) can be rewritten as follows:
According to Eq. (3), the constant C43 is directly related to the cell gap and indirectly to the pretilt angle. Period Λ is related to pretilt, which can be found in Eq. (1), whereas the changed pretilt has a small influence on period Λ, especially in cells with a large cell gap, which can be confirmed in Figs. 10 and 11. Therefore, changes between C43 and C21 are small and result in C43 values that are almost between the maximum and minimum of C21.
3.6 Period of optical grating varies with voltage verified by laser diffraction
According to the grating equation dΛ·sinθ = m·λ, where dΛ is the grating constant (or period of optical grating), θ is the diffraction angle, m is the diffraction order, and λ is the wavelength of incident light. The diffraction spots and morphology of optical grating in the BCN compound are related to the distance between the planar cell and collecting screen with a certain λ. The diffraction spots are different under varying voltages and temperatures because period Λ (i.e., diffraction constant dΛ) changes with voltage U and temperature T-TNI. In addition, the voltage has a large influence on period Λ, which can be confirmed in Figs. 3–4. Thus, the morphology of optical grating that varies with voltage was recorded by the CCD on the POM with two orthogonal polarizers. The corresponding diffraction patterns were obtained under a camera as shown in Fig. 12.
Fig. 12 Morphologies of optical grating reflected in the period Λ and diffraction spots in the fourth cell vary with voltage, where temperature T-TNI = −10 °C; (A) and (P) refer to the analyzer and polarizer, respectively, and are orthogonal to each other. The initial director n indicates the rubbing direction of the PI. The sizes of all photomicrographs are 50 μm × 50 μm. (a) U = 10 V, Λ = ∞; (b) U = 14 V, Λ = 6.71 μm; (c) U = 18 V, Λ = 5.67 μm; (d) U = 22 V, Λ = 4.72 μm; (e) U = 26 V, Λ = 3.80 μm; (f) U = 30 V, Λ = 3.25 μm; (g) U = 34 V, Λ = 2.86 μm; and (h) U = 38 V, Λ = 2.48 μm.
As shown, the left side of the picture shows the morphology in the fourth cell with a BCN compound and the right shows the corresponding diffraction spots. With the increase in voltage (10–14 V), optical grating consists of the FDs and diffraction spots gradually present, as shown in Figs. 12(a) and 12(b). Increasing the voltage continuously, period Λ gradually decreases (14–38 V). The first-order diffraction detaches from the zero-order diffraction. The second-order diffraction dots are present when the contrast between the bright and dark stripes is enhanced (14–18 V) and are out of the camera view ultimately with a decrease in period Λ.
Considering that the BCN compound is a negative liquid crystal, when the BCN LC molecules undergo an external electric field, the director becomes arranged in the direction perpendicular to the electric field. Flexoelectric effect and electric field is one correlation, whereas dielectric effect and electric field is a secondary correlation. Thus, when the BCN compound is under a small voltage, the flexoelectric effect is more obvious than the dielectric effect and FDs are observed by a POM. With the increase in voltage, the dielectric effect is more obvious than the flexoelectric effect, and then the director of additional BCN LC molecules turn to the vertical direction of the external electric field. Thus, the size of a single FD decreases perpendicular to the rubbing direction, which corresponds to the small period Λ.
The increased voltage U alters the diffraction efficiency η and the diffraction spots. Thus, the first-order diffraction intensity in the third and fourth cells was detected using a silicon detector. The first-order diffraction efficiency was calculated by the expression η1 = I1 / I0, where I1 and I0 are the first-order diffraction intensity at different voltages and total intensity at voltage U = 0. Figures 13(a) and 13(b) exhibit how the diffraction efficiency varies with voltage U in the third and fourth cells, respectively. The curves of the diffraction efficiency η1, which vary with voltage U, have an obvious right shift with the increase in temperature T-TNI (i.e., voltages corresponding to the maxima of diffraction efficiency increase with the increase in temperature T-TNI). This phenomenon may result from the enhanced BCN LC molecular thermodynamic movement with the increased temperature. To obtain a similar arrangement of molecules, the increased voltage is indispensable. Figures 13(a) and 13(b) show that the first and second maximum of the diffraction efficiency increase first, and then decrease with the increase in voltage. The situation of the third maximum, which is shown only in Fig. 13(b), is similar to the first and second maximum. The oscillating character of the diffraction efficiency versus voltage curves may be related to the helical twist structure of BCN LC molecules along the electric field [18, 19]. Thus, the appropriate temperature should be selected to obtain the higher diffraction efficiency in a BCN LC compound and a temperature of approximately T-TNI = −20 °C may be appropriate in this study.
Fig. 13 First-order diffraction efficiency η1 of the optical grating induced by the BCN LC in planar cells which varies with voltage U at different temperatures T-TNI. Fig: (a) third and (b) fourth cells.
The first maximum of diffraction efficiency in the fourth cell is larger than that in the third cell. Mutational points exist at a voltage of approximately 24 V in the third cell. These two phenomena may be due to the larger cell gap in the fourth cell than that in the third cell, while the cell gap of the third cell is too small for the transformation of FDs. Although certain numerical calculations of diffraction efficiency [34–36] have been applied to LC compounds, this scenario remains a major theoretical challenge in calculating FDs induced by the BCN LC in the planar cell thus far, because period Λ is related to voltage U and temperature T.
4. Conclusions
The period of the optical grating of FDs generated by the BCN LC 7P-CF2O-ODBP in different planar cells, which varies with externally applied voltage and temperature, has been presented in this letter. Period Λ decreases with the increase in voltage U at different temperatures. Period Λ can increase with the increase in temperature T-TNI and has a linear relationship with temperature T-TNI. This condition explains that the Λ-U curves do not overlap in the same cells at different temperatures. Cell gaps and pretilt angles at the boundaries of the planar cells can influence the period, as well as the voltage and temperature. The increase in pretilt angle can decrease the period of optical grating. The period changes with the cell gap more obviously than the pretilt angle at the cell boundaries; thus, changing the cell gaps will obviously change the range of the period.
Lastly, the variation of period Λ with voltage U was verified by laser diffraction at temperature T-TNI = −10 °C, and optical grating morphology can be continuously controlled by voltage U. This study shows that the BCN compounds have significant potential applications for adjustable period optical grating and can be improved by changing the cell gaps and pretilt angles. To our knowledge, there is no a physical model clearly explain the generation and change of FDs thus far. So, figuring out the generation and change mechanism of FDs can help explain the observed dependences in this study, which is worth researching thoroughly. Note that the nematic temperature of BCN compounds is high at approximately 100 °C. Thus, developing new BCN compounds whose nematic can remain at room temperature is extremely important.
Funding
National Natural Science Foundation of China (NSFC) (11374087, 11504080); Natural Science Foundation of Hebei Province of China (A2014202123, A2017202004); Research Project of Hebei Education Department (QN2014130); Key Subject Construction Project of Hebei Province University.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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