1. INTRODUCTION

Cholesteric liquid crystals (CLCs) with intrinsically periodical director orientation described by a molecular helix have long been a mainstream LC material for developing electrically tunable and/or switchable optically memorable devices owing to their unique optical features and bistable switching [1–5]. In general, CLC in a confined geometry can simply be categorized into three relaxed states—the Grandjean planar (P), focal conic (FC), and fingerprint (FP) textures—according to their helical structures and spectral properties. The P state reveals a uniform standing helix, with its helical axis normal to the substrate plane. A CLC in this state is attractive due to the existence of a reflection (Bragg) band in the spectrum, reflecting circularly polarized light with the same handedness as that of the helix. The FC and FP states are found to have randomly oriented molecular helices and a lying helix along the substrate planes, respectively. Unpolarized light passing through a CLC cell in the FC state would be scattered as a result of the mismatch in the refractive indices between domains, but it could be diffracted or transmitted in the FP state depending on the helical pitch length. Among the three CLC textures, both P and FC are well-known stable states in a planar-aligned cell. For conventional rod-like CLCs, because the molecular reorientation is induced via dielectric coupling between LC molecules and the external field, electrical switching between the two stable states is primarily determined by the field strength (or voltage amplitude) and is independent of the frequency $f$ as long as the dielectric function is a constant [6]. In a special case where dual-frequency LC with the frequency-revertible dielectric property is used as the host material of a CLC, Hsiao et al. proposed to realize direct two-way bistable switching by regulating $f$ of the applied AC voltage with a fixed amplitude, following the conditions of $f<f_c$ for P-to-FC switching and $f>f_c$ for FC-to-P switching, where $f_c$ is the crossover frequency of the dual-frequency LC [7]. The $f$-modulated textural switching has alternatively been realized by using negative-dielectric-anisotropy ($-\mathrm{\Delta }\epsilon$) CLCs doped with ion-rich additives, such as surfactants [8] and ionic chiral liquids [9,10]. In these cases, owing to the enhancement in charge carrier transport, the FC and FP states can be generated by low-frequency voltages via the electrohydrodynamic (EHD) effect, whereas a high-frequency voltage is applied to enable backward switching from either the FC or FP to P state simply by the dielectric effect. Alternatively, several methods aiming to obtain FP textures and control their pattern formation have been proposed based on the photoalignment technique [11–13].

On the other hand, the uniform lying helix (ULH) as a special class of FP states in short-pitch CLCs can be regarded as a uniaxial crystal with the helical axis being the optic axis oriented along a preferred direction in the substrate plane. Such a CLC texture has drawn much attention—from the application point of view—to next-generation displays in that the superior flexoelectric switching offers advantages of wide viewing and fast response, as proposed by Patel and Meyer in 1987 [14]. One of the shortcomings hindering extensive potential applications of the ULH is the difficulty in obtaining a stable and defect-free alignment. Even though various approaches on the basis of electric-field treatment [15–21] or surface alignment modification [22–26] have progressively been developed in attempt to tackle the aligning problem, rod-like CLCs known for the inherently low flexoelectric coefficient are incompatible with realizing ULH with significant flexoelectro-optical responses for desired applications. Incorporating unconventional bent-core LCs into rod-like LCs has been suggested as one of nonsynthesis pathways to the modification of the flexoelectric characteristics of the resultant binary LC mixtures [27–32]. As such, suppressed image flickering in the fringe-field-switching LC mode [27], accelerated response rate in the hybrid-aligned in-plane-switching LC mode [28], and promoted flexoelectro-optical switching in the CLC [29,30] have successively been demonstrated in rod-like/bent-core CLC systems. Moreover, Outram and Elston clarified theoretically the effect of flexoelectric polarization on the dielectric spectrum of a $-\mathrm{\Delta }\epsilon$ rod-like/bent-core CLC mixture and suggested an approach to the formation of ULH alignment by the in-plane electric field [31] and the $f$-modulated textural switching between the P and ULH states by the vertical electric field via the voltage-induced EHD and dielectric effects [32].

In this study, we focus on a binary CLC, comprising $+\mathrm{\Delta }\epsilon$ bent-core and rod-like LCs, in a planar-aligned cell and reveal unusual textural and electro-optical features, having no such counterpart in any known rod-like systems. On account of the considerable increase in flexoelectricity originating from the inherently strong flexoelectric efficacy in the bent-core LC, it is found that the voltage required to sustain the CLC in the homeotropic (H) state becomes a function of the frequency and that the transmittance in the relaxed states after treated by voltage is strongly frequency-dependent, resembling the profile of dielectric dispersion featuring the relaxation in the frequency regime where the flexoelectric polarization is induced. More attractively, apart from the known stable P and FC states, the ULH state with satisfied stability can be readily obtained in the proposed binary CLC by sufficient switching voltage (${\mathit{V}}_{\mathrm{ULH}}<\mathit{V}<{\mathit{V}}_{\mathrm{H}}$) in the frequency regime $f<f_{\mathrm{ULH}}$, where the flexoelectric contribution to the measured dielectric permittivity is substantial.

2. EXPERIMENT

The LC host employed in this study was a binary LC mixture consisting of 55 wt% rod-like LC E7 (Daily Polymer, Taiwan) and 45 wt% bent-core LC dimer $1^{\prime \prime }$, $7^{\prime \prime }$-bis(4-cyanobiphenyl-$4^{\prime }$-yl) heptane (CB7CB) produced by HCCH, China. E7 is a well-known eutectic nematic with clearing temperature of $T_c=58.0°\mathrm{C}$ and positive dielectric anisotropy of $\mathrm{\Delta }\epsilon =14$ (measured at $f=1\mathrm{kHz}$ and temperature of $T=20°\mathrm{C}$). On the other hand, CB7CB as a bent-shaped dimer with a bend angle of $\sim 122°$ is composed of two cyanobiphenyl mesogenic groups, linked by a heptane spacer [$-{({\mathrm{CH}}_2)}_7-$] in its trans conformation [33]. CB7CB in the nematic (N) phase exhibits a relatively low value of dielectric anisotropy ($\mathrm{\Delta }\epsilon =1–2$), but it is best known for the giant flexoelectricity with a flexoelastic coefficient of $e/K=3.67\mathrm{C}·{\mathrm{N}}^{-1}·{\mathrm{m}}^{-1}$ [34]. As identified by means of dielectric spectroscopy and polarizing optical microscopy in this lab, the phase sequence of CB7CB is Iso–115°C–N–101°C–N_TB–70°C–Cr, where Iso denotes the isotropic phase, ${\mathrm{N}}_{\mathrm{TB}}$ is the twist-bend nematic phase exhibiting macroscopically chiral heliconical orientational order on the ten-nanometer scale, and Cr stands for the crystalline phase. Two CLCs—CB7CB-doped CLC and undoped counterpart—were prepared by dispersing 3.5 wt% right-handed chiral agent R5011 ($c_{\mathrm{R}5011}=3.6\mathrm{wt}\%$) into the E7/CB7CB mixture and E7 individually. Each CLC was stirred at the temperature of 120°C to ensure homogeneous blending and then injected into a planar-parallel aligned empty cell (Chipset Technology Co., Ltd.) with cell gap of $d=5.0\pm 0.5\mathrm{\mu m}$ and electrode area of $A=0.25{\mathrm{cm}}^2$. The aligning material used for promoting homogeneous LC alignment is DL-3260, and the rubbing directions treated on the top and bottom substrates are parallel (i.e., the rubbing angle is 0°). Besides, no other conditions were applied to the alignment layer. The phase sequences of thus-made CB7CB-doped and undoped CLCs are Iso–62°C–CLC–16°C–TGBA–13°C–SmA and Iso–54°C–CLC, respectively. The helical pitch length ($P$) and the Bragg central wavelength (${\lambda }_c$) of the undoped CLC cell in the Grandjean planar state (with the helical twisting power of HTP $\sim 115{\mathrm{\mu m}}^{-1}$ of R5011 in E7) are $P=241\mathrm{nm}$ and ${\lambda }_c=394\mathrm{nm}$, as calculated according to the relations of $P={(\mathrm{HTP}\times c_{\mathrm{R}5011})}^{-1}$ and ${\lambda }_c=n\times P$, where $n$ is the average of the extraordinary ($n_e=1.74$) and ordinary ($n_o=1.52$) refractive indices of E7. For the CB7CB-doped CLC, it is indicated, based on our pretested experiments, that the HTP of R5011 in the E7/CB7CB mixture varies linearly from $115{\mathrm{\mu m}}^{-1}$ to $137{\mathrm{\mu m}}^{-1}$ as the CB7CB concentration ($c_{\mathrm{CB}7\mathrm{CB}}$) increases from 0 wt% to 40 wt%. The observed linear correlation between HTP of the chiral dopant and $c_{\mathrm{CB}7\mathrm{CB}}$ ($\le 40\mathrm{wt}\%$) in CB7CB-doped CLC is in good agreement with that reported by Varanytsia and Chien [29]. However, when $c_{\mathrm{CB}7\mathrm{CB}}$ is as high as 45 wt%, we found that the HTP of R5011 at $T=25°\mathrm{C}$ reduces to $121{\mathrm{\mu m}}^{-1}$. This could be explained by the existence of SmA phase at a lowered temperature below the CLC phase, so that the helical pitch and the central wavelength of the bandgap of the 45 wt% CB7CB-doped CLC become dependent of the temperature. As a result, the value of P (${\lambda }_c$) of the 45 wt% CB7CB-doped CLC at the investigated temperature (i.e., $T=25°\mathrm{C}$) is 230 nm (374 nm), which is only 11 nm (20 nm) shorter than that of undoped CLC.

All measurements were performed at an ambient temperature of $\sim 25°\mathrm{C}$. The square-wave voltage applied across the cell thickness was provided by an arbitrary function generator (Tektronix AFG-3022B) in conjunction with an amplifier (TREK Model 603). The voltage-dependent transmission ($\mathit{V}-{\mathit{T}}_r$) curves were obtained by placing the CLC cell between a photodetector and a He–Ne laser source emitting at 632.8 nm in wavelength. The types of CLC textures and their uniformities were examined by means of textural observation with a polarizing optical microscope (BX51, Olympus) in the transmission mode and by measurement of transmission spectra using a high-speed fiber-optic spectrometer (Ocean Optics HR2000+) along with a halogen light source (Ocean Optics HL2000). The dielectric spectra of CLC cells in the frequency range between 20 Hz and 30 kHz were studied using a precision LCR meter (Agilent-E4980A). The probe voltage in the sinusoidal waveform was $0.5{\mathit{V}}_{\mathrm{rms}}$, which is too low to reorient LC molecules.

3. RESULTS AND DISCUSSION

Figure 1 shows the changes in optical transmittance ${\mathit{T}}_r$ with increasing voltage V of the undoped and CB7CB-doped CLC cells. Here, no polarizers were used, and the two cells were initially in the P state prior to the measurements. One can see in Fig. 1(a) that the $\mathit{V}-{\mathit{T}}_r$ curves of the undoped CLC driven by voltages were virtually identical at three different frequencies (i.e., $f=100\mathrm{Hz}$, 1 kHz, and 5 kHz). Such $f$ independence of the electro-optical responses obtained in the undoped CLC by the dielectric coupling of LC molecules with V would be attributable to the invariant dielectric function at these frequencies, which will be confirmed later by means of dielectric spectroscopy in Fig. 3. The voltages for retaining the undoped CLC cell, in particular CLC states, and for inducing their textural transformations can be acquired from the data as given in Fig. 1(a) based on the optical transmittance. The voltages $\mathit{V}<{\mathit{V}}_{\mathrm{P}-\mathrm{FC}}\sim 5{\mathit{V}}_{\mathrm{rms}}$ were too low to alter the helical structure, and the texture of the cell remained in the P state. Because the wavelength of the probe light (i.e., He–Ne laser) at 632.8 nm is far from the reflection (Bragg) band of the CLC, the cell in the P state was transparent with $T_r\sim 80\%$. When V increased from ${\mathit{V}}_{\mathrm{P}-\mathrm{FC}}\sim 5{\mathit{V}}_{\mathrm{rms}}$ to ${\mathit{V}}_{\mathrm{FC}}\sim 12.5{\mathit{V}}_{\mathrm{rms}}$ (i.e., ${\mathit{V}}_{\mathrm{P}-\mathrm{FC}}<\mathit{V}<{\mathit{V}}_{\mathrm{FC}}$), the CLC structure transformed to comprise randomly oriented helical axes, converting the texture from the P into the FC state and leading to a dramatic decrease in transmittance from 80% to as low as $\sim 10\%$ due to light scattering. The FC state was then preserved with a varying degree of optical transmittance, ranging from 10% to 20%, in the voltage range between ${\mathit{V}}_{\mathrm{FC}}\sim 12.5{\mathit{V}}_{\mathrm{rms}}$ and ${\mathit{V}}_{\mathrm{FC}-\mathrm{H}}\sim 42.5{\mathit{V}}_{\mathrm{rms}}$ (${\mathit{V}}_{\mathrm{FC}}<\mathit{V}<{\mathit{V}}_{\mathrm{FC}-\mathrm{H}}$). Once the voltage went higher than ${\mathit{V}}_{\mathrm{FC}-\mathrm{H}}$, the CLC helix became gradually unwound with increasing V and finally reached to the H state at $\mathit{V}>{\mathit{V}}_{\mathrm{H}}=50{\mathit{V}}_{\mathrm{rms}}$, resulting in high $T_r\sim 80\%$ to be comparable to that of the P state. For the CB7CB-containing CLC [Fig. 1(b)], the $\mathit{V}-T_r$ curve at $f=5\mathrm{kHz}$ resembled that of the undoped counterpart with ${\mathit{V}}_{\mathrm{FC}}\sim 12.5{\mathit{V}}_{\mathrm{rms}}$ and ${\mathit{V}}_{\mathrm{H}}=50{\mathit{V}}_{\mathrm{rms}}$, implying that the textural switching at this frequency was solely determined by the voltage-induced dielectric effect. When $f$ decreased from 5 kHz to 1 kHz and to 100 Hz, the voltage interval between $V_{\mathrm{FC}-\mathrm{H}}$ and ${\mathit{V}}_{\mathrm{H}}$ (i.e., ${\mathit{V}}_{\mathrm{FC}-\mathrm{H}}<\mathit{V}<{\mathit{V}}_{\mathrm{H}}$) for the FC-to-H transition was broadened and, in the meantime, the magnitude of ${\mathit{V}}_{\mathrm{H}}$ increased from $50{\mathit{V}}_{\mathrm{rms}}$ at $f=5\mathrm{kHz}$ to $65{\mathit{V}}_{\mathrm{rms}}$ at $f=1\mathrm{kHz}$ and further to $105{\mathit{V}}_{\mathrm{rms}}$ at $f=100\mathrm{Hz}$. Notably, an additional texture known as the ULH with nearly commensurate transmittance ($T_r\sim 75\%$) with that of the H state was generated by voltage between ${\mathit{V}}_{\mathrm{ULH}}=82.5{\mathit{V}}_{\mathrm{rms}}$ and ${\mathit{V}}_{\mathrm{H}}=105{\mathit{V}}_{\mathrm{rms}}$ (${\mathit{V}}_{\mathrm{ULH}}<\mathit{V}<{\mathit{V}}_{\mathrm{H}}$) at $f=100\mathrm{Hz}$ until transiting to the H state at a higher sustaining voltage ($\mathit{V}>105{\mathit{V}}_{\mathrm{rms}}$), as shown in Fig. 1(b). By further taking the optical image of the cell (under crossed polarizers) upon the application of 100 Hz voltage at $\mathit{V}=90{\mathit{V}}_{\mathrm{rms}}$ (i.e., ${\mathit{V}}_{\mathrm{ULH}}<\mathit{V}<{\mathit{V}}_{\mathrm{H}}$) as a proof [inset of Fig. 1(b)], the bright appearance of this texture without defects directly confirms the voltage induction of well-aligned ULH state. Supported by some of the pioneer works, the optical transparency of a CLC in a defect-free ULH state can be as high as that in the H state and in the P state at wavelengths outside the reflection band [17,19,21,35]. Previously, the induction of ULH by low-frequency voltages has been demonstrated via the EHD effect in conventional rod-like CLCs [17,18] and a $-\mathrm{\Delta }\epsilon$ CLC mixture [31]. However, we have established in our previous work that the EHD-induced ULH under the voltage-increasing process occurs in the voltage region of ${\mathit{V}}_{\mathrm{FC}}<\mathit{V}<{\mathit{V}}_{\mathrm{FC}-\mathrm{H}}$ rather than the range of ${\mathit{V}}_{\mathrm{FC}-\mathrm{H}}<\mathit{V}<{\mathit{V}}_{\mathrm{H}}$ in the present study [Fig. 1(b)] and that the magnitude of ${\mathit{V}}_{\mathrm{H}}$ is unaffected by EHD [35]. We therefore believe that the results given by Fig. 1(b) are irrelevant to EHD; instead, they can be attributed to the original dielectric effect together with the promoted flexoelectric effect because of the presence of CB7CB molecules in the composite. This insight can be further confirmed by the $f$ dependence of the CLC texture generations and by dielectric spectroscopy as to be discussed.

 

Fig. 1. Voltage dependence of transmission curves ($\mathit{V}-T_r$) of (a) undoped and (b) CB7CB-doped CLC cells in the increasing voltage process. Here, the unpolarized probe beam was derived from a He–Ne laser source operating at 632.8 nm in wavelength. Inset in (b) is an optical texture (under crossed polarizers) indicating the induction of ULH alignment in the CB7CB-doped cell by 100 Hz voltage of $90{\mathit{V}}_{\mathrm{rms}}$. The abbreviations of P, A, and R denote the transmission axes of the polarizer, analyzer, and the rubbing direction, respectively.

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Deduced from $\mathit{V}-T_r$ curves as measured with increasing V at given frequencies, Fig. 2 depicts the $f$-dependent critical voltages ${\mathit{V}}_{\mathrm{H}}$, ${\mathit{V}}_{\mathrm{ULH}}$, and ${\mathit{V}}_{\mathrm{FC}}$ as required to obtain the corresponding CLC configurations. In considering the coexistence of dielectric and flexoelectric effects manifested by the electric field, ${\mathit{V}}_{\mathrm{H}}$ for total unwinding of the helix in a CLC with $\mathrm{\Delta }\epsilon >0$ can be expressed by [14]

 

Fig. 2. Frequency-dependent critical voltages of ${\mathit{V}}_{\mathrm{H}}$, ${\mathit{V}}_{\mathrm{ULH}}$, and ${\mathit{V}}_{\mathrm{FC}}$ for the inductions of the H, ULH, and FC states in the binary (i.e., CB7CB-doped) CLC cell.

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The values of $f_{\mathrm{di}}$ and $f_{\text{flexo}}$ were further determined in accordance with the flexoelectric contribution to the dielectric permittivity data. Figure 3(a) illustrates the real-part dielectric spectra ${\epsilon }^{\prime }(f)$ of the CB7CB-doped CLC cell in the P, FC, and ULH states. Based on the results of Fig. 2, the stable FC and ULH states (at $\mathit{V}=0{\mathit{V}}_{\mathrm{rms}}$) were obtained by treating the cell in the initial P state with voltages of $\mathit{V}=15{\mathit{V}}_{\mathrm{rms}}$ at $f=5\mathrm{kHz}$ and $\mathit{V}=90{\mathit{V}}_{\mathrm{rms}}$ at $f=100\mathrm{Hz}$, respectively, followed by decreasing the voltages gradually to zero. In the case of the cell in the P state, the flexoelectric polarization can hardly be induced because of the perfect twist deformation with helical axis parallel to the electric field [36]. As a result, the spectrum of the P state showing nearly an invariant value of ${\epsilon }^{\prime }\sim 6.7$ in the frequency range between 20 Hz and 30 kHz is undoubtedly dominated by the orientation of LC molecules, which corresponds to the perpendicular component of the dielectric permittivity (${\epsilon }_{\perp }$). Because the dielectric relaxation in association with the space-charge polarization was not observed in the dielectric spectrum of the P state, the contribution of the EHD effect to the generation of ULH alignment can be ruled out in the E7/CB7CB cell [19]. When the texture was obtained in the FC state with randomly oriented helical axes or in the ULH state with a unidirectional axis parallel to the substrate plane, we monitored a clear dielectric relaxation associated with the flexoelectric polarization in the given spectrum. This finding is explained by the relaxation of splay-bend deformation of LC molecules thanks to the incorporation of CB7CB, which induced a significant flexoelectric coupling with the probe AC field. According to Outram and Elston, the relaxation behavior contributed by flexoelectric polarization in the ${\epsilon }^{\prime }(f)$ spectrum can be characterized by the following equation [31]:

 

Fig. 3. Real-part dielectric spectra ${\epsilon }^{\prime }(f)$ of (a) the CB7CB-doped cell in the P, FC, and ULH states and (b) the undoped counterpart in the P and FC states. The simulated ${\epsilon }^{\prime }(f)$ curves [solid lines in (a)] of the CB7CB-doped cell are plotted with fitting parameters of $f_{\mathrm{R}}=703.1\mathrm{Hz}$, ${\epsilon }_{\mathrm{H}}=9.9$, and ${\epsilon }_{\text{flexo}}=2.1$ for the ULH state and $f_{\mathrm{R}}=702.6\mathrm{Hz}$, ${\epsilon }_{\mathrm{H}}=10.5$, and ${\epsilon }_{\text{flexo}}=2.6$ for the FC state according to Eq. (2).

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The $f$-dependent textural formation in the CB7CB-doped CLC could alternatively be found when the voltage decreased gradually from $\mathit{V}>{\mathit{V}}_{\mathrm{H}}$ to $\mathit{V}=0{\mathit{V}}_{\mathrm{rms}}$. Figures 4(a) and 4(b) show, respectively, the optical spectra (without any polarizer) and the corresponding optical textures (under crossed polarizers) in the field-off state of the CB7CB-doped CLC after treated with a decreasing voltage from $\mathit{V}=120{\mathit{V}}_{\mathrm{rms}}(>{\mathit{V}}_{\mathrm{H}})$ to $\mathit{V}=0{\mathit{V}}_{\mathrm{rms}}$ at designated frequencies. It should be noted that the frequencies labeled in Fig. 4 correspond to the frequency conditions used for the voltage-decreasing process. For the textural observations, the angle $\alpha$ between the rubbing direction (or the helical axis in the ULH state) of the cell and the transmission axis of either of crossed polarizers was chosen at either 0° or 45°. For the undoped CLC with frequency-independent electro-optical responses used in this study, the texture has been proven to be stabilized in the FC state with low transmittance, as the applied voltage drops continuously from ${\mathit{V}}_{\mathrm{H}}$ to zero and the optical texture and its optical transparency are invariant with varying voltage frequency. Interestingly, when performing the same decreasing-voltage treatment to the CB7CB-doped CLC, we found the $f$-dependent variation in transmittance in the field-off state (i.e., $\mathit{V}=0{\mathit{V}}_{\mathrm{rms}}$). Here, the voltage-decreasing process is particularly critical for the uniformity and stability of the ULH texture. When the voltage at $f<f_{\text{flexo}}$ is decreased from ${\mathit{V}}_{\mathrm{H}}$ to an arbitrary voltage amplitude in the regime of ${\mathit{V}}_{\mathrm{ULH}}<\mathit{V}<{\mathit{V}}_{\mathrm{H}}$, the ULH texture can be electrically induced with the helical structure being partially unwound and deformed through the voltage induction of dielectric and flexoelectric effects. Because the FC texture is inherently more stable than the ULH in a planar-aligned CLC cell, switching off the voltage immediately at this time could lead to the change of CLC domains from ULH to FC by instant variation in helical structure. In contrast, decreasing the voltage slowly (from $V_{\mathrm{ULH}}$ to zero) is perspective for supporting uniform reorientation of ULH helices and suppressing the generation of FC domains with decreasing voltage. For the results shown in Fig. 4, the ramping rate of decreasing voltage was set identically to be $2.5{\mathit{V}}_{\mathrm{rms}}$ for 10 s. This parameter ensured the process to be slow enough for obtaining stable and uniform ULH alignment in the field-off state. As shown in Fig. 4(a), the transmittance of the binary CLC cell for the case of $f=100\mathrm{Hz}$ was comparable with that of the voltage-sustained H state, but it decreased as the frequency rose, reaching the minimum and becoming unchanged at $f=5\mathrm{kHz}$ and beyond. Further observations of the optical textures ascertained the state to be FC after the 5 kHz voltage treatment whereas the texture under the condition of $f=100\mathrm{Hz}$ to be the ULH state [Fig. 4(b)]. Note that the uniaxial property of the ULH state generated the excellent dark and bright appearances at $\alpha =0°$ and $\alpha =45°$, respectively. Besides, the contrast between the textures at $\alpha =45°$ and $\alpha =0°$ decreased as the frequency of the perturbing voltage increased from 100 Hz to 5 kHz, indicating that the textures under the conditions of $f=500\mathrm{Hz}$ and $f=1\mathrm{kHz}$ would be constituted by different degrees of combined ULH and FC domains. This implies that, in addition to the known FC state, other possible CLC textures with distinct transmission levels can be generated in the proposed CB7CB-doped CLC by varying the frequency for decreasing-voltage treatment. Except for the homeotropic state, Fig. 4(a) unambiguously shows growing transmittance with increasing wavelength in various CLC states of the CB7CB-doped CLC. This transmittance-wavelength characteristic, which is significant in the FC and FC+ULH states but becomes weakened in the ULH state, can be explained by the Rayleigh scattering due to the existence of CLC helices in multidomains with sizes much smaller than visible light wavelengths [10,20,35].

 

Fig. 4. (a) Transmission spectra (without polarizers) and (b) polarized optical micrographs (under crossed polarizers) of the CB7CB-doped CLC at null voltage after treatments with gradually decreasing voltages from $\mathit{V}=120{\mathit{V}}_{\mathrm{rms}}$ to $0{\mathit{V}}_{\mathrm{rms}}$ at various frequencies. The arrows indicate the transmission axes of the polarizer (P) and analyzer (A) as well as the helical axis (H.A.).

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To examine more clearly the effect of the frequency on the optical characteristics as well as the textural formations in the binary CLC after treating with decreasing voltages, Fig. 5 delineates the $f$-dependent transmission curves ($f-T_r$) at various wavelengths (i.e., 500, 600, and 700 nm) of the cell in the field-off state. The dispersion of the transmittance in the spectrum resembles that of ${\mathit{V}}_{\mathrm{H}}(f)$ in Fig. 2 and the behavior of ${\epsilon }^{\prime }$ ($f$) in Fig. 3. Thus, we adopt the two critical frequencies (i.e., $f_{\text{flexo}}=230\mathrm{Hz}$ and $f_{\mathrm{di}}=2\mathrm{kHz}$), defined on the basis of the results of Fig. 3, to divide the frequency regimes in Fig. 5 as well. In the low-frequency region, where $f<f_{\text{flexo}}\sim 230\mathrm{Hz}$, the transmittances were relatively high ($\sim 75\%$ at 700 nm and $\sim 70\%$ at 500 nm) and nearly invariant at a given wavelength. The stable texture in this frequency region (region I) was undoubtedly the ULH state, formed in the voltage regime between ${\mathit{V}}_{\mathrm{H}}$ and $V_{\mathrm{ULH}}$ during the voltage-decreasing process. At frequencies higher than $f_{\mathrm{di}}=2\mathrm{kHz}$ ($f>f_{\mathrm{di}}$), where the flexoelectric switching was suppressed, one can also find virtually constant transmittances of $\sim 27\%$ at 700 nm and $\sim 12\%$ at 500 nm in frequency region III due to the generation of the FC texture by the dielectric effect. When the frequency was between $f_{\text{flexo}}$ and $f_{\mathrm{di}}$ ($f_{\text{flexo}}<f<f_{\mathrm{di}}$), the texture of the cell exhibited a varying extent of the combination of FC and ULH domains, permitting the transmittance to be tunable by adjusting the frequency condition in region II. In the ULH/FC mixed state, the higher the frequency, the lower the transmittance.

 

Fig. 5. Frequency-dependent transmission curves of the CB7CB-doped CLC at three wavelengths, acquired from transmission spectra of the cell in the unperturbed state after stimulations by gradually decreasing AC voltages from $\mathit{V}=120{\mathit{V}}_{\mathrm{rms}}$ to $0{\mathit{V}}_{\mathrm{rms}}$.

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Referring to the above-mentioned results—Figs. 2 and 5 in particular—one can establish a comprehensive scheme for the tristable switching and for light modulation by voltage frequency in the proposed CB7CB-doped CLC, as illustrated in Fig. 6. Strictly speaking, the P texture is the most stable state (i.e., the ground state) due to the implemented planar surface alignment, whereas the FC and ULH textures are optically metastable, having varying degrees of stability. In this switching scheme, the amplitude of the voltage applied across the cell thickness is fixed at $\mathit{V}=50{\mathit{V}}_{\mathrm{rms}}$, which is equal to the magnitude of ${\mathit{V}}_{\mathrm{H}}$ in frequency region III (i.e., $f>f_{\mathrm{di}}$) (Fig. 2). The texture can be switched from either the stable P or FC state to the ULH state by setting the frequency at 20 Hz (${\mathit{V}}_{\mathrm{ULH}}=45{\mathit{V}}_{\mathrm{rms}}<\mathit{V}=50{\mathit{V}}_{\mathrm{rms}}<{\mathit{V}}_{\mathrm{H}}=105{\mathit{V}}_{\mathrm{rms}}$ at $f=20\mathrm{Hz}$ in region I). The ULH as the optical transparent state with well-aligned helical structure can be stably obtained for more than one day after the removal of the voltage directly from $\mathit{V}=50{\mathit{V}}_{\mathrm{rms}}$ or by decreasing the voltage gradually from $\mathit{V}=50{\mathit{V}}_{\mathrm{rms}}$ to $\mathit{V}=0$. The uniformity and the stability of the ULH structure depend on the concentration of CB7CB and the chiral additive in the binary CLC. The contents of CB7CB and R5011 higher than 45 wt% and 3 wt%, respectively, in the binary CLC are the optimal values for demonstrating defect-free and stable ULH alignment. Reversible switching from the ULH state to P as the light-reflection state and to FC as the light-scattering state with comparably low transmission can be attained by varying the voltage frequency to 5 kHz ($\mathit{V}=50{\mathit{V}}_{\mathrm{rms}}={\mathit{V}}_{\mathrm{H}}$ at $f=5\mathrm{kHz}$ in region III) to first sustain the cell in the H state, followed by turning off the voltage directly and relaxing the voltage gradually to zero, respectively. In addition, when the voltage decreases continuously from $\mathit{V}=50{\mathit{V}}_{\mathrm{rms}}$ to $\mathit{V}=0$, it is specified for the binary CLC that the intensity of transmitted light can be tunable by varying the voltage frequency in between $f_{\text{flexo}}$ and $f_{\mathrm{di}}$ in region II ($f_{\text{flexo}}<f<f_{\mathrm{di}}$) due to the formation of various combinations of mixed FC and ULH polydomains.

 

Fig. 6. Schematic of the driving scheme for the tristable switching among the P, FC, and ULH states and, in turn, the modulation in transmitted light intensity in the stable ULH/FC coexisting state of the binary E7/CB7CB CLC. The applied voltage is fixed at $50{\mathit{V}}_{\mathrm{rms}}$, and the frequency is variable (refer to Figs. 2 and 5).

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4. CONCLUSIONS

Conventional rod-like CLC cells with planar surface alignment usually have the Grandjean planar (P) and the focal conic (FC) as two stable states, and electrical switching between them typically relies on adjusting the voltage amplitude via the dielectric effect. By incorporating 45 wt% bent-core LC dimer CB7CB into the rod-like CLC (E7 as the host in this study), we have proposed an electrical pathway to the generation of uniform lying helix (ULH) texture as the third stable state and demonstrated atypical frequency-modulated textural switching as well as optical properties in this binary CLC mixture based on the frequency-independent dielectric effect as well as the frequency-modulated flexoelectric effect. Owing to the promotion of the flexoelectricity significantly contributed by CB7CB, the dielectric contribution derived from the flexoelectric polarization has been revealed in the real-part dielectric spectra [${\epsilon }^{\prime }(f)$] of the cell in both the FC and the ULH states. On this basis of the flexoelectric contribution to the dielectric permittivity, two critical frequencies of $f_{\text{flexo}}=230\mathrm{Hz}$ and $f_{\mathrm{di}}=2\mathrm{kHz}$ were then defined to specify the virtually invariant and high flexoelectric strength in region I ($f<f_{\text{flexo}}$), reduced flexoelectric strength with rising frequency in region II ($f_{\text{flexo}}<f<f_{\mathrm{di}}$), and suppressed flexoelectric strength in region III ($f>f_{\mathrm{di}}$). Accordingly, we have clarified the minimal voltage for holding the CB7CB-containing CLC in the H state, identically to that of the undoped CLC counterpart, was ${\mathit{V}}_{\mathrm{H}}=50{\mathit{V}}_{\mathrm{rms}}$ in region III, but it increased with decreasing frequency in region II and finally reached a saturated value of $V_{\mathrm{H}}=105V_{\mathrm{rms}}$ in region I. Moreover, the ULH alignment can specifically be obtained in region I ($f<f_{\text{flexo}}$) by treating the CB7CB-doped CLC directly with ${\mathit{V}}_{\mathrm{ULH}}$ or by decreasing the voltage continuously from ${\mathit{V}}_{\mathrm{H}}$ to zero. This was explained by the coupling of the electric field with both dielectric effect and the significantly increased flexoelectric effect, reorganizing CLC helices to lie in the plane perpendicular to the field direction with unidirectional helical axis along the rubbing direction. Attractively, when the cell was first sustained in the H state by a voltage at an arbitrary frequency in region II, our results (Fig. 4) suggested that various metastable states consisting of varying degrees of mixed FC and ULH domains with distinct optical transmissions can be generated by modulating the frequency of the stimulating voltage. As a consequence, the proposed CB7CB-doped CLC with features hardly found in existing rod-like CLC, including the optical tristability, frequency-modulated textural formation, and optical transparency, is of potential for optical and photonic applications as energy-saving devices such as light shutters, light-intensity modulators, and privacy windows. Further attention along this line will be paid to the flexoelectro-optical responses of the CB7CB-doped CLC in the ULH and their frequency dependency with the aim toward the development of fast optoelectronic devices.