The electro-optic Kerr effect in cubic blue phase liquid crystals comprises two components with different characteristic response times: one attributed to the primary (purely electro-optic) effect and another attributed to the secondary, or indirect (photoelastic) effect. Through simultaneous measurement of the polarized reflection spectrum and transmitted phase, we show that the contribution of the secondary electro-optic effect can be as large as 20% of the total change in refractive index, and that it is suppressed in the polymer-stabilized blue phase. Our results show the importance of stabilizing the lattice structure to realize blue-phase devices with fast response.
© 2014 Optical Society of America
Liquid crystalline cholesteric blue phases (BPs) typically appear between the cholesteric phase and the isotropic liquid in a chiral liquid crystal (LC) [1–4]. The cubic orientational order exhibited by BPs I and II makes them interesting both as subjects of soft matter physics [5–8] and as candidate materials for next-generation electro-optic applications [9–12]. It is known that when a field is applied, both the refractive index and Bragg reflection wavelength change approximately with a quadratic dependence on the electric field, and using either of these properties, applications such as tunable lenses, reflectors and flat panel displays have been proposed.
The two effects observed in BPs in response to electric fields, namely, refractive index modulation and shift of Bragg reflection wavelength, are a result of the same phenomenon, i.e., molecular reorientation, manifest at different temporal and spatial scales. Figure 1 illustrates the structure of BP I and the two effects observed. The change in refractive index is mainly due to the local reorientation of shortly correlated nematic domains existing in the BP lattice, and because of the small helical pitch, the effect occurs at submillisecond response times, which is faster than the reorientation dynamics in nematics by about an order of magnitude [13–16]. On the other hand, the shift of the Bragg reflection wavelength is a result of the electrostriction of the lattice, which occurs so as to relax the increase in the elastic energy gained by the local reorientation of the LC molecules. Because the effect involves as many as 107 LC molecules, it has a characteristic time much longer than that of the Kerr effect, ca. several milliseconds to even seconds or more. [17, 18]
In crystal optics, the electro-optic effect (in which an electric field causes a change in the refractive index) is known to comprise two components. In addition to the purely electro-optic component which is referred to as the primary effect, there exists a secondary or a photoelastic effect, in which the refractive index changes as a result of the change in crystal symmetry . While the crystal structure of BPs is much larger than typical crystals composed of atoms, they are not an exception and have been shown theoretically to contain both of these effects . Experimentally, however, many studies to date have not distinguished the two electro-optic effects. Many studies which measure the dynamic response of the Kerr effect have focused on the primary response, which occurs with sub-ms response times, and while some studies have commented on the presence of a slow response in the refractive index [14, 21, 22], quantitative measurements had not been performed. Moreover, in many cases the two effects have been measured independently: the Kerr effect has often been investigated by applying a horizontal electric field and measuring the induced birefringence [14–16, 21–23], while electrostriction has been investigated by applying a vertical field and measuring the shift in the Bragg reflection wavelength [17, 18, 24]. After the discovery of the polymer-stabilization technique to expand the temperature range of BPs , many ‘engineered’ BPs such as those stabilized by nano-particles [26–28] or tailor-made molecules [10, 29–31] have been proposed. It is therefore of much importance to understand the electro-optic response of pristine and engineered BPs.
Here, we investigate the electro-optic effects in pristine and polymer-stabilized (PS) BP I by simultaneously measuring the change in refractive index and the reflection spectrum. We first show that the slow secondary effect can contribute to as much as 20% of the total change in refractive index in a pristine BP, and, in an aim to shed light on the physical mechanism behind the secondary electro-optic effect, present a model which describes the observed phenomena through a coupling parameter between the primary Kerr effect and electrostriction. Secondly, and importantly for device applications, we show that the secondary effect can indeed be removed if electrostriction does not occur. The slow response is not observed in the PSBP, which has a fixed lattice structure that is stabilized by a polymer network. Our results show the importance of stabilizing the lattice structure to realize blue-phase devices with a fast response.
The BP sample used in this study was prepared by adding a chiral dopant [ISO-(6OBA)2, 7 wt%] to a nematic LC mixture (5CB, 46.5 wt% and JC-1041XX, 46.5 wt%) with positive dielectric anisotropy . We also prepared a PSBP by doping two types of monomers (dodecyl acrylate, 4.1 wt% and RM257, 4.2 wt%) and a photoinitiator (DMOAP, 0.8 wt%) in the aforementioned BP sample [25, 32]. Sandwich cells with an approximate thickness of 27 μm, assembled from two pre-cleaned, indium tin oxide- (ITO-) coated glass substrates were filled with the samples. The phase sequences of the samples were determined from polarized optical microscopy to be cholesteric (45.4 °C)/BP I (45.9 °C)/BP II (46.9 °C)/isotropic for the BP (on heating the sample at a rate of 0.1 °C/min) and cholesteric (30.0 °C)/BP I (33.5 °C)/BPII (35.5 °C)/isotropic for the PSBP (on cooling the sample at a rate of 0.2 °C/min). The PSBP sample was polymerized at 32.5 °C by irradiation with UV light (1.66 mW/cm2, 365 nm) for 20 min, after which BP I was stabilized to below −60 °C . Electro-optic measurements were performed at 45.7 °C for BP I and 27.5 °C for PSBP.
A two-beam interference microscope was built on a commercially available upright microscope (Olympus, BX-51), using a He-Ne laser (λ = 632.8 nm) as the probe light source. The setup was also equipped with a white light source and a beam-splitter to enable simultaneous acquisition of polarized reflection spectra . The sample was observed at the edge of the ITO electrodes so that regions both subject to and not subject to the field could be observed: the change in the refractive index was evaluated from the difference in the phase between the two regions. This configuration was employed to reduce errors introduced in the experiment, such as field-induced narrowing of the cell-gap. The resolution of the setup, limited by fluctuations in the fringe originating from environmental vibrations, was δnmin ∼ 10−4. The diameter of the measured spot for spectroscopy was ∼130 μm, and measurement was performed on the (110) platelet of BP I, which was confirmed by Kossel line observations. Application of an electric field oriented the (110) plane of the BP lattice along the electric field, but the azimuthal orientation was random, with domain sizes approximately 100 μm . The experiment was computer-controlled to acquire the interference fringe and the reflection spectrum at 200 ms intervals for 60 s. A rectangular electric field (1 kHz) was applied for 30 s and then removed, and the voltage was incremented after a rest period of 60 s. At this temperature, BP I showed a field-induced transition to the centered-tetragonal BP X at an applied fields above 2.7 V/μm : data were analyzed below this transition threshold.
3. Results and Discussion
Figure 2 shows the transient response of the Bragg wavelength and relative refractive index of the BP sample. The Bragg wavelength shows a slow response with characteristic times of 10 s or more: this is because the shift in the Bragg wavelength is due to an elastic deformation of the lattice involving a large number of LC molecules, and also because we have used a thick cell to obtain a phase difference that is large enough to be detected. The refractive index, on the other hand, shows a fast response because it is caused mainly by reorientation of the nematic director on a scale smaller than the lattice constant. As reported previously in many studies, the response time is on the submillisecond scale and is observed as a step-like response near 0 and 30 s in the time resolution of this measurement [14, 16]. Interestingly, for electric field intensities between 2.11 and 2.57 V/μm, a very slow change in the refractive index is observed, with characteristic times similar to that of electrostriction. The presence of this response was confirmed by making at least five measurements, and was found to be on the order of 10−4, which was as large as 20 % of the total shift in the refractive index (Note that a similar slow response is also observed for 2.72 V/μm at which the sample has turned to BP X. The applied field dependence of the slow response in BP X was found to be different from that of BP I and is out of scope of this paper). In Fig. 2(b), the applied field dependence of the response time of the slow response fitted to a single exponential function, τ, the amount of the slow change in the refractive index, δnslow, and the ratio of the slow change to the total change in refractive index, δnslow/δn, are shown: although the effect was not observable at low fields because of the limitation of our setup, δnslow shows a monotonic increase between applied fields of 2.1 and 2.6 V/μm.
The slow mode in the refractive index has a response time similar to that of electrostriction and can therefore be attributed to the secondary electro-optic effect of BPs. In a BP where the crystal structure is formed by the distribution of the nematic director, the mechanism of the effect can be described physically as follows. The Kerr coefficient has been shown to have a cubic dependence on the BP lattice constant . Thus, the extent to which the local nematic director can reorient is limited by the three-dimensional structure of the BP. As the lattice deforms under an applied field, the director is able to reorient further in the direction of the field, which in turn deforms the lattice further, until equilibrium is reached. The extra reorientation of the LC molecules enabled by lattice deformation contributes to an extra change in the refractive index with a characteristic response time similar to that of electrostriction. Interestingly, the slow response is observed only when the field is applied to the sample and not when the field is removed. This does not contradict the description above, since there is no directional torque acting in the off-switching process. This indicates that the molecules can return instantaneously to a configuration similar to the initial orientation (even though the lattice may be deformed), such that the difference in the refractive index is negligible.
The response of BPs has been modeled theoretically by different groups. The analytical approach that had been employed in earlier studies [3, 20, 34] is now rare and the trend of theoretical BP research has shifted to performing numerical simulations, in both free and confined geometries [5–8, 35–37]. Since BPs are complex materials, performing numerical simulations is no doubt one of the most effective approaches to clarify the reorientation phenomenon. Here, only to prove our point that the slow response in the refractive index can be viewed to result from the coupling of the lattice deformation with the local molecular reorientation, we present a simplified model which introduces a coupling parameter as a measure of the degree of additional director reorientation enabled by lattice deformation. In our model, we consider the averaged free energy density of a BP unit cell, and describe it in terms of the effective (i.e., induced) uniaxial order parameter SBP and lattice deformation u. Since we were not able to observe a noticeable difference in the phase of light being transmitted through (110) domains with different azimuthal orientations, we assume that the BP under the field is uniaxial, and not biaxial . Also, geometric factors such as the cell-gap, domain size, and anchoring that can affect the response time of electrostriction  are not considered: rather, we connect the two modes of responses (each with a distrinct characteristic time) to describe the overall optical response. The model is not rigorous because of the many assumptions employed; however, the coupling parameter we introduce can be an indicator of BP materials suitable for device applications.
The effective (i.e., induced) uniaxial order parameter SBP and electrostriction u of a single unit cell of the BP lattice is described as follows:Eq. (1), θ is the angle between the local director and the electric field, VBP is the volume of the unit cell, d (E) is the periodicity along the 〈110〉 direction at an applied field of E, and d0 is the periodicity without the field, which is related to the cubic lattice constant a0 through the relation . Because of the cubic symmetry of the BP, SBP = 0 and u = 0 at zero applied field: upon application of a field, both values increase monotonically with the field intensity. Assuming that SBP and u experience an elastic restoring force with proportionality constants κ1 and κ2, respectively, we may write the free energy density as follows:
The Euler-Lagrange equations for SBP and u yield the equations of motion for the system, where η1 and η2 are parameters related to the viscosities of each effect.Eq. (3) with Eq. (4), the coupling coefficient c can be determined experimentally by the following expression:
Figure 3 shows the values of c obtained when various electric fields were applied to the sample. The inset shows an example of the theoretical fit reproducing the experimental results for an applied voltage of 2.6 V/μm. For the particular material used, experimental results are reproduced when values of approximately 0.4 are assumed. Furthermore, we performed the same experiments on a BPLC prepared from a different host and chiral dopant (Merck, E44, 40 wt% and CB15, 60 wt%), and obtained values close to 0.2. Our results imply that the secondary electro-optic response is indeed a universal feature in BPs, and that the magnitude of the effect depends on the physical parameters of the LC. For the applications of BPs in flat panel displays or other optical devices, materials with near-zero coupling parameter (i.e., materials showing negligible change in refractive index after the initial response) would need to be realized.
It should be noted that in the above analyses, the responses of electrostriction and refractive index modulation were described using a single exponential function, assuming a single effective elastic constant and viscosity for each. In a realistic BP, however, the elastic constant and viscosity are a complex function of the anisotropic elastic constants (Frank elastic constants) and viscosities (Leslie coefficients) of the host LC, and moreover, would possibly vary with time in an electric field as the director distribution evolves. While satisfactory agreement between theory and experiment is observed in the inset of Fig. 3, assuming multiple time constants for each mode may produce better fits to the data (there is a study which reports the presence of multiple time constants in a PSBP ). Considering the physical origin of the coupling phenomenon, it should not be affected largely by the presence of multiple characteristic times in each effect, but only give rise to an additional change in the refractive index with the same characteristic times as that of electrostriction.
It has been shown recently that the anisotropy of the elastic constant, or more specifically, the ratio of the bend elastic constant to the splay elastic constant K33/K11, has a profound impact on the stability and electrostriction of BPs [31, 40]. Considering that director reorientation results from a competition between the elastic energy of the LC and the dielectric energy of the applied field, it is likely that the coupling parameter also depends on the elastic constants of the host nematic LC and their anisotropy. An important direction of research therefore would be to investigate BPLCs with different material properties and to clarify how different physical properties affect the coupling parameter. We also plan to improve the resolution and precision of the measurement so that important questions such as the field intensity dependence of the coupling parameter can be answered, and molecular design strategies effective in suppressing the slow response can be developed.
Now we show that the slow change in the refractive index can be removed by stabilizing the lattice structure. PSBPs are a class of BP materials in which the lattice structure is frozen by a fine network of polymer templating the periodic structure, but can still show a Kerr-type electro-optic response because of local molecular reorientation of the molecules existing among the polymer network [25, 32]. If the slow response of the refractive index originates in lattice deformation, it should not be observed in the PSBP. Figure 4(a) shows the transient response of the PSBP at various field intensities. As expected, the slow responses of both the Bragg reflection wavelength and refractive index vanish, thereby providing direct evidence that the secondary electro-optic effect, or the coupling of lattice deformation with the electro-optic Kerr effect, is responsible for the slow response.
Interestingly, in contrast to the behavior of the non-polymerized BP sample, the Bragg wavelength shows a small blue shift upon application of a field. This can be attributed to the change in the refractive index of the sample, since Bragg reflection from the (110) plane of a cubic lattice occurs at . A blue shift of the reflected wavelength implies that the refractive index has decreased under the applied field, which is consistent with the refractive index modulation induced by the Kerr effect. Figure 4(b) compares the electric field dependence of the relative refractive index obtained from interferometry and spectroscopy: the two values match almost perfectly, indicating that the small peak shift is indeed due to the Kerr effect.
To conclude, we performed simultaneous measurement of the polarized reflection spectrum and transmitted phase of BPLCs, and showed that the secondary electro-optic effect, which is caused by the additional local molecular reorientation enabled by the deformation of the BP lattice, can contribute to as much as 20% of the total change in refractive index. We introduced a coupling parameter connecting the primary Kerr effect and electrostriction to describe the slow response in the refractive index, and showed that polymer-stabilization of the lattice is effective for suppressing the coupling phenomenon. That the very structure that gives rise to the attractive features of BPs is the cause of the slow response poses a challenge for the practical application of BPs. Recent studies have focused on developing material systems with wide BP stability ranges [10, 26, 27, 29–31], and the dynamics of the electro-optic response is often overlooked. We hope that this study will raise awareness of this subject and promote experimental and theoretical studies targeted at clarifying the mechanisms further and minimizing the effect.
The authors thank JNC Corporation for kindly providing the BP material used in this study. This work was supported by JSPS KAKENHI Grant Numbers 24656015, 23656221, and 23107519. H. Yoshida acknowledges financial support from the JST PRESTO Program.
References and links
1. D. C. Wright and N. D. Mermin, “Crystalline liquids: the blue phases,” Rev. Mod. Phys. 61, 385–432 (1989). [CrossRef]
2. T. Seideman, “The liquid-crystalline blue phases,” Reports on Progress in Physics 53, 659–706 (1990). [CrossRef]
3. H. S. Kitzerow, “The effect of electric fields on blue phases,” Mol. Cryst. Liq. Cryst. 202, 51–83 (1991). [CrossRef]
4. H. Kikuchi, “Liquid crystalline blue phases,” Struct. Bond. 128, 99–117 (2008). [CrossRef]
5. G. P. Alexander and D. Marenduzzo, “Cubic blue phases in electric fields,” Europhys. Lett. 81, 66004 (2008). [CrossRef]
9. W. Cao, A. Munoz, P. Palffy-Muhoray, and B. Taheri, “Lasing in a three-dimensional photonic crystal of the liquid crystal blue phase II,” Nature Mater. 1, 111–113 (2002). [CrossRef]
11. H.-Y. Chen, J.-Y. Chiou, and K.-X. Yang, “Reversible and fast shift in reflection band of a cubic blue phase in a vertical electric field,” Appl. Phys. Lett. 99, 181119 (2011). [CrossRef]
12. S. Yabu, Y. Tanaka, K. Tagashira, H. Yoshida, A. Fujii, H. Kikuchi, and M. Ozaki, “Polarization-independent refractive index tuning using gold nanoparticle-stabilized blue phase liquid crystals,” Opt. Lett. 36, 3578–3580 (2011). [CrossRef]
13. G. Heppke, “Electric field induced variation of the refractive index in cholesteric blue phases,” Mol. Cryst. Liq. Cryst. Lett. 2, 59–65 (1985).
14. P. R. Gerber, “Electro-optical effects of a small-pitch blue-phase system,” Mol. Cryst. Liq. Cryst. 116, 197–206 (1985). [CrossRef]
15. H. Choi, H. Higuchi, and H. Kikuchi, “Electrooptic response of liquid crystalline blue phases with different chiral pitches,” Soft Matter 7, 4252–4256 (2011). [CrossRef]
16. S. Yabu, H. Yoshida, G. Lim, K. Kaneko, Y. Okumura, N. Uehara, H. Kikuchi, and M. Ozaki, “Dual frequency operation of a blue phase liquid crystal,” Opt. Mater. Express 1, 1577–1584 (2011). [CrossRef]
17. H. Gleeson, R. Simon, and H. J. Coles, “Electric field effects and two frequency colour switching in the cholesteric and blue phases of Nematic/Cholesteric mixtures,” Mol. Cryst. Liq. Cryst. 129, 37–52 (1985). [CrossRef]
19. J. F. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices (Oxford University Press, Oxford, 1984), 1.
20. V. E. Dmitrienko, “Electro-optic effects in blue phases,” Liq. Cryst. 5, 847–851 (1989). [CrossRef]
21. F. Porsch, H. Stegemeyer, and K. Hiltrop, “Electric field-induced birefringence in liquid-crystalline blue phases,” Z. Naturforsch. 39a, 475–480 (1984).
22. K.-M. Chen, S. Gauza, H. Xianyu, and S.-T. Wu, “Hysteresis effects in blue-phase liquid crystals,” J. Disp. Technol. 6, 318–322 (2010). [CrossRef]
23. U. Singh and P. H. Keyes, “Measurement of the kerr effect in cholesteric blue phases,” Liq. Cryst. 8, 851–860 (1990). [CrossRef]
24. H. J. Coles and H. F. Gleeson, “Electric field induced phase transitions and colour switching in the blue phases of chiral nematic liquid crystals,” Molecular Crystals and Liquid Crystals Incorporating Nonlinear Optics 167, 213–225 (1989). [CrossRef]
25. H. Kikuchi, M. Yokota, Y. Hisakado, H. Yang, and T. Kajiyama, “Polymer-stabilized liquid crystal blue phases,” Nature Mater. 1, 64–68 (2002). [CrossRef]
26. H. Yoshida, Y. Tanaka, K. Kawamoto, H. Kubo, T. Tsuda, A. Fujii, S. Kuwabata, H. Kikuchi, and M. Ozaki, “Nanoparticle-stabilized cholesteric blue phases,” Appl. Phys. Express 2, 121501 (2009). [CrossRef]
27. E. Karatairi, V. Tzitzios, G. Nounesis, G. Cordoyiannis, J. Thoen, C. Glorieux, and S. Kralj, “Nanoparticle-induced widening of the temperature range of liquid-crystalline blue phases,” Phys. Rev. E 81, 041703 (2010). [CrossRef]
28. J.-M. Wong, J.-Y. Hwang, and L.-C. Chien, “Electrically reconfigurable and thermally sensitive optical properties of gold nanorods dispersed liquid crystal blue phase,” Soft. Mat. 7, 7656–7659 (2011). [CrossRef]
29. A. Yoshizawa, Y. Kogawa, K. Kobayashi, Y. Takanishi, and J. Yamamoto, “A binaphthyl derivative with a wide temperature range of a blue phase,” J. Mater. Chem 19, 5759–5764 (2009). [CrossRef]
30. Y. Kogawa, T. Hirose, and A. Yoshizawa, “Biphenyl derivative stabilizing blue phases,” J. Mater. Chem 21, 19132–19137 (2011). [CrossRef]
31. S. Shibayama, H. Higuchi, Y. Okumura, and H. Kikuchi, “Dendron-stabilized liquid crystalline blue phases with an enlarged controllable range of the photonic band for tunable photonic devices,” Adv. Funct. Mater. 23, 2387–2396 (2013). [CrossRef]
32. Y. Haseba, H. Kikuchi, T. Nagamura, and T. Kajiyama, “Large electro-optic kerr effect in nanostructured chiral Liquid]Crystal composites over a wide temperature range,” Adv. Mater. 17, 2311–2315 (2005). [CrossRef]
33. H. Yoshida, S. Yabu, H. Tone, H. Kikuchi, and M. Ozaki, “Electro-optics of cubic and tetragonal blue phase liquid crystals investigated by two-beam interference microscopy,” Appl. Phys. Express 6, 062603 (2013). [CrossRef]
35. J.-i. Fukuda, M. Yoneya, and H. Yokoyama, “Simulation of cholesteric blue phases using a Landau-de Gennes theory: Effect of an applied electric field,”Phys. Rev. E80, 031706 (2009). [CrossRef]
36. A. Tiribocchi, G. Gonnella, D. Marenduzzo, and E. Orlandini, “Switching dynamics in cholesteric blue phases,” Soft Matter 7, 3295–3306 (2011). [CrossRef]
37. J.-i. Fukuda and S. Zumer, “Field-induced dynamics and structures in a cholesteric-blue-phase cell,” Phys. Rev. E 87, 042506 (2013). [CrossRef]
38. H.-S. Kitzerow, P. P. Crooker, J. Rand, J. Xu, and G. Heppke, “Electrostriction dynamics of blue phase II crystallites,” Journal de Physique II 2, 279–284 (1992). [CrossRef]
39. J. Yan, Y. Chen, S.-T. Wu, S.-H. Liu, K.-L. Cheng, and J.-W. Shiu, “Dynamic response of a polymer-stabilized blue-phase liquid crystal,” Journal of Applied Physics 111, 063103 (2012). [CrossRef]
40. J.-i. Fukuda, “Stabilization of blue phases by the variation of elastic constants,” Phys. Rev. E 85, 020701 (2012). [CrossRef]