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The refractive indices of a class of Blue-Phase liquid crystals (BPLCs) and their temperature dependence have been measured and analyzed. In general, the thermal index gradients in blue phases, BPI and BPII, are both larger than in isotropic liquid state; the index gradient of BPII phase is steeper than that of BPI, and is attributed to the difference between the expansion coefficients of simple and body-centered cubic lattices. Besides their obvious importance in photonics and nonlinear optical processes and applications, the investigation of the phase dependence of the index gradient also provides a useful way for phase identification of BPLCs, namely the second-order and weakly first order phase transitions corresponding to the ISO/BPII transition and the BPII/BPI transition.
©2013 Optical Society of America
1. Introduction
Blue-Phase liquid crystal (BPLC) is a particular phase of chiral nematic in which the director axes are self-assembled in double twisted helix in 3D regular lattices of disclinations (defects); with typical lattice spacing on the order of several hundred nanometers, BPLC thus exhibits selective Bragg reflections (photonic bandgaps) of light in the visible spectrum. As a function of decreasing temperature from the isotropic liquid phase (ISO), BPLC can assume three distinct phases: BPIII, BPII and BPI. While BPIII can exist only in a very limited temperature range (< 1 °C), recent studies [1–3] have resulted in BPLC that exhibit BPII and BPI phases over large temperature ranges. In BPII, the crystalline structure is simple cubic (SC), whereas BPI is body-centered cubic (BCC). Recently, BPLCs have been intensively investigated for displays [4] and photonic applications (e.g. Fabry-Pérot filter [5], nonlinear grating [6], 3D lasers [7] and tunable lenses [8,9]), because of their faster response, optical isotropy (polarization independence) in the absence of applied electric field and surface alignment-free fabrication ease.
For high performance or specialty photonic applications, quantitative characterization of various physical and optical properties such as lattice structure [10,11], density [12], viscosity [13], elasticity [13], Kerr constant [14], and particularly the refractive index are needed. Several methods have also been developed to identify the BP phases and their phase transition temperatures, including polarized optical microscopy observation (especially in reflected light mode, abbreviated R-POM), variations in reflection wavelength [15], Kossel diagram [10], and Differential Scanning Calorimetry (DSC) [16]. In this paper, we present the results of further characterization of a class of BPLC used in recent photonic studies. In particular, we have conducted a detailed measurement of the hitherto unknown temperature dependence of the refractive index along with a discussion of the underlying mechanisms for the various phases (ISO, BPII and BPI).
2. Material preparation and phase sequence identification
The BPLCs utilized in this study were all prepared by blending the left-handed chiral agent S811 into the nematogen E48 but in different mixing ratios. Mixtures containing 35, 40 and 45 wt% S811 are designated as LK-35, LK-40 and LK-45, respectively. In order to confirm that the examined temperature dependence of refractive index would markedly vary with transition temperature, we also employed another BPLC RT-35, which was made with the S811 concentration same as LK-35 but the nematic host different from that of LK-35, consisting of 5CB and E48. Both BPI and BPII are thermodynamically stable in these compounds. Owing to their Bragg reflections in the visible spectrum, the BPI and BPII phases in a bulk sample can be simply distinguished by naked eye observation [Fig. 1(a) ]. To identify their transition temperatures accurately, these BPLCs were infused into glass cells with the same cell gap of 20 μm. Using R-POM images, Kossel diagrams and reflection spectra, we have determined the phase sequences of these samples as listed in Table 1 . Other physical characteristics are depicted in Figs. 1(b)-1(d) using LK-40 as an example. Figure 1(b) show the growth of blue platelets of BPII observed with a microscope with crossed polarizers upon cooling from the isotropic phase [Fig. 1(b)]. On the other hand, Fig. 1(c) depicts the Kossel diffraction pattern observed by means of a Bertrand lens and a monochromatic light in the reflected POM, indicating the lattice direction of [200]. When the temperature fell below the transition temperature of about 40.7 °C, the blue platelets of BPII were increasingly being covered green platelets of BPI, correlating well with the jump in the reflection wavelength shown in Fig. 1(d). As revealed by the Kossel diagram [Fig. 1(c)], these green platelets arise from a lattice direction of [110]. Upon further cooling, these samples eventually transition to the cholesteric phase (N*) with the corresponding R-POM image, reflection wavelength and Kossel diagram.
Fig. 1 Phase identifications through (a) naked eye observation (b) R-POM images, (c) Kossel diagrams and (d) variations in reflection wavelength.
Table 1. Transition Temperatures of the BPLCsa
3. Experimental setup
To investigate the temperature dependence of the refractive index, we have set up a Mach-Zehnder interferometer as shown in Fig. 2 . A 1 mm-thick BPLC cell adhered to a hot stage (Linkam) was inserted into one of the branches. A Helium-Neon laser which operates at a wavelength of 632.8 nm was chosen as a probe beam; note that the color of the chosen probe laser sufficiently far away from that of the blue phase platelets minimize transmission loss (through the BPLC sample) caused by scattering and reflection. While the temperature was gradually changed, the spatially shifting interference signals were being collected by a photodetector that was set behind an iris and the signal was recorded by an oscilloscope. The temperature was determined by the average of the detected temperatures at both sides of the bulk sample.
Fig. 2 The experimental setup for measuring the temperature dependence of the refractive index of BPLCs.
4. Results and discussion
Figure 3 [LK-40 sample with a cooling rate of 0.5 °C/min] depicts the intensity maxima/minima of a spot in the interference pattern caused by the temperature-dependent refractive index and therefore phase shift experienced by the laser in passing through the BPLC sample. Each peak-to-peak shift corresponds to a phase shift of π. Small fluctuations of the plot were results of some environmental vibrations, and thus we have used simple moving average method to smooth out the curve for further analyses. From the phase determination results in Table 1, we can divide the curve into four corresponding parts. Among all phases, the isotropic liquid phase with minimum scattering loss resulted in the largest intensity difference. The peak-to-peak intensity dropped upon cooling down to the BPII due to the multiple scattering by disordered platelet distribution [17]. At the BPII/BPI transition, a strong scattering was observed whence the interference fringes vanish, resulting in a relatively flat line or an indefinable shape of the curve. When the phase transition was about to be complete (~37.7 °C), the pattern of interference resurfaced. At the last part of this curve, as the BPI crosses over to the N*, the interferences became a flat profile again, accompanied by a very strong scattering from the focal conic texture in cholesteric phase. All the phenomena occurred during the heating process as well, differing only in the order of appearance and transition temperature.
Fig. 3 The measured intensity of a spot in the interference pattern as a function of the operating temperature.
These phase shift (Δϕ) measurements allow one to assess the corresponding refractive index change. Note that Δϕ = 2π·d·Δn / λ, where d (~1280 μm) is the path length through a cell, Δn is the index difference and λ is the wavelength (= 0.6328 μm), a phase shift of π therefore corresponds to an index difference of 2.5 × 10−4. The phase shift profiles as recorded in Fig. 3 thus allow one to calculate the temperature dependent, refractive index change in the monophasic region. However, the occurrence of an indefinable part (~38.5 °C) within the BPII/BPI biphasic region makes it hard to determine the index difference between the two ends and the continuity of this phase transition. Therefore, we’ve also measured the exact refractive index values at the high and low temperature sides of the BPII/BPI transition using an Abbe refractometer. Combine the results from the two measuring systems, we thereby acquired the variations in the refractive index of these liquid crystals as plotted in Fig. 4 . It was found that, in monophasic regions, the refractive index increased almost linearly as the temperature dropped, as expected from its principal dependence on the density [12,18,19]. This phenomenon agrees with the average index in liquid crystalline nematic phase as observed by Li et al. [20] As a result, we fitted the experimental data with the following phenomenological equation:
where n is the refractive index at the temperature T, dn/dT is the temperature-dependent index gradient in a specific phase, and nf is a phase-dependent constant term. In the fitting procedure, we abandoned several points in the vicinity of the indefinable (nearly flat) part to ensure that the fitted experimental data are in monophasic region. The experimental data and their fitting results are plotted in Fig. 4, and the index gradients are listed in Table 2 . The exact index of LK-35 cannot be detected due to the upper temperature limit (40 °C) of the Abbe refractometer, so that the data of LK-35 are not included. The slope values in an identical phase among the mixtures are quite close to each other, and any possible trend in the pitch dependence of index gradient may be masked by experimental error margins. However, it is apparently that the index gradients in the Blue phases are generally higher than the isotropic phase, with BPII having the largest gradient value:Fig. 4 Temperature-dependent refractive index of LK-40, LK-45 and RT-35. The open circles are the experimental data; the circles with crosses inside are the points not included in the fittings; the solid lines are the fitting curves using Eq. (1); the arrows point out the phase transition points determined by the experimental curves of the temperature-dependent interference intensity and fitting curves of the temperature-dependent refractive index.
Table 2. Magnitudes of the Refractive-Index Gradients of the BPLCs in Various Phases
This phenomenon is similar to the results when examining the nematics [21] and the cholesterics [18] as compared the average index in mesophase to that in isotropic phase. The difference between the slopes of BPI and BPII is likely due to the fact that the DTC packing of a BCC lattice is more compact than that of a SC lattice, hence stronger binding energy (or weaker thermal expansion) leads to smaller thermal alteration in density and the corresponding refractive index. The smooth change in index gradient shown in the experimental data indicates a continuous (second-order) phase transition from ISO to BPII. However, at the BPII/BPI transition, there exists a jump in the extended fitting lines of refractive index which suggests the occurrence of a weakly first-order transition. These results are consistent with the findings from a DSC investigation on phase transitions of a BPLC reported by Armitage et al. [16]
By means of phase shift detection and calculating the corresponding index variation, the phase transition temperatures were examined and generally in good agreement with the methods mentioned in Fig. 1. The difference in measured transition temperature results from the resolution of our system (0.3-0.6 °C), the inhomogeneous temperature distribution of the sample, the distinction between the boundary conditions of a thick cell and a thin cell, and some experimental inaccuracies caused by environmental vibrations and temperature detection errors. For the use of phase identification, it is better to thicken the cell gap and slow down the cooling rate to enhance the resolution of the index change, and apply a temperature-controlled chamber to provide uniform heating. Nevertheless, it is worth mentioning that the light scattering in blue phases will increase along with the thickening of the sample, and may therefore create more uncertainties in exact determination of the refractive index. Besides, the temperature dependence of refractive index also depends on the probe wavelength [22].
5. Conclusion
In summary, variations in refractive index of BPLCs as a function of temperature have been investigated. In general, liquid crystals in the blue-phases possess larger thermal index gradients than their isotropic phase counterpart. The measured index gradient of BPII is found to be larger than BPI, which may result from the differences in thermal expansion for different types of lattice-like structure, SC and BCC. Within the limit of the resolution of this index-gradient measuring system, the continuous change in refractive index from ISO to BPII indicates a second-order phase transition, while the discontinuity between the index slopes of BPII and BPI within the biphasic region points to a weakly first-order phase transition. These basic information on the refractive indices and their temperature dependence will be important for practical applications of phase identification, thermal-induced nonlinear optics [23] and other blue-phase liquid crystal based photonic devices.
Acknowledgment
The authors gratefully acknowledge the National Science Council of Taiwan, for financial support under Contract No: NSC 99-2119-M-110-006-MY3, NSC 100-2628-E-110-007-MY3, and from the US Air Force Office of Scientific Research.
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