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We report on a first experimental study of the molecular properties of nematic liquid crystals in the terahertz range. In the beginning, we extract the frequency and temperature dependent refractive index and absorption coefficient of the cyanobiphenyls 5CB, 6CB and 7CB from terahertz time domain spectroscopy measurements and investigate the impact of the alkyl chain length on the macroscopic liquid crystal characteristics, focusing especially on the pronounced odd and even effect. Next, we deduce the principle polarizabilities and the order parameter S by applying Vuks’ approximation and Haller’s approach. On this basis, we calculate the main polarizabilities along the longitudinal and transverse axis and link the observed terahertz properties to the molecular structure of the liquid crystals.
©2010 Optical Society of America
1. Introduction
Liquid crystals (LCs) have been the subject of active research for more than a century [1,2]. Without doubt, the most prominent LC based device is the flat panel display, which enjoys an ongoing commercial success and has revolutionized the electronics industry [3,4]. Apart from this mainstream application, the unique properties of LCs also led to the development of many other optical components such as light valves, tunable filters and tunable lenses [4,5]. Recently, first studies of LCs in the millimeter and terahertz (THz) range have been demonstrated [6–8], soon followed by proposals for LC based THz devices. Among them are tunable photonic crystals and Lyot filters as well as switchable dielectric mirrors [9–14]. The cyanobiphenyl (CB) family with its members 5CB, 6CB, and 7CB was one of the first commercially available nematic materials for use in LC displays [15]. Due to its outstanding properties at room temperature, i.e. high chemical stability, strong birefringence and pronounced sensitivity to applied electric fields, it is still commonly employed.
Recently, the authors explored the macroscopic THz properties of the cyanobiphenyls at room temperature [16]. However, room temperature measurements do not give access to the molecular properties of LCs. In this paper, we present temperature and frequency dependent spectroscopic measurements on both the ordinary and the extraordinary axis of the cyanobiphenyls 5CB, 6CB and 7CB. On the basis of this data, we present a first investigation of molecular properties of liquid crystals in the terahertz frequency range studying the order parameter S, the principle and the main polarizabilities. Knowledge of these parameters is mandatory to fully understand the interactions of liquid crystals with terahertz waves and can contribute to the efficient design of THz LC components [17–20].
The remainder of this paper is structured as follows: First, we will investigate the molecular structure of cyanobiphenyls and its implications on the dielectric characteristics of 5CB, 6CB, and 7CB. Afterwards, the experimental setup will be introduced followed by an extensive discussion of the experimental results. From the spectroscopic data, the dielectric material parameters in the THz frequency range will be derived and compared to results from other spectral regions. Furthermore, the anisotropy of the refractive index, the order parameter S as well as the principle and the main polarizabilities will be investigated accompanied by a discussion of the underlying physical mechanisms.
2. Molecular structure of cyanobiphenyls
In contrast to many other commercial LCs, the molecular structure of CBs is well known making them ideal candidates for scientific studies. In general, the cyanobiphenyls consist of a polar cyanid head group (CN), an alkyl chain (H2n + 1 Cn) and a rigid core unit formed by a conjugated phenyl ring system. The investigated molecules have the same basic structure and only differ in the length of the alkyl side chain (Fig. 1). The CBs change their phase with respect to the temperature from a well ordered crystalline over to a less ordered nematic and finally to a disordered, isotropic phase. The corresponding phase transition temperatures TCN and TNI as well as the phase transition temperature TCN in terms of the reduced temperature TCN-TNI are summarized in Table 1.
Table 1. Temperatures for the crystal/nematic (TCN) and the nematic/isotropic phase transition (TNI) as well as the reduced temperature TCN - TNI. Ref [21].
Within the cyanobiphenyls studied here, two fundamental dependencies are observed: First of all, the differing alkyl chain lengths lead to alterations in certain physical properties such as the polarizability and the density, which reflect in the anisotropy of both the absorption and the refractive index. Besides the total chain length, odd and even numbers of C atoms in the alkyl chain result in a different macroscopic behavior [22–25].
From a thermodynamic point of view, the LC system strives towards the lowest energy conformation in order to establish equilibrium. For an odd number of C atoms e.g. in 5CB or 7CB, this lowest energy state is achieved with the tail aligned in a trans-conformation along the principle long axis m. In contrast, CBs with an even number of C atoms e.g. 6CB encounter the lowest energy state when a large angle Θ6CB = 66.1° between the last C atom and the axis m is formed as shown in Fig. 1 [22].
Fig. 1. Molecular structure of 6CB according to Ref [22]. For simplicity, the hydrogen atoms are neglected and the phenyl rings are illustrated in a planar conformation.
The resulting dependency of the molecular properties is known as the odd and even effect [22–25]. In section 4, we will discuss the odd and even effect in the THz range for the anisotropy of both the refractive index n and the absorption coefficient A as well as for the order parameter S.
3. Experimental procedure
The measurements are performed with a standard THz time domain spectrometer over a broad frequency and temperature range [26]. Photoconductive antennas, which are gated by femtosecond pulses generated in a titanium sapphire laser, serve as THz emitters and detectors [27]. The THz beam is guided by four off-axis parabolic mirrors with the LC sample placed in the intermediate focus spot. To avoid water absorption lines in the spectra, the experiments are conducted in a nitrogen atmosphere. The LCs are investigated inside a cuvette with 700 μm thick, THz transparent fused silica walls. The plates are separated by two 1.3 mm copper spacers, which also serve as in-plane electrodes. An electric AC field of Eeff = 37 kV/m with a modulation frequency of 1 kHz is applied to align the LCs [16]. The extraordinary material parameters are measured with the molecular long axis aligned in parallel to the incident THz field, while the ordinary parameters are obtained with the LC’s long axis perpendicular to the incident THz field. In order to measure the material properties at different temperatures, the cuvette’s temperature is stabilized by Peltier elements connected to a thermoelectric controller. The material parameters are extracted from the time domain data using a high accuracy data extraction algorithm, which accounts for the cuvette walls and multiple reflections of the THz pulse at the material boundaries [28,29].
4. Experimental results
In order to better understand the physical interactions of the cyanobiphenyls and THz waves we investigate the frequency dependent dielectric parameters and compare them to other spectral regions. We are especially interested in the birefringence, which in general is a function of (i) the number of molecules per unit volume, (ii) the number of electrons per molecule, (iii) the degree of order, (iv) the resonance anisotropy of electronic transitions, and (v) the wavelength [30–33].
As the basic dispersion behavior of all CBs studied here is very similar [16], we will use 5CB as a representative example. Figure 2 (a) and (b) show the frequency dependent refractive index and the absorption coefficient of 5CB, respectively. Besides the THz data obtained by the authors (right side of the plot), also data from the visible and the near infrared region, which were previously published by S. T. Wu and Li et al., are given for comparison [31,32]. Our data are uncertain in the range of 1%, which is mainly caused by the instability of the laser. As the error bars are too small, they are not shown in the graph.
Fig. 2. Wavelength dependent THz refractive index (a) and absorption coefficient (b) of 5CB at 25.7° C together with measurements from the visible and the IR (cf. Ref [31]. and [32]).
As measured by Li et al., in the short wavelength region, both the extraordinary and the ordinary refractive index exponentially decline with increasing wavelengths. Here, the electronic transitions taking place in the UV still strongly influence the dispersion characteristics. In the NIR region between 700 nm and 1 μm the influence of the electronic transitions weakens resulting in a less dispersive behavior. The mid and far infrared region is dominated by vibrational absorption bands, which cause further resonance dispersion in the vicinity of these modes [31]. At frequencies around 5 THz (λ = 60μm) a torsional absorption band is located [34] (not shown in the graph), which majorly impacts the dispersion characteristics in the lower THz region. As found by the authors, the extraordinary and the ordinary refractive index increase with the wavelength, but the gradient is higher for the ordinary axis. The major absorption band is excited by the ordinary ray, resulting in a stronger dispersion of the refractive index along the corresponding axis [34]. The birefringence, shown in the inset of Fig. 2 (a), continuously decreases over the entire investigated spectral range.
Fig. 3. (a) Refractive index and (b) absorption coefficient at 1.5 THz versus the reduced temperature for 5CB, 6CB, and 7CB.
After having studied the frequency characteristics of the refractive index and the absorption coefficient, we shall now investigate the temperature dependence of these parameters for 5CB, 6CB, and 7CB at a fixed frequency of 1.5 THz as shown in Fig. 3. In order to compare the different CBs, our data are plotted against the reduced temperature T-TNI, where TNI is the phase transition temperature given in Table 1. As expected, the strong polarizability of the LCs in direction of the long molecular axis leads to a positive anisotropy (i.e. ne > no) of the THz refractive index over the entire nematic phase. The pronounced transition between the nematic and the isotropic phase can clearly be seen, whereas the CB’s don’t exhibit a transition between the nematic and the crystalline phase.
The number of C atoms in the alkyl chain increases from five in case of the 5CB to seven in case of the 7CB. Here, 5CB is the densest material with the highest number of molecules per unit volume and it exhibits a higher refractive index both in the ordinary and the extraordinary axis than 7CB for all temperatures. Both 5CB and 7CB possess an odd number of CH2 groups. Hence, the complete tail is aligned along the molecular long axis according to the odd and even rule mentioned in section 2. The two additional CH2 groups in case of 7CB give contributions to the polarizability of the long axis, which results in an enhanced birefringence. Here, Δn5CB = 0.11 and Δn7CB = 0.12 at T-TNI = -10 K.
Although the substance 6CB has the second longest tail and is thus the second densest material, its refractive index does not fit with respect to the refractive indices of 5CB and 7CB. Here, the last CH2 group forms a large angle to the molecular long axis. Therefore, the additional CH2 group primarily contributes to the polarizability of the molecular axis perpendicular to the long axis, resulting in a reduced birefringence of Δn6CB = 0.10. As the birefringence only slightly differs between the CB’s, the reader is referred to section 5. There, the anisotropy is again discussed on the basis of the polarizabilities.
In the absorption coefficient shown in Fig. 3 (b), a pronounced linear dichroism is observed. The ordinary absorption Ao is higher than the extraordinary absorption Ae as THz waves oriented along the ordinary direction excite a broad torsional vibration mode located around 5 THz [34].
5. Molecular properties
Molecular properties, such as the longitudinal and the transverse contributions of the polarizability tensor, are of fundamental interest. However, above absolute zero, the intrinsic thermal energy induces a certain degree of disorder in the LC, so that the spectroscopic measurements cannot directly give access to these molecular characteristics. Yet, if the order parameter S is known, a relation between these fundamental properties and the spectroscopic data can be derived [17,18].
Here, αe and αo represent the principle polarizabilities, which we will calculate from the spectroscopic data in subchapter 5.1, S the order parameter, and γ∥ and γ⊥ the longitudinal and transverse polarizability components of the perfectly ordered molecules. On the basis of Eq. (1) the main polarizabilities γ∥ and γ⊥ can then be estimated using the following equations:
and
Here, the mean polarizability is given by
To obtain the main polarizabilities, γ∥ and γ⊥, we will extract the principle polarizabilities αe and αo as well as the order parameter S of the cyanobiphenyls by applying Vuks’ approximation and Haller’s approach to the refractometric data [35,36].
5.1 Principle Polarizability
In order to link the polarizability of anisotropic materials to their refractive index, Vuks made a generalization of the Lorentz-Lorenz formula [37,38],
where n is the refractive index, α the polarizability, e and o the indices corresponding to the extraordinary and the ordinary polarization, and N the number of molecules per unit volume [39]. He assumed that Eq. (4), which is usually employed to describe isotropic media, also holds true for the mean refractive indices of anisotropic molecules in the crystalline state. On this basis he derived
where is the square of the mean refractive index. Figure 4 shows the principle polarizability of 5CB, 6CB, and 7CB at 1.5 THz calculated according to Vuks’ approximation.
Fig. 4. Principle polarizability of 5CB, 6CB and 7CB at 1.5 THz calculated according to the Vuks’s approximation.
We observe an anisotropy of the polarizability of Δα5CB = 7.7 × 10−24 cm3, Δα6CB = 7.2 × 10−24 cm3, and Δα7CB = 8.8 × 10−24 cm3 at T-TNI = -10 K for 5CB, 6CB, and 7CB respectively. As previously encountered in the birefringence data, 6CB exhibits the smallest anisotropy, while Δα increases with the chain length in case of 5CB and 7CB. This behavior again can be attributed to the odd and even effect discussed in section 2 of this paper.
Fig. 5. Principle polarizability of 5CB, 6CB, and 7CB versus the number of C atoms in the alkyl chain. Data are taken from Fig. 4 at a reduced temperature of T-TNI = 10 K.
Figure 5 shows the polarizabilities for the ordinary and the extraordinary axis and their mean value over the number of C-atoms in the alkyl chain. As expected, the polarizabilities increase with the number of atoms in the alkyl chain. While the mean value increases linearly by approximately 5% per CH2 group, the polarizabilities of the ordinary and the extraordinary axis increase according to the odd and even effect: In case of 6CB, the additional CH2 group mostly contributes to the short molecular axis, whereas the next odd member, alias 7CB, mainly strengthens the long axis.
5.2 Order parameter
The order parameter S is a relevant quantity to characterize the nematic phase. It describes the degree of order and is a measure for the degree of parallel alignment of the LCs. The S parameter equals unity at absolute zero, where all molecules are perfectly oriented. At temperatures above the phase transition TNI, the S parameter equals zero. Here, the molecules are completely disordered [40–42]. As long as thermal energy is left in the molecular system, which means at temperatures above absolute zero, the LC molecules are allowed to move around their center of mass. Hence, even in the nematic phase, despite the presence of an external electric field, only an average preferential orientation results instead of a perfect parallel alignment of all molecules.
To determine the order parameter from the measured data we employ an Haller’s extension to Vuks’s approximation [17,36]. Haller derived that
where Δγ is the difference between the longitudinal and transverse polarizability components of the perfectly ordered molecules and α̅ is the mean polarizability. Figure 6 shows a Haller plot of S · Δγ/α̅ over the normalized reduced temperature τ = (T - TNI)/TNI for 5CB, 6CB, and 7CB. Due to the logarithmic axis, a linear behavior is observed in the low temperature regime [36]. As the order parameter S equals 1 at T = 0 K (equivalent to τ = -1) the intercept of the extrapolated linear fit with the y-axis reveals a graphical solution for the relative polarizability Δγ/α̅. Here, Δγ/ equals 0.378, 0.401 and 0.375 for 5CB, 6CB, and 7CB, repectively.
Now that the relative polarizabilities are known, the S parameters can be calculated according to Eq. (6). Figure 7 illustrates the order parameter S in dependence of the normalized reduced temperature τ for 5CB, 6CB, and 7CB. While the order parameter behaves similar for 5CB and 7CB, it is considerably smaller in case of 6CB. This result is in good agreement with the results obtained by other authors and can be explained by the odd and even effect [43–46]. Molecules with an odd number of C atoms in the alkyl chain exhibit a higher flexibility than molecules with an even number of CH2 groups. This behavior results in a less uniform preferential orientation of the LCs [25].
Fig. 7. Order parameter S of 5CB, 6CB, and 7CB over the normalized reduced temperature τ obtained using Haller’s extension to Vuks’ approximation.
5.3 Main polarizabilities
With αe/o and S derived from Vuks’ approximation and Haller’s approach, we will now calculate the main polarizabilities of the molecules for the short and the long axis using Eqs. (2) and (3). Figure 8 shows the results, which are to our knowledge, the first main polarizabilities of liquid crystals presented in the THz region.
Adding an alkyl chain member increases the polarizability in both, the long and the short molecular axis. As expected from the study of the molecular structure in section 2, odd members induce a larger contribution to the polarizability of the molecular long axis, whereas even member contribute mainly to the molecular short axis due to the odd and even effect.
6. Summary
In conclusion, we studied three liquid crystals, 5CB, 6CB, and 7CB, from the cyanobiphenyl group with terahertz time domain spectroscopy. The frequency dependent refractive index, absorption coefficient, and birefringence between 0.2 and 2.5 THz were presented and discussed in context with data from other spectral regions. We found that the birefringence observed at terahertz frequencies is considerably smaller compared to the birefringence in the optical regime or the infrared.
Besides the frequency dependence, we also studied the temperature dependence of the material parameters. On this basis, we discussed the implications of differing alkyl chain lengths on the dielectric properties of the liquid crystals, especially focusing on the odd and even effect. In addition, the order parameter S and the principal polarizabilities (αe and αo) were determined by applying Vuks’ approximation and Haller’s approach. Finally, the main polarizabilites along the longitudinal and transverse axis were deduced, linking the macroscopic properties to the molecular structure.
Acknowledgment
Nico Vieweg thanks the Studienstiftung des Deutschen Volkes and the Braunschweig International School of Metrology. Furthermore, the authors thank the Merck KGaA for their support.
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