Open Access
Add to BibSonomy
Abstract
The current development of new liquid crystal devices often requires the use of thin cells and new experimental materials. Characterizing these devices and materials with optical methods can be challenging if (1) the total phase lag is small (“thin cells”) or (2) the liquid crystal optical and dielectric properties are only partially known. We explore the limitations of these two challenges for efficient characterization and assessment of new, to the best of our knowledge, liquid crystal devices. We show that it is possible to extract a wealth of liquid crystal parameters even for cells with a phase lag of $\Delta \Phi \approx \pi$, such as E7 liquid crystal in a 1.5 µm cell, using cross-polarized intensity measurements. The reliability of the optical method is also demonstrated for liquid crystals without precise values of dielectric or refractive index coefficients.
Published by Optica Publishing Group under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
1. INTRODUCTION
Thin liquid crystal (LC) cells are popular in industry, for example, paper-thin displays [1], paper-like displays [2], plastic sheet LC displays (LCDs) [3], and ultrathin LCs for augmented reality and virtual reality [4]. Optically thin LC cells with low birefringence are also used as active-matrix LCDs [5] and thin spatial phase modulators for THz applications [6].
Apart from the device thickness, the desired modulation of light is typically achieved by selecting appropriate LCs and alignment layers. LC materials can include, for example, experimental mixtures or, increasingly often, colloidal suspensions of inorganic or plasmonic nanoparticles in LCs [7–12]. The physical, electrical, or optical properties of such mixtures differ from the ones of the LC host, so they need to be recharacterized. This can also be an issue when newly synthesized, experimental LCs are used. Typically, only the birefringence or the dielectric anisotropy is known, but not the actual values of the refractive indices or the dielectric coefficients. This then leads to the question of how this partial knowledge affects the measurement of the other LC properties.
There are several methods for measuring LC properties, such as Fréedericksz transition [13], Rayleigh light scattering [14], free energy perturbations [15], nonlinear effects [16], and birefringence [17,18]. Most of these methods, however, are not able to measure multiple parameters in a single experiment for thin cells. For example, the pretilt angle in 4.5 µm cells was measured with capacitance-voltage methods using previously known LC parameters [19]. In 2.4-µm-thick reflective LC on silicon cells with a thick alignment layer pretilt angle, anchoring energy and effective birefringence were determined by fitting voltage-dependent reflectance curves [20].
Here we show a method based on the use of a single cross-polarized intensity (CPI) measurement of planar LC cells to provide an accurate estimation of the device parameters in optically thin cells with alignment layers of standard thickness (e.g., 20 nm). We investigate the phase lag limit for optically thin cells and also perform error propagation analysis to determine the accuracy of the estimated LC parameters. Our approach allows us to characterize not only the LC material itself, but also the properties of a LC cell or device. We have used the CPI measurement method to reliably determine a range of parameters, such as splay and bend elastic constants, viscosities, cell thickness, pretilt, and polar anchoring energy [21–23], in a single experiment, for several combinations of LCs and alignment layers. However, its correct operation relies on having the values of the LC refractive indices and dielectric coefficients as input parameters. If these parameters are missing or their values are only estimated, the error propagation analysis allows us to determine how reliable the CPI method is in determining the other, core LC parameters.
The structure of this paper is as follows. First, we present the optical setup, experimental techniques, and methods. We then explore the lowest phase lag limit for our CPI-based method in Section 3.A. We characterize LCs with partially known dielectric coefficients and find the errors on the extracted LC parameters that arise from the uncertainty on the refractive indices and the dielectric coefficients in Section 3.B. In the Conclusion, we review our results and discuss the case of geometrically thin, but optically thick cells.
2. EXPERIMENTAL METHOD AND FITTING PROCEDURE
LC cell parameters can be extracted from reliable CPI measurements [17] using an optical multi-parameter analyzer (OMPA) described in detail by Bennett et al. [21,22]. We use planar LC cells with cell gaps of 10–12 µm, filled with nematic LCs, namely, E7, MLC6815, and LC18523 from Merck. The alignment layers consist of polyimide (PI) or the photoconductive polymer polyvinylcarbazole (PVK) doped with fullerene ${{\rm C}_{60}}$, deposited on electrodes made of indium tin oxide (ITO). The LC cells are placed between crossed polarizers at 45° to the optical axis of the LC, and the optical transmission of the system is recorded as a function of the AC voltage amplitude applied to the cell. A diagram of the experimental setup is presented in Fig. 1.
Fig. 1. Schematic representation of the experimental setup. The laser beam is expanded by lenses ${{\rm L}_1}$ and ${{\rm L}_2}$ for analysis of the whole LC cell. The LC sample (LC), which is controlled by an AC voltage source, is placed between polarizers ${{\rm P}_1}$ and ${{\rm P}_2}$. The beam splitter (BS) produces a reference beam detected by the reference photodetector ${{\rm PD}_1}$ and a transmitted beam detected by the main photodetector ${{\rm PD}_2}$. The measurement output is a plot of the CPI as a function of the voltage applied to the cell (inset).
During a computer-controlled CPI measurement, a voltage-dependent CPI trace is collected and subsequently fitted to extract the LC properties. The standard fitting procedure requires a CPI trace with at least one minimum and one maximum to automatically normalize the data between zero (smallest minimum) and one (largest maximum), which is equivalent to having a total phase lag larger than $2\pi$. For cells that are either geometrically thin or contain low-birefringence LCs, i.e., cells with a phase lag smaller than $2\pi$ that we call henceforth “thin cells,” this procedure is modified by measuring separately the minimum and maximum transmitted intensities. To record the minimum intensity, the LC sample is removed, so a configuration of crossed polarizers and no cell is used. The maximum intensity can be measured using parallel polarizers and an empty cell. With this normalization of the CPI trace, the phase lag limit can be reduced to less than $\pi$.
The fitting procedure is based on the Frank–Oseen static theory of nematics, which is used to find the alignment equations of a nematic LC in the electrostatic field. The time evolution of the director field of the LC, and the CPI [see Eq. (3)] are computed in MATLAB as described by Bennett et al. [21,22].
The fitting procedure requires the following LC parameters to be specified: extraordinary and ordinary refractive indices, ${n_e}$ and ${n_o}$, respectively, and dielectric coefficients ${\varepsilon _\parallel}$ and ${\varepsilon _ \bot}$. In this paper, we have determined the dielectric coefficients ${\varepsilon _\parallel}$ and ${\varepsilon _ \bot}$ of MLC6815 for validation purposes by measuring the capacitance of a planar cell before and after it was filled using the auto balancing bridge method [24] and a DC voltage source.
3. RESULTS AND DISCUSSION
The CPI measurement method extracts the LC parameters from the phase lag experienced by a linearly polarized beam as it passes through a LC cell. This is measured as a function of voltage, thus probing and measuring the competing effect of the LC electric coupling and its elastic stiffness. The total phase lag $\Delta \Phi$ of the LC is given by [17]
where $d$ is the cell thickness, $\lambda$ the wavelength of the light propagating through the LC cell, and $\Delta n$ the cell effective birefringence, given by [21,22]Fig. 2. Simulated cross-polarized intensity (top) and phase lag (bottom) of E7 cells with thicknesses of 1, 2, and 10 µm. The LC cell parameters are: elastic constants ${K_1} = 10.9\;{\rm pN}$ and ${K_3} = 17.895\;{\rm pN}$, dielectric coefficients ${\varepsilon _\parallel} = 19.54$ and ${\varepsilon _ \bot} = 5.17$, pretilt ${\theta _0}{= 2^ \circ}$, strong polar anchoring energy, i.e., ${W_p} = 1\; {{\rm J/m}^2}$, and refractive indices ${n_e} = 1.7287$ and ${n_o} = 1.5182$ at $\lambda = 642\;{\rm nm} $.
As discussed in the previous section, the CPI normalization differs in thick and thin cells. In thick cells, where there are at least one maximum and one minimum of the CPI trace, the data can be normalized automatically. In thin cells, additional measurements of the minimum and maximum intensities are needed. One important aspect that needs to be considered here is the reliability of the thin-cell normalization method. The reliability can be compromized further if parameters such as ${n_e}$, ${n_o}$, ${\varepsilon _\parallel}$, and ${\varepsilon _ \bot}$ are only partially known. We address such reliability issues by looking into, separately, the errors on the fitting parameters stemming from either the process of normalization or uncertainty over the values of the dielectric anisotropy and birefringence.
A. Thin-Cell Measurements
Studying optically thin cells allowed us to investigate the limit of the CPI measurement method. In Fig. 3 (top), we present a comparison between the two normalization methods for the case of a low-birefringence 12-µm-thick LC18523 cell with PI and PVK: ${{\rm C}_{60}}$ alignment layers. The CPI traces of the LC18523 cell were measured at three wavelengths: 450, 532, and 642 nm. Using these three wavelengths allowed us to obtain data with three different values of the total phase lag [see Eq. (1)] and hence to compare thin-cell characterization (at 642 nm) to thick-cell characterization regimes (at 450 and 532 nm) for the same cell. In Fig. 3 (top), the CPI traces for the shortest and longest wavelengths are shown, while the 532 nm curve has been omitted for clarity. For the shortest wavelength [empty circles in Fig. 3 (top)], the maximum phase lag is larger than $2\pi$ and the CPI data have both a maximum and a minimum, so automatic normalization is possible. For the longest wavelength (filled circles), there is only one extremum, so we use the thin-cell normalization method.
Fig. 3. Voltage-dependent CPI traces and theoretical fits (top) and normalized phase lag (bottom) for the LC18523 PI PVK: ${{\rm C}_{60}}$ cell at different wavelengths. Parameter values extracted from the CPI traces are presented in Table 1.
In all cases, we obtain very good agreement between the fitting curves and the experimental data [see Fig. 3 (top)]. The parameter values obtained from the OMPA fits for the LC18523 cell at all three wavelengths are presented in Table 1, where the Fréedericksz transition threshold voltage at zero pretilt, ${V_{{\rm th}}}$, is defined as [25]
Here ${K_1}$ is the splay elastic constant, ${\varepsilon _0}$ is the permittivity of free space, and $\Delta \varepsilon = {\varepsilon _\parallel} - {\varepsilon _ \bot}$ is the dielectric anisotropy. The parameters extracted from the fitting, for example, the elastic constants, show good agreement with the published values summarized in Table 2, thus confirming that our method is sufficiently robust to characterize thin cells with a CPI containing just one extremum.
Table 2. Literature Values of LC Parameters for Four Standard LCs: E7, TL205, LC18523, and MLC6815a
Interestingly, at the shortest wavelength used, namely, 450 nm, the anchoring energy is weaker than for longer wavelengths. This small effect can also be seen in the graph of the normalized phase lag [Fig. 3 (bottom)], where at 450 nm, the tail of the normalized phase lag curve is lower than those at longer wavelengths. This result, which is likely to be associated with the increased photoconductivity of the aligning layer, PVK: ${{\rm C}_{60}}$, in the blue region of the spectrum, is currently being studied and will be reported separately. For the purpose of the investigation presented here, we note, however, that the OMPA method is capable of detecting small changes in anchoring even in thin cells.
As a further comparison, the automatic and thin-cell normalization methods were applied to the same CPI experimental data of an MLC6815 cell with one maximum and one minimum. No noticeable difference between the results of the two normalization processes was discovered.
The thin-cell normalization method was tested experimentally only for CPI traces with no minima. Cells that are optically very thin, i.e., with a maximum phase lag smaller than $\pi$, can in principle be measured using the thin-cell normalization of CPI data. In this case, however, due care must be taken to account for reflection and absorption in the LC cell when estimating the maximum intensity. If the cells are geometrically thin as well, their effective birefringence and voltage threshold may depend on cell thickness [31,32], and their response times may be longer under weak anchoring [20,33]. Therefore, it may also be necessary to include the inert layer corrections discussed by Wu and Efron in [32].
B. Partially Characterized Liquid Crystals
New LC mixtures and experimental composite LCs are often accompanied only by their birefringence and dielectric anisotropy values. We explore the impact of this uncertainty in the actual magnitudes of their indices on the reliability of our CPI characterization method. The LC alignment depends on the values of ${\varepsilon _\parallel}$ and ${\varepsilon _ \bot}$, but not on the refractive indices. Only the CPI depends on the refractive indices. More specifically, the phase lag depends primarily on the birefringence and only more weakly on the exact values of the refractive indices; see Eq. (1 )–(3). Therefore, when fitting data with large uncertainties on the refractive indices or dielectric coefficients, we expect relatively small impact from the unknown refractive indices ${n_e}$, ${n_o}$, providing the birefringence $\Delta n = {n_e} - {n_o}$ is known. However, bigger errors are expected when the dielectric coefficients ${\varepsilon _\parallel}$ and ${\varepsilon _ \bot}$ are unknown, even if the dielectric anisotropy $\Delta \varepsilon = {\varepsilon _\parallel} - {\varepsilon _ \bot}$ is known.
On the strength of these observations, we consider first the effect of partly characterized dielectric coefficients. We assume that the refractive indices and dielectric anisotropy, $\Delta \epsilon$, are known, but not the individual values of ${\varepsilon _\parallel}$ and ${\varepsilon _ \bot}$. In this case, one can formulate reasonable guesses for these parameters to fit the data. The results of such estimations are presented in Fig. 4, where the literature value for ${\varepsilon _ \bot}$ is 5.17 for E7, while for MLC6815, we measured ${\varepsilon _ \bot} = 4.3$ at room temperature. Indeed, as can be observed, the fittings with these values of the dielectric coefficient are the closest to the experimental data for each LC; values further away from the true value of ${\varepsilon _ \bot}$ give worse and worse fits, and in some cases, convergence of the fitting algorithm cannot be reached. Based on how close the fitting curves in Fig. 4 are to each other for each LC cell, the quality of the fit cannot be used to determine the values of the dielectric constants. However, we can conclude that the fit is poor if ${\epsilon _ \bot}$ is over- or under-estimated by a factor of at least two. The parameter values obtained from the fitting procedure using different values of ${\epsilon _ \bot}$ are given in Table 3. The results for the elastic constants of E7 using the correct value ${\varepsilon _ \bot} = 5.17$ are well within the accepted range of the literature values given in Table 2. Interestingly, as the results in Table 3 suggest, the elastic constants are somewhat insensitive to the precise values of ${\varepsilon _\parallel}$ and ${\varepsilon _ \bot}$. Reducing ${\varepsilon _ \bot}$ by a factor of two with respect to the accepted value of ${\varepsilon _ \bot}$ gives up to 20% error on the elastic constants, while doubling the accepted value of ${\varepsilon _ \bot}$ gives up to 12% error on the elastic constants. Therefore, even with some uncertainty over the dielectric coefficients, the elastic constants can be estimated reasonably well.
Fig. 4. Theoretical fits using different ${\varepsilon _ \bot}$ values for an E7 cell (top) and an MLC6815 cell (bottom) at 532 nm. The corresponding values of ${\varepsilon _\parallel}$ were obtained using the dielectric anisotropy $\Delta \varepsilon$ reported in the literature and summarized in Table 2. The parameter values extracted from the fitting procedure are presented in Table 3.
Table 3. Fitting Parameters for Different Estimates of ${\varepsilon _ \bot}$ and ${\varepsilon _\parallel} = {\varepsilon _ \bot} + \Delta \varepsilon$ Obtained Using OMPA for E7 and MLC6815 Cells for Fixed Dielectric Anisotropy $\Delta \varepsilon$ a
We now address the more general case of estimating the error on the fitted parameters in terms of the error on the refractive indices and dielectric permittivities. In other words, we consider the case when all optical and electrical LC parameters are known, but only within a certain uncertainty range. How do these uncertainties affect the CPI-based estimate of the other LC parameters? In this analysis, we use standard error propagation formulas, with the added factor that the relation between the known and fitting parameters is known only in implicit form through the fitting procedure. We therefore must use the implicit function theorem to evaluate the derivatives of the fitting parameters in terms of the dielectric permittivities and refractive indices. The details of the method can be found in Appendix A.
The relative errors for four LC cells used to compare the case of unknown refractive indices to that of unknown dielectric permittivities are presented in Tables 4 and 5. The LC parameters used in our calculations are given in Table 2. Comparison between Tables 4 and 5 suggests that measurements of the properties of a LC with refractive indices with 10% error, $\Delta {n_o} = \Delta {n_e} = 0.1 {n_o}$, and fixed birefringence can give reliable results with typical errors below 0.02% for all fitting parameters ${K_1}$, ${K_3}$, $d$, ${\theta _0}$, and ${W_p}$ for the four LCs. These errors are negligible when compared to experimental noise or numerical errors during the fitting procedure. In contrast, measurements of the properties of a LC with dielectric permittivities with 10% error, $\Delta {\epsilon _ \bot} = \Delta {\epsilon _\parallel} = 0.1{\epsilon _ \bot}$, and fixed dielectric anisotropy give errors ranging from 0.02% to 16%. In this case, the errors are significant, so the reliability of the results is considerably worse. The pretilt, followed by the elastic constants, is the most sensitive parameter to errors on the dielectric coefficients, while cell thickness is the least sensitive parameter. Similar behavior is observed for infinitely strong anchoring. These results confirm that unknown refractive indices ${n_e}$, ${n_o}$ and fixed birefringence $\Delta n = {n_e} - {n_o}$ lead to much smaller errors than unknown dielectric coefficients ${\varepsilon _\parallel}$, ${\varepsilon _ \bot}$ and fixed dielectric anisotropy $\Delta \varepsilon = {\varepsilon _\parallel} - {\varepsilon _ \bot}$. This is in agreement with the preliminary analysis at the beginning of this section.
Table 4. Relative Errors on Fitting Parameters for Fixed Birefringence and Absolute Error on Refractive Indices $\Delta {n_o} = \Delta {n_e} = 0.1 {n_o}$ for Four LCsa
Table 5. Relative Errors on Fitting Parameters for Fixed Dielectric Anisotropy and Absolute Error on Dielectric Coefficients $\Delta {\epsilon _ \bot} = \Delta {\epsilon _\parallel} = 0.1 \,{\epsilon _ \bot}$ for Four LCsa
Fig. 5. Absolute error on the splay elastic constant ${K_1}$ divided by the uncertainty on the ordinary refractive index ${n_o}$ as a function of the phase lag at $V = 0 \;{\rm V}$ for E7 cell with varying thicknesses. The LC cell parameters are: elastic constants ${K_1} = 10.9 \;{\rm pN}$ and ${K_3} = 17.895 \;{\rm pN}$, dielectric coefficients ${\varepsilon _\parallel} = 19.54$ and ${\varepsilon _ \bot} = 5.17$, pretilt ${\theta _0}{= 2^ \circ}$, strong polar anchoring energy ${W_p} = 1\; {{\rm J/m}^2}$, and refractive indices ${n_e} = 1.7287$ and ${n_o} = 1.5182$ at $\lambda = 642\;{\rm nm}$.
It is important to note here that when the CPI is close to zero or one (i.e., $\Delta \Phi = m\pi$) at $V = 0$, the CPI is less sensitive to changes in the phase lag. Therefore, there is more freedom for the fitting parameters due to the weaker constraints, and hence, bigger errors are expected. This is confirmed by the error propagation analysis in Appendix A. The dependence of the error of the splay elastic constant on the maximum phase lag, i.e., phase lag at $V = 0$, is shown in Fig. 5 as an example. It clearly indicates that when the maximum phase lag is an integer multiple of $\pi$, i.e., when the CPI at $V = 0$ is close to either zero or one, the error is largest. This behavior is typical of most of the fitting parameters. Therefore, to minimize the error on the fitting parameters, it is important to choose a cell thickness or a light wavelength such that the maximum phase lag is different from an integer multiple of $\pi$.
4. CONCLUSION
In this work, we considered two technologically important constraints for LC devices: thin cells/small phase lag and LCs with incomplete sets of dielectric and refractive indices.
The LC parameters of optically thin cells were successfully extracted from CPI data with a limit of total phase lag of $\Delta \Phi \approx \pi$. The measurement procedure proposed here is also valid for cells that are geometrically very thin, and if their boundary effects become significant [32], effective cell parameters may still be extracted [20].
Furthermore, we have demonstrated the reliability of our approach, using the associated error analysis, to determine elastic constants for LCs with unknown refractive indices and/or unknown dielectric coefficients. In the former case, the uncertainty on the refractive indices does not affect significantly the accuracy of the extracted elastic constants; in the latter case, a reasonably good approximation can be obtained for the elastic constants. The error analysis is based on using the implicit function theorem in the standard error propagation formula. In this paper, we have applied it to a Frank–Oseen model for a planar cell, but it can also be used for twist cells and for cells where flow effects are significant, the latter being modeled using an Ericksen–Leslie theory [34,35]. The combination of thin-cell fitting and error analysis presented here is quite versatile and can serve as a useful aid in efficient assessment of new LCs in technologically relevant geometries.
APPENDIX A: ERROR PROPAGATION
The OMPA fitting procedure returns a set of fitting parameters, ${K_1}$, ${K_3}$, $d$, ${\theta _0}$, and ${W_p}$, by fitting the experimental CPI trace, provided that ${n_e}$, ${n_o}$, ${\varepsilon _\parallel}$, and ${\varepsilon _ \bot}$ are known. Let ${\boldsymbol x}$ and ${\boldsymbol y}$ denote the known and fitting parameter vectors, respectively:
The absolute error on the fitting parameters is given by the standard error propagation formula
Funding
Leverhulme Trust (RPG-2019-055).
Acknowledgment
We thank Tim Sluckin, Tetiana Orlova, Eleni Perivolari, and Elena Ouskova for useful discussions that led to this work.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are available from Ref. [36].
REFERENCES
1. Y. Zhang, J. Sun, Y. Liu, J. Shang, H. Liu, H. Liu, X. Gong, V. Chigrinov, and H. S. Kowk, “A flexible optically re-writable color liquid crystal display,” Appl. Phys. Lett. 112, 131902 (2018). [CrossRef]
2. P. F. Bai, R. A. Hayes, M. Jin, L. Shui, Z. C. Yi, L. Wang, X. Zhang, and G. Zhou, “Review of paper-like display technologies (Invited Review),” Prog. Electromagn. Res. 147, 95–116 (2014). [CrossRef]
3. S. Oka, T. Sasaki, T. Tamaru, Y. Hyodo, L. Jin, S. Takayama, and S. Komura, “9-2: optical compensation method for wide viewing angle IPS LCD using a plastic substrate,” SID Symp. Dig. Tech. Pap. 47, 87–90 (2016). [CrossRef]
4. J. Xiong and S.-T. Wu, “Planar liquid crystal polarization optics for augmented reality and virtual reality: from fundamentals to applications,” eLight 1, 3 (2021). [CrossRef]
5. Y. Iwashita, Y. Umezu, S. Kawakami, K. Takeuchi, T. Kusumoto, S. Takehara, and H. Takatsu, “Liquid crystal mixtures with low birefringence for active-matrix LCD,” Mol. Cryst. Liq. Cryst. 411, 41–48 (2004). [CrossRef]
6. O. Buchnev, N. Podoliak, K. Kaltenecker, M. Walther, and V. A. Fedotov, “Metasurface-based optical liquid crystal cell as an ultrathin spatial phase modulator for THz applications,” ACS Photon. 7, 3199–3206 (2020). [CrossRef]
7. Q. Liu, Y. Yuan, and I. I. Smalyukh, “Electrically and optically tunable plasmonic guest-host liquid crystals with long-range ordered nanoparticles,” Nano Lett. 14, 4071–4077 (2014). [CrossRef]
8. A. Chaudhary, R. Shukla, P. Malik, R. Mehra, and K. Raina, “ZnO/FLC nanocomposites with low driving voltage and non-volatile memory for information storage applications,” Curr. Appl. Phys. 19, 1374–1378 (2019). [CrossRef]
9. A. Chaudhary, P. Malik, R. Shukla, R. Mehra, and K. Raina, “Role of SiO2 optically active mediators to tailor optical and electro-optical properties of ferroelectric liquid crystalline nanocomposites,” J. Mol. Liq. 314, 113580 (2020). [CrossRef]
10. C. Hamon, E. Beaudoin, P. Launois, and E. Paineau, “Doping liquid crystals of colloidal inorganic nanotubes by additive-free metal nanoparticles,” J. Phys. Chem. Lett. 12, 5052–5058 (2021). [CrossRef]
11. Z. Mai, Y. Yuan, J.-S. B. Tai, B. Senyuk, B. Liu, H. Li, Y. Wang, G. Zhou, and I. I. Smalyukh, “Nematic order, plasmonic switching and self-patterning of colloidal gold bipyramids,” Adv. Sci. 8, 2102854 (2021). [CrossRef]
12. N. Brouckaert, N. Podoliak, T. Orlova, D. Bankova, A. F. De Fazio, A. G. Kanaras, O. Hovorka, G. D’Alessandro, and M. Kaczmarek, “Nanoparticle-induced property changes in nematic liquid crystals,” Nanomaterials 12, 341 (2022). [CrossRef]
13. V. Fréedericksz and V. Zolina, “Forces causing the orientation of an anisotropic liquid,” Trans. Faraday Soc. 29, 919–930 (1933). [CrossRef]
14. G.-P. Chen, H. Takezoe, and A. Fukuda, “Determination of Ki (i = 1-3) and µj (j = 2-6) in 5CB by observing the angular dependence of Rayleigh line spectral widths,” Liq. Cryst. 5, 341–347 (1989). [CrossRef]
15. A. A. Joshi, J. K. Whitmer, O. Guzmán, N. L. Abbott, and J. J. de Pablo, “Measuring liquid crystal elastic constants with free energy perturbations,” Soft Matter 10, 882–893 (2014). [CrossRef]
16. B. Klus, U. A. Laudyn, M. A. Karpierz, and B. Sahraoui, “All-optical measurement of elastic constants in nematic liquid crystals,” Opt. Express 22, 30257–30266 (2014). [CrossRef]
17. S.-T. Wu, U. Efron, and L. D. Hess, “Birefringence measurements of liquid crystals,” Appl. Opt. 23, 3911–3915 (1984). [CrossRef]
18. M. Bharath Kumar, M. Awwal Adeshina, D. Kang, Y. Jee, T. Kim, M. Choi, and J. Park, “Enhancement of birefringence in reduced graphene oxide doped liquid crystal,” Nanomaterials 10, 842 (2020). [CrossRef]
19. X. Zhao, T. Li, Z. Tang, Y. Li, Y. Miao, H. Xing, M. Cai, X. Wang, X. Kong, and W. Ye, “Accurate determination on the pre-tilt angle of liquid crystal cell by combining optical and electrical measurement,” Liq. Cryst. 48, 15–22 (2021). [CrossRef]
20. M. Jiao, Z. Ge, Q. Song, and S.-T. Wu, “Alignment layer effects on thin liquid crystal cells,” Appl. Phys. Lett. 92, 061102 (2008). [CrossRef]
21. T. Bennett, M. Proctor, M. Kaczmarek, and G. D’Alessandro, “Lifting degeneracy in nematic liquid crystal viscosities with a single optical measurement,” J. Colloid Interface Sci. 497, 201–206 (2017). [CrossRef]
22. T. Bennett, M. Proctor, J. Forster, E. Perivolari, N. Podoliak, M. Sugden, R. Kirke, T. Regrettier, T. Heiser, M. Kaczmarek, and G. D’Alessandro, “Wide area mapping of liquid crystal devices with passive and active command layers,” Appl. Opt. 56, 9050–9056 (2017). [CrossRef]
23. E. Perivolari, G. D’Alessandro, V. Apostolopoulos, N. Brouckaert, T. Heiser, and M. Kaczmarek, “Two-dimensional snapshot measurement of surface variation of anchoring in liquid crystal cells,” Liq. Cryst. 48, 2086–2096 (2021). [CrossRef]
24. L. M. Blinov, “Magnetic, electric and transport properties,” in Structure and Properties of Liquid Crystals (Springer, 2011), pp. 151–187.
25. I. W. Stewart, The Static and Dynamic Continuum Theory of Liquid Crystals, 1st ed. (CRC Press, 2004).
26. J. Li, C.-H. Wen, S. Gauza, R. Lu, and S.-T. Wu, “Refractive indices of liquid crystals for display applications,” J. Disp. Technol. 1, 51–61 (2005). [CrossRef]
27. C. L. Trabi, C. V. Brown, A. A. T. Smith, and N. J. Mottram, “Interferometric method for determining the sum of the flexoelectric coefficients (e1+e3) in an ionic nematic material,” Appl. Phys. Lett. 92, 223509 (2008). [CrossRef]
28. H. Wang, T. X. Wu, S. Gauza, J. R. Wu, and S. Wu, “A method to estimate the Leslie coefficients of liquid crystals based on MBBA data,” Liq. Cryst. 33, 91–98 (2006). [CrossRef]
29. N. Podoliak, O. Buchnev, M. Herrington, E. Mavrona, M. Kaczmarek, A. G. Kanaras, E. Stratakis, J.-F. Blach, J.-F. Henninot, and M. Warenghem, “Elastic constants, viscosity and response time in nematic liquid crystals doped with ferroelectric nanoparticles,” RSC Adv. 4, 46068–46074 (2014). [CrossRef]
30. M. Warenghem, J. F. Henninot, J. F. Blach, O. Buchnev, M. Kaczmarek, and M. Stchakovsky, “Combined ellipsometry and refractometry technique for characterisation of liquid crystal based nanocomposites,” Rev. Sci. Instrum. 83, 035103 (2012). [CrossRef]
31. S. Wu and U. Efron, “Electro-optic behavior of thin nematic liquid crystal cells,” Proc. SPIE 0613, 172 (1986). [CrossRef]
32. S. Wu and U. Efron, “Optical properties of thin nematic liquid crystal cells,” Appl. Phys. Lett. 48, 624–626 (1986). [CrossRef]
33. X. Nie, R. Lu, H. Xianyu, T. X. Wu, and S.-T. Wu, “Anchoring energy and cell gap effects on liquid crystal response time,” J. Appl. Phys. 101, 103110 (2007). [CrossRef]
34. J. Eriksen, “Conservation laws for liquid crystals,” Trans. Soc. Rheology 5, 23–34 (1961). [CrossRef]
35. F. Leslie, “Some constitutive equations for liquid crystals,” Arch. Ration. Mech. Anal. 28, 265–283 (1968). [CrossRef]
36. D. Bankova, “Characterization of optically thin cells and experimental liquid crystals - Supplementary material,” University of Southampton Institutional Repository, (2022) https://doi.org/10.5258/SOTON/D2125.
