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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 19 — Sep. 13, 2010
  • pp: 19914–19919
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Voltage-assisted photoaligning effect of an azo dye doped in a liquid crystal with negative dielectric anisotropy

Kuang-Yu Yang and Wei Lee  »View Author Affiliations


Optics Express, Vol. 18, Issue 19, pp. 19914-19919 (2010)
http://dx.doi.org/10.1364/OE.18.019914


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Abstract

We demonstrate a method of controlling the pretilt angle by field-assisted photoalignment in a vertically-aligned liquid crystal doped with a dichroic azo dye. By dynamically measuring both the pretilt angle and anisotropy in absorbance, we present detailed alignment variations in both the polar and azimuthal directions with the time of laser exposure that induces photoisomerization of the dye molecules. Significant differences in the resulting pretilt as well as optically-absorptive anisotropy are manifested between a regular photoalignment procedure and that in conjunction with a bias voltage during irradiation. It is found that the field-assisted exposure not only speeds up the photoalignment along the polarization direction of the laser beam but also leads to a higher orientational order.

© 2010 OSA

1. Introduction

Controlling the liquid-crystal (LC) alignment by cell surface treatments as well as external fields is the key principle in many LC photonic devices. Recently, a photoalignment technique has emerged that involves photoisomerization of azo dyes in LC cells [1

1. V. G. Chigrinov, V. M. Kozenkov, and H. S. Kwok, Photoalignment of Liquid Crystalline Materials: Physics and Applications (Wiley, 2008).

]. Once the azo-dye molecules are exposed by polarized light of a specific wavelength and an appropriate intensity, the light–molecule interaction will induce the conformation of the molecules from the trans state to the cis state. The cis-state molecules, which have a greater electrical polarity, will then be diffused and adsorbed onto the substrates, thereby altering the alignment of LCs [2

2. T.-J. Chen and K.-L. Chu, “Pretilt angle control for single-cell-gap transflective liquid crystal cells,” Appl. Phys. Lett. 92(9), 091102 (2008). [CrossRef]

]. There are a good number of studies and hypotheses concerning the models of cis-state dye molecules on the substrates [3

3. L. Komitov, C. Ruslim, Y. Matsuzawa, and K. Ichimura, “Photoinduced anchoring transitions in a nematic doped with azo dyes,” Liq. Cryst. 27(8), 1011–1016 (2000). [CrossRef]

]. They can be briefly classified into two major categories according to their functional effects on LC alignment: planar and vertical alignments. The laser-induced azimuthally reorientational effect in dye-doped LCs (DDLCs) has attracted considerable interest over the last two decades. It is established that, in a specific case where the substrates are layered with a polyimide copolymer doped with a diazodiamine dye, a polarized laser beam can induce LC reorientation perpendicular to the linear-polarization direction [4

4. W. M. Gibbons, P. J. Shannon, S.-T. Sun, and B. J. Swetlin, “Surface-mediated alignment of nematic liquid crystals with polarized laser light,” Nature 351(6321), 49–50 (1991). [CrossRef]

]. On the other hand, a recent study has demonstrated that, depending on the intensity of light, the reorientation can be either parallel or perpendicular to the polarization direction [5

5. E. Ouskova, D. Fedorenko, Yu. Reznikov, S. V. Shiyanovskii, L. Su, J. L. West, O. V. Kuksenok, O. Francescangeli, and F. Simoni, “Hidden photoalignment of liquid crystals in the isotropic phase,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(2 Pt 1), 021701 (2001). [CrossRef] [PubMed]

]. Furthermore, a laser-induced unidirectional ripple structure in parallel to the polarization direction has also been reported. The ripple-groove structure, formed in a LC doped with methyl red (MR) of a specific concentration under prolonged laser irradiation, allows the nematic director to line up with the groove direction [6

6. A. Y.-G. Fuh, C.-C. Liao, K.-C. Hsu, and C.-L. Lu, “Laser-induced reorientation effect and ripple structure in dye-doped liquid-crystal films,” Opt. Lett. 28(14), 1179–1181 (2003). [CrossRef] [PubMed]

]. The surface ripple grooves, consisting of a large number of cis-state molecules with their long axes perpendicular to the groove orientation, are parallel to the linear polarization direction of the exposure light. If the intensity of the pumping light is weak and the concentration of the dye is low, then the amplitude of ripple structure will be small and the orientation of the LC will be dominated by the adsorbed dye, aligning the director perpendicularly to the polarization axis of the incident light beam [7

7. A. Y.-G. Fuh, C.-K. Liu, K.-T. Cheng, C.-L. Ting, C.-C. Chen, P. C.-P. Chao, and H.-K. Hsu, “Chao, and H.-K. Hsu, “Variable liquid crystal pretilt angles generated by photoalignment in homeotropically aligned azo dye-doped liquid crystals,” Appl. Phys. Lett. 95(16), 161104 (2009). [CrossRef]

].

In this paper, we demonstrate the illumination-time dependence of the laser-induced change in pretilt angle and optically-absorptive anisotropy of a MR-doped LC with a negative dielectric anisotropy confined in a cell with an unrubbed vertically-aligning layer on each substrate. The effect of dc voltage applied simultaneously in the course of laser exposure is discussed. Our results supply strong evidence for the illuminated dye to have significant anchoring for the LC molecules along the polarization direction of the laser beam of normal incidence and the voltage-assisted photoalignment results in a higher orientational order.

2. Experimental

We fabricated a vertically-aligned DDLC sample by doping ~1.0 wt% MR (Aldrich) into a negative liquid crystal (ZLI-2806, Merck) and then introducing the mixture by capillary action into a 40-μm-thick empty cell which had two unrubbed vertical-alignment layers of N,N-dimethyl-n-octadecyl-3-aminopropyltrimethoxysilyl chloride (DMOAP) [8

8. F. J. Kahn, G. N. Taylor, and H. Schonhorn, “Surface-produced alignment of liquid crystals,” Proc. IEEE 61(7), 823–828 (1973). [CrossRef]

] coated on the indium–tin-oxide glass substrates. A linearly-polarized Ar+ laser beam of 514.5 nm in wavelength was expanded by means of a lens system. It was normally incident onto the sample, forming a circular light spot of diameter of ~6 mm and intensity of ~300 mW/cm2. Two experimental conditions were considered, one without voltage applied and the other with a 5-V dc bias (i.e., a reduced voltage of 2.8 in this study) across the cell thickness during laser irradiation. In both cases, the cis-state molecules generated by photoisomerization after a period of exposure time were diffused and adsorbed on the substrate. With the cis-state MR ripple structure formed on the exposed substrate and the surfactant alignment layers on both sides, hybrid LC cells with various pretilt angles on the exposed side were yielded [7

7. A. Y.-G. Fuh, C.-K. Liu, K.-T. Cheng, C.-L. Ting, C.-C. Chen, P. C.-P. Chao, and H.-K. Hsu, “Chao, and H.-K. Hsu, “Variable liquid crystal pretilt angles generated by photoalignment in homeotropically aligned azo dye-doped liquid crystals,” Appl. Phys. Lett. 95(16), 161104 (2009). [CrossRef]

]. Dynamic measurements of the pretilt angle; namely, the polar angle measured from the substrate plane, and the reorientation of LC molecules caused by laser exposure were carried out with a typical crystal-rotation method [9

9. J. S. Gwag, S. H. Lee, K.-H. Park, W. S. Park, K.-Y. Han, C. G. Jhun, T.-H. Yoon, J. C. Kim, D.- Song, and D.- Shin, “Simple method for measuring the high pretilt angle of nematic liquid crystals,” J. Appl. Phys. 93(8), 4936 (2003). [CrossRef]

] as well as polarization absorption spectroscopy in the visible spectrum. To evaluate the LC reorientation projected onto the azimuthal plane, we define a value describing the anisotropy in absorbance as ΔA = A ||A , where A || and A represent the peak values of the absorbance in the directions parallel and perpendicular to the polarization direction of the laser beam, respectively. Obviously, if ΔA > 0, then one can tell that the LC director tends to be reoriented along the polarization direction of the coherent exposure light in accordance with the guest–host effect in the DDLC film.

3. Results and discussion

Figure 1
Fig. 1 Variation of the pretilt angle of a DDLC with the exposure time.
displays the exposure-time dependence of the pretilt angle with no voltage applied to the cell during the irradiation process. The angle decreased dramatically from ~90° to ~60° in the first 30 min and continued to fall slowly between 30 and 40 min; it finally saturated at ~55° beyond ~40 min. The photoisomerization of the azo dye MR took place rather rapidly in the beginning of exposure, forming lots of cis-state molecules from their trans state [10

10. C.-Y. Tang, S.-M. Huang, and W. Lee, “Dielectric relaxation dynamics in a dual-frequency nematic liquid crystal doped with C.I. Acid Red 2,” Dyes Pigm. in press., doi:.

]. These molecules were diffused and adsorbed on the aligning layer of the exposed side, providing the LC with a homogeneous anchoring energy against the vertical alignment layers [7

7. A. Y.-G. Fuh, C.-K. Liu, K.-T. Cheng, C.-L. Ting, C.-C. Chen, P. C.-P. Chao, and H.-K. Hsu, “Chao, and H.-K. Hsu, “Variable liquid crystal pretilt angles generated by photoalignment in homeotropically aligned azo dye-doped liquid crystals,” Appl. Phys. Lett. 95(16), 161104 (2009). [CrossRef]

], which, in turn, gave rise to a rapid change in alignment of the LC molecules. Note that there are two kinds of mechanisms to determine whether the orientation of the tilted LC director will tend to be parallel or perpendicular to the polarization of light. According to [7

7. A. Y.-G. Fuh, C.-K. Liu, K.-T. Cheng, C.-L. Ting, C.-C. Chen, P. C.-P. Chao, and H.-K. Hsu, “Chao, and H.-K. Hsu, “Variable liquid crystal pretilt angles generated by photoalignment in homeotropically aligned azo dye-doped liquid crystals,” Appl. Phys. Lett. 95(16), 161104 (2009). [CrossRef]

], if the intensity of the exposure light is weak (<100 mW/cm2) and the dye content is sufficient low (<1 wt%) in the DDLC sample, the amplitude of the ripple structure will be too small to compete with the influence of individually adsorbed dye molecules. The resulting LC orientation will, then, tend to be perpendicular to the polarized axis of the incident light beam. In contrast, if a sufficiently large ripple-groove amplitude is obtained, the torque imposed by the grooves will overcome that by the adsorbed dye, resulting in the LC orientation along the groove direction parallel to the polarization of the light.

Figure 2
Fig. 2 Anisotropy of absorbance ΔA as a function of the exposure time.
depicts the anisotropy in absorbance as a function of the illumination time for the identical sample used to obtain Fig. 1. It reveals that the anisotropy in absorbance was barely observed prior to exposure, presumably due to the vertical alignment of both LC and MR molecules so that A and A were equally contributed by absorption along the short axis of the aligned MR molecules. Once the exposure started, ΔA increased rapidly before it reached saturation as shown in Fig. 2. On the other hand, both time-evolved spectra of A and A decreased monotonically with increasing exposure time since the number of trans MR molecules was reduced by photoisomerization; the increasing difference between the two absorbance curves resulted in the growing ΔA. The comparison between Figs. 1 and 2 sheds light on the direction of the photoalignment which, in competition with the vertical alignment provided by the silane surface-coupling agent DMOAP, was planar and parallel to the polarization direction of the laser beam. In other words, the cis-state azo dye adsorbed on the substrate hindered the initial vertically-orienting force acting on the LC molecules and generated an appreciable magnitude of anisotropic planar anchoring force along the electric field of the exposure light.

The photoisomerization of azo dyes depends on the intensity of exposure light; the process takes place only when the light intensity is above a threshold value. In this situation, the dye molecules transfer to the cis state as the DDLC is continually exposed to the laser beam. However, some cis molecules formed via photoisomerization can be reverted to the trans state spontaneously by the environmental perturbation, say, thermal agitation. Once the rate of the reverse process is comparable to that of the normal photoisomerization, the number of cis-state molecules in the DDLC system will reach a dynamic balance [10

10. C.-Y. Tang, S.-M. Huang, and W. Lee, “Dielectric relaxation dynamics in a dual-frequency nematic liquid crystal doped with C.I. Acid Red 2,” Dyes Pigm. in press., doi:.

], causing both the pretilt angle and ΔA to saturate with the exposure time.

If one applies a voltage greater than the Frederiks transition to the above-mentioned DDLC sample prior to exposure, the characteristic absorptions for both A and A in the blue-to-green spectrum will definitely increase, as illustrated in Fig. 3
Fig. 3 Polarization absorption spectra of the unexposed DDLC sample.
, in that the long axis of the trans MR molecules are aligned with the reoriented host LC possessing a negative dielectric anisotropy. In this case, one would expect approximately identical polarization absorptions since the aligning layers were not rubbed and a multi-domain of the nematic phase was formed. As can be seen from Fig. 3, however, A and A were easily discernable because of our experimental condition to allow the flow direction upon filling the cell to be along the polarization direction of the laser beam. Upon the applied voltage, the nematic director changed its vertical alignment to the planar state, preferably oriented along the filling direction. Consequently, a higher polarization absorbance was obtained in the direction parallel to the linear polarization of the laser beam.

Figure 4
Fig. 4 Exposure-time dependence of the pretilt angle. Labeled in the legend are the voltages applied during laser exposure.
compares the variations of the pretilt angle for the two distinctive bias conditions of 0- and 5-V dc voltages applied during exposure, where the 0-V data were reproduced from Fig. 1. While the anchoring direction provided by the cis-state MR molecules remained unchanged, the raised absorbance of the sample due to the nontrivial dc bias speeded up the photoisomerization and, in turn, increased the rate of the photoaligning process. Differences in pretilt angle between the two cells were remarkable in the exposure-time range between 10 and 25 min. Our choice to use a DC bias of 5 V stemmed from its effectiveness to reorient the LC director so that the long axis of the doped dye molecules became roughly parallel to the polarization of the laser beam, thus enhancing the absorption and promoting the successive photoisomerization. Our fresh LC ZLI-2806 possesses a relatively high resistivity (~1012 Ω-cm) compared with that (1010–1011 Ω-cm) of a typical cyano-based nematic. Therefore, the ionic effect is not as significant as in 5CB or E7. The influence of space charge on the results is not evident.

Figure 5
Fig. 5 Exposure-time dependence of the anisotropy of absorbance.
shows the anisotropy of absorbance for both exposure conditions corresponding to the regular (0 V) irradiation process and bias-assisted (5 V) photoalignment. It reveals that when the exposure by the laser beam lasted continuously, both ΔA increased and then seemed to saturate. Clearly, the sample experiencing voltage-assisted photoalignment exhibited greater anisotropy in absorbance. Note that ΔA became apparently separated (Fig. 5) when the pretilt angles were about to saturate (Fig. 4) to a similar level. Since the optical absorbance of the DDLCs was dominated by the orientation of the trans-state dye molecules and, in turn, by the tilt angle of the LC molecules, the growth of absorbance would also be limited when the decreasing pretilt angle was almost fixed at saturation. However, here the negative LC, aligned homogeneously under applied voltage during exposure, permitted greater optical absorbance in comparison with the cell counterpart exposed at the saturated tilt angle, making the photoisomerization of MR still reactive to produce a larger surface amplitude of the ripple structure made up by the adsorbed cis-state MR molecules [6

6. A. Y.-G. Fuh, C.-C. Liao, K.-C. Hsu, and C.-L. Lu, “Laser-induced reorientation effect and ripple structure in dye-doped liquid-crystal films,” Opt. Lett. 28(14), 1179–1181 (2003). [CrossRef] [PubMed]

], and a greater unidirectional anchoring force was, in turn, obtained based on the Berreman theory [11

11. D. W. Berreman, “Solid surface shape and the alignment of an adjacent nematic liquid crystal,” Phys. Rev. Lett. 28(26), 1683–1686 (1972). [CrossRef]

,12

12. J. Fukuda, M. Yoneya, and H. Yokoyama, “Consistent numerical evaluation of the anchoring energy of a grooved surface,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(1 Pt 1), 011705 (2009). [CrossRef] [PubMed]

]. As a result, a better alignment of both the LCs and the remaining trans MR molecules was induced along the groove of the ripple structure in the polarization direction of the exposure beam, giving a greater ΔA as well as an implied higher orientational order in the azimuthal plane.

It is interesting to examine the temporal stability of electro-optical performance of an exposed DDLC cell. The cell as a LC device exhibited a persistent voltage–transmittance characteristic within a 12-h period. Its long-term stability of pretilt-angle control has yet to be experimentally investigated. It is also of interest to inspect the thermal stability of the DDLC sample. In a simple test, a sample with a pretilt angle of 55.3° changed to 57.7° after 2-h heating at 60°C and then naturally cooling down to the room temperature. Obviously, heat can weaken the effect (i.e., unidirectional planar anchoring) of photoalignment because the sharpness or groove depth of the structure is reduced by heat or/and the cis molecules are reverted to the more stable trans state.

4. Concluding remarks

In summary, we have demonstrated an effective variation of LC alignment due to the voltage-assisted photoaligning properties of a dichroic dye mixed in a homeotropically-aligned LC with a negative dielectric anisotropy. We found that the angle of alignment changes dramatically in the beginning of laser exposure, arising from the vigorous photoinduced isomerization of the azo-dye molecules. As the trans-to-cis transformation gradually reaches saturation, the change in pretilt angle becomes minimal. Moreover, both the tilt angle and azimuthally anisotropic absorption vary in consistence with the exposure time. For the LC cell with a voltage bias applied during exposure in comparison with its reference receiving laser irradiation at null voltage, the greater optical absorbance leads to higher saturated ΔA, higher orientational order as well as faster photoalignment along the polarization direction of the exposure coherent light in the azimuthal plane.

Acknowledgments

The authors acknowledge financial support from the National Science Council of Taiwan under grants NSC 98-2815-C-033-023-M and NSC 98-2112-M-033-004-MY3.

References and links

1.

V. G. Chigrinov, V. M. Kozenkov, and H. S. Kwok, Photoalignment of Liquid Crystalline Materials: Physics and Applications (Wiley, 2008).

2.

T.-J. Chen and K.-L. Chu, “Pretilt angle control for single-cell-gap transflective liquid crystal cells,” Appl. Phys. Lett. 92(9), 091102 (2008). [CrossRef]

3.

L. Komitov, C. Ruslim, Y. Matsuzawa, and K. Ichimura, “Photoinduced anchoring transitions in a nematic doped with azo dyes,” Liq. Cryst. 27(8), 1011–1016 (2000). [CrossRef]

4.

W. M. Gibbons, P. J. Shannon, S.-T. Sun, and B. J. Swetlin, “Surface-mediated alignment of nematic liquid crystals with polarized laser light,” Nature 351(6321), 49–50 (1991). [CrossRef]

5.

E. Ouskova, D. Fedorenko, Yu. Reznikov, S. V. Shiyanovskii, L. Su, J. L. West, O. V. Kuksenok, O. Francescangeli, and F. Simoni, “Hidden photoalignment of liquid crystals in the isotropic phase,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(2 Pt 1), 021701 (2001). [CrossRef] [PubMed]

6.

A. Y.-G. Fuh, C.-C. Liao, K.-C. Hsu, and C.-L. Lu, “Laser-induced reorientation effect and ripple structure in dye-doped liquid-crystal films,” Opt. Lett. 28(14), 1179–1181 (2003). [CrossRef] [PubMed]

7.

A. Y.-G. Fuh, C.-K. Liu, K.-T. Cheng, C.-L. Ting, C.-C. Chen, P. C.-P. Chao, and H.-K. Hsu, “Chao, and H.-K. Hsu, “Variable liquid crystal pretilt angles generated by photoalignment in homeotropically aligned azo dye-doped liquid crystals,” Appl. Phys. Lett. 95(16), 161104 (2009). [CrossRef]

8.

F. J. Kahn, G. N. Taylor, and H. Schonhorn, “Surface-produced alignment of liquid crystals,” Proc. IEEE 61(7), 823–828 (1973). [CrossRef]

9.

J. S. Gwag, S. H. Lee, K.-H. Park, W. S. Park, K.-Y. Han, C. G. Jhun, T.-H. Yoon, J. C. Kim, D.- Song, and D.- Shin, “Simple method for measuring the high pretilt angle of nematic liquid crystals,” J. Appl. Phys. 93(8), 4936 (2003). [CrossRef]

10.

C.-Y. Tang, S.-M. Huang, and W. Lee, “Dielectric relaxation dynamics in a dual-frequency nematic liquid crystal doped with C.I. Acid Red 2,” Dyes Pigm. in press., doi:.

11.

D. W. Berreman, “Solid surface shape and the alignment of an adjacent nematic liquid crystal,” Phys. Rev. Lett. 28(26), 1683–1686 (1972). [CrossRef]

12.

J. Fukuda, M. Yoneya, and H. Yokoyama, “Consistent numerical evaluation of the anchoring energy of a grooved surface,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(1 Pt 1), 011705 (2009). [CrossRef] [PubMed]

OCIS Codes
(160.3710) Materials : Liquid crystals
(230.3720) Optical devices : Liquid-crystal devices
(300.1030) Spectroscopy : Absorption

ToC Category:
Optical Devices

History
Original Manuscript: June 8, 2010
Revised Manuscript: July 22, 2010
Manuscript Accepted: August 7, 2010
Published: September 3, 2010

Citation
Kuang-Yu Yang and Wei Lee, "Voltage-assisted photoaligning effect of an azo dye doped in a liquid crystal with negative dielectric anisotropy," Opt. Express 18, 19914-19919 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-19-19914


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References

  1. V. G. Chigrinov, V. M. Kozenkov, and H. S. Kwok, Photoalignment of Liquid Crystalline Materials: Physics and Applications (Wiley, 2008).
  2. T.-J. Chen and K.-L. Chu, “Pretilt angle control for single-cell-gap transflective liquid crystal cells,” Appl. Phys. Lett. 92(9), 091102 (2008). [CrossRef]
  3. L. Komitov, C. Ruslim, Y. Matsuzawa, and K. Ichimura, “Photoinduced anchoring transitions in a nematic doped with azo dyes,” Liq. Cryst. 27(8), 1011–1016 (2000). [CrossRef]
  4. W. M. Gibbons, P. J. Shannon, S.-T. Sun, and B. J. Swetlin, “Surface-mediated alignment of nematic liquid crystals with polarized laser light,” Nature 351(6321), 49–50 (1991). [CrossRef]
  5. E. Ouskova, D. Fedorenko, Yu. Reznikov, S. V. Shiyanovskii, L. Su, J. L. West, O. V. Kuksenok, O. Francescangeli, and F. Simoni, “Hidden photoalignment of liquid crystals in the isotropic phase,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(2 Pt 1), 021701 (2001). [CrossRef] [PubMed]
  6. A. Y.-G. Fuh, C.-C. Liao, K.-C. Hsu, and C.-L. Lu, “Laser-induced reorientation effect and ripple structure in dye-doped liquid-crystal films,” Opt. Lett. 28(14), 1179–1181 (2003). [CrossRef] [PubMed]
  7. A. Y.-G. Fuh, C.-K. Liu, K.-T. Cheng, C.-L. Ting, C.-C. Chen, P. C.-P. Chao, and H.-K. Hsu, “Chao, and H.-K. Hsu, “Variable liquid crystal pretilt angles generated by photoalignment in homeotropically aligned azo dye-doped liquid crystals,” Appl. Phys. Lett. 95(16), 161104 (2009). [CrossRef]
  8. F. J. Kahn, G. N. Taylor, and H. Schonhorn, “Surface-produced alignment of liquid crystals,” Proc. IEEE 61(7), 823–828 (1973). [CrossRef]
  9. J. S. Gwag, S. H. Lee, K.-H. Park, W. S. Park, K.-Y. Han, C. G. Jhun, T.-H. Yoon, J. C. Kim, D.- Song, and D.- Shin, “Simple method for measuring the high pretilt angle of nematic liquid crystals,” J. Appl. Phys. 93(8), 4936 (2003). [CrossRef]
  10. C.-Y. Tang, S.-M. Huang, and W. Lee, “Dielectric relaxation dynamics in a dual-frequency nematic liquid crystal doped with C.I. Acid Red 2,” Dyes Pigm. in press., doi:.
  11. D. W. Berreman, “Solid surface shape and the alignment of an adjacent nematic liquid crystal,” Phys. Rev. Lett. 28(26), 1683–1686 (1972). [CrossRef]
  12. J. Fukuda, M. Yoneya, and H. Yokoyama, “Consistent numerical evaluation of the anchoring energy of a grooved surface,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(1 Pt 1), 011705 (2009). [CrossRef] [PubMed]

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