Abstract

We investigate the electro-optical response of an in-plane switching (IPS) liquid crystal (LC) device, which is fabricated using the two-easy-axes substrate. The two-easy-axes substrate is fabricated by slightly rubbing the substrate in two different directions. Experimental results indicate that the IPS LC device fabricated using the two-easy-axes substrate has a lower threshold voltage and a faster response time than the traditional IPS LC device, which is fabricated using the unidirectionally rubbed substrate. The weak anchoring condition and the anchoring strengths in two different rubbing directions on the substrate contribute to the fast electro-optical response of the IPS LC device.

©2006 Optical Society of America

1. Introduction

Liquid crystals (LCs) are now widely studied for their promising possibility in display applications. Various display modes are developed according to the applications. [12] Among the display modes, the twisted nematic (TN) mode, first reported more than three decades ago, in their first decade of use, was adopted in watches and electronic calculators due to its advantages of simple structure, small size, low operation voltage and high contrast ratio. One serious problem in these display modes, including the TN mode, lies in the limited viewing angle due to the birefringence effect in LCs. In order to realize larger display area, it is necessary to overcome the limited viewing angle characteristics of liquid crystal displays (LCDs). The multi-domain mode, compensation with birefringence film, the optically compensated bend mode and the in-plane switching (IPS) mode have all been actively studied to overcome this problem. Among them, the IPS mode utilizes homogeneously aligned LCs which respond to an electric field parallel to the substrate. Because no out-of-plane reorientation of the LC molecules occurs as in the TN LCD, optical inversions are not present. Hence, the IPS mode provides much better viewing angle characteristics than any other modes.

The IPS mode, first investigated in the 1970’s, [34] has only seen recent application in desktop displays because of reduced optical throughput, higher drive voltage and slow dynamic response time compared to TN LCDs. Since the IPS mode’s primary application is in desktop displays, and not in power sensitive portable devices, the first two issues can be tolerated. However, the slow dynamic response time prevents true video capability and remains the primary challenge. Many efforts were investigated to improve IPS technology: Ohta et al. attempted to optimize various IPS mode tradeoffs; [56] Wakemoto et al. developed cross-talk free and low-voltage IPS mode devices. [7] Lien et al. simulated various IPS configurations.[8] Yoneya et al. investigated the electro-optical characteristics of the IPS LCDs using weak anchoring effect; [9] and more recently, efforts to reduce response time through viscosity reduction, [1112] the addition of a chiral agent and monomer, [1314] and the rubbing angle optimization, [15] were reported.

The alignment and electro-optical properties of LCs on a surface with two easy axes were recently investigated. Kim et al. reported that LCs became aligned along an axis between the two easy axes, and explained the observed results using the microgroove model. [15] Chung et al. studied the orientation of LCs on a treated substrate with two easy axes, and determined the anchoring energy of the treated surface. [16] Yamaguchi et al. reported a two-easy-axes substrate, which was induced by unidirectional rubbing, and demonstrated a micro-patterning technique using this substrate. [17] Barberi et al. reported a bistable nematic device induced by two independent surface anchoring attractors. [18] Mahajan et al. reported that in a two directionally rubbed substrate, the polyimide (PI) film which had been partially disentangled by the initially rubbing, need only weaker rubbing to be reoriented by the second rubbing. [19] In earlier works, we studied the characteristics of the two-easy-axes cell. The results revealed that the elastic constant and the viscosity of LCs, the rubbing strength ratio between the two different rubbing directions on the substrate dominated the orientation, birefringence and order parameter of LCs. [2021] In this study, the electro-optical response of the IPS LC device fabricated using two-easy-axes substrate were investigated and discussed.

2. Experimental

The following steps were followed to prepare a two-easy-axes substrate. First, an indium-tin-oxide (ITO) substrate was spin-coated with PI AL58 (from Daily Polymer, Taiwan) and rubbed m times slightly and unidirectionally in the x direction. Then, the substrate was rotated through a rubbing angle 90°, and rubbed n times slightly in the y direction. Such a substrate was referred to as an [m, n]-rubbed substrate. The [m, n]-rubbed interdigitated electrode substrate was prepared using the similar process. The interdigitated electrode substrate has an electrode width of 48 µm, an electrode gap of 72 µm, conducted with wet etching technique. The interdigitated electrodes are parallel to x direction. Increasing the number of rubbing was assumed to increase the rubbing strength. The IPS LC cell was made from an [m, n]-rubbed substrate on the top and an [m, n]-rubbed interdigitated electrode substrate on the bottom. The cell thickness was ~6 µm, and the rubbing directions of the top and the bottom substrates were antiparallel. The geometry of the IPS LC cell was shown in Fig. 1. The empty cell was filled with LC ZLI-2806 (from Merck) by capillary action.

 

Fig. 1. Geometry of the [m, n]-rubbed IPS LC device. In the figure, 1st indicates the first rubbing direction and 2nd indicates the second rubbing direction.

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As indicated in our previous works, [2021] in a two-easy-axes cell, LC molecules align along an axis intermediate between the two different rubbing directions. The ratio of the rubbing strength between the two different directions markedly influences the final orientation of LCs. A He-Ne laser with a wavelength of 632.8 nm was used as a probe beam to evaluate the orientation and the electro-optical response of the two-easy-axes IPS LC cell. The deviation angle φ from y axis (the second rubbing direction in an [m, n]-rubbed cell) in which the transmission was minimal, was determined after the sample was rotated between two cross polarizers. The transmission intensity, field-on response time (rise time) and field-off response time (fall time) of the [m, n]-rubbed IPS cell, which was placed between a pair of cross polarizers were measured. For reference, a slightly unidirectionally rubbed IPS cell whose deviation angle was the same as that of the [m, n]-rubbed IPS cell was fabricated and measured. The unidirectionally rubbed IPS cell was referred to as a [0, n]-rubbed IPS cell. In this experiment n was set to be 1.

3. Results and discussion

Figure 2 plots the measured deviation angle φ as a function of m. In this experiment, n is set to be 1. The figure indicates that the deviation angle φ increases with m, suggesting that the LC molecules tend to align in the direction of the final rubbing, but are strongly constrained by a force in the direction of the first rubbing. As shown in the figure, the increase of φ is not linear. The measured deviation angle φ is below and deviated from the solid line connecting the origin and the deviation angle of the [1, 1]-rubbed cell, indicating that when the PI film is more disentangled by the first rubbing, the anchoring strength of the final rubbing is stronger and the LCs are easily reoriented toward the final rubbing direction. Figures 3 and 4 show the transmission versus applied voltage (T-V) curves and the electro-optical responses of the [1, 1] and [0, 1]-rubbed IPS cells, respectively, when the deviation angle is ~8°. As shown in the figures, the threshold voltage and the rise time of the [1, 1]-rubbed cell are lower, and the fall time is higher than those of the [0, 1]-rubbed cell. The decrease in the threshold voltage is not evident in this experiment. The threshold voltage of the cell is defined as the voltage that gives 10% of the maximum transmission of a cell in its T-V curve.

 

Fig. 2. Measured deviation angle of the two-easy-axes IPS LC cell as a function of m. In this experiment, n is set to be 1.

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Fig. 3. Measured transmission versus applied voltage (T-V) curves of the [1, 1] and [0, 1]- rubbed IPS cells. The deviation angle is ~8°.

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Fig. 4. Measured electro-optical responses of the [1, 1] and [0, 1]-rubbed IPS cells. The deviation angle is ~8°.

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In this paper, two possible mechanisms were employed to explain the obtained results. The first one is the two-easy-axes model. As shown in Fig. 1, in an [m, n]-rubbed cell, the LC molecules align in a direction intermediate between the two rubbing directions. The deviation angle is φ. When the applied voltage is turned on, the LC molecules rotate toward perpendicular to the electric field due to the negative dielectric anisotropy of LCs. The reorientation of the LC molecules is constrained by the anchoring strengths in the two different rubbing directions. The [1, 1]-rubbed cell indicates that the 1 time rubbing in the first rubbing direction slightly disentangles the polymer chain and the 1 time rubbing in the second rubbing direction aligns the LC director toward the second rubbing direction. The obtained deviation angle of ~8° indicates that the anchoring strength in the second rubbing direction is apparently higher than that in the first rubbing direction. The anchoring strength in the second rubbing direction impedes the reorientation of LCs. Thus, the threshold voltage and the rise time of the [1, 1]-rubbed cell will be higher, the fall time of the [m, n]-rubbed cell will be lower than those of the [0, 1]-rubbed cell. However, the obtained results of Figs. 3 and 4 indicate that the two-easy-axes model is not suitable to explain the electro-optical response of LCs in an [1, 1]-rubbed IPS cell.

Another possible mechanism is the weak anchoring model. In an [m, n]-rubbed substrate, the multiple rubbings along different directions erase, disentangle and disturb the rubbing grooves and the polymer chains, reducing the anchoring strength and increasing the extrapolation length of the cell. For an IPS LC cell fabricated using the weak anchoring substrate, the threshold voltage Vth is modified as [910]

Vth=π·ld+2bk22ε0Δε,

where k22 is twist elastic constant and Δε is the dielectric anisotropy of the LC mixture; l is the electrode gap, d is the cell thickness and b is the extrapolation length of the substrate. In Eq. (3.1), the extrapolation length b is an increment of effective cell gap that corresponds to nonzero LC twist at the LC/alignment layer interface. This extrapolation length b is related to the twist elastic constant k22 and the in-plane twist anchoring coefficient A at the LC/alignment layer interface in the following equation:

b=k22A.

Therefore, due to the slightly rubbing treatment on the substrate, the reduction of the threshold voltage in an [1, 1]-rubbed cell is attributable to the weak anchoring characteristics of the substrate. Similarly, the rise time τon and fall time τoff of an IPS LC cell fabricated using weak anchoring substrate are also reasonably modified by the extrapolation length b; i.e.,

τon=γ1ε0ΔεE2π2(d+2b)2k22,
τoff=γ1(d+2b)2π2k22.

Equations (3.3) and (3.4) indicate that decreasing the anchoring strength or increasing the extrapolation length b of the substrate reduces the rise time and increases the fall time of the cell. Figure 4 completely conforms to the deduction of Eqs. (3.3) and (3.4). The slightly rubbing treatment barely disturbs PI film and thus results in a weak anchoring substrate. Therefore, in an [1, 1]-rubbed cell, two-easy-axes effect is not apparent. The total response time (rise time and fall time) of the [1, 1]-rubbed cell is the same as that of the [0, 1]-rubbed cell. Figure 5 shows the measured voltage dependent rise times of the [1, 3] and [0, 1]-rubbed IPS LC cells. The deviation angle is ~5°. In an [1, 3]-rubbed substrate, the 1 time rubbing in the first rubbing direction disentangles the polymer chain, and the 3 times rubbing in the second rubbing direction results in a stronger anchoring strength on the [1, 3]-rubbed substrate than that on the [0, 1]-rubbed substrate. Therefore, the rise time of the [0, 1]-rubbed cell is lower than that of the [1, 3]-rubbed cell when the applied voltage is low. As the applied voltage further increases, the applied electric field dominates the reorientation of LCs, the contribution of the surface anchoring to the reorientation of LCs is not apparent, and the rise times of the [1, 3] and [0, 1]-rubbed cells become the same. The threshold voltage and the fall time of the [1, 3]-rubbed cell are the same as those of the [0, 1]-rubbed cell.

 

Fig. 5. Measured voltage dependent rise times of the [1, 3] and [0, 1]-rubbed IPS cells. The deviation angle is ~5°.

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Figure 6 shows the electro-optical responses of the [3, 1] and [0, 1]-rubbed IPS cells. The inset shows the total response times of the [3, 1] and [0, 1]-rubbed IPS cells. The deviation angle is ~20°. The 3 times rubbing in the first rubbing direction significantly disentangles the polymer chain. Therefore, the PI polymer chain is easy to be realigned, indicating that the rubbing strength in the second rubbing direction of the [3, 1]-rubbed substrate is higher than that of the [1, 1]-rubbed substrate. However, the high deviation angle (~20°) in the [3, 1]- rubbed cell indicates that the rubbing strength in the second rubbing direction is only slightly higher than that in the first rubbing direction. As shown in Fig. 6, the rise time of the [3, 1]- rubbed cell is lower than that of the [0, 1]-rubbed cell when the applied voltage is low. This is because that the [3, 1]-rubbed substrate has a weak anchoring strength. Another reason is that because the rubbing strength in the second rubbing direction is only slightly higher than that in the first rubbing direction; the resultant torque that impedes the reorientation of LCs is small when the applied voltage is turned on. As the applied voltage further increases, the contribution of the surface anchoring to the reorientation of LCs is not evident. The rise time of the [0, 1]-rubbed cell becomes lower than that of the [3, 1]-rubbed cell due to the uniform alignment and hence the higher dielectric anisotropy of LCs in the [0, 1]-rubbed cell. The fall time of the [3, 1]-rubbed cell is lower than that of the [0, 1]-rubbed cell. This is a counter-intuitive result in a cell fabricated using a weak anchoring substrate. This is because that the anchoring strength in the second rubbing direction assists the reorientation of LCs when the applied voltage is turned off. Another reason is the frustration of LCs next to the substrate. Unidirectional rubbing causes the LC molecules to align unidirectionally along a given direction. However, following rubbing in two different directions, the LC molecules receive the surface induced anchoring strengths from the rubbing in the two different directions. The energetic competition between the two rubbing directions, associated with rubbing, results in the stochastic random distribution of the LC molecules near the substrate. Therefore, the two directional rubbings hinder the LC molecules from aligning unidirectionally near the substrate,[2022] decreasing the extent of the LC reorientation when the external voltage was applied, [14] and hence decreasing the fall time of the cell. The total response time of the [3, 1]-rubbed cell is lower than that of the [0, 1]-rubbed cell. Figure 7 shows the measured T-V curves of the [3, 1] and [0, 1]-rubbed cells. The slow increase of the T-V curve indicates that LCs in the [3, 1]-rubbed cell are not as uniform as those in the [0, 1]-rubbed cell. Figure 8 shows the optical microscope images of the [0, 1], [1, 1], [1, 3] and [3, 1]-rubbed cells. These images show that rubbing in two different directions frustrates the LC molecules next to the substrate, reducing the effective dielectric anisotropy and generating bright spots in the cell. The obtained results explain that the transmission of the [m, n]-rubbed cell at zero voltage is higher than that of the [0, 1]-rubbed cell at zero voltage. The poor dark state of the [m, n]-rubbed cell at zero voltage can be improved by using the proper alignment material or decreasing the rubbing angle. [22]

 

Fig. 6. Measured electro-optical responses of the [3, 1] and [0, 1]-rubbed IPS cells. Inset shows the total response times of the [3, 1] and [0, 1]-rubbed IPS cells. The deviation angle is ~20°.

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Fig. 7. Measured T-V curves of the [3, 1] and [0, 1]-rubbed cells.

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Fig. 8. Optical microscope images of the [0, 1], [1, 1], [1, 3] and [3, 1]-rubbed cells. In the figure, 1st indicates the first rubbing direction and 2nd indicates the second rubbing direction.

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4. Conclusion

In this paper, we investigate the electro-optical response of an IPS LC device, which is fabricated using the two-easy-axes substrate. The two-easy-axes substrate is fabricated by slightly rubbing the substrate in two different directions. Experimental results indicate that the IPS LC device fabricated using the two-easy-axes substrate has a lower threshold voltage and a faster response time than the traditional IPS LC device, which is fabricated using the unidirectionally rubbed substrate. The weak anchoring condition and the anchoring strength in two different rubbing directions on the substrate contribute to the fast electro-optical response of the IPS LC device. The obtained results also conform to the simulated results of Ref. [14]; i.e., in a [0, 1]-rubbed cell, the rise time of the IPS LC cell is decreased with increasing rubbing angle, and the fall time is independent on the rubbing angle. The rubbing angle of zero indicates that the LC director is perpendicular to the applied electric field.

Acknowledgment

The authors would like to thank the National Science Council of the Republic of China, Taiwan, for financially supporting this research under Contract Nos. NSC 94-2112-M-018-006 and NSC 95-2112-M-018-005-MY3.

References and links

1. E. Lueder, Liquid Crystal Displays, Addressing Schemes and Electro-optical Effects (John Wiley and Sons, Singapore, 2001).

2. S. T. Wu and D. K. Yang, Reflective Liquid Crystal Displays (John Wiley and Sons, Singapore, 2001).

3. T. Shimojyo, “Field-effect LCDs with interdigital electrodes,” Proc. Inf. Disp. 14, 110 (1973).

4. R. A. Soref, “Field effects in nematic liquid crystals obtained with interdigital electrodes,” J. Appl. Phys. 45, 5466 (1974). [CrossRef]  

5. M. Ohta, M. Oh-E, and K. Kondo, “Development of super-TFT-LCDs with in-plane switching mode,” Proc. Asia Disp. 95, 707 (1995).

6. K. Oh-E and Kondo, “Electro-optical characteristics and switching behavior of the in-plane switching mode,” Appl. Phys. Lett. 67, 3895 (1995). [CrossRef]  

7. H. Wakemoto, S. Asada, K. Kato, Y. Yamamoto, T. Tsukane, K. Tsuda, and Y. Yakubo, “An advanced in-plane switching mode TFT-LCD,” SID Digest 28, 929 (1997).

8. A. Lien and C. J. Chen, “2-D simulations of in-plane switching mode LCDs,” SID Digest 27, 175 (1996).

9. M. Yoneya, K. Iwasaki, Y. Tomioka, and K. Kondo, “Cell gap margin enlargement of in-plane switching mode liquid crystal displays using weak-anchoring effects,” Appl. Phys. Lett. 74, 803 (1999). [CrossRef]  

10. P. G. de Gennes, The Physics of Liquid Crystals (Clarendon, Oxford, 1975).

11. K. Kondo, S. Matsuyama, N. Konishi, and H. Kawakami, “Materials and components optimization for IPS TFT-LCDs,” SID Digest 29, 389 (1998). [CrossRef]  

12. D. Klement and K. Tarumi, “Liquid crystal material development of IPS-TFT displays,” SID Digest 28, 393 (1998). [CrossRef]  

13. M. Hasegawa, “Response time improvement of the in-plane-switching mode,” SID Digest 28, 699 (1997).

14. M. J. Escuti, C. C. Bowley, and G. P. Crawford, “Enhanced dynamic response of the in-plane switching liquid crystal display mode through polymer stabilization,” Appl. Phys. Lett. 75, 3264 (1999). [CrossRef]  

15. Y. J. Kim, Z. Zhuang, and J. S. Patel, “Effect of multidirection rubbing on the alignment of nematic liquid crystal,” Appl. Phys. Lett. 77, 513 (2000). [CrossRef]  

16. D. H. Chung, Y. Takanishi, K. Ishikawa, C. J. Yu, S. D. Lee, and H. Takezoe, “Alignment control of a nematic liquid crystal on a doubly treated substrate,” Jpn. J. Appl. Phys. 42, 1686 (2003). [CrossRef]  

17. R. Yamaguchi, Y. Goto, and S. Sato, “Liquid crystal alignment surface with two easy axes induced by unidirectional rubbing,” Appl. Phys. Lett. 82, 4450 (2003). [CrossRef]  

18. R. Barberi, J. J. Bonvent, M. Giocondo, M. Iovane, and A. L. Alexe-Ionescu. “Bistable nematic azimuthal alignment induced by anchoring competition,” J. Appl. Phys. 84, 1321 (1998). [CrossRef]  

19. M. P. Mahajan and C. Rosenblatt, “History-dependent orientational order of rubbed polyimide for liquid-crystal alignment,” Appl. Phys. Lett. 75, 3623 (1999). [CrossRef]  

20. C. Y. Huang, C. H. Lin, J. R. Wang, C. W. Huang, M. S. Tsai, and A. Y. G. Fuh, “Multidirectional rubbed liquid-crystal cells,” J. Appl. Phys. 92, 7231 (2002). [CrossRef]  

21. C. Y. Huang, Y. S. Huang, and J. R. Tian, “Alignment of liquid crystals on two-easy-axis substrate,” Jpn. J. Appl. Phys. 45, 168 (2006). [CrossRef]  

22. C. Y. Huang, Y. S. Huang, J. R. Tian, and S. J. Yeh, “Influence of the two-directionally-rubbed substrate on the alignment of liquid crystals,” Opt. Comm. 265, 241 (2006). [CrossRef]  

References

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  1. E. Lueder, Liquid Crystal Displays, Addressing Schemes and Electro-optical Effects (John Wiley and Sons, Singapore, 2001).
  2. S. T. Wu and D. K. Yang, Reflective Liquid Crystal Displays (John Wiley and Sons, Singapore, 2001).
  3. T. Shimojyo, “Field-effect LCDs with interdigital electrodes,” Proc. Inf. Disp. 14, 110 (1973).
  4. R. A. Soref, “Field effects in nematic liquid crystals obtained with interdigital electrodes,” J. Appl. Phys. 45, 5466 (1974).
    [Crossref]
  5. M. Ohta, M. Oh-E, and K. Kondo, “Development of super-TFT-LCDs with in-plane switching mode,” Proc. Asia Disp. 95, 707 (1995).
  6. K. Oh-E and Kondo, “Electro-optical characteristics and switching behavior of the in-plane switching mode,” Appl. Phys. Lett. 67, 3895 (1995).
    [Crossref]
  7. H. Wakemoto, S. Asada, K. Kato, Y. Yamamoto, T. Tsukane, K. Tsuda, and Y. Yakubo, “An advanced in-plane switching mode TFT-LCD,” SID Digest 28, 929 (1997).
  8. A. Lien and C. J. Chen, “2-D simulations of in-plane switching mode LCDs,” SID Digest 27, 175 (1996).
  9. M. Yoneya, K. Iwasaki, Y. Tomioka, and K. Kondo, “Cell gap margin enlargement of in-plane switching mode liquid crystal displays using weak-anchoring effects,” Appl. Phys. Lett. 74, 803 (1999).
    [Crossref]
  10. P. G. de Gennes, The Physics of Liquid Crystals (Clarendon, Oxford, 1975).
  11. K. Kondo, S. Matsuyama, N. Konishi, and H. Kawakami, “Materials and components optimization for IPS TFT-LCDs,” SID Digest 29, 389 (1998).
    [Crossref]
  12. D. Klement and K. Tarumi, “Liquid crystal material development of IPS-TFT displays,” SID Digest 28, 393 (1998).
    [Crossref]
  13. M. Hasegawa, “Response time improvement of the in-plane-switching mode,” SID Digest 28, 699 (1997).
  14. M. J. Escuti, C. C. Bowley, and G. P. Crawford, “Enhanced dynamic response of the in-plane switching liquid crystal display mode through polymer stabilization,” Appl. Phys. Lett. 75, 3264 (1999).
    [Crossref]
  15. Y. J. Kim, Z. Zhuang, and J. S. Patel, “Effect of multidirection rubbing on the alignment of nematic liquid crystal,” Appl. Phys. Lett. 77, 513 (2000).
    [Crossref]
  16. D. H. Chung, Y. Takanishi, K. Ishikawa, C. J. Yu, S. D. Lee, and H. Takezoe, “Alignment control of a nematic liquid crystal on a doubly treated substrate,” Jpn. J. Appl. Phys.  42, 1686 (2003).
    [Crossref]
  17. R. Yamaguchi, Y. Goto, and S. Sato, “Liquid crystal alignment surface with two easy axes induced by unidirectional rubbing,” Appl. Phys. Lett. 82, 4450 (2003).
    [Crossref]
  18. R. Barberi, J. J. Bonvent, M. Giocondo, M. Iovane, and A. L. Alexe-Ionescu. “Bistable nematic azimuthal alignment induced by anchoring competition,” J. Appl. Phys. 84, 1321 (1998).
    [Crossref]
  19. M. P. Mahajan and C. Rosenblatt, “History-dependent orientational order of rubbed polyimide for liquid-crystal alignment,” Appl. Phys. Lett. 75, 3623 (1999).
    [Crossref]
  20. C. Y. Huang, C. H. Lin, J. R. Wang, C. W. Huang, M. S. Tsai, and A. Y. G. Fuh, “Multidirectional rubbed liquid-crystal cells,” J. Appl. Phys. 92, 7231 (2002).
    [Crossref]
  21. C. Y. Huang, Y. S. Huang, and J. R. Tian, “Alignment of liquid crystals on two-easy-axis substrate,” Jpn. J. Appl. Phys. 45, 168 (2006).
    [Crossref]
  22. C. Y. Huang, Y. S. Huang, J. R. Tian, and S. J. Yeh, “Influence of the two-directionally-rubbed substrate on the alignment of liquid crystals,” Opt. Comm. 265, 241 (2006).
    [Crossref]

2006 (2)

C. Y. Huang, Y. S. Huang, and J. R. Tian, “Alignment of liquid crystals on two-easy-axis substrate,” Jpn. J. Appl. Phys. 45, 168 (2006).
[Crossref]

C. Y. Huang, Y. S. Huang, J. R. Tian, and S. J. Yeh, “Influence of the two-directionally-rubbed substrate on the alignment of liquid crystals,” Opt. Comm. 265, 241 (2006).
[Crossref]

2003 (2)

D. H. Chung, Y. Takanishi, K. Ishikawa, C. J. Yu, S. D. Lee, and H. Takezoe, “Alignment control of a nematic liquid crystal on a doubly treated substrate,” Jpn. J. Appl. Phys.  42, 1686 (2003).
[Crossref]

R. Yamaguchi, Y. Goto, and S. Sato, “Liquid crystal alignment surface with two easy axes induced by unidirectional rubbing,” Appl. Phys. Lett. 82, 4450 (2003).
[Crossref]

2002 (1)

C. Y. Huang, C. H. Lin, J. R. Wang, C. W. Huang, M. S. Tsai, and A. Y. G. Fuh, “Multidirectional rubbed liquid-crystal cells,” J. Appl. Phys. 92, 7231 (2002).
[Crossref]

2000 (1)

Y. J. Kim, Z. Zhuang, and J. S. Patel, “Effect of multidirection rubbing on the alignment of nematic liquid crystal,” Appl. Phys. Lett. 77, 513 (2000).
[Crossref]

1999 (3)

M. J. Escuti, C. C. Bowley, and G. P. Crawford, “Enhanced dynamic response of the in-plane switching liquid crystal display mode through polymer stabilization,” Appl. Phys. Lett. 75, 3264 (1999).
[Crossref]

M. P. Mahajan and C. Rosenblatt, “History-dependent orientational order of rubbed polyimide for liquid-crystal alignment,” Appl. Phys. Lett. 75, 3623 (1999).
[Crossref]

M. Yoneya, K. Iwasaki, Y. Tomioka, and K. Kondo, “Cell gap margin enlargement of in-plane switching mode liquid crystal displays using weak-anchoring effects,” Appl. Phys. Lett. 74, 803 (1999).
[Crossref]

1998 (3)

K. Kondo, S. Matsuyama, N. Konishi, and H. Kawakami, “Materials and components optimization for IPS TFT-LCDs,” SID Digest 29, 389 (1998).
[Crossref]

D. Klement and K. Tarumi, “Liquid crystal material development of IPS-TFT displays,” SID Digest 28, 393 (1998).
[Crossref]

R. Barberi, J. J. Bonvent, M. Giocondo, M. Iovane, and A. L. Alexe-Ionescu. “Bistable nematic azimuthal alignment induced by anchoring competition,” J. Appl. Phys. 84, 1321 (1998).
[Crossref]

1997 (2)

H. Wakemoto, S. Asada, K. Kato, Y. Yamamoto, T. Tsukane, K. Tsuda, and Y. Yakubo, “An advanced in-plane switching mode TFT-LCD,” SID Digest 28, 929 (1997).

M. Hasegawa, “Response time improvement of the in-plane-switching mode,” SID Digest 28, 699 (1997).

1996 (1)

A. Lien and C. J. Chen, “2-D simulations of in-plane switching mode LCDs,” SID Digest 27, 175 (1996).

1995 (2)

M. Ohta, M. Oh-E, and K. Kondo, “Development of super-TFT-LCDs with in-plane switching mode,” Proc. Asia Disp. 95, 707 (1995).

K. Oh-E and Kondo, “Electro-optical characteristics and switching behavior of the in-plane switching mode,” Appl. Phys. Lett. 67, 3895 (1995).
[Crossref]

1974 (1)

R. A. Soref, “Field effects in nematic liquid crystals obtained with interdigital electrodes,” J. Appl. Phys. 45, 5466 (1974).
[Crossref]

1973 (1)

T. Shimojyo, “Field-effect LCDs with interdigital electrodes,” Proc. Inf. Disp. 14, 110 (1973).

Alexe-Ionescu, A. L.

R. Barberi, J. J. Bonvent, M. Giocondo, M. Iovane, and A. L. Alexe-Ionescu. “Bistable nematic azimuthal alignment induced by anchoring competition,” J. Appl. Phys. 84, 1321 (1998).
[Crossref]

Asada, S.

H. Wakemoto, S. Asada, K. Kato, Y. Yamamoto, T. Tsukane, K. Tsuda, and Y. Yakubo, “An advanced in-plane switching mode TFT-LCD,” SID Digest 28, 929 (1997).

Barberi, R.

R. Barberi, J. J. Bonvent, M. Giocondo, M. Iovane, and A. L. Alexe-Ionescu. “Bistable nematic azimuthal alignment induced by anchoring competition,” J. Appl. Phys. 84, 1321 (1998).
[Crossref]

Bonvent, J. J.

R. Barberi, J. J. Bonvent, M. Giocondo, M. Iovane, and A. L. Alexe-Ionescu. “Bistable nematic azimuthal alignment induced by anchoring competition,” J. Appl. Phys. 84, 1321 (1998).
[Crossref]

Bowley, C. C.

M. J. Escuti, C. C. Bowley, and G. P. Crawford, “Enhanced dynamic response of the in-plane switching liquid crystal display mode through polymer stabilization,” Appl. Phys. Lett. 75, 3264 (1999).
[Crossref]

Chen, C. J.

A. Lien and C. J. Chen, “2-D simulations of in-plane switching mode LCDs,” SID Digest 27, 175 (1996).

Chung, D. H.

D. H. Chung, Y. Takanishi, K. Ishikawa, C. J. Yu, S. D. Lee, and H. Takezoe, “Alignment control of a nematic liquid crystal on a doubly treated substrate,” Jpn. J. Appl. Phys.  42, 1686 (2003).
[Crossref]

Crawford, G. P.

M. J. Escuti, C. C. Bowley, and G. P. Crawford, “Enhanced dynamic response of the in-plane switching liquid crystal display mode through polymer stabilization,” Appl. Phys. Lett. 75, 3264 (1999).
[Crossref]

de Gennes, P. G.

P. G. de Gennes, The Physics of Liquid Crystals (Clarendon, Oxford, 1975).

Escuti, M. J.

M. J. Escuti, C. C. Bowley, and G. P. Crawford, “Enhanced dynamic response of the in-plane switching liquid crystal display mode through polymer stabilization,” Appl. Phys. Lett. 75, 3264 (1999).
[Crossref]

Fuh, A. Y. G.

C. Y. Huang, C. H. Lin, J. R. Wang, C. W. Huang, M. S. Tsai, and A. Y. G. Fuh, “Multidirectional rubbed liquid-crystal cells,” J. Appl. Phys. 92, 7231 (2002).
[Crossref]

Giocondo, M.

R. Barberi, J. J. Bonvent, M. Giocondo, M. Iovane, and A. L. Alexe-Ionescu. “Bistable nematic azimuthal alignment induced by anchoring competition,” J. Appl. Phys. 84, 1321 (1998).
[Crossref]

Goto, Y.

R. Yamaguchi, Y. Goto, and S. Sato, “Liquid crystal alignment surface with two easy axes induced by unidirectional rubbing,” Appl. Phys. Lett. 82, 4450 (2003).
[Crossref]

Hasegawa, M.

M. Hasegawa, “Response time improvement of the in-plane-switching mode,” SID Digest 28, 699 (1997).

Huang, C. W.

C. Y. Huang, C. H. Lin, J. R. Wang, C. W. Huang, M. S. Tsai, and A. Y. G. Fuh, “Multidirectional rubbed liquid-crystal cells,” J. Appl. Phys. 92, 7231 (2002).
[Crossref]

Huang, C. Y.

C. Y. Huang, Y. S. Huang, and J. R. Tian, “Alignment of liquid crystals on two-easy-axis substrate,” Jpn. J. Appl. Phys. 45, 168 (2006).
[Crossref]

C. Y. Huang, Y. S. Huang, J. R. Tian, and S. J. Yeh, “Influence of the two-directionally-rubbed substrate on the alignment of liquid crystals,” Opt. Comm. 265, 241 (2006).
[Crossref]

C. Y. Huang, C. H. Lin, J. R. Wang, C. W. Huang, M. S. Tsai, and A. Y. G. Fuh, “Multidirectional rubbed liquid-crystal cells,” J. Appl. Phys. 92, 7231 (2002).
[Crossref]

Huang, Y. S.

C. Y. Huang, Y. S. Huang, J. R. Tian, and S. J. Yeh, “Influence of the two-directionally-rubbed substrate on the alignment of liquid crystals,” Opt. Comm. 265, 241 (2006).
[Crossref]

C. Y. Huang, Y. S. Huang, and J. R. Tian, “Alignment of liquid crystals on two-easy-axis substrate,” Jpn. J. Appl. Phys. 45, 168 (2006).
[Crossref]

Iovane, M.

R. Barberi, J. J. Bonvent, M. Giocondo, M. Iovane, and A. L. Alexe-Ionescu. “Bistable nematic azimuthal alignment induced by anchoring competition,” J. Appl. Phys. 84, 1321 (1998).
[Crossref]

Ishikawa, K.

D. H. Chung, Y. Takanishi, K. Ishikawa, C. J. Yu, S. D. Lee, and H. Takezoe, “Alignment control of a nematic liquid crystal on a doubly treated substrate,” Jpn. J. Appl. Phys.  42, 1686 (2003).
[Crossref]

Iwasaki, K.

M. Yoneya, K. Iwasaki, Y. Tomioka, and K. Kondo, “Cell gap margin enlargement of in-plane switching mode liquid crystal displays using weak-anchoring effects,” Appl. Phys. Lett. 74, 803 (1999).
[Crossref]

Kato, K.

H. Wakemoto, S. Asada, K. Kato, Y. Yamamoto, T. Tsukane, K. Tsuda, and Y. Yakubo, “An advanced in-plane switching mode TFT-LCD,” SID Digest 28, 929 (1997).

Kawakami, H.

K. Kondo, S. Matsuyama, N. Konishi, and H. Kawakami, “Materials and components optimization for IPS TFT-LCDs,” SID Digest 29, 389 (1998).
[Crossref]

Kim, Y. J.

Y. J. Kim, Z. Zhuang, and J. S. Patel, “Effect of multidirection rubbing on the alignment of nematic liquid crystal,” Appl. Phys. Lett. 77, 513 (2000).
[Crossref]

Klement, D.

D. Klement and K. Tarumi, “Liquid crystal material development of IPS-TFT displays,” SID Digest 28, 393 (1998).
[Crossref]

Kondo,

K. Oh-E and Kondo, “Electro-optical characteristics and switching behavior of the in-plane switching mode,” Appl. Phys. Lett. 67, 3895 (1995).
[Crossref]

Kondo, K.

M. Yoneya, K. Iwasaki, Y. Tomioka, and K. Kondo, “Cell gap margin enlargement of in-plane switching mode liquid crystal displays using weak-anchoring effects,” Appl. Phys. Lett. 74, 803 (1999).
[Crossref]

K. Kondo, S. Matsuyama, N. Konishi, and H. Kawakami, “Materials and components optimization for IPS TFT-LCDs,” SID Digest 29, 389 (1998).
[Crossref]

M. Ohta, M. Oh-E, and K. Kondo, “Development of super-TFT-LCDs with in-plane switching mode,” Proc. Asia Disp. 95, 707 (1995).

Konishi, N.

K. Kondo, S. Matsuyama, N. Konishi, and H. Kawakami, “Materials and components optimization for IPS TFT-LCDs,” SID Digest 29, 389 (1998).
[Crossref]

Lee, S. D.

D. H. Chung, Y. Takanishi, K. Ishikawa, C. J. Yu, S. D. Lee, and H. Takezoe, “Alignment control of a nematic liquid crystal on a doubly treated substrate,” Jpn. J. Appl. Phys.  42, 1686 (2003).
[Crossref]

Lien, A.

A. Lien and C. J. Chen, “2-D simulations of in-plane switching mode LCDs,” SID Digest 27, 175 (1996).

Lin, C. H.

C. Y. Huang, C. H. Lin, J. R. Wang, C. W. Huang, M. S. Tsai, and A. Y. G. Fuh, “Multidirectional rubbed liquid-crystal cells,” J. Appl. Phys. 92, 7231 (2002).
[Crossref]

Lueder, E.

E. Lueder, Liquid Crystal Displays, Addressing Schemes and Electro-optical Effects (John Wiley and Sons, Singapore, 2001).

Mahajan, M. P.

M. P. Mahajan and C. Rosenblatt, “History-dependent orientational order of rubbed polyimide for liquid-crystal alignment,” Appl. Phys. Lett. 75, 3623 (1999).
[Crossref]

Matsuyama, S.

K. Kondo, S. Matsuyama, N. Konishi, and H. Kawakami, “Materials and components optimization for IPS TFT-LCDs,” SID Digest 29, 389 (1998).
[Crossref]

Oh-E, K.

K. Oh-E and Kondo, “Electro-optical characteristics and switching behavior of the in-plane switching mode,” Appl. Phys. Lett. 67, 3895 (1995).
[Crossref]

Oh-E, M.

M. Ohta, M. Oh-E, and K. Kondo, “Development of super-TFT-LCDs with in-plane switching mode,” Proc. Asia Disp. 95, 707 (1995).

Ohta, M.

M. Ohta, M. Oh-E, and K. Kondo, “Development of super-TFT-LCDs with in-plane switching mode,” Proc. Asia Disp. 95, 707 (1995).

Patel, J. S.

Y. J. Kim, Z. Zhuang, and J. S. Patel, “Effect of multidirection rubbing on the alignment of nematic liquid crystal,” Appl. Phys. Lett. 77, 513 (2000).
[Crossref]

Rosenblatt, C.

M. P. Mahajan and C. Rosenblatt, “History-dependent orientational order of rubbed polyimide for liquid-crystal alignment,” Appl. Phys. Lett. 75, 3623 (1999).
[Crossref]

Sato, S.

R. Yamaguchi, Y. Goto, and S. Sato, “Liquid crystal alignment surface with two easy axes induced by unidirectional rubbing,” Appl. Phys. Lett. 82, 4450 (2003).
[Crossref]

Shimojyo, T.

T. Shimojyo, “Field-effect LCDs with interdigital electrodes,” Proc. Inf. Disp. 14, 110 (1973).

Soref, R. A.

R. A. Soref, “Field effects in nematic liquid crystals obtained with interdigital electrodes,” J. Appl. Phys. 45, 5466 (1974).
[Crossref]

Takanishi, Y.

D. H. Chung, Y. Takanishi, K. Ishikawa, C. J. Yu, S. D. Lee, and H. Takezoe, “Alignment control of a nematic liquid crystal on a doubly treated substrate,” Jpn. J. Appl. Phys.  42, 1686 (2003).
[Crossref]

Takezoe, H.

D. H. Chung, Y. Takanishi, K. Ishikawa, C. J. Yu, S. D. Lee, and H. Takezoe, “Alignment control of a nematic liquid crystal on a doubly treated substrate,” Jpn. J. Appl. Phys.  42, 1686 (2003).
[Crossref]

Tarumi, K.

D. Klement and K. Tarumi, “Liquid crystal material development of IPS-TFT displays,” SID Digest 28, 393 (1998).
[Crossref]

Tian, J. R.

C. Y. Huang, Y. S. Huang, and J. R. Tian, “Alignment of liquid crystals on two-easy-axis substrate,” Jpn. J. Appl. Phys. 45, 168 (2006).
[Crossref]

C. Y. Huang, Y. S. Huang, J. R. Tian, and S. J. Yeh, “Influence of the two-directionally-rubbed substrate on the alignment of liquid crystals,” Opt. Comm. 265, 241 (2006).
[Crossref]

Tomioka, Y.

M. Yoneya, K. Iwasaki, Y. Tomioka, and K. Kondo, “Cell gap margin enlargement of in-plane switching mode liquid crystal displays using weak-anchoring effects,” Appl. Phys. Lett. 74, 803 (1999).
[Crossref]

Tsai, M. S.

C. Y. Huang, C. H. Lin, J. R. Wang, C. W. Huang, M. S. Tsai, and A. Y. G. Fuh, “Multidirectional rubbed liquid-crystal cells,” J. Appl. Phys. 92, 7231 (2002).
[Crossref]

Tsuda, K.

H. Wakemoto, S. Asada, K. Kato, Y. Yamamoto, T. Tsukane, K. Tsuda, and Y. Yakubo, “An advanced in-plane switching mode TFT-LCD,” SID Digest 28, 929 (1997).

Tsukane, T.

H. Wakemoto, S. Asada, K. Kato, Y. Yamamoto, T. Tsukane, K. Tsuda, and Y. Yakubo, “An advanced in-plane switching mode TFT-LCD,” SID Digest 28, 929 (1997).

Wakemoto, H.

H. Wakemoto, S. Asada, K. Kato, Y. Yamamoto, T. Tsukane, K. Tsuda, and Y. Yakubo, “An advanced in-plane switching mode TFT-LCD,” SID Digest 28, 929 (1997).

Wang, J. R.

C. Y. Huang, C. H. Lin, J. R. Wang, C. W. Huang, M. S. Tsai, and A. Y. G. Fuh, “Multidirectional rubbed liquid-crystal cells,” J. Appl. Phys. 92, 7231 (2002).
[Crossref]

Wu, S. T.

S. T. Wu and D. K. Yang, Reflective Liquid Crystal Displays (John Wiley and Sons, Singapore, 2001).

Yakubo, Y.

H. Wakemoto, S. Asada, K. Kato, Y. Yamamoto, T. Tsukane, K. Tsuda, and Y. Yakubo, “An advanced in-plane switching mode TFT-LCD,” SID Digest 28, 929 (1997).

Yamaguchi, R.

R. Yamaguchi, Y. Goto, and S. Sato, “Liquid crystal alignment surface with two easy axes induced by unidirectional rubbing,” Appl. Phys. Lett. 82, 4450 (2003).
[Crossref]

Yamamoto, Y.

H. Wakemoto, S. Asada, K. Kato, Y. Yamamoto, T. Tsukane, K. Tsuda, and Y. Yakubo, “An advanced in-plane switching mode TFT-LCD,” SID Digest 28, 929 (1997).

Yang, D. K.

S. T. Wu and D. K. Yang, Reflective Liquid Crystal Displays (John Wiley and Sons, Singapore, 2001).

Yeh, S. J.

C. Y. Huang, Y. S. Huang, J. R. Tian, and S. J. Yeh, “Influence of the two-directionally-rubbed substrate on the alignment of liquid crystals,” Opt. Comm. 265, 241 (2006).
[Crossref]

Yoneya, M.

M. Yoneya, K. Iwasaki, Y. Tomioka, and K. Kondo, “Cell gap margin enlargement of in-plane switching mode liquid crystal displays using weak-anchoring effects,” Appl. Phys. Lett. 74, 803 (1999).
[Crossref]

Yu, C. J.

D. H. Chung, Y. Takanishi, K. Ishikawa, C. J. Yu, S. D. Lee, and H. Takezoe, “Alignment control of a nematic liquid crystal on a doubly treated substrate,” Jpn. J. Appl. Phys.  42, 1686 (2003).
[Crossref]

Zhuang, Z.

Y. J. Kim, Z. Zhuang, and J. S. Patel, “Effect of multidirection rubbing on the alignment of nematic liquid crystal,” Appl. Phys. Lett. 77, 513 (2000).
[Crossref]

Appl. Phys. Lett. (6)

K. Oh-E and Kondo, “Electro-optical characteristics and switching behavior of the in-plane switching mode,” Appl. Phys. Lett. 67, 3895 (1995).
[Crossref]

M. Yoneya, K. Iwasaki, Y. Tomioka, and K. Kondo, “Cell gap margin enlargement of in-plane switching mode liquid crystal displays using weak-anchoring effects,” Appl. Phys. Lett. 74, 803 (1999).
[Crossref]

M. J. Escuti, C. C. Bowley, and G. P. Crawford, “Enhanced dynamic response of the in-plane switching liquid crystal display mode through polymer stabilization,” Appl. Phys. Lett. 75, 3264 (1999).
[Crossref]

Y. J. Kim, Z. Zhuang, and J. S. Patel, “Effect of multidirection rubbing on the alignment of nematic liquid crystal,” Appl. Phys. Lett. 77, 513 (2000).
[Crossref]

M. P. Mahajan and C. Rosenblatt, “History-dependent orientational order of rubbed polyimide for liquid-crystal alignment,” Appl. Phys. Lett. 75, 3623 (1999).
[Crossref]

R. Yamaguchi, Y. Goto, and S. Sato, “Liquid crystal alignment surface with two easy axes induced by unidirectional rubbing,” Appl. Phys. Lett. 82, 4450 (2003).
[Crossref]

J. Appl. Phys. (3)

R. Barberi, J. J. Bonvent, M. Giocondo, M. Iovane, and A. L. Alexe-Ionescu. “Bistable nematic azimuthal alignment induced by anchoring competition,” J. Appl. Phys. 84, 1321 (1998).
[Crossref]

C. Y. Huang, C. H. Lin, J. R. Wang, C. W. Huang, M. S. Tsai, and A. Y. G. Fuh, “Multidirectional rubbed liquid-crystal cells,” J. Appl. Phys. 92, 7231 (2002).
[Crossref]

R. A. Soref, “Field effects in nematic liquid crystals obtained with interdigital electrodes,” J. Appl. Phys. 45, 5466 (1974).
[Crossref]

Jpn. J. Appl. (1)

D. H. Chung, Y. Takanishi, K. Ishikawa, C. J. Yu, S. D. Lee, and H. Takezoe, “Alignment control of a nematic liquid crystal on a doubly treated substrate,” Jpn. J. Appl. Phys.  42, 1686 (2003).
[Crossref]

Jpn. J. Appl. Phys. (1)

C. Y. Huang, Y. S. Huang, and J. R. Tian, “Alignment of liquid crystals on two-easy-axis substrate,” Jpn. J. Appl. Phys. 45, 168 (2006).
[Crossref]

Opt. Comm. (1)

C. Y. Huang, Y. S. Huang, J. R. Tian, and S. J. Yeh, “Influence of the two-directionally-rubbed substrate on the alignment of liquid crystals,” Opt. Comm. 265, 241 (2006).
[Crossref]

Proc. Asia Disp. (1)

M. Ohta, M. Oh-E, and K. Kondo, “Development of super-TFT-LCDs with in-plane switching mode,” Proc. Asia Disp. 95, 707 (1995).

Proc. Inf. Disp. (1)

T. Shimojyo, “Field-effect LCDs with interdigital electrodes,” Proc. Inf. Disp. 14, 110 (1973).

SID Digest (5)

H. Wakemoto, S. Asada, K. Kato, Y. Yamamoto, T. Tsukane, K. Tsuda, and Y. Yakubo, “An advanced in-plane switching mode TFT-LCD,” SID Digest 28, 929 (1997).

A. Lien and C. J. Chen, “2-D simulations of in-plane switching mode LCDs,” SID Digest 27, 175 (1996).

K. Kondo, S. Matsuyama, N. Konishi, and H. Kawakami, “Materials and components optimization for IPS TFT-LCDs,” SID Digest 29, 389 (1998).
[Crossref]

D. Klement and K. Tarumi, “Liquid crystal material development of IPS-TFT displays,” SID Digest 28, 393 (1998).
[Crossref]

M. Hasegawa, “Response time improvement of the in-plane-switching mode,” SID Digest 28, 699 (1997).

Other (3)

P. G. de Gennes, The Physics of Liquid Crystals (Clarendon, Oxford, 1975).

E. Lueder, Liquid Crystal Displays, Addressing Schemes and Electro-optical Effects (John Wiley and Sons, Singapore, 2001).

S. T. Wu and D. K. Yang, Reflective Liquid Crystal Displays (John Wiley and Sons, Singapore, 2001).

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Figures (8)

Fig. 1.
Fig. 1. Geometry of the [m, n]-rubbed IPS LC device. In the figure, 1st indicates the first rubbing direction and 2nd indicates the second rubbing direction.
Fig. 2.
Fig. 2. Measured deviation angle of the two-easy-axes IPS LC cell as a function of m. In this experiment, n is set to be 1.
Fig. 3.
Fig. 3. Measured transmission versus applied voltage (T-V) curves of the [1, 1] and [0, 1]- rubbed IPS cells. The deviation angle is ~8°.
Fig. 4.
Fig. 4. Measured electro-optical responses of the [1, 1] and [0, 1]-rubbed IPS cells. The deviation angle is ~8°.
Fig. 5.
Fig. 5. Measured voltage dependent rise times of the [1, 3] and [0, 1]-rubbed IPS cells. The deviation angle is ~5°.
Fig. 6.
Fig. 6. Measured electro-optical responses of the [3, 1] and [0, 1]-rubbed IPS cells. Inset shows the total response times of the [3, 1] and [0, 1]-rubbed IPS cells. The deviation angle is ~20°.
Fig. 7.
Fig. 7. Measured T-V curves of the [3, 1] and [0, 1]-rubbed cells.
Fig. 8.
Fig. 8. Optical microscope images of the [0, 1], [1, 1], [1, 3] and [3, 1]-rubbed cells. In the figure, 1st indicates the first rubbing direction and 2nd indicates the second rubbing direction.

Equations (4)

Equations on this page are rendered with MathJax. Learn more.

V th = π · l d + 2 b k 22 ε 0 Δ ε ,
b = k 22 A .
τ on = γ 1 ε 0 Δ ε E 2 π 2 ( d + 2 b ) 2 k 22 ,
τ off = γ 1 ( d + 2 b ) 2 π 2 k 22 .

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