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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 10 — May. 20, 2013
  • pp: 12135–12144
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Determination of surface nematic liquid crystal anchoring strength using nano-scale surface grooves

Yoonseuk Choi, Hiroshi Yokoyama, and Jin Seog Gwag  »View Author Affiliations


Optics Express, Vol. 21, Issue 10, pp. 12135-12144 (2013)
http://dx.doi.org/10.1364/OE.21.012135


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Abstract

Based on several nano-scale groove models, we propose a new technique to simultaneously determine the azimuthal and polar surface anchoring strengths of nematic liquid crystal (LC). The optical analysis of LCs on a grooved surface made by nanoimprinting lithography was performed on special alignment material, using a typical rubbing process. In our approach, using a polarizing microscope, we can determine the LC alignment exactly as it is in a parallel state, rather than a twisted state. This simple proposed method gives an accurate value of the surface LC anchoring of various surfaces, as well as simultaneously measuring the azimuthal and polar anchoring energy.

© 2013 OSA

1. Introduction

The interfacial properties of liquid crystals (LCs) at various surfaces have been studied continuously so far [1

1. 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]

9

9. H. Yokoyama, Handbook of Liquid Crystal Research, P. J. Collings and J. S. Patel, ed. (Oxford University Press, Oxford, 1997), Chap. 6.

]. This is mainly because the surface anchoring property of nematic LCs on a treated interface is an important parameter from the viewpoints of both basic and applied sciences, such as bio-inspired research and high-tech liquid crystal display devices [9

9. H. Yokoyama, Handbook of Liquid Crystal Research, P. J. Collings and J. S. Patel, ed. (Oxford University Press, Oxford, 1997), Chap. 6.

]. So far, many researchers have investigated the mechanism of the LC molecules’ re-orientation process. It has been concluded that the surface anchoring strength is one of the key parameter [10

10. K. H. Yang, “Freedericksz transition of twisted nematic cells,” Appl. Phys. Lett. 43(2), 171–173 (1983). [CrossRef]

16

16. A. Sugimura, G. R. Luckhurst, and O.-Y. Zhong-can, “Director deformation of a twisted chiral nematic liquid crystal cell with weak anchoring boundaries,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 52(1), 681–689 (1995). [CrossRef] [PubMed]

].

Surface anchoring strength is defined as the energy cost to change the director of the LC from its initial anchoring direction. Many methods of precisely measuring the anchoring strength of LCs have been proposed, including the use of high field [17

17. H. Yokoyama and H. A. van Sprang, “A novel method for determining the anchoring energy function at a nematic liquid crystal-wall interface from director distortions at high,” J. Appl. Phys. 1985, 57, 4520 (1985).

], the measurement of Freedericksz transition [18

18. K. H. Yang and C. Rosenblatt, “Determination of the anisotropic potential at the nematic liquid crystal –to-wall interface,” Appl. Phys. Lett. 43(1), 62 (1983). [CrossRef]

,19

19. C. Rosenblatt, “Temperature dependence of the anchoring strength coefficient at a nematic liquid crystal-wall,” J. Phys. France 45(6), 1087–1091 (1984). [CrossRef]

], surface disclination [20

20. G. Porte, “Tilted alignment of MBBA induced by short-chain surfactants,” J. Phys. France 37(10), 1245–1252 (1976). [CrossRef]

], and of optical transmittance by phase retardation [21

21. Yu. A. Nastishin, R. D. Polak, S. V. Shiyanovskii, and O. D. Lavrentovich, “Determination of nematic polar anchoring from retardation versus voltage measurements,” Appl. Phys. Lett. 75(2), 202–204 (1999). [CrossRef]

24

24. J. S. Gwag, J. Yi, and J. H. Kwon, “Determination of actual surface azimuthal anchoring strength using a wedge-shaped liquid crystal cell,” Opt. Lett. 35(4), 456–458 (2010). [CrossRef] [PubMed]

]. Among these methods, the optical method is the most powerful, due to its versatility and applicability to many LC molecule structures, as well as its ease of use. In our research, we also apply optical analysis to our surface competition system (discussed later in detail) using such merits.

In general, in an LC system, there are two distinguishable contributions to the surface anchoring strength: (1) the polar anchoring strength (the energy which constrains the out-of-plane motion of the LC director normal to a surface), and (2) the azimuthal anchoring strength (the energy which restricts the in-plane motion of the LC director on the surface). The polar anchoring strength has been measured on the basis of the High Field theory, defined by Yokoyama [17

17. H. Yokoyama and H. A. van Sprang, “A novel method for determining the anchoring energy function at a nematic liquid crystal-wall interface from director distortions at high,” J. Appl. Phys. 1985, 57, 4520 (1985).

], while the azimuthal anchoring strength has been characterized by the optical method. We can determine azimuthal anchoring in detail by finding the easy axis of the surface from a minimum optical transmittance in a twisted LC cell based on the intrinsic birefringence of LC material which produces a phase retardation.

However, when the optical transmittance is small, the minimum value is highly dependent on the effect of an optical phase shift caused by the reflection between the isotropic boundary layers and the twisted LC layers. Therefore, we need to consider this complex problem to extract an exact value of the azimuthal anchoring strength by using the optical standard technique [25

25. F. Yang and J. R. Sambles, “The influence of surface reflectivities on measurement of the torsional anchoring strength of nematic liquid crystals,” Jpn. J. Appl. Phys. 37(Part 1), 3998–4007 (1998). [CrossRef]

]. Furthermore, thus far, no simple method has been found which can simultaneously determine these two different anchoring strengths.

In this paper, we present a novel technique to determine surface LC anchoring strengths using rubbed nano-size groove surfaces. To induce an anchoring competition, we used a groove pattern and the mechanical rubbing of the aligning material with mutually orthogonal anchoring directions as shown in Fig. 1
Fig. 1 Schematics showing LC alignment produced by anchoring competition driven from a groove pattern and mechanical rubbing having mutually orthogonal surface anchoring directions. By decreasing the pitch, the LC alignment moves toward the groove direction due to the dominance of the groove contribution over the rubbing-induced anchoring.
. The nano-scale grooves on the bottom substrate have different pitches in respect to the anchoring strength of rubbing (WR), which vary the azimuthal anchoring strength based on the well-known Berreman effect [26

26. D. Berreman, “Solid Surface Shape and the Alignment of an Adjacent Nematic Liquid Crystal,” Phys. Rev. Lett. 28(26), 1683–1686 (1972). [CrossRef]

]. Also, in order to change the azimuthal direction, we have designed a coating and conducted a direct perpendicular rubbing process of the LC alignment layer on the nano-scale groove. In Fig. 1, a tighter pitch of groove, means a higher anchoring strength from the surface, according to the Berreman theory.

Thus, we have designed the surface groove pattern to exhibit three different types of anchoring strengths, compared to the azimuthal anchoring strength of rubbing (WG » WR, WG ≈WR, WG « WR; while WG is the anchoring strength of the grooving effect). As depicted in the figure, the liquid crystal’s alignment near the surface is determined by comparing two different anchoring strengths (WG and WR). Detailed methods, theory and analysis will be discussed in the following section. The bottom line is, using this method, we can simply, successfully and simultaneously determine both the azimuthal anchoring and the polar anchoring strength.

2. Theory

To understand the alignment of LCs on a grooved surface, we must adopt an outline of the theoretical analysis of the LC alignment on a 1-D surface groove. The geometry of the groove pattern with an alignment direction can be defined as [27

27. J. Fukuda, M. Yoneya, and H. Yokoyama, “Surface-Groove-Induced Azimuthal Anchoring of a Nematic Liquid Crystal: Berreman’s Model Reexamined,” Phys. Rev. Lett. 98(18), 187803 (2007). [CrossRef] [PubMed]

]:

z=Asin[q(xcosϕysinϕ)].
(1)

where A is the amplitude of the modulations and q is the wave number, defined as q=2π/λ (where λ is the groove pitch). ϕ is characterized as the azimuthal angle between the surface groove and x-axis, which is the director orientation of the nematic liquid crystal at z=. To designate the average orientation of the nematic liquid crystal, we have used the directorn. The Frank elastic energy, in terms of n, is equated as:

F=12[K1(n)2+K2(n×n)2+K3(n××n)2(K2+K24)(nn+n××n)]dr.
(2)

where K1, K2, K3, and K24 are the splay, twist, bend, and splay-bend surface elastic constants, respectively.

In order to determine stable states of the LC director of the volume with the surface boundary condition of Eq. (1), an Euler-Lagrange equation (with some manipulation) is applied to Eq. (2). We obtain Fukuda’s expression for the free energy density in an undulated surface system, which is formulated as [28

28. J. I. Fukuda, J. S. Gwag, M. Yoneya, and H. Yokoyama, “Theory of anchoring on a two-dimensionally grooved surface,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 77(1), 011702 (2008). [CrossRef] [PubMed]

31

31. J. S. Gwag, J. H. Kwon, M. Oh-e, J. Niitsuma, M. Yoneya, and H. Yokoyama, “Higher-order surface free energy in azimuthal nematic anchoring on nanopatterned grooves,” Appl. Phys. Lett. 95(10), 103101 (2009). [CrossRef]

]:

f=12WG[sin4ϕg1(ϕ)+(K2+K24)K3sin2ϕcos2ϕg1(ϕ)(2(K2+K24)K3g1(ϕ)g2(ϕ)cos2ϕsin2ϕ)],
(3)

f=12WGsin2ϕ.
(4)

Equation (4) seems to be a reasonable expression for defining the surface anchoring strength of LCs on an undulated surface since fsin2ϕ agrees with the Rapini–Papoular form. In fact, Eq. (4) provides an excellent approximation for the case of a surface system producing only one easy axis by a 1-D groove. However, Eq. (4) reaches a limit of phenomenal description in cases where the systems have more than two surface easy axes. In other words, this equation is not applicable to a system which has an anchoring competition. For example, our surface system which has two different anchoring effects (one driven by a periodic 1-D groove while the other is induced by rubbing perpendicular to the groove direction) requires a different model to describe the surface anchoring effect. In this case, the total free energy density with Berreman’s model should be fixed as below due to the mutual orthogonality:

f=12WGsin2ϕ12WAsin2ϕ,
(5)

Here, WA represents the azimuthal anchoring strength due to rubbing, for which we assume the Rapini–Papoular form. In order to find the stable orientations of this rough surface with the anchoring competition, the first derivative of Eq. (5) with respect to ϕ should be taken, and the resultant value becomes zero. Therefore, the derivative can be expressed as follows:

dfdϕ=sinϕcosϕ(WGWA)=0.
(6)

If WGWA, then the ϕ values which satisfy Eq. (6) are only 0 and π/2. However, in the case of WG = WA, all ϕ values satisfy Eq. (6). This does not correspond to rational expectations. Generally, experimental results have continuous ϕ values according to the magnitude of WA. This discrepancy is inevitable as, in the case of a simple elastic constant assumption, fsin2ϕ. In short, Berreman’s groove model, using the one elastic constant approach, cannot be applied to the surface anchoring competition system. Instead, this anchoring competition system, Fukuda’s groove model - with the resultant free energy densityfsin4ϕ, driven without approximation -, should be used.

Therefore, at a surface, when combining the Fukuda groove anchoring model with the Rapini–Papoular form of rubbing anchoring, the total free energy density provided by Eq. (3) can be rewritten as:

f=12WG[sin4ϕg1(ϕ)+(K2+K24)K3sin2ϕcos2ϕg1(ϕ)(2(K2+K24)K3g1(ϕ)g2(ϕ)cos2ϕsin2ϕ)]+12WAcos2ϕ.
(7)

As previously demonstrated, if the first derivative of Eq. (7) with respect to ϕ is taken and rearranged for WA, we can obtain an expression for the stable orientation of the LC director as follows:

WA=WG(12g13(1β1)(β32cos4ϕ+sin4ϕ)+(22β3)sin2ϕg1+β32g2+β3cos2ϕ×(2g1+β3(2g1+12g2(1β2)))+β3sin22ϕ4g13(1β1)),
(8)

where β1=K3/K1, β2=K3/K2, and β2=(K2+K24)/K3. In Eq. (8), if we know the pitch, the amplitude of surface groove, and the elastic constants of the LC, then the azimuthal surface anchoring strength due to rubbing, WA, is determined by experimentally measuring the deviation angleϕ.

On the other hand, when we use K24K2, even though continuity at the lower values of the anchoring strength, WA appears (as shown in Fig. 2), there is no continuity at the higher values of, WA > 1.63 WG. These theoretical results suggest that the surface anchoring energy due to rubbing is proportional to sin4ϕ rather than sin2ϕ, as in the Rapini–Papoular form. In this case, Eq. (8) should be rewritten as:

WA=WG2cos2ϕ{12g13(1β1)(β32cos4ϕ+sin4ϕ)+(22β3)sin2ϕg1+β32g2+β3cos2ϕ×[2g1+β3(2g1+12g2(1β2))]+β3sin22ϕ4g13(1β1)}.
(9)

We intuitively expect that by the new term, 1/2cos2ϕ of Eq. (9), WA will sharply increase as ϕπ/2, and thus will cover the full range of anchoring strengths. We further note that WG = 12K3A2q3, the azimuthal anchoring strength produced from the surface groove, is valid only when the polar anchoring strength WP is exceptionally strong. For a finite value of polar anchoring strength, we can use Faetti’s result in which WG is described as a function of the polar anchoring strength as follows [34

34. S. Faetti, “Azimuthal anchoring energy of a nematic liquid crystal at a grooved interface,” Phys. Rev. A 36(1), 408–410 (1987). [CrossRef] [PubMed]

]:

WG(WP)=12K3A2q3(1+qK32WP)1.
(10)

Note that for WP>>qK3, WG1/2(K3A2q3)corresponds to the elastic Berreman theory. On the other hand, for WP<<qK3, WGA2q2WP is influenced entirely by the polar anchoring strength. By inserting Eq. (10) for Eq. (9), WA becomes a function of WP. Therefore, we cannot determine WA and WP for a single groove surface system since even though the value of ϕ is obtained through observation, two unknown quantities remain, WA and WP. Thus, we use a grooved surface process consisting of two or three kinds of different pitches, as shown in Fig. 1.

In such a case, we can determine the WA and WP for a weak polar anchoring boundary, by solving simultaneous linear equations with the two unknowns produced using Eq. (8). We obtained ϕ1 and ϕ2, produced by the surface grooves with two different pitches using the experimental measurement. Combining Eq. (9) with Eq. (10), we plotted the azimuthal surface anchoring strength as a function of the stable orientation ϕ, as shown in Fig. 3
Fig. 3 Plotting results of the azimuthal surface anchoring strength WA as a function of the stable orientations ϕ using Eq. (9) with Eq. (10). (a) According to changes in polar anchoring strength WP. (b) According to change in groove pitch λ.
. As expected, WA sharply increased as ϕπ/2 and thus we were able to cover the full range of anchoring strengths.

Figure 3(a) plots the azimuthal anchoring strength, WA, as a function of ϕ according to the change in the polar anchoring strength, WP. The plot indicates that ϕ influences the polar anchoring strength at the same WA. In the range of a small deviation in ϕ, the gradient of WA for ϕ is higher than in WP. In the range of a large deviation in ϕ, the gradient is higher for a smaller value of WP.

Figure 3(b) shows WA as a function of ϕ according to the change in the pitch of the groove for a fixed WP. By increasing the pitch, the variation of ϕ is clearly seen due to a decrease ingroove anchoring strength. In general, an LC alignment layer such as typical polyimide, WA can be found in the plotting graph, as shown in Fig. 3(b) by using the deviation angles measured at two different pitches.

3. Experiments

In our experiment, we used the rubbed nano-groove surface structure with a gradient periodic pattern and a functional aligning material to check the competition effect of two different anchorings, by means of an optical method. The functional LC aligning material was specially designed for nanoimprinting lithography (NIL) and gradient LC alignment property by the use of a hybrid-type polyimide solution [35

35. J. S. Gwag, M. Oh-e, M. Yoneya, H. Yokoyama, H. Satou, and S. Itami, “Advanced nanoimprint lithography using a graded functional imprinting material tailored for liquid crystal alignment,” J. Appl. Phys. 102(6), 063501 (2007). [CrossRef]

]. This special material is characterized by the hybridization of two distinct moieties which have significantly different thermo-mechanical properties as well as different surface activities.

For the most part, the bulk layer of this film has the characteristics of epoxy resin while the surface layer is dominantly covered by the polyimide. The polyimide layer (near the upper side) is an excellent LC alignment layer which demonstrates moderately strong anchoring strength (WA and WP) using the typical rubbing process. On the other hand, the layer produced from the epoxy resin and polyester amic acid (the bottom side) is suitable for NIL processing and producing WG. The detailed fabricating and imprinting conditions of this film are described in the previously published paper [35

35. J. S. Gwag, M. Oh-e, M. Yoneya, H. Yokoyama, H. Satou, and S. Itami, “Advanced nanoimprint lithography using a graded functional imprinting material tailored for liquid crystal alignment,” J. Appl. Phys. 102(6), 063501 (2007). [CrossRef]

].

In this research experiment we have used NIL to produce groove patterns on the alignment layer since it has the capability to create micro-topographical patterns with high throughput and high precision through an easy step-and-stamp process. This process can provide the quantitative control of surface anchoring by the use of decalcomania of a master mold with a definite pitch and depth with respect to the LC alignment layer.

4. Results and discussion

Figure 4
Fig. 4 AFM images of the nano-sized groove used in this experiment. The four pitches (200, 400, 800, and 1600 nm) were patterned continuously in order, and the features were transferred by NIL from a mold pattern into a hybrid-type polyimide film.
shows atomic force microscopy (AFM) (SPM-9500J3, Shimadzu Corp., Japan) images of nanosized features which have different pitches which are transferred from the mold patterns. The pitches of the patterns shown in Figs. 4(a)4(d) were 200, 400, 800, and 1600 nm, respectively, with a depth of 120 nm. The rubbing performed in our experiment led to an azimuthally-orthogonal anchoring direction with respect to the anisotropic anchoring direction generated by the groove, thus generating, anchoring competition.

In this situation, the rubbing was performed in the following typical conditions: The roller rotation was 1000 rpm, the rubbing depth was 5 mm, and the stage speed was 30 mm/s. In order to observe the alignment of the LCs more clearly, the counter substrate was coated with a homeotropic LC alignment layer (polyimide SE-1211, Nissan Chemical Corp., Japan).

Figure 5
Fig. 5 Polarizing microscopic images of LC alignments in the LC cell filled with 5CB LC on the rubbed surfaces with 200, 400, and 800 nm pitch grooves. (a) For hybrid-type polyimide film used as an LC alignment layer. (b) For PMMA film used as an LC alignment layer.
shows polarizing microscopic images of the LC cell filled with 4-n-pentyl-4’-cyanobiphenyl (5CB) LC on the rubbed surfaces, with 200, 400, and 800 nm pitch grooves.

Figure 5(a) shows a polarizing optical microscopic image in which the hybrid-type polyimide film was used as an LC alignment layer. The optic axis of a polarizer in a crossed polarizers-configuration was maintained at 45° with respect to rubbing or groove directions.

On the groove of λ = 200 nm, the LC orientation remained along the groove direction, indicating a strong dominance of the groove contribution over the rubbing-induced anchoring. However, by increasing the pitch to reduce the groove anchoring, a conspicuous variation in ϕ was seen to occur.

To validate this method, we have evaluated the anchoring strength in the same manner as the results obtained from another LC alignment layer. Polymethylmethacrylate (PMMA) film has a very weak surface anchoring strength. Figure 5(b) is a polarizing optical microscopic image taken when PMMA film was used as an LC alignment layer, and the optic axis of the polarizer under crossed-polarizers configuration was fixed along the rubbing or the groove directions. To show an indiscernible domain boundary, due to the very low contrast ratio between domains, dotted lines are marked at the domain boundary in Fig. 5(b).

We note that in all pitches, deviation occurs only slightly. A very small deviation occurred at pitches of 800 and 1600 nm, and then deviations of ϕ=2.5° and ϕ=5° occurred, respectively. In the same manner, the evaluated azimuthal and polar anchoring strengths of PMMA from rubbing were WA=1.1×109N/m and WP=1.6×107N/m, respectively. These values are in agreement with the values reported in several other studies.

The value of ϕ was almost constant for rubbings before and after the nanoimprinting. This indicates that information for the rubbed surface before imprinting is not erased by the imprinting process since the rubbing is done by velvet on the groove surface, which is composed of many concave and convex structures. Note that, in the case of PMMA, the information for the rubbed surface before imprinting was erased by the imprinting process. This was due to the low glass transition temperature.

5. Conclusions

In summary, based on the studies of several groove models, we proposed to simultaneously determine the azimuthal surface anchoring strength and the polar surface anchoring strength. Two different anchoring conditions were generated by rubbing, using 1-D surface nano-sized patterns made by NIL. The LC alignment in this method is a parallel state without a twisted state. The optical standard techniques almost always use the twisted LC state influencing the optical phase shift caused by the reflection upon it. When using this method to determine the LC alignment from the transmittance minimum point in the crossed polarizers-configuration, there are very few errors caused by optical noise. Consequently, depending on the surface materials, the proposed method is a good choice for a simple and more accurate measurement of surface LC anchoring.

Acknowledgments

This research was supported by the 2013 Yeungnam University research grant and the Human Resources Development Program (No. 20124030200100) of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korea government Ministry of Trade, Industry and Energy.

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J. S. Gwag, J.-H. Kim, M. Yoneya, and H. Yokoyama, “Surface nematic bistability at nanoimprinted topography,” Appl. Phys. Lett. 92(15), 153110 (2008). [CrossRef]

31.

J. S. Gwag, J. H. Kwon, M. Oh-e, J. Niitsuma, M. Yoneya, and H. Yokoyama, “Higher-order surface free energy in azimuthal nematic anchoring on nanopatterned grooves,” Appl. Phys. Lett. 95(10), 103101 (2009). [CrossRef]

32.

M. Cui and J. R. Kelly, “Temperature dependence of visco-elastic properties of 5CB,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 331(1), 49–57 (1999). [CrossRef]

33.

J. Nehring and A. Saupe, “On the Elastic Theory of Uniaxial Liquid Crystals,” J. Chem. Phys. 54(1), 337–343 (1971). [CrossRef]

34.

S. Faetti, “Azimuthal anchoring energy of a nematic liquid crystal at a grooved interface,” Phys. Rev. A 36(1), 408–410 (1987). [CrossRef] [PubMed]

35.

J. S. Gwag, M. Oh-e, M. Yoneya, H. Yokoyama, H. Satou, and S. Itami, “Advanced nanoimprint lithography using a graded functional imprinting material tailored for liquid crystal alignment,” J. Appl. Phys. 102(6), 063501 (2007). [CrossRef]

OCIS Codes
(120.2040) Instrumentation, measurement, and metrology : Displays
(160.3710) Materials : Liquid crystals
(230.3720) Optical devices : Liquid-crystal devices
(240.0240) Optics at surfaces : Optics at surfaces

ToC Category:
Optical Devices

History
Original Manuscript: April 9, 2013
Revised Manuscript: May 6, 2013
Manuscript Accepted: May 7, 2013
Published: May 10, 2013

Citation
Yoonseuk Choi, Hiroshi Yokoyama, and Jin Seog Gwag, "Determination of surface nematic liquid crystal anchoring strength using nano-scale surface grooves," Opt. Express 21, 12135-12144 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-10-12135


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  31. J. S. Gwag, J. H. Kwon, M. Oh-e, J. Niitsuma, M. Yoneya, and H. Yokoyama, “Higher-order surface free energy in azimuthal nematic anchoring on nanopatterned grooves,” Appl. Phys. Lett.95(10), 103101 (2009). [CrossRef]
  32. M. Cui and J. R. Kelly, “Temperature dependence of visco-elastic properties of 5CB,” Mol. Cryst. Liq. Cryst. (Phila. Pa.)331(1), 49–57 (1999). [CrossRef]
  33. J. Nehring and A. Saupe, “On the Elastic Theory of Uniaxial Liquid Crystals,” J. Chem. Phys.54(1), 337–343 (1971). [CrossRef]
  34. S. Faetti, “Azimuthal anchoring energy of a nematic liquid crystal at a grooved interface,” Phys. Rev. A36(1), 408–410 (1987). [CrossRef] [PubMed]
  35. J. S. Gwag, M. Oh-e, M. Yoneya, H. Yokoyama, H. Satou, and S. Itami, “Advanced nanoimprint lithography using a graded functional imprinting material tailored for liquid crystal alignment,” J. Appl. Phys.102(6), 063501 (2007). [CrossRef]

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