Determination of surface nematic liquid crystal anchoring strength using nano-scale surface grooves

Yoonseuk Choi; Hiroshi Yokoyama; Jin Seog Gwag

doi:10.1364/OE.21.012135

1. Introduction

The interfacial properties of liquid crystals (LCs) at various surfaces have been studied continuously so far [1–9]. 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]. 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 [10–16].

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], the measurement of Freedericksz transition [18,19], surface disclination [20], and of optical transmittance by phase retardation [21–24]. 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], 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]. 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. The nano-scale grooves on the bottom substrate have different pitches in respect to the anchoring strength of rubbing (W_R), which vary the azimuthal anchoring strength based on the well-known Berreman effect [26]. 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.

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.

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Thus, we have designed the surface groove pattern to exhibit three different types of anchoring strengths, compared to the azimuthal anchoring strength of rubbing (W_G » W_R, W_G ≈W_R, W_G « W_R; while W_G 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 (W_G and W_R). 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]:

z = A \sin [q (x \cos ϕ - y \sin ϕ)] .

where A is the amplitude of the modulations and q is the wave number, defined as $q = 2 π / λ$ (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 $z = \infty$ . To designate the average orientation of the nematic liquid crystal, we have used the director $\vec{n}$ . The Frank elastic energy, in terms of $\vec{n}$ , is equated as:

F = \frac{1}{2} \int [K_{1} (\nabla \cdot \vec{n})^{2} + K_{2} {(\vec{n} \cdot \nabla \times \vec{n})}^{2} + K_{3} {(\vec{n} \times \nabla \times \vec{n})}^{2} - (K_{2} + K_{24}) \nabla \cdot (\vec{n} \nabla \cdot \vec{n} + \vec{n} \times \nabla \times \vec{n})] d r .

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 [28–31]:

f = \frac{1}{2} W_{G} [\frac{\sin^{4} ϕ}{g_{1} (ϕ)} + \frac{(K_{2} + K_{24})}{K_{3}} \frac{\sin^{2} ϕ \cos^{2} ϕ}{g_{1} (ϕ)} (2 - \frac{(K_{2} + K_{24})}{K_{3}} \frac{g_{1} (ϕ) g_{2} (ϕ) - \cos^{2} ϕ}{\sin^{2} ϕ})],

where $g_{i} (ϕ)$ and W_G are defined as $\sqrt{\cos^{2} ϕ + (K_{3} / K_{i}) \sin^{2} ϕ}$ (i = 1, 2) and $\frac{1}{2} K_{3} A^{2} q^{3}$ , respectively. Here, $ϕ$ is the angle between the LC director and the direction of the surface grooves. $K_{24} \equiv (K_{1} - K_{2}) / 2$ can be obtained from the molecular theory of Nehring and Saupe. W_G represents the azimuthal anchoring strength due to the groove effect. In the 1-D groove approach, the anchoring strength in Eq. (3) is related only to A and λ ; therefore, it is independent of the shape of the surface groove. If we consider a simplified one-constant case in which K₁ = K₂ = K₃ = K, then Eq. (3) is coincident with Berreman’s groove theory, which is formulated as [26]:

f = \frac{1}{2} W_{G} \sin^{2} ϕ .

Equation (4) seems to be a reasonable expression for defining the surface anchoring strength of LCs on an undulated surface since $f \propto \sin^{2} ϕ$ 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 = \frac{1}{2} W_{G} \sin^{2} ϕ - \frac{1}{2} W_{A} \sin^{2} ϕ,

Here, W_A 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:

\frac{d f}{d ϕ} = \sin ϕ \cos ϕ (W_{G} - W_{A}) = 0.

If W_G ≠ W_A, then the $ϕ$ values which satisfy Eq. (6) are only 0 and π/2. However, in the case of W_G = W_A, 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 W_A. This discrepancy is inevitable as, in the case of a simple elastic constant assumption, $f \propto \sin^{2} ϕ$ . 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 density $f \propto \sin^{4} ϕ$ , 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 = \frac{1}{2} W_{G} [\frac{\sin^{4} ϕ}{g_{1} (ϕ)} + \frac{(K_{2} + K_{24})}{K_{3}} \frac{\sin^{2} ϕ \cos^{2} ϕ}{g_{1} (ϕ)} (2 - \frac{(K_{2} + K_{24})}{K_{3}} \frac{g_{1} (ϕ) g_{2} (ϕ) - \cos^{2} ϕ}{\sin^{2} ϕ})] + \frac{1}{2} W_{A} \cos^{2} ϕ .

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

\begin{array}{l} W_{A} = W_{G} (\frac{1}{2 g_{1}^{3}} (1 - β_{1}) (β_{3}^{2} \cos^{4} ϕ + \sin^{4} ϕ) + \frac{(2 - 2 β_{3}) \sin^{2} ϕ}{g_{1}} + β_{3}^{2} g_{2} \\ + β_{3} \cos^{2} ϕ \times (\frac{2}{g_{1}} + β_{3} (\frac{- 2}{g_{1}} + \frac{1}{2 g_{2}} (1 - β_{2}))) + \frac{β_{3} \sin^{2} 2 ϕ}{4 g_{1}^{3}} (1 - β_{1})), \end{array}

where $β_{1} = K_{3} / K_{1}$ , $β_{2} = K_{3} / K_{2}$ , and $β_{2} = (K_{2} + K_{24}) / K_{3}$ . 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, W_A, is determined by experimentally measuring the deviation angle $ϕ$ .

The relative anchoring strength, W_A / W_G, is plotted, as a function of $ϕ$ , as shown in Fig. 2, where $β_{1} = 1.3$ and $β_{2} = 2.2$ [32] with $β_{3} = 1 / 2 (1 / β_{1} + 1 / β_{2}) = 0.61$ , based on the molecular theory of Nehring and Saupe, and $β_{3} \approx 0.005$ based on our previous result [30,31]. Using the molecular theory of Nehring and Saupe [33] with $K_{24} \equiv (K_{1} - K_{2}) / 2$ in the ranges of W_A > 1.13 W_G and W_A < 0.56 W_G, the LCs are entirely aligned with either the rubbing direction or the groove direction. However, this is not in agreement with our experimental observation, thereby indicating that the anchoring strength W_A, continuously covers the entire range, of the deviation angle $ϕ$ , from 0 to π/2.

Fig. 2 Plotting results of the azimuthal surface anchoring strength, W_A, as a function of the stable orientations $ϕ$ using Eq. (8).

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On the other hand, when we use $K_{24} \approx - K_{2}$ , even though continuity at the lower values of the anchoring strength, W_A appears (as shown in Fig. 2), there is no continuity at the higher values of, W_A > 1.63 W_G. These theoretical results suggest that the surface anchoring energy due to rubbing is proportional to $\sin^{4} ϕ$ rather than $\sin^{2} ϕ$ , as in the Rapini–Papoular form. In this case, Eq. (8) should be rewritten as:

\begin{array}{l} W_{A} = \frac{W_{G}}{2 \cos^{2} ϕ} {\frac{1}{2 g_{1}^{3}} (1 - β_{1}) (β_{3}^{2} \cos^{4} ϕ + \sin^{4} ϕ) + \frac{(2 - 2 β_{3}) \sin^{2} ϕ}{g_{1}} + β_{3}^{2} g_{2} \\ + β_{3} \cos^{2} ϕ \times [\frac{2}{g_{1}} + β_{3} (\frac{- 2}{g_{1}} + \frac{1}{2 g_{2}} (1 - β_{2}))] + \frac{β_{3} \sin^{2} 2 ϕ}{4 g_{1}^{3}} (1 - β_{1})} . \end{array}

We intuitively expect that by the new term, $1 / 2 \cos^{2} ϕ$ of Eq. (9), W_A will sharply increase as $ϕ \to π / 2$ , and thus will cover the full range of anchoring strengths. We further note that W_{G = $\frac{1}{2} K_{3} A^{2} q^{3}$}, the azimuthal anchoring strength produced from the surface groove, is valid only when the polar anchoring strength W_P is exceptionally strong. For a finite value of polar anchoring strength, we can use Faetti’s result in which W_G is described as a function of the polar anchoring strength as follows [34]:

W_{G} (W_{P}) = \frac{1}{2} K_{3} A^{2} q^{3} {(1 + \frac{q K_{3}}{2 W_{P}})}^{- 1} .

Note that for $W_{P} > > q K_{3}$ , $W_{G} \approx 1 / 2 (K_{3} A^{2} q^{3})$ corresponds to the elastic Berreman theory. On the other hand, for $W_{P} < < q K_{3}$ , $W_{G} \approx A^{2} q^{2} W_{P}$ is influenced entirely by the polar anchoring strength. By inserting Eq. (10) for Eq. (9), W_A becomes a function of W_P. Therefore, we cannot determine W_A and W_P for a single groove surface system since even though the value of $ϕ$ is obtained through observation, two unknown quantities remain, W_A and W_P. 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 W_A and W_P for a weak polar anchoring boundary, by solving simultaneous linear equations with the two unknowns produced using Eq. (8). We obtained $ϕ_{1}$ and $ϕ_{2}$ , 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. As expected, W_A sharply increased as $ϕ \to π / 2$ and thus we were able to cover the full range of anchoring strengths.

Fig. 3 Plotting results of the azimuthal surface anchoring strength W_A as a function of the stable orientations $ϕ$ using Eq. (9) with Eq. (10). (a) According to changes in polar anchoring strength W_P. (b) According to change in groove pitch λ.

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

Figure 3(b) shows W_A as a function of $ϕ$ according to the change in the pitch of the groove for a fixed W_P. 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, W_A 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]. 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 (W_A and W_P) 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 W_G. The detailed fabricating and imprinting conditions of this film are described in the previously published paper [35].

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 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.

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.

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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 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.

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.

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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.

In the case of 400 nm, the surface free energy of the LC was minimized when $ϕ = 22 °$ . A very large deviation occurred at a pitch of 800 nm, and then at a deviation of $ϕ = 48 \pm 1 °$ [see Fig. 5(a)]. Consequently, by substituting these two deviation values ( $ϕ_{1} = 22 °$ and $ϕ_{2} = 48 °$ )into Eq. (9) with Eq. (10), we can easily and simultaneously determine the azimuthal anchoring strength W_A and the polar anchoring strength, W_P, of the LCs on the polyimide surface. In this particular case, the evaluated azimuthal and polar anchoring strengths were $W_{A} = 2.76 \times 10^{- 5} N / m$ and $W_{P} = 1.72 \times 10^{- 4} N / m$ , respectively. Compared to previous research, these are considered to be very reasonable values.

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 $W_{A} = 1.1 \times 10^{- 9} N / m$ and $W_{P} = 1.6 \times 10^{- 7} N / m$ , 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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Determination of surface nematic liquid crystal anchoring strength using nano-scale surface grooves

Abstract

1. Introduction

2. Theory

3. Experiments

4. Results and discussion

5. Conclusions

Acknowledgments

References and links

Cited By

Figures (5)

Equations (10)

Optics Express