Gaining Control through Frustration: Two-Fold Approach for Liquid

Jun 17, 2014 - Crystal Three-Dimensional Command Layers ... with experiments, thus making this a proper tool to design multistable 3D command layers...
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Letter pubs.acs.org/NanoLett

Gaining Control through Frustration: Two-Fold Approach for Liquid Crystal Three-Dimensional Command Layers Laura Cattaneo,*,† Jing Zhang,† Marc Zuiddam,‡ Matteo Savoini,† and Theo Rasing† †

Institute for Molecules and Materials, Radboud University Nijmegen, Heyendaalseweg 135, 6525 AJ, Nijmegen, The Netherlands Kavli Nanolab Delft, Technische Universiteit Delft, Lorentzweg 1, 2628 CJ Delft, The Netherlands

‡

S Supporting Information *

ABSTRACT: The alignment of Liquid Crystal (LC) molecules, essential for their applications in optical devices such as displays, is usually controlled by functionalizing their confining surfaces by either patterning or by specific surfactants that induce either parallel or perpendicular molecular arrangement. Inducing a bistable alignment, such as in the new zenithal bistable displays, offers new opportunities in terms of new functionalities and lower energy consumption but a full understanding of such bistable alignment appears still complicated. Here we present a simple phenomenological model that includes surface topography and chemistry. The predicted orientational transitions and bistable states are in excellent agreement with experiments, thus making this a proper tool to design multistable 3D command layers. KEYWORDS: Liquid crystal, topography, chemistry, anchoring control, phenomenological model, bistability

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molecules that cause homeotropic (perpendicular) alignment, which can obviously not be explained by Berreman’s approach. Stöhr et al.10,11 therefore proposed a directional interaction model in which the LC molecules are guided by a “π-like interaction” with the anisotropic polymer coated surface. In the most recent developments in LCD technology, new command surfaces are designed where LC molecules are not anymore aligned by a conventional rubbed polymer layer, but instead by a specifically 3D nanopatterned surface where both chemistry and topology together with the flexoelectric properties of the LC induce two or more alignment states through frustration.8,14−18 The nematic director configurations in these systems have been investigated through numerical methods characterized by 1D14,15 or more recent 2D models,18 where only ref 14 reports a semiquantitative comparison between theory and experiments of a ZBD, while the anchoring energy is assumed to be a fixed effective quantity without considering its actual origin. Here, without loss of generality, we present a very simple macroscopic phenomenological model that incorporates this 2fold approach to control the LC director profile, starting from the well-known Berreman model6 with the introduction of an additional term, the solid−nematic interfacial tension γ, which takes into account the chemical nature of the interactions between the LC molecules and the substrate. The resulting predictions of orientational transitions and bistable frustrated states are in excellent agreement with experiments and can

iquid crystals (LCs) present a fascinating phase of matter that is ubiquitous in biological systems but is mostly known for its applications in devices such as displays (LCDs), for which the control of the interaction between the confining surfaces and the LC molecules is an essential ingredient. Since the pioneering works on surface-induced alignment of LC molecules,1 scientists have been trying to answer the question: why do certain surfaces cause planar and others perpendicular (homeotropic) alignment or no alignment at all? The success of LCDs might indicate that nowadays a thorough understanding has been reached. Though this is certainly true for near-surface LC ordering on flat substrates,2−12 a comprehensive description of the anchoring process on nanopatterned surfaces where both topography and chemistry play a competing role, such as in the new zenithal bistable displays (ZBDs),13−18 appears still a complicated matter. The first theoretical study concerning LC surface anchoring induced by grooved surfaces and the resultant elastic distortion was presented by Berreman in 1972.6 This model has served as a benchmark for the alignment of LC molecules by geometrical constrains and implies that the LC director aligns along the direction of the grooves, with a negligibly small azimuthal distortion. Though qualitatively Berreman’s model describes the parallel alignment of liquid crystal molecules on the industrially available rubbed polyimide coated surfaces, there exist many experimental studies that report LC alignment in clear contrast with Berreman’s picture.19−23 Fukuda et al.9 already demonstrated that this model needs to be substantially modified, incorporating not only the surface anchoring energy but also the bulk twist energy. On the other hand, it has been known for a long time that there are classes of surface © XXXX American Chemical Society

Received: March 28, 2014 Revised: June 2, 2014

A

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wave), and λ is the wavelength (or periodicity of a square wave). Note that for a description of the more square-like profile, higher order terms should be included, however, the good agreement with the experimental results (see below) suggests that this first term is an adequate approximation. In this configuration, we assume that only splay (K1) and bent (K3) deformations play a significant role, neglecting any twist deformation. This assumption has been made in accordance with the classic Berreman model6 and is further justified by the absence of any birefringence from the optical micrographs representing configuration A and B (see Figures 2, 3, and 4).

form a simple but effective tool to design multistable 3D command surfaces that exploit the balance between competing alignment mechanisms. In our model, we describe a 3D corrugated surface of a specific chemical nature. To define the chemical properties of the surface, we introduce the interfacial tension γ, meaning the surface energy per unit area required to bring LC molecules in contact with a particular surface.2,3,24−28 In the case of a solid−nematic interface, γ is anisotropic and related to the orientation of the LC director with respect to the surface plane. We can define such anisotropy as Δγ = |γ⊥ − γ∥|, where γ⊥ and γ∥ define perpendicular (γ⊥) and planar (γ∥) interfacial tensions, corresponding to the condition where the LC-director is oriented perpendicular or planar to the surface, respectively. For the elastic (Berreman) configuration, we chose a patterned substrate characterized by a square-grooved profile with periodicity λ = 500 nm and amplitude A (total depth 2A) between 0 and 250 nm. If we consider a surface that imposes an LC perpendicular alignment (i.e., γ⊥ < γ∥), depending on the relative value of Δγ and the elastic distortion term, four different scenarios can occur, as schematically shown in Figure 1A−D.

Figure 2. Optical microscopy images of four samples viewed in whitelight transmission between crossed polarizer and analyzer and characterized by different total grooves-depths: (A) 150, (B) 200, (C) 250, and (D) 300 nm, respectively. The groove’s orientation is at 45°, highlighted by the orange arrow, and is the same in all four images.

Figure 1. Schematic drawing of LC molecules sectional alignment at the interface of grooved surfaces functionalized with perpendicular aligning molecules. (A) Configuration with lowest interface energy at the cost of elastic distortion energy. Configurations with relaxed elastic distortion energy at the cost of interface energy at the (B) y−z plane and (C) x−y plane and sum of the previous components (D). The differently behaving interfaces are marked by colored LC molecules. (E) Plot of the We − WI energy terms as a function of the total groove amplitude 2A for following three values of the interfacial tension anisotropy: 0.3 × 10−4 (violet line), 0.9 × 10−4 (magenta line), and 2.0 × 10−4Jm2− (pink line), respectively. The elastic constant, k = 13 × 10−12 N (E7, Merck), and the grooves pitch, λ = 500 nm, are constants in this plot. The big arrows indicate the depth 2A at which transitions from configuration (A,B) to (C) are expected.

Figure 3. Optical microscopy images of four samples presenting different grooves amplitude 2A: (A) 150, (B) 200, (C) 250, and (D) 300 nm, respectively. The LC textures are viewed in white-light transmission between crossed polarizer and analyzer and a λ-plate. Relative orientations are marked by black and red arrows. The grooves orientation is kept at −45°with respect to the vertical position.

On the other hand, if γ⊥ ≤ γ∥ (Δγ < 1 × 10−4 Jm−2 15), the LC molecules can reorient against the anchoring energy resulting from the surface chemistry, paying a surface energy cost that can be expressed as

In the case of γ⊥ ≪ γ∥ (Δγ between 2 and 5 × 10−4 Jm−2, see ref 15) the LC molecules will prefer an homeotropic (perpendicular) alignment and will follow the corrugated profile, as shown in Figure 1A. We can describe this corrugation-induced elastic distortion similarly to Berreman’s eq6 2π 3kA2 We ∝ λ3

WI = Δγ ΔS

(3)

where Δγ is the interfacial anisotropy tension and ΔS is the surface area over which the LC molecules reorient. The ΔS and thus WI will depend on the reorientation of the LC molecules. According to eq 3 the interface energy cost in case of a reorientation in the y−z plane of the grooves is (see Figure 1B):

(2)

where k is the isotropic elastic constant K1 = K3 = k, A is the amplitude of the sinusoidal wave (or half depth of a square B

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Figure 4. Optical microscopy images of the 2A = 200 nm sample viewed in white-light transmission between crossed polarizer and analyzer (A) and (D) and a λ-plate (B) and (E). (A,B) Represent the same point as well as (D,E) to underline the darker edges when they frame the vertical aligned zones or appear isolated, respectively. (C,F) Sketches of the LC orientations related to cases (B,E) respectively. These possible configurations are drawn in detail close to the patterned profile together with an averaged view, close to the microscope observation (B,E), sketched on top of the squared profiles with bigger ellipsoids.

⎛ 1 ⎞ 4AΔγ W I(y − z) = 2(2AΔγ )⎜ ⎟ = ⎝λ⎠ λ

give information about the easy axis itself. In contrast, our model describes the surface anchoring energy at the easy axis without considering external force. In fact, we introduce the interfacial tension anisotropy Δγ, that is a property of the local flat interface between an isotropic solid surface and an anisotropic LC that can result either in a random planar (γ∥) or vertical (γ⊥) orientation of the LC molecules. We therefore consider a combination of two anisotropy interactions, the anisotropic molecular LC-surface interaction and the Berreman-like elastic distortion, to formulate a comprehensive macroscopic model able to predict the actual LC director orientation on a specific surface. Figure 1E shows the transformation of these results into a W−2A phase diagram in which, for a given Δγ, the condition for homeotropic and parallel alignment is separated by a phase boundary (see arrows in Figure 1E; see also the Supporting Information, Figure S2). As A increases, the elastic distortion grows faster than the interface anisotropy energies leading to the condition We ∝ (2π3kA2/λ3) > (4AΔγ/λ) > Δγ. As a consequence, a transition of the LC alignment from homeotropic (Figure 1 A,B) to planar (Figure 1C) is expected, where the system relaxes the elastic distortion imposed by the corrugated profile and pays the interface energy cost for reorienting the LC molecules against the surface energy. Thus, a critical grooves-depth AT can be defined as

(4)

Here the first two account for the number of planar surfaces in a unit square wave; (2AΔγ) for the interfacial energy increase in each repeating unit square wave, (1/λ) for the number of unit square waves per unit length. In the case of a reorientation in the x−y plane with the LC perpendicular to the grooves direction (Figure 1C), WI is given by ⎛1⎞ W I(x − y) = 2λΔγ ⎜ ⎟ = Δγ ⎝λ⎠

(5)

If we consider a reorientation in both planes y−z and x−y, along the grooves (Figure 1D), WI becomes W I(y − z)&(x − y) = Δγ +

4AΔγ λ

(6)

W(y−z)&(x−y) I

Because results in the highest energy configuration this implies that LC molecules will never align along the grooves, which is in contrast to Berreman’s prediction. It is important to specify the difference between γ and WI because they refer to different interfaces: γ refers to the interfacial tension from a microscopic point of view, embodying the anisotropic interactions operating at the molecular scale that is corrugation-independent. WI refers to the macroscopic aligning surface summing up all different γ contributions (Figure 1A−D) depending on the microscopic structure of the grooves, resulting in a larger effective surface area due to the presence of the grooves. They coincide only in the case of flat substrates. Up to now, the LC surface energy was commonly defined by the Rapini and Papoular (RP) model24 as the amount of external energy required to change the LC orientation at the surface from an original alignment orientation, also called the surface easy axis. In this model, the initial easy axis is assumed to be known and when an external force is applied, this surface energy generates a stabilizing surface torque proportional to the induced angular deviation. Thus, what the RP gives is an indication of how tightly the LC molecules are anchored to the surface along a specific and known orientation, but it does not

AT ∝

λ 4

(7)

if the transition occurs from the configuration in Figure 1B to Figure 1C (violet and pink arrows), that is, for low and intermediate Δγ values satisfying the condition Δγ < (π3k/8λ). While in case of high Δγ the transition occurs directly from Figure 1A to Figure 1C (green arrow) and AT results are equal to AT ∝ (λ3Δγ/2π3k)1/2. Moreover these critical points are discontinuities similar to first-order phase transitions. As a result, we therefore might expect for a given Δγ and over a finite range of groove-depth 2A, a “mixed-phase regime” or a “bistable” regime in which homeotropic and planar LC alignments are equally stable. This transition between differently frustrated states is a key-element to gain control on the C

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Figure 3 shows four high-magnification images (100× objective, oil immersion) of different samples characterized by grooves amplitude 2A of 150, 200, 250, and 300 nm, respectively. The different colors correspond to different director orientations with respect to the λ-plate slow axis (red arrow in Figure 3A). In all images, the bright magenta color corresponds to a homeotropic orientation of the LC molecules, whereas bluish colors correspond to a planar orientation of the director relative to the λ-plate slow axis, resulting in an additive retardation effect, thus higher-(blue) order interference colors. The sample with 2A = 150 nm results in uniformly magenta transmission (Figure 3A) confirming the LC homeotropic alignment when 2A is below the critical value. The samples with 2A = 200 nm (Figure 3B) and 250 nm present two colors, the magenta vertical-aligned zones and the bluish planar-aligned zones, meaning that the transition from homeotropic to planar occurs through a “mixed-phase regime” within a finite range of 2A. This appears again in perfect agreement with the previous suggestion of a first order-like phase transition with a critical value of 2AT = 250 nm. Furthermore, the grooves are orthogonal with respect to the λ-plate slow axis while the LC director appears parallel to it, because the LC molecules present a blue color due to the λplate. This means that the LC molecules orient perpendicular to the grooves, instead of parallel as in the classical Berreman approach. This clearly demonstrates that it is the combined effect of the LC elastic distortion, induced by the grooves of a surface, and the chemical nature of the surface that accounts for the bistable anchoring behavior. This is confirmed also by the fourth sample with 2A = 300 nm that appears uniformly blue (Figure 3D), that is, homogeneously planar aligned with all the LC molecules perpendicular to the grooves. In both panels B and C of Figure 3, dark blue lines are visible that form frames around the magenta-homeotropic zones or are isolated. Along these darker groove-edges LC molecules present an intermediate polar angle between 0 (vertical alignment) and π/2 (planar alignment), as observable in Figure 4A,D where they appear grayish between crossed analyzer and polarizer. These lines naturally act as domain walls between the vertical and planar zones, where a gradual change of the LC molecules polar angle takes place over a finite length. Depending on the extension of the surrounded zones these domain walls appear as darker lines close to the clearly visible magenta areas (see Figure 4B,C) or as isolated darker edges if the vertical zone is confined in a single spatial period (see Figure 4E,F). Although not treated by our model, at the critical grooves depth the LC orientation can be subjected to a complex pattern of deformations including twist. This observation has been confirmed also by fluorescence confocal microscopy measurements reported in the Supporting Information. The presented counterintuitive results of LC alignment perpendicular instead of parallel to the grooves are in excellent agreement with our theoretical predictions. The latter not only go beyond the classical view of the pioneering Berreman model but also go beyond all the recent theoretical reexaminations,9−12 because those do not consider the intrinsic connection between geometry and chemistry in determining the LC surface anchoring energy. The simplicity and generality of the presented approach for designing LC command layers does not only have fundamental but also practical implications for the development of new

actual anchoring state in such LC devices, engineering a proper 3D grating and a related chemical functionalization. To verify experimentally the predictions of the phenomenological model presented above we prepared different LC cells by sandwiching a 6 μm layer of commercial nematic LC (E7, Merck) with positive dielectric anisotropy between two glasses both treated to give homeotropic LC alignment (dimethyloctadecyl3-(trimethoxysilyl) propylammonium chloride, DMOAP). One of these treated glass surfaces is presenting a square-grooved profile with 2A ranging from 50 to 400 nm with steps of 50 nm, while the grooves periodicity λ, the LC elastic constant k, and the anisotropy surface tension Δγ (DMOAP treatment) are kept constant. Using eq 7 we can estimate the critical depth 2AT of 250 nm considering λ = 500 nm, k = 13 × 10−12 N (Merck data sheet) and an intermediate value ΔγDMOAP ≈ 0.9 × 10−4 Jm−2. Figure 2 shows low-magnification optical microscopy images of four different LC cells observed in white-light transmission between crossed polarizer and analyzer with the grooves oriented at 45° with respect to the vertical position. Each LC cell is characterized by a different 2A of 150 (A), 200 (B), 250 (C), and 300 (D) nm, respectively. There is a clear passage from homeotropic (vertical) to planar alignment occurring at the critical depth 2A between 200 and 250 nm, which is in good agreement with the predicted value of 2AT of 250 nm. Figure 2A can be easily associated with configuration (B) (see Figure 1B) where all the LC molecules are vertically aligned but in view of the resolution of the optical system (∼230 nm) we cannot exclude configuration in Figure 2A, as the diagram in Figure 1E would suggest for grooves of a total depth of 50 nm. At the upper right corner of each image, the boundary between patterned and not patterned parts of the substrate is visible, showing that only the combination of grooves and chemistry can force the LC molecules to reorient, otherwise they maintain the vertical alignment induced by the DMOAP. The observed change in birefringence of the planar aligned areas from Figure 2B to Figure 2D can be qualitatively connected to a change in LC director orientation as the groovedepth increases (Michel-Levy chart). This trend has been confirmed by pretilt angle measurements, where this angle changes from 84 to 25° in the samples with grooves-depths varying from 50 to 400 nm, respectively. See details in the Supporting Information. Moreover it is clear that the transition occurs through the presence of a mixed-phase LC alignment where both homeotropic and planar alignment are equally stable, confirming its first order nature. In few samples, defect lines were observed forming closed loops running along the grooves (not shown). Because their appearance was neither systematic nor groove-depth dependent, we ascribe them to local substrate irregularities induced by the etching step. An alternative and more quantitative comparison between model and experiment can be made by using Δγ as a parameter. For this we used experimental data from ref 29 that again show a perfect agreement with the model predictions. The details can be found in the Supporting Information. To verify the actual orientation of the LC molecules for the planar case, along Figure 1D or perpendicular (Figure 1C) to the grooves, we again use a polarizing microscope with two crossed polarizers but with a first order retardation plate (λplate) inserted between the sample and the analyzer. Having the fast direction making an angle of π/4 with the polarizer, it enables to reconstruct the director azimuthal angle on the grooved substrate. D

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(11) Stöhr, J.; Samant, M. G. J. Electron Spectrosc. Relat. Phenom. 1999, 98, 189−207. (12) Shiyanovskii, S. V.; Glushchenko, A.; Reznikov, Y.; Lavrentovich, O. D.; West, J. L. Phys. Rev. E 2000, 62 (2), R1477. (13) Jones, J. C. J. Soc. Inf. Disp. 2008, 16 (1), 143−154. (14) Parry-Jones, L. A.; Edwards, E. G.; Elston, S. J.; Brown, C. V. Appl. Phys. Lett. 2003, 82 (9), 1476−1478. (15) Spencer, T. J.; Care, C. M.; Amos, R. M.; Jones, J. C. Phys. Rev. E 2010, 82 (2), 021702. (16) Yang, F.; Ruan, L.; Sambles, J. R. J. Appl. Phys. 2000, 88 (11), 6175−6182. (17) Serra, F.; Buscaglia, M.; Bellini, T. Mater. Today 2011, 14 (10), 488−494. (18) Cummings, L. J.; Cai, C.; Kondic, L. Phys. Rev. E 2013, 88 (1), 012509. (19) Chiou, D. R.; Yeh, K. Y.; Chen, L. J. Appl. Phys. Lett. 2006, 88, 133123. (20) Chiou, D. R.; Chen, L. J. J. Phys. Chem. C 2009, 113, 9797− 9803. (21) Gwag, J. S.; Kwon, J. H.; Oh-e, M.; Niitsuma, J. I.; Yoneya, M.; Yokoyama, H. Appl. Phys. Lett. 2009, 95 (10), 103101−103101. (22) Gwag, J. S.; Yi, J.; Kwon, J. H.; Yoneya, M.; Yokoyama, H. J. Appl. Polym. Sci. 2011, 119 (1), 325−329. (23) Choi, Y.; Yokoyama, H.; Gwag, J. S. Opt. Express 2013, 21 (10), 12135−12144. (24) Rapini, A.; Papoular, M. J. Phys., Colloq. 1969, 30 (C4), C4−54. (25) Rosenblatt, C. J. Phys. (Paris) 1984, 45 (6), 1087−1091. (26) Yokoyama, H.; Van Sprang, H. A. J. Appl. Phys. 1985, 57 (10), 4520−4526. (27) Naemura, S. Appl. Phys. Lett. 2008, 33 (1), 1−3. (28) Nobili, M.; Durand, G. Phys. Rev. A 1992, 46 (10), R6174. (29) Hallam, B. T.; Sambles, J. R. Liq. Cryst. 2000, 27, 1207−1211.

concepts of LCDs’ technologies, like ZBDs, where the control of molecular anchoring at the interfaces becomes 3D. Possible improvements can involve, for example, the inclusion of the LC anisotropic elastic constants and its specific flexoelectic properties to achieve a comprehensive description of this multiscale problem.

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ASSOCIATED CONTENT

* Supporting Information S

Additional information on sample preparation and substrate characterization, additional data about pretilt angle LC measurements and fluorescent confocal microscopy and additional analysis on the comparison between our model and reported experimental data. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Author Contributions

L.C. and M.Z. prepared the sample, L.C. and M.S. performed the measurements, J.Z. developed the model idea and T.R. supervised the work. All authors contributed in writing the paper. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge Professor Huub Salemink and Ing. Roel Mattern for their help with the fabrication of the samples. Part of this work was supported by The Netherlands Organization for Scientific Research (NWO) and the EU ITN HIERARCHY.

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ABBREVIATIONS LC, liquid crystal; LCs, liquid crystals; LCDs, liquid crystal displays; ZBDs, zenithal bistable displays; MBBA, methoxy benzilidene butyl analine; 5CB, cyano pentyl byphenil; RP, Rapini and Papoular; DMOAP, dimethyl octadecyl3-(trimethoxysilyl) propyl ammonium chloride

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REFERENCES

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