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Langmuir 2006, 22, 9986-9993
Role of Surface Anchoring and Geometric Confinement on Focal Conic Textures in Smectic-A Liquid Crystals Shahab Shojaei-Zadeh and Shelley L. Anna* Department of Mechanical Engineering, Carnegie Mellon UniVersity, Pittsburgh, PennsylVania, 15213 ReceiVed June 13, 2006. In Final Form: August 16, 2006 A high surface area-to-volume ratio in microchannels increases the importance of surface interactions within them. In layered liquids, such as smectic liquid crystals, surface interactions play an important role in the formation of defect textures. We use 8CB liquid crystal, which is in the smectic-A phase at room temperature, as a model layered liquid. PDMS surfaces can be tuned to be hydrophilic or hydrophobic, and due to the nature of liquid crystalline molecules, we show that this results in planar or homeotropic anchoring conditions, respectively. In a confined system, contrary to the bulk, generated defects cannot grow freely. In the present work, we show that the confinement offered by PDMS microchannels along with the capability of creating mixed anchoring conditions within them results in the formation of particular ordered defect textures through increased surface interactions in smectic-A liquid crystals. Our observations imply that microscale confinement is useful for controlling the size, size distribution, and packing structure of microscale defect structures within these materials. In addition, we show that by placing a droplet of smectic-A liquid crystal on a PDMS surface containing microscale parallel cracks, ordered focal conic defects form between two adjacent cracks. The distance between two adjacent cracks dictates the size of the defects. These observations could lead to useful ideas for exploring new technologies for flexible optical devices or displays that utilize smectic-A liquid crystals.
1. Introduction Microfluidic devices have recently been used to successfully control the formation of microscale structures such as monodisperse emulsions,1-3 foams,4,5 and anisotropic particles.6,7 In these examples, increased importance of surface interactions compared with bulk properties leads to enhanced control over the size, size distribution, and organization of microscale structures within these materials. Other recent studies have reported that confining smectic or lamellar liquid crystals inside microscale geometries can lead to the formation of monodisperse defect textures that are tens of micrometers in size and organize into regular lattices.8-11 The objective of this paper is to explore ways that microchannels and other microscale structures can be used to further control formation of defect textures within liquid crystals. Liquid crystals are elongated molecules for which the long molecular axis locally adopts one common direction in space, usually described by a unit vector (director), n. When anisotropic molecules associate under particular thermodynamic conditions, the molecules can arrange themselves into membranelike sheets, * To whom correspondence should be addressed. Tel: (412) 268-6492. Fax: (412) 268-3348. E-mail:
[email protected]. (1) Anna, S. L.; Bontoux, N.; Stone, H. A. App. Phys. Lett. 2003, 82, 364. (2) Thorsen, T.; Roberts, R. W.; Arnold, F. H.; Quake, S. R. Phys. ReV. Lett. 2001, 86, 4163. (3) Zheng, B.; Tice, J. D.; Ismagilov, R. F. Anal. Chem. 2004, 76, 4977. (4) Garstecki, P.; Gitlin, I.; DiLuzio, W.; Whitesides, G. M.; Kumacheva, E.; Stone, H. A. App. Phys. Lett. 2004, 85, 2649. (5) Xu, S. Q.; Nie, Z. H.; Seo, M.; Lewis, P.; Kumacheva, E.; Stone, H. A.; Garstecki, P.; Weibel, D. B.; Gitlin, I.; Whitesides, G. M. Angew. Chem., Int. Ed. 2005, 44, 724. (6) Dendukuri, D.; Tsoi, K.; Hatton, T. A.; Doyle, P. S. Langmuir 2005, 21, 2113. (7) Okushima, S.; Nisisako, T.; Torii, T.; Higuchi, T. Langmuir 2004, 20, 9905. (8) Choi, M. C.; Pfohl, T.; Wen, Z. Y.; Li, Y. L.; Kim, M. W.; Israelachvili, J. N.; Safinya, C. R. Proc. Natl. Acad. Sci. U.S.A. 2004, 101, 17340. (9) Blanc, C.; Kleman, M. Phys. ReV. E 2000, 62, 6739. (10) Blanc, C.; Kleman, M. Eur. Phys. J. B 1999, 10, 53. (11) Ruan, L. Z.; Sambles, J. R.; Stewart, I. W. Phys. ReV. Lett. 2003, 91.
also called lamellar sheets or smectic layers.12,13 This behavior is observed in diverse systems including small-molecule liquid crystals, concentrated surfactant solutions, block copolymers, and liquid crystalline polymers.14 Smectic layers are flexible and easily distorted but tend to preserve the interlayer spacing, which results in formation of defect domains in the bulk material. Liquid crystalline defect microstructures arise from an interaction between anchoring of molecular orientation near surfaces and bulk elasticity due to intermolecular interactions. For instance, smectic or lamellar liquid crystals typically form complex focal conic textures.15 Focal conic textures can organize into twodimensional hexagonal lattices when the liquid crystalline materials are confined between parallel surfaces with a gap thickness ranging from 10 to 100 µm for both thermotropic and lyotropic systems.10,11,16-18 Further confining these materials within rectangular microchannels leads to lines of equally spaced defect textures.8 Focal conic domains, originally observed by Friedel,15 are a broad class of curvature defects that satisfy the constant layer spacing constraint. In focal conic defects, parallel layers are wrapped around singular lines that form conic sections. The general construction is an ellipse and a hyperbola oriented in perpendicular planes and passing through each other’s focal points; the special case of a toric focal conic defect consists of a circle with a line passing through its center.19,20 Focal conic domains fill bulk thermotropic samples rapidly quenched from (12) de Gennes, P. G.; Prost, J. The Physics of Liquid Crystals, 2nd ed.; Oxford University Press: Oxford, 1993. (13) Chandrasekhar, S. Liquid Crystals, 2nd ed.; Cambridge University Press: Cambridge, 1992. (14) Larson, R. G. The Structure and Rheology of Complex Fluids, 1st ed.; Oxford University Press: New York, 1999. (15) Friedel, G. Annal. Phys. 1922, 18, 273. (16) Fournier, J. B.; Dozov, I.; Durand, G. Phys. ReV. A 1990, 41, 2252. (17) Quilliet, C.; Blanc, C.; Kleman, M. Phys. ReV. Lett. 1996, 77, 522. (18) Blanc, C.; Kleman, M. Mol. Crys. Liq. Crys. 2000, 351, 127. (19) Rosenblatt, C. S.; Pindak, R.; Clark, N. A.; Meyer, R. B. J. Phys. 1977, 38, 1105. (20) Kleman, M.; Lavrentovich, O. D. Soft Matter Physics: An Introduction, 1st ed.; Springer: New York, 2003.
10.1021/la061703i CCC: $33.50 © 2006 American Chemical Society Published on Web 10/21/2006
Focal Conic Textures in Smectic-A Liquid Crystals
the isotropic state,21,22 and they appear in confined smectic-A liquid crystals to relax surface energy anisotropy due to mixed (also known as antagonistic or hybrid) boundary conditions.9,10 Scaling arguments show that a characteristic size for focal conic defects in small-molecule thermotropic liquid crystals is approximately 10-20 µm.9,10 Microfluidic devices thus offer a unique opportunity for controlling the size and packing of this type of defect, whereas in bulk systems one typically has very little control over such parameters. The ability to control defect density, packing, and size could lead to the development of novel applications that exploit the ordered nature of lamellar materials, including optoelectronic devices and displays,11,23,24 templates for information storage media,25,26 and processing of biomaterials to encapsulate drugs.27-30 In this paper, we further explore the idea of reducing the characteristic length scale in order to control the density, packing, and size of defects. We use 8CB liquid crystal, which has been well characterized in the literature,12,13,20 combined with rapid and inexpensive soft lithography methods to fabricate microchannels from poly(dimethylsiloxane) (PDMS) elastomer.31,32 Our first step is to establish methods of controlling molecular anchoring conditions at PDMS surfaces. At surfaces, liquid crystal molecules anchor in a particular orientation due to intermolecular interactions with the substrate molecules. Common configurations are ‘planar’ in which molecules align parallel to the substrate and ‘homeotropic’ in which molecules align perpendicular to the substrate.12,13 These conditions can be controlled by adsorbing hydrophilic or hydrophobic molecules onto the substrate or by rubbing to create fine grooves in the surface, among other methods.33-39 In the present work, a planar anchoring condition on PDMS film is established by oxidizing the PDMS surface in air plasma.40,41 We also show that the hydrophobic PDMS surface with no special treatment imposes a homeotropic anchoring condition, consistent with observations reported elsewhere.33,42 PDMS offers numerous advantages over silicon for fabrication of microchannels in our study. Fabrication of microchannels using PDMS is rapid. Once the microchannel mold is prepared, (21) Fournier, J. B.; Durand, G. J. Phys. II 1991, 1, 845. (22) Chou, N. J.; Depp, S. W.; Eldrige, J. M.; Lee, M. H.; Sprokel, G. J.; Juliana, A.; Brown, J. J. Appl. Phys. 1983, 54, 1827. (23) Busch, K.; John, S. Phys ReV Lett. 1999, 83, 967. (24) Subramanian, G.; Manoharan, V. N.; Thorne, J. D.; Pine, D. J. AdV. Mater. 1999, 11, 1261. (25) Cavallini, M.; Biscarini, F.; Leon, S.; Zerbetto, F.; Bottari, G.; Leigh, D. A. Science 2003, 299, 531. (26) Park, C.; Yoon, J.; Thomas, E. L. Polymer 2003, 44, 6725. (27) Roux, D.; Cheneviera, P.; Pott, T.; Navailles, L.; Regev, O.; Monval, O. M. Curr. Med. Chem. 2004, 11, 169. (28) Freund, O.; Mahy, P.; Amedee, J.; Roux, D.; Laversanne, R. J. Microencap. 2000, 17, 157. (29) Bernheim-Grosswasser, A.; Ugazio, S.; Gauffre, F.; Viratelle, O.; Mahy, P.; Roux, D. J. Chem. Phys. 2000, 112, 3424. (30) Diat, O.; Roux, D. J. Phys. II 1993, 3, 9. (31) McDonald, J. C.; Duffy, D. C.; Anderson, J. R.; Chiu, D. T.; Wu, H. K.; Schueller, O. J. A.; Whitesides, G. M. Electrophoresis 2000, 21, 27. (32) Whitesides, G. M.; Stroock, A. D. Phys. Today 2001, 54, 42. (33) Gupta, V. K.; Abbott, N. L. Langmuir 1999, 15, 7213. (34) Hallam, B. T.; Sambles, J. R. Liq. Cryst. 2000, 27, 1207. (35) Jerome, B. Rep. Prog. Phys. 1991, 54, 391. (36) Newsome, C. J.; O’Neill, M.; Farley, R. J.; Bryan-Brown, G. P. Appl. Phys. Lett. 1998, 72, 2078. (37) Skaife, J. J.; Brake, J. M.; Abbott, N. L. Langmuir 2001, 17, 5448. (38) Kim, S. R.; Shah, R. R.; Abbott, N. L. Anal. Chem. 2000, 72, 4646. (39) Kawata, Y.; Takatoh, H.; Hasegawa, M.; Sakamoto, M. Liq. Cryst. 1994, 16, 1027. (40) Yaroshchuk, O.; Kravchuk, R.; Dobrovolskyy, A.; Lee, C. D.; Liu, P. C.; Lavrentovich, O. D. Mol. Cryst. Liq. Cryt. 2005, 433, 1. (41) Shahidzadeh, N.; Merdas, A.; Urbach, W.; Arefi-Khonsari, F.; Tatoulian, M.; Amouroux, J. Langmuir 1998, 14, 6594. (42) Brake, J. M. Imaging of Phenomena at an Aqueous-Liquid Crystal Interface through Changes in the Orientation of Liquid Crystals. Ph.D. Thesis, University of Wisconsin-Madison, Wisconsin, 2003.
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Figure 1. Cross-section of a typical PDMS microchannel.
we can produce many microchannels by repeated use of the mold within a short time. The ability to control anchoring conditions on PDMS surfaces without using surface chemical treatments also facilitates the use of mixed anchoring conditions within a given microchannel. Moreover, the optical transparency of PDMS permits the observation of liquid crystal textures within closed (4-sided) microchannels. The organization of this paper is as follows. In Section 2, we describe the materials and methods we use in the present work. Section 3 describes our experimental observations, including the establishment of desired surface anchoring on PDMS surfaces, the formation of defect textures in microchannels with mixed anchoring conditions, and the use of surface cracks on PDMS films to generate and control focal conic defects. We discuss preliminary efforts to model the size of the observed defects in Section 4, and finally we offer concluding remarks in Section 5. 2. Materials and Methods We use small-molecule thermotropic liquid crystals in the present study. In particular, we select from the widely used cyanobiphenyl (nCB) family.12,13,20 8CB (4-octyl-4′-cyanobiphenyl), a smectic-A liquid crystal at room temperature, is used in our observations of defect textures confined between surfaces, and 5CB (4-pentyl-4′cyanobiphenyl), a nematic liquid crystal at room temperature, is used to establish anchoring conditions at PDMS surfaces. Both 8CB and 5CB were purchased from Frinton Laboratories, Inc. and used as received. PDMS films and microchannels are fabricated using soft lithography techniques.32 PDMS elastomer (Sylgard 184, Dow Corning) is prepared by thoroughly mixing and degassing a 10:1 ratio of liquid silicone base and a curing agent. The un-cross-linked mixture is then poured into a flat plastic Petri dish (for smooth films) or over a microchannel mold and cured overnight in a 60 °C oven. The as-prepared PDMS surface is hydrophobic (contact angle θc ) 117° ( 1.5°). To generate hydrophilic surfaces, we oxidize PDMS films in air plasma (Harrick Scientific) and obtain a contact angle of θc ) 6° ( 1.5°.43 Oxidized surfaces are used immediately to avoid loss of hydrophilicity. To achieve parallel, straight microchannel walls, we use a directwrite laser (Heidelberg DWL66) to generate high-resolution chromeon-glass masks. We subsequently fabricate our molds in a class 100 clean room using these high-resolution masks and using SU8 negative tone photoresist on a silicon wafer. A typical microchannel cross section is shown in Figure 1. Polarized light microscopy is used to observe the liquid crystal textures, using an inverted microscope (Nikon Eclipse TE2000U) with a polarizer and analyzer inserted in the optical path. The microscope stage is fitted with a heating/cooling stage (Linkam PE100-NI, temperature control to within 0.1°C) and a digital camera (PixeLink, PL-A661). Atomic force microscopy (AFM) measurements are performed on a scanning probe microscope (Digital Instruments Nanoscope) in tapping mode using a sharpened tip (half (43) Delamarche, E.; Geissler, M.; Bernard, A.; Wolf, H.; Michel, B.; Hilborn, J.; Donzel, C. AdV. Mater. 2001, 13, 1164.
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Figure 2. 5CB droplet on as-prepared PDMS surface: (a) image between crossed polarizers, (b) bright field image. Scale bar is 70 µm. cone angle of less than 10° at the apex and a tip radius of less than 7 nm as reported by the manufacturer, Nanoscience).
3. Observations 3.1. Establishing Anchoring Conditions on PDMS Surfaces. In this section, we examine the anchoring conditions on treated PDMS surfaces. We first consider as-prepared films with no additional surface treatment. Several complementary methods are used to verify that these surfaces impose homeotropic anchoring in which the liquid crystal molecules orient perpendicular to the surface. To show this, we first place a droplet of 5CB liquid crystal on an as-prepared PDMS surface. Within the droplet, the resulting director field minimizes free energy contributions from interfacial forces and the bulk elasticity of the liquid crystal arising from the local variations in director orientation.12,20 The resulting optical texture observed when the droplet is placed between crossed polarizers provides information about the director field. The bright field image and corresponding optical texture of a 5CB droplet placed on an as-prepared PDMS film are shown in Figure 2. The classic Maltese cross pattern suggests that the director field is normal to the film surface (homeotropic anchoring) since the polarization is not changed along directions parallel to either the polarizer or the analyzer. This result was confirmed via detailed numerical simulations and experiments reported by Gupta and Abbott.33 On the basis of similar arguments, one should also observe that the intensity profile of transmitted light does not change upon rotation of a liquid crystal droplet between crossed polarizers when the surface imposes homeotropic anchoring.42 Figure 3 shows the intensity profile across the diameter of a 5CB droplet on an as-prepared PDMS surface at different rotation angles of the sample, with fixed orientation of the polarizer and analyzer. Intensity values for all angles are normalized by the maximum intensity value observed at 90°. Finally, we also verify that untreated PDMS films impose homeotropic anchoring by demonstrating a texture called a pseudo-isotropic texture.44 Normally the isotropic liquid crystal phase appears dark when observed between crossed polarizers due to the random orientation of the director field in this phase. The polarization image of a nematic liquid crystal sandwiched between two surfaces that impose homeotropic anchoring also appears dark because the director field is parallel to optic axis and the polarization of the light is not changed as it passes through the sample. This configuration is called the pseudo-isotropic state. In this state, bright spots can be observed near bubbles or pieces of dust where the director field is disturbed. If one disturbs the director field by perturbing the surface, the intensity dramatically increases due to sudden reorientation of the director field, but this intense flash fades over time and the system returns (44) Demus, D. Textures of Liquid Crystals, 1st ed.; Deutscher Verlag fur Grundstoffindustrie: Leipzig, Germany, 1978.
Figure 3. Normalized intensity variation across the diameter of a 5CB droplet on as-prepared PDMS surface at various rotation angles. The relative orientation of the polarizer and the analyzer is fixed at 90°, and the sample is rotated around the optic axis at different angles relative to the polarizer orientation.
to the pseudo-isotropic state. This sequence of events is observed with 5CB sandwiched between two as-prepared PDMS surfaces. We have also observed the pseudo-isotropic state for 8CB sandwiched between two parallel PDMS surfaces with homeotropic anchoring. On the basis of these simple tests, we infer that as-prepared PDMS surfaces impose homeotropic anchoring on both 5CB and 8CB liquid crystal molecules.45 We also note that liquid crystal molecules exhibit homeotropic anchoring at an air interface. In addition to as-prepared PDMS films, we also observe anchoring conditions imposed by oxidized PDMS films. Plasma surface treatment, which ionizes a gas using RF excitation, is commonly used to improve surface properties of polymer films.46 Previous studies have reported that exposing a PDMS surface to oxygen plasma changes its surface polarity from hydrophobic to hydrophilic43 and leads to planar anchoring of liquid crystal molecules.40 Shahidzadeh et al.41 showed that, by changing the plasma gas and duration of exposure, one could change the anchoring condition from planar to homeotropic. In the present work, we expose as-prepared PDMS surfaces to 2 min of air plasma. As before, we examine the resulting anchoring condition by placing a 5CB droplet on the surface and observing its optical texture between crossed polarizers. Comparing the observed texture with those presented by Gupta and Abbott,33 we infer that an oxidized PDMS surface imposes planar anchoring on 5CB liquid crystal molecules. We note that, with no additional treatment, this planar anchoring is not uniform, i.e., in-plane liquid crystal molecules are not uniformly oriented in the same direction. The droplet textures shown in Figure 4a correspond to a PDMS surface treated with air plasma. Figure 4b shows a schematic illustration of the molecules in contact with such a surface. Patterns shown in Figure 4a and b are consistent with patterns reported by Gupta and Abbott for nonuniform planar anchoring.33 We further influence the in-plane orientation by applying a large uniaxial strain (45) Dierking, I. Textures of Liquid Crystals, 1st ed.; John Wiley and Sons: New York, 2003. (46) Pizzi, A.; Mittal, K. L. Handbook of AdhesiVe Technology; M. Dekker: New York, 1994; p 680.
Focal Conic Textures in Smectic-A Liquid Crystals
Figure 4. Texture of a 5CB droplet between crossed polarizers on an oxidized PDMS film: (a) surface exposed to 2 min air plasma, (b) schematic illustration of molecules in contact with the surface exhibiting nonuniform planar anchoring corresponding to image (a), (c) surface exposed to 2 min air plasma and subsequently stretched under uniaxial strain (γ ≈ 30%), (d) schematic illustration of molecules in contact with the surface exhibiting uniform planar anchoring corresponding to image (c). Scale bar is 20 µm.
(γ ≡ (l - lo)/lo ≈ 30%) on the oxidized PDMS film. Here lo is the initial length of the sample and l is the stretched length. Stretching the film post-oxidization leads to uniform planar anchoring, an effect that has been reported previously on rubbed and stretched polymer films.47,48 Figure 4c corresponds to an oxidized and stretched PDMS surface. Figure 4d shows a schematic illustration of molecules in contact with such a surface. The observed textures in Figure 4b and d are consistent with those reported by Gupta and Abbott for uniform planar anchoring.33 We note that it is also possible to use chemical surface treatments to achieve planar and homeotropic anchoring on PDMS films, as has been reported by numerous authors on different surfaces.8,42,45 For example, we have coated PDMS surfaces with polyethyleneimine (PEI, Aldrich) to obtain planar anchoring,8 and we observe optical textures similar to that shown in Figure 4. We have treated the surface with octadecyltrichlorosilane (OTS, Aldrich) to produce homeotropic anchoring,42 and we observe optical textures similar to that shown in Figure 2. Therefore, we have established several possible routes to controlling the liquid crystal molecular anchoring conditions at PDMS surfaces. These multiple methods offer significant flexibility when combined with rapid and inexpensive soft lithography fabrication techniques,32 allowing the possibility to create arbitrary mixedboundary conditions within open or closed microchannels. In the present work, we take advantage of the intrinsic homeotropic anchoring conditions imposed by as-prepared and untreated PDMS and the ease and speed with which planar anchoring conditions can be created by exposure to air plasma. In the next section, we use these methods to create PDMS microchannels with mixed anchoring conditions and examine the influence of the antagonistic boundary conditions combined with microscale confinement on defect textures arising in smectic liquid crystals. 3.2. Defect Textures Formed Due to Antagonistic Anchoring Conditions in Microchannels. In this section, we show the effect of mixed anchoring conditions within a microchannel on (47) Yamaguchi, R.; Sato, S. Jpn. J. Appl. Phys., Part 2 1996, 35, L117. (48) Aoyama, H.; Yamazaki, Y.; Matsuura, N.; Mada, H.; Kobayashi, S. Mol. Cryst. Liq. Cryst. 1981, 72, 127.
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Figure 5. Smectic liquid crystal 8CB confined inside a rectangular microchannel with mixed anchoring conditions (Case 1). Three microchannel walls impose planar anchoring, while the fourth wall imposes homeotropic anchoring. (a) image viewed between crossed polarizers, (b) schematic diagram of the anchoring conditions at the wall compared with the viewing plane in (a). The widths of the microchannels are 80, 60, 40, and 20 µm from left to right. The depth of the microchannel is fixed at 10 µm. Sample is at room temperature. Scale bar is 10 µm.
the defect texture of a smectic-A liquid crystal. We consider two cases of mixed anchoring conditions in microchannels. In the first case, three walls impose planar anchoring while the fourth wall imposes homeotropic anchoring. In the second case, the situation is reversed: three walls impose homeotropic anchoring while the fourth wall imposes planar anchoring. Figure 5a shows a series of optical textures of smectic-A 8CB liquid crystal confined within closed rectangular PDMS microchannels with successively decreasing widths (80, 60, 40, and 20 µm) and a fixed depth of 10 µm. To prevent flow-induced anchoring, we inject 8CB liquid crystal in the isotropic phase into these microchannels. After cooling the injected samples into the smectic-A regime (T ≈ 26 °C), we observe the resulting textures using polarized light microscopy. A schematic illustration of the corresponding anchoring conditions is shown in Figure 5b. The objects in these images are toroidal textures characteristic in smectic-A liquid crystals.8 We note several features in the images of Figure 5. First, these observed textures are consistent with results reported by Choi and co-workers using open silicon microchannels with chemical surface treatments.8 We observe that the objects fill the entire channel width, with an average texture diameter of about 10 µm, which is close to the smaller dimension (in this case, the depth) of the microchannel. In wider channels, the textures form relatively disordered lattices and are polydisperse in size. Near the walls, some textures appear to be cut in half, consistent with planar anchoring at the sidewalls. As the channel width decreases to 20 µm, the textures become increasingly uniform and they organize more regularly along the microchannel axis. These findings suggest that the ordering and packing of these objects is directly controlled by the width of the microchannels, while the channel depth controls their size.8 The second case of mixed anchoring conditions is shown in Figure 6. Following the same procedure for injecting 8CB in the isotropic state, we observe the formation of two distinct bands along the microchannel sidewalls, with toroidal textures appearing along the center of the channel. The
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Figure 7. (a) 10 µm deep channel tilted slightly (θ ≈ 2.5°) out of the horizontal (x-y) plane about its long axis reveals the presence of “focal conic trees” structure, as indicated within the white box (b) schematic illustration of a focal conic tree. The width of the microchannel is 40 µm. Scale bar is 10 µm.
Figure 8. Top view of focal conic defects in a 10 µm deep 20 µm wide microchannel when only sidebands are present observed via (a) bright field and (b) polarization microscopy. Scale bar is 10 µm.
Figure 6. Smectic 8CB liquid crystal confined inside a rectangular microchannel with mixed anchoring conditions (Case 2). Three microchannel walls impose homeotropic anchoring, while the fourth wall imposes planar anchoring. The domain is filled with toroidal textures. The microchannel width is 20 µm in (a) and 10 µm in (b), while the depth is fixed at 3.8 µm. Reducing the width of the channel results in a decreasing number of focal conic textures at the center. In (c) and (d), the depth has increased to 10 µm. Here the structure of the sidebands are similar to the previous case but can be seen more clearly. Reducing the width of the channel from 60 µm in (c) to 40 µm in (d) results in a decreasing number of focal conic textures at the center. (e) Schematic diagram of the anchoring conditions at the walls compared with the viewing plane in (a)-(d). Samples are observed at room temperature. Scale bars are 5 µm in (a) and (b) and 10 µm in (c) and (d).
microchannels are 3.8 µm deep in Figure 6a and b and 10 µm deep in Figure 6c and d. As the microchannel width decreases from 20 µm in Figure 6a to 10 µm in Figure 6b, the toroidal texture in the center disappears and it appears that only sidebands are present. As we increase the microchannel depth to 10 µm, the structure of the sidebands becomes more clear, as shown in Figure 6c and d. A schematic illustration of the corresponding anchoring conditions is shown in Figure 6e. To obtain additional information about the banded textures formed on the sidewalls, we slightly tilt (θ ≈ 2.5°) the microchannel out of its horizontal (x-y) plane about its long axis. We observe the presence of discrete segments of nearly uniform size. Each segment contains a singular line along its center, which then splits into multiple singular lines along the microchannel wall. These structures are highlighted in the white box shown in Figure 7a. Our observations are consistent with a ‘focal conic tree’ structure described by Fournier21 in which a principal conic structure splits into multiple smaller conic structures as the mouth of the cone grows. This is shown schematically in Figure 7b. This splitting occurs in order to further relax the anisotropy in the interfacial energy due to different anchoring conditions at neighboring sidewalls. These observations appear to be consistent with the structure of parabolic focal conic defects, in which the singular lines consist of two
Figure 9. (a) Cracks produced on a PDMS surface by first exposing the surface to air plasma and subsequently stretching the film (γ ≈ 10%). In this figure, to observe cracks with an optical microscope, we have covered the PDMS surface with 8CB liquid crystal in the isotropic phase. (b) AFM image of typical arrays of cracks in tapping mode using a sharpened tip. (c) Average height profile of the outlined region shown in (b).
parabolas that each pass through the focal point of the other. In our images, one of the parabolas lies parallel to the viewing plane while the other lies perpendicular to the viewing plane, spanning the depth of the microchannel. As further support for this hypothesis, we also note that in both bright field and polarization images of these structures, the sideband textures appear as a row of “wishbones”. Following our earlier hypothesis that these structures are parabolic focal conic textures, this would imply that each wishbone shows the intersection of two parabolas through their focal point. As an example, Figure 8 shows bright field and polarized light images of a sideband texture that clearly shows the wishbone structure. The structure of parabolic focal conic textures can satisfy the homeotropic anchoring condition at the sidewalls, as shown schematically in Figure 11. These observations and our hypothesis for the structure of the defects in this configuration allow us to infer the geometrical configuration of smectic-A layers within the closed microchannel, which can subsequently be used to
Focal Conic Textures in Smectic-A Liquid Crystals
Figure 10. (a) Bright field image of a droplet of 8CB in the smectic-A phase on a cracked PDMS surface (inset shows corresponding polarized light image). (b) Size of defects as a function of crack spacing. A transition from multiple defect domains forming across the width of a crack to a single defect domain per crack width occurs when the spacing between cracks falls below 12 µm.
Figure 11. (a) Polarization image of a single row of focal conic texture in a microchannel, 40 µm wide × 4 µm deep with mixed anchoring conditions. A corresponding illustration of the orthogonal projection of smectic-A layers for the pattern in (a) is shown in the box. A schematic illustration of the cross-sectional view of the smectic layers at cross-section AA is shown in (b). The thick black lines depict the singular lines that consist of two parabolas oriented normal to each other and each passing through the focal point of the other.
derive a model based on minimization of free energy to predict the size of the observed textures. The basis for such a model will be described in Section 4. 3.3. Defect Textures Formed Due to Cracks on PDMS Surfaces. In this section, we demonstrate a fast and inexpensive method to create a two-dimensional network of toroidal textures without using microchannels. We create fine cracks on the surface
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of oxidized PDMS films by applying a small uniaxial tensile strain (γ ≈ 10%). We show here that these surface cracks are capable of controlling the size of toroidal textures. Zhu and co-workers49 detail a procedure for generating an array of fine cracks. In brief, an as-prepared PDMS substrate is treated with RF air plasma to generate a thin, silica-like layer on the surface.43 This thin film is significantly more brittle than the bulk PDMS, leading to the formation of parallel cracks in the brittle surface layer upon application of a uniaxial strain to the bulk film. Cracks form in a direction perpendicular to that of the applied strain. Due to the very small thickness of this silica layer (approximately 1 nm),50 we expect that the formation of cracks exposes the bulk hydrophobic PDMS inside the cracks. Figure 9a shows a bright field image of an array of parallel cracks formed on an oxidized PDMS surface. To observe these cracks using an optical microscope, the surface is coated with 8CB liquid crystal in the isotropic phase. Depending upon the amplitude of the applied strain, it is possible to produce a broad range of spacings between cracks.49 Investigation of the resulting cracks via tapping mode AFM, as shown in Figure 9b and c, reveals their groovelike structure and sub-micrometer dimensions. For AFM imaging, we use a PointProbe Plus (Nanoscience) tip with a half cone angle of less than 10° at the apex and a tip radius of less than 7 nm. Thus, we expect the AFM images to reproduce the actual surface with reasonably good fidelity. To examine the influence of cracks on the smectic liquid crystal defect texture, we dispense a small droplet of 8CB onto a cracked PDMS surface while the substrate temperature is above the temperature corresponding to the nematic-isotropic phase transition in order to avoid any flow-induced alignment. The substrate is then cooled to room temperature, and the resulting texture is observed under crossed polarizers. We observe the formation of toroidal defect textures due to the antagonistic boundary conditions: planar anchoring at the PDMS surface and homeotropic anchoring at the liquid crystal-air free surface.42 We notice that, near the edge of the droplet where the liquid layer is thinner than at its center, toroidal textures form and align between the cracks, as shown in Figure 10a. If the spacing between two adjacent cracks is large enough, we observe the formation of multiple rows of toroidal defects. Reducing the spacing leads to a single row of defects between two cracks, which we refer to as ‘single domain’. By measuring the spacing between two adjacent cracks and the effective size of the observed toroidal objects, we show that the transition from a single domain, where the size of toroidal objects is comparable to the spacing between cracks, to multiple domains occurs at a crack spacing of around 12 µm as shown in Figure 10b. The effective size of a toroidal object, Dt, is defined as the square root of the projected area of the object Dt ) xAt. The area At is measured using the ‘polygon selections’ function in the ImageJ (NIH) software. Unfortunately, we cannot measure the sample thickness directly, but we can estimate the height of the drop as follows. We have measured the contact angle of a sessile drop of 8CB on a similar surface to that depicted in Figure 10a, and this value is θc ) 16.5° ( 1.7°. Assuming that the drop shape is a wedge for the distance from the drop edge (∼63 µm) over which we have measured defect sizes, we obtain a maximum thickness of t ) 17.8 ( 2 µm for the film. From our images, it appears that as the film thickness increases toward the center of the drop, the defects do not sense (49) Zhu, X. Y.; Mills, K. L.; Peters, P. R.; Bahng, J. H.; Liu, E. H.; Shim, J.; Naruse, K.; Csete, M. E.; Thouless, M. D.; Takayama, S. Nat. Mater. 2005, 4, 403. (50) Chou, N. J.; Tang, C. H.; Paraszczak, J.; Babich, E. Appl. Phys. Lett. 1985, 46, 31.
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the presence of surface cracks anymore. We believe that, when the film is thin enough to feel the presence of the cracks, then the spacing between the cracks governs the defect diameter. As an example see the two adjacent tapered regions in the lower right corner of the bright field image of Figure 10a. If we assume that the film thickness is constant along a horizontal line, we then observe that there are defects formed under approximately the same film thickness but which have different diameters. When the film is thin enough and the crack separation controls the domain size, then we observe that as the crack width increases beyond a critical width of ∼12 µm, a transition from a single domain to multiple domains between two cracks occurs. In addition we note that our estimated film thickness and measured defect diameters fall within the range of predicted and measured values reported in Designolle et al. for a similar configuration.51
4. Modeling of Defect Textures In an effort to establish a model capable of predicting the size of the observed defects in confined microchannels, we note that the literature to date is limited to two particular cases: (a) the confinement of lyotropic liquid crystals between two parallel plates9,10 and (b) the formation of defects in thermotropic liquid crystals on a flat surface.21,22 To our knowledge, no model has been presented that considers the effect of total confinement of a thermotropic liquid crystal within microchannels. To build such a model, one can follow a similar procedure to that reported in Blanc and Kleman,10 calculating the total free energy based on contributions from bending energy, wall energy, and interfacial energy. By minimizing the resulting total free energy density function, the defect size under equilibrium conditions can then be found. This approach requires a clear understanding of the layer configuration in three dimensions. At this point, our ability to experimentally obtain threedimensional images of the director field is limited; however, we can infer a plausible configuration by considering known anchoring conditions and our observed two-dimensional images of tilted and plan views of the microchannel defect textures, shown in Figures 6-8. A proposed schematic representation of the layer arrangement for mixed boundary conditions corresponding to case 2 is shown in Figure 11. Modeling the total energy of this system is beyond the scope of the present paper and is the subject of future work. However, we note that by tuning the parameters in the Blanc and Kleman model for defects between parallel plates10 we can obtain an initial estimate for the defect size observed in closed microchannels. To show this, we focus in Figure 6a and c. In the case of a close-packed hexagonal lattice, Blanc and Kleman9 show that the defect size, ω, can be expressed as 3
ω)
x
1 2Φ1x
(1)
In this equation, Φ1 ) ∆σλ/12K is the ratio of interfacial energy anisotropy to bending energy, where ∆σ is the difference in interfacial tension between the two surfaces, K is the splay elastic constant, λ is the penetration depth, x ) h/Rλ is a dimensionless thickness with R, a constant, varying between 20 and 30,10 and h is the film thickness. The defect radius, a, is given by a ) h tan(ω). Substitution of x and Φ1 into eq 1 and subsequently ω into defect radius a gives a dimensional equation, (51) Designolle, V.; Herminghaus, S.; Pfohl, T.; Bahr, C. Langmuir 2006, 22, 363.
(x ) 3
a ) h tan
6RK h∆σ
(2)
In this equation, the depth of the microchannel, h, and splay elastic constant, K, are known, and we can thus only vary R or ∆σ. In Figure 6a, h ) 3.8 µm. For 8CB, the value of the splay constant, K ) 8 × 10-12 J/m.12 By taking values of R ) 15 and ∆σ ) 0.001, we obtain a defect diameter of 2a ) 5 µm, which is comparable to the measured value of 2ameasured ) 5.6 ( 0.5 µm from Figure 6a. As the film thickness increases, a hierarchy of defect formation is observed in experiments and is captured in the model of Blanc and Kleman.9 The authors show that for a texture similar to the one shown in Figure 6c in which smaller defects form between larger ones, the defect size between two parallel plates can be expressed by 3
ω)
x
1 + 2β 2Φ1x(1 + 2β4)
(3)
Definitions of the parameters in this equation are the same as in the previous equation, and β ) 0.1547 is the Apollonius number.9 The corresponding dimensional equation is thus given by
(x 3
a ) h tan
6RK(1 + 2β)
h∆σ(1 + 2β4)
)
(4)
Again, we notice that, in this equation, the depth of the microchannel, h, splay elastic constant, K, and Apollonius number, β, are fixed and we can only change R or ∆σ. Since the anchoring conditions are identical in Figure 6a and c, it is reasonable to assume that the interfacial tension anisotropy value ∆σ ) 0.001 is also identical. However, to obtain good agreement with experimental observations, we must increase R by a factor of 2 to a value of R ) 30. This yields a defect diameter of 2a ) 12.9 µm, for h ) 10 µm, which agrees well with the measured value of 2ameasured ) 13.0 ( 0.5 µm. Thus, by fitting values of R to the model presented by Blanc and Kleman, we have shown that we can obtain relatively good agreement with the observed defect sizes in at least two cases, despite the fact that confinement within the microchannel has not been taken into account. We note that the physical interpretation of R is not intuitive and that a consistent model for our configuration will need to appropriately account for the additional confinement and the dependence upon the geometry of the microchannel.
5. Conclusions Most reported studies of the confinement of layered liquid crystals between surfaces containing mixed anchoring conditions have been conducted on flat surfaces or between two parallel surfaces.16,21,22 Mixed anchoring conditions can drive texture distortions involving both layer dilation and curvature. These can result in the formation of focal conic defects in particular. In this work, we confine the smectic-A liquid crystal 8CB in PDMS-based microchannels with mixed anchoring conditions at the four channel walls in order to control the size and ordering of these defects. We first show that as-prepared PDMS surfaces impose homeotropic anchoring on liquid crystal molecules. Exposing such a surface to air plasma changes this anchoring to a nonuniform planar one.
Focal Conic Textures in Smectic-A Liquid Crystals
At first, we expose three walls of an open PDMS microchannel to air plasma and then cover the top with an as-prepared PDMS film. In this way, three adjacent walls impose planar anchoring and the fourth wall imposes homeotropic anchoring. In this configuration, we observe the formation of focal conic defects similar to those observed by Choi and co-workers8 in open silicon microchannels. We then alter the anchoring configuration such that three adjacent walls impose homeotropic anchoring and the fourth wall imposes planar anchoring. We observe a different texture in this case: toroidal focal conic textures along the middle of the channel surrounded by two sidebands, as shown in Figure 6a. Continuously decreasing the width of the channel results in a reduction in the number of rows of focal conic defects along the center. We show that the objects in the sidebands are also consistent with a focal conic texture, but in this case, we hypothesize that the sidebands consist of parabolic focal conic defects with the parabolic singular lines oriented normal to each other. In thicker channels, these sideband structures form focal conic trees.21 From these two cases, we conclude that, by confining the smectic-A molecules inside microchannels containing mixed anchoring conditions, one can create a variety of ordered patterns of focal conic textures. The size and ordering of these defects are geometry driven, while their pattern is determined by the surface anchoring conditions. In a separate effort to create planar arrays of focal conic defects with controlled dimensions on a flat surface, we place a thin film of smectic-A liquid crystal at the surface of an oxidized and stretched PDMS film. Again, focal conic defects form because of the presence of mixed anchoring conditions: planar anchoring at the PDMS surface and homeotropic anchoring at the liquid crystal-air free surface. We show that the spacing between two
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adjacent cracks governs the size of the focal conic defects as shown in Figure 10. To predict the size of focal conic defects in closed microchannels, we use models based upon minimizing free energy due to curvature and surface effects that have been proposed previously for smectic liquid crystal between two parallel plates.9,10 By tuning the physical parameters in these models, we find that the predicted size of focal conic defects in closed microchannels are in reasonably good agreement with the measured values from our experiments despite obvious limitations of the model. To propose a more general model, it is necessary to fully characterize the three-dimensional director field inside the microchannel. This characterization is the subject of future work. This work has shown that microscale confinement and surface chemistry can be used to create various ordered arrays of defect textures in smectic-A liquid crystals. The results of this work could aid in the fundamental understanding of defect formation and defect-solid boundary interaction in layered liquids. From a practical point of view, ordered arrays of defects could be used to create novel optical systems based on the unusual optical properties of liquid crystalline materials and the defect textures within them. Acknowledgment. This research was supported by the Pennsylvania Infrastructure Technology Alliance and the National Science Foundation under Grants No. CTS-0527909 and CTS0547432 (CAREER). We thank Professor Nicholas Abbott for providing the OTS recipe and Burak Aksak for assistance with the AFM imaging. We thank Professors Lynn Walker, Steve Garoff, Cristina Amon, and Tomek Kowalewski for helpful discussions. LA061703I
