Anchoring of a Nematic Liquid Crystal on a Wettability Gradient

Abstract Image. We have studied the anchoring of the nematic liquid crystal 5CB (4'-n-pentyl-4-cyanobiphenyl) as a function of the surface wettability...
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Langmuir 2006, 22, 9753-9759

9753

Anchoring of a Nematic Liquid Crystal on a Wettability Gradient Andrew D. Price and Daniel K. Schwartz* Department of Chemical and Biological Engineering, UniVersity of Colorado, Boulder, Colorado 80309 ReceiVed June 29, 2006. In Final Form: August 24, 2006 We have studied the anchoring of the nematic liquid crystal 5CB (4′-n-pentyl-4-cyanobiphenyl) as a function of the surface wettability, thickness of the liquid crystal layer, and temperature by measuring the birefringence of a hybrid aligned nematic cell where the nematic material was confined between octadecyltriethoxysilane-treated glass surfaces, with one surface linearly varying in its hydrophobicity. A homeotropic-to-tilted anchoring transition was observed as a function of the lateral distance along the hydrophobicity gradient, typically in a region corresponding to a water contact angle of ∼64°. The effect of the nematic layer thickness was measured simultaneously by preparing a wedge cell where the thickness varied along the direction perpendicular to the wettability. The detailed behavior of the onset of birefringence was found to be consistent with a dual-easy-axis model that predicts a discontinuous anchoring transition from homeotropic to planar. The anchoring was independent of temperature, except within 1 °C of the nematic-to-isotropic transition temperature (TNI). As the temperature approached TNI, the tendency for planar anchoring gradually increased relative to that for homeotropic anchoring.

Introduction Nematic liquid crystals (LCs) confined between surfaces exhibit an equilibrium director configuration highly dependent on the chemical and physical properties of the surfaces.1 The aligning effect of surfaces is of both fundamental and applied interest; control of interfacial behavior is involved in almost all LC-based technologies.2 Typically surfaces are physically or chemically treated to fix the nematic director either homeotropic (perpendicular) to the surface or approximately planar (parallel) to the surface. Such alignment layers are typically discovered empirically; the molecular-scale details of surface anchoring are not completely understood. For example, aliphatic self-assembled monolayers are often used to induce homeotropic alignment, while a clean glass surface induces approximately planar alignment. The anchoring of LCs at surfaces is a mesoscopic phenomenon, with the surface influencing the director orientation over tens of micrometers. A number of prior studies have investigated the case of hybrid aligned nematic (HAN) liquid crystal cells which anchor the nematic director with a different pretilt angle, θ, at each surface,3-6 where θ ) 0° refers to homeotropic alignment of the LC director; i.e., the director is parallel to the surface normal. It is important to note that the director angle at a particular surface (often called the “pretilt angle”) is not a local phenomenon; the director configuration is determined by a global minimization of free energy which considers the free energy at each surface as well as the elastic energy in the nematic phase. One may characterize the interaction between a surface and a nematic by a free energy with two important parameters: the preferred nematic anchoring angle for that surface (called the “easy axis”) and the anchoring strength. These parameters are characteristic of the particular surface. A typical form of the anchoring free energy for a surface is * To whom correspondence should be addressed. Phone: (303) 7350240. Fax: (303) 492-4341. E-mail: [email protected]. (1) Dierking, I. Textures of liquid crystals; Wiley-VCH: Weinheim, Germany, 2003. (2) Jerome, B. Rep. Prog. Phys. 1991, 54, 391-451. (3) Barbero, G.; Barberi, R. J. Phys. (Paris) 1983, 44, 609-616. (4) Priezjev, N. V.; Skacej, G.; Pelcovits, R. A.; Zumer, S. Phys. ReV. E 2003, 68, 041709. (5) Zhao, W.; Wu, C. X.; Iwamoto, M. Phys. ReV. E 2002, 65, 031709. (6) Carbone, G.; Rosenblatt, C. Phys. ReV. Lett. 2005, 94, 057802.

F(θ) ) (W/2) sin2(φ - θ) where W is the anchoring strength, φ is the preferred anchoring angle (easy axis), and θ is the actual angle of the director at the surface (pretilt angle).3 In the limit of completely decoupled surfaces (infinite nematic thickness or W f ∞), the actual anchoring angle will adopt the preferred one at each surface. However, in general this is not the case, and the global minimization of free energy often results in a director configuration where the actual anchoring angle deviates significantly from the easy axis, particularly at surfaces with weaker anchoring strengths. This presents a challenge in terms of data interpretation, because while the pretilt angle is relatively easy to determine from birefringence measurements, the fundamental parameter of interest for a given surface chemistry is the easy axis, which is nontrivial to determine because it is coupled to the anchoring strength. While this distinction has been recognized for some time,3,7,8 many researchers have simplified the analysis by making an approximation of infinitely strong anchoring.9,10 This simplification is not valid in general and may lead to misinterpretation. As we show explicitly in this paper, the measured anchoring angle is quite sensitive to the experimental configuration, e.g., cell thickness, and is not, in general, simply a function of the surface chemistry. HAN cells containing a surface continuously varying in its polar anchoring energy create an interesting case in which θ varies from 0° to some larger θ. Several approaches have been used to create such continuously varying surfaces. Seo11 used rubbed polyimide surfaces to anchor 4-n-pentyl-4′-cyanobiphenyl (5CB) with θ dependent on the rubbing strength. For a polyimide surface without side chains, θ increased with the rubbing strength. Sinha et al.12 also used a rubbed polyimide surface with θ for 5CB controllable from 0° to nearly 40°. Their work showed the birefringence as measured by optical retardation for HAN cells (7) Matsumoto, S.; Kawamoto, M.; Mizunoya, K. J. Appl. Phys. 1976, 47, 3842-3845. (8) Barbero, G.; Simoni, F. Appl. Phys. Lett. 1982, 41, 504-506. (9) Fonseca, J. G.; Hommet, J.; Galerne, Y. Appl. Phys. Lett. 2003, 82, 58-60. (10) Clare, B. H.; Efimenko, K.; Fischer, D. A.; Genzer, J.; Abbott, N. L. Chem. Mater. 2006, 18, 2357-2363. (11) Seo, D. S. Liq. Cryst. 1999, 26, 1615-1619. (12) Sinha, G. P.; Wen, B.; Rosenblatt, C. Appl. Phys. Lett. 2001, 79, 25432545.

10.1021/la061885g CCC: $33.50 © 2006 American Chemical Society Published on Web 10/06/2006

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remained near zero until a critical rubbing strength was reached, at which point the birefringence rapidly increased. Although the rubbing strength is presumably correlated to a physical property of the surfaces, this has proven difficult to characterize directly. In another approach Zhang et al.13 lithographically fabricated a surface with alternating horizontal and vertical corrugations. With decreasing periodicity of the corrugations, θ decreased from 90° to nearly 40°. In work more closely related to the current approach, several researchers have prepared surfaces with a gradient of surface chemistry by plasma polymerization,14 plasma treatment of selfassembled monolayers (SAMs),9 or diffusion-controlled grafting of partially fluorinated silanes.10 No rubbing was employed in these experiments. In all cases, a transition was observed from homeotropic alignment to tilted and eventually approximate planar alignment on the gradient surface. The transition had varying degrees of abruptness, but intermediate pretilt angles were observed in all cases. The researchers attempted to correlate the measured pretilt angle with the surface chemistry, suggesting that they were truly interested in the fundamental easy axis of the surface. However, they did not consider the effect of coupling to the opposite surface through the elasticity of the nematic director field, as described above. As we show explicitly in this paper, the apparent form of the anchoring transition is sensitive to the entire LC cell; e.g., the transition may appear to be more or less abrupt for the same gradient surface depending on other parameters such as the cell thickness. In this paper we report on a photochemically degraded alkylsiloxane self-assembled monolayer to create a surface linearly varying in its wettability. This chemically treated surface allows us to directly correlate the response of 5CB in a HAN cell to the surface energy (as probed by the wettability, for example). Our results indicate that the pretilt angle at the degraded surface rapidly increases at a critical surface energy which corresponds to a water contact angle of about 64°. The increase of birefringence is broadened due to the homeotropic anchoring on the opposite, undegraded surface and the elasticity of the nematic director field. However, a direct comparison with theoretical models that describe anchoring transitions from homeotropic to planar reveals that the observations are consistent with a model predicting a discontinuous anchoring transition and fundamentally inconsistent with a model that predicts a continuous transition. Our chemically treated surface also allows us to easily visualize temperature-dependent effects on the alignment of 5CB. In agreement with prior light scattering studies,15,16 the birefringence at a given location increases as the temperature approaches the nematic-to-isotropic transition temperature, TNI, suggesting that the tendency for planar alignment dominates that for homeotropic alignment near the transition. Experimental Details LC Cell Preparation. Borosilicate glass slides were cleaned with a fresh piranha solution composed of 30% aqueous H2O2 and concentrated H2SO4 (1:3, v/v) for 1 h at 70 °C. Warning: piranha solution reacts strongly with organic compounds and should be handled with extreme caution; do not store solution in closed containers. An octadecyltriethoxysilane (United Chemical Technologies, Inc.) self-assembled monolayer (OTES-SAM) was de(13) Zhang, B. S.; Lee, F. K.; Tsui, O. K. C.; Sheng, P. Phys. ReV. Lett. 2003, 91, 215501. (14) Watanabe, R.; Nakano, T.; Satoh, T.; Hatoh, H.; Ohki, Y. Jpn. J. Appl. Phys., Part 1 1987, 26, 373-376. (15) Mertelj, A.; Copic, M. Phys. ReV. Lett. 1998, 81, 5844-5847. (16) Vilfan, M.; Copic, M. Phys. ReV. E 2003, 68, 031704.

Price and Schwartz posited on the glass following the procedure described by Walba et al.17 This produced a surface with a water contact angle of ∼95° as measured optically by the sessile drop method with a goniometer, sufficient to induce homeotropic alignment of the LC. A coated slide was cut in half, with one of the halves being treated to produce a linearly degraded OTES-SAM. The degraded surface was produced by positioning the glass slide adjacent to a mercury pen lamp (UVP) for approximately 6 min. The irradiation intensity decreased along the length of the slide with increasing distance from the lamp. A linearly degraded surface and a nondegraded surface of OTESSAMs on glass were separated by 20 µm thick copper spacers and filled with 4-n-pentyl-4′-cyanobiphenyl (EMD Chemicals) via capillary action. The thickness of the LC cells averaged 20 µm with a standard deviation of 0.5 µm as determined by measuring interference fringes for monochromatic light transmitted through reference cells of air. For wedge cell preparation, the spacers were placed on one end of the glass slide in line with the hydrophobicity gradient such that the thickness of the cell varied from 0 to 20 ( 0.5 µm. The temperature of the LC was heated above its nematicisotropic transition temperature and then slowly lowered back to room temperature. The temperature of the sample was controlled using an Instec temperature-controlled stage mounted onto the rotating stage of the microscope. The temperature of the sample was accurate to within 0.1 °C. Optical Measurements. Birefringence measurements were made by measuring the wavelength retardation of polarized light traveling through our optical cells using a Berek compensator positioned above the sample in a polarizing light microscope. Contact Angle Measurements. Contact angle measurements were made at 1.5 mm intervals along the degraded slide using the sessile drop method and a custom-built contact angle goniometer. Atomic Force Microscopy. Samples of OTES-SAMs on polished Si wafers were imaged using a Nanoscope III AFM-E (Digital Instruments/Veeco). UV-irradiated samples were exposed to 254 nm light for 6 min. AFM images were collected in contact mode with silicon nitride tips (k ) 0.12 N/m).

Birefringence Calculations To calculate the birefringence within a HAN cell, one must know the nematic director profile across the LC layer. This profile is the result of three competing forces that govern the director orientation in the nematic LC layer. Two of these forces involve the anchoring of the nematic director at the top and bottom surfaces; the remaining force is attributed to the elastic property of the nematic LC director. The conventional description of nematic anchoring is due to Rapini and Papoular18 (RP), who proposed a surface energy expression given by gs ) (W/2) sin2(φi - θ) where W is the anchoring strength, θ is the direction of the nematic director at the surface, and φi is the easy-axis direction for surface i. A value of φi ) 0 indicates homeotropic anchoring, while φi ) π/2 indicates planar anchoring. This expression must be applied at each bounding surface. The behavior of the nematic between surfaces is described by continuum Frank elasticity theory.19 For the HAN cell, splay (K1) and bend (K3) must be considered. Although K3 is typically about 20% greater than K1,20 it is common to adopt the “one-constant” approximation, i.e., K1 ) K3 ) K,15 to permit an analytical solution of the director profile. At room temperature we took K ) 9.0 × 10-12 N for 5CB. In summary, to determine the equilibrium director configuration, one must know the Frank constants (17) Walba, D. M.; Liberko, C. A.; Korblova, E.; Farrow, M.; Furtak, T. E.; Chow, B. C.; Schwartz, D. K.; Freeman, A. S.; Douglas, K.; Williams, S. D.; Klittnick, A. F.; Clark, N. A. Liq. Cryst. 2004, 31, 481-489. (18) Papoular, M.; Rapini, A. J. Phys., Colloq. 1970, 4, 27-28. (19) Gennes, P.-G. d.; Prost, J. The physics of liquid crystals, 2nd ed.; Clarendon Press, Oxford University Press: Oxford, New York, 1993. (20) Dunmur, D.; Fukuda, A.; Luckhurst, G. R. Physical properties of liquid crystals: nematics; INSPEC Institution of Electrical Engineers: London, 2001.

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and the anchoring strength and easy axis at each of the bounding surfaces. For surfaces with more than one preferred anchoring direction, the RP expression can be extended to create a “dual-easy-axis” free energy of anchoring.6 In this paper, we consider two such expressions: A 2 B sin θ + cos2 θ 2 2

(1)

A 2 B C sin θ + cos2 θ + sin4 θ 2 2 4

(2)

f1 ) f2 )

Here, A is a Rapini-Papoular coefficient for homeotropic anchoring and B is an analogous coefficient representing a competing preference for planar anchoring. The third term is typically added to expression 2 to produce a continuous anchoring transition21 as we show below. For either expression, the alignment angle (easy axis) is determined by minimizing the free energy over the range 0 g θ e π/2, and the anchoring strength, W, is determined from the expression

The equilibrium pretilt angle at surface 1 (homeotropic) must satisfy the boundary condition 2L1

(21) Shioda, T.; Wen, B.; Rosenblatt, C. Phys. ReV. E 2003, 67, -. (22) Blinov, L. M.; Kabayenkov, A. Y.; Sonin, A. A. Liq. Cryst. 1989, 5, 645-661. (23) Seo, D. S.; Matsuda, H.; Ohide, T.; Kobayashi, S. Mol. Cryst. Liq. Cryst. 1993, 224, 13-31. (24) Rasing, T.; Musevic, I. Surfaces and interfaces of liquid crystals; Springer: Berlin, New York, 2004.

(3)

as given by BB, where L1 is the extrapolation length for surface 1; L1 ) K/W1. At surface 2, the appropriately generalized expression becomes 2L2

dθ - sin[2(φ2 - θ)] ) 0 dz

(4)

where φ2 is the preferred anchoring direction for surface 2 and L2 is the extrapolation length for surface 2; L2 ) K/W2. Following BB, these boundary conditions are solved when 2θ2 ) sin-1

W ) [d2f/dθ2]df/dθ)0 To connect these expressions to our experimental situation, we postulate that the degraded SAM surfaces with a linear wettability profile can be represented by a situation where the coefficient A decreases linearly from a value of A0 to zero while B increases linearly from zero to B0 over the same region. One finds that both expressions 1 and 2 have a minimum at θ ) 0 (homeotropic) when A > B, and in both cases the strength of the homeotropic anchoring decreases continuously to zero as A - B goes to zero. Both expressions also predict a transition in the easy axis of the anchoring angle, φ, at B ) A; however, the nature of the transition is distinctly different for the two models. Expression 1 has a discontinuous transition; i.e., for B > A, the free energy minimum shifts abruptly to φ1 ) π/2 (planar anchoring) with an anchoring strength given by W1 ) B - A. The minimum of expression 2, on the other hand, increases continuously from φ ) 0 for B > A. In particular, the minimum is found at φ2 ) sin-1[(B - A)/C]1/2 with an anchoring strength given by W2 ) [(B - A) - (B - A)2/C]. Though C is usually empirically determined, we adopt the simplifying assumption that C ) B so that the easy axis tends toward π/2 as A goes to zero. Since both models predict a homeotropic-to-planar transition at B ) A, we have chosen to graph the calculated effective birefringence values versus the parameter B - A, which we call the “excess planar anchoring energy”. For our calculations, it was necessary to determine numerical values of the coefficients A0 and B0. B0, the planar anchoring strength on a clean glass surface, was determined experimentally from the birefringence of a HAN cell to be B0 ) 8.0 × 10-6 J/m2. Values for A0, the homeotropic anchoring energy for aliphatic surfaces, vary in the literature22-24 but generally have a strong anchoring about 2 orders of magnitude greater than the planar anchoring of glass surfaces. We adopted a value of A0 ) 4.0 × 10-4 J/m2. Because this value of A0 is much greater than B0, the calculations are relatively insensitive to the exact value of A0. In particular, a variation of A0 causes a slight shift of the position of the homeotropic-to-tilted transition along the wettability gradient. Barbero and Barberi3 (BB) calculated the director field profile, θ(z), for a HAN cell by minimizing the free energy at all points along the z-axis, where z is the surface normal direction and the two anchoring surfaces are at z ) -d/2 and z ) d/2. Their calculations assumed that surface 1 preferred hometropic anchoring and surface 2 preferred planar anchoring. We generalize their approach to the situation where surface 2 has an arbitrary preferred anchoring angle φ2.

dθ - sin 2θ ) 0 dz

[

]

L1 d sin[2(φ2 - θ2)] + sin[2(φ2 - θ2)] L2 L2

[

L1 1 θ1 ) sin-1 sin[2(φ2 - θ2)] 2 L2

]

(5)

(6)

where θ1 ) θ(-d/2) and θ2 ) θ(d/2) refer to the values of θ that solve eqs 3 and 4. The transcendental eq 5 was solved using the solver in Microsoft Excel. Once θ2 was known, θ1 could be calculated directly from eq 6. Qualitatively, the solution is a bent director configuration that can exist only when the cell is thicker than the critical thickness, dc ≡ |L1 - L2|. In thinner cells the director field is uniform, with the nematic director parallel to the easy axis of the surface with the stronger anchoring. As a result, thin cells or regions with a strong anchoring coefficient at one surface and a weak anchoring coefficient at the other surface exhibit a uniform nematic director throughout the cell.3 Once the boundary conditions are known, the effective birefringence of the LC cell can be calculated directly. Birefringence, ∆n, is defined as the difference between the extraordinary, ne, and ordinary, no, indices of refraction. For a HAN cell the effective birefringence, nj, is given by nje - no, where no ) n⊥ and nje )

∫

1 θ2 - θ 1

n|n⊥ dθ

θ2

θ1

(n| cos θ + n⊥2 sin2 θ)1/2 2

2

(7)

where n⊥ is the index of refraction perpendicular to the optical axis, n| is the index of refraction for 5CB parallel to the optical axis. Refractive indices used were 1.5326 and 1.729 for n⊥ and n|, respectively.20 Expression 7 was integrated numerically using Mathematica.

Results and Discussion Substrate Characterization. The wettability of the OTESSAM surface was significantly impacted by UV irradiation. Exposure of an OTES-SAM to UV irradiation caused the initially hydrophobic surface to become increasingly hydrophilic with UV irradiation as shown in Figure 1 by contact angle measurements using deionized water (18.2 MΩ). In this figure we see an approximately linear dependence of the cosine of the contact angle and the distance from the UV light source. According to Young’s equation γLV cos φ ) γSV - γSL, where φ is the contact angle, γSL and γSV are the interfacial tensions between the solid/ liquid and solid/vapor interfaces, respectively, and γLV is the surface tension of water.25 We can conclude from Young’s equation that the interfacial energy of the surface and therefore wettability increased approximately linearly with UV irradiation.

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Figure 1. Wettability of a degraded OTES-SAM as a function of the distance from the UV light source.

Price and Schwartz

Figure 3. Polarizing microscope images of a 20 µm thick HAN cell with one surface linearly increasing in its wettability. The increasing wettability creates a region of rapidly increasing effective birefringence as shown by the magnified section in the white box.

Figure 2. AFM images of an OTES-SAM on silicon before (image a) and after (image b) UV irradiation for 6 min. The irradiated sample had a contact angle of ∼42°. The data scale for surface features runs from 0 to 10 nm. The lack of surface features in the UV-irradiated image supports a stepwise degradation of the OTE alkyl chain.

UV irradiation is believed to degrade the OTES-SAM-treated surface in a stepwise manner by C-C bond cleavage of the alkyl chain by reaction with UV-generated, short-lifetime, oxygencontaining radicals.26 Featureless AFM images, Figure 2, of an OTES-SAM on silicon both before and after UV irradiation support this proposed “lawnmower” mechanism of SAM degradation. In comparison, cleavage of the C-Si, Si-O, or O-C bonds in the headgroups would have left islands of bare or nearly bare silicon as occurs in UV-mediated ozonolysis of alkanethiol SAMs.27 Birefringence as a Function of the Wettability and LC Layer Thickness. Liquid crystals in the nematic phase exhibit birefringence except when the director is aligned parallel to the direction of light propagation (homeotropic alignment) or the projection of the nematic director on the surface is at an azimuthal angle parallel to either the polarizer or the analyzer. The varying wettability of one surface in the HAN cell caused the effective birefringence to increase along the axis of increasing wettability. This increase in effective birefringence was visualized by examining the HAN cell between a set of crossed polarizers. Figure 3 shows a representative image where a transition from homeotropic to tilted anchoring occurs at a distinct location along the surface gradient. The transition typically occurred at a location where the water contact angle was ∼64°. The magnified section of the image from the white box shows a region of rapidly changing birefringence. The wettability of the UV-exposed surface was least over the dark area at the bottom of the figure, increasing in the direction of the arrow, as indicated. The color variation follows the sequence predicted for increasing effective birefringence as represented by the Michel-Levy diagram.28 (25) Israelachvili, J. N. Intermolecular and surface forces, 2nd ed.; Academic Press: London, San Diego, 1991. (26) Ye, T.; Wynn, D.; Dudek, R.; Borguet, E. Langmuir 2001, 17, 44974500.

Figure 4. Wedge HAN cell viewed between crossed polarizers with thickness increasing from 0 to 20 µm at the right edge. Effective birefringence patterns vary as a result of the cell thickness and surface energy of the OTES-SAM UV-irradiated surface.

Although neither surface strongly anchored the azimuthal angle of the liquid crystal at a molecular level, prior studies have shown that the director aligns parallel to a region of rapidly increasing birefringence, twist deformation requiring the least energy.29 Away from the transition the azimuthal angle of the LC in birefringent regions for HAN cells aligned as in Figure 3 was measured to be 80° ( 10° with the horizontal. We speculate that the application of the gradient provided a broken macroscopic azimuthal symmetry, but it is not immediately clear why this direction should be chosen. Figure 4 shows a HAN wedge cell with surface energy increasing in the vertical direction and LC thickness increasing in the horizontal direction as indicated by the arrows. The thickness increases from 0 to 20 µm along the length of the horizontal axis. The transition to zero effective birefringence from a positive effective birefringence value was approximately independent of the cell thickness. However, the bands of color suggest that the transition from homeotropic to birefringent becomes more abrupt as the cell thickness increases. Figure 5 shows data that quantify this effect: (a) effective birefringence vs surface energy (where the surface energy is represented by the cosine of the water contact angle on the UV-exposed surface) and (b) pretilt angle vs surface energy for 5, 10, and 15 µm thick HAN cells. The effective birefringence saturated at a magnitude of ∼0.075, corresponding to a pretilt angle of 75°. This is in (27) Poirier, G. E.; Herne, T. M.; Miller, C. C.; Tarlov, M. J. J. Am. Chem. Soc. 1999, 121, 9703-9711. (28) Robinson, P. C.; Davidson, M. W. http://www.microscopyu.com/articles/ polarized/michel-levy.htm. (29) Lee, B. W.; Clark, N. A. Science 2001, 291, 2576-2580.

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Figure 6. Effective birefringence of HAN cells calculated as described in the text. The horizontal axis refers to the excess planar anchoring energy, B - A, of one surface of the HAN cell. The other bounding surface uniformly induces homeotropic alignment. Dotted lines are calculated effective birefringence values using the surface energy model represented by eq 1 in the text. This model predicts a discontinuous transition of the easy angle with excess planar anchoring energy. Solid lines are calculated effective birefringence values using the surface energy model represented by eq 2 in the text. This model predicts a continuous transition of the easy angle with excess planar anchoring energy.

Figure 5. (a) Effective birefringence measurements and (b) pretilt angle of the liquid crystal directors along the gradient surface for HAN cells of varying thicknesses. The cosine of the contact angle is directly related to the surface energy via Young’s equation. For clarity, error bars are omitted. The typical uncertainties are (2° for the contact angle, (0.003 for the effective birefringence, and (2° for the pretilt angle.

agreement with the effective birefringence measured for a reference cell with one OTES-SAM surface (homeotropic) and one clean glass surface. According to Figure 5, the thickness of the HAN cells has two important implications on the effective birefringence: (1) An equivalent surface energy results in greater effective birefringence for the thicker cell, prior to saturation. (2) The rate of effective birefringence increase is greater for the thicker cell. In particular, these data are explicit evidence of the fact that the anchoring angle is not simply a function of the local surface energy. In fact, it is intuitively reasonable that the anchoring transition would be more abrupt for thicker LC layers, because in thicker layers the gradient of the director field is smaller, resulting in lower elastic energy; i.e., the coupling to the opposite surface is weaker. The apparent relative offset along the contact angle axis in Figure 5 is on the order of the experimental uncertainty related to correlating the lateral distance with the contact angle from one experiment to another. Figure 6 shows calculated effective birefringence values for HAN cells with one surface that induces homeotropic alignment and another surface with a linear variation in anchoring properties. Specifically, the second surface is described by a linearly decreasing Rapini-Papoular coefficient, A, and linearly increasing planar anchoring coefficient, B. As described in a previous section, there is no effective birefringence for the case A > B, so for convenience the effective birefringence is plotted against the excess planar anchoring energy, B - A. Calculated effective

birefringence values are shown for HAN cells with LC layer thickness ranging from 5 to 15 µm and for the two surface energy models described previously. The model described by eq 1 displays a transition from a uniform director configuration to a bent director configuration at the critical thickness of a given cell. This results in a relatively abrupt onset of effective birefringence which eventually saturates for sufficiently thick LC layers. This model appears to be qualitatively consistent with the experimental observations, with regard to both the abruptness of the transition and the dependence on the LC layer thickness. These calculations demonstrate that a discontinuous transition of the easy axis in a HAN cell does not necessarily result in an abrupt change in birefringence. The abruptness of the apparent condition depends on the relative anchoring strengths of the two surfaces, the LC thickness, the nematic elasticity, etc. Obviously, it is also true that the experimental observation of a gradual change in birefringence does not necessarily imply a gradual, continuous transition of the easy axis. The model described by eq 2, on the other hand, predicts a very gradual onset of effective birefringence, and the maximum value of effective birefringence is relatively low. The differences due to the cell thickness are extremely subtle. These features are qualitatively inconsistent with our experimental observations. Clearly, the behavior of this model is pathological as B - A approaches B; in particular, the anchoring strength vanishes as the equilibrium anchoring angle approaches π/2. However, this is the lowest order free energy that results in a continuous anchoring transition, and it exhibits representative behavior for small values of the excess planar anchoring energy. It is instructive to compare these observations with previous work on anchoring transitions. Crawford et al. investigated 5CB anchoring within a tube that was modified by adsorption of a surfactant. They found that polar anchoring was directly related to the number of carbons in the aliphatic chains, n; in particular, the anchoring was planar for n < 7 and homeotropic for longer chain lengths.30 Although the structure of the modified surfaces (30) Crawford, G. P.; OndrisCrawford, R. J.; Doane, J. W. Phys. ReV. E 1996, 53, 3647-3661.

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was not characterized in detail in this work, the basic conclusion would appear to be consistent with our results. In contrast, anchoring angles intermediate between homeotropic and planar have been observed in some cases, particularly where surface treatment involved rubbing. For example, Walba et al. prepared OTES-SAM surfaces by UV irradiation followed by unidirectional rubbing.31 They reported the observation of intermediate pretilt angles in some cases. Fonseca et al.9 and Clare et al.10 both observed homeotropicto-planar transitions on gradient SAM surfaces created in different ways, but not involving rubbing. In the former case, the authors claimed an abrupt transition, while a gradual transition was claimed in the latter. Both observed intermediate values of effective birefringence; however, a direct comparison is difficult because the length scales on which the two experiments were reported were dramatically different, and more importantly, in both cases the measured pretilt angle was reported for only a single optical cell configuration. In our opinion, it is difficult to extract unambiguous information about the easy axis from these data. Although polymer surfaces are obviously quite different from SAM surfaces, it is also worth noting that Sinha et al.12 observed a continuous range of pretilt angles as a function of the rubbing strength on the polyimide surface. While it is clear that rubbing introduces a broken azimuthal symmetry, the specific structural consequences of rubbing remain mysterious. Even the characteristic length scale is unknown. Nevertheless, it is tempting to speculate that mechanical rubbing results in a specific surface structure that supports pretilt angles intermediate between planar and homeotropic. Temperature Effects. Figure 7a shows a region of rapid effective birefringence increase for a 20 µm thick HAN cell. This image at 32.0 °C was obtained well below the TNI for 5CB and is representative of images at lower temperatures. In fact, a 1 °C increase in temperature to 33.0 °C (Figure 7b) resulted in a barely noticeable change in the effective birefringence. However, at 34.0 °C (Figure 7c), the LC was within 1 °C of the TNI and subtle increases in effective birefringence were visible for regions of comparable surface energy; i.e., the bands of effective birefringence shift slightly upward to regions of lower surface energy (larger contact angle). Incremental 0.1 °C increases in temperature (Figure 7d-k) continued to cause an increase in the effective birefringence as well as a spreading of the color bands, indicating a more gradual change in the effective birefringence, in comparison to the more abrupt change in the effective birefringence when the temperature was well below the TNI. At a temperature of 34.9 °C, Figure 7l, which is higher than the TNI, 5CB loses its effective birefringence, entering the isotropic phase. Vilfan and Copic16 have demonstrated using dynamic light scattering measurements that the polar anchoring coefficients for 5CB show a smoothly decreasing behavior with increasing temperature. This behavior was well described by an empirical model derived by Faetti et al.32 They concluded that the anchoring strength is dependent on the square of the surface order parameter and varies with temperature as

W ) [a + b(TNI - T)1/2]2 where a and b are fitting parameters having units of J1/2/m and J1/2/(m K1/2), respectively, and are dependent on the interactions (31) Walba, D. M.; Liberko, C. A.; Shao, R. F.; Clark, N. A. Liq. Cryst. 2002, 29, 1015-1024. (32) Faetti, S.; Gatti, M.; Palleschi, V.; Sluckin, T. J. Phys. ReV. Lett. 1985, 55, 1681-1684.

Price and Schwartz

Figure 7. Changes in the birefringence for a 20 µm thick HAN cell as the temperature approaches the nematic-to-isotropic transition. The temperatures of the images are (a) 32.0 °C, (b) 33.0 °C, (c) 34.0 °C, (d) 34.1 °C, (e) 34.2 °C, (f) 34.3 °C, (g) 34.4 °C, (h) 34.5 °C, (i) 34.6 °C, (j) 34.7 °C, (k) 34.8 °C, and (l) 34.9 °C.

between the LC and the surface. This suggests that all anchoring energies decrease as TNI is approached. Our observations suggest, therefore, that the homeotropic anchoring energy decreased more rapidly, resulting in an increased tendency for planar anchoring in the vicinity of TNI.

Conclusions We have demonstrated a direct correlation between a measurable surface property, surface wettability, and its effect on the anchoring of nematic 5CB by observing the effective birefringence of hybrid aligned nematic cells where one bounding surface had a continuous wettability gradient. As the surface wettability of the alignment layer increased, the LC underwent an anchoring transition from a uniform to bent director configuration when the critical extrapolation length became less than the thickness of the HAN cell. The uniform director was characterized by homeotropic anchoring at both surfaces, while the bent director was characterized by homeotropic anchoring at the unexposed OTES-SAM surface and a nonzero tilt angle at the UV irradiated surface. Along the axis of increasing surface wettability, the bent director segment showed an initial sharp increase in the effective birefringence followed by a more gradual approach to a saturated value governed by the refractive properties of 5CB. The abruptness of the transition was dependent on the thickness of the cell due to the elasticity of the nematic director field. The principal features of the transition and its dependence on the surface energy and LC layer thickness were compared with effective birefringence calculations based on a continuum elastic theory using “dual-axis” models for surface anchoring energy based on competing tendencies for homeotropic and planar anchoring. The experimental observations, in particular the

Anchoring of a Nematic LC on a Wettability Gradient

abruptness of the transition and the dependence on the layer thickness, were qualitatively consistent with a model that predicts a discontinuous homeotropic-to-planar transition; they were fundamentally inconsistent with a model predicting a continuous anchoring transition. This suggests an approximate correspondence between the anchoring energy and interfacial free energies. These observations may also provide useful insight into the effective birefringence patterns observed in nematic LC cells as amphiphilic molecules adsorb at an aqueous/LC interface. In this situation, a continuous increase in the effective birefringence is observed with time, as the interfacial concentration of surfactant increases and the monolayer of amphiphilic molecules becomes increasingly packed at the interface (see the Supporting Information). Finally, studies near the nematic-to-isotropic transition show that the increase in the effective birefringence observed can be

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attributed to a smooth decrease in the magnitude of the anchoring energies, a greater decrease experienced by the strong homeotropic anchoring than by the weak planar anchoring energy. Acknowledgment. This work was supported by the National Science Foundation (Award No. CHE-0349547) and the Liquid Crystal Materials Research Center (NSF MRSEC, Award No. DMR-0213918). A.D.P. acknowledges support from a Department of Education GAANN fellowship. Supporting Information Available: Polarizing microscope images showing the dynamic evolution of the effective birefringence due to surfactant adsorption at the aqueous/nematic interface. This material is available free of charge via the Internet at http://pubs.acs.org. LA061885G