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Nematic Pancakes Revisited Ulysse Delabre,†,‡ Ce´line Richard,† Geoffroy Gue´na,†,‡ Jacques Meunier,‡ and Anne-Marie Cazabat*,†,‡ UniVersite´ Pierre et Marie Curie, and Laboratoire de Physique Statistique de l’Ecole Normale Supe´ rieure, 24 rue Lhomond, 75231 Paris Cedex 05, France ReceiVed December 20, 2007. In Final Form: January 31, 2008 The spontaneous spreading of the 5CB nematic liquid crystal on solid substrates has been extensively studied in the last years both at the microscopic1-4 and macroscopic5-6 scales. The remarkable feature at the microscopic scale is the presence of a discontinuity in the thickness profile of the films. On the other hand, the spreading dynamics of macroscopic drops is quite specific. The drop first spreads like a simple liquid, and then progressively faster, while a remarkable bell-shaped profile develops at the bottom.5-6 How the behaviors at the various scales are linked is an open question. Any answer requires reconsidering these wetting experiments deeper into the context of nematic films. More specifically, the anchoring of molecules at the interfaces7-8 and the competition between nematic elasticity9 and anchoring10 must be discussed quantitatively. For the thinnest films, the problem proves to be more complex than expected and contradictory data are found in the literature. Therefore, we decided to complete our previous studies with further experiments using another compound of the cyanobiphenyls series, the 6CB in the nematic phase, and also on liquid substrates, water and glycerol. These new data confirm that the description of the thinnest nematic films is not yet fully understood.
1. Introduction The spontaneous spreading of the 5CB (4′-n-pentyl-4cyanobiphenyl) nematic liquid crystal on solid substrates has been extensively studied in the last years both at the microscopic1-4 and macroscopic5-6 scales. After a first experiment on “rough” silicon wafers (bearing an obliquely evaporated silica layer),1 “smooth” wafers (bearing an amorphous layer of natural oxide) have been used as substrates. Complete wetting is achieved by the nematic phase. The remarkable feature at the microscopic scale is the presence of a discontinuity in the thickness profile of the films. At the edge of spreading droplets, the profile drops off abruptly at a well-defined thickness hN toward a flat film of thickness hT (Figure 1).1-4,6 When the liquid crystal is spin-coated on the wafer, and if the deposited average thickness is between hT and hN, then a coexistence between two flat films with same thicknesses hT and hN is observed.2-3 These thicknesses do not depend significantly on temperature far from the nematicisotropic transition, where hT ≈ 3.5 nm, while hN ≈ 20-30 nm.1-4 The spreading dynamics of macroscopic drops is quite specific. The drop radius R first increases with time t according to the * To whom correspondence should be addressed. E-mail:
[email protected]. † Universite Pierre et Marie Curie. ‡ Laboratoire de Physique Statistique de l’Ecole Normale Supe ´ rieure. (1) (a) Barberi, R.; Scaramuzza, N.; Formoso, V.; Valignat, M. P.; Bartolino, R.; Cazabat, A. M. Europhys. Lett. 1996, 34, 349. (b) Valignat, M. P.; Villette, S.; Li, J.; Barberi, R.; Bartolino, R.; Dubois-Violette E.; Cazabat, A. M. Phys. ReV. Lett. 1996, 77, 1994. (2) (a) Vandenbrouck, F.; Bardon, S.; Valignat, M. P.; Cazabat, A. M.; Phys. ReV. Lett. 1998, 81, 610. (b) Vandenbrouck, F.; Valignat, M. P.; Cazabat, A. M. Phys. ReV. Lett. 1999, 82, 2693. (c) Vandenbrouck, F. Ph.D. Thesis, University Pierre et Marie Curie, Paris, 2001. (3) (a) Van Effenterre, D.; Ober, R.; Valignat, M. P.; Cazabat, A. M. Phys. ReV. Lett. 2001, 87, 125701. (b) Van Effenterre, D. Ph.D. Thesis, University Pierre et Marie Curie, Paris, 2002. (4) Bardon, S.; Ober, R. R.; Valignat, M. P.; Cazabat, A. M.; Daillant, J. Phys. ReV. E 1999, 59, 6808. (5) Poulard, C.; Cazabat, A. M. Langmuir 2005, 21, 6270. (6) Poulard, C.; Voue´, M.; De Coninck, J.; Cazabat, A. M. Colloids Surf. 2006, A282, 240.
Figure 1. For thin films and antagonist anchoring at interfaces, there is a range of forbidden thicknesses below the NI transition (a)3, which reveals as a thickness discontinuity for spreading drops (b)1-4 or spin-coated films (c)2-3.
usual R ∝ t1/10 law, then the spreading accelerates while the drop takes a remarkable bell shape.5-6 The distortion of the drop profile is visible for thicknesses up to several micrometers. The same dynamics is observed on “hydrophobic” (bearing only siloxane bridges) and “hydrophilic” (partially hydroxylated)
10.1021/la703981q CCC: $40.75 © 2008 American Chemical Society Published on Web 03/07/2008
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fanchoring )
W1 2 W2 2 sin (θ1 - Θ1) + sin (θ2 - Θ2) ) 2 2 W1 W2 2 cos2 θ1 + sin θ2 (1) 2 2
The distortion of the structure has a cost in energy, which depends on the film thickness. The simplest model uses one elastic constant K.9,12-13 Let z be the film thickness. The free elastic energy per unit area of a distorted film is Figure 2. Structure of the distorted nematic film: the director stays in the zy plane. The actual anchoring angles are θ1 at the lower interface and θ2 at the upper interface. The arrows show the local orientation of the nematic director across the film.
substrates, which are both completely wetted by the liquid crystal. In contrast, instability patterns at the interface of spreading drops can be observed only on hydrophilic silica.5 They are due to the heterogeneity of this surface, which induces local changes in the orientation of the liquid crystal molecules in its vicinity. These defects interact with the flow. The wavelength of the instabilities is on the order of 3-5 µm. How the specific behaviors at the various scales are linked is an open question. The thickness discontinuity is observed around 20-30 nm, while the distortion of the profile extends over micrometers. Similarly, the wavelength of instabilities is around 3-5 µm, while the spatial scale of the substrate heterogeneity is much smaller. Writing models requires the knowledge of the structure of the films over the complete range of thickness. What is well characterized at the present time is the film of thickness hT far from the nematic-isotropic transition (Figure 1). Combined investigations using ellipsometry and X-ray reflectometry4,11 reveal that the film is a trilayer, formed by a monolayer of molecules which are almost flat on the substrate, and a bilayer on top of it. Polarized AFM measurements show that the monolayer has a strong dipole moment, i.e., that the molecular dipole moments are locally parallel.11 In the bilayer, the dipole moment is small, i.e., the dipoles are antiparallel. The structure of the film is controlled by short-range interaction. Far from the nematic-isotropic transition, thickness hN is around 30 nm on “rough”1 and around 20 nm on “smooth” silicon wafers.2-3 The structure of the film is controlled by longer range effects and anchoring conditions, i.e., the preferred orientation of the molecules at interfaces. In the present case, the free interface promotes homeotropic anchoring (perpendicular to the interface), while the anchoring is preferentially planar at the silica surface (in the plane of the interface). These antagonist anchoring conditions tend to distort the structure. More precisely, anchoring contributions must be introduced in the free energy of the films. A simple expression has been proposed by Rapini and Papoular.10 Let θ1 (respectively θ2) be the actual anchoring angle at the substrate (respectively at the free interface), see Figure 2. The preferred anchoring angle at the substrate is Θ1 ) π/2 (planar anchoring). The preferred anchoring angle at the free interface is Θ2 ) 0 (homeotropic anchoring). The anchoring energy per unit area is written as (7) Barbero, G.; Barberi, R. J. Phys. (Paris) 1983, 44, 609. (8) Ziherl, P.; Podgornik R.; Zumer, S. Phys. ReV. Lett. 2000, 84, 1228. (9) Frank, F. C. Discuss. Faraday Soc. 1958, 25, 19. (10) Rapini, A.; Papoular, M. J. Phys. (Paris) 1969, Colloq.30 C4. (11) Bardon, S. Ph.D. thesis, University Pierre et Marie Curie, Paris, 1999.
K(θ1 - θ2)2 felastic ) 2z
(2)
If the anchoring energies are finite, the cost of elastic distortion is too large for very thin films. More precisely (see Appendix 1 for details), the structure is distorted only if z g hc ) |L1 L2|, where L1 ) K/W1 and L2 ) K/W2 are the anchoring extrapolation lengths at the substrate and at the free interface, respectively.7,12-13 For z < hC, the strongest anchoring imposes its orientation within the whole film, which is no longer distorted. The dynamic equations are different below and above hC. The anchoring transition at z ) hC is second-order. Our experiments on silicon wafers have been interpreted assuming that the film of thickness z ) hN is distorted (hN g hC), with L2 ) 0 (infinitely strong anchoring at the free interface) and therefore L1 ) hc.1-4 The thickness discontinuity at the microscopic scale is accounted for provided the energy of the undistorted phase is larger than the one of the trilayer (see Appendix 2). For the description of a spreading drop, the anchoring and elastic contributions (eqs 1 and 2) are merely included in the free energy of the film. This allows us to write dynamic equations in a well-defined frame. A different interpretation of the thickness discontinuity has been proposed by Ziherl et al.8 Now, hN is supposed to be less than hC. Therefore, the film with thickness z ) hN is fully homeotropic. In this picture, the fluctuations of the nematic director (the “pseudo-Casimir” effect) induce a repulsive interaction in the film, just like elasticity does in the distorted structure. In a drop, the homeotropic film becomes distorted at some thickness z ) hC larger than hN. Choosing one model or the other should be easy because elastic constants and anchoring energies can be measured. The elastic constants of 5CB are well known and of the order 10-11 N.14 The difficulty comes from the anchoring energies, where scattered and contradictory values are found in the literature. There is a general agreement on the fact that the anchoring is stronger at the air interface than at the silica interface. All measurements of W2 at the air interface agree also on values around 10-5 N‚m-1 at ambient temperature.15-18 In contrast, data for anchoring energy W1 of 5CB on silica are scattered, between 1.25 × 10-5 19 and 15 × 10-5 N‚m-1,20 and surprisingly large, considering that they should be smaller than W2. (12) de Gennes, P. G.; Prost, J. The physics of liquid crystals, 2nd ed.; Clarendon Press: Oxford, 1993. (13) Oswald, P.; Pieranski, P. Les cristaux liquides; Gordon and Breach Science Publishers, 2000; tome 1. (14) Bunning, J. D.; Faber, T. E.; Sherrell, P. L. J. Phys. (Paris) 1981, 42, 1175. (15) Lavrentovich, O. D.; Pergamenshchik, V. M. Mol. Cryst. Liq. Cryst. 1990, 179, 125. (16) (a) Lavrentovich, O. D.; Pergamenshchik, V. M. Phys. ReV. Lett. 1994, 73, 979. (b) Lavrentovich, O. D.; Pergamenshchik, V. M. Int. J. Mod. Phys. B 1995, 9, 2389. (17) Sparavigna, A.; Lavrentovich, O. D.; Strigazzi, A. Phys. ReV. E 1994, 49, 1344. (18) Perez, E.; Proust, J. E.; Terminassian-Saraga, L.; Mauer, E. Colloid Polym. Sci. 1977, 255, 1003. (19) Van Sprang, H. A.; Aartsen, R. G. Mol. Cryst. Liq. Cryst. 1985, 123, 355. (20) Yokohama, H.; Kobayashi, S.; Kamei, H. J. Appl. Phys. 1987, 61, 4501.
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The model1-4 assumes L2 ) 0 and L1 ) hc e 20-30 nm. With K ) 10-11 N this means W2 > > W1 g 3-5 × 10-4 N‚m-1, which is not plausible. The model proposed by Ziherl8 does not give better results. As a matter of fact, the model requires a small value of hC ≈ 35 nm because the Casimir contribution to the disjoining pressure is of the order kT/hC3 and becomes negligible for larger hC. A way to reach low values of hC with reasonable values of the anchoring energies is to assume that the anchoring energies are very close: W2 ≈ W1, then L2 ≈ L1 and hc , L1, L2. With K ) 10-11 N and W ≈ 10-5 N‚m-1, one finds L ≈ 1 µm. In order to obtain hC ≈ 30 nm ≈ 3.10-2L, the anchoring energies must be the same within 3%, which again is not very plausible. As a matter of fact, the same behavior is observed on smooth and rough silica (hN is somewhat larger on rough silicas30 nm instead of 20 nmsbut the order of magnitude is the same). Other groups became recently aware of the same difficulty:21-22 SFA experiments performed on 5CB films with antagonist anchoring show that the distorted state is still observed for thicknesses as low as 20 nm. More precisely, antagonist anchoring of 5CB on mica muscovite surfaces has been obtained by coating one of them with a monolayer of CTAB.21 The anchoring is planar on bare mica and strong, of the order of 10-4 N‚m-1. On the CTAB monolayer, the anchoring is homeotropic, with the usual value ∼10-5 N‚m-1, and the usual extrapolation length ∼1 µm. No anchoring transition has been observed for surface separations as small as 20 nm: the fully planar structure corresponding to the stronger anchoring is actually excluded by the birefringence measurements.21 These contradictory results mean that further experiments with different compounds and substrates are required, but also that eqs 1-2, providing a relevant continuous description of thick films, might probably be reconsidered for thinner ones. The present work has been conducted in two directions: First, the previous experiments with 5CB on smooth silicon wafers have been completed with new ones using the 6CB compound. Second, experiments have been performed on liquid substrates, water or glycerol, with the same compounds. 2. Experimental Section 2.1. Materials and Methods. The liquid crystals investigated are the 5CB, 6CB, and 8CB (for control measurements) from SigmaAldrich (purity 98%) used as received. The transition temperatures are Tsolid-nematic ) TSN ) 24 °C for 5CB,
14.5 °C for 6CB
Tsolid-smectic ) TSSm) 21.5 °C for 8CB Tsmectic-nematic ) TSmN ) 33.5 °C for 8CB Tnematic-isotropic ) TNI ) 35.3 °C for 5CB,
29 °C for 6CB, 40.5 °C for 8CB
The room temperature is 22 ( 1 °C, far from the NI transitions. Surface induced melting is the rule for the nCB, which means that we do not have to consider the occurrence of a SN or SSm transition. 2.1.1. Solid Substrates. Complementary ellipsometric measurements have been performed with the 6CB in order to check the presence of the thickness discontinuity at the microscopic scale. The substrates are smooth, oxidized silicon wafers (crystallographic plane 100, p-doped, purchased from Siltronix), cleaned by oxygen plasma. The Brewster angle ellipsometer is phase-modulated, and an (21) Zappone, B.; Richetti, P.; Barberi, R.; Bartolino, R.; Nguyen, H. T. Phys. ReV. E, 2005, 71, 041703. (22) Israelachvili, J. Private communication, 2007.
additional lens allows us to reach a spatial resolution of ∼50 µm without loss of thickness resolution. 2.1.2. Liquid Substrates. Pure water (18.2 MΩ.cm) and glycerol from Sigma-Aldrich (purity 99%) are used as the liquid substrates. Experiments with glycerol must be conducted in closed boxes with desiccant in order to avoid uptaking atmospheric water. Films are obtained by depositing drops of solutions of the nCB in chloroform, nitromethane, or hexane. The drop rapidly covers the whole liquid surface, and the evaporation of the solvent leaves a more or less homogeneous film of (insoluble) liquid crystal on the liquid. Solutal Marangoni flows occur if the differences in surface tension between liquid crystal and solvent are large (hexane) and lead to poorer quality films. While the 6CB spreads on water and glycerol, the 5CB spreads only on glycerol. On water, small droplets with finite contact angle are obtained. The techniques used are Brewster angle microscopy (BAM) on a large Langmuir trough with mobile barriers on water,23-24 and mere observations under microscope on water and glycerol. The BAM is convenient to visualize monolayers or very thin insoluble films. The area available is changed by shifting the barriers. The light source is the green line of an argon laser at 514.5 nm. The advantage of the technique is its very high sensitivity.23-24 However, it does not provide thickness values and is relatively tricky. The anchoring of 5CB on glycerol and water is planar, and the energy per unit area on both liquids is around 10-5 N‚m-1 at ambient temperature, close to the one at the air interface, although larger.15-18 This means that films thicker than hC are distorted and that films thinner than hC are completely planar. No data are available for the 6CB, but we expect it to behave like the 5CB. 2.2. Experimental Results. 2.2.1. Experiments on Silicon Wafers. Ellipsometric measurements on smooth wafers confirm that the 6CB behaves like the 5CB on these substrates. We observe a thickness discontinuity at the edge of the drops, where a film of thickness hN ≈ 25 nm (20 nm for the 5CB) coexists with a trilayer (hT ≈ 4, 3.5 nm for the 5CB). 2.2.2. Experiments on Liquid Substrates: the BAM Experiment. We performed the BAM experiment on water with the nematic 6CB (the 5CB does not wet), and also with the smectic phase of 8CB in order to check the experimental procedure against known data.25 Previous work by Mann with the smectic 8CB on water using the BAM technique has shown a trilayer (identified from film pressure measurements) in coexistence with flat smectic domains of different thickness.25 We obtained identical images, which validates our procedure of film preparation. For the 6CB compound, our aim was to compress slowly the surface layer in order to reach the coexistence between two films of uniform thickness (Figure 1). We expected the thinner one to be the trilayer observed on solid substrates (or at least a compact structure controlled by short range interactions) and the thicker one to be the counterpart of the film of thickness hN. However, the images obtained during compression invariably showed flat nematic domains with different thicknesses. Compression/dilatation cycles did not help significantly. Although water is a fluid, the exchange of molecules with a compact layer is apparently very long. This means that the system is in metastable equilibrium, with large energy barriers and that the mere compression of the surface layer does not allow us to reach the coexistence between two films of well-defined thickness. Moreover, the typical image in Figure 3 shows that a complex structure is present even for a given flat domain. Therefore, a more exploratory study is needed, for which the BAM is not the most convenient technique. As a matter of fact, a microscope gives good images because of the large index contrast between air, liquid crystal, and water or glycerol. Therefore, we decided to investigate films of different thickness by varying the amount deposited in Petri glass dishes by a mere observation under microscope. (23) He´non, S.; Meunier, J. ReV. Sci. Instrum. 1991, 62, 936. (24) Mann, E. K.; He´non, S.; Langevin, D., Meunier, J. J. Phys. II (France) 1992, 2, 1683. (25) de Mul, M. N. G.; Mann, J. A. Langmuir 1994, 10, 2311.
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Figure 3. BAM: Isolated domain of 6CB nematic crystal surrounded by the trilayer on a water substrate. The speckle-like modulations are caused by the presence of a thicker domain out of the field of the camera, which scatters light. Width of the image ) 500 µm. Wavelength of the stripes ≈ 7 µm. The film thickness z ≈ 70 nm (z is not measurable with the BAM experiment; it has been estimated using the trend λ ≈ 100 z for z < 0.3 µm).
2.2.3. Experiments on Liquid Substrates: ObserVation under Microscope. As expected, the two liquid crystals give very similar results, and so does the 6CB either on water or glycerol. The mere difference is that experiments are more difficult with water because of the larger mobility of the films. (i) For relatively thick films, with chloroform or nitromethane as solvent, and if surfactants are avoided during the cleaning procedure, the film far from the walls is flat and homogeneous. For films thicker than 0.5 µm, typically the usual defects of the nematic phases are present. Thinner films show remarkable instability patterns (Figure 4a) with periodic stripes, first reported by Lavrentovich and Pergamenshchik.15 As the wavelength of the stripes depends on the film thickness, these patterns provide useful information on the film. This is why we studied them systematically. (ii) For smaller amounts deposited, film thickness and wavelength decrease. The shortest wavelengths we measure are ∼2 µm (Figure 4b). For such small amounts of deposited molecules, even if great care is taken, flat domains of different thickness coexist with a much thinner film which is now the “continuous phase” on the liquid surface. For very dilute samples, the edge of that film can be seen, which suggests that the structure is probably a trilayer. For a given thickness, the stripe pattern is the same for an isolated domain as for an extended homogeneous film. Although coexistence in this case does not mean equilibrium between domains and trilayer, the behavior of the domains does not depend on their size, as long as it is much larger than the wavelength of the stripe pattern. Figure 5 illustrates the variety of coexisting thicknesses in a purposely “bad” film. What is surprising is that these domains do not merge within hours. (iii) As the films do not evenly cover the surface (even for continuous films, there is an accumulation of liquid crystal in the meniscus contacting the Petri dish), the thickness is estimated using the Newton scale of colors. The wavelength of the stripe patterns is plotted versus the film thickness in Figure 6a. Figure 6b compares our data for 5CB on glycerol to the ones obtained earlier by Lavrentovich et al.16 and Sparavigna et al.17 The largest values of thickness do not agree well with the ones in ref 16, although this is the range where the Newton scale should give relatively precise results (in ref 16, the thickness is deduced from the amount deposited, but the corrections needed because of the presence of thickness heterogeneity are not reported). Anyway,
the interesting part of the diagram for the present discussion is toward low thicknesses. The shortest wavelengths in our experiment are ∼2 µm, in agreement with ref 17 and with an older paper by Lavrentovich et al.15 (where the wavelength was measured but not the film thickness). They were previously observed close to holes where the thickness of the film is not constant,15,17 but we were able to obtain them over extended flat domains (Figure 4b). In that range, the determination of thickness by the Newton scale is not possible. This is why the data taken from ref 17 are especially valuable (wavelength ≈ 2 µm for a thickness ≈ 20 nm). (iv) The stripe patterns correspond to azimuthal variations of the nematic director, which no longer belongs to the initial plane (see Figure 2). Due to the complex elastic behavior of a nematic phase, the in-plane and out-of-plane (azimuthal) fluctuations of the local orientation are coupled. Provided the planar anchoring is stronger, static director instabilities may develop and give rise to stripe patterns. Models have been proposed by Pergamenshchik,16,26,28 and Sparavigna et al.,17,27 see Appendix 3. The complexity of the equations and the numerical solution given in ref 16 do not allow us to figure out a simple physical picture of the process. Noticeably, if one may understand that there is an upper thickness threshold for the presence of stripes, the corresponding value of the wavelength is zero or infinite depending on the exact values of elastic constants, which are not known. The experiment shows that the wavelength diverges at the threshold. However, specific care must be taken to study large wavelengths because squared structures easily replace stripes in the thickest films (Figure 7). No model is available for these squared structures. Investigating the divergence more precisely will require a specific study. (v) Note that a domain is considered as “flat” when the thickness is constant at a scale larger than the wavelength of the stripe pattern. It is possible that the stripes cause some thickness modulations (this is the case in the presence of a magnetic field parallel to the free surface of a nematic12). Our way to measure thickness using the Newton scale of colors does not allow a safe estimate of this effect in structurally complex films. (26) Pergamenshchik, V. M. Phys. ReV. E 1993, 47, 1881. (27) Sparavigna, A.; Komitov, L.; Stebler, B.; Strigazzi, A. Mol. Cryst. Liq. Cryst. 1991, 207, 265. (28) Pergamenshchik, V. M. Phys. ReV. E, 1998, 58, R16.
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Figure 5. Small domains on a continuous trilayer, illustrating the variety of observable thicknesses after some stirring of the sample. 6CB on water, size of the images: top, 1200 µm × 920 µm; bottom, 610 µm × 460 µm.
glycerol substrates (also on water for 6CB) show a thickness discontinuity between a thinner film (known as to be a trilayer on silica) and a thicker one, which has a thickness hN around 20-30 nm. Figure 4. (a) Continuous films with stripe patterns. Top: 6CB on glycerol, Size of image: 280 µm × 210 µm. Wavelength of the stripes ≈ 5.6 µm, film thickness ≈ 60 nm Bottom: 5 CB on glycerol, size of the image: 610 µm × 460 µm. Wavelength of the stripes ≈ 40 µm, film thickness 300 nm (from Newton colors). (b) Short wavelength stripe pattern: average wavelength 2.5 µm 6 CB on glycerol, size of the image 113 µm × 85 µm Film thickness ≈ 25 nm. 2.2.4. Experiments on Liquid Substrates: Conclusion. Although the measurements are not very precise, it is clear that the value 2-2.5 µm is a threshold for the wavelength, corresponding to a threshold in thickness around 20-25 nm. There is no intermediate thickness between the one of the thinnest striped film and the trilayer. Therefore, we assume that the thinnest striped film is the film of thickness hN. As far as film coexistence and thicknesses are concerned, the behavior of 5CB and 6CB on liquid substrates is quantitatively similar and also similar to the one on silica. In contrast, stripe patterns are specific of the liquid substrates that are used. As a matter of fact, they are observable only if the planar anchoring is stronger. This is the case for our liquid substrates, but not on silica, where actually no stripe pattern was ever observed.
3. Discussion 3.1. Summary of the Experimental Findings. The situation is as follows. Films of 5CB and 6CB deposited on silica or on
On solid substrates (silica,1-4 present work, or mica21), experiments suggest that the film with thickness hN is distorted. As a matter of fact, no qualitative change is seen when the thickness decreases, either on drop profiles1-4 or with the SFA,21 where birefringence measurements exclude the nondistorted structure. However, the small value of hN leads to either implausibly large or implausibly close anchoring energies, at least if the usual equations for anchoring and nematic elasticity (eqs 1-2) are used. On liquid substrates, the situation is the same. Here, the models15-16 proposed for the description of stripe patterns use eq 1 for the anchoring energy, i.e., the film is distorted, and the general expression (eq A1, 1) for the elastic energy is used. Even with very close values for the anchoring energies, the model cannot account for stripe patterns in films thinner than 0.15 µm (see Appendix 3). However, experiment shows no qualitative change in the pattern at lower thickness: the wavelength is merely shorter, the wavelength/thickness ratio being approximately constant (∼100). 3.2. Reconsideration of the Equations. The eqs 1, 2, and (A1, 1), which are the base of the analysis, are written in a continuous description of the medium with coarse-grained variables.
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Figure 7. Square structures. 5CB on glycerol, size of the images: 238 µm × 173 µm Top : film thickness ) 80 nm, Bottom : film thickness ) 120-150 nm. Figure 6. (a) Stripe wavelength as a function of film thickness for the different systems. Scattering of points gives an idea of the reproducibility of measurements. (b) Stripe wavelength as a function of film thickness for 5CB on glycerol (semilog plot). Black circles, our data; triangles, some data by Lavrentovich and Pergamenshchik; star, data by Sparavigna et al. Full line: λ ) 100h.
About the anchoring, it is worth noting that the relation hc ) |L1 - L2| results from the specific shape of the anchoring energy:7,10
fA )
W1 2 W2 2 sin (θ1 - Θ1) + sin (θ2 - Θ2) 2 2
(3)
The sin2 shape was chosen because of the periodicity, sin2 (θ - Θ) (or sin 2(θ2 - Θ2) being the first term of a development in sin 2n(θ - Θ). The occurrence of higher order terms far from the NI transitions is already well admitted.29-31 They account for conic anchorings which disappear when the transition is approached because of the smoothing out of the interaction potential and are often used in displays.29 Therefore, the shape of the actual anchoring energy depends on the scale30-31 and in consequence on the film thickness. Equation 1 (or 3) is probably not relevant to describe anchoring in very thin films. In fact, the mere definition of the nematic director and therefore anchoring angle when the film thickness is not very large compared to the molecule size causes a problem. The same remark (29) Faget, L. Ph.D. thesis, Ecole Normale Supe´rieure de Cachan, 2003. (30) Fournier, J. B.; Galatola, P. Phys. ReV. Lett. 1999, 82, 4859. (31) Fournier, J. B.; Galatola, P. Europhys. Lett. 2005, 72, 403.
can be made for the elastic energy (eq 2 in the main text and eq A1, 1 in Appendix 1).
4. Conclusion The present study does not provide answers for the general description of nematic films under antagonist anchoring conditions, but it gives some guidelines for forthcoming analyses: (i) Films thicker than typically 0.15-0.20 µm can be safely described using the continuous equations for distorted nematic phases. Noticeably, for any description of the macroscopic dynamics on silica substrates, an elastic term must be included in the disjoining pressure. Considering the interface instability pattern on hydrophilic wafers, the important role of azimuthal contributions on liquid substrates suggests to include them in the analysis when surface disorder comes into play.32 (ii) For very thin films, the apparent contradictions provide evidence of the failure of the continuous descriptions. The validity of the Rapini-Papoular shape for the anchoring energy10 is already under debate.21-22,29-31 The relevance of the Frank elastic energy9 at small scale is also questionable. Stripes observed on films as thin as 20 nm on liquid substrates suggest that the local ordering is more complex than a small azimuthal perturbation on a 2D structure, like in Figure 2. A biaxial order at very low thickness has actually been proposed for the interpretation of SFA experiments below 20 nm.21 (iii) In conclusion, a proper description of the structure of the films thinner than 50 nm typically is required. Elastic and anchoring contributions need to be rewritten when the film thickness is not much larger than the molecule size. The “long (32) Ben Amar, M.; Cummings, L. Phys. Fluids, 2001, 13, 1160.
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range” van der Waals interaction must be included.8,33-34 The most difficult task is to write properly the short-range structural contribution which controls the trilayer. This term is crucial for a quantitative description of the observed thickness coexistence. This is a real theoretical challenge where numerical methods will possibly be needed. The accumulation of experimental data in the present study allows us to show that the problem is not specific of a given nematic compound on a given substrate, but that a fundamental question is asked.
last term comes into play for very thin (but still mesoscopic) films and accounts for van der Waals interaction. In the present case
f ) γNS + γNA +
W1 W2 2 cos2 θ1 + sin θ2 + 2 2 K(θ1 - θ2)2 + PvdW(z) (A1, 5) 2z
which can be rewritten as
Appendix 1: Nematic Elasticity, Description of a Distorted Nematic Film7,9,12-13 In the continuous description, the density of elastic free energy for a nematic phase with local director b n can be written as
K11 K22 K33 2 2 (divn b)2 + (n b.botn) r (n b.∧botn) r + + 2 2 2 K24 2 div(n b divn b+b n .∧botn) r + K13 div (n b divn b) (A1, 1) 2
ω j)
The three first terms are respectively associated with splay, twist, and bend deformations, respectively, and are commonly known as the Frank elastic energy.9 The last two are often referred to as “surface terms” and come into play only for thin films and finite anchoring energies. The elastic constants of 5CB can be found in the literature and are of the order of 10-11 N. Measured values of K11 are given by Bunning et al. (6.3 10-12 N at 22 °C14). The values used in the models are between 6.3 10-12 N16 and 10-11 N.31 In many cases, a simplified form is used, where the two last terms are ignored, and the same value assumed for the three remaining constants:
ω j)
K K K 2 2 (divn b)2 + (n b.botn) r b.∧botn) r + (n (A1, 2) 2 2 2
Let us briefly summarize the relations controlling the behavior of a distorted nematic film which thickness z is known, like the one in Figure 2. The director stays in the (z, y) plane. The deformation is a combination of splay and bend. Let θ1 (respectively θ2) be the actual anchoring angle at the substrate (respectively at the free interface). After integration over the film thickness, the elastic free energy per unit area takes the simple form:
K(θ1 -θ2)2 felastic ) 2z
(A1, 3)
Let Θ1 ) π/2 (respectively Θ2 ) 0) be the preferred anchoring angle at the substrate (respectively at the free interface). The total free energy per unit area f of the film can be written as
f ) γNS + γNA +
W1 2 W2 2 sin (θ1 - Θ2) + sin (θ2 - Θ2) + 2 2 2 K(θ1 - Θ2) + PvdW(z) (A1, 4) 2z
Here γNS and γNA are, respectively, the nematic-substrate and the nematic-air interfacial tensions, W1and W2 are the anchoring energies per unit area as introduced by Rapini-Papoular.10 The (33) Rajteri, M.; Barbero, G.; Galatola, P.; Oldano, C.; Faetti, S. Phys. ReV. E 1996, 53, 6093. (34) Sarlah, A.; Zumer, S. Phys. ReV. E 2001, 64, 051606.
f ) γNS + γNA + PN(z) + PVdW(z)
(A1, 6)
The anchoring angles are determined at a given thickness by minimization of the free energy as a function of the actual anchoring angles. If the van der Waals contribution is ignored, the relations are as follows7
K -W1 cos θ1 sin θ1 + (θ1 - θ2) ) 0 z K W2 cos θ2 sin θ2 - (θ1 - θ2) ) 0 (A1, 7) z and noticeably
W1 sin 2θ1 ) W2 sin 2θ2
(A1, 8)
The distorted state is the stable solution provided z g hc ) |L1 - L2|, where L1 ) K/W1 and L2 ) K/W2 are the anchoring extrapolation lengths at the substrate and at the free interface, respectively.7,12-13 For z < hC, the stable state is the undistorted one corresponding to the strongest anchoring. For very strong anchorings, the anchoring angles are merely Θ1 and Θ2 and L1, L2, hc f 0.
Appendix 2. Thickness Coexistence at Full Equilibrium Presently, the film thickness is not known, but it results from the equilibrium condition. The equilibrium condition between coexisting films is that the surface free enthalpies per unit area are equal. Using the vocabulary of wetting,1 this corresponds to a “pancake” (the thicker film), which is the final state of a drop of constant volume spreading on a “substrate” (substrate + trilayer). The “spreading parameter” is
S ) γTS - (γNS + γNA) > 0
(A2, 1)
where γTS is the surface tension of the substrate covered by the trilayer. This gives a condition for the thickness z. More precisely, for the film ffilm ) γNS + γNA + PN(z) + PVdW(z) and for the trilayer fT ) γTS. We assume that the trilayer has a defined structure, i.e., fT ) γTS ) cst. Let x be the fraction covered by the film. Then, f ) (1 - x)fT + xffilm has to be minimal as a function of thickness and anchoring angles, with the condition (1 - x)hT + xz ) cst, which is easily taken into account using Lagrange multipliers. One obtains
∂P with P ) PN(z) + PVdW(z) ≈ PN(z) ∂z (A2, 2)
S ) P - (z - hT)
Nematic Pancakes ReVisited
S)
Langmuir, Vol. 24, No. 8, 2008 4005
W1 W2 2 K (θ1 -θ2)2 cos2 θ1 + sin θ2 + 2 2 z hT K (θ1 - θ2)2 (A2, 3) 2z z
The relations (A1, 7-8) are still valid
K -W1 cos θ1 sin θ1 + (θ1 - θ2) ) 0 z K W2 cos θ2 sin θ2 - (θ1 - θ2) ) 0 z The solution is z ) hN, provided z g |L1 - L2|. Note that the condition z g |L1 - L2| must be obeyed whatever the way the film is obtained, including spreading, which was missed in ref 1. Comment on the Structure of the Free Energy. The variable part of the free energy of the nematic mesoscopic film has been written as
P(z) ) PN(z) + PVdW(z)
(A2, 4)
The van der Waals contribution to the free energy can be calculated for the 5CB-glycerol system35 and has been plotted on Figure 8, together with the values for anchoring and elastic energies taken from ref 16. The second-order anchoring transition takes place at 0.1 µm (100 nm), and the van der Waals contribution becomes significant below 40 nm typically. This is why it is ignored in the model.16 The van der Waals contribution in Figure 8 is quite similar to the one calculated by Ziherl et al.8,34 on silica, i.e., it is negative (on a silicon wafer, there would be also a weak positive contribution at larger thickness). It is worth noting that the mesoscopic free energy curve in Figure 8 leads neither to thickness coexistence nor to complete wetting. Another contribution is needed at a shorter range. The discussion above rests on experimental observations and takes as given both thickness coexistence and complete wetting. The short-range part of the interaction is not discussed. It is merely assumed that, far from the nematic-isotropic transition, the thinner film has a fixed free energy per unit area, independent of the thickness of the thicker film. In order to demonstrate theoretically that coexistence takes place between films as thin as 20 and 4 nm, respectively, it would be necessary to calculate the complete shape of the energy (or of the disjoining pressure) in that range of thickness. As the film structure is not known, this is a difficult task where numerical studies may help. Film coexistence has previously been investigated very close to the nematic-isotropic transition. As the coexisting films are both much thicker, the model assumes the thinner film to be isotropic and the thicker one to be nematic,36 in which case the free energy can be written explicitly. This is not the case here.
K11 K22 K33 2 2 + + (divn b)2 + (n b.botn) r (n b.∧botn) r 2 2 2 K24 2 div(n b divn b+b n .∧botn) r + K13 div (n b divn b) (A1, 1) 2
ω j)
To account for the deformations of an initially distorted film like the one in Figure 2, three more elastic constants come into play, the twist elastic constant K22, the saddle-splay elastic constant K24, and the splay-bend elastic constant K13.16,27 Due to the complexity of the problem and the uncertainty about the values of the constants, no precise model is available. However, a few points emerge. (i) Stripes are observed only if the planar anchoring is stronger. (ii) For z > hc (distorted nematic film), “long wavelength” stripes are expected. The control parameter is K24 (possibly also K13). The wavelength λ is at least 100 times larger than the film thickness. This is the case in experiments (see Figure 6). (iii) Several models for the function λ(z) are available.15-16,26 They agree on the presence of a maximum threshold for the thickness, which in the 5CB-glycerol system is of the order of 0.5 µm. The wavelength at the threshold may diverge or vanish depending on the exact value of the elastic constants. Experiment shows unambiguously a divergence of the wavelength. (iv) The models fail to account for stripes in films thinner than typically 0.15 µm. The authors assumed that thinner striped films
Appendix 3: Stripe Patterns on Liquid Substrates15-17,26-27 The general expression of the elastic energy density is now needed. (35) Israelachvili, J. N. Intermolecular and surface forces; Academic Press: London, 1985. (36) Van Effenterre, D.; Valignat, M. P.; Roux, D. Europhys. Lett. 2003, 64, 526.
Figure 8. Contributions to mesoscopic free energy per unit area versus film thickness (in µm) for the system air-5CB-glycerol. Top: energy vs log(z), Bottom : energy vs z. Values of extrapolation lengths from Lavrentovich et al. L1 ) 0.76 µm, L2 ) 0.86 µm, and K ) 10-11 N.
4006 Langmuir, Vol. 24, No. 8, 2008
could not be homogeneous. Then, the large change in local thickness would induce azimuthal anchoring.16-17 However, we obtained flat films with thickness of the order of 20-30 nm, see Figure 4b. (v) For z < hc (homogeneous planar film), “short wavelength” instabilities may arise, controlled by the K22 term.26 These distortions resemble more those produced by an external field,
Delabre et al.
and the wavelength λ is of the order of the film thickness.26 Such small wavelengths cannot be observed with a mere microscope. Acknowledgment. We gratefully thank the “Fe´de´ration des syste`mes complexes” of the University Pierre et Marie Curie (director Professor Martine Benamar) for financial support. LA703981Q
