pubs.acs.org/Langmuir © 2010 American Chemical Society
Quantitative Spreading Kinetics of a Three Molecular Layer Liquid Patch Olivier Noel, Jean-Luc Buraud, Laurent Berger, and Dominique Ausserre* Laboratoire de Physique de l’Etat Condens e, “Molecular landscapes and biophotonics” group, Universit e du Maine, Avenue Olivier Messiaen, 72000 Le Mans, France Received December 14, 2009. Revised Manuscript Received March 9, 2010 The late stage kinetics of the spreading of a smectic nanodrop on a solid surface was investigated by direct and real time imaging of a three molecular layer patch using the SEEC microscopy. Experimental data do not conform to the only available theory, which covers only weakly stratified liquids. A new model is proposed, in remarkable agreement with experiments, in which the spreading mechanism appears to be a quasi-static process ruled by solid/liquid interactions, 2D Laplace pressure, and separate edge and surface permeation coefficients.
Finite extended molecular films on solid substrates present considerable interest in both technological and scientific research. On the technological side, their control is detrimental in many applications such as surface patterning, cell adhesion, microfluidics, and biochips. On the fundamental side, they are particularly interesting as they provide finite 2D or quasi-2D thermodynamic systems.1-5 They can be obtained by the deposition of a nanodrop in air, which evolves toward a molecular film by wetting a solid surface. In practice, the molecules used to make surface patterns are nonsymmetric and present, at least at the solid-liquid interface, some amphiphilic character. Indeed, because of the ordering of the first molecular layers induced by the solid surface, every liquid becomes more or less stratified in the vicinity of a surface,6 and this determines its wetting behavior.7-10 Therefore, understanding the wetting of solid surfaces by stratified liquids is a very important task, and it has been the object of many studies.11-14 The most extensively studied system is the combination of 4-noctyl-40 -cyanobiphenyl (8CB) liquid crystal with oxidized silicon wafers. It may be considered as the model system of the field. Experimental studies on this system were mainly conducted using ellipsometry,14 atomic force microscopy (AFM),12 or X-ray reflectivity.13 They have shown that the liquid crystal partially or totally wets a clean silicon oxide surface, depending on the drop size.15 This behavior was observed in the whole range extending from the solid/smectic transition temperature, 21.5 °C, up to about the nematic/isotrope transition temperature, 40.5 °C.16 Even when the free surface of a macroscopic 8CB drop presents *Corresponding author. E-mail:
[email protected].
(1) De Gennes, P. G.; Cazabat, A. M. C. R. Acad. Sci. 1990, 310, 1601–1606. (2) Lazar, P.; Schollmeyer, H.; Riegler, H. Phys. Rev. Lett. 2005, 94, 116101-1– 116101-4. (3) De Jeu, W. H.; Ostrovskii, B. I.; Shalaginov, A. N. Rev. Mod. Phys. 2003, 75, 181–235. (4) Oshanin, G.; De-Coninck, J.; Cazabat, A.-M.; Moreau., M. Phys. Rev. E 1998, 58, R20–R23. (5) Wasan, D. T.; Nikolov, A. D. Nature 2003, 423, 156–159. (6) Israelachvili, J. N.; McGuiggan, P. M. Science 1988, 241, 795–800. (7) De Gennes, P. G. Rev. Mod. Phys. 1985, 57, 827–863. (8) Heslot, F.; Fraysse, N.; Cazabat, A. M. Nature 1989, 338, 640–642. (9) Forcada, M. L.; Mate, C. M. Nature 1993, 363, 527–529. (10) Bardon, S.; et al. Faraday Discuss. 1996, 104, 307–316. (11) Lucht, R.; Bahr, C. Phys. Rev. Lett. 2000, 85, 4080–4083. (12) Xu, L.; Salmeron, M.; Bardon, S. Phys. Rev. Lett. 2000, 84, 1519–1522. (13) Daillant, J.; Zalczer, G.; Benattar, J. Phys. Rev. A 1992, 46, R6158–R6161. (14) Bardon, S.; et al. Phys. Rev. E 1999, 59, 6808–6818. (15) Benichou, O.; et al. Adv. Colloid Interface Sci. 2003, 100-102, 381–398. (16) Vandenbrouck, F. Thesis Universite-de-Paris VI, Paris France, Paris, 2001.
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a finite contact angle with the solid surface, the molecular structure of the edge is made up of a sequence of flat terraces with a decreasing thickness in the outward direction, lying on a surface precursor film.12 It is composed itself of a well-defined bilayer lying on top of a more complex monolayer. In the smectic range (up to 33.5 °C), the thickness of the former was consistently reported to be 3.3 nm, which is close to the bulk period 3.2 nm.15 The latter is described as being in a dense state in the region where it is covered with at least one additional bilayer and as having a density that smoothly decreases to zero outward into the region where it is exposed to the vapor phase (see Figure 1). It was interpreted as a 2D gas phase diffusing along the surface. X-ray reflectivity showed that whatever its density, its physical thickness has a constant value of 1.2 nm, slightly smaller than the bulk halfperiod.13 Complementarily, ellipsometry, which is sensitive to the optical thickness of the monolayer, demonstrated that the surface density is vanishing in the gas phase.14 When the drop volume is very small, the entire drop is terraced and complete wetting is observed.15 At some stage of the spreading kinetics, the structure reduces to a single bilayer patch lying on the surface monolayer. Here starts what we may call the late stage spreading, where the last bilayer shrinks and vanishes while the surface monolayer spreads all over the solid. As far as spreading kinetics is concerned, the literature remains very poor. Especially for late stage kinetics, only few experimental results were reported, which were not consistent with available theories.13,15-17 One reason for this is the difficulty to image molecular layers with a sufficient sensitivity and resolution, in real time, and over large fields. However, the quantitative study of the late stage kinetics is particularly relevant in order to understand the spreading mechanisms at a molecular level. Here we report for the first time such quantitative spreading kinetics obtained from real time and high-resolution optical microscopy measurements. To our knowledge, the only theory dealing with the spreading kinetics of terraced liquids is the one proposed by de Gennes and Cazabat in ref 1. We show that this theory cannot reproduce our results, which is not surprising since it was developed in order to describe the behavior of weakly stratified liquids. Then we present a new phenomenological model in perfect agreement with our experimental data. This model brings new insights into the field. (17) Betelu, S.; Law, B. M.; Huang, C. C. Phys. Rev. E 1999, 59, 6699–6707. (18) Ausserre, D.; Abou-Khachfe, R. Langmuir 2007, 23, 8015–8020. (19) Ausserre, D.; Valignat, M. P. Nano Lett. 2006, 6, 1384–1388.
Published on Web 03/19/2010
DOI: 10.1021/la904704u
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Figure 2. Bilayer evolution as observed: (a) upper sequence, at T = 22 °C; (b) lower sequence, at T = 26 °C. Time is progressing from left to right in each line.
Figure 1. (a) Optical microscopy image of a 8CB nanodrop in the late stage. (b) Scheme of the internal structure of the drop: a bilayer on the top of a monolayer; the monolayer is receding outward; arrows represent the various material flows. (c) 3D view of the optical thickness profile of the drop.
All the experiments reported below were conducted using the recent surface-enhanced ellipsometric contrast (SEEC) technique,18,19 which allows direct and real time visualization of molecular layers on surfaces, with a much higher resolution than ellipsometry or X-ray reflectivity. This technique commonly provides height resolution better than 0.1 nm, so that direct and real time visualization of molecular layers of 8CB becomes possible. Wetting experiments were conducted using 4-n-octyl40 -cyanobiphenyl (8CB, 99.5%, Merck) on oxidized silicon surfaces, for different temperatures ranging from 21.5 to 33.5 °C. In this temperature range, 8CB is in a smectic phase. Optical observations were achieved through a polarization microscope (Leica DMRX PlanApo Pol Fluotar 20 0.5 objective lens). The solid support, a thin silicon wafer covered with a 106 nm thick oxide layer, was cleaned with UV/ozone for 30 s and then was set on a heating stage with (0.5 °C of accuracy. In order to install stable and reproducible experimental conditions, the sample was enclosed in a glovebag in which the relative humidity was kept constant at 30 ( 5% during the experiment. Pure 8CB was deposited onto the support using a homemade nanospotter inserted into the glovebag. Microscope images were recorded through a tri CCD camera (Optronics DEI 750). Image processing was limited to shadow corrections. The typical size of the bilayer at the beginning of the late stage is some hundreds of micrometers, and the typical duration of an experiment is 1 h. In Figure 1a, we reproduce the image of the drop as it is seen directly from the microscope during the late stage. The bright patch is the bilayer, with radius R2; the diffuse corona (radius r > R2) is the monolayer in the gas phase. Figure 1b presents the schematic view of the internal drop structure that was described above. The defect line topology appearing at the bilayer edge in Figure 1b was inferred as a possible one. Another structure is possible in which the dislocation line is shifted half a period downward. The important point to notice is that the phenomenological model proposed in the present paper is not conditioned by the exact structure of the edge defect. The different arrows in the figure represent the various material flows induced by spreading. Horizontal arrows demonstrate 2-dimensional intralayer flows. Because of the radial drop shape, they have a radial symmetry. The vertical arrows demonstrate the permeation flux, a molecule per molecule material transfer from the bilayer toward the 6016 DOI: 10.1021/la904704u
monolayer. The curved shape arrow figures how the molecules of the upper part of the bilayer, with zero radial velocity, pass into the lower part of the bilayer during bilayer shrinking. Figure 1c (the “Monument Valley Butte”) shows a topographic view of the drop right after the late stage structure was attained. The intensity values of Figure 1a were converted into height values Z by inferring that image intensity is a regular function of the optical thickness of the liquid crystal.18 This function was approximated by a second-order polynomial that was determined using the values known from previous experiments,14,15 i.e., 4.5 nm for the trilayer thickness, 1.2 nm for the monolayer thickness at the junction with the bilayer edge, and zero thickness on the bare substrate. Since the drop shape during spreading remains circular, it was easy to calculate the average radial thickness profile Z(r) in the drop from the topographic data. From there, we were able to probe the time evolution of the total volume of the drop, including the gas monolayer. Within a 20% error margin, we found this volume to remain constant for the entire experiment. It confirms the results already reported by other groups13,15 that in ambient conditions evaporation is negligible over some days. Once this point is clarified, it becomes possible to elucidate the kinetic law that governs material flow from the last bilayer into the surface monolayer, until the bilayer vanishes completely. Hereafter, we report our experimental observations and discuss them in the frame of classical and new theoretical models. Figure 2 presents two series of top view images of 8CB bilayers that were arbitrarily extracted from kinetic experiments conducted at two different temperatures: right above the solid/ smectic transition temperature (22 °C) and further away (26 °C). Both sequences start when spreading of the initial stratified droplet becomes reduced to the trilayer structure. The monolayer corona, which corresponds to a smooth intensity profile extending far beyond the image frame, is not visible. Shrinking of the bilayer occurs due to material permeation1,20 from the bilayer into the monolayer. Two different behaviors were observed. In the upper sequence (a), the bilayer stays dense while shrinking. At the end of the process, the last molecules to disappear are located at the center of the initial disk. The domain evolution is completely deterministic. In the lower sequence (b), nucleation and growth of holes is observed in the bilayer, in addition to shrinking.13 Moreover, holes are not uniformly distributed over the layer but preferentially located in the outer part of the disk. Ostwald ripening and holes coalescence lead to the formation of one or several complex bilayer domains before completely vanishing. These patterns are conditioned by the random location of the holes that nucleate and, therefore, are intrinsically unpredictable. In contrast, the deterministic evolution observed at low temperature allows direct comparison with (20) Helfrich, W. Phys. Rev. Lett. 1969, 23, 372–374.
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theory. Then, we chose to focus on the low temperature regime as a priority. The late stage spreading kinetics of stratified liquids was predicted by de Gennes and Cazabat1 (dGC). Their model was originally intended to describe the spreading behavior of weakly layered liquids, for which layering is due to presmectic effects induced by the solid surface plane. In this model, the driving force of the spreading is the attractive interaction profile between the liquid and the solid. This interaction includes a long-range contribution which is a decreasing function of the distance from the solid and a short-range contribution affecting only the first molecular layer. On the other hand, the limiting forces are twofold: The first one is resistance to molecular permeation across the interface between adjacent layers, a thermally activated process. The second one is friction between adjacent layers, which are slipping against each other (see Figure 1c). Assuming quasiisotropic molecular diffusion in the liquid, dissipation in monolayer slipping becomes so high that the permeation flow is confined to a thin permeation ribbon near the bilayer edge. The width of this ribbon, which remains a small portion of the interlayer interface, was referred to as the permeation length ξ. It is related to the various (resistive) interlayer friction coefficients ζij and to the (accelerating) permeation coefficient C12 between the bilayer and the first monolayer by the relationship ξ-2=(ζ01 þ 4ζ12)(c12/l ) , where l l1=2l2 is a thickness which is common to the two layers of the model. From this theory, the kinetic law obtained for a two layer system is dR2 ¼ dt
W2 -W1 R2 ζ01 R2 ln R1
ð1Þ
where Wi is the interaction potential of the i index layer with the solid, ζ01 is the friction coefficient between the solid and the liquid, and R1 and R2 are respectively the radii of the innermost and outermost layer from the solid. One might notice that dR2/dt is independent of ξ, provided that ξ , R2. However, our experimental results cannot be directly compared with eq 1 for two obvious reasons. The first one is that our system is composed of one monolayer and one bilayer, the latter being terminated at the edge by a dislocation line, characteristic of amphiphilic molecules, while in the dGC model, two identical layers were considered. The assumption that the monolayer and the entire bilayer have the same thickness is far from our experimental reality since we actually have 2l2 . l1. However, it was shown in ref 17 that taking into account the various layer thicknesses affects only prefactors in the kinetic law R2(t). The second one is that in our case the geometry of the liquid layers (see Figure 1) imposes R1 = R2, which is not compatible with eq 1. Indeed, molecular diffusion in the gas region of the monolayer (r > R2) is faster than convective motion in the liquid region (r < R2). Therefore, the gas region has no impact on the liquid region kinetics. One could also overcome this difficulty and adapt the dGC model to the present system, but then we would encounter three much more serious limitations. The first one is the basic assumption of the dGC model ξ/R2 , 1. This assumption is true for isotropic liquids, but it breaks down with highly stratified liquids such as smectic liquid crystals because in such liquids the diffusion coefficient is so anisotropic that a layer has enough time to equilibrate before being significantly affected by permeation. The second one is that the two-dimensional Laplace pressure in a layer was ignored, making the model inappropriate for nanodrops. The third one is that it was not considered that permeation Langmuir 2010, 26(8), 6015–6018
toward the surface monolayer could be easier from the defect line at the bilayer edge than from rest of the bilayer. Fully revisiting the dGC theory is beyond the scope of the present work. However, a phenomenological description that overcomes these three limitations is proposed hereafter. In this model, we consider two driving forces for spreading. One is the molecular interactions, which result in a pressure difference (W2 - W1)/Σ, Σ denoting the area per molecule. The other one is line tension, which results in the Laplace pressure difference (τ2 - τ1)/R2, τi denoting the line tension at the liquid edge of layer i. Each of them generates material permeation at two levels: at the plane interface between the bilayer and the monolayer and at the line interface between the bilayer edge and the monolayer. These two levels must be distinguished because the energy barrier that the molecules must overcome in order to jump from the bilayer to the monolayer is a priori lower at the bilayer edge than in the rest of the bilayer domain. Resulting fluxes correspond to the different combinations of driving forces and exchange channels. Hence, we get the kinetic law for bilayer shrinking: dS2 τ2 -τ1 W2 -W1 ¼ -R ½S2 þ λP2 þ ð2Þ dt R2 Σ Here S2 and P2 hold for area and perimeter of the bilayer. Dimensional coefficients R and Rλ are proportional to surface and edge permeation coefficients, respectively. Their ratio is characterized by the length λ. When writing eq 2, we implicitly assumed that the hydrodynamic contribution to the 2D pressure is negligible when compared to other terms. In other words, we considered that the evolution of the two layers is quasi-static and that it is regulated by permeation. This is the essence of the approximation ξ/R2 . 1, which is the exact opposite of the dGC model approximation. The kinetic law for R2 given by eq 2 has the following form: -T
dR2 L1 L2 ¼ þ ðL1 þ L2 Þ þ R2 dt R2
ð3Þ
It is ruled by the three parameters T=(2/R)[Σ/(W2 - W1)], L1 = [(τ2 - τ1)Σ]/(W2 - W1), and L2=2λ. The first one, T, is a natural time scale for the experiment. L1 and L2 are two characteristic lengths to be compared to R2 in order to identify the dominant driving force and/or dominant permeation mechanism. When R2 , L1, i.e., when the drop is small enough, the main driving force for spreading becomes Laplace pressure. On the other hand, when R2 , L2, permeation is dominated by the edge rather than by the surface. When R2 , L1 or R2 , L2, the third term in eq 3 is negligible during the entire kinetic experiment. If R2 , L1 and R2 , L2, the second term becomes also negligible. The relative values of L1 and L2 condition the overall shape of the kinetic law R2(t). By looking at eq 3, one readily sees that the lengths L1 and L2 play exactly symmetric roles. Integral form of eq 3 is L1 R2 L2 R2 t -tf þ ¼ ln 1 þ ln 1 þ ð4Þ L2 -L1 L1 L1 -L2 L2 T where tf refers to the end of the kinetics and is known from the experiment. However, this form is valid only if L1 6¼ L2 .When L1 =L2 L, eq 4 is replaced by R2 R2 t -tf þ ¼ -ln 1 þ L L þ R2 T
ð5Þ
It is the continuous limit of eq 4. Notice that eq 3 then becomes DOI: 10.1021/la904704u
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Figure 3. Bilayer radius R2 as a function of time t. Left side: (black þ) experimental values; (red -) values calculated using the best parameters in eq 4 or 5; (green -) values calculated using the best parameters in eq 1 and assuming a dense liquid monolayer. Right side: (red þ) radius versus time distance Δt between experimental and calculated data.
-T(dR2/dt)=(L2/R2) þ 2L þ R2, exhibiting the two dependent crossover lengths L and 2L. Figure 3 shows on the same graph the experimental evolution of the bilayer radius and the evolution expected from eq 4 after adjusting T, L1, and L2 to their best values, the two latter parameters being indiscernible although a priori different. It leads to T=6840 s, L1=(50 þ ε) μm, and L2=(50 - ε) μm, with ε very small. Using these parameter values, the agreement between the model and the experimental data is excellent. Fitting the experimental data with eq 5 gives T=6840 s and L=50 μm with similar quality. The L value is within the experimental range of R2 (0.5-100 μm), showing that none of the three terms in eq 3 can be neglected.
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As said previously, eq 1 does not apply to the present geometry. However, since it is the standard reference, we tried to extend its use by assuming, as it was done in ref 1, that the monolayer stays a dense 2-dimensional liquid everywhere on the surface, i.e., by stating that the sum of the monolayer and bilayer volumes remains constant. The results obtained in that way are very poor. They are reported using the green curve in Figure 3 for comparison. Considering the symmetry of eqs 2 and 3 with respect to L1 and L2 and considering also that the best fitting values of L1 and L2 are identical, one may wonder if the two parameters should not always merge into a single one. Were it the case, the entire kinetic problem would be ruled by a single characteristic length L. This question cannot be clarified without going deeper into the theory. At this level, one should consider that the relationship L1=L2 may be accidental. From our understanding, there are two important lessons in this work. The first is the role of 2D Laplace pressures in stratified nanodrop kinetics. We believe that considering these forces may even help to understand some static properties of these liquids which are still unclear. For instance, the transition from partial to complete wetting, depending on drop size, might be understood as an instability driven by Laplace pressures. The second is that the spreading kinetics of stratified liquids can be treated in first approximation as a quasi-static phenomenon, which renders its model representation much simpler than previously thought. For instance, a direct consequence is that residual evaporation, as it could be encountered with materials other than 8CB, would not qualitatively change the kinetic laws established in the present work. Finally, it is worth recalling that such questions can be envisaged only because the SEEC microscopy technique allows direct imaging of molecular films. Acknowledgment. This work was entirely supported by ANR under Project PNANO-07-050. We also thank Stephane Joly (Nemoptics) for his help with the materials and Marilyn Twell for her help with the English.
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