pubs.acs.org/Langmuir © 2009 American Chemical Society
Superoleophobic Behavior Induced by Nanofeatures on Oleophilic Surfaces Stella M. M. Ramos,*,† Abdenacer Benyagoub,‡ Bruno Canut,§ and Cecile Jamois§ †
Universit e de Lyon, F-69000, France; Univ. Lyon 1 Laboratoire PMCN; CNRS UMR 5586; F-69622 Villeurbanne cedex, France, ‡CIMAP (ex-CIRIL-GANIL), CEA-CNRS-ENSICAEN-Universit e de Caen, e de Lyon, F-69000, France; Univ. Lyon 1, Institut des UMR 6252 F-14070 Caen cedex 5, France, and §Universit Nanotechnologies de Lyon; CNRS UMR 5270; F-69622 Villeurbanne cedex, France Received September 24, 2009. Revised Manuscript Received November 24, 2009 The control of surface wetting properties to produce robust and strong hydrophobic and oleophobic effects on intrinsically oleophilic surfaces is at the heart of many technological applications. In this paper, we explore the conditions to observe such effects when the roughness of the substrate is of fractal nature and consists of nanofeatures obtained by the ion track etching technique. The wetting properties were investigated using eight different liquids with surface tensions γ varying from 18 to 72 mN m-1.While it is observed that all the tested oils readily wet the flat substrates, it is found that the contact angles are systematically exalted on the rough surfaces even for the liquids with very low surface tension. For liquids with γ g 25 mN m-1 an oleophobic behavior is clearly induced by the nanostructuration. For liquids with γ < 25 mN m-1, although the contact angle is enhanced on the nanorough surfaces, it conserves its oleophilic character (θ* lower than 90). Moreover, our experiments show that even in the case of hexane, liquid having the lowest surface tension, the homogeneous wetting (Wenzel state) is never reached. This high resistance to liquid impregnation is discussed within the framework of recent approaches explaining the wetting properties of superoleophobic surfaces.
1. Introduction When a droplet of a liquid is placed on a very rough surface, it takes a state that minimizes its free energy.1 Thus, the contact angle increases if an intrinsically hydrophobic surface (with contact angle θ0 > 90) is roughened. Two classical models proposed by Wenzel2 and Cassie and Baxter3 describe the state of the drop by the respective equations: cos θW ¼ r cos θ0 cos θCB ¼ -1 þ φS ð1 þ cos θ0 Þ where θ0 is the equilibrium (Young’s) contact angle on the smooth surface, θ*W and θ*CB are the apparent contact angles on the rough surface, r* is the roughness factor (i.e., the ratio between the true surface area and the apparent one), and φS is the fraction of solid in contact with the liquid. The Wenzel state (which will be denoted in the following as the W state) corresponds to a homogeneous wetting where the liquid completely penetrates into the roughness valleys, increasing thus the total area of the liquid-solid interface. The Cassie-Baxter state (denoted in the following as the CB state) corresponds to a composite wetting where the liquid phase is suspended on the top of the asperities with a gas phase trapped in between. For surfaces of high roughness, this gas phase that usually comprises air and vapor of the liquid induces both a higher contact angle and a very small contact angle hysteresis, leading to a super-repellent state. Such surfaces have attracted much attention during the past decade due *Corresponding author: e-mail
[email protected]; Ph þ33-4-72-431218; Fax þ33-4-72-431592.
(1) Krupenkin, T. N.; Taylor, J. A.; Schneider, T. M.; Yang, S. Langmuir 2004, 20, 3824–3927. (2) Wenzel, R. N. Ind. Eng. Chem. 1936, 28, 988–994. (3) Cassie, A. B. D.; Baxter, S. Trans. Faraday. Soc. 1944, 40, 546–551. (4) Zhao, B.; Moore, J. S.; Beebe, D. J. Science 2001, 291, 1023–1026. (5) Blossey, R. Nat. Mater. 2003, 2, 301–306. (6) Quere, D. Physica A 2002, 313, 32–46.
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to their usefulness in many areas4-7 as well as for understanding the fundamental mechanisms underlying the wetting behavior of a liquid.8-11 From a practical point of view, there is obviously much interest to generalize this repellency effect to surfaces that initially (without any chemical coating) would be wetted by water and/or oil liquids. In this way, recent investigations12-14 have evidenced that microtextured surfaces consisting of cavities with re-entrant curvatures were able to repel both water and oil, inducing thus the superhydrophobic and superoleophobic phenomena on intrinsically hydrophilic and oleophilic substrates. Indeed, the capillary force produced at the liquid-air interface inside the re-entrancy is able to prevent liquid (water or oil) from entering the indent; the drop is thus typically in the Cassie-Baxter state. This configuration, however, only appears as a metastable state, and the understanding of the mechanisms responsible for this metastability has elicited numerous theoretical, numerical, and experimental investigations.12-21 In general, the link between the wetting properties of a structured surface and those of an unstructured surface (i.e., in a flat topographical form) of the (7) Kim, S. J.; Bang, I. C.; Buongiorno, J.; Hu, L. W. Appl. Phys. Lett. 2006, 89, 153107. (8) Patankar, N. A. Langmuir 2004, 20, 7097–7102. (9) Patankar, N. A. Langmuir 2004, 20, 8209–8213. (10) Marmur, A. Langmuir 2003, 19, 8343–8348. (11) Biben, T.; Joly, L. Phys. Rev. Lett. 2008, 100, 186103. (12) Tuteja, A.; Choi, W.; Ma, M.; Mabry, J. M.; Mazzella, S. A.; Rutledge, G. C.; McKinley, G. H.; Cohen, R. E. Science 2007, 318, 1618–1622. (13) Cao, L.; Price, T. P.; Weiss, M.; Gao, D. Langmuir 2008, 24, 1640–1643. (14) Ahuja, A.; Taylor, J. A.; Lifton, V.; Sidorenko, A. A.; Salamon, T. R.; Lobaton, E. J.; Kolodner, P.; Krupenkin, T. N. Langmuir 2008, 24, 9–14. (15) Zimmermann, J.; Rabe, M.; Artus, G. R. J.; Seeger, S. Soft Matter 2008, 4, 450. (16) Marmur, A. Langmuir 2008, 24, 7573–7579. (17) Cao, L.; Hu, H. H.; Gao, D. Langmuir 2007, 23, 4310–4314. (18) Han, D.; Steckl, A. J. Langmuir 2009, 25, 9454–9462. (19) Joly, L.; Biben, T. Soft Matter 2009, 5, 2549–2557. (20) Reyssat, M.; Yeomans, J. M.; D. Quere, D. Europhys. Lett. 2008, 81, 26006. (21) Kusumaatmaja, H.; Blow, M. L.; Dupuis, A.; Yeomans, J. M. Europhys. Lett. 2008, 81, 36003.
Published on Web 12/15/2009
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Figure 1. Wetting diagram for a structured surface. The solid, the dashed, and the dotted lines correspond to the Wenzel state, the Cassie-Baxter state, and the metastable Cassie-Baxter state, respectively. Inset: schematic representation of these three states.
same substrate is plotted on a wetting diagram. As displayed in Figure 1, this diagram summarizes the different behaviors that can be adopted by a liquid drop on a rough surface. The interception of the lines separating the three states characterizes the transition point. Despite the great number of investigations and the modern micro- and nanofabrication techniques allowing to process surfaces with controlled repellent properties, the major part of surfaces repelling water are usually attracting to oil, and processing surfaces able to prevent both water and oil from penetrating the texture still constitutes a true challenge. In this paper we investigate the ability of superhydrophobic nanostructured surfaces to repel oil liquids and thus to induce a superoleophobic effect on intrinsically oleophilic substrates. Starting from SiO2 films processed at a nanometric scale in the same way as in our previous studies,22 we report in a first part the morphological characteristics of the processed substrates extracted from SEM and AFM observations. The second part is devoted to the investigation of their wetting properties with different oils. We show that for liquids with γ g 25 mN m-1 an oleophobic behavior is induced by the nanostructuration of oleophilic surfaces and that even for liquids with the lowest surface tension the homogeneous wetting is never observed. Our experimental data are interpreted within the framework of recent approaches proposed to describe the wetting behavior of superoleophobic surfaces.
2. Experimental Section 2.1. Surface Processing. Figure 2 schematically shows the different steps followed to process the nanostructured SiO2 surfaces. We started with silicon wafers of Æ100æ orientation coated with a 400 nm thermally grown amorphous silica (a-SiO2) film. These films were then tailored by the ion track etching technique. In this way, a random distribution of nanometric damaged zones (latent tracks) was initially created by irradiating the samples at normal incidence with 208Pb ions of 500 MeV energy and a fluence of 7 109 cm-2 delivered by the GANIL accelerator (Caen). Prior to the etching, the irradiated samples were cleaned in hot water with Micro90 (Fisher-Bioblock, (22) Ramos, S. M. M.; Canut, B.; A. Benyagoub, A. J. Appl. Phys. 2009, 106, 024305.
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Figure 2. Schematic process to fabricate the nanorough superhydrophobic surfaces. The process began with ion irradiation of a SiO2 film thermally grown on Æ100æ silicon wafer. The irradiated samples were chemically etched in HF aqueous solution. The etching is anisotropic in the irradiated zones. The tailored surfaces were then grafted with PFOTS molecules. A contact angle of 158 and a contact angle hysteresis of 4 were measured with deionized water, indicating that the processed surfaces are superhydrophobic. France) detergent in an ultrasonic bath for ∼30 min and then rinsed thoroughly in deionized water. The cleaned samples were then chemically etched for 17 min at room temperature in hydrofluoric acid (HF) aqueous solution of 1% volumic concentration. Immediately after the etching process, the samples were again washed in deionized water and dried in a N2 flux. Because the etching velocity is higher along the tracks than in the pristine material, holes are formed. As reported in previous works,22-25 the size of the structures resulting from anisotropic etching by HF is determined basically by the limited ratio between the etch rate (23) Sigrist, A.; Balzer, R. Helv. Phys. Acta 1977, 50, 49–64. (24) Spohr, R. In Ion Tracks and Microtechnology, Basic Principles and Applications; Bethge, K., Ed.; Vieweg: 1990. (25) Canut, B.; Blanchin, M. G.; Ramos, S.; Teodorescu, V.; Toulemonde, M. Nucl. Instrum. Methods B 2005, 245, 327–331.
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Figure 3. SEM cross section (top) and planar view (bottom) of a structured sample (a). AFM micrograph showing a 3D view (b) and surface profile (c) of a processed SiO2 film; surface roughness power spectrum of the processed sample (d). H is the Hurst exponent related to the fractal dimension by Df = 3 - H. The small arrows indicate the indents in the inner walls of the holes. VT of the tracks (i.e., the damaged material) and the etch rate VB of the pristine film which are respectively 13.5 nm/min and 4.2 nm/min for a-SiO2 films. According to studies based on transmission electron microscopy (TEM) observations, a quite good estimation of the hole depth h* is obtained by using the equation h*=(VT - VB)t.25 Thus, for an etching time of 17 min a hole depth h*=158 nm is expected. The structured surfaces were, in a last step, hydrophobized by grafting perfluorooctyltrichlorosilane (PFOTS) molecules. For this purpose, the previously cleaned substrates were exposed to PFOTS vapor at low pressure for about 10 h according to the procedure usually adopted in the self-assembled monolayer production.26 In order to provide a reference surface, the PFOTS molecules were also grafted on a bare sample. The quality of the monolayer grafting was checked by comparing the static contact angle to literature values (105-115).27,28 2.2. Characterization Procedure. The samples were characterized using atomic force microscopy (AFM), scanning electron microscopy (SEM), and contact angle measurements. AFM measurements were carried out under ambient conditions using a Nanoscope III Digital Instruments microscope operating in tapping mode. The samples were probed at three different locations, and the images, acquired on scanned zones of two different sizes (25 and 100 μm2), were flattened to eliminate the experimentally obtained bowing effects. The SEM observations were performed using an Inspect F instrument by FEI operated at 30 kV (26) Gorham, J. M.; Stover, A. K.; Fairbrother, D. H. J. Phys. Chem. C 2007, 111, 18663–18671. (27) Singh, R. A.; Yoon, E.-S.; Han, H.-G.; Kong, H. Wear 2007, 262, 130–137. (28) Bhushan, B.; Kasai, T.; Kulik, G.; Barbieri, L.; Hoffmann, P. Ultramicroscopy 2005, 105, 176–188.
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with small spot size, in order to achieve high resolution imaging without disturbances due to sample charging. The sessile drop method was used to characterize the wetting properties of the processed surfaces. For this purpose, different fluids were used: deionized water, ethylene glycol, and various alkanes (hexadecane, dodecane, decane, octane, heptane, and hexane) with surface tension γ varying from 18 to 72 mN m-1. Droplets of tested fluids of ∼0.6 μL volume were used to measure the static contact angle θ*. The advancing θA and receding θR angles were recorded during the expansion and contraction of the droplets by placing a needle in the liquid and continuously supplying and withdrawing liquid through the needle. The contact angle hysteresis defined as Δθ=θA - θR was thus determined. The measurements were performed optically with an accuracy of (2, using at least five drops put on different locations.
3. Results and Discussion Morphological Characterization of Surface Structures. Figure 3 displays SEM and AFM images obtained on the processed samples. From the cross-section view presented in Figure 3a it appears that, as expected from the literature,25 the etching process resulted in conically shaped holes. The diameter and the depth measured from SEM observations are of ∼100 and ∼148 nm, respectively. From the top of this picture illustrating the sample surface, we can observe that the etching process also created a shallow scalloped finish in the nontracked regions of the sample. A similar effect seems to occur in the inner wall of the conical structures, where few indents of a nanometric size can be identified. The SEM resolution of our images is not sufficiently high to discuss here the fine characteristics of such walls. DOI: 10.1021/la9036138
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However, previous studies dealing with the morphology of etched tracks evidenced a screw structure for the internal wall of the cones.29,30 A more quantitative analysis of the hole structures was performed from the AFM micrographs, an example of which is shown in Figure 3b,c with the corresponding surface profile. The surface morphology is characterized by a random distribution of holes having a basal diameter, DAFM, of 108 ( 8 nm and separated by an average distance √λ=120 ( 3 nm. The judicious choice of λ (given by λ = 1/ N, where N is the irradiation fluence) and the etching time results that a very thin wall, with an average thickness given by λ - DAFM, is formed between two adjacent holes. A “forest” of distorted vertical structures with an average width of 12 nm and a height of ∼50 nm is observed. However, it appears from SEM observations that the values measured by AFM are clearly underestimated. This effect can be understood by considering that AFM has limitations to accurately measure hole depths. In fact, these measurements are strongly influenced by the shape of the tip probe, which may not reach some deep objects in the 3D profiles, the maximal measurable depth being limited by the contact between the tip support and the hole side wall. In our case, the hole depth h* is expected to be ∼158 nm, and consequently the aspect ratio of these surfaces, defined as Α = h*/(λ - D), can reach a value of ∼13. As shown previously,22 this high surface aspect ratio is a key parameter to ensure a superhydrophobicity state on the nanostructured surfaces. Considering that an adequate characterization of rough morphologies is very important to better understand the wetting properties of surfaces, we completed the characterization of the processed samples by rms roughness measurements and fractal analysis of the AFM images. Qualitatively, the rms accounts for the long-wavelength roughness while the fractal dimension Df describes the short-wavelength roughness. A rms roughness of 14.8 nm was measured on a scanned zone of 100 μm2. The fractal dimension Df of these surfaces was determined from the analysis of the power spectrum density of the surface roughness which allows us to determine the Hurst exponent H related to the fractal dimension by Df=3 - H. Figure 3d shows the calculated power spectrum density C(q) in a logarithmic scale. The value of the Hurst exponent is 0.32. This value agrees well with those obtained in previous studies22,31 dealing with films of two different thicknesses (400 and 1500 nm) processed in a similar way and where Df values were found in the 2.58-2.69 interval. The effects of fractality on wetting were also discussed in these reports. Wetting Properties. First, let us precise that the behavior of a water droplet on the present substrates was studied in our previous report.22 The water drop exhibits a static contact angle and a contact angle hysteresis of θ* = 158 and Δθ = 5, respectively. These substrates are thus superhydrophobic. Here, we concentrate our study on the wetting behavior of oil drops deposited on these peculiar surfaces. The contact angles measured with different liquids are reported in Table 1. Figure 4a shows typical drops of three different alkanes (hexadecane, dodecane, and decane) deposited on both smooth and nanostructured surfaces. As one can see in the figure, the tested liquids exhibit a static contact angle well below 90 on the (29) Musket, R. G.; Yoshiyama, J. M.; Contolini, R. J.; Porter, J. D. J. Appl. Phys. 2002, 91, 5760–5764. (30) Bergamini, F.; Bianconi, M.; Cristiani, S.; Gallerani, L.; Nubile, A.; Petrini, S.; Sugliani, S. Nucl. Instrum. Methods B 2008, 266, 2475–2478. (31) Ramos, S. M. M.; Benyagoub, A.; Canut, B.; De Dieuleveult, P.; Messin, G. Eur. Phys. J. B 2008, 62, 405–410.
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Ramos et al. Table 1. Surface Tension γ, Typical Measured Values of Advancing θA and Receding θR Contact Angles for Different Liquids, and Contact Angle Hysteresis Δθa smooth surfaces γ (mN m-1)
θA (deg)
θR (deg)
Δθ (deg)
rough surfaces θA (deg)
θR (deg)
Δθ (deg)
water 72.0 113 99 14 161 155 6 ethylene 48.0 86 68 18 128 118 10 glycol hexadecane 27.5 66 51 15 120 109 11 dodecane 25.3 63 50 13 105 95 10 decane 23.8 56 42 14 94 85 9 octane 21.6 55 43 12 85 74 11 heptane 20.1 41 35 6 76 65 11 hexane 18.4 39 34 5 71 60 11 a The contact angle measurements were done optically with an accuracy of (2.
flat substrates. However, on rough surfaces the droplets exhibit high contact angles, near 120 in the case of the hexadecane. The evolution of advancing θA and receding θR contact angles as a function of the surface tension of the tested liquids is presented in Figure 4b. Although all the tested oils readily wet the flat samples, the contact angles are systematically exalted on the rough surfaces even in the case of liquids with the lowest surface tensions such as heptane and hexane. Two regimes are clearly identified in the figure. For liquids with γ g 25 mN m-1, the nanostructuration drives a wetting inversion leading intrinsically oleophilic surfaces to repel oil and thus to acquire an oleophobic behavior. For liquids with γ < 25 mN m-1, the surface keeps its oleophilic character (with θA lower than 90) despite a significant increase (of about 80%) of the contact angle values induced by the nanostructuration. It is worth noting that the surface tension value separating these two regimes results here from an experimental observation. However, as described in a previous study,22 the repellent properties of similar nanostructured surfaces were found to be strongly influenced by the geometry (diameter and depth) of the holes as well as by the surface aspect ratio. This repellency can be understood in terms of the balance between the surface tension force and the air resistance to compression inside the holes. Although in this previous work our investigations were limited to water droplets, one can reasonably expect to observe a similar effect with oil liquids. Therefore, the experimental results reported in Figure 4b evidence that, using a controlled nanodesign method based on the ion track etching technique, the processed surfaces are able to resist the impregnation phenomenon by liquids with surface tension varying in a wide range. In addition, it appears from the data of Table 1 that except for heptane and hexane the contact angle hysteresis on rough surfaces is reduced to ∼10, a value lower than the values measured on the smooth substrates (12-18 range). This effect combined with the contact angle exaltation evidence that the oil drop is sitting on a composite surface that comprises solid and trapped air and is thus in the CB state. For heptane and hexane the situation is different; the contact angle hysteresis is slightly larger on processed than on smooth surfaces. This result suggests that some Wenzel patches should have emerged at a local scale on the nanostructured samples. This effect is further explored in the form of the general wetting diagram shown in Figure 5 which reports a plot of cos θ* on the rough surfaces as a function of the cos θ0 on the corresponding smooth surfaces. Two different behaviors can be noticed in this diagram. For -0.5 < cos θ0 < þ0.5, a strong enhancement of the metastability is clearly observed. The Cassie-Baxter branch Langmuir 2010, 26(7), 5141–5146
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Figure 4. Typical photographs of alkane drops with a radius of 0.5 mm on flat (a) and nanostructured (b) surfaces. Advancing (θA) and receding (θR) contact angles of various liquids measured on both flat (green squares) and nanorough surfaces (blue circles) as a function of the liquid surface tension (c).
Figure 5. Contact angle cosines on a nanostructured surface versus contact angle cosines on a flat substrate.
reaches the lower right quadrant illustrating that a super-repellent behavior can be achieved on surfaces intrinsically oleophilic. This wetting transition can be understood in the case of surfaces with a re-entrant geometry (or with overhanging structures),12-14 where the liquid-gas interface inside the indented walls prevents liquid from entering the indent. In our surfaces, this re-entrant behavior can be attributed to the presence of the small features in the inner walls of the holes (as suggested by SEM observations), which should act as a local re-entrancy retaining thus the interface. Moreover, the fractal morphology of our surface roughness could also contribute to entrap the liquid drop in a metastable state.32 For þ0.5 < cos θ0 < 1.0, a drastic slope change is observed in the experimental curve at cos θ0 ∼ 0.5. This shows that when the surface tension of alkanes decreases (below 25 mN m-1), the liquid drop begins to depart from the metastable Cassie-Baxter state, without reaching the Wenzel state. This situation is different from that experimentally observed on previously investigated re-entrant surfaces,14,17-21 where at a well-defined value of cos θ0 (usually in the range 0.4-0.617,18) the liquids in the metastable CB state abruptly switch to a minimum-energy Wenzel state in which (32) Synytska, A.; Ionov, L.; Grundke, K.; Stamm, M. Langmuir 2009, 25, 3132–3136.
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the surface is fully wetted. In the present case, a gas layer persists between the liquid and the solid surface. The switch from a heterogeneous to a homogeneous wetting does not occur. However, due probably to the highly heterogeneous nature of the roughness, a mixed Wenzel-CB state emerges progressively. Energetic considerations14 indicate that on the processed surfaces the morphology is such that the strength of the energy barrier separating the metastable CB state from the homogeneous wetting should be high enough to retain the liquid in the metastable state. Ours results can be described within the framework of a recent free-energy approach developed by Joly and Biben19 to account for the superoleophobic behavior of a model re-entrant surface and the mechanisms leading to the loss of this property. This transition was attributed to a local condensation phenomenon of the liquid at the internal corners of the cavity. Although only a partial impregnation seems to occur in our systems, we think that a similar mechanism may be responsible for the behavior of the oil drops on our substrates. Evidently, in the case of real surfaces with a fractal roughness the situation is more intricate. For instance, having a volume of ∼0.6 μL, the liquid drop covers about 106 hole structures. We can reasonably consider that if a similar spontaneous filling of the holes with the liquid phase may occur, the probability that such a process occurs simultaneously on all covered cavities is extremely low because obviously the holes do not have exactly the same size and shape. The gradual lost of the oleophobic character observed in our samples can thus be ascribed to an average effect in the filling of the various holes by the liquid phase. In addition, the fractal roughness combined with a re-entrant geometry should increase the number of sites able to promote the retention of the small gas pockets in the inner walls of the holes. The emergence of local wetting effects and consequently the complete impregnation of the nanofeatures by the liquid phase should thus be delayed. This effect could explain the resistance of the processed surfaces to a total liquid impregnation. The nonobservation of a full wetting regime on our surfaces remains however an open question. To clarify this point, we plan to perform new experiments taking into account the influence of the surrounding pressure on the contact angle values. These experiments should contribute to a better understanding of the mechanisms leading to the loss of the superoleophobicity properties. DOI: 10.1021/la9036138
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4. Conclusion In this study we have investigated the behavior of oil droplets deposited on superhydrophobic surfaces obtained by combining a controlled structuration at a nanometric scale of SiO2 layers, via the ion track etching technique and a simple functionalization with PFOTS molecules. For this purpose, eight different fluids with surface tensions varying from 18 to 72 mN m-1 were used. We have shown that two regimes depending on the surface tension of the fluid are clearly identified. We propose to qualitatively relate them to the mechanisms of impregnation (or not) of a reentrant structure, initially filled with gas, by a liquid phase. The wetting inversion leading intrinsically oleophilic surfaces to repel oil was ascribed to a re-entrant geometry of the surface features. The air pockets should be stored in the re-entrancy arising from the nanoindents created in the inner wall of the hole structures.
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On the other hand, this result evidence that surfaces with a reentrant geometry can be constructed by the ion track etching process. A homogeneous wetting state is never attained. The resistance of our surfaces to the impregnation phenomenon even by liquids with low surface tension is a more complex phenomenon. It is probably due to a favorable conjunction of different factors giving rise to surface characteristics (e.g., fractal roughness, collective effects, etc.) that prevents the formation at a local scale of wetting zones which should be the starting point to the liquid invasion. Nevertheless, for a quantitative comparison with theoretical and numerical approaches, complementary experiments are still required. Acknowledgment. It is a pleasure to thank T. Biben and L. Joly for helpful discussions.
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