Langmuir 2007, 23, 1723-1734
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Water Wetting Transition Parameters of Perfluorinated Substrates with Periodically Distributed Flat-Top Microscale Obstacles Laura Barbieri, Estelle Wagner, and Patrik Hoffmann* Ecole Polytechnique Fe´ de´ rale de Lausanne (EPFL), AdVanced Photonics Laboratory, CH-1015 Lausanne, Switzerland ReceiVed June 22, 2006. In Final Form: NoVember 2, 2006 Superhydrophobicity is obtained on photolithographically structured silicon surfaces consisting of flat-top pillars after a perfluorosilanization treatment. Systematic static contact angle measurements were carried out on these surfaces as a function of pillar parameters that geometrically determine the surface roughness, including pillar height, diameter, top perimeter, overall filling factor, and disposition. In line with thermodynamics models, two regimes of static contact angles are observed varying each parameter independently: the “Cassie” regime, in which the water drop sits suspended on top of the pillars (referred to as composite), corresponding to experimental contact angles greater than 140-150°, and the “Wenzel” regime, in which water completely wets the asperities (referred to as wetted), corresponding to lower experimental contact angles. A transition between the Cassie and Wenzel regimes corresponds to a set of well-defined parameters. By smoothly depositing water drops on the surfaces, this transition is observed for surface parameter values far from the calculated ones for the thermodynamic transition, therefore offering evidence for the existence of metastable composite states. For all studied parameters, the position of the experimental transition correlates well with a rough estimation of the energy barrier to be overcome from a composite metastable state in order to reach the thermodynamically favored Wenzel state. This energy barrier is estimated as the surface energy variation between the Cassie state and the hypothetical composite state with complete filling of the surface asperities by water, keeping the contact angle constant.
Introduction Wetting of liquids on solid surfaces is a research topic of fundamental interest1 with a variety of technological implications. Microfluidics for biotechnology, thin film technology, lubrication, textiles, self-cleaning, and anti-snow-sticking surfaces are just few examples of applications where this field has a significant impact. In particular, great interest is devoted to so-called superhydrophobic surfaces,2-6 i.e., those that exhibit both high water contact angles (close to 180°) and low hysteresis (drops roll off very easily even at small inclination). First examples of such surfaces were found in nature, such as lotus leaves,7 and the mechanisms involved in these superhydrophobic properties are still under investigation.8 The behavior of a liquid drop on a solid surface depends mainly on two dominant solid properties: the surface energy, determined by the chemical nature of the topmost molecular layer of the considered solid, and the surface roughness. On a “flat” selfassembled monolayer of perfluorolauric acid (CF3-(CF2)11COOH) adsorbed on a platinum foil, Hare got the lowest critical surface energy (γc) ever reported, γc ) 5.6 mN/m,9,10 resulting in the highest contact angle of 120°. Many techniques were * To whom correspondence should be addressed. Phone: +41-21-693 60 18. Fax: +41-21-693 37 01. E-mail:
[email protected]. (1) de Gennes, P.-G.; Brochard-Wyart, F.; Que´re´, D. Capillarity and Wetting phenomena: Drops, Bubbles, Pearls, WaVes; Springer-Verlag: New York, 2004. (2) Blossey, R. Nat. Mater. 2003, 2, 301. (3) Callies, M.; Que´re´, D. Soft Matter 2005, 1, 55. (4) Martines, E.; Seunarine, K.; Morgan, H.; Gadegaard, N.; Wilkinson, C. D. W.; Riehle, M. O. Nano Lett. 2005, 5, 2097. (5) Nakajima, A.; Hashimoto, K.; Watanabe, T. Monatsh. Chem. 2001, 132, 31. (6) Que´re´, D. Nat. Mater. 2002, 1, 14. (7) Neinhuis, C.; Barthlott, W. Ann. Bot. 1997, 79, 667. (8) Otten, A.; Herminghaus, S. Langmuir 2004, 20, 2405. (9) Hare, E. F.; Shafrin, E. G.; Zisman, W. A. J. Phys. Chem. 1954, 58, 236. (10) Zisman, W. A. In Contact Angle, Wettability, and Adhesion; Advances in Chemistry Series No. 43; American Chemcial Society: Washington, DC, 1964; p 1.
developed to prepare low-surface-energy coatings, by alkyl- or perfluoroalkyl-silanization of SiO2 surfaces (references in reference 11). We demonstrated that the controlled gas-flow reaction of perfluorodecyltrichlorosilane on SiO2 surfaces results in a robust monolayer coating, with a critical surface energy for spreading of 13 mN/m at a rms (root-mean-square) roughness of 0.3 nm.11 The latter information is crucial because surface roughness was recognized long ago to have an even more pronounced effect on hydrophobicity than surface energy,12-15 and superhydrophobicity appears on rough substrates with different surface chemistries.16 Depending on the surface fabrication technique, surface roughness inducing superhydrophobicity might vary from irregular or random obstacles,5,14,17-23 regularly microtextured,4,16,24-31 fractal or hierarchical roughness,32,33 to nanorod or nanotube forests.34-39 (11) Barbieri, L.; Kulik, G.; Hoffmann, P.; Gaillard, C.; van der Lee, A.; Mathieu, H.-J.; Pfeffer, M. Submitted for publication. (12) Wenzel, R. N. Ind. Eng. Chem. 1936, 28, 988. (13) Bartell, F. E.; Shepard, J. W. J. Phys. Chem. 1953, 57, 211. (14) Dettre, R. H.; Johnson, R. E. In Contact Angle, Wettability, and Adhesion; Advances in Chemistry Series No. 43; American Chemcial Society: Washington, DC, 1964; p 136. (15) Herminghaus, S. Europhys. Lett. 2000, 52, 165. (16) Oner, D.; McCarthy, T. J. Langmuir 2000, 16, 7777. (17) Erbil, H. Y.; Demirel, A. L.; Avci, Y.; Mert, O. Science 2003, 299, 1377. (18) Hozumi, A.; Takai, O. Thin Solid Films 1997, 303, 222. (19) Miwa, M.; Nakajima, A.; Fujishima, A.; Hashimoto, K.; Watanabe, T. Langmuir 2000, 16, 5754. (20) Morra, M.; Occhiello, E.; Garbassi, F. Langmuir 1989, 5, 872. (21) Nakajima, A.; Abe, K.; Hashimoto, K.; Watanabe, T. Thin Solid Films 2000, 376, 140. (22) Shibuichi, S.; Onda, T.; Satoh, N.; Tsujii, K. J. Phys. Chem. 1996, 100, 19512. (23) Tadanaga, K.; Katata, N.; Minami, T. J. Am. Ceram. Soc. 1997, 80, 3213. (24) Bico, J.; Marzolin, C.; Que´re´, D. Europhys. Lett. 1999, 47, 220. (25) Yoshimitsu, Z.; Nakajima, A.; Watanabe, T.; Hashimoto, K. Langmuir 2002, 18, 5818. (26) He, B.; Patankar, N. A.; Lee, J. Langmuir 2003, 19, 4999. (27) Lafuma, A.; Que´re´, D. Nat. Mater. 2003, 2, 457. (28) Jopp, J.; Gru¨ll, H.; Yeruslami-Rozen, R. Langmuir 2004, 20, 10015. (29) Ou, J.; Perot, B.; Rothstein, P. Phys. Fluids 2004, 16, 4635.
10.1021/la0617964 CCC: $37.00 © 2007 American Chemical Society Published on Web 01/03/2007
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The contact angle θflat that a drop of a certain liquid forms in contact with an ideally flat, homogeneous, and rigid surface is expressed by Young’s equation (1805):40
cos θflat )
γSV - γSL γLV
(1)
where γSV, γSL, and γLV are the interfacial free energies per unit area of the solid-vapor, solid-liquid, and liquid-vapor interfaces, respectively. Two models describe the contact angle between a liquid drop and a rough (non-flat) surface, depending on the configuration adopted by the drop: The first model was proposed by Wenzel (1936),12 who describes a complete wetting of a liquid drop on the rough surface asperities (referred to by us as the wetted regime). The corresponding so-called “Wenzel apparent contact angle” θwet is expressed by the equation
cos θwet ) r cos θflat
(2)
where r is the Wenzel roughness factor, defined as the ratio between the actual area of the rough surface and the geometric projected area, which is always larger than 1. Consequently, the equation indicates that in the wetted regime, the surface roughness enhances the hydrophilicity of hydrophilic surfaces (i.e., θflat < 90°) and the hydrophobicity of hydrophobic surfaces (i.e., θflat > 90°). The second model was proposed by Cassie and Baxter (1944),41 who describe a liquid drop as suspended on top of open grid arrangement of pillars (referred to by us as the composite regime). The “Cassie apparent contact angle” θcomp is expressed by the following equation:
cos θcomp ) f cos θflat - (1 - f)
(3)
where f is the Cassie roughness factor, defined as the fraction of the solid-liquid interface at the drop-surface contact base, and (1- f) is the fraction of the solid-air interface at the same drop-substrate contact base. These models, which can be derived by the surface-energy minimization of a hemispherical drop sitting on a rough substrate under a certain number of assumptions42 (see next section) are used in almost all scientific studies in the field, as they roughly describe all experimental data. In order to elucidate the fundamental principles of superhydrophobicity, systematic studies on model rough surfaces have been initiated more than 60 years ago;13 since then, a wide range of surfaces with different sizes and shapes of regular obstacles, grooves, etc. have been examined. In recent years, various techniques have been used to produce such regular superhydrophobic surfaces in a well-controlled way (30) Callies, M.; Chen, Y.; Marty, F.; Pe´pin, A.; Que´re´, D. Microelectron. Eng. 2005, 78-79, 100. (31) Lee, J.; He, B.; Patankar, N. A. J. Micromech. Microeng. 2005, 15, 591. (32) Onda, T.; Shibuichi, S.; Satoh, N.; Tsujii, K. Langmuir 1996, 12, 2125. (33) Jeong, H. E.; Lee, S. H.; Kim, J. K.; Suh, K. Y. Langmuir 2006, 22, 1640. (34) Li, H.; Wang, X.; Song, Y.; Liu, Y.; Li, Q.; Jiang, L.; Zhu, D. B. Angew. Chem. 2001, 113, 1793. (35) Feng, L.; Li, S. H.; Li, Y. S.; Li, H. J.; Zhang, L. J.; Zhai, J.; Song, Y. L.; Liu, B. Q.; Jiang, L.; Zhu, D. B. AdV. Mater. 2002, 14, 1857. (36) Lau, K. K. S.; Bico, J.; Teo, K. B. K.; Chhowalla, M.; Amaratunga, G. A. J.; Milne, W. I.; McKinley, G. H.; Gleason, K. K. Nano Lett. 2003, 3, 1701. (37) Fan, J.-G.; Tang, X.-J.; Zhao, Y.-P. Nanotechnology 2004, 15, 501. (38) Tsoi, S.; Fok, E.; Sit, J. C.; Veinot, J. G. C. Langmuir 2004, 20, 10771. (39) Krupenkin, T. N.; Taylor, J. A.; Schneider, T. M.; Yang, S. Langmuir 2004, 20, 3824. (40) Young, T. Philos. Trans. R. Soc. London 1805, 95, 65. (41) Cassie, A. B. D.; Baxter, S. Trans. Faraday Soc. 1944, 40, 546. (42) Patankar, N. A. Langmuir 2003, 19, 1249.
in the micrometer scale range, including photolithography and silanization of silicon wafers,16,29 siloxane sol-gel molding and silanization,24 Si-wafer dicing and silanization,25 PDMS molding,26,28,31 molding and UV-curing of a mixture of perfluoro- and non-perfluoro-acrylates,27 photolithography with deep reactive ion etching inducing Teflon-like polymer passivation,30 and especially in the nanometer range, by e-beam lithography or UV-photolithography, and etching.39 Many recent publications address some fundamental wetting questions. How can the contact angle of a drop on a surface be predicted, and which model (Wenzel or Cassie) should be used? Already long ago, the experimental variation of some surface roughness parameters had been shown to induce a change of wetting regimes between composite and wetted modes.14 The thermodynamic models show that Cassie and Wenzel contact angles are connected respectively to two free energy minima of a drop on a rough surface,28,42,43 with usually one of the two configurations being energetically favored. A thermodynamic criterion (eq 15 in the next section), expressed as a function of surface geometrical parameters, allows one to determine which regime is thermodynamically favored.44 However, experience shows that drops on rough surfaces do not always assume the configuration with the absolute minimum energy, and in particular, metastable composite states have been frequently observed when deposing smoothly drops on the surface.26,27,42 This metastability was associated with the idea of an energy barrier that needs to be overcome to reach the wetted state.28,42,45 Experimentally, it was proven that providing energy to the drop, either by applying a force to it27 or by dropping it from some height26 or by dragging it on the rough surface,30 no longer leads to a metastable composite state but to a wetted one. It was also proven that such metastable states are not observed when the drop is formed on the surface by condensation from the vapor phase.27 The reasons and details of the energetic transitions from a metastable composite state to a wetted state are unknown, but it has been proposed that the energy barrier can be estimated as the energy required to fill the surface asperities with water, while keeping the composite contact angle45 constant (see the calculation in the next section). This assumption, which until now has been tested by Patankar45 for only a few experimental data of Yoshimitsu,25 is applied here to several experimental series, each one considering one selected geometrical roughness parameter. A completely different approach (from thermodynamics consideration) has been proposed by Extrand46 to predict advancing and receding contact angles, based on the assumption that contact angles manifest themselves as fractional contributions along the contact line of the different surfaces. He formulated two fundamental criteria that must be met in order for a surface to show superhydrophobicity:47 the first is referred to as the “contact line density criterion” (obtained by balancing the drop weight by the surface forces along the contact line, as suggested earlier48), and the second is referred to as the “asperity height criterion” (obtained by assuming a curvature to the liquid-air surface in the roughness). These criteria were tested with reasonable agreement on selected experiments from the literature,47 from He,26 Yoshimitsu,25 Oner,16 and Bico.24 However, we believe that more systematic investigations are needed to validate any model based on still to be proven assumptions and that is lacking a convincing theoretically (43) Marmur, A. Langmuir 2003, 19, 8343. (44) Bico, J.; Tordeux, C.; Que´re´, D. Europhys. Lett. 2001, 55, 214. (45) Patankar, N. A. Langmuir 2004, 20, 7097. (46) Extrand, C. W. Langmuir 2002, 18, 7991. (47) Extrand, C. W. Langmuir 2004, 20, 5013. (48) Dettre, R. H.; Johnson, R. E. Soc. Chem. Ind. Monogr. N. 1967, 25, 144.
Cassie-Wenzel Transition: Static Wetting Study
foundation, such as thermodynamic derivation. Additionally, quantitative comparison with experimental results is, in our opinion, not so convincing, as discussed in the Supporting Information. How should hysteresis be interpreted? As it is very difficult to experimentally study ideal surfaces with absolute homogeneity and smoothness at the atomic level, hysteresis remains a highly discussed topic. Theoretically, the existence of an inherent hysteresis was proposed.46 On the other hand, it was observed that hysteresis does not exceed a few degrees on surfaces prepared carefully to be smooth and homogeneous.49,50 On rough surfaces, hysteresis often reaches several tens of degrees. This has been explained by the energy barrier that needs to be overcome in order to displace the drop contact line from one surface asperity to the next.16,49,51,52 Pillar microtextured surfaces should therefore show low hysteresis6,16 because a discontinuous contact line should offer little resistance to motion. Experimentally, it was observed that hysteresis is much lower for drops in the composite regime than in the wetted one.53 Which is the ideal roughness type and scale to promote superhydrophobicity? Pillar textured surfaces can exhibit superhydrophobic properties, but the best shape, size, and disposition of the pillars are still unclear. In the literature several propositions were explored. For maximum contact angles and minimum hysteresis, micropillar surface densities between 3 and 8% were proposed;30 for robust composite states, simultaneous shrinkage of pillar size and spacing was theoretically predicted.54 Identical Cassie and Wenzel contact angles were also proposed,26 although hysteresis would depend on the wetted regime.31 Most promising are probably biomimetic hierarchical surfaces, on which nanoroughness is added to microroughness.33,55-57 Our work contributes to the understanding of the superhydrophobic mechanisms. We provide a systematic experimental study of the wetting behavior of periodic flat-top pillar structured surfaces. The transition between the wetted and composite regime as function of different roughness parameters is studied and analyzed. We discuss the few cases of such transitions reported in literature16,25,28 in the Supporting Information, but the aim of these studies was neither to explain nor to precisely determine the position of the transitions. Water static contact angle measurements on periodically structured hydrophobic surfaces, with systematic and independent variation of six geometric surface parameters (the pitch, or minimum spacing between two consecutive pillars; the surface disposition mode; the pillar diameter; the pillar height; the pillar shape; and a so-named “scaling criterion”), are presented here. The results are compared to simple thermodynamic models, detailed in the theoretical background. Dynamic contact angle measurements and hysteresis, which are known to be important for surface hydrophobicity characterization,50 were also carried out on our substrates and are reported independently58 for the (49) Drelich, J.; Miller, J. D.; Good, R. J. J. Colloid Interface Sci. 1996, 179, 37. (50) Chen, W.; Fadeev, A. Y.; Hsieh, M. C.; Oner, D.; Youngblood, J.; McCarthy, T. J. Langmuir 1999, 15, 3395. (51) Johnson, R. E.; Dettre, R. H. In Contact Angle, Wettability, and Adhesion; Advances in Chemistry Series No. 43; American Chemcial Society: Washington, DC, 1964; p 112. (52) Li, W.; Amirfazli, A. J. Colloid Interface Sci. 2005, 292, 195. (53) He, B.; Lee, J.; Patankar, N. A. Colloids Surf. A 2004, 248, 101. (54) Extrand, C. W. Langmuir 2006, 22, 1711. (55) Patankar, N. A. Langmuir 2004, 20, 8209. (56) Shirtcliffe, N.; McHale, G.; Newton, M. I.; Chabrol, G.; Perry, C. C. AdV. Mater. 2004, 16, 1929. (57) Zhu, L.; Xiu, Y.; Xu, J.; Tamirisa, P. A.; Hess, D. W.; Wong, C.-P. Langmuir 2005, 21, 11208. (58) Barbieri, L.; Wagner, E.; Hoffmann, P. Manuscript in preparation.
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sake of clarity. As a matter of fact, we show here that equilibrium contact angles are sufficiently well described by simple thermodynamics, while arguments linked to the contact line, which are useful to explain dynamic measurements, will be discussed elsewhere.
Theoretical Background The equations of Young, Wenzel, and Cassie can be derived by minimization of the free energy of the considered liquid dropsubstrate system under consideration at equilibrium with the following assumptions:28,42,43 (i) the drop is hemispherical and its contact line is circular;59 (ii) it is large compared to roughness asperity size; (iii) the volume of liquid in the asperities is negligible with respect to the total liquid volume; (iv) gravity effects and line tension contribution are neglected; (v) the liquid-air interface below the drop is flat and at a constant height; (vi) the surfaces delimiting the main substrate roughness (i.e., top, lateral, and bottom surfaces) are considered atomically flat; and (vii) all molecular and microscopic interactions, especially at the level of the three-phase contact line, are ignored. Such a derivation will be presented below, adapted specifically to the rough surface types considered in this work, and the validity of the different assumptions will be discussed. The system is composed of a volume V of water and a surface Stotal of rough substrate. The model surface consists of periodic distributions of identical flat-top obstacles. The characteristic geometrical parameters of these surfaces are the following: pitch p (minimum distance between two consecutive obstacles), disposition factor A (statistical number of obstacles per unit surface area p2), obstacle height h, obstacle top-surface area s, perimeter of the horizontal obstacle section L. The parameters p and A as defined here allow us to describe all possible statistical regular obstacle distributions, and are directly correlated to the number of obstacles per surface N, by the relation N ) A/p2. The scanning electron microscope (SEM) and optical microscope images of a typical substrate are shown in Figure 1. We study millimetric drops on microscale sized pillars, therefore we can assume that the obstacles are small compared to the drop base surface and that the water meniscus between obstacles is flat. A nanoroughness on the side of the structured obstacles necessarily exists, due to the fabrication process, but it is neglected in this work. Consequently, the Wenzel factor r and the Cassie roughness factor f can be respectively expressed as
r)1+ f)
A hL p2
A s p2
(4) (5)
The free energy of the system should include an interfacial energy (59) The assumption of the ideal drop base circularity is an approximation, and the two wetted and composite wetting regimes should be also distinguished, since in one case the contact line is continuous, and in the other is fragmented. Some authors reported experimental observations that show the limits of this approximation.39 Theoretically, some simulations on regular square heterogeneous domain surfaces60 demonstrated that when increasing the drop size with respect to the heterogeneity, the continuous drop base line evolves from a polygon to a circle. During our experiments we have never noted effects due to the directional dependence of the measured contact angle. This can be attributed to the fact that for wetted regimes the drops were effectively much larger than the typical asperity size, and for composite drops the contact angle variation, eventually present along the three-phase contact line, in the considered range of asperity size is not detectable by the adopted contact angle measurement method, looking at the drop from different positions around its vertical axis. In our study, the apparent contact angle assumed by a millimeter drop on a microstructured surface has always appeared stable along the whole drop-substrate contact interface perimeter.
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measurements confirm theoretical predictions proving that its contribution is negligible.65 Therefore, the free energy may be restricted to the sum of the interfacial energies, which is expressed as
Esurf ) γLVSLV + γSVSSV + γLSSLS
(6)
where γij are the interfacial energies between i and j, Sij are the corresponding interfaces between the phases L, V, and S, for liquid, vapor, and solid, respectively. Introducing Sext (the external drop surface), Sbase (the geometric drop base surface), Stotal (the total solid sample surface), x (the penetration depth of water in the asperities), and y (the ratio of the true interface vapor-liquid meniscus in the asperities to the horizontal section)ssee Figure 2sthe surface energy in a composite regime is
[
Esurf(composite) ) γLV[Sext + (1 - f)ySbase] + γLS fSbase +
Figure 1. Model rough substrates. (A) SEM micrograph (tilt angle 20°) of a typical microstructured Si surface showing the hexagonal arrangement of 10 µm diameter cylindrical pillars (h ) 40 µm, p ) 30 µm). The main defined roughness parameters (diameter d, height h, pitch p, and more generally, the top-perimeter L and the top-surface s) are shown in the inset. (B) Top view optical microscope images of the three selected surface geometrical arrangements of cylindrical posts (hexagonal, square, honeycomb). The obstacle disposition factors A are also reported; they correspond to the statistical number of obstacles (cylinders) per unit surface (i.e., p2).
]
[
A x 2LSbase + γSV r(Stotal - Sbase) + (1 - f)Sbase + p A LSbase(h - x) ) γLVSext + CcompSbase + StotalrγSV (7) p2
]
with
[(
Ccomp ) - γLV f 1 +
Lx cos θflat + (f - 1)y s
)
]
(8)
and that in a wetted regime is
Esurf(wetted) ) γLVSext + γLSSbaser + γSV(Stotal - Sbase)r + A γSV r(Stotal - Sbase) + (1 - f)Sbase + 2LSbase(h - x) ) p γLVSext + CwetSbase + StotalrγSV (9)
[
]
with
Cwet ) - γLVr cos θflat
(10)
and cos θflat given by Young’s equation. The general free energy can consequently be expressed as function of the measured contact angle θ by the following equation
1 (2 - 3 cos θ + cos3 θ)2/3 [2 γLV(1 - cos θ) + C(x, y) sin2 θ] + γSVStotalr (11)
Esurf(θ, x, y) ) π Figure 2. Surface energy minimization. Top images: the different interfaces (liquid-vapor, solid-vapor, liquid-solid) are highlighted, for both the composite and wetted case. Bottom image: the notations and formulae used in the hemispherical drop approximation are reported.
term, a potential energy term, and a line tension term,61 as proposed in the literature.62 The variation of the potential energy of the drop depending on the position of its mass center63 is a weak term, which will not be considered in this work, as it only induces a variation of a few degrees in the contact angle estimation, especially if the drop is assumed to be hemispherical. The influence of this last assumption is more critical.64 Line tension was a very controversial concept for a long time, but recent (60) Brandon, S.; Haimovich, N.; Yeger, E.; Marmur, A. J. Colloid Interface Sci. 2003, 263, 237. (61) Iliev, S. D.; Pesheva, N. C. Langmuir 2003, 19, 9923. (62) Swain, P. S.; Lipowsky, R. Langmuir 1998, 14, 6772. (63) Sakai, H.; Fujii, T. J. Colloid Interface Sci. 1999, 210, 152. (64) Chatterjee, J. J. Colloid Interface Sci. 2003, 259, 139.
(3Vπ )
2/3
where C(x, y) is either Ccomp or Cwet, and cos θ is in the range [-1, 1]. Changing variable with X ) cos θ and calculating the derivative of the function with respect to X gives 2 ∂Esurf 3V 2/3[γLVX + C(x, y)](X + 1) (X, x, y) ) 2π ∂X π [(X - 1)2(X + 2)]5/3
( )
(12)
The sign of this expression is negative with increasing X until X ) Xext ) -C(x, y)/γLV or X ) -1, and then it becomes positive. Therefore, the function Esurf exhibits a minimum in θ for cos θ ) Xext or θ ) π, and the lower θ, the lower the surface energy. We present below the solutions for cos θ ) Xext, but it should not be forgotten that when this solution is not in the range [-1, (65) Pompe, T.; Herminghaus, S. Phys. ReV. Lett. 2000, 85, 1930.
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Figure 3. Influence of different geometrical parameters. Calculated Cassie (black) and Wenzel (gray) contact-angle curves as function of selected surface-roughness parameters: (A) height (h) of cylindrical flat-top pillars in square arrangement; (B) diameter (d) of cylindrical posts in square arrangement; (C) perimeter (L) of the top-surface pillars in square arrangement; (D) obstacle disposition factor (A). The formula giving the parameter value of the thermodynamic transition point is reported in each graph.
1], the derivation also gives θ ) π, as limited cases in both wetted and composite regimes. In the composite case, the minimum surface energy Esurf (θ, x, y) is obtained for
(
cos θcomp min ) f 1 +
Lx cos θflat - (1 - f)y s
)
(13)
This equation is a generalized Cassie’s equation, with f from eq 5, expanded by the liquid penetration depth x and the curvature y of the penetrating liquid. Similar equations have already been derived62,66 with different interpretations of the coefficients.43,67,68 By increasing either x or y, the minimum composite contact angle θcomp min increases, which shows that the minimum energy is obtained for x ) 0, y ) 1. Partial filling of the roughness structure is thermodynamically less favorable, and does not take place, which agrees with experimental observations,39,69 but contradicts others’ assumptions.25,47 In the wetted case, the minimum surface energy Esurf(θ, x, y) is obtained for
cos θwet min ) r cos θflat
(14)
The Cassie configuration corresponds to the lowest energy state in the composite case (eq 13), while the Wenzel configuration corresponds to the lowest energy state in the wetted case (eq 14). Therefore, two energy minima are obtained for the system, one for the composite case (Cassie), without a filling of the asperities (Figure 2A), and one for the wetted case (Wenzel), corresponding to a complete filling of the asperities (Figure 2B). The absolute interfacial energy minimum corresponds to the lower minimum contact angle of both. Therefore, a thermodynamic criterion that can be employed to define which of the two regimes is more favored can be derived by equating the Cassie and Wenzel contact angle.44 Defining the contact angle on the flat surface that delimits the two regimes on a rough surface, the equation thus obtained
cos θflat )
1-f f-r
(15)
expresses the condition for which the energy for both Cassie and Wenzel state are equal, and it therefore enables us to calculate the point where the thermodynamic transition between the wetted and composite regime occurs. For instance, the Cassie state is thermodynamically favored if cos θcomp > cos θwet, that is, when cos θflat < (1 - f)/(f - r).
which is Wenzel’s equation, with r from eq 4.
(67) De Coninck, J.; Ruiz, J.; Miracle-Sole, S. Phys. ReV. E 2002, 65-03613.9, 1.
(66) Hitchcock, S. J.; Carroll, N. T.; Nicholas, M. G. J. Mater. Sci. 1981, 16, 714.
(68) Nakae, H.; Inui, R.; Hirata, Y.; Saito, H. Acta Mater. 1998, 46, 2313. (69) Kijlstra, J.; Reihs, K.; Klamt, A. Colloids Surf. A 2002, 206, 521.
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Figure 4. Thermodynamic study of the composite and wetted energy states as function of the pitch. Image A shows the calculated Cassie and Wenzel contact-angle curves as function of the pitch, for a square distribution of flat-top cylindrical pillars, showing the thermodynamic intersection th between the two regimes. B and C show energy calculations, B for a pitch of 15 µm, where Cassie is the energy minimum state, and C for a pitch of 50 µm, where Wenzel is the energy minimum state. D is a “zoom” of the “Cassie angle” region delimited by the dashed circle in C. Graphical representation of the liquid drop state on the structured surfaces are linked by a flash to the corresponding point in the graph. Energy calculations are carried out for a 3 µL drop.
Examples of the theoretical evolution of the contact angle as function of one geometrical parameter, keeping the others constant, are shown in Figure 3 and Figure 4A for both the Cassie and Wenzel model. All parameters were chosen to fit the carried out experiments, with a constant static contact angle on the flat substrate θflat ) 110°. Smaller contact angles in the Cassie state, i.e., smaller total energy states, are obtained for smaller values of the variables diameter (d), disposition factor (A), and pitch (p), until the crossing to the Wenzel curve appears (see parts B and D of Figure 3, respectively, and part A of Figure 4). The intersection of the lines, corresponding to the value where both states are energetically equal, is evidenced in each graph by th, and the related mathematical expression (from eq 15) is also reported. However, as already mentioned in the introduction, experience shows that drops gently deposited on rough surfaces do not always assume the wetted configuration, even when this represents the absolute minimum energy. This is easily understandable from
a thermodynamic perspective, as explained for one example in Figure 4. This figure presents in A the contact angle curves as function of pillar distance (pitch), keeping all other parameters constant; and in B and C (and D) the energy curves are given as a function of contact angle for two different pitch values, falling respectively into the “Cassie” thermodynamic region for B, and in the “Wenzel” thermodynamic region for C (and D), calculated using eq 13 (with y ) 1, varying x) and eq 14. The surface energies presented in Figure 4B-D of a “drop-structured substrate” system are calculated for a drop of 3 µL volume, and includes a reference term E0 related to the substrate surface under consideration (here, a disc of 3 mm radius). With the fixed geometrical roughness parameters from Figure 4A the isoenergetic pitch would be ∼26 µm, with a contact angle of 157° (solution of eq 15). Smaller pitch values, i.e., 15 µm as presented in Figure 4B, results in an energy difference of about 100 nJ between the favored Cassie composite state and the corresponding wetted state for the reference 3 µL drop.
Cassie-Wenzel Transition: Static Wetting Study
Figure 5. Pitch influence. (A) (b) Static Contact Angle (SCA) values of ∼3 µL drops softly deposited on the micrometer-structured surfaces as function of pitch, for a square disposition (A ) 1) of cylindrical pillars (d ) 10 µm, h ) 40 µm). The Cassie-Wenzel transition is indicated by the dashed arrow. (0) Advancing contact angle values measured keeping the needle that forms the drop very close to the surface. (B) Plot of the thermodynamic energy calculations. The black line (left scale) represents the energy difference between the Cassie and the Wenzel states as function of pitch, and the gray line (right scale) represents the energy barrier to fill the asperity with increasing the pitch, in the range where the Cassie regime is metastable (eq 16). The symbol 9 represents the energy barrier at the Cassie-Wenzel equal energy point. The lowest energy barrier, marked with “X”, falls in the experimental range where the transition occurs. (C) Images of different volume drops (V ) 3.5, V ) 5.5, and V ) 10.3 µL) showing the stability of the composite metastable regime on the micrometer-scructured sample with p ) 100 µm.
When Wenzel is the thermodynamically favored state, for example at 50 µm pitch (see Figure 4C), the energy curves show that, from the initial spherical drop in air to the smoothly deposited drop on the substrate (right part of Figure 4C), the system can easily enter into the metastable composite state shown in the zoom in Figure 4D, a state associated with a local energy minimum. Moving from this state to the thermodynamic favored Wenzel state requires that an energy barrier to overcome, due to the filling of the asperities with water (see Figure 4C,D). A transition energy barrier of about 1.5 nJ between the Cassie and
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Figure 6. Geometrical arrangement influence. (A) Experimental SCAs measured on samples with three different geometrical arrangements and disposition factors (hexagonal, A ) 1.547; square, A ) 1; honeycomb, A ) 0.770) of cylindrical posts (d ) 10 µm, h ) 40 µm) as function of pitch for a ∼3 µL water volume drop are presented with the corresponding theoretical Cassie and Wenzel curves (Hex: black line; Sq: gray line; Hc: dashed line). Arrows indicate the Cassie-Wenzel transition for the different dispositions. (B) The energy barrier diagram between the Cassie and Wenzel state from eq 16 presents the correct order predicted by thermodynamics calculations, and the minima of the energy barrier (full dots) are in the experimental transition region. In part C all the data for the different dispositions are presented in function of the pillars surface density (N ) A/p2). The dashed rectangle in part C shows the very narrow gap of N where all transitions fall (comprised between 0.77 and 1 pillars per (100 × 100) µm2).
Wenzel state is obtained for the reference 3 µL drop by calculating the energy required to fully fill the asperities, assuming that the contact angle (here the Cassie contact angle corresponding to the lowest metastable energy state) remains constant in this transformation45 (see Figure 4D).
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Figure 7. Influence of pillar diameter. (A) SCAs of a ∼3 µL drop on square dispositions of cylindrical pillars as function of pillar diameter (constant p ) 50 µm and h ) 20 µm). The Wenzel regime corresponds to small diameters and the Cassie regime to pillar diameters above 4 µm. The experimental Cassie-Wenzel transition is marked by the dashed double arrow. (B) Energy diagram: energy difference between composite and wetted state (black line, left scale), and energy barrier to fill the asperities in the range where the Cassie regime is metastable (gray line, right scale, from eq 16). The symbol 9 represents the energy barrier at the Cassie-Wenzel equal energy point. Again the experimental transition (delimited by the two dashed vertical gray lines) occurs close to the calculated energy barrier minimum (X). comp Using eq 11 to evaluate the surface energy Esurf , the barrier energy is
comp ∆Ebarrier ) Ecomp surf (θcomp, x ) h) - Esurf (θcomp, x ) 0) (16)
As the variations of surfaces we are dealing with are extremely small, it is convenient to correct for the liquid volume in the asperities for this energy calculation. The corrected drop radius R(θ, x) is found solving the third polynomial equation:
π (2 - 3 cos θ + cos3 θ) R(θ, x)3 + 3 (1 - f)π sin2 θ x - V ) 0 (17) and the corrected volume is introduced in eq 11. Experimental Section Periodically Structured Substrate Preparation. Silicon wafers (4 in. diameter, P/N doped, oriented, 525 µm thick) were structured by standard photolithography techniques. A direct laser writing apparatus (Heidelberg DWL200) transferred the computer predefined design on the photoresist (Shipley Microposit S1800 series) coated wafer with about 1 µm precision. After irradiation and
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Figure 8. Cylindrical obstacle height influence. (A) Experimental water SCAs of ∼3 µL drops, and the theoretical Cassie and Wenzel curves, for samples with square disposition of cylindrical pillars as function of pillar height (constant p ) 30 µm and d ) 10 µm). The arrow indicates the Cassie-Wenzel transition. (B) Energy barrier to pass from the Cassie metastable composite state to the Wenzel wetted state is reported (gray line, right scale, from eq 16), proving that the experimental transition occurs in a region where this energy is very low. The symbol 9 represents the energy barrier at the Cassie and Wenzel equal energy point. (C) A side view schematic presentation of the contact between water and the structured surfaces at scale, for the two smallest considered heights (h ) 1 µm and h ) 1.8 µm), showing the wetted and the composite regime observed respectively for h ) 1 µm and h ) 1.8 µm. For the square arrangement, the largest distance b between two consecutive obstacles, to be correlated with the meniscus curvature, is given by b ) x2p - d. development, the wafers were etched using deep reactive ion etching DRIE (Alcatel 601E) with fluorine based reagents, for different times (10 s to 6 min) depending on the desired height of the structures. Resist stripping (Microposit Remover 1165), and oxygen plasma (Branson IPC 2000) exposure prepared the substrate for the chemical coating process. Chemical Surface Modification. The micrometer-structured silicon substrates were silanized with perfluorodecyltrichlorosilane in a sophisticated gas flow system resulting in reproducible homogeneous highly hydrophobic surfaces.11 The obtained average pure water contact angle at 23 ( 1 °C for all the silanized Si substrates (on flat regions) are 110 ( 3° for the static contact angle, 116 ( 3° for the advancing, and 104 ( 3° for the receding contact angle.
Cassie-Wenzel Transition: Static Wetting Study
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Figure 9. Obstacle section perimeter influence. (A) Schematic top-view of the six selected shapes of the pillar surface, with constant area and different perimeters L. In the two rows, from left to right, the shape is circular, square, triangular, four-edged star, six-edged star, and eight-edged star. (B) As an example, a SEM image (tilt angle 15°) of a six-edged star pillar is reported. (C) Water SCAs on substrates with square distribution of pillars with same surface area and height, and different section shapes, as function of the section perimeter. Vdrop ∼ 3µL, s ) 78.5 × 10-12 m2 and h ) 40 µm. Results are presented for three different pitches p ) 100, 110, and 120 µm, which correspond to the pitch values around the pitch for which the Cassie-Wenzel transition occurs when the square distribution of cylindrical pillars with identical s and h (see Figure 5) is studied in function of the pitch. (D) Thermodynamic analysis for p ) 110 µm, showing the evolution of the energy barrier to pass from the Cassie metastable state to the thermodynamically favored wetted Wenzel state. In agreement with experimental data, it is observed that the energy barrier increases with increasing obstacle perimeter. (E) Calculation of the energy barriers for the three different experimental pitches in function of the obstacle perimeter, showing that the energy barrier as function of perimeter is lower for lower pitch value. Contact Angle Measurements. Sessile drop wetting measurements were performed with a Digidrop, GBX apparatus, in a controlled atmosphere of water vapor almost saturated N2 (relative humidity ∼90%), at controlled room temperature (23 ( 1 °C) with puriss. p.a. water (Fluka, Switzerland) used as probe liquid. For static contact angle measurements, water drops are formed at the tip of homemade perfluorosilanized glass capillaries. They detach from the tip, and fall onto the substrate, when they reach precise volumes between 2.5 and 6 µL, depending on the tip size. Different capillaries (i.e., different drop sizes) were used especially on samples with structures around the metastable to stable Cassie-Wenzel transition. The distance of the specific capillary from the substrate was kept constant and at a value as to minimize the kinetic energy of the falling drop. Digitized images of the drop-substrate system were recorded when the drop external surface stopped to vibrate (i.e., ∼3-5 s after the drop-substrate contact occurred, depending on the drop size). Each reported static contact angle value is obtained averaging at least six measurements, corresponding to the left and right angles of three drops deposited at different spots on the same substrate. Static contact angle estimations were performed by a computer program for drop shape analysis method based on the Laplace equation.70-72 All the reported static contact angle measurements are within (3° of the averages. (70) Li, D.; Cheng, P.; Neumann, A. W. AdV. Colloid Interface Sci. 1992, 39, 347. (71) Rotenberg, Y.; Boruvka, L.; Neumann, A. W. J. Colloid Interface Sci. 1983, 93, 169. (72) Hoorfar, M.; Neumann, A. W. J. Adhes. 2004, 80, 727.
Results and Discussion In the rest of the text, SCA stands for static contact angle. Influence of Number of Pillars (N) per Surface Area. Figure 5 presents the results of water SCA measurements performed as function of the pitch (p ) 15-150 µm), for cylindrical pillars (d ) 10 µm and h ) 40 µm) in a square arrangement. The SCA values (full dots) reported in Figure 5A are in good agreement with the Cassie and the Wenzel models. With increasing pitch, a sharp transition in SCA values (full dots) from the Cassie to Wenzel regime appears when passing from p ) 100 µm to p ) 110 µm, when the drops are deposited softly on the surface. The sharpness of the transition clearly shows that no states with partial filling of the asperities with water are observed, but only the extreme cases of the Cassie (x ) 0) or Wenzel regime (x ) h) are observed. When pushing the drop into the structures with the needle close to the substrate, the dynamic contact angles (in Figure 5A open squares represent the advancing contact angle) show the transition for p ∼ 30 µm, which corresponds to the thermodynamic transition, given theoretically by the intersection of Cassie and Wenzel curves.73 Therefore, the composite states observed after soft deposition of the drop on the substrate for 30 µm < p < 100 µm are metastable composite states, corresponding to the theoretical situation depicted in Figure 4 (C and D). To predict the limit of stability of these composite states, we calculated the energy barrier from
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Figure 11. SCAs as function of Cassie’s roughness parameter f. Summary of all results presented in the previous graphs, presented as function of f, i.e., surface fraction of solid-liquid contact at the drop-solid interface. The Cassie theoretical curve is also reported for the series with squared distribution of 10 µm diameter pillars. All experimental transitions take place for f around 0.01, a much smaller f value, in a much smaller f range, than the theoretical thermodynamic ones, delimited by the two vertical dashed lines in the graph.
Figure 10. Scaling experiments. (A) Scheme illustrating the criterion adopted to prepare the samples of the “scaling experiments”. (B) SCA measurements of ∼3 µL drop as function of p/d for five different pillar diameters (3 µm < d < 20 µm), keeping constant h ) 2d and varying p. Arrows evidence the Cassie-Wenzel transition position for the different samples series. The larger the diameter, the lower the p/d value at which the Cassie-Wenzel transition takes place. (C) Energy barrier for the five series of measurements as function of p/d parameter, showing that the energy barrier minimum falls at smaller p/d when d is higher, and vice versa, which is in agreement with experimental data. The minima of the energy barriers are marked by X.
the Cassie state to the Wenzel state by complete filling of the asperities,45 according to eq 16. This thermodynamic analysis is reported in Figure 5B. While the difference between the free energy of the Cassie and Wenzel states (black curve referring to the left axis, and calculated by eqs 11, 8, and 10) increases with increasing pitch, the estimation of the energy barrier that a drop in the metastable composite regime has to overcome in order to reach the thermodynamically favored wetted state (gray curve referred to the right axis, and calculated by eq 16) decreases with increasing p, from 1.26 × 10-5 mJ at the equal energy state, to even negative values at the experimental transition, and presents a minimum marked with X. The precision of the calculated values should not be overinterpreted, in view of the large number of assumptions used, especially the neglecting of gravity, both in terms of drop distortion from a hemispherical cap and of potential energy. But, qualitatively, the graph shows that the stability of the metastable regime decreases with increasing pitch, and in particular, the experimental transition belongs to a pitch range (delimited by the two vertical dashed lines), in agreement with the minimum value position of the energy barrier curve. In the literature, it has been proposed that the transition occurs when the gravitational energy compensates the energy barrier: 45,74 we would rather think that the gravitational energy should be taken into account in the calculation of free energy and contact angle, and that the transition occurs when the drop vibration energy balances the energy barrier. For comparison, an energy barrier of 1 × 10-5 mJ corresponds to the kinetic energy of a 3 µL water drop falling freely from 340 µm, while an energy barrier of 1 × 10-7 mJ is balanced by the kinetic energy of the same drop falling from 3.4 µm. This order of magnitude and the reduced robustness of the metastable states with increasing pitch are aspects that we are currently investigating. A water drop size effect on the transition from the metastable composite state to the wetted state was not observed for drop (73) At p ) 30 µm we observe a drastic change in the hysteresis values (difference between advancing and receding contact angle) measured with the needle close to the substrate. It passes from values in the order of 20° for p ) 20 µm to a value as large as 120° for p ) 30 µm. The corresponding very low receding angle confirms the complete liquid penetration among the asperities. This proves the wetted regime for the drops formed on the samples with p g 30 µm and the composite regime for smaller pitches. (74) Dupuis, A.; Yeomans, J. M. Langmuir 2005, 21, 2624.
Cassie-Wenzel Transition: Static Wetting Study
volumes between 2.5 µL and 10 µL (see Figure 5C), contradicting published assumptions,25 but supporting recent experimental results.27 The flattening of big drops due to gravity changes the ratio Sbase/Sext with respect to the hemispherical drop, but maintains the drop in the composite regime, with high contact angle. Figure 6A shows water SCA measurements as function of the pitch for the three defined geometrical arrangements: honeycomb “Hc” (A ) 0.770), square “Sq” (A ) 1), and hexagonal “Hex” (A ) 1.547). The transitions from the Cassie to Wenzel regimes are observed at slightly different pitches: first, for the honeycomb arrangement between 90 and 100 µm, then for the square arrangement between 100 and 110 µm, and finally for the hexagonal arrangement between 110 and 120 µm. The corresponding barrier energy between the metastable composite state and the thermodynamically favored wetted state (eq 16) as function of the pitch is reported in Figure 6B, for the three disposition factors A under consideration, together with the observed experimental gap for the transition. The experimental transitions are observed in the pitch order predicted by thermodynamics calculations, being the minima of the three energy barrier curves all in the region of the experimental transition range. Presentation of these results as function of the asperity surface density N (i.e. N ) A/p2) reveals that the different Cassie-Wenzel transitions in Figure 6A fall into the same asperity surface density range. This similarity in the N values at the experimental Cassie-Wenzel transition confirms that the drops are not sensitive to the distribution of micrometric obstacles (provided these obstacles are homogeneously and regularly distributed on the surface), and supports the validity of the axis-symmetric drop-shape assumption. Even though the drop may sit only on a limited number of pillars (calculations give between 5 and 6 pillars of 10 µm diameter with pitch of 100 µm under a 3 µL drop with a contact angle of 170°), no effect was observed on the static wetting regime (i.e., if composite or wetted) that potentially could be attributed to the different three-phase contact line distortions caused by different obstacle surface arrangements. To summarize, both pitch and disposition factor influence the Cassie-Wenzel transition, in agreement with thermodynamic models. The different geometrical arrangements considered in our study are sufficiently regular and symmetric to avoid observable differences in the drop wetting behavior. Consequently, the Cassie-Wenzel transition can be analyzed in terms of asperity surface density N. In the case considered here (cylindrical pillars 10 µm diameter, 40 µm height) the transition is observed for 0.77 < N < 1 pillars per (100 × 100) µm2. Influence of d, Cylindrical Pillar Diameter. The influence of pillar diameter in case of cylindrical obstacles is presented in Figure 7. SCA measurements on a series of samples with square arrangement (A ) 1) of cylindrical pillars with constant height and pitch (h ) 20 µm, p ) 50 µm) but with different pillar diameter (3 µm < d < 20 µm) are presented in Figure 7 (A), together with the corresponding energy barrier graph (B). The measured SCAs are in good agreement with the Cassie or Wenzel models. The transition composite-wetted regime appears between 4 and 5 µm, in a region close to the calculated energy barrier minimum. Since the thermodynamic equal energy diameter (i.e., the intersection between the two curves) corresponds to d ) 40.3 µm, composite metastable states are again observed, and estimation of the energy barrier to reach the Wenzel thermodynamically favored state (Figure 7B) shows that the transition is observed in the region of d where the energy barrier is again very low.
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Influence of h, Cylindrical Pillar Height. In Figure 8, water SCA values are reported as function of the pillar height, for a series of samples with square distribution of 10 µm diameter cylindrical pillars with constant pitch p ) 30 µm, and height h comprised between 1 and 40 µm. For this whole height range, the Wenzel wetted state is calculated to be the thermodynamic favored regime; however, almost all measurements show the SCA corresponding to the metastable composite states. In particular, being the composite case independent of the pillar height, the Cassie curve results a straight line at about 160° (see Figure 8A). Thermodynamic estimation of the energy barrier to overcome the metastable composite states, and to reach the thermodynamically favored wetted states, shows that this energy decreases to zero with decreasing pillar height (see Figure 8B). The transition to the Wenzel regime is observed for the minimum considered asperity height (h ) 1 µm). However, this observed transition should be related to the unavoidable vibrations of the liquid-vapor menisci among the asperities at the macroscopic liquid-solid interface associated to drop deposition. Due to such vibrations, the liquid can possibly touch the bottom of the asperities, thus generating the thermodynamically stable Wenzel state. To illustrate this, Figure 8C shows a real scale image of the situation of the water in, and on top of, the asperities. In such a figure the distance between two consecutive obstacles, to be correlated with the maximum meniscus curvature, is the largest distance b for a regular square arrangement of obstacles, and is given by the expression b ) x2p - d. The composite state observed for a pillar height of 1.8 µm (right image, Figure 8C) shows that the meniscus in the cavities presents a radius of curvature larger than that calculated assuming a contact angle at the asperity edges equal to the advancing angle on the flat silanized surface (116°), as proposed by several authors.25,45,47 As a matter of fact, a minimum pillar height of 3.7 µm would be required in this case in order to keep the drop in the composite state (assuming a spherical meniscus shape). However, in agreement with Yoshimitsu’s experimental data,25 during dynamic contact angle measurements on small height structures, we observed that heavier drops reach the Wenzel state more easily, which we attribute to larger amplitude of the menisci vibration due to drop inertia.58 Influence of L, Pillar Top-Surface Perimeter. To investigate the influence of pillar top-surface perimeter, samples were prepared with a square distribution of obstacles, with constant height (h ) 40 µm), and constant pillar top-surface area (s ) 78.5 µm2, which is the area of a 10 µm diameter cylindrical pillar) but with varying obstacle-section shape, which results in different top-surface perimeters L (see Figure 9A). Maintaining a constant pillar surface while varying the pillar perimeter is necessary in order to rigorously determine whether the contact-line length and corrugation play a role in the robustness of metastable composite states, a control that was unfortunately not carried out in previous studies.16 Figure 9B shows the scanning electron microscope tilted (15°) image of the six-edged star structure as an example. Three selected pitches, p ) 100, 110, and 120 µm, are chosen in the Cassie-Wenzel transition range observed for square distribution of cylindrical pillars (see Figure 5A), and the resulting SCAs are reported in Figure 9C as function of the perimeter length L. For p ) 100 µm, all contact angle values correspond to the metastable composite regime, while for p ) 120 µm, all drops are observed in the wetted thermodynamically favored state. Only for the intermediate p ) 110 µm, a transition between the metastable Cassie regime into the Wenzel regime is observed between the triangular shape to the four-edged star shape. This
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transition range, delimited by two gray dashed lines in the energy analysis of Figure 9D, corresponds again to a low value of the estimated energy barrier with increasing perimeter, to pass from the metastable Cassie composite state to the thermodynamically favored Wenzel wetted state. The energy barrier calculations for the three considered pitches as function of the perimeter (Figure 9E) show that the Cassie-Wenzel transition should occur at a smaller perimeter for a smaller pitch and at a higher perimeter for a larger pitch. However we want to point out that the experimental Cassie-Wenzel transition occurs at the change of the pillar top-surface shape from a convex to a concave structure. Although our thermodynamic interpretation seems in agreement with the experimental results without considering this aspect, we cannot exclude that the variation of the contact line on the asperities from a convex to a concave structure could have a non-negligible effect on the Cassie-Wenzel transition position, independently from its length. Further investigations in this direction would be very useful. We can nevertheless conclude that our experimental results support the idea that increasing the complexity of the pillar shape enhances the robustness of the composite metastable state. Influence of Pillar Absolute Size. The influence of obstacle absolute size, keeping f and r parameters constant, was also studied. Five series of samples were prepared varying simultaneously diameter and height of cylindrical pillars in square disposition, using the relations: d ) i and height h ) 2i, for i ) 3, 5, 10, 14, and 20 µm. Each sample series, one for every i value, consisting of eight different pitches, was selected to have the same series of r and f parameters, using the following relations: f ) (Aπ/4)(d/p)2, and r ) 1 + 2Aπ(d/p)2. Figure 10A reports a scheme that explains the criterion adopted to prepare these samples, referred to as “scaling criterion”. Keeping the f and r roughness factors constant means that the Cassie and Wenzel theoretical models predict exactly the same series of contact angle values for all five series as function of p/d, and therefore the identical energy value for the Cassie and Wenzel intersection point, given by the expression
(dp)
) th
x
Aπ 1 - 7 cos θflat 4 1 + cos θflat
from eq 15. Figure 10B presents the experimental SCAs, as well as Cassie and Wenzel theoretical curves. The Cassie and Wenzel curves fit correctly the data, and again, transitions from composite to wetted regimes are observed at pitch/diameter values much larger than the equal energy state at the intersection of the Cassie and Wenzel curves. Concerning the Cassie-Wenzel transition, we observe that the smaller the pillar diameter, the higher the p/d value at which it occurs, and vice versa. Calculation of the energy barrier according to eq 16 shows a good correlation with these data, as all the energy barrier minima roughly correspond to the experimental transition regions (see Figure 10C). This nicely demonstrates that for identical r and f values, reducing the pillar size in the micrometric range increases the extent and the robustness of the metastable composite states. Influence of f, Cassie Roughness Parameter. Summarizing all the experimental SCAs of this work in Figure 11 as function of f (i.e., surface fraction of the liquid-solid contact at the dropsubstrate interface) reveals that the transition between the metastable Cassie regime and the Wenzel regime always occurs for low and similar values of f, all comprised between 0.0035 (75) Brakke, K. A. Experimental Mathematics 1992, 1, 141.
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and 0.0140 (apart from the transition as function of h, which should mostly depend on another phenomenon, i.e., meniscus curvature due to drop vibration, as discussed earlier). This means that metastable composite drops can sit on 99.6% of air, or, otherwise said, that only 0.4% of the macroscopic drop-solid interface holds the drop in a superhydrophobic state! The calculated thermodynamic transition between Cassie and Wenzel states for the surface geometries realized in this work lies in the range of f between 0.0873 and 0.1939 (vertical dashed line in Figure 11), corresponding to a solid-liquid interface fraction at the drop-substrate contact surface between 8.7 and 19.4%, consequently far away from the observed transitions, and comprised in a much larger f gap. Dynamic contact angle measurements of these metastable states will be presented elsewhere,58 showing that they also exhibit a low hysteresis value, while robustness of these states is still under investigation. In particular, it should be determined to which extent the rough approximation of the energy barrier between the metastable Cassie state and the Wenzel state, which was demonstrated here to be qualitatively good, is also quantitatively correct. In this direction, we believe that the use of a powerful computational method that minimizes drop energy75 with fewer assumptions should be considered.
Conclusion Superhydrophobic surfaces with a periodic controlled roughness on the micrometer scale were prepared, and the effect of six different geometrical parameters (minimum distance between pillars, pillar geometric distribution mode, cylindrical pillar diameter, pillar height, pillar shape, and scaling criterion) were investigated by static water contact angle measurements at 23 °C. The contact angle series measured were in good agreement with behavior predicted by simple thermodynamic models, based on free energy minimization. In particular, a transition between the Cassie and Wenzel regimes is observed for a well-defined range of each geometrical parameter. When millimetric water drops were softly deposited on the prepared surfaces, this transition did not correspond to the thermodynamic one, (i.e., where the two regimes present identical contact angle values and consequently the same energy level), but to a point where the system has enough energy to overcome the energy barrier between its initial state and the thermodynamically favored wetted regime. Although the details of the Cassie-Wenzel transition remain unknown, its position correlates nicely with a calculation of the energy required to fill the asperities, at constant Cassie angle constant. To improve quantitative predictions of the transition, more realistic parameters should be applied, by reducing the drastic assumptions. Acknowledgment. This work was financially supported by the Swiss KTI/CTI TopNano21 program. The authors express their gratitude to Dr. Frederick Fenter for the English revision. Supporting Information Available: Test of Extrand’s model47 applied to our experiments, and comments on other literature data.25,16,28 This material is available free of charge via the Internet at http://pubs.acs.org. LA0617964
