pubs.acs.org/Langmuir © 2010 American Chemical Society
Correlation between Superhydrophobicity and the Power Spectral Density of Randomly Rough Surfaces :: Houssein Awada,† Bruno Grignard,‡ Christine Jer^ome,‡ Alexandre Vaillant,§ Joel De Coninck,§ † ,† Bernard Nysten, and Alain M. Jonas* †
Institute of Condensed Matter and Nanosciences - Bio & Soft Matter (IMCN/BSMA), Universit e Catholique de Louvain, Croix du Sud 1 box 4, 1348 Louvain-la-Neuve, Belgium, ‡Centre d’Etude et de Recherche sur les Macromol ecules (CERM), Universit e de Li ege, Belgium, and §Laboratoire de Physique des Surfaces et des Interfaces (LPSI), Universit e de Mons, Belgium Received September 2, 2010
We show experimentally and analytically that for single-valued, isotropic, homogeneous, randomly rough surfaces consisting of bumps randomly protruding over a continuous background, superhydrophobicity is related to the power spectral density of the surface height, which can be derived from microscopy measurements. More precisely, superhydrophobicity correlates with the third moment of the power spectral density, which is directly related to the notion of Wenzel roughness (i.e., the ratio between the real area of the surface and its projected area). In addition, we explain why randomly rough surfaces with identical root-mean-square roughness values may behave differently with respect to water repellence and why roughness components with wavelength larger than 10 μm are not likely to be of importance or, stated otherwise, why superhydrophobicity often requires a contribution from submicrometer-scale components such as nanoparticles. The analysis developed here also shows that the simple thermodynamic arguments relating superhydrophobicity to an increase in the sample area are valid for this type of surface, and we hope that it will help researchers to fabricate efficient superhydrophobic surfaces based on the rational design of their power spectral density.
Introduction Superhydrophobic surfaces (i.e.,, roughened hydrophobic surfaces displaying an apparent contact angle with water larger that ∼150, a small contact angle hysteresis, and an almost zero rolloff angle for water droplets) have been attracting a great amount of attention over the last 15 years owing to their potential use in microfluidics or in the fabrication of self-cleaning coatings.1,2 Generally, such surfaces consist of microprotrusions and/or nanoprotrusions coated by, or made of, a hydrophobic material. When placed over this type of surface, water droplets may either penetrate the porosity and be pinned on the surface (the so-called Wenzel regime) or sit over protrusions and a mattress of trapped air, in which case superhydrophobicity ensues (the so-called Cassie-Baxter regime). The many articles published regarding these surfaces can be essentially divided into three categories: articles presenting a methodology for preparing such surfaces,3-11 (sometimes polemical) *Corresponding author. E-mail:
[email protected]. (1) de Gennes, P.-G.; Brochard-Wyart, F.; Quere, D. Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves.: Springer: New York, 2004. (2) Li, X.-M.; Reinhoudt, D.; Crego-Calama, M. Chem. Soc. Rev. 2007, 36, 1350–1368. (3) Soeno, T.; Inokuchi, K.; Shiratori, S. Appl. Surf. Sci. 2004, 237, 543–547. (4) Zhai, L.; Cebeci, F. C.; Cohen, R. E.; Rubner, M. F. Nano Lett. 2004, 4, 1349–1353. (5) Hosono, E.; Fujihara, S.; Honma, I.; Zhou, H. S. J. Am. Chem. Soc. 2005, 127, 13458–13459. (6) Jisr, R. M.; Rmaile, H. H.; Schlenoff, J. B. Angew. Chem., Int. Ed. 2005, 44, 782–785. (7) Zhang, G.; Wang, D.; Gu, Z.-Z.; M€ohwald, H. Langmuir 2005, 21, 9143– 9148. (8) Gao, L.; McCarthy, T. J. J. Am. Chem. Soc. 2006, 128, 9052–9053. (9) 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. (10) Hsu, S.-H.; Sigmund, W. M. Langmuir 2010, 26, 1504–1506. (11) Desbief, S.; Grignard, B.; Detrembleur, C.; Rioboo, R.; Vaillant, A.; Seveno, D.; Voue, M.; De Coninck, J.; Jonas, A. M.; Jer^ome, C.; Damman, P.; Lazzaroni, R. Langmuir 2010, 26, 2057–2067.
17798 DOI: 10.1021/la104282q
articles considering the reasons that a droplet settles in the Wenzel or Cassie-Baxter regime (or any other possibility),12-19 and articles dealing with the geometrical requirements that must be met by protrusions on heterogeneous surfaces to promote superhydrophobicity.21-32 This letter falls into the last two categories and focuses on randomly rough surfaces that, compared to regularly rough surfaces, have been considered much less and in ways that differ from the present approach either in aim or in mathematical details.33-35 Regularly rough surfaces are experimentally obtained by lithography procedures, giving rise to protrusions that are very well defined and can be modeled by trigonometric analysis. Such (12) Bico, J.; Thiele, U.; Quere, D. Colloids Surf., A 2002, 206, 41–46. (13) Gao, L.; McCarthy, T. J. Langmuir 2007, 23, 3762–3765. (14) McHale, G. Langmuir 2007, 23, 8200–8205. (15) Panchagnula, M. V.; Vedantam, S. Langmuir 2007, 23, 13242–13242. (16) Bormashenko, E. Colloids Surf., A 2008, 324, 47–50. (17) Whyman, G.; Bormashenko, E.; Stein, T. Chem. Phys. Lett. 2008, 450, 355–359. (18) Gao, L.; McCarthy, T. J. Langmuir 2009, 25, 7249–7255. (19) Marmur, A.; Bittoun, E. Langmuir 2009, 25, 1277–1281. (20) Lafuma, A.; Quere, D. Nat. Mater. 2003, 2, 457–460. (21) Wolansky, G.; Marmur, A. Colloids Surf., A 1999, 156, 381–388. (22) Yoshimitsu, Z.; Nakajima, A.; Watanabe, T.; Hashimoto, K. Langmuir 2002, 18, 5818–5822. (23) Patankar, N. A. Langmuir 2003, 19, 1249–1253. (24) Patankar, N. A. Langmuir 2004, 20, 7097–7102. (25) Marmur, A. Langmuir 2003, 19, 8343–8348. (26) Marmur, A. Langmuir 2004, 20, 3517–3519. (27) Patankar, N. A. Langmuir 2004, 20, 8209–8213. (28) Extrand, C. W. Langmuir 2005, 21, 10370–10374. (29) Zheng, Q.-S.; Yu, Y.; Zhao, Z.-H. Langmuir 2005, 21, 12207–12212. (30) Bittoun, E.; Marmur, A. J. Adhes. Sci. Technol. 2009, 23, 401–411. (31) Boinovich, L.; Emelyanenko, A. Langmuir 2009, 25, 2907–2912. (32) Kwon, Y.; Patankar, N.; Choi, J.; Lee, J. Langmuir 2009, 25, 6129–6136. (33) Palasantzas, G.; De Hosson, J. Th. M. Acta Mater. 2001, 49, 3533–3538. (34) Duparre, A.; Flemming, M.; Steinert, J.; Reihs, K. Appl. Opt. 2002, 41, 3294–3298. (35) Flemming, M.; Coriand, L.; Duparre, A. J. Adhes. Sci. Technol. 2009, 23, 381–400.
Published on Web 11/08/2010
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surfaces are of interest in applications such as microfluidics, where the area of active surfaces is limited. However, larger-scale surfaces such as windows or wall panels are not likely to be produced by lithographic means but by some process involving the random deposition of particles, phase separation, abrasion, dewetting, or a similar methodology, which creates randomly rough surfaces that can be analyzed only by statistical means. Here we show that for single-valued,36 isotropic, homogeneous, randomly rough surfaces, superhydrophobicity is related to the power spectral density of the surface height, which can be derived from microscopy measurements or from simulations. More precisely, superhydrophobicity correlates with the third moment of the power spectral density, which is directly related to the notion of Wenzel roughness. In addition, we explain why randomly rough surfaces with identical root-mean-square (rms) roughness σ may behave differently with respect to water repellence and why roughness components with a wavelength larger than ∼10 μm are of limited importance or, as stated otherwise, why superhydrophobicity usually requires submicrometer components. Importantly, the letter is essentially intended to propose a view to stimulate further research but does not aim to provide an exhaustive treatment.
Superhydrophobicity When deposited on a moderately rough surface, a liquid droplet settles in the Wenzel regime (apparent contact angle θw), in which case it impregnates the valleys of the roughness and “feels” the total external area of the material, or in the Cassie-Baxter regime (apparent contact angle θcb), in which case it rests over a composite surface made of air-filled cavities and material protrusions. In the Wenzel regime, the apparent contact angle θw on the rough surface is related to the contact angle θs of the liquid over a smooth surface of identical chemical nature by1 cos θw ¼ rw cos θs
ð1Þ
in the valleys of the roughness profile.1,20 This practically occurs when the apparent contact angle θw computed from eq 1 becomes larger than ∼150, corresponding to a critical rw* of ∼1.5-2.5 for surfaces made of perfluoroalkanes or alkanes (θs ≈ 120 or 110, respectively). Therefore, a knowledge of the Wenzel roughness of a surface should help predict whether a water droplet will settle in the Wenzel or Cassie-Baxter regime, even though it is important to stress that a superhydrophobic surface is not in the Wenzel regime. This is where the power spectral density of the surface enters the picture.
Power Spectral Density of a Surface The power spectral density P 1(s) of a randomly rough surface is related to the average amplitude of sine waves of reciprocal wavelength (or spatial frequency) s = 1/λ required to reconstruct by summation the height profile of the surface.39,40 Consider a single-valued, isotropic, homogeneous, randomly rough surface described by the random variable z(rB), where B r is a vector lying in the average plane of the surface and z(rB) =h(rB) - h0 is the height deviation at location B r , with h(rB) being the local height and h0 = < h(rB) > being the average height of the surface. The local variable z can be conveniently measured by atomic force microscopy (AFM) or optical profilometry (OP) over some sampling surface A0. Here, we limit ourselves to the consideration of surfaces that are isotropic and homogeneous (as defined above) because most randomly rough surfaces prepared by the methods listed in the Introduction are of this type. The single-valued nature of the surface36 is an important supplementary requirement: hidden re-entrant interfaces cannot be easily detected by microscopy and would considerably complicate the treatment; they are therefore excluded from our analysis. Mathematically, the power spectral density of this sample image is obtained by circularly averaging the squared magnitude of the bidimensional Fourier transform of z, z~(sB), according to41
where rw, known as the Wenzel roughness or Wenzel factor, is the ratio between the real area A of the surface (assumed here to be horizontal) and its area A0 projected over the horizontal plane, rw = A/A0. The Wenzel roughness is larger than 1 because of the extra surface area arising from bumps and holes. Note that eq 1 is readily obtained by minimizing the free energy of the liquid droplet in contact with the surface,12 in a manner similar to what can be done for a thermodynamically rigorous derivation of Young’s equation.37,38 It is strictly valid provided the roughness computed along the triple line is equal to the one computed over the contact area of the droplet, which will correspond to our definition of homogeneity for now on. This definition definitively ensures that the recent polemical discussion13-15,18,19 of the behavior of droplets on surfaces heterogeneous on a larger scale does not need to concern us in the sequel. When θs g 90 and the considered liquid is water (i.e., when the smooth surface is hydrophobic), eq 1 indicates that surfaces with larger rw display larger apparent contact angles θw with water. For a critical value of the Wenzel roughness rw*, the free energy of the droplet in the Wenzel regime becomes larger than it would have in the Cassie-Baxter regime. Under such circumstances, thermodynamics drives the droplet in a “fakir” state, in which the apparent contact angle is controlled by the amount of air trapped
where A0 is the projected area of the image and φ is the azimuthal angle in the plane of the image. Because the experimental sample image contains information only for wavelengths smaller than twice the side of the image and larger than twice the pixel size, it is necessary to sample the surface at different magnifications and to combine the power spectral densities of the resulting images in order to obtain P 1(s) over a wide range of reciprocal wavelengths s. In contrast to the simple rms roughness σ that is often quoted for rough surfaces, P 1(s) provides information not only on the vertical range of the roughness as does σ but also on the lateral range of the height fluctuations of the surface. For instance, a peak in P 1(s) at sp would testify to the existence of a preferred distance sp-1 between successive bumps or holes on the surface. Here, however, what interests us more is the possibility to compute from P 1(s) geometrical factors that are of direct relevance in deciding whether a surface should be superhydrophobic. The rms
(36) A single-valued surface is a surface for which any line drawn perpendicular to the average interface intersects the air/material interface only once. Multivalued surfaces are, for example, surfaces exhibiting re-entrant roughness, such as those made of a mat of horizontal fibers. (37) Johnson, R. E., Jr. J. Phys. Chem. 1959, 63, 1655–1658. (38) Gray, V. R. Chem. Ind. 1965, 969–978.
(39) Press, W. H.; Teukolsy, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes in C: The Art of Scientific Computing, 2nd ed.; Cambridge University Press: Cambridge, U.K., 1992. (40) Pratt, W. K. Digital Image Processing; Wiley: New York, 1991. (41) Stone, V. W.; Jonas, A. M.; Nysten, B.; Legras, R. Phys. Rev. B 1999, 60, 5883–5894.
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P 1 ðsÞ ¼ ¼
1 2π 1 2π
Z
2π
j~zðs, φÞj2 dφ
0
Z 0
2π
Z 1 j zðrBÞ expð2πjsB 3 B r Þ drBj2 dφ A0 A0
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ð2Þ
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Scheme 1. Methodology of the Preparation of the Randomly Rough Al Surfaces and the Structure of Fluorinated Block Copolymer 1 Used to Hydrophobize the Surfaces
roughness of a surface, σ, is trivially computed from P 1(s) using Parseval’s theorem42 Z Z ¥ 1 2 2 σ ¼ lim z ðrBÞ drB ¼ 2π sP 1 ðsÞ ds ð3Þ A0 f ¥ A0 A 0 0 showing that σ2 is the first moment of P 1(s). Likewise, the Wenzel roughness can be derived from P 1(s) in a first-order approximation as follows11,33 Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A 1 ¼ lim 1 þ jrzðrBÞj2 drB A0 f ¥ A0 A0 f ¥ A0 A 0 Z 1 1 1 þ jrzðrBÞj2 drB lim A0 f ¥ A0 A 2 0 Z ¥ Z 1 1 ¥ ¼ 1þ ð2πsÞ2 j~ zðsBÞj2 dsB ¼ 1 þ ð2πsÞ3 P 1 ðsÞ ds 2 0 2 0 ð4Þ rw ¼ lim
where 3 is the gradient operator. Higher-order approximations have been proposed,33 but these can still be expressed as polynomial series of the third moment of P 1(s) and will therefore not be used here for the sake of simplicity.
Experimental Section Fabrication of the Samples. Track-etched polycarbonate (PC) membranes were obtained from it4ip (Seneffe, Belgium) and had pore diameters of 0.4, 1, 2, and 3 μm, pore densities of 2 106, 5 106, 2 107, and 5 108 cm-2, and a thickness of 20 μm. The membranes were glued onto their periphery on clean Si wafers and placed in a Vacotec e-gun evaporator. Al was evaporated on the membranes at different rates (0.1, 1, 2.5, and 5 nm/s) up to a thickness of 1 μm, and the samples were glued upside down on Si wafers before the PC membranes were dissolved in methylene chloride. The Al surfaces were then covered by spin coating a 5 g/L solution of the fluorinated block copolymer 1 (Scheme 1) in trifluorotoluene (Aldrich) at 3000 rpm for 30 s. When performed on a flat surface of evaporated Al, this procedure leads to a film thickness of 20 nm as measured by X-ray reflectometry. (42) Sentenac, A.; Daillant, J. In X-ray and Neutron Reflectivity: Principles and Applications; Daillant, J., Gibaud, A., Eds; Springer: Berlin, 1999; Chapter 2. (43) Grignard, B.; Jer^ome, C.; Calberg, C.; Detrembleur, C.; Jer^ome, R. J. Polym. Sci., Part A: Polym. Chem. 2007, 45, 1499–1506. (44) Grignard, B.; Jer^ome, C.; Calberg, C.; Jer^ome, R.; Detrembleur, C. Eur. Polym. J. 2008, 44, 861–871.
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Fluorinated block copolymer 1 was synthesized as described before.43,44 Only samples free of macroscopic defects were retained for the analysis presented in this letter. Optical Profilometry. The topography was measured at various magnifications using a Wyko NT1100 optical profilometer. The images obtained at different magnifications (10, 25, and 50 and 100 for some samples,) were processed to compute a merged power spectral density as described before.11,41,45 The spectral power densities were cut for reciprocal wavelengths larger than 4 or 2.27 μm-1, depending on whether 100 magnification was used. This corresponds to optical resolutions of, respectively, 250 and 440 nm, which are well in the range of the physical optical resolution of the experiments. For one especially smooth sample (sample 1 in Figure 2), the data was obtained by AFM; in this case, the maximum reciprocal wavelength was half the Nyquist frequency of the AFM image of higher magnification. Assessment of Superhydrophobicity. The superhydrophobic character of the samples was assessed by first attempting to deposit a 5 μL droplet of water on the horizontal surfaces using the needle on the syringe of a Kr€ uss DSA100 contact angle goniometer. When deposition failed because of the lack of affinity of the surface with water, a droplet of 11 μL was tried because gravity allowed such a heavier droplet to detach from the needle. If this 11 μL droplet rolled directly off the surface without allowing any measurement of the static contact angle to be performed, then this surface was classified as superhydrophobic. The roll-off angle in this case is lower than 1. All other surfaces were classified as nonsuperhydrophobic. For superhydrophobic surfaces, attempts to measure the apparent contact angles using the method of Johnson and Dettre invariably provided values larger than ∼150 and a hysteresis between advancing and receding angles on the order of ∼1. In contrast, for nonsuperhydrophobic surfaces, it was possible to measure an apparent static contact angle, which was invariably smaller than ∼150.
Results and Discussion Equation 4 provides us with a link between the Wenzel roughness rw and the power spectral density P 1(s). A water droplet should settle in the fakir state on a perfluoroalkane-based superhydrophobic surface when rw g rw* ≈ 1.5 (or rw g rw* ≈ 2.5 for alkane-based surfaces). To test this criterion, a series of model rough hydrophobic surfaces were prepared by evaporating Al on track-etched polycarbonate (PC) membranes using a wide (45) Bollinne, C.; Cuenot, S.; Nysten, B.; Jonas, A. M. Eur. Phys. J. E 2003, 12, 389–396.
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Figure 1. SEM images of two samples prepared by replicating the porosity of track-etched PC membranes. (a) Superhydrophobic sample (reference 5 in Figure 2) obtained from a membrane with 2 106 cm-2 pores of 3 μm diameter using an Al evaporation rate of 1 nm/s. (b) Nonsuperhydrophobic sample (reference 4 in Figure 2) obtained from a membrane with 5 106 cm-2 pores of 3 μm diameter using an Al evaporation rate of 0.1 nm/s.
set of membrane characteristics (pore size and pore density) and different Al evaporation rates (Scheme 1). The Al replicas were recovered by dissolving PC in dicholoromethane, and a diblock copolymer was spin coated onto the rough Al surfaces (polymer 1 in Scheme 1). The copolymer consists of one carboxylic acid chain end used to anchor the polymer on the surface of Al, of a vinylacrylate block that was designed to cross link the polymer if needed (this was not done here) and of a second perfluorinated block that should segregate to the air surface. The presence of components of low surface energy at the air interface was checked by measuring the contact angle of water on a flat sample of polymer 1 spin coated on an Al layer evaporated on a Si wafer; the contact angle of water on this flat surface was 118 and was stable over a period of more than 3 months. Because the preparation of track-etched membranes first involves the irradiation of the membrane by heavy ions, which is a stochastic process in nature, the replication process leads to ideal randomly rough surfaces; actually, because two ions can successively hit the membrane at the same location, there is not even an excluded volume effect with respect to the location of the Al protrusions that replicate the porosity of the membrane. Scanning electron microscopy (SEM) images of typical model Al surfaces are given in Figure 1. The Al model surfaces were characterized by optical profilometry at different magnifications, from which their power spectral densities P 1(s) were obtained. Attempts to widen the range of reciprocal wavelengths s by AFM were unsuccessful because of the large amount of roughness in the samples. The experimental data were complemented by literature data taken from the work of Duparre and co-workers on superhydrophobic kohlrabi and lotus leaves35 and on a rough hydrophobic (but not superhydrophobic) surface.34 Power spectral densities were also computed for the flat, block-copolymer-covered Al surface sample and for a thin film of polystyrene (PS) spin coated over a Si wafer, for which AFM data could be obtained and were thus added to P 1(s). The experimental P 1(s) are displayed in Figure 2. Blue curves are indicative of superhydrophobic surfaces (i.e., curves for which the deposited water droplet rolls off the surface upon deposition as described in the Experimental Section). The curves in red are for surfaces that have apparent contact angles below ∼150 and do require the surface to be tilted for the droplet to roll off. The natural leaves (kohlrabi and lotus, samples 14 and 15 in Figure 2) are located at the top of the graph, whereas flat surfaces such as the Al layer evaporated on a Si wafer (sample 3 in Figure 2) are at Langmuir 2010, 26(23), 17798–17803
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Figure 2. Power spectral density versus reciprocal wavelength for a series of rough surfaces. Curves in blue indicate superhydrophobic surfaces, curves in red indicate nonsuperhydrophobic surfaces, and curves in black indicate nonsuperhydrophobic surfaces consisting of holes in a continuous background instead of protrusions over a continuous background. Sample codes: (1) PS film spin coated on a Si wafer; (2) ZrO2 layer coated with an alkane thiol, taken from Figure 4 of ref 34; (3) layer of Al evaporated on a Si wafer and then coated with polymer 1; (4-13) Al samples prepared by replicating the porosity of track-etched PC membranes, coated with polymer 1 (see Supporting Information for details); (14) kohlrabi leaves, taken from Figure 10 of ref 35; (15) lotus leaves, taken from Figure 10 of ref 35. The dashed lines for samples 14 and 15 are extrapolated power laws as proposed in ref 35.
the bottom. This is because P 1(s) is proportional to the square of the rms roughness. The model Al surfaces of the present study (samples 4-13 in Figure 2) draw a band where red and blue mix, indicating that these surfaces mark the limit where superhydrophobicity begins to exist (hence the interest in these model surfaces). It is, however, not trivial to decide from a casual inspection of the P 1(s) of these model Al samples the reason that a surface is or is not superhydrophobic. Therefore, the Wenzel roughness was computed. More precisely, a roughness function R w(ξ) was defined by the integration of (2πs)3 P 1(s) from 0 to a cutoff reciprocal wavelength ξ R w ðξÞ ¼
1 2
Z
ξ
ð2πsÞ3 P 1 ðsÞ ds
ð5Þ
0
so that the Wenzel roughness is rw =1 þ R w(¥) according to eq 4. This function, which represents to a first-order approximation the relative increment in the area of a surface due to its roughness when probed with a stylus of radius on the order of 1/ξ, is plotted in Figure 3 using the same color code as before. For the integration in eq 5, the P 1(s) values were linearly extrapolated to s = 0, which is a valid approximation because data at low s values carry a small weight in the total integral. Data at large s values were extrapolated with a simple power law (dashed lines), which is in agreement with the general behavior of the power spectral density of rough surfaces at large s and is the equivalent of scattering laws from self-affine surfaces. Also shown in Figure 3 are two horizontal axis lines corresponding to the critical values of rw* - 1 as mentioned before, which are ∼0.5 or ∼1.5 depending on whether the surface is made of perfluoroalkanes or alkanes. As can be seen in Figure 3, superhydrophobic surfaces are surfaces for which R w reaches the critical rw* - 1 value for reciprocal DOI: 10.1021/la104282q
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Figure 3. Wenzel roughness function R w(ξ) versus reciprocal wavelength ξ. Blue curves are for superhydrophobic surfaces, red curves are for nonsuperhydrophobic surfaces, and black curves are for surfaces consisting of holes in a continuous background instead of the reverse morphology. The sample numbering is identical to that in Figure 2. The left and right curves differ only by their logarithmic or linear vertical axis. The horizontal axis lines indicate the critical values above which superhydrophobicity should appear for alkane or perfluoroalkane rough surfaces. Note that sample 1 is polystyrene, samples 2, 14, and 15 are alkane-rich surfaces, and the others are perfluorinated surfaces. Dashed lines are power law extrapolations of the experimental data.
wavelengths shorter than ∼10 μm-1 (i.e., for roughness components with wavelengths longer than about 100 nm). For surfaces consisting of protrusions over a continuous background, there is only one exception to this (sample 8), testifying to the existence of secondary factors also controlling superhydrophobicity. By contrast, surfaces consisting of a porous continuous film (black curves in Figure 3) do not respect this rule, as will be discussed later. It is enlightening to analyze the behavior of R w with reciprocal wavelength ξ. For all samples, R w is well below the critical values required for superhydrophobicity when considering only reciprocal wavelengths below ∼0.1 μm-1 (wavelengths g ∼10 μm). In other words, the increase in area resulting from roughness components with wavelength longer than ∼10 μm can be neglected. This is because the Wenzel roughness is related to the third moment of P 1(s), which means that large bumps are not likely to increase the area of a surface significantly. This conclusion is general and explains why superhydrophobicity often requires a contribution from submicrometer-scale components such as nanoparticles in order to reach large Wenzel roughnesses. The shape factor of larger particles, that is, the Fourier transform of their shape, extends over a smaller range of reciprocal wavelengths than the shape factor of smaller particles. The contribution of large particles to the integral in eq 5 is thus limited, and a further increase in Wenzel roughness is possible only by adding smaller particles, which add roughness components in the higher range of reciprocal wavelengths. This is certainly the reason that many superhydrophobic surfaces exhibit a so-called double roughness, as originally noted for lotus leaves and discussed extensively later on.27,46-48 The surfaces that are not superhydrophobic have an R w(s) that probably still increases with s outside the probed range and thus might reach the critical value of roughness for a sufficiently large s. However, the roughness power spectrum P 1(s) decreases exponentially with s, meaning that the amplitude of the roughness components is rapidly decreasing with s for our randomly rough surfaces. This effect will be even stronger because of the smooth(46) Barthlott, W.; Neinhuis, C. Planta 1997, 202, 1–8. (47) Herminghaus, S. Europhys. Lett. 2000, 52, 165–170. (48) Hipp, B.; Kunert, I.; D€urr, M. Langmuir 2010, 26, 6557–6560.
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ing or damping effect that the polymer coating has on roughness components of small wavelength: very small particles adsorbed on a surface will be essentially buried in the hydrophobic coating after their deposition, contrary to the situation for larger particles. This is actually demonstrated by the experimental observation that the power spectral density of a rough Al surface (measured by AFM) is damped by a factor of 0.2 in the 4-40 μm-1 range after being coated with a polymer layer (Supporting Information). Therefore, for wavelengths shorter than ∼100 nm (s g 10 μm-1), the contribution of roughness components will be damped in the final superhydrophobic surface, at least for the specific samples in this study. These arguments explain why roughness components with wavelengths in the range of ∼100 nm-10 μm are likely to be more important for our superhydrophobic randomly rough surfaces. The present analysis also explains why the rms roughness σ is of little interest when analyzing a superhydrophobic surface. Indeed, eq 4 shows that rw depends on the third moment of P 1(s), whereas the rms roughness is related to the first moment of P 1(s) (eq 3). Therefore, one may have surfaces of identical rms roughness that behave very differently with respect to superhydrophobicity. Hence, a complete assessment of the correlation between roughness and superhydrophobicity certainly requires a measurement of the third moment of P 1(s). The analysis could easily be extended to droplets of different sizes by appropriately varying the lower limit in the integral of eq 5. It is indeed known that superhydrophobicity may depend on the size of droplets; here, however, we have limited ourselves to macroscopic droplets, for which the lower limit in the integral of eq 5 can be approximated as zero. As a final comment, it should be noted that the arguments developed here are valid only for surfaces displaying protrusions over a continuous background, not for reverse surfaces made of holes in a continuous film. Among the samples tested, a small number had such reverse morphology (black curves in Figures 2 and 3). These samples were invariably not superhydrophobic, irrespective of their power spectral density. This is because the computation of P 1(s) does not differentiate between a surface and its negative according to Babinet’s theorem, whereas a droplet of water definitely feels the difference in the amount of trapped Langmuir 2010, 26(23), 17798–17803
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air, which should certainly be larger than 50% for superhydrophobicity.
Conclusions The power spectral densities of a set of randomly rough samples was computed, from which a Wenzel roughness function R w(s) arising from components of reciprocal wavelengths of up to s was obtained. This function represents the relative increment of sample area resulting from the bumps and holes in the roughness. Samples for which R w(s) reaches a critical value, depending on the chemical nature of the surface, below s = ∼10 μm-1 are superhydrophobic, which corresponds to expectations based on surface thermodynamics. Because the Wenzel roughness is the third moment of the power spectral density, long wavelengths in the spectrum, above ∼10 μm, are of lesser importance: the area increment brought about by features of long wavelengths is hardly significant. This is a general conclusion applying to most surfaces and explains why nanoparticles often have to be added in order to reach the critical value of Wenzel roughness. In addition, for our samples, very short wavelengths, below ∼100 nm, do not contribute much to the Wenzel roughness. This is because the power spectral density decreases exponentially with reciprocal wavelength, an effect further amplified by the deposition of the hydrophobic coating on the rough surface. The analysis developed here also shows that the simple thermodynamic arguments
Langmuir 2010, 26(23), 17798–17803
relating superhydrophobicity to an increase in the sample area are valid for homogeneous, isotropic, single-valued, randomly rough surfaces displaying a morphology of protrusions standing over a background (as opposed to holes in a film). The proper parameter to consider is thus not the rms roughness, which is the first moment of the power spectral density, but the Wenzel roughness, which is its third moment. Obviously, kinetic factors should also enter the balance, making the picture somewhat more complex. Nevertheless, we hope that the present analysis will help researchers to fabricate efficient superhydrophobic surfaces on the basis of the rational design of their power spectral density. Acknowledgment. We acknowledge the help of Sandrine Lenoir for the synthesis of the polymer. B.N. is senior research associate of the F.R.S.-FNRS. Financial support was provided by the Belgian Federal Science Policy (IAP-PAI P6/27), the F.R.S.FNRS, and the Wallonia Region (corronet 03/1/5557). Supporting Information Available: Complementary data on copolymer 1 spin coated on a flat sample of evaporated Al (X-ray reflectometry and X-ray photoelectron spectroscopy). Smoothing effect of the polymer layer in the high range of reciprocal wavelengths. Correspondence between the sample number code and processing conditions. This material is available free of charge via the Internet at http:// pubs.acs.org.
DOI: 10.1021/la104282q
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