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When Wenzel and Cassie Are Right: Reconciling Local and Global Considerations Abraham Marmur* and Eyal Bittoun Department of Chemical Engineering, Technion - Israel Institute of Technology, 32000 Haifa, Israel ReceiVed August 15, 2008. ReVised Manuscript ReceiVed December 5, 2008 The condition under which the Wenzel or Cassie equation correctly estimates the most stable contact angle is reiterated and demonstrated: these equations do hold when the drop size is sufficiently large compared with the wavelength of roughness or chemical heterogeneity. The numerical demonstrations somewhat mimic recent experiments that seemingly refuted the Wenzel and Cassie equations and show that these experiments were performed only for drops of sizes similar in order of magnitude to the wavelength of roughness or chemical heterogeneity. Under such conditions, the Wenzel and Cassie equations are a priori not expected to be valid. It is also explained that both the local equilibrium condition at the contact line and the global equilibrium condition involving the wetted area within the contact line are necessary and complementary.
Introduction The useful application of wetting phenomena depends on understanding the effects of roughness and chemical heterogeneity.1-34 Pioneering papers by Wenzel,1 Cassie, and Cassie and Baxter2,3 laid the foundations for understanding wetting on rough and chemically heterogeneous surfaces; however, a statement has recently been made by Gao and McCarthy25 to the effect that these equations are wrong. The reason given was that the experimental data presented in the literature “indicate that contact angle behavior (advancing, receding, and hysteresis) is determined by interactions of the liquid and the solid at the three-phase contact line alone and that the interfacial area within the contact perimeter is irrelevant.” A few publications have responded to this statement. McHale28 suggested that the Wenzel and Cassie equations do apply when local values of the roughness ratio and of the area fractions that define chemical heterogeneity are used instead of global values; he also proposed that the original Wenzel and Cassie equations apply only when the surface is similar and isotropic everywhere. Nosonovsky29 also argued that the Wenzel and Cassie equations should be modified to involve local values of the roughness ratio and the area fractions of chemical heterogeneity. Panchagnula and Vedantam30 similarly claimed that the area fractions in the Cassie equation need to be calculated only for the immediate vicinity of the contact line in order to get meaningful results, but they did not comment on the Wenzel equation. These responses will be discussed below together with the present reasoning. The main problem with the above statement25 is that it stems from experiments performed with drops that were too small, ignoring the indications of existing theoretical understanding.13,16,21 It was rigorously proven13 and numerically demon* Corresponding author. E-mail:
[email protected]. Fax: +972-4829-3088. (1) Wenzel, R. N. J. Ind. Eng. Chem. 1936, 28, 988. (2) Cassie, A. B. D.; Baxter, S. Trans. Faraday Soc. 1944, 40, 546. (3) Cassie, A. B. D. Discuss. Faraday Soc. 1948, 3, 11. (4) Good, R. J. J. Am. Chem. Soc. 1952, 74, 5041. (5) Johnson, R. E., Jr.; Dettre, R. H. J. Phys. Chem. 1964, 68, 1744. (6) Huh, C.; Mason, S. G. J. Colloid Interface Sci. 1977, 60, 11. (7) Boruvka, L.; Neumann, A. W. J. Chem. Phys. 1977, 66, 5464. (8) Drelich, J.; Miller, J. D. Langmuir 1993, 9, 619. (9) Marmur, A. AdV. Colloid Interface Sci. 1994, 50, 121. (10) Marmur, A. Colloids Surf., A 1998, 136, 209. (11) Wolansky, G.; Marmur, A. Langmuir 1998, 14, 5292. (12) Swain, P. S.; Lipowsky, R. Langmuir 1998, 14, 6772.
strated16 that the Wenzel and Cassie equations are approximations that become valid when the size ratio of the drop to the wavelength of roughness or chemical heterogeneity is sufficiently large. It is obvious that a theory can hold only when the assumptions underlying it are valid. Therefore, it may be more useful (though less provocative) to explain when and why the Wenzel and Cassie equations are right. This is done in the present letter first by way of numerical examples that demonstrate the essential points and then by reiterating and somewhat generalizing the theoretical considerations underlying these equations. It will be shown that local considerations (regarding the contact line) and global considerations (involving the interfacial area within the contact line) do not contradict but rather complement each other. In addition, it will be argued that meaningful measurement and definition of contact angles for a realistic, general case of roughness and chemical heterogeneity are possible only for relatively large drops. For such drops, the Wenzel and Cassie equations apply without any modification.
Numerical Examples A few numerical examples that somewhat mimic the experiments done17,25 are shown below. For simplicity, the studied model system is 2D. It consists of a 2D drop on a smooth solid surface with a 2D pattern of chemical heterogeneity (Figure 1). The system is surrounded by a gaseous environment, and the effect of gravity is assumed to be negligible. The heterogeneity pattern consists of a periodic arrangement of stripes of two types of chemistries. The stripes and the drop are infinitely long in the direction perpendicular to the paper. The liquid-air interface is circular because of the lack of gravity. An axisymmetric geometry also could have been chosen; however, assuming an axisymmetric heterogeneity pattern seems to be less consistent with reality than a self-consistent 2D case. In any case, the qualitative results would have been exactly the same because the only difference is in some geometric factors. This system was chosen not only for its obvious simplicity but also for deeper reasons. A drop at equilibrium has to conform to two thermodynamic conditions: (a) its local mean curvature has to match the local pressure difference across the liquid-air interface according to the Young-Laplace equation and (b) the local, actual contact angle (CA) must conform to the local value
10.1021/la802667b CCC: $40.75 2009 American Chemical Society Published on Web 01/07/2009
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(Figure 1b). The first term on the RHS of eq 1 is the liquid-air surface energy PUD (surface tension multiplied by the length of the circular arc). The second term is the difference between the solid-liquid and solid-air interfacial energies PUD, assuming that the interfacial tensions vary along the x direction only and that the drop must sit symmetrically with respect to the chemical heterogeneity pattern (otherwise it must move to a symmetrical position at equilibrium). It is convenient to transform eq 1 into a dimensionless form in order to discuss ratios rather than absolute values r*
∫
G 2r*θ G* ≡ ) - 2 cos θY(x * ) dx* σll sin θ 0
Figure 1. Two-dimensional wetting system. (a) A relatively small drop on the inner hydrophilic domain, with the most stable CA being θY1. (b) A relatively small drop on the most inner hydrophobic domain, with the most stable CA being θY2. (c) A relatively large drop whose contact line is in between domains and the most stable CA approaches the Cassie angle.
of the Young CA (assuming that line tension is negligible21).7,11,12 The contact line of a 3D drop on a chemically heterogeneous surface has the freedom to adapt its local shape to conform to both conditions. In contrast, the shape of the contact line of a 2D drop is fixed (it must be a straight line); therefore, the reconciliation of the local requirement at the contact line and the global energy considerations is more challenging. Thus, it may be even more convincing to demonstrate the validity of the Cassie equation with respect to the more difficult case of a 2D drop. Gravity is neglected because it affects the shape of a drop but not its actual CAs.21 In addition, line tension is neglected because it is important only for extremely small drops.21 To identify the equilibrium states of the system, its Gibbs energy was calculated as a function of all possible CAs for various drop sizes. The equilibrium CAs of the system were determined by identifying the minima in the Gibbs energy curve. The main emphasis is on the effect of the size ratio of the drop to the wavelength of chemical heterogeneity on the equilibrium CAs. This effect clearly explains the seeming contradiction between the experimental observations17,25 and the Cassie equation. The case of the Wenzel equation will be discussed later but not numerically demonstrated because it has been rigorously dealt with in the most general way.13 The Gibbs energy per unit depth (PUD), G, of the system of a 2D drop is given by9 r
G ) σl
∫
2rθ + 2 [σsl(x) - σs(x)] dx sin θ 0
(1)
where r is the “radius” of the drop base (half its width; Figure 1a), θ is the geometric CA (the CA as an independent geometric variable, not necessarily at equilibrium21), σl and σs are the surface tensions of the liquid and the solid, respectively, σsl is the solid-liquid interfacial tension, and x is a distance coordinate along the surface, measured from the center of the drop base (13) Wolansky, G.; Marmur, A. Colloids Surf., A 1999, 156, 381. (14) Henderson, J. R. Mol. Phys. 2000, 98, 677. (15) Kamusewitz, H.; Possart, W. Appl. Phys. A: Mater. Sci. Process. 2003, 76, 899. (16) Brandon, S.; Haimovich, N.; Yeger, E.; Marmur, A. J. Colloid Interface Sci. 2003, 263, 237–243. (17) Extrand, C. W. Langmuir 2003, 19, 3793–3796. (18) Marmur, A. Langmuir 2003, 19, 8343.
(2)
where l is the wavelength of heterogeneity (Figure 1c; to be specifically defined for each case), r* ≡ r/l, x* ≡ x/l, and θY is the Young CA at each location x* on the solid surface. Under the constraint of a constant drop volume PUD, V (which is actually the cross-sectional area of the drop), the variables r* and θ are related by the simple geometric formula
(r * )2 ) V*
2 sin2 θ 2θ - sin 2θ
(3)
where V* ≡ V/l2 is the dimensionless drop volume PUD. Substituting r* from eq 3 into eq 2 enables the calculation of G* versus θ for any prescribed V*. The apparent (equilibrium) CAs are the geometric CAs associated with minima in the Gibbs energy curve. It should be noticed that on a smooth surface the apparent and actual CAs are identical. The specific model surface assumed for the present calculations consists of stripes of two chemistries (Figure 1), mimicking in terms of area fractions and water CAs one of the experimental models used by Gao and McCarthy:25
θY(x * ) ) 22 for 0 < x* e 0.125 113 for k + 0.125 < x* e k + 0.875 k ) 0, 1, 2... (4) 22 for k + 0.875 < x* e k + 1.125 k ) 0, 1, 2...
{
The values of these two Young CAs are approximately the averages of the receding and advancing CAs of each type of chemistry, as measured by Gao and McCarthy,25 and the area fractions (25 and 75%) are the same as one set of values used by them. The Cassie CA for this surface is θC ) 93.5°, as calculated from the well-known Cassie equation3
cos θC ) f1 cos θY1 + f2 cos θY2
(5)
where fi is the area fraction occupied by chemistry of type i. The hydrophilic stripe was chosen to be at the center for the present examples, similar to the experimental situation.25 Figure 2a shows that when the drop is sufficiently small relatively to the wavelength of heterogeneity there may be a single minimum in the Gibbs energy. The relative size of the drop in these examples is measured by
R ≡ (√4V ⁄ π)/l ) √4V*/π
(6)
which is the ratio of the diameter of the cylindrical (2D) drop before touching the solid to the wavelength of surface heterogeneity. For the present model, this wavelength is the combined width of the two stripes (i.e., l ) 1). When the drop is sufficiently small to be completely within the inner, hydrophilic stripe of the surface (e.g., R ) 0.05), the only minimum in G* is indeed at θ ) 22° ) θY1 (the numerical error is less than 0.01%). When the drop is somewhat larger and its contact line sits within the hydrophobic stripe (e.g., R ) 0.5), there is also a single minimum
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Figure 2. Gibbs energies and contact angles for 2D drops on a smooth, chemically heterogeneous surface. θY1 ) 22° and θY2 ) 113°. f1 ) 0.25. (a-c) Dimensionless Gibbs energies vs geometric CAs; the lowest minimum for each curve is marked by a full circle; the dotted vertical lines mark the Cassie CA for this case. (d-f) Young CA, θY, vs dimensionless distance along the solid surface, x*, and the geometric CA, θ, vs the dimensionless radius of the drop base, r*. In all plots, the relative drop size, R, is given by the number near each curve.
in G* that occurs at θ ) 113° ) θY2. For the intermediary drop size shown in Figure 2a (R ) 0.15), the only equilibrium is at θ ) 80.6°, in between the two Young CAs. This is possible because in reality the junction between a hydrophilic and a hydrophobic domain cannot correspond to a mathematical step function but must represent some smooth transition between the domains over a very short distance. Thus, all apparent CAs between θY1 and θY2 are, in principle, feasible at the junction. Obviously, these three apparent CAs are very different from the Cassie CA for this surface (θC ) 93.5°). Figure 2b shows three cases of higher values of R: 1, 2.5, and 5. In these cases, the Gibbs energy curves show a few minimum points. The lowest minimum is the most stable one, and the others are metastable. The most stable apparent CA, θms, (or, in short, most stable CA) is of interest here because this is the CA that is supposed to be approximated by the Cassie equation. The most stable CAs in this Figure turn out to be 86.2, 113, and 80.8° for the above three values of R, respectively. These are still different from the Cassie CA but somewhat closer to it than the CAs in the previous examples. Figure 2c shows the case of a relatively much larger drop, R ) 700, for which there exist many minima, and θms ) 94.3, which is quite close to θC. A schematic representation of some of the above situations is shown in Figure 1. It presents the cases of a relatively small drop on the inner hydrophilic domain (Figure 1a), on the most inner hydrophobic domain (Figure 1b), and of a relatively large drop whose contact line is in between domains (Figure 1c). The corresponding most stable CAs are θY1, θY2, and a CA that is approaching the Cassie angle. Thus, it is clear that the multiplicity of minima in the Gibbs energy plays an essential role in determining the most stable CA. The origin of this phenomenon can be quite simply understood, as follows.10 From a thermodynamic point of view, the drop may be at equilibrium (metastable or most stable) at any position for which the curvature is constant (no-gravity case), and the actual CA (identical in this case with the apparent CA) equals the local Young CA.21 However, as shown by eq 3, the geometrical constraint of a constant volume imposes a relationship between r* and θ. Thus, only the states that simultaneously fulfill the thermodynamic as well as geometrical conditions, as shown in Figure 2d-f, can actually be allowed.10 In this Figure, the periodic
lines (steps) present the Young CAs at each location, x*, on the solid surface. The descending lines show the geometric CAs as they depend on the position of the contact line, r*, according to eq 3, for various values of R (which represent the dimensionless drop sizes). Each intersection between a thermodynamic (Young CA) line and a geometric line leads to a minimum or a maximum point in the energy curve,10 as can be confirmed by comparing each vertical pair of plots in Figure 2. For the present discussion, only the minimum points, which represent metastable equilibrium states, are of interest. Figure 2d thus shows that for a small value of R (0.05, 0.15, or 0.5) there may indeed be only a single intersection between the thermodynamic and geometric lines. Figure 2e shows that for larger values of R (1, 2.5, and 5) there are more intersection points because geometric lines for larger volumes have lower slopes (in absolute value). It is geometrically quite clear that for a given value of R, at most one intersection point may occur at an apparent CA of either θY1 or θY2 (or one for each). Thus, when there are many intersection points, most of them occur at intermediate values of the apparent CA. Figure 2f shows the case of a much larger value of R (700), for which the number of intersections indeed becomes large, and as mentioned above, the most stable CA is relatively close to the Cassie value. Figure 3 summarizes and complements the above numerical results by showing the dependence of θms on R. The value of θms oscillates around the Cassie value; however, it is clear that θms approaches the value of the Cassie CA when the relative drop size, R, becomes sufficiently large (in these cases, for R > ∼700).
Discussion For the sake of clarity, the understanding gained from 2D systems will be discussed first, followed by a discussion of 3D situations. As is well known, the curve of Gibbs energy versus the geometric CA for a 2D system with an ideal surface has a single minimum, whereas for a rough or chemically heterogeneous surface it displays multiple minima.5 Every minimum corresponds to a situation where the actual CA equals the local Young CA. Obviously, every minimum for a rough or chemically heterogeneous surface is associated with a different apparent CA. The lowest (global) minimum corresponds to the most stable CA,
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Figure 3. Ratio of the most stable CA to the Cassie CA as it depends on the relative drop size for 2D drops on a smooth, chemically heterogeneous surface. θY1 ) 22° and θY2 ) 113°. f1 ) 0.16 (-) or 0.44 (---).
and all other minima correspond to metastable states.21 The most stable CA is the one to which the Wenzel or Cassie equation refers. The range of metastable, apparent contact CAs constitutes the hysteresis range. It is important to emphasize that neither the Wenzel nor the Cassie equation has anything to do with the range of CA hysteresis or with its extremes, the advancing and receding CAs.21 These CAs are usually much easier to measure than the most stable CA. However, we do not yet have any theory for interpreting them in terms of interfacial tensions. Applying the Wenzel or Cassie equation to the advancing or receding CA contradicts current theoretical understanding. In contrast, the most stable CA is amenable to theoretical interpretation, and a methodology for its measurement is under development.34 Thus, the following discussion focuses only on the most stable CA. The main question under discussion is whether the most stable CA is determined exclusively by interactions at the three-phase contact line or whether the solid-liquid interfacial area is also relevant. This is a paraphrase of the question raised by Gao and McCarthy,25 which is necessary because they did not make the essential distinction between actual, apparent (in general), and most stable (apparent) CAs. The answer is that both the contact line and the wetted area are relevant. The interactions at the contact line affect every metastable apparent CA. This is so because every metastable equilibrium state is determined by the actual CA along the contact line being equal to the local Young CA at the same point.7,11,12 However, the identification of one of these apparent CAs as the most stable one is affected by the nature of the wetted area, as explained below. To determine the most stable CA, the global minimum in the Gibbs energy needs to be identified. Unfortunately, there is no general, mathematical way to single out the global minimum of a function that has multiple minima except for checking all minima one by one. Therefore, a general equation for the most stable CA cannot exist. The intuitive approach taken by Wenzel1 and Cassie and Baxter2,3 was to define a seemingly uniform surface with average properties that represent the roughness or chemical (19) Zhang, J.; Kwok, D. Y. J. Colloid Interface Sci. 2005, 282, 434. (20) Iwamatsu, M. J. Colloid Interface Sci. 2006, 294, 176. (21) Marmur, A. Soft Matter 2006, 2, 12. (22) Martines, E.; Seunarine, K.; Morgan, H.; Gadegaard, N.; Wilkinson, C. D. W.; Riehle, M. O. Langmuir 2006, 22, 11230. (23) Bormashenko, E.; Bormashenko, Y.; Stein, T.; Whyman, G.; Pogreb, R.; Barkay, Z. Langmuir 2007, 23, 4378. (24) Bormashenko, E.; Pogreb, R.; Whyman, G.; Erlich, M. Langmuir 2007, 23, 12217.
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heterogeneity. Though they may not have looked at it from such an angle, their concept actually implies replacing the real surface with an equivalent ideal one. Thus, the Gibbs energy on this equivalent surface has a single minimum that can be easily identified and is associated with the Wenzel or Cassie CA.32 This intuitive, conceptual foundation inevitably classifies the Wenzel and Cassie equations as approximate ones. However, as mentioned above, it was discovered13,16 that the approximate Wenzel and Cassie CAs approach the most stable CAs as the drop size is increased in comparison with the wavelength of roughness or chemical heterogeneity. Intuitively, this observation can be explained by saying that when the relative drop size is sufficiently large the surface appears to be uniform to the drop; therefore, the Wenzel or Cassie averaging approach is justified. It does not have to be truly uniform, as suggested by McHale.28 True uniformity means that the wavelength of roughness or chemical heterogeneity approaches zero; therefore, the relative drop size approaches infinity. In reality, the approximation predicted by the Wenzel or Cassie equation may also be sufficiently good when the relative drop size is not as large. How large is large enough is not yet known, but from the present and previous16 examples, it appears that a relative size of about 3 orders of magnitude may be satisfactory. The above numerical examples clearly demonstrate this dependence on the relative size of the drop and in this way explain the experimental results reported in the literature.17,25 In the experiments of Gao and McCarthy,25 the base diameter of the drop was larger than the diameter of the second chemistry spot by at most a factor of 2.5, and in the experiments of Extrand,17 the two length scales were also very similar. The equivalent case in the above examples would be an R value on the order of magnitude of 1 (l* ) 1; width of inner stripe ) 0.25, therefore the base drop diameter is at most 0.625 and the diameter of the drop before touching the surface is on the order of magnitude of 1). Indeed, the above examples show that when the relative size of the drop is that small the Cassie equation is not valid. It should again be emphasized that for relatively small drops there is no general equation and each case has to be individually studied. For example, the most stable CA of a relatively small drop (low R) could be either θY1 or θY2 as shown in Figure 2a,d. Whereas these two cases can be predicted on the basis of the suggestion of using Cassie averaging along the contact line,28-30 no general rule can be formulated for the drop with R ) 0.15, for example, that sitting at the junction between the two stripes. The important point to remember is that the Cassie equation is not supposed to be valid for relatively small drops.16 Therefore, claiming the Cassie equation to be wrong in general is misleading. Exactly the same arguments may be applied to the Wenzel equation.13 Moreover, initial experimental data show that the Wenzel equation is indeed valid for sufficiently large drops.34 In addition, the above examples demonstrate how local considerations and global considerations indeed complement each other: the equilibrium CA that the drop makes with the surface (actual or apparent, which are identical in this case) must equal the local Young CA at the position of the contact line; however, there are many possible such (metastable) CAs for which this condition is fulfilled. The special CA that renders the system lowest in Gibbs energysthe most stable CAsis determined by the nature of the interfacial area within the contact line. Using the terminology presented above, we find that the number of metastable states and the identity of the most stable state depend on the intersections between the thermodynamic line and geometric line. These definitely depend on the details of the solid-liquid interfacial area.
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For a smooth but chemically heterogeneous surface, the apparent CA is identical to the actual CA. Therefore, the following question can be asked regarding the above numerical examples: how can the most stable apparent CA be different from either θY1 or θY2 if the fundamental equilibrium condition is that the actual CA must equal the local Young CA at every point? The answer, as known for a long time5 and re-explained above, is that the contact line in the most stable state may be located at the junction between the two domains. At this junction, all apparent CAs between θY1 and θY2 are, in principle, feasible. For a rough surface, the apparent and actual CAs may be very different, depending on the local inclination of the solid surface. However, a similar phenomenon occurs for rough surfaces: the contact line tends to be located on roughness ridges (the “equivalents” of junctions between chemistries), where large variations in the apparent CA are possible for the same Young CA.5,6,13 The experiments done by Gao and McCarthy with textured surfaces25 may appear at first sight to contradict the above conclusions because the drop was large compared with the texture scale. In these experiments, a rough spot was formed within a smooth surface of the same chemistry, and a smooth spot was formed within a rough surface of the same chemistry (Figure 3b,c in ref 25). The drop size was indeed large compared with the roughness scale but of the same order of magnitude as that of the spot. Therefore, the criterion of the drop being sufficiently large compared with the wavelength of heterogeneity is actually not fulfilled. The surfaces in these experiments are characterized by two scales of heterogeneity: microheterogeneity (roughness wavelength on the order of magnitude of micrometers), and macroheterogeneity (transition from a rough surface to a smooth one over a wavelength of millimeters). The overall wavelength of heterogeneity must be a combination of the contributions of the two scales, and obviously it is determined mainly by the macroheterogeneity scale. While the drop is sufficiently large compared with the microheterogeneity scale, it still has the same order of magnitude on the macroheterogeneity scale. Therefore, the experimental observation that neither the Wenzel nor the Cassie equation holds for this system actually supports the theoretical conclusion that these equations are not supposed to be valid for such systems. For 3D systems, the situation is more complex; however, the qualitative conclusions are exactly the same as for the 2D systems. For such systems, the metastable equilibrium states are still determined by the actual CA along the contact line being equal (25) Gao, L.; McCarthy, T. J. Langmuir 2007, 23, 3762. (26) Gao, L.; McCarthy, T. J. Langmuir 2007, 23, 13243. (27) Han, T.-Y.; Shr, J.-F.; Wu, C.-F.; Hsieh, C.-T. Thin Solid Films 2007, 515, 4666. (28) McHale, G. Langmuir 2007, 23, 8200. (29) Nosonovsky, M. Langmuir 2007, 23, 9919. (30) Panchagnula, M. V.; Vedantam, S. Langmuir 2007, 23, 13242.
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to the local Young CA at the same point.7,11,12 However, the definition of an apparent CA is not straightforward because the drop is not necessarily axisymmetric and the apparent CA itself may vary along the contact line. For a lack of axisymmetry, the drop volume does not uniquely define its shape, and the Gibbs energy cannot be expressed as a function of a single variable such as the geometric CA. Moreover, the apparent CA cannot be uniquely defined by measurement because it may be different from each angle of observation. Thus, the case of a nonaxisymmetric drop is not amenable to any general treatment. However, during the 3D analysis of the Wenzel and Cassie equations,13,16 it was also found out that the drop approaches axisymmetry when its relative size becomes sufficiently large. Most of the drop becomes axisymmetric, while deviations from axisymmetry are limited to the close vicinity of the contact line. Thus, by extrapolating the shape of the axisymmetric part of the drop all the way to the apparent solid surface, a unique definition of the apparent CA can be achieved (theoretically as well as practically).13 This is an extremely important point because it enables the measurement and theoretical analysis of such drops. Therefore, only relatively large drops are practically useful for the measurement and interpretation of CAs on realistically rough or chemically heterogeneous systems. The most stable CA of relatively small drops can be neither measured nor theoretically predicted. For this reason, there is also no real need for the suggested generalization of the Wenzel or Cassie equation;28-30 for sufficiently large drops, the original form is quite satisfactory. To summarize, local conditions at the contact line determine the actual CAs, and global considerations regarding the solid-liquid interfacial area determine the most stable apparent CA. This CA is the one predicted by the Wenzel or Cassie equation if the drop is sufficiently large compared with the wavelength of roughness or chemical heterogeneity.13,16 Moreover, meaningful measurements of CAs on rough or chemically heterogeneous surfaces requires sufficiently large drops in order to avoid errors due to nonaxisymmetry.13,16,34 This makes the Wenzel and Cassie equations in their original form adequately useful. The experiments that seemingly refuted these equations17,25 were done with drop sizes of the same order of magnitude as the wavelength of heterogeneity; therefore, no generalization can be drawn from them. As is the case with any theory, the Wenzel or Cassie equation should be expected to hold only under the assumptions underlying them. LA802667B (31) Dorrer, C.; Ruehe, J. Langmuir 2008, 24, 1959. (32) Marmur, A. Langmuir 2008, 24, 7573–7579. (33) Que´re´, D. Annu. ReV. Mater. Res. 2008, 38, 71. (34) Meiron, T. S.; Marmur, A.; Saguy, I. S. J. Colloid Interface Sci. 2004, 274, 637.
