pubs.acs.org/Langmuir © 2009 American Chemical Society
An Attempt to Correct the Faulty Intuition Perpetuated by the Wenzel and Cassie “Laws” Lichao Gao and Thomas J. McCarthy* Polymer Science and Engineering Department, University of Massachusetts, Amherst, Massachusetts 01003 Received April 21, 2009. Revised Manuscript Received May 4, 2009 We respond to a recent report in this journal that criticizes our experiments, which disproved the Wenzel and Cassie theories. The criticism is that we measured contact angles “with drops that were too small, ignoring the indications of existing theoretical understanding.” We take a step back to give an explanation of what we believe to be the reason that the “existing theoretical understanding” is wrong. We explain that the teaching of surface science has led generations of students and scientists to a misunderstanding of the wetting of solids by liquids. This continues as evidenced by this recent criticism and numerous recent papers. We describe several demonstrations that were designed to help teachers, students, and scientists overcome the widespread learning disability that is rooted in their faulty intuition and to help them regard wetting from the perspective of lines and not areas.
We reported in 2007 the results of experiments that were designed with the objective of methodically disproving the theories of Wenzel and Cassie.1 Prior to our report, we and others2-11 had questioned and criticized these theories, pointing out that they were inconsistent with data. Our statements, however, were weak and uniformly ignored as many hundreds of papers concerning superhydrophobicity were published that citied and/or used the Wenzel and Cassie theories casually, assuming that they are valid and even “laws.” We used the direct and deliberately provocative title “How Wenzel and Cassie Were Wrong,” tried to make our points clear, strong, and definitive, and used the first paragraph and Figure 1 of the paper to explain and justify the use of the provocative title. Results from multiple experiments on multiple samples at multiple length scales were reported. Much of the data were superfluous and redundant, all of the data were predictable (if you ignored Wenzel’s and Cassie’s theories and used a simple contact line perspective), and all of it made the same point over and over again, that the Wenzel/Cassie perspective is not consistent with facts. We made the statement1 that our results, as well as others that we refer to “indicate that contact angle behavior (advancing, receding and hysteresis) is determined by interactions between the liquid and the solid at the three phase contact line alone and that the interfacial area within the contact perimeter is irrelevant” (the Wenzel and Cassie theories are based on the region that we refer to as irrelevant). Our statement has generated controversy12-15 and was recently quoted directly and vehemently criticized in this journal.16 *E-mail:
[email protected]. (1) Gao, L.; McCarthy, T. J. Langmuir 2007, 23, 3762. (2) Pease, D. C. J. Phys. Chem. 1945, 49, 107. (3) Bartell, F. E.; Shepard, J. W. J. Phys. Chem. 1953, 57, 455. (4) Extrand, C. W. Langmuir 2003, 19, 3793. (5) Chen, W.; Fadeev, A. Y.; Hsieh, M. C.; Oner, D.; Youngblood, J.; McCarthy, T. J. Langmuir 1999, 15, 3395. (6) Fadeev, A. Y.; McCarthy, T. J. Langmuir 1999, 15, 3759. (7) Youngblood, J. P.; McCarthy, T. J. Macromolecules 1999, 32, 6800. (8) Oner, D.; McCarthy, T. J. Langmuir 2000, 16, 7777. (9) Gao, L.; McCarthy, T. J. Langmuir 2006, 22, 2966. (10) Gao, L.; McCarthy, T. J. Langmuir 2006, 22, 5998. (11) Gao, L.; McCarthy, T. J. Langmuir 2006, 22, 6234. (12) McHale, G. Langmuir 2007, 23, 8200. (13) Nosonovsky, M. Langmuir 2007, 23, 9919. (14) Panchagnula, M. V.; Vendantam, S. Langmuir 2007, 23, 13242. (15) Gao, L.; McCarthy, T. J. Langmuir 2007, 23, 13243. (16) Marmur, A.; Bittoun, E. Langmuir 2009, 25, 1277.
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Our goals in this letter are three: (1) to directly address the recent criticism16 of the statement that we made (italicized above) from the perspective of the roles of experiment and theory in progressing an understanding of wetting, (2) to give our best explanation of the origin and basis of the faulty understanding of contact angle that is held by the majority of researchers using and/or studying wetting and that is perpetuated by the Wenzel and Cassie theories, and (3) to describe simple experiments (demonstrations) with results that are obvious but contrary to what would be predicted using the Wenzel and Cassie theories.
Roles and Responsibilities of Experiment and Theory We comment on the nature of the work that we reported1 (experiment-based), the nature of the criticism of this work that was recently published16 (theory-based), and we try to emphasize that contact angle analysis is an example of a situation where a little experience can upset a lot of theory.17 We do not refer to our results in quantitative terms here but refer the reader to our publication1 to gain an appreciation for the excess of consistent data that was reported. We measured advancing and receding water contact angles with differently sized water drops on surfaces containing spots of different sizes within the contact lines of the sessile drops. We also reported contact angles inside the spots. The spots were smoother, rougher, or chemically different from the field of the surfaces that surrounded them. The length scales of the spots and drops spanned the dimensions of (were both smaller and larger than) normal contact angle analysis. We used the most common commercial goniometer (originally designed by Zisman at the Naval Research Laboratory) and spanned the range of drop volume that could be conveniently analyzed with this instrument. The vast majority of contact angle values in the literature were measured using drops of a volume near the low end of this range. The larger drops that we used were flattened by gravity, and although contact angle values of drops of this size are consistent with those of smaller drops, drops of this volume (puddles) are essentially never used to measure contact angles. The length scale (17) We make this statement on the basis of the responses of hundreds of Ph.D. students, faculty, and colleagues as to the results of experiments of the sort shown in ref 1. We use the word experience in the context of the definition “the accumulation of knowledge and skill that results from direct participation.”
Published on Web 06/11/2009
DOI: 10.1021/la901416m
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of the topography used for roughness was chosen to be in the range that dramatically affects the contact angle.8 Chemistries of the two components were chosen to maximize differential wetting.18 The controversial statement recently criticized and quoted above is consistent with every single datum reported. Wenzel’s and Cassie’s theories are consistent with none of them, could not have predicted any of them, and cannot be used for any meaningful analysis of them. The data are experimental results;facts with error analysis that Wenzel’s and Cassie’s theories are inconsistent with. Our statement under criticism should not be viewed as bold or contentious and should not have been considered controversial. It is merely a straightforward conclusion made from the analysis of data that we still believe is the most appropriate first sentence for the Conclusion and Comments section of our paper. (It is the only conclusion we made, hence the singular use in the heading of this section). This is the normally accepted role of the experimental scientist. We did not offer a new theory that addresses contact line tortuousness, contact line continuity, or chemical composition fluctuations at contact lines. In fact, we do not believe that one is necessary nor do we believe that any would be useful; we think the Wenzel and Cassie theories are just fine in this regard. We made the comment that “Wenzel’s and Cassie’s equations are valid only to the extent that the structure of the contact area reflects the ground-state energies of contact lines and the transition states between them.” Perhaps we should not have included the word “only” because clearly there are many situations where the contact area reflects the contact line structure and the Wenzel and Cassie equations will be fortuitously consistent. We deliberately prepared samples where this was not the case. We stated, “We do not advocate never using Wenzel’s and Cassie’s equations.” The double negative was deliberate. We have used and will use these equations and advocate doing so. The recent criticism16 is from the perspective of theory. There are many statements made in this paper that we take objection to, and we specifically address four of them below. These four were not chosen to direct criticism at their authors but to emphasize the different vantages of experimentalists and theorists. (1) “The main problem with the above statement is that it stems from experiments performed with drops that were too small, ignoring the indications of existing theoretical understanding.” (“above statement” refers to the italics in the first paragraph above.) The drops that were used to measure contact angles were point blank not too small but, as described above, were the same size, smaller, and larger than the size used for over 99% of the sessile drop contact angle values ever reported. Many of the drops we studied were much larger than the drops that Wenzel and Cassie used. Photographs of drops (in Figures 4 and 5 of the paper1 that is criticized) show that some were so large that they were severely distorted by gravity, much larger than stable raindrops. We reported that it was necessary to “zoom out” a video camera to record images of two of the drops (Figure 4e,f of the paper1 that is criticized). We measured contact angles of puddles (cannot be described as drops) but did not report them because they were less demanding tests of the Wenzel and Cassie theories and would have diluted our arguments. Anyone with experience measuring experimental contact angles would not call these drops too small but would likely think the opposite.
(18) Fadeev, A. Y.; McCarthy, T. J. Langmuir 1999, 15, 7238.
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We admit ignorance of the “existing theoretical understanding” that we are ignoring, but it does not apply to reality: raindrops, raincoats, and everyone who measures contact angles ignore it as well. We advise continuing to ignore it if you have the goal of solving any problem involving real solid/liquid interfaces. (2) “It was rigorously proven and numerically demonstrated 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.” For many decades, researchers, analytical personnel, and production line workers have made meaningful measurements of contact angle with no knowledge of or ability to control the wavelength of surface structure (chemistry or topography). They have done this using drops significantly smaller than those criticized as being too small, often on surfaces of which they knew little concerning the composition or topography, let alone their wavelengths.19 Our criticism of the theories is one of concept, not dimensions or ratios of dimensions. Researchers should not use larger drops, not be concerned with the wavelength of surface roughness or chemical heterogeneity, and not be concerned with the ratio of drop size to wavelength. Contact angle data can and should often be analyzed using the Wenzel and Cassie equations. They will yield meaningful analyses “to the extent that the structure of the contact area reflects the ground-state energies of contact lines and the transition states between them”,1 independent of the size of the drop. It may have been “rigorously proven and numerically demonstrated” that the Wenzel and Cassie equations become valid with very large drops and small-wavelength roughness or chemical heterogeneity, but they (fortuitously) work as well as they are going to with small drops. A practical issue, apparent to people experienced in contact angle analysis, is that when the “wavelength of roughness or chemical heterogeneity” is too great, advancing and receding contact angles cannot be measured and what is known as stick-slip occurs. Increasing the drop volume does not help overcome this problem. (3) “It may be more useful (though less provocative) to explain when and why the Wenzel and Cassie equations are right.” We agree with this sentiment (not, however, the use of the word “right” to describe theories; “consistent” is the proper choice of words), had this sentiment when we wrote the paper that these authors criticize, and stated in this paper, on the basis of experimental results, the conditions under which these equations are useful and the understanding that should be had when using them. We disagree, however, that it is in any way “useful” to use theory to explain “when and why...equations are right,” particularly when it leads to useless requirements of impractical drop size and surfaces of particular structural wavelength. Experimentalists are forced to ignore these theoretical requirements because they do not apply to real analytical situations, for instance, actual raindrops or the multiple length scales of topography and leaf size of a pesticide-coated crop. If researchers were to heed the less provocative advice of this theory and use drops of the size recommended (conditions under which the Wenzel and Cassie equations may have been rigorously proven or numerically simulated to be “right”), then they would not be able to analyze many surfaces because the objects would be too small and the instruments would need to be replaced with those of different design. They would be provoked to not measure (19) We note that there is a commercially successful, portable hand-held goniometer that can be connected to a laptop and that uses very small drops. It measures advancing and receding angles using a built-in micropump that feeds 0.5 μL increments of liquid. See www.pocketgoniometer.com.
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contact angles, and hence the Wenzel and Cassie equations would not be used. For several years before we submitted the paper1 that was criticized, we wrestled with the issues (benefits and detriments) of pointing out why useful equations are wrong. After these years of “tip-toeing around” Wenzel and Cassie, thinking that they have done more good than harm, the field of superhydrophobicity was born, and the number of people affected by these equations increased exponentially. We were trying to be both responsible and useful and still believe that the provocative tone of the title will contribute to this usefulness. We do not doubt that this paper,16 which is critical of our work “may be more useful” to understanding wetting from an important-to-some theoretical perspective, but prefer to remain in reality. These authors are far from alone in clinging to their faith in Wenzel and Cassie. Likely for reasons that we discuss below, there is a consensus among researchers in the field, abundantly demonstrated by their publications and their citations to Wenzel and Cassie, that “the Wenzel and Cassie equations are right.” In fact, they are considered by some to be laws. References 20-22 and 23-25 are six recent examples of where Wenzel’s theory and Cassie’s theory, respectively, have been referred to as “Wenzel’s Law” and “Cassie’s Law.”20-25 (4) “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.” We address much of this fourth statement in our comments concerning the first three and refute the second sentence in it with demonstrations described below. The words “meaningful” and “realistic” are used although drops are required that are so large that they are sphere sections only in places with a gravitational constant much lower than the one on earth. This fourth statement purports that essentially all of the contact angle data measured on earth are meaningless measurements, including Wenzel’s and Cassie’s. It makes it clear that the “current theoretical understanding” is at odds with meaningful and realistic issues of contact angle and wetting. We close this section by emphasizing that we believe it is our role, as experimentalists who have disproved the Wenzel and Cassie theories, to advocate using them, justify our use of them, and advise using the same experimental techniques and drop sizes that for decades have given insight into surfaces with all wavelengths of structure. We performed the experiments1 that have been criticized16 because we believed that it was both important to and our role to forcefully advocate the analysis of contact angle data from the perspective of the contact line and not the perspective of the contact area. We show examples below that demonstrate this importance. The theorists who have criticized our experiments have apparently felt it their role to defend the Wenzel and Cassie theories and identify when and why they are right even when their conclusions suggest that Wenzel’s and Cassie’s experiments were not done “correctly.” It is the experimentalist’s responsibility to state that their suggestions are ridiculous and that their statements hold up to neither experiment nor common sense. From the perspective of meaningful argu(20) Kusumaatmaja, H.; Vrancken, R. J.; Bastiaansen, C. W. M.; Yeomans, J. M. Langmuir 2008, 24, 7299. (21) Patrı´ cio, P.; Pham, C.-T.; Romero-Enrique, J. M. Eur. Phys. J. 2008, E26, 97. (22) Kuo, C. S.; Tseng, Y. H.; Li, Y. Y. Chem. Lett. 2006, 35, 356. (23) Iwamatsu, M. J. Colloid Interface Sci. 2006, 297, 772. (24) Martic, G.; Blake, T. D.; De Coninck, J. Langmuir 2005, 21, 11201. (25) Mykhaylyk, T. A.; Evans, S. D.; Hamley, I. W.; Henderson, J. R. J. Chem. Phys. 2005, 122, 104902.
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ment, the logical counter to our statement would have been the design of a surface that is predicted to, or experimentally demonstrated to, exhibit contact angle data that are more consistent with the contact area structure than the contact line structure. This would be a valid criticism of our statement italicized in the first paragraph.
Origin of Faulty Intuition Another comment that we made in the Conclusion and Comments section of our paper1 concerning the Wenzel and Cassie theories is, “They support the incorrect concepts that contact area is important and interfacial free energies dictate wettability.” This faulty view is prevalent, held by most people educated in surface science and part of the instinct/intuition used by most in addressing problems in surface science. It is indeed what they were taught, but it is wrong. We believe that the principal origin of this widespread misunderstanding is 2-fold: First, from the analysis of a statement in Young’s “An Essay on the Cohesion of Fluids”26 using the perspective of thermodynamics and second, from the way surface tension has been taught (and learned) for the last century using soap bubbles, soap films, and stretched elastic membranes as models. Surface tension and surface free energy are discrete, distinct, and different quantities.27 Surface tension is a tensor that acts perpendicularly to a line on a surface and is a force per unit length (dyn/cm). Experimentally, we can understand surface tension from the perspective of the force required to start peeling a certain width of adhesive tape from a surface or the force that the contractile surface of a sessile drop makes at a contact line; the units of dyn/cm are intuitive. Surface free energy is a scalar nondirectional property of an area of a surface and is energy per unit area (erg/cm2). We understand surface energy in terms of the work required to make more surface area (bring molecules from the bulk to the surface);28 the units of erg/cm2 are intuitive. Because these quantities are mathematically equivalent at equilibrium, people have regarded them as interchangeable.29 People who have jobs designed around thermodynamics have used this equivalence both to attempt to determine useful thermodynamic quantities from contact angle data and to derive equations that may be useful in predicting or interpreting contact angle data. It is not intuitive how surface free energy relates to the contact angle or the forces at a three-phase contact line. Young’s 1804 statement26 concerns the balance of forces between what he understood as particles. Young did not know about molecular structure, covalent bonds, metallic bonding, dipoles, or hydrogen bonding. He viewed liquids and solids as collections of particles that attract one another and “produce the effect of a uniform tension of the surface.” He “assumed as consonant both to theory and observation, that the contractile force of the common surface of two substances, is proportional, (26) Young, T. Phil. Trans. R. Soc. London 1805, 95, 65. This essay has been digitized and is available at www.google.com/books . (27) Gray, V. R. Chem. Ind. 1965, 23, 969. This paper should have been cited in ref 1 and a number of our other publications, but we were unaware of it until recently. This forgotten work was last cited in 1997 (once) and before that in 1986 (once). This paper makes a mathematical argument for the necessity of distinguishing surface tension and surface energy. The points that we make here and in ref 1 were well appreciated by Gray in 1965 . (28) Squeezing a spherical drop between perfectly hydrophobic (θ = 180°)38,39 surfaces is an intuitively obvious way to picture surface free energy. The drop changes shape from a section of a sphere, increasing the surface area. Bulk molecules must move from the bulk to the surface, and work is required to do this. Figure 3 in ref 33 and Figure 3 in ref 39 show photographs of the compression of water drops using surfaces with 180° contact angles. (29) Adamson, A. W.; Gast, A. P. Physical Chemistry of Surfaces; Wiley Interscience: New York, 1997.
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other things being equal, to the difference of their densities.” His insight is stunning even after 200 years and obvious genius. Young did not confuse forces with energies and could not have. He did not know of surface free energy or of thermodynamics. Gibbs and Helmholtz were not yet born (nor was Dupre), and Carnot was 8 years old. Young did not write an equation but states clearly, using the word “force” multiple times, what can be expressed in equation form as eq 1, where FSV, FLV, and FSL are the forces Young ascribes to the “cohesion of superficial particles” at the surfaces of the solid and liquid and the common surface of the solid and liquid. F SV ¼ F LV cos θ þ F SL
ð1Þ
This equation is not derived or proven nor does it need to be; it is the simple balance of forces in a plane operating on a line. This equation is valid when the solid surface is smooth, rough, clean, dirty, homogeneous, or heterogeneous. Today we think of these forces, compare them, and balance them using dyn/cm units. Equation 2, which is most commonly called and accepted as (has become) Young’s equation, where γSV, γSL, and γLV are surface free energies, is Young’s statement from the perspective of the contact angle; however, forces have been substituted by energies. γSV - γSL γLV
cos θ ¼
ð2Þ
This confusing substitution of the directional forces at a line envisioned by Young with nondirectional interface properties (energy per unit area, erg/cm2) cannot be attributed to Young but to Wenzel. Equation 1 (Young’s statement) is no more valid because eq 2 can be derived using thermodynamic arguments. Equation 3, the Wenzel equation, is the equation incorrectly attributed to Young (eq 2) with γSV and γSL multiplied by Wenzel’s roughness coefficient R, which is defined as the ratio of the contour surface area to the projected surface area. cos θ ¼
RðγSV - γSL Þ γLV
ð3Þ
We quote directly from Gray’s neglected paper27 regarding Wenzel: “he confused surface tension and surface free energy to such an extent as to use an area roughness coefficient.” Gray goes on to state, “This confusion has been perpetuated by subsequent workers.” Cassie was one of them. Equation 4, the Cassie or Cassie-Baxter equation, is again the misrepresented statement of Young (eq 2) for a solid containing two components with different surface free energies. f1 and f2 are defined as the area fractions of the two components. cos θ ¼
f 1 ð1 γ SV -1 γ SLÞ f 2 ð2 γ SV -2 γ SLÞ þ γLV γLV
ð4Þ
Gray’s paper, which addresses the issue of confusing force with energy, may be more convincing to some than the arguments we make here. We feel that Gray, however, did not give Wenzel and Cassie sufficient credit. Their papers are insightful, profound, and demonstrably abundantly useful. We cannot provide more useful theories or more precise equations. It is, however, a fact that they have contributed to faulty intuition. We see the second aspect of the origin of faulty intuition to be the practice of teaching and learning surface science using soap bubbles, soap films, and elastic membranes as models. These are 7252 DOI: 10.1021/la901416m
no doubt useful teaching and learning tools, but care needs to be taken with their use because they can lead and have led to the belief that interfacial areas affect wetting. There are numerous examples of this teaching in the literature, and we cite four. We chose these not to be critical but because we highly recommend them as a result of the insight of their authors. Boys, in his classic “Soap Bubbles and the Forces Which Mould Them,”30 which is based on lectures given in 1889-1890, refers repeatedly to the “elastic skin” of water: “it acts as if it were an elastic skin made of something like very thin india-rubber, only that it is perfectly and absolutely elastic.” The last paragraph of his first lecture begins, “The chief result that I have endeavored to make clear today is this. The outside of a liquid acts as if it were an elastic skin, which will, as far as it is able, so mold the liquid within it that it shall be as small as possible.” This invokes the image of a water balloon to most students. In a 1969 review31 of capillarity, Schwartz states, “A liquid-fluid surface behaves like a stretched elastic membrane in that it tends to contract.” Adamson and Gast in the latest edition of the text29 that has trained surface scientists since 1960 uses the example of a soap film stretched across a wire frame with one movable side to explain how: “Although referred to as a free energy per unit area, surface tension may equally be thought of as force per unit length.” These authors favor using surface free energy over surface tension “because of its connection to thermodynamic language” and state that “the two terms are used interchangeably in this book.” de Gennes, Brochard-Wyart, and Quere on page 1 of their 2004 text32 state, “A liquid surface can be thought of as a stretched membrane characterized by a surface tension that opposes its distortion.” These authors also use the soap film model with one movable side (glass rods instead of wires) and on page 4 state, “If the frame is tilted, it is even possible for the mobile rod to climb up the incline, only to fall back down suddenly the moment the liquid membrane is pierced.” These two statements by Quere et al., taken together, clearly support the faulty intuition that events at interfaces, away from the contact line, will affect the contact angle. Figure 1 shows an image of the mechanical balance of contracting skins, stretched membranes, or elastic membranes described by these authors and envisioned by most students. (a) Liquid/vapor, (b) solid/liquid, and (c) solid/vapor tensions balance one another (d) to generate an equilibrium contact angle. Figure 1e shows the common depiction of vectors that are used to represent forces in Young’s statement. That three elastic membranes, working to minimize their areas in the configuration of Figure 1d, should form an equilibrium contact angle is intuitive to most people and is the way that they view (and have learned) Young’s equation. Absent from this image are the particles that Young envisioned26 with short-range attractive forces. Absent also from this image is the perspective of Schwartz31 who states, concerning forces, “Physically, they operate in each phase within a few molecular diameters of the other two phases. Neither the state nor the geometry of the phase interfaces in the regions remote from the line boundary has any direct effect on the contact angle.” Absent as well from the image in Figure 1d is the perspective that we had hoped the italicized statement in the first paragraph of this paper would bring. (30) Boys, C. V. Soap Bubbles and the Forces That Mould Them; Society for Promoting Christian Knowledge: London, 1896. There are a number of editions of this book, and the style, wording, and figures differ. The text quoted is from the 1896 version that has been digitized and is available online at www.google.com/ books. (31) Schwartz, A. M. Ind. Eng. Chem. 1969, 61, 10. We were unaware of this neglected paper until recently. It has not been cited by papers concerned with wetting . (32) deGennes, P.-G.; Brochard-Wyart, F.; Quere, D. Capillarity and Wetting Phenomena; Springer: New York, 2004.
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Figure 1. (a) Liquid-vapor, (b) liquid-solid, and (c) solid-vapor elastic membranes (d) in mechanical equilibrium at a contact line. (e) The vectors generally used to represent the forces in Young’s statement.
It is not difficult to envision why people without these perspectives would use the Wenzel and Cassie theories (particularly if they learned them as laws), believing that the area roughness and relative area fractions affect contact angle. Nor is it difficult to envision these people believing that they can determine surface roughness or area fractions from contact angle data. Without these perspectives, one might believe that piercing, perforating, roughening, or chemically changing the solid-liquid interface under a sessile drop (Figure 2a,b) would change the contact angle. Using this faulty logic, the contact angle should also change if either of the other interfaces is adulterated (Figure 2b,c).
Figure 2. Wenzel and Cassie theories suggest that the contact angle defined by (a) interfacial free energies would change upon adulterating the (b) liquid-solid, (c) liquid-vapor, or (d) solidvapor interfaces.
Demonstrations The experiments that we reported in the recently criticized paper1 were our calculated best efforts at disproving the Wenzel and Cassie theories. We describe here four experiments, more accurately called demonstrations, that more graphically remake the point of the italicized statement in the first paragraph above, however with much less experimental rigor or quantitative detail. The results of these demonstrations should be obvious, predictable, and intuitive. Our intentions are to adjust people’s instinct concerning wetting to one of the lines rather than areas and help them purge themselves of their faith in Wenzel and Cassie. Demonstration 1. Figure 3a is a photograph of 19 brass Phillips head screws countersunk in a 9 12 1.3 cm block of poly(tetrafluoroethylene) (PTFE). This PTFE sample exhibited contact angles of θA/θR = 113°/89° (measured well away from the screws), and the tops of the screws exhibited contact angles of θA/θR=∼25°/
