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Comments Comment on How Wenzel and Cassie Were Wrong by Gao and McCarthy
In a recent article, Gao and McCarthy1 explored the possibility that the theories due to Wenzel2 and Cassie3 may be more restrictive than previously envisioned or even being utilized today. The importance of the behavior in the vicinity of the three-phase contact line has been acknowledged in the literature.4-6 By this comment, we wish to delineate the progression of thought in Cassie’s3,7 work to demonstrate that his theory is not in contradiction with the experiments of Extrand8 and Bartell and Shepard.9 In addition, we wish to demonstrate that surface energy minimization arguments that form the basis of Cassie theory are essentially analogous to contact line kinetics in the absence of hysteresis. Cassie’s3 analysis of drop contact angles on a general heterogeneous surface consisting of two materials is based on the energy minimization principle. Beginning his derivation, he writes,3 “If a unit geometrical area of a surface has an actual surface area σ1 of contact angle θ1 and an area σ2 of contact angle θ2, the energy E gained when the liquid spreads over the unit geometrical area is E ) σ1(γS1A - γS1L) + σ2(γS2A - γS2L), where γS1A and γS1L are interfacial solid-air and solid-liquid tensions for the σ1 areas, and γS2A and γS2L are interfacial solidair and solid-liquid tensions for the σ2 areas.” For a smooth heterogeneous surface, σ1 + σ2 ) 1. It must be noted that E in the above statement refers to the net change δE in the system free energy during the advancing event (and not the absolute value of the drop free energy), given by the difference between the energy σ1γS1A + σ2γS2A which “is gained by the destruction of the solid-air interfacial area”, and the energy σ1γS1L + σ2γS2L which “is expended in forming the solid-liquid interface over the same area”.7 The well-known Cassie equation for a heterogeneous surface follows from this statement. The contact angle predicted by this equation requires the surface area fractions, σ1 and σ2, of the solid heterogeneous surface that is about to be wetted by the advancing three-phase contact line as inputs and not those of the entire drop footprint. * To whom correspondence should be addressed. Telephone: (931) 372 6143. Fax: (931) 372 6340. E-mail:
[email protected]. † Tennessee Technological University. ‡ National University of Singapore. (1) Gao, L.; McCarthy, T. J. Langmuir 2007, 23, 3762-3765. (2) Wenzel, R. N. Ind. Eng. Chem. 1936, 28 (8), 988-994. (3) Cassie, A. B. D. Discuss. Faraday Soc. 1948, 3, 11. (4) Nosonovsky, M. Langmuir 2007, 23, 9919-9920. (5) de Gennes, P. G. ReV. Mod. Phys. 1985, 57, 827. (6) McHale, G. Langmuir 2007, 23 (15), 8200-8205. (7) Cassie, A. B. D.; Baxter, S. Trans. Faraday Soc. 1944, 40, 546-551. (8) Extrand, C. W. Langmuir 2003, 19 (9), 3793-3796. (9) Bartell, F. E.; Shepard, J. W. J. Phys. Chem. 1953, 57, 455. (10) Vedantam, S.; Panchagnula, M. V. Phys. ReV. Lett. 2007, 99, 176102.
In their article, the authors cite experiments from Extrand8 and Bartell and Shepard9 to suggest that Cassie theory predictions based on the overall surface area fractions of the component materials “are meaningless” and that it is due to a fault with Cassie theory. Let us consider the case of the experiments presented by Extrand8 with an island of etched Perfluoroalkoxy (PFA) (material 2) on a smooth PFA (material 1) surface. (Similar arguments can also developed to explain the experiments of Bartell and Shepard.9) When the contact line is wholly resident on the smooth PFA surface (with the etched PFA island entirely under the drop footprint) and acquires an incremental unit geometrical wetted area by advancement, all of the area acquired by the drop footprint is of material PFA. The net energy change for this event is E ) γS1A - γS1L. Comparing this to the general equation for E from Cassie,3 σ1 ) 1 and σ2 ) 0. When these values are used, the Cassie equation yields the expected contact angle which is the advancing contact angle of smooth PFA. In contrast, an inappropriate choice of surface area fractions for this case would be based on the total geometric surface area fractions under the drop, given by σ2 ) (area of etched PFA)/(total solid-liquid interfacial area) and σ1 ) 1 σ2 (as in Table 1 in ref 1). The disagreement between Cassie theory and Extrand’s8 experiment does not arise from a fault with Cassie theory but from an incorrect choice of surface area fractions. As a general rule of how the Cassie equation can be used, we would like to propose that, in keeping with the intent of the energy minimization principle, surface area fractions that the contact line will experience as it advances need to be used with the Cassie equation. In instances where agreement with the Cassie equation has been reported, it arises from the fact that the contact line experiences the same area fractions as the overall surface area fractions.10 The authors write, “We do not advocate never using Wenzel’s or Cassie’s equations, but they should be used with the knowledge of their faults.” We wish to point out that a correct application of Cassie’s equation in terms of area fractions in the neighborhood of the advancing three-phase contact line yields the desired results. Mahesh V. Panchagnula*,† and Srikanth Vedantam‡
Department of Mechanical Engineering, Tennessee Technological UniVersity, CookeVille, Tennessee 38501, and Department of Mechanical Engineering, National UniVersity of Singapore, Singapore 117576 ReceiVed July 21, 2007 In Final Form: September 7, 2007
10.1021/la7022117 CCC: $37.00 © 2007 American Chemical Society Published on Web 11/15/2007
LA7022117
