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2009, 113, 17275–17277 Published on Web 09/11/2009
Wetting of Flat and Rough Curved Surfaces Edward Bormashenko Applied Physics Department, Ariel UniVersity Center of Samaria, POB 3, 40700 Ariel, Israel ReceiVed: June 4, 2009; ReVised Manuscript ReceiVed: July 2, 2009
Wetting of flat and rough curved surfaces is discussed within the general framework of the variational approach. Explicit expressions for Young’s and apparent contact angles are obtained by the use of the transversality conditions imposed on the appropriate variational problem. The Young, Wenzel, Cassie-Baxter, and Boruvka-Neumann equations are justified from this point of view. The redefined Young’s and apparent contact angles are insensitive to external fields. 1. Introduction The wetting of solid surfaces is ubiquitous and a technologically important phenomenon. The foundations of interface science were laid by Thomas Young, “When the attraction of the particles of a fluid for a solid is less than their attraction for each other, there will be an equilibrium of the superficial forces, if the surface of the fluid makes with that of the solid a certain angle.”1 Since 1805, the notion of the equilibrium (or Young’s) contact angle has served as a natural characteristic of wetting.2-7 Despite the problems connected with the experimental establishment of the Young angle due to the effect of hysteresis, it remains useful for describing wetting-related phenomena.6,7 The famous Young’s equation predicting equilibrium contact angles was grounded only recently on thermodynamic arguments.8-11 The same is true for the extensions of the Young’s equation for rough substrates, known as the Wenzel and Cassie-Baxter equations, predicting so-called apparent contact angles.8-13 The variational analysis supplies an effective mathematical framework for the treatment of wetting problems.14-17 I demonstrate that a certain modification of the variational analysis, that is, the application of “transversality conditions”, used first by Marmur for analysis of wetting, allows the effective treatment of complicated problems when the wetting of rough and curved surfaces is analyzed.18,19 2. Results and Discussion 2.1. Wetting of Curved Surfaces (the 2D Problem). For the sake of simplicity, I start with the 2D wetting problem, where a cylindrical drop extended uniformly in the y direction is under discussion (Figure 1 depicts the cross section of such a drop). I consider the symmetrical liquid drop deposited on the curved solid substrate described by the given function f(x) and exerted to some external potential U(x,h). Thus, the free energy of the drop in this general case is given by the integral11,14-16
Φ(h, h') )
∫-aa [γ√1 + (dh/dx)2 + (γSL - γSA)√1 + (df/dx)2 + U(x, h)]dx (1) 10.1021/jp905237v CCC: $40.75
Figure 1. Scheme of the section of a cylindrical drop deposited on a curved substrate.
where h(x) is the local height of the liquid surface above the point x of the substrate and γ, γSL, and γSA are the surface tensions at the liquid/air, solid/liquid, and solid/air (vapor) interfaces, respectively (the profile of the droplet h(x) is assumed to be a single-valued and even function). The condition in eq 2 of the constant area S has also to be taken into account
S)
∫-aa [h(x) - f(x)]dx ) const
(2)
Note that this is equivalent to the constant volume requirement in the case of cylindrical “drops” (extended in the y direction; h is independent of y). Equations 1 and 2 reduce the problem to minimization of the functional
Φ)
∫-aa Φ˜ (h, h')dx
(3)
˜ ) γ√1 + h'2 + (γSL - γSA)√1 + f'2 + Φ U(x, h) + λ(h - f) (4) where λ is the Lagrange multiplier to be deduced from eq 2. I do not analyze the shape of the drop but focus on effects occurring at the three-phase line (“triple line”). It has to be emphasized that the variational problem with free end points is solved, that is, it is suggested now that the endpoints of the drop x ) a, -a are not fixed. Thus, the conditions of transversality have to be considered.20 Let us suppose that the end points are free to move along the line f(x). Without the loss of the generality, we suggest that the curve f(x) and the entire 2009 American Chemical Society
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J. Phys. Chem. C, Vol. 113, No. 40, 2009
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Figure 2. The Wenzel-like wetting of a curved surface.
Figure 3. The Cassie-like wetting of a curved surface.
problem are symmetrical around the vertical axis. Thus, the transversality condition at the end point a yields20
where tan θ* ) -h′(a) and θ˜ - θ* is the redefined apparent contact angle. Equation 10 reminds us of the well-known Wenzel formula.13 Let us suppose that the surface under the drop is flat but consists of n sorts of materials randomly distributed over the substrate (see Figure 3). This corresponds to assumptions of the Cassie-Baxter wetting model.12 Each material is characterized by its own surface tension coefficients γi,SL and γi,SA, and by the fraction Ri in the substrate surface, R1 + R2 + ... + Rn ) 1. Analogously to the above treatment, we simply put into eq 4
˜ +Φ ˜ h'′ (f' - h')]x)a ) 0 [Φ
(5)
˜ h′′ denotes the h′ derivative of Φ ˜ . Substitution of eq 4 where Φ into the transversality condition (eq 5) and taking into account h(a) ) f(a) and U(a,h(a)) ) 0 gives rise to
[
γ√1 + h'2 + (γSL - γSA)√1 + f'2 +
γh'(f' - h')
√1 + h'2
]
)0 x)a
(6)
˜ ) γ√1 + h'2 + (√1 + f'2) Φ
n
∑ Ri(γi,SL - γi,SA) + 1
U(x, h) + λ(h - f) (11)
Simple transformations yield
[√ γ
1 + h'f' 1 + h'2
]
+ (γSL - γSA)√1 + f'2
)0
(7)
x)a
Taking into account h′(x ) a) ) -tan θ˜ , where tan θ˜ is the slope of the liquid-air interface at x ) a, and f ′(x ) a) ) -tan θ, where tan θ is the slope of the solid substrate in x ) a, (θ˜ < π/2) immediately gives
cos(θ˜ - θ) )
γSA - γSL γ
(8)
The well-known Young’s equation is recognized. It is reasonable to define the equilibrium (Young’s) contact angle as θ˜ - θ. The redefined Young’s angle is insensitive to an external field, meeting the conditions: U ) U(x,h), U * U(h′), and U(a,h(a)) ) 0. 2.2. Wetting of Rough Curved Surfaces. Now consider, following Wenzel (ref 13), the curved rough surface (Figure 2) characterized by the roughness Rf > 1, that is, by the ratio of the length of the real curve in contact with liquid to the lenght of the arc f(x) confined between -a and a (remember that the droplet is cylindrical). This means that the area of the liquid-solid interface is increased, and instead of eq 4, we have ˜ the minimized functional Φ
˜ ) γ√1 + h'2 + Rf(γSL - γSA)√1 + f'2 + Φ U(x, h) + λ(h - f) (9) The transversality condition (eq 5), transformations akin to eqs 6 and 7, and taking into accounth(a) ) f(a) and U(a,h(a)) ) 0 yield
cos(θ˜ - θ*) ) Rf
γSA - γSL γ
(10)
The transversality condition (eq 5) and the transformations similar to eqs 6 and 7 give rise to the corrected Cassie-Baxter apparent contact angle θ˜ - θ* n
cos(θ˜ - θ*) )
∑ Ri(γi,SA - γi,SL) 1
γ
(12)
It is noteworthy that apparent contact angles are also insensitive to the external fields, satisfying the demands defined above. The use of the notion “apparent contact angle” (as well as the use of the Wenzel and Cassie-Baxter approximations) is justified on the scale much larger than the characteristic scale of surface roughness.21-23 2.3. 3D Curved Surfaces. For the sake of simplicity, I will suppose that in the 3D case, both the droplet and the solid substrate are symmetrical around the vertical axis; thus, the free energy functional Φ to be minimized assumes the form
Φ)
∫0a Φ˜ (h, h', x)dx
(13)
˜ (h, h', x) ) 2πγx√1 + h'2 + 2πx√1 + f'2(γSL - γSA) + Φ U(x, h) + 2πλx(h - f) (14) (λ is the Lagrange multiplier). Substitution of eq 14 into the transversality condition (eq 5) immediately gives rise to the Young’s eq 8. Taking into account the roughness of the substrate yields the Wenzel and Cassie-Baxter equations, as it was demonstrated in the previous section. 2.4. Taking into Account the Line Tension. Now let us consider the effect of the elasticity of the triple line.2 The elasticity of the triple line results in the linear tension coefficient ˜ (h,h′,x) assumes the form Γ.2,24,25 The expression Φ
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J. Phys. Chem. C, Vol. 113, No. 40, 2009 17277
˜ (h, h', x) ) 2πγx√1 + h'2 + 2πx√1 + f'2(γSL - γSA) + Φ U(x, h) + 2πλx(h - f) + 2πΓ (15)
Professor Gr. Kresin for his inestimable help in variational analysis. The author is thankful to Professor A. Marmur for helpful discussions. The author is thankful to Mrs. Yelena Bormashenko for her help in preparing this manuscript.
Substitution of eq 15 into the transversality condition (eq 5) yields
References and Notes
cos(θ˜ - θ) )
γSA - γSL Γ γ γa
(16)
Equation 16 represents the known Boruvka-Neumann formula considering the effect of the linear tension.24,25 Actually, accurate introduction of “line tension” for curved surfaces is connected with additional problems (the line tension depends on the contact angle and on the curvature of the surface), neglected in this paper and discussed in ref 26. I also neglected mathematical problems connected with a rigorous definition of the cases when the three-phase line is free to move along a solid/air interface. This definition is not trivial for inhomogeneous surfaces and will be cleared up in a future investigation. 3. Conclusions The paper generalizes results reported by the author to the case of unpinned drops deposited on the flat and rough curved solid substrates. The transversality conditions imposed on the variational problem of minimizing the free-energy functional yield the known Young’s, Boruvka-Neumann, Wenzel, and Cassie-Baxter equations. The redefined Young’s, Wenzel, and Cassie-Baxter contact angles are independent of external fields of a quite general form exerted on the droplet. Acknowledgment. The author is grateful to Dr. G. Whyman for extremely fruitful discussions. The author is thankful to
(1) Young, Th. Philos. Trans. R. Soc. London 1805, 95, 65–87. (2) de Gennes, P. G.; Brochard-Wyart, F.; Que´re´ D. Capillarity and Wetting Phenomena; Springer: Berlin, Germany, 2003. (3) Erbil, H. Y. Surface Chemistry of Solid and Liquid Interfaces; Blackwell Publishing: Oxford, U.K., 2006. (4) Marmur, A. Soft Matter 2006, 2, 12–17. (5) Nosonovsky, M. J. Chem. Phys. 2007, 126, 224701. (6) Tadmor, R.; Yadav, P. S. J. Colloid Interface Sci. 2008, 317, 241– 246. (7) Chibowski, E. AdV. Colloid Interface Sci. 2007, 133, 51–59. (8) Bico, J.; Thiele, U.; Que´re´, D. Colloids Surf., A 2002, 206, 41–46. (9) Letellier, P.; Mayaffre, Al.; Turmine, M J. Colloid Interface Sci. 2007, 314, 604–614. (10) Whyman, G.; Bormashenko, Ed.; Stein, T. Chem. Phys. Lett. 2008, 450, 355–359. (11) Bormashenko, Ed. Colloids Surf., A 2009, 345, 163–165. (12) Cassie, A. B. D. Discuss. Faraday Soc. 1948, 3, 11–16. (13) Wenzel, R. N. Ind. Eng. Chem. 1936, 28, 988–994. (14) Herminghaus, St.; Brochard, Fr. C. R. Phys. 2006, 7, 1073–1081. (15) Wolansky, G.; Marmur, A. Langmuir 1998, 14, 5292–5297. (16) Berim, G. O.; Ruckenstein, E. J. Phys. Chem. B 2004, 108, 19330– 19338. (17) Minkov, E.; Novick-Cohen, A. Eur. J. Appl. Math. 2001, 12, 367– 393. (18) Marmur, A. Colloids Surf., A 1998, 136, 81–88. (19) Marmur, A. Langmuir 1996, 12, 5704–5708. (20) Gelfand, I. M.; Fomin, S. V. Calculus of Variations; Dover: Mineola, NY, 2000. (21) Marmur, A.; Bittoun, E. Langmuir 2009, 25, 1277–1281. (22) Bormashenko, E. Colloids Surf., A 2008, 324, 47–50. (23) Nosonovsky, M. Langmuir 2007, 23, 9919–9920. (24) Pompe, T.; Herminghaus, S. Phys. ReV. Lett. 2000, 85 (9), 1930– 1933. (25) Boruvka, L.; Neumann, A. W. J. Chem. Phys. 1977, 66, 5464. (26) Marmur, A.; Krasovitski, B. Langmuir 2002, 18, 8919–8923.
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