pubs.acs.org/Langmuir © 2009 American Chemical Society
A Variational Approach to Wetting of Composite Surfaces: Is Wetting of Composite Surfaces a One-Dimensional or Two-Dimensional Phenomenon? Edward Bormashenko Ariel University Center of Samaria, Applied Physics Department, Department of Chemistry and Biotechnology Engineering, POB 3, 40700, Ariel, Israel Received July 8, 2009. Revised Manuscript Received August 2, 2009 Accurate treatment of the variational problem of wetting a composite surface clarifies the physical character of wetting. It is shown that only the area adjacent to a triple line exerts an influence on the apparent contact angle. This conclusion is true for flat composite surfaces and also for rough surfaces governed by Cassie and Wenzel wetting regimes. A surface density of defects in the vicinity of the triple line dictates the apparent contact angle.
1. Introduction The wetting of composite surfaces has been subjected to intensive and hot scientific discussion, started recently by Gao and McCarthy in their paper, entitled “How Wenzel and Cassie Were Wrong”. The discussion was concentrated on the question, Is the wetting of a composite surface a one-dimensional (1D) or two-dimensional (2D) affair?1-6 Physics consists of measurements, formulas and “words”.7 Thus, accurate wording is as important as carefully performed experiments and correct formulas. What do we mean when the 1D or 2D nature of wetting is discussed? Actually, two very different questions are often mixed, i.e., (1) Is an apparent contact angle (APCA) governed by a linear or surface fraction of defects on rough surfaces?8 (2) Is the APCA governed by the entire surface underneath a drop, or it is dictated by the area adjacent to the triple (three-phase) line?1-6 Accurate variational treatment of the problem allows separating these questions and leads to the conclusion that APCAs are influenced by the three-phase adjacent area only. This conclusion is supported by recent experimental findings.9-14 The same variational treatment eliminates the contradiction between the force- and energy-based approaches, which, as might appear at first sight, lead to different results. Indeed, the analysis of forces acting on the three-phase line supports the 1D scenario of wetting, whereas the thermodynamic calculation takes into account the entire surface underneath the drop.15,16 I will show here that an
accurate thermodynamic calculation clears up the apparent contradiction.
2. Results and Discussion I treat the problem of the wetting of a composite surface as a variational problem with a free boundary.17-19 For the sake of simplicity, I start with the wetting problem where the cylindrical drop is under discussion (the cross-section of the drop is presented in Figure 1). In the most general case when the “gradient surface” is treated,20 the free energy per unit length of the cylindrical drop could be written as 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2 dh 4γ 1 þ þ γSL ðxÞ - γSA ðxÞ þ UðhÞ5dx Gðh, h Þ ¼ dx -a 0
Z
a
ð1Þ where h(x) is the height of the liquid surface above the substrate, γ, γSL(x), and γSA(x) are the surface tensions at the liquid/air, solid/liquid, and solid/air (vapor) interfaces, respectively, and U(h) is the external field (say, gravity). Contrasting with the analysis performed in ref 17, I now treat the general case, i.e., where the substrate is inhomogeneous. Condition 2 of the constant area S also has to be taken into account: Z S ¼
(1) Gao, L.; McCarthy, T. J. Langmuir 2007, 23, 3762–3765. (2) Gao, L.; McCarthy, T. J. Langmuir 2009, 25, 7249–7255. (3) McHale, G. Langmuir 2007, 23, 8200–8205. (4) Panchagnula, M. V.; Vedantam, S. Langmuir 2007, 23, 13242–13242. (5) Nosonovsky, M. Langmuir 2007, 23, 9919–9920. (6) Marmur, A Langmuir 2009, 25, 1277–1281. (7) Duff, M. L.; Okun, L. B.; Veneziano, G. J. High Energy Phys. 2002, 3, 023. (8) Extrand, C. W. Langmuir 2002, 18, 7991–7999. (9) Extrand, C. W. Langmuir 2003, 19, 3793–3796. (10) Extrand, C. W. Langmuir 2004, 20, 5013–5018. (11) Tadmor, R. Langmuir 2004, 20, 7659–7664. (12) Bormashenko, E.; Pogreb, R.; Stein, T.; Whyman, G.; Erlich, M.; Musin, A.; Machavariani, V.; Aurbach, D. Phys. Chem. Chem. Phys. 2008, 27, 4056–4061. (13) Bormashenko, E.; Pogreb, R.; Whyman, G.; Bormashenko, Ye.; Erlich, M. Langmuir 2007, 23, 6501–6503. (14) Bormashenko, E.; Pogreb, R.; Whyman, G.; Erlich, M. Langmuir 2007, 23, 12217–12221. (15) Yamamoto, K.; Ogata, S. J. Colloid Interface Sci. 2008, 326, 471–477. (16) Whyman, G.; Bormashenko, E.; Stein, T. Chem. Phys. Lett. 2008, 450, 355– 359.
Langmuir 2009, 25(18), 10451–10454
a -a
hðxÞdx ¼ const
ð2Þ
which is equivalent to the constant volume requirement in the case of cylindrical “drops”. Equations 1 and 2 reduce the problem to minimization of the functional: Z G ¼
a -a
~ h0 Þdx Gðh,
ð3Þ
(17) Bormashenko, E. Colloids Surf., A 2009, 345, 163–165. (18) Gelfand, I. N.; Fomin, S. V. Calculus of Variations; Dover: New York, 2000. (19) Marmur, A. Colloids Surf., A 1998, 136, 81–88. (20) McHale, G.; Elliott, S. J.; Newton, M. I. Shirtcliffe, N. J. Superhydrophobicity: Localized parameters and gradient surfaces. In Contact Angle Wettability and Adhesion; Mittal, K. L., Ed.; Koninlijke: Brill, The Netherlands, Vol. 6, 2009; pp 219-233.
Published on Web 08/12/2009
DOI: 10.1021/la902458t
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Figure 1. The cross-section of the cylindrical drop deposited on a gradient surface.
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G~ ¼ γ 1 þ h0 2 þ γSL ðxÞ - γSA ðxÞ þ UðhÞ þ λh
ð4Þ
where λ is the Lagrange multiplier to be deduced from eq 2. The end points are free to move; thus, the transversality condition at the end point a yields ðG~ - h0 G~h0 Þx ¼a ¼ 0
ð5Þ
~ h0 denotes the h0 derivative of G. ~ Substitution of formula 4 where G into the transversality condition 5 and taking into account h(a) = 0, U(h = 0) = 0 gives rise to ! pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 02 γh ¼ 0 ð6Þ γ 1 þ h0 2 þ γSL ðxÞ - γSA ðxÞ - pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ h0 2 x ¼a Simple transformations yield ! 1 γ ðaÞ - γSL ðaÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ SA 2 γ 0 1 þ h x ¼a
γSA ðaÞ - γSL ðaÞ γ
G ¼ 0
ð8Þ
a
þ 2π b
ðγ1SL - γ1SA Þxdx
þ πb
2
ðγ2SL - γ2SA Þ
ð9Þ
where U(h, x) describes the external field, superscripts 1 and 2 are related to the substrate and spot, respectively (see Figure 1), and the profile of the droplet h(x) is assumed to be a single-valued and even function. It has to be stressed that the end points are free to move along axis x, whereas the radius of the spot b is fixed. Thus, it is clear that the second term in eq 9 is a variable, whereas the third one is constant and could be omitted or redefined. The constant energy has no physical manifestation; only energy changes are important. Without loss of generality, we can shift 10452 DOI: 10.1021/la902458t
2
Z
a
b
ðγ1SL - γ1SA Þxdx þ πb2 ðγ1SL - γ1SA Þ
3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi dh 42πγx 1 þ þ Uðh, xÞ þ 2πðγ1SL - γ1SA Þxdx5 dx ð10Þ
It is clear that the free energy variation of the droplet deposited on the composite substrate equals the variation of free energy of the droplet deposited on the flat substrate: δG = δG ; however, expression 10 is much more convenient for mathematical treatment, and it allows immediate application of transversality conditions for a variational problem with free end points.17-19 The conservation of volume yields Z a 2πxhðxÞdx ¼ λ ¼ const: ð11Þ V ¼ 0
Equations 10 and 11 reduce the problem to minimization of the functional: Z a ~ h0 , xÞdx G ¼ ð12Þ Gðh, 0
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ h0 , xÞ ¼ 2πγ 1 þ h0 2 þ 2πðγ1 - γ1 Þx Gðh, SL SA þ Uðx, hÞ þ 2πλxh
3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi dh 42πγx 1 þ þ Uðh, xÞ5dx dx Z
a
ð7Þ
2 a
Z þ 2π
0
We conclude that only the values of surface tensions at the end points govern the contact angle, and the dependencies γSL(x) and γSA(x) have no influence on it. Now I consider the two-component composite flat surface introduced by Gao and McCarthy and comprising the round spot of the radius b (i.e., chemical heterogeneity); this particular case has given rise to hot debates lately.1-6 I consider the liquid drop of the radius a deposited on this surface in the axisymmetric way depicted in Figure 2. The free energy of the drop is given by eq 1: Z
the zero level of the free energy of the droplet, and the free energy could be redefined as follows: 2 3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi Z a dh 42πγx 1 þ þ Uðh, xÞ5dx G ¼ dx 0
¼
Taking into account h0 (x = a) = -tan θ, where θ is the equilibrium (Young) contact angle, immediately gives cos θ ¼
Figure 2. The drop deposited on the composite surface, comprising a “spot” with the radius b.
ð12aÞ
where λ is the Lagrange multiplier to be calculated from eq 11. For the variational problem with free end points, the transversality condition 5 holds. Substitution of formula 12a into the transversality condition 5, taking into account h(a) = 0, U(a,h(a)) = 0, and h0 (x = a) = -tan θ, where tan θ is the slope of the liquid-air interface at x = a, gives rise to the well-known Young equation (for details, see ref 11): cos θ ¼
γ1SA - γ1SL γ
ð13Þ
It is clear that the spot has no influence on the contact angle, and the discrepancy with the force-based approach is avoided. Langmuir 2009, 25(18), 10451–10454
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Letter
Figure 3. Composite Cassie-like surfaces of different kinds.
The external field U = U(h, x) also does not exert influence on the contact angle. Now the most delicate point has to be considered. All our treatment is valid when δx , a - b, namely, the boundary is far from the spot, and it could be moved freely. The question is: what is the precise meaning of the expression “far from the spot”? From the physical point of view it means that the macroscopic approach is valid when a three-phase line is displaced, namely, a b g 100 nm; when this condition is fulfilled, particles located on the triple line do not “feel” the spot, i.e., the long-range force influence is negligible.21 It should be stressed that APCA is essentially macroscopic notion; hence, all our discussion holds the macroscopic approach. Obviously, physical and rigorous mathematical approaches are different in this case.5 Now I discuss more complicated composite Cassie-like surfaces such as those depicted in Figure 3, when a solid substrate comprises two species of solids, characterized by various γSL.21-23 It is important that there is no general approach to the Cassie-like wetting. It has been already well understood by Johnson and Dettre that Figure 3A,C demonstrates very different kinds of surface heterogeneity.24 When the droplet is deposited axisymetrically onto the composite surface depicted in Figure 3A, the 2D scenario of wetting occurs independently of the heterogeneity scale. The three-phase line, when displaced, covers both species of solids, and the transversality conditions for appropriate variational problem yield the well-known Cassie-Baxter equation:21-23
radius of the drop. If δr is ∼100 nm and less, the displacement of the boundary will cover both kinds of solid species, and eq 14 will work, i.e., a 2D scenario of wetting takes place. It could be recognized that the linear fraction of species is irrelevant in this case. Actually, more complicated heterogeneities than that presented in Figure 3 occur in nature and engineering, and a fresh look at the validity of the Cassie equation is necessary for specific composite surfaces.8 It is also noteworthy that the extension of the Cassie-Baxter eq 14 to surfaces comprising air pockets is not trivial.25 The Wenzel-like wetting is analyzed in a similar way. Figure 4 depicts a drop deposited on a composite surface characterized by a variable roughness; the roughness of the central spot with the radius b equals Rf2, whereas the roughness of the area adjacent to the triple line equals Rf1 (see Figure 4). The free energy of the drop is given by 2 3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi Z a 42πγx 1þ dh þ Uðh, xÞ5dx þ 2πRf1 G ¼ dx 0
cos θ ¼ f1 cos θ1 þ f2 cos θ2
The last term in formula 15 is constant, and transformations akin to expression 10 yield
ð14Þ
where f1 and f2 denote the fractional surfaces occupied by each one of the species, θ1 and θ2 are the Young angles inherent for the species, and θ* is the APCA.21-23 It should be stressed that, again, only the area adjacent to the triple line governs the APCA. If the Cassie-like surface comprises the central spot depicted in Figure 3B, the spot far spaced from the triple will have no influence on the APCA. It should be mentioned that, for the surfaces displayed in Figure 3A,B the surface and linear fractions (as measured along the three-phase line) occupied by species coincide. The situation on the composite surface depicted in Figure 3C is much more complicated. Inner and outer stripes located far from the three-phase line do not exert an impact on the APCA. What about the stripes close to the triple line? If the characteristic scale δr is much greater than 100 nm (see Figure 3C), the Cassie-Baxter equation fails, because the displacement of the boundary in the variational problem will cover only one kind of species, which will dictate the APCA. In this case, everything depends on the initial
Figure 4. Wenzel-like wetting of the composite substrate.
Z
a b
G ¼
ðγSL - γSA Þxdx þ Rf2 πb2 ðγSL - γSA Þ
2
Z
a
¼ 0
3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi dh 42πγx 1þ þ Uðh, xÞ þ 2πRf1 ðγSL - γSA Þxdx5 dx ð16Þ
~ introduced according The conservation of volume 11 gives G, to formula 17: pffiffiffiffiffiffiffiffiffiffiffiffiffi ~ h0 , xÞ ¼ 2πγ 1þ h0 2 þ 2πRf1 ðγSL - γSA Þx Gðh, þ Uðx, hÞ þ 2πλxh
Langmuir 2009, 25(18), 10451–10454
ð17Þ
Substitution of 17 into the transversality condition 5, taking into account h(a) = 0, U(a,h(a)) = 0, and h0 (x = a) = -tan θ* gives cos θ ¼ Rf1
(21) de Gennes, P. G.; Brochard-Wyart, F.; Quere, D. Capillarity and Wetting Phenomena; Springer: Berlin, 2003. (22) Cassie, A. B. D.; Baxter, S. Trans. Faraday Soc. 1944, 40, 546–551. (23) Cassie, A. B. D. Discuss. Faraday Soc. 1948, 3, 11–16. (24) Johnson, R. E.; Dettre, R. H. J. Phys. Chem. 1964, 68, 1744–1750.
ð15Þ
γSA - γSL ¼ Rf1 cos θ γ
ð18Þ
(25) Bormashenko, E. Colloids Surf., A 2008, 324, 47–50.
DOI: 10.1021/la902458t
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Again, when a - b g 100 nm, only roughness in the area adjacent to the triple line dictates the APCAs. Both the Cassie and Wenzel APCAs are insensitive to external fields U = U(h, x).
Conclusions This letter demonstrates that physical and chemical heterogeneities located far from the three-phase line have no influence on the APCA for both Cassie and Wenzel wetting regimes. This is also true for composite flat surfaces. On the heterogeneous surfaces, APCAs are governed by a surface density of defects
10454 DOI: 10.1021/la902458t
located in the vicinity of the three-phase (triple) line. Young and apparent contact angles are insensitive to external fields. Acknowledgment. The author is grateful to Dr. G. Whyman, Professor O. Gendelman, and Professor A. Marmur for fruitful discussions. The author is thankful to Professor Gr. Kresin for his inestimable help in variational analysis. The author is indebted to anonymous reviewers for extremely instructive remarks. The author is thankful to Mrs. Yelena Bormashenko and Mrs. Albina Musin for their help in preparing this manuscript.
Langmuir 2009, 25(18), 10451–10454
