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Langmuir 2006, 22, 1400-1402
Underwater Superhydrophobicity: Theoretical Feasibility Abraham Marmur* Department of Chemical Engineering, Technion, Israel Institute of Technology, 32000 Haifa, Israel ReceiVed October 18, 2005. In Final Form: December 9, 2005 The possibility of underwater superhydrophobicity is theoretically analyzed. Thermodynamic equilibrium and stability conditions are formulated, and the design goal is defined as minimizing the solid-liquid contact area. It is shown that for sufficiently high roughness ratios, underwater superhydrophobicity may be feasible and thermodynamically stable. In addition, some generic design optimization considerations are demonstrated.
Introduction The phenomenon of superhydrophobicity and its possible applications have recently attracted much attention.1-16 Many attempts have been made to design surfaces that mimic the Lotus leaf effect, from which water drops completely roll off under a very low driving force (such as created by a small inclination angle in a gravitational field). Possible applications include, for example, easy drop transport on nanochips or self-cleaning paints and windows. A hydrophobic surface can be turned into a superhydrophobic one if its surface is roughened in a proper way, which is not yet completely understood.1,3 Wetting on rough surfaces may assume either of two regimes: homogeneous wetting,17 where the liquid completely penetrates into the roughness grooves (Figure 1a), or heterogeneous wetting,18 where air (or another fluid) is trapped underneath the liquid, inside the roughness grooves (Figure 1b). The transition between these regimes plays a major role in superhydrophobicity,3,19 since it is the heterogeneous wetting regime that minimizes the solid-liquid contact area.1 Most theories and applications so far have focused on a (water) drop on a solid surface, in air. However, superhydrophobicity may also be extremely useful for underwater systems: minimizing the wetted area of immersed surfaces may greatly reduce drag, as well as the rate of biofouling.20-23 Therefore, the objective of the present * Fax: 972-4-829-3088. E-mail:
[email protected]. (1) Marmur, A. Langmuir 2004, 20, 3517-3519. (2) Erbil, H. Y.; Demirel, A. L.; Avci, Y.; Mert, O. Science 2003, 299, 13371380. (3) Marmur, A. Langmuir 2003, 19, 8343-8348. (4) He, B.; Patankar, N. A.; Lee, J. Langmuir 2003, 19, 4999-5003. (5) Lundgren, M.; Allan, N. L.; Cosgrove, T. Langmuir 2003, 19, 7127-7129. (6) Feng, L.; Song, Y.; Zhai, J.; Liu, B.; Xu, J.; Jiang, L.; Zhu, D. Angew. Chem. 2003, 115, 824-826. (7) Lafuma, A.; Quere, D. Nat. Mater. 2003, 2, 457-460. (8) Yoshimitsu, Z.; Nakajima, A.; Watanabe, T.; Hashimoto, K. Langmuir 2002, 18, 5818-5822. (9) Li, S.; Li, H.; Wang, X.; Song, Y.; Liu, Y.; Jiang, L.; Zhu, D. J. Phys. Chem. B 2002, 106, 9274-9276. (10) Duparre, A.; Flemming, M.; Steinert, J.; Reihs, K. Appl. Opt. 2002, 41, 3294-3298. (11) Thieme, M.; Frenzel, R.; Schmidt, S.; Simon, F.; Hennig, A.; Worch, H.; Lunkwitz, K.; Scharnweber, D. AdV. Eng. Mater. 2001, 3, 691-695. (12) Herminghaus, S. Roughness-induced nonwetting. Europhys. Lett. 2000, 52, 165-170. (13) Oner, D.; McCarthy, T. J. Langmuir 2000, 16, 7777-7782. (14) Ball, P. Nature 1999, 400, 507-508. (15) Tadanaga, K.; Katata, N.; Minami, T. J. Am. Ceram. Soc. 1997, 80, 10401042. (16) Neinhuis, C.; Barthlott, W. Ann. Bot. 1997, 79, 667-677. (17) Wenzel, R. N. Ind. Eng. Chem. 1936, 28, 988-994. (18) Cassie, A. B. D.; Baxter, S. Trans. Faraday Soc. 1944, 40, 546-551. (19) Johnson, R. E.; Dettre, R. H. In Contact Angle, Wettability, and Adhesion; Advances in Chemistry Series 43; American Chemical Society: Washington, DC, 1964. (20) Bers, A. V.; Wahl, M. Biofouling 2004, 20, 43-51. (21) Scardino, A. J.; de Nys, R. Biofouling 2004, 20, 249-257.
Figure 1. Wetting regimes of a rough surface: (a) homogeneous wetting, (b) heterogeneous wetting.
communication is to discuss the feasibility of underwater superhydrophobicity (UWSH).
Theory The system is schematically shown in Figure 1: a rough plate is in contact with a liquid. Theoretically, the plate is infinite in size and the amount of liquid is infinite. In this figure, only one side of the plate is shown to be in contact with the liquid, however, the analysis below is not limited to this case. The roughness is considered to be uniform across the plate surface. The purpose of the present theoretical analysis is to find out if, and under what conditions, the heterogeneous wetting case (Figure 1b) may exist. This is done by searching for the minimum in the Gibbs surface energy of the system, G, which is given by3
G ) σlAl + σslAsl + σsAs
(1)
In this equation, σ stands for surface or interfacial tension, A stands for surface or interfacial area, and the subscripts s and l stand for the solid and liquid, respectively. Equation 1 is based on the common assumption that the interfacial tension equals the Gibbs energy per unit area. This assumption is reasonable for a system of pure components.24 When the heterogeneous wetting case exists, the curvature of the liquid-air interface inside the grooves is determined by the difference between the atmospheric pressure of the air in the grooves and the pressure in the liquid. If the effect of gravity on the liquid pressure may be neglected, the liquid-air interface, for simplicity, may be assumed approximately flat. However, in some possible applications, such as the coating of a ship hull, the plate may be vertically dipped into water. In such cases, it (22) Carman, M. L.; Estes, T. G.; Feinberg, A. W.; Schumacher, J. F.; Wilkerson, W.; Wilson, L. H.; Callow, M. E.; Callow, J. A.; Brennan, A. B. Biofouling 2006. (23) Hoipkemeier-Wilson, L.; Schumacher, J. F.; Carman, M. L.; Gibson, A. L.; Feinberg, A. W.; Callow, M. E.; Finlay, J. A.; Callow, J. A.; Brennan, A. B. Biofouling 2004, 20, 53-63. (24) Defay, R; Prigogine, I; Bellemans, A. Surface Tension and Adsorption; Longmans: London, 1966; p 61. (25) Wolansky, G.; Marmur, A. Langmuir 1998, 14, 5292-5297.
10.1021/la052802j CCC: $33.50 © 2006 American Chemical Society Published on Web 01/24/2006
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Langmuir, Vol. 22, No. 4, 2006 1401
is useful to find out to what extent the assumption of a flat liquid-air interface is valid, as follows. The pressure difference across a meniscus inside the roughness grooves is of the order of magnitude of 2σl/R, where R is a typical mean radius of curvature of the meniscus. This pressure difference must equal the hydrostatic pressure difference Fgh, where F is the density of the liquid, g is the gravitational acceleration, and h is the immersion depth. Thus, the typical mean radius of curvature is estimated by R ≈ 2σl/Fgh ≈ 10-5/h m. The liquid-air interface can be considered flat if this radius of curvature is much larger than the typical scale of the roughness. For a depth of 1 m, R is about 10 µm, so it is large compared with a typical roughness scale of 1 µm. For larger depths, the following analysis is valid if the roughness scale is submicrometer. Assuming a flat liquid-air interface, the surface area of this interface is given by
Al ) (1 - f)A
(2)
where f is the fraction of the projected solid area that is wet by the liquid (f ) 1 implies complete penetration of the liquid into the roughness grooves; f ) 0 implies zero contact between the liquid and the solid) and A is the total projected area of the solid. The solid-liquid interfacial area is
Asl ) rf fA
(3)
where rf is the roughness ratio of the wet part of the solid surface. The roughness ratio is defined, in general, as the ratio between the true surface area of the solid and its projected (nominal) area. The solid-air surface area is expressed by
As ) r1-f(1 - f)A
(4)
where r1-f is the roughness ratio of the dry part of the solid. rf and r1-f are related by
rf f + r1-f(1 - f) ) r
(5)
where r is the roughness ratio that characterizes the solid surface as a whole. Substituting eqs 2-5 into eq 1, the Gibbs energy equation becomes
G/A ) σl(1 - f) + σslrf f + σs(r - rf f)
(6)
Using the definition of the Young contact angle
cos θY )
σs - σsl σl
(7)
the dimensionless Gibbs energy per unit area, G*, can be expressed as
G* ≡
G ) 1 - f - φ cos θY + σ/s r Aσl
(8)
where φ ≡ rf f, and σ/s ≡ σs/σl. The possible equilibrium states of the system are defined by the minima in G*. For a local extremum
dG* dφ ) -1 cos θY ) 0 df df
(9)
Thus, a local extremum that determines an equilibrium location
of the liquid interface inside the roughness grooves is defined by
dφ ) -(cos θY)-1 df
(10)
This is actually equivalent to stating that the Young contact angle has to apply locally inside the groove, so it conforms to previous knowledge.3,25 To ensure that this local extremum is a minimum, and therefore a stable or a metastable equilibrium state, the following condition must be fulfilled
d2φ d2G* ) cos θY > 0 df 2 df 2
(11)
This condition is the same as that required for a drop on a rough surface,3 however, for a different mathematical reason. To determine whether a local equilibrium state is metastable or stable, its Gibbs energy has to be compared in each specific case with that of all other local equilibrium states and with the Gibbs energies at the borders of f. Especially, it is of interest to compare the energy at the local minimum with that of homogeneous wetting, for which the liquid completely penetrates into the roughness grooves (f ) 1). From eq 8
G*(fe) - G*(1) ) 1 - fe + (r - φe)cos θY
(12)
where fe and φe are the equilibrium values of f and φ. This difference must, of course, be negative for a stable local minimum. If eq 10 cannot be fulfilled, then a border minimum should be looked for. The values of f must be within the range fo e f e 1, where fo ) 0 if the roughness asperities are extremely sharp at their tops and larger than zero if they are flat at the top (such as in the case of pillars on a surface). In any case, φ(fo) ) fo, since rf ) 1 at the top of the asperity if it is flat (fo > 0) and φ(fo) ) fo ) 0 if it is sharp. The state with the lower energy will, of course, be the equilibrium state, as can be conveniently determined from the sign of the difference between the two energies
G*(fo) - G*(1) ) 1 - fo(1 + cos θY) + r cos θY (13) Results and Discussion Superhydrophobicity in air is characterized by a high apparent contact angle of a water drop and a low roll-off angle. These criteria cannot be applied to UWSH, since they are meaningless for a plate dipped into a liquid. The only direct criterion that can be applied for UWSH is the area fraction of the solid that is wet at equilibrium, φe. Thus, the goal of designing a surface for UWSH is to make φe as small as possible. Therefore, it makes sense to study first the possibility of a stable equilibrium at the extreme case of the lowest possible value of φe, which is fo. Practically, this means that the liquid touches only the tops of the roughness asperities, without penetrating into the roughness grooves. This may be achieved if the asperities have a shape that either enables eq 10 to be fulfilled at their tops (e.g., pillars that are flat at the top) or excludes the possibility of fulfilling this equation. The stability of this state is assured if the difference in energies, expressed by eq 13, is negative. This implies that
r>-
(
)
1 1 + fo 1 + ≡ rmin cos θY cos θY
(14)
where rmin is the minimum required value of the roughness ratio of the surface, to achieve stable superhydrophobicity at fo. Also, by definition, r > 1. It is easy to show that these two conditions
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Figure 3. Rough surface made of adjacent, two-dimensional, semicircular protrusions.
Figure 2. Minimum required roughness ratio in order to ensure a stable equilibrium state, for which the liquid-air surface is at the tops of the roughness asperities. The numbers in the figure indicate the values of fo.
can be simultaneously fulfilled only if cos θY < 0 (θY > π/2), so the surface must be hydrophobic. Now, for such a surface, the first term on the right-hand side of eq 14 is positive, while the second is negative. Thus, the first term increases the required roughness ratio of the surface, while the second term decreases it. This means that a nonzero value of fo relaxes to some extent the roughness requirements; however, an increase in fo increases the solid-liquid contact area, which is counterproductive. Figure 2 quantitatively demonstrates the predictions of eq 14, for the relevant range of contact angles (the achievable value of θY is usually less than 120°; a realistic value is 110°). This figure shows that stable UWSH at fo is, in principle, possible, if the roughness ratio can be made sufficiently high (preferably about 3 for θY ) 110°). The effect of increasing fo, in addition to being counterproductive, does not help much in reducing the required roughness ratio. To get a practical perspective, such roughness ratios are very difficult to get with random roughening processes; however, structured surfaces may produce such r values. For example, a rough surface made of square pillars has a roughness ratio of (1 + 4βfo), where β is the ratio of the height of the pillar to its width. Thus, for example, if β ) 2, fo needs to be 0.25 in order to obtain r ) 3. The design optimization in this case needs to take into account the benefits and drawbacks of increasing β (e.g., decreasing mechanical strength) vs increasing fo () φe) (increasing drag or biofouling rate), to achieve the required r.
It is of interest to also study the possibility of equilibrium partially inside the roughness grooves. This possibility cannot be studied in general, since the results depend on the specific details. However, the main issues can be demonstrated using a simple case, such as a surface composed of adjacent twodimensional semicircular protrusions (see Figure 3). For this rough surface,3 f ) sin R, where R is the angle shown in Figure 3 and rf ) R/sin R. The solution of eq 10 is R ) π - θY, which is feasible for π/2 e θY e π, since R e π/2. For this range of θY, eq 11 is always fulfilled, so the system is at an energy minimum for fe ) sin θY. For θY ) 110°, fe ) 0.94, φe ) 1.22, and eq 12 shows that this state is stable compared with the homogeneous wetting case. Thus, this specific example is characterized by a φe value that is indeed lower than the corresponding value for homogeneous wetting (π/2), however is higher than 1, therefore undesirable. In conclusion, the above analysis shows that UWSH is, in principle, feasible, and may be thermodynamically stable. It depends on finding a practical way of producing solid surfaces of high roughness ratio that keep the water surface as close as possible to the tops of the roughness asperities. The final optimization of the surface design must take into account many factors in addition to the above equilibrium considerations; however, the latter determine the essential criteria for underwater superhydrophobicity. Acknowledgment. The author acknowledges support from the AMBIO Project (NMP-CT-2005-011827) funded by the European Commission’s Sixth Framework Program. Views expressed in this publication reflect only the views of the author and the Commission is not liable for any use that may be made of information contained therein. LA052802J
