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The Lotus Effect: Superhydrophobicity and Metastability Abraham Marmur* Department of Chemical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel Received December 16, 2003. In Final Form: March 15, 2004 To learn how to mimic the Lotus effect, superhydrophobicity of a model system that resembles the Lotus leaf is theoretically discussed. Superhydrophobicity is defined by two criteria: a very high water contact angle and a very low roll-off angle. Since it is very difficult to calculate the latter for rough surfaces, it is proposed here to use the criterion of a very low wet (solid-liquid) contact area as a simple, approximate substitute for the roll-off angle criterion. It is concluded that nature employs metastable states in the heterogeneous wetting regime as the key to superhydrophobicity on Lotus leaves. This strategy results in two advantages: (a) it avoids the need for high steepness protrusions that may be sensitive to breakage and (b) it lowers the sensitivity of the superhydrophobic states to the protrusion distance.
Introduction Natural superhydrophobicity has recently gained much attention and inspired mimetic attempts.1-15 In many plants, notably the Lotus flower, leaves utilize superhydrophobicity as the basis of a self-cleaning mechanism: water drops completely roll off the leaf, carrying undesirable particulates.15 Possible man-made applications range from nanochips to self-cleaning windows. To turn a hydrophobic surface into a superhydrophobic one, its surface must be made rough in a way that is just beginning to unravel.2 Two criteria define superhydrophobicity: a very high water contact angle (Figure 1a) and a very low roll-off angle, defined as the inclination angle at which a water drop rolls off the surface (Figure 1b). While the former criterion has been thoroughly theoretically studied,2,16-18 the latter has been theoretically discussed only to a very limited extent with respect to superhydrophobicity.7,19,20 * E-mail:
[email protected]. (1) Erbil, H. Y.; Demirel, A. L.; Avci, Y.; Mert, O. Science 2003, 299, 1337-1380. (2) Marmur, A. Langmuir 2003, 19, 8343-8348. (3) He, B.; Patankar, N. A.; Lee, J. Langmuir 2003, 19, 4999-5003. (4) Lundgren, M.; Allan, N. L.; Cosgrove, T. Langmuir 2003, 19, 7127-7129. (5) Feng, L.; Song, Y.; Zhai, J.; Liu, B.; Xu, J.; Jiang, L.; Zhu, D. Angew. Chem. 2003, 115, 824-826. (6) Lafuma, A.; Quere, D. Nat. Mater. 2003, 2, 457-460. (7) Yoshimitsu, Z.; Nakajima, A.; Watanabe, T.; Hashimoto, K. Langmuir 2002, 18, 5818-5822. (8) Li, S.; Li, H.; Wang, X.; Song, Y.; Liu, Y.; Jiang, L.; Zhu, D. J. Phys. Chem. B 2002, 106, 9274-9276. (9) Duparre, A.; Flemming, M.; Steinert, J.; Reihs, K. Appl. Opt. 2002, 41, 3294-3298. (10) Thieme, M.; Frenzel, R.; Schmidt, S.; Simon, F.; Hennig, A.; Worch, H.; Lunkwitz, K.; Scharnweber, D. Adv. Eng. Mater. 2001, 3, 691-695. (11) Herminghaus, S. Roughness-induced non-wetting. Europhys. Lett. 2000, 52, 165-170. (12) Oner, D.; McCarthy, T. J. Langmuir 2000, 16, 7777-7782. (13) Ball, P. Nature 1999, 400, 507-508. (14) Tadanaga, K.; Katata, N.; Minami, T. J. Am. Ceram. Soc. 1997, 80, 1040-1042. (15) Neinhuis, C.; Barthlott, W. Ann. Bot. (London) 1997, 79, 667677. (16) Wenzel, R. N. Ind. Eng. Chem. 1936, 28, 988-994. (17) Cassie, A. B. D.; Baxter, S. Trans. Faraday Soc. 1944, 40, 546551. (18) Johnson, R. E.; Dettre, R. H. Contact Angle, Wettability, and Adhesion; Advances in Chemistry Series 43; American Chemical Society: Washington, DC, 1964. (19) Roura, P.; Fort, J. Langmuir 2002, 18, 566-569.
Figure 1. A drop on a rough surface: (a) the contact angle, θ; (b) the roll-off angle, R; (c) the homogeneous wetting regime; (d) the heterogeneous wetting regime.
Wetting on rough surfaces may assume either of two regimes: homogeneous wetting16 (Figure 1c), where the liquid completely penetrates the roughness grooves, or heterogeneous wetting17 (Figure 1d), where air (or another fluid) is trapped underneath the liquid inside the roughness grooves. The transition between these regimes18 has a major role in superhydrophobicity.2 The apparent contact angle on a rough surface in the homogeneous regime, θW, is given by the Wenzel equation16
cos θW ) r cos θY
(1)
where θY is the ideal Young contact angle and r is the roughness ratio, defined as the ratio of the true area of the solid surface to its projection area. It is clearly seen that, if a surface is hydrophobic (θY > 90°), roughness (r > 1) makes θW higher than θY. (20) Miwa, M.; Nakajima, A.; Fujishima, A.; Hashimoto, K.; Watanabe, T. Langmuir 2000, 16, 5754-5760.
10.1021/la036369u CCC: $27.50 © 2004 American Chemical Society Published on Web 04/01/2004
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The apparent contact angle in the heterogeneous regime, θCB, is given by the Cassie-Baxter (CB) equation17
cos θCB ) rf cos θY + f - 1
(2)
In this equation, f is the fraction of the projected area that is wet and rf is the roughness ratio of the wet area. The conditions for the existence of each regime have been recently formulated in full, in terms of the geometrical features of the roughness.2 It is important to note that the wetting regime that yields the lowest contact angle is the more stable one from a thermodynamic point of view, since the Gibbs energy turns out to be a monotonically increasing function of the contact angle.2 Since nature has developed very efficient superhydrophobic systems, it is useful to learn from and mimic its design. Therefore, the objective of the present communication is to theoretically analyze the underlying superhydrophobicity mechanism of the Lotus leaf. To achieve this purpose, a simple, approximate substitute to the consideration of roll-off is proposed. This approach complements the calculations of contact angles by the Wenzel and CB equations and is utilized to elucidate the preferred superhydrophobic states for a model system that resembles the Lotus leaf surface. Theory The problem of drop roll-off from rough surfaces is very complicated from a theoretical point of view. It is closely linked with the complex problem of contact angle hysteresis, since without hysteresis the roll-off angle would be zero. Because of the difficulties involved, the roll-off problem for superhydrophobic surfaces and its relationship with contact angle hysteresis have been studied so far only to a very limited extent7,19,20 (the theory of roll-off from heterogeneous smooth and homogeneous rough surfaces has been studied much more in depth21-24). Here, it is proposed to circumvent the theoretical difficulties of calculating the roll-off angle by using the criterion of a very low wet contact area (solid-liquid interfacial area) as a simple first-order substitute for the roll-off angle criterion. This is based on the intuitive notion that the lower the wet area, the lower the roll-off angle, since the force that keeps the liquid in contact with the solid is lower. This point of view may not necessarily be accurate for low contact angles but seems very reasonable for high contact angles, for which the wet contact area is small to begin with. For the homogeneous wetting regime, it can be shown by simple geometrical arguments that the wet area, aW, is given by
aW ) 32/3π1/3V 2/3F(θW)-2/3r sin2 θW
(3)
Figure 2. The model system: (a) paraboloids of revolution; (b) paraboloids of revolution arranged in a square lattice. The numbers indicate the steepness, a (see eq 6). The radius of the base of the paraboloid, xm, determines the distance between them (the square is of unit size). Two curves are shown for a ) 5 to demonstrate the effect of different values of xm.
presented for a simple model of a rough surface that resembles that of the Lotus leaf. Electron microscope pictures of the Lotus leaf surface structure15 show protrusions that roughly look like paraboloids of revolution (Figure 2a). They also demonstrate a fine structure on top of these protrusions. The role of these fine structures will be discussed elsewhere. For simplicity, the present calculations of the contact angle and wet area of a drop assume a model surface structure consisting of a square arrangement of paraboloids of revolution (each paraboloid is assumed to be located at the center of a square, Figure 2b). The shape of a paraboloid of revolution is described by
y ) ax2
where x is the radial coordinate and y is the vertical coordinate measured downward from the apex (Figure 2a). The higher the value of the constant parameter a, the steeper is the shape of the paraboloid. The side of the square is considered to be of unit length; thus, x and y are dimensionless lengths normalized with respect to the side of the square. The dimensionless radius of the base of the paraboloid is xm. The value of xm determines the distance between the protrusions (the lower the value of xm, the higher the distance) and must be smaller than 0.5. The present calculations also imply that the drop is much larger than the scale of the protrusions.2 On the basis of simple geometry, the roughness ratio of this model system is
where V is the volume of the drop and the function F is given by
F(θ) ≡ (2 - 3 cos θ + cos3 θ)
(4)
For the heterogeneous regime, the wet area, aCB, can be calculated from
aCB ) 32/3π1/3V 2/3F(θCB)-2/3rff sin2 θCB
(5)
In the following, the contact angles and the wet area are (21) Macdougall, G.; Ockrent, C. Proc. R. Soc. London 1942, 180A, 151-173. (22) Furmidge, C. G. L. J. Colloid Sci. 1962, 17, 309-324. (23) Dimitrakopoulos, P.; Higdon, J. J. L. J. Fluid Mech. 1999, 395, 181-209.
(6)
r ) 1 + πx2m
[
(1 + 4a2x2m)3/2 - 1 6a2x2m
-1
]
(7)
(This equation is based on the expression for the surface area of a paraboloid of revolution, whose base radius is xm and height is ax2m, taking into account the fact that the area of the unit square around the paraboloid is flat and horizontal). Using the value of r from this equation, the Wenzel contact angle for the homogeneous wetting regime can be calculated from eq 1. Then, the wet area can be calculated from eq 3. (24) Kim, H.-Y.; Lee, H. J.; Kang, B. H. J. Colloid Interface Sci. 2002, 247, 372-380.
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Figure 3. The Wenzel (thin lines) and CB (thick line) contact angles vs paraboloid steepness for various protrusion distances. The numbers indicate the dimensionless radius of the paraboloid base. Note that the CB curve does not depend on the protrusion distance.
The parameters defining the heterogeneous wetting regime can also be calculated for the model system. Of key importance in determining the existence and details of the heterogeneous regime is the product φ ≡ rff. This is given for the present model by
φ)
π 4 1 + a2f π 6a2
[(
)
3/2
-1
]
(8)
The existence of the heterogeneous regime depends on if the sign of the second derivative, d2φ/df 2, is positive.2 Indeed, it can be shown that for the present model system it is always positive. Then, the location of the solid-liquidair contact line within the roughness grooves is determined from the condition that the local contact angle between the liquid and the protrusion surface must be the Young contact angle. This is determined by the value of the first derivative,2 dφ/df, and can be shown to yield
f)
π (1/cos2 θY - 1) 4a2
(9)
Substituting eqs 8 and 9 into eq 2, the CB contact angle for the heterogeneous wetting regime can be calculated. Then, the wet area can be calculated from eq 5. Results and Discussion It is important to keep in mind that the following discussion is based on the geometry of paraboloids of revolution as a simplified model of the surface structure of the Lotus leaf. Therefore, some of the following conclusions may not necessarily be universal. However, much can be learned from this example regarding the design of superhydrophobic surfaces. All calculations presented here were done for θY ) 110° (the highest contact angle on existing smooth surfaces is ∼115°). For the homogeneous wetting regime, Figure 3 clearly shows (thin lines) the interplay between the effect of the steepness, a, and the effect of the protrusion distance on the Wenzel contact angle. As the distance gets larger (smaller xm), the protrusions need to be steeper in order to reach a sufficiently high Wenzel contact angle. Figure 4 shows (thin line) that the wet area monotonically decreases when the Wenzel contact angle increases (in this figure, the dimensionless wet area is shown normalized with respect to the drop volume, AW ≡ aW/V 2/3, in order to eliminate the effect of the latter). Therefore, as the protrusion distance gets larger, the protrusions need to be steeper also in order to decrease the wet area in the homogeneous domain. However, it should be kept in mind that steep protrusions might not be desirable, since they are more vulnerable to mechanical breakage. Thus,
Figure 4. The dimensionless wet area in the homogeneous (thin line) and heterogeneous (thick line) regimes vs the Wenzel and CB contact angles, respectively. The two curves do not depend on the protrusion distance.
designs that do not require steep protrusions should be preferred. The contact angle, θCB, and the normalized wet area, ACB ≡ aCB/V 2/3, for the heterogeneous wetting regime are shown in Figures 3 and 4 by the thick lines. It is interesting and important to notice that the CB contact angle as well as the wet area does not depend on the protrusion distance (this is true, of course, when this distance is not too large). This is an extremely useful feature of the system, which allows robustness in design. However, it is not necessarily universal, as evidenced by some man-made systems that behave differently.12 It is also of great interest to realize that the wet area in the heterogeneous wetting regime is much smaller than that in the homogeneous regime, even for the same apparent contact angle (as long as θCB, θW < 180°, since then the wet area is zero for both). Intuitively, the heterogeneous wetting regime is in general associated with a smaller wet area than the homogeneous one. However, it is not at all a priori clear that this is true even for the same contact angle. The above results lead to the conclusion that the heterogeneous wetting regime is practically preferred by nature as the superhydrophobic state on Lotus leaves for several related reasons: (a) for most combinations of steepness and protrusion distance, the CB contact angle is higher than the Wenzel one; (b) the CB contact angle is insensitive to the protrusion distance and mildly sensitive to the steepness; and (c) the heterogeneous wetting regime yields a much lower wet area even when the Wenzel and CB contact angles are equal. However, as already stated, the thermodynamically stable state is the one with the lower contact angle; therefore, when the heterogeneous regime offers the higher contact angle, it must be metastable. This was also evidenced by some ingenious experiments3,6 with artificial systems, where it was shown that the same system could have two different contact angles. The above conclusions are based on the wet area criterion as a substitute for the roll-off angle criterion. It is hoped that future calculations of the rolloff angle for superhydrophobic surfaces will justify the use of this simple, intuitively appealing criterion. In summary, the present calculations indicate that nature uses metastable states as the key to superhydrophobicity on Lotus leaves. This strategy avoids the need for high steepness protrusions that may be more prone to erosion and breakage. In addition, due also to the specific shape of the Lotus leaf protrusions, this strategy lowers the sensitivity of the superhydrophobic state to the protrusion distance. Consequently, the metastable superhydrophobicity of the Lotus leaf is relatively insensitive to mild variations in the surface roughness design. LA036369U
