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Transition between Superhydrophobic States on Rough Surfaces Neelesh A. Patankar Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, B224, Evanston, Illinois 60208-3111 Received March 15, 2004. In Final Form: May 27, 2004 Surface roughness is known to amplify hydrophobicity. It is observed that, in general, two drop shapes are possible on a given rough surface. These two cases correspond to the Wenzel (liquid wets the grooves of the rough surface) and Cassie (the drop sits on top of the peaks of the rough surface) formulas. Depending on the geometric parameters of the substrate, one of these two cases has lower energy. It is not guaranteed, though, that a drop will always exist in the lower energy state; rather, the state in which a drop will settle depends typically on how the drop is formed. In this paper, we investigate the transition of a drop from one state to another. In particular, we are interested in the transition of a “Cassie drop” to a “Wenzel drop”, since it has implications on the design of superhydrophobic rough surfaces. We propose a methodology, based on energy balance, to determine whether a transition from the Cassie to Wenzel case is possible.
1. Introduction Surface roughness amplifies hydrophobicity.1,2 This is frequently seen in nature1 and has been demonstrated for microfabricated rough surfaces.3-6 Roughness-induced superhydrophobicity is considered a viable option for surface-tension-induced drop motion in microfluidic devices. Another application is inspired by superhydrophobic plant leaves. Water drops are almost spherical on some plant leaves and can easily roll off, cleaning the surface in the process.1 This is usually referred to as the Lotus effect. It is brilliantly exhibited by the leaves of the Lotus plant. There are numerous applications of artificially prepared “self-cleaning” surfaces. Drops can exist in multiple equilibrium states on rough surfaces.7 It is now known that there are typically two prominent states in which a drop can reside on a given rough surface4-6 (Figure 1). The drop either sits on the peaks of the rough surface or it wets the grooves (to be referred to as a wetted contact), depending on how it is formed. The drop that sits on the peaks has “air pockets” along its contact with the substrate; hence, it will be termed a “composite” contact. The apparent contact angle of the drop that wets the 8 grooves, θw r , is given by Wenzel’s formula
cos θw r ) r cos θe
(1)
where r is the ratio of the actual area of liquid-solid contact to the projected area on the horizontal plane and θe is the equilibrium contact angle (which is typically a value between the advancing and receding contact angles) of the liquid drop on the flat surface. We do not consider the separate cases of advancing and receding angles on a (1) Barthlott, W.; Neinhuis, C. Planta 1997, 202, 1-8. (2) Hazlett, R. D. J. Colloid Interface Sci. 1990, 137, 527. (3) Onda, T.; Shibuichi, N.; Satoh, N.; Tsuji, K. Langmuir 1996, 12, 2125-2127. (4) Bico, J.; Marzolin, C.; Que´re´, D. Europhys. Lett. 1999, 47, 220226. (5) He, B.; Patankar, N. A.; Lee, J. Langmuir 2003, 19, 4999. (6) Yoshimitsu, Z.; Nakajima, A.; Watanabe, T.; Hashimoto, K. Langmuir 2002, 18, 5818. (7) Johnson, R. E.; Dettre, R. H. Adv. Chem. Ser. 1963, 43, 112. (8) Wenzel, T. N. J. Phys. Colloid Chem. 1949, 53, 1466.
Figure 1. Two states of water drops on the same rough surface. The left side shows a composite drop, while the right side shows a wetted drop (courtesy of J. Lee and B. He).
surface. The apparent contact angle of a drop that sits on the roughness peaks, θcr, is given by Cassie’s formula9
cos θcr ) rw φs cos θe + φs - 1
(2)
where φs is the area fraction on the horizontal projected plane of the liquid-solid contact and rw is the ratio of the actual area to the projected area of liquid-solid contact. This will be referred to as the Cassie or composite drop. If φs ) 1, we have rw ) r, in which case the Cassie formula becomes the Wenzel formula. A Cassie drop shows significantly less hysteresis10 compared to a Wenzel drop due to low resistance from the air pockets. Hence, it is imperative that a composite contact is achieved for applications such as self-cleaning surfaces or surface-tension-induced droplet motion. The rough surface should be such that it prefers to form a composite contact rather than have the liquid wet its grooves. Our recent theoretical analysis11 has shown that, of the two possible states of a drop on a rough surface, one has lower energy (or a lower apparent contact angle). Geometric parameters of the surface roughness determine whether a Cassie or a Wenzel drop has lower energy. We verified our predictions by performing experiments.5 On the basis of this analysis, we can propose that, to ensure the formation of a composite contact, the surface roughness parameters should be chosen such that the Cassie drop (9) Cassie, A. B. D. Discuss. Faraday Soc. 1948, 3, 11. (10) Lafuma, A.; Que´re´, D. Nat. Mater. 2003, 2, 457. (11) Patankar, N. A. Langmuir 2003, 19, 1249.
10.1021/la049329e CCC: $27.50 © 2004 American Chemical Society Published on Web 07/22/2004
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has lower energy than the Wenzel drop. This may be regarded as a conservative design condition. Previous experiments have shown that even if the Wenzel drop has lower energy a Cassie drop can be obtained if, for example, the drop is formed by depositing it gently on the rough surface.5 Available experimental data4-6 imply that work has to be done in order to transition the Cassie drop to a Wenzel drop of lower energy. Various means of doing this have been reported, for example, depositing the drop from some height,5 pushing the drop,4 or using a heavy drop.6 This observation suggests that, in the experiments above, the Cassie and Wenzel states represent local energy minima separated by an energy barrier. Due to this barrier, the Cassie drop can be observed even if it has higher energy than the Wenzel drop. In our previous work, we briefly discussed the nature of this energy barrier for the particular roughness geometry of square pillars in a periodic array.11 In this paper, we will consider this issue in detail. We will consider the effect of various surface geometries on the nature of the energy barrier between the Cassie and Wenzel states. This will lead to a methodology, based on energy balance, to determine whether a transition from the Cassie to Wenzel case is possible. In this paper, we will formally present the theoretical arguments to predict transition between the superhydrophobic states, namely, the Cassie and Wenzel states. The theoretical arguments will be applied to the experimental results reported in the literature. In particular, we will consider the experiments of Yoshimitsu et al.6 2. Energy Balance to Determine the Condition for Transition For the applications of our interest, the objective is to have a Cassie drop or a composite contact. Hence, we will focus on the transition of a Cassie drop to a Wenzel drop and on how that could be avoided. One underlying assumption in this work is that the drop size is much bigger than the size of the surface roughness. This assumption allows us to use the Cassie and Wenzel formulas for the apparent contact angles. We will first consider, in section 2.1, a roughness made of square pillars in a periodic array. In section 2.2, we will consider the implications of other roughness geometries. 2.1. Surface Roughness of Periodically Placed Pillars. Let the rough surface be made of square pillars of size a × a, height H, and spacing b arranged in a regular array (Figure 2). The contact angle, θe, on the flat surface is assumed given. The geometric parameters in eqs 1 and 2 are
rw ) 1 1 ((b/a) + 1)2 H r ) 1 + 4φs a
φs )
(
)
}
(3)
Consider a Cassie drop on a rough substrate. The energy, Gc, of the drop is represented by
Gc ) Scσlv - cos θcrAc
(4)
where Sc is the area of the liquid-vapor contact of a Cassie drop, Ac is the area of contact with the substrate projected on the horizontal plane, σlv is the liquid-vapor surface energy per unit area (or surface tension), and we assume
Figure 2. Side and top views of a roughness geometry of periodically placed pillars (filled black) of square cross section. The pillar cross-sectional size is a × a. In the figure, we show one period.
gravity to be negligible. In our discussion, the liquid can be considered to be water and the vapor is air. If gravity is negligible, then the drop can be assumed to be a spherical cap. Substituting appropriate expressions for Sc and Ac, eq 4 becomes11
Gc 3
x9πV2/3σlv
) (1 - cos θcr)2/3(2 + cos θcr)1/3
(5)
where V is the drop volume. The left-hand side denotes nondimensional energy. If the Cassie drop transitions to a Wenzel drop, the new energy, Gw, of the drop is given by eq 5 but with θcr 11 replaced by θw r . It can be easily verified that the righthand side of eq 5 is a monotonically increasing function of θ for 0° e θ e 180°. Hence, if θcr > θw r , then the Cassie drop will have higher energy than the Wenzel drop and vice versa. The liquid-vapor surface area, Sw, and the projected area of contact, Aw, of the Wenzel drop are different from those of the Cassie drop. We consider the case where θcr > θw r , that is, the Cassie drop is at higher energy. This implies Ac < Aw for a drop of given volume, V.11 We ask the following question: Is it necessary that a Cassie drop that has higher energy will always transition to a Wenzel drop at lower energy? In an attempt to answer this question, we will first hypothesize how the energy of the drop might change if it transitions from a Cassie to a Wenzel drop. Consider a microscopic view of the contact between the drop and the rough surface. For a Cassie drop, the liquidvapor interface hangs from corner to corner at the top of the grooves (Figure 3a). This interface does have a curvature of the order of the radius of the drop, but it can be ignored at the scale of the surface roughness (note that we have assumed the drop size to be much larger than the roughness size). The above conclusions are consistent with the assumption of a rounded corner with a very small radius of curvature.12 The local contact angle of the interface at the corner is equal to θe to satisfy local equilibrium.12 It can be easily verified that any hydrophobic (>90°) contact angle can find an equilibrium position at some point on a 90° corner.12 The actual details of the transition from composite to wetted contact are not well understood. Different pos(12) Oliver, J. F.; Huh, C.; Mason, S. G. J. Colloid Interface Sci. 1977, 59, 568.
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GB1 ) Gy)0 ) Gc - (r - 1) cos θeσlvAc
(7)
Even if the Cassie drop, on a surface roughness made of pillars, is transitioning to a Wenzel state at lower energy, it has to go through a higher intermediate energy state. Hence, energy must be provided to the drop to enable transition. The transition can be enabled, for example, by depositing the drop from some height,5 by pushing the drop,4 or simply due to its own weight.6 A different estimate for the energy barrier can be obtained by considering the Wenzel drop but with the liquid-solid contact yet to be formed at the bottom of the valleys. The barrier energy of this state, GB2, is given by
GB2 ) Gw + (1 - φs)(1 + cos θe)σlvAw
Figure 3. (a) Side view of the liquid on top of the pillars. The liquid interface hangs from edge to edge of the pillars. (b) An intermediate state in which the liquid is entering the valley as a drop transition from a Cassie to a Wenzel state. (c) An intermediate state during transition where the liquid has not yet wetted the bottom of the valleys.
sibilities may be hypothesized. Consider a possibility where the liquid-vapor interface, initially at the top of the groove, starts moving down; that is, it starts wetting the sides of the pillars (Figure 3b). The transition is in general a nonequilibrium process where the local contact angle is not necessarily equal to the equilibrium value. The surface energy will change during transition. Assuming that the filling of the grooves happens below the projected area, Ac, of the Cassie drop, the energy at an intermediate stage is given by
(
Gy ) Gc - 1 -
y (r - 1)cos θeσlvAc H
)
(6)
where y is the location of the interface from the bottom of the groove (Figure 3b) and Gy is the drop energy at that state. As the liquid fills the grooves, Sc and Ac should change because some volume is moving out of the spherical liquid cap above the substrate. This in turn should change the energy, Gc, but we will assume these changes to be negligible compared to the surface energy changes (the second expression on the right-hand side of eq 6) due to the wetting of the grooves. This assumption is justified because the surface area per unit volume is much larger in the grooves compared to the spherical liquid cap. Gc and Ac are therefore assumed constants in eq 6. Equation 6 implies that the energy of the drop at intermediate states is larger than Gc for θe > 90°. The maximum value is reached at y ) 0 when the liquid has filled the grooves but the liquid-solid contact at the bottom of the valley is yet to be formed (Figure 3c). When the liquid wets the bottom of the valley, the corresponding change in energy is given by -(1 + cos θe)σlv(1 - φs)Ac; that is, the energy of the system decreases. The liquid would then proceed to wet a greater area of the substrate (Aw > Ac) to eventually reach the equilibrium shape of a Wenzel drop at the energy Gw. Since we are assuming θcr > θw r , we have Gc > Gw, as argued above. The maximum energy state (Gy at y ) 0) among all the intermediate states can be used to obtain an estimate of the barrier energy for the transition of a Cassie drop to a Wenzel drop, GB1.
(8)
The actual wetting of the grooves may not occur as hypothesized here. It is quite possible that the liquid does not wet the grooves all at once, as supposed above, but does so in parts. If that is the case, then the energy barrier will be less than that estimated in eqs 7 and 8. Another way of looking at this is the following. At the end of the transition, there is a net decrease in the energy of the drop equal to Gc - Gw (since we are considering a lower energy Wenzel state). We could then assume that part of this energy is available to overcome the energy barrier estimated above. We have assumed that gravity does not play a significant role in determining the shape of the drop either in the Cassie state or in the Wenzel state. This assumption is reasonable if the drop radius, R, is ,acap, where acap ) (σlv/Fg)1/2 is the capillary length of a liquid of density F and g is the gravitational acceleration. For water, acap ) 2.7 mm; a spherical water drop of radius acap weighs 82 mg. Thus, the water drop should be smaller than 82 mg for the gravity effects to be of little significance in determining the shapes of the Cassie and Wenzel states. We can restate the above condition in terms of a nondimensional parameter. We define the Bond number as Bo ) (V/Vcap)2/3 ) (m/mcap)2/3, where m is the mass of the drop and mcap is the mass of a drop of radius acap. V’s denote the corresponding volumes. Bo denotes the square of the ratio of the length scale of the drop to the capillary length scale. The drop shape is almost spherical if Bo , 1. Even if gravity plays an insignificant role in determining the drop shape, it can play an important role in the transition from a Cassie to a Wenzel drop. This will be discussed next. At small Bo values, the effect of gravity is to make the drop shape only slightly nonspherical. This lowers the center of mass of the drop by some height, δ, compared to a perfectly spherical drop. The corresponding decrease in the potential energy of a drop, that has a superhydrophobic contact with the surface, can be estimated as13
mgδ ∼
(mg)2 σlv
(9)
where g is the gravitational acceleration. Next, we estimate the potential energy change when a drop transitions from a Cassie to a Wenzel state. Since we are considering a lower energy Wenzel state, which has a lower apparent contact angle, the center of mass of the Wenzel drop is lower. The change in the gravitational potential energy during transition is given by (13) Mahadevan, L.; Pomeau, Y. Phys. Fluids 1999, 11, 2449.
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{4Vπ (R
4 c
mg(Hc - Hw) ) mg
Rc ) Rw )
{ {
sin4 θcr - R4w sin4 θw r) -
(Rc cos θcr - Rw cos θw r )}
3V π(1 - cos θcr)2(2 + cos θcr) 3V
π(1 - cos
2 θw r ) (2
} }
1/3
1/3
+ cos
θw r)
}
Patankar
(10)
where Hc and Hw are the heights of the center of mass of the Cassie and Wenzel drops (assumed to be spherical), respectively. Rc and Rw are the radii of curvature of the Cassie and Wenzel drops, respectively, and V is the drop volume. The R cos θ term in eq 10 is dominant. The scale of mg(Hc - Hw) is at least mgV1/3. Using eqs 9 and 10, we conclude that mgδ/mg(Hc - Hw) ∼ Bo , 1; that is, the change in the gravitational potential energy due to the transition is much more compared to mgδ. Thus, gravity is important for transition even if it does not significantly change the spherical shape of a drop. All or part of the gravitational potential energy during transition may be available to overcome the energy barrier. We formulate various conditions for the transition of a Cassie drop to a Wenzel drop. These are listed below. (1) The barrier energy is GB1, and the source of energy to overcome the energy barrier is gravitational:
mg(Hc - Hw) g GB1 - Gc
(11)
(2) The barrier energy is GB1, and the source of energy is gravitational and Gc - Gw:
mg(Hc - Hw) + Gc - Gw g GB1 - Gc
(12)
(3) The barrier energy is GB2, and the source of energy is gravitational:
mg(Hc - Hw) g GB2 - Gc
(13)
(4) The barrier energy is GB2, and the source of energy is gravitational and Gc - Gw:
mg(Hc - Hw) + Gc - Gw g GB2 - Gc
(14)
GB1 - Gc and GB2 - Gc are given by eqs 7 and 8, respectively. The base area of a Cassie drop is Ac ) πR2c sin2 θcr, and that of a Wenzel drop is Aw ) πR2w sin2 θw r , where Rc and Rw are given in eq 10. Gc - Gw can be found using eq 5. In all the criteria, above, we have included the gravitational term. We could have set up a criterion where the only gain in energy comes from Gc - Gw. In that case, the transition would be independent of the drop mass, a prediction contrary to the observations of Yoshimitsu et al.6 (to be discussed below). Therefore, we did not set up such criteria. In setting up the above criteria, we are not quantifying the viscous dissipation that will consume some energy during the transition process. To test the above criteria, we will consider the experiments of Yoshimitsu et al.6 They studied the effect of surface geometry on the hydrophobic states of drops. They fabricated surfaces that had a roughness geometry of periodically placed square pillars. In their experiments, θe ) 114°. There were two cases that are of interest to us: (A) They considered a 1 mg drop on substrates with pillar dimensions given by a ) 50 µm and b ) 100 µm. The height of the pillars was changed by fabricating different surfaces. They reported that a pillar height of 35 µm was
Figure 4. Plots of the theoretical estimate of the critical mass (in milligrams) of a drop, for the transition from a Cassie to a Wenzel state, as a function of H/a for case A of the experiments of Yoshimitsu et al.6 (a) Plots according to the four criteria hypothesized for transition. (b) A close-in view of the plot in part a along with the experimental data points.
required to obtain a Cassie drop. For lesser heights, a Wenzel drop was formed. (B) In another set of experiments, they considered a substrate with a ) 50 µm and b ) 50 µm. Two pillar heights, 14 µm and 53 µm, were considered. They observed that all drops (they had drops up to a maximum of 35 mg) were in the Cassie state on the surface with a 53 µm pillar height. On the surface with a 14 µm pillar height, they observed that a 7 mg drop was in the Cassie state while 12 mg and heavier drops were in the Wenzel state. We will now check the transition criteria (eqs 11-14) by comparing them with the above observations. Equations 11-14 can be used to obtain the critical mass for the drop to transition to the Wenzel state. We plot the critical mass as a function of H/a with the rest of the parameters given. If the mass of the drop is greater than the critical mass at a given value of H/a, then the theory predicts transition to the Wenzel state; below the critical mass, a Cassie drop should be formed. Figure 4 shows a plot of the critical mass versus H/a for case A where b/a ) 2 (from eq 3, φs ) 0.11) and θe ) 114°. All the criteria show a general trend where the critical mass increases with H/a. As H/a increases, more energy is required to wet the deeper grooves; that is, the energy barrier is larger. A heavier drop is required to overcome the larger energy barrier. This trend continues such that the critical mass tends to infinity as H/a f 2.92. At H/a ) 2.92, we have θcr ) θw r ; that is, the energies of the Wenzel and Cassie drops are the same (Gc ) Gw). There is no change in the gravitational potential energy (Hc ) Hw). Thus, there is no source of energy to overcome the energy barrier which is nonzero. A transition is thus not possible. For H/a > 2.92, the Cassie drop is at lower energy. The center of mass of the Wenzel state is higher due to its
Transition between Superhydrophobic States
Figure 5. Plots of the theoretical estimate of the critical mass (in milligrams) of a drop, for the transition from a Cassie to a Wenzel state, as a function of H/a for case B of the experiments of Yoshimitsu et al.6
higher apparent contact angle. Both mg(Hc - Hw) and Gc - Gw are negative (i.e., the respective energies increase) and cannot act as the source of energy to overcome the barrier. Once again, transition to the Wenzel state is not possible unless there is a different energy source. Therefore, in Figure 4, we show values of H/a only up to 2.92 (not including 2.92). The theory has assumed that gravity does not influence the Wenzel and Cassie drop shapes significantly. As argued above, only those results in Figure 4 where Bo , 1, that is, where the critical mass, m, is ,mcap ()82 mg), are consistent with this assumption. Figure 4a shows the trend of theoretical predictions based on all criteria. The critical mass based on criterion 3 is much larger than any other criteria. The values are also much higher than the experimental observations. Criterion 3, thus, does not appear realistic. This is not surprising, since the energy barrier is based on the Wenzel drop. Criterion 3 is based on the contact area Aw which is larger than Ac for the parameters being considered here. This results in a larger value for the energy barrier. Figure 4b shows a close-in view of the plot in Figure 4a. The critical mass predicted by criteria 2 and 4 is zero up to a certain value of H/a. This is because Gc - Gw is sufficient to overcome the energy barrier. In criterion 1, we assume that Gc - Gw is not available for transition; hence, the critical mass is zero only at H/a ) 0. Figure 4b also shows two experimental data points from Yoshimitsu et al.,6 both of which are drops with a mass equal to 1 mg. A Wenzel drop was formed on a substrate with H/a ) 0.2 (H ) 10 µm), and a Cassie drop was formed when H/a ) 0.72 (H ) 36 µm). Thus, a mass of 1 mg was claimed to be critical for H/a ) 0.7 (H ) 35 µm). Criterion 2 predicts a Wenzel drop for both the data points. It underestimates the critical mass. Criteria 1 and 4 appear to provide the most reasonable predictions, criterion 1 being in best agreement with the observations of Yoshimitsu et al.6 In Figure 5, we compare the observations in case B with the theoretical predictions. Here, b/a ) 1 (from eq 3, φs ) 0.25) and θe ) 114°. For this case, the critical mass tends to infinity as H/a f 1.09. At H/a ) 1.09, we have θcr ) θw r . For H/a ) 1.06 (H ) 53 µm), the critical mass is very high; that is, a Cassie drop should be formed according to all criteria. This is in agreement with the data of Yoshimitsu et al.6 who observed a Cassie drop for all cases. Their maximum drop mass was 35 mg. For H/a ) 0.28 (H ) 14 µm), a 7 mg drop was in the Cassie state while a 12 mg drop was in the Wenzel state. Both these data points are shown in Figure 5. While none of the criteria predict this exactly, once again criterion 1 is closest to the observation.
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Figure 6. Side and top views of a roughness geometry of periodically placed pillars with inclined side walls.
Very little is understood of the actual transition process, but the above comparisons suggest that criterion 1 probably provides the best predictions. This conclusion is, however, based on limited data. More experiments are desirable to understand the transition better. The main conclusion from the above analysis is that, even if the Wenzel state is at a lower energy, it does not necessarily mean that a drop will transition to that state. It will do so only if it can overcome the energy barrier. Criterion 1 may be used to predict whether the transition is possible. This information is useful in designing superhydrophobic self-cleaning surfaces where a Cassie drop is more desirable since it has less hysteresis. Analyses such as the one above can be envisaged for the cases of pushing the drop4 and releasing the drop from some height5 to achieve transition. Similarly, different geometries can also be analyzed. 2.2. Other Surface Roughness Geometries. Most of the experimental data on the Wenzel and Cassie states on fabricated surfaces have considered a pillar geometry.4-6 As discussed in the previous section, in this case, there is an energy barrier between the Cassie and Wenzel states for hydrophobic contact (θe > 90°). This is argued theoretically and evidenced by experimental observation. We now ask the question if an energy barrier will exist between the Cassie and Wenzel states for any surface geometry. Specifically, we will once again consider a Cassie state with higher energy compared to the Wenzel state. We wish to explore if these two states are necessarily separated by an energy barrier for any surface geometry. If an energy barrier is not present, then a Cassie drop will always transition to the lower energy Wenzel state. It will become evident from the discussion below that there can be examples (in addition to the particular case considered below) where, in fact, an energy barrier may not be present. This is consistent with the conclusions of Marmur.14 The methodology of Marmur14 can be used to analyze the energy change for different roughness geometries. Figure 6 shows a particular geometry where the surface roughness is similar to that in section 2.1 except that the pillars now have an inclined surface instead of vertical faces. The geometric parameters are as defined in the figure. Note that a Cassie drop will be formed only if a liquid-vapor interface can hang from pillar-to-pillar. This interface can find equilibrium at the top corners of the pillars only if θs e θe e 180°.12 For other values of θe, a Cassie drop is not possible and the valley will always be filled. In the following discussion, we will consider a case (14) Marmur, A. Langmuir 2003, 19, 8343.
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where θs ) θe. We will show that the Wenzel drop is at lower energy, in this case, compared to the Cassie drop. Then, we will propose that there is no energy barrier between the two states. When θs < θe, the reader can verify that there is an energy barrier between the Wenzel and Cassie states similar to the case in section 2.1 above. Assume that a Cassie drop is formed such that the liquid-vapor interface hangs from the top corners of the pillars (Figure 6). Since θs ) θe, the liquid-vapor interface can also be in equilibrium at other locations along the pillar height, but we will consider these possibilities only during the filling of the valleys. The apparent contact angles in the Cassie and Wenzel states are given by substituting the following in eqs 1 and 2
(
rw ) 1
1+
φs )
(
2H a tan θs
)
2
2
((b/a) + 1)
(
(1 + cos θs) H H r) 1+4 1+ a sin θs ((b/a) + 1)2 a tan θs
))
}
(15)
where we have made θs ) θe for the particular case we are considering, H is the pillar height, and a and b are as defined in Figure 6. Equation 15 reduces to eq 3 when θs ) 90°. Using the expressions in eq 15, it follows that, for c the particular case when θs ) θe, we get θw r < θr for b/a * 0; that is, the Wenzel drop has lower energy than the Cassie drop. For b/a ) 0, the Wenzel and Cassie drops c have the same energy; that is, θw r ) θr . We will consider b/a * 0. As was hypothesized before, consider a transition from the Cassie to the Wenzel state where the liquid-vapor interface, initially at the top of the pillar, starts moving down; that is, it starts wetting the sides of the pillars. Assuming that the filling of the valleys happens below the projected area, Ac, of the Cassie drop, the energy change at an intermediate stage is given by
Gy - Gc ) (cos θs - cos θe)σlvAs ) 0; ∴ θs ) θe
(16)
where y is the location of the interface from the bottom of the groove, As is the corresponding surface area of the slanted faces of the pillars that gets wetted in the filling process, and Gy is the drop energy at that state. Equation 16 implies that the energy of the drop at intermediate states is unchanged. The energy remains unchanged until the liquid-vapor interface reaches the bottom of the valley. When the liquid wets the bottom of the valley, the corresponding change in energy is given by -(1 + c c , where Avalley is the area of the valley cos θe)σlvAvalley under a Cassie drop. The energy of the system decreases. The liquid would then proceed to wet a larger projected area of the substrate (Aw > Ac) to eventually reach the equilibrium shape of a Wenzel drop at the energy Gw. 11 Since we have θcr > θw r , it implies Gc > Gw. Similar to the estimate in section 2.1, we can estimate the barrier energy by GB1 ) Gy)0. According to eq 16, we get GB1 ) Gc; that is, there is no energy barrier if one
assumes the above as the transition process that best represents the real situation. As before, a different estimate for the energy barrier can be obtained by considering the Wenzel drop but with the liquid-solid contact yet to be formed at the bottom of the valleys. The barrier energy of this state, GB2, is given by w GB2 ) Gw + (1 + cos θe)σlvAvalley
(17)
w is the area of the valley under a Wenzel drop. where Avalley It can be shown that GB2 > Gc; that is, there is an energy barrier if the transition process is represented by this hypothesis. The results from section 2.1 suggest that GB1 and not GB2 is a better estimate of the intermediate energy during transition. Hence, it is likely that the problem being considered will not have an energy barrier. A Cassie drop will always transition to a lower energy Wenzel state. This proposition should be tested with suitable experiments. c w and Avalley are equal to zero. When b/a ) 0, both Avalley In this situation, the Cassie and Wenzel drops will not only have the same energy as discussed above, the energy barrier between the two states will be zero according to both eqs 16 and 17. Note that if the Cassie and Wenzel states have the same energy this does not necessarily imply that there is no energy barrier, as is evident from the case considered in section 2.1.
Conclusion In this paper, we investigate the transition of a higher energy Cassie drop to a lower energy Wenzel drop. Previous experiments suggest that, for the case of a roughness geometry made of pillars, there is an energy barrier between the two states. Hence, work must be done to cause the transition. We propose a methodology, based on energy balance, to determine whether a transition from a Cassie to a Wenzel drop is possible. We consider a roughness geometry of square pillars. The theoretical predictions are compared with the experimental results of Yoshimitsu et al.6 The comparison is helpful in understanding the transition process. The most probable mechanism is that the decrease in the gravitational potential energy during the transition helps in overcoming the energy barrier. The best estimate for the energy barrier is obtained by considering an intermediate state in which the water fills the grooves below the contact area of a Cassie drop but the liquidsolid contact is yet to be formed at the bottom of the valleys. Similar analysis can be extended to other roughness geometries. The main conclusion from the above analysis is that, even if the Wenzel state is at a lower energy, it does not necessarily mean that a drop will transition to that state. It will do so only if it can overcome the energy barrier. We also argue that an energy barrier will not exist between the Cassie and Wenzel states for all roughness geometries. We consider an example where an energy barrier may not be present. This discussion is consistent with the conclusions of Marmur.14 Acknowledgment. This work has been supported by a DARPA (SymBioSys) grant (Contract No. N66001-01C-8055) with Dr. Anantha Krishnan as the monitor. LA049329E
