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On the Modeling of Hydrophobic Contact Angles on Rough Surfaces Neelesh A. Patankar Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, B224, Evanston, Illinois 60208-3111 Received September 26, 2002. In Final Form: November 14, 2002 The apparent contact angle of a drop on a rough surface is often modeled using either Wenzel’s or Cassie’s formulas. Previous experiments are not conclusive regarding which formula to use and when. This information is critical in designing a superhydrophobic substrate for applications in microscale devices. A drop on a rough substrate can occupy multiple equilibrium states. These equilibrium states denote respective local minima in energy. The particular shape that a drop attains depends on how the drop is formed. We propose a design procedure to develop a rough superhydrophobic substrate that accounts for the multiple equilibrium drop shapes. The theory is expected to work well to maximize the advancing contact angle of a drop. It is noted in the end that appropriate models for the receding contact angles on rough substrate must be investigated further before appropriate design procedures, which will maximize the receding contact angle or minimize hysteresis (i.e., minimize the difference between the advancing and receding contact angles), are developed. We discuss a model for the receding contact angle, based on the limited data in the literature.
1. Introduction It is known that the wettability of a surface is a function of its roughness. Nonwetting liquids exhibit superhydrophobicity on a rough surface. It has been demonstrated that surfaces with micromachined structures can have similar effects.1,2 This phenomenon has many applications. In particular it is considered a viable option for surface tension induced drop motion (Figure 1) in microfluidic devices. Our goal is to develop a systematic procedure to design a rough substrate that will maximize superhydrophobicity for a given liquid-solid combination. This investigation focuses on stationary drops on rough substrates. The earliest work on this problem can be attributed to Wenzel3 and Cassie.4 They provided expressions for the apparent contact angle, based on certain average characteristics of the roughness. In Wenzel’s approach it is assumed that the liquid fills up the grooves on the rough surface (Figure 2a). We shall refer to this as the wetted contact with the rough surface. From energy considerations it can be shown that the apparent contact angle θrw is given by
cosθrw ) 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 of the liquid drop on the flat surface. In Cassie’s approach it is assumed that the liquid forms a composite surface on the rough substrate (Figure 2b), i.e., the liquid does not fill the grooves on the rough surface. We shall refer to this as the composite contact with the (1) Onda, T.; Shibuichi, N.; Satoh, N.; Tsuji, K. Langmuir 1996, 12 (9), 2125-2127. (2) Bico, J.; Marzolin, C.; Quere, D. Europhys. Lett. 1999, 47 (2), 220-226. (3) Wenzel, T. N. J. Phys. Colloid Chem. 1949, 53, 1466. (4) Cassie, A. B. D. Discuss. Faraday Soc. 1948, 3, 11.
Figure 1. Surface tension induced drop motion between two parallel plates. Higher contact angle (superhydrophobicity) on the left side of the drop induces higher curvature of the drop surface resulting in higher pressure. This causes the drop motion to the right.
Figure 2. (a) A drop fills the grooves of the rough surface. (b) The drop sits on the crests of the rough pattern forming a composite surface.
rough substrate. In this case
cos θrc ) φs cos θe + φs - 1
(2)
where θrc is the apparent contact angle assuming a composite surface and φs is the area fraction of the liquidsolid contact. Previous experiments are not conclusive regarding which formula to use and when. This information is critical in designing a superhydrophobic substrate for applications in microscale devices. In this paper we address this issue and propose a procedure to design a superhydrophobic rough substrate for a given liquid-solid combination. Only
10.1021/la026612+ CCC: $25.00 © 2003 American Chemical Society Published on Web 01/17/2003
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Figure 4. Formation of a drop by introducing the liquid from the bottom of the substrate. Figure 3. The apparent contact angle as predicted by the Wenzel and Cassie theories for a given rough surface.
the hydrophobic liquid-solid contacts are considered in this paper. In section 2 we present some discussion based on previous experimental results which indicate the presence of multiple equilibrium shapes of a drop on a rough substrate. In section 3 we will discuss the energy of the different equilibrium shapes. The design procedure will be proposed in section 4. Issues pertaining to different models for advancing and receding contact angles will be discussed in section 5 and conclusions in section 6. 2. Experimental Section Results Onda et al.1 prepared fractal surfaces and compared the measured values of the apparent contact angles with the predictions from the wetted surface theory. Some agreement with theory was claimed. Bico et al.2 prepared substrates of specified surface roughness and compared the observed apparent angle with theory. They claimed a good agreement between the composite surface formula and the observed apparent advancing contact angle. Although,this list of experiments reported in the literature is not claimed to be comprehensive, they highlight the key modeling issues. We shall briefly discuss these in the following. Consider a rough substrate with given values of r and φs. Figure 3a shows the plot of the apparent contact angle as a function of θe according to Wenzel’s formula. Only the hydrophobic angles are shown given the focus of this work. Since r > 1, the apparent contact angle is 180° when cos θe ) -1/r. Hence, Wenzel’s theory predicts that the apparent angle is 180° for θe > cos-1(-1/r). Onda et al.1 compared their experimental results for drops on fractal surfaces with this theory (they modified the expression to account for adsorption, but qualitatively, the plot remains similar to that in Figure 3a) and claimed some agreement. Similarly, one can plot the predicted apparent angles as a function of θe (Figure 3b) according to Cassie’s formula. Here the prediction is qualitatively different from that in Figure 3a. According to Cassie formula the apparent contact angle changes sharply at θe ) 90° and is 180° only when θe ) 180°. Bico et al.2 reported good agreement between theory and experiments and hence proposed that Wenzel’s formula be used in the hydrophilic region (θe < 90°) and Cassie’s formula be used in the hydrophobic region (θe < 90°). Bico et al.2 reported another interesting observation. They found that a drop whose apparent contact angle agreed with Cassie’s theory, when pressed physically, resulted in a drop shape with an apparent contact angle in agreement with Wenzel’s theory! The apparent contact angle changed from 170° to 130° in their case. This clearly indicates that both the equilibrium shapes (according to Wenzel and Cassie theories) are possible. It also suggests that the applicability of Wenzel’s or Cassie’s theory will depend on how the drop is formed. As a result one cannot assume Cassie’s formula when designing a superhydrophobic substrate because a transition from one equilibrium to another can cause a significant change in the apparent angle; 40° in the case above. If the transition occurs, then the surface may no longer be considered superhydrophobic (130° is hardly a superhydrophobic contact). The transition from one apparent angle to another can also be induced by motion, e.g., in surfacetension-induced drop motion.
Indeed, both Wenzel and Cassie formulas must be considered when designing a superhydrophobic substrate. A “robust” superhydrophobic surface will be the one where the apparent contact angle does not change when a drop makes a transition from a wetted surface to a composite surface, i.e., when cos θrw ) cos θrc. Second, the value of apparent contact angle should be as close to 180° as possible. One can thus design a substrate that fulfills these objectives. In the next section we will compare the energy of the drop in the two equilibrium conditions mentioned above. We will then develop an appropriate criterion to design robust superhydrophobic substrates for a given value of θe.
3. Energy Analysis Consider an experimental setup where a liquid is dispensed from a narrow tube to form a drop on the substrate (Figure 4). Assume that the effective energy per unit area of liquid-substrate contact with respect to the dry substrate is (σls - σvs)eff. Thus if the substrate is rough and a composite contact is formed between the liquid and the substrate then (σls - σvs)eff/σlv ) -φs[(σvs - σls)/σlv] + (1 - φs) ) -φs cos θe + (1 - φs), where σls is the liquidsolid surface energy per unit area, σvs is the vapor-solid surface energy per unit area, and σlv is the liquid-vapor (vapor is the surrounding air) surface energy per unit area. For a wetted contact we have (σls - σvs)eff/σlv ) r[(σls - σvs)/σlv] ) -r cos θe. The change in energy from the initial state (no drop) to the final state (a drop formed, see Figure 4), assuming that the diameter of the dispensing tube is negligible compared to the drop size, is given by
G ) Sσlv - cos θrA cosθr ) (σvs - σls)eff/σlv
(3)
where S is the area of the drop surface (liquid-vapor), A is the area of contact with the substrate projected on the horizontal plane, and gravity is ignored. For a composite contact we have θr ) θrc and for a wetted contact θr ) θrw. The final equilibrium state i.e., the values of S and A should be such that G is minimized for a given value of θr. Although well-known, we will formally show here for sake of clarity that this energy minimizing shape is such that the contact angle is θr. Let ψ be the contact angle of some drop that is formed on the substrate. We have to show that G is minimized when ψ ) θr. The problem statement is
Minimize:
G ) 2πa2(1 - cos ψ) - πa2 sin 2 ψ cos θr σlv (4)
subject to the volume constraint
g)V-
πa3 (1 - cos ψ)2(2 + cos ψ) ) 0 3
(5)
where appropriate expressions for S and A are substituted,
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a is the radius of the drop with a contact angle ψ, and V is the drop volume. The above problem statement can be rewritten as
minimize:
E)
G + λg σlv
(6)
where λ is the Lagrange multiplier. g ) 0, ∂E/∂a ) 0, and ∂E/∂ψ ) 0 give three equations for the three unknowns a, λ, and ψ.
2 cos ψ cos θr + 2cos θr - 4 ∂E ) 0 w aλ + ) 0 (7) ∂a (1 - cos ψ)(2 + cos ψ) and
2(cos ψ cos θr - 1) ∂E ) 0 w aλ + )0 ∂ψ a(1 - cos ψ)(1 + cos ψ)
(8)
Eliminating aλ from eqs 7 and 8 we obtain cos ψ ) cos θr, i.e., ψ ) θr for angles between 0° and 180°. g ) 0 then gives the value of a for a given value of V. Note that for hydrophobic substrates (θr > 90°), G is always positive. This implies that energy increases during the drop forming process i.e., work must be done. The above indicates that the equilibrium shape is such that the increase in energy is minimized for a given value of θr. A rough substrate can have at least two possible values of θr, namely, θrc corresponding to a composite contact and θrw corresponding to a wetted contact. Hence a drop of a given volume can attain two shapes corresponding to ψ ) θrc and ψ ) θrw, depending on the type of contact it forms with the substrate. Both these shapes represent local minimum in energy as shown in the derivation above. It remains to be analyzed which of these two equilibria has the lower energy. The energy of a drop of given volume in equilibrium on a substrate is given by
G 3
x9πV σlv 2/3
) (1 - cos θr)2/3(2 + cos θr)1/3
Figure 5. Prediction by Wenzel and Cassie formulas. For θe < θc, the wetting contact gives a lower apparent contact angle, and for θe > θc, the composite contact gives a lower apparent contact angle.
(9)
where appropriate expressions for S and A are substituted. The left-hand side denotes nondimensional energy. It can be easily verified that the right-hand side is a monotonically increasing function of θr for 0° e θr e 180°. As a result, an equilibrium drop shape with lower value of the apparent contact angle θr will have lower energy. If θrc > θrw, then the drop forming a composite contact will have higher energy than that of the drop with wetting contact and vice versa. Consider now the experimental results of Bico et al.,2 mentioned in the previous section, where they observed two equilibrium shapes for the same drop. In that case θrc ) 170° and θrw ) 130°, implying that the drop with wetted contact has lower energy. Yet the drop with a composite contact was formed first and had to be pressed down (in other words energy had to be added) to form a drop with wetting contactsan equilibrium position with lower energy according to the above analysis. The fact that the drop with a composite contact is formed first can be explained by noting that if a drop is formed by dispensing the liquid from above (as was done by Bico et al.2) then the equilibrium shape corresponding to a composite contact is encountered first (physically). Hence this is the shape that is obtained by Bico et al.2 in their experiments leading to their suggestion that Cassie’s formula should be applied for the hydrophobic case (Figure
Figure 6. (a) Filled-up grooves just before the liquid-solid contact is formed in the valleys. (b) Filled-up grooves after the liquid-solid contact is formed.
3b). Formation of a drop with a composite contact does not imply that the corresponding energy is the global minimum (although it is indeed a local minimum). This is evident in Figure 5 where we plot the apparent contact angle as a function of θe according to Cassie and Wenzel formulas. When θe ) θc, we have θrw ) θrc. Consider a substrate in one of the experiments by Bico et al.2 where r ) 1.3 and φs ) 0.05. For this case θc ) 139.5°. If we have a liquid with θe < 139.5°, then, on this substrate, a drop with a wetting contact has lower energy whereas for a liquid with θe > 139.5° a drop with a composite contact has lower energy. It is evident that the drop need not necessarily form a shape that has global minimum energy. The determining factor is how the drop is formed. This is consistent with the experimental results of Bico et al.2 Why then do we need to do work to transition the drop from θrc ) 170° to θrw ) 130° as in the experiments of Bico et al.?2 The drop is in fact going to a lower energy state. This can be explained in terms of an energy barrier between the two equilibrium positions. The concept of energy barrier between multiple equilibrium positions, for drops on rough substrates, was discussed by Johnson and Dettre.5 Here we discuss the energy barrier between drops shapes with composite and wetted contact with the substrate. Consider the transition from composite to wetted contact on the microscopic scale. For a composite liquid-substrate contact the effective energy of contact is (σls - σvs)effc/σlv ) -cos θrc ) -φs cos θe + (1 - φs), as seen before. The liquid must start filling the valleys or grooves of the substrate as the transition occurs. The actual details of the transition are not well understood but it is likely that some intermediate state will have higher energy than that corresponding to a composite or a wetted contact. As an example, we consider a state shown in Figure 6 where the liquid has filled the grooves but the liquid-solid contact at the bottom of the valley is yet to be formed. The effective energy in this intermediate state is (σls - σvs)effint/σlv ) (5) Johnson, R. E.; Dettre, R. H. Adv. Chem. Ser. 1964, 43, 112.
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Figure 7. Top view of one period of a roughness geometry of square pillars. The pillar cross-sectional size is a×a. A quarter of the pillar cross section is shown at each corner of the figure above.
Figure 8. A qualitative plot of the apparent contact angle, according to Wenzel and Cassie formulas, as a function of geometric parameters for a given value of θe.
-cos θrint ) -(r - 1 + φs)cos θe + (1 - φs). As soon as the liquid wets the bottom of the valley, the effective energy drops to the value corresponding to the wetting contact, i.e., (σls - σvs)effw/σlv ) -cos θrw ) -r cos θe. It is easily verified that θrint is always greater than θrc or θrw, irrespective of the relative magnitudes of θrc and θrw. This implies that this intermediate state, if it exists, represents an energy barrier. As a result, work may be required, as evidenced by the pressing of the drop by Bico et al.,2 to enable transition across the energy barrier. We will next discuss the procedure to design a robust superhydrophobic substrate that takes the above energy analysis into consideration. 4. Designing a Superhydrophobic Substrate We shall explain the design procedure by considering a geometry of square pillars of size a × a, height H, and spacing b arranged in a regular array (Figure 7). The contact angle θe on the flat surface is assumed to be given. What should be the value of b/a and H/a so that it will result in a robust superhydrophobic surface (as defined at the end of section 2) for a given value of θe? In Figure 8 we show a qualitative plot of the predicted values of the apparent contact angle, according to Wenzel’s and Cassie’s formulas, as a function of the geometric parameters of the surface roughness. The expressions are
cosθrc ) B(1 + cos θe) - 1
(
cos θrw ) 1 +
)
4B cos θe (a/H)
where
B)
1 (b/a + 1)2
(10)
It is seen that θrc, as expected, is independent of a/H. The
value of cos θrw at b/a ) 0 depends on a/H. Location of the intersection of the Wenzel and Cassie curves (where θrc ) θrw) go closer to -1 (see Figure 8) as the value of a/H becomes small, i.e., for tall slender pillars. The lower energy and higher energy segments of the Wenzel and Cassie curves are marked in Figure 8. The intersection point between the Wenzel and Cassie curves denotes the maximum value of the apparent contact angle (for a given value of a/H) among all the possible lower energy states. To obtain a robust substrate (i.e., minimize the difference between θrc and θrw), we need the point of intersection of the Wenzel and Cassie curves as the design condition. The more one moves away from the design condition, the greater the change in the apparent contact angle when the drop transitions from a composite contact to a wetted contact. To ensure superhydrophobicity the point of intersection should be as close to -1 as possible. This is ensured by using the smallest possible value of a/H. This value is usually determined by the capability of the fabrication process. In conclusion, the design procedure recommends slender pillars; then, given the smallest possible value of a/H, the design procedure further recommends that the periodic spacing (b/a)des between the pillars renders the substrate insensitive to whether a wetted contact is formed or a composite contact is formed. It is clear that a forest of nanopillars with appropriate spacing will be an excellent superhydrophobic surface. Nanofabrication process can enable very small values of a/H resulting in highly superhydrophobic surfaces according to the above theory. It is straightforward to apply the above procedure to different kinds of roughness geometries. A different objective function can lead to another design procedure. Other examples of objective functions could be to develop a superhydrophobic substrate for a range of values of θe or to develop a substrate such that the energy barrier is maximized (thus minimizing the possibility of transition). 5. Advancing and Receding Contact Angles Until now we have not discussed advancing and receding contact angles. Often, the flat surface itself does not have a unique static contact angle θe. There is usually an advancing contact angle θeadv and a receding contact angle θerec on a flat surface depending on whether a given drop is formed by an advancing front (i.e., by increasing the drop volume in steps) or by a receding front (i.e., by decreasing the drop volume), respectively. The above theory is expected to work well for advancing angles on rough substrates where we simply put θe ) θeadv in all the formulas above. This is in agreement with previous experimental results (see Bico et al.2). However, Bico et al.2 found that neither Wenzel’s or Cassie’s theory could predict the observed receding contact angle if we substitute θe ) θerec in the corresponding formulas. Bico et al.2 had a surface with pillars (similar to the one considered in Figure 7 above except that the pillar’s were circular in cross-section) with φs ) 0.05, θerec ) 100° and r ) 1.3. This gives θrw,rec ) 103° by Wenzel’s formula and θrc,rec ) 163.5° by Cassie’s formula. The observed angle was 155°, in disagreement with both the predicted values. This discrepancy can probably be explained as follows. Assume that a composite surface initially exists between the liquid and the rough surface (as in the case of experiments of Bico et al.2). As the contact line recedes, assume that it leaves behind a thin film of liquid on peaks of the pillars instead of leaving behind a dry surface (Figure 9). This possibility was also proposed by Roura and Fort.6 (6) Roura, P.; Fort, J. Langmuir 2002, 18 (2), 566.
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The presence of advancing and receding contact angles introduce one more possible objective function: to minimize hysteresis, i.e., to minimize the difference between the advancing and receding contact angles. Appropriate models for the receding contact angles must be investigated further before appropriate design procedures that account for such objective functions can be developed.
Figure 9. A receding composite surface on an array of pillars. (a) A dry surface is left behind as the liquid recedes. (b) A thin layer of liquid is left behind on top of the pillars as the liquid recedes.
From energy considerations it can be shown that the apparent contact angle in this case is given by
cosθrfilm,rec ) 2φs - 1
(11)
Equation 11 predicts θrfilm,rec ) 154.2° for φs ) 0.05 which is in excellent agreement with the observed value of 155° in the experiments of Bico et al.2 At this stage, the experiment by Bico et al.2 is the only controlled experiment, that we know of, to compare this theory with. It is not understood how the receding contact angle should be modeled if a wetted contact exists initially instead of the composite contact as considered above. More experiments are required to determine the conditions under which the above analysis is valid.
6. Conclusions In this paper we consider the modeling of hydrophobic contact angles on rough substrates. Wenzel (wetted contact) and Cassie (composite contact) theories present two possible equilibrium drop shapes on a rough substrates. Of these possibilities the drop shape with a lower apparent contact angle has the lower energy. The drop may not attain an equilibrium shape with the lowest energy. The drop formation process is important in determining the equilibrium drop shape. The drop can transition from one equilibrium to another provided it can overcome the energy barrier between the two states. A procedure to design a rough superhydrophobic substrate that takes these details into account is proposed. We discuss a model for the receding contact angle on a rough substrate based on the limited data in the literature. It is noted in the end that appropriate models for the receding contact angles on rough substrate must be investigated further before appropriate design procedures, which minimize hysteresis (i.e., minimize the difference between the advancing and receding contact angles), are developed. Acknowledgment. This work has been supported by DARPA (SymBioSys) grant (Contract N66001-01-C-8055) with Dr. Anantha Krishnan as monitor. LA026612+
