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Mimicking the Lotus Effect: Influence of Double Roughness Structures and Slender Pillars Neelesh A. Patankar Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, B224, Evanston, Illinois 60208-3111 Received June 3, 2004 Surface roughness is known to amplify hydrophobicity. The apparent contact angle of a drop on a rough surface is often modeled using either Wenzel’s or Cassie’s formulas. These formulas, along with an appropriate energy analysis, are critical in designing superhydrophobic substrates for applications in microscale devices. In this paper we propose that double (or multiple) roughness structures or slender pillars are appropriate surface geometries to develop “self-cleaning” surfaces. The key motivation behind the double structured roughness is to mimic the microstructure of superhydrophobic leaves (such as lotus). Theoretical analysis similar to that presented in the paper can be used to obtain optimal geometric parameters for the rough surface. The calculation procedure should result in surface geometries with excellent water repellent properties.
1. Introduction Hydrophobicity is amplified by surface roughness. Largely enhanced hydrophobicity is termed as superhydrophobicity. This is demonstrated by natural as well as micromachined surfaces.1-6 This phenomenon has many applications. It is considered a viable option for surface tension induced drop motion in microfluidic devices. Other application, considered to be of significant technological impact, is to make “self-cleaning surfaces”slike some leaves (e.g., lotus) that remain clean despite its surroundings. Superhydrophobic leaves are excellent model surfaces that have “self-cleaning” properties. The surfaces of such leaves are found to have roughness1 (Figure 1). Water drops do not spread and form beads on these leaves. These beadlike droplets easily roll off the surface of these leaves (exhibiting minimal hysteresis) thus cleaning it in the process. In this paper we propose with appropriate justification that double (or multiple) roughness structures or slender pillars are appropriate surface geometries to develop “selfcleaning” surfaces. The key motivation behind the double structured roughness is to mimic the microstructure of superhydrophobic leaves (such as lotus) (Figure 1). The primary focus of this work is to explain theoretically what is achieved by a double structured roughness, e.g., seen in lotus leaves. To this end we consider a model problem and show that it helps in achieving a high contact angle drop that also has less hysteresis. We will show that the same objectives can also be served by a slender (i.e., high aspect ratio) pillar geometry. Theoretical analysis similar to that presented in the paper can be used to obtain optimal geometric parameters for a rough surface. The earliest theoretical work to model the apparent contact angle of a drop on a rough surface can be attributed to Wenzel7 and Cassie.8 In Wenzel’s approach it is assumed (1) Barthlott, W.; Neinhuis, C. Planta 1997, 202, 1. (2) Onda, T.; Shibuichi, N.; Satoh, N.; Tsuji, K. Langmuir 1996, 12 (9), 2125. (3) Hazlett, R. D. J. Colloid Interface Sci. 1990, 137, 527. (4) Bico, J.; Marzolin, C.; Que´re´, D. Europhys. Lett. 1999, 47 (2), 220. (5) O ¨ ner, D.; McCarthy, T. J. Langmuir 2000, 16, 7777. (6) He, B.; Patankar, N. A.; Lee, J. Langmuir 2003, 19 (12), 4999. (7) Wenzel, T. N. J. Phys. Colloid Chem. 1949, 53, 1466.
Figure 1. (a) Microscopic image of the surface of a lotus leaf depicting the “double” rough structure. (b) A composite mercury drop on top of the leaf. Figures courtesy Professor W. Barthlott.
that the liquid fills up the grooves on the rough surface. 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
cos θrw ) r cos θe
(1)
where r is 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. (8) Cassie, A. B. D. Discuss. Faraday Soc. 1948, 3, 11.
10.1021/la048629t CCC: $27.50 © 2004 American Chemical Society Published on Web 08/06/2004
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Patankar
In Cassie’s approach it is assumed that the drop settles on the peaks of the roughness geometry (Figure 1b); i.e., the liquid does not fill the grooves on the rough surface. We shall refer to this as the composite contact with the 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. The Wenzel (wetted contact) and Cassie (composite contact) theories present two possible equilibrium drop shapes on a given rough substrate. Previous results6,9-11 have confirmed that indeed a droplet can be in either the wetted or the composite state on a rough surface depending on how it is formed. Energy analysis reveals that one of these two states has lower Gibbs free energy than the other.10 The geometric parameters of the rough surface determine which of these two states has lower energy.10 In this paper we will extend the results from our previous work10 to analyze the microstructure of superhydrophobic leaves such as lotus and to propose some appropriate geometries for self-cleaning objectives. 2. Experimental Observations of Superhydrophobic Leaves Water repellency of plant surfaces has been known for a long time. Water droplets are almost spherical on these leaves and can easily roll off (exhibiting minimal hysteresis) thus cleaning the surface in the process. This is usually referred to as the lotus effect because it is brilliantly exhibited by the leaves of the lotus plant (Nelumbo nucifera, Figure 1a). Barthlott and Neinhuis1 collected experimental data by photographing the microstructure of rough water repellent leaves. Their seminal work revealed for the first time that the interdependence between surface roughness, reduced particle adhesion, and water repellency is the keystone in the self-cleaning mechanism of many biological surfaces. In their experiments the plant leaves were artificially contaminated with various particles and subsequently subjected to artificial rinsing by sprinkler or fog generator. In the case of water repellent leaves, the particles were removed completely by water droplets that rolled off the surfaces, independent of their chemical nature or size. Water repellent leaves exhibit various surface sculptures.1 Of particular interest is the double structured rough surface like that of the lotus leaf (Figure 1a). The fine scale rough structure (200 nm to 1 µm), visible as a “hairy” surface of the leaf, is made of epicuticular wax crystalloids. The coarse scale rough structure (∼20 µm) is visible as bigger protrusions on the surface. The surface material is thus made of wax, which is not highly hydrophobic, yet a water drop on lotus leaf type surfaces typically show an apparent contact angle of about 160°. Clearly, nature seems to have optimized the roughness, possibly in the form of double structure, to achieve substantial amplification of hydrophobicity. It is the primary focus of this paper to quantitatively study the effect of such double structures by considering a model problem. It is evident from Figure 1b that a drop, placed on the double structured rough surface of the leaf, sits on top of the bigger bumps (i.e., forms a composite drop).1 The drop (9) Lafuma, A.; Que´re´, D. Nat. Mater. 2003, 2 (7), 457. (10) Patankar, N. A. Langmuir 2003, 19 (4), 1249. (11) Bico, J.; Thiele, U.; Que´re´, D. Colloids Surf., A 2002, 206, 41.
easily rolls off the surfaces of the leaves thus showing very little hysteresis. It has been observed experimentally9 that a composite drop has much less hysteresis than a wetted drop. It seems apparent, therefore, that the objective of the double structured roughness is not only to amplify the hydrophobicity but also to ensure that a composite drop is formed so that hysteresis is minimized. In the next section we will consider a model geometry to understand the above observations. In this paper we do not focus on the chemistry of reduced particle adhesion on a self-cleaning surface. 3. Mimicking the Lotus Effect A few comments are in order before we present our theoretical analysis. A liquid contact on a surface typically has an advancing contact angle corresponding to an advancing front and a receding contact angle corresponding to a receding front. The value of θe (see eqs 1 and 2) obtained by simply depositing a drop on a surface is usually between the advancing and receding values. In the interest of simplicity, we will consider θe in our analysis. Correspondingly, θrw and θrc (eqs 1 and 2) are not the advancing or receding contact angles on the rough surface but are intermediate values. Previous experimental results have shown that θrw and θrc are good representations of the hydrophobicity of a rough surface.4,6,10 The advancing and receding contact angles on rough surfaces do however affect the ability of a drop to roll off the surface. Experiments have shown9 that a composite drop rolls off easily (i.e., has much less hysteresis or the difference between the advancing and receding contact angles is small) compared to a wetted drop on a rough surface. In our analysis, we will not explicitly model the advancing and receding contact angles on a rough surface. Nevertheless, it will be evident from our presentation below that we will indirectly account for the phenomenon of hysteresis by requiring that a composite drop is preferred by a self-cleaning rough surface so that hysteresis is minimized. In Figure 2a we depict a model fine scale roughness structure made of square pillars arranged in a regular array. This fine scale roughness in turn forms the surface of the coarse scale roughness which is also modeled to have the same geometry as the fine scale structure, i.e., a regular array of square pillars. We shall refer to the fine scale structure as the first generation and the coarse scale structure as the second generation of roughness geometry. Note that the model geometry is considered for the simplicity of exposition and yet captures the key physical insights. The same analysis can be repeated for more complex geometriessthe key conclusion would still remain the same. At the first generation, let the square pillars be of size a1 × a1 and height H1. Let the periodic spacing of the regular array be b1 (Figure 2a). If a drop is placed on this surface (without the coarse scale roughness features), in general, two drop shapes corresponding to the wetted and composite cases are possible. The apparent contact angles are given by10
cos θrc ) A1(1 + cos θe) - 1
(
cos θrw ) 1 +
4A1 (a1/H1)
)
cos θe
where
A1 )
1 ((b1/a1) + 1)2
(3a) (3b)
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Figure 3. Plot of the apparent contact angles for wetted and composite drops as a function of roughness geometry of the first generation. All states for which cos θ is less than -1 are physically unrealizable according to the theoretical model used here.
Figure 2. A model roughness geometry for theoretical analysis. (a) The fine scale roughnesssthe first generation. (b) The fine scale roughness forms the surface of the coarse scale pillarss the second generation of roughness. (c) The pillar geometry at both scales is assumed periodic. The top view of one period is shown. The pillar cross-sectional size is a × a. In our analysis, we use subscripts “1” and “2” to denote the geometric parameters for the first and second generation, respectively.
θe is the equilibrium contact angle of the liquid drop on a flat surface made of the surface material. If we consider water and the surface material to be epicuticular wax (as in the case of lotus leaves), then θe ) θwax. The equilibrium contact angle for water on cuticular wax is given as 100°.12 The advancing contact angles of water measured on paraffin wax are in the range of 108-111°.13 Our objective here is to show that double roughness structure greatly amplifies the apparent contact angle. To this end we will consider a weakly hydrophobic surface and assume θe ) 95° for the purposes of our calculation. A larger value of θe will result in even more amplification of the apparent contact angle compared to that obtained in the calculations below. At the second generation of roughness, let the square pillars be a2 × a2 and of height H2. Let the periodic spacing be b2. Once again if a drop is placed on this doubly rough substrate (Figures 1a or 2b), water might fill up the grooves of the coarse scale geometry or the drop might sit on top of the “coarse” pillars (e.g., see Figure 1b). The corresponding apparent angles can be found using the same formulas as eq 3 with subscripts “1” replaced by “2”. Also, θe should be replaced by the apparent contact angle of a drop on the first generation of rough surface. To mimic the Lotus effect we need: 1. A composite drop formed on the coarse scale roughness. This ensures that the drop has minimum hysteresis and may roll-off easily. A wetted drop exhibits much more hysteresis (about 10 times) as compared to a composite drop even if the apparent contact angles are same.9 Note that we do not impose any condition on whether water wets the fine scale grooves or not. If a composite drop is formed on the coarse scale structure, the contact area of the drop with the fine scale structure is small. Hence, (12) Holloway, P. J. J. Sci. Food Agric. 1969, 20, 124. (13) Jan˜czuk, B.; Biau`opiotrowicz, T. J. Colloid Interface Sci. 1989, 127, 189.
even if water wets the fine scale structure it is not expected to have a significant effect on hysteresis. 2. We need a composite drop with the apparent contact angle as high as possiblesto obtain superhydrophobicity. Of course, a wetted drop is also possible. Given that the lotus leaf type surfaces are known to form only composite drops,1 we will assume that the composite state should represent the global minimum in energy for self-cleaning surfaces. Hence, it must be ensured that the geometric parameters of the rough surface are such that the composite drop has lower energy than the wetted drop. This implies that the apparent contact angle of the composite drop should be less than the apparent contact angle of the wetted drop.10 In such a case, even though the wetted drop has a larger apparent contact angle, it should be avoided because it leads to more hysteresis.9 In the following we will analyze how the double rough structure helps in achieving the objectives identified above. Specifically, as an example, we will set the goal of amplifying the contact angle from θe ) 95° to 160°. 3.1. Designing the Fine Scale Structure. Let us first consider the effect of the fine scale structure (first generation of roughness) on the apparent contact angle of the drop. To this end, we assume that the coarse scale structure (second generation of roughness) is not present. Figure 3 shows a plot of the apparent contact angles for wetted and composite drops as a function of the roughness geometry of the first generation. Since our objective is to achieve an apparent angle of 160°, it is depicted as the “intended line” corresponding to cos(160°) in the plot. As indicated by eq 3, the plot for the composite case depends only on b1/a1 (Figure 3). The wetted case depends on both b1/a1 and H1/a1. In Figure 3 we have depicted the wetted case for three values of H1/a1 plotted as a function of b1/a1. We note that cosine of the apparent angle θ is also a measure of energy of the drop10
G ) (1 - cos θ)2/3(2 + cos θ)1/3 (9π) V2/3σlv 1/3
(4)
where G is the energy of the drop, V is the drop volume, and σlv is the liquid-air surface tension. The larger the value of cos θ (i.e., the smaller the value of θ), the lower the energy of the drop. Although the cosine of an angle can never be less than -1, it is well-known that such values do fall out of Wenzel equations 1 and 3b if the surface is very rough. Physically this implies that it is energetically expensive to wet the grooves of the surface.10 A wetted equilibrium drop shape is not possible per the current theoretical model. Only a
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composite drop is possible in these conditions. Alternate wetted drop models have been proposed2 that essentially lead to a contact angle of close to 180° whenever the Wenzel equation gives cos θrw < -1. A wetted drop with a contact angle of 180° is also not realistic since the drop barely contacts the substrate and hence could not possibly wet the grooves. Thus we will regard a wetted drop physically unrealizable whenever cos θrw < -1. 3.1.1. Designing a Substrate without the Need for Double Structure. Let us first say, we wish to achieve objectives 1 and 2 stated above without the need for a double roughness structure. According to Figure 3, a composite drop will have an apparent angle of 160° if b1/ a1 is 2.89 (the point of intersection of the composite case curve and the intended line). To ensure that this composite case has lower energy than the wetted case, the apparent angle predicted by the wetted case, for b1/a1 ) 2.89, must be larger than 160°. For this to be true, eq 3 (also see Figure 3) predicts that we need H1/a1 > 37. If H1/a1 ) 37, then both the composite and wetted drops have the same apparent angle (Figure 3) and therefore have the same energy at b1/a1 ) 2.89 (however the hysteresis for the wetted case will be much larger). Smaller values of H1/a1 will imply that, for a substrate with b1/a1 ) 2.89, the wetted case will be at lower energy since it predicts a lower apparent contact angle (this is evident for the case with H1/a1 ) 5, depicted in Figure 3). If a drop transitions to the wetted case under such conditions, then the superhydrophobic behavior is lost in addition to the increase in hysteresis. We also note that merely having H1/a1 > 37 need not be enough to ensure a robust hydrophobic surface. To prevent the composite drop from transitioning to a higher energy wetted case, due to external “disturbances”, the energy difference between these two states should be increased. This implies that cos θ as predicted from the Wenzel equation should be much smaller than cos(160°), preferably less than -1 so that the wetted drop is physically unrealizable. For b1/a1 ) 2.89 this will be ensured if H1/a1 > 40. These are extremely high pillar aspect ratios to be achieved using microfabrication technology. Nanotechnology does provide such possibilities where, e.g., nanopillars can have aspect ratios as high as 1000. Suppose we have a substrate decorated by nanopillars with H1/a1 ) 100. As seen above, the periodic spacing of the pillars should be b1/a1 ) 2.89 to get an apparent angle of 160° for a composite drop on the substrate. A smaller value of b1/a1, i.e., a more compact assembly of nanopillars, is undesirable because it will reduce the hydrophobicity of the surface. Increasing b1/a1 will marginally increase the hydrophobicity as seen from Figure 3, but this comes at a price. Even if the composite drop continues to have lower energy for larger values of b1/a1, the energy difference between the wetted and composite cases decreases. The value b1/a1 cannot be increased indefinitely because at a “critical” value the composite and wetted cases will have the same energy. For H1/a1 ) 100, this “critical” value of b1/a1 ) 5.26 (using eq 3). The corresponding apparent angle is 168°. For b1/a1 > 5.26, the composite drop has higher energysa condition to be avoided. Thus slender or high aspect ratio pillars with appropriate spacing can be excellent water repellent surfaces. 3.1.2. Designing the First Generation of a Double Structure. High aspect ratios are not always reasonable to fabricate. In this case the required objective can be achieved in two steps. Let us assume that the maximum possible aspect ratio based on a fabrication process is H1/ a1 ) 5. The objective to be realized in the first generation
Patankar
Figure 4. Plot of the apparent contact angles for wetted and composite drops as a function of roughness geometry of the second generation. The apparent contact angle due to the first generation is taken to be 133.3°. All states for which cos θ is less than -1 are physically unrealizable according to the theoretical model used here.
should be to amplify the apparent contact angle as high as possible. The best way to ensure that is to pick the value of b1/a1 such that the apparent contact angle is maximized along the lower energy segments.10 This point corresponds to the intersection of the composite and wetted curves. For the parameters we have (b1/a1)des ) 0.7 and the corresponding apparent contact angle θint ) 133.3° (see Figure 3). The apparent angle is thus amplified from 95° to 133.3° in the first generation. The liquid may have composite or wetted contact with the first generation of roughness because both configurations have the same energy. Even if a wetted contact is formed with the first generation, it need not imply more hysteresis for a drop on the double rough structure. That is because, if a composite drop is formed on the coarse scale structure, the contact area of the drop with the fine scale structure will still be small. Next, we will consider the effect of the second generation of roughness. In that analysis we will assume that the surface of the coarse scale structure is made of an “equivalent material” with a equilibrium contact angle θe ) θint. 3.2. Designing the Second Generation of Roughness. Figure 4 shows the plot of the apparent contact angles for wetted and composite drops as a function of roughness geometry of the second generation. The composite and wetted case curves are plotted using eq 3 but for the second generation. In the formulas, θe is replaced by θint. The wetted case is drawn for H2/a2 ) 1, values typically observed in superhydrophobic leaves.1 The “intended line” is also shown. To achieve the design objectives, we select b2/a2 such that the composite drop has an apparent angle of 160°. At the point of intersection of the composite case curve and the intended line, b2/a2 ) 1.28. Let us say we choose H2/a2 ) 1. We must now ensure that, for b2/a2 ) 1.28 and H2/a2 ) 1, the energy of the wetted case is higher than that of the composite case. From Figure 4 it is evident that this condition is satisfied (cos θrw ) -1.21)sin fact the wetted drop is physically unrealizable. It should be noted that it was not possible to satisfy this condition in the first generation at such low aspect ratio of the pillars. It has become possible in the second generation because of the amplification of the apparent contact angle in the first generation of roughness. To emphasize this point further, we will consider the example of lotus in the following. Consider the geometry of the lotus leaves. Let us assume that the coarse scale roughness is the only one that causes amplification of the contact angle i.e., the fine scale roughness plays no role in causing superhydrophobicity.
Mimicking the Lotus Effect
The typical geometric parameters of the coarse scale roughness are b2/a2 ∼ 0.5-1 and H2/a2 ∼ 1,1 similar to the values chosen in Figure 4. We realize that the coarse scale roughness geometry of lotus is paraboloid instead of square pillars. The arguments to follow will nevertheless provide a reasonable estimate. Since we assume the fine scale roughness to be inconsequential, we have θint ) θwax ) 110°. We have assumed an optimistic value of θwax. If we choose b2/a2 ∼ 0.5, then we get θrc ) 135° and θrw ) 161°, whereas if b2/a2 ∼ 1, then θrc ) 147° and θrw ) 133°. Clearly, these estimates do not explain a composite drop of apparent contact angle of 160° on the coarse scale roughness of lotus leaves.1 In addition, transition to a wetted drop is possible especially when b2/a2 ∼ 1, where the wetted drop has lower energy than the composite drop. In experiments composite drops are formed consistently.1 Therefore, we propose that the fine scale most likely plays a significant role in the lotus effect. Our theoretical analysis above indicates that high aspect ratios of wax crystals and/or the double roughness structure could be contributing to the amplification of the apparent contact angle. In addition, these geometries also help in ensuring that only a composite drop is formed by making the wetted drop physically unrealizable. 4. Conclusions In this paper we show that double (or multiple) roughness structures or slender pillars are appropriate surface geometries to develop “self-cleaning” surfaces. The key motivation behind the double structured roughness is to mimic the microstructure of superhydrophobic leaves (such as lotus).
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The calculation process presented in this paper can be used to obtain optimal surface geometries to fabricate selfcleaning substrates. The analysis can also be generalized for a fractal surface or multiple roughness structures. We have used nondimensional parameters in our analysis. We have not considered the influence of gravity on drop shapes. Thus, the theory is applicable at millimeter and smaller length scales as long as the contact angles can be meaningfully defined. This is supported by previous experiments.4,6,9,10 Note that at length scales on the order of a nanometer the theory used here may not be applicable. The main conclusion of our analysis is that double roughness structures or slender pillars help in amplifying the apparent contact angle. More importantly, it also helps in making the composite drop energetically much more favorable. In fact the wetted drop can be made physically unrealizable with appropriate choice of parameters. This will ensure that a composite is always formed. A composite drop is critical in ensuring that the drop can roll off easily from the rough surfacesan important property for selfcleaning objectives. Acknowledgment. This work has been supported by DARPA (SymBioSys) grant (Contract No. N66001-01-C8055) with Dr. Anantha Krishnan as monitor. The author thanks Professor David Que´re´ for insightful discussion and Professor W. Barthlott for providing pictures in Figure 1. LA048629T
