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Contact Line Shape on Ultrahydrophobic Post Surfaces Christian Dorrer and Ju¨rgen Ru¨he* Laboratory for the Chemistry and Physics of Interfaces, Department of Microsystems Engineering, UniVersity of Freiburg, Georges-Ko¨hler-Allee 103, D-79110 Freiburg, Germany ReceiVed September 5, 2006. In Final Form: December 22, 2006 In this work, we investigate the configuration of the contact line of a water drop lying on an ultrahydrophobic post surface using the numerical algorithm Surface EVolVer. For the special situation of Cassie wetting, we propose a modified definition of the contact line as the line in space where the meniscus starts to curve upward out of the plane of the composite surface. In our simulations, it is found that the contact line is very strongly distorted, indicating a strong tendency of the drop to “ball up” in those areas where it is not in contact with the solid surface. The distortion of the contact line corresponds to a pronounced deformation of the liquid-air interface around the base of the drop. We discuss the consequences of this distortion for the definition and practical measurement of the contact angle on ultrahydrophobic surfaces.
Introduction and Theory On ultrahydrophobic surfaces, impinging water drops assume a ball-like shape and quickly roll off even at very low tilting angles. In most cases, ultrahydrophobicity is associated not only with a specific surface chemistry, but also with surface roughness. The size scale and structure of the roughness features is such that, for a given surface chemistry, drops are suspended on top, with air trapped underneath. Drops can thus be seen as resting on a carpet composed of solid and air. For such a situation, Cassie’s original theory describes the static contact angle by the following expression:1
cos θr ) φ cos θe + φ - 1
(1)
where φ is the solid fraction, that is, the fraction of the drop footprint area in contact with solid, θr is the contact angle on the rough surface, and θe is the equilibrium contact angle on the smooth material. Over the past years, surfaces equipped with regularly arranged microposts have established themselves as model systems for the wetting of ultrahydrophobic surfaces.2-9 One aspect that has been studied is the question of the contact angle hysteresis in the Cassie state. In early attempts, the Cassie equation was used to derive the dynamic contact angles from the corresponding values measured on the smooth material; however, strong discrepancies with experimental results were observed.6 More recently, the influence of the contact line structure on the dynamic contact angles has been emphasized. It has been suggested that, in the Cassie state, the contact line can jump back and forth between metastable states more easily, leading to a destabilization of the liquid meniscus.4,10 Other models have implicitly assumed a straight contact line, introducing criteria such as the fraction of * Corresponding author. E-mail:
[email protected]. (1) Cassie, A. B. D.; Baxter, S. Trans. Faraday Soc. 1944, 40, 546. (2) Yoshimitsu, Z.; Nakajima, A.; Watanabe, T.; Hashimoto, K. Langmuir 2002, 18, 5818. (3) Bico, J.; Marzolin, C.; Que´re´, D. Europhys. Lett. 1999, 47 (2), 220. (4) O ¨ ner, D.; McCarthy, T. J. Langmuir 2000, 16, 7777. (5) Gao, L.; McCarthy, T. J. Langmuir 2006, 22, 2966. (6) He, B.; Patankar, N. A.; Lee, J. Langmuir 2003, 19, 4999. (7) Patankar, N. Langmuir 2003, 19, 1249. (8) Ishino, C.; Okumura, K.; Quere, D. Europhys. Lett. 2004, 68, 419. (9) Patankar, N. Langmuir 2004, 20, 7097. (10) Chen, W.; Fadeev, A. Y.; Hsieh, M. C.; O ¨ ner, D.; Youngblood, J. P.; McCarthy, T. J. Langmuir 1999, 15, 3395.
the contact line supported by solid.11,12 In a recent work, we have investigated the influence of the geometric parameters of an ultrahydrophobic post surface on the dynamic contact angles.13 On a flat, homogeneous surface, a drop of liquid will assume a circular footprint. This results in the typical spherical cap shape for which the interfacial energy of the droplet is minimized. In contrast, on rough or heterogeneous surfaces, this is no longer the case: the footprint of the drop is distorted from the ideal circular shape by the underlying surface structure. This effect has been studied for a number of chemically heterogeneous surfaces; for example, the “faceting” of drops has been observed.14 Also, pinning on wettability defects has been shown to lead to substantial deformations of the contact line.15-17 Several authors have looked at drops on surfaces composed of alternating hydrophilic/hydrophobic stripes, where the contact line was found to undulate with the periodicity of the stripe pattern.18-22 In a theoretical work, Schwartz and Garoff computed the shape of the contact line on a flat surface with patches of varying wettability and found deviations from the straight line configuration.23,24 Schwartz has recently presented a simulation of the problem, finding a distortion of the drop base.25 In summary, the cited studies underline the fact that local variations in the wettability of a surface give rise to a distortion of the contact line: hydrophilic regions are wetted preferably, leading to a more forward position of the contact line; in hydrophobic regions, in contrast, the contact line is held back in a more receded position. Ultrahydrophobic post surfaces can be seen as a special case of chemical heterogeneity: drops are resting on a carpet composed of solid and air, spanning over the gaps between the roughness (11) Extrand, C. W. Langmuir 2002, 18, 7991. (12) Extrand, C. W. Langmuir 2004, 20, 5013. (13) Dorrer, C.; Ru¨he, J. Langmuir 2006, 22, 7652. (14) Cubaud, T.; Fermigier, M. Europhys. Lett. 2001, 55, 239. (15) Cubaud, T.; Fermigier, M.; Jenffer, P. Oil Gas Sci. Technol. 2001, 56, 23. (16) Marsh, J. A.; Cazabat, A. M. Phys. ReV. Lett. 1993, 71, 2433. (17) Paterson, A.; Fermigier, M.; Jenffer, P.; Limat, L. Phys. ReV. E 1995, 51, 1291. (18) Pompe, T.; Herminghaus, S. Phys. ReV. Lett. 2000, 85, 1930. (19) Drelich, J.; Wilbur, J. L.; Miller, J. D.; Whitesides, G. M. Langmuir 1996, 12, 1913. (20) Gau, H.; Herminghaus, S.; Lenz, P.; Lipowsky, R. Science 1999, 283, 46. (21) Lenz, P.; Lipowsky, R. Phys. ReV. Lett. 1998, 80, 1920. (22) Brinkmann, M.; Lipowsky, R. J. Appl. Phys. 2002, 92, 4296. (23) Schwartz, L. W.; Garoff, S. J. Colloid Interface Sci. 1985, 106, 422. (24) Schwartz, L. W.; Garoff, S. Langmuir 1985, 1, 219. (25) Schwartz, L. W. Langmuir 1998, 14, 3440.
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features. The contact angle with air is generally assumed to be 180° (this is the assumption leading to eq 1); the wettability contrast is therefore very pronounced. We thus expect that the liquid will try to recede in those regions where the meniscus is spanning over air. In contrast, on the solid parts of the surface, the meniscus will remain in a more advanced position. It can be expected that this effect will give rise to a considerable deformation of the overall liquid-air interface in the region around the drop base. Additionally, it is likely that, specifically on post surfaces, geometric pinning on post edges will have an influence on the movement of the meniscus over the surface. The description of the behavior of Cassie drops on microstructured surfaces is inconclusive as far as the precise shape of the contact line is concerned: several authors have investigated, for example, the wetting morphology inside rectangular channels, concentrating, however, on drops in the Wenzel state (i.e., drops penetrating the surface structure).26 Patankar et al. computed the overall shape of drops sitting above and inside rectangular grooves.27 Porcheron et al. used a lattice gas model to investigate the coexistence of Wenzel and Cassie states on a post surface.28 Also on the microscopic scale, Zhang et al. concentrated on the simulation of the motion of a droplet between two superhydrophobic walls, reporting a stick-jump behavior.29 However, these authors restricted themselves mainly to side views of the liquidair interface, as did Dupuis et al.30 Lundgren et al. performed molecular dynamics simulations of drops on post surfaces with the dimensions of the posts and drops in the nanometer range, where high Laplace pressures lead to a penetration of the post structure by the liquid.31 Chatain et al. recently used the Surface Evolver to study the shape of drops on surfaces with posts and holes; however, the surface geometry and the relation in size between the drops and the roughness features were very different from the post surfaces in question here.32 In this work, we numerically compute the shape of the contact line on ultrahydrophobic surfaces with microscale posts. The distortion of the overall liquid-air interface is taken into account. To gain insight on the mechanisms of droplet motion, we look at how the shape of the contact line changes as the meniscus moves over the surface. In the following section, the simulation method is first briefly discussed. We then proceed to present the simulation results.
Simulation The Surface Evolver. For the numerical computation, the public domain algorithm Surface Evolver (SE), developed by Brakke in the early 1990s,33 was chosen. The SE offers the possibility to study the shape of surfaces under the influence of quantities such as surface tension and gravity. Applications include simulations of the water-air interface, but also include investigations of other problems such as, for example, the geometry of solder joints.34 In the SE, an initial surface is specified by the user: this includes the definition of the simulation geometry, the specification of the respective interfacial energies, and the setting (26) Seemann, R.; Brinkmann, M.; Kramer, E.; Lange, F.; Lipowsky, R. Proc. Natl. Acad. Sci. U.S.A. 2005, 102, 1848. (27) Patankar, N.; Chen, Y. Nanotechnology 2002, 1, 116. (28) Porcheron, F.; Monson, P. A. Langmuir 2006, 22, 1595. (29) Zhang, J.; Kwok, D. Y. Langmuir 2006, 22, 4998. (30) Dupuis, A.; Yeomans, J. M. Langmuir 2005, 21, 2624. (31) Lundgren, M.; Allan, N. L.; Cosgrove, T. Langmuir 2003, 19, 7127. (32) Chatain, D.; Lewis, D.; Baland, J.-P.; Carter, W. C. Langmuir 2006, 22, 4237. (33) Brakke, K. Exp. Math. 1992, 1, 141. (34) Martino, P.; Freeman, G.; Racz, L.; Szekely, J. Predicting solder joint shape by computer modeling. In Proceedings of the 44th Electronic Components and Technology Conference,Washington, DC, 1994.
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Figure 1. Geometry of a quadratic post surface. The shadowed area indicates one-half of a possible unit cell. Dotted lines indicate planes of symmetry. Symmetries can be found at a 90 or 45° orientation to the geometry of the post surface.
of boundary conditions. The surface is discretized and approximated by a mesh of triangular facets. Each facet is defined by edges and vertices. Subsequently, the shape of the surface is iteratively “evolved” toward the free energy minimum by a gradient descent method. In each iteration step, the forces acting on each vertex are calculated from the free energy gradient. Taking boundary conditions into account, the vertices are then moved accordingly. For mathematical details, the reader is referred to Brakke’s SE Manual.35 In the SE, a continuum model is implemented; the finite extension of the liquid-air interface is therefore neglected. The wetting interactions are assumed to be completely described by surface energies, and the contact angle follows as a secondary quantity. While gravity is included, other effects such as the line tension τ are not taken into account. However, it has been pointed out that, due to the small value of τ, its influence can be neglected at length scales larger than the nanometer range (our geometry is on the size scale of a few tens of micrometers).18 As has been shown, the local contact angle is then given by the equilibrium (i.e., Young) contact angle on the respective material.36,37 Simulation Model. In our particular model, only a part of the overall meniscus shape was simulated in order to reduce the computational complexity. The quadratic post surface is characterized by several planes of symmetry, as shown in Figure 1. Making use of this symmetry allowed us to restrict our analysis to one-half of the unit cell. The relevant geometric parameters of the surface (i.e., the post width d and the post spacing s) are also indicated in Figure 1. In the starting position, the liquid meniscus was assumed to be resting on the surface as shown in Figure 2, which means we restricted ourselves to a slice of the water front resting on the post surface (drop diameter ) 2 mm). We simulated the shape of the liquid-air interface for two orientations of the water front to the post structure (Figure 3): (i) for the case where the water front was orientated at an angle of 90° to the post surface and (ii) for the case where the orientation angle was 45°. Both situations are extremes that will only appear at a few places for a given droplet. However, all other configurations can be seen as combinations of these two cases. We therefore think that this approach represents a useful approximation. Comparison with a Model Surface. In a preliminary step, the contact line shape on a flat surface composed of alternating (35) Brakke, K. Surface EVolVer Manual; Susquehanna University: Selinsgroove, PA, 1998. (36) Wolansky, G.; Marmur, A. Langmuir 1998, 14, 5292. (37) Marmur, A. Soft Matter 2006, 2, 12.
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Figure 2. Schematic depiction of a droplet slice resting on a post surface for an orientation of the water front relative to the post geometry of 90° (not to scale). σLV denotes the liquid-vapor interface. (a) In this region, we can adhere to the usual definition of the contact line as the line in space where the three phases (liquid, air, and solid) meet. (b) Where the meniscus spans over the gap between two posts, we adopt a modified definition where the contact line is defined as the line in space where the meniscus starts to curve out of the z ) 0 plane. Figure 4. Shape of the meniscus lying on a surface composed of alternating hydrophilic/hydrophobic stripes. Top: Simulation. The areas appearing in light gray are the result of mirroring the simulation results on the respective symmetry plane. Bottom: Microscope image of the contact line on a model surface prepared in our laboratory.
Figure 3. Drop lying on a post surface (not to scale). We look at the contact line shape in those places where the meniscus is oriented at 90 and 45° to the post geometry.
hydrophilic/hydrophobic stripes was simulated. In parallel, a model surface was prepared by dip-coating a thin film of polystyrene-co-methacryloyloxybenzophenone onto silicon slides and crosslinking the polymer layer by UV irradiation. The film thickness was around 10 nm (as determined by ellipsometry). Three hundred micrometer-wide stripes of the polymer were generated by photoablation through an aluminum mask. (A detailed description of the surface modification process can be found elsewhere.)13,38 The contact line shape of a water front lying on the surface was observed through an optical microscope. Images were taken for the case of a drop that had receded over the surface: the contact angles were θRec ) 70 ( 2° on the polymer and θRec ) 10 ( 2° in those areas where the polymer had been ablated. (For comparison, the static angles were 90 ( 3° and 15 ( 4°, respectively.) In the simulation, identical parameters were assumed. Figure 4 shows the simulation results together with an optical micrograph of the contact line. Good agreement between the experimental data and the simulation is found.
Results and Discussion Before presenting the simulation results for (now threedimensional) post surfaces, it becomes necessary to comment on one important aspect: on a flat surface, the contact line is defined as the line in space where the three phases (i.e., liquid, air, and solid) meet. In this definition, for an ultrahydrophobic post surface, we have multiple closed contact lines where the drop is in contact with the individual roughness features, that is, around the top of each wetted post. In contrast, where the meniscus spans over the void between two posts, there can only be a two-phase contact (i.e., between liquid and air; see index b in Figure 2). The usual (38) Mock, U.; Michel, T.; Tropea, C.; Roisman, I.; Ru¨he, J. J. Phys.: Condens. Matter 2005, 17, 595.
definition of the contact line is therefore meaningless for a description of the movement of the meniscus in these regions. In this work, we suggest a modified definition: the drop is assumed to lie flatly on a composite (i.e., Cassie) surface composed of solid and air. Now, the contact line is defined as the line in space where the interface of the drop starts to curve upward out of the plane of the composite surface (Figure 2). Mathematically speaking, letting z ) z(x,y) be the free surface describing the interface of the drop, we position the coordinate system in such a way that the drop underside (i.e., the footprint) is defined by z(x,y) ) 0. The contact line is given by the line in space where z(x,y) ) 0 and grad(z(x,y)) * 0. This definition includes those parts of the contact line where it crosses over the posts (index a in Figure 2) as well as those parts where it spans over the gap between two posts (index b in Figure 2). We think that the assumption of the drop lying flatly on the composite surface is valid for the following reasons: First, for hydrophobic solids, as in our case, the meniscus cannot move down into the post structure since the advancing contact angle is not reached on the side of the posts; from an energetic point of view, such a movement of a drop into a post structure has been shown to be associated with an increase in the free energy for hydrophobic solids (in our case, θe ) 120°).8,9 Second, according to the condition of constant Laplace pressure within the drop, the curvature of the meniscus should be constant around the entire drop interface. Calculations based on this assumption show that, for example, for a drop 2.7 mm in diameter, the meniscus between two posts spaced at 32 µm sags by only ∼50 nm, which can be safely neglected. First results were obtained for a surface decorated with posts 32 µm wide and 32 µm apart. The equilibrium contact angle on the solid parts of the surface, θe, was set to 120°. Figure 5A shows the liquid meniscus resting on a post. For the further discussion, we adopt a depiction of the contact line as illustrated in Figure 5B, that is, the observer looks at the contact line from above. In Figure 6, the silicon posts are indicated by quadratic outlines, while the rest of the surface is composed of air. In those regions where the surface is meshed (upper part), the liquid-air interface is curving away from the post structure (z(x,y) > 0). Where the surface appears white (below the contact line), the interface is lying in the z ) 0 plane (z(x,y) ) 0). The contact line
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Figure 5. (A) Side view of the liquid meniscus resting on a micropost (the post has been cut in half). σLV denotes the liquid-air interface. Local contact angles are 180° (index a), between 120 and 180° (index b), or 120° (index c). (B) For further analysis, the shape of the meniscus is projected onto the xy-plane. Arrow indicates the direction of view.
Figure 6. (a) Shape of the contact line on an ultrahydrophobic post surface. The observer looks into the negative z-direction. Where the area appears meshed, the meniscus is curving away from the plane of the composite surface. σLS denotes the liquid-solid interface, and σLV denotes the liquid-air interface. (b) Shape of the contact line for an orientation angle of 45°.
separates the two regions, indicating where the meniscus leaves the z ) 0 plane. We restricted our simulation to the section of the contact line between symmetry planes S1 and S2 (appearing
Dorrer and Ru¨he
in dark gray); the sections to the left of S1 and to the right of S2 (appearing in light gray) are the result of mirroring the obtained shape on the respective symmetry planes. 90° Orientation Angle (Figure 6a). The contact line is only slightly curved where the meniscus is in contact with the posts. A shape similar to the situation for a flat heterogeneous surface is observed. As the left and right post edges are reached, however, the contact line suddenly reaches backward, first following the contour of the posts. It then detaches from the solid material in the lower left/right post corner, and even subsequently curves into the area behind the posts. 45° Orientation Angle (Figure 6b). In this case as well, the posts in the first row are not completely wetted (upper post corner). The contact line follows the contour of the posts, then curves into the space between the posts of the first and second rows. Contact with the posts in the second row is made. The contact line thus undulates between posts in the first and second rows. Where the contact line spans over air, the local contact angle is 180° (index a in Figure 5A). Where it crosses over the posts, the local contact angle is 120° (index c in Figure 5A). Where the contact line is resting against a wettability boundary (i.e., a post edge), the local contact angle assumes a value between 120 and 180° (pinning of the contact line, index b in Figure 5A). It is interesting to note that regions of 120° contact angle (i.e., on top of the posts) are separated from regions of 180° contact angle (i.e., on air) by regions where the contact line is pinned. This allows for a gradual change in the local contact angle from 120 to 180° and vice versa. The liquid-air interface of the drop above the z ) 0 plane is distorted in a complex way in order to satisfy these local contact angles. Under the influence of variations in the Laplace pressure, the distortion of the interface levels off with increasing z-position. To summarize, the results in Figures 5 and 6 seem very plausible: in those regions where the drop is in contact with air, it has a strong tendency to dewet from the composite surface and to “ball up”; this leads to a falling back of the contact line. On the other hand, the contact line rests in a more advanced position where the liquid is in contact with the solid surface. Comparing our results to the Surface Evolver simulations on macroscopic posts carried out by Chaitain et al.,32 we note that, for our case, where the posts are small compared to the drop size, the deformation of the contact line remains restricted to the microscale. In a second step, we looked at how the shape of the contact line was changed as the x-position of the center of gravity of the liquid mass with respect to the substrate was varied. The post spacing and post width were again 32 µm, while the equilibrium contact angle on the posts remained 120°. AdVancing Motion. Figure 7a illustrates the situation where the drop has been advanced by 200 µm from the position corresponding to the shape in Figure 6. The contact line has receded from the region below and between the posts. On the posts itself, the contact line has advanced to the front edge, where it now remains pinned at the 90° corner. (The meniscus cannot move down the front of the posts as long as the advancing contact angle is not reached with respect to this plane.) As an overall result of the advancing motion, the length and degree of distortion of the contact line have been considerably decreased. The length of the contact line continued to shrink as the drop was further advanced (another 200 µm): in Figure 7b, an angle approaching 180° has been reached in the pinned sections of the contact line. Receding Motion (Figure 7c). As the drop receded from the post (by 30 µm compared to the situation in Figure 6), the solid-
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As a final remark, we would like to stress the fact that the measurement of a contact angle on ultrahydrophobic surfaces is often compromised for practical reasons: on a composite surface, the local contact angle varies along the length of the contact line. Its value can either be θe, 180°, or anything in between in those sections where the contact line is pinned. As the contact line undulates strongly, the local contact angle is in many places not accessible to a direct optical measurement. We have pointed out above that the liquid-air interface of the drop is distorted in order to satisfy these various local contact angles. With variations in the Laplace pressure acting as a counterforce, this distortion levels off toward an average curvature as we get further away from the substrate. A uniform (in the sense of constant around the entire drop perimeter) contact angle can thus only be defined at a certain distance above the substrate. From Figure 5, we estimate this distance as being in the range of a few micrometers for the microscale surface features studied here.
Conclusions Figure 7. (a) Shape of the contact line for the situation where the drop has been advanced in the positive x-direction. (b) The meniscus has been even further advanced. (c) The meniscus has been moved in the negative x-direction (receding motion).
liquid interfacial area was considerably decreased, indicating a destabilization of the meniscus on the post. Several years ago, Extrand introduced the concept of the contact line density (defined as the length of solid perimeter per unit area that could support a drop) as the crucial parameter for determining the contact angle.11,12 He suggested that “contact angles are determined by interactions at the contact line, not by those within the interfacial contact area”.39 In our simulations we find that, corresponding to the distortion of the contact line, the entire interface of the droplet close to the substrate is distorted in a complex way so as to satisfy the local contact angles on the composite surface. This in fact supports the conclusion that the contact angle is actually not so much dependent on the ratio of surface energies underneath the entire droplet, but much more behaves as a function of the different local contact angles the drop experiences along the length of its contact line. The overall contact angle is then an average value that follows as a result of the deformation of the drop base. Our simulations reveal that it is actually not so trivial to predict the contact line density for a given geometry, because (i) the contact line is so strongly distorted and (ii) depending on the position of the drop, pinning on roughness features can lead to deviations in the local contact angle from the equilibrium value on the respective material. (39) Extrand, C.W. Langmuir 2003, 19, 3793.
The conformation of the contact line of a drop of water resting on an ultrahydrophobic post surface has been numerically computed. The simulation results suggest that the contact line is in fact not a straight line, but very strongly distorted. This is in contrast to what is implicitly assumed in several other publications dealing with the wetting of ultrahydrophobic post surfaces. The distortion of the contact line indicates a strong tendency of the drop to “ball up” in those regions where it is not in contact with the solid parts of the surface. During the motion of the meniscus over the post surface, the shape and length of the contact line changes significantly. Local contact angles can deviate from the equilibrium values due to pinning effects. To conclude, we believe that our results are relevant not only for the specific system studied in this work, but for the characterization of microstructured ultrahydrophobic surfaces in general. The distortion of the contact line corresponds to various local contact angles that appear along its length, while an average contact angle can only be defined at a certain distance above the substrate. It is important to note that, with the commonly used low-magnification goniometers, an exact characterization of the local contact angles in the plane of the composite surface is problematic and that the exact value that is measured for the macroscopic angle will probably depend on the details of the optics and software algorithms used for the interpretation of the images. Acknowledgment. It is our pleasure to thank Jan Lienemann and Andreas Greiner for valuable discussions. LA062596V
