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Contact Angles on Spherical Surfaces C. W. Extrand* and Sung In Moon Entegris, Inc., 3500 Lyman BouleVard, Chaska, Minnesota 55318 ReceiVed April 7, 2008. ReVised Manuscript ReceiVed June 4, 2008 In this paper, we explore the influence of curved surfaces on contact angles. Small liquid drops were deposited at the apex of spheres. Liquid was added to advance the contact line (or withdrawn to cause recession). As drop volume increased, the contact line advanced outward and downward. With the addition of each increment of liquid, the contact line encountered a steeper slope and showed progressively larger apparent advancing contact angles. Observed apparent contact angles could be explained in terms of intrinsic contact angles and surface orientation. We found that if curvature and geometry were correctly accounted for, the classic Gibbs relation held. The experimental approach and analysis used here for estimating intrinsic wettability from curved surfaces could easily be integrated into automated contact angle measurement systems.
Introduction The interest in the wetting of structured surfaces has exploded in the past five years.1 Many investigators have used the classic Cassie-Baxter2 and Wenzel3 equations to describe the observed contact angles. Unfortunately, their success in correctly predicting wetting behavior has been mixed because these theoretical constructs are fundamentally flawed. First, they incorrectly assume that contact angles are determined by liquid-solid interfacial area. They are not. Wetting is controlled by interactions at the contact line.1,4–6 Second, these equations do not fully account for geometry of the surfaces, i.e., they neglect the contribution of feature shape. Although often ignored, long ago Gibbs proposed a simple geometric relation that describes the interaction of liquids with sharp edges.7 Consider the conical frustum (i.e., truncated cone) depicted in Figure 1. The intersection between the flat, horizontal top of the frustum and its sloped side is defined by the edge angle R. In Figure 1(a), a small liquid drop is deposited on top of the frustrum. The drop width is less than its diameter of the top surface. Thus, the drop exhibits an intrinsic advancing contact angle of θa,o. As liquid is added to the drop, the contact line advances to the asperity edge and is pinned, Figure 1(b). With the addition of more liquid, the contact angle relative to the horizontal plane increases. Eventually the drop establishes a critical apparent advancing contact angle (θa) on the side of the frustrum, Figure 1(c). This is why structured surfaces can show higher apparent contact angles than their smooth counterparts and the basic phenomenon that leads to very high contact angle ultralyophobic surfaces. If additional liquid is added, the drop * To whom correspondence should be addressed. E-mail: chuck_extrand@ entegris.com. Phone: 1-952-556-8619. (1) Gao, L.; McCarthy, T. J., How Wenzel and Cassie Were Wrong. Langmuir 2007, 23, 3762-3765. (2) Cassie, A. B. D.; Baxter, S. Trans. Faraday Soc. 1944, 40(21), 546–551, Wettability of porous surfaces. (3) Wenzel, R. N. Ind. Eng. Chem. 1936, 28(8), 988–994, Resistance of Solid Surfaces to Wetting by Water. (4) Pease, D. C. J. Phys. Chem. 1945, 49(2), 107–110, The Significance of the Contact Angle in Relation to the Solid Surface. (5) Bartell, F. E.; Shepard, J. W. J. Phys. Chem. 1953, 7(4), 455–458, Surface Roughness as Related to Hysteresis of Contact Angles. II. The Systems Paraffin-3 Molar Calcium Chloride Solution-Air and Paraffin-Glycerol-Air. (6) Extrand, C. W. Langmuir 2003, 19(9), 3793–3796, Contact Angles and Hysteresis on Surfaces with Chemically Heterogeneous Islands. (7) Gibbs, J. W., On the Equilibrium of Heterogeneous Substances. In The Collected Works of J. Willard Gibbs; Yale University Press: New Haven, CT, 1961; Vol. 1, pp 326-327.
Figure 1. Sequential addition of liquid to a small drop atop a conical frustrum with an edge angle of R. (a) A small drop is deposited on the flat top of the frustrum. The drop width is less than the diameter of the top surface and exhibits its intrinsic advancing contact angle (θa,o). (b) As liquid is added, the contact line advances to the edge and is pinned. (c) With the addition of more liquid, the drop establishes a critical apparent advancing contact angle (θa) on the side of the frustrum. (d) With yet more liquid, the drop collapses.
collapses, Figure 1(d). A smaller residual volume with a lower contact angle may remain. According to Gibbs, the apparent advancing contact angle (θa) should increase due to interaction with the sharp edge such that
θa ) θa,o + (π - R)
(1) 8
In the late 1970s, Mason and colleagues performed classic experiments using conical frustra that demonstrated these expressions are correct. These relations have been further validated using lyophobic or textured frustra that suspend larger liquid volumes.9,10 They also have been used to describe soft surfaces,11 (8) Oliver, J. F.; Huh, C.; Mason, S. G. J. Colloid Interface Sci. 1977, 59(3), 568–581, Resistance to Spreading of Liquids by Sharp Edges. (9) Extrand, C. W. Langmuir 2005, 21(23), 10370–10374, Modeling of Ultralyophobicity: Suspension of Liquid Drops by a Single Asperity. (10) Zhang, J.; Gao, X.; Jiang, L. Langmuir 2007, 23(6), 3230–3235, Application of Superhydrophobic Edge Effects in Solving the Liquid Outflow Phenomena.
10.1021/la801091n CCC: $40.75 2008 American Chemical Society Published on Web 07/22/2008
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The sphere has a diameter of 2R. The chord that intersects the contact line and passes orthogonally through the vertical axis of the sphere has a length of 2a. The spreading angle φ between the contact line and vertical axis can be estimated as
φ ) sin-1(a ⁄ R)
(2)
Since the effective edge angle and spreading angle of the sphere are supplementary angles
R)π-φ
(3)
Equations 1, 2 and 3 can be combined to produce an expression that relates θa,o to three easily measurable quantities, θa, a and R Figure 2. Schematic depiction of a liquid drop resting on top of a solid sphere.
θa,o ) θa-sin-1(a ⁄ R)
rough surfaces12 as well as super lyophobic13–15 and super lyophilic16 surfaces. In each of the cited cases, the simple relation given in eq 1 assumes that the facets of the surface features are rectilinear, where the contact angle variation occurs at a welldefined juncture between two intersecting surface planes. While structured surfaces with rectilinear features are relatively simple to model, there are many structured surfaces with spherical or cylindrical features, both natural and synthetic, where eq 1 is not applicable. Examples include the surface of the Lotus leaf, which is covered with microscopic hemispherical protuberances17 as well as synthetic surfaces covered with spherical particles18 or cylindrical fibers.19 Therefore, in this study, we examined how interaction with curved surfaces affects contact angles. In the spirit of Mason, small drops of liquid were deposited onto the tops of polymer spheres of different size and composition. Liquid was sequentially added or removed to initiate movement of the pinned contact line. The relations between liquid volume, sphere size, apparent contact angles on the sides of the sphere and the “true” contact angles from the corresponding flat, horizontal surface were noted and compared. Analysis. Consider a sessile liquid drop of volume V on a solid sphere, as depicted in Figure 2. Assume that the drop was deposited such that the liquid is symmetrically centered, i.e., the vertical “z” axis of the sphere passes through the apex of the drop. Also, assume that the drop has spread along the curved surface to produce an intrinsic advancing contact angle, θa,o, which is the angle that one would expect if the liquid were deposited on a flat, horizontal surface of the same composition. Due to the curvature of the sphere, the apparent advancing contact angle, θa, measured relative to the horizon is greater than the intrinsic value, θa,o.
Experimental Details
(11) Extrand, C. W.; Kumagai, Y. J. Colloid Interface Sci. 1996, 184(1), 191– 200, Contact Angles and Hysteresis on Soft Surfaces. (12) Shuttleworth, R.; Bailey, G. L. J. Discuss. Faraday Soc. 1948, 3(1), 16– 22, Spreading of a Liquid over a Rough Surface. (13) Extrand, C. W. Langmuir 2002, 18(21), 7991–7999, Model for Contact Angles and Hysteresis on Rough and Ultraphobic Surfaces. (14) Extrand, C. W. Langmuir 2004, 20(12), 5013–5018, Criteria for Ultralyophobic Surfaces. (15) Extrand, C. W. Langmuir 2006, 22(4), 1711–1714, Designing for Optimum Liquid Repellency. (16) Extrand, C. W.; Moon, S. I.; Hall, P.; Schmidt, D. Langmuir 2007, 23 Superwetting of Structured Surfaces. (17) Barthlott, W.; Neinhuis, C. Planta 1997, 202(1), 1–8, Purity of the Sacred Lotus, or Escape from Contamination in Biological Surfaces. ¨ ner, D.; Youngblood, J.; (18) Chen, W.; Fadeev, A. Y.; Hsieh, M. C.; O McCarthy, T. J. Langmuir 1999, 15(10), 3395–3399, Ultrahydrophobic and Ultralyophobic Surfaces: Some Comments and Examples. (19) Gao, L.; McCarthy, T. J. Langmuir 2006, 22(14), 5998–6000. “Artificial Lotus Leaf” Prepared Using a 1945 Patent and a Commercial Textile.
(4)
The wetting liquids used in the experiments were 18 MOhm deionized (DI) water and ethylene glycol (Sigma-Aldrich, anhydrous, 99.8%). The surface tensions of the liquids are, respectively, 73 and 47 mN/m.20,21 Solid spheres, composed of polytetrafluoroethylene (PTFE) and polycarbonate (PC), were purchased from McMaster Carr. Their diameters (2R) ranged from 3.2 mm (1/8 in.) to 25.4 mm (1 in.). Intrinsic advancing and receding contact angles (θa,o and θr,o) were measured directly from flat PTFE and PC surfaces with the same chemical composition as the spheres. The flat PTFE surface was made by machining away a section from a 25.4 mm PTFE sphere and then polishing with 600 mesh sand paper. PC was flattened by placing a 12.7 mm PC sphere between Al sheets and slowly applying 4500 kg of force via a platen press (PHI Bench Design Hydraulic Compression Press). Prior to use, spheres and flats were cleaned with DI water and allowed to dry. Spheres and flats were immobilized using double-sided adhesive tape. Small liquid drops were gently extruded from a one-milliliter, glass syringe (M-S, Tokyo, Japan). Syringe plunger displacement was converted to liquid volume, V. On the flat surfaces, a small volume of liquid was deposited, liquid was added or removed to advance or retract the contact line and then a drop shape analyzer (Kru¨ss DSA10) was used to measure θa,o or θr,o. For measurements of advancing contact angles on spheres, each experiment began by depositing a 1 µL drop at the apex of a sphere and then immediately adding another 1 µL to the first drop to advance the contact line. Additional liquid was sequentially deposited in 2 µL increments. At each 2 µL increment (2 µL, 4 µL, 6 µL,...), an image was captured. Apparent contact angles (θa) were measured using the drop shape analyzer. Contact line chords (2a) and sphere diameters (2R) were estimated from images using Image-Pro Plus software. For receding contact angle measurements on spheres, drops with volumes of 10-20 µL were deposited on a sphere and then liquid was withdrawn from the drop until the contact line retreated. Values of θr, 2a and 2R were measurement as described above. Intrinsic contact angles were computed from measurements on spheres using eq 4. Standard deviation of the measured θa, θr, θa,o and θr,o values was (2°. For 2R and 2a, it was (0.02 mm. Using standard error propagation techniques,22 the uncertainty in the calculated values of the intrinsic contact angles from the curved surfaces was estimated to be ( 3-4°. All measurements were performed at room temperature (25 °C). (20) Weast, R. C., Handbook of Chemistry and Physics, 73rd ed; CRC Press: Boca Raton, FL, 1992. (21) Adamson, A. W., Physical Chemistry of Surfaces, 5th ed.; Wiley: New York, 1990. (22) Taylor, J. R., An Introduction to Error Analysis, 2nd ed., University Science Books: Sausalito, CA, 1997.
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Figure 3. Sessile water drops on PTFE. (a) A 2 µL water drop on the horizontal surface of a flat portion of a PTFE sphere. (b) A small water drop (2 µL) on a much larger sphere (2R ) 25.4 mm).
Figure 5. Apparent advancing contact angles (θa), contact chord lengths (2a) and intrinsic advancing contact angle (θa,o) versus drop volume (V) for water drops on PTFE spheres of various diameters (2R). Figure 4. Deposition and sequential growth of a water drop on a 6.4 mm PTFE sphere.
Results and Discussion Figure 3(a) shows a drop on a flattened, horizontal PTFE surface. If water was deposited on the flat PTFE and added sequentially, the contact angle advanced with an intrinsic contact angle of θa,o ) 108°. If water was withdrawn until the contact line receded, then θr,o ) 87°. Alternatively, intrinsic contact angles were estimated from very small drops (V ) 2 µL) on large spheres (2R ) 25.4 mm), Figure 3(b). For example, a 2 µL water drop on a 25.4 mm PTFE sphere exhibited a relatively small contact chord, 2a ) 1.51 mm. Thus, φ was negligible (only 3°) and the portion of the sphere covered by the drop was effectively flat. Here, water on PTFE yielded θa,o ) 110° and θr,o ) 90°, which agreed well with the measurements from the machined PTFE flat. Although still somewhat hydrophobic, PC interacted more strongly with water than PTFE and consequently produced lower intrinsic contact angles, θa,o ) 89° and θr,o ) 51°. Likewise, ethylene glycol, with a surface tension lower than water, also yielded lower intrinsic contact angles on PTFE, θa,o ) 89° and θr,o ) 78°. These values agree well with those published in the literature.23,24 Figure 4 shows growth of a water drop on a 6.35 mm PTFE sphere. A drop was deposited on the apex of the sphere and then water was added sequentially to the existing drop. As drop volume increased, the contact line advanced outward and downward. In contrast to frusta that have two intersecting surface planes on which liquids can establish their contact lines, the curved surface of the spheres provides a spectrum of potential wetting planes. Indeed with the addition of each increment of liquid, the contact line encountered a steeper slope and the liquid showed a progressively larger contact angle. In Figure 5, values of θa and 2a from measurements of water on PTFE are plotted against drop volume (V) for spheres ranging in diameter (2R) from 3.2 mm to 12.7 mm. Both θa and 2a (23) Fox, H. W.; Zisman, W. A. J. Colloid Sci. 1950, 5(6), 514–531, The Spreading of Liquids on Low Energy Surfaces. I. Polytetrafluoroethylene. (24) Wu, S., Polymer Interface and Adhesion.; Marcel Dekker: New York, 1982.
Figure 6. Apparent advancing contact angles (θa), contact chord lengths (2a) and intrinsic advancing contact angle (θa,o) versus drop volume (V) for water drops on PC spheres of various diameters (2R).
increased with drop volume. The smaller the sphere, the more pronounced the changes in apparent contact angles for a given volume of liquid. For example, a 10 µL water drop on a 3.2 mm PTFE sphere produced a large apparent advancing contact angle of θa ) 142°. In such a case, the size of the drop rivaled that of the sphere, where 2a ) 1.81 mm and φ ) 34°. In contrast, the same size water drop on a much larger 12.7 mm PTFE sphere exhibited a much smaller apparent advancing contact angle, θa ) 117°. Values of θa,o calculated with eq 4 are also shown in Figure 5. Note that calculated θa,o values were invariant with respect to V and 2R. The solid line corresponds to the average of all calculated intrinsic values of water on PTFE spheres, θa,o ) 107°. The agreement between the measured θa,o values from the flat PTFE surface (108°) and calculated θa,o values from the curved PTFE surfaces (107°) was excellent. Thus, water advanced
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Figure 7. Apparent advancing contact angles (θa), contact chord lengths (2a) and intrinsic advancing contact angle (θa,o) versus drop volume (V) for EG drops on PTFE spheres of various diameters (2R).
along the curvature of the PTFE spheres exhibiting its θa,o at each contact location. Figures 6 and 7 show data of θa, 2a and θa,o versus V for water on PC spheres and ethylene glycol on PTFE spheres. These liquid-solid combinations behaved similarly to water on PTFE. Values of θa and 2a increased with V. As drop size grew, smaller spheres exhibited more pronounced contact angle changes. θa,o values were independent of V and 2R. The solid lines represent the average of the calculated intrinsic values. Water on PC spheres had an average calculated θa,o value of 90°. For ethylene glycol on PFTE spheres, the average calculated θa,o value was 91°. Again, agreement between the measured θa,o values from the flat surfaces (89° for water on PC and 89° for ethylene glycol on PTFE) and calculated θa,o values from the curved ones (90° for water on PC and 91° for ethylene glycol on PTFE) was excellent. Receding Contact Angles. Intrinsic values of receding contact angles also were successfully estimated on spheres by measuring θr and 2a, then applying the following expression
θr,o ) θr-sin-1(a ⁄ R)
(5)
which has the same form as eq 4. For example, if 10 µL of water was deposited on a 12.7 mm PTFE sphere and liquid was slowly withdrawn, the contact line remained pinned as the apparent contact angle decreased until the volume of drop was reduced
to 6 µL, then the contact line began to recede. At this point, θr ) 96° and 2a ) 2.41 mm. (Here, the spreading angle was φ ) 11°.) From eq 5, we estimated the intrinsic receding angle to be θr,o ) 85°. This value agreed well with the θr,o value measured directly from the flattened PTFE surface, θr,o ) 87°. Note that θr > θr,o due to the curvature of the sphere. Experimental Challenges. While these measurements are conceptually quite simple, there are several experimental challenges. One challenge is maintaining symmetry; another is contact angle estimation from two tangent lines. As for symmetry, it is difficult to keep the liquid drop perfectly centered on the z-axis of the sphere. After initial deposition, the orientation of the sphere often had to be adjusted so that the z-axis of the sphere was directed through the apex of the drop. Sequential addition, if not made from directly above the drop, also tended to move the drop off axis. Therefore, many experimental attempts were often made to acquire properly oriented drops for data acquisition. These minor deviations can be observed in Figure 4. Let us turn to angle estimation from two lines. In principle, θa,o values could be measured from two intersecting tangent linessone at the air-liquid interface and the other at the sphere surface. Attempts to manually measure θa,o values at the intersection between tangent lines led to greater uncertainty (>10%) than computed values from automated measurements of θa and 2a (∼5%). Scheme for Automated Measurement. We suggest that automated measurement systems could be enhanced to measure intrinsic wettability from spherically proportioned particles or asperities. Consider the following example involving advancing contact angles. After depositing a liquid drop on the apex of a particle and advancing the contact line, an image could be captured, then θa and 2a values could be estimated using resident software. After inputting 2R, θa,o could be automatically computed using eq 4.
Conclusions As liquids spread over the curved surfaces of spheres, their apparent advancing contact angles increased. The apparent angles changed via a simple geometric relation. If corrected for the curvature, the contact angles on the curved surfaces were the same intrinsic values as those observed on flat, horizontal surfaces. Thus, the Gibbs relation holds for spherical surfaces, just as it does for rectilinear structures. Acknowledgment. We thank Entegris management for supporting this work and allowing publication. Also, thanks to M. Acevedo, L. Monson and J. Pillion for their suggestions on the technical content and text. LA801091N
