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Modeling of Ultralyophobicity: Suspension of Liquid Drops by a Single Asperity C. W. Extrand* Entegris, Inc., 3500 Lyman Blvd., Chaska, Minnesota 55318 Received May 16, 2005. In Final Form: August 16, 2005 With the aim of understanding the underlying physical phenomenon associated with utlralyophobic (or super repellent) surfaces, model studies have been performed on single asperities of different size and shape. A small liquid drop was deposited on top of each model asperity, and liquid was sequentially added. If the advancing contact angle was sufficiently large, it was possible to suspend large drops atop asperities with an apparent contact angle approaching 180°. If more and more liquid was added, eventually the suspended drops collapsed. Roughening the surface of the asperities further bolstered suspension. Using an analysis that accounts for both capillary forces and the influence of gravity, the critical suspension volume was correctly predicted for each liquid/asperity combination.
Introduction Studies of ultralyophobic (or super repellent) surfaces first appeared in the scientific literature in the 1930s.1 While this subject has received continued attention since that time,2-6 interest has intensified in recent years7-26 due to the potential use of ultralyophobicity in a variety of applications, such as self-cleaning surfaces and microfluidic devices.27 In efforts to reveal the underlying physical phenomenon, a number of investigators have prepared model surfaces covered with periodic arrays of regularly shaped asperities (square pillars, cylindrical rods, pyramids, etc.),16-18,21,24 rendered these surfaces hydrophobic, and then examined their wetting behavior, looking for high contact angles (140-180°) and low drop * To whom correspondence should be addressed. E-mail:
[email protected]. (1) Wenzel, R. N. Ind. Eng. Chem. 1936, 28, 988. (2) Cassie, A. B. D.; Baxter, S. Trans. Faraday Soc. 1944, 40, 546. (3) Bartell, F. E.; Shepard, J. W. J. Phys. Chem. 1953, 57, 211. (4) Johnson, R. E. Jr.; R. H. Dettre, R. H. Adv. Chem. Ser. 1964, 43, 112. (5) Dettre, R. H.; Johnson, R. E., Jr. Adv. Chem. Ser. 1964, 43, 136. (6) Holloway, P. J. Pestic. Sci. 1970, 1, 156. (7) Kunugi, Y.; Nonaka, T.; Chong, Y.-B.; Watanabe, N. J. Electroanal. Chem. 1993, 353, 209. (8) Onda, T.; Shibuichi, S.; Satoh, N.; Tsujii, K. Langmuir 1996, 12, 2125. (9) Shibuichi, S.; Onda, T.; Satoh, N.; Tsujii, K. J. Phys. Chem. 1996, 100, 19512. (10) Barthlott, W.; Neinhuis, C. Planta 1997, 202, 1. (11) Tadanaga, K.; Katata, N.; Minami, T. J. Am. Ceram. Soc. 1997, 80, 1040. (12) Tadanaga, K.; Katata, N.; Minami, T. J. Am. Ceram. Soc. 1997, 80, 3213. (13) Hozumi, A.; Takai, O. Thin Solid Films 1997, 303, 222. (14) Chen, W.; Fadeev, A. Y.; Hsieh, M. C., O ¨ ner, D.; Youngblood, J. P.; McCarthy, T. M. Langmuir 1999, 15, 3395. (15) Youngblood, J. P.; McCarthy, T. M. Macromolecules 1999, 32, 6800. (16) Bico, J.; Marzolin, C.; Que´re´, D. Europhys. Lett. 1999, 47, 220. (17) O ¨ ner, D.; McCarthy, T. M. Langmuir 2000, 16, 7777. (18) Yoshimitsu, Z.; Nakajima, A.; Watanabe, T.; Hashimoto, K. Langmuir 2002, 18, 5818. (19) Extrand, C. W. Langmuir 2002, 18, 7991. (20) Patankar, N. A. Langmuir 2003, 19, 1249. (21) He, B.; Patankar, N. A.; Lee, J. Langmuir 2003, 19, 4999. (22) Marmur, A. Langmuir 2003, 19, 8343. (23) Marmur, A. Langmuir 2004, 20, 3517. (24) Krupenkin, T. N.; Taylor, J. A.; Schneider, T. M.; Yang, S. Langmuir 2004, 20, 3824. (25) Extrand, C. W. Langmuir 2004, 20, 5013. (26) Patankar, N. A. Langmuir 2004, 20, 7097. (27) Blossey, R. Nat. Mater. 2003, 2, 301.
retention as evidence of super repellency. These model surfaces are relatively complex due to the number and the microscopic dimensions of the asperities. The physicochemical phenomena that enable these complex surfaces to exhibit ultralyophobicity also should apply to a single macroscopic asperity. The use of a single macroscopic asperity allows for fast, simple fabrication and experimental evaluation of potential designs. Therefore, in an attempt to further elucidate ultralyophobic behavior, single-model asperities were constructed and their ability to suspend liquid drops was investigated theoretically and experimentally. Most of the model asperities made for this study had smooth surfaces. Alternatively, a few were constructed to have a rough surface with the intent of mimicking the structural hierarchy found in many naturally repellent flora or fauna. Theoretical Basis Consider the model asperity shown schematically in Figure 1. A small drop is deposited atop the asperity, and liquid is sequentially added. If the contact diameter of the drop is less than the diameter of the asperity and the top of the asperity is smooth and horizontal, then the drop exhibits an intrinsic advancing contact angle of θa,o.28 As liquid is added, the contact line advances to the asperity edge and is pinned. With the addition of more liquid, the contact angle increases and the liquid eventually establishes its intrinsic advancing contact angle on the side of the asperity. As it happens, the apparent contact angle, θa, relative to the horizon increases.29 If the surface tension is directed downward at the contact line, when θa reaches a critical value, θa,c, the contact line will advance and the liquid will flow down the side of the asperity. This relation between θa,c, θa,o, and the geometry of a surface feature (28) The local advancing contact angle of a liquid on a smooth portion of a solid surface, regardless of the solid orientation relative to the horizon, will be referred to as an “intrinsic” contact angle, θa,o. θa,o describes the essential or fundamental molecular interactions between the liquid and solid. (29) The apparent advancing contact angle, θa, refers to the perceived angle of the liquid contact relative to the horizon, regardless of the local orientation and/or roughness of the solid surface. If the solid surface is smooth and horizontal, then θa,o ) θa. Otherwise, if a solid surface is rough and/or the contact line resides on a nonhorizontal portion, then θa,o * θa.
10.1021/la0513050 CCC: $30.25 © 2005 American Chemical Society Published on Web 10/01/2005
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subtended by the top of the asperity is R, the rise angle of the side of the asperity is ω, and
R ) 180° - ω
Figure 1. Sequential addition of liquid to a small drop atop a single model asperity. (a) A small drop is deposited on a smooth, flat-topped, cylindrical asperity. The drop width is less than the diameter of the asperity and thus exhibits an intrinsic advancing contact angle of θa,o. (b) As liquid is added, the contact line advances to the asperity edge and is pinned. With the addition of more liquid, the apparent contact angle, θa, relative to the horizontal plane, increases and eventually establishes its intrinsic advancing contact angle on the side of the asperity. (c) If the vertical component of the surface tension is directed downward, the drop will collapse when the contact angle reaches its apparent advancing value. (d) Otherwise, if the vertical component of the surface tension is directed upward, the drop will be suspended atop the asperity and the apparent contact angle will approach 180° as the volume of the drop increases.
(2)
For the frustoconical asperity depicted in Figure 2, R * ω * 90°. Alternatively, if R ) ω ) 90°, then the asperity is cylindrical. Because the drop in Figure 2 has established its intrinsic advancing contact angle on the side of the asperity, the surface tension is directed upward. Assume the drop is in a critical or equilibrium state where the addition of more liquid will cause collapse. Here, a geometric criterion alone is not sufficient. One must also account for surface and body forces. For the purposes of analysis, the maximum volume of a suspended drop, V, can be described as the sum of three parts: a cylindrical volume, Vcyl, a cap volume, Vcap, and an annular volume, Vann,
V ) Vcyl + Vcap + Vann
(3)
The liquid associated with the cylindrical volume and cap volume is supported by the surface area of the asperity top and can be described simply from geometry as
Vcyl ) πa2hcyl
(4)
and
Vcap ) (1/6)πh(1 - hcyl/h)[3a2 + h2(1 - hcyl/h)2]
(5)
where h is the maximum height of the liquid between the drop apex and the asperity top and hcyl is the height at the edge of the liquid cylinder, as depicted in Figure 2. The annular volume, Vann, is distorted by the downward pull of gravity. While it may be difficult to describe geometrically, it can be determined from the interplay of body and surface forces.33 The body force, F, pulling down on Vann depends on the density of the liquid, F,
F ) FgVann Figure 2. A cross-section of a suspended drop on a single model asperity showing angles, critical dimensions, and critical volume.
θa,c ) θa,o + ω
where g is the acceleration due to gravity. The surface force can be calculated19,25 as the product of the asperity top perimeter and the vertical component of the liquid surface tension, γ,
f ) -2πaγ cos(θa,o+ ω - 90°)
was first suggested by Gibbs30,31 in the 1870s,
(1)
and experimentally verified by Mason and colleagues in the 1970s.32 Otherwise, if the surface tension at the edge of the asperity is directed upward, the drop will be suspended atop the asperity and θa will increase as liquid is added, reaching 180°. With the continued addition of liquid, eventually the suspension force provided by the surface tension will be surpassed by the weight of the liquid, the contact line will move rapidly downward, and the suspended drop will collapse. Figure 2 shows the cross-section of a drop suspended atop a model frustoconical asperity with a smooth surface. The top diameter of the asperity is 2a. The edge angle (30) Gibbs, J. W. The Collected Works of J. Willard Gibbs; Yale University Press: New Haven, CT, 1961; Vol. 1, p 326. (31) A similar expression was suggested by: Shuttleworth, R.; Bailey, G. L. J. Discuss. Faraday Soc. 1948, 3, 16. (32) Oliver, J. F.; Huh, C.; Mason, S. G. J. Colloid Interface Sci. 1977, 59, 568.
(6)
(7)
The maximum annular volume, Vann, that can be suspended occurs where
F)f
(8)
and therefore can be determined by combining eqs 6-8 and rearranging to give
Vann ) -2πa(γ/Fg)cos(θa,o + ω - 90°)
(9)
Finally, eqs 4, 5, and 9 can be substituted into eq 3 to yield an expression for estimating the maximum volume of liquid that can be suspended on top of a single “ultra(33) The weight of the annular volume is suspended along the contact line by the vertical component of the surface tension, analogous to capillary rise in a tube. There are a number of additional forces in this problem that balance each other and therefore need not be addressed. For example, the liquid over the asperity exerts a downward force on the asperity, and the asperity in turn exerts an upward force on the liquid column. There also are forces acting along the interface between the liquid and solid, as well as along the free surface of the liquid. However, the sum of these free surface forces, if not zero, is negligible.
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Table 1. Geometric Parameters and Critical Volumes for Suspended Water Drops on Smooth PFA Asperities, θa,o ) 108° a V (µL) asperity
ω (deg)
2a (mm)
h (mm)
hcyl (mm)
measured
calcd
cylindrical frustoconical
90 80
3.08 ( 0.01 3.15 ( 0.05
3.17 ( 0.02 3.11 ( 0.02
2.62 ( 0.04 2.46 ( 0.05
42.9 ( 1.3 30.9 ( 1.2
43.6 32.0
a ω is the rise angle of the smooth asperity; 2a is the diameter of the asperity top; h is the critical height of the drop from its apex to the asperity top; hcyl is the critical height of the drop along the cylindrical plane define by the asperity top; V is the critical suspension volume. See Figure 2 for further explanation of the various parameters.
lyophobic” asperity,
V ) -2πa(γ/Fg)cos(θa,o + ω - 90°) + πh{a2hcyl/h + (1/6)(1 - hcyl/h)[3a2 + h2(1 - hcyl/h)2]} (10) Note that this expression is only valid where the liquid surface tension at the contact line is directed upward. (This condition requires that the first term in eq 10 is positive or, more specifically, that θa,o + ω - 90° > 0.) Materials and Methods Single asperities of various shape and size were machined from PFA rod stock (DuPont Teflon poly(tetrafluoroethylene)PerFluoroAlkoxy copolymer, Entegris part no. 1112-059), PEEK rod stock (Victrex polyetheretherketone, 11/2 in. diameter from Minnesota Plastics), and a glass rod. PFA and PEEK rod stock were machined into small-diameter cylinders (ω ) 90°) and conical frusta (truncated cones) with ω ) 35°, 75°, and 80°. Surface roughness of the PFA and PEEK asperities was controlled with cutter shape, cutter orientation, and cutting speed. Glass rods were mounted in a small aluminum base; the tops of the rods were flattened and polished using a carbide grinder with a 150-grit diamond-coated wheel. The polished glass rods were coated with a Teflon AF 1600 solution (Teflon AF is an amorphous perfluoropolymer from DuPont) and dried for 1 h under vacuum. The liquids were 18MQ deionized water, formamide (AlfaÆsar, ACS, 99.5+%), and n-hexadecane (Alfa-Æsar, 99%). Using a 1-mL glass syringe (M-S, Tokyo, Japan), liquid was deposited on top of each model asperity and then added sequentially. Intrinsic contact angles, θa,o, were measured on top of each asperity. Syringe plunger displacements were converted to volumes. Drop shape, drop dimensions, and contact angles were monitored with a Kru¨ss drop shape analyzer (DSA10). Critical values of V, h, and hcyl were noted at the point of collapse. Values of F and γ used in calculations were taken from the literature.34
Results and Discussion When a drop was deposited atop a model asperity and then liquid was sequentially added, the drop advanced with a constant θa,o value. Water gave the following values, θa,o ) 108° on PFA, θa,o ) 117° on AF-coated glass, and θa,o ) 75° on PEEK, in agreement with previous reports.35-40 The AF film, with a greater concentration of CF3 groups, was more hydrophobic than PFA, as demonstrated by a larger θa,o value. For formamide on AF-coated glass, θa,o was 104°. (34) Handbook of Chemistry and Physics, 73rd ed.; Weast, R. C., Ed.; CRC Press: Boca Raton, FL, 1992. (35) Fabretto, M.; Sedev, R.; Ralston, J. In Contact Angle, Wettability and Adhesion; Mittal, K. L., Ed., VSP: Boston, 2003; Vol. 3, p 161. (36) Ellison, A. H.; Fox, H. W.; Zisman, W. A. J. Phys. Chem. 1956, 57, 622. (37) Johnson, R. E. Jr.; Dettre, R. H. J. Colloid Sci. 1965, 20, 173. (38) Neumann, A. W.; Haage, G.; Renzow, D. J. Colloid Interface Sci. 1971, 35, 379. (39) Penn, L. S.; Miller, B. J. Colloid Interface Sci. 1980, 78, 238. (40) Bismarck, A.; Kumru, M. E.; Springer, J. J. Colloid Interface Sci. 1999, 217, 377.
Figure 3. Water drops on smooth model PFA asperities (2a ) 3.1 mm) just before collapse. (a) A water drop on a frustoconical PFA asperity (ω ) 35°) and (b) a water drop on a cylindrical PFA asperity (ω ) 90°).
With the addition of sufficient liquid, the contact line of a drop was pinned at the asperity edge and the apparent contact angle increased. For frustoconical asperities with a shallow rise angle, the contact line remained pinned until the θa,o value was established on the sloping side of the asperity. Figure 3a shows a photograph of a water drop that has reached its critical configuration on a frustoconical PFA asperity with a shallow rise angle. In this case, the vertical component of the surface tension was directed downward and with addition of more water, the contact line advanced causing the drop to move downward. Measurements from several experimental trials produced an average θa,c value of 143° ( 2°, which agreed well with the value predicted by eq 1, θa,c ) 143°. In contrast, if a liquid established its intrinsic contact angle on the side of the asperity such that the surface tension was directed upward, the drops were suspended. With the addition of sufficient liquid, θa values approached 180°. Figure 3b shows a water drop suspended atop a cylindrical PFA asperity (ω ) 90°). The drop has reached a critical volume where the upwardly directed surface force equals the downward pull of gravity on the unsupported annular volume. Adding more water to the drop would cause collapse. Note that the shape of the drop is not spherical but has been flattened by its own weight. Table 1 lists asperity angles, dimensions, and critical volumes for water suspended on a cylindrical PFA asperity (ω ) 90°). The total volume of the suspended water drop was estimated with eq 10 and agreed well with the experimentally measured one, V ) 43 µL. Table 1 also includes data for a frustoconical asperity, ω ) 80°. The cylindrical and frustoconical asperities have the same composition and top diameter but differ in their ability to suspend drops. With a smaller rise angle, the frustoconical asperity (ω ) 80°) was not able direct the vertical component of the liquid surface tension upward to the same extent as the cylindrical asperity (ω ) 90°) and therefore suspended a smaller volume of water. Table 2 lists critical dimensions and critical volumes for water drops suspended on AF-coated cylindrical asperities of various diameters. Everything else being equal, the larger the diameter of the asperity, the smaller the fraction of unsupported liquid. Again, eq 10 correctly predicted V values. The fraction of liquid that was suspended (Vann) varied with asperity diameter. Consider
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Table 2. Geometric Parameters and Critical Volumes for Suspended Water Drops on Smooth, AF-coated Cylindrical Asperities, r ) ω ) 90° and θa,o ) 117° a V (µL) 2a (mm)
h (mm)
2.03 ( 0.03 3.57 ( 0.01 4.09 ( 0.02 5.07 ( 0.01
2.91 ( 0.04 3.57 ( 0.05 3.78 ( 0.04 4.09 ( 0.04
hcyl (mm)
measured
Vann (µL) calcd
calcd
2.50 ( 0.04 29.2 ( 0.5 30.0 3.07 ( 0.05 71.2 ( 1.2 70.7 3.09 ( 0.03 91.5 ( 1.3 88.1 3.38 ( 0.04 128.8 ( 4.2 128.7
21.2 37.4 42.8 53.1
a
2a is the diameter of the smooth asperity top; h is the critical height of the drop from its apex to the asperity top; hcyl is the critical height of the drop along the cylindrical plane defined by the asperity top; V is the critical suspension volume. See Figure 2 for further explanation of the various parameters.
Figure 5. Angles and dimensions of a frustoconical asperity with secondary features: (a) side view and (b) enlarged side view of the secondary features.
Figure 6. A magnified image of the secondary features on the side of a frustoconical asperity, R1 ) 105°, ω1 ) 75 °, R2 ) 75°, and ω2 ) 25 °. The white bar represents 0.1 mm.
Figure 4. Scanning electron micrograph of the surface of a Lotus leaf (Nelumbo nucifera). Photograph kindly provided by Prof. W. Barthlott.
the 3.6 mm AF-coated cylindrical asperitysit suspended a water drop with total volume of 71 µL. From the terms of eq 10, the supported volume (Vcyl + Vcap) was 33 µL while the unsupported annular volume, Vann, was 38 µL.41 Table 3 lists critical dimensions and critical volumes for water and formamide drops suspended on an 3.57 mm AF-coated cylindrical asperity. The V value for formamide was 63% of the water value. As compared to water, the greater F of formamide along with its lesser γ and θa,o combined to reduce its suspension force. Nevertheless, agreement between the experimental and predicted V values was good. Attempts to suspend drops of hexadecane were unsuccessful. In all cases, hexadecane drops collapsed as soon as the intrinsic advancing contact angle was established on the side of the asperity. For example, on AF-coated glass rods, the θa,o value measured on horizontal top surface was 59° ( 2°. As hexadecane was added, drops advanced to the top edge and were pinned. With the addition of more hexadecane, the apparent contact angle increased until θa ) 148° ( 2°, according to eq 1, and then drops collapsed. Some researchers have attempted to mimic the super repellency exhibited by plants.10,23,26 Figure 4 shows the surface of the Lotus leaf (Nelumbo nucifera), which is super water repellent (θa ) 160°) and exhibits self-cleaning properties.10 The surface of the Lotus leaf, like the leaves of many water repellent plants, is covered with protrusions that have a structural hierarchy. The primary structure of these protrusions takes the form of prolate hemispheroids. In turn, the protrusions are covered with wax crystalloids that project orthogonally outward, forming a secondary structure that consists of a myriad of sharp (41) The first term on the right side of eq 10 describes the suspended, annular volume, while the second term accounts for the supported volume.
edges. [The same wax substances on a smooth surface produce water contact angles that are 90° is not a requirement for super repellency.1,3,16 This work has focused on the interaction of capillary forces and gravity. Other body or external forces, such as inertia, also could be important. (43) Lafuma, A.; Que´re´, D. Nat. Mater. 2003, 2, 457.
Conclusions In order for a single asperity to suspend a liquid drop, two conditions must be met. First, the liquid must establish an advancing contact angle at the asperity edge such that the surface tension at the contact line is directed upward. Second, the vertical component of the surface tension must be of sufficient magnitude to suspend unsupported liquid against the downward pull of gravity (or other forces). Suspension is not determined solely by asperity size and intrinsic contact angle but also by surface tension, asperity angles, liquid density, and external forces. If an asperity has a hierarchical structure, the secondary features determine wetting properties. Local changes in contact angles can be described geometrically, as first proposed by Gibbs and later demonstrated by Mason and colleagues. Acknowledgment. I thank J. M. Miller for helpful discussions that led to this study and for suggesting that the asperities of ultralyophobic surfaces may support the mass of liquid above them. Also, thanks to Entegris management for supporting this work and allowing publication and D. Schmidt and P. Hall for fabricating the model asperities. LA0513050
