Model for Contact Angles and Hysteresis on Rough and Ultraphobic

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Model for Contact Angles and Hysteresis on Rough and Ultraphobic Surfaces C. W. Extrand* Entegris Inc., 3500 Lyman Boulevard, Chaska, Minnesota 55318 Received March 21, 2002. In Final Form: June 17, 2002 Very rough surfaces can suspend small water drops, prevent wetting, and cause contact angles to approach 180°. A criterion based on contact line density is proposed for predicting the conditions that produce these ultrahydrophobic surfaces, where, above a critical value, drops are suspended by asperities. Critical values of the contact line density can be calculated from contact angles, asperity shape, and information about the contact liquid, such as density and surface tension. The criterion was found to correctly predict suspension for several model surfaces prepared by lithography techniques. Apparent contact angles from suspended and collapsed drops also were modeled by accounting for rough edges and employing a linear average of contact angle along the perimeter of the drop, rather than an area average of cosine. This linear model suggests that, for suspended drops, both advancing and receding angles should increase. Alternatively, for drops that have collapsed over surface asperities, advancing contact angles should increase, while receding angles should decrease. These findings agree with the observations of a number of investigators that have suggested that asperity height may be less important than asperity shape in determining wetting.

Introduction Wetting is important in many industrial processes, such as cleaning, drying, painting, coatings, adhesion, and pesticide application. The most common method of evaluating wetting is contact angle measurements. Although these measurements are simple to perform, a variety of factors can complicate their interpretation. For example, surfaces usually show two stable values. Consider the case of a sessile drop, as depicted in Figure 1. If additional liquid is added, the contact line advances. Each time motion ceases, the drop exhibits an advancing contact angle (θa). Alternatively, if liquid is removed from the drop, the contact angle decreases to a receding value (θr) before the contact line retreats. This difference between θa and θr is referred to as contact angle hysteresis (∆θ),1

∆θ ) θa - θr

(1)

Hysteresis can arise from molecular interactions between the liquid and solid or from surface anomalies, such as roughness or heterogeneities.1 Among these factors, none are capable of exerting a greater influence on apparent contact angles than roughness. The effect of roughness on contact angles has been explored by many * [email protected]. (1) Extrand, C. W. In Encyclopedia of Surface and Colloid Science; Hubbard, A., Ed.; Marcel Dekker: New York, 2002; p 2414. (2) Wenzel, R. N. Ind. Eng. Chem. 1936, 28, 988. (3) Cassie, A. B. D.; Baxter, S. Trans. Faraday Soc. 1944, 40, 546. (4) Shuttleworth, R.; Bailey, G. L. J. Discuss. Faraday Soc. 1948, 3, 16. (5) Bikerman, J. J. J. Colloid Sci. 1950, 5, 349. (6) Good, R. J. J. Am. Chem. Soc. 1952, 74, 5041. (7) Bartell, F. E.; Shepard, J. W. J. Phys. Chem. 1953, 57, 211. (8) Bartell, F. E.; Shepard, J. W. J. Phys. Chem. 1953, 57, 455. (9) Shepard, J. W.; Bartell, F. E. J. Phys. Chem. 1953, 57, 458. (10) Allan, A. J. G.; Roberts, R. J. Polym. Sci. 1959, 39, 1-8. (11) Johnson, R. E., Jr.; Dettre, R. H. Adv. Chem. Ser. 1964, 43, 112. (12) Dettre, R. H.; Johnson, R. E., Jr. Adv. Chem. Ser. 1964, 43, 136. (13) Johnson, R. E., Jr.; Dettre, R. H. In Surface and Colloid Science; Matijevic, E., Ed.; Wiley: New York, 1969; Vol. 2, p 85. (14) Tamai, Y.; Aratani, K. J. Phys. Chem. 1972, 76, 3267. (15) Eick, J. D.; Good, R. J.; Neumann, A. W. J. Colloid Interface Sci. 1975, 53, 235. (16) Huh, C.; Mason, S. G. J. Colloid Interface Sci. 1977, 60, 11.

Figure 1. Small, sessile drop on a solid surface. (a) As liquid is added, the contact line advances. Each time motion ceases, the drop exhibits an advancing contact angle (θa). (b) Alternatively, if liquid is removed from the drop, the contact angle decreases to a receding value (θr), and then, the contact line retreats.

investigators.2-41 In most cases, if placed on a rough surface, a drop will collapse over the asperities, completely engulfing them, as shown in Figure 2a. Here, asperities can lead to large values of hysteresis,7-10,17-19,31,41,42 where substantial forces may be required to initiate drop movement.43 Less frequently, for very rough surfaces, drops can be suspended atop asperities, leaving air (or vapor) between them.3,11-13,21,25,26,30-36,38-41 This suspension produces very large apparent contact angles that can approach 180°, as shown in Figure 2b. As these ultraphobic44 surfaces exhibit little hysteresis, suspended drops roll off them with the slightest perturbation, like water off the back of a duck.3 One of the first theoretical treatments of rough surfaces was given by Wenzel,2 who used thermodynamic arguments to modify the Young equation to include a roughness (17) Oliver, J. F.; Huh, C.; Mason, S. G. J. Adhes. 1977, 8, 223. (18) Oliver, J. F.; Huh, C.; Mason, S. G. Colloids Surf. 1980, 1, 79. (19) Oliver, J. F.; Mason, S. G. J. Mater. Sci. 1980, 59, 431. (20) Bayramli, E.; van de Ven, T. G. M.; Mason, S. G. Can. J. Chem. 1981, 59, 1954. (21) Washo, B. D. Org. Coat. Appl. Polym. Sci. Proc. 1982, 47, 69. (22) Cox, R. G. J. Fluid Mech. 1983, 131, 1. (23) Busscher, H. J.; van Pelt, A. W. J.; de Boer, P.; de Jong, H. P.; Arends, J. Colloids Surf. 1984, 9, 319. (24) Andrade, J. D.; Smith, L. M.; Gregonis, D. E. In Surface and Interfacial Aspects of Biomedical Polymers; Andrade, J. D., Ed.; Plenum Press: New York, 1985; Vol. 1, p 249.

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Figure 2. Small, sessile drops on rough surfaces. (a) A collapsed drop on a rough surface; impaled by the asperities, the contact area is composed solely of a liquid/solid interface. (b) A suspended drop on an ultraphobic surface; the apparent contact area is a composite of liquid/vapor and liquid/solid interfaces.

factor (σ) that relates the apparent (θi) and true contact angle (θi,0),

cos θi ) σ cos θi,0

(2)

where i ) a for advancing angles, i ) r for receding angles, and σ is the ratio of the actual area (A) and the projected area (A0),

σ ) A/A0 g 1

(3)

Alternatively, Cassie and Baxter3 derived a simple expression for ultraphobic surfaces that links the apparent contact angle (θi) to the value of the solid surface (θp,i),

cos θi ) Rp cos θp,i - Rb

(4)

where Rp is the fractional area of wetted solid surface, Rb is the fractional area between the asperities, and

Rp + Rb ) 1

(5)

These two models often do not accurately predict contact angle variation.7-10,17-19,31,41 The Cassie-Baxter equation was initially derived for a specific surface geometry and then was generalized. Applying it without accounting for (25) Sacher, E.; Kelmberg-Sapieha, J.; Schreiber, H. P.; Wertheimer, M. R.; McIntyre, N. S. In Silanes, Surfaces, and Interfaces; Leyden, D. E., Ed.; Gordon and Breach: New York, 1986; p 189. (26) Morra, M.; Occhiello, E.; Garbassi, F. Langmuir 1989, 5, 872. (27) Schulze, R.-D.; Possart, W.; Kamusewitz, H.; Bischof, C. J. Adhes. Sci. Technol. 1989, 3, 39. (28) Hazlett, R. D. J. Colloid Interface Sci. 1990, 137, 527. (29) Lin, F. Y. H.; Li, D.; Neumann, A. W. J. Colloid Interface Sci. 1993, 159, 86. (30) Kunugi, Y.; Nonaka, T.; Chong, Y.-B.; Watanabe, N. J. Electroanal. Chem. 1993, 353, 209. (31) Miller, J. D.; Veeramasuneni, S.; Drelich, J.; Yalamanchili, M. R.; Yamauchi. G. Polym. Eng. Sci. 1996, 36, 1849. (32) Onda, T.; Shibuichi, S.; Satoh, N.; Tsujii, K. Langmuir 1996, 12, 2125. (33) Shibuichi, S.; Onda, T.; Satoh, N.; Tsujii, K. J. Phys. Chem. 1996, 100, 19512. (34) Tadanaga, K.; Katata, N.; Minami, T. J. Am. Ceram. Soc. 1997, 80, 1040. (35) Tadanaga, K.; Katata, N.; Minami, T. J. Am. Ceram. Soc. 1997, 80, 3213. (36) Hozumi, A.; Takai, O. Thin Solid Films 1997, 303, 222. (37) Swain, P. S.; Lipowsky, R. Langmuir 1998, 14, 6772. (38) Chen, W.; Fadeev, A. Y.; Hsieh, M. C.; O ¨ ner, D.; Youngblood, J. P.; McCarthy, T. M. Langmuir 1999, 15, 3395. (39) Youngblood, J. P.; McCarthy, T. M. Macromolecules 1999, 32, 6800. (40) Bico, J.; Marzolin, C.; Que´re´, D. Europhys. Lett. 1999, 47, 220. (41) O ¨ ner, D.; McCarthy, T. M. Langmuir 2000, 16, 7777. (42) Extrand, C. W.; Gent, A. N. J. Colloid Interface Sci. 1990, 138, 431. (43) MacDougall, G.; Okrent, C. Proc. R. Soc. London 1942, 180A, 151. (44) The term “ultraphobic” as used here is meant to describe any surface that is sufficiently rough that it suspends liquid drops and is a generalization of the terms ultrahydrophobic and ultralyophobic.

Figure 3. Schematic depiction of a surface covered by a hexagonal array of square asperities of width x and height z. (a) Plan view. The square grid has a linear dimension of 1/2y. The dashed line defines a single unit cell. The crosshatched areas are the tops of the asperities. (b) Side view. The angle subtended by the top edges of the asperities is φ, and the rise angle of their sides is ω. As drawn here, φ ) ω ) 90°.

details of the surface can lead to errors. Neither the Cassie-Baxter equation nor the Wenzel equation includes the influence of sharp edges. Also, wetting behavior generally is determined by interaction at the fluid/liquid/ solid triple line and not by the liquid/solid interfacial area,8,37,41,45 so it should not be surprising that these areabased concepts have not always been successful. Although more elaborate models have been proposed,11,15,16,20,22,29,37 they suffer from many of the same pitfalls. Moreover, they are mathematically complex and difficult to apply. In the past, one of the biggest impediments to a more thorough understanding has been the lack of good experimental data on model surfaces. Although recent work40,41 has addressed this lack of data, the ability of current models to predict contact angles on rough surfaces, ultraphobic or otherwise, is inadequate. In this paper, a criterion for determining drop suspension or collapse is proposed along with models for estimating subsequent contact angles. Attributes of the criterion and the models are presented, and then, their ability to correctly predict experimental observations on rough surfaces with regular, periodic asperities is investigated. Theoretical Basis Let’s begin by considering the model surfaces constructed by O ¨ ner and McCarthy41 shown schematically in Figure 3. Surface asperities take the form of hexagonal arrays of square posts of width x and height z. The angle subtended by the top edge of the asperities is φ, the rise angle of the side of the asperities is ω, and

φ ) 180° - ω

(6)

As drawn in Figure 3, φ ) ω ) 90°. The dashed line contains a single unit cell, where the area density (δ) of asperities is

δ ) 1/2y2

(7)

Contact line density (Λ), which is the length of the asperity perimeter per unit area that could potentially suspend a liquid drop, is defined as the product of the feature perimeter (p) and δ,

Λ ) pδ

(8)

For the hexagonal arrays of square posts described in (45) Pease, D. C. J. Phys. Chem. 1945, 49, 107.

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advancing value (θa,0) on the sides of the asperities. The contact angle on the side of the asperities (θs) is related to the true advancing value by φ (or ω),46,47

θs ) θa,0 + 90° - φ ) θa,0 + ω - 90°

(12)

If drops are sufficiently small that gravity does not affect their shape and they retain spherical proportions, then the apparent contact area (A) can be written in terms of the apparent advancing contact angle (θa) and the drop volume (V),

A ) π1/3(6V)2/3{[(1 - cos θa)/sin θa]{3 + [(1 cos θa)/sin θa]2}}-2/3 (13) Equating the body and surface forces,

F)f

(14)

and then combining eqs 8 and 10-14 gives rise to a critical value of contact line density (Λc) for small drops,

Λc ) -FgV1/3{[(1 - cos θa)/sin θa]{3 + [(1 cos θa)/sin θa]2}}2/3/[(36π)1/3γ cos(θa,0 + ω - 90°)] (15)

Figure 4. Magnified side view near the contact line of a suspended advancing drop. (a) The drop exhibits its true advancing value (θa,0) on the side of the asperity. (b) If liquid is injected into the drop, its leading edge will bulge forward, contacting a neighboring asperity. As the contact line advances, the contact line crosses the flat top of the asperities with a “true” advancing contact angle (θa,0) equal to that exhibited on an equivalent featureless surface. (c) When the contact line reaches the edge of the asperity, it is pinned. (d) The apparent advancing contact angle (θa) increases until the drop regains its true advancing value (θa,0) on the side of the asperity.

Figure 3, Λ is

Λ ) 4x/y2

(9)

Criterion for Suspension or Collapse. A criterion can be derived to determine if a drop will be suspended by a rough surface by examining the interplay of body and surface forces. Body forces (F) associated with gravity are determined from the density (F) and volume (V) of the liquid drop,

F ) FgV

(10)

where g is the acceleration due to gravity. On the other hand, the surface forces (f) depend on the surface tension of the liquid (γ), its apparent contact angle with the side of the asperities with respect to the vertical plane (θs), the perimeter of each asperity (p), the area density of the asperities (δ), and the apparent contact area of the drop (A),

f ) -pδAγ cos θs

(11)

Figure 4 shows the contact line of a liquid drop on our model rough surface. Suspended drops exhibit their true

If Λ > Λc, drops will be suspended by the asperities, producing an ultraphobic surface. Otherwise, if Λ < Λc, drops will collapse over the asperities and the contact interface will be solely liquid/solid. Contact Angles of Suspended Drops. Assume apparent contact angles (θi) manifest themselves as fractional contributions along the contact line,48-52

θi ) λpθp,i + λbθb,i

(16)

where θp,i is the contact angle of the liquid on the asperities, θb,i is the contact angle between the asperities, λp is the linear fraction of the contact line on the asperities, λb is the linear fraction of the contact line between the asperities, and

λ p + λb ) 1

(17)

If the drops are suspended atop of the asperities, then the apparent contact area between the drop and featured solid surface will be composite in nature, with interfaces of both liquid/solid and liquid/vapor. Figures 4 and 5 show magnified side views of such suspended drops near the contact line. Contact lines between asperities are assumed to be straight, and contact interfaces, to be flat. On the basis of the LaPlace pressure of microliter-sized drops, the curvature between micron-sized features should be negligible. It certainly is true when θa,0 ) 90°. However, (46) Johnson, R. E., Jr.; Dettre, R. H. In Surface and Colloid Science; Matijevic`, E., Ed.; Wiley: New York, 1969; Vol. 2, p 126. (47) Wapner, P. G.; Hoffman, W. P. Langmuir 2002, 18, 1225. (48) Gaines, G. L., Jr. J. Colloid Sci. 1960, 15, 321. (49) Brockway, L. O.; Jones, R. L. Adv. Chem. Ser. 1964, 43, 275. (50) Bain, C. D.; Evall, J.; Whitesides, G. M. J. Am. Chem. Soc. 1989, 111, 7155. (51) Extrand, C. W. Langmuir 1993, 9, 475. (52) Extrand, C. W. J. Colloid Interface Sci. 1998, 207, 11.

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subtended by the edge of the asperity,53-55

θp,a ) θa,0 + (180° - φ)

(19)

Combining eqs 6 and 16-19 produces a working equation for estimating apparent advancing contact angles (θa) for suspended “Cassie” drops:

θa ) λp(θa,0 + ω) + (1 - λp)θair

(20)

Figure 5 shows the receding contact line of a suspended drop. As the contact line retreats, the drop moves across the tops of asperities with an angle that is equal to the “true” receding value (θr,0). When the contact line reaches the edge of an asperity, it is pinned. Here, the receding contact line behaves differently than the advancing line. The receding contact angle (θp,r) decreases as the contact line unsuccessfully tries to establish its “true” receding angle on the side of the asperity. In doing so, the contact line is pinched off and ruptures. The receding contact line jumps to the next asperity, and the process starts again. Thus,

θp,r ) θr,0

(21)

Combining eqs 6, 17, 18, and 21 gives an expression for the apparent receding contact angle of suspended drops:

θr ) λpθr,0 + (1 - λp)θair

(22)

Contact angle hysteresis (∆θ) for suspended drops comes from eqs 1, 20, and 22, Figure 5. Magnified side view near the contact line of a suspended receding drop. (a) As the drop retreats, the contact line crosses the flat tops of the asperities with the “true” receding contact angle (θr,0). (b) When the contact line reaches the edge of the asperity, it is pinned. (c) The apparent receding contact angle (θr) decreases as the contact line tries to establish its “true” receding angle on the side of the asperity. In doing so, the contact line is pinched off and ruptures. (d) The receding contact line jumps to the next asperity, and the process starts again.

contact angles established on the sides of asperities by suspended drops may impart some local curvature. Since the contact angle of the free surface between air and liquid (θair) is θair ) 180°, then

θb,i ) θair ) 180°

(18)

Figure 4 shows the advancing case of a suspended liquid drop. Suspended drops exhibit their true advancing value (θa,0) on the sides of the asperities. If liquid is injected into the drop, its leading edge will bulge forward, contacting a neighboring asperity. The contact line then moves across the tops of the asperities with its “true” advancing value (θa,0). At the edge, the contact line is pinned. The contact angle increases until the true advancing value (θa,0) is attained on the side of the asperity. As this occurs, the leading edge of the drop bulges forward and the process starts again. The difference between the apparent advancing contact angle relative to the horizontal plane of the surface (θp,a) and the true advancing angle depends on the angle

∆θ ) λp(∆θ0 + ω)

(23)

where ∆θ0 is the hysteresis of the corresponding smooth surface,

∆θ0 ) θa,0 - θr,0

(24)

Because of physical limitations imposed by the surface, θa and θr of suspended drops cannot be >180°. For surfaces covered with a hexagonal array of square posts depicted in Figure 3, the linear fraction of the contact line on the asperities (λp) is

λp ) (x/y)/{[5/4 - 2(x/y) + (x/y)2]1/2 + x/y}

(25)

and the linear fraction of the contact line between the asperities (λb) is

λb ) 1 - λ p

(26)

Contact Angles of Collapsed Drops. For drops that collapse, the entire contact area will be comprised of a liquid/solid interface where the contact line will encounter flat areas and sharp corners associated with the base of the asperities. Between the asperities, the drop will exhibit the same advancing and receding contact angles as those on an otherwise equivalent flat, featureless surface:

θb,a ) θa,0

(27)

(53) Gibbs, J. W. The Collected Works of J. Willard Gibbs; Yale University Press: New Haven, Connecticut, 1961; Vol. 1, p 326. (54) Coghill, W. H.; Anderson, C. O. Sci. Am. Suppl. 1918, 86, 52. (55) Oliver, J. F.; Huh, C.; Mason, S. G. J. Colloid Interface Sci. 1977, 59, 568.


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and

θb,r ) θr,0

(28)

At the base of the asperities, contact angles will be distorted by sharp corners. The advancing angle along the asperity base (θp,a) will increase,4

θp,a ) θa,0 + ω

(29)

while the receding angle along the asperity base (θp,r) will decrease,

θp,r ) θr,0 - ω

(30)

Combining eqs 16, 17, 27, and 29, one arrives at an expression for apparent advancing contact angles (θa) of collapsed “Wenzel” drops,

θa ) λp(θa,0 + ω) + (1 - λp)θa,0

(31)

Similarly, from eqs 16, 17, 28, and 30, the equivalent expression for apparent receding angles (θr) is

θr ) λp(θr,0 - ω) + (1 - λp)θr,0

(32)

and from eqs 1, 31, and 32, contact angle hysteresis (∆θ) for collapsed drops is

∆θ ) ∆θ0 + 2λpω

(33)

Because of physical limitations imposed by the geometry of the surfaces, for advancing drops (eq 31), θa,0 + ω e 180°, and for receding drops (eq 32), θr,0 - ω g 0°. Results and Discussion Suspended or Collapsed. The reported size of asperities required to create ultraphobic surfaces has been controversial.38 The problem lies in defining surfaces solely in terms of asperity height or spacing. While they are important considerations, other parameters also play a role. In order for liquid drops to be suspended, surface forces acting around the perimeter of asperities must be greater than body forces, or in terms of the contact line density (Λ) criterion presented here, this requires that Λ > Λc. Note that asperity height (z), which correlates directly to average or root-mean-square roughness, does not enter into estimates of Λ or Λc (eqs 8 and 15). Rather, there are a number of other factors that determine if a drop is suspended or collapses. From eq 15, if the liquid density (F) or volume (V) increases, higher Λc values are required to suspend drops. Larger values of φ or θa also necessitate a greater Λc. On the other hand, if γ or θa,0 increases, a lower Λc will suffice. While the influence of F, γ, and θa,0 is straightforward, the dependence of Λc on φ, θa, and V merits further discussion. The edge angle is extremely important in determining ultraphobicity. In order for a surface to support a liquid drop, φ must be sufficiently small (and θa,0 sufficiently large) to direct the surface forces upward against the body force (a positive Λc value); otherwise, drops will collapse, regardless of Λ. (If both the surface and body forces are directed downward, Λc will be negative.) Marshalling the surface forces against gravity requires that θa,0 - φ + 90° > 90° (or θa,0 + ω - 90° > 90°). This is not, however, a sufficient condition for drop suspension. Not only must the surface forces be directed upward, but they also must be of greater magnitude than the body

Figure 6. Schematic depiction of asperities with a square top of width x and height z with various edge and rise angles. (a) φ > 90° and ω < 90°. (b) φ < 90° and ω > 90°.

forces. If the rise angle were decreased such that ω < 90°, Figure 6a, asperities would be less capable of suspension. Conversely, increasing ω, as depicted in Figure 6b, would enhance the ability of asperities to suspend drops. Although estimating Λc requires knowledge of θa, eqs 15 and 20 can be combined to eliminate θa and this would allow a priori estimates of Λc, where Λ and ω are known. From eq 15, Λc is expected to increase weakly with drop volume, where drops are small. Achieving a higher θa requires greater Λc, because as θa goes up, a smaller contact area is supporting the same mass of liquid. The free surface of water and organic liquids begins to distort under the influence of gravity, where V > 10 µL. As drops flatten, the contact area per volume increases and Λc should decrease. (An expression for estimating Λc for large, flat drops is given in the Appendices.) If the contact area of the solid surface is constrained by walls and liquid is continually added, the height of the liquid will begin to rise. Eventually the pressure head will be great enough to collapse a suspended liquid. (See the Appendices for details.) Other body forces could cause suspended drops to collapse, such as the inertia associated with the impact of a falling drop or rapid injection of a liquid into an existing sessile drop. Contact Angles of Suspended Drops. Both advancing and receding contact angles are expected to increase for ultraphobic surfaces (eqs 20 and 22). The magnitude of variation depends on λp and ω. θa should increase with ω, while θr should rise with decreasing λp. Hysteresis should increase with asperity contact (larger λp values) and with the severity of the edge angles (larger ω values). The magnitude of hysteresis for suspended drops may be less than that for a corresponding smooth surface. For example, if a hydrophobic surface of square asperities (Figure 3) with ω ) 90°, θa,0 ) 110°, and ∆θ0 ) 10° has minimal contact with a suspended drop, say λp ) 0.01, then ∆θ ) 1°. Contact Angles from Collapsed Drops. As surfaces become rougher, θa and ∆θ for collapsed drops are expected to increase with λp or ω; on the other hand, θr should decrease. In the case of a surface composed of parallel grooves with λp ) 1, eq 33 reduces to a modified form of the Shuttleworth-Bailey equation (eq 1),

∆θ ) ∆θ0 + 2ω

(34)

(As originally proposed, eq 34 did not include a term for inherent hysteresis, ∆θ0.) Comparison with Experimental Data: Suspended or Collapsed. The most comprehensive studies on model rough surfaces were done by O ¨ ner and McCarthy.41 They prepared hexagonal arrays of square posts on silicon wafers with φ ) ω ) 90° (Figure 3) using photolithography and then treated them with one of three hydrophobic silane agents: dimethyldichlorosilane (DS), octyldimethylchlorosilane (OS), or heptadecafluoro-tetrahydrodecyl-dimethylchlorosilane (FS). Table 1 lists the dimensional parameters for some of these surfaces and their ability to

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Table 1. Geometric Parameters for Suspended or Collapsed Water Drops on Hydrophobic Surfaces Covered with Hexagonal Arrays of Square Asperitiesa-c x (µm)

y (µm)

x/y (µm)

δ (µm-2)

Λ (m-1)

λp

Rp

σ

suspended or collapsed

2 8 16 32 64 128 8 8 8

4 16 32 64 128 256 23 32 56

0.5 0.5 0.5 0.5 0.5 0.5 0.35 0.25 0.14

6.3 × 10-2 3.9 × 10-3 9.8 × 10-4 2.4 × 10-4 6.1 × 10-5 1.5 × 10-5 1.9 × 10-3 9.8 × 10-4 3.2 × 10-4

5.0 × 105 1.3 × 105 6.3 × 104 3.1 × 104 1.6 × 104 7.8 × 103 6.1 × 104 3.1 × 104 1.0 × 104

0.41 0.41 0.41 0.41 0.41 0.41 0.30 0.22 0.13

0.25 0.25 0.25 0.25 0.25 0.25 0.13 0.069 0.023

14 4.1 2.6 1.8 1.4 1.2 2.5 1.8 1.3

suspended suspended suspended suspended collapsed collapsed suspended suspended collapsed

a x is asperity width and y is the width of each unit cell. δ is the area density of asperities (eq 7). Λ is the contact line density (eq 9). λp and Rp are contact line and contact area fractions occupied by the asperities (eqs 25 and 40). σ is the Wenzel roughness factor (eq 44). b z ) 40 µm and ω ) 90°. c Reference 41.

suspend water drops. (Expressions for estimating the fractional area contacting asperities (Rp) and the Wenzel roughness factor (σ) are given in the Appendices.) As the three treatments gave similar water contact angles on smooth surfaces (DS θa,0 ) 107°, θr,0 ) 102°; OS θa,0 ) 102°, θr,0 ) 94°; FS θa,0 ) 119°, θr,0 ) 110°), their model rough surfaces showed similar abilities to suspend small drops. It appears that the controlling parameter for suspension or collapse is indeed contact line density (Λ). The critical value of the contact line density (Λc) for these water/surface combinations resided between 1 × 104 and 3 × 104 m-1. Where Λ > Λc, drops were suspended; otherwise, if Λ < Λc, drops collapsed. This conclusion is in general agreement with calculated Λc values. According to eq 15, the critical value for small, spherical water drops on DStreated surfaces was Λc ) 3 × 104 m-1. Values of Λc were slightly higher for the OS-treated surfaces and a bit lower for the FS-treated surfaces. The validity of the critical contact line density criterion to predict suspension or collapse was further tested using data from Bico and co-workers.40 Figure 7 shows their surfaces. They were molded on silicon wafers using a tetramethylothrosilicate sol-gel and then treated with a fluorosilane (θa,0 ) 118°, θr,0 ) 100°). The geometric parameters for these surfaces with stripes, posts, or cavities40 are listed in Table 2. Water drops were suspended by the stripes and posts but collapsed into the cavities. Critical values of contact line density (Λc) were calculated for these surfaces with θa,0 ) 118° and θa ) 170°. Predictions of suspension or collapse were in agreement with experimental observations. For the stripes and circular posts, where ω ) 90°, Λc ) 1 × 104 m-1, Λ > Λc, and the stripes and posts did indeed suspend small water drops. On the other hand, for the cavities, where ω ) 60°, Λc ) -1 × 105 m-1, Λc < Λ, and drops collapsed. It can be argued that the surfaces with stripes, posts, and cavities all have posts with cavities between them. Therefore, why did they behave differently? All had similar λp values, but the stripes and posts had larger ω values than the cavities. For the stripes and posts, ω values were sufficiently large that surface forces were directed upward, opposing gravity (a large, positive Λc value); therefore, drops on these two surfaces were suspended. The more gradual slope (or smaller ω) of the cavities produced a negative value of Λc, implying surface forces were directed downward and drops collapsed. To this point, the discussion has focused on water. It was not mentioned by the authors, but it seems unlikely that the model surfaces of Oner and McCarthy or Bico and co-workers could have suspended lower energy liquids.26 This does not imply that ultraphobic surfaces cannot be constructed for such liquids. In an earlier study,

Figure 7. Schematic depictions of surfaces with microscopic stripes, circular posts, or circular cavities.40 (a) Plan view and (b) side view of a striped surface. (c) Plan view and (d) side view of a surface with circular posts. (e) Plan view and (f) side view of a surface with circular cavities. Crosshatched areas have a greater elevation.

McCarthy and co-workers38 prepared surfaces covered with spherical PTFE particles that suspended liquids other than water. Their success presumably was due to undercuts (or sufficiently large ω values) on the bottom side of the particles. On smooth fluorocarbon (CF2) surfaces, hexadecane exhibits an advancing contact angle of θa,0 ≈ 50°.56-59 For a fluorocarbon-treated surface described in Figure 3 with ω ) 90°, hexadecane drops (F ) 773 kg/m3,56 γ ) 27 mN/ m60) would likely collapse because surface forces would be (56) Johnson, R. E., Jr.; Dettre, R. H. J. Colloid Sci. 1965, 20, 173. (57) Ellison, A. H.; Fox, H. W.; Zisman, W. A. J. Phys. Chem. 1953, 57, 622. (58) Neumann, A. W.; Haage, G.; Renzow, D. J. Colloid Interface Sci. 1971, 35, 379. (59) Penn, L. S.; Miller, B. J. Colloid Interface Sci. 1980, 78, 238.


Langmuir, Vol. 18, No. 21, 2002 7997

Table 2. Geometric Parameters for Suspended or Collapsed Water Drops on Hydrophobic Surfaces with Microscopic Stripes, Circular Posts, or Shallow Circular Cavitiesa,b feature

x (µm)

y (µm)

z (µm)

ω (deg)

δ (µm-2)

Λ (m-1)

λp

Rp

σ


stripes circular posts circular cavities

1 1 2

4 4 3

0.7 1.5 0.5

90 90 60

6.3 × 10-2 6.3 × 10-2 1.1 × 10-1

5 × 105 2 × 105 7 × 105

0.25 0.34 0.24

0.25 0.05 0.65

1.35 1.29 1.35

suspended suspended collapsed

a x is feature width and y is the size of the square grid. ω is the rise angle of the asperity. δ is the area density of asperities. Λ is the contact line density. λp and Rp are contact line and contact area fractions occupied by the protruding features. σ is the Wenzel roughness factor. Equations for calculating these surface parameters can be found in Table 6 of the Appendices. b Reference 40.

Table 3. Measured and Calculated Contact Angles for Suspended Water Drops on DS-Treated, Hydrophobic Surfaces Covered with Hexagonal Arrays of Square Asperitiesa,b measured

Cassie equation

linear equations

x (µm)

y (µm)

x/y

θa (deg)

θr (deg)

∆θ (deg)

θa (deg)

θr (deg)

∆θ (deg)

θa (deg)

θr (deg)

∆θ (deg)

2 8 16 32 8 8

4 16 32 64 23 32

0.5 0.5 0.5 0.5 0.35 0.25

176 173 171 168 175 173

141 134 144 142 146 154

35 39 27 26 29 19

145 145 145 145 155 162

143 143 143 143 154 161

2 2 2 2 1 1

180 180 180 180 180 180

148 148 148 148 157 163

32 32 32 32 23 17

a Measured contact angles from ref 41. x is feature width, and 1/ y is the size of the square grid. Values of the contact angles for a smooth 2 DS-treated surface were θa,0 ) 107° and θr,0 ) 102°. θa and θr values were calculated using θa,0, θr,0, and parameters from Table 1. Values listed under “Cassie equations” were computed with eqs 4 and 5. Values for the “linear equations” were calculated using eqs 20 and 22. Hysteresis values (∆θ) were determined with eq 1. b z ) 40 µm and φ ) ω ) 90°.

Table 4. Measured and Calculated Contact Angles for Water Drops on Hydrophobic Surfaces with Microscopic Stripes, Circular Posts, or Shallow Circular Cavitiesa surface feature


θa (deg)

stripes circular posts circular cavities

suspended suspended collapsed

165 170 138

measured θr (deg) ∆θ (deg) 132 155 75

33 15 63

Cassie/Wenzel equation θa (deg) θr (deg) ∆θ (deg) 150 167 129

143 164 117

8 3 12

linear equations θa (deg) θr (deg) ∆θ (deg) 180 180 133

160 153 81

20 27 52

a Measured contact angles from ref 40. x is feature width and y is the size of the square grid. Contact angles for striped surfaces were measured parallel to the stripes. Water contact angles for the smooth fluorosilane-treated surface were θa,0 ) 118° and θr,0 ) 100°. θa and θr values for the Wenzel, Cassie, and linear equations were calculated using θa,0, θr,0, and parameters from Table 2 as follows: for suspended drops, Cassie values were calculated with eqs 4 and 5, while “linear” values were computed from eqs 20 and 22; for collapsed drops, Wenzel values were determined from eq 2, while linear values came from eqs 31 and 32. Hysteresis values (∆θ) were calculated with eq 1.

directed downward, Λc < 0. However, one can imagine modifications that would enable suspension. If the asperities were undercut to reduce the edge angle to φ ) 40° (Figure 6b), Λc would increase to 5 × 104 m-1 and a hexagonal array of square posts with x ) 8 µm and y ) 16 µm (Λ ) 1.3 × 105 m-1) should suspend hexadecane. Comparison with Experimental Data: Contact Angles for Suspended Drops. The linear models proposed here provide reasonable estimates of the contact angles measured on the various model surfaces. Table 3 shows measured and calculated contact angles for suspended water drops on hydrophobic DS-treated surfaces consisting of hexagonal arrays of square posts. Contact angles for an equivalent smooth surface were θa,0 ) 107° and θr,0 ) 102°. Suspension of small water drops caused a substantial increase in both apparent contact angles, θa and θr. Values were independent of asperity heights ranging between z ) 20 µm and z ) 140 µm. Rather, variation was related to asperity width (x) and spacing (y). Where x/y ) 0.5, suspended drops showed θa ) 172° ( 3°, θr ) 140° ( 4°, and ∆θ ) 32° ( 6°. The predicted values (θa ) 180°, θr ) 148°, and ∆θ ) 32°) were in fair agreement with the measured ones. (θa values calculated with eq 20 were between 187° and 183° but are listed in Table 3 as 180° because of the physical limitations imposed by the surfaces.) This linear approach also correctly predicted a decline in ∆θ with decreasing x/y. (60) Handbook of Chemistry and Physics, 73rd ed.; Lide, D. R., Ed.; CRC Press: Boca Raton, Florida, 1992.

The linear expressions also did a reasonably good job of estimating the contact angles of water drops suspended on OS- and FS-treated square asperities as well as on FS-treated stripes and circular posts, Table 4. In some cases, the Cassie-Baxter equation gave acceptable predictions of θa or θr but always greatly underestimated ∆θ. Comparison with Experimental Data: Contact Angles for Collapsed Drops. Table 5 lists measured and calculated contact angles for collapsed water drops on DS-treated, hydrophobic surfaces covered with hexagonal arrays of square asperities. Note that, because of interaction with the sharp edges, measured θa values increased and θr values decreased, which led to a significant rise in ∆θ as compared to the case of an equivalent smooth surface, where ∆θ0 ) 5°. Water contact angles predicted by the linear equations agreed fairly well with the measured values, Table 5. The linear equations also gave a reasonable estimate of the contact angles of collapsed water drops on a hydrophobic surface covered with circular cavities, Table 4. In contrast, the Wenzel equation predicts modest increases in both θa and θr, leading to a gross underestimation of ∆θ. As with suspended drops, agreement between measured values and those predicted by the linear equations potentially could be improved by better accounting for contact line contours and variations in the contact angles near asperities. Several factors may lead to the divergence between measured and calculated θi values. It is possible that in

7998


Extrand

Table 5. Measured and Calculated Contact Angles for Collapsed Water Drops on Hydrophobic Surfaces on DS-Treated, Hydrophobic Surfaces Covered with Hexagonal Arrays of Square Asperitiesa,b measured

Wenzel equation

linear equations

x (µm)

y (µm)

x/y

θa (deg)

θr (deg)

∆θ (deg)

θa (deg)

θr (deg)

∆θ (deg)

θa (deg)

θr (deg)

∆θ (deg)

64 128 8

128 256 56

0.5 0.5 0.14

139 116 121

81 80 67

58 36 54

114 110 112

107 104 105

7 6 6

137 137 116

65 65 91

73 73 26

a Measured contact angles from ref 41. x is feature width, and 1/ y is the size of the square grid. Values of the contact angles for the 2 equivalent smooth DS-treated surface were θa,0 ) 107° and θr,0 ) 102°. θa and θr values were calculated using θa,0, θr,0, and parameters from Table 1. Values listed under “Wenzel equations” were calculated with eq 2. Values for the “linear equations” were calculated from eqs 31 and 32. Hysteresis values (∆θ) were calculated with eq 1. b z ) 40 µm and φ ) ω ) 90 °.

Table 6. Equations for Determining Geometric Parameters for Surfaces with Microscopic Stripes, Circular Posts, or Shallow Circular Cavitiesa,b feature

δ

Λ

λp

Rp

σ

array of square posts stripes circular posts circular cavities

1/2y2 1/y2 1/y2 1/y2

4x/y2 2/y πx/y2 πx/y2

(x/y)/{[5/4 - 2(x/y) + (x/y)2]1/2 + x/y} x/y 1/ π(x/y)/[1 + (1/ π - 1)(x/y)] 2 2 (1 - x/y)/[1 + (1/2π - 1)(x/y)]

2(x/y)2/[7/4 + (x/y)2] x/y (π/4)(x/y)2 1 - (π/4)(x/y)2

1 + 5/2xz/y2 1 + 2z/y 1 + πxz/y2 1 + πxz/y2

a δ is the area density of features. Λ is the contact line density. λ and R are linear and area fractions occupied by the protruding features, p p and σ is the Wenzel roughness factor. b Figure 7.

some cases drops were neither fully suspended nor fully collapsed. Discrepancies also may have arisen from the assumption that the contact line between asperities is straight and exhibits a constant angle. For the surfaces discussed here, the error from assuming straight contact lines was probably small, since θi,0 values were all relatively close to 90°. (The greater the difference between the true angles and 90°, the greater the error in the estimates of the linear fractions.) Agreement could possibly be improved by better accounting for line shape and any variations in the contact angles in the vicinity of the asperities. Furthermore, interaction with asperities and the pull of gravity61 can slightly flatten the bottom of small spherical drops, creating a finite contact area, which may lead to an underestimation of the apparent angles for suspended drops, particularly for θa values. This flat bottom may bias the observer toward reporting θa < 180°, when, in fact, θa may be ) 180°.62 Conclusions A contact line density criterion for estimating the suspension or collapse of liquid drops on rough surfaces has been derived by balancing body and surface forces. This criterion suggests that the potential for a rough surface to suspend liquid drops cannot be determined solely from the height or spacing of asperities. Additional parameters such as asperity slope, contact angles, liquid density, and surface tension must be considered. As for contact angles, it is not the absolute magnitude of roughness that determines variations, but the slope of surface asperities and the linear fraction of the contact line. Local changes in contact angles can adequately be described geometrically, as first proposed by Gibbs. For suspended drops, both advancing and receding angles should be greater than those on a corresponding flat surface. Alternatively, for drops that have collapsed over surface asperities, advancing contact angles and hysteresis should increase, while receding angles should decrease. The equations derived here for estimating contact angles on ultraphobic and rough surfaces describe surfaces with (61) Aussillous, P.; Que´re´, D. Nature 2001, 411, 924. (62) While the apparent macroscopic θa value cannot exceed 180°, it is conceivable that local interactions of the contact line with steep, sparsely scattered asperities may lead to an average value of θa that is slightly >180° at the microscopic level.

regular geometry. Applying these equations directly to random or poorly characterized surfaces may prove difficult. That said, they constitute a set of design rules for creating ultraphobic or liquid-retaining surfaces. Appendices Critical Contact Line Density for Large Drops. An expression similar to eq 15 can be derived for very large, flat drops. Assuming that drop height (h) can be approximated as13

h ) [2(γ/Fg)(1 - cos θa)]1/2

(35)

the critical contact line density for large drops is

Λc ) -[(1/4)(Fg/γ)(1 - cos θa)]1/2/cos(θa,0 + ω - 90°) (36) Everything else being equal, a lower Λ is required to suspend large, flat drops as compared with small, spherical ones because the mass for large drops is spread over a bigger area. For example, if water drops are placed on a surface where ω ) 90°, θa,0 ) 120°, and θa ) 170°, suspension of a 1-µL drop would require Λ > 7 × 103 m-1, while suspension of a large, flat drop would occur where Λ > 4 × 102 m-1. Height (or Pressure) to Collapse a Suspended Liquid. An expression for estimating the height (h) that would cause the collapse of a liquid column suspended by asperities can be estimated as follows. The body force associated with the liquid column is

F ) FghA

(37)

Combining this expression with eqs 8, 11, 12, and 14 gives

h ) -(Λγ/Fg) cos(θa,0 + ω - 90°)

(38)

If gently added to a surface covered with a hexagonal array of square posts where x ) 8 µm, y ) 16 µm, Λ ) 1.3 × 105 m-1, ω ) 90°, and θa,0 ) 120°, water would be suspended until its height reached 0.48 m. If additional water were added, the pressure head would be sufficient to cause collapse, forcing water into the spaces between asperities.


Langmuir, Vol. 18, No. 21, 2002 7999

Similarly, the pressure (∆P) to collapse a suspended liquid is

∆P ) -Λγ cos(θa,0 + ω - 90°)

(39)

For water on the hypothetical surface described above, a pressure of 4.7 kPa would be required to cause collapse. Fractional Areas and Wenzel Roughness Factors for Surfaces Covered with a Hexagonal Array of Square Posts. For a surface covered with a hexagonal array of square posts, the fractional area occupied by asperity tops (Rp) is

Rp ) 2(x/y)2/[(7/4) + (x/y)2]

(40)

and the fractional area between asperities (Rb) is

Rb ) 1 - Rp

(41)

The Wenzel factor (σ) is a ratio of the actual (A) to projected

surface area (A0),

σ ) A/A0

(42)

where the actual surface area is the sum of the projected area and the area of the asperity sides (As). Therefore, σ can be written as

σ ) (A0 + As)/A0 ) 1 + As/A0

(43)

For a hexagonal array of square asperities,

σ ) (A0 + As)/A0 ) 1 + (5/2)xz/y2

(44)

Geometric Parameters for Surfaces with Stripes, Circular Posts, or Circular Cavities. Table 6 shows equations for calculating the area density (δ), contact line density (Λ), contact line fraction (λp), contact area fraction (Rp), and Wenzel roughness factor (σ) of surfaces with stripes, circular posts, or circular cavities, Figure 7. Parameters for surfaces with square posts, Figure 3, are included for comparison. Acknowledgment. I thank Entegris management for supporting this work and allowing publication. Also, thanks are extended to T. M. McCarthy for providing additional information about his experimental work on ultraphobic surfaces. LA025769Z

Model for Contact Angles and Hysteresis on Rough and Ultraphobic

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