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Criteria for Ultralyophobic Surfaces C. W. Extrand† Entegris Inc., 3500 Lyman Boulevard, Chaska, Minnesota 55318 Received December 31, 2003. In Final Form: March 29, 2004 Very rough surfaces can suspend small liquid drops and produce very large contact angles. This behavior often is referred to as ultralyophobicity or super repellency. It is proposed that two criteria must be met to invoke ultralyophobicity: a contact line density criterion and asperity height criterion. The proposed criteria were tested using experimental data available in the literature and were found to correctly predict suspension of small water drops on model rough surfaces with a wide variety of asperity shapes, sizes, and spacing.
Introduction If a small liquid drop is deposited on a rough surface, in most cases it will spread, engulfing surface asperities. Figure 1a shows such a surface where the contact liquid has penetrated the spaces between the asperities. Here, interaction of the liquid with the asperities can lead to large values of contact angle hysteresis1-11 and, consequently, substantial forces may be required to initiate drop movement.12 Less frequently, drops are suspended atop asperities, leaving air (or vapor) between them.2,13-26 This suspension produces very large apparent contact angles (140°-180°) that are characteristic of super repellent or ultralyophobic surfaces, Figure 1b. Studies of ultralyophobic surfaces first appeared in the scientific literature in the 1940s.2 While this subject has received continued attention since that time, most in†
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(1) Wenzel, R. N. Ind. Eng. Chem. 1936, 28, 988. (2) Cassie, A. B. D.; Baxter, S. Trans. Faraday Soc. 1944, 40, 546. (3) Shuttleworth, R.; Bailey, G. L. J. Discuss Faraday Soc. 1948, 3, 16. (4) Bikerman, J. J. J. Colloid Sci. 1950, 5, 349. (5) Bartell, F. E.; Shepard, J. W. J. Phys. Chem. 1953, 57, 211. (6) Oliver, J. F.; Huh, C.; Mason, S. G. Colloids Surf. 1980, 1, 79. (7) Busscher, H. J.; van Pelt, A. W. J.; de Boer, P.; de Jong, H. P.; Arends, J. Colloids Surf. 1984, 9, 319. (8) Extrand, C. W.; Gent, A. N. J. Colloid Interface Sci. 1990, 138, 431. (9) Miller, J. D.; Veeramasuneni, S.; Drelich, J.; Yalamanchili, M. R.; Yamauchi, G. Polym. Eng. Sci. 1996, 36, 1849. (10) O ¨ ner, D.; McCarthy, T. M. Langmuir 2000, 16, 7777. (11) Tanaguchi, M.; Belfort, G. Langmuir 2001, 18, 6465. (12) MacDougall, G.; Okrent, C. Proc. R. Soc. (London) 1942, 180A, 151. (13) Johnson, R. E., Jr.; Dettre, R. H. Adv. Chem. Ser. 1964, 43, 112. (14) Dettre, R. H.; Johnson, R. E., Jr. Adv. Chem. Ser. 1964, 43, 136. (15) Johnson, R. E., Jr.; Dettre, R. H. In Surface and Colloid Science; Matijevic´, E., Ed.; Wiley: New York, 1969; Vol. 2, p 85. (16) Morra, M.; Occhiello, E.; Garbassi, F. Langmuir 1989, 5, 872. (17) Kunugi, Y.; Nonaka, T.; Chong, Y.-B.; Watanabe, N. J. Electroanal. Chem. 1993, 353, 209. (18) Onda, T.; Shibuichi, S.; Satoh, N.; Tsujii, K. Langmuir 1996, 12, 2125. (19) Shibuichi, S.; Onda, T.; Satoh, N.; Tsujii, K. J. Phys. Chem. 1996, 100, 19512. (20) Tadanaga, K.; Katata, N.; Minami, T. J. Am. Ceram. Soc. 1997, 80, 1040. (21) Tadanaga, K.; Katata, N.; Minami, T. J. Am. Ceram. Soc. 1997, 80, 3213. (22) Hozumi, A.; Takai, O. Thin Solid Films 1997, 303, 222. (23) Chen, W.; Fadeev, A. Y.; Hsieh, M. C.; O ¨ ner, D.; Youngblood, J. P.; McCarthy, T. M. Langmuir 1999, 15, 3395. (24) Youngblood, J. P.; McCarthy, T. M. Macromolecules 1999, 32, 6800. (25) Bico, J.; Marzolin, C.; Que´re´, D. Europhys. Lett. 1999, 47, 220. (26) Yoshimitsu, Z.; Nakajima, A.; Watanabe, T.; Hashimoto, K. Langmuir 2002, 18, 5818.
Figure 1. Small, sessile drops on rough surfaces. (a) A collapsed drop on rough surface; impaled by the asperities, the contact area is composed solely of a liquid/solid interface. (b) A suspended drop on an ultralyophobic surface.
vestigators have focused on the relation between contact angles and surface geometry.2,5,9,13,18-20,23-26 More recent investigations have targeted the conditions that lead to ultralyophobicity.10,27-29 In one of these studies, a contact line density criterion was proposed and used to successfully predict suspension of water drops on model rough surfaces.27 This criterion alone, however, did not adequately describe the recently published findings of Yoshimitsu and colleagues.26 Therefore, in this work, an additional asperity height criterion for ultralyophobicity is proposed. After derivation and explanation of the contact line density and asperity height criteria, their ability to correctly predict experimental observations on model rough surfaces with regular, periodic asperities was investigated. Theoretical Basis In order for surfaces to show ultralyophobic behavior, three conditions must be met. First, interaction of a liquid with asperities must direct surface forces at the contact line upward. Second, the surface forces must be of sufficient magnitude to suspend the liquid against the downward pull of gravity (or other body forces). Third, the asperities must be tall enough that liquid protruding between them does not contact the underlying solid. The first and second conditions can be described quantitatively by a contact line density criterion.27 The third can be addressed by an asperity height criterion. Consider the model rough surface shown schematically in Figure 2. Surface asperities take the form of a regular array of square pillars. The pillars have a top width of x and height of z. The angle subtended by the top edge of (27) Extrand, C. W. Langmuir 2002, 18, 7991. (28) Patankar, N. A. Langmuir 2003, 19, 1249. (29) He, B.; Patankar, N. A.; Lee, J. Langmuir 2003, 19, 4999.
10.1021/la036481s CCC: $27.50 © 2004 American Chemical Society Published on Web 05/13/2004
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Figure 4. A magnified side view of a suspended liquid drop on a model ultralyophobic surface. The liquid exhibits its true advancing value, θa,o, on the sides of the pillars. Figure 2. Schematic depiction of a surface covered by a regular array of square pillars of top width, x, and height, z. (a) Plan view. Each unit cell has a linear dimension of y. The crosshatched areas are the tops of 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; φ ) ω ) 90°.
A magnified side view of the composite interface is shown in Figure 4. Suspended drops protrude between the pillars and exhibit their true advancing value, θa,o, on the sides of the asperities.30 The greatest linear distance, 2b, between adjacent pillars, which factors into the protrusion depth of a suspended liquid, is
2b ) 21/2(y - x)
(6)
Contact Line Density Criterion for Small Drops. A contact line density criterion can be derived for a small, liquid drop on a rough surface by examining the interplay of body and surface forces.27 Body forces, F, associated with gravity are determined from the density, F, and the unsupported volume, Vu, of the liquid drop,
F ) FgVu
Figure 3. A small liquid drop suspended on an ultralyophobic surface consisting of a regular array of square pillars. (a) Side view. (b) Plan view. The apparent contact area is a composite of gas/liquid and liquid/solid interface.
the asperities is φ, the rise angle of the side of the asperities is ω, and
φ ) 180° - ω
(1)
As drawn in Figure 2, φ ) ω ) 90°. The area density, δ, of asperities is
δ ) 1/y2
(2)
Contact line density, Λ, which is the length of asperity perimeter per unit area that could potentially suspend a liquid drop, is defined as the product of δ and the feature perimeter, p,
Λ ) pδ
(3)
For the regular array of square pillars described in Figure 2, Λ is
Λ ) 4x/y2
(4)
Figure 3 shows side and plan views of a liquid drop on a model ultralyophobic surface covered with a regular array of square pillars. The drop is suspended atop the pillars and exhibits a very large apparent advancing contact angle, θa. Its apparent contact area (defined by the contact diameter D) is composite in nature, consisting of both gas-liquid and liquid-solid interface. The fraction of liquid-solid contact area within this composite interface, Rp, can be estimated as
Rp ) (x/y)2
(5)
(7)
where g is the acceleration due to gravity. The unsupported or suspended volume can be approximated from the total volume of the liquid drop, V, and the fractional area of contact between the liquid and the pillar tops, Rp, as
Vu ) V - RphA
(8)
where h is drop height and A is its apparent contact area. On the other hand, the surface forces, f, depend on surface tension of the liquid, γ, the contact angle on the side of the asperities with respect to 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
(9)
The contact angle on the side of the asperities, θs, is related to θa,o by ω,
θs ) θa,o + ω - 90°
(10)
The contact line density has a critical value, Λc,
Λ ) Λc
(11)
where the body and surface forces are equal,31
F)f
(12)
Combining eqs 3 and 7-12 leads to a critical value of Λc for small drops,
Λc ) -FgV(1 - RphA/V)/Aγ cos(θa,o + ω - 90°)
(13)
(30) The true advancing contact angle, θa,o, is the value measured on a chemically equivalent, smooth, horizontal surface. (31) It is assumed that the ultralyophobic surface is sufficiently porous so that gases are not trapped between asperities and in turn do not contribute to the suspension forces.
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the cross section of the liquid protrusion can be described as a segment of a circle, then d can be calculated as
d ) b tan(θd/2)
(19)
θd ) θa,o + ω - 180°
(20)
where
If the protrusion depth equals the critical value of asperity height, zc,
zc ) d Figure 5. A magnified side view of a suspended liquid protruding to a depth, d, between two model asperties. The asperities have a height of z and a maximum distance between them of 2b. The liquid exhibits its true advancing value, θa,o, on the sides of the asperities. θd is the angle between the horizontal plane and the protruding liquid.
If sufficiently small, drops retain spherical proportions, which allow both drop height and apparent contact area, A, to be written in terms of the apparent advancing contact angle, θa, and the drop volume, V,27
h ) [(3V/π)(1 - cos θa)/(2 + cos θa)]1/3
(14)
A ) π1/3(6V)2/3{tan(θa/2)[3 + tan2(θa/2)]}-2/3
(15)
and
Substituting eqs 14 and 15 into eq 13 allows estimation of Λc values without knowledge of drop dimensions,
Λc ) -FgV1/3(1 - k){tan(θa/2)[3 + tan2(θa/2)]}2/3/ [(36π)1/3γ cos(θa,o + ω - 90°)] (16) where k is a correction factor that accounts for liquid supported by pillars or asperities,
k ) Rp[96(1 - cos θa)/(2 + cos θa)]1/3{tan(θa/2)[3 + tan2(θa/2)]}-2/3 (17) If Rp is small and θa is large, then k is effectively zero and eq 16 reduces to
Λc ) -FgV1/3{tan(θa/2)[3 + tan2(θa/2)]}2/3/ [(36π)1/3γ cos(θa,o + ω - 90°)] (18) Error in Λc values from neglecting k is discussed in the Appendix.32 Critical Asperity Height. Because suspended drops establish true advancing contact angles on the sides of asperities, surface curvature will cause liquid to protrude downward between asperities as depicted in Figures 4 and 5. Figure 5 shows a detailed view of a liquid protrusion. The protrusion depth, d, depends on the distance between the asperities, 2b, and the depth angle, θd, between the horizontal plane and the liquid protrusion. Assuming that (32) Equation 18 is a simplified form of the Λc expression derived in ref 27.
(21)
then liquid will just touch the floor between asperities, causing collapse. Merging eqs 19-21 gives an expression for estimating the critical asperity height,
zc ) b tan[(θa,o + ω - 180°)/2]
(22)
Results and Discussion Attributes of the Model. The reported size of asperities required to create ultralyophobic surfaces has been controversial.23 The problem lies in defining surfaces solely in terms of asperity height or spacing. While these are important considerations, other parameters such as liquid density, surface tension, contact angles, and asperity shape, also play a role. From eq 18, if F, V, or θa increase, higher Λc will be required to suspend drops. On the other hand, if γ, ω, or θa,o increase, a lower Λc will suffice. The edge angle is extremely important in determining ultralyophobicity. In order for a surface to support a liquid drop, ω (and θa,o) must be sufficiently large to direct the surface forces upward against the body force (a positive Λc value). Marshalling the surface forces against gravity requires that θa,o + ω - 90° > 90°. In terms of the contact line density criterion, Λ must be > Λc. If the surface and body forces are both directed downward, Λc will be negative and drops will collapse regardless of the magnitude of Λ. High contact line density alone is not a sufficient condition for ultralyophobicity. Not only must Λ > Λc, but also asperity height must be greater than the protrusion, z > d (or z > zc). Otherwise, the protruding liquid will contact the base of the solid surface, instigating collapse. Consider a designed ultrahydrophobic surface covered with a regular array of square pillars where x ) 8 µm, y ) 16 µm, z ) 40 µm, ω ) 90°, θa,o ) 120°, and θa ) 170°. The contact line density of this model surface is Λ ) 1.3 × 105 m-1 (eq 4). On the other hand, eq 18, which captures the interplay of surface and body forces, suggests that the minimum Λ value for suspension of small water drops is Λc ≈ 1 × 104 m-1. From eqs 19 and 20 water suspended atop these square pillars would protrude downward 1.5 µm. Because Λ > Λc and z > zc, this surface would be expected to suspend small, gently deposited water drops. Comparison with Experimental Data. In this section, results from experimental studies on model ultralyophobic surfaces10,25,26,29 are used to test the proposed criteria. Patankar and co-workers29 created model rough surfaces consisting of regular arrays of square pillars, depicted in Figure 2, by casting polydimethyl siloxane (PDMS) in photoresist molds. The square pillars all had the same width, height, and shape (x ) 23 µm, z ) 28 µm, and ω ) 105°). Spacing between them was varied from y ) 32 µm to y ) 104 µm. Table 1 shows various geometric parameters and apparent advancing contact angles for water drops on their PDMS surfaces. The true advancing contact angle, θa,o, on the smooth PDMS surface was 118°. If gently deposited, water drops (5-10 µL) were suspended
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Table 1. Geometric Parameters and Apparent Advancing Contact Angles for Suspended Water Drops on Ultrahydrophobic Polydimethyl Siloxane (PDMS) Surfaces Covered with Regular Arrays of Square Pillars, x ) 23 µm, z ) 28 µm, ω ) 105°, and θa,o ) 118° a δ (µm-2)
y (µm)
10-4
9.8 × 6.2 × 10-4 3.8 × 10-4 2.2 × 10-4 9.3 × 10-5
32 40 51 67 104
Λ (m-1)
Rp
9.0 × 5.8 × 104 3.4 × 104 2.0 × 104 8.5 × 103
0.53 0.34 0.20 0.12 0.052
104
2b (µm) d (µm) θa (deg) 13 24 41 63 113
1.3 2.5 4.3 6.6 12
147 147 148 150 153
a (Bo, Patankar, and Lee, ref 29.) y is the width of each unit cell; δ is the area density of asperities, eq 2; Λ is contact line density, eq 4; and Rp is the fractional contact area between the liquid and pillar tops, eq 6. 2b is the greatest distance between asperities and d is the depth a suspended liquid protrudes between asperities, eq 19. θa is the apparent advancing contact angle.
Table 2. Geometric Parameters and Apparent Advancing Contact Angles for Suspended or Collapsed Water Drops on Silicon Surfaces Covered with Regular Arrays of Square Pillars, x ) 50 µm, y ) 150 µm, and ω ) 90° a z (µm)
δ (µm-2)
Λ (m-1)
Rp
10 36 148 282
4.4 × 10-5 4.4 × 10-5 4.4 × 10-5 4.4 × 10-5
8.9 × 103 8.9 × 103 8.9 × 103 8.9 × 103
0.11 0.11 0.11 0.11
2b d θa suspended (µm) (µm) (deg) or collapsed? 141 141 141 141
15 15 15 15
138 155 151 153
collapsed suspended suspended suspended
a Surfaces were rendered hydrophobic with a fluorosilane, θ a,o ) 114°. (Yoshimitsu et al., ref 26). z is asperity height; δ is the area density of asperities, eq 2; Λ is contact line density, eq 4; and Rp is the fractional contact area between the liquid and pillar tops, eq 6. 2b is the greatest distance between asperities and d is the depth a suspended liquid protrudes between asperities, eq 19. θa is the apparent advancing contact angle.
on these pillared surfaces and showed large θa values. From eqs 18 and 22 the critical contact line density and asperity height criteria were Λc ) 1.0 × 103 to 1.7 × 103 m-1 and zc ) 1.3 to 12 µm. Since Λ and z values for these surfaces (Table 1) were greater than their respective critical values, these surfaces would have been expected to show ultrahydrophobicity, as indeed they did. Yoshimitsu and colleagues26 also examined the wettability of regular arrays of square pillars. They used a diamond saw to sculpt silicon wafers and then treated them with a fluorosilane to produce super repellency. Table 2 lists data for their surfaces. Pillar width and spacing were held constant at x ) 50 µm and y ) 150 µm, corresponding to Λ ) 8.9 × 103 m-1. Pillar height was varied between z ) 0 µm and z ) 282 µm. The critical Λ value from eq 18 for 1-µL water drops was Λc ) 1.6 × 103 m-1. Even though Λ > Λc, not all of these surfaces showed ultrahydrophobic behavior. Water was suspended atop
Figure 6. Schematic depiction of a surface covered by a hexagonal array of square pillars of top 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 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°.
the tallest pillars and exhibited very large contact angles, θa ) 155°. However, drops collapsed over the shortest pillars, z ) 10 µm producing a lower apparent value, θa ) 138°. Water would have been expected to protrude as deep as 15 µm between the pillars. For the shortest pillars (z ) 10 µm), this depth likely allowed suspended drops to touch the floor between the pillars and thereby triggered collapse. O ¨ ner and McCarthy10 prepared hexagonal arrays of square pillars on silicon wafers with ω ) 90° (Figure 6) using photolithography and then treated them with hydrophobic silane agents. Table 3 lists dimensional parameters and apparent contact angles of dimethyldichlorosilane (DS) treated surfaces. (The expressions used to determine δ, Λ, Rp, and 2b values of the various surfaces are summarized in Table 4.) The hydrophobic surfaces created by O ¨ ner and McCarthy had pillar heights that were greater than the penetration depth of the liquid, that is, z > zc; therefore, suspension likely was determined by contact line density. Based upon examination of the experimental data, the critical value of the Λ for these water/surface combinations resided between 1 × 104 and 3 × 104 m-1. This observation is in general agreement with Λc values calculated from eq 18, Λc ) 3 × 104 m-1. The criteria were further tested using data from Bico and co-workers.25 Figure 7 depicts their model surfaces with stripes, pillars, or cavities that were molded on silicon wafers using a tetramethylothrosilicate sol-gel and then treated with a fluorosilane (θa,o ) 118°). The geometric parameters and apparent contact angles for these surfaces
Table 3. Geometric Parameters and Apparent Advancing Contact Angles for Suspended or Collapsed Water Drops on Hydrophobic Silicon Surfaces Covered with Hexagonal Arrays of Square Asperities, z ) 40 µm and ω ) 90° a x (µm)
y (µm)
x/y (µm)
δ (µm-2)
Λ (m-1)
Rp
2b (µm)
d (µm)
θa (deg)
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
0.25 0.25 0.25 0.25 0.25 0.25 0.13 0.069 0.023
3.6 14.4 28.8 57.7 115 231 14.4 24.2 36.9
0.3 1.1 2.2 4.3 8.6 17.2 1.1 1.8 2.8
176 173 171 168 139 116 175 173 121
suspended suspended suspended suspended collapsed collapsed suspended suspended collapsed
a The surfaces were treated with a hydrocarbon silane (DS), θ ¨ ner and McCarthy, ref 10). x is pillar top width and y is the a,o ) 107°. (O width of each unit cell. δ is the area density of asperities, eq 2. Λ is contact line density, eq 4. Rp is the fractional contact area between the liquid and pillar tops. 2b is the greatest distance between asperities and d is the depth a suspended liquid protrudes between asperities, eq 19. θa is the apparent advancing contact angle.
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Table 4. Equations for Determining Parameters of Surfaces with Various Asperity Geometriesa feature
δ
Λ
regular array of square pillars hexagonal array of square pillars stripes circular pillars circular cavities
1/y2
4x/y2
1/y2 1/y2 1/y2 1/y2
4x/y2 2/y πx/y2 πx/y2
Rp
2b
2(x/y)2/[7/4 + (x/y)2] x/y (π/4)(x/y)2 1 - (π/4)(x/y)2
- x) [(5/4)x2 - 3xy + 2x2]1/2 y-x 21/2y - x x
(x/y)2
21/2(y
a δ is the area density of features. Λ is contact line density. R is the fractional contact area between the liquid and pillar tops. 2b is p the greatest depth a liquid protrudes between the asperities.
Table 5. Geometric Parameters for Suspended or Collapsed Water Drops on Hydrophobic Surfaces with Microscopic Stripes, Circular Pillars, or Shallow Circular Cavitiesa feature
x (µm)
y (µm)
z (µm)
ω (deg)
δ (µm-2)
Λ (m-1)
Rp
2b (µm)
d (µm)
θa (deg)
suspended or collapsed?
stripes circular pillars 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 0.05 0.65
3 4.7 2
0.4 0.6 -0.02
165 170 138
suspended suspended collapsed
a (Bico, Marzolin, and Que ´ re´, ref 25). 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 contact line density. Rp is the fractional contact area between the liquid and pillar tops. 2b is the greatest distance between asperities. d is the distance a suspended liquid protrudes between asperities. θa is the apparent advancing contact angle. Equations for calculating these surface parameters can be found in Table 4.
Figure 8. The difference between eqs 16 and 18 for various values of the fractional contact area, Rp, and apparent advancing contact angle, θa, according to eqs 17, 23, and 24.
to compute Λc values from eq 13. Values of Λc from eq 13 generally agreed with those from eq 18. For example, eq 13 predicts Λc ) 1.0 × 103 to 1.7 × 103 m-1 for 1-µL drops on Yoshimitsu’s ultrahydrophobic surfaces.26 The value calculated with eq 18 was Λc ) 1.6 × 103 m-1. This work has focused on the interaction of surface forces and gravity. Other body forces, such as the inertia, also could be important.27 It has been reported that allowing drops to fall from small heights29 or pressing down on suspended drops25 can cause collapse. Conclusions
Figure 7. Schematic depictions of surfaces with microscopic stripes, circular pillars, or circular cavities. (a) Plan view and (b) side view of a striped surface. (c) Plan view and (d) side view of a surface with circular pillars. (e) Plan view and (f) side view of a surface with circular cavities. Cross-hatched areas in plan view have greater elevation.
are listed in Table 5. Critical values of contact line density, Λc, and pillar height, zc, were calculated with θa,o ) 118° and θa ) 170°. Predictions of suspension or collapse were in agreement with experimental observations. For stripes and circular pillars, Λ > Λc, and as would be expected, water drops were suspended. On the other hand, the more acute rise angle of the cavities led to a negative Λc value and accordingly drops collapsed. (z > zc for all three surfaces.) Where available, drop dimensions (D and h) were measured from images of suspended water drops and used
Criteria developed and explored in this paper constitute a set of design rules for creating ultralyophobic surfaces. In order for a liquid-solid combination to show super repellency, a contact line density and asperity height criterion must be met. Surface forces acting around the perimeter of asperities must be greater than body forces and directed upward. Also, asperities must be tall enough that liquid protruding between them does not contact the base of the solid, causing the liquid to be drawn downward, leading to collapse. Quantitatively, this requires that values of contact line density and asperity height determined from surface geometry must exceed critical values calculated from properties of the liquid, solid, and their interfacial interactions. If Λ > Λc and z > zc, then the liquid will be suspended by the asperities, producing an ultralyophobic surface. If Λ < Λc or z < zc, then the liquid will collapse and the contact interface will be solely liquid/ solid. These criteria were tested using data from several experimental studies of model rough surfaces. In all cases, they correctly predicted, either suspension or collapse.
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Acknowledgment. I thank Entegris management for supporting this work and allowing publication. Also, thanks to T. M. McCarthy for providing details of their experimental work on ultralyophobic surfaces and to J. M. Miller for insightful discussions and helpful comments. Appendix The relative error in not accounting for the liquid volume that is supported above the pillars can be estimated as
∆Λ/Λc ) (eq 18 - eq 16)/(eq 16)
(23)
If eqs 16 and 18 are substituted into eq 23 and then
simplified, one arrives at
∆Λ/Λc ) 1/(1/k - 1)
(24)
where k is given by eq 17. Values of ∆Λ/Λc calculated for various values of Rp and θa using eqs 17 and 24 are plotted in Figure 8. If Rp is small and θa is large, then the difference between Λc values from eqs 16 and 18 is small. Therefore, in most cases, ignoring the supported liquid volume is inconsequentialsless than 5% for most of the experimental data examined in this study. LA036481S
