Langmuir 2007, 23, 1867-1871
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Retention Forces of a Liquid Slug in a Rough Capillary Tube with Symmetric or Asymmetric Features C.W. Extrand* Entegris Incorporated, 3500 Lyman BouleVard, Chaska, Minnesota 55318 ReceiVed August 28, 2006. In Final Form: NoVember 9, 2006 On surfaces with asymmetric “sawtooth” features, liquid slugs or drops tend to move preferentially in one direction. In this theoretical study, the imbalance of capillary forces that leads to directionally biased wetting is examined. Capillary tubes with symmetric and asymmetric sawtooth features were used to estimate the ratios of retention forces in opposing directions. Our analysis suggests that the difference between the retention force in one direction versus the other can be maximized by increasing feature asymmetry and minimizing inherent hysteresis of the materials of construction. This work has implications for small channels or surfaces of fluid-handling components found in microfluidic devices and fuel cells.
Introduction In recent years, a number of investigators have published work on surfaces with ratchetlike features or asymmetric surface roughness that cause liquid drops or slugs to move preferentially in one direction.1-6 This type of surface could potentially enhance the performance of a broad array of fluid-handling components or systems, such as pipettes, nozzles, microfluidic devices, and fuel cells. While methods for producing directionally biased wetting have been explored, the capillary forces underlying this phenomenon have received less attention. Therefore, in this theoretical study, capillary tubes were used to examine the forces that lead to the anisotropic wetting behavior associated with ratchetlike surfaces. This wetting behavior is described qualitatively in Figure 1 for a liquid slug (or cylinder) in a capillary tube. Figure 1a shows a stationary slug of liquid in a smooth capillary tube. Even though the surface is featureless, heterogeneities, contaminants, or simply molecular interactions at the contact lines may lead to liquidsolid adhesion7-11 that pin the slug. Therefore, an external force, Fi, must be applied to dislodge it. The retention force that impedes movement, f0, acts equally in both directions. Figure 1b shows * E-mail:
[email protected]. (1) Ajdari, A.; Mukamel, D.; Peliti, L.; Prost, J. Rectified Motion Induced by AC Forces in Periodic Structures. J. Phys. I France 1994, 4 (10), 1551-1561. (2) Gorre, L.; Ioannidis, E.; Silberzan, P. Rectified Motion of a Mercury Drop in an Asymmetric Structure. Europhys. Lett. 1996, 33 (4), 267-272. (3) Marquet, C.; Buguin, A.; Talini, L.; Silberzan, P. Rectified Motion of Colloids in Asymmetrically Structured Channels. Phys. ReV. Lett. 2002, 88 (16), 168301. (4) Buguin, A.; Talini, L.; Silberzan, P. Ratchet-like Topological Structures for the Control of Microdrops. Appl. Phys. A 2002, 75 (2), 207-212. (5) Linke, H.; Alema´n, B. J.; Melling, L. D.; Taormina, M. J.; Francis, M. J.; Dow-Hygelund, C. C.; Narayanan, V.; Taylor, R. P.; Stout, A. Self-Propelled Leidenfrost Droplets. Phys. ReV. Lett. 2006, 96 (15), 154502. (6) Smith, K. A.; Alexeev, A.; Verberg, R.; Balazs, A. C. Designing a Simple Ratcheting System to Sort Microcapsules by Mechanical Properties. Langmuir 2006, 22 (16), 6739-6742. (7) MacDougall, G.; Ockrent, C. Surface energy relatios in liquid/solid systems. I. The adhesion of liquids to solids and a new method of determining the surface tension of liquids. Proc. R. Soc. (London) 1942, 180A (1), 151-173. (8) Bartell, F. E.; Ray, B. R. Wetting Characteristics of Cellulose Derivatives. I. Contact Angles Formed by Water and by Organic Liquids. J. Am. Chem. Soc. 1952, 74 (3), 778-783. (9) Schwartz, A. M. Contact Angle Hysteresis: A Molecular interpretation. J. Colloid Interface Sci. 1980, 75 (2), 404-408. (10) Chen, Y. L.; Helm, C. A.; Israelachvili, J. N. Molecular Mechanisms Associated with Adhesion and Contact Angle Hysteresis of Monolayer Surfaces. J. Phys. Chem. 1991, 95 (26), 10736-10747. (11) Extrand, C. W. Contact Angles and Their Hysteresis as a Measure of Liquid-Solid Adhesion. Langmuir 2004, 20 (10), 4017-4021.
Figure 1. Cylindrical capillaries with various surface topographies. (a) A smooth inner surface. (b) An inner surface with symmetric sawtooth features. (c) An inner surface with asymmetric features.
a capillary tube with symmetrically shaped sawtooth features. The retention force of this roughened tube is greater than the otherwise equivalent smooth tube.12 Because the features are symmetrically shaped the retention force, f1, again acts equally in both directions. Figure 1c shows a capillary tube with asymmetric sawtooth features. As with the previous case, the retention forces will be greater than those associated with the equivalent smooth tube. However, the retention force is not equal in both directions, f2 * f3. In the case depicted in Figure 1c, the retention force acting to the right, f2, is less than the one acting to the left, f3. Therefore, the external force that must be applied to initiate slug movement to the left is smaller than the force required to move it to the right.
Analysis The liquid slugs depicted in Figure 1 are free from the influence of external forces. If an external force is applied, the fluidliquid interfaces distort with both interfaces deflecting in the direction of the applied force. The leading edge of the slug may advance slightly, but the slug, as a whole, does not move. This external force could arise from inclination of the tube or fluid pressure applied to one end of the tube, for example. Capillary forces present at the contact line of the leading and trailing edges will anchor the slug, preventing movement, until the applied external force exceeds a critical value, Fi. Figure 2 shows a detailed side view of a stationary liquid slug in a capillary tube (12) Extrand, C. W.; Gent, A. N. Retention of Liquid Drops by Solid Surfaces. J. Colloid Interface Sci. 1990, 138 (2), 431-442.
10.1021/la0625289 CCC: $37.00 © 2007 American Chemical Society Published on Web 01/05/2007
1868 Langmuir, Vol. 23, No. 4, 2007
Extrand
Figure 2. Side view a liquid slug in a capillary tube of radius R. The slug is exposed to an external force, Fi, that is equal to the retention force, fi, that prevents movement. At this critical condition where Fi ) fi, the drop exhibits an advancing contact angle, θa, at the leading contact line and a receding value, θr, at the trailing contact line.
of radius R that has reached its critical configuration where the critical external force equals that of the retention force, fi. The distorted interfaces of the liquid slug exhibit an advancing contact angle, θa, at the leading contact line and a receding value, θr, at the trailing contact line. If the external force is further increased such that Fi > fi, the slug will begin to move. The magnitude of the retention force, fi, that resists incipient motion within a cylindrical capillary tube of radius R is determined by the fluid-liquid interfacial tension, γ, and the advancing and receding contact angles,13,14
fi ) kγR(cos θr - cos θa)
(1)
k ) 2π
(2)
Figure 3. Enlarged side view of the contact line of a slug interacting with a surface asperity or feature where the external force is directed to the right. The horizontal dashed lines lie in the same plane as the surface. The rise angle of the feature is ωi. (a) An advancing contact line with an apparent contact angle of θa. (b) A receding contact line with an apparent contact angle of θr.
where
If the surface is smooth, then the expression for retention force can be written in terms of inherent contact angles, θa,0 and θr,0,15
f0 ) kγR(cos θr,0 - cos θa,0)
(3)
Roughness generally increases retention of liquid drops and slugs. This increase arises from contact angle changes due to their geometric interaction with surface asperities or features. Consider the contact line of a slug interacting with a surface asperity or feature as depicted in Figure 3. The surface feature is two-dimensional and thus extends indefinitely in the normal plane. The horizontal dashed lines define the plane of the surface. The rise angle of the feature from the surface plane is ωi. Figure 3a shows a contact line advancing across the feature with an apparent advancing contact angle of θa. The liquid exhibits its true advancing value, θa,0, on the face of the feature. The difference between the apparent advancing contact angle, θa, relative to the horizontal plane of the surface and true advancing angle, θa,0, depends on the rise angle subtended by the feature, ωi. Interaction with the feature causes θa to increase such that
θa ) θa,0 + ωi
(4)
as first described by Gibbs16 and experimentally verified by Mason and co-workers.17 Figure 3b shows a contact line retreating across the same feature with an apparent receding contact angle of θr. The liquid exhibits its true receding value, θr,0, on the face of (13) West, G. D. On the Resistance of the Motion of a Thread of Mercury in a Glass Tube. Proc. R. Soc. (London) 1912, 86A (7), 20-25. (14) Yarnold, G. D. The Motion of a Mercury Index in a Capillary Tube. Proc. Phys. Soc. 1938, 50 (4), 540-552. (15) Most real surfaces show contact angle hysteresis. In many cases, hysteresis is due to some sort of imperfection. On a smooth surface, this imperfection may come from heterogeneities or contaminants. The analysis presented here is still valid if inherent hysteresis is zero. (16) Gibbs, J. W. On the Equilibrium of Heterogeneous Substances. In The Collected Works of J. Willard Gibbs; Yale University Press: New Haven, CT, 1961; Vol. 1, pp 326-327. (17) Oliver, J. F.; Huh, C.; Mason, S. G. Resistance to Spreading of Liquids by Sharp Edges. J. Colloid Interface Sci. 1977, 59 (3), 568-581.
Figure 4. Enlarged side view of sawtoothed surface features. (a) A symmetric feature with a rise angle of ω1. (b) An asymmetric feature with rise angles of ω2 and ω3.
the feature. In contrast to the advancing case, interaction with the feature causes θr to decrease
θr ) θr,0 - ωi
(5)
(The form of the equations given here was first proposed by Shuttleworth and Bailey.18) In Figure 3, the contact lines are shown arbitrarily positioned near the midpoint of the feature faces. The diagrams were drawn in this manner to facilitate the line and symbol placement. However, on actual textured surfaces, the contact lines are likely to be pinned on the feature edges, especially at the receding interface. To keep the analysis relatively simple, assume that the roughness on the inner surface of the capillary tubes takes the form of a sawtooth or ratchet pattern depicted in Figure 4. Also assume that the roughness in these tubes is radially symmetric. Or in other words, the teeth extend around the entire perimeter without variation in their cross-sectional shape.19 If the sawtooth features are axially symmetric with a rise angle of ω1 as portrayed in Figure 4a, then the retention force, f1, acts equally in both directions. An expression for estimating f1 comes from the combination of eqs 1, 4, and 5, (18) Shuttleworth, R.; Bailey, G. L. J. Spreading of a Liquid over a Rough Surface. Discuss. Faraday Soc. 1948, 3 (1), 16-22. (19) Equations 4 and 5 are true for surface features that have a constant profile around the entire diameter of the capillary tube. If the shape of the sawtooth features shows radial variation or periodicity, then a different form of eqs 4 and 5 must be used in the analysis.
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f1 ) kγR[cos(θr,0 - ω1) - cos(θa,0 + ω1)]
(6)
Similarly, if the inner surface of the capillary has the asymmetric roughness represented in Figure 4b, then an expression can be written in terms of ω2 and ω3 that describes the retention force differential, ∆f, or the difference between f3 and f2,
∆f ) f3 - f2 ) kγR[cos(θr,0 - ω3) - cos(θa,0 + ω3) cos(θr,0 - ω2) + cos(θa,0 + ω2)] (7) Due to geometric limitations, θr,0 - ωi must be g0° and θa,0 + ωi must be e180°. A number of trigonometric functions can be applied to separate terms and simplify these expressions. As described in the Appendix, a general form of the retention force, fi,
fi/kγR ) 2 sin
[21 (θ
r,0
1 + θa,0) sin ωi + ∆θ0 2
] (
)
(8)
can be framed in terms of inherent hysteresis, ∆θ0,
∆θ0 ) θa,0 - θr,0
(9)
and rise angles, ωi, where i ) 0 for a smooth surface, with ω0 ) 0; i ) 1 for a surface with symmetric sawtooth features where ω1 > 0; and i ) 2 or 3 for a surface with asymmetric features, ω2 * ω3. The retention force differential from eq 7 that describes a tube with asymmetric roughness can be rewritten as
∆f ) f3 - f2 ) 2kγR sin
[21 (θ
r,0
]
+ θa,0) ×
1 1 sin ω3 + ∆θ0 - sin ω2 + ∆θ0 2 2
[ (
)
)] (10)
(
Equation 8 also can be used to configure the retention forces as ratios. Two are given here. The first ratio contrasts retention forces of a capillary tube with symmetric sawtooth features to the corresponding tube with a smooth surface,
1 1 f1/f0 ) sin ω1 + ∆θ0 /sin ∆θ0 2 2
(
) (
)
(11)
The second describes the retention force ratios of a tube with asymmetric sawtooth features where the retention forces are diametrically opposed,
1 1 f3/f2 ) sin ω3 + ∆θ0 /sin ω2 + ∆θ0 2 2
(
) (
)
(12)
Results and Discussion Surfaces with anisotropic wetting behavior have been called force-free2 or self-propelled.5 However, these terms can be a bit misleading because in most cases electrical,2-4 thermal,5 or mechanical energy20 must be introduced to initiate movement of a liquid drop or slug. This point can be demonstrated with a cup of water and a plastic drinking straw with a flexible elbow. A slug can be introduced into the straw by dipping one end into the water. If the end of the straw is sealed, when the straw is withdrawn from the water and then oriented horizontally, a slug will sit motionless in the straw, even after the seal is removed. Now with both ends open to the atmosphere, an external force can be applied to initiate movement, for example, by inclining the straw. The slug will remain stationary until the body force (20) Daniel, S.; Chaudhury, M. K.; de Gennes, P.-G. Vibration-Actuated Drop Motion on Surfaces for Batch Microfluidic Processes. Langmuir 2005, 21 (9), 4240-4248.
Figure 5. Ratio of the retention forces of a surface with symmetric sawtooth features relative to the corresponding smooth surface, f1/f0, versus rise angle, ω1, for different levels of inherent hysteresis, ∆θ0.
associated with the slug exceeds its retention forces, then the slug will begin to move. The impeding force that must be overcome to move the slug along the smooth portions of the straw is present in both directions. That is to say, either the right end or left end of the straw must be tipped to trigger motion. When the liquid slug encounters the ridges of the articulated elbow, resistance to movement increases. Because the ridges were asymmetric, the flow resistance is no longer the same in both flow directions. If tipped vertically in one direction, a slug of a given volume will remain anchored in the articulated portion of the straw, but if tipped vertically in the other direction, the slug will move. The analysis presented earlier allows us to quantitatively compare these retention forces. Figure 5 shows the ratio of the retention forces of a capillary tube with symmetric sawtooth features relative to the corresponding smooth tube for liquid/ solid combinations with different levels of inherent hysteresis, ∆θ0. Retention forces usually are present, even if the surface of the tube is smooth, i.e., |f0| is almost always >0. As ω1 increases and the surface of the tube becomes rougher, the relative retention force increases. For smaller ω1 values, the increase in f1/f0 is linear. Or, in more mathematical terms, if ω1 + (1/2)∆θ0 , 1, then eq 11 reduces to
f1/f0 ≈ 2ω1/∆θ0 + 1
(13)
Both Figure 5 and eq 13 suggest that the larger the inherent hystersis in the system, the less influence the features will exert on the retention force. For example, we compare two capillary tubes of equal diameter, one with a smooth inner surface and the other with sawtooth features where ω1 ) 10°. If the liquid and material of construction show an inherent hysteresis of ∆θ0 ) 10°, then the retention force ratio is much larger (f1/f0 ) 2.97) than if ∆θ0 ) 30° (f1/f0 ) 1.63). Figure 6 shows retention force ratios of a tube with asymmetric sawtooth features, f3/f2, plotted against ω2 for various ω3 values. Inherent hysteresis of the system is assumed to be ∆θ0 ) 10°. For a tube with ratchetlike structure where ω2 ) 45° and ω3 ) 90°, f3/f2 ) 1.30. The greater the difference between ω2 and ω3, the greater the disparity of the retention force in the two opposing directions; if ω2 ) 10° and ω3 ) 90°, f3/f2 ) 3.85. In principle, a strong directional bias could also be created using shallow rise angles. If ω2 ) 1° and ω3 ) 10°, then f3/f2 ) 2.48. Alternatively, if ω2 ) 1° and ω3 ) 90°, then f3/f2 > 9. However, creating practical surfaces with very shallow rise angles may prove difficult. Figure 7 demonstrates the variation in retention force ratios of a surface with asymmetric sawtooth features, f3/f2, versus rise
1870 Langmuir, Vol. 23, No. 4, 2007
Figure 6. Ratio of the retention forces of a surface with asymmetric sawtooth features, f3/f2, versus rise angle, ω2, for ω3 different values. Inherent hysteresis is ∆θ0 ) 10°.
Extrand
PTFE described above (θa,0 ) 108°, ∆θ0 ) 10°, ω2 ) 10°, and ω3 ) 90°), if the tube diameter were shrunk to 20 µm, then the pressure differential to move a water slug in one direction versus the other would be ∼20 kPa. Thus, this type of construction could be useful in valves or gates of microfluidic devices. The size of the features should be relatively unimportant for static slugs.22 Small features should produce the same directionally biased wetting behavior as large ones. Once a slug begins to move, however, smaller features may have an advantage over larger ones. Smaller features would be less likely to disrupt flow. If too large, surface features could increase turbulence, thereby increasing flow resistance.23 While the analysis here is aimed at capillary tubes with circular cross sections, it also applies to noncircular tubes and sessile drops. For these alternative geometries, eq 1 would differ only in the prefactor k. Sessile drops are discussed further in the Appendix. The analysis given here has been limited to static phenomena. With the onset of flow, additional forces must be considered, such as those that arise from liquid viscosity,13,14 inertia, or vertical displacement.
Concluding Remarks
Figure 7. Ratio of the retention forces of a surface with asymmetric sawtooth features, f3/f2, versus rise angle, ω2, for different levels of inherent hysteresis, ∆θ0. In this example, ω3 ) 90°.
angle, ω2, for different levels of inherent hysteresis, ∆θ0. In this example, ω3 ) 90°. As with the symmetric features, the retention force ratios are diminished by increases in ∆θ0. For instance, let us assess a capillary tube with asymmetric sawtooth features where ω2 ) 10° and ω3 ) 90°. If the inherent hysteresis is ∆θ0 ) 10°, then retention force ratio of f3/f2 ) 3.85. If ∆θ0 increases to 30°, then f3/f2 falls to 2.29. To this point in the discussion, we have focused on the relative magnitude of the retention force ratios. Let us turn to a practical example to gauge the absolute magnitude of the retention forces. Consider a water slug in a horizontal PTFE tube with a radius of R ) 1 mm. The remainder of the tube volume is occupied by air. Assume the smooth PTFE surface exhibits an advancing contact angle of θa,0 ) 108° 21 and inherent hysteresis of ∆θ0 ) 10°. Without features, the retention force that impedes slug displacement would be f0 ) 77 µN. Therefore, incipient motion of the water slug in either direction would require an external force >77 µN. If a fluid-liquid combination with a lower interfacial tension were employed, the retention forces would be less. By introducing asymmetric sawtooth features with ω2 ) 10 and ω3 ) 90° on the inner surface of the PTFE tube, ∆f ) 672 µN. However, there is a penalty to be paid for creating a directional bias; the external force needed to move the slug in either direction will be greater than the value for the smooth tube, f2 ) 228 µN and f3 ) 900 µN. Since the smaller retention force, f2, is directed to the right, the slug will move more easily to the left, Figure 4b. If the tube were made sufficiently small, the pressure differential to initiate movement could be quite large. For the (21) Fox, H. W.; Zisman, W. A. The Spreading of Liquids on Low Energy Surfaces. I. Polytetrafluoroethylene. J. Colloid Sci. 1950, 5 (6), 514-531.
To initiate movement of a liquid slug in a small diameter capillary tube, a minimum external force must be applied to overcome the retention force associated with interfacial tension acting at the contact lines. The magnitude of the retention force increases with the fluid-liquid interfacial tension and surface roughness. If the surface roughness consists of symmetric features, then the increased resistance to incipient motion acts equally in both directions along the axis of the capillary. However, if the features are asymmetric, then the retention force is less in one direction. The greater the asymmetry of the surface features, the greater the disparity of the retention force in the two opposing directions. With the appropriate design of surface features, the retention force ratio could exceed 5×. Liquid-solid combinations that show minimal inherent contact angle hysteresis would be expected to show greater rectification.
Appendix Derivation of Alternative Retention Force Expressions and Retention Force Ratios. A number of trigonometric functions can be implemented to frame the forces associated with asymmetric features solely in terms of rise angles, ωi, and inherent hysteresis, ∆θ0. Let us begin with a general form of eqs 3 and 6
fi/kγR ) cos(θr,0 + ωi) - cos(θa,0 - ωi)
(14)
where i ) 0 for a smooth surface, with ω0 ) 0; i ) 1 for a surface with symmetric sawtooth features where ω1 > 0; and i ) 2 or 3 for a surface with asymmetric features, ω2 * ω3. The following angle-difference and angle-sum relations,24
cos(θr,0 - ωi) ) cos θr,0 cos ωi + sin θr,0 sin ωi (15) and
cos(θa,0 + ωi) ) cos θa,0 cos ωi - sin θa,0 sin ωi (16) (22) If the liquid slug is shorter than the length of a single feature, then the analysis presented here probably is not valid. Otherwise, it should be fine. (23) Bennett, C. O.; Myers, J. E. Momentum, Heat, and Mass Transfer, 3rd ed.; McGraw-Hill: New York, 1982; pp 169-171. (24) Beyer, W. H. Standard Mathematical Tables, 27th ed.; CRC Press: Boca Raton, FL, 1984.
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1 1 f1/f0 ) sin ω1 + ∆θ0 /sin ∆θ0 2 2
(
Figure 8. Side view of a sessile liquid drop at the critical condition where Fi ) fi. The drop exhibits advancing and receding contact angles, θa and θr.
can be used to separate the rise angle terms from the contact angle terms. Substituting eqs 15 and 16 into eq 14 and rearranging gives
fi/kγR ) cos ωi(cos θr,0 - cos θa,0) + sin ωi(sin θr,0 + sin θa,0) (17) Next, by employing function-sum and function-difference relations,24
cos θr,0 - cos θa,0 ) -2 sin
[21 (θ
r,0
] [21 (θ
+ θa,0) sin
r,0
]
- θa,0) (18)
) (
1 1 f3/f2 ) sin ω3 + ∆θ0 /sin ω2 + ∆θ0 2 2
(
) (
r,0
] [21 (θ
+ θa,0) cos
r,0
- θa,0)
]
(19)
k ≈ 1.5
][
(
)
(
)] (20)
Once again, an angle-difference relation,24
1 1 sin ωi - - ∆θ0 ) sin ωi cos - ∆θ0 2 2
[ (
)]
(
(
)
can be applied to further simplify eq 20,
[21 (θ
r,0
1 + θa,0) sin ωi + ∆θ0 2
] (
(22)
Acknowledgment. I thank Entegris management for supporting this work and allowing publication. Also, thanks to C. Metzger, L. Monson, and S. I. Moon for their suggestions on the technical content and text. LA0625289
)
1 cos ωi sin - ∆θ0 (21) 2
fi/kγR ) 2 sin
(1)
If the drop elongates due to the external force, then k increases.27
1 1 (θ + θa,0) sin ωi cos - ∆θ0 2 r,0 2 1 cos ωi sin - ∆θ0 2
[
(12)
where in the case of a sessile drop, 2R is the drop width and the prefactor, k, depends on the shape of the contact line. If the contact line of the sessile drop is circular,12,27-30 then
to group the contact angles, eq 17 becomes
fi/kγR ) 2 sin
)
Retention of Sessile Drops. Figure 8 shows the side view of a stationary sessile drop under the influence of an external force, Fi. The external force distorts the shape of the drop, causing it to lean forward. The drop has reached a critical state where the retention and external force are equal, fi ) Fi. In this critical configuration the contact angle of the drop at the leading edge exhibits advancing value, θa, and the trailing edge a receding value, θr. If Fi were increased slightly, the drop would move. The general equation that describes the retention forces associated with a liquid slug in a capillary also describes the retention forces associated with sessile drops,12,25-28
fi ) kγR(cos θr - cos θa)
[21 (θ
(11)
The second describes the retention force ratios of a surface covered with asymmetric sawtooth features where the retention forces are diametrically opposed,
and
sin θr,0 + sin θa,0 ) 2 sin
)
)
(8)
Finally, from eq 8, we can write two retention force ratios in terms of inherent hysteresis and rise angle. The first describes the ratios of retention forces of a rough surface covered with symmetric sawtooth features relative to the corresponding smooth surface,
(25) Kawasaki, K. Study of Wettability of Polymers by Sliding of Water Drop. J. Colloid Sci. 1960, 15 (5), 402-407. (26) Furmidge, C. G. L. Studies at Phase Interfaces. I. The Sliding of Liquid Drops on Solid Surfaces and a Theory for Spray Retention. J. Colloid Sci. 1962, 17 (4), 309-324. (27) Extrand, C. W.; Kumagai, Y. Liquid Drops on an Inclined Plane: The Relation between Contact Angles, Drop Shape, and Retentive Force. J. Colloid Interface Sci. 1995, 170 (2), 515-521. (28) ElSherbini, A. I.; Jacobi, A. M. Retention Forces and Contact Angles for Critical Liquid Drops on Non-horizontal Surfaces. J. Colloid Interface Sci. 2006, 299 (2), 841-849. (29) Brown, R. A.; Orr, F. M., Jr.; Scriven, L. E. Static Drop on an Inclined Plate: Analysis by the Finite Element Method. J. Colloid Interface Sci. 1980, 73 (1), 76-87. (30) Carre´, A.; Shanahan, M. E. R. Drop Motion on an Inclined Plane and Evaluation of Hydrophobic Treatments to Glass. J. Adhes. 1995, 49, 177185.
