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Contact Line and Contact Angle Dynamics in Superhydrophobic Channels Junfeng Zhang† and Daniel Y. Kwok*,‡ Department of Biomedical Engineering, School of Medicine, Johns Hopkins UniVersity, Baltimore, Maryland 21205, and Nanoscale Technology and Engineering Laboratory, Department of Mechanical Engineering, Schulich School of Engineering, UniVersity of Calgary, Calgary, Alberta, T2N 2N4, Canada ReceiVed December 13, 2005. In Final Form: March 29, 2006 The dynamics of the wetting and movement of a three-phase contact line confined between two superhydrophobic surfaces were studied using a mean-field free-energy lattice Boltzmann model. Principle features of superhydrophobic surfaces, such as trapped vapor/air between rough microstructures, high contact angles, reduced contact angle hysteresis, and low resistance to fluid flow, were all observed. Movement of the three-phase contact line over a well-patterned superhydrophobic surface displays a periodic stick-jump-slip behavior, while the dynamic contact angle changes accordingly from maximum to minimum. Two regimes were found for the flow velocity as a function of surface roughness and can be related directly to the balance between driving force and flow resistance. This work provides a better understanding of dynamic wetting and fluid flow behaviors over superhydrophobic surfaces and hence could be useful in related applications.
I. Introduction Wettability of a liquid in contact with a solid surface is an important property in many natural and industrial processes. Clearly, its property depends on the chemical nature and physical topography of the surface.1 Numerous experiments have shown that rough hydrophobic surfaces with microstructures can distinctly increase contact angles and reduce contact angle hysteresis. This phenomenon is termed superhydrophobicity.2-5 Superhydrophobic surfaces also exhibit lower resistance to liquid droplets moving over them.6 Contradictory to extensive experimental work, theoretical and numerical analysis on this subject is relatively inadequate. Furthermore, interpretation of such experimental contact angles in the literature3,4,7-9 normally relies a priori on the well-known Cassie equation for heterogeneous surfaces10 and the Wenzel equations for rough surfaces.11 However, it is well-known from thermodynamic analysis1,12-14 that both the Cassie and Wenzel angles on chemically heterogeneous smooth surfaces and chemically homogeneous rough surfaces, respectively, cannot be easily accessible from experimental contact angles. Because of the complex physical nature of the interfaces and hydrodynamics involved, analytical study for the dynamics of * To whom correspondence should be addressed. Tel: (403) 210-8428; fax: (403) 282-8406; e-mail:
[email protected]. † Johns Hopkins University, Baltimore. ‡ University of Calgary. (1) Neumann, A. W.; Spelt, J. K. Applied Surface Thermodynamics; Marcel Dekker: New York, 1996. (2) Nakajima, A.; Hashimoto, K.; Watanabe, T. Monatsh. Chem. 2001, 132, 31. (3) Bico, J.; Marzolin, C.; Quere, D. Europhys. Lett. 1999, 47 (2), 220. (4) Lafuma, A.; Quere, D. Nat. Mater. 2003, 2, 457. (5) Jopp, J.; Grull, H.; Yerushalmi-Rozen, R. Langmuir 2004, 20 (23), 10015. (6) Kim, J.; Kim, C. Proc. 2002 IEEE Conf. MEMS 2002, 479-482. (7) He, B.; Lee, J.; Patankar, N. A. Colloids Surf., A2004, 248, 101. (8) Quere, D.; Lafuma, A.; Bico, J. Nanotechnology 2003, 14, 1109. (9) Ishino, C.; Okumara, K.; Quere, D. Europhys. Lett. 2004, 68 (3), 419. (10) Cassie, A. B. D. Discuss. Trans. Faraday Soc. 1948, 3, 11. (11) Wenzel, R. N. J. Phys. Chem. 1949, 53, 1466. (12) Johnson, R. E., Jr.; Dettre, R. H. AdV. Chem. 1964, 43, 112. (13) Johnson, R. E., Jr.; Dettre, R. H. J. Phys. Chem. 1964, 68, 1744. (14) Neumann, A. W. AdV. Colloid Interface Sci. 1974, 4, 105.
wetting is not an easy task, but fundamental understanding could be inferred from either quantitative or qualitative simulations. Several numerical studies on droplets and interfaces have been conducted by molecular dynamics,15,16 the lattice Boltzmann method ,17 and the diffuse interface method.18 However, they all assumed the surfaces to be smooth and homogeneous and have no direct relationship to superhydrophobic surfaces. Recently, droplet wetting behaviors were simulated by means of a lattice Boltzmann model (LBM);19 however, the physics of contact line dynamics and its relation with dynamic contact angles were not examined. In hydrodynamic systems with superhydrophobic surfaces where the chemical nature and physical roughness of the surface are equally important to the phenomenological contact angle and fluid flow, their effect on the dynamics of wetting, the movement of the three-phase contact line, and flow velocity remain unclear. Thus, our aim is to examine the hydrodynamic system in a two-dimensional (2D) channel made from superhydrophobic surfaces using a mean-field free-energy LBM recently proposed.20 For simplicity, the chemical heterogeneity of the surface is not considered here.
II. Mean-Field Free-Energy LBM Scheme According to the mean-field version of van der Waals’ theory, the total free-energy for a fluid system can be expressed as21-24
F)
∫dr{ψ[F(r)] + 21F(r)∫dr′φff(r′ - r)[F(r′) - F(r)] +
}
F(r)V(r) (1) (15) (16) (17) (18) (19) (20) (21) Oxford, (22) (23) (24)
Thompson, P. A.; Robbins, M. O. Phys. ReV. Lett. 1989, 63 (7), 766. Qian, T.; Wang, X.-P.; Sheng, P. Phys. ReV. E 2003, 68, 016306. Zhang, J.; Kwok, D. Y. Langmuir 2004, 20 (19), 8137. Seppecher, P. Int. J. Eng. Sci. 1996, 34 (9), 977. Dupuis, A.; Yeomans, J. Langmuir 2005, 21, 2624. Zhang, J.; Li, B.; Kwok, D. Y. Phys. ReV. E 2004, 69, 032602. Rowlinson, J.; Widom, B. Molecular Theory of Capillary; Claredon: 1982. Sullivan, D. E. J. Chem. Phys. 1981, 74 (4), 2604. Zhang, J.; Kwok, D. Y. J. Phys. Chem. B 2002, 106, 12594. Zhang, J.; Kwok, D. Y. Langmuir 2003, 19 (11), 4666.
10.1021/la053375c CCC: $33.50 © 2006 American Chemical Society Published on Web 04/26/2006
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where ψ(F) is a local free-energy with respect to the bulk density F. The second term is a nonlocal term taking into account the free-energy cost of variations in density; φff(r′ - r) is the interaction potential between two fluid particles located at r′ and r with respect to a given coordinate system. This term can be reduced to that of a square-gradient approximation when the local density varies slightly.22,25 The third term represents the contribution of an external potential energy V(r) to the freeenergy F. Both integrations are taken over the entire space. With this expression of free-energy, we follow the procedures described by Yang et al.26 and define a nonlocal pressure as
P(r) ) F(r)ψ′[F(r)] - ψ[F(r)] + 1 F(r) dr′φff(r′ - r)[F(r′) - F(r)] (2) 2
∫
For a bulk fluid with uniform density, the nonlocal integral term disappears, and eq 2 reverts to the equation of state of the fluid:
P ) Fψ′(F) - ψ(F)
(3)
Here, we briefly describe the implementation of these results to an LBM algorithm. In general, after discretization in time and space, the lattice Boltzmann equation (LBE) with a BhatnagarGross-Krook (BGK) collision term can be written as
1 fi(x + ei, t + 1) - fi(x, t) ) - [fi(x, t) - fieq(x, t)] (4) τ where the distribution function fi(x, t) denotes particle population moving in the direction of ei at a lattice site x with a time step t; τ is a relaxation time; and fieq(x, t) is a prescribed equilibrium distribution function depending on the lattice structure employed. The macroscopic density F and velocity u can be calculated from the distribution function fi given by
F)
∑i fi,
field related to F by
F(x, t) ) - ∇Φ(x, t)
(10)
To obtain the Navier-Stokes equation with a pressure term similar to that given by eq 2, we set an artificial Φ as follows:
Φ(x, t) ) F(x)ψ′[F(x)] - ψ[F(x)] + 1 F(x) dx′φff(x′ - x)[F(x′) - F(x)] - cs2F(x) (11) 2
∫
The above equations set up a complete LBM scheme with the mean-field free-energy function.20 It should be noted that there exists another so-called free-energy LBM in the literature,19,29 which, however, does not physically incorporate the solid-fluid interactions.20,22
III. Simulation Setup and Numerical Implementation Following refs 20, 29, and 30, we employ a van der Waals fluid model to express the free-energy of bulk fluid:
F ψ(F) ) FkT ln - aF2 1 - bF
where a and b are the van der Waals constants, k is the Boltzmann constant, and T is the absolute temperature. In this study, we selected a ) 9/49, b ) 2/21, and the scaled temperature kT ) 0.52. The choice of this set of parameters was based on previous studies.20,29 In this work, a D2Q9 (two dimensions, nine lattice velocities; see Figure 1a) lattice structure was employed. The nine discrete lattice velocities are expressed as
e0 ) 0
( i -2 1π, sin i -2 1π), i ) 1-4 2i - 9 2i - 9 e ) x2(cos π, sin π), i ) 5-8 4 4 ei ) cos
(5)
and
i
Fu )
∑i fiei
(6)
∑i fiei + τF
(7)
and employ the u here to calculate the equilibrium distribution function fieq [e.g., eq 14].27,28 Redefining the fluid momentum Fv to be an average of the momentum before collision and that after yields
Fv )
1
∑i fiei + 2F
(8)
Following the Chapman-Enskog procedure, a Navier-Stokes equation with the equation of state
P ) cs2F + Φ
(9)
can be obtained, where cs is the sound speed and Φ is the potential (25) Widom, B. J. Stat. Phys. 1978, 19, 563. (26) Yang, A. J. M.; Fleming, P. D.; Gibbs, J. H. J. Chem. Phys. 1976, 64 (9), 3732.
(13)
31,32 and the equilibrium distributionfeq i is given as
2 2 feq 0 )F 1- u 3
(
However, if an external force F(x, t) exists, we can modify eq 6 to reflect the momentum change as
Fu )
(12)
[91 + 31e ·u + 21(e ·u) - 61u ], i ) 1-4 1 1 1 1 ) F[ + e ·u + (e ·u) - u ], i ) 5-8 (14) 36 12 8 24
feq i )F feq i
)
2
i
i
i
i
2
2
2
The fluid-fluid interaction potential φff can be reduced to a single number K:
{
K, |x′ - x| ) 1 φff(x′ - x) ) K/4, |x′ - x| ) x2 0, otherwise
(15)
where K represents the interaction strength among the nearest (27) Shan, X.; Chen, H. Phys. ReV. E 1994, 49, 2941. (28) Buick, J. M. Lattice Boltzmann Methods in Interfacial Wave Modelling. Ph.D. Thesis, The University of Edinburgh, U.K., 1997. (29) Swift, M. R.; Osborn, W. R.; Yeomans, J. M. Phys. ReV. Lett. 1995, 75 (5), 830. (30) Angelopoulos, A. D.; Paunov, V. N.; Burganos, V. N.; Payatakes, A. C. Phys. ReV. E 1998, 57 (3), 3237. (31) Yang, Z. L.; Dinh, T. N.; Nourgaliev, R.; Sehgal, B. R. Int. J. Therm. Sci. 2000, 39, 1.
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Figure 1. A schematic of the simulation setup for (a) a D2Q9 lattice structure, (b) a rough surface model, and (c) the simulation domain. The advancing θa and receding θr angles can be obtained from the corresponding ends of the three-phase contact line.
neighboring particles, and the nonlocal integral term can be replaced by a summation over the neighbors of a site x. In this work, K for the liquid was set to -0.02. A negative value implies an attraction between two fluid particles, and its magnitude affects the calculated liquid-vapor interfacial tension. The interfacial tension for such a fluid system was found to be γ ) 0.03423 by measuring the pressure difference across the curved interface of a bubble (or droplet).20 With the presence of a solid surface, the solid-fluid interactions should also be considered. The orientation of a liquid-vapor interface near a three-phase contact point would consequently be influenced, and various contact angles can be obtained.33 Here, we implement the solid-fluid potential as an attractive force between a solid (xs) and a liquid (xf) site:
{
KwF(xf)(xs - xf),
|xs - xf| ) 1
Fs ) KwF(xf)(xs - xf)/4x2, |xs - xf| ) 1 0, otherwise
(16)
The coefficient Kw is positive for attractive forces and can be adjusted to obtain different wettability (contact angles). The larger the Kw, the stronger the solid-liquid attractions and hence the solid-liquid work of adhesion, resulting in a smaller contact angle. Here, Kw was selected as 0.07 to produce a relatively larger contact angle of 127.8 ° on a smooth surface. Numerical experiments showed that, to ensure the interface isotropy, the gradient of φ in eq 10 should be calculated through the following finite difference scheme:
[∇φ(x)]1 )
1 [φ(x + e5) + 4φ(x + e1) + φ(x + e8) 12 φ(x + e6) - 4φ(x + e3) - φ(x + e7)]
[∇φ(x)]2 )
1 [φ(x + e5) + 4φ(x + e2) + φ(x + e6) 12 φ(x + e8) - 4φ(x + e4) - φ(x + e7)] (17)
Here, the specific weight factors (1 and 4) may come from the (32) Qian, Y. H.; d’Humieres, D.; Lallemand, P. Europhys. Lett. 1992, 17, 479. (33) Zhang, J.; Kwok, D. Y. J. Colloid Interface Sci. 2005, 282 (2), 434.
lattice weights by projecting the four-dimensional (4D) facecentered hypercubic (FCHC) lattice to this D2Q9 model.31,34 Figure 1b displays how a rough surface was modeled with solid blocks separated by void spaces. Various roughness can be obtained by scaling the fraction between the solid blocks and the void spaces. Simulations of the movement of the three-phase contact line were conducted over a 2400 × 251 lattice domain (Figure 1c). The general no-slip bounce-back scheme was employed for the solid-fluid interfaces, and periodic boundary conditions were applied in the horizontal direction.35 Fluid motion was induced by means of a body force g along the channel.
IV. Results and Discussion A. Contact Angle and Contact Line Dynamics. Figure 2 displays the advancing and receding contact angles together with their corresponding three-phase contact line positions over a superhydrophobic channel. The rough surface was modeled using Ns ) 25, Nf ) 5, and Nh ) 24, as described in Figure 1b, and a body force g ) 0.5 × 10-7 along the horizontal direction was used to drive the flow. Unlike those found on a flat surface,17 the three-phase contact line and contact angle exhibit a periodic varying behavior as a result of the regularly patterned superhydrophobic surface. The periodicities of the contact line and contact angle appear to correspond very well. Here, the minimum advancing (132.6°) and maximum receding (133.3°) angles appear to have values similar to the Cassie angle (132.6°) from eq 18. This is merely a coincidence, and has not been observed on other surfaces. For clarity, we have enlarged two periods from Figure 2a,b and displayed the advancing three-phase contact line and contact angle in Figure 3. Here, we selected a typical time period between two minimum advancing contact angles (t1 ) 221400 and t3 ) 261200, θ1 ) θ3 ) 132.6 °, dashed lines) and separated this period into two parts at t2 ) 255800, where the contact angle is a maximum θ2 ) 154.1°. Snapshots of the advancing liquidvapor interface during the period (t1-t3) are provided in Figure 4. It can be seen that, because of the hydrophobic nature of the surface, the Laplace pressure prevents the liquid from penetrating into the gaps (approximately 1 lattice unit only), and vapor is trapped. This is the most important character of superhydrophobic (34) Kang, Q.; Zhang, D.; Chen, S. Phys. Fluids 2002, 14 (9), 3203. (35) Succi, S. The Lattice Boltzmann Equation; Oxford University Press: Oxford, 2001.
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Figure 2. The contact line position versus simulation time step for (a) advancing and (c) receding front, and dynamic (b) advancing and (d) receding contact angles.
Figure 3. A segment of Figure 2a,b for a selected time period (t1-t3). The maximum contact angle occurs at t2 as the three-phase contact line sticks at t2 and slips into t3.
surfaces, and our simulation captured it well. Going from panel a to panel d in Figure 4, the contact angle increases gradually from 132.6 to 154.1° during the first part of the period (t1-t2), while the contact line advances very slowly for only 8.5 lattice units. This corresponds to the sticking of the three-phase contact line on the right corner of a solid block, since the void gap can be thought of as a more hydrophobic surface (with a 180° contact angle) when compared to that of the solid surface (with a 127.8° contact angle). As more and more energy is accumulated in the system, the three-phase contact line finally touches the left corner of the next solid block and suddenly jumps onto the block surface (Figure 4f). The contact angle then quickly relaxes to its minimum, while the contact line slips over the surface for about 21.5 lattice units until it reaches the next corner of the block at t3 (Figure 4). To elucidate the sticking and jumping processes, we have
also shown in Figure 4h-l the various snapshots that were enlarged near the three-phase contact point. Movement of the contact line during t1-t3 corresponds to 30 lattice units and is consistent with the surface pattern that we have selected, Ns + Nf ) 30. Similar analysis can also be examined for the receding end, with similar results. The results reported above are similar to those observed experimentally elsewhere36-39 on the dynamics of the wetting and movement of the three-phase contact line over physically rough or chemically heterogeneous surfaces. (36) Fabretto, M.; Ralston, J.; Sedev, R. J. Adhes. Sci. Technol. 2004, 18 (1), 29. (37) Gleiche, M.; Chi, L.; Gedig, E.; Fuchs, H. ChemPhysChem 2001, 2 (3), 187. (38) Kwok, D. Y.; Neumann, A. W. AdV. Colloid Interface Sci. 1999, 81, 167. (39) Tavana, H.; Yang, G.; Yip, C. M.; Appelhans, D.; Zschoche, S.; Grundke, K.; Hair, M. L.; Neumann, A. W. Langmuir 2006, 22, 628.
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Figure 4. Snapshots of the three-phase contact line advancing (from left to right) over a superhydrophobic surface. Panels a-g (80 × 60 lattice units) display a complete period of the advancing contact point, while panels h-l (20 × 15 lattice units) show the detailed pictures during the stick-jump process. Simulation time step t and simultaneous contact angle θ are also shown in each figure. Note that the time intervals between every two successive snapshots are not the same.
B. Roughness Effect. As can be seen in Figure 2, both the advancing and receding contact angles change with the contact line positions, instead of constant values on flat surfaces.17 To study the roughness effect on contact angles quantitatively, we averaged the maximum and minimum advancing angles (154.1 and 132.6° in Figure 2b) to obtain a nominal advancing contact angle (143.4° in Figure 5a at r ) Nf/(Ns + Nf) ) 0.167). A similar treatment was also applied to the receding angles. The choice of this averaging reflects our intention to follow literature experimental measurements as closely as possible, and does not imply that such procedures are correct, as there exist a larger number of metastable angles during such a stick-jump-slip process. Typically, experimental contact angles reported on superhydrophobic surfaces in the literature40,41 are averaged over time or space instead of using the instantaneous angles. As an estimation, droplet velocity was obtained through a linear fit to the contact line positions in Figure 2. These results are plotted in Figure 5 versus the surface void ratio r using two body forces g ) 0.5 × 10-7 (circles) and g ) 1 × 10-7 (squares) to drive the flow. Here, we fixed the surface pattern as a sum of Ns + Nf ) 30. Thus, a value of r ) 0 represents a flat surface with no void space, and a larger r value means a rougher surface. (40) Suh, K. Y.; Jon, S. Langmuir 2005, 21 (15), 6836. (41) Ren, S.; Yang, S.; Zhao, Y.; Yu, T.; Xiao, X. Surf. Sci. 2003, 546 (2-3), 64.
From our simulation, the maximum contact angle on a superhydrophobic surface is about 177° at r ) 0.5. For comparison purposes, the contact angles predicted by the Cassie equation
cos θC ) (1 - r) cos θ0 - r
(18)
is shown in Figure 5a (dashed line) by considering the contact area between the droplet and the surface to be compose of pure solid and vapor patches. θC is called the Cassie angle, and θ0 is the contact angle on a chemically identical flat surface (r ) 0) and is hence equal to 127.8° in this work. The results from the combined Cassie-Wenzel equation
cos θCW ) (1 - r)R cos θ0 - rβ
(19)
are also displayed in the same Figure (dotted line) by taking into account small liquid penetration and a curved liquid-vapor interface between the gaps (in Figure 4). θCW is the combined Cassie-Wenzel angle. Following Wenzel’s approach,11 R ) (Ns + 2h)/Ns is the ratio of the real solid-liquid contact area to the nominal value, where h is the penetration depth and was set to h ) 1 here (see Figure 4). β is the ratio of a curved interface length to the gap width and has a value of 1.0764 for our surface model where θ0 ) 127.8°. It can be found in Figure 5a that the
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Figure 5. (a) Contact angles versus the void ratio r: filled symbols ) advancing; open symbols ) receding. (b) Three-phase contact line velocity versus the void ratio r: squares represent results using a driving force g ) 1 × 10-7; circles represent results using g ) 0.5 × 10-7. Contact angles predicted from the Cassie (eq 18) and combined Cassie-Wenzel (eq 19) equations are also shown in panel a as dashed and dotted lines, respectively.
contact angle calculated from these equations (dashed and dotted lines) increases with the void ratio r. When r > 0.77, contact angles from the combined Cassie-Wenzel equation (eq 19) reach the 180° limit. We wish to point out that these equations determine the contact angle to be a pure function of the surface topological structure, and hence no difference is anticipated between the advancing and receding angles. The predicted contact angles are also unique and independent of the three-phase contact line position. This prediction is in direct contradiction to those obtained from our simulation. The problems associated with both the Cassie and Wenzel equations are indeed well-known within the context of the thermodynamic model developed elsewhere.14 The advancing angles (solid symbols) from the LBM simulation increase with increasing r up to a limit (at about r ) 0.5) and then decrease slightly for two different driving forces, while the receding angles (open symbols) from the simulation are found to increase with r monotonically. These results are in qualitative agreement with experimental observations.42 The Cassie and combined Cassie-Wenzel equations also predict all angles to be identical, independent of the three-phase contact line velocities, while this is clearly not the case in the simulation results. We expect our simulation results to be more realistic since the three-phase contact line should be prone to the hydrodynamic effect, and this is clearly illustrated in Figure 5. Advancing angles obtained with a body force g ) 1 × 10-7 (squares) are larger than those using g ) 0.5 × 10-7 (circles); the inverse is true for the receding angles where the latter yields larger receding angles. This results in a larger droplet velocity under g ) 1 × 10-7 since a higher contact line velocity would induce a larger advancing and smaller receding contact angle.17,43 We would like to point out that contact angle phenomena are complicated: even for the simple surface model that we have (42) Johnson, R. E., Jr.; Dettre, R. H. Surface and Colloid Science; Matijevic, E., Ed.; John Wiley: New York, 1969; Vol.2, pp 85-154. (43) Blake, T. D. Wettability; Berg, J. C., Ed.; Marcel Dekker: New York, 1993; pp 251-309.
employed, a complex stick-jump-slip behavior can be easily observed. Hence, naive interpretation of experimental contact angles on rough surfaces can be misleading.44 The droplet velocities in Figure 5b display two regimes: (1) decrease with increasing void ratio r, and then, after a moderate value of r, (2) increase with increasing void ratio r. This behavior can be easily understood by considering the balance between driving force and flow resistance. The driving force here is the body force g, which is constant for all surfaces with different r. However, flow resistance depends on two components: (1) contact angle difference between the two droplet ends (i.e., contact angle hysteresis) and (2) interfacial friction from contacting the channel surfaces. As the void ratio r increases, interfacial friction decreases as more vapor is trapped in the void space; this has been observed from experiment.6 However, the resulting contact angle hysteresis increases faster in the low r regime and decreases slower in the high r regime. Therefore, in the low r region, contact angle hysteresis dominates, resulting in a lower droplet velocity. After r increases to a higher value (r > 0.5), contact angle hysteresis decreases, and the interfacial friction also becomes smaller; hence, higher liquid velocities on superhydrophobic surfaces are observed. A recent experimental study6 demonstrated similar flow behaviors on superhydrophobic surfaces. With the current fabrication technologies, it is a relatively easy task to engineer surfaces with a much larger void ratio r, and hence the velocity and efficiency of channel flow can be greatly enhanced for the same driving force if superhydrophobic surfaces are used.
V. Summary We have employed a lattice Boltzmann method to study the dynamics of the wetting and movement of the three-phase contact line in a superhydrophobic channel. Principle characters of superhydrophobic surfaces, including trapped vapor/air between (44) Zhang, J.; Grundke, K.; Kwok, D. Y. Langmuir 2003, 19 (24), 10457.
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rough microstructures, high contact angles, reduced contact angle hysteresis, and low resistance to liquid motion, were all observed. The three-phase contact line exhibits a periodic stick-jumpslip behavior as it moves over a superhydrophobic surface where the dynamic advancing and receding contact angles vary accordingly. The contact angles and hysteresis were found to change with the surface roughness in a similar manner with those from other experimental studies. The Cassie and combined Cassie-Wenzel equation can only predict the general increasing trend but fail to describe the actual contact angle behavior. The three-phase contact line velocity changes with the surface void
Zhang and Kwok
ratio in two regimes and has been explained in terms of the balance between driving force and flow resistance in terms of contact angle hysteresis and interfacial friction. Acknowledgment. This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the University of Calgary startup fund. J.Z. acknowledges financial support from the NSERC postdoctoral fellowship at the Johns Hopkins University and helpful discussion with R. Mohmammadi. LA053375C
