Water Drop Friction on Superhydrophobic Surfaces - American

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Article pubs.acs.org/Langmuir

Water Drop Friction on Superhydrophobic Surfaces Pontus Olin,*,† Stefan B. Lindström,§ Torbjörn Pettersson,† and Lars Wågberg*,†,‡ †

Fibre and Polymer Technology, KTH Royal Institute of Technology, Teknikringen 56, SE-100 44 Stockholm, Sweden Wallenberg Wood Science Centre, KTH Royal Institute of Technology, Teknikringen 56, SE-100 44 Stockholm, Sweden § Mechanics, Department of Management and Engineering, The Institute of Technology, Linköping University, SE-581 83 Linköping, Sweden ‡

S Supporting Information *

ABSTRACT: To investigate water drop friction on superhydrophobic surfaces, the motion of water drops on three different superhydrophobic surfaces has been studied by allowing drops to slide down an incline and capturing their motion using high-speed video. Two surfaces were prepared using crystallization of an alkyl ketene dimer (AKD) wax, and the third surface was the leaf of a Lotus (Nelumbo Nucifera). The acceleration of the water droplets on these superhydrophobic surfaces was measured as a function of droplet size and inclination of the surface. For small capillary numbers, we propose that the energy dissipation is dominated by intermittent pinning−depinning transitions at microscopic pinning sites along the trailing contact line of the drop, while at capillary numbers exceeding a critical value, energy dissipation is dominated by circulatory flow in the vicinity of the contacting disc between the droplet and the surface. By combining the results of the droplet acceleration with a theoretical model based on energy dissipation, we have introduced a material-specific coefficient called the superhydrophobic sliding resistance, bsh. Once determined, this parameter is sufficient for predicting the motion of water drops on superhydrophobic surfaces of a general macroscopic topography. This theory also infers the existence of an equilibrium sliding angle, βeq, at which the drop acceleration is zero. This angle is decreasing with the radius of the drop and is in quantitative agreement with the measured tilt angles required for a stationary drop to start sliding down an incline.

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INTRODUCTION Surfaces with extreme water-repelling properties, superhydrophobic surfaces, are of great importance in many application areas such as water-proofing of textiles,1 water-repellent automotive parts,2 self-cleaning windows,3 prevention of biofouling,4 and drag reduction in microchannels5 and on larger scale objects. Superhydrophobicity is usually defined by the following criteria:6 the apparent contact angle between a static water droplet and the surface must be large (>150°), the contact angle hysteresis must be small and droplets must be able to slide off the surface at a low tilt angle. Superhydrophobic surfaces can be manufactured by varying methods7,8 and are typically composed of materials with low surface energy and the surface topography includes microscopic protrusions. The tips of these protrusions are wetted by the water drop, but most of the nominal contact area remains dry, keeping the wetting energy low. The aim of this work is to develop a model for the motion of water drops on superhydrophobic surfaces. We thus seek a constitutive equation for the frictional force between a water drop and a superhydrophobic surface. In addition, we propose a robust experimental method for determining the parameters governing this constitutive relationship. Millimeter-sized drops moving down inclined, superhydrophobic wax surfaces at low © 2013 American Chemical Society

velocities (v < 1 m/s) have been studied using high-speed video. The surfaces exhibit a hierarchical roughness topography,9 with surface protrusions in the size range of nanometers up to several micrometers. Generally, the equilibrium contact angle of a droplet on a flat surface is described by the Young equation,10which is based on the minimization of interfacial energy. This equation has been modified for rough surfaces11 and for heterogeneous wetting states,12 by taking into account the difference between the wetted area and the nominal surface area. Superhydrophobicity develops under certain conditions,13,14 when surface protrusions prohibit or postpone complete wetting, so that a cushion of air is left between the drop and the surface,15 thus exceedingly reducing the apparent wetting energy. Surface roughness on several length scales is important for robust superhydrophobicity.16,17 The Lotus leaf exemplifies a naturally occurring superhydrophobic surface that has previously been extensively studied.18,19 Since the definition of superhydrophobicity is based on the static and dynamic contact angles, the most common method Received: March 26, 2013 Revised: May 29, 2013 Published: May 30, 2013 9079

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surfaces have been evaluated: two based on AKD, a hydrophobic wax commonly used in the paper-making industry, crystallized on silica wafers by two different methods, and the third was a leaf of a Lotus plant (Nelumbo nucifera). The results have been used to test whether the drops are rolling or sliding down the incline and to refute or corroborate different hypotheses for the drop friction mechanism.

for characterizing superhydrophobic surfaces is to measure these contact angles using the sessile drop method.20,21 One complication associated with this method is that the contact angles are affected by drop size due to gravitation-induced deformation.3 Moreover, the computed contact angle strongly depends on the analysis, mainly due to difficulties in finding the true three-phase contact line and the true tangent plane of the drop at this line. Dynamic contact angles have also been evaluated by measuring the advancing and receding contact angles of drops sliding down tilted surfaces.22 However, the static hydrophobic properties do not yield sufficient information regarding water-repelling kinetics, such as sliding acceleration or velocity23 and, consequently, the existence of a stable heterogeneous wetting state and a high apparent contact angle do not necessarily imply a low roll-off angle and superhydrophobicity.24 The interior of the nominal contact area seems to be irrelevant to contact angle behavior, and the interactions between the liquid and the solid at the three-phase contact line appear to determine the advancing and receding contact angles.25 Several studies have subsequently investigated the contact time,26 wetting transitions,27−30 number of bounces,31 and restitution coefficient32 for droplets bouncing on superhydrophobic surfaces. Sakai et al. have recently described a method for determining the sliding acceleration and internal fluidity of drops on such surfaces.33 A drop moving on a superhydrophobic surface usually exhibits a slipping motion rather than a rolling motion and a small fraction of wetted area is important to achieve superhydrophobicity.34 Other studies have shown that slipping motion and rolling motion are usually present simultaneously.35 It has also been proposed that the shape and initial movement of small drops are controlled by the surface tension of the air−liquid interface and liquid−solid interactions in the immediate proximity of the contact line.36 Different mechanisms for the kinetic friction force acting on a drop moving across a superhydrophobic surface have been previously proposed. A scaling law for the speed of a nonwetting drop moving on an inclined plane has been introduced based on the viscous dissipation near the contacting disc between the drop and the surface.37 It has also been suggested that the retentive force against the drop motion is governed by the contact angle hysteresis,38 as well as by the length and microscale continuity of the three-phase contact line.39 Roughness on multiple length scales, so-called hierarchical roughness, may increase or decrease the total length of the pinned contact line.40 Recently, the retentive forces have been measured experimentally for a contact line pinned to a grid of surface protrusions,41 and these forces have been calculated by using molecular dynamic simulations.42 Since superhydrophobic surface properties are dependent on both surface chemistry and surface topography, it is furthermore essential to characterize the surface structures. Characterization of surface topography is usually performed by stylus profilometry,43 noncontact optical profilometry,44 or atomic force microscopy (AFM),45,46 but it is difficult to use the same method for characterizing both the small scale, less than 1 μm, and the large scale, larger than 1 μm, of surface roughness. Combinations of different methods are therefore necessary to achieve a proper characterization of the surface structure. In this work, the velocity and acceleration of water drops on inclined superhydrophobic surfaces under the influence of gravity have been studied. Three different superhydrophobic

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MATERIALS AND METHODS

Sample Preparation. The silica wafers (p-doped with Boron), purchased from MEMC Electronic Materials (Novarra, Italy) were cut into 10 × 50 mm pieces and rinsed with the following sequence of solvents: Milli-Q water, ethanol, and Milli-Q water. The silica substrates were subsequently dried in a N2(g) flow and cleaned by plasma treatment in a PDC-002 plasma cleaner (Harrick Scientific, Pleasantville, NY) for three minutes at 30 W immediately prior to use. AKD (Figure 1) wax granules with a mean particle diameter of

Figure 1. Structure of alkyl ketene dimer (AKD), where R1 and R2 represent alkane groups with 16 or 18 carbons in their backbone. approximately 4 mm supplied by EKA chemicals (Bohus, Sweden) were used as received. AKD is commonly used to make paper hydrophobic and contains a mixture of C-16 to C-18 alkane chains. The fresh Lotus leaves, supplied by Bergianska Trädgården (Stockholm, Sweden), were cut into 10 × 50 mm pieces and used without further treatment. They were studied within 2 h after removal from the plant, due to the rapid withering process. These samples are henceforth abbreviated Lotus. AKD was applied to the substrates using two different methods: drop-coating from an AKD/heptane solution or spraying using the rapid extraction of supercritical CO2 solution (RESS) process.47 For the drop-coated samples, heptane supplied by Sigma-Aldrich (Munich, Germany) was heated in a water bath to 50 °C and AKD was added to a concentration of 10 wt %. The silica wafers were drop-coated with this solution and allowed to dry at ambient temperature. These samples are abbreviated heptane-AKD. Spraying was conducted with a pre-expansion pressure of 250 bar, a temperature of 70 °C, and a spraying distance of 1.5 cm, as previously described.47 The sprayed samples are abbreviated RESS-AKD. Evaluation of Samples. Height maps and surface roughness were evaluated using a Microprof 200 (FRT − Fries Research & Technology GmbH, Bergisch Gladbach, Germany) noncontact optical profilometer. The instrument is based on confocal microscopy with chromatic aberration. White light is refracted through a lens onto the sample surface, and the spectrum of the reflected light is analyzed and translated into a height position. The in-plane resolution is 2 μm and the height resolution is approximately 3 nm with a working range of ±150 μm. Areas of 500 × 500 μm were analyzed with a sampling distance of 1 μm in the x and y directions. One measurement was performed for each sample type. The surface roughness from these measurements (RMS roughness, Sq) is taken to be44

Sq =

1 n

n

∑ zi2 i=1

(1)

where zi is the surface height relative to the mean plane at point i. An area of 200 × 200 μm was analyzed, and the sampling distance was 1 μm in the x and y directions. Three measurements were performed on each sample type, and the mean value was taken. The details of the surface structures were evaluated using field emission scanning electron microscopy (Hitachi S-4800 FE-SEM, Hitachi, Japan). The samples were coated with a 5 nm thick Au/Pd 9080

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coating with the aid of a sputter coater (Cressington 208 HR, Cressington Scientific Instruments Ltd., Watford, U.K.) to improve the image quality. Advancing and receding contact angles were measured using a CAM200 contact angle meter (KSV Instruments, Helsinki, Finland) with an automatic dispenser. A 5 μL water drop was deposited on the surface and a stainless steel needle (external diameter 0.4 mm) was inserted into the drop. The drop volume was increased by injecting water at a rate of 0.2 μL/s. When the volume had reached 20 μL, the injection was stopped and switched to suction. This suction, with a rate of 0.2 μL/s, resulted in a reduction of the droplet volume. The contact angles were calculated using Laplace fitting every 2 s, during both injection and suction. Contact angles were measured at three different positions on each surface and an average value was calculated from these measurements. The sliding angles of water drops were determined using a custommade tilting stage with an accuracy of 0.1°. Drops with volumes ranging from 3 to 30 μL were deposited on a horizontal surface, which was then tilted at a rate of approximately 0.1°/s. The angle at which the drop started to move down the incline was recorded. The sliding acceleration of water drops was measured using a highspeed camera (IDT N4M-S3) with a fixed focal length objective (Cosmicar 50 mm/F1.4), attached to a table with adjustable inclination (Figure 2). This setup allowed the stage to be tilted up

Article

RESULTS The RMS roughness Sq for the three samples are compiled in Table 1. The roughness for the Heptane-AKD and Lotus are Table 1. Summary of the Measured Surface Propertiesa surface RESS-AKD heptaneAKD Lotus

θa (deg)

θr (deg)

cos θr − cos θa

Sq (μm)

160.6 ± 2.3 154.8 ± 3.1

149.6 ± 3.5 151.0 ± 4.3

0.08 0.09

1.23 ± 0.12 5.33 ± 0.45

154.2 ± 2.9

148.2 ± 3.4

0.05

4.92 ± 0.36

θa is the advancing contact angle, θr is the receding contact angle, cosθr − cosθa is a measurement of the contact angle hysteresis, and Sq is the RMS roughness. The errors represent one standard deviation. a

comparable, 5.33 and 4.92 μm, respectively, while the RESSAKD exhibits a much lower roughness of 1.23 μm. Topography maps obtained from the optical profilometer are shown in Figure 3. The Lotus leaf displayed randomly distributed microbumps with a spacing distance of approximately 10 μm (Figure 3a). This is in good agreement with previous findings.48 The Heptane-AKD (Figure 3b) had a rugged microstructure with large agglomerates with a larger interspacing than that of the Lotus. The RESS-AKD surface (Figure 3c) had a smoother structure, with almost no aggregated structures on the microscale, which is consistent with its lower measured roughness. FE-SEM micrographs are presented in Figure 4. A pronounced nanoroughness superimposed on the microroughness can be seen in all the samples, yielding a hierarchical topography that is vital for superhydrophobicity.49 In Figure 4, panels b and c, overhang structures are also observed. This leads to re-entrant curvature that improves the superhydrophobic properties.50 The results of the contact angle measurements are also listed in Table 1. All the samples have a macroscopically large water contact angle and low contact angle hysteresis, implying superhydrophobicity. The hysteresis is significantly lower for the Heptane-AKD and Lotus surfaces, 3.8° and 6.0°, respectively, indicating superior superhydrophobicity compared to the RESS-AKD surface with a hysteresis of 11°. This is in agreement with the sliding angle data, summarized in Figure 5, with a larger sliding angle for the RESS-AKD surface than for the Heptane-AKD and Lotus surfaces. A sample of the results of the sliding acceleration measurements at a surface inclination of 10° can be seen in Figure 6; t = 0 and s = 0 corresponds to the first image where the entire drop is visible. Note that ṡ(t = 0) ≠ 0 . A seconddegree polynomial is fitted to the data for each drop size. The excellent fit of the second-degree polynomial (Figure 6) indicates that the acceleration is essentially constant within the time frame of the measurement. The sliding acceleration increases with increasing drop size and increasing tilt angle and below a certain limiting drop size and tilt angle, a negative acceleration is observed, indicating constant retarding motion. The sliding acceleration is plotted as a function of drop radius for different tilt angles in Figure 7. The scatter in the acceleration data appears to increase with decreasing acceleration. This may be due to the lower kinetic energy of the drops which in turn leads to a larger sensitivity toward surface defects that can pin the three-phase contact line.

Figure 2. Schematic description of the experimental setup for measurement of drop acceleration.

to an angle of 45° while ensuring that the optical axis of the camera remained perpendicular to the sample surface. The sample was illuminated by a light-emitting diode (LED) light source (IDT 7 LED). Water drops with different volumes between 3 and 40 μL were deposited on the tilted surface, at tilt angles between 5° and 20°, and the movement of the drops was recorded at a frame rate of 3000 frames per second (fps) and a shutter speed between 60 and 130 μs. Fewer samples were taken from the Lotus leaf for practical issues, since the leaf started to wither and curl within two hours of sample preparation. The recorded images were analyzed with custom-made LabView software, yielding the time development of the x and y coordinates of the drop center. The radius of the drop was estimated by fitting a circle to the drop perimeter for each image frame. The sliding acceleration was determined by fitting a second-degree polynomial to the position data and calculating the second derivative with respect to time. 9081

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Figure 3. Topography maps of (a) Lotus, (b) heptane-AKD, (c) RESS-AKD obtained from optical profilometry. The scan area is 500 × 500 μm, and the length of the z axis is 200 μm. Figure 4. Scanning electron micrographs of (a) Lotus, (b) heptaneAKD, and (c) RESS-AKD.

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The surfaces with a smaller contact-angle hysteresis and a higher surface roughness (i.e., Lotus and Heptane-AKD) generally exhibit a higher acceleration for a given set of test parameters than the surface with a large contact angle hysteresis and low roughness (i.e., RESS-AKD) (Table 1 and Figure 7). This indicates that a certain microscale roughness is required for a sufficiently large air cushion to be present below the drop. The effects of the nanoscale roughness seen in the SEM micrographs (Figure 4) could not be evaluated due to the limited resolution of the optical profiling instrument, again showing the difficulty to collect the different length scales of the surface structure with one measurement technique.

DISCUSSION Droplet deformation. Consider a drop of a Newtonian fluid with viscosity, η, density, ρ, and surface tension, γ, with respect to air. If the volume of the drop is V, the nominal radius is given by R=

⎛ 3V ⎞1/3 ⎜ ⎟ ⎝ 4π ⎠

(2)

In a gravitational field of magnitude g, the capillary length is defined by 9082

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Figure 5. Experimental sliding angle values as a function of drop volume; the error bars represent one standard deviation.

Figure 7. Drop acceleration as a function of drop size and surface inclination for (a) Lotus, (b) Heptane-AKD, and (c) RESS-AKD.

⎛ γ ⎞1/2 Sc = ⎜ ⎟ ⎝ ρg ⎠

(3)

As previously described,37 for a sufficiently small drop [i.e., for a Bond number, Bo = (R/S c)2 ≪ 1], the surface tension forces will dominate over the gravitational forces, creating an almost spherical droplet (Figure 8a). For very large drops, Bo ≫ 1, the shape becomes sheetlike with a thickness on the order of S c (Figure 8c). The capillary length for water in air is S c = 2.7 mm. The drops studied herein are in the order of S c and thus attain the shapes of deformed spheres (Figure 8b). Energy Balance. As the drop moves down a surface with inclination β, the rate of dissipation of mechanical energy (Q̇ ),

Figure 6. Drop position, s, plotted against time for water drops on (a) Lotus, (b) Heptane-AKD, and (c) RESS-AKD. These surfaces have an inclination of 10°.

Figure 8. Schematic drawing of water drops of different bond numbers on superhydrophobic surfaces: (a) Bo ≪ 1, (b) Bo ≅ 1, and (c) Bo ≫ 1. 9083

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the rate of change of potential energy (Ẇ ), and the rate of change of kinetic energy (K̇ ) are balanced: Q̇ + Ẇ + K̇ = 0 (4)

behind the receding contact line, energy can only be dissipated due to the internal flow of the drop.37 With u⃗ representing the velocity field inside the droplet and Ω denoting the interior of the drop, the viscous dissipation becomes

The position coordinate s, introduced in presenting Figure 6, represents the distance traveled down the incline (Figure 9a).

Q̇ V = η

∫Ω (∇u⃗)2 dV

(8)

As previously described,37 a spherical droplet would perform rigid rotation, yielding Q̇ V = 0, while a drop of a small but finite Bo becomes slightly deformed, locating the dissipation to the vicinity of the contacting disc of radius, r ∼ Bo1/2R. The typical velocity is |u⃗| ∼ [(rṡ)/R], with a gradient |∇u⃗| ∼ (ṡ/R) within a region of volume ∼r3, so that eq 8 infers that ⎛ ṡ ⎞ Q̇ V ∼ η⎜ ⎟r 3 = ηRs 2̇ Bo2/3 ⎝R⎠

If viscous losses within the droplet at the length scale of the contacting disc is the dominant mechanism for dissipation (i.e., Q̇ = Q̇ V), and if Bo ≲ 1, eq 7 becomes

Figure 9. A drop of intermediate bond number rolls down a tilted superhydrophobic surface of inclination β. (a) Side view. (b) Top view with the contacting disc of radius r. The shaded band represents the area 2rs traversed by this contacting disc. (c) A close-up of the asperities of the surface with transferred fluid. (d) The receding contact line progresses through pinning−depinning transitions.

ηs ̇ s̈ = sin β − C1 Bo1/2 gk γ

Q̇ ρVs ̇

(5)

In the case of sliding motion, we interpret F = Q̇ /ṡ as the frictional force. For rolling motion, we obtain s̈ =

Q̇ ⎞ 5⎛ ⎜g sin β − ⎟ 7⎝ ρVs ̇⎠

(6)

Equations. 5 and 6 can be expressed in the common form l 2Q̇ s̈ = sin β − c gk γVs ̇

(10)

with C1 ∼ 1 a nondimensional constant. Here, we identify the capillary number, Ca = [(ηṡ)/γ]. Drop Motion on a Rough Surface. When a drop of low capillary number traverses a rough surface, its contact line becomes arrested at a local energy minima, and an energy barrier (EA) must be overcome (by e.g., thermal or mechanical activation)52 in order for the contact line to progress. This energy barrier can be significantly reduced if the contact line is able to deform between the pinning sites.53 For a drop of small kinetic energy K ≪ EA, the energy barrier is sufficient for arresting the drop, and its center of mass displays erratic motion controlled by activation events.52 In this case, the continuous model for drop motion, eq 7, breaks down. When the kinetic energy of the drop is large, K ≫ EA, the velocity of the center of mass is a smooth function of time, with small, superimposed fluctuations due to the frustrated motion of the drop’s contact line, which is still governed by pinning− depinning events. This smooth macroscopic motion together with discontinuous contact line motion is enabled by shape fluctuations. In this regime, the velocity of the drop is differentiable with respect to time, permitting the use of eq 7. Finally, when the capillary number becomes sufficiently large, the shape fluctuations are suppressed by viscous forces, so that both the drop and its contact line moves smoothly across the rough surface. With the assumption that the contacting disc is mostly an air−liquid interface, and thus, its interfacial energy per unit area is essentially γ, we can compute the energy released during each depinning event. If the pinning site interseparation is λ, the energy released during depinning is proportional to the difference in drop surface area before and after the event (Figure 9d).

The potential energy of the drop is W = −ρVgs sin β. The kinetic energy depends on translation and rotation due to internal flow within the drop. For an aspherical drop, rotations are restricted, so that the kinetic energy is mainly translational K = (1/2)ρVṡ2. If the drop is almost spherical (Figure 8a), the internal flow develops until the drops moves similarly to a rotating rigid body to minimize dissipation,37 yielding K = (1/ 2)ρVṡ2 + (1/2)I(ṡ/R)2, where rolling without slipping is assumed. If the mass moment of inertia, I, of the drop is approximated by that of a sphere I = (2/5)ρVR2, we obtain K = (7/10)ρVṡ2. For sliding motion typical of a large drop (Figure 8, panels b and c), eq 4 becomes s ̈ = g sin β −

(9)

(7)

with k = 5/7 for rolling and k = 1 for sliding motion. Drop Motion on a Smooth Surface. A drop rolling or sliding down a chemically homogeneous, smooth surface at a sufficiently small capillary number,51 Ca = [(viscous forces)/ (interfacial forcesa)] = [(μṡ)/γ] ≪ 1 will not change its shape, so that the interfacially stored elastic energy will be constant during its motion. The capillary number represents the relative effect of the viscous forces versus the surface tension acting across an interface between two immiscible fluids. Assuming that the drop does not alter the surface energy of the substrate

ΔQ d = γ ΔA ∼ γλ Λ(1 + cos θr̂ )

(11)

with Λ < 2r the width of the depinned contact line segment and θ̂r the dynamic, receding contact angle at the pinning site at the time of a depinning event. Note that θ̂r should not be confused with the macroscopic receding contact angle, θr, normally measured by standard methods such as the sessile drop 9084

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method. This energy difference ΔQd is identified with the energy scale of the barriers encountered by the contact line: EA ∼ γλΛ ≲ γλr ∼ γλRBo1/2. Moreover, using that K ∼ ρR3ṡ2, we find that the regime of smooth drop motion, K ≫ EA, is conservatively estimated by ρR3s 2̇ ≫ γλR Bo1/2 ⇔ Ca 2Bo1/2 ≫ C2 ,

C2 =

η 2 gλ γ2 (12)

At each depinning event, the energy ΔQd is converted into interfacial energy of water droplets transferred onto the pinning sites (Figure 9c) and into kinetic energy of the fluid as manifested by surface waves and flow inside the drop. Importantly, because θ̂r is very large, the impulse delivered to the drop during a depinning event is directed almost entirely along the surface normal, so that the stored elastic energy is effectively dissipated; only a small fraction of the released energy can be converted into net translation of the drop. In the following, we employ the approximation that all energy released during a depinning event is dissipated or converted into interfacial energy. A total of ∼[(rs)/(λΛ)] ∼ [(RBo1/2s)/(λΛ)] depinning events are required to advance the drop a distance s (Figure 9b), yielding the total reduction in mechanical energy Qd =

ΔQ dR Bo1/2s λΛ

∼ γ(1 + cos θr̂ )R Bo1/2s

Figure 10. Two regimes of drop friction appear in the domain 0 < Bo ≲ 1, 0 < Ca ≲ 1: Friction dominated by pinning−depinning transitions and friction dominated by viscous losses of the macroscopic flow inside the deforming drop.

sn̈ =

(13)

(14)

with C3 ∼ 1 a nondimensional constant. Regimes of Drop Friction. Two different mechanisms for the loss of mechanical energy were identified above, leading to different expressions for the drop acceleration down the incline. The circulatory flow within the drop due to the no-slip boundary condition (BC) at the contacting disc leads to drop motion as described by eq 10, while the pinning−depinning transitions lead to motion described by eq 14. By a comparison of the friction terms on the right-hand-sides of eqs 10 and 14, it is clear that viscous dissipation due to the no-slip BC becomes important when CaBo ≳ 1 + cosθ̂r, while the losses due to pinning−depinning transitions dominate friction when CaBo ≪ 1 + cosθ̂r. Also, to ensure that the drop motion is smooth, we must have Ca2Bo1/2 ≫ C2. The two regimes related to macroscopic flow and pinning− depinning, respectively, are illustrated in Figure 10. If a drop starts from rest, with Ca = 0, its friction will always be due to pinning−depinning transitions of the contact line initially. As it picks up speed, Ca increases and friction ultimately becomes dominated by losses due to the macroscopic flow circulation within the drop. In order to test the validity of eq 10, [s̈/(gk)] − sin β is plotted against [(ηṡBo1/2)/γ] for k = 1 and k = 5/7, respectively. Here, ṡ and s̈ are calculated for each measurement using the central difference method: sṅ =

sn + i − sn − i 2iΔt

i 2Δt 2

(16)

where sn is the distance traveled at measurement n, Δt is the time between each measurement, and i is a positive integer. Here, we have chosen i = 50 to achieve a small scatter in s̈. Plots for three different drop sizes on a heptane-AKD surface inclined 10° are presented in Figure 11. It is not possible to fit a straight line intersecting the origin to the data, showing that eq 10 does not describe the drop motion, which infers that dissipation is not governed by circulatory flow within the drop. The same conclusion could be drawn for the RESS-AKD and the Lotus (not shown). In order to test the validity of eq 14, [s̈/(gk)] − sin β is plotted against Bo−1/2 for k = 1 and k = 5/7, respectively (Figure 12). Values for Bo < 2.5 are lacking for the Lotus surface due to the rapid withering process described in the sample preparation section. The slope of the trend lines represents C3(1 + cos θ̂r). The assumption k = 1 yields a proportionality, while k = 5/7 does not, indicating that the drops move with a slipping/sliding motion, governed by pinning−depinning transitions of the three-phase contact line, rather than with a rolling motion governed by macroscopic flow within the drops. Furthermore, oscillations in the air−water interface of the drop, possibly arising from the depinning events, can be observed in the high-speed video footage (see the Supporting Information). In order to verify that all drops are within the pinning− depinning regime, the experimental Ca numbers are plotted against the Bo numbers (Figure 13). The lines,

In the regime of smooth drop motion, Ca2Bo1/2 ≫ C2, for a drop with Bo ≲ 1 and friction arising from the loss of mechanical energy during pinning−depinning of the contact line, eq 7 gives s̈ = sin β − C3(1 + cos θr̂ )Bo−1/2 gk

sn + i − 2sn + sn − i

CaBo = C3(1 + cos θr̂ )

(17)

represent our semiempirical estimate of the crossover between the pinning−depinning regime to the macroscopic flow regime. All experimental measurements are within the regime where drop friction is dominated by pinning−depinning transitions. This is expected for the low drop velocities measured (ṡ < 1 m/ s). For comparison, the crossover velocity for a drop with a 1 mm radius on a Lotus leaf surface is approximately 7 m/s and even higher on the other surfaces. In order to ensure that the drop motion is smooth for all drops, Ca2 is plotted against Bo1/2 (Figure 14). All values reside well above the trend line, C2 = Ca2Bo1/2, indicating that the drop motion is smooth and, consequently, differentiable for all experiments. In the calculation of C2, the pinning site interseparation, λ, is estimated to be 10 μm based on the

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Figure 11. [s̈/(gk)] − sin β plotted against [(ηṡBo1/2)/γ] with k = 1 (△) and k = 5/7 (□) for droplets with radii (a) 0.841 mm, (b) 1.21 mm, and (c) 2.34 mm on a heptane-AKD surface with an inclination of 10°.

Figure 12. [s̈/(gk)] − sin β plotted against Bo−1/2 with k = 1 (△) and k = 5/7 (□) for (a) Lotus, (b) heptane-AKD, and (c) RESS-AKD. The slopes of the trend lines represent C3(1 + cosθ̂r).

typical distance between the protrusions seen in the FE-SEM micrographs of the surfaces (Figure 4). It is difficult to measure C3 and (1 + cosθ̂r) separately. Consequently, a parameter named the superhydrophobic sliding resistance is introduced: bsh = C3(1 + cos θr̂ )

(18)

Combining eqs 14 and 18 and employing our empirical finding that k = 1, gives the acceleration: bsh (19) Bo The experimental values of bsh are 0.073 ± 0.008, 0.028 ± 0.005, and 0.012 ± 0.005 for RESS-AKD, heptane-AKD, and Lotus, respectively. A comparison between experimental acceleration values and calculated acceleration values using eq 19 is shown in Figure 15. With the current approach, it is also possible to define an equilibrium sliding angle, βeq, by setting s̈ = 0 in eq 19 and solving for β, s ̈ = g (sin β − bsh Bo−1/2),

Ca ≪

Figure 13. Ca plotted against Bo for Lotus, heptane-AKD, and RESSAKD. The lines represent CaBo = C3(1 + cos θ̂r), the approximate crossover between pinning−depinning and macroscopic flow. 9086

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Figure 14. Ca2 plotted against Bo1/2 for Lotus, heptane-AKD, and RESS-AKD surfaces. The solid line represents C2 = Ca2Bo1/2 for a pinning site interseparation λ = 10 μm.

Figure 15. Comparison between measured acceleration values and calculated acceleration values on surfaces with 15° inclination.

s ̈ = g (sin βeq − bsh Bo−1/2) = 0 ⇒ βeq = arcsin(bsh Bo−1/2) (20)

The equilibrium sliding angle denotes the minimum tilt angle required for a nonstationary droplet to spontaneously slide off the surface. The sliding angle, βslide, denotes the tilt angle required for a stationary droplet to spontaneously slide off the surface. This implies that βslide ≥ βeq, analogous to the case of static dry friction compared to kinetic dry friction. A comparison between the experimentally measured sliding angle and the equilibrium sliding angle is presented in Figure 16. There is a good agreement between the experimental data and the calculated equilibrium sliding angle for the heptaneAKD surface and the RESS-AKD surface, while there is a relatively large discrepancy for the Lotus surface. This is arguably due to the larger macroscopic roughness on the Lotus leaf compared to the AKD-coated silica substrates; the droplet is able to come to rest in a depression on the leaf surface where the local tilt angle of the surface can differ by several degrees from the measured tilt angle. It is also possible to rank surfaces with respect to superhydrophobicity using bsh; a lower value indicates superior superhydrophobic properties, implying that the Lotus is the most superhydrophobic surface followed by the heptane-AKD and the RESS-AKD. In comparison, by using the value of the contact angle hysteresis (cos θr − cos θa, Table 1) as a measure of superhydrophobicity, the conclusion would be that the RESS-AKD surface is more superhydrophobic than the heptane-AKD surface. This discrepancy may be due to an inherent physical difference between the static and the dynamic

Figure 16. Experimentally measured sliding angle and calculated equilibrium sliding angle using eq 20 for (a) Lotus, (b) heptane-AKD, and (c) RESS-AKD.

evaluation of superhydrophobicity, or alternatively, due to the difficulty in measuring the water contact angle correctly. Finally, and importantly, we may interpret our findings as a constitutive law: multiplying eq 19 by the mass m = ρV of the drop renders ma = mg sin β − bshmg Bo−1/2

(21)

where the second term on the right-hand side is interpreted as a frictional force (F) following the constitutive equation F = bsh Bo−1/2mg

(22)

This frictional force clearly takes a different form than Amontons’ law.

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CONCLUSIONS We have introduced a new surface parameter called the superhydrophobic sliding resistance bsh, which enters into a constitutive law for drop friction on superhydrophobic surfaces: F = bshBo−1/2mg. In the specific case of drops sliding down an incline, this expression for friction infers the existence of an 9087

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equilibrium sliding angle βeq = arcsin(bshBo−1/2) at which the friction force balances the gravitational pull. The superhydrophobic sliding resistance is measured using high-speed video and can be used to predict the sliding motion of drops on any plane or curved surface geometry through the use of the constitutive law. It also allows for qualitative ranking of superhydrophobic surfaces; a surface with a lower bsh parameter is more hydrophobic than a surface with higher bsh parameter. The equilibrium sliding angle denotes the tilt angle required for a drop with radius r to attain zero acceleration on a superhydrophobic surface. The equilibrium sliding angle is smaller than the angle at which a stationary drop is observed to start moving. This is due to aging of the contact line, resulting in a stronger pinning effect for a stationary drop than for a drop in motion.

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ASSOCIATED CONTENT

S Supporting Information *

High-speed video sequence recorded at a frame rate of 3000 frames per second, showing the presence of oscillations in the air−water interface of a drop sliding on a superhydrophobic RESS-AKD surface with a 20° inclination. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*P.O.: e-mail, [email protected]; tel, +46-8-790 8102. L.W.: e-mail, [email protected]; tel, +46-8-790 8294. Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Funding

This work was supported by the Swedish Foundation for Strategic Research (SSF, Grants 2005:0073/13 and RMA08− 0044). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank the Swedish Foundation for Strategic Research (SSF) for funding, Innventia AB for assistance with the topography measurements, EKA Chemicals for supplying AKD, Dr. Irene Rodriguez Meizoso at Lund University for supplying sprayed surfaces, and Bergius Botanic Garden for supplying fresh Lotus leaves.

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