Measurement and Scaling of Hydrodynamic Interactions in the

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Measurement and Scaling of Hydrodynamic Interactions in the Presence of Draining Channels Rohini Gupta and Joel̈ le Fréchette* Department of Chemical and Biomolecular Engineering, Johns Hopkins University, Baltimore, Maryland 21218, United States S Supporting Information *

ABSTRACT: Central to the adhesion and locomotion of tree frogs are their structured toe pads, which consist of an array of 10 μm hexagonal epithelial cells separated by interconnected channels that are 1 μm wide and 10 μm deep. It has been proposed that the channels facilitate the drainage of excess fluid trapped between the toe pads and the contacting surface, and thus reduce the hydrodynamic repulsion during approach. We performed direct force measurement of the normal hydrodynamic interactions during the drainage of fluid from the gap between a structured and a smooth surface using surface force apparatus. The structured surface consisted of a hexagonal array of cylindrical posts to represent the network of interconnected channels. The measured hydrodynamic drainage forces agree with the predictions from Reynolds’ theory for smooth surfaces at large separations. Deviations from theory, characterized by a reduction in the hydrodynamic repulsion, are observed below some critical separation (hc), which is independent of drive velocity. We employ a scaling analysis to establish the relationship between structural features (channel depth, width, and post diameter) and the critical separation for the onset of deviations. We find agreement between our experiments and the scaling analysis, which allows us to estimate a characteristic length scale that corresponds to the transition from the fluid being radially squeezed out of the nominal contact area to being squeezed out through the network of interconnected channels.

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INTRODUCTION The locomotion mechanisms employed by tree frogs under flooded conditions could offer the ultimate solution for the need of strong, reversible, reusable, tunable, and water-tolerant adhesives. Central to the adhesion and locomotion of tree frogs are their structured toe pads, which consist of an array of 10 μm hexagonal epithelial cells separated by a network of 1 μm wide and 10 μm deep interconnected channels that end in mucus secreting glands. On dry surfaces, the toe pads enhance adhesion via deformation of the soft epithelial cells to improve foot-surface conformity and secretion of watery-mucus into the area of contact to promote capillary or hydrodynamic interactions.1−11 Under flooded conditions, however, capillary interactions are expected to be negligible given the absence of any free fluid interface. The mechanisms for tree frog adhesion and locomotion under flooded conditions, and by extension the role played by the structured toe pads, have been the subject of speculations. It has been proposed that the channels facilitate the drainage of excess fluid trapped between the toe pads and the contacting surface, and thus reduce the hydrodynamic repulsion during approach.12,13 In fluids, hydrodynamic drainage forces arise when two surfaces move relative to each other. The fluid’s resistance to motion leads to drag: repulsion when the surfaces are driven toward or adhesion when they are separated away from each other.14 In the case of tree frogs, a reduction in the © 2012 American Chemical Society

hydrodynamic repulsion would, therefore, enable the toe pad to come into a more intimate contact with the surface, which would lead to better grip. This effect is somewhat analogous to how tire treads prevent hydroplaning in the case of rolling and fluid flow dominated by inertia.12,13 It has also been proposed that the epithelial cells in the toe pad may deform, leading to channels being closed during retraction for adhesion.12,13 Recent experiments with structured surfaces inspired by bush cricket tarsal pads (or tree frog toe pads) also suggest that the presence of channels that run perpendicular to the crack may lead to crack arrest and elastic relaxation at the edge of the structures.15,16 The coupling of these mechanisms makes it challenging to assess their individual contribution in facilitating tree frog adhesion and locomotion under flooded conditions, especially when combined with the complexity associated with the different modes of approach and detachment (normal or peeling3,10). While the subject of how surface structures can be employed to enhance adhesion is not new, the additional contribution due to hydrodynamic interactions is often overlooked. More specifically, the role and importance of the interconnected channels in modulating the hydrodynamic interactions has not been investigated directly. Received: August 30, 2012 Revised: September 25, 2012 Published: September 25, 2012 14703

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a fluid. In the limit when the surface separation (h) is much smaller than the radii of the cylinders, the hydrodynamic forces (FH) (adhesion and repulsion) within Reynolds’ lubrication theory are given by:35

Recent designs of biomimetic systems toward improved fluid-mediated adhesion have shown significant promises. For example, adhesion of a hexagonally packed array of cylindrical micropillars inspired by beetle foot-hair has been shown to be significantly enhanced in the presence of a thin viscous film, likely due to capillary and hydrodynamic interactions caused by the individual liquid bridges present on each of the micropillars.17,18 Structured surfaces inspired by bush cricket tarsal pads (or tree frog toe pads) also have been shown to offer better friction than smooth surfaces in wet conditions.19 Yet in all of these systems the enhanced adhesion likely had contributions from a variety of surface forces such as capillary, hydrodynamic, and van der Waals interactions, as well as crack arrest and elastic relaxation of the material. Therefore, the individual contribution of the reduction in hydrodynamic interactions in fluid-mediated adhesion, especially in the presence of a network of interconnected channels, could not be assessed independently. The question of how surface structures alter hydrodynamic drainage forces is a general problem that is not limited to tree frog adhesion, especially for micro- and nanoscale surface features. Nanoscale roughness, for instance, has been shown to have a complex influence on the local shear rate and may result in slip at the solid−liquid interface or no-slip at shifted plane away from the surface.20−30 Moreover, micro- and nanoscale resonators in fluid environment suffer viscous losses and losses due to collisions of individual fluid molecules, which must be minimized to increase their quality factor.31,32 The quality factor has been shown to be sensitive to the effective surface area due to surface roughness, which must be accounted for in the design criteria to improve the crystal resonator response when operating in fluid environments.33 In porous media, hydrodynamic interactions are important in the three-dimensional fluid flow and solute transport through a fractured crack during oil recovery.34 Here, we show how an interconnected network of draining channels alters normal hydrodynamic interactions. We highlight the interplay between channel depth, width, and post diameter by using specifically designed structured rigid surfaces with micrometer-size features that are driven toward a smooth surface across a viscous fluid. The hydrodynamic interactions are measured using the surface force apparatus (SFA) and analyzed within Reynolds’ continuum approach in the lubrication limit. Our results are relevant for the design of biomimetic systems inspired by tree frog toe pads, slip associated with surface roughness, hydrofracture and micromodels for oil recovery, and minimum detectable mass and quality factor estimation of micro- and nanoresonators.

FH = −

6πηR GRH dh h dt

(1)

where dh/dt is the rate of change of surface separation or instantaneous velocity, which is negative for approach and positive for retraction thus setting the sign convention for the system, and RH and RG (∼2 cm) are the harmonic and geometric means of radii of the two cylinders in crossedcylinder geometry. The hydrodynamic interactions diverge as the surface separation asymptotically approaches zero and can be significantly strong and long-ranged. Normal hydrodynamic interactions between two circular disks have also been referred to as Stefan adhesion in the literature.14 Normal and shear hydrodynamic interactions play a key role in applications such as lubrication, colloidal stability, and microfluidics, and have been extensively studied using atomic force microscopy ( A FM ) 2 1 , 2 7 , 3 0 , 3 6 − 4 6 and surfa ce force appa ra tus (SFA).20,35,47−52 The pioneering study of the drainage of thin films using the SFA reported by Chan and Horn35 has established the validity of Reynolds’ theory for drainage of nonpolar Newtonian fluids between atomically smooth mica for surface separations larger than 50 nm. In the direct measurement of hydrodynamic interactions, one of the surfaces is mounted on a cantilever driven at a constant speed toward an opposing surface across a viscous fluid. The hydrodynamic interactions cause the cantilever to deflect such that the instantaneous surface velocity is less than the drive velocity. In their analysis, inertial or acceleration effects as well as frictional and cantilever drag forces are neglected such that the sum of surface and hydrodynamic drainage forces is balanced by the restoring spring force. (The cantilever drag, however, often cannot be neglected for the measurements performed using the AFM.36) Deviations for surface separation ranging from 50 to 5 nm could be accounted for by incorporating a shear plane located 1.3 nm away from each of the surfaces.35 For surface separations smaller than 5 nm, artifacts such as hard wall (∼3 nm) and step-like (∼0.75 nm) features due to finite size of molecule were observed,35 which is consistent with other reports in the literature that have shown that confinement alters the molecular organization, viscosity, and mechanical, tribological, and phase behavior of fluid leading to structural or oscillatory solvation forces when surface separation is of the order of about 10 molecular diameters or less.53−60 Hydrodynamic repulsion has also been employed in the investigations of attractive surface forces (van der Waals, capillary, and electrostatic interactions) to overcome the jump into contact at short separation during approach.61−63 The no-slip assumption at the fluid−solid interface has been reported in the literature to not be applicable for all fluid−solid systems. Vinogradova has presented the analytical solutions of the Navier−Stokes equations within the lubrication approximation while allowing for slip at either both or one of the solid−fluid interfaces in the form of a correction factor that incorporates a finite slip length,64 and slip at shifted plane.27 Normal and shear hydrodynamic interactions have been studied extensively to probe how the slip behavior of the solid−liquid interface is affected by surface wettability,64 surface roughness,20,21,27,30 trapped nanobubbles or a gaseous film,25,65

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REYNOLDS’ LUBRICATION THEORY: VALIDITY AND REFINEMENTS The hydrodynamic repulsion when two surfaces are driven toward or adhesion when they are separated away from each other, across an incompressible Newtonian fluid in the continuum regime, can be described within the lubrication approximation using Reynolds’ theory. Fluid inertia and body forces are neglected when simplifying the Navier−Stokes equations within the lubrication approximation. The theory also assumes no-slip boundary condition at both fluid−solid interfaces, and independence of fluid viscosity (η) with respect to confinement at very small surface separations.35 Measurements of hydrodynamic drainage forces with the surface force apparatus (SFA) employ two cross-cylinders interacting across 14704

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adsorbed species,51,52 fluid viscosity and polarity,26 and local shear rates.46,50 It is, however, tricky to decouple the influence of these strongly interdependent factors.26 Drive velocity, fluid viscosity, and cantilever spring constant have also been recently reported to influence the magnitude of slip length, which may reconcile the contrasting results reported in the literature.42

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k(h − h initial − vt ) = −

6πηR GRH dh h dt

(2)

Surface Preparation. Muscovite mica pieces (Ruby, ASTM V-1, S&J Trading) were cleaved in a laminar hood and placed on a larger and thicker freshly cleaved mica backing sheet. 50 nm of silver (99.999% purity, Alfa Aesar) was deposited on the cleaved mica pieces (thickness = 2−8 μm) via thermal evaporation (Kurt J. Lesker Nano38) at a rate of 3−4 Å/s. Smooth HDT-Coated Silver (Bottom Surface). The silvered mica pieces on backing sheet were rendered hydrophobic (water contact angle ∼107°) by immersing them in 1 mM HDT (92%, Aldrich) solution in ethanol (200 proof) overnight. After immersion in the thiol solution overnight, the backing sheet is removed and rinsed thoroughly with ethanol to remove any excess reagents. The mica piece with HDT-coated silver is glued (Epon 1004 epoxy) mica side down (the mica sheet is not part of the interferometer) onto the cylindrical disk (radius of curvature, R ∼ 2 cm), which is then mounted on the cantilever connected to the microstepping motor. Smooth or Structured SU-8 (Top Surface). The silvered mica piece is used for support and handling of both smooth and structured SU-8 surfaces. The mica is not part of the interferometer, but its silver film serves as one of the two reflective layers. For the fabrication, backing sheet with one of the silvered mica pieces is taped to a glass slide. Negative photoresist SU-8 2007 (MicroChem) is spin-coated onto the silver side at 5500 rpm for 1 min. The mica piece is then removed and placed on a freshly cleaved mica backing sheet followed by baking at 65 °C for 3 min, 95 °C for 5 min, and 65 °C for 3 min. The SU-8 2007 is then exposed to ultraviolet radiations at 140 mJ/cm2, followed by postexposure bake identical to the previous baking step to form a contiguous layer of smooth SU-8. To form structured SU-8, thinner SU-8 2000.5 or 2002 (MicroChem) of appropriate thickness is spincoated onto the existing SU-8 layer followed by baking (same procedure as earlier). The second thinner layer is exposed to ultraviolet radiations at 100 mJ/cm2 through a chrome-on-glass mask, which is followed by the postexposure bake and then by developing for 3 min to yield structured SU-8. Excess SU-8 developer is removed using isopropanol, and the sample is dried with compressed air. The silvered mica piece with smooth or structured SU-8 is glued (Epon 1004 epoxy) mica side down onto cylindrical disk (radius of curvature, R ∼ 2 cm), followed by hard baking at 150 °C for 30 min. The cylindrical disk is then screwed onto the fixed (immobile) top mount.

EXPERIMENTAL DETAILS

Surface Force Apparatus (SFA). The SFA is employed to measure hydrodynamic forces as fluid is drained out of the nominal contact between two rigid crossed-cylinders (geometrically equivalent to a sphere-plane) with radius ∼2 cm, and submerged into a bath filled with silicone oil (Xiameter PMX 200 Silicone Fluid, η = 48 mPa·s or 50 cSt). The intrinsic length scale associated with the hydrodynamic interactions between two crossed-cylinders in the SFA is of the order of micrometers, implying that the SFA is perfectly suited for probing the influence of the microscale network of interconnected channels. The AFM would, instead, probe the interactions of an isolated channel or post. In the SFA, the surface separation is estimated from the wavelengths of the fringes of equal chromatic order (FECO)66−69 resulting from multiple beam interferometry (MBI).70,71 The shape of the FECO reflects the geometry of the interacting surfaces: parabolic fringes are obtained for two crossed-cylinders and can be used to calculate the radii of curvature of the interacting cylinders. The microscale structural features on the surface that mimic the tree frog toe pad are also visible on the FECO. The wavelengths at the vertex of the parabolic fringes are used to estimate the surface separation at the point of closet approach for a sphere-plane configuration. The surface separation for structured surfaces is measured at the top of a post at the point of closet approach. To determine surface separation, we use the multilayer matrix method72,73 combined with the fast spectral correlation algorithm designed by Heurberger.74 We found that this approach was particularly suitable to asymmetric optical filters and allowed for fast and accurate data analysis using multiple fringes at once. This method requires the prior knowledge of SU-8 and hexadecanethiol (HDT) thicknesses, which were obtained from the contact fringes. Nanoscale surface roughness associated with both HDT-coated silver and SU-8 will cause a small shift in the position of the optical fringes,75,76 but this effect is not incorporated here in the determination of the SU-8 thickness from interferometry measurements. The interaction between the two crossed-cylinders is calculated from the deflection of a soft cantilever spring (k = 165 N/m). In the dynamic measurements, the top surface is kept immobile and the bottom surface is mounted on the cantilever (initially at rest) and is driven toward the top surface at constant velocity, v (negative for approach and positive for retraction), ranging from 25 to 700 nm/s via microstepping motors. Anhydrous calcium chloride is placed inside the SFA chamber to remove moisture from the ambient air. Force Measurement and Analysis. We follow the approach proposed by Chan and Horn35 such that raw experimental data consist of the instantaneous surface separation as a function of time, and, because of the drag, is different from the separation predicted from the drive (dh/dt < v). A quasi-static force balance between the hydrodynamic drainage forces and the restoring spring force results in a first-order differential equation (eq 2), where hinitial is the surface separation at time t = 0 s. Equation 2 is solved numerically using a mixed fourth- and fifth-order Runge−Kutta method (ode45 in MATLAB) to obtain theoretical predictions for the surface separation as a function of time for different driving functions. At a given time, the difference between surface separation and the separation predicted from the constant velocity drive multiplied by the spring constant is a measure of the hydrodynamic drainage forces as a function of surface separation. In our experimental system, nonhydrodynamic forces such as van der Waals, capillary, electrostatic, and steric interactions should be negligible and are, therefore, not included in our analysis and in eq 2. Similarly, inertial or acceleration effects (low Reynolds’ number regime) and frictional and cantilever drag forces are also ignored.35

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RESULTS AND DISCUSSION Design Considerations. Our design considerations are focused toward eliminating any contributions due to chemistry and other surface forces in the system, allowing us to probe only the role played by the network of interconnected channels in reducing the hydrodynamic drainage forces. The epoxybased negative photoresist SU-8 is chosen as the material for the structured surfaces because of its rigidity (E = 2 GPa after baking) such that we can rule out the role of elastohydrodynamic deformation in the forces measured. Smooth and structured SU-8 surfaces are easy to fabricate using standard photolithography procedure and offer advantageous wetting properties and transparency. Silver is chosen to reduce the number of layers in the optical filter and is rendered hydrophobic to reduce contributions due to van der Waals interactions. Both of the surfaces are submerged in the silicone oil to eliminate any contributions from capillary interactions such that the hydrodynamic drainage forces alone should balance the restoring spring force. The silicone oil chosen here is a nonpolar, chemically inert, and Newtonian polymer melt with a viscosity of 48 mPa·s. Hydrodynamic drainage forces scale with the product of drive velocity and fluid viscosity: working with 48 mPa·s viscosity oil allows us to operate at 14705

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Table 1. Topographical Features of the Different Structured SU-8 Surfaces Investigated in This Work surface control A B C D

channel depth (D) (μm) 0.30 0.42 0.80 1.26

post diameter (d) (μm) 65.0 6.5 6.5 65.0

channel width (w) (μm) 6.5 6.5 3.0 6.5

periodicity (D1 = d + w) (μm)

surface coverage

ho (μm)

71.5 13.0 9.5 71.5

100% 75% 23% 43% 75%

0.14 0.34 0.55 0.57

hc (μm) 0.64 1.34 1.61 1.68

± ± ± ±

0.15 0.13 0.27 0.14

Figure 1. (a) Schematic for the topographical features and (b) a representative scanning electron micrograph (at tilt angle of 40°) of structured SU-8 Surface D. (c) Schematic for the SFA configuration employed here showing a structured SU-8 interacting with smooth hydrophobic silver across viscous silicone oil.

larger separation (100 nm to 2 μm), which falls in a region wellrepresented by Reynolds’ theory for the interactions between unstructured SU-8 and silver. Structured Surfaces. The hydrodynamic interactions caused by the drainage of oil from the gap between a structured SU-8 (Surface A) and a hydrophobic silver film measured for five different drive velocities are shown in Figure 3. We observe that the measured drainage forces agree with Reynolds’ theory for smooth surfaces at large separations, and deviations, characterized by a reduction in the hydrodynamic repulsion, are observed below some critical separation, hc. Here, hc is determined as the surface separation at which the difference between the experimental hydrodynamic drainage forces and those predicted by Reynolds’ theory for smooth surfaces exceeds the experimental error for each individual run. (See Supporting Information Figure S2 for an example.) The experimental error manifests itself as small deviations (within ±1−15 mPa) at large separations, where the experimental hydrodynamic drainage forces agree with Reynolds’ theory for smooth surfaces. For Surface A, we obtain a critical separation of 0.64 ± 0.15 μm, as shown by the dashed line in Figure 3. As can be seen in Figure 3, the presence of surface structure leads to significant deviations from Reynolds’ theory that are observed at separation larger than those corresponding to deviations observed for the interactions between smooth surfaces (Figure 1). We also observe that, for a given surface pattern, the critical separation for onset of deviations is independent of the drive velocity (Figure 3), which is consistent with fluid flow at low Reynolds’ number. Alternatively, the no-slip boundary condition in Reynolds’ theory for smooth surfaces can be applied at a plane located at the bottom of the channel. The resulting hydrodynamic drainage forces provide a complementary lower bound to the predictions from Reynolds’ theory and have been shown for reference in the Supporting Information (see Figure S1). Three different surface structures were employed to understand how the critical surface separation for the onset of deviations from Reynolds’ theory for smooth surfaces

slower drive velocities while still being able to scale our results for the case of tree frogs (viscosity of water being 1 mPa·s). To study how a network of interconnected channels alters the hydrodynamic drainage forces, we created four surfaces with different topographical features (see Table 1). The structured SU-8 consists of a hexagonal array of cylindrical posts (surface coverage of 91% with respect to that for hexagonal). These four surfaces allowed us to investigate systematically the role of channel depth, width, post diameter, and surface coverage in modulating hydrodynamic drainage forces. A schematic illustrating the topographical features of structured SU-8 investigated in this work along with a representative scanning electron micrograph and SFA configuration of structured SU-8 (Surface D) interacting with smooth hydrophobic silver across viscous silicone oil are shown in Figure 1. Control Experiments. We performed control experiments to verify that the hydrodynamic interactions associated with the drainage of viscous silicone oil from the gap between an unstructured (or smooth) SU-8 and a hydrophobic silver surface are well-represented by Reynolds’ theory. Shown in Figure 2a is a representative curve for the surface separation as a function of time that shows good agreement with the predictions from Reynolds’ lubrication theory. The force curve associated with the separation as a function of time is obtained from the spring deflection and is shown in Figure 2b. While the agreement between Reynolds’ theory and our measurements is excellent for separations greater than 50 nm, we do observe deviations at short-range as discussed in the previous section. The structure and finite size of the fluid molecules manifest themselves as a shear wall away from each of the surfaces located at a distance comparable with the radius of gyration of the polymer, and step-like features with step size comparable with the width of dimethylsiloxane chain (see inset in Figure 2b).48 Here, our efforts focus on deviations caused by structural features much larger than those due to surface roughness (see Table 1 for values). As discussed in the next section, these deviations from Reynolds’ theory manifest themselves at much 14706

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Figure 3. Hydrodynamic drainage forces between a structured SU-8 (Surface A) and a smooth hydrophobic silver film measured at different drive velocities (−48, −94, −136, −182, and −445 nm/s). The solid lines represent the predictions from Reynolds’ theory for smooth surfaces. The critical separation for the onset of deviations, hc, is shown by the dashed line. The different values for hc obtained at different drive velocities are shown in the inset; the solid line represents the mean, and the dashed lines represent the standard deviation (hc = 0.64 ± 0.15 μm).

between the geometrical features of our surfaces and the characteristic length scale for the onset of deviations from Reynolds’ theory for smooth surfaces. Persson proposed a simple scaling to determine a separation of length scales where the influence of channels would be evident for the hydrodynamic drainage forces. If experimentally confirmed, the scaling analysis would enable us to isolate the regimes where the fluid squeezed out from contact region between two rigid bodies would preferentially drain through the network of interconnected channels versus where it would drain radially outward from the contact region. In his analysis, the drainage of an incompressible fluid between two rigid bodies in the absence and presence of a network of interconnected channels is compared. In the absence of surface structures, all of the fluid squeezed out of the nominal contact region of diameter Do between two smooth rigid bodies is expelled radially outward such that fluid mass conservation leads to πD2o(dh/dt) ∼ −πDohv′. The order-of-magnitude scaling obtained from Navier−Stokes equations relating the fluid pressure (p) and average fluid velocity (v′) in the contact region is given by (p/ Do) ∼ (ηv′/h2), which when combined with the fluid mass conversation above leads to the following:

Figure 2. Control experiments with smooth surfaces. (a) Surface separation as a function of time and (b) associated force curve obtained during the drainage of viscous silicone oil from the gap between an unstructured SU-8 and a hydrophobic silver surface (v = −253 nm/s). The dashed line represents the separation predicted from the constant velocity drive, and solid lines represent predictions from Reynolds’ theory for smooth surfaces. The inset in (b) shows deviations from theory observed for separations smaller than 50 nm.

depends on the structural features (channel depth, width, and post diameter). Shown in Figure 4 are the force curves obtained for three different structured SU-8: Surfaces A, B, and C (see Table 1). In all three curves, we observe that the hydrodynamic drainage forces agree with Reynolds’ theory for smooth surfaces at large separations with deviations observed below some critical separation, hc. The critical separation for the onset of deviations, however, is different for the three surfaces investigated (see dashed lines in the figure and Table 1 for numerical values hc). Interestingly, as seen from Figure 4 and Table 1, we observe that no single structural feature alone determines the onset of deviations from Reynolds’ theory. Let us consider, for example, the surface coverage, which is a function of channel width and post diameter: the critical separation for the onset of deviations corresponding to the smallest surface coverage (23%) lies between that for 43% and 75% surface coverage (Figure 4 and Table 1). The same can be said of the other parameters in the problem. Therefore, if the characteristic length scale for onset of deviations is related to the structural features, it must be related to some combination of channel depth, width, and post diameter. We use a scaling analysis proposed by Persson,12,13 but modified for our system specifications, to find a relationship

p dh ∼ − 2 h3 dt ηDo

(3)

In the case of a network of channels, we consider the limit where all of the fluid squeezed out of the nominal contact region of diameter Do between two smooth rigid bodies is assumed to go through the channels and be expelled from the periphery of the network of channels such that fluid mass conservation leads to πD2o(dh/dt) ∼ −(πDo/D1)Dwv″. Here, (πDo/D1) is the number of open channels at the periphery, D1 is the channel periodicity (defined as D1 = w + d), d is the post diameter, and w and D are the channel width and depth, respectively (therefore, Dw is the area of the channel orthogonal to the periphery). The order-of-magnitude scaling obtained from Navier−Stokes equations relating the fluid 14707

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Consequently, during drainage, no influence of channels should be experienced for h ≫ ho, and channels would influence the fluid flow for h ∼ ho (see Table 1 for the calculated ho for the four structured surfaces investigated in this study). We note that the original analysis proposed by Persson was instead for a system where channel depth is larger as compared to channel width.12,13 Similar conclusions could be drawn by treating the structured surface as an effective porous media. The presence of draining channels would, therefore, lead to three fluid flow regimes as illustrated in Figure 5. (a) Far-field: The fluid squeezed out is expelled radially outward from the contact region between two macroscopic surfaces as if there are no channels present, and the resistance to flow agrees with the hydrodynamic drainage forces given by Reynolds’ theory for smooth surfaces for h ≫ ho. (b) Intermediate-field: The resistance to flow is dominated by the fluid flow through the network of channels such that the fluid squeezed out is expelled preferentially from the periphery of the network of channels for h ∼ ho. (c) Near-field: The fluid squeezed out is expelled radially outward from the contact region between each of the isolated posts and the contacting surface, and the resistance to flow is dominated by the hydrodynamic interactions of the individual posts for h ≪ ho. In the context of our measurements, we would expect that the transition from far to intermediate field will be gradual and would occur at a separation h > ho. This behavior is consistent with our experiments where we find that the onset of deviations from Reynolds’ theory for smooth surfaces occurs at h = hc > ho (Figures 3, 4, and 6). In our experiments, the near-field regime overlaps with other short-range effects (microscopic surface roughness, finite size of the fluid molecule) such that we cannot distinguish its contribution from the force measured. The nearfield regime would lead to a strong short-range force that would start to take effect at h ∼ 10 nm. As a guide, the force for this near-field regime can be seen in Figure S3a for the linear combination of hydrodynamic interactions for the total number of posts available either on the entire surface or within a region of interaction with radius r ≈ √2Rh ∼ μm. To verify if the scaling argument applies to our experimental system, we created two surfaces with completely different geometries (C and D) but with the same ho (Table 1). We then compared the force curves obtained with these two surfaces to see if they deviate from Reynolds’ theory for smooth surfaces at the same separation (hc). As seen in Figure 6, we indeed observe that for two completely different structured SU-8 surfaces that have been designed to have the same characteristic length scale, ho, the critical separation for onset of deviations, hc, is the same, implying that they are correlated. Other evidence indicating that the critical separation obtained for the deviations from Reynolds’ theory for smooth surfaces (hc) is due to a transition from the fluid being radially squeezed out of the nominal contact area to being squeezed out through the network of channels is the linear relationship between the experimentally obtained hc and the calculated ho, as shown in Figure 7. hc as a function of ho is a straight line with slope greater than 1 implying that hc > ho, which is consistent with the conclusions of the scaling argument that no influence of channels is experienced for h ≫ ho and channels influence the fluid flow for h ∼ ho. We note that the scaling analysis presented here was developed by Persson for two rigid

Figure 4. The hydrodynamic interactions for three different structured surfaces: (a) Surface A (v = −445 nm/s), (b) Surface B (v = −126 nm/s), and (c) Surface C (v = −220 nm/s). The critical separation for the onset of deviations, hc, is shown by the dashed line and is different for the three surfaces. The solid lines represent predictions from Reynolds’ theory for smooth surfaces, and the insets show data for smaller separation range.

pressure (p) and average fluid velocity (v″) in the channel is given by (p/Do) ∼ (ηv″/D2) (for channel depth small as compared to channel width, which is the case in our study), which, when combined with the fluid mass conversation above, would lead to the following: p wD3 p dh ∼− 2 ∼ − 2 ho3 dt ηDo D1 ηDo

(4)

This scaling, when compared to that for drainage for smooth surfaces (eq 3), leads to separation of length scales such that ho = D(w/D1)1/3. ho, therefore, refers to a characteristic length scale where fluid squeezed out from the contact region is expelled only from the periphery of the network of channels. 14708

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Figure 5. Three fluid flow regimes predicted from the scaling analysis: fluid being expelled (a) radially outward from the contact region in the absence of channels as indicated by solid arrows, (b) preferentially from the periphery of the network of interconnected channels, indicated by dashed arrows, and (c) radially outward from the contact region between each of the isolated posts and the contacting surface indicated by solid arrows.

Figure 7. Experimentally obtained hc as a function of calculated ho along with a linear correlation between the two.

separation for onset of deviations (Figure S3a). We have also considered correction factors to account for no-slip at shifted plane and finite slip at one or both of the solid−liquid interfaces. Shift distance and slip length are the fitting parameters in the models proposed by Vinogradova27,64 and are assumed to be of the order of channel depth (D). We observe that of the three corrections considered (Figure S3b− d), correction for no-slip applied to a plane located at some distance inside the channel (Figure S3b) seems to be the most promising, but must be considered in further detail. Needless to say, accurate prediction of the force curves for separations below hc requires a more rigorous analysis of the problem at hand, which is beyond the scope of this work, and will be addressed in future work. Our results can be directly extended to the tree frog system to understand the implications of having a network of interconnected channels on its toe pads because the characteristic length scale ho is found to be independent of drive velocity and fluid viscosity. For geometrical features of tree frog toe pad, the characteristic length scale is estimated to be of the order of 1 μm, implying that the influence of channels is long-ranged such that they enable tree frogs to come in close contact with flooded surfaces, thus facilitating their adhesion to and locomotion on flooded surfaces. Nonuniform pressure distribution in the contact region may result in elastohydrodynamic deformation35 of the soft epithelial cells in tree frog system. Elastohydrodynamic deformation would modify the surface profile and, consequently, alter the pressure distribution and the resulting hydrodynamic interactions,77−80 and must be

Figure 6. Two different structured SU-8 surfaces with channel depth, width, and post diameter such that the characteristic length scale, ho, is equal: (a) Surface C (v = −220 nm/s) and (b) Surface D (v = −244 nm/s). The two surfaces present the same critical separation for the onset of deviations hc. The solid lines correspond to the predictions from Reynolds’ theory for smooth surfaces, and the insets show data for smaller separation range.

bodies,12,13 but the general conclusions seem to be applicable for crossed-cylinder geometry where one of the cylinders is mounted on a cantilever. Alternate forms of eq 1 are employed to see if they would predict the force curves at separations below hc (see the Supporting Information for the data fitted with the alternate forms of eq 1). We first consider the case where each of the cylindrical posts is treated as an isolated circular disk for which the hydrodynamic repulsion is added linearly for the total number of posts available either on the entire surface or within a region of interaction with radius, r ≈ √2Rh ∼ μm, neither of which agrees with the measured force curve below the critical 14709

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accounted for in the understanding of locomotion mechanisms employed by tree frogs under flooded conditions. It has also been proposed that deformation of the epithelial cells in the toe pad may lead to channels being closed during retraction for adhesion.12,13

AUTHOR INFORMATION

Corresponding Author

*Tel.: (410) 516-0113. Fax: (410) 516-5510. E-mail: [email protected].

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Notes

The authors declare no competing financial interest.

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CONCLUSIONS The normal hydrodynamic interactions associated with the drainage of viscous silicone oil from the gap between specifically designed structured rigid surfaces and a hydrophobic silver surface were measured using the SFA and were analyzed within Reynolds’ continuum approach in the lubrication limit. The structured surfaces consisted of hexagonal arrays of micrometer-size cylindrical posts aimed to mimic the design found on tree frog toe pads. Our results demonstrate that in the absence of any contributions due to chemistry, deformation, and other surface forces, a network of interconnected channels results in significant reduction in hydrodynamic repulsion as compared to that predicted by Reynolds’ theory for smooth surfaces. Such a reduction in hydrodynamic repulsion has also been observed in force measurements performed with a microstructured surface (consisting of a square array of pillars) using AFM.81 For all of the structures investigated in our work, the measured hydrodynamic forces agreed with Reynolds’ theory for smooth surfaces at large separations, and deviations, characterized by a reduction in the hydrodynamic repulsion, were observed below critical separation, hc. For a given surface pattern, the critical separation for onset of deviations was observed to be independent of the drive velocity, which is consistent with fluid flow at low Reynolds’ number. Four different structured surfaces were investigated to probe and highlight the interplay between structural features (channel depth, width, and post diameter) and the critical separation for the onset of deviations. We found that a single structural feature was not sufficient to predict the critical separation for the onset of deviations from Reynolds’ theory for smooth surfaces, and instead a combination of channel depth, width, and post diameter was necessary. We employed a scaling analysis proposed by Persson, but modified for our system specifications, to establish a characteristic length scale, ho, where fluid squeezed out from the contact region is expelled only from the network of channels at the periphery such that no influence of channels should be experienced for h ≫ ho and channels would influence the fluid flow for h ∼ ho. We found that the experimentally obtained hc as a function of the calculated ho was a straight line with slope greater than 1, indicating that hc > ho and the critical separation obtained for the deviations from Reynolds’ theory for smooth surfaces (hc) was indeed due to a transition from the fluid being radially squeezed out of the nominal contact area to being squeezed out through the network of channels.

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ACKNOWLEDGMENTS This material is based upon work supported by the Office of Naval Research − Young Investigator Award (N000141110629) and by the National Science Foundation under Grant CMMI-0709187. Acknowledgment is also made to the Donors of the American Chemical Society Petroleum Research Fund for partial support of this research under Grant 51803-ND5. We would also like to thank Profs. German Drazer and Patricia McGuiggan for helpful discussions, and Mark Koontz for the scanning electron micrograph.

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REFERENCES

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ASSOCIATED CONTENT

* Supporting Information S

Hydrodynamic drainage forces for no-slip boundary condition applied at the bottom of the channel, estimation of the critical separation for onset of deviations, and approximations for the force curves below the critical separation for onset of deviations. This material is available free of charge via the Internet at http://pubs.acs.org. 14710

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