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hydrophilic, hydrophobic, and superhydrophobic surfaces in de-ionized ... believed to be because the increasing hydrophobicity favors nanobubble formation.
Atomic force microscopy measurement of boundary slip on hydrophilic, hydrophobic, and superhydrophobic surfaces Yuliang Wang

Nanoprobe Laboratory for Bio- and Nanotechnology and Biomimetics (NLB2), The Ohio State University, 201 W. 19th Avenue, Columbus, Ohio 43210-1142 and School of Mechanical Engineering, Harbin Institute of Technology Harbin, 150001, People’s Republic of China

Bharat Bhushana兲

Nanoprobe Laboratory for Bio- and Nanotechnology and Biomimetics (NLB2), The Ohio State University, 201 W. 19th Avenue, Columbus, Ohio 43210-1142

Abdelhamid Maali Centre de Physique Moleculaire Optique et Hertzienne, University Bordeaux I, 351 cours de la Liberation, F-33405 Talence, France

共Received 5 November 2008; accepted 28 January 2009; published 29 June 2009兲 Reduction in drag is important in fluid flow applications. So called boundary slip, a measure of relative fluid velocity at the solid-fluid interface, affects the drag. The slip is a function of the degree of hydrophobicity. In this study, boundary slip was studied through slip length measurements on hydrophilic, hydrophobic, and superhydrophobic surfaces in de-ionized water with atomic force microscopy. On the hydrophilic surface, the experimental data are consistent with no-slip boundary conditions. However, boundary slip is observed on hydrophobic and superhydrophobic surfaces. Experimental results obtained with different squeezing velocities show that the slip length is independent of squeezing velocity. Moreover, the degree of boundary slip is observed to increase when the surface was changed from the hydrophobic surface to the superhydrophobic one. The increasing degree of boundary slip from a hydrophobic surface to a superhydrophobic surface is believed to be because the increasing hydrophobicity favors nanobubble formation. Nanobubbles with a diameter of about 150 nm and a height of about 6 nm were observed on the hydrophobic surface and were verified through observation of nanobubble coalescence. © 2009 American Vacuum Society. 关DOI: 10.1116/1.3086637兴

I. INTRODUCTION In fluid dynamics, it is generally assumed that the relative velocity between a solid wall and liquid is zero at the solidliquid interface, which is so called the no-slip boundary condition.1,2 However, this assumption has been widely debated for hydrophobic surfaces, and fluid film exhibits a phenomenon known as slip, which means that the fluid velocity near the solid surface is not equal to the velocity of the solid surface.3–7 It is generally recognized that there is no boundary slip on hydrophilic surfaces, while boundary slip occurs on hydrophobic surfaces. Boundary slip at the solid-liquid interface reduces drag in fluid flow, which is particularly important in various fluid flow applications, including micro-/nanofluidic based biosensors.8 The boundary slip is usually quantified by a slip length, which is defined as the ratio of the velocity to velocity gradient in a direction orthogonal to the interface. Early experimental results show that liquids may undergo slip when flowing over nonwettable surfaces.9,10 Further squeezing experiments with surface force apparatus11–13 共SFA兲 or atomic force microscopy14 共AFM兲 and particle image velocimetry15 techniques reported slip length on hydrophobic surfaces, and no slip was observed on hydrophilic surfaces.11,13–17 These a兲

Author to whom correspondence should be addressed; electronic mail: [email protected]

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experimental results are consistent with the molecular dynamic 共MD兲 simulation results performed by Barrat and Bocquet,18 who obtained 30 molecular diameters slip length on a surface of contact angle 140°, while no slip was observed on hydrophilic surfaces. However, boundary slip is also reported on hydrophilic surfaces in other experiments, such as SFA,19 AFM,20,21 PIV,22 and total internal reflectionfluorescence recovery after photobleaching method.23,24 These unexpected data have something to do with the sample preparation and data analysis. For example, electrostatic double-layer force and Stokes friction exist in experimental data and need to be subtracted. Nanobubbles are known to be present on hydrophobic surfaces.25–32 Nanobubbles can exist for several hours32,33 and are stable not only under ambient conditions but also under enormous reduction of the water pressure down to ⫺6 MPa.34 Under mechanical perturbation of the AFM tip, coalescences of nanobubbles can occur.29,31,33 By tracking the movement of nanobubbles and the consequent coalescence during scanning with tapping mode AFM 共TMAFM兲, Bhushan et al.31showed that the spherical features formed on hydrophobic surfaces are indeed nanobubbles. The presence of nanobubbles at the solid-liquid interface will decrease the solid-liquid interaction and hence will increase boundary slip as indicated by MD simulations35 and theoretical analyses.36–38

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©2009 American Vacuum Society

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Wang, Bhushan, and Maali: AFM measurement of boundary slip

Two factors are thought to be important in boundary slip study: surface wettability and surface roughness. To date, as reported earlier, the impact of surface wettability on boundary slip is generally explored by measuring slip length on hydrophilic and hydrophobic surfaces. Regarding surface roughness, it is found to impact boundary slip in different ways for hydrophilic and hydrophobic surfaces. If the liquid flows over a fully wetting surface, the roughness decreases.39,40 However, if the liquid partially wets the surface, roughness favors the formation of gas pockets, resulting in a large slip length.35,41 Although nanobubbles are thought to be one reason for boundary slip on hydrophobic surfaces, experimental evidence has not been reported. Additionally, boundary slip on superhydrophobic surfaces with AFM has not been studied. The impact of the surface topography of the hydrophobic and superhydrophobic surfaces in these experiments also needs to be studied. Moreover, the velocity dependence of slip length in a squeezing experiment with AFM is still not clear. In this research, the boundary slip studies are carried out on surfaces with different wettabilities through slip length measurement by performing squeezing experiments with AFM. The slip length is obtained on hydrophobic and superhydrophobic surfaces by taking the surface topography into consideration. The velocity dependence of slip length is explored by conducting squeezing experiments with different squeezing velocity. The effect of nanobubbles on boundary slip is also studied by detecting nanobubble existence on hydrophobic surfaces. II. EXPERIMENTAL The measurement technique is introduced first, followed by the methodology for calculation of hydrodynamic forces. The fabrication procedure for hydrophobic and superhydrophobic surfaces used for boundary slip study is presented next. A. Measurement of hydrodynamic force and nanobubble imaging

To measure hydrodynamic forces, squeezing experiments were performing using a Multimode III AFM 共Veeco兲 共Ref. 42兲 with a probe with a sphere of large radius attached to it. A soda lime glass sphere 共9040, Duke Sci. Corp., Palo Alto, CA兲 with a diameter of 41.7⫾ 0.7 ␮m was glued to the end of a silicon nitride rectangular cantilever 共ORC8, Veeco兲 using epoxy 共Araldite, Bostik, Coubert, France兲. The side view of the cantilever and glued sphere is shown in Fig. 1. The stiffness of the cantilever was calibrated as 0.16⫾ 0.02 N / m via thermal noise method.43 The thermal power spectrum of the cantilever was obtained using a lock-in amplifier 共model 7280, AMETEK Inc. Oak Ridge, TN兲. The diameter of the sphere is larger than that of earlier experiments20,21 and is beneficial to increase hydrodynamic forces and to minimize the effect of the cantilever itself on deflection signal response. The liquid used in this study is de-ionized 共DI兲 water because a more accurate value of the viscosity of the DI water is known than other aqueous solutions, especially since JVST A - Vacuum, Surfaces, and Films

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FIG. 1. Optical microscopy image of the side view of the probe obtained by gluing a 41.7 ␮m glass sphere at the end of a rectangular cantilever.

density continuously changes for other solutions, which may change the viscosity and introduce error. The value of viscosity for water used in this study is ␩ = 851.5 ␮Pa at 300K.44 In squeezing experiments, the sample surfaces are driven relative to the sphere by a piezotube 共PZT兲 with a constant velocity 共28 ␮m / s for slip length measurement and 56 ␮m / s for the velocity dependence study of slip length, referred to as low and high velocities, respectively兲. The cantilever deflection signal induced by hydrodynamic force and PZT displacement is obtained simultaneously, and these data are used for the calculation of hydrodynamic force. In addition, nanobubble images can be stably obtained in DI water on hydrophobic surfaces, which is useful to explore the nanobubble effect on boundary slip. As in our previous experiment,31 a modified tip holder45 was used to conduct liquid experiments. A horizontal slot was carved out in the opening of a commercially available tip holder for nonfluid use above the piezoelement in order to insert a glass slide. When the liquid is added between the glass slide and the substrate, the liquid meniscus is formed between the glass and sample surface for fluid imaging. TMAFM is used to image sample surfaces. An oscillating tip intermittently contacts the sample surface with much a lighter average force than contact mode AFM.42 Thus, the technique is widely used to study soft and fragile materials. A silicon cantilever rotated force-modulation etched silicon probe 共Digital Instruments兲 with a tip radius ⬍10 nm and stiffness of 3 N/m 共quoted by the manufacturer兲 was used to image surfaces both in air and DI water with TMAFM. The free vibration amplitudes at the resonance frequencies of 73 and 27 KHz are about 8 and 7 nm for air and water, respectively. During imaging in air and DI water, a setpoint of 95% is chosen to minimize the load applied on sample surfaces. The setpoint is defined as the ratio of vibration amplitude in contact to the vibration amplitude in liquid before the engagement. Corresponding average normal load during imaging can be obtained by multiplying cantilever stiffness to the average cantilever deflections, which are about 0.10 and 0.09 nN normal load for scanning in air and DI water, respectively. Additionally, the setpoint of 85% free amplitude is chosen to get nanobubble coalescence in DI water, which corresponds to about 0.26 nN normal load in DI water.

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FH = f ⴱ where fⴱ =

FIG. 2. 共Color online兲 Schematic of a sphere approaching and retracting relative to a surface and velocity profiles of fluid flow with and without boundary slip. The definition of slip length b characterizes the degree of boundary slip at solid-liquid interface. The arrows above and below the solid surface represent directions for fluid flow and relative movement of sample surface to the sphere, respectively.

B. Methodology for calculation of hydrodynamic forces

As shown in Fig. 2, when a sphere with radius R approaches or retracts relative to a solid plane surface, liquid between the sphere and the surface will be squeezed out or squeezed into the gap. Near the solid-liquid interface, the liquid velocity profiles will vary with different boundary conditions at the solid-liquid interface, as shown on the right side of Fig. 2. If there is no slip at the solid-liquid interface, the liquid velocity will gradually reduce to zero. Otherwise, there will be relative velocity between liquid flow and solid surface when boundary slip exists at the interface. Under the assumption of the no-slip boundary condition, the hydrodynamic force acting on the sphere is given as FH = 共6␲R2␩ / D兲共dD / dt兲, where ␩ is the viscosity of the liquid, D is the closest separation distance between the sphere and the solid surface, and dD / dt is the velocity of the sphere approaching to the surface. In order to take into account boundary slip at the interface, Vinogradova46 developed a relationship for hydrodynamic force. She applied the following boundary condition at solid-liquid interface:

vw,b = b

冏 冏 ⳵ vw ⳵z

,

共1兲

6␲R2␩ dD , D dt

共2兲

再 冋冉 冊 冉 冊 册冎

1 6D 1+ 4 4b

1+

4b D ln 1 + −1 4b D

.

共3兲

Note here that the Vinogradova correction f ⴱ assumes a constant slip length b that does not depend on the distance and shear rate. Using Eqs. 共2兲 and 共3兲, slip length b can be obtained by fitting the hydrodynamic force FH obtained in the squeezing experiment. For our case, the sphere used is hydrophilic, and there is no-slip length at the sphere-liquid interface, while there is slip length at solid-liquid interface for both hydrophobic and superhydrophobic surfaces. The use of Eqs. 共2兲 and 共3兲 is consistent with the combination of the sphere and sample surfaces in our experiments. In order to fit slip length b, nonlinear least-square fitting method is used to fit hydrodynamic force data with Eqs. 共2兲 and 共3兲. The fitting is conducted with a numerical computing software MATLAB 6.5. In Eqs. 共2兲 and 共3兲, the viscosity ␩ and radius R are known or can be measured. Separation distance D and velocity dD / dt can be obtained with experimental data. Therefore, the only parameter we need to fit is slip length b. Before fitting, one should do some processing of experimental data to eliminate the influence of several factors: cantilever deflection impact on actual separation distance and squeezing velocity, as well as impact of electrostatic doublelayer force between the sphere and solid surface and the Stokes friction component21 on cantilever deflection. In order to obtain actual separation distance, the cantilever vertical deflection was added to the PZT vertical displacement. The actual squeezing velocity is then obtained from the real sphere-surface separation distance by differentiating the real sphere-surface separation distance. To measure electrostatic double-layer force, the experiments were conducted at two very low velocities, 0.04 and 0.08 ␮m / s, in order to minimize the hydrodynamic force so that the force measured is dominated by the electrostatic force. The cantilever deflection signal obtained from two different velocities can be given as S1 = Shydro_1 + Selec ,

共4兲

S2 = Shydro_2 + Selec ,

共5兲

z=0

where vw and vw,b are liquid velocity in liquid and at the solid-liquid interface, respectively, and b is slip length. By solving the continuity equation and the Navier–Stokes equation 共also known as the momentum equation兲 of fluid flow in the gap, she came up with a modified model for hydrodynamic force for the case of slip on one of the two interfaces. Assuming there is slip length b at liquid and the solid plane interface, while there is no boundary slip at liquid-sphere interface, J. Vac. Sci. Technol. A, Vol. 27, No. 4, Jul/Aug 2009

where Selec is the cantilever deflection signal generated by electrostatic double-layer force, and S1 and S2 are the total cantilever deflection signals obtained with 0.04 and 0.08 ␮m / s velocities, respectively. Shydro_1 , Shydro_2 are the cantilever deflection signals generated by pure hydrodynamic force with 0.04 and 0.08 ␮m / s velocities. Hence, there is the following relationship for hydrodynamic components:

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Shydro_2 = 2Shydro_1 .

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共6兲

By combining Eqs. 共4兲–共6兲, the electrostatic double-layer component can be subtracted through the operation of Selec = 2S1 − S2. In order to measure the Stokes friction component, the sphere was retracted to be far away from the surface 共over 20 ␮m兲, where the cantilever deflection is independent of PZT displacement, and the cantilever deflection signal obtained can be regarded as generated by Stokes friction. After the actual separation distance is obtained, and the electrostatic double layer and Stokes friction components are subtracted from the total cantilever deflection signal, the component generated by the hydrodynamic force with respect to separation distance can be obtained. The hydrodynamic force can then be obtained by multiplying the obtained cantilever deflection by the cantilever stiffness. Processing of data to obtain hydrodynamic force was conducted using a commercial software KALEIDAGRAPH 3.5 共Synergy Software, Reading, PA兲.

FIG. 3. 共Color online兲 AFM images of hydrophilic, hydrophobic, and superhydrophobic surfaces in air with 95% setpoint of free amplitude, corresponding to about 0.1 nN normal force.

in Fig. 3. The rms roughnesses of hydrophilic, hydrophobic, and superhydrophobic surfaces are 0.2, 11, and 178 nm, respectively 共Table I兲. III. RESULTS AND DISCUSSION A. Slip length measurement and velocity dependence

C. Sample surfaces

Mica was taken as the hydrophilic surface. The contact angle measured on the surface is zero. One hydrophobic surface and one superhydrophobic surface are used in this study. Smooth epoxy substrates were first prepared using a liquid epoxy resin 共Epoxydharz L®, No. 236349, Conrad Electronics, Hirschau, Germany兲 and hardener 共Harter S, Nr 236365, Conrad Electronics, Hirschau, Germany兲 by a conventional molding method. The hydrophobic and superhydrophobic surfaces were created by self-assembly of alkane n-hexatriacontane 共CH3共CH2兲34CH3兲 共purity of ⱖ99.5%, Sigma-Aldrich, USA兲 and Lotus wax 共nonacosane-10,15diol and nonocosan 10-ol兲 deposited by thermal evaporation. The detailed process was described by Bhushan et al.47 and Koch et al.48 Here we briefly describe the fabrication process. The smooth epoxy substrates were placed in a vacuum chamber at 30 mTorr 共4 kPa pressure兲, 2 cm above a heating plate loaded with 500 ␮g of n-hexatriacontane and 2000 ␮g of Lotus wax. The n-hexatriacontane and Lotus wax were evaporated by heating them up to 120 ° C. After coating, the specimens with n-hexatriacontane were placed in a desiccator at room temperature for three days for crystallization of the alkanes to generate the platelet nanostructure.47 After that, the specimen was heated in an oven 共85 ° C, 3 min兲 and then immediately cooled down 共5 ° C兲 to interrupt the recrystallization process to generate the hydrophobic surface. A contact angle of 91⫾ 2.0° was measured on the hydrophobic surface. The specimens with Lotus wax were stored for seven days at 50 ° C in a crystallization chamber and exposed to a solvent 共20 ml of ethanol兲 in vapor phase. A tubule nanostructure was produced on the specimen surface.48 The specimens with Lotus wax had contact angles of 167⫾ 0.7°. After fabrication, the hydrophilic, hydrophobic, and superhydrophobic surfaces were scanned in air with TMAFM over 5 ⫻ 5 ␮m2 scan area at the setpoint of 95%, as shown JVST A - Vacuum, Surfaces, and Films

By conducting a squeezing experiment, cantilever deflection as a function of PZT vertical displacement on hydrophilic, hydrophobic, and superhydrophobic surfaces was obtained, as shown in Fig. 4共a兲. Then the experimental data were processed as mentioned in the Sec. II B. The hydrodynamic force obtained after processing is shown in Fig. 4共b兲. The hydrodynamic force gradually increases with decreasing separation distance for each surface. More importantly, the hydrodynamic force is smaller on the hydrophobic surface than on the hydrophilic surface. Similarly, the hydrodynamic force on the superhydrophobic surface is smaller than on the hydrophobic surface. The measured hydrodynamic force on the hydrophilic surface is fitted with the formula FH = 共6␲R2␩ / D兲共dD / dt兲, assuming there is no boundary slip on both the sphere and mica surfaces, as shown in Fig. 5共a兲. The experiment data fit well with a theoretical no-slip boundary condition, which verifies a no-slip boundary condition on the hydrophilic surface. Regarding hydrophobic and superhydrophobic surfaces, corresponding treatments were carried out before fitting the data to eliminate the influence of surface topography. As shown in Fig. 3, the hydrophobic and superhydrophobic sample surfaces are rough. Unlike the mica surface, the water among asperity structures on these surfaces will reduce hydrodynamic force when the sphere contacts the surfaces. Therefore, surface roughness must be taken into considerTABLE I. rms 共AFM, scan size= 5 ⫻ 5 ␮m2兲, contact angle, and slip length of three different surfaces. The variation represents ⫾1 standard deviation.

Surfaces

rms 共nm兲

Peak-to-mean distance 共nm兲

Contact angle 共deg兲

Slip length 共nm兲

Hydrophilic Hydrophobic surface Superhydrophobic surface

0.2 11 178

0.4 34 185

⬃0 91⫾ 2.0 167⫾ 0.7

⬃0 44⫾ 10 257⫾ 22

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FIG. 4. 共Color online兲 Comparison of 共a兲 measured cantilever deflection vs PZT displacement and 共b兲 calculated hydrodynamic forces as a function of separation distance on the hydrophilic, hydrophobic, and superhydrophobic surfaces with squeezing velocity of 28 ␮m / s.

ation when conducting squeezing experiments on a rough surface with AFM. In other words, the position at which the sphere contacts with the surfaces cannot be taken as a solidliquid interface. In this study, we take the mean surface as a virtual plane where the solid-liquid interface is located, called virtual solid-liquid interface. The average of peak-tomean distance for each scan line of the whole area shown in Fig. 3 is taken as the distance offset between the actual contact position and virtual solid-liquid interface. For the hydrophobic and superhydrophobic surfaces, 34 and 185 nm offset to the separation distance are separately obtained by calculation. After adding the relevant offset value to the corresponding separation distances, the measured hydrodynamic force data on hydrophobic and superhydrophobic surfaces are fitted with Eqs. 共2兲 and 共3兲. The fitted curves are plotted in Figs. 5共b兲 and 5共c兲 with solid curves for hydrophobic and superhydrophobic surfaces, respectively. About 44 and 257 nm slip lengths are obtained on hydrophobic and superhydrophobic surfaces, respectively, which implies the boundary slip increases with increasing hydrophobicity of solid surfaces. J. Vac. Sci. Technol. A, Vol. 27, No. 4, Jul/Aug 2009

FIG. 5. 共Color online兲 Hydrodynamic force at the squeezing velocity of 28 ␮m / s as a function of separation distance, and corresponding fitted curves on the 共a兲 hydrophilic, 共b兲 hydrophobic, and 共c兲 superhydrophobic surfaces. In the case of 共b兲 hydrophobic and 共c兲 superhydrophobic surfaces, experiments were also performed at the velocity 56 ␮m / s. Two times hydrodynamic force with squeezing velocity 28 ␮m / s is plotted to compare to that obtained with squeezing velocity 56 ␮m / s for each surface. Two curves agree with each other, which demonstrates the velocity independence of slip length measurement on both the hydrophobic and superhydrophobic surfaces in our study.

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FIG. 7. Schematic of a sphere-flat system with nanobubbles distributed on the flat surface. The presence of nanobubbles on the flat surface changes the velocity profile between the sphere and the plane surface, which results in an increase of slip length. The nanobubbles with a height Hb are approximated by a gas layer with effective thickness hb

FIG. 6. 共Color online兲 AFM images of hydrophilic and hydrophobic surfaces in DI water with 95% and 85% setpoint of free amplitude, corresponding to about 0.09 and 0.26 nN normal forces, respectively. Nanobubbles with typical diameter of 150 nm are observed on the hydrophobic surface. No change is observed on hydrophilic surface between high and low load scannings. However, nanobubble coalescence is observed on the hydrophobic surface after high load scanning.

Moreover, to explore the velocity dependence of slip length in squeezing experiments, a high squeezing velocity of 56 ␮m / s was applied on both hydrophobic and superhydrophobic surfaces, as shown in Figs. 5共b兲 and 5共c兲, respectively. In Figs. 5共b兲 and 5共c兲, Flow and Fhigh represent the hydrodynamic forces obtained with squeezing velocities of 28 and 56 ␮m / s for each surface. For each figure, two times Flow is plotted to compared to Fhigh. The two curves agree with each other well for each surface, which implies slip length is independent of squeezing velocity in a squeezing experiment with AFM, at least with a squeezing velocity up to 56 ␮m / s. B. Correlation of nanobubbles with slip length

As mentioned in Sec. I, nanobubbles are generally thought to be the reason for boundary slip at the solid-liquid interface. To check whether nanobubbles are present on the surfaces, the three surfaces were imaged with TMAFM in DI water. A featureless image was obtained on the mica surface with 95% setpoint of cantilever free oscillation, as shown in Fig. 6 共left兲. Then, a higher load 共85% setpoint兲 was applied to image the surface. No change was observed. However, for the hydrophobic surface, spherical objects were observed over the whole area, as shown in Fig. 6 共right兲. The diameter and height of the objects are about 150 and 6 nm, respectively. To verify the objects are nanobubbles, 85% setpoint of free oscillation amplitude was then applied. Similar to our JVST A - Vacuum, Surfaces, and Films

previous experiment,31 bigger nanobubbles with less density over the surface are observed at a lower setpoint. Here one can see that although the hydrophobic surfaces are rough we can distinguish nanobubbles from the roughness. However, on the superhydrophobic surface, we cannot get nanobubble images due to the high value of roughness. Tretheway and Meinhart37 calculated the slip length for fluid flow between two infinite parallel plates by modeling the presence of either a depleted water layer or nanobubbles as an effective air gap at the wall. They reported that the slip length increases with an increasing value of air gap thickness, assuming that air covers the wall continuously. For an intermittent surface coverage of nanobubbles, the slip length increases with increasing nanobubble height and surface fraction covered by nanobubbles. A schematic of nanobubble impact on boundary slip in squeezing experiments is shown in Fig. 7, where hb is an effective thickness of the air gap induced by nanobubbles. When a gas layer exists between a solid surface and a liquid, the slip length generated by the discontinuity of viscosity at the liquid-gas interface is given as46 b=

冉 冊

␩w − 1 hb . ␩a

共7兲

The viscosities for water and air at a temperature of 300 K ␩w = 851.5 ␮Pa s and ␩a = 18.6 ␮Pa s, are about respectively.44 Therefore, slip length b may be lead to about 45 times air gap thickness hb due to high value of ␩w / ␩a. Now let us compare the experimental observation with the theoretical analysis of Eq. 共7兲. To generate a slip length of 44 nm on the hydrophobic surface, approximately 1.0 nm thickness air gap hb is needed based on Eq. 共7兲. Because the nanobubbles actually discretely distributed over the sample surface, but do not act as a continuous air gap, here we roughly treat the effective air gap hb thickness as a function of nanobubble height Hb, given as h b = H b␾ ,

共8兲

where ␾ is the percentage of the sample surface covered by nanobubbles. By simply using 6 nm for nanobubble height

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Wang, Bhushan, and Maali: AFM measurement of boundary slip

on the hydrophobic surface, the corresponding ␾ needed on the hydrophobic surface is about 17%. For the nanobubble image on the hydrophobic surface shown in Fig. 6 共right兲, about 20% value of coverage is obtained, which is close to the value 17% calculated with Eq. 共8兲. Regarding the superhydrophobic surface, unfortunately, we cannot get a corresponding value for the superhydrophobic surface due to the high roughness as we mentioned earlier. To summarize, theoretical analysis shows that nanobubbles favor boundary slip. The existence of nanobubbles on the hydrophobic surface is verified by conducting TMAFM in water. The increasing slip length from hydrophobic to superhydrophobic surfaces is expected to be because of increasing hydrophobicity. IV. CONCLUSION Boundary slip studies were carried out through slip length measurement on hydrophilic, hydrophobic and superhydrophobic surfaces in DI water. On the hydrophilic surface, the hydrodynamic force is consistent with a no-slip boundary condition. However, about 44 and 257 nm slip lengths are obtained on the hydrophobic and superhydrophobic surfaces, of which the contact angles are about 91° and 167°, respectively. The increasing boundary slip from the hydrophobic to the superhydrophobic surface is believed to be because of increasing hydrophobicity, which favors the generation of nanobubbles. Additionally, high squeezing velocity experiments were conducted on both hydrophobic and superhydrophobic surfaces. The results show that the slip length is independent of squeezing velocity in squeezing experiments with AFM. By imaging the hydrophobic surface in DI water using TMAFM, spherical objects with a diameter of about 150 nm and height of about 6 nm were observed, which are confirmed to be nanobubbles through a higher load TMAFM image. The coverage of the nanobubbles on the hydrophobic surface is about 20% of the surface, which is close to the value 17% expected to generate 44 nm slip length according to a theoretical analysis. ACKNOWLEDGMENTS Yuliang Wang acknowledges financial support from Chinese Scholarship Council. The authors are grateful to Yong Chae Jung for the preparation of hydrophobic and superhydrophobic samples. S. G. G. Stokes, Trans. Cambridge Philos. Soc. 9, 8 共1851兲. G. K. Batchelor, An Introduction to Fluid Dynamics 共Cambridge University Press, Cambridge, England, 1970兲. 3 S. Goldstein, Modern Development in Fluid Dynamics 共Clarendon, Oxford, 1938兲. 4 S. Goldstein, Annu. Rev. Fluid Mech. 1, 1 共1969兲. 5 E. Lauga, M. P. Brenner, and H. A. Stone, Handbook of Experimental Fluid Dynamics 共Springer, New York, 2005兲. 1 2

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