pubs.acs.org/Langmuir © 2009 American Chemical Society
Evidence of the No-Slip Boundary Condition of Water Flow between Hydrophilic Surfaces Using Atomic Force Microscopy Abdelhamid Maali,† Yuliang Wang,‡,§ and Bharat Bhushan*,‡ †
Centre de Physique Moleculaire Optique et Hertzienne, University Bordeaux I 351 cours de la Liberation, F-33405 Talence, France, ‡Nanoprobe Laboratory for Bio- & Nanotechnology and Biomimetics (NLB2) The Ohio State University, 201 West 19th Avenue, Columbus, Ohio 43210-1142, and §Mechanical Engineering, Harbin Institute of Technology, Harbin, 150001, P.R. China Received August 7, 2009. Revised Manuscript Received September 4, 2009 In this study we present measurements of the hydrodynamic force exerted on a glass sphere glued to an atomic force microscopy (AFM) cantilever approaching a mica surface in water. A large sphere was used to reduce the impact of the cantilever beam on the measurement. An AFM cantilever with large stiffness was used to accurately determine the actual contact position between the sphere and the sample surface. The measured hydrodynamic force with different approach velocities is in good agreement with the Taylor force calculated in the lubrication theory with the no-slip boundary conditions, which verifies that there is no boundary slip on the glass and mica surfaces. Moreover, a detailed procedure of how to subtract the electrostatic double-layer force is presented.
The no-slip boundary condition (BC)1,2 at a solid-liquid interface is at the center of our understanding of fluid mechanics. In the no-slip BC, the fluid velocity is assumed to be equal to the velocity of the surface. However, this assumption has been widely debated for hydrophobic surfaces. Fluid flow exhibits a phenomenon known as boundary slip, which means that the fluid velocity near the solid surface is not equal to the velocity of the solid surface.3-5 The degree of boundary slip is usually quantified by a slip length, which is defined as the ratio of the velocity to the velocity gradient in a direction orthogonal to the solid-liquid interface. The development of experimental devices and techniques make it possible to control systems at the micro/nanoscale and hence opens the way to investigate BC in micro/nanofluidics. The experimental results obtained using a surface force apparatus (SFA)6 and particle imaging velocimetry (PIV)7,8 techniques demonstrate an apparent slip of Newtonian liquids near hydrophobic solid surfaces. In addition to SFA and PIV techniques, contact mode atomic force microscopy (AFM) has also been used to study the BC in micro/nanofluidics. In contact AFM method, spheres with diameters ranging from 10 to 14 μm are glued to the end of AFM cantilevers.9-12 Then the spheres approach sample surfaces with a certain velocity, like in the *Corresponding author. E-mail:
[email protected]. (1) Batchelor, G. K. An Introduction to Fluid Dynamics; Cambridge University Press: Cambridge, England, 1970; p 149. (2) Stokes, S. G. G. Trans. Cambridge Philos. Soc. 1851, 9, 8. (3) Goldstein, S. Annu. Rev. Fluid Mech. 1969, 1, 1. (4) Lauga, E.; Brenner, M. P.; Stone, H. A. Handbook of Experimental Fluid Dynamics; American Chemical Society: New York, 2005. (5) Maali, A.; Bhushan, B. J. Phys.: Condens. Matter 2008, 20, art. no. 315201. (6) Cottin-Bizonne, C.; Cross, B.; Steinberger, A.; Charlaix, E. Phys. Rev. Lett. 2005, 94, art. no. 056102. (7) Tretheway, D. C.; Meinhart, C. D. Phys. Fluids 2002, 14, L9. (8) Lasne, D.; Maali, A.; Amarouchene, Y.; Cognet, L.; Lounis, B.; Kellay, H. Phys. Rev. Lett. 2008, 100, art. no. 214502. (9) Craig, V. S. J.; Neto, C.; Williams, D. R. M. Phys. Rev. Lett. 2001, 8705, art. no. 054504. (10) Bonaccurso, E.; Kappl, M.; Butt, H. J. Phys. Rev. Lett. 2002, 88, art. no. 076103. (11) Bonaccurso, E.; Butt, H. J.; Craig, V. S. J. Phys. Rev. Lett. 2003, 90, art. no. 144501. (12) Neto, C.; Craig, V. S. J.; Williams, D. R. M. Eur. Phys. J. E 2003, 12, S71. (13) Georges, J. M.; Millot, S.; Loubet, J. L.; Tonck, A. J. Chem. Phys. 1993, 98, 7345. (14) Chan, D. Y. C.; Horn, R. G. J. Chem. Phys. 1985, 83, 5311.
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drainage experiments.13,14 During this process, the cantilever deflection signal and separation distance between the spheres and sample surfaces are recorded and used to get the hydrodynamic force, which is then used to get the slip length. Early AFM experiments on BC study have reported boundary slip on hydrophilic surfaces.9-12 However, other experimental results using AFM,15-17 SFA,6 fluorescence correlation spectroscopy,18 and fluorescence cross-correlation19 show that there is no boundary slip on hydrophilic surfaces. The contradiction may be due to the processing of experimental data in the AFM measurement of boundary slip. Several factors should be taken into consideration: cantilever deflection impact on actual separation distance and squeezing velocity, impact of electrostatic doublelayer force10 between the sphere and solid surface, treatment of surface roughness during data processing,16 as well as the contribution of the cantilever beam to the hydrodynamic force.15 As mentioned earlier, although BC has been studied with AFM, contradiction exists as to whether there is boundary slip on hydrophilic surfaces. More efforts need to be made to investigate the actual BC on hydrophilic surfaces, which is both theoretically and practically important in micro/nanofluidics. In the BC experiments using AFM, the diameters of the spheres used in the AFM experiment are generally less than 20 μm, which is smaller than the width of the AFM cantilevers and is not good for hydrodynamic force measurement. Additionally, the stiffness of the AFM cantilevers is generally less than 0.2 N/m. In this case, it is difficult to tell at which point the spheres and sample surface make hard contact. When the hydrodynamic force is large, one cannot even get hard contact. Moreover, electrostatic force is found to affect hydrodynamic force measurement. The detailed procedure of eliminating electrostatic force effect needs to be studied. In this study, the BC is studied between two hydrophilic surfaces with the contact AFM method, and the no-slip BC is (15) Vinogradova, O. I.; Yakubov, G. E. Langmuir 2003, 19, 1227. (16) Vinogradova, O. I.; Yakubov, G. E. Phys. Rev. E 2006, 73, art. no. 045302. (17) Honig, C. D. F.; Ducker, W. A. Phys. Rev. Lett. 2007, 98, art. no. 028305. (18) Joly, L.; Ybert, C.; Bocquet, L. Phys. Rev. Lett. 2006, 96, art. no. 046101. (19) Vinogradova, O. I.; Koynov, K.; Best, A.; Feuillebois, F. Phys. Rev. Lett. 2009, 102, art. no. 118302.
Published on Web 09/15/2009
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verified. In order to minimize the influence of liquid squeezing by the body of the cantilever beam on measurements, a sphere with larger radius is used to increase the distance between the cantilever beam and the sample surfaces.20 Additionally, larger spheres are expected to increase the useful signal during experiments and improve the signal-to-noise ratio. In contrast to the previous AFM measurements,9-12 here we use a stiffer cantilever in order to get a accurate hard contact position between the surfaces. Moreover, the procedure of how to eliminate the impact factors, especially the electrostatic double-layer force, on the boundary slip measurement will be demonstrated in detail. When a sphere approaches a sample surface, liquid between the sphere and the surface will be squeezed out the gap, and the hydrodynamic force that opposes the approaching movement is applied on the sphere. Under the assumption of the no-slip BC, the hydrodynamic force exerted on the sphere with radius of R can be calculated using the expression known as the Taylor expression: Fsphere = (6πR2η/D)(dD/dt), where η is the viscosity of the liquid, D is the closest separation distance between the sphere and the solid surface, and dD/dt is the approach velocity of the sphere to the surface. For the case where there is a slip length b on the sphere and on the sample surface, the expression of the hydrodynamic force needs to be corrected, given as 21 Fsphere ¼
6πR2 η dD f ðDÞ D dt
ð1Þ
where the correction factor f *(D) is given as D f ðDÞ ¼ 3b
"
# D 6b 1þ ln 1 þ -1 6b D
ð2Þ
Note here that the Vinogradova correction factor f *(D) assumes a constant slip length b that does not depend on the distance and shear rate. Using eqs 1 and 2, slip length b can be obtained by fitting the hydrodynamic force Fsphere obtained in the squeezing experiment. In this study, the contact AFM measurement of boundary slip was performed using a Multimode III AFM (Veeco) with a probe with a sphere of large radius attached to it. A soda lime glass sphere (9040, Duke Scientific Corp., Palo Alto, CA) with a diameter of 42.4 ( 0.8 μm was glued to the end of a silicon nitride rectangular cantilever (ORC8, Veeco) using epoxy (Araldite, Bostik, Coubert, France). The stiffness of the cantilever was calibrated as 1.5 ( 0.1 N/m via the thermal noise method after the sphere was glued at the end of the cantilever.22 Soda lime glass is hydrophilic with a contact angle of 20.3 ( 1.5° measured on a soda lime glass plate with the sessile drop method. The mica surface is also hydrophilic with a contact angle of about zero. The liquid used in this study is deionized (DI) water. The sample surface approaches the sphere by a piezotube (PZT) with velocity varying from 0.4 to 56 μm/s. After the deflection signal as a function of PZT displacement signal is obtained in squeezing experiments, the actual separation distance between the sphere and the mica surface is obtained by adding the deflection signal to the PZT displacement signal. After that, the actual squeezing velocity v is then obtained from the time (20) With the cantilever and the sphere that we have used in our experiments, the ratio between the beam contribution to the sphere contribution is calculated using the Vinogradova calculation ( Vinogradova, O. I.; Yakubov, G. E. Langmuir 2003, 19, 1227), and it is found that it is smaller than 0.001 on the whole range of distance studied in our experiments. (21) Vinogradova, O. I. Langmuir 1995, 11, 2213. (22) Matei, G. A.; Thoreson, E. J.; Pratt, J. R.; Newell, D. B.; Burnham, N. A. Rev. Sci. Instrum. 2006, 77, art. no. 083703.
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Figure 1. The measured cantilever deflection signal as a function of separation distance at approach velocities of 0.4 and 0.8 μm/s, calculated electrostatic double-layer component, and pure hydrodynamic components. The electrostatic component is obtained with eq 6, and the pure hydrodynamic components for 0.4 and 0.8 μm/s are calculated with eqs 3 and 4, respectively.
derivative of the actual sphere-mica surface separation distance. As stated earlier, the electrostatic force should be taken into consideration in the boundary slip study. The electrostatic force is believed to be independent of approach velocity. In order to get the electrostatic force and minimize the hydrodynamic force so that the force measured is dominated by the electrostatic force, two very low approach velocities, 0.4 and 0.8 μm/s, were first applied. Then experiments at larger velocities of 28 and 56 μm/s are performed, at which hydrodynamic forces are dominating to demonstrate that there is no slip on the mica surface. The cantilever deflection signal obtained from velocities of 0.4 and 0.8 μm/s can be given as: S1 ¼ Shydro1 þ Selec
ð3Þ
S2 ¼ Shydro2 þ Selec
ð4Þ
where Selec is the cantilever deflection signal generated by electrostatic double layer force, S1 and S2 are the total cantilever deflection signals obtained with 0.4 and 0.8 μm/s approach velocity, respectively, and Shydro_1, Shydro_2 are the cantilever deflection signals generated by pure hydrodynamic force with 0.4 μm/s and 0.8 μm/s velocity. Hence, there is the following relationship for hydrodynamic components: Shydro2 ¼ 2Shydro1
ð5Þ
By combining eqs 3, 4, and 5, the electrostatic double-layer component can be subtracted through the following expression: Selec ¼ 2S1 - S2
ð6Þ
The cantilever deflection as a function of separation distance, as well as electrostatic force obtained with eq 6 and hydrodynamic components for both 0.4 and 0.8 μm/s are shown in Figure 1. The hydrodynamic components are obtained through eqs 3 and 4. One can see that, because the velocities are very low, the hydrodynamic forces for 0.4 and 0.8 μm/s are very small compared with the electrostatic component, which means the measured DOI: 10.1021/la902934j
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Figure 3. Velocity/hydrodynamic force as a function of separation distance at approach velocities of 28 and 56 μm/s. The inset shows the boxed data, which is fitted at a separation distance less than 100 nm.
Figure 2. The measured cantilever deflection signal as a function of separation distance of both approach and retract movement at approach velocities of (a) 28 and (b) 56 μm/s, and corresponding signals after the electrostatic double-layer components are subtracted. The insets show the actual approach velocity obtained from the time derivative of the actual sphere-mica surface separation distance.
deflection signal is dominated by electrostatic double-layer force when the approach velocity is low. Additionally, the hydrodynamic component for the velocity of 0.8 μm/s is approximately two times that of 0.4 μm/s. Once the electrostatic force is determined, approach velocities of 28 and 56 μm/s are applied to study boundary slip on the mica surface. The deflection signal as a function of separation distance for both approach and retract movement and the corresponding signal after subtraction of electrostatic force for approach velocities of 28 and 56 μm/s is shown in Figure 2. Insets show the actual approach velocity from the time derivative of the actual sphere-mica surface separation distance. The components of cantilever deflection generated by Stokes friction force are obtained when the cantilever is far from the mica surface. After the electrostatic double-layer and Stokes friction components are subtracted from the cantilever deflection signal, the hydrodynamic force is obtained for both approach velocities of 28 and 56 μm/s by multiplying the deflection signal by the cantilever stiffness. To better distinguish the no-slip BC case from the slip one, it is better to plot v/Fsphere versus the distance. If there 12004 DOI: 10.1021/la902934j
is no slip at the interface, the signal will be a linear line that intercepts the distance axis at the origin (v/Fsphere = D/6πR2η). If there is a slip, the signal can be extrapolated by a line that has the same slope of the line corresponding to the no-slip BC, but shifted by a distance equal to the slip length:6,17 v/Fsphere = (D/6πR2η)(1/f *(D)) ≈ (D þ b)/6πR2η. The data of v/Fsphere versus the distance are shown in Figure 3. One can see that the data for different approach velocities is a line having the same slope and intercept the distance axis at a zero value. The inset shows the data and the fitting using eq 1 and 2 for separation distance less than 100 nm for both 28 and 56 μm/s velocities. Slip lengths of 1.6 ( 1.5 and 0.7 ( 1.0 nm are obtained for approach velocities of 28 and 56 μm/s, respectively. The very small value of the slip length, which is within the error bars, is in contradiction with the previous measurements.9-12 The increasing shear force due to the reduced separation distance is in good agreement with the Taylor expression (eq 1 with f * = 1), which is derived from the lubrication theory with the no-slip BC. The discrepancy between these experimental results of no-slip BC and BC with slippage reported in the literature exist probably because of experimental setup and data processing. According to our experience, we find that, when hydrodynamic force is large (corresponding to a larger approach velocity than that used in this paper), it is difficult to determine the actual contact position through the cantilever deflection signal. It may lead to error in determination of the actual separation and actual approach velocity. Additionally, reasonable deformation of the cantilever is required to make sure the cantilever works at linear regime. Therefore, stiffer cantilevers are desirable even with sacrifice of resolution in force. Moreover, improper treatment of electrostatic force will also lead to error. For example, if cantilever deflection signal with low approach velocity is taken as the electrostatic component and the hydrodynamic component is ignored, the obtained hydrodynamic component at high velocity will be lower than the actual one, which will lead to an increase of measured boundary slip. Moreover, the experimental results presented in this paper show that, for the different velocities, the data are superimposed as expected, which is in contradiction with the shear ratedependent boundary slip detected on hydrophilic surfaces reported by Craig et al. 9 and in agreement with experimental results Langmuir 2009, 25(20), 12002–12005
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reported by Vinogradova and Yakubov 16 and by Cottin-Bizonne et al.6 As mentioned by Vinogradova and Yakubov,16 errors in the experimental determination of dD/dt in eq 1 are probably the reason for the shear-dependent boundary slip. In conclusion, we have presented contact AFM measurements of the hydrodynamic drag force exerting on a glass sphere approaching a mica surface in DI water. A large sphere was used
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to reduce the impact of the cantilever beam on the measurement. An AFM cantilever with large stiffness was used to accurately determine the actual contact position between the sphere and the sample surface. The measured hydrodynamic force with different approach velocities is in good agreement with the Taylor force calculated in the lubrication theory with the no-slip BC, which verifies there is no boundary slip on glass and mica surfaces.
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