No-Slip Boundary Condition for Weak Solid−Liquid ... - ACS Publications

ARTICLE pubs.acs.org/JPCC

No-Slip Boundary Condition for Weak Solid-Liquid Interactions Adam P. Bowles, Christopher D. F. Honig, and William A. Ducker* Department of Chemical Engineering, Virginia Tech, Blacksburg Virginia 24061, United States

bS Supporting Information ABSTRACT: Lubrication forces between hydrophobized glass surfaces in alkane liquids have been measured using the atomic force microscopy colloid probe technique and compared to Reynolds lubrication theory with the aim of understanding the nature of the solid-liquid boundary condition for flow. Forces in n-hexadecane are consistent with the no-slip boundary condition. Forces in n-pentane and n-hexane are better fit when a small slip length of about 5-20 nm is included in the model. The error estimating slip length is about 10 nm, which is much greater than the error in measuring the deflection of an AFM cantilever. We rationalize the finite slip length for n-pentane and n-hexane using an activation model for shear-driven molecular motion, which predicts that only molecules with an activation energy less than about 5 kT will have their diffusion affected by applied shear. The difference in slip length between short (pentane) and long (hexadecane) molecules is then explained by the fact that the short alkanes experience a weaker attraction to the solid.

’ INTRODUCTION The Navier-Stokes equations successfully describe the laminar flow of liquids using the no-slip boundary condition at the solid-liquid interface. The no-slip boundary condition states that the velocity of the liquid immediately adjacent to the solid is the same as the velocity of the solid. Recent interest in nanotechnology and an improved ability to study flow in very narrow channels (micrometer to nanometer) has stimulated interest in whether the no-slip boundary condition is an exact property of solid-liquid interfaces or simply an approximation that is “good enough” under macroscopic conditions but fails in thin films. The Navier-Stokes equations are a continuum approach that is inappropriate for films less than a few liquid diameters in thickness. For example, when a single layer of “liquid” molecules separates two solids and one solid translates relative to the other, the liquid molecules cannot remain stationary relative to both solids, and so the concept of slip or no-slip is inapplicable. Thus, the interesting thickness regime for slip spans from several nanometers (beyond molecular) to several micrometers, where the film is thin enough that small deviations of the boundary condition could, in principle, be detected. It is customary in the literature to invoke the concept of a slip length, b, to quantify deviations from the no-slip boundary condition ∂vx ¼ 0Þ ð1Þ ¼b vðz x ∂z

velocity would be zero. A large body of this literature claims that the no-slip boundary condition is only an approximation that fails in thin films. This literature has been reviewed;1,2 here, we note two observations: (1) The experimental techniques, principally atomic force microscope (AFM) or surface forces apparatus (SFA) lubrication measurements and particle imaging velocimetry, are often used at the limits of significance or resolution. (2) The simulations of shear in confined channels that show slip lengths greater than a few liquid diameters are usually under extremely high strain rates. These rates are much greater than those examined by experiment. In this paper, we investigate the solid-liquid boundary condition via the AFM colloid probe technique,3,4 in which a sphere is driven against a plate while the force is measured.5-9 The velocity of the flow is not measured directly; the boundary condition is determined through comparison of the measured lubrication force to the calculated force with the slip length as a fitted parameter. Two methods of determining the lubrication force are employed: (1) measuring force directly from the deflection of a cantilever and (2) measuring the damping of thermal oscillations. Work in our laboratory has compared forces from AFM squeeze films to Reynolds lubrication theory for aqueous sucrose7 and liquid polydimethylsiloxane.9 The experiments show, within error, a no-slip boundary condition (slip length = 0 ( 2). At this point, we consider the best opportunity to potentially observe slip in a Newtonian liquid to be when the strength of interaction

Here, v is the velocity of the fluid, x is an axis parallel to the plane of the solid, and z is the direction normal to the solid. It is assumed that (∂νx)/(∂z) is a constant. In other words, the slip length is the extrapolated distance inside the solid where the

Received: November 5, 2010 Revised: March 18, 2011 Published: April 13, 2011

r 2011 American Chemical Society

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The Journal of Physical Chemistry C

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Table 1. Physical Properties of Alkanes at 1 atm10,11 fluid

viscosity (mPa s)

boiling point (°C)

n-pentane

0.22 at 25 °C

36

n-hexane

0.31 at 25 °C

69

0.25 at 55 °C n-hexadecane

3.03 at 25 °C

287

between the liquid and the solid is minimal. Therefore, we investigated the possibility of slip for short-chain alkanes interacting with hydrophobized glass. The solid and liquid interact primarily via weak van der Waals forces in this case. The hydrophobization also simplifies the experiment by hindering the formation of water films, which would complicate the analysis. The n-alkanes investigated are listed with their properties in Table 1.10,11 A similar experimental system has recently been investigated by McBride and Law.12 They conclude that the slip length increases with the viscosity of the liquid, but the measured changes in slip length are very small, and it is not clear they that are significant. Ruckenstein and Rajora 13 postulated that, in some cases, apparent slip measurements may be explained by the presence of a gas film between the liquid and the solid. The existence of such a film would not imply that slip occurs at the solid-liquid interface, but it would provide a mechanism to explain results in terms of a low-viscosity film at the surface. To investigate the possibility of a gas film, we examine lubrication forces in hexane far from and near its boiling point. Our aim is to determine the effect of low and high vapor pressure on the same fluid.

’ THEORY

Figure 1. Effect of activation energy on slip length using model of Ruckenstein and Rajora, calculated using eq 4. Measurable slip lengths are expected for systems with bonding energies less than about 5kbT. The calculation is shown for a reasonable range of surface-bonding site spacings. Other parameters used in this calculation are vo = 5.54 1014 s-1, T = 300 K, A = 1.617 10-19 m-2, and η = 2.24 10-4 Pa s.

in our experiments and is not constant. It decreases with separation between our sphere and plate and is also a function of z and radial position.14 We will use the maximum value of the shear stress at a separation of 5 nm to estimate the maximum value of the slip length. Even for the highest strain rates used in our experiments (∼2.8 107 s-1), Fδ , kT. In other words, the applied shear stress is a small perturbation compared to the thermal energy. This leads to a simplification of eq 2 that suggests the slip length is strain rate independent under these conditions δ2 Aηνo E exp b¼ kb T kb T

Ruckenstein’s Model for Slip. Ruckenstein and Rajora13

described a simple quantitative estimate of the slip length based on a model in which surface liquid molecules jump from one potential well to another. The purpose of our paper is to examine a condition where the Ruckenstein model would predict slip and see whether slip indeed occurs. The model considers a onedimensional solid with a periodic array of potential wells separated by barriers of energy, E. Thermal energy (kbT) allows the molecules to hop between wells. The authors then postulate that this diffusion may be biased by the application of a shear stress. For Ruckenstein and Rajora’s model, the slip length is given by δν0 E Fδ exp b¼ sinh ð2Þ kb T 2kb T γ_ where γ_ is the strain rate and νo is the vibrational frequency of the molecule. F is the force applied by the external bias, and δ is the distance between neighboring potential wells. (1/2)Fδ is approximately the energy to escape a well. This model predicts that a greater slip will be achieved by decreasing the binding energy of the liquid for the solid or by increasing the driving energy to hop from one position to another. The difficulty occurs in interpreting or estimating the various terms in eq 2. Following on previous work from our group, we calculate F from the shear stress at the solid, τ07 F ¼ τo A; τo ¼ ηγ_

ð3Þ

where A is the area of the molecule and η is the bulk dynamic viscosity. Unfortunately, the shear stress is not measured directly

ð4Þ

This model predicts a positive correlation between b and η, which is also the conclusion of McBride and Law, based on their experiments.12 Equation 4 suggests that the best condition to observe a finite slip length is at the minimum activation energy. One possibility is to employ a liquid composed of molecules with a diameter much larger than the roughness of the solid, so that the liquid can translate without falling into energetic minima on the solid, i.e., the potential surface would look smooth. However, the energy to create a vacancy in the liquid would be greater for a larger molecule. The other possibility, and the subject of this paper, is to examine weak interactions with the solid. The effects of the activation energy and the spacing between molecules according to eq 4 are shown in Figure 1 for reasonable parameters. Significant slip is expected for activation energies less than 5 kT, and the slip increases for greater bond site spacing. Use of the theory of Ruckenstein and Rajora is distinct from any attempt to correlate slip lengths with the liquid contact angle on the solid in air.15,16 The contact angle does depend on the liquid-solid interfacial tension but also depends on the liquid-vapor and the solid-vapor interfacial tensions. However, the contact angle may be used to estimated the work of adhesion of the liquid to the solid via the Young-Dupre equation and hence provide a rough approximation of E. The contact angle may be used to estimate the work of adhesion of the liquid to the solid via the Young-Dupre equation. Here, we examine flow adjacent to a solid where the contact angle is 8614

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ESD, near resonance is given by19 ESD ¼

Figure 2. Schematic of the experiment.

only 11° yet, the solid-liquid interfacial tension and the interactions between the solid and the liquid are also small. The contact angle is also not a direct measure of the activation energy to move from one site to another. Reynolds Lubrication Theory. Brenner calculated the lubrication force, F, acting on a sphere as it approaches normal to a flat plate for low Reynolds number, incompressible, Newtonian fluids using the no-slip boundary condition17 R 1 R þ ln F ¼ - 6πηRv þ 0:9713 ð5Þ h 5 h where R is the radius of the sphere, v is the sphere velocity, and h is the sphere-plate separation at the apex of the sphere. These parameters are defined in Figure 2. In the limit R . h, only the first term is important. A smaller force is obtained if slip occurs. Vinogradova calculated the force acting on a sphere for finite slip length (b on each solid) using the same assumptions as Brenner in the limit R > h18 R F ¼ - 6πηRv f ; h h h 6b 1þ f ¼ ln 1 þ -1 ð6Þ 3b 6b h for b 6¼ 0. Damped Thermal Oscillation Model. The force acting on an AFM cantilever is usually determined from its deflection using Hooke’s law, but it can also be determined from dynamic properties.19-22 Under ambient conditions the cantilever oscillates due to thermal excitation. These oscillations are commonly used to calibrate AFM cantilever spring constants23 but can also be used to determine the forces on a cantilever as a function of separation from a plate. Following Honig et al.,19 we measure spectral energy density and determine the damping from a fit to the data. We use an equation of motion m

d2 h dh þ DðhÞ þ k 3 d ¼ Ftherm þ FðhÞ dt 2 dt

ð7Þ

where m is the effective mass of the cantilever, t is the time, d is the deflection of the cantilever, Ftherm is the thermal driving term, and F(h) are any separation-dependent forces such as electrostatic or van der Waals forces, which in the current experiments are negligible. D(h) is the damping coefficient of the system, which contains the information about the lubrication force. To solve eq 7, we assume m, D(h), F(h), and k are frequency independent. We also assume that m and k are independent of separation. Under these conditions, the energy spectral density,

B 2 þ C 2 !2 f DðhÞ 1þ ð2πf Þ2 f0 k

ð8Þ

where f is frequency, fo is the resonant frequency of the cantilever at a specific separation, and the variables B and C are lumped parameters. In this fit B, C, fo, and D(h) are adjustable parameters. The damping coefficient is the sum of all damping; we subtract an estimate of the damping due to the cantilever and probe at large separation to obtain the lubrication force contribution to the damping, Dlub, which is given by the Vinogradova equation (eq 6) 1 v 1 h ¼ ¼ Dlub F 6πηR 2 f

ð9Þ

The slip length is identified by plotting the experimental values of 1/Dlub as a function of separation and fitting eq 9 with a single value of the slip length.

’ EXPERIMENTAL METHODS Instrumentation. The AFM employed to measure lubrication forces and to image the plates was an MFP-3D (Asylum Research) with a closed fluid cell (BioHeater Closed Fluid Cell). During experiments this fluid cell was flooded with n-pentane, n-hexane, or n-hexadecane (99þ% purity; Alfa Aesar) that was not degassed. The fluid cell materials were stainless steel, Viton, PEEK, and Teflon, which are reported to be compatible with n-alkanes.24 All fluid cell components were sonicated (VWR B2500A-DTH; VWR International) in ethanol (200 Proof; Decon Laboratories) and then DI water (Elix 3 followed by Synergy UV finishing; Millipore) for 20 min each prior to assembly and use. A Cypher AFM (Asylum Research) was utilized to image the colloidal particles employed as probes. Probe Fabrication and Characterization. The D-lever on an ORC8 cantilever chip (nominal k = 0.05 N/m; Veeco Metrology) was used for most experiments. The spring constant was determined by an in situ calibration of the force but was also checked by the thermal calibration procedure of Hutter and Bechhoefer.23 The in situ calibration was performed at high approach speed on n-hexadecane. The high speed and high viscosity gave a large magnitude of force. An in-situ calibration incorporates effects such as cantilever tilt, cantilever torque, and probe position, as described in McBride and Law.25 The lubrication measurements entailed ramping the cantilever toward a plate at a velocity of 333 μm/s. A plot of the inverse damping (1/D) versus separation, h, was generated from these experiments. We analyzed 500 nm e h e 1000 nm since at larger separations the inverse damping plot becomes increasingly noisy and at smaller separations the lubrication force is in principle affected by b. In this case, b ≈ 0, so smaller separations could have been used. The probe radius, R, can be accurately determined as described shortly. The spring constant was first measured by the thermal method. Then, to obtain a more precise estimate of the spring constant, we selected a range of spring constants near the measured value and calculated the best-fit viscosity by minimizing the sum of the squared residuals between the v/F data and the Vinogradova equation. We used the spring constant that provided the literature value of the viscosity. This value of spring 8615


The Journal of Physical Chemistry C constant also occurred at the minimum in the sum of the residuals squared, which shows that the literature value of the viscosity is also the best fit to the data. The fitted spring constant was within 20% of the value obtained by the thermal method. At large separations (h . b) the slope of the inverse damping curve is inversely proportional to η (see eq 9), so by fitting only at h . b, it is not necessary to allow for changes in b. After publication of work on optimizing the spring constant by McBride and Law paper,25 we also checked the spring constants by their method, i.e., we assumed the literature viscosity, and obtained the spring constant by minimzing the sum of the squared residuals between the measured force and the theoretical force. This gave the same optimal spring constant as our method. (An example of the residuals as a function of spring constant is shown in the Supporting Information.) We also examined whether the fitted k values that we obtained produced a flat residual between our v/F and Vinogradova theory and found that the residuals were approximately constant with separation. The average spring constant for the five cantilevers employed in this work is k = 0.07 ( 0.03 N/m as determined by the lubrication force. Another class of cantilever was used in one of the experiments: the C-lever of a MSCT cantilever chip (nominal k = 0.01 N/m; Veeco Metrology). The measured spring constant of this cantilever was k = 0.0125 N/m. Colloidal probes were mounted at the end of these cantilevers using a heat-setting epoxy (Epon Resin 1004F; Hexion Specialty Chemicals) and a custom-built mounting apparatus. The probes were borosilicate glass spheres supplied by Duke Scientific. The roughness of the spheres was determined by contact imaging of a cantilever-mounted sphere using a TGT01 grating (NT-MDT). Particles with asperities greater than 1 nm above the apex of the sphere were rejected as were spheres with rms roughness greater than 2 nm. The radius of each particle was measured by optical microscopy with an error of about 0.2 μm. The particle radii were all about 10 μm. Silanization Procedure. Following the probe mounting and quality control steps the cantilever chip was UV irradiated for 90 min using a UV/Ozone Pro Cleaner (BioForce Nanosciences). This acted to clean the surface and engender reactive sites for silanization. A monolayer of trimethylchlorosilane (99þ% purity, Aldrich Chemical) was vapor deposited by adding 1-2 mL of TMCS to a 1.5 L reaction vessel under ambient conditions. The cantilever was left in the quiescent chamber for a period of 2-3 h. After removal, the sphere again underwent imaging with a TGT01 grating to identify if the probe had been fouled. The same selection criteria for asperities and general roughness were used, resulting in rejection of additional probes. Plate Preparation and Characterization. The plates used in these experiments were FisherFinest Premium Cover Glass slips (Thickness No. 1; Fisher Scientific) coated with TMCS. AFM images taken prior to silanization suggest very uniform surfaces with rms roughnesses < 1 nm. The plate was exposed to a H2O/O2 plasma (Plasma Prep III cleaner, SPI Supplies) at ∼200 mTorr for 2-3 min. This step serves the same purpose as the UV-ozone step for the AFM cantilevers (cleaning and creating reactive sites). However, the plasma is more aggressive than the UV-ozone cleaner, and experience within our lab suggests that AFM cantilevers can be warped when plasma cleaned. Following plasma cleaning, substrates are placed within the silanization reaction chamber with the AFM cantilevers and removed at the same time. A sacrificial piece of substrate is used to assess the quality of the TMCS film via a contact angle study.

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Figure 3. AFM contact-mode image of a chlorotrimethylsilane (TMCS)-coated glass plate in air. The small islands (white spots) appear after TMCS coating and are independent of the fluid above the plate. They may indicate intermittent surface coverage by the TMCS.

These contact angle studies are performed on a FTA125 Contact Angle Analyzer (First Ten Angstroms) using DI water. In addition, the uniformity of the TMCS films are examined using contact and tapping AFM imaging under dry, wet, and rinsed conditions. Water contact angles on the TMCS-coated plates were (85 ( 2)° advancing and (74 ( 4)° receding (errors represent standard deviations). This agreed with the results found in the literature for a similar TMCS deposition that claimed uniform deposition.19,26 AFM images of TMCS-coated surfaces using contact and tapping imaging in both air and n-pentane reveal an rms roughness of 0.3 ( 0.1 nm. However, examination of these images reveals ubiquitous lumps (of ∼400 pm height, ∼30 nm diameter, and less than 50% surface coverage) that were not present before silanization. An example of these features is shown in Figure 3. It is unclear whether the coating is heterogeneous or if the features represent small lumps of TMCS on top of a uniform film. We do not attribute these lumps to adsorbed nanobubbles because they are present in air. The features are thin enough that surface roughening caused by them does not affect our ability to determine the zero of separation. The n-alkanes wet the TMCS-coated surfaces. Contact angles with n-pentane, n-hexane, and n-hexadecane as the wetting liquid on these TMCS films (enclosed in a small chamber so the effect of evaporation is minimized) reveal advancing contact angles of about 11° for each surface. Conventional Force Measurement Description. Standard force curve measurements were performed at ambient temperature (22-25 °C) over a separation range of 10 μm and at ramp speeds of 100-2000 μm/s. The probe velocity was calculated at all separations using the methodology of Honig and Ducker to generate the 1/Dlub plots for determination of the slip length,7,9 but experiments are sometimes described in terms of the nominal drive velocity, which varied during a run. Six to ten force curves were acquired at each drive velocity during an AFM experiment. Multiple experiments were performed for the different fluids using independent substrates, probes, fluid reservoirs, and cleaning steps for the surfaces to demonstrate repeatability. The AFM 8616


The Journal of Physical Chemistry C was operated with the closed loop control for the z-piezo active to reduce the acceleration and the concomitant ringing at high velocities. Forces were measured for a series of n-alkanes: n-pentane, n-hexane, and n-hexadecane, which provides a range of viscosity, boiling point, and intermolecular forces. A variety of piezo drive speeds are used, but only those at 333 μm/s are shown in the figures. For all conventional force curves, the color red will denote the approach branch and blue will indicate the retract portion. Analysis proceeds by first converting deflection to force and then extracting the lubrication force, Flub, from the total force, Ftotal. Low-velocity force curves are performed prior to every data set with a given cantilever for two reasons. The first reason is to obtain the detector voltage to cantilever deflection calibration. At high velocities the contact position becomes ambiguous due to the larger lubrication forces. To eliminate this ambiguity, the slope of the contact (or constant compliance) region4 is determined from the slow data and applied to the higher speed curves. The average of the approach and retract constant compliance was used to eliminate the effects of friction.27 Thus, the results “in contact” in the fast run may have a slight error in force due to a change in constant compliance, but this contact error is not relevant to our measurement of the lubrication force because friction will not occur when the solids are separated by a thin film. The second reason for the slow force curves is to quantify quasistatic forces.9 The main quasi-static force arises from static charge, which was removed by irradiation with a Staticmaster (SPI Supplies) device. Normally, other forces were negligible, but occasionally, we observed contamination (as evidenced by a sawtooth retract curve), in which case the experiment was abandoned. “Virtual deflection”, an apparent quasi-static force, is an artifact that occurs in the MFP-3D AFM that manifests as approximately linear force gradients (with respect to piezo position) superimposed upon both branches of AFM data. This gradient appears approximately constant from run to run. This effect is minimized by adjusting the laser and diode belts to relieve tension. To reduce the error associated with virtual deflection, data is only analyzed in regions where the hydrodynamic force is much larger than the virtual deflection. In a few cases, the virtual deflection is subtracted from the deflection data for demonstrative purposes. In addition to the quasi-static forces there remains a velocitydependent force due to drag on the cantilever and sphere as they move through the fluid far from the plate. This force can be differentiated from virtual deflection because the approach and retraction branches of the force curve exhibit approximately equal and opposite forces, even far from the plate. An estimate of the drag force was obtained by multiplying the velocity by the DFAR obtained with the thermal method (at >50 μm separation). This cantilever drag contribution is subtracted from the total force measured. In practice, cantilever drag is significant at large separations, but near contact it is less relevant. All data presented in the Results section already have DFAR subtracted. Parameter Selection for Conventional Force Curve Measurements. Before acquiring meaningful data using the conventional force curve technique, operating ranges for R, v, and k had to be identified. These parameters were usually selected to maximize the deflection signal from the photodiode detector since the lubrication forces for low molecular weight n-alkanes are small. From eq 6, we desire the largest available probe radius,

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Figure 4. Inverse damping coefficient (Dlub) as a function of separation for TMCS-coated glass in n-pentane measured by the thermal damping method. The circles indicate the experimental data, and the line is the best fit to the Vinogradova model, with a slip length of 22 nm. DFAR ≈ 3.8 10-7 m/Ns.

but in practice, this was limited to R = 15 μm because we could not find larger spheres that were sufficiently smooth. Also, interpretation of data for larger spheres is more complex because of an increasing torque arm on the cantilever.28 The spring constant was chosen as a compromise between the need for a weak cantilever, to obtain maximum deflection, and keeping the deflection small enough to remain in the linear portion of the photodiode response. The linear region corresponds very roughly to deflections between (200 nm. Velocity was maximized to increase the lubrication force with a limit of ∼2 mm/s. At these velocities, ringing of the cantilever occurred following the breaking of contact between the probe and the plate. Thermal Oscillation Measurement Description. The thermal experiments were performed immediately preceding or following the standard lubrication experiments. Force curves were performed at low speeds (20 nms-1) with high data collection rates (50 kHz), so thermal oscillations could be observed. A ramp size of 1.0 μm was used for these experiments.

’ RESULTS Damped Thermal Oscillation Experiments. As described earlier, two methods are employed to measure the lubrication force: the damping of a thermally driven cantilever during lowvelocity force curves (∼0.02 μm/s), and “conventional” measurement of the deflection during high-velocity force curves (100-2000 μm/s).7 The thermal oscillation technique is described first; it was used to measure the slip length associated with n-pentane films. We focused our attention on the range 200 nm < h < 1000 nm to avoid regions where damping is large and the ESD peak becomes difficult to identify. The raw data is analyzed using the following procedure. First, thermal oscillations are measured at large separations (>50 μm) to obtain the damping in the absence of lubrication effects, DFAR, and this was subtracted from measured D values to obtain Dlub. The probe was then ramped toward the sample as for a conventional force curve. The recorded data was binned, and the ESD was calculated for each bin (Δh ≈ 50 nm). D was fitted to each bin using eq 8 over the range 1-25 kHz. Binning introduces an error because the damping varies over the range of the bin. The best fit to the Vinogradova equation for the data in Figure 4 gave a slip length of 22 nm per surface, whereas the average 8617



Figure 5. Force curves with the cantilever drag subtracted and the separation calculated for pentane (grayscale, dashed line) and hexadecane (colored, solid line). The magnitude of the lubrication forces is greater in hexadecane because the viscosity is 13 times greater.

Figure 6. Inverse damping versus separation for pentane and hexadecane films. Circles represent approach data, and squares represent retract data. The data is fitted over 5-150 nm separation by the Vinogradova model. The best fits indicate a slip length of 7 nm for the pentane (grayscale, dashed line) and a slip length of -1 nm for the hexadecane (colored, solid line). (Inset) Same pentane data over a larger range, suggesting the slip length is small and analyzing the data on the range 5-150 nm is valid.

value from four different experiments (different sphere, cantilever, and plate) and 8 curves per measurement was 20 ( 9 nm per surface. Again, we assume the slip is equal on each solid. The 9 nm error includes 7 nm of random error in the determination of DFAR and 6 nm of random error due to measurements variation for Dlub. There is the possibility of an additional 3 nm systematic error in determining the zero of separation. The thermal noise drive is always applied to the colloid probe, so it sets the baseline for all other applied shears. To roughly estimate the extent of the thermally imposed shear, we examine the shear imposed by the thermal motion at the resonant frequency, where the deflection amplitude is maximum. As described above, the strain is a function of both the separation and the position in the film at each separation. At 250 nm separation, the maximum strain from this motion is about 1000 s-1. This is about the same as the shear rate imposed by a 25 μm s-1 drive in the conventional method below. The drive rate that we use during the thermal experiment is very slow, so the shear rate from the drive is only about 0.8 s-1.

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Conventional Force Curves. Examples of force-separation curves for n-pentane and n-hexadecane are shown in Figure 5. A repulsive force (positive) is seen on approach and an attractive force (negative) on retract. This hysteresis scales with velocity, and together they suggest a hydrodynamic force. Qualitatively, the force curves from all Newtonian liquids look similar, but greater hydrodynamic forces are measured in more viscous solutions. Figure 6 shows the same data as Figure 5 but plotted as v/F, or damping coefficient. A v/F plot will be linear, with the data passing though the origin when there is no slip (see eq 9). Figure 6 shows that for n-hexadecane, the v/F plot is nearly linear and passes through the origin, demonstrating that the slip length is approximately zero. The average slip length obtained from fits to the Vinogradova solution for 5-150 nm separation for each curve in several experiments is -1 ( 3, as shown in Table 2. Analysis of v/F plots for n-pentane and n-hexane are complicated by the fact that the lubrication force is so much smaller, and therefore, effects such as virtual deflection are a larger percentage of the total measured deflection. The Figure 6 inset shows a v/F plot for n-pentane; the best fit to the Vinogradova equation for data in the range 5-500 nm gives a slip length of 3 nm on approach and 8 nm on retract. This fit makes it clear that the slip length is not large, but it includes much data (150-500 nm) where virtual deflection and any residual far field viscous forces may be a large fraction of the total signal. Having established that the slip length is small, we focus our fit on data in the range 5-150 nm, where the relative errors are smaller because the lubrication force is larger. The data from 296 force curves for both the approach and the retract branches is summarized in Table 2; the errors are 95% confidence intervals for each liquid. While the hexadecane showed no slip length, the n-pentane and n-hexane show a small slip length. The slip lengths in n-pentane are in the range 3-15 nm, depending on the velocity. The random errors (1-8 nm) are shown in the table. There is an additional (3 nm of systematic error. Thus, the thermal and conventional methods give similar results. We also do not resolve any difference in slip length as a function of temperature for hexane (Table 2). The difference in slip length between the short and the long alkyl chain is small and only just resolved, so before concluding that there is a finite fluid velocity at the pentane-solid interface, it is necessary to examine alternate hypotheses. The first is that the difference is caused by surface roughness and that the slip lengths simply represent fluid flow between asperities when the asperities are in contact. This conclusion does not account for the difference between pentane and hexadecane because measurements for all liquids are on the same plates. An alternative is the slip length, caused by errors measuring zero separation. Accurate determination of probe-plate contact is difficult, and inaccuracies can arise from several sources. First, noise in the raw data complicates determination of contact and can result in an error, of up to 3 nm on any individual force measurement (in these experiments). However, as noise is random the effect on the average slip length should be small. Second, poor calibration could artificially increase or decrease the slip length of our system.9 Here, we have not used data at high deflections where the deflection detector becomes nonlinear. In any case, we observe greater slip lengths for smaller lubrication forces, so the calibration does not explain the slip length. The third alternative hypothesis is that the slip length is due to the artifact of virtual deflection. To check this, we made a correction for virtual deflection. We calibrated the virtual deflection 8618



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Table 2. Summary of Slip Length Measurements Using Conventional AFM Methoda n-pentane (nm)

n-hexane, 25 °C (nm)

100

3(1

10 ( 3

9(3

0(2

333

7(3

7(4

11 ( 3

-1 ( 3

1000

15 ( 8

12 ( 3

13 ( 3

-5 ( 3

2000

15 ( 4

velocity (μm/s)

a

n-hexane, 55 °C (nm)

n-hexadecane (nm)

Fits to data in range 5-150 nm separation.

Figure 7. Effect of the virtual deflection. Circles represent approach data, and squares represent retract data. The open icons denote data without the correction, while the filled icons indicate data that has a virtual deflection correction applied. The correction narrows the gap between the approach and retract curves and makes the curves more similar and more linear.

Table 3. Effect of Virtual Deflection Correction on Slip Length in n-Pentane Fits to Data in the Range of 5-150 nm Separation velocity (μm/s)

no VD correction

VD correction

100

3(1

6(1

333

7(3

9( 1

1000

15 ( 8

18 ( 8

2000

15 ( 4

13 ( 3

on a slow run and subtracted the effect from the high velocity runs, assuming it was only a function of the piezo drive position. The virtual deflection correction is approximately linear with drive position, so a linear correction was made. The virtual deflection correction is shown in Figure 7. The correction looks significant; it brings the approach and retract portions closer together, which is more physically reasonable than the data without the correction. However, as shown in Table 3, the virtual deflection correction makes little difference to the fitted slip lengths for n-pentane. The slip length for n-hexadecane is more weakly affected by the virtual deflection correction because the forces are so much larger. Thus, we conclude that there is a small but finite slip length in n-pentane and that the slip length in hexadecane is significantly smaller (zero). Dependence of Slip Length on Strain Rate. In the range of strain rates investigated (up to about 3 107 s-1 in parts of the flow), we do not find that the magnitude of slip depends on the strain rate. Table 2 appears to show a slight variation in slip length with strain rate for pentane at low strain rates, but because of the low viscosity, the forces are weak and the error is large at low

strain rate, so this result is not significant. Molecular dynamics simulations show a strain rate dependence of slip but only at strain rates that are greater than currently measurable by AFM with films that are greater than a few nanometers in thickness.29-31 Therefore, our work helps to bring experiment into line with simulation. Using the Ruckenstein and Rajora model and our rough estimates for the parameters in the system, the strain rate should be around 1011 s-1 before a strain rate-dependent slip is measurable in n-pentane, so the strain rates in our experiments are about 104 times too low. Presence of Vapor. Apparent slip has also been hypothesized to arise from the existence of nanobubbles adsorbed to the solid (the gas gap). The existence of a continuous gas phase at the interface would insert a low-viscosity, η2, phase between the solid and liquid, producing a slip length η1 b¼t -1 ð11Þ η2 where t is the thickness of the gas film and η1 is the viscosity of the liquid. An adsorbed gas layer of air (η ≈ 0.0185 mPa s) would need to be 0.9 nm thick to yield b = 10 nm, as observed in some of our experiments. The presence of nanobubbles and films is well documented,32-36 but these are discontinuous films with a low coverage. It is not clear that a sparse, discontinuous film will have much effect if the liquid is constrained by a no-slip boundary condition over the rest of the surface. In the experiments on n-hexane described here, we investigated the effect of changing the temperature of n-hexane from 44° to 14° below the normal boiling point. Through this change in temperature, the vapor pressure changes from ∼150 to ∼490 mmHg.10,11 By increasing the vapor pressure, we expected to increase the opportunity to form an n-alkane gas film at the solid-liquid interface. However, we did not observe a significant change in the slip length with this change in vapor pressure, so we are unable to provide evidence for the gas gap. In addition, no evidence of gas adsorption was obtained from tapping or contact AFM imaging. Effect of Activation Barrier. The primary purpose of this paper is to investigate whether a change in the activation barrier energy between sites for liquid molecules on the solid affects the slip length. The Ruckenstein and Rajora model predicts there will be an inverse exponential dependence on the barrier energy. Qualitatively, our experiments show an increase in slip length for a smaller molecule (n-pentane) over a larger molecule (nhexadecane) and smaller molecules of the same type do experience weaker attractive forces to the solid, which is consistent with the model. From this point, we make a very approximate quantitative estimate of parameters in the theory to estimate the predicted magnitude of the slip length. First, using eq A12 in Chan and Horn,14 we determine that the maximum shear rate at the highest approach velocity in our experiments is 3 107 s-1. 8619



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Then we estimate the “work of shear” per molecule following previous work from our group7 Fδ ¼ ηγ_ max Aδ 2

ð12Þ

We estimate a bonding site spacing of δ = 0.25 nm and an alkane footprint A = 1.617 10-19 m2,37 which gives Fδ/2 ≈ 2.58 10-25 J, justifying the approximation in eq 4 that the work , kT . We estimate ν0 from the relaxation frequency of n-pentane dipoles in the IR of 5.540 1014 s-1.38 The most difficult estimate is the activation energy. The activation energy must be smaller than the adhesion energy for pentane to the silica in vapor, which is 1.08 10-20 J for crystalline silica (obtained from inverse gas chromatography experiments).39 Using these parameters gives a predicted slip length of 23 nm for the n-pentane system. The solid-liquid adhesion energy could also be estimated from the Young-Dupre equation.40 Since the activation energy is lower than the adhesion energy, the actual slip length should be longer that the above prediction. In view of the very rough approximations used for parameters, the “agreement” with the measured slip length of 5-20 nm is reasonable. It is also important to note that the Ruckenstein model predicts a measurable slip length even for activation energies that are greater than kT. To estimate the slip length of hexadecane with the Ruckenstein and Rajora model, we use the following parameters η = 3.03 mPa s,10,11 A = 4.494 10-19 m2,37 and E = 5.04 10-20 J (by extrapolation from Papirer39) to obtain an estimated slip length of 0.1 nm for hexadecane. This is too small to measure, which is consistent with observation.

’ CONCLUSION The Navier-Stokes boundary condition was investigated for thin films of three alkanes using AFM. We observe very small slip lengths (5-20 nm) for n-pentane and n-hexane and zero slip length for n-hexadecane. As expected from the fluctuationdissipation theorem, analysis of the thermal fluctuations produced the same result (within error) as analysis of the conventional force curves are near the resolution of the technique because the error in measuring the slip length in current AFM colloid-probe lubrication experiments is much larger (∼8 nm random error and ∼3 nm systematic error) than the error in measuring deflection (∼0.1 nm). These experiments are qualitatively consistent with the theory of Ruckenstein and Rajora, in which the mechanism for “partial slip” in liquids is that the applied strain increases the probability of a molecule to advance in the direction of the strain and decreases the probability to move against the strain. Because the effect of the applied strains is very small, it is necessary to have a very weakly bound molecule for significant “partial slip” to occur. Our experiments semiquantitatively agree with the theory’s prediction that the slip length becomes significant when the activation energy for motion of the liquid in the direction of shear is less than about 5 kT and that the slip length is independent of strain rate at the rates accessible in our experiments. The theory predicts an increase in slip length with viscosity, which is not observed, but this effect is likely obscured in the pentane/hexadecane comparison by the dominating effect of the activation energy for binding: both parameters change when n-hexadecane is substituted for n-pentane. We were not able to resolve a change in slip length with the vapor pressure of the liquid. Finally, it is important to

reiterate that all the slip lengths observed here are very small and have a large relative error.

’ ASSOCIATED CONTENT

bS

Supporting Information. The spring constants used for the calculation of data were fitted to obtain the literature value of the viscosity of the hexadecane (see Experimental Methods). The Supporting Information is a plot of the sum of squared residuals between the experimental lubrication force and the theoretical lubrication force as a function of spring constant. The spring constant in this instance was determined to be 0.064 Nm-1 from the viscosity calibration. This material is available free of charge via the Internet at http://pubs.acs.org.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT This work was funded by the National Science Foundation CBET-0828163. ’ REFERENCES (1) Neto, C.; Evans, D. R.; Bonaccurso, E.; Butt, H.-J.; Craig, V. S. J. Rep. Prog. Phys. 2005, 68, 2859–2897. (2) Lauga, E.; Brenner, M. P.; Stone, H. A. Handbook of Experimental Fluid Dynamics; Springer: New York, 2008. (3) Ducker, W. A.; Senden, T. J.; Pashley, R. M. Nature 1991, 353, 239–241. (4) Ducker, W. A.; Senden, T. J.; Pashley, R. M. Langmuir 1992, 8, 1831–1836. (5) Bonaccurso, E.; Kappl, M.; Butt, H.-J. Phys. Rev. Lett. 2002, 88, 076103-1–4. (6) Cottin-Bizonne, C.; Jurine, S.; Baudry, J.; Crassous, J.; Restagno, F.; Charlaix, E. Eur. Phys. J. E 2002, 9, 47–53. (7) Honig, C. D. F.; Ducker, W. A. J. Phys. Chem. C 2007, 111, 16300–16312. (8) Honig, C. D. F.; Ducker, W. A. Phys. Rev. Lett. 2007, 98, 028305-1–4. (9) Honig, C. D. F.; Ducker, W. A. J. Phys. Chem. C 2008, 112, 17324–17330. (10) Speight, J. G. Lange’s Handbook of Chemistry; McGraw-Hill: New York, 2005. (11) Lange, N. A.; Forker, G. M. Lange’s Handbook of Chemistry, 10th ed.; McGraw-Hill: New York, 1967. (12) McBride, S. P.; Law, B. M. Phys. Rev. E 2009, 80, 060601-1–4. (13) Ruckenstein, E.; Rajora, P. J. Colloid Interface Sci. 1983, 96, 488–491. (14) Chan, D. Y. C.; Horn, R. G. J. Chem. Phys. 1985, 83, 5311–5324. (15) Blake, T. D. Colloids Surf. 1990, 47, 135–145. (16) Cottin-Bizonne, C.; Steinberger, A.; Cross, B.; Raccurt, O.; Charlaix, E. Langmuir 2008, 24, 1165–1172. (17) Brenner, H. Chem. Eng. Sci. 1961, 16, 242–251. (18) Vinogradova, O. I. Langmuir 1995, 11, 2213–2220. (19) Honig, C. D. F.; Sader, J. E.; Mulvaney, P.; Ducker, W. A. Phys. Rev. E 2010, 81, 056305-1–11. (20) Maali, A.; Bhushan, B. Phys. Rev. E 2008, 78, 027302-1–4. (21) Ducker, W. A.; Cook, R. F. Appl. Phys. Lett. 1990, 56. (22) Ducker, W. A.; Cook, R. F. J. Appl. Phys. 1990, 67. (23) Hutter, J. L.; Bechhoefer, J. Rev. Sci. Instrum. 1993, 64, 1868–1873. 8620



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