Tribology Letters 7 (1999) 147–152

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Friction and slip of a simple liquid at a solid surface Remmelt Pit, Hubert Hervet and Liliane L´eger Laboratoire de Physique de la Mati`ere Condens´ee, Coll`ege de France, 11 place Marcelin Berthelot, 75232 Paris Cedex, France

We report a novel experimental technique using total internal reflection – fluorescence recovery after photobleaching (TIR-FRAP) to probe the velocity of a liquid near a wall with a resolution of the order of 100 nm. As an example of use, we have investigated the boundary condition of the liquid velocity during lubricated friction and studied the influence of a classical additive (stearic acid) in a base oil (hexadecane), and demonstrate that simple Newtonian fluids can develop slip at the wall. Keywords: slip, lubrication, friction, lyophobic, liquid–solid interface

1. Introduction

2. SFA, QCM and MD

Pioneering work at the Cavendish Laboratory in Cambridge, England, after WWII has encouraged new methods for studying friction, and our knowledge and understanding is growing very fast today [1]. One usually distinguishes solid friction where the static and dynamic friction laws stated by Coulomb still remain valid, and lubricated friction where a liquid supports the strain between two solid surfaces. Fluid dynamics is then introduced to predict a friction law [2]. However, in most cases where lubrication is crucial, pressure, temperature and thickness of the liquid film, and surface conditions (roughness and chemical nature) of the opposing solids affect the usual bulk properties of the liquid. The question is how can we study the influence of these parameters on lubrication. Our main interest in this study is to understand how interactions between the solid surface and the liquid influence the velocity boundary condition. A standard lubricant includes a base oil (short alkanes of 12–20 carbons), oiliness additives (surfactants such as acids, alcohols or glycerol) and extreme pressure additives which usually contain a sulfur or a phosphate group. The former additives are known to adsorb onto the surface at room temperature to form an organized monolayer which prevents surfaces from touching each other [3]. The latter react with the surface at higher temperature (rather than higher pressure) to play a similar role. Both modify surface properties and bulk surface interactions. In order to get more precise information on the hydronamics of lubrication a novel experimental technique has been set up, allowing characterization of the liquid velocity near a wall in a simple shear flow. We first discuss previous results and complementary techniques investigating liquid–solid friction. Then, we describe the details of our technique and show how experimental curves may be analyzed through scaling. Finally, the possibilities of the technique are illustrated with one example and compared with numerical simulations for quantitative analysis.

Amongst others, two experimental methods allow a direct investigation of the interplay between the liquid lubricant and the surface on the properties of friction: the surface force apparatus (SFA) [4,5] and the quartz-crystal microbalance (QCM) [6]. In the SFA a liquid film is trapped between two mica sheets. The film thickness can be adjusted from several nanometers down to only a few a˚ngstroms while the contact diameters are of the order of 10–100 µm depending on the load. The mica surfaces may be modified either chemically (e.g., self-assembled monolayers), or physically (e.g., by sputtering on a layer of gold), or both. The SFA can be used to measure normal forces upon approach or “friction” forces by monitoring the stress response of one surface to an oscillatory motion of the other surface. It does not, however, give direct information on liquid motion within the lubricated gap. Moreover, SFA generally look at very confined films (less than 10 nm), which are known to have properties very different from a bulk liquid. In the QCM a layer of liquid molecules is adsorbed on a crystal which performs mechanical vibrations. The liquid molecules slip on the surface as would for example marbles on a vibrating solid. Vibrational properties of the quartz are then related to the mass of the liquid, and to the frictional properties of these molecules on the surface. It is not clear to what extent the friction of a monolayer or even multilayers of an adsorbed liquid in contact with its vapor phase is representative of liquid–solid friction within a lubricated system. Finally, molecular dynamics (MD) simulations should be mentioned as a theoretical tool which allows investigation of several experimental conditions which could be difficult to analyze with either of the above techniques [7].

 J.C. Baltzer AG, Science Publishers

3. Slip or no-slip velocity boundary condition? One of the fundamental parameters needed to solve fluid mechanics equations is the knowledge of the boundary con-

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Figure 1. Velocity boundary condition and slip length.

dition of the fluid velocity. In his “Lectures on Physics” Feynman wrote: “It turns out – although this is not at all self-evident – that in all circumstances where it has been experimentally checked, the velocity of a fluid is exactly zero at the surface of a solid” [8]. However, for non-Newtonian fluids such as polymers, several experiments have proven the existence of true slip at the wall (as opposed to shear thinning near the wall with conservation of no-slip at the surface). The extent of slippage may in these systems be stress dependent [9,10]. It is clear that if one goes from a situation of no-slip to a situation of slip, the stress transmitted to the surface will vary accordingly. One convenient way to characterize the flow at the wall is by introducing the slip length b as illustrated in figure 1. Note that b is taken positive when the liquid velocity at the wall is positive. Negative values of b correspond to the presence of a stagnant layer of thickness b at the wall or to back-flow near the wall. The no-slip boundary condition yields correct results in calculations only if the gap between the two surfaces is much larger than b. For polymers, b may reach 10–100 µm. Such values are not uncommon in lubricated contact. Another way to describe liquid–solid friction is to introduce a friction coefficient k linking the tangential shear stress σ to the velocity at the wall vS : σ = kvS . And since for a Newtonian fluid of viscosity η dy vS σ=η =η , dz z=0 b we find η . b Hence a constant slip length is equivalent, at constant viscosity, to a constant friction coefficient. In the case of Newtonian fluids, experiments at the macroscopic scale have not been able to prove a breakdown of the no-slip boundary condition, but on a microscopic scale, both experimental work (SFA, QCM, flow through capillaries) and MD simulations have suggested the possibility of slippage for simple liquids. Nevertheless, in the case of SFA experiments, it seems that slippage is mainly due to the fact that the molecules within the gap are subjected to a strain rate larger than the inverse of their natural relaxation time [11]. For simple liquids in the bulk this k=

yields a critical shear rate higher than 107 s−1 , whereas typical values in lubricated contact are often less than 106 s−1 . MD simulations have investigated corresponding situations and similarly found a very high critical shear rate [12]. In non-confined liquids, the interaction between the wall and the fluid was introduced. In that case, negative as well as positive slip lengths have been found. Wall-induced structure of the liquid on a strong attracting surface strongly reduces the mobility of the first molecular layers. Conversely, when the interaction potential is weak, wall and fluid become decoupled and true slip occurs [13,14]. QCM experiments suggest real slip of liquid molecules on an atomically flat surface [15]. Persson uses corresponding friction coefficients of water on silver to evaluate a slip length in shear flow [16] and finds b = 20 µm. Einzel et al. suggest that roughness reduces b to an effective value of the order of 100 nm [17]. This would explain why such high slippage lengths have never been measured. Churaev et al. studied the flow of water through hydrophobic glass capillaries and extrapolated a slip length of order 30 nm [18]. However, to be sensitive to such length scales, the capillaries need to have a radius of comparable magnitude, which is difficult to obtain. Furthermore, proper grafting of silanes in such thin capillaries has proven to be another challenge. This might explain the large scatter in their results [19].

4. Velocimetry by total internal reflection fluorescence recovery after photobleaching (TIR-FRAP) Let us first define our objective: we want a direct measurement of the velocity near the solid wall of a simple fluid under shear, with two requirements: (a) the flow should not be perturbed, (b) the probed thickness should be no greater than 100 nm. The first condition is fulfilled using fluorescent molecules of the same size as the liquid, and at trace level (5 ppm). If well chosen, these probes can be photobleached locally by a strong laser illumination. Photobleaching is an irreversible reaction which renders the probes nonfluorescent. This allows a subsequent tracking of the photobleached probes in a background of fluorescent probes. To obtain spatial resolution the solid–liquid interface is illuminated at an angle larger than the critical angle, so that total internal reflection occurs. An exponentially decaying evanescent wave thus penetrates the liquid and excites the remaining fluorescent probes locally. 4.1. Material The surface of study is sapphire (Al2 O3 ) as a model for the oxidized surface of aluminum. A disk of radius 50 mm and thickness 5 mm was supplied by Melles Griot. The roughness measured by X-ray reflectivity and AFM ˚ rms. Sapphire is birefringent (δn = 0.008). Its is 4 A

R. Pit et al. / Friction and slip of a simple liquid at a solid surface

optical axis is orthogonal to the faces. Prior to the experiment, the surface was cleaned by use of a Piranha solution (50%H2 O2 + 50%H2 SO4 – the reaction is very exothermic and removes all organic material), rinsed with tridistilled water (18 MΩ cm), dried with nitrogen and cleaned during 2 h by a combination of UV and oxygen treatment (UV/ozone) [20] followed by rehydration with a flow of oxygen passing through tridistilled water. This is believed to yield a clean aluminol-covered surface. The liquid is hexadecane (Aldrich 99%). The optical penetration depth for this system is λ = 80 nm at Θ = 58◦ (nsol = 1.7785, nliq = 1.4335), which gives the required resolution. ˚ rms). The opposite surface is polished silica (5 A The fluorescent probe is 4-dihexadecylamino-7-nitrobenz2-oxa-1,3-diazole (NBD dihexadecylamine) at a weight concentration of 5 × 10−6 (dihexadecylamine was preferred over hexadecylamine to prevent adsorption of the probe on polar surfaces through the amine). The additive we used here is stearic acid (n-octadecanoic acid Aldrich 99%) at a molar concentration of 1%. Experiments are done at 20 ◦ C.

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Figure 2. Schematic of the setup.

The lubricant is trapped on a 5 mm wide ring, of inner radius 29 mm, between the lower disk of silica and the upper disk of sapphire. The lower window is pinned on a rotating axis. Parallax of the rotating silica disk is corrected by use of Mylar spacers and checked by projecting a reflected laser beam on a far wall. The deviation is reduced to less than 0.01 mrad by this adjustment. This means a wobbling of the driving surface of less than 5 µm. The gap is set to 190 µm by use of three micrometers carrying the upper window. Without slip, shear rate is then equal to the velocity of the driving surface divided by the gap size.1 TIR-FRAP measurements of the velocity are done at a radius of 31.5 mm. Altogether, experimental adjustments introduce a relative error of less than 8% in the set shear rate. Accessible shearing rates are between 100 s−1 (diffusion limit) and 10000 s−1 (mechanical limit).

457.9 nm at 400 mW) is split by a quartz window at 45◦ . The transmitted beam (∼90% power) is then directed by mirrors through a spherical lens L1 (f = 30 cm) into the sample. It traverses the gap vertically so that t(Φ) is the relevant time. This is the photobleaching beam. The reflected beam (∼10% power depending on polarization) is directed into the liquid sample at an angle of 58◦ . This beam is polarized horizontally by use of a NICOL prism and focused through two cylindrical lenses L2 and L3 to produce an elliptical spot of diameters 60 µm in the shear direction (ft = 30 cm) and 30 µm in the perpendicular direction (fr = 7 cm). The isotropic fluorescence emission is collected onto a photomultiplier through a condenser C (f = 15 mm) positioned at 2f to minimize distortion and spatially filtered at the focal point by a 200 µm diameter pinhole. A band-pass optical filter rejects the 457.9 nm laser beam and transmits the 520 nm centered fluorescence. The two separated beams are positioned relative to the photomultiplier so that the centers of the photobleaching and reading spots coincide within a 2 µm accuracy. The signal is processed through a current/tension amplifier and fed into an acquisition card. Finally, three electronic shutters allow one to block either of two beams and protect the photomultiplier during the high power photobleaching pulse. A personal computer controls the whole acquisition system.

4.3. Optical and electronic setup

4.4. Experimental procedure

The main difficulty of FRAP in simple liquids is the rapid diffusion of the probe. Two length scales are of importance: the penetration depth of the evanescent wave (λ = 100 nm) and the laser beam diameter Φ = 60 µm. Applying the simple Einstein–Stokes diffusion equation x2 = 2Dt, we find two characteristic times t(λ) ≈ 50 µs and t(Φ) ≈ 20 s (the coefficient of diffusion D is estimated to be 10−10 m2 /s). Therefore, the following optical setup has been chosen (figure 2): an initial laser beam (argon

Acquisition of the velocity at the wall is done in three steps, as illustrated in figure 3:

4.2. Shearing setup

1

If there is slip at the wall, the true shearing rate is equal to the velocity of the driving surface divided by (gap + slip length). Nevertheless, since the gap is set to 190 µm, this correction is not relevant for slip length smaller than 1 µm.

(1) The low power evanescent wave beam excites the fluorescent probes to give a reference intensity value. (2) The high power vertical beam photobleaches the probes for a short time (50 ms). During this time, the photomultiplier is protected. (3) The low power evanescent wave beam excites remaining fluorescent probes. At first, intensity is low, but the flow of the liquid brings new probes into the reading zone, while pushing photobleached probes out. The intensity thus recovers back to the reference value.

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Figure 4. FRAP curves at 1000 and 2000 s−1 at different times of incubation. Figure 3. Procedure of TIR-FRAP velocimetry.

During this procedure, shearing is permanent. The setup allows easy multiple acquisitions to improve the signal to noise ratio. Furthermore, measurements are always done shearing in both directions to make sure that the two beams do coincide. Addition of both curves is relatively independent of an eventual asymmetry. The fluorescence recovery will be faster if the shearing rate is higher. This is easily verified. The interesting feature is that a faster recovery also occurs at a given shearing rate if there is slip at the wall. Note that the fluorescence recovery curve is not sensitive to the velocity at the wall only: it takes into account the photobleaching rate, as well as diffusion, convection and eventually slip of the liquid. In the next paragraph we will show how all FRAP curves can be fitted on a single curve by proper scaling. 4.5. Scaling of intensity and of time Within the range of shear rates investigated, it appears that all FRAP curves follow a universal curve. This is mainly due to the fact that the characteristic diffusion time t(λ) ≈ 50 µs is very small compared to the characteristic fluorescence recovery time of order of 25 ms. First, the intensity can be scaled as follows: minimum intensity right after photobleaching is set to 0, while the reference intensity is set to 100. This simply means that what we are interested in is the dynamics of recovery. Furthermore, this minimizes the influence of an eventual change in photobleaching rate due to variations of the laser power. Finally, all curves are reduced to a single universal one by time scaling: time is multiplied by (effective shearing rate)0.68 . The exponent value of 0.68 is purely experimental. For a non-diffusing liquid this should be equal to 1. If there is no slip at the wall, the effective shearing rate is equal to the set shearing rate. If on the other hand there is slip at the wall, the effective shearing rate is higher than the set value. At this point it is fair to note that this new experimental setup does not discriminate true slip from a change in the velocity profile near the wall for simple liquids with small molecules (for polymer, the penetration

depth λ = 100 nm is of the same size as a polymer coil so one can measure dynamics of the first layer). It is for that reason that we talk about effective shear rate. We will see further down how we can link this effective shear rate to a slip length by reproducing FRAP curves with simulations. 4.6. Results and discussion The following example illustrates the effect of stearic acid on the boundary condition of hexadecane flowing on aluminum oxide. Figure 4 shows the FRAP curves at two different shear rates (1000 and 2000 s−1 ), as a function of incubation time (0, 26, 96 and 116 h). For sake of simplicity we show the curves with a scaling of the intensity only. At time t = 0 (about 30 min after the liquid is put in contact with the surface), FRAP curves are identical to those obtained without additive in the liquid. At longer times, FRAP curves evolve: the recovery becomes faster. Time scaling to a universal curve requires an effective shear rate of 1000 and 2000 s−1 at time t = 0 and 1300 and 2500 s−1 at longest time t = 116 h and t = 98 h, respectively. The above results clearly indicate that the velocity profile at the wall is affected by the presence of stearic acid. An increase of the effective shear rate of about 30% is observed. Within experimental error, no dependence on shear rate was found. Several experimental studies [21,22], have shown that, in a 1% solution, stearic acid adsorbs onto the hydrophilic surface through its polar head within a few days, and self-assembles as a dense monolayer exhibiting methyl groups towards the liquid. During incubation, the surface interacting with the bulk liquid therefore changes from the initial aluminum oxide to a monolayer of surfactants, whereas the liquid remains unchanged. Our interpretation is as follows: at time t = 0, hexadecane interacts with the bare surface, hydrophilic after cleaning. Hexadecane wets this surface and we suppose the attractive forces to cause the first molecular layer of hexadecane to adhere to the surface imposing a no-slip boundary condition. Since the surface energy of hexadecane is γ = 27.6 mJ/m2 and that of a stearic acid monolayer is about 21 mJ/m2 (depending on the density of the monolayer) hexadecane dewets

R. Pit et al. / Friction and slip of a simple liquid at a solid surface

such a self-assembled monolayer (capillary forces prevent the liquid from leaving the gap). The typical contact angle should be of order 40◦ at maximum coverage [21]. Subsequent analysis of the sapphire surface at the end of the experiment yielded a contact angle of 25◦ only, and the surface energy was found to be higher than expected (24.8 mJ/m2 ), indicating that the monolayer was not complete. In any case, bulk liquid–solid surface interactions are very weak once the stearic acid has adsorbed onto the surface, and the layer of fluid immediately adjacent to this new surface is less bound. Since the penetration of hexadecane in this monolayer is also hindered spatially, these results suggest the existence of true slip at the wall. The influence of shear rate on slippage is difficult to assess experimentally outside the explored range: at lower values where a static friction limit might exist, the time of recovery becomes essentially dominated by the process of diffusion. Conversely, at high shear rates it would be interesting to approach the critical shear rate observed with the SFA apparatus, but for hexadecane this is too high for our setup.

5. Simulation and quantitative slip length In principle, our experiment can easily be simulated: the equation we have to integrate is a combination of flow, diffusion and photobleaching. In two dimensions, the concentration c(x, z, t) of fluorescent probes is set by the following equation: ∂c ∂2c ∂c + vX − D 2 + kB c = 0. ∂t ∂x ∂z Convection is taken along the x-axis. The velocity is supposed to be linear with slip length b: V X (z) = γ(z ˙ + b)X, where γ˙ is the set shear rate. Diffusion is taken along the z-axis only. This is consistent since the diffusion time in the x direction is of the order of 20 s, as seen above, whereas experimental recovery times are smaller than 0.2 s. Photobleaching is supposed to be a first-order reaction with constant kB = 10 s−1 (measured independently) modulated with a Gaussian beam along x of beam diameter Φ = 60 µm for t = 0 to t = tB . For t = tB to t = tend photobleaching is switched off by setting kB = 0. The equation is solved using a finite difference algorithm. At each time step the evanescent wave induced fluorescence intensity, proportional to the concentration of fluorescent probes, is the sum of all cells multiplied by the beam profile (gaussian along x and evanescent along z). Note that the chosen optical setup with an elliptical evanescent spot and a cylindrical photobleaching beam makes comparison with a two-dimensional simulation more realistic. Using experimental values for all parameters, FRAP curves have been reproduced to a great accuracy for all shearing rates explored. On a wetting surface we suppose

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Figure 5. Effective slip length and shear rate; hexadecane + stearic acid (1%) on Al2 O3 .

a no-slip boundary condition (b = 0). This may not be true. Rather than try and adjust simulated curves on experimental curves we studied how slip changed the simulated curves. We find again that all curves can be scaled to a universal curve by using the same rules as discussed above, introducing an effective shear rate, but this effective shear is different from the set shear rate if slip length b is non-zero. Thus the slip length b can be correlated to the effective shear rate for each set shear rate. In the case of the example discussed above, we found that the adsorption of stearic acid increased the effective shear rate near the wall by about 30%. An identical increase is found in the simulations when the slip length is set to 300 nm. Figure 5 shows the evolution of slip length and equivalently of the increase in shearing rate with time of incubation. We studied how this slip length could be affected by changing parameters in the simulations. Keeping D, kB , Φ within reasonable values, this affected the exact value, but not the order of magnitude. 6. Conclusion We have demonstrated the possibility to use TIR-FRAP to probe the velocity at the wall of a simple liquid. In a first example, we find that the chemical modification of a sapphire surface by adsorption of stearic acid modifies the boundary condition of shear flow of hexadecane at ambient temperature, atmospheric pressure and relatively small shear rates. We use numerical simulations to describe this effect as a breakdown of the usual no-slip boundary condition, with a slip length large compared to molecular sizes. This result could have dramatic consequences in fluid mechanics calculations for boundary lubrication. Subsequent study of other liquids such as water or squalane, and other additives or surfaces is presently underway. Acknowledgement This work is part of a global research program “Molding of materials ∼ Tool-metal-lubricant contact”, supported by

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