Tribology Letters 7 (1999) 147–152
147
Friction and slip of a simple liquid at a solid surface
Remmelt Pit, Hubert Hervet and Liliane Léger
Laboratoire de Physique de la Matière Condensée, Collège 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 ångstroms
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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R. Pit et al. / Friction and slip of a simple liquid at a solid surface
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
is 4 Å rms. Sapphire is birefringent (δn = 0.008). Its
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.
The opposite surface is polished silica (5 Å rms).
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.
149
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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R. Pit et al. / Friction and slip of a simple liquid at a solid surface
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) = γ̇(z + b)X,
where γ̇ 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
151
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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R. Pit et al. / Friction and slip of a simple liquid at a solid surface
CNRS, Irsid (Usinor) and Pechiney Recherche, involving
l’Université de Paris-Sud Orsay (LMS), Collège de France
(PMC), ECL (LTDS), INPT (IMF), INSA de Lyon (LMC),
ENSMP (CEMEF), SCA.
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