EUROPHYSICS LETTERS
1 September 1999
Europhys. Lett., 47 (5), pp. 562-567 (1999)
Humidity effect on static aging of dry friction
J. Crassous, L. Bocquet, S. Ciliberto and C. Laroche
Laboratoire de Physique, Ecole Normale Supérieure de Lyon
46 Allée d’Italie, 69007 Lyon, France
(received 22 March 1999; accepted in final form 2 July 1999)
PACS. 46.55+d – Tribology and mechanical contacts.
PACS. 68.10Jy – Kinetics (evaporation, adsorption, condensation, catalysis, etc.).
PACS. 68.35Gy – Mechanical and acoustical properties; adhesion.
Abstract. – We report an experiment about the aging properties of the static coefficient of
friction under moisture. Depending on the material used, aging properties may be totally driven
by the presence of moisture. This may be understood by taking into account the capillary
condensation of liquid bridges between surfaces. A simple model describing the time-dependent
construction of adhesive forces between surfaces leads to a logarithmic, humidity-dependent
aging effect. The estimated magnitude for the aging rate obtained within this description is in
agreement with experimental data.
Introduction. – Solid friction is widely studied both theoretically and experimentally. However, the microscopic mechanisms which produce the macroscopic experimental observations
are not yet completely understood. Among these experimental results we recall the dependence
of the friction coefficient µ on the velocity and on the contact time. Specifically it is very well
known that the static friction coefficient µs increases in a quasi-logarithmic way with the time
tw during which the two surfaces have been kept into contact before trying to move them:
µs (tw ) = µ0 + α ln(tw ). The effect is material independent and has been explained assuming
a viscoplasticity of the bumps on the solid surfaces in contact.
This quite reasonable explanation neglects an important experimental result. Indeed it has
been reported that humidity influences the value of α, which vanishes in rock-rock experiments
under null humidity [1]. The purpose of this letter is to discuss several experimental results
showing that humidity plays a crucial role in the contact aging and that the rate of aging
strongly depends on the humidity level.
Experimental procedure. – A schematic drawing of the experimental set-up is shown in
fig. 1. A slide of 40 mm × 40 mm can be displaced on an inclined plane. The angle θ between
the inclined plane and the horizontal one can be changed by means of an electrical motor. The
angle θ is measured by an angular encoder giving a resolution of ±0.5◦ . The position of the
slide on the inclined plane is measured by an inductive position detector with a resolution of
0.5 µm. The apparatus is placed inside a 40 l sealed glass container equipped with an adjustable
opening. In order to obtain a humidity lower than the laboratory humidity (typically 40%)
c EDP Sciences
°
563
j. crassous et al.: humidity effect on static aging of dry friction
0.80
Hygrometer
0.75
Adjustable Opening
Pv
0.70
tan ( θw )
Inductive Sensor
0.65
0.60
0.55
0.50
θ
0.45
4
Water
Fig. 1
6
8
10
ln (t w )
Fig. 2
Fig. 1. – Schematic drawing of the experimental set-up. A mobile slider is placed on an adjustable
plate. After a waiting time tw the angle θ of the plate is increased until the slider begins to move,
and the corresponding angle θw is measured.
Fig. 2. – Logarithmic aging of friction coefficient for Bristol-Bristol experiment. As explained in the
text, the aging properties are best analyzed by plotting tan(θw ) as a function of nn(tw ). Humidity is
46% (circles), 34% (squares), 20% (triangles) and 8% (diamonds). Dotted lines represent least-square
fits of experimental data.
a dessicator (P2 O5 ) is placed in the box. A small fan is included in the enclosure. After
few hours the humidity in the enclosure reaches its stationary value and is measured with
an electronic hygrometer with a precision of ±0.1%. Changes of vapor pressure are obtained
by modifying the opening size. For humidity greater than ambient humidity, the dessicator
is replaced by a beaker of distilled water. Experiments are performed at room temperature
T = 23 ± 1◦ C. During one day of experiment, the humidity drift is less than 1%.
In order to precisely measure the dependence of µ on humidity we use the following
procedure. Hygrometry is changed in the evening and experiments begin the following morning.
During the night the slide surface is not in contact with that of the inclined plane. Thus the
two surfaces are immersed for a long time in the air of the box, with a well-controlled humidity.
After at least 12 hours of such a surface treatment, the mobile surface is laid on the previously
inclined plane, and the chronometer is started. After a time interval, ranging from 10 to
6000 s, the inclination is increased. The starting of the mobile surface is recorded by the
inductive sensor. At this instant the angle θw and the waiting time tw are measured. About
ten nonconsecutive waiting times are recorded in one day of experiment. The experiments at
fixed humidity are repeated several times in order to reduce the statistical errors.
Three different symmetrical couples of materials are investigated: Bristol board, Teflon and
Glass. Bristol board is 0.2 mm thick and is fixed with double-face scotch (paper glue does not
change behavior). The weight of the mobile plate is 98 g. Isotropy of paper has been checked
by optic visualization. Teflon surfaces were first roughened by hand lapping with #800 silicon
abrasive paper, and the surfaces were carefully cleaned with soapy water, rinsed with alcohol,
and dried. The weight of the mobile plate is 112 g. Glass surfaces consist of 3 mm thick
window glass and the weight of the mobile plate is 400 g. They were cleaned in the same way
as Teflon surfaces.
Experimental results. – For a fixed humidity, measurements of static friction coefficients
µs = tan θ are done for waiting times tw ranging typically from 10 to 6000 s. In this range of
times, we observe a quasi-logarithmic aging of the coefficient µs with the waiting time tw , as
564
EUROPHYSICS LETTERS
25
25
25
20
20
20
15
10
15
10
5
5
0
0.0
0
0.0
0.2
0.4
P v / P sat
0.6
Slope α
30x10- 3
Slope α
30x10- 3
Slope α
30x10- 3
15
10
5
0.2
0.4
0.6
0
0.0
Pv / P sat
0.2
0.4
0.6
0.8
P v / P sat
Fig. 3. – Variations of the slope α with the relative humidity Pv /Psat for three different materials:
Bristol (triangles), glass (circles) and Teflon (diamonds). The full triangle in Bristol experiment
corresponds to the slope measured in ref. [2]. Error bars are the standard deviation of the slope. The
dotted line for Bristol and glass experiments is the least-square fit of α(Pv /Psat ) using expression (2).
Experiments on Teflon do not exhibit variation of aging properties with vapor pressure.
previously reported in various experiments. Figure 2 represents measurements of tan(θw ) as a
function of ln(tw ) for four different values of humidity in the Bristol-Bristol experiment. For
each humidity a logarithmic aging of static coefficient is observed for waiting times accessible
during the experiment.
As shown in this figure, the slope of the straight line, i.e. the “aging rate” is found to be
strongly dependent on the vapor pressure. The slope tends to vanish at small values of humidity
and increases systematically with the vapor pressure. This observation is in agreement with a
previously reported observation on rock-rock friction [1].
In order to analyze the dependence of the slope on the vapor pressure, we fit the measurements done at a fixed humidity with a logarithmic law
tan(θ(tw )) = tan(θ0 ) + α(Pv /Psat ) ln(tw )
with the time tw expressed in seconds. The standard deviation on the slope determination is
typically 10% of the nominal value of the slope. Figure 3 shows the slope α as a function of the
relative humidity Pv /Psat for the three different systems, for relative humidity ranging from 10
to 70%. The variation of the aging rate with the humidity is clearly material dependent. For the
Bristol-Bristol and the glass-glass experiment, the aging rate strongly increases with the vapor
pressure. On the contrary, the aging rate in the Teflon-Teflon experiment is comparatively
weak and appears to be independent of the relative humidity.
Model. – The humidity dependence of aging properties of the static coefficient µs suggests
that the aging properties are due not only to the visco-plastic properties of material. Moreover,
the influence of humidity vanishes with the Teflon surfaces, which are not wetted by water [3].
Effects of ambient humidity on the magnitude of the friction coefficient have been frequently
reported from pioneering works [4] up to now [5, 6]. More recently, experimental studies
on rock-rock friction [1] and on stability angle of granular media [6] have shown a connection
between the logarithmic aging of friction coefficient and ambient humidity. Those observations
lead to the idea that aging properties are related to the ability of water to be present at the
surface of the sliding surfaces. Surface force experiments [7] have clarified the influence of
humidity on adhesion effects. In the presence of an undersaturing vapor, a liquid bridge forms
between two surfaces in contact, and produces an attractive capillary force between them.
In order to understand how such a force can induce a variation of friction coefficient, we
j. crassous et al.: humidity effect on static aging of dry friction
565
µm
a)
nm
e
b)
c)
ao
Fig. 4. – Description of the contact between surfaces. a) The contact between two surfaces consists
of micrometric contact areas between them. b) Model of a microcontact consisting of nanometric
spherical bumps or radius a0 separated by a distance e. c) Those bumps are possible sites for a liquid
bridge formed by capillary condensation of the vapor.
consider first the equilibrium of force of two solids in contact in the presence of adhesive force
between surfaces. Let N and T be the projection of the external load respectively normal and
tangential to the contact plane between surfaces. It is now well established that the contact
area between two solid surfaces in contact is not the total surface area, but instead consists
of small areas of micrometric size. Let Ar be the real surface of contact between surfaces.
On the one hand, the tangential force needed in order to move tangentially the two solid
surfaces is T = σAr , where σ is the tensile strength of the material [5]. On the other hand, the
real contact area is proportional to the normal load N [8, 5], the constant of proportionality
χ = N/Ar being related either to the hardness [5], or to the Young modulus [8] of the material,
without changing the following description.
In the presence of adhesion effects, the normal force between surfaces is increased by the
adhesion force and the normal load becomes N + Fadh , where Fadh is the total adhesive
force between surfaces. Equilibrium of tangential forces just before sliding leads to T =
(σ/χ)(N + Fadh ). With N and T due to the weight of mobile plate, and assuming that
Fadh ¿ N , we obtain for the maximum angle of stability θ
µ
¶
Fadh
σ
1+
.
(1)
tan(θ) ≈
χ
χAr
As mentioned previously, the adhesive force is due to the presence of liquid bridges which
connect the surfaces in contact. For an evaluation of its magnitude, we need to know the
number of liquid bridges and the average adhesion force due to the presence of a liquid bridge.
Under our humidity conditions, the curvature of the liquid-vapor interface as given by the
Kelvin equation [3] is nanometric, and only formation of liquid bridges of nanoscopic size can
take place. So, in order to evaluate the number of liquid bridges, we need to describe the contact
between solid surfaces at this scale, in other words in the interstitial space between micrometric
bumps (fig. 4a). However, the magnitude of the aging effect reported in our experiment and in
other systems at laboratory humidity is not very sensitive to the material used [9], and to the
normal load [10]. On this basis, we can infer that a rough determination of this effect must
not be strongly dependent on a precise description of the contact between surfaces.
566
EUROPHYSICS LETTERS
As discussed above, the size of liquid bridges is nanometric, so we cannot consider that the
micrometric asperities are smooth at the corresponding scale. So, we consider a model for the
nanometric surface roughness of a micrometric asperity which consists of nanometric spherical
bumps of radius a0 with a density of 1/4a20 as shown in fig. 4b. Bumps are separated by a
nanometric gap e, and let φ(e)de be the probability that the gap between bumps lies between
e and e + de. We assume that separations from 0 to a nanometric distance λ are equiprobable:
φ(e) = 1/λ, for e < λ and 0, otherwise.
The total number of bumps coupled in front of each other is Ar /4a20 . A couple of asperities
is a possible site for a liquid bridge, if their separation is less than two times the radius of
curvature of the liquid-vapor interface req . So the total number of possible sites for a liquid
bridge is (Ar /4a20 )(2req /λ).
Two different states for a couple of bumps are possible. They may be filled with a liquid
bridge or not. The filled situation is stable and the empty is only metastable [11]. So, in
order to nucleate the stable liquid bridge, the system needs to overcome an energy barrier
∆E corresponding to the formation of a critical nucleus. This energy may be expressed as
∆E = ρl v∆µ, where ∆µ = µsat − µg = kT ln(Psat /Pv ) is the difference between the chemical
potential of the vapor and the chemical potential of a saturated vapor, ρl is the density of the
liquid phase, and v is the volume of the critical nucleus [6]. Assuming an activated process for
overcoming this barrier, the time t needed to condense a liquid bridge in the interstitial space
between bumps can be written as t = t0 exp[∆E/kT ], where t0 is a microscopic time. So at a
waiting time tw , all possible sites with ∆E < kT ln(tw /t0 ) are filled with a liquid bridge, with
a nucleation volume v ≈ ea20 . The fraction of filled sites is (kT /ρl ∆µ)(1/λa20 ) ln(tw /t0 ).
With a force per filled site fadh = 2πγa0 [7] where γ is the liquid-vapor surface tension, and
a total number of sites Ar /4a20 , the adhesion force at a time tw is
Fadh = (kT /ρl ∆µ)(1/λa20 ) ln(tw /t0 )(2πγa0 )(Ar /4a20 ).
Using eq. (1), one obtains
σ
tan(θw (tw )) ≈
χ
µ
¶
α0
1+
ln(tw /t0 )
ln(Psat /Pv )
(2)
with
α0 =
πγ
.
2χρl a30 λ
(3)
Let us note that the adhesion force is proportional to the real area Ar , and thus to the
“bare” normal load N . As a consequence, the aging coefficient α is independent of N in this
interpretation, as experimentally observed [10].
Discussion of results. – Figure 3 shows the least-square fits of α(Pv /Psat ) using expression (2). On the one hand, in the experiment with the Bristol card, the variation of the slope
with relative humidity is well reproduced by this equation, with α0 = 1.5 × 10−2 . On the
other hand, in the experiment with glass, the fit leads to α0 = 9.8 × 10−3 . The “noisy” values
of slope measurement in the glass-glass system might be understood with a rough estimate of
the number of contact points. Under a load W ∼ 4 N, and with a hardness H ∼ 1 GPa, the
real surface contact Ar = W/H ∼ 10−9 m2 . Assuming a radius of microcontact ∼ 10 µm [8],
it corresponds to a number of microcontacts between surfaces of the order of 10. This rough
estimate leads to a small number of microcontacts, resulting in a bad statistical average of
frictional properties [12]. This bias is removed using the Bristol card, which is a material
known to have a very regular behavior [2] attributed to a large number of contact points
between surfaces.
j. crassous et al.: humidity effect on static aging of dry friction
567
A comparison between α0 values used to fit experimental data and eq. (3) requires information about the geometrical parameters describing surface roughness. Since capillary
condensation can only take place at the nanoscopic scale, one has a0 ∼ λ ∼ nm. With a
typical value of hardness χ = H ∼ 1 GPa this give for water α0 ∼ 3 × 10−3 . This order of
magnitude appears to be in agreement with the experimental results. However, we remark
that our estimate is only an indication of the magnitude of the total adhesive force between
surfaces.
Conclusion. – The above-described mechanism of capillary condensation of liquid bridges
permits to explain the logarithmic aging of the static friction coefficient without using viscoplastic effects. It is supported by the observation that the magnitude of aging evolves with the
ambient pressure of water. A very rough description of the roughness shows that the expected
magnitude of this effect is in agreement with the measurements. However this mechanism is
irrelevant in the Teflon experiment, where a logarithmic aging is however measured. Viscoplastic effects appear in this case to be the dominant mechanism for logarithmic aging. We
would like to point out that our interpretation is in agreement with the logarithmic increase
of the real area of contact Ar as measured by Dieterich et al. [13]. As discussed above, the
real area is proportional to the normal load, which is now increased by the supplementary
adhesion force Fadh (t). A logarithmic dependence of Ar with the waiting time is thus expected
in return.
***
This work has been partially supported by the Region Rhone Alpes “Contrat Thematique
Materiaux”.
REFERENCES
[1] Dieterich J. and Conrad G., J. Geophys. Res., 89 (1984) 4196.
[2] Heslot F., Baumberger T., Perrin B., Caroli B. and Caroli C., Phys. Rev. E, 49
(1994) 4973.
[3] Adamson A. W., Physical Chemistry of Surfaces, fifth edition (Wiley-Interscience) 1990.
[4] Dowson D., History of Tribology (Longman, London) 1979.
[5] Bowden F. P. and Tabor D., The Friction and Lubrication of Solids (Clarendon Press, Oxford) 1950.
[6] Bocquet L., Charlaix E., Ciliberto S. and Crassous J., Nature, 396 (1998) 735.
[7] Israelachvili J. N., Intermolecular and Surfaces Forces (Academic, London) 1985.
[8] Greenwood J. A. and Williamson J. B. P., Proc. R. Soc. London, Ser. A, 243 (1966) 300.
[9] Baumberger T., Solid State Commun., 102 (1997) 175.
[10] Dokos S. J., J. Appl. Mech., 13 (1946) 148.
[11] Dobbs H. T., Darbellay G. A. and Yeomans J. M., Europhys. Lett., 18 (1992) 439.
[12] Johansen A., Dimon P., Ellegaard C., Larsen J. S. and Rugh H. H., Phys. Rev. E, 48
(1993) 4779.
[13] Dieterich J. and Kilgore B., Pageophys., 143 (1994) 283.