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Wu-Bavouzet2007.pdf
The Journal of Adhesion, 83:761–784, 2007
Copyright # Taylor & Francis Group, LLC
ISSN: 0021-8464 print=1545-5823 online
DOI: 10.1080/00218460701586178
Stick-Slip: Wet Versus Dry
F. Wu-Bavouzet
J. Clain-Burckbuchler
A. Buguin
P.-G. De Gennesy
F. Brochard-Wyart
Institut Curie, Centre de Recherche, Université Paris 6, CNRS UMR
168, Paris, France
A rubber lens (polydimethylsiloxane) is pressed against silanated or bare glass
plates (Johnson-Kendall-Roberts (JKR) contact). As the plate slides with a velocity
U, we measure the friction on the lens using a ‘‘macro Atomic Force Microscope
(AFM)’’, where the cantilever is a thin rectangular glass rod and the tip is the
rubber lens. We observe the contact area via optical interferometry.
In air for ‘‘hard’’ lenses (Young’s modulus E 1 MPa), we find smooth sliding on a
model substrate, and a transition to stick-slip on a hysteretic substrate above a
threshold velocity, VM. For soft lenses (E 0.1 MPa), we observe Schallamach
waves and stick-slip depending on normal force and the plate’s velocity, U. When
immersed in a liquid (silicone oils, water-glycerol mixtures), the contact remains
dry at low velocities, but is invaded by a liquid film above a critical velocity, Uc.
For hard lenses we observe smooth sliding and high friction below Uc, and low
friction above Uc. For soft lenses, we find wet Schallamach waves for U < VM
and stick-slip instabilities at large velocities. In the stick-slip regime, the contact
is wet in the slip phase, and dewets in the stick phase. We measure the period of
the stick-slip cycle as a function of the liquid viscosity.
We interpret the stick-slip process by the formation and rupture of adhesive bonds
(between the rubber polymer chains and active sites on the glass). Using a recent
model, we can explain most of the data for the stick-slip period and slip threshold
velocity.
Keywords: Lubrication; Rubber=solid friction; Sliding instabilities; Stick-slip
y
Deceased.
Received 1 February 2007; in final form 20 June 2007.
One of a Collection of papers honoring Liliane Léger, the recipient in February 2007
of The Adhesion Society Award for Excellence in Adhesion Science, Sponsored by 3m.
Address correspondence to Francoise Brochard-Wyart, Institut Curie, Centre de
Recherche, CNRS UMR 168, Université Paris 6, F75248 Paris, France. E-mail: [email protected]
761
762
F. Brochard-Wyart et al.
NOMENCLATURE
Fn, Ft
Ft0
a
A
U
v
r(v)
f
VM
Vm
E
W
Won, Woff
S ¼ cSR–(cSL–cLR)
cij
R
Uc
g
g0
h0
K
h
hA , hR
n
son
sR
s0
l
Vc ¼ l=son
Vtip
Vs
r0
Fc
fv
Ub
w, t, L
normal, tangential forces
rupture force
contact radius
contact area
glass plate velocity
rubber sliding velocity
shear stress of the sliding lens r(v) ¼ Ft=A
friction coefficient
velocity corresponding to a maximum of r(v)
velocity corresponding to a minimum of r(v)
rubber lens Young’s modulus
adhesion energy
advancing, receding adhesion energy
liquid, L, spreading parameter at soft interface
solid, S, rubber, R
interfacial tension
curvature radius of the rubber lens
forced wetting velocity
liquid viscosity
monomer viscosity
elastic length
spring constant of the adhesive lens
(K ¼ 16 Ea=9)
contact angle
advancing, receding contact angles
density of adhesive bonds
healing time
Rouse time of reticulated polymer chains
microscopic relaxation time
maximum extension of bond at rupture
critical unbinding velocity
crack tip velocity
slip velocity
static rupture stress
static rupture force
bond rupture force
bond binding energy
cantilever dimensions
Stick-Slip: Wet Versus Dry
763
1. INTRODUCTION
The friction of rubbers on dry or wet solid substrates is clearly of major
practical importance and has been studied in considerable detail for
engineering purposes [1–6]. However, there is still a lack of
understanding of the various slippage regimes observed, and the new features when the rubber in air is immersed in a liquid. We present here:
. Some new experiments with controlled mechanical conditions;
. Comparisons with a simple theoretical model.
We concentrate on the case of polydimethylsiloxane (PDMS)
rubbers in contact with:
. A silanated substrate;
. A poorly silanated substrate, where holes in the molecular carpet
give rise to local strong adhesion and hysteresis;
. Bare glass, which is chemically heterogeneous.
We use hard and soft rubber lenses, of Young’s modulus E 1 MPa
and E 0.1 MPa, respectively. The mechanical conditions (Figure 1)
are an adaptation of the classical test analyzed long ago by Johnson,
Kendall, and Roberts (JKR) [7]. The rubber is a flat=convex lens (of
FIGURE 1 Experimental set-up. The PDMS lens is pressed against the substrate, which moves at velocity U. The contact is observed with a microscope.
764
F. Brochard-Wyart et al.
radius R 1 mm) squeezed against the flat support by a normal force, Fn;
we monitor the contact radius, a, by optical interference microscopy. Let
us first recall the basics for static experiments (no sliding):
. In ideal conditions, the adhesive energy, W (energy per unit contact
area), is the same for loading or unloading. Using the JKR theory
[7], we can extract from the plots a(Fn), the energy, W, and the
Young’s modulus, E, of the rubber.
. In many practical cases we must distinguish a Won (for loading
processes) and a Woff (for unloading processes).
We have performed these experiments with our lenses in both the
‘‘dry’’ and ‘‘wet’’ conditions (in the latter case the lens is immersed in
a liquid). For hard lenses on a silanated glass plate, we find
Woff ¼ Won, but hysteresis appears on poorly silanated or bare glass
substrates. For soft lenses we see hysteresis on all substrates, with
two contributions: one coming from rubber chains ‘‘zipping off’’ the
solid surface, and one related to silanisation defects on the glass plate.
Let us now turn to sliding experiments, where we impose a slippage
velocity at the glass=lens interface, by moving the glass plate at
velocity, U (Figure 1), and measuring the tangential force, Ft.
Our aim is to characterize different sliding modes (smooth=
unstable) versus:
. The nature of the substrate;
. The softness of the rubber;
. The squeezing force, F;
for dry contacts and for contacts immersed in a liquid.
For dry contacts, we shall define a characteristic velocity, VM, associated
with a transition between steady and unstable (stick-slip) sliding.
In a liquid (water, water-glycerol mixtures, fluorinated silicone
oils), we expect similar regimes, plus a wetting transition at U > Uc,
where a liquid invades the contact. This film lubricates the contact
and the friction coefficient, l ¼ Ft= Fn, drops by a factor of order ten
[8]. The scaling law for Uc is [9] Uc W=gðh0 =RÞ1=3 , where g is the
liquid viscosity, h0 is the elastic length [7] (h0 ¼ W=E), and R is the
lens radius. Uc results from a competition between U (forced wetting)
and the natural dewetting of the liquid film. By tuning Uc through the
viscosity of the liquid we shall expect different successive regimes:
. For VM < Uc, smooth sliding=stick-slip=lubrication;
. For VM > Uc, smooth sliding=lubrication.
Stick-Slip: Wet Versus Dry
765
Our aim here is to compare the sliding of the rubber lens in air,
and immersed in a liquid. We shall focus on the stick-slip sliding
instabilities, which can be coupled to wetting and dewetting of the contact if the slip velocity becomes larger than Uc. We first describe the
experimental set-up and the fabrication of rubber lenses, the surface
treatments, and their characterization. Then, we study and interpret
slippage in air. In the last section, we describe the characteristics of
sliding in a liquid. We extend the model of stick-slip of lenses in air
to lenses in liquid, where wetting and stick-slip are coupled.
2. MATERIAL AND METHODS
The elastomer used for the flat=convex lenses is polydimethylsiloxane
(PDMS 170, Dow Corning Corp., Midland, USA) supplied in two liquid
parts. Part A contains vinyl endcapped oligomeric PDMS chains: part
B consists of a cross linker and a catalyst for the reticulation reaction.
Millimetric droplets of A50:B50 and A85:B15 mixtures (w=w) are
deposited on a silanated glass slide, and then cured for 48 h at 65C.
The Young’s moduli, E, for hard (A50:B50) and soft (A85:B15)
flat=convex lenses are 1 MPa and 0.1 MPa, respectively, estimated
from JKR tests, or measured directly by compression of a PDMS cylinder, which gives for a hard lens, a Young’s modulus value E 1 MPa.
The substrates used are:
. A silanated microscope glass slide (SIT84, ABCR, Karlsruhe,
Germany), obtained by grafting a monolayer of octadecyltrichlorosilane onto the surface following a classical procedure [10,11];
. A poorly silanated microscope glass slide. Atomic Force Microscope
(AFM) images [12] show holes in the dense molecular silane layer
of size 100 nm;
. A bare glass slide. This high energy surface is rapidly contaminated
by aerosols, which should lower its surface energy.
We have characterized these substrates by the measurement of advancing and receding contact angles, Dh ¼ hA hR of clean water droplets. We
find Dh ¼ 16 3 on silanated glass (hA ¼ 112 2 ; hR ¼ 96 2 ) and
Dh ¼ 30 3 on poorly silanated or bare glass (for a poorly silanated surface, hA ¼ 93 2, hR ¼ 63 2). This contact angle hysteresis is due to
the chemical defects (adhesive patches) [12] and is used as a quality test
of surface treatments [13]. The silanated glass (Dh < 20) is characterized
by a small hysteresis whereas for ‘‘poorly silanated’’ glass Dh > 20.
The liquids used are:
. Water-glycerol mixtures leading to high adhesion energies as in air.
We vary the water mass proportions (95% (kinematic viscosity,
766
F. Brochard-Wyart et al.
n ¼ 1.1 cS, optical index n ¼ 1.34), 80% (n ¼ 1.7 cS, n ¼ 1.36), 20%
(n ¼ 50 cS, n ¼ 1.44) and 4% (n ¼ 624 cS, n ¼ 1.47). Mixtures of
optical index close to the PDMS index (1.40) have not been used
because the contact cannot be observed;
. Fluorinated silicone oils [polyfluoromethylalkylsiloxane (PFAS),
ABCR, Karlsruche, Germany], leading to low adhesion energies.
We can vary the viscosities from 80 cS to 10,000 cS, depending on
molecular weight.
The experimental set-up is shown in Figure 1. The contact between
the silanated glass slide and the PDMS lens immersed in the liquid is
observed by reflection interference contrast microscopy (RICM)
(microscope Axiovert 135, Zeiss, Le Pecq, France). The interference
fringes are clearly visible, and give a measurement of the contact
radius, a (Figure 1), and the lens profile. Typical size for a is about
50 mm, measured with a resolution of 2 mm.
The PDMS lens is pressed against the solid substrate with a normal
force, FN, which is maintained constant via a piezoelectric crystal. The
microscope glass slide lays on a motorized stage (continuous current
motor, 860A-2, Newport, Irvine, USA). Sliding is imposed by the translation of the stage with respect to the PDMS lens, which stays at rest
in the laboratory reference frame. The sliding velocity is measured by
an optical sensor (Renishaw, Wotton-Under-Edge, UK) and ranges from
100 to 800 mm s.
Our home-made force apparatus [8] to measure the normal and
tangential forces, Fn and Ft, applied to the lens is similar to an
AFM. The PDMS lens (the tip) is stuck under a cantilever which is
a thin glass fibre of dimensions 20 mm 1 mm 200 mm. A laser diode
reflects on the cantilever covered with a thin gold layer (50 nm), and
points on a four-quadrant photo-detector.
A normal force leads to a cantilever bending and, consequently, to a
vertical deflection of the laser beam. Similarly, when a tangential force
is exerted on the PDMS lens, the cantilever twists and the laser beam
is deflected horizontally.
A data acquisition interface (Keithley, Cleveland, USA) records the
sliding velocity, the piezoelectric crystal voltage, and the photo-detector output signals. The latter gives the normal and tangential forces
with two calibrations.
The output signal is proportional to the laser beam deflection on the
quadrants, and, thus (for normal forces), to the vertical displacement,
Dz, of the cantilever extremity. Once the proportionality coefficient
between photo-detector voltage and Dz is measured, the normal force,
Fn, is derived from the cantilever stiffness: Fn ¼ knDz. The coefficient kn
Stick-Slip: Wet Versus Dry
767
is measured by the dynamic Cleveland method [14] in air, but remains the
same in the liquids used (as it only depends on the cantilever dimension
and material). Similarly, for the tangential forces, we measure the proportionality coefficient between photo-detector voltage and the cantilever
twist, and the stiffness, kt, is given by: kt ¼ kn ð5=9ÞðL=ðh þ t=2ÞÞ2 , where
h is the lens thickness (see Figure 1), and L and t are the cantilever length
and thickness, respectively [15,16]. Typically, kn ¼ 10 N m1 and
kt=kn 800 for millimetric lenses. The ranges of force obtained are of
the order of 100 mN for Fn and 1–100 mN for Ft.
Before each sliding experiment, we perform a JKR test. It consists
of a discrete loading=unloading cycle of the lens, which is performed
by steps of 2.5 mN. The contact radius, a, and the normal force, Fn,
applied on the lens are measured at each step. We wait one minute
between two steps to be sure that the system reaches its equilibrium
state. We deduce the Young’s modulus; E, and the adhesion energy,
W, for the lens by using the JKR relation:
a3=2
1
Fn
6pW 1=2
3=2 þ
¼
;
ð1Þ
16E=9 a
16E=9
R
where for compression W ¼ Won, and for decompression W ¼ Woff. The
slope of a3=2=R versus Fn=a3=2 yields E and the extrapolation to zero
force gives W. The results of several JKR tests for hard lenses, compressed on a silanated substrate in air (see Figure 2a) and in fluorinated oil give E ¼ 0.95 0.05 MPa and Won ¼ 42 1 mN/m in air
and Won ¼ 4.7 0.8 mN/m in oil. For a soft lens we find E ¼ 0.10 0.05 MPa and the same values for Won.
For a hard lens on a silanated substrate, the compression and
decompression curves are superimposed: Woff ¼ Won (see Figure 2a)
whereas, in the case of the same lens on a poorly silanated surface,
we observe hysteresis (see Figure 2b), with Woff Won ¼ 30 mN/m.
We also have hysteresis for a soft lens on a silanated substrate.
Hysteresis can have several origins:
. Adsorption of rubber chains on active sites of the substrate—bonds
created during the compression phase between the two surfaces do
not modify significantly the adhesion energy, Won, but lead to a
higher unbinding energy, because energy is lost in the stretching
of these bonds before rupture;
. Free chains extraction—surface chains of the lens can interact with
the substrate [17,18]. During the decompression, they will be
extracted to return to their original position. This costs energy;
. Bulk dissipations in the lens during the decompression phase for
soft rubbers.
768
F. Brochard-Wyart et al.
FIGURE 2 (a) Hard lens on silanated glass in air. a versus Fn during a JKR
test; no hysteresis is observed. The loading and unloading curves are superimposed; plotting a3=2=R versus Fn=a3=2 yields E and Won. (b) Hard lenses on
a poorly silanated substrate in air; we observe hysteresis.
3. SLIDING OF RUBBER LENSES IN AIR
We study the friction of hard and soft lenses on three types of substrate:
silanated, poorly silanated, and bare glass slides. Two parameters
(normal force and sliding velocity) have been varied.
Stick-Slip: Wet Versus Dry
769
3.1. Friction on Model Substrates: Hard Lenses
With hard lenses on silanated slides no hysteresis is observed in JKR
tests. During a sliding experiment, the normal force is applied using a
piezoelectric crystal and we move the glass plate at velocity U. The
tangential force and the lens position are recorded. From the latter,
we derive the sliding velocity v of the rubber lens on the substrate.
3.1.1. Contact Observation During Sliding
When the sliding starts at time t ¼ 0 s, the contact sticks on the substrate (see Figure 3a, for U ¼ 430 mm=s). At t ¼ 240 ms, corresponding
to a displacement of about ten micrometers, the contact area, A,
decreases and becomes approximately an ellipse, whereas the lens
slides smoothly on the glass: in steady state, the sliding velocity, v,
of the lens is equal to the substrate velocity U. When the substrate
slides slowly, the contact keeps its original circular shape: it follows
the JKR laws. As the velocity U increases, the contact area becomes
smaller. This effect is mostly visible at low normal loads and is
explained by Vorvolakos [19] by a progressive loss of adhesive interactions, and a transition from JKR adhesive to Herzian non-adhesive
behaviour. To test this hypothesis, an equivalent radius, a~, is defined
by A ¼ pa~2 . Figure 4a shows the plot of a~3=2 versus Fn =a~3=2 . The plots
for different velocities have the same slopes as expected, leading to
a measure of E 0.9 MPa. The extrapolation to zero force gives the
effective adhesion energy, W, versus velocity U (see Figure 4b).
This decrease of the effective adhesion energy with velocity may
have at least two sources:
. A decrease of the binder density, n, discussed later. W ¼ nUb with
fixed Ub and nðUÞ ¼ n 0 =ð1 þ ðU=Vc ÞÞ n 0 ð1 ðU=Vc ÞÞ. We get from
Figure 4b, Vc ¼ 1.2 mm/s;
. The application of a tangential force also leads to a decrease of the
static contact area. However, a calculation by Savkoor (for a static
force) does predict only small changes in the contact area [20].
3.1.2. Tangential Stress r(U)
The tangential stress r(U) ¼ Ft=A increases linearly with U
(Figure 5a). r(U) discussed in Sec. 4 has two contributions associated
to the rupture of molecular bonds and to passive friction. It can be written as r ¼ r0 þ fU in the limit of small binding density. Experimentally, we find on silanated glass r0 ¼ 5 kPa and f ¼ 108 Pa s/m. r0 is
proportional to the density n 0 of binders and to the force of fc kT=D
to break a bond (kT is the thermal agitation, D a molecular length).
770
F. Brochard-Wyart et al.
FIGURE 3 (a) Smooth sliding of a hard lens on silanated glass in air. The
sliding velocity is U ¼ 430 mm/s and the normal force applied on the lens is
Fn ¼ 430 mm/s. The substrate slides from the bottom to the top. (b) Stick-slip
of a hard lens on a poorly silanated substrate in air. (c) Stick-slip of a soft lens
on a poorly silanated substrate in air. (d) Schallamach wave propagation in air
(soft bead on a poorly silanated substrate). The sliding velocity U is 380 mm/s
and the normal force Fn is 360 mN.
With D > lnm, we find r0 > 1012 m2 . Our value for f is comparable
with the friction coefficient measured by Bureau on PDMS grafted
Si wafers [21]. There are two contributions to f associated with viscous losses in the rubber sliding on microscopic asperities [5], and
molecular friction of polymer chains bound to active sites [21].
We shall see now that on a poorly silanated glass, or bare glass, n 0
is much larger. In this limit, one must include the progressive
Stick-Slip: Wet Versus Dry
771
FIGURE 4 Hard lenses sliding smoothly on a silanated substrate: (a) Variations of ã3=2=R in function of Fn=ã3=2 in air. The plots for different velocities
have the same slope. (b) Adhesion energy Wdyn versus sliding velocity U in air.
detachment of binders n ¼ n 0 =ð1 þ U=VcÞ (equations 5 and 6). r(U)
becomes a decreasing function of U, leading to stick-slip instabilities.
3.2. Friction on Hysteretic Substrates
The two types of lenses (hard and soft) have been used. The experiments lead to various sliding behaviours (see Table 1), including
stick-slip instabilities [22] and Schallamach waves [23].
FIGURE 5 (a) Variations of the stress rt versus sliding velocity U in air. (b) Tangential force Ft for smooth sliding on silanated and poorly silanated substrates in
air. The normal force Fn is 380 mN and the sliding velocity U is 270 mm/s.
772
F. Brochard-Wyart et al.
TABLE 1 Friction Regimes Observed for Hard and Soft Lenses on Different
Substrates in Air
Substrate
‘‘Hard’’ bead ! Decreasing crosslinker proportion ! ‘‘soft’’ bead
Silanated glass
Poorly silanated glass
Bare glass
Sliding ! Sliding=stick-slip
Sliding=stick-slip ! Waves=stick-slip
Stick-slip ! Waves=stick-slip
3.2.1. Hard Lenses
We describe now the sliding of the same hard lens on a substrate
which is now poorly silanated, or on bare glass, both characterized
by a hysteresis of JKR loading-unloading tests.
On poorly silanated glass, we observe a sliding transition from
stable to unstable friction. Figure 6a shows a diagram of various
regimes.
At low velocities U < VM, the lens slips smoothly. We show a typical
plot of Ft when we move the lens forward and backward, and the same
plot on a silanated substrate: we clearly see that the friction force is
enhanced (see Figure 5b). Because the friction coefficient is the same,
we conclude that r0 is at least ten times as large.
Above VM, a stick-slip regime starts. We show in Figure 3b the oscillations of the contact area versus time over a cycle at a velocity, U,
above VM and the tangential forces Ft in Figure 7a. Stick-slip is a periodic phenomenon with two phases. During the first phase, the PDMS
lens sticks on the solid substrate, whereas the tangential force, Ft,
increases linearly with time. In the second phase, the lens slips and
Ft decreases (see Figure 7a).
FIGURE 6 Phase diagram of sliding regimes (a) for hard lenses (b) for soft
lenses on poorly silanated glass in air.
Stick-Slip: Wet Versus Dry
773
FIGURE 7 Stick-slip on a poorly silanated substrate in air. (a) For hard
lenses, fluctuations of the tangential force Ft for different sliding velocities.
The plots have been translated for a better reading. For soft lenses, (b) tangential force Ft during stick-slip cycles, (c) tangential force Ft during Schallamach
waves.
FIGURE 8 Stick-slip frequency versus sliding velocity in air, for lenses of
different radii.
774
F. Brochard-Wyart et al.
We have studied stick-slip at different velocities U. As shown in
Figure 8, the stick-slip frequency increases linearly with the substrate
velocity. The slope decreases when the lens radius increases, but is
independent of the normal force.
3.2.2. Soft Lenses
Two types of behaviours are shown in the phase diagram of
Figure 6b. With soft lenses, we do not observe smooth sliding at low
velocities, but Schallamach sliding waves [23]. At a threshold velocity,
VM, we see a transition from Schallamach waves to stick-slip.
A series of pictures of the contact during a stick-slip cycle are shown
in Figure 3c. Soft lenses are much more deformed than hard ones (see
Figure 3b). During the stick phase, micro-ripples appear on the
contact front and the contact area shrinks.
A Schallamach wave going through the contact is shown in
Figure 3d. The sliding wave propagates from the back towards the
front of the contact: the propagation of the fold is the displacement
mechanism of the lens. When U varies from 50 to 400 mm/s, the wave
velocity increases with the sliding velocity and varies from 2000 to
6000 mm/s. Micro-ripples are observed on the rim.
The tangential force can be seen on Figures 7b and 7c at a transition
between the two regimes. During the first phase (Figure 7b), the
oscillations amplitude is higher and the frequency smaller, showing
that the energy dissipated at each cycle is the same.
4. INTERPRETATION OF SLIDING INSTABILITIES IN AIR:
THE ROLE OF MOLECULAR BRIDGES
We start this section by a simple description of stick-slip instabilities
(derived in more detail elsewhere [24]). We then discuss our data of
friction in air using this framework.
4.1. Model of Stick-Slip
In the reference frame of the glass plate, the lens is pulled by an engine
which moves at velocity U (see Figure 9a). The lens is attached to the
engine by a spring of elastic constant K: the spring is the lens itself,
and K depends on the size, a, of the contact [25]: K ¼ ð16=9Þ Ea.
The tangential force, Ft, acting on the lens is Ft ¼ KX ¼ K(Ut x),
where x is the position and dx=dt ¼ v is the local sliding velocity of
the lens.
As the engine starts to move, the force increases. The lens sticks
(v ¼ 0) up to a critical force, Ft max, where the contact breaks.
Stick-Slip: Wet Versus Dry
775
FIGURE 9 (a) The lens is pulled by an engine at velocity U. The spring
constant is K. (b) Stress variations r in function of sliding velocity U.
4.1.1. Rupture Force Ft
max
The rupture force is not the JKR transverse unbinding force
Ft JKR (WoffEa3)1=2, calculated long ago by Savkoor [20], and discussed
more recently by Johnson [26]. Ft JKR corresponds to the force required
to open a type II adhesive fracture assuming that there is no barrier
due to bond breaking at the contact. In fact, what is observed is a much
higher force! The lens sticks to the glass by molecular bridges and
there is a barrier energy which prevents the rupture at Ft JKR. The
relevant parameter is the rupture stress, r0, to break the bridges:
r0 ¼ n 0 fv
ð2Þ
where n 0 is the equilibrium density of bridges, and fv Ub =l the
molecular force to break one bridge (Ub is the binding energy, l is
the bond rupture length).
The rupture force is:
Ft0 ¼ n 0 fv pa2
ð3Þ
as proposed long ago by Tabor [27].
This force is much larger than Ft JKR because it is proportional to
the square of the macroscopic length a, while Ft JKR a3=2.
The rupture force Ft
max
is thus equal to Ft0 [Eq. (3)].
4.1.2. Sliding: Friction Stress r(v)
When the lens slides on the substrate at velocity v ¼ dx=dt, the
friction stress can be written as:
r ¼ nfv þ fv
ð4Þ
776
F. Brochard-Wyart et al.
The first term corresponds to the energy dissipated to break the
bridges, and the second term is the friction of the rubber on the glass,
characterized by a friction coefficient f.1
The density, n, of bridges is n 0 at equilibrium, but as the lens moves
faster and faster, the density n(v) decreases. A kinetic equation for n
can be written as:
dn n 0 n vn
¼
dt
sm
l
ð5Þ
The first term is the chemical rebinding rate and the last term
the forced bind breaking rate. In steady sliding (v constant)
nðvÞ ¼ n 0 =ð1 þ ðv=VC ÞÞ, where VC ¼ l=son.
Typically, if we take l 1 nm, and for son the Rouse time of reticulated chains sR ¼ s0N2, we find VC 1 mm/s for N ¼ 102, s0 ¼ 10 10s,
in reasonable agreement with the data shown in Figure 4b.
The stress r(v) is then:
Ft
r0
þ fv
¼ rðvÞ ¼
pa2
1 þ ðv=VC Þ
ð6Þ
where f describes passive friction in the absence of bonds.
If r0 =VC > f; rðvÞ decreases at small v up to v ¼ Vm, and increases
again (Figure 9b). Slippage is unstable for v < Vm, which corresponds
to the minima of r(U).
Remark: we assume that fv ¼ Ub=l is a constant. In fact, as pointed
out first by Schallamach [4], fv depends slightly upon the pulling rate,
v, at which the breaking force is applied, fv ¼ ðkT=aÞ lnðv=V1 Þ, with
V1 ¼ V0 eUb=kT . This leads to an increase of r(v) at very low velocities.
This explains why the hard lenses on poorly silanated glass slide
smoothly at U < VM corresponding to the maxima of r(U); a stick-slip
transition occurs at VM. For VM < U < Vm, we expect stick-slip. For
U > Vm, we return to smooth sliding.
4.1.3. Stick-Slip Cycles for Soft Systems
The description of stick-slip in rubbers is simple because inertia is
negligible. We assume that the healing time, son, is very small. Then,
in the diagram r(v), the system follows the cycle ABCD (see Figure 9b).
4.1.3.1. Stick phase
. The force Ft increases linearly with time Ft ¼ Ft m þ KUt, where
Ft m ¼ rmpa2 is the residual stress at the minimum velocity, Vm.
1
If N is the number of units between cross links in bulk rubber, we can estimate
f gR =D where nR n0N and D N1/2a is the mesh size of the rubber.
Stick-Slip: Wet Versus Dry
777
. When Ft reaches Ft0, the contact starts to break: we move from A to
B. If we consider the local events, we find that a mode II fracture
occurs and advances with a velocity Vtip. The prediction of Ref. 24 is:
Vtip E
f
ð7Þ
Eq. (7) is derived in Ref. 24 from a transformation of elastic energy
into viscous losses.
4.1.3.2. Slip phase. The characteristic time of the slip phase is
TS a=Vtip. During this phase, the force is observed (and predicted)
to decrease linearly with time Ft ¼ Ft max ðFt max Ft min ÞVtip t=a.
To conclude, the model of stick-slip leads to the following
predictions:
. The rupture force Ft 0 ¼ r0pa2;
. The stick phase controls the period T of the stick-slip instability;
KUT ¼ Ft max Ft min ;
. The frequency of the stick-slip f ¼ T 1 is proportional to U;
f ¼
16
EUa
16
EU
Þ
;
9 ðFt max Ft min Þ
9p ðr0 rm Þa
ð8Þ
where r0 and rm are shown in Figure 9b, and Ft max is the rupture
force Ft 0 [Eq. (3)];
. The duration of the force drop is DTS ¼ af=E and the force decreases
linearly with time.
4.2. Discussion
From the plot of Ft (t) in the stick-slip regime, we can derive:
. The elastic constant of the spring, K;
. The tangential static detachment force, the stick-slip frequency, f,
and the slip velocity, Vs.
4.2.1. Elastic Modulus K
In the stick phase, Ft ¼ KUt þ Ft min. K is related to the contact radius,
a, and to the lens Young’s modulus E. For a hard lens and a ¼ 50 mm, we
measured K ¼ 100 N=m. For a soft lens and a ¼ 100 mm, K ¼ 14 4 N=m. We verify that K 16Ea=9, which means that the rigidity of
our system can be tuned either by the Young’s modulus, E, or by a
through the applied normal force.
778
F. Brochard-Wyart et al.
4.2.2. Tangential Detachment Force
Theoretically, Ft R ¼ r0pa2. From the maximum of Ft at each cycle,
we derive r0 for hard and soft rubbers on poorly silanated glass.
We find respectively, r0 hard ¼ 0.15 0.05 MPa and r0 soft ¼ 0.10 0.05 MPa.
4.2.3. Stick-Slip Frequency f
We observe that the period of the stick-slip is imposed by the stick
phase, which is much longer than the slip phase. According to Eq. (7),
the plot of f ¼ T 1 versus U is linear, and the slope is
4.0 104 0.4 104 m 1 for a R ¼ 1.04 mm hard lens. It leads to
Ft max–Ft min ¼ 3.6 mN and r0 – rm ¼ 0.18 MPa. For a soft lens, we also
find that f(U) is linear, with a slope of 10 103 10 103 m 1 leading
to Ft max–Ft min ¼ 2.4 mN, in agreement with data shown in Figure 7b,
and to r0–rm ¼ 0.05 MPa.
We see a dependence of f with R, but not with Fn [see Eq. (8)]:
E=a E4=3 =ðWR2 þ Fn RÞ1=3 . In the range of normal force explored
(Fn < 100 mN) the first term is dominant and we expect that the slope will
decrease with R as R2=3, in agreement with the data shown in Figure 8.
4.2.4. Stick-Slip Velocity Vs
From the Figure 7a for hard lenses, we see that the stick time
depends on U, while the very short slip time is nearly constant. We
can estimate Vs ¼ a=ts 5 103 m/s, which is of the same order of
magnitude as E=f ¼ 102 m/s. For a soft lens, we estimate
Vs 4 5 10 3 m/s.
4.2.5. Shape of the Curve Ft (t)
In the theoretical model [24], the force increases linearly in the stick
phase, and decreases linearly in the slip phase. We do see clearly a saw
tooth shape in Figure 7a.
5. SLIDING OF RUBBER LENS IMMERSED IN A LIQUID
We now study the friction for rubber lenses immersed in a non-wetting
liquid. The spreading parameter S ¼ cSR (cSL þ cLR) is negative (in
the opposite case, S > 0, the liquid is a lubricant, and a thin film wets
the contact: the friction is very low, and the sliding is always smooth).
In a wet JKR test where the lens is immersed in the liquid, on loading,
one measures the thermodynamic Dupré energy Won ¼ S. The presence
of the liquid leads to a broad range of adhesive energy, ranging from a few
mN/m in fluorinated silicone oils to 50 mN/m in water. For hard lenses in
silicone oils, Won ¼ 6.3 0.5 mN/m, whereas Won ¼ 4.0 0.5 mN/m for
Stick-Slip: Wet Versus Dry
779
the soft lenses [8]. The ratio S=E defines the elastic length, h0, which can
be measured directly from the ‘‘penny’’ shape of droplets intercalated
between the rubber and the substrate [28, 29]. For the silicone oils, one
finds for hard lenses h0 ¼ 60 10 Å, leading to jSj ¼ 7 mN/m, and for
water h0 ¼ 429 25 Å, leading to jSj ¼ 40 mN/m. There is a good agreement between S measured by capillarity and the JKR test.
When the lens is pressed against the plate a liquid film is squeezed
at the interface. The film is removed by drainage and dewetting: a dry
patch nucleates, and grows [30]. Some droplets are left at the interface
after the dewetting, and their shape allows us to measure S directly.
They are pulled out of the contact by the JKR pressure field, leading
to a perfect circular dry contact.
We describe now what happens when the substrate is moved at a
velocity U. The new feature is a forced wetting transition at a critical velocity, Uc, where the contact is lubricated (see Figure 10). Uc
FIGURE 10 The three lubrication regimes. For low velocities, the contact
remains dry. For intermediate velocities, the liquid invades partially the
contact. In oil, we observe two stationary dry contacts, whereas in water=
glycerol mixtures, liquid waves move in the direction of U (and in the opposite
direction of Schallamach waves shown in Figure 3). For high velocities, the
contact is completely invaded.
780
F. Brochard-Wyart et al.
can be deduced from a competition between forced wetting induced
by the shear, U, and dewetting. This leads to Uc ¼ S=g ðh0 =RÞ1=3 , in
good agreement with the experiments of Martin [9]. The adhesive contact is lost at Uc, this is the origin of hydroplaning of cars on wet roads.
We focus first on the slippage in the dry regime U < Uc, but we shall
see that a periodic wetting of the contact may appear in the stick-slip
regime if Vs > Uc.
5.1. Sliding on Non-hysteretic Substrate
We use a hard rubber lens, and a silanated substrate immersed in silicone
oil, or in water. We find that Ft is larger in water than in air and much
smaller in oil but the shear stress r ¼ Ft=A is the same as in air. The areas
verify Awater > Aair > Aoil, this corresponds to a decrease of adhesion
energy W0 water 50 > W0 air 40 > W0 oil 5 mN/m and this explains
the huge variation of the friction forces. We conclude that sliding is
smooth, and the rubber-substrate friction is the same as in air.
For the silicone oil used (g ¼ 500 gwater), the forced wetting velocity
occurs at Uc ¼ 20 mm/s. At Uc we observe a wetting transition between
dry and lubricated contact. At the transition, in a very short range of
velocities U, we have a semi-stable lubricated regime shown in Figure
10. The tangential force measured by Clain [8] drops by a factor of
order ten at the wetting transition.
For water, Uc 1 mm/s and the contact remains dry in the range of
velocities studied here.
For viscous water-glycerol mixtures, we see a lubrication transition
at a velocity which depends on glycerol concentration. At Uc, (370 mm/s
for a mixture of viscosity g ¼ 40 gwater) when we transit from dry to wet
friction, we observe a semi-lubricated regime with waves of liquid
going through the contact (in a direction opposite to Schallamach
waves) shown in Figure 10.
5.2. Sliding on Hysteretic Substrates
5.2.1. Hard Rubber Lens
. In fluorinated silicone oils: On a poorly silanated plate, with a
silicone oil of viscosity 400 cS, we observe the same behaviour as on
silanated glass: smooth sliding and lubrication of the contact. The
stick-slip regime is masked by the wetting transition, because Uc < Vm.
. In water: On a silanated plate, a hard rubber lens immersed in
water slides smoothly at all velocities, U. On a poorly silanated
plate, we observe a sliding instability for U > VM. Below VM, the
sliding is smooth. Above VM we see a stick-slip regime. The shape
Stick-Slip: Wet Versus Dry
781
of the contact during a stick-slip period is shown on Figure 3c. VM is
the same in water and in air. The difference is that small water
droplets are carried into the contact, and remain trapped.
. In water-glycerol mixtures: With water-glycerol mixtures, we
can increase progressively the viscosity. For gmixture < 10gwater
are, we observe the three regimes: smooth sliding, stick-slip, and
lubrication transitions. As soon as gmixture > 10gwater, Uc < Vs and
we lose the stick-slip regime.
5.2.2. Soft Lenses
. With silicone oils and a poorly silanated substrate, we see only a
lubricated regime, even for U < Uc, in the range of velocities
studied. The film penetrates the front part of the contact, while
the stick part is fluctuating, showing stick-slip instabilities. We
interpret this behaviour by a forced wetting of the contact induced
by the slip. Because the viscosity is large, and jSj small, dewetting
[30] is slow and the contact cannot dewet during the short period,
T, of stick-slip.
. With water, for both silanated and poorly silanated surfaces, we
observe ‘‘wet’’ Schallamach waves, and stick-slip above a threshold
velocity. We show in Figure 11 the contact zone during a stick-slip
period. We see the black contact becoming suddenly bright, showing
a short appearance of a water film induced by the slip. This film
leads to a drop in the friction, and the fluctuations of the tangential
force have a larger amplitude. Because jSj is large and g small, the
water film is able to dewet very fast. The period of the stick-slip
cycle versus U is shown in Figure 12. It does not depend on Fn.
. With water-glycerol mixtures, for both silanated and poorly silanated surfaces, jSj is large, but g can increase from gwater to
103gwater. Uc and the velocity of dewetting decreases. Our aim
FIGURE 11 Stick-slip of a soft lens on poorly silanated glass in water.
The sliding velocity U is 176 mm=s.
782
F. Brochard-Wyart et al.
FIGURE 12 Stick-slip frequency f versus sliding velocity U for a soft lens
immersed in water-glycerol mixtures (the substrate is poorly silanated).
was to see if the period of the stick-slip depends on the mixture viscosity. We observe stick-slip instabilities at low velocities, and a wetting transition at a velocity which decreases as the proportion of
glycerol in the mixture increases. On the other hand, the viscosity
of the mixture has no effect at all on the stick-slip frequency: all data
fall on the same linear curve.
To conclude, for wet stick-slip observed with pure water or waterglycerol mixtures, a liquid film can be squeezed in the contact during
the slip phase, and dewets very fast. The rupture force remains
the same, as for stick-slip in air. But the amplitude of the force
fluctuations increases a lot, showing that friction is reduced in the
wet phase. The frequency of the stick-slip is not sensitive to the viscosity, because the dewetting is very fast, and the rupture force the
same. For all water-glycerol mixtures, including pure water,
the frequencies, f(U), follow the same linear relation. The slope of
the stick-slip frequency versus sliding velocity is 6.3 103 0.4 m 1.
This leads to Ft max–Ft min ¼ 3.7 mN, which is similar to what is
observed in air, and to r0 rm 0.07 MPa.
6. CONCLUSION
We have studied the friction of soft lenses on model and hysteretic substrates and we observed dry friction in air, and wet friction in a liquid.
The differences between advancing Won and receding Woff adhesion
energy is a measure of contact hysteresis and is due to the binding
Stick-Slip: Wet Versus Dry
783
of polymer chains on active sites of the substrate. When the rubber is
detached, elastic energy is stored in the stretched polymer chains,
which increases Woff.
On model surfaces, we observe a smooth sliding. The friction forces
are not the same in air and in liquid because the adhesion energy
increases from fluorinated oil, to air, to water. On the other hand,
the stress, r ¼ Ft=A, with A the contact area, is the same: this means
that the contact is dry, and not modified by the surrounding liquid.
Above a critical wetting velocity, we observe a transition from a dry
to a lubricated contact. The friction decreases by a factor of order
ten. This transition is the source of hydroplaning for cars.
On hysteretic substrates, we observe a smooth sliding only at low
velocities, U < VM. Above VM, we see stick-slip instabilities. The
contact area and the tangential force vary periodically. The stick-slip
frequency increases linearly with U. We interpret the sliding instabilities by the presence of active sites on the substrate, which bind
polymer chains of the rubber. Using a recent model, we explain the
amplitude of the oscillating tangential force, the period of the stick-slip
cycle, and the velocity of decohesion from the stick state. For soft lenses,
we find that the sliding at low U is replaced by Schallamach waves.
We compare dry and wet friction. For lenses immersed in a liquid, we
see the same regime as for lenses in air, as long as VM < Uc. The difference is that now Schallamach waves are wet (the rim is full of liquid),
and stick-slip is coupled to wetting: the slip velocity, Vs, is greater than
Uc, and a liquid film is deposited at each period. This film is seen by interferential microscopy, and also by an increase of the amplitude of tangential forces fluctuations. If VM > Uc, we observe only smooth sliding. At
Uc, in all cases, the contact is lubricated and the sliding becomes smooth.
In all cases studied here, we observe a periodic stick-slip, with a linear
relation between frequency and velocity. From the slope of f(U) and the
size of the contact, we can extract the variation, Ft max–Ft min, of the
tangential force simple (RICM) observations allow to derive tangential
forces.
In all these experiments reported here, the substrate and the
rubber lens are smooth at atomic scales. In the near future, we will
study the role of substrate roughness.
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