Slippery but tough - the rapid fracture of lubricated frictional interfaces

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by  Elsa Bayart

Slippery but tough - the rapid fracture of lubricated frictional interfaces
E. Bayart,1 I. Svetlizky,1 and J. Fineberg1
1The Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem, Israel
(Dated: April 25, 2016)
We study the onset of friction for rough contacting blocks whose interface is coated with a thin
lubrication layer. High speed measurements of the real contact area and stress fields near the
interfacerevealthatpropagatingshearcracksmediatelubricatedfrictionalmotion. Whilelubricants
reduce interface resistances, surprisingly, they significantly increase energy dissipated, Γ, during
rupture. Moreover, lubricant viscosity affects the onset of friction but has no effect on Γ. Fracture
mechanics provide a new way to view the otherwise hidden complex dynamics of the lubrication
layer.
6
1
PACSnumbers: 46.55.+d,46.50.+a,62.20.Qp,81.40.Pq
0
2

r Lubricationofsolidsurfacesisgenerallyusedtoreduce by LEFM. While reducing static friction by facilitating
p
frictionalresistancetoslidingmotionandtopreventma- rupturenucleation,wewillshowthat,surprisingly,lubri-
A
terial wear [1]. Effects of fluids on the frictional proper- cants make solid contacts effectively tougher, increasing

2 ties of an interface are of particular significance in geo- thefractureenergyoftheinterface(thedissipatedenergy
2 physics, since tectonic faults are generally lubricated by per unit crack extension). Moreover, while the macro-

interstitial water or melted rocks [2–5]. Along spatially scopic frictional resistance of the interface depends on
]
t extended multi-contact interfaces, which are considered the lubricant viscosity, the fracture energy does not. We
f
o here, much fundamental understanding of the collective use this to demonstrate that nucleation and propagation
s mechanisms responsible for the reduction of friction due of frictional ruptures are independent processes.
.
t to lubrication is still lacking [6–8]. While the sliding dy-
a
m namicsoflubricatedsystemsisanactivefieldofresearch a FN b 0.6 dry
[9–11], the mechanisms mediating their transition from
d- stick to slip remain largely unexplored. At the micro- y 10 (cid:3015) bmoixuenddary
L = 150 mm 0 (cid:1832)
n scopic level, stick-slip mechanisms have been discussed x m ⁄0.3
o m (cid:1832)(cid:3020)
for decades [12, 13]. Within single contacts, confined m
c m
[ lubrication layers, typically at nanometric sizes, exhibit 0 F 0
enhanced strength [14–16]. 3 S 0 100 200

2 t(s)
Alongspatiallyextendedroughinterfaces,therealcon-
v
tactareaisdefinedbyalargeensembleofsinglecontacts
5 FIG.1: Experimentalsetupandstick-slipbehavior. (a)Nor-
8 (asperities) that couple contacting elastic blocks. The malF andshearF forcesareappliedtocontactingPMMA
N S
0 real contact area, A, is generally orders of magnitude blocks. Shear is applied uniformly via translation of a rigid
0 smaller than the apparent one [1, 17, 18]. Here we con- stage. Strain gage rosettes measure the 3 components of the
0 sider rough surfaces in the boundary lubrication regime, 2D-strain tensor at 14 locations along and 3.5 mm above the
2. where the contacting surfaces are covered by a thin lu- interface, while the real contact area is measured optically.
0 bricant layer [19]. The discrete asperities in this regime (b) Loading curves, FS/FN vs time, are plotted for typical
6 experiments,withFN =4000N: dry(solidblueline),bound-
stillbeartheentirenormalload;theyarenotentirelyim-
1 arylubricated(dashedgreenline)and,forcomparison,inthe
: mersedinthefluidlayerasinthefulllubricationregime. mixedlubricatedregime(dottedredline). Thelubricantused
v
The mixed lubrication regime is an intermediate region, is a hydrocarbon oil (TKO-77).
i
X wherethenormalloadispartiallybornebysolidcontacts
r and partially by the liquid layer. We describe experiments where two blocks of
a
In dry friction, the onset of motion is mediated by poly(methylmethacrylate) (PMMA) are first pressed to-
rupture fronts propagating along the frictional interface gether with normal forces, F , of 2500 < F < 7000N.
N N
[20, 21]. These fronts are true singular shear cracks; the Shear forces, F , are then applied uniformly, as the bot-
S
strain fields during their propagation are well-described tom block is translated via a rigid stage, until stick-
byLinearElasticFractureMechanics(LEFM)[22]. Fric- slip motion initiates (Fig. 1). A detailed description
tionalrupturearrestisalsogovernedbythesameframe- of the setup is given in [22]. PMMA has a rate-
work [23, 24]. dependent Young’s modulus 3 < E < 5.6GPa and Pois-
Here, we examine the mechanisms coming into play son ratio ν = 0.33. PMMA’s Rayleigh wave speed
p
when motion initiates within lubricated interfaces, in the is c = 1255 ms−1 for plane strain conditions. Top
R
boundarylubricationregime. Wefirstfindthatinterface and bottom blocks have respective x×y×z dimensions
rupture still corresponds to the shear cracks described 150×100×5.5mm and 200×30×30mm. The contact-
2
a b x10‐3 x10‐3 x10‐3
1 A/A0 0.3 lubricant 0.2 0
1
ms) (cid:2013)(cid:3051)(cid:3052) (cid:2013)(cid:3052)(cid:3052) (cid:2013)(cid:3051)(cid:3051)
( Δ Δ Δ
t
0.5 dry -0.5
0 0 0
0 150 -40 0 40 -40 0 40 -40 0 40
x (mm) x-x (mm)
tip
FIG. 2: Singular interfacial shear cracks govern friction initiation. (a) The spatio-temporal evolution of the contact area
A(x,t) of a typical lubricated interface (hydrocarbon oil, TKO-77). Each line is a snapshot in time of A(x,t), normalized by
A =A(x,0) immediately prior to the event. Here, a rupture accelerates to a propagation velocity c =0.92c . The rupture
0 f R
tip,x (t)arethelocationswhereA(x,t)dropssharply. (b)Variationofthestrainfield∆ε (x−x )withthedistancefrom
tip ij tip
rupturetips,x ,forrupturespropagatingalongdry(blueline)andlubricated(greenline,rupturepresentedin(a))interfaces.
tip
In both, the applied normal stress was (cid:104)σ (cid:105) = 7±0.5 MPa and strains were measured at x = 77mm, where c ∼ 0.3c .
yy f R
Black solid lines are fits to the LEFM solution for y = 3.5 mm (Eq. 1). The only fitting parameter is the fracture energy;
Γ =2.6±0.3Jm−2 for the dry and Γ =23±3Jm−2 for the lubricated interfaces.
dry lub
ing flat surfaces of the top and bottom blocks were (top) tion coefficient (i.e. the shear force threshold). The am-
optically smooth and (bottom) with a surface roughness plitudes of the force drops, however, are larger than for
of 0.5µm r.m.s. Experiments were all performed with dry friction. In the boundary lubrication regime, this
the same two blocks, to negate any effects due to surface pattern is extremely robust, and is independent of the
preparation or roughness. During each sliding event, an nature and quantity of the lubricant. For completeness,
arrayof14straingagesrecordedthe3componentsofthe atypicalloadingcurveinthemixedlubricationregimeis
2D-strain tensor, ε , 3.5 mm above the interface, each included in Fig. 1b, where F /F thresholds are further
ij S N
at 106 samples/s. Corresponding stresses, σ , are calcu- reduced. Motion in this regime is not addressed here.
ij
lated from ε after accounting for the viscoelasticity of As in dry friction, each sliding event in the bound-
ij
PMMA (see [24]). In parallel, the real area of contact, ary lubrication regime is preceded by propagating rup-
A(x,t), was measured at 1000 x 8 locations at 580000 ture fronts that break the solid contacts forming the in-
frames/s, using an optical method based on total inter- terface, as shown in Fig. 2a. Macroscopic sliding only
nalreflection(see[22])whereincidentlightonlytraverses occurs when a front traverses the entire interface [24–
the interface at contacts, and is otherwise reflected. 27]. For steady rupture fronts moving at a velocity c ,
f
ε (x,t)=ε (x−c t). Usingthisandtheopticallyiden-
Experiments of lubricated friction were performed us- ij ij f
tified location of the rupture tip, x (t), we converted
ingsiliconeoilswithkinematicviscosities,ν=5,100and tip
ε (x,t) to spatial measurements ε (x−x ) [22]. As
104 mm2s−1 and a hydrocarbon oil (TKO-77, Kurt J. ij ij tip
in the example of Fig. 2b (blue line), rupture fronts in
Lesker Company) of ν ∼ 200 mm2s−1. Lubricants were
dryfrictionareshearcrackswhosestressfieldvariations,
applied to either or both of the contacting surfaces and
∆σ (r,θ), are quantitatively described by LEFM, with
then wiped. Our results are not appreciably affected by ij
respect to the crack tip (r =0) [22]:
the wiping procedure (number of wipes, application or
not between experiments) or by the cleaning (soap, wa-
K (c )
terandisopropanol). PMMAandthelubricantsusedare ∆σ (r,θ)= √II f ΣII(θ,c ), (1)
ij ij f
2πr
nearly index-matched: PMMA-1.49, TKO-77-1.48, and
silicone oils-1.42. Hence, under total internal reflection,
where ΣII(θ,c ) is a universal angular function and the
incident light will be totally transmitted where gaps be- ij f
coefficient, K (c ), is called the stress intensity factor
II f
tween asperities are filled with liquid. As light could be
[28]. ∆σ expresses the stress changes between the ini-
ij
transmitted via capillary bridges across contacting sur-
tially applied and residual stresses along the frictional
faces, we only consider relative variations in light inten-
crack faces. ∆σ are related to measured strain varia-
ij
sity. At the onset of motion, the observed contact area
tions∆ε viathedynamicYoung’smodulusandPoisson
ij
variations (see below), demonstrate that air, not lubri-
ratio of PMMA. LEFM relates K to the fracture en-
II
cant, fills the gaps between contacts. This provides vali-
ergy, Γ, the energy dissipated per unit crack advance;
√
dation that the experiments take place in the boundary
K ∝ f(c ) Γ, where f(c ) is a known universal func-
II f f
lubrication regime.
tion [28].
When sheared, the lubricated system undergoes stick- InFig.2bwecomparemeasurementsof∆ε (x−x )
ij tip
slip motion (Fig. 1b). Drops of F in the loading curves during rupture front propagation for dry and lubricated
S
correspond to slip events with macroscopic relative dis- interfaces. Following Eq. 1, fitting the three strain com-
placement of the blocks. The lubricant layer affects the ponentsprovidesadynamicmeasurementofK [22,28]
II
macroscopicfrictionalresistance,reducingthestaticfric- and,therefore,ameasurementofΓ. AsFig.2bexplicitly
3

shows, the agreement between measured ∆εij(x−xtip) 3 a 0 b
for the lubricated interface and the LEFM solution is ex- a) (cid:2026)(cid:3031)(cid:3045)(cid:3045)(cid:3032)(cid:3052)(cid:3046) Δ(cid:1827)
cinetlleernfat.ceHaernecseh,eraurpcturarcekssp.roSpuarpgaritsiinngglayl,oΓn,gfaorlutbhreicsaatmede (MP(cid:3052)2 (cid:2026)(cid:3021)(cid:3045)(cid:3012)(cid:3032)(cid:3046)(cid:3016) A(cid:4667)/(cid:2868)-0.6 (cid:1850)(cid:3030)
appliednormalload, isanorderofmagnitudegreaterfor (cid:2026)(cid:3051) (cid:2026)(cid:3046)(cid:3045)(cid:3036)(cid:3032)(cid:3039)(cid:3046) x hdyrdyrocarbon A(cid:3398)
1 silicone (cid:4666) -1
the lubricated interface, Γ , than for the dry one, Γ .
lub dry -40 0 40 -10 0 10
FortheexamplespresentedinFig.2b,Γ =23±3Jm−2 x-x (mm) x-x (mm)
lub tip tip
while Γdry = 2.6 ± 0.3 Jm−2. In Fig. 3a we present c (cid:2026)(cid:3043)(cid:3032)(cid:3028)(cid:3038)8
(cid:3021)(cid:3012)(cid:3016)
∆ε (x−x )fordryandlubricated(hydrocarbon)inter-
ij tip √
faces, when rescaled by 1/ Γ. We find that the rescaled Pa) (cid:2026)(cid:3046)(cid:3043)(cid:3036)(cid:3032)(cid:3039)(cid:3028)(cid:3038)
dry and lubricated strain fields are indeed identical. (M(cid:2026)(cid:3031)(cid:3043)(cid:3045)(cid:3032)(cid:3052)(cid:3028)(cid:3038)4
WhatdeterminesΓ? Indryfriction,whencontactsare (cid:3051)(cid:3052) (cid:2026)(cid:3031)(cid:3045)(cid:3045)(cid:3032)(cid:3052)(cid:3046)
(cid:2026) (cid:2026)(cid:3045)(cid:3032)(cid:3046)
plastically deformed, Γ grows linearly with the normal (cid:3021)(cid:3012)(cid:3016)
(cid:2026)(cid:3045)(cid:3032)(cid:3046)
load [1, 24]. Extracting Γ from the rescaling procedure, 0 (cid:3046)(cid:3036)(cid:3039)
Fig. 3b shows that Γ indeed remains proportional to the 0 (cid:1856)(cid:3030)(cid:3031)(cid:3045)(cid:3052) 4 (cid:1856)(cid:3030)(cid:3046)(cid:3036)(cid:3039) (cid:1856)(cid:3030)(cid:3021)(cid:3012)(cid:3016) 8
slip(m)
averagenormalstress,(cid:104)σ (cid:105),intheboundarylubrication
yy
regime. Moreover, the value of Γ is unaffected by the lu- FIG. 4: Measurements of σres, X , σpeak and d . (a) Shear
xy c xy c
bricant viscosity; Γ is constant for viscosity variations of stress as a function of the distance from x for ruptures
tip
5 < ν < 104mm2s−1 in silicon oils. We do, however, propagating(c ∼0.3c )alongadry(bluecrosses)andlubri-
f R
find that Γ strongly depends on the lubricant composi- cated interfaces with hydrocarbon oil (green diamonds) and
tion; TKO-77 has values of Γ about 3 times larger than silicone oil with ν = 104mm2s−1 (red circles). Γ =2.6, 9
and 23 Jm−2 for respectively dry, silicone and hydrocarbon
in all of the silicon oils used. For a given c , increased
f
oils for (cid:104)σ (cid:105) = 7 MPa. The blue and green plots are mea-
values of Γ induce increased shear stress drops during yy
surements presented in Fig. 2b, where stain variations are,
rupture propagation. The increased shear force drops in
instead, presented in terms of the absolute stress values. (b)
loading curves (e.g. Fig. 1b) are partially caused by this Reduction of the contact area A−A , normalized by the to-
0
largestressdrop,withtheremainderduetomotionafter tal drop in A, ∆A=A −A , as a function of the distance
0 res
the rupture passage. from x for the three experiments in (a). The dissipative
tip
zone size X is defined as the length scale where a 60% drop
c
Γa0.06x10‐3 b dsirliycone 5 mm2s‐1 oafnd∆dAcoacrceuersst.im(ca)teSdhweairthsitnretshsevlsinselaipr sdliispt-awnecaekewnhinergemσoxpeydaekl
/ 30 ssiilliiccoonnee  110040 mmmm22ss‐‐11 [29]. Respectively for dry and lubricated with silicone and
Δ(cid:2013)/Γ(cid:3051)(cid:3052)0.004 -2(J m)20 ehtyhdyrloecnaer bgolyncol ha4tir.yo6edn,r25oo..4vc1,aerra1bntaohdnned8bo.2l1iul.sM7ei(MnPgtarPeeraaef,nancd,peresdea,dckr)aeshrsteiardet2usc.sha4ele,sds4σt.ar4xpreeyesaaasknes,dsp,u7rsdoienµvfigimdnEe.esdqIt.nihtn2ee,g(dararra)ye-,
(cid:3052) 10
(cid:3052) (lubricated) fracture energy.
(cid:2013)
Δ 0

Γ 0 0
3 5 7 9 11
/  > (MPa)
(cid:3051)(cid:3051) yy rupture occurs when the shear stress on the interface
(cid:2013)
Δ a‐legend: (cid:3407)(cid:2026)(cid:3052)(cid:3052)(cid:3408) reaches a maximal value, σpeak. Slip is then initiated
-0.16 7 MPa (dry) 7 MPa xy
-40 0 40 10 MPa 5.7 MPa and σ is reduced to the residual value σres over a slip
x-x (mm) 8.3 MPa 4.4 MPa xy xy
tip distance d . While simple, this model contains the main
c
features of the regularized dissipative zone, and d pro-
FIG. 3: Dependence of the fracture energy with normal c
vides an accurate estimate of the sliding distance, typi-
stress. (a) Comparison of ∆ε (x − x ) for dry and lu-
ij tip √
cally the asperity size. The fracture energy is expressed
bricated experiments, when normalized by Γ, for different
normal loads. Units are (Pam)−1/2. Superimposed are the as Γ= 12(σxpeyak−σxreys)dc. More sliding occurs after rup-
dryexperimentinFig.2band5lubricated(TKO-77)experi- ture passage, dissipating more energy. Therefore, the
mentswherecf ∼0.3cR with(cid:104)σyy(cid:105)asinthelegend,yielding energy dissipated by the rupture is only part of the total
Γdry = 2.6Jm−2 and ΓTKO = 12.3, 18.2, 23, 25 and 29.5 energy dissipated during a slip event. An increase of Γ
Jm−2. (b) Γ is measured by fitting the strain field with the
can be induced by increased values of either σpeak or d ,
LEFM solution (as in (a)) for both dry and lubricated inter- xy c
or a decrease of σres.
faces vs F . All Γ vary linearly with F , Γ is independent xy
N N
ofthelubricantviscositywhilehighlydependentonlubricant As Fig. 4a shows, σres is indeed strongly reduced by
xy
composition. thelubricant. Themagnitudeofthereductionrelativeto
the dry interface depends on the nature of the lubricant.
Why does the lubricant increase Γ? We consider the ItisgreaterforsiliconeoilthanforTKO-77. Wecannot
simplest (linear slip-weakening) description of the dis- measure σpeak directly, as our strain gages are located
xy
sipative zone near a rupture tip [29]. In this model, abovetheinterface[22]. Thelinearslipweakeningmodel
4

2 a(cid:2026)(cid:3051)(cid:3043)(cid:3052)(cid:3045)(cid:3032) b (cid:2026)(cid:3051)(cid:3045)(cid:3052)(cid:3032)(cid:3046) 5 mm2s‐1 and therefore, an interface’s “static” frictional strength.
(cid:4667) 100 mm2s‐1 Hence, nucleation is the key in understanding initial in-
a
P 104mm2s‐1 terfacial strength, although processes determining how
M
(cid:4666)(cid:3052) 1 and at what stress levels nucleation takes place remain
(cid:2026)(cid:3051) enigmatic [30, 31]. Fig. 5 demonstrates that the lubri-
0
cantviscositydirectlyaffectstheinitiallyimposedstress,
0 50 100 150 50 100 150
x (mm) x (mm) σpre, needed to nucleate the rupture. Using silicone oils
xy
c d ofdifferentviscosities,inexperimentsperformedwiththe
)S -1s) same normal stress profile, Fig. 5 reveals that the higher
p( c(m f νp,enthdeolnowνe.rσσpxrperyed.eOtenrmthineeostthheershtaantidc,fσrixrceytsiodnoceosenffioctiednet-
xy
µ =F /F . Hence, as Fig. 5c shows, µ is significantly
0.1 0.3 0.5 0 50 100 150 s S N s
 x (mm) dependent on ν, as suggested by earlier studies [1, 32].
S
As Γ and σres are ν−independent (Figs. 3 and 5b),
xy
FIG. 5: Dependence of the macroscopic frictional resistance LEFM predicts that the only effect of σpre should be on
xy
wσriteshfνo.rPinrotefirlfeascoefs(lau)btrhiceaitneidtiawlitσhxpryseilaicnodn(ebo)irlseshidauvainlgstνre=sse5s therupturedynamics. Alargerσxprye yieldsfasterrupture
xy fronts [28] as verified in both dry friction [33] and in ice-
(diamonds),100(circles)and104 (squares)mm2s−1. Normal
quakes [34]. This is born out by the examples shown in
stressdistributionsareidenticalforthethreeexperimentsand
Γ=6.7±0.3Jm−2. σres doesnotdependontheviscosityof Fig. 5d; for the same σyy profile, the higher ν, the lower
thesiliconeoil. (c)Hisxtoygramsofthestaticfrictioncoefficient σxprye,andtheslowertherupturefront. Ourresultsimply
µ ofslidingeventsforlubricatedinterfaceswithsiliconeoils that the reduction of µ is purely the result of rupture
S s
ofdifferentviscosity. Around50eventsareconsideredforeach front nucleation at a reduced threshold, due to higher ν.
ν. Higher ν yield lower frictional resistance. (d) 3 examples Lower initial stresses do not prevent interfacial rupture
of c (x) for the lubricants in (a) and (b). The larger ν, the
f propagation as long as the elastic energy stored by block
slowerthefront. Dashedlinedenotesc . Symbolsandcolors
R deformation is sufficiently above the energy dissipated
in (a-d) as in (b).
during the rupture process (Fig. 4c).
We have shown that, in the boundary lubrication
provides us with a way to accurately estimate σpeak by regime,interfacialresistanceisreducedduetofacilitated
xy
measuringthesizeofthedissipativezoneX ,thedistance nucleation of the rupture front (Fig. 5), while contacts
c
behindthecracktipoverwhichcontactsarebeingbroken become tougher (Fig. 3). The increase of Γ is explained
[29]: byastrengtheningofthecontacts(increasedσpeak)cou-
xy
pled to reduced σres, with resultant increases in slip
(cid:115) xy
9π ΓE distances d . The reduced σres and larger d may re-
σpeak =σres+ (2) c xy c
xy xy 32(1−ν2)X sult from the fact that, while in motion, the lubricant
p c
layer facilitates slip. The increased value of Γ and σpeak,
xy
Xc is the scale over which A(x) drops from its initial however, are both new and intriguing observations. In
to residual value. In Fig. 4b we compare Xc for dry and ourexperiments,onroughmulticontactinterfaces,pres-
lubricatedexperiments(see[22]fordetails). Wefindthat sures at a single contact reach the yield stress of PMMA
Xc is not significantly affected by the lubricant layer; its (∼500 MPa) [18], suggesting that nanometric lubricant
value (for cf ∼0.3cR) is approximately 3 mm. Inserting layerscouldwellbetrappedbetweenasperities. Atthese
this value in Eq. 2, we see that σpeak is not reduced extreme conditions the contribution of capillary bridges
xy
by the lubricant and is even significantly increased when is negligible to both the frictional resistance and Γ (see
TKO-77 is used (Fig. 4c). The dynamic measurements Supp. Mat.). At a microscopic level, the physics of lu-
of the stress drop, σpeak −σres coupled with Γ yield a bricatedsinglecontactinterfacesarebothinterestingand
xy xy
quantitativeestimateofd . Measurementsofσpeak,σres, puzzling. Experimentsrevealthatfluidlubricationlayers
c xy xy
Γ and d are presented Fig. 4c. These measurements confinedtonanometricscalescantransitiontosolids[14–
c
indicate that increases in Γ are therefore explained by 16]. Otherrecentexperimentsonsimilarsystemssuggest
increased stress drops, coupled to larger slip distances. that the nature of fluid layers does not change, but high
Wehaveseenthat,intheboundarylubricationregime, systemstiffnessresultsfromcouplingofthefluidtoelas-
while the contacts become tougher, requiring a larger tic deformation of the surrounding medium [35]. While
amount of energy (Γ) to break (Fig. 3), the macroscopic our rough, multi-contact system is far from these ideal
frictional resistance is actually reduced (Fig. 1b). These cases,itisinterestingthatlubricantstrengtheningindeed
intriguing results are not contradictory; we show here takes place. Obtaining a fundamental understanding of
that rupture nucleation and dissipation are independent the dynamics of the lubrication layer and its associated
processes. dissipative properties in such a disordered system is an
Rupture nucleation determines the initial stress levels important and interesting challenge.
5
WeacknowledgesupportfromtheJamesS.McDonnell 374, 607 (1995).
Fund Grant 220020221, the European Research Council [17] J. Greenwood and J. Williams, Proc. R. Soc. Lond. A
(Grant No. 267256) and the Israel Science Foundation 295, 300 (1966).
[18] J. H. Dieterich and B. D. Kilgore, Tectonophysics 256,
(Grants76/11and1523/15). E.B.acknowledgessupport
219 (1996).
from the Lady Davis Trust. We thank G. Cohen for
[19] B.Hamrock,S.Schmid,andB.Jacobson,Fundamentals
comments.
of fluid film lubrication (CRC Press, 2004).
[20] S. M. Rubinstein, G. Cohen, and J. Fineberg, Nature
430, 1005 (2004).
[21] K. Xia, A. J. Rosakis, and H. Kanamori, Science 303,
1859 (2004).
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