Formation and rupture of capillary bridges in atomic scale frictionItay Barel, Aleksander E. Filippov, and M. Urbakh Citation: J. Chem. Phys. 137, 164706 (2012); doi: 10.1063/1.4762863 View online: http://dx.doi.org/10.1063/1.4762863 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v137/i16 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

THE JOURNAL OF CHEMICAL PHYSICS 137, 164706 (2012)

Formation and rupture of capillary bridges in atomic scale frictionItay Barel,1 Aleksander E. Filippov,2 and M. Urbakh1

1School of Chemistry, Tel Aviv University, 69978 Tel Aviv, Israel2Donetsk Institute for Physics and Engineering of NASU, 83144 Donetsk, Ukraine

(Received 29 July 2012; accepted 4 October 2012; published online 31 October 2012)

While formation of capillary bridges significantly contributes to the adhesion and friction at micro-and nanoscales, many key aspects of dynamics of capillary condensation and its effect on frictionforces are still not well understood. Here, by analytical model and numerical simulations, we addressthe origin of reduction of friction force with velocity and increase of friction with temperature, whichhave been experimentally observed under humid ambient conditions. These observations differ sig-nificantly from the results of friction experiments carried out under ultrahigh vacuum, and disagreewith predictions of thermal Prandtl-Tomlinson model of friction. Our calculations demonstrate whatinformation on the kinetics of capillary condensation can be extracted from measurements of frictionforces and suggest optimal conditions for obtaining this information. © 2012 American Institute ofPhysics. [http://dx.doi.org/10.1063/1.4762863]

INTRODUCTION

Formation of capillary bridges is a common phenomenonin nature that significantly contributes to the adhesion andfriction of biological and artificial micro- and nanoscale ob-jects. Examples are the flow of granular materials,1 adhe-sion of insects or Geckos to surfaces,2 friction and stictionof micro- and nanoelectromechanical systems.3 It has beenfound that condensation of capillary bridges may be responsi-ble for frictional ageing that is manifested as logarithmic in-crease of static friction with time during which two surfacesare held in stationary contact.4 Capillary bridges play a crucialrole in the operation of atomic force microscopy (AFM) un-der humid ambient conditions and their formation often dom-inates the measured forces.5–9 Recent investigations of depen-dencies of frictional force on hold time, �t, velocity, V, andtemperature, T, suggested that capillary condensation is ther-mally activated.4–8 It has been found that under ambient con-ditions the nanoscopic friction force decreases linearly withlnV at low velocities, and increases with T.5–7 These observa-tions disagree with predictions of thermal Prandtl-Tomlinsonmodel of friction10–12 and with results of experiments carriedout under ultrahigh vacuum, which show a logarithmic in-crease of friction force with velocity and reduction of frictionwith temperature.13, 14 In order to explain the observed loga-rithmic decrease of friction force with velocity a phenomeno-logical model has been proposed5 that is based on the ther-mally activated mechanism of nucleation of capillary bridgesin a gap between asperities on the contacting surfaces. Themodel suggested that higher velocities correspond to shortercontact times and hence to smaller number of water bridgesformed between the tip and substrate, and as a result the fric-tion force decreases with V.

Despite importance of water bridges in frictional phe-nomena and the growing efforts in the field, many key aspectsof dynamics of capillary condensation and its effect on fric-tion forces are still not well understood. The present studyaddresses this problem through an analytical model and nu-

merical simulations. The results obtained allow to explain ex-perimental observations5–7 and show what information on thekinetics of capillary condensation can be extracted from mea-surements of friction forces.

MODEL

In our model and simulations we consider a rigid tip withmass M and center-of-mass coordinate X that interacts withthe underlying surface through an array of contacts represent-ing capillary bridges. The tip is pulled along the surface with aconstant velocity V, through a linear spring of spring constant,K (see Fig. 1). In accordance with recent observations,5–7

we assume that nucleation of capillary bridges occurs pref-erentially between asperities at the tip and substrate surfaces,whose contact radius, a0, is much smaller than the AFM tipradius, R. The asperities are uniformly distributed over the tipand the substrate surfaces.

Nucleation of capillary bridges is a thermally activatedprocess,4–8 and its rate, kon, can be described by the equation

kon(ri) = ω0on exp[−�Eon(ri)/kBT ], (1)

where ω0on, �Eon(ri) are the attempting frequency and the bar-

rier height for the bridge nucleation at the i-th asperity, and ri

is a lateral distance between this asperity and the tip apex. Theenergy barrier is proportional to the liquid volume needed tonucleate the liquid bridge, v = a2

0h (ri), where a20 is a typi-

cal nucleation area and h is the gap between the surfaces atthe nucleating site. Once a liquid bridge is condensed, it fillslocally the volume surrounding the nucleating site, until theKelvin equilibrium condition for the radius of curvature met.

Considering the system geometry that imitates a hemi-spherical AFM tip of radius R above a sample surface (seeFig. 1), we find that the average gap, h, grows with a distancefrom the tip apex, ri, as h (ri) = h0

(1 + r2

i /2Rh0), where h0

is a minimal gap between the tip and the surface. The distribu-tion of gaps leads to a broad distribution of activation energy

0021-9606/2012/137(16)/164706/6/$30.00 © 2012 American Institute of Physics137, 164706-1

164706-2 Barel, Filippov, and Urbakh J. Chem. Phys. 137, 164706 (2012)

FIG. 1. Schematic sketch of a model geometry.

barriers for capillary condensation, which is given by the fol-lowing equation:

�Eon (ri) = �E0on

(1 + r2

i /2Rh0), (2)

where �E0on is a minimal value of the barrier for nucleation.

As long as a bridge is intact, it responds elastically tothe applied force, and it can be modeled as an elastic springwith stiffness, κ .15 Under the action of the pulling force thebridges are stretched in the lateral direction with the velocity,which is equal to the velocity of the tip, X, and therefore, thetip experiences the surface force, Fcap = −∑Ns

i fi , where fi= κ(li(t) − h(ri)), li(t) is the time-dependent bridge length,h(ri) is the bridge length at moment of nucleation and a sumis taken over all bridges.

When the force, fi, acting on the i-th bridge exceeds thethreshold value, fc, corresponding to a maximum force that acapillary bridge can hold the bridge detaches from the tip. Itshould be noted that the height of the potential barrier for rup-ture of condensed bridges is much higher than kBT,16, 17 andbecause of this the effect of temperature on rupture processesis negligible.

Here for the estimation of the threshold force fc we usethe value of the maximum adhesion force (the capillary force)between the asperity and the tip that is given by the equation

fc = 2πγ a0 (cos θT + cos θS) , (3)

where γ is the liquid surface tension, a0 is the average ra-dius of asperity, and θT and θS are the contact angles at the tipand the substrate, respectively. Equation (3) should be appliedwith caution at microscopic scales,16–18 because it relies onmacroscopic concepts. It should be also noted that the max-imum force that a capillary bridge can hold in the lateral di-rection before it breaks (adhesion force) can be different fromthe maximum adhesion force in vertical direction, fc.

In the framework of the model described above themotion of the driven tip is described by the followingequation:

MX + ηX − Fcap + K(X − V t) = 0, (4)

where η is a damping coefficient responsible for the dissipa-tion of the tip kinetic energy to phonons and other degrees offreedom which are not considered explicitly here. In a widerange of parameters the results of calculations are indepen-dent of the value of η.

The instantaneous lateral spring force, which is themain observable in friction experiments, reads as F = −K(X− V t), and its time average is equal to the friction force 〈F〉.

RESULTS AND DISCUSSION

The essential difference of the model described abovefrom the previous works,19–21 where friction was describedin terms of rupture and reattachment of nanoscale contacts, isa broad distribution of nucleation barriers given by Eq. (2).This leads to a long time scale dependence of number of con-densed bridges on the time of contact between the tip and thesubstrate and to the effect of frictional ageing. In order to cal-culate an average number of bridges, N(t), which are formedduring the time t, we introduce a density of bridges, n(r,t),nucleated at the distance r from the tip apex. This density sat-isfies the equation

d

dtn (r, t) = kon (r) (n0 − n (r, t)) , (5)

where n0 is the surface density of nucleating sites. The so-lution of Eq. (5) reads as n(r, t) = n0[1 − exp ( − kon(r)t)].Using this equation for n(r,t) and kon(r) defined by Eq. (1), wearrive at the following expression for N(t):

N (t) = 2πn0

∫ ∞

0n(r, t)rdr=N0

∫ t

0

1

t ′[1− exp(−t ′/τ )]dt ′

= N0[E1(t/τ ) + ln(t/τ ) + γ ], (6)

where E1(z) is the exponential integral, 1/τ

= ω0on exp[−�E0

on/kBT ] is the inverse time needed fornucleation of smallest bridges, N0 = 2πn0h0R

kBT

�E0on

is anumber of bridges condensed at t � τ , and γ � 0.5572 isEuler-Mascheroni constant.22 At short and long times N(t)exhibits the following asymptotic behaviors:

N (t) = N0 ×{

t/τ, t < τ

ln (t/τ ) + γ, t > τ. (7)

A logarithmic increase of number of condensed bridges withtime has been already suggested in previous studies, whichconsidered the effect of capillary condensation on friction atnano and macro scales.4, 5

The mean field description of N(t) presented above al-lows to determine the effect of capillary condensation on ve-locity and temperature dependencies of friction. When the tipis pulled with velocity V, the force, fi, acting on a water bridgegrows with a time-dependent rate KV/N (t), and the bridgewill be ruptured at the time, t0, for which fi = fc. This yieldsthe following equation for t0:∫ t0

τ0

dt

N (t)= fc

KV, (8)

where τ 0 is a time needed for nucleation of one bridge, N(τ 0)= 1, and it can be estimated as τ 0 ≈ τ (N0)−1. Then consid-ering a stick-slip regime of motion, the time averaged frictionforce, 〈F〉, can be calculated as

〈F 〉 ≈ 1

2KV (t0 − τ0) . (9)

The latter estimation follows from consideration of the stick-slip force traces. After the transition to sliding (at t = 0) thereis a short time interval (0 < t < τ ) needed for condensation ofthe first bridge, during which the force is negligible, and af-ter that the force grows linearly with the slope approximately

164706-3 Barel, Filippov, and Urbakh J. Chem. Phys. 137, 164706 (2012)

equals to KV until the time t0, when the bridges are ruptured.It should be noted that Eqs. (8) and (9) have been derivedunder the assumption that all condensed bridges are rupturedsimultaneously. The validity of this assumption will be testedbelow using stochastic simulations.

At low pulling velocities or relatively high temperatures,for which a large number of bridges are condensed during thestick phase of motion (N(t0) � N0), the friction force can bewritten as

〈F 〉 = fcN0(

fcN0

KV τ

), (10)

where is a scaling function, which depends only on one pa-rameter, ζ = fcN0

KV τ, and increases with increasing ζ . It should

be noted that both N0 and τ (1/τ = ω0on exp[−�E0

on/kBT ],N0 = 2πn0h0R

kBT

�E0on

), are functions of temperature, and thusEq. (10) provides a direct link between the temperature andvelocity dependencies of friction. In particular, in the range ofparameters, where 〈F〉 grows linearly with T, the average fric-tion force decreases approximately logarithmically with ve-locity, 〈F 〉 � const − ln V . This result can be obtained sub-stituting Eq. (7) for N(t) in Eq. (8) and neglecting the terms ofthe form of ln ln[ fcN0

KV τ], which are weakly dependent on tem-

perature and velocity.In the region of high velocities or low temperatures,

where only few bridges are condensed during the stick phaseof motion (1 ≤ N(t0) < N0) we get

〈F 〉 ≈ 1

2fc

(KV τ

fcN0

) (exp

(fcN0

KV τ

)− 1

). (11)

Figures 2 and 3 show results of numerical calculationsusing Eqs. (8) and (9). We found the same characteristicfriction-velocity and friction-temperature curves as observedin the experiments,5–7 which exhibit: (i) decrease of the fric-tion force with velocity and increase with temperature, (ii)two different regimes of velocity dependence of friction forhigh and low pulling velocities, (iii) increase of critical ve-locity, Vc, for a transition between the two velocity regimeswith T. In accordance with experimental observations,5–7 atlow velocities 〈F〉 shows a logarithmic-like decrease with V,and it levels off at high velocities.

The critical velocity, Vc, is defined by the equationfcN0

KVcτ≈ 1, that gives

Vc ≈ πfcn0h0Rω0onkB

K�E0on

T exp[−�E0

on/kBT]. (12)

Equation (12) shows that the critical velocity increaseswith T.

The above results have been obtained in the frame-work of mean field description, which is based on consid-eration of average number of condensed bridges, N(t), andassumes that all condensed bridges are ruptured simultane-ously. This is an approximation, and more accurate approachrequires a consideration of stochastic dynamics of nucle-ation and rupture of capillary bridges that is governed bythe nucleation rates kon(ri) and coupled to the motion of tipin Eq. (4).23 Results of stochastic simulations are presentedin Figs. 4 and 5, and they show velocity and temperature

FIG. 2. Dependencies of the average friction force on the pulling ve-locity for three different temperatures calculated within the mean fieldapproximation, Eqs. (8) and (9). Parameter values: fc = 0.8nN, n0 = 2.5× 1018m−2, �E0

on = 4.2 × 10−20J, ω0on = 106s−1, K = 6 N/m, R

= 30 nm, h0 = 1 nm .

dependencies of friction force, and of time-averaged num-ber of condensed bridges. One can see that the observedvariations of friction force with V and T are entirely deter-mined by the corresponding variations of number of con-densed bridges. The results of simulations shown in Figs. 4and 5 are qualitatively similar to those obtained within themean field approach and presented in Figs. 2 and 3. How-ever, for the same values of system parameters the frictionforces obtained in the stochastic simulations are significantlyhigher (almost two-times) than those derived in the meanfield approximation. This discrepancy results from complexdynamics of rupture and reattachment of capillary bridges,which has been observed in simulations and neglected in theapproximate description. We found that part of condensedbridges is ruptured at the values of the applied force, whichare considerably lower than the value needed to initiate aslip of the tip, and then they are nucleated again during

FIG. 3. Dependencies of the average friction force on temperature for threedifferent velocities calculated within the mean field approximation, Eqs. (8)and (9). Parameter values as in Fig. 2.

164706-4 Barel, Filippov, and Urbakh J. Chem. Phys. 137, 164706 (2012)

FIG. 4. Results of stochastic simulations for velocity dependence of theaverage friction force (a) and the average number of condensed bridges(b) calculated for three different temperatures. Parameter values: M = 5× 10−11kg, η = 2 × 10−5kg/s, fc = 0.8nN, n0 = 2.5 × 1018m−2,�E0

on

=4.2×10−20J, ω0on =107s−1, K = 6N/m, k = 1N/m, R = 30 nm, h0

= 1 nm .

the same stick interval. This effect leads to an enhancementof friction force peaks in the stick-slip series compared tothose calculated in Eqs. (8) and (9).

The proposed stochastic model enables us to describe allfeatures of friction-velocity and friction-temperature curves,which have been observed experimentally under humid am-bient conditions. The simulation parameters, fc, �E0

on, n0,have been chosen to fit qualitatively the experimental data,5–7

and their values are close to those suggested by experimentalstudies. However, there is one important point, which requiresadditional consideration. The kinetics of capillary condensa-tion is characterized by a long-scale, logarithmic increase ofnumber of condensed bridges with time that results in fric-tional ageing.4 This behavior is very different from the kinet-ics of attachment of molecular contacts with a narrow distri-bution of nucleation barriers, �Eon � �E0

on, whose impacton nanoscale friction has been discussed recently.19–21 In thelatter case

N (t) = Nmax (1 − exp (−t/τ )) , (13)

where Nmax is the maximal number of contacts, and the ma-jority of contacts is formed during the time interval of theorder of τ . The important question is: do the measurementsof velocity and temperature dependencies of friction allowto distinguish between the two kinetics? In order to answerthis question we show in Fig. 6 the friction-velocity curves,which have been calculated assuming that all surface contacts

FIG. 5. Results of stochastic simulations for temperature dependence of theaverage friction force (a) and average number of condensed bridges (b) cal-culated for three different velocities. Parameter values as in Fig. 4.

are identical and their condensation follows the law describedby Eq. (13). As one can see these curves exhibit the samecharacteristic features as the 〈F (V )〉 curves calculated for thecase of capillary condensation. The same similarity has beenfound for the 〈F(T)〉 curves calculated for the two different ki-netic laws of contact formation. The only essential conditionfor observation of friction-velocity and friction-temperaturecurves similar to those in Figs. 2–6 is that formation of

FIG. 6. Results of stochastic simulations for velocity dependence of the av-erage friction force calculated for the ensemble of bridges with equivalentnucleation barriers. Parameter values: �E0

on = 5 × 10−20J, Nmax = 80,other as in Fig. 4.

164706-5 Barel, Filippov, and Urbakh J. Chem. Phys. 137, 164706 (2012)

FIG. 7. Maximum friction force as a function of the hold time calculated forthe case of capillary condensation (purple) and for identical contacts (green).The red and blue lines represent polynomial fitting curves to the calculateddata. Inset shows the lateral force versus time calculated for three values ofthe hold time �t. Pulling velocity, V = 400 nm/s, and other parameters asin Figs. 4 and 6.

contacts (bridges) is a thermally activated process, while ki-netics of rupture is temperature-independent, or only slightlyinfluenced by T. This is different from the major assumptionof the standard thermally activated model of friction,10–12 ac-cording to which the rupture of nanoscale contacts (barriercrossing) is a thermally activated process. The above consid-eration shows that measurements of velocity and temperaturedependencies of friction do not allow to draw definite conclu-sions on contribution of ageing processes to friction.

Nevertheless, friction measurements can provide an effi-cient way to study kinetics of bridge formation. This can bedone using slide-hold-slide experiments,24 which are similarto slide-hold-slide tests on rocks,25, 26 where the external driveis stopped for a certain hold time, �t, and then reinitiated withthe same pulling velocity, V. Figure 7 shows results of calcu-lations of the maximum force, FS, following specified holdperiods. The presented forces have been found by averagingover hundred realizations. The maximum force is larger thanthe time-averaged friction force corresponding to the same ve-locity by an amount �F = Fs − 〈F〉, called friction drop. Forcapillary condensation of bridges FS, grows linearly with thelogarithm of the hold time showing a strong effect of ageingon nanoscopic friction that results from a broad distributionof activation energy barriers for capillary condensation. In thecase of identical contacts significant deviations from the log-arithmic dependence are observed already for �t ≥ 3τ (seeFig. 6). Thus, our simulations demonstrate that slide-hold-slide experiments can provide important information on ki-netics of bridge formation and mechanism of frictionalageing.

Figure 7 shows that in the case of capillary condensationthe calculated relative friction drop, �F/〈Fs〉 increases aboutten times as the hold time increased by two orders of magni-tude. This result is consistent with recent nanoscopic frictionmeasurements,24 which found ten times increase of relativefriction drop, when the hold time increased by two orders ofmagnitude. However, it should be noted that in the cited ex-

periments the large ageing effect has been observed also forlow humidities showing that the capillary condensation is notthe only mechanism for frictional ageing. The magnitude ofthe ageing effect for the nanoscale single asperity contactsthat has been found in experiments24 and in our simulationsis much larger than for macroscopic multi-asperity frictionexperiments.25–27 Further experimental and theoretical studiesare needed to address the mechanisms that account for this.

SUMMARY

Capillary bridges play a crucial role in the operation ofatomic force microscopy under humid ambient conditions andtheir formation often dominates the measured forces. It hasbeen found that under ambient conditions the nanoscopic fric-tion force decreases with pulling velocities and increases withtemperature.5–7 These observations differ significantly fromthe results of friction experiments carried out under ultrahighvacuum, and disagree with predictions of thermal Prandtl-Tomlinson model of friction. Here, by analytical model andnumerical simulations, we demonstrate that the observed fric-tion phenomena are caused by the fact that formation of cap-illary bridges is a thermally activated process, while kineticsof their rupture is temperature-independent, or only slightlyinfluenced by T. Our calculations show that measurements ofvelocity and temperature dependencies of friction force do notallow to draw definite conclusions on contribution of ageingprocesses to friction. However, we found that this informationcan be provided by slide-hold-slide measurements.

ACKNOWLEDGMENTS

We are grateful to R. W. Carpick, C. Greiner, P.-E.Mazeran, and O. Noel for helpful discussions. A.E.F. ac-knowledges hospitality at the School of Chemistry, Tel AvivUniversity. The work was supported by DIP (German-IsraeliProject Cooperation Program) and the Israel Science Founda-tion (1109/09).

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