Friction Through Dynamical Formation and Rupture of Molecular Bonds 2004 Physical Review Letters

8/16/2019 Friction Through Dynamical Formation and Rupture of Molecular Bonds 2004 Physical Review Letters

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Friction through Dynamical Formation and Rupture of Molecular Bonds

A. E. Filippov,1 J. Klafter,2 and M. Urbakh2

1 Donetsk Inst itute for Physics and Engineering of NASU, 83144, Donetsk, Uk raine2School of Chemistry, Tel Aviv University, 69978 Tel Aviv, Israel

(Received 23 November 2003; published 30 March 2004)

We introduce a model for friction in a system of two rigid plates connected by bonds (springs) and

experiencing an external drive. The macroscopic frictional properties of the system are shown to bedirectly related to the rupture and formation dynamics of the microscopic bonds. Different regimes of motion are characterized by different rates of rupture and formation relative to the driving velocity. Inparticular, the stick-slip regime is shown to correspond to a cooperative rupture of the bonds. Moreover,the notion of static friction is shown to be dependent on the experimental conditions and time scales.The overall behavior can be described in terms of two Deborah numbers.

DOI: 10.1103/ PhysRevLett.92.135503 PACS numbers: 81.40.Pq, 46.55.+d

There has been a g rowing number of attempts to under-stand the microscopic origin of frictional forces betweensurfaces in relative motion [1–4]. One of the importantareas of investigations that has been somewhat over-

looked is that of strongly irreversible tribological pro-cesses, which include cold welding, material mixing, andtribochemical and triboelectrical effects. So far only afew experimental investigations [5–7] and moleculardynamics simulations [8–11] have been performed tostudy these strongly irreversible phenomena at a micro-scopic scale.

Recent experimental studies of dynamics of cold welds[5] and adhesive boundary lubrication [6] have suggestedthat macroscopic friction might originate from the for-mation and rupture of microscopic bonds (junctions) thatform between surfaces in close vicinity. Furthermore,these findings indicate that stick-slip motion is connected

to a collective behavior of the bonds [5]. However, themechanism underlying this collective behavior is an openissue. In this Letter we propose a microscopic model thatestablishes a relationship between the dynamics of for-mation and rupture of individual bonds and macroscopicfrictional phenomena. We suggest mechanisms of slidingand stick-slip motions that have been observed in tribo-logical experiments with adhesive boundary lubricatedsurfaces [6] and cold welding [5]. Here we go beyond theelastic response of the embedded system to includestrongly nonlinear rupture effects that contribute essen-tially to energy dissipation. Closely related models havebeen proposed earlier [12,13], but the detailed micros-copy, its relationship to the macroscopic plate motion,and, in particular, the conclusions are all different.

Figure 1 presents the model investigated in the Letter.The model includes two rigid plates connected by bonds(junctions) that spontaneously break and then reformupon a contact. We model the bonds (N of them) by elasticsprings each with a force constant and a rest length l0.The top plate with mass M and center-of-mass coordinateX is pulled with a linear spring of spring constant K . The

spring is connected to a stage t hat moves with a constantvelocity V . The motion of the driven plate can be de-scribed by the following equation:

M X X

_X X F b K X Vt 0; (1)

where

F b XN

i1

qi f xi (2)

is the force due to t he interaction between the bonds andthe driven plate, and is a damping coefficient. Theparameter qi characterizes the state of the individualbonds: qi 1 corresponds to the state of an intact bondthat connects the two plates, a nd qi 0 corresponds to aruptured bond that is attached to only one of the plates.The interaction between the plate and the intact bond i isgiven by the elastic force f i li l0 whose projec-tion in the direction of motion equals f xi f ixi=li. Hereli and xi are the length of the bond and its extension in theX direction correspondingly, li x

2i h

21=2, and h isthe distance between plates.

The frictional dynamics in the model is governed bytwo competing processes: (i) bond formation—creationof an interplate junction that tends to inhibit sliding, and(ii) bond rupture—detachment of a spring from one of the plates, a process that helps sliding. As long as a bond is

FIG. 1. Schematic sketch of a model setup.

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intact, it is stretched in the lateral direction with thevelocity that is equal to the velocity of the top plate, _xxi _X X , while a ruptured bond relaxes to its equilibrium state.

The dynamics of bond stretching and contraction can bedescribed by the following equation:

_xx i qi _X X 1 qixi; (3)

where is a relaxation constant that describes the ap-proach of a spring to its equilibrium length. Notice thatbond stretching is renewed every time the contact reformsand, thus, the lengths and elastic forces are different fordifferent bonds.

Equations (1)–(3) should be supplemented by an equa-tion that governs rupture and formation of bonds anddefines the time dependence of the state parameter, qi.This equation can be written as

qit t qit qiti tkoff

1 qiti tkon; (4)

where t is a time step in our calculations, i is a randomvariable chosen from the interval 0; 1, and z is aHeaviside step function that accounts for a stochasticrupture (formation) of a bond that occurs for i <tkoff on. The rates of these processes are given by koff and kon correspondingly.

A bond rupture can be considered as a thermally as-sisted escape from a bound state over an activation bar rierEli, which is spring-length dependent and diminishesas the applied elastic force, f i, increases (the bond isstretched) [14–17]. Below we use t wo approximationsfor the r upture rate, koff li, which are appropriate forthe cases of weak bonds (the bond energy is only slightly

larger than kBT ) a nd of strong bonds (the bond energy kBT ) [14–17]. In the first case the time-dependent disso-ciation rate ta kes the form [16]

koff li k0 expf ix; (5)

while for the strong bonds we use [17]

koff li k01 f i=f c1=2expfU 01 f i=f c

3=2 U 0g:

(6)

Here k0 is the spontaneous rate of bond dissociation in theabsence of the external force, and x is the distancebetween the minimum and the maximum of the reactionpotential, U 0 is the depth of the potential, f c is the criticalforce at which the potential barrier vanishes and unbind-ing occurs in the absence of t hermal fluctuations, and 1=kBT . Equation (5) assumes that the steady increase inthe pulling force produces a small constant bias thatreduces the height of the potential barrier. Equation (6)takes into account that for a high barrier a bond ruptureoccurs preferentially when the bond is close to its slippagecondition.

The formation, or rebinding, process is characterizedby the rate kon. We assume for simplicity that t his rate isnot affected by the bond length. However, we assume thatthe probability of a bond formation depends on t he ‘‘age’’of the contact, [1,18]. This age, , is defined as t he timeduring which the free end of the bond is exposed to acontact area t hat moves with respect to the bond. Toinclude the effect of the contact age we introduce a

characteristic time scale of the contact time, 0, neededfor a bond formation and assume that the rate of bondformation depends continuously on the time difference 0,

kon k0ong 0=: (7)

Here k0on is the rate of bond formation for an immobilecontact, g is a ‘‘smeared’’ stepwise function (g 0 for 0 and g 1 for > 0), and is a width of thefunction g. The contact time, , is inversely proportionalto the velocity of the driven plate, a= _X X , where a is atypical length scale of the contact. The time scale 0

defines a critical velocity, V 0 a=0, above which thebond formation becomes improbable. We find that t ak inginto account the contact time in the process of bondformation is a necessary condition to obtain stick-slipmotion within our model.

We now discuss results obtained for a large number of bonds, N > 300, where the measurable frictional forcesare proportional to N . In all figures the distances, time,velocities, and forces are presented in units of h, M=,h=M , and kBT=h, respectively.

Our simulations demonstrate that the model exhibitsthree different regimes of motion: two sliding regimesand a stick-slip motion between them (see Figs. 2 and 3).

These regimes can be easily distinguished in Fig. 3(a),which shows the time-averaged maximal and minimalvalues of the spring forces, F K X Vt, as a functionof the driving velocity. These values coincide in thesliding regimes (Sl), but differ in the velocity intervalcorresponding to the stick-slip regime (SS). The dynami-cal behavior depicted in Fig. 3(a) has been recently ob-served in surface forces apparatus experiments withadhesive surfactant surfaces [6].

Low-velocit y sliding regime.— Figure 2(a) shows a timeseries of the spring force a nd the corresponding f ractionof intact bonds, int, in the very low-velocity sliding state.This state corresponds to a close-to-equilibrium situationwhere the rupture of bonds is determined by a sponta-neous (thermal) bond dissociation rather than by theeffect of shear induced stress that leads to a reductionof the activation barrier E. This regime is limited tovelocities for which the spontaneous rate of bond disso-ciation, k0, is higher than the velocity dependent rate of decrease of the activation bar rier, kbar V x, whichyields the condition V < V c k0=x. The very low-velocity sliding demonstrates that observation of static


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friction depends on the competing external and internaltime scales. The definition of static friction should there-fore include the experimental time scales.

Our calculations show that the low-velocity slidingactually corresponds to an atomic-scale stick-slip motionof individual bonds. This behavior is clearly seen inFig. 4(a), which presents the time dependence of an in-dividual bond length and of t he ensemble averaged lengthof the intact bonds. Another characteristic feature of thesliding regime is the essential absence of correlationbetween individual rupture events, which is again amanifestation of thermal bond dissociation. Figure 4(a)also demonstrates that in spite of strong fluctuations inindividual bond length, the ensemble averaged bondlength remains constant as a function of ti me, and itonly slightly exceeds the equilibrium bond length.

The model includes two channels of energy dissipation:the first one is related to the rupture and the consequentrelaxation of the bonds, and the second is the viscousdissipation, which is given by the term _X X in Eq. (1).The bold solid line in Fig. 3(b) shows the velocity depen-dence of the kinetic frictional force, F k

Rt0 K X

Vt dt=t, t ! 1. This gives the net energy dissipatedper a unit length passed by t he top plate. The r upture andviscous components of the energy dissipation are shown

by the dashed and thin solid lines, respectively. We seethat in the low-velocity sliding regime the energy dissi-pation is completely dominated by the rupture processes.Then the kinetic frictional force can be estimated taking

into account that, being in the intact state, a bond elon-gates on average by a length V=k0. The latter givesF k VN hinti=k0, where hinti is a time-averaged frac-tion of the intact bonds. In this regime hinti does notdepend of the driving velocity and is close to its equilib-rium value, kon=kon koff .

Stick-slip motion.— In t he interval of driving velocitiesV 0 V V c the model exhibits stick-slip behavior as isseen in Figs. 2(b), 2(c), and 3(a). For the velocities V V c where processes of spontaneous a nd shear inducedbond dissociation compete, kbar k0, we observe an er-ratic stick-slip motion. The stick-slip motion becomesregular under the condition V 0 > V V c, where the

rupture is completely determined by the effect of shearstress on the height of t he activation bar rier [see Fig. 2(c)].A transition from a sliding to a stick-slip motion is

FIG. 3. Velocity dependence of time-averaged frictionalforces in the case of weak bonds. (a) Maximal (closed circles)and m inimal (open circles) spring forces calculated withi n themicroscopic model for weak bonds. (b) The net kinetic friction(bold line), rupture (dashed line), and viscous (thin line)components of the friction force. Parameter values as inFig. 2. The symbols Sl and SS indicate sliding and stick-slipregions correspondingly.

FIG. 2. Time series of the spring force (top panels in eachfigure) and the fraction of intact bonds (bottom panels) calcu-lated for the model of weak (a),(b) and strong (c) bonds.(a) V 0:5, (b),(c) V 2. Parameter values: N 300, M /N , 10, K 102N , h 0:25N , 0:4=M , l0 h x 1, k0 k

0on 0:1, V 0 0:5, 0:1; in (c) we used

U 0 10, f c 10, and k0 0:1expU .

FIG. 4. Time series of an individual bond elongation (thinline) and of the ensemble-averaged bond elongation (bold).(a) V 0:5 corresponding to uncorrelated rupture and(b) V 2 corresponding to collective rupture. Parameter val-ues as in Fig. 2.


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accompanied by a decrease of the time-averaged fractionof intact bonds that leads to a decrease of the rupturecontribution to the energy dissipation. This effect is bal-anced by a growth of the viscous component of the energydissipation [see Fig. 3(b)]. As a result, the kinetic frictiondepends only slightly on the velocity in the stick-slipregime.

In contrast to the low-velocity sliding regime where

rupture events are uncorrelated, the stick-slip motion ischaracterized by a cooperative behavior of bond subsys-tem that is clearly visible in Figs. 2(b), 2(c), and 4(b). Asmore and more bonds break, the force on the remainingones increases and the bond rupture becomes synchro-nized. The cooperative rupture of bonds is followed bya time window of a ‘‘slip’’ motion where rebinding issuppressed because of the rather high velocity of thesliding top plate, _X X V 0 a=0. This simple type of cooperativity arises even in the absence of direct inter-action between bonds. Recently, a correlation betweenmacroscopic frictional properties and a collective behav-ior of microscopic bonds has been observed experimen-

tally [5].The time-averaged maximal spring force, F max, that

corresponds to a synchronized bond rupture can be cal-culated analytically using a dependence of the rate of rupture, koff , on the external force, Eqs. (5) and (6). Forthe cases of weak and strong bonds, we get the followingdependencies of F max on the driving velocity, F max /j lnV j and F max / j lnV j

2=3, respectively. Similar scalinglaws have recently been proposed for i nterpretation of t heresults of frictional forces measurements [19– 23] andsingle molecular rupture experiments [16,17]. A possibil-ity of experimental discrimination between the two scal-ing laws is actively discussed in the literature [21,22].

High -velocit y sliding.— For V V 0 a transition fromstick-slip motion to the high velocity smooth slidingoccurs [see Fig. 2(a)]. In this region bond formationbecomes impossible due to the short contact time, <0, and friction is completely determined by viscousdissipation so that F V . It should be noted that lowand high velocity sliding regimes differ by the rate of energy dissipation (N hinti=k0 vs ).

To summarize, we have proposed a model that estab-lishes relationships between macroscopic frictional phe-nomena and the dynamics of formation and rupture of microscopic bonds. The dynamical response of the systemcan be characterized by two Deborah numbers, D1

e

V=V c and D2e V=V 0, which describe a competition

between t he rates of bond rupture a nd formation andthe rate of external drive. In the low-velocity slidingregime D1e ; D

2e

1; D2e < 1, while in the high velocity sliding regimeD1e ; D

2e > 1. And now to a point of caution. Other

models such as the Tomlinson, single particle, andFrenkel-Kontorova type models [4,24,25], althoughphysically very different, lead to similar dynamical re-sponses of the driven plate. Namely, force measurementsalone do not allow one to establish a microscopic mecha-nism of friction.

Financial support by the Israel Science foundation(Grant No. 573/00) is gratefully acknowledged. A. E. F.

thanks the ESF ‘‘Nanotribology’’ Program for financialsupport.

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Friction Through Dynamical Formation and Rupture of Molecular Bonds 2004 Physical Review Letters

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