Origin of Stick-Slip Motion in Boundary Lubrication

PETER A. THOMPSON AND MARK 0. ROBBINS

Molecular dynamics simulations of atomically thin, fluid films confined between twosolid plates are described. For a broad range ofparameters, a generic stick-slip motionis observed, consistent with the results of recent boundary lubrication experiments.Static plates induce crystalline order in the film. Stick-slip motion involves periodicshear-melting transitions and recrystllization of the film. Uniform motion occurs athigh velocities where the film no longer has time to order. These results indicate thatthe origin of stick-slip motion is thermodynamic instability of the sliding state, ratherthan a dynamic instability as usually assumed.

ONE OF THE GENERIC PHENOMENA

observed when solids slide overeach other is stick-slip motion. In-

stead of moving smoothly, the solids alter-nately stick together and slip past each oth-er. This oscillatory motion produces most ofthe squeaks heard in our daily lives and canpersist even in the presence oflubricants (1).The usual explanation of stick-slip motion

(1) is based on the well-known observationthat the frictional force F on static objects islarger than that on sliding objects. Then, ifF is a single-valued function of velocity v, itmust decrease at small v. In the regimewhere dF/dv < 0, uniform motion is dynam-ically unstable because decreasing the forceincreases the velocity. The velocity will oscil-late between 0 and a value where dF/dv > 0.Measurements have been made for a numberofsystems (1, 2), but it is difficult to confirmthe existence of a region where F(v) de-creases because of the inherent instability.Another difficulty has been obtaining well-characterized, reproducible, experimentalsystems.The latter difficulty has been overcome in

recent measurements on atomically flat micaplates separated by fluid films only a fewmolecules thick (3-5). This is a prototypicalexample of boundary lubrication, an impor-tant topic in tribology (1). Very precisemeasurements of stick-slip motion and itsdependence on velocity, load, and fluidcharacter have been made (4, 5). Theseresults have motivated new theoretical ef-forts toward understanding the microscopicorigin of stick-slip behavior.We describe here molecular dynamics

simulations of boundary lubrication, whichreproduce many features of experiments (1-

5). Analysis of structure in the films revealsthat stick-slip motion is associated with peri-odic phase transitions between ordered stat-ic and disordered kinetic states. Temporal orspatial averaging of the force produces de-creasing F(v) curves like those assumed inthe dynamic instability model of stick-slipmotion. However, this confuses cause andeffect. The frictional force in the sliding stateis a monotonically increasing function of v.The real origin of stick-slip motion is ther-modynamic instability of the sliding statebelow a critical force (and correspondingvelocity) rather than a dynamic instabilitydue to dF/dv < 0.To model the experimental system, we

considered two solids separated by a thinfilm offluid molecules of mass m interactingwith a Lennard-Jones (L) potential:

V(r) = 4E[(r/r)12 - (air)6]

where e, ar, andTv /moc2/E are characteris-tic energy, length, and time scales for theinteraction. An LJ potential with modifiedparameters evf and u,wf was also used tomodel interactions between the fluid mole-cules and discrete wall atoms. The details ofthis model have been presented in an earlierpaper (6) in which we examined the influ-ence of ewf and wall geometry on the struc-ture and flow of fluids sheared betweensolids separated by a distance h - 12a. For awide range ofparameters, we found that oneor more layers offluid molecules crystallizedat the solid surface. This tendency to form asolid wetting phase increased with the de-gree of corrugation in the wall-fluid poten-tial and with the degree to which the naturalspacing of fluid molecules matched that ofthe wall. For most of the simulations de-scribed below, walls were face-centered cu-bic (fcc) solids with (11 1) surfaces anddensity equal to that of the fluid. This face

Walls | h Fluid

Fig. 1. Sketch of the simulation geometry. Solidwalls are held together by a constant normal loadL. The top wall is pulled by a spring connected toa stage moving at constant velocity v. Periodicboundary conditions are imposed in the x and ydirections.

most closely resembles the effective closed-packed structure of mica surfaces (7).The wall separation is typically much

smaller than 12a in boundary lubrication.Proximity to two walls amplifies the tenden-cy to order, and the crystallized regions mayspan the film. For instance, at values of Effor which a single solid layer formed on eachplate at large h, we find full crystallization ath - 4'r. Similar results have been observedin experiments (3-5, 8) and earlier MonteCarlo studies (9).To mimic boundary lubrication experi-

ments, we performed simulations in a con-stant load ensemble where h was allowed tovary (10). A constant temperaturekbT = 1. Ie (kb is the Boltzmann constant)was maintained by coupling to a heat bath(6). At this T and nominal densityp = 0.81r-c3, the bulk phase is a slightly

NtI-1,

.Q 1.8

1.7

° 0.8cro

a 0.4co

2 4 6At/l10t

Fig. 2. Time profiles of (A) the force per unit areaf, (B) the wall displacement X, (C) the wallspacing h, and (D) the Debye Waller factor,during stick-slip motion for a system with v = 0.1UhT-1, k = 7.2 mvK2, Ek = 2, and crf/cr = 1.The simulation contains 288 wall and 144 fluidatoms. The walls are fcc solids with (111) sur-faces, and the shear direction is (100).

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Department of Physics and Astronomy, The Johns Hop-kins University, Baltimore, MD 21218.

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compressed liquid about 30% above itsmelting temperature.As in experiments, the top plate was

coupled through a spring with force con-stant k to a stage moving at constant velocityv (Fig. 1). Initially, the force per unit areaapplied by the spring, f_ F/A, is zero andthe top plate is at rest. As the stage movesforward, the spring stretches andfincreases.If the film is a liquid, the top plate acceler-ates until the steady-state f balances theviscous dissipation. If the film has crystal-lized, stick-slip behavior is observed as illus-trated in Fig. 2A. Initially, the film respondselastically. The top plate remains stuck, andfincreases linearly. Eventually f exceeds theyield stress of the film f, and the top platebegins to slide. Since the frictional force isless in the sliding state, the plate acceleratesto catch up with the stage and f decreases.At some lower value fi, the top plate sticksonce more, and the process repeats. Duringthe sliding phase, the plate jumps forwardby an amount Ax - (fs - fm)A/k, approxi-mately 20 to 30a for the case illustrated inFig. 2B. These forward jumps are typicallymuch larger (-1 ,um) in experiments (3-5)because weaker spring constants are used(11).

Figure 2C shows the plate separation h asa function of time. Since shearing the filmincreases the effective repulsion between flu-id molecules, sliding plates separate at con-stant normal load. For the stick-slip motionillustrated in Fig. 2, h increases by -10%during each slip. The resolution of experi-ments done thus far has not been sufficient

2

0

2

E 02

2

o2 4 6

AtO/100TFig. 3. Time profiles of f at various v for thesystem of Fig. 2: (A) v = 0.1, (B) v = 0.2, (C) v= 0.3, and (D) v = 0.5.

to observe changes of this order ( 1 A), butthey had been anticipated on the basis of the"cobblestone" model (4).The relaxation behavior observed in ex-

periments indicates that substantial rear-rangement of the molecules occurs (5). Wehave quantified the time dependence of or-der in the film by computing the two-dimensional (2-D) structure factor S(q) ofthe molecules in layers adjacent to the walls.The periodic wall potential induces 8-func-tion peaks in S at wave vectors q equal to the2-D reciprocal lattice vectors Gi. In crystal-lized films, S(GI)/S(O) = e-W 1, wheree-w is the Debye-Waller factor for the short-est reciprocal lattice vector G1. Figure 2Dshows that e-w is close to 1 in the staticstate, indicating a good solid. However, inthe sliding state, S(G1)IS(O) drops below0.2, a value consistent with liquid structureand dynamics (6, 12). Snapshots of instanta-neous configurations of atoms also showthis loss of order.These observations clearly indicate that

the film undergoes solid-liquid phase transi-tions as it goes from static to sliding states.Analogous transitions termed shear-meltingtransitions occur when bulk solids aresheared (13). Recent theory (14) and experi-ment (15) indicate that the ratio of meltingstress to shear modulus fmeIt/c44 is -0.03 to0.06, depending on the potential and crystalorientation. For low-temperature LJ crystalsat the density used in our simulations, c44 -

61 m/nrr2. Thus, the expectedfnel1 - 1.8 to3.6 m/ur2 is consistent with the observedyield stress f, - 2.1 M/err2.

If we interpret stick-slip motion in termsof a first-order shear-melting transition, it isnatural to identify spinodal points corre-sponding to the forces fu and fe, where thesolid and fluid phases, respectively, are nolonger even metastable. One expects these tocorrespond to fs and fm in the appropriatelimits. To determine fe, we performed oursimulations with constant force applied tothe top plate. Starting from a sliding state,we slowly decreased f until the plates stucktogether. For the system shown in Fig. 2,sliding ceased for f < fe = 0.82 M/(FT2(Ve 0.46 oKr-1). Recrystallization shouldinitiate at fe, but the observed repinningforce fm is lower. The difference depends onthe amount ofdamping. In an underdampedsystem fm < fe because f decreases rapidlypast fe at a shear rate that is still large.Experimental systems are usually over-damped. In this case, fdecreases slowly andfm fe.The velocity dependence of the stick-slip

behavior is illustrated in Figs. 3 and 4.Figure 3 shows simulation results at fourvelocities, and Fig. 4 shows the generic Fversus v relation. In general, we find a

9 NOVEMBER 1990

2.0 1-.

E 1.5N-

1.0 F

0.5

t

0 0.4 0.8 1.2V /Irl

Fig. 4. Force versus v for the system described inFig. 2. The dashed line and squares denote themaximum and mean yield stress during stick-slipmotion, respectively. Circles denote the time-averaged forcef. The solid line indicates constant-velocity results. The arrow indicates the point(fe,ve) at which constant fsimulations crystallize.

unique yield stress fs in the limit of small v.However, for a system with (111) solidsurfaces, two values corresponding to fccand hexagonal close-packed ordered phasesof the film are observed (Fig. 3A). As vincreases, the maximum value of fS remainsfixed, but variations in f, increase substan-tially. Intermittent periods without stick-slipmotion become more frequent (Fig. 3, B, C,and D). These changes occur because thefilm does not have enough time to fullyorder during each stick. Consequently, themean yield stress decreases as indicated bythe squares in Fig. 4. Similar results arefound in experiments (5). Examination ofS(q) shows a direct correlation between theamount of disorder in the film and theobserved decrease in f, Above a criticalvelocity vc, the film remains in a liquid stateand slides smoothly.An experiment without sufficient time

resolution would average the stick-slip be-havior. Similar averaging occurs in a largesystem where different regions ofthe surfacemay stick and slip at different times. Thecircles in Fig. 4 indicate the time-averagedf(v) from our simulations. As v increasesfrom 0, there is a linear decrease offdue toincreasing disorder in the stuck state and thecorresponding decrease in fs. Above vc, fincreases monotonically.This curve (Fig. 4) is strikingly similar to

the results of earlier measurements (1),which did not resolve stick-slip motion.Interpreting it as an intrinsic F(v) curve, onewould conclude that stick-slip motionshould occur for v < vc because ofthe nega-tive slope. However, this confuses cause andeffect. Constant v simulations show thatF(v) is monotonically increasing in the dis-ordered kinetic state (Fig. 4). The decreasein F results from decreases in the yield stressdue to increasing disorder.We have shown results and presented

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discussion for a single geometry. The behav-ior reported here is quite universal inboundary lubrication by simple fluids. Wehave also studied different values of ef, cf,and normal load; different crystal faces andshear directions; solid walls with muchhigher density than the fluid; amorphouswalls generated by the rapid quenching of afluid state; and even the effect of introduc-ing vacancies in the film. In each case,similar stick-slip behavior occurs. Whatchanges are the values of fs, fn, and vc, andthe plate separation hc at which stick-slipbehavior starts. For example, we find thatincreasing the degree of corrugation in thewall-fluid potential increases fn, vc, and h,When the walls are amorphous, the order

induced in the fluid is no longer crystalline.Instead, a glassy structure that minimizesthe wall-fluid interaction is observed. Thisleads to larger fluctuations in fs, but thequalitative behavior is similar. Glassy statesmay also be formed by long-chain moleculesbetween ordered plates because the relax-ation of these molecules is slow (8). Theinterplay between molecular structure andordering during stick-slip motion is an im-portant issue for future work.Another open question is whether stick-

slip motion is caused by analogous mecha-nisms in other systems. The ordered staticand disordered sliding states need not corre-spond to solid and fluid structure in anintermediate film. Instead, in the case ofsolid-on-solid sliding, they may correspondto states with and without chemical bondsor elastic deformations that increase thecoupling between plates. For example, stick-slip motion of a weak solid over a hard solidleaves behind patches of the weak material(1), suggesting periodic welding of the sur-faces followed by cracking. These and otherpossibilities remain to be explored.

REFERENCES AND NOTES

1. See, for example, F. P. Bowen and D. Tabor, Frictionand Lubrication (Oxford Univ. Press, Oxford, 1958);E. Rabinowicz, Friction and Wear ofMaterials (Wiley,New York, 1965).

2. C. Scholz, P. Molnar, T. Johnson, J. Geophys. Res.77, 6392 (1972).

3. J. N. Israelachvili, P. M. McGuiggan, A. M. Ho-mola, Science 240, 189 (1988); A. M. Homola, J. N.Israelachvili, M. L. Gee, P. M. McGuiggan, J.Tribol. 3, 675 (1989).

4. P. M. McGuiggan, J. N. Israelachvili, M. L. Gee, A.M. Homola, Mater. Res. Soc. Symp. Proc. 140, 79(1989).

5. M. L. Gee, P. M. McGuiggan, J. N. Israelachvili, A.M. Homola, J. Chem. Phys. 93, 1895 (1990).

6. P. A. Thompson and M. 0. Robbins, Phys. Rev. A41, 6830 (1990).

7. R. Erlandsson, G. Hadziioannou, C. Mate, G.McClelland, S. Chiang, J. Chem. Phys. 89, 5190(1988).

8. J. Van Alsten and S. Granick, Phys. Rev. Lett. 61,2570 (1988); Langmuir 6, 876 (1990); Tribol.Trans. 33, 436 (1990).

9. M. Schoen, C. L. Rhykerd, Jr., D. J. Diestler, J. H.

Cushman, Science 245, 1223 (1989).10. The simulations were performed in a constant num-

ber, load, and temperature ensemble. The number offluid molecules N is not fixed in experiments. How-ever, the time scale of stick-slip events is too short toallow significant transport from the bulk.

11. In the limit of infinitely stiff springs, the forwardjump may reduce to a single lattice constant as seenin simulations of solid-on-solid sliding by U. Land-man, W. D. Luedtke, and A. Nitzan [Surf. Sci. 210,L177 (1990)].

12. In particular, the shear plane in bulk films is veryclose (<ar) to the solid wall for eW -~0.2 (6).

13. B. J. Ackerson and N. A. Clark, Phys. Rev. Lett. 44,1005 (1980); ibid. 46, 123 (1981).

14. M. J. Stevens, M. 0. Robbins, J. F. Belak, inpreparation.

15. P. M. Chaikin, J. M. diMeglio, W. D. Dozier, H. M.Lindsay, D. A. Weitz, in Physics of Complex andSupermolecular Fluids, S. Safran and N. A. Clark, Eds.(Wiley-Interscience, New York, 1987), pp. 65-81.

16. We thank J. N. Israelachvili, P. M. McGuiggan, andM. J. Stevens for useful conversations. Supportedthrough NSF grant DMR 85-53271 and allocationsat the Pittsburgh Supercomputing Center. The workof M.O.R. was supported by the Sloan Foundationand through NSF grant PHY 82-17853 at theInstitute for Theoretical Physics, where this projectwas conceived. The work of P.A.T. was supportedby the Johns Hopkins University Applied PhysicsLaboratory.

8 May 1990; accepted 10 August 1990

Pressure Dependence of Elastic Wave Velocity for ,0-Mg2SiO4 and the Composition of the Earth's Mantle

GABRIEL D. GWANMESIA, SALLY RIGDEN, IAN JACKSON,ROBERT C. LIEBERMANN

The pressure dependence of the elastic wave velocities for hot-pressed, elasticallyisotropic polycrystals of the P (modified spinel) phase of magnesium orthosilicate(Mg2SiO4) has been determined at room temperature to 3 gigapascals (GPa) byultrasonic pulse interferometry. Pressure derivatives of the bulk (dK/dP = 4.8) andshear (dG/dP = 1.7) moduli derived from the travel times ofthe compressional (P) andshear (S) waves clearly demonstrate that the velocity contrast between the olivine and Pphases ofMg2SiO4 decreases with increasing pressure. When combined with plausiblevalues for the (as yet unmeasured) temperature derivatives, these new data can be usedto calculate the contrast in P and S wave velocities across an olivine-,B phasetransformation occuaring at pressure-temperature conditions corresponding to about400 kilometers depth in the earth. The seismologically observed contrasts AV in bothP and S wave velocities constrain the percentage of orthosilicate in a model mantle ofuniform chemical composition for appropriate relative magnitudes ofthe temperature(1) derivatives ofthe bulk and shear moduli for the ( phase. Allowed combinations oforthosilicate content (percent), dK/dT, and dG/dT (both in gigapascals per Kelvin) fora pair of recent seismological models with AVp = AVs = 4.6% include (65, -0.018,-0.020), (55, -0.015, -0.018), and (45, -0.012, -0.016).

K CONTINUING CHALLENGE FOREarth scientists has been to deter-ne the relative contributions of

isochemical phase transformation and chem-ical stratification to the discontinuous in-creases of seismic wave velocity with depththat characterize the transition zone (400 to670 km depth) of the earth's mantle. Adetailed knowledge of the chemical compo-sition of the earth's mantle is vital in placingconstraints on the early formation of theearth and on its thermal and chemical evolu-tion.

Pioneering studies (1) suggested thatphase transformations might play an impor-

G. D. Gwanrmesia and R. C. Liebermann, MineralPhysics Institute and Department of Earth and SpaceSciences, State University of New York at Stony Brook,Stony Brook, NY 11794.S. Rigden and I. Jackson, Research School of EarthSciences, Australian National University, Canberra,A.C.T., 2601, Australia.

tant role in explaining the velocity and den-sity structure of the transition zone. Subse-quently it was demonstrated that the domi-nant minerals of the upper mantle (olivineand pyroxene) transform to more close-packed crystal structures under the pressure-temperature conditions of the transitionzone (2) and that these transformations arelikely to produce relatively sharp seismicdiscontinuities (3). In particular, the discon-tinuous increase in seismic wave velocitynear 400 km depth has been attributed atleast in part to the transformation of(Mg,Fe)2SiO4 olivine to the m phase (4).More quantitative assessment of the na-

ture of the 400-km discontinuity has beenfacilitated by measurements under ambientconditions of the elastic properties of single-crystal P-Mg2SiO4; these studies revealedthat there is a -13% contrast in elastic wavevelocities between the olivine and ,3 phases(5, 6). However, the conclusions of recent

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