conditions for static friction between flat crystalline surfaces

8
Conditions for static friction between flat crystalline surfaces Martin H. Mu ¨ ser Institut fu ¨r Physik, Johannes Gutenberg Universita ¨t, 55099 Mainz, Germany Mark O. Robbins Department of Physics and Astronomy, Johns Hopkins University, Baltimore, Maryland 21218 ~Received 15 June 1999! The conditions for the presence of static friction between two atomically smooth crystalline surfaces are investigated. Commensurate and incommensurate walls are studied. While two commensurate walls always pin at zero lateral force and positive pressures, incommensurate walls only pin if mobile atoms are present in the interface between the surfaces or if the solids are particularly soft. Surprisingly, static friction can be observed between rigid surfaces, either commensurate or incommensurate, that are separated by a freely diffusing fluid layer. I. INTRODUCTION Recent studies have revealed interesting transitions in the shear response of many fluids when they are confined be- tween crystalline surfaces that are only a few nanometers apart. 1–10 Even though thick films are simple fluids at the imposed pressure and temperature, a yield stress or static friction is observed when the film thickness is decreased to a few molecular diameters. This is generally assumed to reflect a transition to a solid-like state of the film due to the bound- ing walls. In some experiments there is a continuous diver- gence of the viscosity and relaxation time that is typical of a bulk glass transition. 9 Other experiments show a sharp onset of the yield stress that is more akin to a first-order liquid to crystal phase transition. 10 Simulations have found both types of transition depending on factors such as the relative size of wall and fluid atoms and the molecular structure of the fluid. 4,5 They also reveal that solid films transform back into a fluid state when the yield stress is exceeded, explaining the stick-slip motion observed in some experiments. 2,11 There is, however, an important difference between most simulations 1–6 and experiments. 7–10 Surfaces used in com- puter simulations are typically commensurate, i.e., share common periodicities. In fact, most simulations use identi- cal, aligned crystals for the two walls. Many also set the number of atoms between the surfaces to an integer multiple of the number of atoms in one surface layer, facilitating the formation of ideal crystals. This cannot reflect typical experiments—even those between nominally identical sur- faces. The reason is that the crystallographic orientation of the surfaces is rarely controlled, and any small orientational misfit between otherwise identical surfaces makes them incommensurate. 12,13 There is no well-defined crystalline state of the film that can simultaneously lock into registry with two incommensurate surfaces. Some of the interactions between the wall and fluid atoms must be frustrated, and the dynamics and statistical properties of the system may be al- tered qualitatively due to this frustration. The contrast between commensurate and incommensurate cases can be illustrated by considering a submonolayer film of molecules between two identical crystalline surfaces in the limit of high confining pressures where the hard-sphere in- teractions between wall and film atoms dominate. If the sur- faces are aligned and translated so that all surface atoms are directly above each other, they create a periodic array of large openings that can accommodate film molecules. Any relative displacement of the commensurate walls greatly re- duces the volume of these openings and is resisted by the hard-sphere repulsion between wall and film atoms. This can be expected to prevent translation of one wall relative to the other until a yield stress is exceeded. Note that individual film molecules should still diffuse freely because there is a finite activation energy for motion between openings. This diffusion does not affect the equilibrium positions of the walls because all openings are equivalent. The situation is very different when the walls are made incommensurate by a relative rotation. Because the walls share no common periodicity, all possible relative positions of atoms on the two surfaces are sampled with equal prob- ability. Each opening is slightly different and all displace- ments of the walls produce the same distribution of open- ings. It is well known that this symmetry under translation can lead to a vanishing yield stress and free diffusion of the walls in the absence of a thin intervening film. 12,13 However, recent computer simulations 14 indicate that a film can pin incommensurate surfaces together, providing a natural expla- nation for the observation of static friction in experiments. The simple picture for submonolayer films is that mol- ecules search out the best set of openings for the given wall positions. They then resist translation of the walls because these openings will be constricted by any translation. Al- though there is an equivalent set of openings after transla- tion, these may be far away and only reached via a complex, coordinated reshuffling of the molecules with a large activa- tion free energy. This type of pinning is much more subtle than that be- tween commensurate walls, and one may wonder whether it persists in the thermodynamic limit. For example, diffusion of individual molecules necessarily takes them to inequiva- lent openings where they produce a different force on the walls. This will lead to a small displacement of the walls in any finite system, and accumulated motion of individual film PHYSICAL REVIEW B 15 JANUARY 2000-I VOLUME 61, NUMBER 3 PRB 61 0163-1829/2000/61~3!/2335~8!/$15.00 2335 ©2000 The American Physical Society

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Page 1: Conditions for static friction between flat crystalline surfaces

PHYSICAL REVIEW B 15 JANUARY 2000-IVOLUME 61, NUMBER 3

Conditions for static friction between flat crystalline surfaces

Martin H. MuserInstitut fur Physik, Johannes Gutenberg Universita¨t, 55099 Mainz, Germany

Mark O. RobbinsDepartment of Physics and Astronomy, Johns Hopkins University, Baltimore, Maryland 21218

~Received 15 June 1999!

The conditions for the presence of static friction between two atomically smooth crystalline surfaces areinvestigated. Commensurate and incommensurate walls are studied. While two commensurate walls always pinat zero lateral force and positive pressures, incommensurate walls only pin if mobile atoms are present in theinterface between the surfaces or if the solids are particularly soft. Surprisingly, static friction can be observedbetween rigid surfaces, either commensurate or incommensurate, that are separated by a freely diffusing fluidlayer.

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I. INTRODUCTION

Recent studies have revealed interesting transitions inshear response of many fluids when they are confinedtween crystalline surfaces that are only a few nanomeapart.1–10 Even though thick films are simple fluids at thimposed pressure and temperature, a yield stress or sfriction is observed when the film thickness is decreasedfew molecular diameters. This is generally assumed to refla transition to a solid-like state of the film due to the bouning walls. In some experiments there is a continuous divgence of the viscosity and relaxation time that is typical obulk glass transition.9 Other experiments show a sharp onsof the yield stress that is more akin to a first-order liquidcrystal phase transition.10 Simulations have found both typeof transition depending on factors such as the relative sizwall and fluid atoms and the molecular structure of tfluid.4,5 They also reveal that solid films transform back ina fluid state when the yield stress is exceeded, explainingstick-slip motion observed in some experiments.2,11

There is, however, an important difference between msimulations1–6 and experiments.7–10 Surfaces used in computer simulations are typically commensurate, i.e., shcommon periodicities. In fact, most simulations use idencal, aligned crystals for the two walls. Many also set tnumber of atoms between the surfaces to an integer mulof the number of atoms in one surface layer, facilitatingformation of ideal crystals. This cannot reflect typicexperiments—even those between nominally identical sfaces. The reason is that the crystallographic orientationthe surfaces is rarely controlled, and any small orientatiomisfit between otherwise identical surfaces makes thincommensurate.12,13 There is no well-defined crystallinstate of the film that can simultaneously lock into regiswith two incommensurate surfaces. Some of the interactibetween the wall and fluid atoms must be frustrated, anddynamics and statistical properties of the system may betered qualitatively due to this frustration.

The contrast between commensurate and incommenscases can be illustrated by considering a submonolayerof molecules between two identical crystalline surfaces in

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limit of high confining pressures where the hard-sphereteractions between wall and film atoms dominate. If the sfaces are aligned and translated so that all surface atomdirectly above each other, they create a periodic arraylarge openings that can accommodate film molecules. Arelative displacement of the commensurate walls greatlyduces the volume of these openings and is resisted byhard-sphere repulsion between wall and film atoms. Thisbe expected to prevent translation of one wall relative toother until a yield stress is exceeded. Note that individfilm molecules should still diffuse freely because there isfinite activation energy for motion between openings. Tdiffusion does not affect the equilibrium positions of thwalls because all openings are equivalent.

The situation is very different when the walls are maincommensurate by a relative rotation. Because the wshare no common periodicity, all possible relative positioof atoms on the two surfaces are sampled with equal prability. Each opening is slightly different and all displacments of the walls produce the same distribution of opings. It is well known that this symmetry under translatiocan lead to a vanishing yield stress and free diffusion ofwalls in the absence of a thin intervening film.12,13However,recent computer simulations14 indicate that a film can pinincommensurate surfaces together, providing a natural exnation for the observation of static friction in experiments

The simple picture for submonolayer films is that moecules search out the best set of openings for the givenpositions. They then resist translation of the walls becathese openings will be constricted by any translation.though there is an equivalent set of openings after trantion, these may be far away and only reached via a compcoordinated reshuffling of the molecules with a large actition free energy.

This type of pinning is much more subtle than that btween commensurate walls, and one may wonder whethpersists in the thermodynamic limit. For example, diffusiof individual molecules necessarily takes them to inequilent openings where they produce a different force onwalls. This will lead to a small displacement of the wallsany finite system, and accumulated motion of individual fi

2335 ©2000 The American Physical Society

Page 2: Conditions for static friction between flat crystalline surfaces

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2336 PRB 61MARTIN H. MU SER AND MARK O. ROBBINS

molecules might cause gradual diffusion of the walls tremains relevant in the thermodynamic limit.

In this paper we will investigate the conditions that detmine whether two surfaces are pinned together in the tmodynamic limit. The most sensitive technique is to consithe case where no force is applied in the direction tangento the walls, and to measure the relative displacement ofwalls as a function of time. Finite systems will always shodiffusion at sufficiently long times due to thermal activatiojust as the magnetic moment of finite systems will alwachange sign. Hence it is crucial to consider the scaling ofmobility of the walls with system size in order to determithe behavior in the thermodynamic limit.

We present results for commensurate and incommerate walls as a function of temperature and normal pressIn each case we compare results for bare walls to resultswalls separated by a submonolayer film. We find that bcommensurate and incommensurate walls remain pineven when the film molecules diffuse freely. One may expthat incommensurate walls undergo a transition from pinto unpinned with increasing temperature or decreasing psure, but it is difficult to identify the transition with availablsystem sizes. A general argument is presented that commsurate walls are always pinned even as the film becomnearly ideal gas, although the pinning force may becoexponentially small. Thus observation of static friction btween two commensurate crystals need not imply thatintervening film is solid as is often assumed.

In the next section we describe the model used insimulations and the averaging techniques. Section III psents results for commensurate and incommensurate wwith and without monolayer films. Our conclusions are sumarized in Section IV.

II. METHOD

In this study, we have used the same model as in Ref.The walls are@111# surfaces of an fcc crystal, and therefohave a triangular lattice structure. Atoms in the walls acoupled to their equilibrium lattice sites by springs of stinessk. In the limiting case of rigid walls, the coupling iconsidered infinitely strong,k5`, and the atoms are constrained to their equilibrium positions. Periodic boundaconditions are applied in the plane of the walls. The coornate system is chosen so thatx and y are in the plane of thewalls andz is normal to them.

The molecules between the walls are short chains, econtaining six monomers. All monomers interact with eaother and with wall atoms via a truncated Lennard-Jopotential,

V~r !54e@~s/r !122~s/r !6#1Vc , ~2.1!

where r is the separation, ande and s are characteristicenergy and length scales, respectively. Wall atoms fromposing surfaces interact via the same Lennard-Jones potial. The potential is cut off atr c and shifted byVc so thatV(r c)50. Adjacent monomers on a chain interact viaadditional FENE potential15

VCH~r !52~1/2!kR02 ln@12~r /R0!2#, ~2.2!

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whereR051.5s andk530e/s2. All quantities are expressein units of s, e, and the massm of one monomer. Thecharacteristic time istLJ[Ams2/e. Unless stated otherwiser c521/6s.

The lattice sites of the bottom wall are fixed, while the twall is allowed to move under the combined influence of tforce from the coupling springs and an external force on elattice sitefW . A constant external forcef z is applied normalto the top wall, and the tangential components of the exteforce fW i are set to zero to allow free diffusion in the planethe walls. We choose a mass of the top wallMw that is onlyhalf the combined mass of one layer of atoms. This choallows the top wall to respond more rapidly than if wchoose a more physical mass of several layers. Transstate theory and corrections to it show that the rate at whenergy barriers are crossed scales asA1/Mw to leadingorder.16 By using a lighter mass we speed the calculatiowithout changing the qualitative behavior.

Both the diffusion of the top wall and the diffusion oindividual monomers are monitored. The mean-squaredplacements alongx and y are calculated separately, and thaveraged to get dx2(t)&, the mean-squared displacemealong a single coordinate after a timet. The results are alsoaveraged over at least eight independent intervals of lengt.

The equations of motion are integrated using a fifth-orpredictor-corrector method with time stepDt50.005tLJ . ThetemperatureT is controlled by coupling the monomers, anwall atoms if mobile, to a Langevin thermostat.17 The fric-tional force in the Langevin equation is2gmv wherev isthe instantaneous velocity andg is the damping rate. We usg52tLJ

21 so that the motion of the particles is well into thunderdamped regime. A small additional damping with r0.05g was added to the center of mass of the top wall. Thdampings fix the free diffusion of the top wall in the absenof any interactions with the bottom wall atD05kBT/2.05gMw , wherekB is Boltzmann’s constant. Notethat the denominator gives the ratio of damping forcevelocity for uniform motion of the top wall.

III. RESULTS

We will present results for the simplest choices of comensurate and incommensurate walls. In all cases theand bottom walls have the same structure and nearneighbor spacing,d51.209s. The size of the surfaces wilbe expressed in terms of the numberN of atoms per layer ofwall atoms. The areaA is given byA5NA3d2/2. Thus toconvert between the force on each atom and a pressurshear stress, values offW should be divided by 1.266s2.

In the commensurate case the walls are perfectly alignas if a single crystal had been cut in two at the interface. Tincommensurate case corresponds to rotating the top wa90° relative to the bottom wall. However, the walls must abe distorted slightly in order to conform to the same perioboundary conditions. This means that the walls are not trincommensurate, but the residual commensurability doesappear to influence the results. As discussed below,amount of the distortion and the difference from ideal incomensurate surfaces decrease with increasing system siz

For each choice of walls we will first consider the limitin

Page 3: Conditions for static friction between flat crystalline surfaces

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PRB 61 2337CONDITIONS FOR STATIC FRICTION BETWEEN FLAT . . .

case of rigid walls with no molecules in between. Thenconstraint of rigidity is relaxed, and finally molecules aintroduced between the surfaces. The presence of amonolayer film is enough to move the walls far enough apthat there are no direct interactions between them. Any pning must be mediated through interactions with the film

The key question is to determine whether there is atential that pins the lateral position of the top wall relativethe bottom wall. We define the following time-dependemeasureD(t) for the mobility of the top wall at timet:

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whereD0 is the free diffusion constant of the top wall andincluded to remove the trivial decrease in diffusion dueincreasing wall mass asN increases. The top wall is pinned

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tends to zero. IfD remains finite, the wall is unpinned anthe productDD0 can be interpreted as the long time diffsion constant of the top wall.

A. Commensurate walls

1. Bare surfaces

If the surfaces are infinitely rigid, the only degreesfreedom are associated with the location of the centermass of the top wall. Each atom on the surface of thewall ‘‘feels’’ exactly the same potential and force from aoms in the bottom wall. The total force on the center of mcoordinate is a periodic function of the lateral displacemthat grows linearly withN. In the thermodynamic limit, aninfinite activation energy is needed to displace the wall athere can be no diffusion.

Relaxing the constraint of infinite rigidity does not chanthe linear scaling of activation energy withN, although itmay lower the prefactor. In our system, decreasingk allowsatoms to translate relative to the center of mass coordinaorder to lower their energy. There is no change in the grostate energy, but the energy of transition states is decreaAt finite temperature, thermal displacements further decrethe activation free energies. Due to the strictly harmonicture of the springs, the motions of the independent atomsincoherently to yield a single effective Langevin noise adamping term on the center of mass. Thus the problem minto diffusion of a single, damped particle in a periodic ptential U, where U depends on the temperature, pressuandk.

Diffusion in a periodic potential has been studied in grdetail by a number of authors.18,19 The general trends arillustrated by the results for the simple case of diffusion oparticle in a one-dimensional sinusoidal potential with acvation energy DF. In this case the diffusion constansatisfies18

D5D0@ I 0~DF/2kBT!#22 ~3.3!

whereI 0 is the modified Bessel function. WhenDF/2kBT issmall, the particle diffuses almost freely:

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In the opposite limitDF/2kBT@1, the diffusion is activated:

D'D0~DF/2kBT!exp~2DF/kBT!. ~3.5!

Since DF increases linearly with the area of the wall, thmotion will always be activated for large enough systesizes, and will vanish exponentially in the thermodynamlimit. However, one may need very large system sizesreach this limit for reasonable parameters.

Figure 1 shows the mean-squared displacement of thewall ^dx2(t)& as a function of time forf z51.2e/s, kBT50.8e, and the indicated wall sizes. The mean-squaredplacements are multiplied byN to remove the trivial dependence on wall mass. This collapses the data at early timwhere the walls move ballisticallydx2(t)&}t2. At longertimes the smallest system shows a simple crossover to dsive motion, dx2(t)&52Dt. The value of the diffusion con-stant is nearly equal to the value for free diffusion,D0. As Nincreases, a plateau develops between the ballistic and dsive regions, and the diffusion constant decreases. ByN5144 the wall is completely pinned over the length of tsimulation, although any finite system will eventually difuse. These results are just what would be expected from~3.3! with DF}N. For small systems,DF/2kBT may be somuch less than unity that free diffusion is observed. Hoever asN increases, the motion becomes activated, andDdrops precipitously.

Figure 2 illustrates this behavior for a number of normforces. The diffusion constant is plotted as a function ofnumber of atoms in a wall layerN for k5100es22 andkBT50.8e. The free diffusion constantD0 is indicated by asolid line. In each case,D was evaluated from the timet1 todiffuse a distances along one of the coordinates parallelthe walls using the relationD5s2/2t1.

At the lowestf z andN, D decreases roughly as 1/N and isnearly equal to the value for free diffusion. AsN increases,there is a rapid drop inD, indicating a crossover to activatebehavior. The success of Eq.~3.3! in describing this cross-over is illustrated by fits to the results forf z50.1 and 0.3e/s~broken lines!. Increasingf z moves the crossover to actvated behavior to lowerN, indicating thatDF rises withf z aswell asN. This is entirely consistent with the linear relatio

FIG. 1. Mean-squared displacement of top wall along a sincoordinate, dx2(t)&, as a function of timet for commensurate wallsof the indicated sizes. Displacements are multiplied byN to removethe trivial dependence on wall mass. Heref z51.2e/s and kBT50.8e.

Page 4: Conditions for static friction between flat crystalline surfaces

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2338 PRB 61MARTIN H. MU SER AND MARK O. ROBBINS

between static friction and pressure that is found for comensurate surfaces20 and for incommensurate surfaces seprated by adsorbed layers.14

The main lesson to be learned from Fig. 2 is that whileis easy to determine that a system is pinned, there issimple way to prove that a system is unpinned in the thmodynamic limit. Relatively large systems can appear tofuse freely, even under conditions where systems becpinned in the thermodynamic limit. The reason is thatactivation energy forf z50.1e/s is small, which is due to thelarge average distance between the walls^Dz&51.214. Thisvalue exceeds the value for the interaction cutoff radiusr c521/6. Thus the probability of an atom in the top wahaving a nonzero interaction with the bottom wall at agiven instant is very small.

2. Submonolayer lubrication

The two commensurate walls considered here con32332 atoms in each surface. The film in between consof 42 chain molecules each containing six monomers. Tcorresponds to 1 monomer for every 4 atoms on each wand is roughly 1/4 of an equilibrium monolayer. The bottowall is fixed and the normal force on each atom in thewall f z510e/s. This corresponds to a normal pressurepz

57.96e/s3. The tangential forcefW i50W , andkBT50.8e. Forcomparison, we note that the triple point of monomers wlong-range interactions (r c→`) is at kBT50.7e.

In Fig. 3, the motion of the top wall is compared to thmotion of individual monomers in the chain molecules. Tinterpretation of the dynamics of the monomers in Fig. 3as follows. For timest,531022tLJ the monomers are in thballistic regime^dx2(t)&}t2. For times 0.5,t/tLJ,103, themonomers exhibit subdiffusive behavior that indicates thare initially trapped near a single energy minimum in tperiodic potential provided by wall atoms.21 At t5103tLJ , amonomer has typically moved 1s, which approximately cor-responds to the distance between two equivalent minimthe periodic potential. At longer times the motion of thmonomers approaches diffusive behavior, where the mesquared displacement grows linearly with time. Thusmonomers act like particles in a lattice gas, hopping betwminima in the wall potential.

During the entire length of this simulation the top waremained stuck in one minimum and no diffusive behav

FIG. 2. Diffusion constant calculated from the time to moves as a function of the number of atoms per wallN at the indicatedvalues off zs/e andkBT50.8e. A solid line shows the free diffu-sion constantD0. Dashed and dot-dashed lines show fits to Eq.~3.3!for f z50.1 and 0.3e/s, respectively.

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could be observed. Oscillations around the equilibrium potion led to a mean-squared displacement that saturate5.031025s2. This and the temperature can be used to emate the effective spring constant for the local free eneminimum of keff515.6e/s2 per wall atom. An estimate othe static frictionFs in thex direction can then be obtainedone makes the simplest assumption for the form of the podic variation of free energy withx, leavingy unconstrained.The period is given by the distance alongx to the nearestequivalent minimum. For a triangular surface this is halfthe nearest-neighbor spacingd. Using a single Fourier component to represent the free energy we haveF(x)52F1 cos(4px/d). The static friction is given by the maximum force, i.e., the maximum of the first derivative ofF.The maximum of the second derivative giveskeff . Using thisand the value ofkeff from above, we obtainFs5keffd/4p51.5e/s. This agrees quite well with the actual frictioforce ofFs'1.4e/s that we obtained in an independent ruHowever, we note that our arguments are too rough to expthis level of agreement, because geometrical factorshigher harmonics in the free energy have been left out.

The walls never approached close enough to interactdi-rectly. Hence, the pinning of top and bottom wall was meated by the film in between, which was freely diffusing inlattice-gas-like state. This result may seem rather countetuitive. The observation of a yield stress in surface forapparatus experiments7–10 is often assumed to imply that ththin film confined between the surfaces has entered a sstate. This clearly need not be the caseif the two surfaces arealigned into a commensurate configuration. More generathe ability of crystals to resist shear does not depend olack of diffusion, but rather the presence of long-range orthat produces Bragg peaks. In certain crystals, e.g., ioconductors, the diffusion of some species can be quite raAs long as the density modulation measured by the Brpeaks remains, the system can resist shear.

A simple argument shows that two commensurate wshould be pinned in the thermodynamic limit at allT and f z .The reason is that the periodic potential of a single winduces a commensurate density modulation parallel towall in an adjacent film.22 The magnitude of the densitmodulation will decrease exponentially with distance frothe wall, but remains finite. If there is a commensurate wat some distanceh, the energy will necessarily depend on th

FIG. 3. Mean-squared displacement along a single coordin^dx2(t)&, of individual monomers and of top wall as a functiontime t at f z510e/s and kBT50.8e. The walls are commensuratandN532332.

Page 5: Conditions for static friction between flat crystalline surfaces

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registry between the density modulation and the wall lattiThus there will always be a periodic force that pins the wand that grows linearly with system size. However, the fenergy barrierDF will decrease exponentially withh, andthe size of the system must increase proportionately in oto reach the thermodynamic limit. Our conclusions are csistent with simulations by Curryet al.,6 who consideredfilms that were several layers thick and found a periopinning force even when there was rapid diffusion within tfilm.

Figures 4 and 5 show that simulations with afixedsystemsize exhibit anapparenttransition from pinned to depinneas f z decreases orT increases. The mean-squared displament after 2000tLJ is plotted as a function off z or T. At lowf z or high T the mean-squared displacement is consiswith the value for free diffusion, dx2&'10s2. Thus forthese parameters and at this system sizeDF is much lessthankBT @Eq. ~3.3!#. However, as noted for dry commensrate walls, this is not enough to establish that the sysremains unpinned in the thermodynamic limit. IfDF di-verges in the limitN→`, the system will be pinned in thethermodynamic limit. Our results are consistent withDF}N at all f z and T. Simulations with largerN show a con-sistent shift of the apparent transition to free diffusionlower f z and higherT.

B. Incommensurate walls

1. Bare surfaces

Models of the friction between two surfaces with differelength scales are generically referred to as Fren

FIG. 4. Mean-squared displacement of top wall along a sincoordinate, dx2(t)&, after time t52000tLJ as a function off z atkBT50.8e. The walls are commensurate andN532332.

FIG. 5. Mean-squared displacement of top wall along a sincoordinate, dx2(t)&, after timet52000LJ as a function ofT at f z

50.75e/s. The walls are commensurate andN532332.

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Kontorova ~FK! models and many examples have bestudied.12,13In the limit of infinite, perfectly rigid walls it canbe shown that the free energy barrier for sliding motionexactly zero if the walls are incommensurate.12,13 Thus thetop wall will diffuse freely in the thermodynamic limit. Fi-nite systems with periodic boundary conditions, like thoconsidered here, can never be perfectly incommensurate.order of commensurability can be measured by the smaintegerq that allows the ratio of lattice constants to be epressed asp/q, where p is also an integer. Theoreticacalculations13,23 suggest thatDF vanishes exponentially faswith increasingq. If the highest possible value ofq is chosenfor each system size, then the total value ofDF shouldvanish in the thermodynamic limit at least as fastc1N exp(2Ac2N), where theci are constants. Thus ousimulations should show the same behavior as truly incomensurate systems in the thermodynamic limit.

As the constraint of perfect rigidity is relaxed, it becompossible for two incommensurate walls to lock into a comon periodicity.12,13There is a transition at a critical value othe ratio of the strength of the intersurface potential tointernal stiffness of the walls. This would correspond to tratio DF/Ns2k in our simulations. The critical value depends on the shape of the potential and the ratio of latconstants, and has mostly been determined for odimensional systems.

To illustrate this behavior we consider two incommensrate walls of size 31336 atoms and vary the wall stiffnessk.One wall is rotated by 90° with respect to the other waThen small strains are applied to make the resulting surfasquare so that they share the same periodic boundary cotions. Unlike the other simulations presented here, we uslong cutoff radius,r c52.2s, for the Lennard-Jones potentiabetween atoms on different walls. One consequence of ththat there is an effective normal force on each atom duethe adhesion of the surfaces that is of ordere/s. We used asmall external forcef z50.1e/s and a low temperaturekBT50.1e so that thermal fluctuations are small.

In Fig. 6, the mean-squared displacement of the top wis plotted against time. Fork<10e/s2 the top wall is pinnedand the mean-squared displacement saturates at a veryfraction of a lattice constant. Direct observations of atompositions show that atoms have undergone large rearraments from their initial lattice sites in order to lock togeth

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FIG. 6. Mean-squared displacement of top wall along a sincoordinate, dx2(t)&, as a function of timet for different couplingsk of wall atoms to their equilibrium positions. HerekBT50.1e,f z50.1e/s, and r c52.2s. The walls are incommensurate andN531336.

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2340 PRB 61MARTIN H. MU SER AND MARK O. ROBBINS

in a free energy minimum. For allk>25e/s2, the wallsfollow nearly identical curves, and the asymptotic behavis consistent with the free diffusion constantD0. Note thatthere is a smooth crossover from ballistic to diffusive motiwith no subdiffusive regime like that found for monomebetween commensurate walls~Fig. 3!. This indicates thatthere is no potential well that temporarily locks the surfactogether.21 Our results for commensurate systems show tthe above findings are not enough for us to conclude thattop wall would remain depinned in the thermodynamic limat k>25e/s2. However, in contrast to the commensurasystems, increasing the system size does not changevalue ofk where the transition occurs. If there was a finDF that scaled withN this shift would be evident.

We can use the Lindemann criterion to estimate whakshould be in order to model a Lennard-Jones crystal. In oto have an rms displacement of 10% of the nearest-neighspacing at the triple point (kBT50.7e), we must havek'140e/s2. This is well into the range of values where wfind free diffusion. In order to see pinning for realistic valuof k, the interaction between the walls must be increarelative to that within the walls~i.e.,k). This can be done byincreasing the normal force.

Note that our use of springs connected to lattice sites isEinstein approximation to an elastic crystal and doestreat long-wavelength elastic deformations accurately. Hoever, the displacements required to lock two lattices togehave relatively short wavelengths, and simulations with maccurate elastic models yield the same transition frompinned to pinned with an increasing ratio betweenstrength of inter- and intrawall interactions.12,13

2. Submonolayer lubrication

As above, two identical walls were made incommensurby a rotation of 90°, and then strained slightly to fit squaperiodic boundary conditions. As for the commensurate cthe film contained about 1 monomer for every 4 atomseach wall layer or about 1/4 of a monolayer. Unless othwise noted, the walls contained 31336 atoms each, and therwere 46 film molecules containing six monomers each.chose to consider the most difficult case for pinning bsurfaces, completely rigid walls (k5`).

We first compare the diffusion of the top wall to thatindividual monomers. Figure 7 shows results for the sa

FIG. 7. Mean-squared displacement along a single coordin^dx2(t)&, of individual monomers and of top wall as a functiontime t at kBT50.8e and f z510e/s for incommensurate walls withN531336.

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parameters (f z510e/s, kBT50.8e) as the commensuratsystem of Fig. 3. Note that individual monomers are muless mobile at long times when between the incommensuwalls. Presumably this is because commensurate wallsduce long channels of relatively wide openings between pallel lines of atoms on opposing walls. Incommensurwalls produce a more random environment with fewer laopenings between atoms on opposing walls.

The incommensurate walls themselves move furthermore rapidly than commensurate walls. As a result therno clear time separation between the motion of monomand walls in the incommensurate case at thisN. Over thetime interval shown the top wall appears pinned, becausemotion is subdiffusive, and the total distance moved is lthan 10% of a lattice constant. The monomers have amoved less than a lattice constant and exhibit subdiffusmotion.

Figure 8 shows how the diffusion of the top wall changwith decreasing normal force. As in Fig. 7 the walls moballistically up to a timet'1. The distance traversed increases asf z decreases. At longer times, motion is subdiffsive and the curves are roughly parallel on a log-log plot. Ff z54, the mean-squared displacement can be describedpower law ^dx2(t)&}ta, for at least three decades of timAn exponent ofa50.19460.004 is obtained.

As in the commensurate case, there appears to be asition from pinned to depinned asf z decreases orT in-creases. Figures 9 and 10 show the mean-squared disp

e, FIG. 8. Mean-squared displacement of top wall along a sincoordinate, dx2(t)&, as a function of timet for two different nor-mal forces andkBT50.8e. Bottom and top wall are incommensurate andN531336. The straight line is a fit to a power lawta witha50.19460.004.

FIG. 9. Mean-squared displacement of top wall along a sincoordinate, dx2(t)&, after timet52000tLJ as a function off z atkBT50.8e. The walls are incommensurate andN531336.

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PRB 61 2341CONDITIONS FOR STATIC FRICTION BETWEEN FLAT . . .

ment after 2000tLJ as a function off z and T, respectively.Note that the displacement changes over a somewhat brorange than in the corresponding figures for the commenrate walls~Figs. 4 and 5!. The transition is also at a lowerTand higherf z , indicating that it is more difficult to pin theincommensurate walls. This is consistent with our studiesthe static friction,14 which is roughly proportional toDF. Wefound that submonolayer films between commensurate wgave three to five times larger friction forces than incommsurate walls under similar conditions.14

To test whether the transition from pinned to depinnedreal, we performed simulations with larger walls at the safilm density. Figure 11 compares results for walls contain31336 atoms and 62372 atoms at f z52e/s and kBT50.8e. Note that the monomer diffusion is nearly identicat the two system sizes, and shows a clear diffusive reg~slope of one! at the longest times. This implies that thenergy landscape that monomers move through is not inenced by system size. In contrast, there is a strikingdependence in the dynamics of the top wall. The trivial sdependence of the free diffusion constant,D0}N, has beenremoved by multiplying the mean-squared displacementhe larger wall by four. This collapses results for differesizes in the ballistic regime (t,10tLJ). At larger times the

FIG. 10. Mean-squared displacement of top wall along a sincoordinate, dx2(t)&, after time t55000 tLJ as a function of tem-perature for two different system sizes atf z52e/s. The results forthe larger wall were multiplied by a factor of four to remove ttrivial dependence on wall mass.

FIG. 11. Mean-squared displacement along a single coordin^dx2(t)&, for the top wall~squares! and monomers~circles! at twodifferent system sizes withf z52e/s and kBT50.8e. The mean-square displacement of the larger top wall has been multipliedfactor of four to be compatible with the small system in the ballisregime, and the larger system contained four times as manymolecules~184!.

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smaller wall shows a simple crossover from ballistic to dfusive motion, while the larger wall shows subdiffusive bhavior and seems to stop moving at long times.~The step inthe data at a few hundredtLJ is a result of a rare, relativelylarge displacement of the wall.! This is the same type obehavior that was seen for increasing system size in cmensurate systems. It indicates that there is a finiteDF thatgrows with N causing motion to stop asNDF rises abovekBT. One can conclude that the incommensurate wallspinned in the thermodynamic limit for these parameters.

The apparent transition between pinned and depinstates continues to shift to higherT ~Fig. 10! and lower f zwith increasingN for the largest systems we have been ato study. However, we have no analog of the argumentcommensurate walls that suggests that incommensurate wshould be pinned at allT and f z . The density modulationproduced by one wall will be incommensurate with the oposite wall and produce no net energy shift. Locking btween the two surfaces must enter as a higher-order sustibility that is difficult to calculate. Forf z!kBT/s the wallsmove far apart and the molecules form an increasingly idgas. It seems reasonable that the depinning force wouldish in the thermodynamic limit under these conditions, bthis remains an open question.

IV. CONCLUSIONS

We have performed a systematic study of the conditiofor pinning of commensurate and incommensurate waThe case of bare walls is relatively straightforward and hbeen considered previously. However, examination ofscaling of the diffusion with system sizeN, temperatureT,and normal forcef z provides a useful benchmark for oustudies of submonolayer films. Bare commensurate wallsalways pinned by a periodic potential that grows with systsize. However, relatively large systems can appear unpinif the potential is small enough. Incommensurate wallscompletely unpinned until they become so deformable tthey can rearrange by distances of orders to accommodatethe opposing wall.

Commensurate walls remain pinned when a submolayer film is introduced between them. A general argumwas given that this pinning should always exist, and oresults show that even when the film becomes a gas the wdo not diffuse in the thermodynamic limit. However, thpinning is very weak and one has to go to large system sto detect it. Curry and coworkers have also seen pinningcommensurate walls by diffusing films.6

Introducing a submonolayer film can pin incommensurwalls, even when they are completely rigid. In the Introdution we noted that diffusion of a monomer to an inequivalesite should cause a displacement of the top wall in any finsystem, and wondered whether such displacements mighcumulate into diffusion of the top wall. However, the monmer will in general diffuse to a site that minimizes its enerfor the given position of the top wall, and thus add to tpotential energy barrier that pins it. Our results are consiswith a pinning potential that is linear inN, just as in thecommensurate case. Independent studies of the static fricas a function of system size confirm this linear relation.14

The behavior of incommensurate walls in the thermod

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2342 PRB 61MARTIN H. MU SER AND MARK O. ROBBINS

namic limit is very important to the study of static frictionsince contacting surfaces are almost never commensuOur studies confirm previous work12,13 in showing that bareincommensurate walls are very unlikely to exhibit static frtion in the thermodynamic limit. Surfaces are also very ulikely to be bare, especially if exposed to air. Our resuclearly show that even a small fraction of a monolayertween the surfaces is enough to produce static friction inthermodynamic limit. A particularly surprising result is ththe monolayer itself need not be in a crystalline or glastate. As shown in Fig. 11, molecules may undergo neunhindered diffusion and still produce static friction ov

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large enough areas. It remains to be seen whether theretransition to a depinned state as the layer is made thickelowering pressure, increasing temperature, or introducmore molecules between the surfaces.

ACKNOWLEDGMENTS

We thank Gang He for useful discussions. Support frthe National Science Foundation through Grant No. DM9634131 is gratefully acknowledged. M.H.M. is grateful fsupport through the Israeli-German D.I.P. Project No. 3101.

,

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