�22� that friction would be in direct proportion to the surfaceenergy �. This explains why in many experiments strongeradhesion usually corresponds to a higher friction.

In the above discussions the only normal force on thesliding molecule results from the attractive interactions.When externally applied load or pressure is to be considered,the energy balance has to be modified as

F � �d = �E + Fext � �D �26�

where �E is the change in system energy defined in Eq �21�,the second term in the right hand represents the work doneby external force during the upward motion of the molecule.The shear stress is

�C =��

�+

�D

�d·

Fext

A= C1 + C2Pext �27�

Equation �27� provides important information on thenature of wearless friction, which deserves a further discus-sion.

The wearless friction consists of two components,namely the adhesion term C1 and the pressure term C2Pext.Similar expressions for the shear stress, e.g., �C=�0+�P,were also derived by other investigators �26,27�, where �0represents the contribution from adhesion and � is referredto as the friction coefficient.

The adhesion term is a primary contribution in mostcases of atomic-scale friction since a typical external pres-sure is in the order of 10 MPa while the internal van derWaals pressure is estimated as 1 GPa when using a typicalHamaker constant. The adhesion term should be also con-sidered as a function of the applied normal load because it isinversely proportional to the mean space D0, which is depen-dent on the external pressure.

If the surfaces are damaged during sliding so that weardebris and multi-asperity contacts are involved in the pro-cess, the mechanism of friction will be substantially differ-ent from what we discussed for wearless friction.

In summary, sliding can be regarded as a process duringwhich interfacial atoms would experience a series of stick-slip motions, similar to the jump in and out in the adhesioncase, and it is the energy loss in this approach/separationcycle that determines the level of friction.

5 The Nature of Static Friction

5.1 Molecular Origin of Static FrictionFigure 29 schematically shows the energy barrier of a systemin contact and its change with applied lateral force. For twosurfaces in static contact, interfacial atoms or molecules are

pinned in potential energy minima �Fig. 29�a��. If a lateralforce is applied on one of the surfaces in an attempt to ini-tiate motion, the force biases the potential energy curve sothat the energy barrier, preventing the trapped atoms fromescaping the energy minima, decreases with increasing force�Fig. 29�b��. Eventually, the atoms will start to move once theenergy barrier drops to zero �Fig. 29�c��.

From a microscopic point of view, the change of systemstate from rest to sliding is a process of pinning/depinningtransition. The interfacial atoms in a sliding system may notdepin at the same time. When a part of atoms are depinned,the system will reorganize itself into a new state that requiresa larger force to depin. Therefore, static friction is defined asthe largest force when all the pinning states disappear.

Studies based on the Frenkel-Kontorova model revealthat static friction depends on the strength of interactionsand structural commensurability between the surfaces incontact. For surfaces in incommensurate contact, there is acritical strength, bc, below which the depinning force be-comes zero and static friction disappears, i.e., the chainstarts to slide if an infinitely small force F is applied �cf. Sec-tion 3�. This is understandable from the energetic point ofview that the interfacial atoms in an incommensurate sys-tem can hardly settle in any potential minimum, or the en-ergy barrier, which prevents the object from moving, can bealmost zero.

Solid contacts are incommensurate in most cases, ex-cept for two crystals with the same lattice constant in perfectalignment. That is to say, a commensurate contact will be-come incommensurate if one of the objects is turned by acertain angle. This is illustrated in Fig. 30, where open andsolid circles represent the top-layer atoms at the upper andlower solids, respectively. The left sector shows two surfacesin commensurate contact while the right one shows thesame solids in contact but with the upper surface turned by90 degrees. Since the lattice period on the two surfaces,when measured in the x direction, are 5�3 Å and 5 Å, respec-tively, which gives a ratio of irrational value, the contact be-comes incommensurate.

In reality, static friction is always observed regardless ofwhether the surfaces in contact are commensurate or not.This raises a new question as to why the model illustrated inFig. 29 fails to provide a satisfactory explanation for the ori-gin of static friction.

Another mechanism of static friction suggests thatwhen two surfaces are pressed together under a normal load,the atoms or molecules at the interface will rearrange them-selves to minimize the energy and to form localized junc-tions called cold welding, which is often observed in contacts

Fig. 29—A diagram of the energy barrier, �a� for a system in static contact, �b� and �c� variations of the barrier with applied lateral force.

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between clean metal surfaces in vacuum. Cold-welded junc-tions provide one explanation for static friction, but ruptureof the junctions will inevitably cause structural damage andwear. This brings us back to the old question, raised earlier inthis chapter, which was how to explain the static friction thatoccurs without wear.

It has been proposed recently �28� that static frictionmay result from the molecules of a third medium, such asadsorbed monolayers or liquid lubricant confined betweenthe surfaces. The confined molecules can easily adjust or re-arrange themselves to form localized structures that are con-formal to both adjacent surfaces, so that they stay at the en-ergy minimum. A finite lateral force is required to initiatemotion because the energy barrier created by the substrate-medium system has to be overcome, which gives rise to astatic friction depending on the interfacial substances. Themodel is consistent with the results of computer simulations�29�, meanwhile it successfully explains the sensitivity offriction to surface film or contamination.

Another possibility associates with the thermodynamicnature of the system. It has been recognized that the energybarrier diminishes for the incommensurate contacts be-cause the potential energies from two surfaces in contact arecombined together and canceled with each other. However,local and instantaneous energy barriers randomly distrib-uted over time and space may appear due to thermodynamicfluctuations. It is thus possible that friction may result froma nonzero time and spatial average of the instantaneous en-ergy barriers.

5.2 Static Friction and Stick-Slip MotionFrom the point of view of system dynamics, the transitionfrom rest to sliding observed in static friction originatesfrom the same mechanism as the stick-slip transition in ki-netic friction, which is schematically shown in Fig. 31. Thesurfaces at rest are in stable equilibrium where interfacial at-oms sit in energy minima. As lateral force on one of the sur-faces increases �loading�, the system experiences a similarprocess as to what happens in the stick phase that the surface

starts to slide when the energy barrier disappears. The initia-tion of surface motion in this sense is a result of system insta-bility, the same as the occurrence of slip. The similarity al-lows us to employ the models in studying wearless kineticfriction, to examine the performances of static friction, forexample, the creation of creeps.

In the experiments of static friction where two solidswere nominally at rest, a small displacement between thesurfaces, or a creep, was observed, prior to rapid slip that oc-curred at F=Fs. The creep length d0 was reported to be in arange of 1 �m �30,31�. The formation of creep was conven-tionally interpreted as a result of pulling and breaking theadhesive junctions. However, the presumption of regardingplastic deformation or cold-welding as the sole cause ofstatic friction is fundamentally inaccurate. What is to be em-phasized here is that the criterion for two surfaces undershearing to hold together without macroscopic motion, or toslide with each other, simply depends on the favorable en-ergy conditions of the system. In other words, the occur-

Fig. 30—Change in commensurability of two surface in contact, on the left is a commensurate contact, on the right the contact becomesincommensurate when the upper body being turned by 90 degrees, the open and solid circles denote the surface atoms of the upper andlower bodies, respectively.

Fig. 31—A schematic plot of friction versus time, illustrating theclimbing of the friction coefficient followed by a sudden drop atthe time of slip.

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rence of stick-slip does not have to be interpreted in terms ofthe formation or rupture of adhesive junctions, but is allabout which state is more favorable for the system to realizethe energy minimization.

Although a system under shear may not exhibit a macro-scopic motion during the stick process, there will be a smallrelative displacement between the sticking surface atomsand the substrate. This can be demonstrated by an exampleshown in Fig. 32 where the oscillator stands for a surfaceatom that sticks at position x, and the supporting block rep-resents the substrate where force F is applied. The coordi-nate difference between the oscillator and the support is �x0−x� that increases with the growing force. At the momentright before the atom starts to jump, the value of �x0−x�reaches the maximum, which corresponds to the creeplength of static friction. The creep length in this case is com-parable to the period of the potential function of the oppositesurface, which has been demonstrated by the force curvegiven in the inserted panel of Fig. 32.

A similar process can be observed at the asperity level, asshown in Fig. 33, where a lateral force F pulls the upper solidforward by a distance, u, while the asperity attached to thesolid body remains in contact with the lower asperity. Thevalue of u at the moment when the asperity is suddenlypulled out of contact gives rise to creep length of static fric-tion. By referring to the force curve shown in the insertedpanel of Fig. 33, the creep distance for this system is esti-mated to be similar with the asperity dimension in the slid-ing direction, which is in agreement with the measuredcreep length, �1 �m, as reported in Ref. �30�.

So far we have compared the static friction with thestick-slip transition. In both cases the system has to choosebetween the states of rest and motion, depending on whichone is more favorable to the energy minimization. On theother hand, the differences between the two processes de-serve a discussion, too. In stick-slip, when the moving sur-face slides in an average velocity V, there is a characteristictime, tc=d0 /V, that defines how long the two surfaces can

Fig. 32—Static friction and creep in an atomic system: the creep distance is defined as the maximum value of �x0-x�, inserted panel showsthe variation of lateral force.

Fig. 33—Static friction between asperities, inserted panel shows variation of lateral force.

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hold together, or the “age” of the contact. This is the timeneeded for the interfacial molecules to relax and to reestab-lish the contact. For this reason, stick-slip will disappear un-der a large sliding velocity because time is too short to re-sume the contact population. In static friction, on the otherhand, the surfaces usually are set at rest for a long time be-fore being pulled apart so that the contact age in most casescan be calculated from the time of −�. This would lead to twoconsequences. First, static friction, the maximum forceneeded to initiate sliding, is usually higher than the forcepeak appeared in stick-slip, mainly due to the longer contacttime. Secondly, instability in static friction occurs at the be-ginning of sliding, possibly followed by a steady sliding,while the stick-slip will proceed in a periodic manner pro-vided that the dynamical set up of the system remains un-changed. Nevertheless, one has to keep in mind that the con-tact time in some cases is an important factor to beaccounted for in the study of static friction.

5.3 Static Friction in Lubricated ContactsThe mechanisms of static friction and stick-slip motion, asdiscussed in the last section, are supposed to be a good de-scription of dry friction. Another case, perhaps more generalin engineering practices, to be addressed in this section is lu-bricated sliding where liquid lubricant, consisting of a fewmolecule layers, is confined between two solid walls. Bothexperimental and theoretical studies indicate, as we havediscussed in Chapter 5, that there are substantial changes inrheology of the confined lubricant, and the liquid may tran-sit practically to a solid-like state when film thickness be-comes molecularly thin �32,33�.

As a lateral force drags one solid surface along the x di-rection, the solidified thin films experience an increasingshear stress until reaching a certain limit �a when the lubri-cant yields and starts to flow again like a liquid. The solid sur-face pulled by the lateral force then starts to move forwardwith a certain sliding velocity. If the solid slows down, thestress will decrease correspondingly but the lubricating filmmay remain at liquid state until it meets the second criticalvalue, �c, when the lubricant returns to solid state. In thismodel, the static friction, or transition from rest to motion isinterpreted as a result of yield of solidified lubricating films,and stick-slip is regarded as a process of periodic transitionbetween melting and freezing phases �24�.

On the basis of experimental observations and com-puter simulations, Persson �34� proposed a stress-velocityrelation, �= f�v�, as schematically reproduced in Fig. 34,which describes how shear stress changes with sliding veloc-ity. Assume the system begins from a stationary state withboth sliding velocity and shear stress being zero. After a lat-eral force is applied on one surface and pulls it along the slid-ing direction, the shear stress on the confined lubricant risesup but the surface remains stationary, i.e., the shear stressincreases along a vertical line of V=0 in Fig. 34. When thegrowing shear stress goes to a critical value, �a, the solidifiedlubricant yields to the stress and starts to flow. As a result, thevelocity of the solid surface increases rapidly from 0 to va.For a further increase of velocity, the stress will go up alongthe oblique line BE, following the viscous law of fluid trac-tion. If the sliding slows down, however, the lubricant doesnot immediately resume the solid state, but remains as liq-

uid so that the stress decreases along the line BC until itmeets the point C where the sliding velocity and correspond-ing stress are denoted as vb and �c, respectively. As the veloc-ity decreases continuously, the lubricant will go through astate known as “granular fluid” in the range of vc�v�vb,where the shear stress keeps almost constant. At the point Dthe lubricant returns completely to solid state.

At the beginning of sliding, the system is accelerated be-cause the driven force must excess the resistance from lubri-cating film. For this reason, the system actually jumps fromA to the point B�, instead of B, to gain a shear stress lowerthan the critical value �a. This phenomenon, so called“velocity-weakening” has been regarded widely in the litera-tures as the cause for instability and stick-slip motion in lu-bricated systems.

Finally, it deserves to be mentioned that considerablenumbers of models of static friction based on continuummechanics and asperity contact were proposed in the litera-ture. For instance, the friction at individual asperity was cal-culated, and the total force of friction was then obtainedthrough a statistical sum-up �35�. In the majority of suchmodels, however, the friction on individual asperity was esti-mated in terms of a phenomenal shear stress without involv-ing the origin of friction.

6 Summary1. For solid surfaces interacting in air, the adhesion forces

mainly result from van der Waals interaction and capil-lary force, but the effects of electrostatic forces due tothe formation of an electrical double-layer have to be in-cluded for analyzing adhesion in solutions. Besides, ad-hesion has to be studied as a dynamic process in whichthe approach and separation of two surfaces are alwaysaccompanied by unstable motions, jump in and out, at-tributing to the instability of sliding system.

2. The conventional interpretation for the origin of frictionin terms of material damage or wear is not satisfactorilyconvincing for it greatly underestimates the energy lossin sliding. We have demonstrated in this chapter that awearless friction may originate from unstable motion ofinterfacial atoms, molecules, or asperities, resulting invibrations that are consequently damped through pho-non emission or electron excitation, and finally dissi-pated into heat.

Fig. 34—Shear stress on boundary films versus sliding velocity �re-produced after Ref. �33��.

184 PHYSICS AND CHEMISTRY OF MICRO-NANOTRIBOLOGY �

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