A thermodynamic model of sliding frictionLasse Makkonen Citation: AIP Advances 2, 012179 (2012); doi: 10.1063/1.3699027 View online: http://dx.doi.org/10.1063/1.3699027 View Table of Contents: http://aipadvances.aip.org/resource/1/AAIDBI/v2/i1 Published by the AIP Publishing LLC. Additional information on AIP AdvancesJournal Homepage: http://aipadvances.aip.org Journal Information: http://aipadvances.aip.org/about/journal Top downloads: http://aipadvances.aip.org/features/most_downloaded Information for Authors: http://aipadvances.aip.org/authors

Downloaded 06 Oct 2013 to 137.99.26.43. All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license.See: http://creativecommons.org/licenses/by/3.0/

AIP ADVANCES 2, 012179 (2012)

A thermodynamic model of sliding frictionLasse Makkonena

VTT Technical Research Centre of Finland, Box 1000, 02044 VTT, Espoo, Finland

(Received 2 December 2011; accepted 12 February 2012; published online 22 March 2012)

A first principles thermodynamic model of sliding friction is derived. The modelpredictions are in agreement with the observed friction laws both in macro- andnanoscale. When applied to calculating the friction coefficient the model provides aquantitative agreement with recent atomic force microscopy measurements on a num-ber of materials. Copyright 2012 Author(s). This article is distributed under a CreativeCommons Attribution 3.0 Unported License. [http://dx.doi.org/10.1063/1.3699027]

I. INTRODUCTION

Friction is one of the most important problems in physics and engineering. It has been studiedfor many centuries, and many mechanisms that may cause a friction force have been identified. Forexample, sliding of solids often involves work spent in forming grooves and wear particles. Theseprocesses can be modeled in different scales by basic mechanics. However, nanoscale measurementshave shown that friction is observed even in the absence of wear, and the fundamental origin offriction is still controversial.1–4

Numerous models of the origin of sliding friction have been developed and recently reviewed.5–8

These include the classical adhesion theory,9 the related adhesion hysteresis theory10 and treatmentswhere the separation is due to cracking.11, 12 The atomic scale models include the idea of phononicfriction13, 14 and a number of sophisticated molecular dynamics simulations.5, 15

In the present theories friction is supposed to originate either by assumed contact dynamicsor atomic scale interfacial processes within the contact area. In this paper it is shown that surfacethermodynamics alone provide a solution for a friction force. This force is derivable from thefollowing fundamentals alone a) definition of sliding, b) definition of real contact, c) laws ofthermodynamics. When combined with simple contact mechanics, this first-principle model predictsthe observed friction laws and the material dependence of friction in agreement with nanoscalemeasurements.

II. DYNAMICS OF SLIDING

A. Defining the contact

Upon sliding of solid objects, only a very small part of the apparent contact area is in contact. Thecontacts may be viewed in various scales from the milliscale to atomic dimensions, as is illustratedin Fig. 1. Obviously, in any theoretical treatment of friction, it is necessary to define precisely whatis meant by a contact and its scale.15, 16

The theory presented here is based on the thermodynamic surface energy per unit area. Thisconcept, outlined by Gauss17 and Gibbs,18 refers to the excess energy of a surface with respect toan atomically complete contact (as in the bulk). Therefore, this treatment requires considering thegrowth and disappearance of the real contacts, i.e., the contact areas containing atoms that interactacross the interface. Such nanoscale real contact areas can be defined e.g. as containing atoms

aE-mail: lasse.makkonen@vtt.fi, phone +358 020 722 4914, fax +358 020 722 7007.

2158-3226/2012/2(1)/012179/9 C© Author(s) 20122, 012179-1

Downloaded 06 Oct 2013 to 137.99.26.43. All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license.See: http://creativecommons.org/licenses/by/3.0/

012179-2 Lasse Makkonen AIP Advances 2, 012179 (2012)

MACROSCALE MILLISCALE MICROSCALE NANOSCALE

FIG. 1. Contact at various scales. The area within the small square in the middle of an image is magnified in the next imageto the right. The smallest real contacts considered in this theory are shown in the image at far right. The points within the realcontacts schematically show atoms.

1

V

BA

2

1

V

DC

2

FIG. 2. Schematic illustration of the growth (left) and disappearance (right) of a nano-contact between two sliding solids.At A surface 1 disappears and at B surface 2 is replaced by interface 1,2. At C interface 1,2 is replaced by surface 2 and atD surface 1 is formed. The surface energies are denoted by γ 1, γ 2 and γ 1,2 respectively. As γ 1 and γ 2 are higher than γ 1,2,a growing contact (left) releases surface energy both at A and B while a diminishing contact (right) requires energy to formsurface both at C and D.

separated by less than some distance.16 They are schematically illustrated on the right hand side ofFig. 1. These contacts represent the smallest possible contact scale.

B. Defining sliding

While the total area of the real contacts is stable during sliding, at any arbitrary time, somecontacts are growing and some are diminishing in area. Figure 2 illustrates this process schematically.In Fig. 2 solid 1 (red) is moving to the right with respect to solid 2 (green). In this process at Asurface 1 disappears and at B surface 2 is replaced by interface 1,2. At C interface 1,2 is replaced bysurface 2 and at D surface 1 is formed. Upon sliding the contact on the left grows and the contacton the right diminishes. There may also be contacts which do not change in area. This would bethe case when a block 1 is riding entirely on a block 2 which is longer. At such a contact, surface 2disappears at the front edge and is formed at the back edge.

Thus, sliding of solids may be defined as the process where a surface of one kind is convertedto a surface of another kind, and then again back to the original surface. These surface conversionprocesses take place at the edges of the real nano-contacts as shown in Fig. 2.

Clearly, the number of the contact edges where surface is formed is the same as that of thecontact edges where surface disappears. Therefore, regardless of the detailed contact morphology,the total rate of formation of real surface area is

d A/dt = 1/2N l V (1)

Downloaded 06 Oct 2013 to 137.99.26.43. All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license.See: http://creativecommons.org/licenses/by/3.0/

012179-3 Lasse Makkonen AIP Advances 2, 012179 (2012)

where N is the number and l the mean width of the real contacts and V is the sliding velocity.Simultaneously, real surface area disappears at the same rate, keeping the total contact area stable.

III. SURFACE THERMODYNAMICS OF SLIDING

A. The power of surface conversion

Surface energy γ is defined as the energy per unit area that is required to form surface.17–20 It isnoteworthy that γ defines an excess surface property regardless of how the surface is formed.18 Atthe atomic scale, γ represents the integral of the imbalance of the atomic forces in perpendicular tothe interface and macroscopically, γ is a measurable property of a material’s surface.19, 20

Because γ exists on all solids, generation of new surface in a sliding process at the contact edges(see Fig. 2, right) inevitably involves consumption of energy at them. The total power P involved is,based on Eq. (1)

P = 1/2N l V(γ1 + γ2 − γ1,2

)(2)

where the surface energies of the interfaces are denoted by γ 1, γ 2 and γ 1,2 respectively. The samepower P is involved in releasing surface energy at those contact edges where surface disappears(Fig. 2, left).

B. The effect of forming surface

At the diminishing contacts energy is consumed in forming surface at the rate P given by Eq. (2).The first law of thermodynamics requires that free energy must be taken from the environment atthe same rate. At the diminishing contacts the only source of free energy is the kinetic energy of thesliding motion. Thus,

P = F V (3)

where F is the total frictional force produced by the diminishing contacts. From Eqs. (2) and (3) itfollows that

F = 1/2N l(γ1 + γ2 − γ1,2

)(4)

C. The effect of disappearing surface

At the growing contacts surface energy is released at the rate given by Eq. (2). The firstlaw of thermodynamics requires that, at equilibrium, this energy must either be transferred to theenvironment as heat or consumed so that the system is doing mechanical work.

Consider the local thermodynamic system around point A in Fig. 2. When solid 1 moves aninfinitesimal distance to the right, the geometry of this system remains the same. Note that thissituation differs from a contact line of a liquid on a solid where the system can do work by adjustingthe contact angle.21 In a solid-solid contact in Fig. 2 the state of the local thermodynamic systemaround A does not change, except that its temperature may rise due to the incoming flux of surfaceenergy. By principle of minimum energy, a system can do work only when its state changes towardsa smaller internal energy. Since this does not happen at A (or B) upon sliding, the growing contactscannot do work to support the motion.

It is noteworthy that the situation in Fig. 2 is fundamentally different from that considered inthe adhesion theories of friction in which pulling off surfaces in perpendicular to them is assumed.When a gap is formed by pulling and the surfaces are then brought back together, the system cando work and minimize the internal energy by snapping on. Thus, in separating and rejoining thesurfaces the net loss of mechanical energy reduces to the adhesion hysteresis. In Fig. 2, in contrast,there is no gap to be closed so that this system cannot reduce its internal energy. In other words, thesurface energy in the thermodynamic system of a sliding contact is not “free energy”. Note that acomponent of the internal energy being free or not is, not a property of a surface, but a property of athermodynamic system. It depends on the constraints of the system.

Downloaded 06 Oct 2013 to 137.99.26.43. All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license.See: http://creativecommons.org/licenses/by/3.0/

012179-4 Lasse Makkonen AIP Advances 2, 012179 (2012)

The thermodynamic argument submitted above is fundamental. That the contact edges wheresurface disappears cannot do work to support the sliding motion underlines the origin of the irre-versible nature of friction. Because this aspect is crucial in understanding the thermodynamics offriction, another equally compelling argument is submitted below based on the force vectors.

Mechanical work is defined by the following line integral

Wc =∫c

Fdx (5)

where c is the path of the curve traversed by the object, F is the force vector and x is the positionvector. Thus, the calculation of Wc is path-dependent and cannot be differentiated to give F dx.According to Eq. (5) a non-zero force can do zero work. A simple example of this is where theforce is always perpendicular to the direction of motion, making the integrand always zero. This isthe case in a sliding process, since the atomic force imbalance across an interface is defined as avector in perpendicular to it. The consequence of Eq. (5) can also be stated as a fundamental ruleof mechanics: Only a component of a force parallel to the velocity vector of an object can do workon that object. The net atomic forces across interfaces of solid materials do not have a componentparallel to a sliding motion and are, therefore, unable to do mechanical work to support it.

In conclusion, consideration of the local thermodynamic state of the system and of the forcesacting upon the system both show that the growing contacts cannot do mechanical work on thesliding object. Consequently, when the sliding motion forces surface energy to be released at thegrowing contacts, this energy must be converted to heat. The power of the resulting frictional heatingis given by Eq. (2).

IV. THE FRICTION COEFFICIENT

Because the growing contacts cannot do mechanical work to support the sliding motion, thetotal thermodynamic friction force is given by Eq. (4). It is noteworthy that the friction force ata single diminishing contact l (γ 1 + γ 2 - γ 1,2) is independent of the normal force Fn. The totalmacroscopic friction force, given by Eq. (4) on the other hand, depends on Fn, because the normalforce affects the number of contacts N by changing the contact area at larger scales (see Fig. 1).

The aim of this paper is not to discuss the detailed contact mechanics, but to demonstrate afundamental thermodynamic friction mechanism. Therefore, the simplest contact model wherebythe materials yield so that the contact pressure pn equals the indentation hardness H of the softermaterial22, 23 is adopted here. The hardness H depends on the indentation type and scale but stronglycorrelates with the yield strength of the material both in the micro- and nanoscale. Taking the totalreal contact area as A = N d l, where d is the characteristic length of the real contacts, this results in

Fn = N d l H (6)

Combining Eqs. (4) and (6) yields for the friction coefficient μ = F/Fn

μ =(γ1 + γ2 − γ1,2

)2 d H

(7)

When the slider and the substrate are of the same material the friction coefficient becomes

μ = γ

d H(8)

V. DISCUSSION

A. Agreement with the classical friction laws and macroscale experiments

It has been shown15 that a friction force is proportional to the contact area at all length scales aslong as the contact area is defined analogously to this model, i.e. as the sum of the real contact areas.In view of Eq. (4) this suggests that the contact dimension d may be considered as independent of

Downloaded 06 Oct 2013 to 137.99.26.43. All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license.See: http://creativecommons.org/licenses/by/3.0/

012179-5 Lasse Makkonen AIP Advances 2, 012179 (2012)

Fn. Accordingly Eq. (7) agrees with Amontons’ Law, i.e., μ is independent of Fn. The theory alsoagrees with Coulomb’s Law, i.e. μ is independent of V.

According to Eq. (4) there is no frictionless motion of solids in contact since d is finite and γ ispositive for all insoluble interfaces. In the extreme case of a complete microscale contact the theorypredicts a negligible frictional force, as observed for adsorbed monolayers.1

When the slider and the substrate are of the same material (eq. (8)), the material property whichpredicts μ in this theory is the ratio of surface energy to hardness γ /H. This ratio has in a number ofexperimental studies been found to correlate with μ better than any other material property.2, 24–27

Considering that d is of nanoscale, Eq. (7) agrees with the finding28, 29 that μ of polymersdecreases with increasing molecular chain length and alignment in parallel to the sliding direction.The observation that μ is different on different faces of a crystalline material30, 31 is also in agreementwith this theory, since γ depends on the crystal face.

The classical adhesion theory of friction9 predicts μ = τ /H, where τ is the shear strength ofthe junctions. The concept of shear strength is defined as the force measured prior to failure andis, therefore, a static parameter. The friction mechanism discussed here, on the contrary, is purelykinetic, as it is directly derived from the dynamics of sliding.

Equation (7) may be presented in the form

μ = Wa

2 d H(9)

where Wa = γ 1 + γ 2 - γ 1,2 can be identified as the classical definition of the work of adhesion.Thus, while this theory does not assume any separation of the sliding surfaces perpendicularly,it nevertheless predicts that the thermodynamic work of adhesion Wa correlates with the frictioncoefficient. This prediction is corroborated by experimental studies.26–28

B. Relation to nanoscale models and experiments

As discussed in connection with Fig. 2, nanocontacts constantly grow and disappear. Thefrictional force produced by the diminishing contacts is, therefore, periodic at the scale of thecontact spacing. Thus, even in the absence of separation, elastic deformations and phase changeprocesses, this theory predicts an apparent close to nanoscale stick-slip, whose frequency increaseslinearly with increasing sliding velocity. This particular prediction is corroborated by empiricalstudies.32, 33

The molecular dynamics simulations of friction, see e.g.15, 34 consist of a system of particlesinteracting via prescribed interaction potentials. The connection of the theory presented here withthese models is that the surface energy γ represents the integral of the interaction potentials fromthe bulk of the solid to the surface. For example, by applying the Lennard-Jones potential, the ther-modynamic surface energy can be expressed as a function of the attractive constant, the equilibriumdistance between the molecules, and the molecular density.35, 36 On the other hand, the surface energyis a measurable macroscopic quantity.19, 37 Thus, this theory provides a connection between atomicand macroscopic descriptions of friction.

The thermodynamic theory presented here does not involve assumptions on how kinetic energyis dissipated at the atomic scale. Accordingly, this model of friction does not contradict with themolecular scale explanations, such as phononic friction.13, 38 However, the thermodynamic frictionand frictional heating processes discussed here occur at the edges of the real contacts. This mayaffect the interpretation of frictional laws at the nanoscale15, 16 because it implies that the frictionalforce needs to defined in terms of the total width of the real contacts rather than their total area.

The theory predicts no dependence on the normal load for a nanocontact the size of whichapproaches that of the real contacts. This agrees with observations.39, 40

C. Quantitative verification

Quantitative verification of the theory requires an estimate of the characteristic real contactlength d since it is not known per se. Considerations of the observed spectrum of contacts and

Downloaded 06 Oct 2013 to 137.99.26.43. All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license.See: http://creativecommons.org/licenses/by/3.0/

012179-6 Lasse Makkonen AIP Advances 2, 012179 (2012)

FIG. 3. Friction coefficient of various materials predicted by the theory versus that measured against a silicone tip ofan AFM.45 The theoretical values are based on assuming a characteristic real contact length of 1 nm and obtaining thesurface energy production in Eq. (7) from the AFM adhesion measurements of the same samples (see Appendix). The linearcorrelation coefficient is 0.96. The line shows the 1:1 relationship.

atomic clustering at interfaces suggest a smallest possible stable contacts radius of 1 - 10 nm.41

Since all real contacts need to be accounted for in this theory (see Fig. 1), it is reasonable here toapply d at the lower limit of this scale. Upon sliding, the contact length is, in the mean, half of itsstable maximum value (see Fig. 1), so that d = 1 nm can be used as the first estimate for testing thetheory quantitatively.

Using this value of d for diamond, with γ of 5 Nm-1 and H of 10 GPa, gives μ ∼ 0.5. Forstainless steel with γ of 2.5 Nm-1 and nanohardness of 5 GPa42 the prediction is μ ∼ 0.5. Forpolymers γ is of the order of 0.03 Nm-1 and nanohardness of the order 0.3 GPa43 giving μ ∼ 0.1.These estimates are in the correct range of measured friction coefficients.39, 42, 43

Further verification of the theory is hampered by a possible role of other frictional mechanisms inmacroscale experimental data. However, Eq. (9) makes it possible to quantitatively verify the theoryby recent wearless atomic-force-microscopy (AFM) adhesion and friction measurements.44, 45 Themethod of using the data in reference 45 in verifying the theory is explained in the Appendix.The results of this verification are shown in Fig. 3 which shows excellent quantitative agreement ofthe theory using d = 1 nm. The linear correlation coefficient between the theoretical and measuredfriction coefficients is 0.96, although the same d is used for all the materials. This demonstrates thepotential of the model and strongly supports the significance of the thermodynamic explanation ofsliding friction proposed here.

Adhesion hysteresis has been widely used in explaining the material dependence of friction.In the adhesion hysteresis model it is assumed that frictional energy dissipation occurs throughthe same mechanism as adhesion energy dissipation (as occurs during a single loading-unloadingcycle).46 This differs from the purely dynamic theory presented here, in which the dissipation is dueto pure sliding and occurs even in the absence of adhesion hysteresis.

For comparison of the results of the two theories, Fig. 4 shows a plot of the measured adhesionhysteresis vs. measured friction for the same AFM data45 as in Fig. 3. Figures 3 and 4 show thatthe theory presented here, not only provides quantitatively reasonable results, but also explains thematerial dependence of the friction coefficient better than the adhesion hysteresis theory.

Downloaded 06 Oct 2013 to 137.99.26.43. All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license.See: http://creativecommons.org/licenses/by/3.0/

012179-7 Lasse Makkonen AIP Advances 2, 012179 (2012)

FIG. 4. Measured adhesion hysteresis versus the friction coefficient measured against a silicone tip of an AFM.45 The linearcorrelation coefficient is 0.84. Modified from reference 45.

D. Static friction and rolling friction

The frictional force in Eq. (4) is independent of the sliding velocity and, thus, does not vanishwhen V goes to zero. Hence, kinetic and static friction coefficients are the same based on thethermodynamic reasoning. However, this reasoning is not fully extendable to static friction becausethe nanoscale structure of the interface may change while in contact.46 Consequently, the state ofthe system may change when sliding begins so that this theory cannot be applied.

Upon sliding, kinetic energy is consumed at all diminishing real contacts within the apparentcontact area, as discussed above. In contrast, at any arbitrary time upon rolling, the vast majority ofthe nanoscale contacts are stable. In rolling, only the real contacts at the back edge of the apparentmacroscale contact form new surface by being lifted apart and new contacts are formed by snap-ononly at the front edge. Therefore, the ratio of the frictional forces in sliding motion and rollingmotion on an apparent contact with a length D can be approximated by d/D. Here d is of the order ofnanometers and D is typically in a millimeter range. Hence, the thermodynamic model of friction,presented here, readily explains why rolling friction is negligible when compared to sliding friction.

VI. CONCLUSIONS

As outlined in this paper, sliding of solids can be defined by the formation and disappearanceof surfaces. By laws of thermodynamics, this inherently involves consuming kinetic energy anddissipating surface energy at the real contacts. Consequently, friction is generated even when nowear or separation occurs in perpendicular to the interface and when no storing of elastic energy andatomic scale mechanisms within the real contacts are accounted for.

The thermodynamic analysis presented in this paper provides an explanation for the origin of theirreversibility of friction. The key issue is that at the diminishing real contacts kinetic energy must betaken from the motion to form surfaces, whereas at the growing contacts the released surface energycannot be spent in mechanical work because the local thermodynamic system at them is constrainedin such a way that it cannot do mechanical work in parallel to the sliding interfaces.

Downloaded 06 Oct 2013 to 137.99.26.43. All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license.See: http://creativecommons.org/licenses/by/3.0/

012179-8 Lasse Makkonen AIP Advances 2, 012179 (2012)

Qualitatively, the predictions of this theory agree with major experimental evidence. The theoryalso explains why friction laws are different in macro-scale and nanoscale and predicts the verylarge frictional force observed in nanoscale measurements. The theory outlines the importance ofthe contact morphology at the nanoscale. In the extreme case of a complete macroscale contact, i.e.very large d, the theory predicts a negligible frictional force, as observed for adsorbed monolayers.Based on the theory, friction can be manipulated if one can find ways to affect the real contactlength d.

It needs to be emphasized that the friction theory derived here considers processes at the scaleof the real nanoscale contacts only. Thus, it is in no discrepancy with the postulated atomic scalemechanisms within the real contacts or with various proposed friction and wear mechanisms at alarger scale.

ACKNOWLEDGMENTS

I thank K. Homberg, R.M. Nieminen, L. Penn, C.A. Knight, E.P. Lozowski, K. Kolari, M.Tikanmaki and I. Havukkala for comments and the Academy of Finland and Jenny and Antti WihuriFoundation for financial support.

APPENDIX

1. Verification of the theory by the nanoscale measurements of Szoszkiewicz et al.44,45

Szoszkiewicz et al.44 measured adhesion hysteresis and friction forces by atomic force mi-croscopy (AFM) on eight different materials. They state that “all the samples should display sim-ilar wearless, adhesion dominated, and not lubricated solid-solid friction”. The thermodynamictheory presented in this paper describes friction specifically under such conditions. Since themeasurements44 include data, not only on the adhesion hysteresis, but also the work of adhesion,they provide the means to quantitatively verify the theory.

Adhesion energies and friction forces were measured44 by a silicone cantilever on the samesamples. Details of the measurements, calibration and the sample materials are given in references 44and 45. As to their sample materials, Szoszkiewicz et al. write “. . . we believe that apart from its highroughness, an inhomogeneous coverage of copper by its oxides is critical to experimental data”.Copper was, therefore excluded from their comparison, leaving seven materials for verification:calcite, mica and five metallic materials, zinc titanium, iron, aluminum and lead.

The experimental friction coefficients are here calculated from the friction force given45 inTable 1, column 4 in using the applied normal force of Fn = 100 nN. The friction coefficientspredicted by the theory are based on taking the same characteristic contact length of d = 1 nm forall the materials (see discussion in the paper).

The prediction of this theory from Eqs. (6) and (7) is

μ = W A

2 d Fn(A1)

where A is the total real contact area in the friction experiment. The work of adhesion W as a materialparameter can be obtained from the measured apparent adhesion energy Wa by the AFM tip as

W = At Wa

Aa(A2)

where At is the tip-sample contact area and Aa is the total area of the nanoscale contacts in theadhesion measurement. Inserting Eq. (A2) in Eq. (A1) and considering that the yield limit is thesame in the adhesion and friction measurements

Fn

A= Fa

Aa(A3)

Downloaded 06 Oct 2013 to 137.99.26.43. All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license.See: http://creativecommons.org/licenses/by/3.0/

012179-9 Lasse Makkonen AIP Advances 2, 012179 (2012)

where Fa is the normal force applied in the adhesion measurement, gives

μ = Wa Aa

2 d Fa(A4)

As discussed in reference 45 the measured Wa and the Young modulus Y make it possible tocalculate the tip-sample contact area Aa at the applied static normal load Fa by the Johnson-Kendall-Roberts-Sperling (JKRS) contact mechanics47 as

Aa = 32/3π5/3 R4/3 [(Fa/ (3π RWa)) + 1]2/3 (Wa/Y )2/3 (A5)

Using Fa = 10 nN and tip radius of curvature of R = 100 nm, applied in the measurements aswell as the data for Wa and Y in45 (Table 1, columns 1 and 3), and inserting Aa from Eq. (A5) inEq. (A4), provides the theoretical friction coefficient shown in Fig. 3.

1 D. A. Kessler, Nature 413, 260 (2001).2 K. Holmberg, A. Matthews, and H. Ronkainen, Tribol. Inter. 31, 107 (1998).3 J. Krim, Surface Sci. 500, 741 (2002).4 M. Urbakh and E. Meyer, Nature Mater. 9, 8 (2010).5 I. Szlufarska, M. Chandross, and R. W. Caprick, J. Phys. D: Appl. Phys. 41, 123 (2008).6 C. V. Suciu and H. Goto, Tribol. Inter. 43, 1091 (2009).7 M. Amiri and M. M. Khonsari, Entropy 12, 1021 (2010).8 O. M. Braun, Tribol. Lett. 39, 283 (2010).9 F. P. Bowden and D. Tabor, The Friction and Lubrication of Solids (Clarendon Press, Oxford, 1950).

10 M. J. Nosonovsky, J. Chem. Phys. 126, 224701 (2007).11 E. Gerde and M. Marder, Nature 413, 285 (2001).12 B. De Celis, Wear 116, 287 (1987).13 G. A. Tomlinson, Phil. Mag. 7, 905 (1929).14 Z. Xu and P. Huangm, Wear 262, 972 (2007).15 Y. Mo, K. T. Turner, and I. Szlufarska, Nature 457, 1116 (2009).16 S. Cheng and M. O. Robbins, Tribol. Lett. 39, 329 (2010).17 C. F. Gauss, Commun. Soc. Regiae Sci. Gottingen Rec. 7, 30 (1830).18 J. W. Gibbs, Thermodynamics (Dover Publishers, New York, 1876).19 A. W. Adamson, Physical Chemistry of Surfaces (Interscience Publ., New York, 1960).20 A. W. Neumann and J. K. Spelt, Applied Surface Thermodynamics, 2nd ed, (Marcel Dekker, New York, 2010).21 Y. L. Chen, C. A. Helm, and J. N. Israelachvili, J. Phys. Chem. 95, 10736 (1991).22 K. L. Johnson, Contact Mechanics (Cambridge University Press, Cambridge, 1985).23 N. P Suh, Tribophysics (Prentice-Hall, New Jersey, 1986).24 E. Rabinowich, J. Appl. Phys. 32, 1440 (1961).25 J.-M. Dubois, P. Brunet, W. Costin, and A. Merstallinger, J. Non-Crystalline Solids 334&335, 475 (2004).26 M. Minn and S. K. Sinha, Wear 268, 1030 (2010).27 P. Samyn, G. Schoukens, J. Quintelier, and P. De Baets, Tribol. Inter. 39, 575 (2006).28 N. Maeda, N. Chen, M. Tirrell, and J. N. Israelachvili, Science 297, 379 (2002).29 F. Fan, X. Li, and T. Miyashita, Thin Solid Films 348, 238 (1999).30 Y. Enomoto and D. Tabor, Nature 283, 51 (1980).31 R. Perez, Nature Mater. 8, 857 (2009).32 A. Ghatak, K. Vorvolakos, H. She, D. L. Malotky, and M. K. J. Chaudhury, J. Phys. Chem. B 104, 4018 (2000).33 P. A. Thompson and M. A. Robbins, Science 250, 792 (1990).34 O. M. Braun, Tribol. Lett. 39, 283 (2010).35 S. Wu, Polymer Interface and Adhesion (Marcel Dekker, New York, 1982).36 L. Makkonen, J. Phys. Chem. B 101, 6196 (1997).37 L. Makkonen, Langmuir 16, 7669 (2000).38 J. Krim, Am. J. Phys. 70, 890 (2002).39 G. J. Germann, S. R. Cohen, G. Neubauer, G. M. McClelland, H. Seki, and D. Coulman, J. Appl. Phys. 73, 163 (1993).40 A. Socoliuc, R. Bennewitz, R. Cnecco, and E. Meyer, Phys. Rev. Lett. 92, 134301 (2004).41 B. N. J. Persson, Sliding Friction: Physical principles and applications (Springer, Berlin, 2000).42 M.-L. Zhu and F.-Z. Xuan, Mater. Sci. Eng. A 527, 4035 (2010).43 X. P. Liu and J. Zhang, Key Eng. Mater 295, 9 (2005).44 R. Szoszkiewicz, B. Bhushan, B. D. Huey, A. J. Kulik, and G. Gremaud, J. Chem. Phys. 122, 14708 (2005).45 R. Szoszkiewicz, B. Bhushan, B. D. Huey, A. J. Kulik, and G. Gremaud, J. Appl. Phys. 99, 014310 (2006).46 H. Yoshizawa, Y.-L. Chen, and J. Israelaschvili, J. Phys. Chem. 97, 4128 (1993).47 D. Maugis, Contact, Adhesion and Rupture of Elastic Solids (Springer, Berlin, 1999).

Downloaded 06 Oct 2013 to 137.99.26.43. All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license.See: http://creativecommons.org/licenses/by/3.0/