18Atomic-Scale Friction

Studies Using ScanningForce Microscopy

18.1 Introduction 18.2 The Scanning Force Microscope as a Tool for

Nanotribology 18.3 The Mechanics of a Nanometer-Sized Contact18.4 Amontons’ Laws at the Nanometer Scale18.5 The Influence of the Surface Structure on Friction

Frictional Contrast Caused by Local Changes in the Chemical Composition • Friction of Surfaces Possessing Identical Chemical Composition • Direction Dependence of Sliding Friction

18.6 Atomic Mechanism of Friction18.7 The Velocity Dependence of Friction18.8 Summary

18.1 Introduction

Although frictional forces are familiar to everyone from daily life experiences, little progress has beenmade in finding an exact physical description of friction since the following phenomenological frictionlaws were established by Amontons (1699) and Coulomb (1783):

1. Friction is proportional to the load (i.e., to the applied normal force).2. Friction is independent of the (apparent) contact area.3. Sliding friction is independent of the sliding velocity.

These laws hold surprisingly well in the case of dry friction or Coulomb friction, if there is no lubricationbetween the two interacting bodies. Nevertheless, they could not be derived from first principles up tonow. Moreover, the understanding of the fundamental mechanisms of friction on the atomic scale ispoor since most macroscopic and microscopic frictional effects are normally dominated by the influenceof wear, plastic deformation, lubrication, surface roughness, and surface asperities. Macroscopic frictionexperiments (see Figure 18.1a), especially if performed under ambient conditions are therefore difficultto analyze in terms of a universal theory. Moreover, the complexity of the phenomena prevented theobservation of pure wearless friction for many years, thus disabling all attempts to understand the originsof friction on the molecular and atomic scale.

In the last decades, however, the field of nanotribology was established by introducing new experi-mental tools which opened the nanometer and the atomic scale to tribologists. One of the basic ideas of

Udo D. Schwarz University of Hamburg

Hendrik HölscherUniversity of Hamburg

nanotribology is that for a better understanding of friction in macroscopic systems, the frictional behaviorof a single asperity contact should be investigated first. Macroscopic friction could then possibly beexplained with the help of statistics, i.e., by adding the interactions of a large number of individualcontacts, which together form the macroscopic roughness of the interface between the two bodies whichare in relative motion (see Figure 18.1b for illustration). The first big step toward such experiments atthe atomic level was taken when the surface force apparatus was equipped with special stages enablingthe measurement of lateral forces between two molecularly smooth surfaces sliding against each other(Israelachvili and Tabor, 1973; Briscoe and Evans, 1982). The biggest step concerning the reduction ofthe asperity size was accomplished with the development of the friction force microscope (FFM) (Mateet al., 1987). The FFM is derived from the scanning force microscope (SFM) (Binnig et al., 1986) andenables the local measurement of lateral forces by moving a sharp tip, representing (approximately) apoint contact, over a sample surface.

In this chapter, we intend to describe how the FFM can contribute to the understanding of the originand nature of the fundamental laws of friction by the investigation of atomic-scale frictional effects. Asshown in the following sections, friction at the nanometer scale manifests itself in a significantly differentmanner than the Coulomb friction at the macroscopic scale. The friction laws (1) and (2) are found tobe not valid for point contact friction, since the frictional forces on the nanometer scale are proportionalto the actual area of contact, which is generally not proportional to the load (Section 18.4). On the otherhand, friction law (3) could be confirmed for moderate sliding velocities also on this scale (Section 18.7).The analysis of the frictional behavior of such approximate point contacts in terms of a simple mechanicalmodel exhibits a two-dimensional nature of friction on the atomic scale, which is determined by a “stick-slip” type movement of the foremost tip atoms (Section 18.6). Molecular-scale effects involving wear orlubrication, however, will be discussed in the next chapter.

18.2 The Scanning Force Microscope as a Tool for Nanotribology

The principle of scanning force microscopy is rather simple and resembles that of a record player. A forcemicroscope (Figure 18.2) detects forces acting between a sample surface and a sharp tip which is mountedon a soft leaf spring (the cantilever). A feedback system which controls the vertical z-position of the tipon the sample surface keeps the deflection of the cantilever (and thus the force between tip and sample)constant. Moving the tip relative to the sample in the x–y-plane of the surface by means of piezoelectricdrives, the actual z-position of the tip is recorded as a function of the lateral x–y-position with (ideally)

FIGURE 18.1 (a) The well-known experimental setup to confirm the frictional laws (1) and (2) of dry friction onthe macroscopic scale. (b) The apparent contact area observed on the macroscopic scale consists of many individualsmall asperities. A nanometer-sized single asperity contact (“point contact”) can be realized in a scanning forcemicroscope.

sub-Ångström precision. The obtained three-dimensional data represent a map of equal forces; thenecessary conversion of the cantilever deflection into the normal force is basically performed by applyingHook’s law. The data can be analyzed and visualized through computer processing. A more detaileddescription of the method and the generally used instrumentation is given, e.g., by Meyer and Heinzel-mann (1992), Wiesendanger (1994), Bhushan and Ruan (1994), or Schwarz (1997).

The experimental setup for the deflection detection measurement, which is most frequently used inSFM studies on friction, is the beam deflection scheme sketched in figure 18.2a (Marti et al., 1990; Meyerand Amer, 1990). With this detection scheme, not only the deflection, but also the torsion of the cantilevercan be measured (see the caption of Figure 18.2a for further details). Force microscopes, which can recordbending and torsion of the cantilever simultaneously, are often referred to as friction force microscopes(FFMs) or lateral force microscopes (LFMs).

What are the specific advantages of friction force microscopy in comparison with other methods? Aswe have seen in the introduction, macroscopic friction experiments are often difficult to analyze due tothe complexity of the experimental system. For the unambiguous analysis of physical phenomena, theexperiments performed to clarify their nature are preferably arranged as simply as possible. Friction forcemicroscopy is an especially suitable tool for nanotribological investigations because it meets this require-ment quite well:

1. Measurements within the purely elastic regime prevent plastic deformation or wear of tip or sample.Consequently, all experiments presented in this chapter were performed at loading forces whichwere sufficiently low such that no effects due to plastic deformation could be observed.

2. The contact area of the sliding tip with the sample surface is very small (typically some nm2), thusenabling the localized detection of frictional forces.

3. The apparent and the actual contact area are identical. Therefore, problems occurring from surfaceroughness and surface asperities can be neglected if the experiments are performed on atomicallysmooth surfaces.

4. If measured under controlled conditions (e.g., in ultra-high vacuum or in an argon atmosphere),an influence of adsorbates on the measured friction can be reduced or completely avoided.

FIGURE 18.2 (a) Principle of the scanning force microscope. Bending and torsion of the cantilever are measuredsimultaneously by measuring the lateral and vertical deflection of a laser beam while the sample is scanned in thex–y-plane. The laser beam deflection is determined using a four-quadrant photo diode: (A+B)–(C+D) is a measurefor the bending and (A+C)–(B+D) a measure for the torsion of the cantilever, if A, B, C, and D are proportional tothe intensity of the incident light of the corresponding quadrant. (b) The torsion of the cantilever (middle) is solelydue to lateral forces acting in the x-direction, whereas both forces acting normal to the surface (Fz) as well as actingin plane in the y-direction (Fy) cause a bending of the cantilever (bottom).

Finally, two more points should be noted: (1) During scanning, the torsion of the cantilever isexclusively induced by lateral forces acting in the x-direction (Fx), whereas for corrugated samples, thebending of the cantilever is mainly caused by forces acting perpendicular to the sample surface in thez-direction (Fz). Nevertheless, a bending of the cantilever induced by lateral forces acting in the y-direction(Fy) cannot be excluded (see Figure 18.2b, bottom). For atomically flat surfaces, Fujisawa et al. (1993a,b;1995) pointed out that the variation of Fz is so small that the measured signal of an SFM equipped withthe beam deflection method is mostly determined by lateral forces. Therefore, on the atomic scale, thelateral forces both in scan direction and perpendicular to the scan direction are measured. (2) The correctcalibration of the bending and the torsion of the cantilever in terms of lateral forces is much more difficultthan the conversion of the cantilever deflection into the normal force, which can simply be performedby applying Hook’s law. Many parameters such as the cantilever dimensions, the elastic moduli of thecantilever material, the tip length, the position on the cantilever backside where the laser beam is reflectedand the sensitivity of the four-quadrant photo diode have to be known. Different procedures for thequantitative analysis of lateral force microscopy experiments are described, e.g., by Marti et al. (1993),Bhushan and Ruan (1994), Lüthi et al. (1995), Putman et al. (1995), Ogletree et al. (1996), and Schwarzet al. (1996a).

18.3 The Mechanics of a Nanometer-Sized Contact

Since it is the attempt of nanotribology to reconstruct the macroscopic frictional behavior from thefrictional properties of an individual nanometer-sized contact, it is clear that a knowledge of the mechan-ics of such small single asperity contacts is essential. For example, the exact contact area of tip and sampleas a function of the applied loading force has to be known in order to compare the data obtained byscanning force microscopy with theoretical models, as we will see in Section 18.4. Scanning force micros-copy, however, does not allow an independent measurement of the contact area. In order to circumventthis restriction, it is a common approach in SFM to use tips with a geometrically well-defined apex andto calculate the effective contact area (and other important parameters such as deformation or stiffnessof the contacts) on the basis of continuum elasticity theory. A detailed description of contact mechanicsis given by Johnson (1994).

Here, we will restrict ourselves to the mathematically simplest and most frequently used case of a sphericaltip which is in contact with a flat surface (the so-called Hertzian contact, Figure 18.3), in spite of the factthat there also exist extensions to other geometries (see, e.g., Carpick et al., 1996). Without adhesion, thecontact area of such a Hertzian contact can be determined using the Hertzian theory (Hertz, 1881), whichwas already developed in 1881. It predicts a contact area-load dependence of the following form:

(18.1)

FIGURE 18.3 (a) Macroscopic tip–sample contact; Fl denotes the externally applied loading force. The contactarea-load dependence is difficult to describe due to the irregular shape of both tip and sample surface. (b) Geometryof a Hertzian contact (see text). R = radius of the tip apex; a = radius of the contact area.

ARF

Kl= π

2 3

Here, R is the radius of the tip, Fl the loading force, and

(18.2)

the “effective elastic modulus” (with E1,2 and ν1,2 as Young’s moduli and Poisson’s ratios of sphere andflat, respectively). Other useful quantities are, e.g., the contact radius a of the interface region

(18.3)

the vertical displacement δ of the two bodies in contact

(18.4)

and the pressure or stress distribution p(r) within the contact region

(18.5)

Since the early 1970s, however, theories have been developed which include attractive forces (Johnsonet al., 1971; Derjaguin et al., 1975; Muller et al., 1983; Maugis, 1987; Fogden and White, 1990; Maugis,1992; Maugis and Gauthier-Manuel, 1994). The general mathematical formulation of the contact area-load dependence of a Hertzian contact including adhesion cannot be solved analytically (Fogden andWhite, 1990; Maugis, 1992). Nevertheless, depending on how the attractive forces act, approximationscan be derived.

In order to decide which of the above-mentioned approximations applies under which conditions,Tabor (1977) introduced the nondimensional parameter Φ

(18.6)

(γ: surface energy of sphere and flat [for equal surfaces] and z0: equilibrium distance in contact), whichrepresents basically a measure for the magnitude of the elastic deformation outside the effective contactarea compared with the range of the surface forces. When Φ is large (Φ > 5 [Johnson, 1997]), theJohnson–Kendall–Roberts (JKR) theory (Johnson et al., 1971) applies, which has been confirmed in manyexperiments (see, e.g., Israelachvili et al. [1980] and Homola et al. [1990]). If Φ < 0.1, however, contactsshould be appropriately described by the analysis of Derjaguin, Muller, and Toporov (DMT) (Derjaguinet al., 1975; Muller et al., 1983). Nevertheless, since the small value of Φ indicates that there is onlynegligible elastic deformation outside the effective contact area, the influence of attractive forces can infurther simplification be considered by introducing an effective normal force Fn = F1 + F0, where F0 isthe sum of all acting attractive forces. Replacing F1 with Fn in Equation 18.1 leads to

(18.7)

KE E

= − + −

−4

3

1 112

1

22

2

1

ν ν

aRF

Kl3 =

δ = =a

R

F

Kal

2

p rKa

R

r

a

F

a

r

al( ) =

π−

=π

−

3

21

3

21

2

2

2

Φ =

9

4

2

203

1 3

R

K z

γ

AR F F

K

RF

Kl n= π+( )

= π

0

2 3 2 3

This model will be referred to as the Hertz-plus-offset model in the following to distinguish between theoriginal and more precise DMT formalism and the present further simplification. Detailed mathematicalderivations of Equation 18.7 for the case of capillary forces were published by Fogden and White (1990)and Maugis and Gauthier-Manuel (1994).

By introducing typical values for contacts as they occur in SFMs into Equation 18.6 (R = 30 nm, γ =25 mJ/m2 [typical for van der Waals surfaces], K = 50 GPa, and z0 = 3 Å), we find a value for Φ of ≈0.09. Therefore, Equation 18.7 should represent the correct contact area-load dependence for typicalcontacts found in SFMs. A more detailed discussion of the contact mechanics of a nanometer-sizedHertzian contact is given by Schwarz et al. (1996a, 1997a).

However, Equation 18.7 only applies for tips with exactly spherical apex. Using SFM tips as suppliedby the manufacturer, such a shape is only accidentally realized, and thus all kinds of power laws A ∝ Fn

with n ranging from n ≈ 0.4 to n ≈ 1.2 have been found (Schwarz et al., 1997a). Consequently, tips ofexactly defined spherical tip apex and a tip radius which is known with nanometer accuracy are mandatoryfor quantitatively reproducible friction force measurements.

Mainly two different methods to prepare such well-defined tips have been introduced in the past. Thefirst method is to heat a very sharp tip in high vacuum (Binh and Uzan, 1987; Binh and Garcia, 1992).Surface diffusion will then induce migration of atoms from regions of higher curvature to regions oflower curvature. Another method was realized by covering doped single-crystalline silicon tips (apexradii 5 to 15 nm without coating) with a layer of amorphous carbon in a transmission electron microscope(Schwarz et al., 1997b). Molecules from the residual gas are ionized in the electron beam and acceleratedto the tip end. There, the molecules spread out evenly due to their charge, forming a well-defined sphericaltip end. With this method, tips with radii from 7 nm up to 112 nm could be successfully produced. Threeexamples of tips prepared using the second method described above are shown in Figure 18.4.

18.4 Amontons’ Laws at the Nanometer Scale

Amontons’ macroscopic friction laws (1) and (2) can be condensed in the well-known equation

(18.8)

where Ff denotes the observed friction force. µ represents a value which is constant for a given materialcombination in a wide range of applied normal forces Fn and which is usually referred to as the frictioncoefficient. Nevertheless, for the description of the frictional behavior of materials at the nanometer scale,it is practical to introduce the mean friction per unit area (the so-called shear stress S)

FIGURE 18.4 Three examples of tip specially prepared in a transition electron microscope with spherical tip apexesof (a) 21 ± 5 nm, (b) 35 ± 5 nm, and (c) 112 ± 5 nm radius.

F Ff n= µ

(18.9)

Combining Equations 18.7 and 18.9, we find

(18.10)

with µ = πS/K2/3.There have already been many reports in the literature when the frictional force has been measured

as a function of the normal force using an FFM (Mate et al., 1987; Mate, 1993; Hu et al., 1995; Bhushanand Kulkarni, 1995; Carpick et al., 1996; Meyer et al., 1996; Lantz et al., 1997; Schwarz et al., 1997a;Enachescu et al., 1998). In most of these reports, the frictional force showed a strongly nonlinear depen-dence on the normal force. Figure 18.5 features results published by Schwarz et al. (1997a), which wereacquired in argon atmosphere on amorphous carbon with tips showing exactly spherical tip apexes (tipradii of 17 ± 5 nm [a] and 58 ± 10 nm [b], respectively). The fits according to Equation 18.10 showexcellent agreement between the experimental results and theory. The dashed lines illustrate the deviationof the present friction law from the macroscopic linear law Equation 18.8.

It is interesting to note that not only can the relationship Ff ∝ F 2/3n be confirmed, but also the Ff ∝

R2/3-dependence. Comparing Figure 18.5a with 18.5b, much higher friction is observed in (b); the fric-tional force at Fn = 10 nN, for example, is about three times larger (≈15 nN) than the frictional force ofonly 5.5 nN observed in (a). However, the calculated numerical value of µ = 0.17 ± 0.07 GPa1/3 for thesecond measurement is in good agreement with the value of µ = 0.17 ± 0.09 GPa1/3 obtained in the firstmeasurement.

Summarizing the main results obtained from the different research groups on a large variety ofmaterials, it was found that the friction as a function of the normal load showed good agreement ofexperimental data and theoretical fits using Equation 18.9 combined with contact mechanical modelsappropriate for the specific tip-sample contact, if S is set constant. This finding leads to the followingremarkable consequences:

FIGURE 18.5 The frictional force Ff as a function of the normal force Fn, measured on amorphous carbon withgeometrically well-defined spherical tips in argon atmosphere (Schwarz et al., 1997a). The data presented in (a) wereobtained using a tip with a radius of R = 17 ± 5 nm; the data in (b) using a tip with R = 58 ± 10 nm. Both curvesare in excellent agreement with fits according to Equation 18.10 (solid lines); the dashed lines illustrate the deviationfrom the linear macroscopic model (Equation 18.8). The offsets of the solid lines from the zero point of the normalforce are caused by the experimental uncertainty in the determination of the zero point by means of force-vs.-distancecurves.

S F A= f

F SR

KF R Ff n n= π

= µ2 3

2 3 2 3 2 3˜

• The observed frictional force Ff is in opposition to law (2) proportional to the crack area A (seeEquation 18.9). In the special case of a Hertzian contact, this leads to a Ff(Fn)-dependence of Ff ~F 2/3

n .

• The shear stress S depends only on the materials used and on environmental conditions such astemperature and humidity, but not on the mean contact pressure p = Fn/A.*

• The independence of the shear stress S from the mean contact pressure p = Fn/A is in flagrantcontradiction to Amontons’ law (1).

• The continuum elasticity theory also applies at the nanometer scale.

Since the continuum elasticity theory also applies at the nanometer scale, Equation 18.7 can be usedto calculate the actual contact area of an SFM tip and the sample surface. Typical normal forces Fn

occurring in an SFM during measurement are between 1 nN and 100 nN. This results in contact areasof a few nm2, with the effect that only some tens or some hundreds of atoms are in direct contact. Thedeformations of tip and sample are then fully elastical, i.e., fully reversible. Therefore, with an SFM, itis indeed possible to study wearless friction of a quasipoint contact.

Another outcome from the above considerations is the finding that it is obviously useless to determine“friction coefficients” µ (see Equation 18.8) from point contact measurements, since these values willheavily depend on the specific geometry of the tip–sample contact. For a classification of the microscopicfrictional properties of materials, Schwarz et al. (1997a) have therefore proposed the factor µ = πS/K2/3

(see Equation 18.10), which combines the frictional and elastic properties of tip and sample and whichcan be regarded as an effective friction coefficient for point contact-like single-asperity friction in the caseof a nanometer-sized Hertzian contact. The definition of such a coefficient is advantageous for two mainreasons: (1) If materials with identical intrinsic frictional properties (i.e., the materials show the samefriction per unit area) but different Young moduli are examined in measurements performed withidentical tips and normal forces, higher friction will be found on the softer material due to the largercontact area. Materials which have the same µ, however, will always show the same friction during anexperiment. (2) If the same sample is investigated with tips featuring different apex radii, lower frictionwill be found in the experiment carried out with the sharper tip. Nevertheless, consideration of thegeometry by the calculation of µ will lead to identical values of µ, as we have seen in Figure 18.5.

How can the above results now be correlated with the macroscopic laws of friction? As we have seenin the introduction, it is the general attempt of nanotribology to conclude from the frictional propertiesof an individual nanocontact to the behavior of macroscopic bodies by the statistical adding of theinteractions of a large number of individual contacts with represent the macroscopical roughness of thecontact interface. This means for the given situation that the contradiction between Amontons’ laws (1)and (2) and their corresponding counterparts on the nanometer scale could be entirely eliminated if theeffective contact area of such a macroscopic contact increased proportionally to the externally appliedloading force. Then, the observed frictional force would also increase linearly with the load, accordingto the nanoscopic friction law found above. Such a linear Ff(Aeff)-dependence for macroscopically flat,but microscopically rough surfaces has indeed been demonstrated by Greenwood (1967, 1992). Interest-ingly, the apparent contact area of the two bodies does not matter in this case.

18.5 The Influence of the Surface Structure on Friction

In the preceding sections, we dealt with the mechanical and tribological properties of nanometer-sizedcontacts. Questions concerning the fundamental mechanisms of friction at the atomic scale, however,

*The same result has been found by Carpick et al. (1997) and Enachescu et al. (1998) with different approaches,which both circumvent the assumption of a specific contact mechanical model. Additionally, experiments performedwith the surface force apparatus and consequently much larger contact areas A also show the same behavior(Israelachvili and Tabor, 1973; Homola et al., 1990).

are still unanswered. Since we focus in this chapter on the analysis of wearless friction on atomically flatsurfaces without defects, it is reasonable to expect that the origin for the different friction observed onthe different materials can be found in the local arrangement of the atoms at the tip–sample contact area.

If this expectation holds, it should be possible to distinguish areas of different chemical compositionby lateral force microscopy. This ability is sometimes called “chemical imaging”; examples for suchinvestigations will be discussed in Section 18.5.1. Another interesting consequence of the above expec-tation is that the friction should depend on the direction with which the tip profiles the sample surface.This frictional anisotropy is especially easy to verify experimentally if a material exhibiting an asymmetricsurface potential is used as a sample, as presented in Section 18.5.3. Finally, we will see in Section 18.5.2that even small conformational (i.e., purely geometrical) changes within a surface unit cell give rise todifferent friction in lateral force maps.

18.5.1 Frictional Contrast Caused by Local Changes in the Chemical Composition

The ability of the FFM to provide a contrast between surfaces of different chemical composition is notonly important for tribologists, but also of considerable general interest for users from the whole fieldof scanning force microscopy, where contrast mechanisms are needed which provide information inaddition to the pure topography. Since the lateral forces acting on the FFM tip are not independent ofthe topography,* atomically or molecularly flat terraces give the most unambiguous results. Consequently,chemical contrast has first been demonstrated for Langmuir–Blodgett films, which can easily be preparedand investigated under atmospheric conditions and exhibit large, molecularly flat surfaces. See, forexample, Overney et al. (1992), or Meyer et al. (1992).

However, in this chapter we will restrict the discussion to the atomic-scale frictional effects at crystallinesurfaces. In order to check the frictional behavior for such systems, Lüthi et al. (1995) partially coveredthe surface of an NaCl(001) single crystal with 1 to 6 monolayers of AgBr (Figure 18.6a), which has thesame crystalline structure as NaCl. In the corresponding FFM investigations (Figure 18.6b), they founda strong frictional contrast between the AgBr islands and the NaCl substrate due to the different chemicalcomposition of the two materials.

FIGURE 18.6 (a) Topography and (b) lateral force map of an AgBr thin film deposited on NaCl(001) at roomtemperature (image size: 1.4 µm × 1.4 µm). The AgBr islands are 1 to 6 monolayers high and partially cover theNaCl substrate. In the lateral force map, the AgBr islands are revealed as areas with about ten times higher frictionthan the corresponding friction observed on the NaCl surface. (From Meyer, E., Lüthi, R., Howald, L., andGüntherodt, H.-J., 1995. Friction force microscopy, in Forces in Scanning Probe Methods, Güntherodt, H.-J., Ansel-metti, D., and E. Meyer (Eds.), Kluwer Academic Publishers, Dordrecht, The Netherlands, 285. With permission.)

*Scanning “up hill” causes more torsion of the cantilever than scanning “down hill”; see, for example, Grafströmet al. (1993), Fujisawa et al. (1993b), Ruan and Bhushan (1994a), or Aimé et al. (1995).

According to the same principle, basically all areas of different chemical composition can be distin-guished in force microscopical investigations. Fompeyrine et al. (1998), for example, could differentiateareas terminated by SrO from areas terminated by TiO2 on an SrTiO3 single crystal. This ability of theSFM is sometimes called “chemical imaging.” It should be noted, however, that chemical imaging doesnot mean chemical identification. The reason for this has been discussed in Section 18.4. If the contactmechanics of the tip–sample contact is not exactly known, the results obtained are difficult to interpretwithout additional information. In order to circumvent this restriction at least partially, chemicallymodified tips have been used to sensitize the tip to certain functional groups which have been patternedon suitable sample surfaces. See, for example, Frisbie et al. (1994) or Sasaki et al. (1998). A theoreticalstudy on the bases of “chemical force microscopy” on organic monolayers has been published recentlyby Fujihira and Ohzono (1999).

18.5.2 Friction of Surfaces Possessing Identical Chemical Composition

After what has been discussed above, it seems promising to analyze the influence of differences in thestructure of surfaces possessing identical chemical composition on the friction observed by a point probe.Probably the first observation on this issue was published by Ruan and Bhushan (1994b). They observedgraphite(0001) areas with unusually high friction, which could be identified as areas exhibiting graphiteplanes of different orientations (other than [0001]) as well as amorphous carbon. Comparative quanti-tative studies for different carbon compounds have been performed by Mate (1993) and Schwarz et al.(1997a) on diamond, graphite, C60 thin films, and amorphous carbon. High frictional forces have beenfound especially on the C60 thin films, whereas friction nearly vanishes on graphite. These examplesdemonstrate that even on materials that consist of the same chemical species, large variations in theobserved friction might occur.

As a consequence, it follows that it is not the specific atomic species which form a crystal, but factorssuch as the crystalline structure, the charge distribution, and/or the binding conditions that might decidethe frictional properties of a material. The “chemical differentiation” postulated in Section 18.5.1 istherefore based more on structural and electronical differences between the materials than on the factthat they represent chemically different species.

This issue will be further illustrated with the example of the ferroelectric material guanidiniumaluminum sulfate hexahydrate (GASH) in the following. Since the Curie point of GASH is above itsdecomposition temperature, the crystals always show the ferroelectric phase. The surfaces of positivelyand negatively charged domains not only have the same chemical composition, the surface atoms arealso bound in the exact same manner on both domains. The difference is that the surface undergoes aconformational (i.e., geometrical) change of the surface structure if the polarity of a certain surface areais flipped. Certain aluminum ions which are octahedrally coordinated by water molecules stick on thepositive domain more out from the surface (by 0.5 Å) than they do no the negative domain, resultingin a different surface corrugation. Bluhm et al. (1998) showed in a combination of voltage-dependentfriction force microscopy experiments and electrostatic force microscopy experiments that the domaincontrast observed in FFM images is not caused by the electrostatic interaction between a charged tip andthe electric field of the sample, but is explained by the geometrical differences exhibited by the surfacesof domains with opposite polarity (Figure 18.7). This example elegantly demonstrates that even smallchanges in the arrangement of the atoms located at the sample surface influence the friction occurringduring the sliding of nanocontacts.

18.5.3 Direction Dependence of Sliding Friction

In the above sections, we have seen that friction is very sensitive to the structure of the surface. In everydaylife, it is often observed that friction depends on the sliding direction. This phenomenon can be observedrubbing by one’s hand over certain pieces of cloth. Another example is that of a saw; there, how much

one has to pull or push depends on the direction. It is now instructive to investigate how this mechanismalso applies on the atomic scale.

The basic idea of this issue is illustrated in Figure 18.8 by contrasting the behavior of a tip moving(a) in a symmetric and (b) in an asymmetric surface potential. The asymmetric surface potential isrepresented by a sawtooth-like structure. Intuitively, one might assume that a sharp tip which profilessuch an asymmetric potential would experience a larger lateral force at the steep sides of the sawtooth than

FIGURE 18.7 (a) Topographical image of GASH(0001) (image size: 30 µm × 30 µm). The surface is featurelessand without any steps. (b) Friction force map of the same surface spot as in (a) (slight zoom-out; image size: 40 µm ×40 µm). A well-expressed contrast is visible. (c) Image acquired with electrostatic force microscopy at the samesurface area as in (a), showing the structure and absolute sign of the ferroelectric domains. By comparison of (b)and (c), it can be concluded that the contrast in the friction image represents the domain structure of GASH.

FIGURE 18.8 Principle of FFM contrast formation. (a) On materials that have no directional dependence of friction(i.e., the friction coefficient µ is the same in the forward and backward scan directions), the contrast between surfaceareas exhibiting different µ is reversed when scanning forward compared to scanning backward since the FFM signalis a measure for the torsion of the cantilever. At the surface steps, peaks occur in the FFM signal due to some torqueof the tip. (b) FFM tip probing the frictional force of a surface with asymmetric surface potential, illustrated by asawtooth-like structure. The surface structure on the lower terrace is rotated by 180° along the surface normalcompared to the structure on the upper terrace. The friction coefficient of one single terrace is, e.g., µ→ in the forwardscan direction, whereas it changes to µ← for the backward scan direction due to effects as described in the text; i.e.,the friction coefficient is direction-dependent. The contrast recorded in the FFM signal is the same for both directions.The direction dependence, however, has no influence on the torque of the FFM tip at the surface steps.

on the less inclined sides.* Therefore, lower friction is expected for the upper terrace in comparison withthe lower terrace for the “forward” scan direction. Additionally, the frictional contrast between the upperand the lower terrace should vanish if the tip is moved along the grooves of the sawtooth, i.e., perpen-dicular to the sketched figure.

For the first time, effects due to the atomic-scale asymmetry of a surface potential were reported byOverney et al. (1994) for a lipid bilayer on oxidized Si(100). A similar domain contrast was also foundby Gourdon et al. (1997) and Liley et al. (1998) with a thiolipid monolayer on mica. As a possibleexplanation, the observed contrast was considered to be caused by different molecular tilts within theindividual domains. On crystalline surfaces, however, such effects can be studied in a more controlledmanner. Here, we will use the ferroelectric material triglycine sulfate (TGS) as an example; similar resultshave recently been published by Shindo et al. (1999) for alkaline earth sulfate crystals.

Figure 18.9 displays a perspective view on two neighboring terraces separated by a b/2 step within anegative domain. The glycine molecules form a sawtooth-like pattern perpendicular to the c-axis. Themost remarkable feature of the surface structure in this context is that the arrangement of the moleculeson the upper terrace is rotated by 180° compared to the structure on the lower terrace. This sawtooth-like surface structure is very similar to the case considered above, and an asymmetry in friction is thereforeexpected.

Measurements by Bluhm et al. (1995) fully confirm the expected behavior. Figure 18.10 shows therelative frictional contrast between neighbored terraces (i.e., the difference of the friction on the upperand on the lower terrace) in arbitrary units as a function of the sample rotation angle; 0° is orthogonalto the a-axis. Due to a 105° angle between the a and the c-axis of the TGS crystal, the sample orientationis for 165° and 345° parallel to the grooves of the sawtooth structure. For these angles, the frictionalcontrast vanishes as predicted. A detailed discussion of the frictional anisotropy on TGS(010), includinga simple theoretical model, can be found by Schwarz et al. (1996b).

18.6 Atomic Mechanism of Friction

Although the experiments described above were performed with sliding distances in the micrometerrange, it is clear that the origins for the studied effects can only be found on the atomic scale. InSection 18.2, we have seen that it is possible to measure frictional forces with a scanning force microscopedown to the atomic scale. The analysis of such measurements gives interesting insight into the basicmechanism of friction.

FIGURE 18.9 Perspective view of the (010) surface of a negative domain exhibiting a surface step of half of theunit cell height.

*Such a correlation between surface slope and lateral force can indeed not only be observed on the macroscopicscale, but has also been demonstrated at length scales of some 10 or 100 nanometers (Fujisawa et al., 1993b; Grafströmet al., 1993; Bhushan et al., 1994; Ruan and Bhushan, 1994a), making our assumption even more substantial.

Many issues observed on the atomic scale can be illustrated by using a simple but instructive mechanicalmodel based on the so-called Tomlinson (Tomlinson, 1929) or independent oscillator (IO) model (Prandtl,1928; McClelland, 1989; Helman et al., 1994). A schematic view of this simple spring model is shown inFigure 18.11. A point-like tip is coupled elastically to the main body M with a spring possessing a springconstant cx in the x-direction, and it interacts with the sample surface via a periodic potential Vint(xt),where xt represents the actual position of the tip. During sliding, the body M is moved with the slidingvelocity νM in the x-direction. All energy dissipation — independent of the actual dissipation channel(phonons or electronic excitations) — is considered by a simple velocity-dependent damping term (–γx

.xt).

Within this model, the point-like tip represents the average of the real tip–sample contact or singleasperity contact, where up to hundreds of atoms might be involved (see Section 18.4). In principle, thereal tip–sample contact could also be treated as a system consisting of many individual atomically sharptips interacting through springs, leading to more complex models of friction. However, we will restrictourselves to the simple model introduced above, which has proven to describe successfully the tip motionfor many materials (Table 18.1).

The resulting equation of motion for the tip in the interaction potential is given by

(18.11)

FIGURE 18.10 Relative frictional contrast for neighbored terraces on the negative domain of TGS(010) as a functionof the sample rotation angle.

FIGURE 18.11 (a) A simple model for a tip sliding on an atomically flat surface based on the Tomlinson model.A point-like tip is coupled elastically to the body M by a spring with spring constant cx in the x-direction; xt representsits position within an external potential V(xt) with periodicity a. If xt = xM, the spring is in its equilibrium position.For sliding, the body M is moved with the velocity νM in the x-direction. (b) A schematic view of the tip movementin a sinusoidal interaction potential. If condition (18.13) holds, the tip shows the typical stick-slip-type movement,i.e., it jumps from one potential minimum to another. (c) If the tip moves with stick-slips over the sample surface,the lateral force Fx manifests as a sawtooth-like function.

m x c x xV x

xxx x x

˙ ˙int

t M t

t

t

t= −( ) −∂ ( )

∂− γ

where mx is the effective mass of the system, xM = νMt the equilibrium position of the spring, and γx thedamping constant. The solution of this differential equation is the path of the tip xt(t). The lateral forceFx to move the tip in the x-direction can be calculated from Fx = cx(xM – xt), whereas the friction force Ff

is defined as the averaged lateral force (Fx).If the sliding velocities are very low, analytical solutions of Equation 18.11 can be derived. In this case,

the tip will always be in its stable equilibrium position, and Equation 18.11 can be solved for xt = 0 and.xt = 0:

(18.12)

From this equation, a path xt(xM) can be determined.* For a stiff spring, there is only one solution forall xM, resulting in a continuous tip movement and vanishing friction, since the averaged lateral force(Fx) is zero (Figure 18.12). However, the tip movement changes dramatically if the condition

(18.13)

is fulfilled. Now the tip moves discontinuously in a stick-slip-type motion over the sample surface andjumps from one potential minimum to another. This specific movement of the tip results in a sawtooth-like function for the lateral force Fx (see Figures 18.11c and 18.12). Since the averaged lateral force (Fx)is nonzero in this case, the friction force Ff is necessary to move the body M in the x-direction.

To simulate experimental data obtained with a scanning force microscope, it is important to rememberthat an SFM can measure the lateral forces along and perpendicular to the scan direction (Section 18.2).Therefore, it is necessary to extend the one-dimensional model introduced above to two dimensions ina manner that considers explicitly the tip movement perpendicular to the scan direction (Gyalog et al.,1995). In the further analysis, we will show that this two-dimensional movement of the tip is significantand thus has a considerable effect on SFM images.

The extension of the model to two dimensions leads to the following coupled second-order differentialequation

(18.14)

TABLE 18.1 Some Examples of Materials where the Tomlinson or Independent Oscillator Model Has Been Successfully Applied to Simulate Experimental Data

Sample Reference

Graphite(0001) Sasaki et al., Phys. Rev. B, 54, 2138 (1996)Toussaint et al., Surf. Interface Anal., 25, 620 (1997)Hölscher et al., Phys. Rev. B, 57, 2477 (1998)

KBr(001) Lüthi et al., J. Vac. Sci. Technol. B, 14, 1280 (1996)MoS2(001) Hölscher et al., Surf. Sci., 375, 395 (1997)β-MoTe2(001) Hölscher et al., Phys. Rev. B, 59, 1661 (1999)NaF(001) Hölscher et al., Europhys. Lett., 39, 19 (1996)Organic Monolayers Ohzono et al., Jpn. J. Appl. Phys., 37, 6535 (1999)

*An alternative way to determine the path of the tip is to calculate the minimum of its energy (McClelland, 1989).

c x xV x

xx M t

t

t

−( ) =∂ ( )

∂int

cV

xx < − ∂∂

2

2int

mint

m x c t xV x y

xxx x x

˙,

˙int

t M t

t t

t

t= −( ) −∂ ( )

∂−ν γ

(18.14)

which describes the movement of the tip in the tip–sample potential of an idealized SFM (Hölscher et al.,1997). The simulation of experimental data is performed by moving the support M with constant velocityνM in the “forward” direction (i.e., from left to right) as well as the “backward” direction (i.e., from rightto left) continuously along a certain line in the x-direction while yM is held constant. Calculating theposition of the tip (xt, yt) from Equation 18.14, the lateral forces Fx = cx(xM – xt) and Fy = cy(yM – yt) canbe determined as a function of the support position (xM, yM). Then, after increasing yM, a new scan lineis computed parallel to the previous line.

The application of the model to a realistic sample system (and thus the movement of a sharp single-asperity tip in general) is illustrated in the following with the example of the MoS2(001) surface, wherethe simulated results are compared with experimental data of Fujisawa et al. (1994, 1995). For thecalculations, it is necessary to choose a suitable tip–sample interaction potential Vint(xt, yt), which has tobe used to solve Equation 18.14. The exact tip–sample interaction potential is difficult to determine, sincethe precise structure of the tip is usually unknown. However, it can be shown that all basic features ofSFM measurements are reproduced by the simulation as long as Vint reflects the translational symmetry

FIGURE 18.12 The tip movement in the Tomlinson model, illustrated with the example of the simple sinusoidalpotential displayed in (a). If the tip moves in such a potential, its path can be determined from the solution ofEquation 18.12. The graphical solution of this equation is shown in (b) for two different cases. For a stiff spring,

there is only one point of intersection between cx(xM – xt) (dotted lines in [b]) and (solid line). Consequently,

the obtained path xt(xM) and the lateral force Fx — displayed by dotted lines in (c) and (d) — are continuous, with

the effect that the lateral force averages to zero and no friction occurs. If condition (18.13) holds, however, there ismore than one point of intersection for certain support positions xM (dashed lines in [b]). This leads to instabilities,

which manifest as “jumps” of the tip, as shown in (c) and (d). The occurrence of more than one intersection in (b)means that more than one value xt is possible for the same support position xM (dashed line in [c]). Thus, xt jumps

for increasing xM instantaneously from one stable value to the next (solid line with arrows), leading to a discontinuous

lateral force which looks like a sawtooth (solid line with arrows in [d]). The averaged lateral force Ff is now nonzero.

Vsinxt∂δxt

---------------

m y c y yV x y

yyy y M y

˙,

˙int

t t

t t

t

t= −( ) −∂ ( )

∂− γ

of the sample surface. A comparison with the surface structure of the MoS2(001) sample surface displayedin Figure 18.13 shows that this requirement is met by

(18.15)

where ax = 3.16 Å and ay = 5.48 Å; the actual value of V0 depends on the loading force.With this potential, numerical solutions of Equation 18.14 were computed. The resulting images and

line plots presented here were obtained with the following set of parameters: V0 = 1 eV, cx = cy = 10 N/m,mx = my = 10–8 kg, νM = 400 Å/s, γx = γy = 10–3 Ns/m ~ 2 (critical damping is assumed). Theseparameters have typical magnitude, but qualitatively similar results are obtained with other choices.

Figure 18.14 shows a comparison of lateral force images calculated using the two-dimensional IO-model with experimental data acquired in the [100]- and [120]-directions. The good agreement dem-onstrates that the presented method has the capability to simulate experimental lateral force maps.Moreover, it suggests that the model is able to describe the basic mechanism of atomic-scale friction,although it is based on comparably simple assumptions.

A better insight into the scan process allows Figure 18.15a, where both lateral forces Fx and Fy areplotted as a function of the support position xM for the [100]-direction and scan positions yM = 0.0, 0.2,1.0, 1.3, 1.4, 1.7, 2.5, and 2.7 Å (from left to right) according to the definition of yM given in Figure 18.13.The lateral force in scan direction Fx(xM) exhibits a sawtooth-like shape, whereas the force perpendicularto the scan direction, Fy(xM), looks like a step function. All features of the calculated forces are in goodagreement with experimental data, which is shown for comparison in Figure 18.15b.

The origin of this behavior is analyzed in Figure 18.16. Figure 18.16a shows the paths of the tip in thepotential of VMoS2

(xt, yt) for different values of yM. For yM = 0 Å, no movement of the tip perpendicularto the scan direction can be observed, as expected from Figure 18.15. However, with increasing yM-values,the path of the tip becomes zigzag. A smaller area of the potential is shown in Figure 18.16b, where thepath of the tip for yM = 1.4 Å and 0.7 Å is plotted “time-resolved” by points of equal time distance with∆t = 62.5 µs. It can be seen that the tip only moves slowly as long as it is in the areas which are framedby white lines. These areas are obtained by calculating the surface areas where the tip has a stable positionfor very slow scan velocities (νM → 0) (Gyalog et al., 1995). If the tip leaves these “areas of stability,” itbecomes unstable and jumps into the next “area of stability.” This explains the stick-slip-type behaviorof the lateral force Fx as well as the “step-function”-like shape of the force Fy.

FIGURE 18.13 A schematic view of the MoS2 surface structure. The spheres represent the positions of the sulfatesurface atoms. The dashed line marks the position yM = 0 used in Figure 18.15.

V x y Va

xa

yx y

MoS t t t t2, cos cos( ) = π

π

0

2 2

cx mx⁄

One consequence of this behavior is that the maximum of Fx might appear at different positions thanthe maximum values of Fy (see Hölscher et al., 1998), as it was first observed experimentally by Ruanand Bhushan (1994a). Another consequence is that a large part of the surface is skipped by the tip and,therefore, no information about these skipped surface areas can be extracted from the acquired images.This latter behavior has been experimentally verified for mica by Kawakatsu and Saito (1996); a visual-ization of this issue based on the MoS2 data presented in Figures 18.14 and 18.15 is shown in Figure 18.16c.Here, the “position probability density” of the tip during the scan is plotted, i.e., only the parts of thesurface that show a high density of black points have contact with the tip for a significant time; the whiteareas are skipped by the tip.

FIGURE 18.14 A comparison between the simulated (top) and measured (bottom) lateral force images for the scandirections indicated in Figure 18.13. The movement of the support was in positive x-direction (“forward”); the scansize was 25 Å × 25 Å. (Data from Fujisawa, S., Kishi, E., Sugawara, Y., and Morita, S. (1995), Atomic-scale frictionobserved with a two-dimensional frictional-force microscope, Phys. Rev. B, 51, 7849-7857.)

FIGURE 18.15 A comparison between the calculated (a) and measured (b) lateral forces Fx and Fy for a scan in the[100]-direction and different support positions yM as defined in Figure 18.13. (Data from Fujisawa, S., Kishi, E.,Sugawara, Y., and Morita, S. (1995), Atomic-scale friction observed with a two-dimensional frictional-force micro-scope, Phys. Rev. B, 51, 7849-7857.)

a)

b)

4

2

0

-4

-225xM [A

o

]

Fy

[nN

]F

x [n

N]

LFM (fx)

4.8x

10-7

N (

8.8A

o )

1.1x

10-1

0 N

(1.

5Ao )

AFM (fY ) 25 A

o

At the end of this section, the question arises of how far the situation described above correspondswith the tribological behavior of a nanocontact which is not connected to a cantilever. By the comparisonof experiment and simulation, however, it can be demonstrated that elastic deformations between theatoms at or close to the tip–sample contact area are responsible for the stick-slip movement of the tip;astonishingly the cantilever has only a minor influence. Additionally, in molecular dynamics simulations,atomic stick-slips as found in the above analysis are found to take place as a direct consequence of theinterplay between the surface forces and interatomic interactions within tip and sample (Landman et al.,1989; Harrison et al., 1992, 1993a; Schimizu et al., 1998). Even the zigzagging of the tip atoms in theinteraction potential of the surface has been reproduced (Harrison et al., 1993a,b; Shluger et al., 1999).Thus, it can be concluded that the scanning force microscope indeed measures the “real” frictionalbehavior of a nanocontact.

18.7 The Velocity Dependence of Friction

So far, we have discussed different phenomenological aspects as well as theoretical concepts of frictionon the nanometer scale, but we still have to find an explanation for Coulomb’s law (3) on the independenceof sliding friction from the sliding velocity νM. As already mentioned in the introduction, this law hasalso been confirmed on the nanometer scale within a wide velocity range for crystalline surfaces (Huet al., 1995; Bouhacina et al., 1997; Zwörner et al., 1998).* Figure 18.17 shows two examples of experi-mental data sets for amorphous carbon and diamond. It is found that the friction force Ft is nearlyconstant for the velocities that can be reached with an SFM.

To get a basic idea of the mechanism which causes the velocity independence of the frictional forces,we again apply the simple mechanical model introduced in Section 18.6. For this purpose, we areinterested in the solutions of Equation 18.11 for moderate scan velocities. The qualitative behavior withinthe one-dimensional case is shown for a sinusoidal potential Vint(xt) = V0sin(2πxt). All data presentedhere are obtained with the parameters V0 = 1.0 eV, a = 3 Å, cx = 10 N/m (condition [13] holds for these

FIGURE 18.16 (a) The interaction potential V(xt, yt) (image size: 25 Å × 25 Å) and the calculated paths of the tip(xt(t), yt(t)) in this potential for the support positions yM = 0.0, 0.2, 0.7, 1.0, and 1.4 Å (from top to bottom; for thedefinition of yM see Figure 18.13). If yM ≠ 0, the tip performs a zigzag movement. (b) A detail of (a) (image size:10 Å × 10 Å). The contours of the “areas of stability” (see text) are marked by white lines. In order to show that thetip stays in these areas most of time, the path of the tip is drawn for yM = 1.4 Å (top) and 0.7 Å (bottom) by pointsseparated by equal time intervals ∆t = 62.5 µs. (c) Visualization of the tip position during the scanning of the sample.All calculated paths (54 lines per period) are plotted as in (b). Consequently, the density of points can be interpretedas the “position probability density” of the tip on the surface during the scan. The “areas of stability” are framed byblack lines. Only parts of the stable areas — the dark areas — are in contact with the tip for a significant time.

*There is, however, experimental evidence that friction has a logarithmic or even more complicated dependenceon the sliding velocity for noncrystalline overlayers (e.g., polymeric, self-assembled, or Langmuir–Blodgett-typelayers), probably due to a certain viscosity of the layers due to a liquid-like structure (Liu et al., 1994; Koinkar andBhushan, 1996; Bouhacina et al., 1997). Additionally, it should be mentioned that there have been reports whereeven on crystalline surfaces, a slight logarithmic dependence of the friction on the sliding velocity was observed bySFM (Liu et al., 1994; Koinkar and Bhushan, 1996).

parameters, i.e., stick-slips will occur), mx = 10–10 kg (with this effective mass, the system [tip and samplein contact] has a typical resonance frequency of 1/(2π) ≈ 50 kHz, which is typical for SFMexperiments), and γx = 2 (critical damping). With this set of parameters, we calculate the numericalsolutions of Equation 18.11 and determine the lateral forces Fx as well as the frictional force Ff for differentsliding velocities (νM = 10 nm/s … 100 µm/s).

In Figure 18.18, the calculated lateral force Fx is displayed for different sliding velocities νM. It is obviousthat the obtained results change only little within the wide range of sliding velocities νM = 10 nm/s …10 µm/s. Only for high sliding velocities, where the solution of Equation 18.11 is dominated by thevelocity-dependent damping term (see Fx for 100 µm/s in Figure 18.18), the lateral force increasessignificantly. Consequently, since the friction force is the average of the lateral force, Ff is approximatelyindependent of the sliding velocity in a wide range.

What is now the reason for the relative independence of the lateral forces on the sliding velocities atmoderate values of νM? This issue is illustrated in Figure 18.11b. We have seen above that the tip usuallymoves in a stick-slip-type motion over the sample surface. With this type of motion, the tip stays in theminima of the interaction potential most of the time where it slides very slowly or “sticks.” Therefore,almost no energy is dissipated in this “stick”-state, since we assumed that the damping is proportionalto the sliding velocity. In the “slip”-state, however, the tip “jumps” from one to another minimum. Duringthis jump, the tip reaches very high peak velocities and dissipates significant amounts of energy due tothe velocity-dependent damping mechanism.

It is now important to note that the maximum velocity reached during the slip depends for low valuesof νM in first approximation only on the spring constant cx, the height and the shape of the potential,the chosen damping constant, and the effective mass of the system, but not on the sliding velocity νM.As a consequence, the total amount of energy dissipated during sliding does not change significantly aslong as νM is much smaller than the slip velocity of the tip. From Figure 18.18, we obtain a slip velocityof about 60 µm/s for sliding speeds up to 10 µm/s.* For larger sliding velocities, the mechanism of energydissipation through the “stick-slip” effects breaks down, as can be seen for νM = 100 µm/s.

However, despite its success, the model introduced above is still unsatisfactory from an atomistic view,since it is fully phenomenological in the way that it does not give any insight into the dissipationmechanism involved. It is clear that the energy dissipation process will be much more complicated in areal physical system than that considered here; more complete theories can be found in, for example,Sokoloff (1990, 1993) or Colchero et al. (1996).

FIGURE 18.17 The friction forces Ff as a function of the sliding velocity νM measured with a silicon tip on(a) amorphous carbon, and (b) diamond. Ff is independent of the sliding velocity in good approximation.

*The numerical value of the slip velocity obtained within this simple mechanical model is determined by the one-dimensional interaction potential and the chosen numerical values of the parameters, which partially might vary ina very large range without changing the obtained results qualitatively. Consequently, the slip velocity of 60 µm/s isnot a prediction for an experimental measurement and could actually be much higher without contradiction of thegeneral mechanism discussed here.

cx mx⁄cx mx⁄

18.8 Summary

In this chapter, we have reviewed some of our present understanding of atomic-scale friction gained fromscanning force microscopy experiments. Our analysis, however, has always been restricted to fundamentalaspects of wearless Coulomb friction (i.e., dry friction); effects due to boundary lubrication and wettingof surfaces were not considered.

It has been shown that the main advantage of the SFM in this context is its ability to enable detailedstudies of the friction of “point contacts” (which have been found to exhibit, in reality, contact areas ofsome nm2, depending on the applied load, the elastic properties of tip and sample, and the tip radius)with high spatial resolution as a function of the contact geometry, the applied load, the effective contactarea, the sliding velocity, and the surface structure of the profiled sample. Remarkably, it has been foundthat some phenomena manifest similarly on the nanometer scale compared to their macroscopic behavior,whereas others show a very different behavior.

Most strikingly, friction on the nanometer scale is proportional to the effective contact area, contraryto the macroscopic case. Consequently, the concept of the “friction coefficient,” which is independent ofthe actual contact geometry, fails on this scale. In the corresponding experiments, contact pressures ofup to 1 GPa are easily reached. Such pressures are already above the yield stress of many materials andtherefore seem very high at first sight, but might match the “realistic” pressures of individual nanocontactswithin a macroscopical contact far better than lower contact pressures. However, despite these high

FIGURE 18.18 The lateral force Fx and the tip velocity νt for different sliding velocities νM = 10 nm/s … 100 µm/s.

pressures and the small dimensions of the individual contact with only some tens or some hundreds ofatoms in intimate contact, contact mechanics, which has been derived from continuum elasticity theory,has proved to be still valid.

Another interesting point is that the shear stress is independent of the mean contact pressure — atleast in the case of nanometer-sized Hertzian or approximately Hertzian contacts. Again, very high shearstresses of up to 1 GPa are found, orders of magnitude more than typically obtained from other frictionexperiments. These high values can possibly be accounted for as representing approximately the theoreticalshear stress of the contact, which is about 1/30 of the shear modulus of the contact (Cottrell, 1953) andtherefore has about the correct order of magnitude.

Much more difficult to answer is the question of which factors determine whether a given contactshows high or low friction. In many experiments, it has been demonstrated that friction is very sensitiveto even small changes in the surface structure and might additionally show a strong dependence on thesliding direction relative to the crystal lattice. This feature has been used frequently to distinguish betweenchemically different areas of samples or to obtain a contrast between individual domains that differ dueto structural effects, but has not yet resulted in significant progress in our understanding of why certainsurfaces have higher friction than others.

Nevertheless, studying such issues, the atomic-scale movement of the individual atoms which are inintimate contact has been clarified to a certain degree. By comparison with theoretical models, it can beshown that the tip atoms move in a “stick-slip”-type motion over the sample surface, “jumping” fromone minima of the tip–sample interaction potential to the next. Since they try to avoid passing over theposition of a sample atom which represents a maximum of the interaction potential, their paths oftenresemble a zigzag curve. Due to this specific behavior, it follows that the “atomically resolved” images ofcontact force microscopy represent solely the periodicity of the potential minima of the interactionpotential, which does not generally match the atomic structure of a sample for nontrivial unit cells.

Another intention of this chapter was to bridge the gap between the macroscopic and the micro-scopic/nanoscopic scale. As we have seen in the introduction, it is one of the basic aims of nanotribologyto explain the macroscopic frictional behavior of materials and contacts with fundamental processesoccurring on the atomic scale. And it seems indeed possible to derive Amontons’ and Coulomb’s lawsof friction from nanoscopic effects. Greenwood (1967, 1992) showed that the effective contact areabetween macroscopically flat bodies, which nevertheless exhibit a statistical microscopical roughness,increases linearly with the load, which eliminates any contradiction between the corresponding macro-scopic and the nanoscopic friction laws. Moreover, the relative independence of friction on the slidingvelocity on both the macroscopic and the nanoscopic scale might be explained with the stick-slip-typemotion of the atoms in the tip–sample interaction potential. As long as the relative velocity of the twobodies is much lower than the intrinsic “slip velocity” of the atoms, the dissipated energy will beindependent of variations in the sliding velocity.

To conclude, nanotribology is a young field which has developed rapidly during the last one or twodecades. Considerable progress has been made during this time in our understanding of fundamentalfrictional processes on the nanometer scale, as we hope to have demonstrated above, and first attemptsto connect the atomic-scale effects with macroscopic phenomena seem to be successful. However, in spiteof this progress, there are still many more open than answered questions. For example, it is still impossiblewith our present knowledge to predict the frictional properties of a specific contact from first principles.Nevertheless, the optimization of the frictional properties of surfaces for a given purpose will be increas-ingly important in the future for the further miniaturization of motors and other moving parts in, forexample, hard discs or micro electromechanical systems. A further unsolved question is how energy isdissipated during wearless friction. It seems that the excitation of phonons represents the main channelfor energy dissipation, even though electronic processes might be equally important in the case ofconducting materials. Thus, the exact control of the dissipation channels might be a prerequisite for thetailoring of surfaces with given frictional properties — and simultaneously offers plenty of room fororiginal and innovative nanotribological research.

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