Bhushan B. Handbook of Micro/Nano Tribology, Second Edition

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6.4.3.5 Energy Dissipation and Friction

The model presented explains the observed lateral force curve and the fact that energy is dissipated during

the scanning motion of the tip. However, the central physical question still remains: exactly how and

where is this energy dissipated?

If the force constant c

in Equation 6.27 is assumed to be simply the lateral force constant of the

cantilever, then this energy is dissipated only in the macroscopic cantilever. From a fundamental point

of view, this would be rather disappointing. If friction is to be modeled on an atomic scale, it would be

more appealing if the energy is lost within the microscopic tip–sample contact. This should also be the

case if friction in an SFFM setup is to be considered as a model system for the friction of a single asperity

contact. To address the question of where and how the energy is dissipated, we recall the spring model

presented above: in a complex system energy is stored in each different subsystem. Each of these sub-

systems can be described by a spring with a (differential) force constant c

and the ratio of energy stored

in this system is proportional to 1/c

. If an instability occurs, the energy stored in each subsystem is

converted into kinetic energy which is ﬁnally dissipated. Most energy is dissipated in the subsystem with

the lowest force constant. In an SFFM setup elastic energy is built up during the scanning motion not

only in the cantilever, but also in the tip–sample contact. Each of these subsystems can in turn be divided

into further subsystems. The cantilever stores energy not only in the torsion and bending mode, but also

in the bending of the tip itself (Fujihira, 1997; Lantz et al., 1997), and the energy stored in the tip–sample

FIGURE 6.34 Critical force curves (see text). Top: for the case of isotropic coupling of the tip to its support (that

is, the force constants of the tip in the x- and y-direction are equal). Left: hard lever, right: soft lever. Bottom: for

the case of anisotropic coupling of the tip to its support. Left: hard lever, right: soft lever. When the separation ∆ is

moved so that the scan line crosses a critical curve, the tip–sample system becomes unstable and the tip snaps into

a new position on the sample. (From Gyalog, T. et al. (1995), Europhys. Lett. 31(5–6), 269–274. With permission.)

contact is due to deformation of the tip as well as of the sample surface. Therefore, the force constant is

in fact an effective force constant c

eff

which can be decomposed into different force constants according to

(6.30)

where c

bend

is the force constant describing the lateral bending of the cantilever, c

tors

the force constant

describing its lateral twisting, c

tip

the force constant describing the (macroscopic) bending of the tip,

tip

cont

the force constant describing the microscopic deformation of the tip, and correspondingly, c

sample

cont

the

force constant describing the microscopic deformation of the sample.

Depending on the value of these force constants energy can therefore be dissipated mainly in the

macroscopic degrees of freedom of the cantilever or in the microscopic degrees of freedom of the

tip–sample contact. Since the stiffness of the tip–sample contact depends on the normal force (see

Equations 6.17 and 6.18), for sufﬁciently low loads the lateral force constant will be determined mainly

by this local stiffness and correspondingly energy is dissipated mostly in the microscopic degrees of

freedom of the tip–sample sample (Colchero and Marti, 1995; Colchero et al., 1996b). This dependence

of the local stiffness on the normal force has been veriﬁed experimentally for the case of an Si

tip on

mica (Carpick et al., 1997). Energy dissipation due to the intrinsic local stiffness of the tip–sample contact

is the more fundamental process, since it makes it possible to model friction of a single asperity contact.

Continuum theories have been applied to study the friction of such a single asperity contact (Johnson,

1996, 1997a,b). According to those models, once a critical stress is reached, complete sliding of the

tip–sample contact will occur and therefore all energy stored in its elastic deformation will be released.

By contrast, due to the atomic corrugation within the interface, the model discussed above predicts that

a new stable position will be found after the tip has slipped about one lattice constant. In this case only

part of the elastic energy stored in the elastic deformation is released. It would be interesting to try to

combine atomistic and continuum models.

By computing the positions of all the atoms of tip and sample before and after the slip, the exact shape

of the elastic wave generated in the contact can in principle be calculated. This wave propagates out of

the contact area into the tip and the sample and is ﬁnally thermalized. In a phonon picture this process

can be interpreted as creating (a superposition of) phonons in the tip–sample contact during the slip

event, which are then radiated into the bulk solids of the tip and sample, where they are ﬁnally scattered

and thermalized. In the end, the elastic energy built up during the scanning motion of the tip — or

within any of the multiple asperities in a macroscopic tribological contact — is converted into heat,

which is what is usually associated with friction.

The ideas discussed in the present section are a qualitative description of what we consider to be the

most fundamental processes associated with energy dissipation in an SFFM setup. However, for a detailed

and quantitative analysis of friction, much more elaborate models and exact theories involving molecular

dynamics simulations and ﬁrst principles calculations have to be developed.

6.5 Summary

The most striking and impressive performance of SFFM is imaging the lateral force with the atomic

periodicity of the surface. The ﬁrst interpretation is that these images correspond to the friction force as

an atomically sharp tip passes over the surface atoms. However, this simple view is incorrect, since both

experimental and theoretical arguments show that in a typical SFFM setup the radius of the tip–sample

contact is considerably larger than the atomic periodicity observed. Many experiments are described in

this chapter to illustrate this point. “True” atomic resolution of the lateral force, as demonstrated, for

example, by imaging defects or steps, has not yet been reported. “Atomically resolved” friction images

should therefore be interpreted with great care. The question of how this apparent atomic resolution can

be achieved has still not been settled. Different mechanisms have been proposed, but none seems to apply

11 1 1 111 1

cc c

ccccc

xxx

eff lever cont

bend tors tip

cont

tip

cont

sample

=+=++++ ,

generally and others are simply a mathematical restatement of the problem. On the other hand, if an

atomically corrugated potential between tip and sample is assumed a priori, a simple mechanical model

of the SFFM setup can explain the features observed in SFFM, i.e., the shape of the friction loop as well

as the stick-slip motion with the atomic periodicity of the surface.

From a fundamental point of view, the question could be posed as whether atomic resolution of the

friction force is reasonable at all. Evidently, mechanical friction of a single isolated atom is not deﬁned

at all. “True” atomic resolution of the friction force in an SFFM would require an interface with a single

atom between the sliding surfaces, but it is not clear that such a one-atom contact is a stable conﬁguration

that can be realized experimentally. And even in this hypothetical case, friction would occur between the

interfaces consisting of many atoms. Correspondingly, a large number of atoms are always involved in

friction.

The question of whether true atomic resolution is possible or not is indeed intriguing. However, from

a pragmatic point of view it is rather irrelevant. Compared with other tribological instruments, SFFM

achieves a resolution which is order of magnitudes higher; lateral and normal forces can be measured

with nano- and even piconewton precision and with nanometer spatial resolution, and the topography

of the surface can be determined with (sub)atomic resolution. Moreover, an SFFM can measure simul-

taneously these three tribologically most relevant quantities. The amount of new tribological phenomena

that can be addressed is therefore immense. We have described recent progress in two important ﬁelds,

boundary lubrication and the friction of a single asperity contact.

In air, as well as in many tribological applications, surfaces are covered by a thin liquid ﬁlm which

modiﬁes their tribological properties. Liquid menisci may form increasing the adhesive force between

the sliding surfaces. The physical and chemical interactions between the surfaces are affected by the

presence of water which can act as a lubricant. The tribology of the contact can also be changed

deliberately by covering the surface with a few monolayers of organic materials, that is, by boundary

lubrication. SFFM has resolved the structure of these ﬁlms and already shown that one monolayer of LB

ﬁlms or self-assembling monolayers can signiﬁcantly reduce friction and wear on a nanoscale.

The other important matter addressed in this chapter is the tribology of a single-asperity contact. The

behavior of such a contact is rather well understood by now. Essentially, the shear stress within a single-

asperity contact seems to be constant. Therefore, the friction is proportional to the contact area, as in

the case of macroscopic friction. By contrast, the corresponding friction–load relation is nonlinear and

determined by the tip geometry and the area–load relation as predicted by continuum theories. Further

experiments have to show up to what limit the shear stress can really be assumed constant.

Typical for SFFM are high pressures in the contact easily exceeding 1 GPa and thus the yield pressure

of many materials. In the context of tribology this is not a disadvantage, since high pressures are usually

inherent to friction; therefore, those occurring in an SFFM contact are necessary in order to study

tribology under “normal” conditions. On the other hand, to investigate some fundamental aspects of

friction a system with much lower interaction is easier to understand. In tribology, the friction of an

atomically thin ﬁlm sliding on the moving surface of a quartz microbalance represents such a low-

interaction system. A lot has been learned, theoretically as well as experimentally, from this system. It

seems that the dominant factor for energy dissipation is phononic excitation even though electronic

excitation is important in the case of conducting substrates. This is in agreement with the corresponding

processes in SFFM, where energy dissipation is assumed to occur as elastic waves — phonons — are

scattered into the sample and the tip during the sudden stick-slip transition.

Summarizing, nanotribology is a young and emerging ﬁeld that is maturing fast, as the experiments

described show. Due to the never-ending trend to miniaturization, understanding friction on a nanoscale

will become of increasing importance, since as the length scale is reduced, friction forces become stronger

relative to other forces. On the other hand, the development of simple systems such as the (single-asperity

contact) SFFM, the quartz microbalance, but also the surface force apparatus, is important for the progress

of basic research in tribology, since these well-deﬁned and conceptually simple systems are needed for

the classical interplay between theory and experiment: developing theories that can be tested with simple

model systems. Possibly, if a bridge between nano- and macroscopic friction is found, nanotribology

might improve our understanding of classical macroscopic friction and thus help to improve the efﬁciency

and lifetime of everyday devices.

We hope to have convinced the reader that friction on an atomic scale is not only a intriguing ﬁeld,

but also an increasingly important one.

Acknowledgments

This work was supported by the Ministerio de Educación y Cultura (proyecto CICYT, Modalidad C

Referencia PB 95-0169), by the Swiss National Science Foundation and the Kommission zur Förderung

der Wissenschaftlichen Forschung. We thank all the colleagues who have generously supplied images

from their work. We are indebted to many people for stimulating discussions, valuable suggestions, and

proofreading of the manuscript. E. Paetz, R. M. Overney, J. Frommer, A. M. Baró, N. Agaït, M. Salmeron,

M. Luna, E. Barrena, T. Bonner, W. Gutmannsbauer, L. Howald, R. Lüthi, H. Haefke, M. Rüetschi, H.-

J. Güntherodt, and H. Rudin, among others.

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Bhushan, B. “Microscratching/Microwear, Nanofabrication/Nanomachining...”

Handbook of Micro/Nanotribology.

Ed. Bharat Bhushan

Boca Raton: CRC Press LLC, 1999