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The traditional description of atomic-scale friction, as investigated in Friction force microscopy, in terms of mechanical stick-slip instabilities appears so successful that it obscures the actual mechanisms of frictional energy dissipation. More sophisticated theoretical approach, which takes into account damping explicitly, reveals the existence of some hidden, unexplained problems, like the universal nearly-critical damping and unexpectedly high value of the dissipation rate. In this paper, we combine analysis in the framework of nonequilibrium statistical mechanics with simple atomistic modeling to show that the hidden problems of atomic scale friction find their origin in the nontrivial character of energy dissipation that is non-local and dominated by memory effects, which have not been addressed before in the context of dry, atomic-scale friction.

The problem of friction is of enormous practical importance. However, fundamental understanding of the origin of dissipative surface forces on microscopic level is still far from being satisfactory. Only recently, with the application of atomic force microscopes to friction (friction force microscopy―FFM) atomic- scale access has been acquired to the origin of dissipative surface forces, boosting the rapidly developing field of nanotribology [

There are many physical parameters which can be important for atomic scale friction, but two of them seem really crucial. One is the corrugation of the effective (mean force) contact potential: the lateral barriers faced by the slider are responsible for the appearance of stick-slip motion and sizable friction experienced. The other is the inherent dissipation rate, which describes how rapidly the energy invested into the system is losing irretrievably to the outside world. These two key parameters are never known in advance, not only because of uncontrolled contact conditions but also in view of nontrivial physics involved. This is in contrast to practically all other system parameters, which can be taken directly from experiments (like mass and flexibility of the tip and cantilever) or reasonably estimated. In our earlier work [

The traditional description of atomic-scale friction in terms of mechanical stick-slip instabilities appears so successful that it obscures the actual mechanisms of energy dissipation. The reason is that the observed friction force in the stick-slip regime is nearly completely determined by the system parameters (flexibility and contact potential corrugation) and it is practically independent of the dissipation rate, provided the latter is constrained to a certain range, the origin of which has never been explained. As was shown in our recent works [

The mere fact that atomic stick-slip patterns are readily observed in experiments implies that the inherent dissipation rate is close to the characteristic frequency of the measuring system (nearly critical damping). Similar assumption is needed to explain some more subtle aspects of sliding dynamics, like the rare occurrence of long slips. Although nearly critical damping is suggested by experiments, it is absolutely unclear what could be physical reasons for that. Moreover, it seems to defy common sense. Indeed, it implies that the dissipation rate is mainly determined by the characteristic frequency (i.e. by the flexibility and mass) of the measuring system but it is only weakly dependent on the contact conditions and the substrate where energy is dissipated. Certainly, one would expect just the opposite.

Another problem is due to the observation that the typical (nearly critical) value of the dissipation rate needed to explain FFM experiments turns out to be several orders of magnitude higher than we can expect on the basis of data collected for surface diffusion (see, e.g., [

These observations force us to critically reconsider traditional views on frictional energy dissipation. In what follows, we propose that the hidden problems of atomic scale friction find their origin in the nontrivial character of the dissipation that is non-local and dominated by memory effects, which?as far as we know?have not been addressed before in the context of dry, atomic-scale friction.

In the theory of atomic scale friction (see, e.g., review papers [

(mass M of the moving object is introduced here by formal reasons to provide η with the convenient dimensionality of frequency). In view of the fluctuation- dissipation theorem, this implies that thermal noise is white (δ-correlated) and memory effects can be neglected. The force (1) is sometimes called viscous force, by a formal analogy with Stock’s force in liquids, but actually this analogy is poor. To our knowledge, however, the applicability of this approximation to dry friction has never been properly justified. In fact, the above mentioned problems (i.e. too a high value of η and its closeness to the characteristic frequency of the measuring system, as follows from FFM experiments) indicate that the traditional approach (1) is strongly oversimplified.

To clarify the question, we first address some general statements of Non- equilibrium statistical mechanics, see, e.g., [_{diss}, sometimes called friction) and random (F_{ran}) forces experienced by the particle are due to the dissipation of the particle’s energy and momentum to the substrate bath and thermal fluctuations in the bath, respectively. F_{diss} is not necessarily linear in the particle’s velocity

with

with _{B} the Boltzman constant.

The familiar approximation (1) corresponds to the limiting case when, by some reasons, the random force correlations are extremely short and memory effects can be neglected, so that

If memory effects are important, the usage of the traditional linear approximation (1) inevitably implies that the description is coarsened, i.e. averaged over certain time intervals τ*. In this case the damping factor in (1) has the meaning of mean dissipation rate,

In the stick-slip regime of FFM friction, the characteristic time intervals are related with the period of the tip vibration in a surface potential well, which, in turn, is directly related with the flexibility and mass of the measuring system. This can provide a clue [

With very few exceptions [

Friction is concerned with the coupling between the relative motion of two bodies and their internal degrees of freedom. A variety of coupling mechanisms exist, depending on the nature of the solids (e.g. insulator, conductor, semiconductor), sliding velocity and other parameters. There are phononic, electronic, electromagnetic and some other mechanisms (see, e.g., our earlier review paper [

For solid systems, it is suggested to use the advantage of the normal mode presentation for vibrational degrees of freedom. Considering the evolution of a particle (or a system of particles) interacting with a solid, one can start with the full system of dynamical (Newtonian) equations of motion for the particle(s) and for all atoms in the solid. The equations are coupled, in view of interatomic interactions. Rewritten in terms of normal phonon modes, equations of motion for the solid in harmonic approximation (when there is no direct coupling between phonon modes) turn out to be relatively simple in their form and they can be formally solved analytically, for given positions of the particle under interest at current moment of time and in the past. After substituting the solutions into the equation of motion of the particle, the later can be reduced to generalized Langevin equation with memory, with the dissipative force of the form (2) and random force obeying the fluctuation-dissipation theorem (3). Such a procedure has been used in a number of works [

An important conclusion of this derivation is that the non-retrievable loss of the particle’s energy and momentum is not only due to atoms in the close vicinity to the contact, as one could think, but also atoms in bulk of the solid contribute to friction significantly. Notice that earlier the role of bulk solid atoms was anticipated [

These conclusions are general as they do not need any specification of interatomic interactions and structure of the solid. More detailed analysis is cumbersome and it will be addressed elsewhere [

Our approach is based on the concept of a dynamical deformation pattern. When an object (the tip apex) is in contact with the surface, there is a certain deformation of the substrate lattice. This deformation pattern follows the object upon its motion along the surface. Dynamical deformation of the lattice is an essentially dissipative process, as it is accompanied by the creation of phonons. Such is, in particular, the known mechanism of attenuation of low frequency sound in solids [

At first glance, the effective size of DDP can seem very small, since the (static) deformation of the substrate lattice rapidly decays with distance n from the contact (in units of atomic spacing a), basically as n^{−2} when na is large compared to the contact size. However, the number of solid atoms at distance n from the contact scales as n^{2}, and hence these two trends are to compensate each other. Consequently, DDP can be―in principal―very large, and its actual effective size should be determined by some other reasons. Apparently, this is the system history that determines which atoms of the solid have already been involved in collective motion with the slider and which not. The picture can be essentially different depending on type of the slider motion, e.g. for its translational motion along the surface and for an oscillatory one. In this paper we concentrate at the latter, since it is just the case directly related with FFM measurements.

In the stick-slip regime, the FFM tip vibrates in a surface potential well and slips to the neighboring (typically) well at a critical value of the external force exerted by the cantilever [

The problems are met in the framework of the traditional?coarsened? description (1) when η should be treated as the mean dissipation rate (5). Now our task is to estimate it taking explicitly into account the non-local and memory character of dissipation.

For an oscillator with frequency ν_{os} the dynamical deformation pattern includes atoms which have been involved in concerted motion during halve the period of its vibration. The effective radius (n_{max}) of DDP can be estimated as the distance travelled with sound velocity c during this time interval, n_{max} ~ c/(2ν_{os}a). For typical frequency much smaller than the atomic frequency, _{a} moving with induced velocity _{a} and ν_{os} in terms of masses and stiffness of the corresponding springs, and introducing the mean dissipation rate for the oscillator,

with k_{a} the characteristic stiffness of atomic bonds and K_{os} stiffness of the oscillator spring.

Physics behind the result (6) seems clear: the oscillator frequency ν_{os} simply determines the rate with which the system delivers mechanical energy to the contact, while the dimensionless factor Ck_{a}/K_{OS} determines the part of energy which is invested into the substrate. The absence of any other physical parameter in (6) reflects the fact that at velocities small compared to sound velocity, _{os}. In fact, this similarity is remarkable, since halve wave length of such a wave corresponds to the effective size of DDP considered above.

Although the estimates performed are simple, they capture the basic aspects of non-local and memory character of frictional energy dissipation, and they lead to a principal result. Expression (6) reveals direct coupling of the (mean) dissipation rate with the slider frequency, thus explaining the paradox of critical damping. Moreover, the obtained value of _{a} (~1 N/m), so that damping of the rapid FFM mode with K_{OS} ~ k_{a} is predicted by (6) to be close to or somewhat smaller than critical (by definition η_{crit} = 4πν). This is just what is needed to explain atomic periodicity of FFM scans and the rare occurrence of long slips [

Krylov, S.Yu. and Frenken, J.W.M. (2017) Non-Local and Memory Character of Frictional Energy Dissipation on Atomic Scale. Engineering, 9, 14-21. http://dx.doi.org/10.4236/eng.2017.91002