# On the Non-trivial Origin of Atomic-Scale Patterns in Friction Force Microscopy

## Abstract

Friction between two surfaces is due to nano- and micro-asperities at the interface that establish true contact and are responsible for the energy dissipation. To understand the friction mechanism, often single-asperity model experiments are conducted in atomic-force microscopes. Here, we show that the conventional interpretation of the typical results of such experiments, based on a simple mass-spring model, hides a fundamental contradiction. Via an estimate of the order of magnitude of the dissipative forces required to produce atomic-scale patterns in the stick-slip motion of a frictional nano-contact, we find that the energy dissipation must be dominated by a very small, highly dynamic mass at the very end of the asperity. Our conclusion casts new light on the behavior of sliding surfaces and invites us to speculate about new ways to control friction by manipulation of the contact geometry.

## Keywords

Friction force microscopy Atomic-scale friction Energy dissipation Stick-slip motion## 1 Introduction

Friction, the familiar force experienced both on the macroscopic scale in everyday life and on the scale of atoms and molecules, is directly related to the dissipation of energy. One may hope that a full understanding of the phenomenon will ultimately result in complete control of friction, which can be employed, e.g., to eliminate undesired forms of friction and associated wear in engines and other mechanical applications. In this context, special attention is given to the origin of friction [1, 2, 3 and refs. therein].

One of the key instruments in the experimental investigation of friction at the nano-scale is the friction force microscope (FFM) [2], in which the tip–surface contact serves as a model for a single-asperity contact. Two-dimensional maps of the lateral force experienced by the tip in an FFM-experiment often display a clear, atomic-scale periodicity, as is illustrated in Fig. 1a and b. Such patterns reflect the stick-slip (SS) motion that the tip executes between nearby minima in the corrugated tip–substrate interaction potential.

Here we use the mere observation of such SS-behavior to draw conclusions about both the kinetic energy and the energy dissipation rate that are in play during a typical atomic slip event in an FFM. We demonstrate that the amount of energy one should reasonably expect to be dissipated by the atomic-size contact is some six orders of magnitude lower than what is typically assumed. This completely invalidates the traditional interpretation of the observed slip dynamics. Instead, we propose a more sophisticated dissipation mechanism that involves the dynamics of a tiny mass at the very end of the tip apex. Our mechanism is consistent with the experimental observations. In addition, it provides more general insight in sliding and friction and may open the possibility to predict the friction behavior of both typical and non-typical contact geometries.

## 2 Basic Considerations

*x*-direction

*y*-direction, where \(\left( {{x_{{\text{tip}}}},{y_{{\text{tip}}}}} \right)\) is the position of the tip in the surface plane. \({m_{{\text{eff}}}}\) represents the effective mass of the tip and the atomically corrugated tip–substrate interaction potential is given by \({V_{{\text{int}}}}\). The spring force resulting from the difference between the positions of support and tip is governed by an effective spring constant \({k_{{\text{cant}}}}\), which is a combination of the stiffness of the cantilever of the FFM and the stiffness of the tip–surface contact and can be deduced directly from the slope of the measured lateral force trace in Fig. 1b [7]. In addition to the two static forces, one due to the interaction with the substrate and the other due to the spring, the tip also experiences two dynamic forces. The first, \(\xi\), is due to thermal noise, while the second, \(\gamma {\dot {x}_{tip}}\), is the velocity-dependent dissipation. For the present study, the dissipation term establishes the most important term in the equation; it is through this term that friction is introduced. Note that the values of the dissipation rate \(\gamma\) and the amplitude of the noise term \(\xi\) cannot be ‘chosen’ independently, as they are intimately connected via the fluctuation–dissipation theorem.

Most numerical calculations that are used to reproduce FFM-experiments apply the ‘recipe’ of Eq. 1. Typically, the effective ‘tip’ mass is assumed to be in the order of 10^{−11} kg, which corresponds to the combination of the tip and a sizable fraction of the cantilever that moves together with the tip. The dissipation rate \(\gamma\) is almost always used as the fitting parameter to reproduce the experimental data, which results in a typical value of 10^{−6} kg s^{−1}, corresponding to tip motion that is in the order of critically damped [3, 4, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. Although this approach produces atomic patterns, it does not lead to a physical understanding of the resulting \(\gamma\)-values, for example in terms of the microscopic dissipation mechanism.

## 3 Dissipation as a Sum of Atomic Contributions

Note that Eq. 3 does not imply that dissipation would be restricted to the contact atoms. It does, however, pay tribute to the fact that the mechanical excitations in the sliding interface that lead to friction certainly originate from the contact, even when dissipation of these excitations occurs further away in the substrate and the tip body. Thereby, it is the contact area, i.e., the number of atoms in contact \({N_{{\text{cont}}}}\), that determines how much mechanical energy is temporarily stored in the contact and subsequently dissipated in a slip event. In Appendix A, we provide a further justification for the assumed proportionality between the friction force and \({N_{{\text{cont}}}}\). By relating the sliding friction force directly to the microscopic, true area of contact [26], to the actual, interfacial velocity, and to the atomic-scale dissipation rate, we propose to put the phenomenological Amontons–Coulomb law on a truly fundamental footing for the case of wearless friction.

A typical value for \({N_{{\text{cont}}}}\) for FFM and contact-AFM experiments is 10. For a typical atomic mass number of 74 amu (tungsten) and a typical atomic or short-wavelength lattice vibration frequency of 10^{12} Hz, this leads to a dissipation rate of the contact of 10^{−11} kg s^{−1}, which is some five orders of magnitude lower than the values required to reproduce atomic patterns in FFM-calculations. The two assumptions that have gone into Eq. 3 may be crude, but a more refined description will not remove the blatant discrepancy.

At this point, we draw the conclusion that the values adopted for the dissipation parameter in most FFM-calculations are unphysically high. Correspondingly, calculations within the same model for realistic \(\gamma\)-values would result in highly underdamped motion and could never reproduce a regular SS-pattern with lattice periodicity.

## 4 Strong Dissipation Implies Ultralow Mass

In order to remedy the phenomenal discrepancy encountered above, we return to Eq. 1. The only remaining element that can be completely incorrect in the employed, typical estimates is the assumed value of the effective mass \({m_{{\text{eff}}}}\) [18]. What does this mass stand for and how does its value affect the friction behavior? The effective mass represents the number of dynamic atoms \({N_{{\text{dyn}}}}\) that effectively move at the same velocity as the contact and it is usually associated with the entire tip plus a part of the FFM-cantilever, hence its large, typical value of 10^{− 11} kg. Since the maximum velocity \({\dot {x}_{{\text{tip}}}}\) that the tip apex can acquire during a slip event in the absence of damping (cf. Equation 1) is inversely proportional to the square root of the effective mass, this mass also influences the maximum friction force \({F_{{\text{diss}}}}\) that the tip apex can experience in this motion (cf. Equation 3). We should thus expect a simple scaling between the maximum dissipative force and the effective mass of the type \(F_{{{\text{diss}}}}^{{{\text{max}}}} \propto 1/\sqrt {{m_{{\text{eff}}}}}\). So, the required increase in the friction force by five orders of magnitude translates directly into a reduction in the effective mass by at least ten orders of magnitude with respect to the value typically assumed. Obviously, we can no longer associate this with the entire tip and part of the FFM-cantilever, but see that this small mass can only stand for a very small portion of the tip, namely its very apex. We will refer to this tiny mass as the dynamic mass \({m_{{\text{dyn}}}}\) and will be forced to describe its dynamics and that of the rest of the tip and cantilever with separate equations of motion (see below).

In Eq. 4, the total number of dynamic atoms \({N_{{\text{dyn}}}}\) is expressed in terms of the dynamic mass and the atomic mass. It should, of course, be larger than or equal to the number of contacting atoms \({N_{{\text{cont}}}}\).

Equation 4 provides us with a straightforward condition that needs to be satisfied in order to ‘see atoms’ in FFM-measurements. It is cast in the form of a relation between the number of tip atoms that are in direct contact with the substrate and the total number of atoms that can be considered to effectively move rigidly with these contact atoms, including the contact atoms themselves. In practice, both spring coefficients, \({k_{{\text{at}}}}\) and \({k_{{\text{dyn}}}}\), are in the order of 1 N/m, so that the ratio \({k_{{\text{at}}}}/{k_{{\text{dyn}}}}\) is approximately unity. This provides us with a unique way to estimate the typical dynamic mass in FFM-experiments, based on the mere fact that atomic patterns are often observed in these experiments, which implies that the inequality of Eq. 4 is often satisfied. For a typical, 10-atom contact (\({N_{{\text{cont}}}}=10\)), \({N_{{\text{dyn}}}}\) has to be below 100 atoms, corresponding to a maximum dynamic mass of approximately 10^{−23} kg. This is about 12 orders of magnitude lower than what is typically assumed, obviously satisfying the coarse, 10-orders-of-magnitude estimate that we arrived at above. The qualitative picture that goes with these small numbers is illustrated in Fig. 2, which shows the situation in which a significant fraction of the elastic deformation is concentrated in the very apex of the FFM-tip.

## 5 Model Calculations

### 5.1 General

The essential feature of our new description is formed by the high velocities to which the ultralow dynamic mass is accelerated, which enable efficient energy dissipation. We used numerical calculations for an extensive test of this scenario. To this end, we describe the combined system of the support, cantilever, flexible tip and small dynamic tip apex as a classical 2-mass-2-spring configuration [3, 27, 28, 29, 30]. One mass is the low, dynamic mass \({m_{{\text{dyn}}}}\), associated with the very tip apex. It is connected via the effective spring \({k_{{\text{dyn}}}}\) to a large mass that corresponds to the rest of the tip and part of the cantilever. This large mass is, in turn, connected via the typical cantilever spring coefficient to the rigid support. As described in Appendix B, we employ a fully two-dimensional version of this model in the form of 2 × 2 coupled Langevin equations, each similar to Eq. 1. These are solved numerically, with special attention given to the enormous difference in the characteristic frequencies of the two pairs of coupled oscillators in this model. As in the real FFM-experiment, the displacement of the cantilever mass is monitored to construct the two-dimensional, lateral force maps.

### 5.2 Numerical Results

Typical results of these model calculations are shown in Fig. 3. In order to keep calculation times manageable, we used a relatively large contact size of \({N_{{\text{cont}}}}\;=\,100\) atoms, corresponding to a damping rate of \(\gamma \,=\,{10^{ - 10}}\) kg/s. Different values were chosen for \({N_{{\text{dyn}}}}\), in order to explore overdamped, critically damped and underdamped situations. The numerical results are in complete agreement with our qualitative description. Close to critical damping, more or less regular SS-motion is observed, which is reflected in patterns with clear atomic-scale periodicity.

Underdamping of the tip apex motion by only a few orders of magnitude (Fig. 3a) leads to a complete loss of atomic periodicity. As a result of the lack of damping, slip events frequently extend over multiple lattice spacings, which ruins the regularity.

Actually, the calculations show that also overdamping destroys the regular SS-signature in the FFM-patterns. This is caused by the fact that high damping goes hand in hand with a high level of thermal fluctuations, as a direct consequence of the fluctuation–dissipation theorem. Still, the loss of regular SS-patterns at overdamping may seem surprising. After all, at high damping all fluctuations are damped in a very short time. As a consequence, thermal activation of ‘early slips’ becomes ineffective, so that all slip events start very close to points of mechanical instability. In combination with the suppression of multiple slips, this might make one expect the SS-patterns to become very regular. However, such an expectation would be wrong. After all, the points of mechanical instability are determined by the force balance of Eq. 1 (or better, Eqs. 5 and 6) and the strong force fluctuations that accompany the strong damping introduce strong variations in the precise locations at which the slip events commence. This establishes the second mechanism by which the SS-patterns become more chaotic. This trend can be recognized in Fig. 3d.

Only in a limited window around critical damping, the dissipation rate is high enough to avoid frequent multiple slip events and the statistical force fluctuations are low enough to leave the pattern of slip positions well defined. As Eq. 4 indicates, calculations for other values of \({N_{{\text{cont}}}}\) give similar results, when \({N_{{\text{dyn}}}}\) is changed accordingly.

Figure 4 summarizes our understanding in the form of a friction ‘phase’ diagram. Depending on the number of atoms in the contact \({N_{{\text{cont}}}}\) and the total number of atoms in the dynamic part of the tip \({N_{{\text{dyn}}}}\), the diagram goes from underdamped motion (upper left corner) with multiple slips, to overdamped behavior (lower right) with strong fluctuations. Only the intermediate regime around critical damping, \(D\,=\,1\) in Fig. 4, is characterized by FFM-patterns with clear atomic periodicities. Note that \({N_{{\text{dyn}}}}\) cannot be lower than \({N_{{\text{cont}}}}\), so that the far lower right of the diagram is unphysical. For further details and trajectories of the tip and the cantilever, we refer to [31].

## 6 Summary and Discussion

Stick-slip motion in friction force microscopy with clearly visible atomic periodicity is only possible by virtue of the high-speed motion that is carried out by a small and highly dynamic mass at the very end of the tip apex. The speed needs to be high enough to make the motion close to critically damped. Damping that would be either much higher or much lower would destroy the periodic patterns via excessively strong fluctuations or frequent slips of the tip over multiple lattice spacings. In this study, we arrived at this insight through estimates of the order of magnitude of the dissipative forces, contributed by the atoms in the contact and we formulated a condition on the effective shape of the tip apex, in Eq. 4. By integrating a set of four coupled Langevin equations, two for the *x*- and *y*-coordinates of the ‘regular’ tip-plus-cantilever combination and two for the *x*- and *y*-coordinates of the tiny dynamic mass at the end of the tip, we numerically explored the conditions that the tip needs to satisfy, which is summarized in the ‘phase’ diagram of Fig. 4.

The conclusion drawn here highlights a hitherto unrecognized signature of the flexibility and ultralow mass of the tip apex and the resulting, highly dynamic character of its motion. In several earlier publications [e.g., 3,27–30], we concentrated on cases where the atomic corrugation of the interaction potential between tip and substrate is relatively modest, so that the high rate, with which the rapidly moving tip is ‘attempting’ to overcome the energy barriers, can lead to a delocalization of the contact and a corresponding, extreme lowering of the friction force—an effect that we have nick-named “thermolubricity.” In the present manuscript, we focused on the increase in dissipation, associated with the high velocity that the tip acquires during a slip event, which is relevant for turning the sliding into nearly ideal atom-scale stick-slip motion, in case of a more pronounced corrugation of the interaction potential.

In view of the strict conditions found here for \({N_{{\text{cont}}}}\) and \({N_{{\text{dyn}}}}\), it may seem miraculous that many FFM-experiments yield force maps with clear, atomic-scale SS-motion. However, the tips used in AFM- and FFM-experiments have geometries that make them naturally fall in the central, green part of the diagram. In other words, even though we now understand that it is far from trivial that an FFM-experiment would be sensitive at all to the lattice periodicity, it is the typical shape of the employed tips that leads to a nearly critically damped motion of the tip apex, once it is brought in contact with the surface. We stress that the naïve, ideal FFM setup (Fig. 1c), equipped with a fully rigid tip apex, should always perform highly underdamped motion (upper left, far outside the diagram of Fig. 4) and should never produce atomic-scale stick-slip patterns.

What we have found here for a single asperity, moving over atomic distances, may also apply to frictional energy dissipation in practical contacts, consisting of large numbers of micrometer-scale contact regions. Both the geometry of each contact apex and the elasticity with which it is connected to the rest of the moving body determine the amount of energy each contact can dissipate. We speculate that when the connection between apex and body is made more rigid, e.g., by making the contacting surfaces nearly perfectly flat, the motion of the apices should become less strongly damped and the average friction force should reduce. The other extreme would be to pattern surfaces deliberately, for example in the form of nano-pillar arrays, to pre-define \({N_{{\text{dyn}}}}\) and \({k_{{\text{dyn}}}}\) in order to make the apex motion highly overdamped, which would maximize the friction force.

## Footnotes

## Notes

### Acknowledgements

The authors gratefully acknowledge support from the European Research Council through the ERC Advanced Grant project Science F(r)iction and from the Netherlands Organisation for Scientific Research (NWO), through the FOM-Program on Fundamental Aspects of Friction. Prof. André Schirmeisen (Justus-Liebig University, Gießen, Germany) is gratefully acknowledged for providing us with the data of panels (a) and (b) in Fig. 1.

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