Appl. Phys. A 66, S263–S267 (1998)
Applied Physics A Materials Science & Processing Springer-Verlag 1998
The velocity dependence of frictional forces in point-contact friction O. Zwörner, H. Hölscher∗ , U. D. Schwarz, R. Wiesendanger Institute of Applied Physics and Microstructure Research Center, University of Hamburg, Jungiusstr. 11, D-20355 Hamburg, Germany (E-mail:
[email protected]) Received: 25 July 1997/Accepted: 1 October 1997
Abstract. The velocity dependence of point-contact friction is studied by means of friction force microscopy of different carbon compounds (diamond, graphite, amorphous carbon). The measured frictional force is found to be constant over a wide range of sliding velocities [nm/s to µm/s]. This result is substantiated by a simple mechanical model, where the frictional forces are shown to be constant at sliding velocities well below the slip velocity.
In this study, we examine the validity of law (iii) for point contacts by using a friction force microscope. It is found that the frictional forces are independent of the sliding velocity in a wide range. This result is substantiated by a simple mechanical model based on the Tomlinson [6] and the independent oscillator [7, 8] models. 1 Experiment 1.1 Samples
Although frictional phenomena are familiar from daily life and the study of friction is one of the oldest topics in physics because of its technological relevance [1], there has been only little success in deriving an exact description of friction since the time the following phenomenological laws of dry friction were established by Amontons and Coulomb 200 years ago. These laws are: (i) the frictional force is proportional to the normal load; (ii) the frictional force is independent of the (apparent) contact area of the sliding surfaces; (iii) sliding friction is independent of the sliding velocity. These three laws of friction hold surprisingly well at the macroscopic scale, but cannot be explained by first principles. Nevertheless, with the advent of new experimental tools such as the surface force apparatus, the quartz crystal microbalance, and the friction force microscope, the expanding field of nanotribology has been established where the tribological properties of contacts with well-defined geometries are studied on the nanometer scale [2]. By using the friction force microscope (FFM) [3], which is an extension of the scanning force microscope (SFM) [4], it is possible to examine the frictional properties of an (approximate) point contact at the nanometer scale. The observed frictional behavior differs significantly from the behavior expected from the macroscopic friction laws introduced above. In particular, it is found that frictional forces are proportional to the true area of contact, which is generally not proportional to the loading force [5]. Consequently, the laws (i) and (ii) are no longer valid at the nanometer scale. ∗ Corresponding
author
In order to investigate the velocity dependence of frictional forces in point-contact friction, three different modifications of carbon were studied as examples. (i) A diamond film deposited by chemical vapor deposition on a silicon (001) substrate which exhibited crystallites with flat, {100} oriented surfaces. The surfaces of individual crystallites were typically several 100 nm2 up to somewhat more than 1 µm2 . (ii) Highly oriented pyrolytic graphite (HOPG) with 99.99% purity [9]. The sample was cleaved each time right before measurement along its (0001) plane by using Scotch tape. (iii) Films of amorphous carbon prepared by electron-beam evaporation of highly purified graphite (purity better than 99.99% [9]) on a mica substrate. The mica substrate was held at room temperature during evaporation; films with thicknesses of ≈ 400 Å were produced. The corrugation of these films was found to be less than 1 nm. More details about the samples, including a detailed discussion of their frictional behavior at different loads, can be found in [5]. 1.2 Experimental setup A commercial atomic force microscope [10] working in the beam-deflection mode [11] and operated in ambient air was used to record the velocity dependence of frictional forces between these materials and the tip of a rectangular silicon
4 2
amorphous carbon
0 0.0
0.5 1.0 1.5 2.0 Scan velocity [µm/s]
40 30 20 10 0
d)
diamond 0
5 10 Scan velocity [µm/s]
15
60 40 20
amorphous carbon 0
b)
e)
0
5
10
15
20
25
Scan velocity [µm/s]
Frictional force [nN]
6
Frictional force [nN]
8
80
20 15 10 5
diamond 0 0.0
c)
0.5
1.0
1.5
2.0
Scan velocity [µm/s]
8
6 4 2 0 -2 -4
HOPG
-6
Frictional force [nN]
Frictional force [nN]
a)
10
Frictional force [nN]
Frictional force [nN]
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4 0 -4
HOPG -8
0
1 2 3 Scan velocity [µm/s]
f)
0
1
2
3
4
Scan velocity [µm/s]
Fig. 1a–f. The measured frictional forces as a function of the sliding velocity. These forces are found to be independent of the sliding velocity. The exact experimental parameters can be found in Table 1
Table 1. Experimental parameters of the data sets presented in Fig. 1. The sliding distance was 1 µm in each direction of the friction loop for all measurements Material
Fig. No.
No. of data points
vmin [µm/s]
vmax [µm/s]
Fn /nN
F fric /nN
1a 1b 1c 1d 1e 1f
150 89 80 60 170 236
0.1 0.2 0.2 0.78 0.1 0.02
2.0 24.4 2.0 12.2 3.39 3.81
5.2 83.4 3.3 33.0 26.3 8.1
6.0 44.9 12.5 24.4 0.01 0.1
amorphous carbon amorphous carbon diamond diamond HOPG HOPG
cantilever [12]. Prior to measurements, the tips were exposed to the electron beam of a transmission electron microscope to round their tip ends and obtain geometrically well-defined, spherical tip apexes. After exposure, the apex radii could be determined very accurately with nanometer precision from the electron micrographs. The exact procedure for the preparation of the spherical tip apexes is described in [13]. Within an individual set of experiments, we have analyzed ten friction loops for each sliding velocity according to the calibration procedure described in [14], each resulting in an individual data point. The sliding distance within the individual friction loops was 1 µm in each direction in all cases. The forces Fn normal to the plane of contact were held constant for each set of data points shown below. The normal forces were calibrated to be zero at the point where the cantilever leaves the surface (jump-off point) [14].
R/nm 62 23 23 62 62 35
± ± ± ± ± ±
4 2 2 4 4 3
and 24.4 µm/s, i.e.about 2 orders of magnitude were covered. Normal forces between 3.3 nN and 83.4 nN were applied, and tips with apex radii between 23 nm and 62 nm were used. The exact experimental parameters are given in Table 1. It is found that under all these different experimental conditions and for all materials investigated, the frictional force Ffric is constant to a good approximation. Additionally, it should be noted that the “negative friction” observed for several data points in the case of HOPG is due to noise in the experimental setup during the measurement. This can occur because friction is negligibly small for HOPG in a large range of loading forces [5] and obviously also in a large range of sliding velocities. The average frictional force, however, was always positive. 2 A simple model of friction
1.3 Experimental results
2.1 The model
Figure 1 shows the velocity dependence of frictional forces of amorphous carbon, diamond and HOPG. In these measurements, the sliding velocities were varied between 0.02 µm/s
The velocity independence of the frictional forces can be illustrated by using a simple mechanical model based on the Tomlinson [6] and independent oscillator model [7, 8].
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b)
M
“static” case, it can be shown that the tip movement changes dramatically if the condition
vM
cx < −
cx
slip
V(xt) xM
}
xt
x
a
Fig. 2. a A simple model for a tip sliding on an atomically flat surface. xt represents the position of the tip, which is coupled elastically to the body M by a spring with spring constant cx in the x direction and to an external potential V(xt ) with the periodicity a. If xt = xM , the spring is in its equilibrium position. For sliding, the body M is moved with the velocity vM in the x direction. b A schematic view of the tip movement in the interaction potential. If condition (2) holds, the tip shows the typical “stick-slip”-type movement, i.e. it “jumps” from one potential minimum to another
A schematic view of the model is shown in Fig. 2. A tip is coupled elastically to the main body M with a spring possessing a spring constant cx in the x direction and interacts with the sample via the periodic interaction potential V(x t ), where x t represents the position of the tip. Sliding is performed by moving the microscope body with the sliding velocity vM . All energy dissipation during sliding – whether it is due to phonons or due to electronic excitations – is considered by a simple damping term that is proportional to the sliding velocity. This simplification of the atomic-scale energy dissipation process is justified by the assumption that there is no energy dissipation if the tip sticks (vt ≈ 0), but energy dissipation will occur as soon as the tip moves. Within the present model, the point-like tip represents the average of the real tip–sample contact where up to thousands of atoms can be involved. In principle, the problem of the tip motion on a flat sample can be treated as a system consisting of a finite number of coupled or uncoupled oscillators, leading to more complex models of friction [15]. Nevertheless, models similar to the one presented above have been successfully applied to explain the motion of an SFM tip on a sample surface [16–19]. With these assumptions, the equation of motion for the tip in a sinusoidal interaction potential becomes 2π 2π sin x t − γx x˙t , (1) m x x¨t = cx (x M − x t ) − V0 a a
∂2 V . ∂x t2
(2)
is fulfilled [8, 16]. Then, the tip moves discontinuously in a “stick-slip”-type movement over the sample surface and friction occurs. Otherwise, no friction occurs within this static approach. For large velocities (vM 0), the tip movement is dominated by the viscous damping term and the frictional force occurring is proportional to the sliding velocity (Ffric ≈ γx vM ) [8]. 2.2 Calculation procedure At this point we are interested in the solutions of (1) for moderate sliding velocities (vM > 0). For the explicit calculation, a suitable set of realistic parameters m x , cx , V0 , a, and γx has to be found. Unfortunately, exact values of these parameters cannot be deduced from experiments; they can only be estimated. Nevertheless, the qualitative results do not change within a wide range of the parameters if the right order of magnitude is chosen. All data presented here is obtained with the following set of parameters: V0 = 1.0 eV, a = 3 Å, cx = 10 N/m [condition (2) holds for these parameters, i.e. stick-slip will occur], m x = 10−10 kg [with this
12 vM = 10 nm/s vM = 1 µm/s vM = 10 µm/s vM = 100 µm/s
10 8
Fx [nN]
a)
6 4 2
0 140 120
vt [µm]
100
where m x is the effective mass of the system, x M = vM t the equilibrium position of the spring, a the lattice spacing, and γx the damping constant. The solution of this differential equation is the path of the tip x t (t). The lateral force Fx to move the tip in the x direction can be calculated from Fx = cx (x M − x t ), whereas the frictional force Ffric is identified as the lateral force averaged over time hFx i [16] or the dissipated energy per unit length [8]. In the case of slow (vM → 0) or large (vM 0) sliding velocities, analytical approximations of (1) can be derived. If the position of body M is changed with very low scan velocities (vM → 0), the tip will be always in its stable equilibrium position and (1) can be solved for (x¨t = 0, x˙t = 0). For this
80 60 40 20 0 −20
0
3
6
9
12
15
18
21
xM [Å] Fig. 3. The lateral force Fx and the tip velocity vt for different sliding velocities vM = 10 nm/s to 100 µm/s
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effective √ mass, the system has a resonance frequency of 1/(2π) cx /m x ≈ 50 kHz, √ which is typical for SFM experiments [20]], and γx = 2 cx m x (critical damping is assumed). With this set of parameters, we calculate the numerical solutions of (1) and determine the lateral forces Fx and the frictional force Ffric for different sliding velocities. 2.3 Results and discussions In Fig. 3, the calculated lateral force Fx is displayed for different sliding velocities vM . It is obvious that the results obtained change only little within the wide range of sliding velocities vM = 10 nm/s to 10 µm/s. For higher sliding velocities (see Fx for 100 µm/s in Fig. 3), the lateral force increases significantly since the solution of (1) is now dominated by the velocity-dependent damping term, as already mentioned in Sect. 2.1 for vM 0. Consequently, since the frictional force is the average of the lateral force over time, Ffric is approximately independent of the sliding velocity in a wide range, as shown in Fig. 4. What is now the reason for the relative independence of the lateral forces from the sliding velocities at moderate values of vM ? This issue is illustrated in Fig. 2b. We have seen above that the tip usually slides with a “stick-slip”-type movement over the sample surface. With this type of movement, the tip stays for most of the time in the minima of the interaction potential where it slides very slowly or “sticks”. Therefore, almost no energy is dissipated in this “stick” state since we assumed that the damping is proportional to the sliding velocity. In the “slip” state, however, the tip “jumps” from one minimum to another. During this “jump”, the tip reaches very high peak velocities and dissipates significant amounts of energy because of the assumed velocity-dependent damping mechanism. The consequence of the issues discussed above is that the total amount of energy dissipated during sliding does not
change significantly as long as the sliding velocity vM is much smaller than the slip velocity of the tip. From Fig. 3, we obtain a slip velocity of about 60 µm/s for sliding speeds up to 10 µm/s. For larger sliding velocities, the mechanism of energy dissipation through the “stick-slip” effect breaks down, as can be seen for vM = 100 µm/s, and Ffric is proportional to γx vM (cf. Fig. 4). It is important to mention that the numerical value of the slip velocity obtained within this simple mechanical model (60 µm/s) is determined by the mathematical forms of the used velocity-dependent damping and the one-dimensional interaction potential, and finally by the numerical values of the chosen parameters. Qualitatively similar results will be obtained with other choices. Consequently, the resulting slip velocity is not a prediction for an experimental measurement and could actually be much higher without contradicting the general mechanism proposed in the present model. 3 Summary We have presented a study on the velocity dependence of point-contact friction. It was found that frictional forces are independent of the sliding velocity even on the nanometer scale for all experimentally realised sliding velocities. Consequently, friction law (iii) holds within a wide range of sliding velocities although the macroscopic friction laws (i) and (ii) are no longer valid on the nanometer scale. This behavior can be illustrated by a simple mechanical model, which shows that the frictional forces are independent of the sliding velocity as long as the slip movement of the tip is faster than the sliding velocity. Acknowledgements. We are indebted to A. Schwarz, W. Allers, J. Müller, H. Bluhm, and K.L. Johnson for helpful discussions, and to K. Schiffmann for supplying the diamond sample. Financial support from the Graduiertenkolleg “Nanostrukturierte Festkörper” and the Deutsche Forschungsgemeinschaft (Grant No. Wi 1277/2-2) is gratefully acknowledged.
100 Ffric γx vM
References g pin dam
1
us
10
visc o
Ffric [nN]
Ffric(vM −> 0) + γx vM
10
−2
10
−1
0
10 10 vM [µm/s]
1
10
2
10
3
Fig. 4. The frictional force Ffric = hFx i as a function of the sliding velocity vM . For low sliding speeds, the frictional force is determined by the energy dissipation occurring during the slip. Therefore, Ffric is nearly constant in this range. With increasing sliding velocity, the frictional force is just proportional to vM owing to the viscous damping Ffric ∼ γx v M which is indicated by the dashed line
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