Reinterpretation of velocity-dependent atomic friction: Influence of the inherent instrumental noise in friction force microscopes

We have applied both the master equation method and harmonic transition state theory to interpret the velocity-dependent friction behavior observed in atomic friction experiments. To understand the discrepancy between attempt frequencies measured in atomic force microscopy experiments and those estimated by theoretical models, both thermal noise and instrumental noise are introduced into the model. It is found that the experimentally observed low attempt frequency and the transition point at low velocity regimes can be interpreted in terms of the instrumental noise inherent in atomic force microscopy. In contrast to previous models, this model also predicts (1) the existence of a two-slope curve of velocity dependence and (2) the decrease of critical velocity with temperature, which provides clues for further experimental verification of the influence of instrumental noise in friction measurements.

Figure 1

Graphical depiction of the one-dimensional PT model. In this model, an AFM tip having nanoscale dimensions (represented by brown circles) is dragged by a support to slide on an atomically flat and periodic substrate. The total potential energy of the system is represented by the gray curve and evolves with time as the tip slides over the surface. The twisting motion of the cantilever is depicted by the spring attached between the support and the AFM tip. The support provides the scanning motion of the tip along the surface.

Figure 2

(a) Numerical result showing the variation of the friction force as a function of sliding velocity and temperature when only thermal noise is considered. (b) Numerical result showing the variation of the friction force as a function of velocity and temperature when both thermal noise and instrumental noise are considered. The instrumental noise associated with an effective temperature of $T_{\mathrm{eff}}=1200$ K at a frequency of $f_{\mathrm{in}}$ = 10 kHz is used.

Figure 3

Velocity dependence of atomic friction with both thermal and instrumental noise at different temperatures. Instrumental noise has the parameters $f_{\mathrm{in}}=10$ kHz and $T_{\mathrm{eff}}=1200$ K, while the temperature, or the thermal noise, is allowed to vary at a constant attempt frequency. A transition point exists in the friction vs velocity curve as indicated by an arrow, where the dominant friction reducing mechanism transitions from mechanical noise to thermal noise. The attempt frequency for thermal activation $\left(f_{\mathrm{att}}\right)$ in all cases was 1 GHz.

Figure 4

Velocity dependence of atomic friction when both thermal an instrumental noise are accounted for. (a) The frequency of the instrumental noise varied (0.1, 1, 10, and 100 kHz), but a constant $T_{\mathrm{eff}}$ = 2000 K maintained. Arrows mark the points at which the frequency of the mechanical noise matches the frequency at which the tip traverses the surface atoms. (b) The amplitude of the instrumental noise varied ($T_{\mathrm{eff}}=0$, 1200, and 2000 K), but a constant frequency of 10 kHz maintained. In both (a) and (b), the real temperature $\left(T\right)$ is 300 K and the attempt frequency $\left(f_{\mathrm{att}}\right)$ is 1 GHz.

Figure 5

Variation of friction with temperature in the presence of instrumental noise whose amplitude depends on the sample temperature. The instrumental noise has a frequency $\left(f_{\mathrm{in}}\right)$ of 10 kHz, $f_{\mathrm{att}}=1$ GHz, and the amplitude varies such that at $T>300$ K, $T_{\mathrm{eff}}=0$ K and for $T<300$ K, $T_{\mathrm{eff}}=4(300-T)$. Other parameters used in this thermally activated PT model are the same as in the other graphs.