Reexamination of Damping in Sliding Friction

Friction is responsible for about one-third of the primary energy consumption in the world. So far, a thorough atomistic understanding of the frictional energy dissipation mechanisms is still lacking. The Amontons’ law states that kinetic friction is independent of the sliding velocity while the Prandtl-Tomlinson model suggests that damping is proportional to the relative sliding velocity between two contacting objects. Through careful analysis of the energy dissipation process in atomic force microscopy measurements, here we propose that damping force is proportional to the tip oscillation speed induced by friction. It is shown that a physically well-founded damping term can better reproduce the multiple peaks in the velocity-dependent friction force observed in both experiments and molecular dynamics simulations. Importantly, the analysis gives a clear physical picture of the dynamics of energy dissipation in different friction phases, which provides insight into long-standing puzzles in sliding friction, such as velocity weakening and spring-stiffness-dependent friction.

Figure 1

Schematic illustration of (a) an AFM used to measure friction force as its tip slides over a ${\mathrm{MoS}}_2$ substrate. (b) A simplified friction model for the friction force measurement with an AFM, which describes the energy dissipation process in sliding friction. (c) The measured friction force (dots) and the best fitting with the classical PT model (solid lines). (d) The measured friction force and the fitting with the proposed phononic friction (PF) model. The scan size in the experiments is $5\times 5\mathrm{\mu }{\mathrm{m}}^2$. The periodic length along the sliding direction is determined to be $a=0.316\mathrm{nm}$. Note that $f_r$ in (c) and (d) stands for the torsional resonant frequency ($f_r$) of the cantilever, which depends on the sliding velocity because the contact stiffness varies with the sliding velocity. This implies that the tip-substrate system at each sliding velocity has a specific resonant frequency $f_r$.

Figure 2

The instantaneous friction force calculated from the phononic friction model [Eq. (6)] and its corresponding FFT spectrum at four representative sliding velocities: (a),(b) $v_s=12\mathrm{\mu }\mathrm{m}/\mathrm{s}$, (c),(d) $v_s=31\mathrm{\mu }\mathrm{m}/\mathrm{s}$, (e),(f) $v_s=89\mathrm{\mu }\mathrm{m}/\mathrm{s}$, (g),(h) $v_s=109\mathrm{\mu }\mathrm{m}/\mathrm{s}$. The insets in the right panel [(b) and (d)] stand for the FFT spectrum of the ringing oscillation. In the FFT figure, the green dotted lines represent the washboard frequency $f_{\mathrm{wb}}$ and $n{f}_{\mathrm{wb}}$ ($n=1,2,3\dots$) at the corresponding velocity.

Figure 3

Energy dissipation in the slip (gray dots), oscillation (green dots), and stick (orange dots) phases at various sliding velocities. Total dissipated energy is also presented with black dots for indication.