Atomic stick-slip friction as a two-dimensional thermally activated process

Widely recognized as a thermally activated process, atomic stick-slip friction has been typically explained by Prandtl-Tomlinson model with thermal activation. Despite the limited success, theoretical predictions from the classic model are primarily based on a one-dimensional (1D) assumption, which is generally not compatible with real experiments that are two-dimensional (2D) in nature. In this letter, a theoretical model based on 2D transition state theory has been derived and confirmed to be able to capture the 2D slip kinetics in atomic-scale friction experiments on crystalline surface with a hexagonal energy landscape. Moreover, we propose a reduced scheme that enables extraction of intrinsic interfacial parameters from 2D experiments approximately using the traditional 1D model. The 2D model provides a theoretical tool for understanding the rich kinetics of atomic-scale friction or other phenomena involving higher dimensional transitions.

Figure 1

(a) A schematic showing the simulation setup. (b) Lateral force $f_x$ (upper panel) and axial force $f_y$ (lower panel) images for $\theta ={15}^{\circ }$. The gray dots visualize the periodic sites. (c) Lateral force $f_x$ (upper panel) and axial force $f_y$ (lower panel) traces for the white dashed lines marked in (b). The red circles indicate the extracted slip forces. (d) 1D histograms of the slip force $f_x$ for $\theta =0^{\circ }$ (black), $\theta ={15}^{\circ }$ (red) and $\theta ={30}^{\circ }$ (green). (e) 2D histogram of slip force for $\theta =0^{\circ }$ (left), $\theta ={15}^{\circ }$ (mid) and $\theta ={30}^{\circ }$ (right). The dotted lines are visual guidelines to highlight the distribution shapes.

Figure 2

(a) Zoom-in view of the energy landscape near a minimum point. The dashed curve represents $\left\{{\mathbf{x}}_{\mathrm{c}}\right\}$. (b) A contour plot showing the energy barrier V as a function of $\mathbf{f}$ calculated by Eq. (1). The dashed curve represents $\left\{{\mathbf{f}}_{\mathrm{c}}\right\}$. (c) A contour plot showing the probability $p$, at which a slip does not occur. The dashed curve represents $\left\{{\mathbf{f}}_{\mathrm{c}}\right\}$. (d) 2D histogram of slip forces for $\theta =0^{\circ }$ (left panel), $\theta ={15}^{\circ }$(middle panel) and $\theta ={30}^{\circ }$(right panel). The dashed curves represent $\left\{{\mathbf{f}}_{\mathrm{c}}\right\}$ and the dotted lines are guidelines to visually show the trend. (e) 1D histograms of slip force $f_x$ if all slip events in (d) are counted.

Figure 3

(a) Reduced critical slip force $f_{\mathrm{c}}$ as a function of $\theta$ when $f_y=0$. The lower panel shows an enlarged view between 0 to 60º. (b) Energy corrugation shape factor $\beta$ as a function of $\theta$ when $f_y=0$. The lower panel shows an enlarged view between 0 to 60º. (c) Reduced histograms of $f_x$ when $f_y=0$ for $\theta =0^{\circ }$ (black), $\theta ={15}^{\circ }$(red) and $\theta ={30}^{\circ }$(green) from 2D theoretical prediction. (d) Reduced histograms of $f_x$ when slip events with $f_y$ in the range of $0\pm 0.01\mathrm{nN}$ for $\theta =0^{\circ }$(black), $\theta ={15}^{\circ }$(red) and $\theta ={30}^{\circ }$(green) from 2D Langevin simulation.

Figure 4

(a) A schematic showing the experimental setup. (b) 2D slip force histograms for $\theta =0^{\circ }$(left panel), $\theta ={15}^{\circ }$(middle panel) and $\theta ={30}^{\circ }$(right panel). The dashed lines are guidelines to visually highlight the trends. (c) Histograms of slip force $f_x$ for $\theta =0^{\circ }$(black), $\theta ={15}^{\circ }$(red) and $\theta ={30}^{\circ }$(green) when all slip events are counted. (d) Reduced histograms of $f_x$ when slip events with $f_y$ in the range of $0\pm 0.1\mathrm{nN}$ for $\theta =0^{\circ }$(black), $\theta ={15}^{\circ }$(red) and $\theta ={30}^{\circ }$(green).