A generalized friction law depicting the thermal effects at chemical bonding interface

Fig. 1Simulation setup and calculation of $T_{\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}$ . (a) Simulation model of two rough surfaces. Blue balls represent hydrogen atoms, while orange and red ones represent carbon atoms in the upper and lower substrates, respectively. (b) Temperature distributions of a typical friction simulation with $T_{\mathrm{s}\mathrm{u}\mathrm{b}}$ = 300 K, $F_{\mathrm{N}}$ = 675 nN, and $v$ = 100 m/s. (c) $T_{\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}$ as a function of $T_{\mathrm{s}\mathrm{u}\mathrm{b}}$ under the same $F_{\mathrm{N}}$ and $v$ as in (b).

Fig. 2Friction simulation results. (a) $F_{\mathrm{f}}$ , (b) $N_{\mathrm{i}\mathrm{b}}$ , and (c) $F_{\mathrm{i}\mathrm{b}}$ as a function of $T_{\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}$ with $F_{\mathrm{N}}$ = 675 nN and $v$ = 100 m/s.

Fig. 3 $F_{\mathrm{i}\mathrm{b}}$ as a function of $T_{\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}$ under different $v$ values predicted by PTT model. Parameters $F_{\mathrm{c}}$ and $b$ are 2.06 nN and 0.78, respectively.

Fig. 4Validation of load and velocity effects. (a) $N_{\mathrm{i}\mathrm{b}}$ as a function of $F_{\mathrm{N}}$ with $T_{\mathrm{s}\mathrm{u}\mathrm{b}}$ = 300 K and $v$ = 100 m/s. (b) $F_{\mathrm{i}\mathrm{b}}$ as a function of $v$ with $T_{\mathrm{s}\mathrm{u}\mathrm{b}}$ = 300 K and $F_{\mathrm{N}}$ = 675 nN; note the use of a logarithmic coordinate for $v$ (abscissa).

Fig. 5AFM friction experimental results of DLC. (a) Experimental setup for friction measurements. All experiments were performed under UHV conditions using a DLC-coated AFM tip against a DLC-coated silicon disk that was in contact with a temperature-controllable stage. The inset of (a) shows a typical topography image of post-friction DLC surface (with a root mean squared roughness of approximately 0.8±0.1 nm) at a temperature of 255.0±0.1 K, an applied normal force ( $F_{\mathrm{N}}$ ) of 6.8 nN, and a scanning velocity ( $v$ ) of 500.0 nm/s; corresponding measurements of lateral force signals and friction force ( $F_{\mathrm{f}}$ ) are given in (b). The blue line indicates the scanning direction from left to right, while red one corresponds to the opposite direction. (c) Measured $F_{\mathrm{f}}$ as a function of temperature under constant $v$ of 50.0 nm/s and different $F_{\mathrm{N}}$ values of 1.4–15.0 nN. (d) Measured $F_{\mathrm{f}}$ as a function of temperature under constant $F_{\mathrm{N}}$ of 6.8 nN and different $v$ of 650–10,000 nm/s.

Fig. 6Friction simulation results of a flat-surface contact model. (a) Flat-surface contact model of two hydrogen-passivated DLC substrates. (b, c) $F_{\mathrm{f}}$ results as a function of $T_{\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}$ under different $v$ of 1 and 10 m/s, respectively. $F_{\mathrm{N}}$ is set to 75 nN for all simulations. Gray region indicates approximate range of $T_{\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}$ , from which mean and error bars of $T_{\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}$ can be estimated.

Fig. 7Model vs. data. (a) MD simulation and MM model fitting results of $F_{\mathrm{i}\mathrm{b}}$ – $T_{\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}$ for rough-surface contact. (b) Summarized results of $T_{\mathrm{m}\mathrm{a}\mathrm{x}}$ as a function of sliding velocity. Reference data of Si/Si and Si/NaCl were taken from Refs. [ 30 , 31 ], respectively.