Velocity dependence of atomic-scale friction: A comparative study of the one- and two-dimensional Tomlinson model

We present a comparative analysis of the velocity dependence of atomic-scale friction for the Tomlinson model, at zero and finite temperatures, in one and two dimensions, and for different values of the damping. Combining analytical arguments with numerical simulations, we show that an appreciable velocity dependence of the kinetic friction force $F_{\mathrm{fric}}$, for small scanning velocities $v_s$ (from $1\mathrm{nm}∕\mathrm{s}\text{to}\mu \mathrm{m}∕\mathrm{s}$), is inherent in the Tomlinson model. In the absence of thermal fluctuations in the stick-slip regime, it has the form of a power-law $F_{\mathrm{fric}}-F_0\propto v_s^{\beta }$ with $\beta =2∕3$, irrespective of dimensionality and value of the damping. Since thermal fluctuations enhance the velocity dependence of friction, we provide guidelines to establish when thermal effects are important and to which extent the surface corrugation affects the velocity dependence.

Figure 1

Frictional force $F_{\mathrm{fric}}$ as a function of sliding velocity $v_s$ in the 1D Tomlinson model, plotted on a linear (a) and on a log-log scale (b) for $V_0=1\mathrm{eV}$, $m={10}^{-10}\mathrm{kg}$, $K_x=10\mathrm{N}∕\mathrm{m}$, $a_x=0.316\mathrm{nm}$ $({\tilde{V}}_0=7)$, and $\eta =2\sqrt{K_x∕m}\simeq 6.3\times {10}^5{\mathrm{s}}^{-1}$. The increase of $F_{\mathrm{fric}}$ for small velocities is hidden using a log-log scale. The dotted line in (a) is a power-law fit to the data of the form $F_{\mathrm{fric}}-F_0\propto v_s^{2∕3}$ for $v_s<2\mu \mathrm{m}∕\mathrm{s}$.

Figure 2

Tip position as a function of support position in the 1D Tomlinson model for different values of the scanning velocity (from left to right $v_s=1.5$, 15, 300, $750\mathrm{nm}∕\mathrm{s}$, $1.5\mu \mathrm{m}∕\mathrm{s}$, $\eta =2\sqrt{K_x∕m}$, and ${\tilde{V}}_0=7$. The inset is a blow up of the region around the first slip event.

Figure 3

Slip time as a function of scanning velocity in the 1D Tomlinson model for critical damping and ${\tilde{V}}_0=7$. The points connected by the solid line are obtained by numerical simulations, while the dotted line is a power-law fit to the data of the form $\delta t_s\propto v_s^{-1∕3}$.

Figure 4

Frictional force $F_{\mathrm{fric}}$ as a function of sliding velocity $v_s$ in the 1D Tomlinson model for ${\tilde{V}}_0=7$ and different values of the damping parameter: from bottom to top $\eta ∕\left(\sqrt{k_x∕m}\right)=0.4$, 1.5, 2, 10, 100. The dotted lines are fit to the numerical data of the form $F_{\mathrm{fric}}-F_0\propto v_s^{\beta }$, with $\beta =2∕3$. In the most underdamped case (lower line) the friction force is lower because the tip performs jumps of two lattice parameters.

Figure 5

Tip velocity as a function of support position in the 1D Tomlinson model for different scanning velocities (from left to right $v_s=1.5$, 15, 300, $750\mathrm{nm}∕\mathrm{s}$) in the overdamped case $(\eta =100\sqrt{k_x∕m})$ and for ${\tilde{V}}_0=7$. The horizontal line is the value $2\pi V_0∕\left(m\eta a_x\right)$.

Figure 6

Lateral force as a function of tip position for two values of the damping parameter: critically damped $\eta =2\sqrt{k_x∕m}$ (solid line) and underdamped $\eta =0.4\sqrt{k_x∕m}$ (dashed line). The reduced corrugation is ${\tilde{V}}_0=7$ and the scanning velocity $v_s=300\mathrm{nm}∕\mathrm{s}$. Notice the presence of jumps with periodicity $2a_x$ in the underdamped case. The upper and lower horizontal lines indicate the friction force for $\eta =2\sqrt{k_x∕m}(F_{\mathrm{fric}}=2.33\mathrm{nN})$ and $\eta =0.4\sqrt{k_x∕m}(F_{\mathrm{fric}}=1.01\mathrm{nN})$, respectively. The dotted line represents $F_x=(2\pi V_0∕a_x)\sin (2\pi x∕a_x)$, as obtained from Eq. (26a).

Figure 7

Friction force as a function of scanning velocity in 1D (upper curve) and 2D Tomlinson model, for critical damping, ${\tilde{V}}_0=7$ and different values of $y_s$ (from bottom to top $y_s=0.274$, 0.137, 0.1, and $0.05\mathrm{nm}$).

Figure 8

Trajectory of the tip in the 2D Tomlinson model for critical damping, ${\tilde{V}}_0=7$ and $v_s=7.5\mathrm{nm}∕\mathrm{s}$. The circles connected by the solid line indicate the positions of the tip in the $xy$ plane during the dynamics. The regions where the distribution of points is denser are the sticking domains, where the tip stays predominantly for most of the time. Note that the tip slips from one sticking domain to the other following a zig-zag pattern around the scanning direction (indicated by the dashed line, $y_s=0.137\mathrm{nm}$).

Figure 9

Lateral force as a function of tip position in the 1D Tomlinson model for critical damping, $T=300\mathrm{K}$ and ${\tilde{V}}_0=7$, for different scanning velocities (nonsolid lines from bottom to top $v_s=1.5$, 15, 300, $750\mathrm{nm}∕\mathrm{s}$). The solid line represents $F_x=(2\pi V_0∕a_x)\sin (2\pi x∕a_x)$, as obtained from Eq. (26a) (see also Fig. 6). The inset shows a blow up of the region around a slip event.

Figure 10

Velocity dependence of friction force in the 1D Tomlinson model at $T=0$ (upper curve) and $T=300\mathrm{K}$ (lower curve) for critical damping and ${\tilde{V}}_0=7$. The solid line is a fit of the data for $T=300\mathrm{K}$, using Eq. (42) in the small velocity regime $(v_s<2\mu \mathrm{m}∕\mathrm{s})$.

Figure 11

Velocity dependence of friction force in the 1D (upper curve) and 2D (lower curve) Tomlinson model for $T=300\mathrm{K}$, critical damping and ${\tilde{V}}_0=7$. The solid lines are fits to the data using Eq. (42) in the small velocity regime $(v_s<2\mu \mathrm{m}∕\mathrm{s})$.

Figure 12

Total potential energy $V_{\mathrm{tot}}$ as a function of tip position $x$ for three values of the cantilever position $x_s$ (from bottom to top $x_s=0.287$, 0.382, $0.413\mathrm{nm}$). The horizontal lines indicate the values of the minimum $\left(V_{\min }\right)$ and the maximum $\left(V_{\max }\right)$ of the potential for each curve. The potential barrier is $\Delta E=V_{\max }-V_{\min }$. The upper curve corresponds to $\Delta E=0$, while the middle curve to the case where $\Delta E\simeq k_{B}T$.

Figure 13

Friction force as function of scanning velocity for ${\tilde{V}}_0=2$, ${\tilde{V}}_0=4$, and ${\tilde{V}}_0=7$. The filled circles connected by the dotted lines are the data for $T=0$, while the open circles connected by the dashed lines correspond to the data for $T=300\mathrm{K}$. The solid lines are fits to the data at $T=300\mathrm{K}$, according to Eq. (44), with exponent $\alpha =0.37$ for ${\tilde{V}}_0=2$, $\alpha =0.56$ for ${\tilde{V}}_0=4$, and $\alpha =0.67\simeq 2∕3$ for ${\tilde{V}}_0=7$. The minimum value of the scanning velocity in the plot is $v_s=1.5\mathrm{nm}∕\mathrm{s}$.