Two-dimensional simulation of superlubricity on NaCl and highly oriented pyrolytic graphite

The friction between an atomically sharp tip and a solid surface (NaCl and highly oriented pyrolytic graphite) is analyzed theoretically in the framework of a modified Tomlinson model in two dimensions. Lateral forces are studied as a function of temperature, load, and magnitude of actuation. The actuation leads to a reduction in friction and allows one to enter a dynamic superlubricity regime. In addition, our model is able to describe other ultralow friction states as static superlubricity and thermolubricity. We find a good agreement between the calculations and the experimental results.

Figure 1

(Color online) Potential-energy surface of (a) NaCl and (b) HOPG with a side length of 1.2 nm. The unit-cell parameter is 0.564 nm for NaCl and $0.246\times 0.426{\text{nm}}^2$ for HOPG. In NaCl only one sort of atom is visible, depending on the tip apex. HOPG has a trigonal lattice because only the potential minima are detected (in the experiment).

Figure 2

(Color online) Contour plot of the potential $V_{\text{NaCl},\text{HOPG}}\left(x,y\right)$ for (a) NaCl with $E_0=1\text{eV}$ and (b) HOPG with $E_0=0.5\text{eV}$. The horizontal dashed line indicates the scanning line with (a) $y_{\text{sup}}=0.2a$ and (b) $0.4a$. Gray regions correspond to ${\lambda }_{1,2}>0$. A superposition of the gray regions with the tip trajectory (red), where $\partial V/\partial y=0$, leads to the analytical stick-slip motion of the tip and is in very good agreement with the numerical solution at 0 K (black points connected by lines).

Figure 3

(Color online) Calculated lateral force map of NaCl (forward scan direction) for different spring constants in the $y$ direction, (a) $k_{y\text{-eff}}=1$, (b) 2, and (c) 10 N/m, whereas the spring constant in the $x$ direction $k_{x\text{-eff}}=2\text{N}/\text{m}$ is retained. The movement of the tip is heavily damped; thus an overshooting or jump over two lattice constants is avoided and the stick-slip behavior is sustained. The lateral force maps reveal different patterns; however a good agreement with experiment is only achieved when $k_{x\text{-eff}}\approx k_{y\text{-eff}}$. The amplitude of the corrugation potential is $E_0=1\text{eV}$.

Figure 4

(Color online) Average friction force ${\overline{F}}_L$ for NaCl versus amplitude $E_0$ of the corrugation potential. The amplitude $E_0$ increases with the externally applied load in the experiment and shows a linear relation to the average friction force ${\overline{F}}_L$. However below a critical threshold of $E_0$, the linear relation goes over into an ultralow friction state where the average friction force ${\overline{F}}_L\approx 0$. In two dimensional the tip avoids passing the maximum of the corrugation potential, which leads to a zig-zag movement, and thus the average friction force ${\overline{F}}_L$ is lowered compared to a one-dimensional simulation.

Figure 5

(Color online) Temperature dependence of the average friction force ${\overline{F}}_L$ for NaCl for different corrugation amplitudes $E_0$. Above a critical temperature $T_{\text{critical}}$, which depends on $E_0$, the system passes into the thermolubricity regime, where the average friction force ${\overline{F}}_L\approx 0$. The activation rate theory is in good agreement at lower temperatures (fitting curves) but cannot be applied to the thermolubricity regime, where the average friction force nearly vanishes.

Figure 6

(Color online) Tip paths at $T=0\text{K}$ on (a) NaCl with $E_0=1\text{eV}$ and (b) HOPG with $E_0=0.5\text{eV}$ in the case of actuation with $\alpha =0.9$. The support paths with (a) $y_{\text{sup}}=0.2a$ and (b) $0.4a$ are indicated as dashed lines. The stability regions for $E_0^{+}$ are mapped dark gray, while the corresponding stability regions for $E_0^{-}$ cover the whole image (light gray). Analytic tip trajectories for the extreme cases $E_0^{+}$ are plotted as blue lines and for $E_0^{-}$ as red lines. Between these lines, the numerically calculated tip path (black dots) spatially oscillates and thus jumps are suppressed. For visualization reasons, the actuation frequency is lowered to $f=500\text{Hz}$.

Figure 7

(Color online) (a) Calculated lateral force map on NaCl using $E_0=2\text{eV}$ (forward scan direction). In the upper part of the image is $\alpha =0$, whereas in the lower part actuation with $\alpha =0.9$ is switched on. Typical line scans (black: forward; red: backward) on NaCl (c) without actuation showing a friction loop and (d) with actuation showing no hysteresis. (b), (e), and (f) are the analog lateral force map line scan profiles without and with actuation for HOPG using $E_0=1\text{eV}$.

Figure 8

(Color online) Average friction forces ${\overline{F}}_L$ for three different corrugation amplitudes $E_0$ on NaCl as a function of normalized actuation amplitude $\alpha$. For larger corrugation amplitude $E_0$, a stronger actuation is required to enter the superlubricity regime. For sufficiently small corrugation potential amplitude $\left(E_0=0.5\text{eV}\right)$, the system is already in the superlubricity regime independent of actuation.