Figure 1

Illustration of the modified Tomlinson model used in our calculations. A rigid flake consisting of $N$ atoms (here $N=24$) is connected by an $x$ spring and a $y$ spring to the support of the microscope. The support is moved in the $x$ direction. The substrate is modelled as an infinite single layer of rigid graphite.Reuse & Permissions

Figure 2

$0.246\mathrm{nm}\times 0.426\mathrm{nm}$ rectangular unit cell of the potential energy surface (PES) that describes the interaction between a single carbon atom and the outermost layer of the graphite substrate. The potential has minima of $-3V_0$ and maxima of $1\frac{1}{2}{V}_0$ (here, $V_0=0.032\mathrm{eV}$). Solid and dashed contour lines in the PES denote positive $(V⩾0)$ and negative $(V<0)$ potential energy values, respectively. The contour lines are separated by $0.01\mathrm{eV}$.Reuse & Permissions

Figure 3

Symmetric flakes used in the calculations, consisting of (a) 6, (b) 24, (c) 54, (d) 96, and (e) 150 atoms.Reuse & Permissions

Figure 4

Total potential energy surfaces and lateral force images $(1.0\mathrm{nm}\times 0.426\mathrm{nm})$, calculated in the forward $x$ direction for a symmetric, 96-atom flake, for misfit angles $\Phi =0°$ (a,b), $\Phi =7°$ (c,d), and $\Phi =30°$ (e,f). The grey scale in the lateral force images corresponds to the range $[-1.04,0.63]\mathrm{nN}$. For this range, (b) has maximal contrast. Solid and dashed contour lines in the PES denote positive $(V⩾0)$ and negative $(V<0)$ energy values, respectively. The contour lines in (a), (c), and (e) are separated by $0.12\mathrm{eV}$, $0.012\mathrm{eV}$, and $6.2\times {10}^{-4}\mathrm{eV}$, respectively. The grey areas in the potential energy contour plots denote positions that were visited by the flake. The black lines denote pathways of the flake during single scan lines of the support in the $x$ direction, at $y_m=0.104\mathrm{nm}$, $y_m=0.212\mathrm{nm}$, and $y_m=0.284\mathrm{nm}$ [indicated with black dashed lines in (a,c), and with white dashed lines in (e)]. In (a), the pathway that belongs to the scan line at $y_m=0.212\mathrm{nm}$ is colored dark grey to distinguish it from the upper and lower pathways.Reuse & Permissions

Figure 5

Calculated friction loops for a symmetric 96-atom flake at rotation angles $\Phi =0°$, 7°, and 30° at (a) $y_m=0.104\mathrm{nm}$, (b) $y_m=0.212\mathrm{nm}$, and (c) $y_m=0.284\mathrm{nm}$. The solid lines show the force in the forward $x$ direction, the dotted lines show the force in the backward $x$ direction. In all three panels, the forward and backward forces coincide within the resolution of the plot for $\Phi =30°$ (lowest-amplitude curves) and $\Phi =7°$ (intermediate-amplitude curves). Here, only for $\Phi =0°$ the forward and backward curves are visibly separated and energy is dissipated.Reuse & Permissions

Figure 6

Calculated friction as a function of pulling direction for three different orientations of a 96-atom flake: $\Phi =0°$, 4°, and 30°.Reuse & Permissions

Figure 7

Friction as a function of the orientation angle for different symmetric flakes ranging in size from 6 to 150 atoms.Reuse & Permissions

Figure 8

Width of the friction peaks (FWHM) in Fig. 7 versus flake diameter. The dotted curve is the simple geometrical estimate of Eq. (5).Reuse & Permissions

Figure 9

Three asymmetric flakes consisting of (a) 78, (b) 56, and (c) 30 atoms.Reuse & Permissions

Figure 10

Total potential energy surfaces and lateral force images $(1.0\mathrm{nm}\times 0.426\mathrm{nm})$, calculated in the forward $x$ direction for an asymmetric, 56-atom flake, for orientation angles $\Phi =12°$ (a,b) and $\Phi =30°$ (c,d). Solid and dashed contour lines in the PES denote positive $(V⩾0)$ and negative $(V<0)$ energy values, respectively. The contour lines are separated by (a) $6.2\times {10}^{-3}\mathrm{eV}$ and (c) $3.1\times {10}^{-3}\mathrm{eV}$. The grey areas in the potential energy contour plots denote positions that were visited by the flake when scanning in the $x$ direction. The grey scale in the lateral force images corresponds to a force range $[-1.04,0.63]\mathrm{nN}$, equal to that of Fig. 4.Reuse & Permissions

Figure 11

Friction as a function of the orientation angle for three different asymmetric flakes with (a,d) 56 atoms, $V_0=0.0093\mathrm{eV}$; (b) 30 atoms, $V_0=0.017\mathrm{eV}$; (c) 78 atoms, $V_0=0.0068\mathrm{eV}$. (a,b,c) are calculations for a pulling direction $\Theta =0°$, (d) is calculated at $\Theta ={10}^{°}$.Reuse & Permissions

Figure 12

Calculated effective PES for an asymmetric 56-atom flake for rotation angles $\Phi =63.5°$ (a) and $\Phi =116.5°$ (b). The grey areas denote positions that were visited by the flake when scanning in the forward $x$ direction. Solid and dashed contour lines in the PES denote positive $(V⩾0)$ and negative $(V<0)$ energy values, respectively. The contour lines are separated by $6.2\times {10}^{-2}\mathrm{eV}$.Reuse & Permissions

Figure 13

The data points in both panels show the average friction force versus the rotation angle measured in Ref. 21. The curve through the data points shows the calculated friction force from the Tomlinson model for (a) a symmetric 96-atom flake and (b) an asymmetric 78-atom flake. The calculations were performed for a pulling direction $\Theta =70°$ (see text).Reuse & Permissions