Figure 1

Top view of the geometry of a graphite flake of 24 atoms on the substrate, in a commensurate orientation (left, mismatch angle $\varphi =0$) and incommensurate orientation (right, $\varphi =30°$). The open circles represent substrate atoms, while the closed circles are flake atoms. The scan lines used in this paper are along the $x$ axis and shown from top to bottom: scan line 1 (solid line), 2 (dashed line), 3 (dotted line), and 4 (dot-dashed line). The scan lines are separated by a distance $a/4$. Due to the symmetry of the lattice, the range between scan lines 1 and 4 fully describes all scan lines in this direction. The scan line at a distance $a/4$ below scan line 4 is again equivalent to scan line 3. Note that in a symmetric hexagonal flake, the center of mass does not correspond to the position of an atom.Reuse & Permissions

Figure 2

(Color online) Three typical trajectories for a 24-atom flake subjected to $F_{\text{load}}=20\text{nN},v_{\text{s}}=32\text{m}/\text{s}$. All three converge to stable periodic orbits at approximately constant $\varphi \approx {\varphi }_0$. The trajectories converging to ${\varphi }_0\approx 0°$ and ${\varphi }_0\approx 26°$ are for scan line 3, but have different initial angular velocity, and the trajectory converging to ${\varphi }_0\approx 30°$ is for scan line 4. From left to right and top to bottom: (a) the mismatch angle as a function of time, (b) the position as a function of time once the trajectories have converged to the periodic orbits, (c) the trajectories on the surface for the same interval, and (d) the friction force $F_s$.Reuse & Permissions

Figure 3

Examples of more complicated trajectories of a 24-atom flake: (a) a periodic trajectory with a longer period, in this case 6 lattice periods, for $F_{\text{load}}=30\text{nN}$ at scan line 3 and (b) a chaotic trajectory for $F_{\text{load}}=40\text{nN}$ at scan line 1. All other conditions are the same as for the trajectories plotted in Fig. 2.Reuse & Permissions

Figure 4

A bifurcation diagram of the stable periodic orbits as a function of the parameter $y$ for $N=24$ and $F_{\text{load}}=20\text{nN}$. The data was obtained by doing a large number of simulations with a wide range of initial conditions. The plotted points are the set of final angles. Clearly visible between scan lines 3 and 4 are the points at which the ${\varphi }_0\approx 26°$ periodic orbit becomes unstable and the ${\varphi }_0\approx 30°$ periodic orbit becomes stable.Reuse & Permissions

Figure 5

The plot of Fig. 4 repeated for (a) various sixfold symmetric flakes of (b) 54, (c) 96, (d) 150, and (e) 216 atoms. Larger flakes have more stable periodic orbits. The lone point in the bifurcation diagram for $N=96$ near scan line 4 at ${\varphi }_0\approx 12°$ indicates that the ${\varphi }_0\approx 12°$ periodic orbit is still stable there, but has such a small basin of attraction that the spacing between the initial conditions used to calculate this bifurcation diagram is not fine enough to detect it.Reuse & Permissions

Figure 6

(Color online) The potential energy $V\left(\mathbf{r},\varphi \right)$ of a 24-atom flake as a function of the mismatch angle $\varphi$ and position $x$ along the trajectory of the support, for constant $z$ at the average value belonging to a load of 20 nN, and $y$ corresponding to scan lines (a) 1, (b) 2, (c) 3, and (d) 4. Due to the symmetries of the system given in Eqs. (14, 15, 16), the dependence on $\varphi$ is determined by the behavior between 0 and $30°$.Reuse & Permissions

Figure 7

The (a) offset $U\left(\varphi \right)$ and amplitude (b) $W\left(\varphi \right)$ of the potential $V\left(x,\varphi \right)$ as a function of $\varphi$ for the same case displayed in Fig. 6. The region near $\varphi =30°$ is enlarged separately for scan lines (c) 1, (d) 2, (e) 3, and (f) 4. $U$ and $W$ were obtained from a Fourier transform of $V$ with respect to $x$ over 492 points for each $\varphi$. The extrema of $U$ and $W$ coincide at $\varphi ={\varphi }_0$, which implies the existence of an invariant manifold $\varphi ={\varphi }_0,\omega =0$ with ${\varphi }_0=0°,26°$ or $30°$.Reuse & Permissions

Figure 8

The (a) offset $U\left(\varphi \right)$ and (b) amplitude $W\left(\varphi \right)$ for a flake of 216 atoms. The extrema of $U$ coincide with the maxima of $W$, and the nodes of $W$ correspond to a constant value of $U$. There are more extrema than for the flake of 24 atoms, and therefore more stable and unstable incommensurate periodic orbits.Reuse & Permissions

Figure 9

(Color online) Cross sections of the phase space, including the basin of attraction of the incommensurate stable periodic orbits, which are at $\varphi \approx 26°,\omega =0$, for scan lines (a) 1, (b) 2, (c) 3, and $\varphi \approx 30°,\omega =0$ for (d) scan line 4. The flake has 24 atoms and $F_{\text{load}}=20\text{nN},v_s=32\text{m}/\text{s}$. The final state of the flake is plotted as a function of the initial orientation and angular momentum. The initial position and velocity have been chosen such that the stable incommensurate periodic orbit intersects with the cross section, in $\omega =0$. The colors indicate to which periodic orbit the initial conditions converge: red incommensurate ${\varphi }_0∊\phantom{[}⟨0°,30°\phantom{⟩}]$, blue incommensurate ${\varphi }_0∊\phantom{⟨}[30°,60°\phantom{]}⟩$, purple with blue commensurate ${\varphi }_0=60°$, purple incommensurate ${\varphi }_0∊\phantom{[}⟨60°,90°\phantom{⟩}]$, black incommensurate ${\varphi }_0∊\phantom{⟨}[90°,120°\phantom{]}⟩$, red with black commensurate ${\varphi }_0=120°$, green with red commensurate ${\varphi }_0=0°$, green incommensurate ${\varphi }_0∊\phantom{⟨}[-30°,0°\phantom{]}⟩$, cyan incommensurate ${\varphi }_0∊\phantom{[}⟨-60°,-30°\phantom{⟩}]$, yellow with cyan commensurate ${\varphi }_0=-60°$, yellow incommensurate ${\varphi }_0∊\phantom{⟨}[-90°,60°\phantom{]}⟩$, gray incommensurate ${\varphi }_0∊\phantom{[}⟨-120°,-90°\phantom{⟩}]$. The incommensurate periodic orbit at ${\varphi }_0\approx 30°$ is indicated in blue, as it visits both $\phantom{[}⟨0°,30°\phantom{⟩}]$ and $\phantom{⟨}[30°,60°\phantom{]}⟩$ in one period.Reuse & Permissions

Figure 10

The final orientation of a 24-atom flake (mapped onto the interval $\left[0°,30°\right]$), which was initially in the stable incommensurate periodic orbit is plotted as a function of temperature after a long, but finite time, with $F_{\text{load}}=20\text{nN},v_{\text{s}}=32\text{m}/\text{s}$ for scan lines (a) 1, (b) 2, (c) 3, and (d) 4. For every temperature, 250 realizations are plotted. Enough time has elapsed for the system to decay to the static state distribution.Reuse & Permissions

Figure 11

The plot of Fig. 10 for scan line 2 repeated for flakes of (a) 96 and (b) 216 atoms. The stable periodic orbits of large flakes are more robust against temperature, because the moment of inertia grows as $N^2$.Reuse & Permissions

Figure 12

(Color online) The (a) orientation $\varphi$, positions (b) $x$ and (c) $y$, and (d) friction $F_{\text{s}}$ as a function of support position $x_{\text{s}}$ for various support velocities and $N=24$, $F_{\text{load}}=20\text{nN}$, scan line 2. As the velocity decreases, the fluctuations in $\varphi$ and $y$ increase and the system behaves less one-dimensionally. For sufficiently low $v_s$, the system can no longer be described by the simplified model.Reuse & Permissions

Figure 13

The plot of Fig. 4 repeated for load force equal to (a) 0, (b) 10, (c) 30, and (d) 40 nN.Reuse & Permissions

Figure 14

(Color online) The cross section for scan line 3 in Fig. 9 repeated with the commonly used two-dimensional (2D) interaction potential. The basin of attraction is different in shape and size.Reuse & Permissions

Figure 15

The plot of Fig. 4 repeated for (a) various threefold symmetric flakes of (b) 33 and (c) 69 atoms.Reuse & Permissions