Superlubric to stick-slip sliding of incommensurate graphene flakes on graphite

We calculate the friction of fully mobile graphene flakes sliding on graphite. For incommensurately stacked flakes, we find a sudden and reversible increase in friction with load, in agreement with experimental observations. The transition from smooth sliding to stick-slip and the corresponding increase in friction is neither due to rotations to commensurate contact nor to dislocations but to a pinning caused by vertical distortions of edge atoms also when they are saturated by hydrogen. This behavior should apply to all layered materials with strong in-plane bonding.

Figure 1

FFM measurements of the load dependence of the friction force between a graphene flake on an HOPG graphite surface over a region of the graphite to which the flake is incommensurate. Every data point was the average over five separate measurements. The sudden increase of friction at loads in excess of $\sim$40 nN was reproducible and fully reversible.

Figure 2

(a) Model of the rigid tip and mobile flake. Each flake atom with coordinate ${\overline{\mathbf{r}}}_i$ is attached with a spring to a support point ${\overline{\mathbf{r}}}_i^0$. (b) Flake with $\varphi ={30}^{\circ }$ on the substrate. The scan lines over which the center of mass of the support moves, from top to bottom: A, B, C, D. The period $a$ is indicated.

Figure 3

Friction and friction coefficient $\mu$ as a function of load for commensurately oriented hexagonal flakes of different sizes as indicated.

Figure 4

Friction as a function of load for several scan lines and flake sizes of a flake incommensurately oriented with $\varphi ={30}^{\circ }$ (left). The line labeled “H” shows the friction of a C${}_{54}$H${}_{16}$ flake along scan line D.

Figure 5

Lateral force as a function of the support position for a low (blue, gray) and a high (red, black) load for the A, B, C, D scan lines (top to bottom) and different sizes (left to right). The total load $L$ is indicated in the figure. Notice the change from smooth to stick-slip behavior.

Figure 6

Two periods of the trajectory of the center of mass (left) and rotation angle vs the tip position (right). Simulations are for the 54 atom flake with $L=80$ nN or $L/N$=1.48 nN (top), for a 96-atom flake with $L=140$ nN or $L/N=1.46$ nN (middle) and for a 150-atom flake with $L=220$ nN or $L/N=1.47$ nN (bottom). Since the flakes are coupled to the support with springs, the center of mass can deviate from the center of mass of the support (shown in black).

Figure 7

54-atom flake along scan line D: (a) Energy as a function of the tip position for a rigid flake at a fixed height $z=2.97$ Å and 2.80 Å for $L=20$ nN and $L=40$ nN respectively. (b) Different contributions to the energy as a function of the tip position for a deformable flake which is relaxed from its ideal configuration for each tip position, for $L=20$ nN. (c) Same as in (b) for $L=40$ nN. (d) Distance between the center of mass of the relaxed flake and the center of mass of the tip. (e) Variation of the total energy from that at $x_{\mathrm{tip}}=0$ for a pulled flake for several loads. (f) The average distance to the substrate for the edge and nonedge atoms for $L=20$ nN. (g). Same as in (f) for $L=40$ nN.

Figure 8

Two minimum-energy configurations in the trajectory of the 54-atom flake along scan line D for $L=40$ nN. The color shows $\Delta z=z_i-z_{CM}$ where ${}_{CM}$ refers to the center of mass of the flake: red is closer to the substrate, blue is farther away. The outer atoms are clearly more mobile.

Figure 9

Lateral force $F_x$ as a function of the tip position for a 96-atom flake along scan line C for $L=100$ nN or $L/N\approx$ 1 nN calculated by MD at 10 K. The constant lines give the resulting friction at 10 K and the one at 300 K ($F_x$ not shown). The asterisk corresponds to the minimum-energy configuration shown in the right figure. The red (dark) color indicates atoms closer to the substrate as in Fig. 8 with minimum at $\Delta z=-0.165$ Å.