Surface-Phonon-Induced Rotational Dissipation for Nanoscale Solid-State Gears

Compared to nanoscale friction of translational motion, the mechanisms of rotational friction have received less attention. Such motion becomes an important issue for the miniaturization of mechanical machinery that often involves rotating gears. In this study, molecular-dynamics simulations are performed to explore rotational friction for solid-state gears rotating on top of different substrates. In each case, viscous damping of the rotational motion is observed and found to be induced by the pure van der Waals interaction between the gear and the substrate. The influence of different gear sizes and various substrate materials is investigated. Furthermore, the rigidities of the gear and the substrate are found to give rise to different dissipation channels. Finally, it is shown that the dominant contribution to the dissipation is related to the excitation of low-frequency surface phonons in the substrate.

Figure 1

A schematic illustration of a diamond-based solid-state gear with radius $r=5$ nm rotating with angular velocity ${\omega }_0$ on top of a $\alpha$-cristobalite ${\mathrm{Si}\mathrm{O}}_2$ (001) substrate.

Figure 2

The scheme of the 5-nm diamond-based solid-state gear on top of the (a) $\alpha$-cristobalite ${\mathrm{Si}\mathrm{O}}_2$ with size 23.0 nm $\times$ 15.3 nm $\times$ 1.39 nm and separation distance 2.5 Å, (b) amorphous ${\mathrm{Si}\mathrm{O}}_2$ with dimension 28.2 nm $\times$ 16.9 nm $\times$ 1.86 nm and separation distance around 1–2 Å, and (c) four-layer graphene with size 21.8 nm $\times$ 12.6 nm $\times$ four monolayers and separation distance 2 Å.

Figure 3

The (a) angular velocity (exponential fits indicated by dashed lines) and (b) normalized angular velocity of the 5-nm diamond gear on top of $\alpha$-cristobalite ${\mathrm{Si}\mathrm{O}}_2$ undergoing a relaxation within the simulation time $t=100$ ps with different initial angular velocities $\omega =0.1$, $0.15$, and $0.2$ rad/ps, respectively. The highlighted yellow region (from $0$ to $50$ ps) is used for extracting the velocity relaxation time $\tau$ by fitting to an exponential function $\propto e^{-t/\tau }$.

Figure 4

The normalized angular velocity of the diamond-based solid-state gear with initial angular velocity ${\omega }_0=0.2$ rad/s for different gear radii $r=3$, $4$, and 5 nm on top of (a) crystalline ${\mathrm{Si}\mathrm{O}}_2$, (b) amorphous ${\mathrm{Si}\mathrm{O}}_2$, and (c) four-layer graphene, respectively.

Figure 5

The normalized angular velocity with initial angular velocity $\omega =0.2$ rad/ps for different cases of the available degrees of freedom—rigid surface, rigid gear, and both deformable—for (a) crystalline ${\mathrm{Si}\mathrm{O}}_2$, (b) amorphous ${\mathrm{Si}\mathrm{O}}_2$, and (c) graphene.

Figure 6

(a) The relaxed-layer thickness dependence of the angular-velocity relaxation. (b,c) Kinetic energy spectra of the substrate (b) without and (c) with gear rotation within 50 ps.

Figure 7

The three-dimensional topography of the intensity from the kinetic energy spectrum $K_j({\nu }_0)$ with a contribution around the following frequencies (with bandwidth $\mathrm{\Gamma }=1$ THz): (a) $\nu =2$ THz without rotation, (b) $\nu =13$ THz without rotation, (c) $\nu =2$ THz during rotation, and (d) $\nu =13$ THz during rotation.

Figure 8

The vertical distribution of the average kinetic energy spectrum $K_{{\nu }_0}(z)$ of the substrate with the relaxed-layer thickness 14 Å, where $z$ is the position relative to the bottom of the crystalline ${\mathrm{Si}\mathrm{O}}_2$ substrate. The blue and red lines show $K_{{\nu }_0}(z)$ for ${\nu }_0=2$, and ${\nu }_0=13$ THz, respectively; the dashed line and solid line represent $K_{{\nu }_0}(z)$ before and during the gear rotation, respectively.