Nanoscale sliding friction versus commensuration ratio: Molecular dynamics simulations

The pioneer work of Krim and Widom [Phys. Rev. B 38, 12184 (1988)] unveiled the origin of the viscous nature of friction at the atomic scale. This generated extensive experimental and theoretical activity. However, fundamental questions remain open like the relation between sliding friction and the topology of the substrate, as well as the dependence on the temperature of the contact surface. Here we present results, obtained using molecular dynamics, for the phononic friction coefficient $\left({\eta }_{\mathit{ph}}\right)$ for a one-dimensional model of an adsorbate-substrate interface. Different commensuration relations between adsorbate and substrate are investigated as well as the temperature dependence of ${\eta }_{\mathit{ph}}$. In all the cases we studied ${\eta }_{\mathit{ph}}$ depends quadratically on the substrate corrugation amplitude, but is a nontrivial function of the commensuration ratio between substrate and adsorbate. The most striking result is a deep and wide region of small values of ${\eta }_{\mathit{ph}}$ for substrate-adsorbate commensuration ratios between $\approx 0.65$ and 0.9. Our results shed some light on contradictory results for the relative size of phononic and electronic friction found in the literature.

Figure 1

The one-dimensional model.

Figure 2

Typical behavior of the adatom center-of-mass velocity $\left(v_{\mathrm{c.m.}}\right)$ during a Langevin molecular dynamics run, before and after the force is applied. The temperature is $k_{B}T=0.05\epsilon$, and the applied force is $F={10}^{-3}\epsilon ∕r_0$; in this case, $a∕b=0.95$ and$u_0=0.04\epsilon$. $v_0=\sqrt{\epsilon ∕m}$ and $t_0=r_0\sqrt{m∕\epsilon }$.

Figure 3

(Color online) Center-of-mass velocity $v_{\mathrm{c.m.}}$ as a function of the external force $F$ for different substrate corrugation values and for two commensuration ratios: (a) $a∕b=0.96$ and (b) $a∕b=0.6$, at temperature $k_{B}T=0.1\epsilon$. The resulting $\eta$ coefficients are 0.0068 (•), 0.010 (∎), 0.0198 (▴), 0.0051 (◻), 0.0052 (◻), and 0.0057 (▵), all in units of ${\omega }_0=\sqrt{\epsilon ∕\left(m{r}_0^2\right)}$. $F_0=ϵ∕r_0$.

Figure 4

Total friction coefficient $[{\omega }_0=\sqrt{\epsilon ∕\left(m{r}_0^2\right)}]$ for some commensuration ratios $a∕b$ and for a temperature $k_{B}T=0.1\epsilon$: (a) “strong”-friction coefficients, the lines are fittings to ${\eta }_e+9.45u_0^2$ (solid line) and ${\eta }_e+2.64u_0^2$ (dashed line); (b) “weak”-friction coefficients, the lines are fittings to ${\eta }_e+1.05u_0^2$ (solid line) and ${\eta }_e+0.47u_0^2$ (dashed line).

Figure 5

Coefficient $c$ of the phononic friction $({\eta }_{\mathit{ph}}=c{u}_0^2)$ for several commensuration ratios. (a) Results obtained with external force $F$ in the range $(0.0001–0.001)\epsilon ∕r_0$ and three different temperatures. (b) Results obtained with external force $F$ in the range $(0.001–0.002)\epsilon ∕r_0$ and four different temperatures. $c_0={\left(m{r}_0^2{\epsilon }^3\right)}^{-1∕2}$.

Figure 6

Temperature dependence of the coefficient $c$ for two values of the commensuration ratio $a∕b$, corresponding to the “strong”-friction cases. $c_0={\left(m{r}_0^2{\epsilon }^3\right)}^{-1∕2}$.

Figure 7

(Color online) Comparison of methods for obtaining the total friction coefficient, represented as a function of corrugation $u_0$: circles are results from the fluctuation-dissipation method, squares are from the steady-state method, and the lines are quadratic fittings. $k_{B}T=0.05\epsilon$, $a∕b=0.55$, and ${\omega }_0=\sqrt{\epsilon ∕\left(m{r}_0^2\right)}$.

Figure 8

(Color online) Normalized phonon-frequency spectrum for two different type of commensuration ratio values, one of the (a) weak-friction type $(a∕b=0.85)$ and the other of the (b) strong-friction type $(a∕b=0.95)$. In both cases the corrugation amplitude is $u_0=0.04\epsilon$ and the temperature is $T=0.05\epsilon ∕k_B$. The frequency $\omega$ is presented in units of ${\omega }_0=\sqrt{\epsilon ∕\left(m{r}_0^2\right)}$.

Figure 9

Snapshots of the displacements of the particles as function of time (in units of $t_0=r_0\sqrt{m∕\epsilon }$) relative to their equilibrium position in the center-of-mass system. In order to emphasize visualization of the structural change we have diminished the distances between particles down to $0.075r_0$. Snapshots (a) and (b) correspond to a weak-friction case $(a∕b=0.75)$, while snapshots (c) and (d) correspond to a strong-friction case $(a∕b=0.95)$; snapshots (a) and (c) were taken before the application of force, while (b) and (d) were taken after that.

Figure 10

Comparison between our results for the $c$ parameter and the data of Braun et al. of mobility as a function of commensuration ratio. Present work results are at $k_{B}T=0.05\epsilon$ ($\approx 1\mathrm{meV}$ for Xe), while the results of Braun et al. are at $k_{B}T=2.5\mathrm{meV}$ (a) and $k_{B}T=5\mathrm{meV}$ (b). Curves (a) and (b) of Braun et al. are curves (1) and (2), respectively, of Fig. 2 from Ref. 14. $c_0={\left(m{r}_0^2{\epsilon }^3\right)}^{-1∕2}$ and mobility is defined as $m{\eta }_{\mathit{el}}v∕F$. The scale was adjusted for the comparison between them.