Size effects in friction of multiatomic sliding contacts

We studied the multiatomic contact effect of a friction force microscope at a finite temperature using the dynamical simulation of the Frenkel-Kontrova-Tomlinson model with a finite contact size. The friction force depends crucially on both the lattice mismatch between the tip of a friction force microscope and the sample surface and the strength of the lateral coupling between atoms in the tips. In the case of strong coupling, the friction force depends strongly on both the lattice mismatch and the tip size: there exists a magic size at which the friction force is reduced dramatically due to suppression of the effective corrugation of the surface potential to drag the tip. In the case of weak coupling, a decrease of the friction force with increasing temperature is enhanced as the tip size increases, irrespective of the lattice mismatch. This is caused by the enhanced thermal fluctuation for the multiatomic contact. The correlation among atoms in the multiatomic contact is also discussed.

Figure 1

The relation between the effective amplitude $U^{\prime }$ of the corrugation and the lattice mismatch $b/a$ at three values of $N$. The horizontal line represents the threshold of $U^{\prime }=k_{\text{tip}}{a}^2/{\left(2\pi \right)}^2$ for occurrence of the stick-slip motion. Here, $U=0.55k_{\text{tip}}{a}^2$ is assumed.

Figure 2

The relation between the time-averaged friction force $⟨F⟩$ and the tip size $N$ in the lattice-matched case of $a=b$ for several values of $k_2/k_{\text{tip}}$ at (a) $T=0$ and (b) $T=0.013k_{\text{tip}}{a}^2/k_B$.

Figure 3

The relation between the time-averaged friction force $⟨F⟩$ and the tip size $N$ in the lattice-mismatched case of $b/a=0.887$ for several values of $k_2/k_{\text{tip}}$ at (a) $T=0$ and (b) $T=0.013k_{\text{tip}}{a}^2/k_B$.

Figure 4

The time $t$ dependence of $x_i$ for $1\le i\le 14$, $x_G$, and $F$ with $R=vt$ for (a),(b) $k_2/k_{\text{tip}}=1$, and (c),(d) $k_2=0$, respectively. Here, $N=14$, $b/a=1$, and $T=0.013k_{\text{tip}}{a}^2/k_B$.

Figure 5

The time $t$ dependence of $x_i$ for $1\le i\le 9$, $x_G$, and $F$ with $R=vt$ for (a),(b) $k_2/k_{\text{tip}}=1$, and (c),(d) $k_2/k_{\text{tip}}=50$, respectively. Here, $N=9$, $b/a=0.887$, and $T=0.013k_{\text{tip}}{a}^2/k_B$.

Figure 6

$\left(x_1-x_i\right)$ as a function of $x_1$ at $N=14$ for $b/a=1$. (a) $k_2/k_{\text{tip}}=1$, $T=0.0026k_{\text{tip}}{a}^2/k_B$, (b) $k_2/k_{\text{tip}}=1$, $T=0.013k_{\text{tip}}{a}^2/k_B$, (c) $k_2/k_{\text{tip}}=0$, $T=0.0026k_{\text{tip}}{a}^2/k_B$, and (d) $k_2/k_{\text{tip}}=0$, $T=0.013k_{\text{tip}}{a}^2/k_B$.

Figure 7

$\left(x_1-x_i\right)$ as a function of $x_1$ at $N=9$ for $b/a=0.887$. (a) $k_2/k_{\text{tip}}=1$, $T=0.0026k_{\text{tip}}{a}^2/k_B$, (b) $k_2/k_{\text{tip}}=1$, $T=0.013k_{\text{tip}}{a}^2/k_B$, (c) $k_2/k_{\text{tip}}=50$, $T=0.0026k_{\text{tip}}{a}^2/k_B$, and (d) $k_2/k_{\text{tip}}=50$, $T=0.013k_{\text{tip}}{a}^2/k_B$.

Figure 8

$\left(x_1-x_i\right)$ as a function of $i$ at $N=14$ for $b/a=1$. (a) $k_2/k_{\text{tip}}=1$, $T=0.0026k_{\text{tip}}{a}^2/k_B$, (b) $k_2/k_{\text{tip}}=1$, $T=0.013k_{\text{tip}}{a}^2/k_B$, (c) $k_2/k_{\text{tip}}=0$, $T=0.0026k_{\text{tip}}{a}^2/k_B$, and (d) $k_2/k_{\text{tip}}=0$, $T=0.013k_{\text{tip}}{a}^2/k_B$.

Figure 9

$\left(x_1-x_i\right)$ as a function of $i$ at $N=9$ for $b/a=0.887$. (a) $k_2/k_{\text{tip}}=1$, $T=0.0026k_{\text{tip}}{a}^2/k_B$, (b) $k_2/k_{\text{tip}}=1$, $T=0.013k_{\text{tip}}{a}^2/k_B$, (c) $k_2/k_{\text{tip}}=50$, $T=0.0026k_{\text{tip}}{a}^2/k_B$, and (d) $k_2/k_{\text{tip}}=50$, $T=0.013k_{\text{tip}}{a}^2/k_B$.

Figure 10

The standard deviation of $\left(x_1-x_i\right)$ as a function of $i$ in two cases of (a) $N=14$ at $b/a=1$ and (b) $N=9$ at $b/a=0.887$. Four cases of $k_2/k_{\text{tip}}=1$ at $T=0.0026k_{\text{tip}}{a}^2/k_B$ and at $T=0.013k_{\text{tip}}{a}^2/k_B$, $k_2/k_{\text{tip}}=0$ at $T=0.0026k_{\text{tip}}{a}^2/k_B$ and at $T=0.013k_{\text{tip}}{a}^2/k_B$ are plotted in (a), corresponding to Figs. 6a, 6b, 6c, 6d, respectively. Four cases of $k_2/k_{\text{tip}}=1$ at $T=0.0026k_{\text{tip}}{a}^2/k_B$ and at $T=0.013$, $k_2/k_{\text{tip}}=50$ at $T=0.0026k_{\text{tip}}{a}^2/k_B$ and at $T=0.013k_{\text{tip}}{a}^2/k_B$ are plotted in (b), corresponding to Figs. 7a, 7b, 7c, 7d, respectively.

Figure 11

The relation between the time-averaged friction force $⟨F⟩$ and the temperature $T$ for (a) the lattice-matched case of $b/a=1$ at $N=14$ and (b) the lattice-mismatched case of $b/a=0.887$ at $N=9$ for several values of $k_2/k_{\text{tip}}$. The solid line in (a) and the broken lines in (a) and (b) represent the predicted behavior in the ramped creep regime (Ref. 12) for $k_2=\infty$ and $k_2=0$, respectively.

Figure 12

The time $t$ dependence of (a) $x_i$ for $1\le i\le 14$ and (b) $x_G$ and $F$ with $R=vt$ for $k_2=0$. Here, $N=14$, $b/a=1$, and $T=0.0065k_{\text{tip}}{a}^2/k_B$.