Nonlinear friction of a damped dimer sliding on a periodic substrate

The nonlinear sliding friction of a dimer over a substrate is studied within a one-dimensional model, consisting of a vibrating dimer (two masses connected by a spring), internally damped, sliding over a sinusoidal potential. Molecular dynamics simulations show that the friction force has an approximate $v^{-3}$ dependence if the velocity is sufficiently large, and that there is a striking periodic variation of the proportionality coefficient with the ratio of the length of the dimer to the substrate wavelength. The nonlinear velocity dependence was predicted earlier for a Langevin model of an adsorbed layer in the presence of strong external force. We study it here in detail in the transient regime and without external force. We obtain the dependence on key parameters (internal dissipation, dimer mass, substrate corrugation, and length ratio), and examine the validity of the friction law. A semianalytical expression is suggested which confirms the numerical observations in the high-velocity regime.

Figure 1

Evolution of the center of mass coordinate $x_{+}$ in the absence of dissipation. Plotted is $x_{+}$ vs time $t$ from Eqs. (1) for two different initial velocities, below and above the upper chaos threshold, respectively; $x_{+}$ is expressed in units of $b$, $t$ is in units of $b\sqrt{m∕u_0}$, and $v_0$ is in units $\sqrt{u_0∕m}$. In the present case $a=1.4b$ and $k=2u_0∕b^2$.

Figure 2

Evolution of the center of mass velocity $v_{+}\left(t\right)$ in the presence of damping applied to the internal coordinate. Plotted is $v_{+}\left(t\right)$ in units of $\sqrt{u_0∕m}$ vs time $t$ in units of $b\sqrt{m∕u_0}$. Parameters here are $a∕b=0.25,k{b}^2∕u_0=1,$ and $\gamma b\sqrt{m∕u_0}=4$. Two insets show the detail of oscillations at times much smaller (left lower inset) and larger (right upper inset) than the stopping time. Solid lines in the inset show the numerical results while the dashed lines represent our proposed friction law.

Figure 3

Evidence for our proposed power law $\left(\alpha =3\right)$. Shown is the relation between the apparent stopping time $t_s$ and the initial velocity $v_0$, for $a∕b=2$ and different sets of $\gamma$, and $k$. Velocity and the stopping time are given in units of $\sqrt{u_0∕m}$ and $b\sqrt{m∕u_0}$, respectively; $\gamma$ is in units of $\sqrt{(u_0∕m)}∕b$ and $k$ is in units of $u_0∕b^2$. The lines are a linear regression of the log-log relation given consistently by an exponent $\approx 4$, thus $\alpha \approx 3$.

Figure 4

Periodic dependence of the friction coefficient $\eta$ on the ratio of the dimer length to substrate wavelength. Plotted is $\eta$ in units of $\gamma {\left(u_0∕m\right)}^2$ vs $a∕b$.

Figure 5

Internal coordinate $x=x_{-}-a∕2$ vs time $t$. Note the relatively small amplitude of the oscillation supporting the $x⪡a∕2$ assumption; $x$ is in units of $b$, time is in units of $b\sqrt{m∕u_0}$, and parameters are the same as in Fig. 2.

Figure 6

Comparison between the analytic prediction of our proposed friction law 4 (solid thin line without oscillations) and numerical simulations for the center of mass velocity $v_{+}\left(t\right)$ vs time $t$. Parameters are $a∕b=0.5$ and $\gamma b\sqrt{m∕u_0}=1$. Solid thick line and broken line are for $k{b}^2∕u_0$ equal to $0.02$ and $10$, respectively. Velocity is expressed in units of $\sqrt{u_0∕m}$ and time is in units of $b\sqrt{m∕u_0}$. Initial velocity $v_0$ is $5\sqrt{u_0∕m}$.