Exactly Quantized Dynamics of Classical Incommensurate Sliders

We report peculiar velocity quantization phenomena in the classical motion of an idealized 1D solid lubricant, consisting of a harmonic chain interposed between two periodic sliders. The ratio $v_{\mathrm{c}.\mathrm{m}.}/v_{\mathrm{ext}}$ of the chain center-of-mass velocity to the externally imposed relative velocity of the sliders stays pinned to exact “plateau” values for wide ranges of parameters, such as slider corrugation amplitudes, external velocity, chain stiffness, and dissipation, and is strictly determined by the commensurability ratios alone. The phenomenon is explained by one slider rigidly dragging the kinks that the chain forms with the other slider. Possible consequences of these results for some real systems are discussed.

Figure 1

(a) Average drift velocity ratio $w=v_{\mathrm{c}.\mathrm{m}.}/v_{\mathrm{ext}}$ of the chain as a function of its spring stiffness $K$ for different length ratios ($r_{+}$, $r_{-}$): commensurate ($3/2$, $9/4$), golden mean (GM) ($\varphi$, ${\varphi }^2$), spiral mean (SM) ($\sigma$, ${\sigma }^2$) ($\sigma \sim 1.3247\dots$ is the solution of ${\sigma }^3=\sigma +1$), and (${\varphi }^{-1}$, $\varphi$). The plateau labeling is explained in the text. Here $\gamma =0.1$, $v_{\mathrm{ext}}=0.1$, and periodic boundary conditions [5] are used. The ($\varphi$, ${\varphi }^2$) $1/1$ plateau value is $w=0.381966\dots$, identical to $1-{\varphi }^{-1}$ to eight decimal places. (b) The main plateau speed $w$ as a function of $r_{+}$. (c) A sketch of the model.

Figure 2

Snapshots of the distance between neighbor lubricant particles in the chain $x_i-x_{i-1}$ at three successive time frames. All parameters as in Fig. 1, except $r_{-}\simeq 10.36$, and $K=10$ (inside the main plateau). (a) $r_{+}=1.031$ (kink density $\delta /a_{+}=0.031$); (b) $r_{+}=0.995$ (antikink density $|\delta |/a_{+}=0.005$).

Figure 3

Time evolution of a particle velocity ${\dot{x}}_i$, and of the chain c.m. velocity $v_{\mathrm{c}.\mathrm{m}.}$ (fluctuations rescaled by a factor 50), for the GM (a) and SM (b) cases of Fig. 1. Amplitudes of the Fourier spectrum of ${\dot{x}}_i(t)$ (c) and of $v_{\mathrm{c}.\mathrm{m}.}(t)$ (d) for $r=\sigma$. Individual particle spectra have identical amplitudes, and differ only in the phases, which leads to a remarkable cancellation in the $v_{\mathrm{c}.\mathrm{m}.}$ power spectrum. Here $K=1$, $\gamma =0.1$, and $v_{\mathrm{ext}}=0.1$.

Figure 4

Dynamical depinning force $F_c$ for the GM plateau (a), extracted by a slow adiabatic increase of the force $F$ applied to all particles, until intermittencies appear, signaling collective slips (b). The critical behavior of the average velocity for $K\to K_c$ (c), (d), showing the typical square-root singularity (dotted line) associated to type-I intermittencies. Here $\gamma =0.1$, and $v_{\mathrm{ext}}=0.1$ were used.