Influence of substrate potential shape on the dynamics of a sliding lubricant chain

We investigate the frictional sliding of an incommensurate chain of interacting particles confined in between two nonlinear on-site substrate potential profiles in relative motion. We focus here on the class of Remoissenet-Peyrard parametrized potentials $V_{\mathrm{RP}}(x,s)$, whose shape can be varied continuously as a function of $s$, recovering the sine-Gordon potential as a particular case. The observed frictional dynamics of the system, crucially dependent on the mutual ratios of the three periodicities in the sandwich geometry, turns out to be significantly influenced also by the shape of the substrate potential. Specifically, variations of the shape parameter $s$ affect significantly and not trivially the existence and robustness of the recently reported velocity quantization phenomena [A. Vanossi et al., Phys. Rev. Lett. 97, 056101 (2006)], where 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 the dynamical parameters.

Figure 1

Schema of the model with three characteristic length scales $a_{+}$ (spatial period of the static substrate), $a_0$ (the equilibrium spacing of the harmonic chain representing the lubricant film), and $a_{-}$ (spatial period of the advancing substrate). The substrate corrugation is modeled by the RP potential, which is illustrated below for three values of the deformation parameter. Top to bottom: $s=-0.6$ (narrow valleys), 0 (pure sine), and $0.6$ (broad valleys).

Figure 2

Ratio $V_{\mathrm{CM}}/V_{\mathrm{ext}}$ of the steady-state average chain center-of-mass velocity to the top speed, as a function of the chain stiffness $K$, for three values $s=-0.6$, 0.0, and 0.6 of the potential shape parameter. The length ratios are taken as approximants to the golden mean $r=377/233$, $r^{\prime }=233/144$; the simulation involves $N_0=377$ particles; the damping and top driving speed are $\gamma =0.1$, $V_{\mathrm{ext}}=0.1$. The dashed line marks $w_{\mathrm{plateau}}\simeq 0.382$.

Figure 3

Left panels: Time evolution of the velocity of a particle $V_i$ (solid) and the chain overall center-of-mass velocity $V_{\mathrm{CM}}$ (dashed) for three different shapes of the RP substrates potentials, and for stiffness $K=100$ ($s=0$) and $K=400$ ($s=\pm 0.6$), representative of the quantized plateau. Right panels: The corresponding Fourier analysis of the velocity signal, representing the power spectral decomposition of the motion.

Figure 4

The boundaries $K_{\mathrm{Min}}$ and $K_{\mathrm{Max}}$ of the quantized velocity plateau as a function of the chain stiffness $K$, as a function of the potential shape parameter $s$. The model conditions are the same as in Fig. 2.

Figure 5

Antikink geometry, for $r=25/36<1$: The average center-of-mass velocity $V_{\mathrm{CM}}/V_{\mathrm{ext}}$ of the lubricant chain as a function of its stiffness $K$. Note the negative-velocity plateau, representing a leftward lubricant motion. The other parameters are $V_{\mathrm{ext}}=0.1$, $r^{\prime }=36/11$, and the model is simulated taking $N_{+}=144$, $N_0=100$, and $N_{-}=44$, for deformations $s=-0.6$, $0.0$, $0.3$, and $0.6$. The dashed line marks $w_{\mathrm{plateau}}=-0.44$.

Figure 6

Normalized center-of-mass velocity $V_{\mathrm{CM}}/V_{\mathrm{ext}}$ as a function of the stiffness $K$, for three values of the deformation parameter. The simulations are carried out for $N_0=377$ particles in $N_{+}=233$ bottom wells with $N_{-}=180$, 144, and 72, producing coverage $\Theta =0.8$, 1, and 2, respectively. Note the shrinkage of the plateau for $\Theta \ne 1$, quite substantially so for $\Theta =0.8$.