Dynamic hysteresis of a confined lubricant under shear

A nonlinear model inspired by the tribological problem of a thin solid lubricant layer between two sliding periodic surfaces is used to analyze the novel phenomenon of hysteresis at pinning or depinning around a moving state rather than around a statically pinned state. The cycling of an external driving force $F_{\text{ext}}$ is used as a simple means to destroy and then to recover the dynamically pinned state previously discovered for the lubricant center-of-mass velocity. Depinning to a freely sliding state occurs either directly, with a single jump, or through a sequence of discontinuous transitions. The intermediate sliding steps are reminiscent of phase-locked states and stick-slip motion in static friction, and can be interpreted in terms of the appearance of traveling density defects in an otherwise regular arrangement of kinks. Repinning occurs more smoothly, through the successive disappearance of different traveling defects. The resulting bistability and multistability regions may also be accessed by varying mechanical parameters other than $F_{\text{ext}}$. The hysteretic phenomena are confined to the underdamped dynamics, and the overdamped dynamics of the same model is generally not hysteretic, much like in static friction.

Figure 1

(Color online) Normalized velocity of the center of mass, $v_{\text{cm}}∕v_{\text{ext}}$, as a function of the chain stiffness $K$ in units of $F_{+}∕a_{+}$, for $v_{\text{ext}}=0.1{\left(F_{+}{a}_{+}∕m\right)}^{1∕2}$, $\gamma =0.1{\left(F_{+}m∕a_{+}\right)}^{1∕2}$, and $r_{+}=\left(1+{\pi }^{1∕4}\right)∕2$. Crosses: one-to-one kink coverage $\theta =1$ $\left(r_{-}\simeq 7.036\right)$; diamonds: commensurate kink coverage $\theta =5∕4$ $\left(r_{-}\simeq 8.795\right)$; circles: incommensurate kink coverage $\theta =\left(1+{10}^{1∕2}\right)∕3$ $\left(r_{-}\simeq 9.762\right)$. Dashed line: the quantized-plateau velocity ratio of Eq. (2). Note the logarithmic scale in the abscissa. Inset: a sketch of the driven three-length-scale confined model.

Figure 2

(Color online) Hysteresis in the $v_{\text{cm}}-F_{\text{ext}}$ characteristics for a confined chain of intermediate spring stiffness $\left(K=5F_{+}∕a_{+}\right)$, and length ratios $r_{+}=\left(1+{\pi }^{1∕4}\right)∕2$, $\theta =5∕4$ $\left(r_{-}\simeq 8.795\right)$. The behavior is shown for fast ($v_{\text{ext}}=0.1$, upper panel) and slow ($v_{\text{ext}}=0.01$, lower panel) drive. Adiabatic increase and decrease of $F_{\text{ext}}$ (in steps of ${10}^{-3}{F}_{+}$) are denoted by crosses and circles, respectively. Characteristic hysteretic multistep features appear. Here $\gamma =0.1{\left(F_{+}m∕a_{+}\right)}^{1∕2}$, and a chain of $N=387$ lubricant particles is simulated.

Figure 3

(Color online) The $\left(K,F_{\text{ext}}\right)$ phase diagram illustrating the unpinning-repinning transitions for $r_{+}=\left(1+{\pi }^{1∕4}\right)∕2$, $\gamma =0.1$, $v_{\text{ext}}=0.1$, and for (a) one-to-one kink coverage $\theta =1$; (b) commensurate kink coverage $\theta =5∕4$; (c) incommensurate kink coverage $\theta =\left(1+{10}^{1∕2}\right)∕3$. The white areas have perfect plateau dynamics, the dotted region indicates quasifree sliding. Simulations done with $N=387$ particles for (a) and (b) and with $N=781$ particles for (c).

Figure 4

(Color online) Soft chain $\left(K=5\right)$. Four snapshots of a 60-particle section of the lubricant chain and $\left(+∕-\right)$ substrates (lower and upper sinusoids), at successive times separated by 14 time units ${\left(a_{+}m∕F_{+}\right)}^{1∕2}$. The horizontal direction represents distance, the dots the particle positions $x_i$. Vertical displacements of dots measure the distance $x_i-x_{i-1}$ of a given particle to its left neighbor: on this scale, the horizontal solid and dashed lines indicate the average interparticle distance $a_0$, and the $(+)$ lattice parameter $a_{+}$, respectively. The snapshots refer to $r_{+}=\left(1+{\pi }^{1∕4}\right)∕2$, $\theta =1$, $F_{\text{ext}}=0.08136$ (decreasing), and illustrate the crossing of the critical line $F_c^{+\downarrow }$, with the recovery of the plateau state obtained through the disappearance of the last bi-kink–no-kink defect. The other parameters are $\gamma =0.1$, and $v_{\text{ext}}=0.1$.

Figure 5

(Color online) Stiff chain $\left(K=50\right)$. Four successive snapshots of the substrates and lubricant chain, separated by time intervals of nine time units ${\left(a_{+}m∕F_{+}\right)}^{1∕2}$. All notations and parameters are the same as in Fig. 4, except for $K=50$, $F_{\text{ext}}=0.00685$ (decreasing), which falls in the region immediately above the critical line $F_c^{+\downarrow }$, before the recovery of the plateau state.

Figure 6

(Color online) Hysteresis loop at the plateau edge in the $v_{\text{cm}}-K$ characteristics for a confined chain of length ratios $r_{+}=\left(1+{\pi }^{1∕4}\right)∕2$, $\theta =1$ $\left(r_{-}\simeq 7.036\right)$. Adiabatic increase and decrease of $K$ are denoted by crosses and circles, respectively. Here $\gamma =0.1{\left(F_{+}m∕a_{+}\right)}^{1∕2}$.

Figure 7

(Color online) Driving-velocity dependency of the dynamical depinning and repinning forces $F_c^{+\uparrow }$ $F_c^{+\downarrow }$, $F_c^{-\downarrow }$ $F_c^{-\uparrow }$ [shifted upward by the trivial $F_{\text{w}}\propto v_{\text{ext}}$ contribution, Eq. (4)]. $v_{\text{ext}}$ is measured in units of ${\left(F_{+}{a}_{+}∕m\right)}^{1∕2}$, the chain stiffness $K=50F_{+}∕a_{+}$, and all other parameters are as in Fig. 6.

Figure 8

(Color online) Typical plateau arrangements of the $\theta =5∕4$ commensurate soft $K=5$ (a) and hard $K=50$ (b) chain: a regular arrangement of bi-kinks (one every four kinks). The conventions and all other parameters are the same as in Fig. 4, but for $F_{\text{ext}}=-F_{\text{w}}$.

Figure 9

(Color online) $\theta =\left(1+{10}^{1∕2}\right)∕3$ incommensurate soft chain $K=5$ (a) and hard chain $K=50$ (b): irregular alternation of kinks and bi-kinks. Pinning is realized for the soft chain, while even with $F_{\text{ext}}=-F_{\text{w}}$, so that $v_{\text{cm}}\simeq v_{\text{plateau}}$, the $K=50$ hard chain is unpinned, with the defects slowly drifting along. The conventions and all other parameters are the same as in Fig. 4.

Figure 10

(Color online) (a) Hysteresis loop at the plateau edge in the $v_{\text{cm}}-v_{\text{ext}}$ characteristics for a confined chain of length ratios $r_{+}=\left(1+{\pi }^{1∕4}\right)∕2$, $\theta =1$ $\left(r_{-}=7.036\right)$. Adiabatic increase and decrease of $v_{\text{ext}}$ are denoted by crosses and circles, respectively. Here $\gamma =0.1{\left(F_{+}m∕a_{+}\right)}^{1∕2}$ and $F_{\text{ext}}=0$, which corresponds to the dashed path of Fig. 7. (b) No hysteresis is observed in the overdamped regime $\left[\gamma =1.0{\left(F_{+}m∕a_{+}\right)}^{1∕2}\right]$ along the same path.