Static Friction on the Fly: Velocity Depinning Transitions of Lubricants in Motion

The dragging velocity of a model solid lubricant confined between sliding periodic substrates exhibits a phase transition between two regimes, respectively, with quantized and with continuous lubricant center-of-mass velocity. The transition, occurring for increasing external driving force $F_{\mathrm{ext}}$ acting on the lubricant, displays a large hysteresis, and has the features of depinning transitions in static friction, only taking place on the fly. Although different in nature, this phenomenon appears isomorphic to a static Aubry depinning transition in a Frenkel-Kontorova model, the role of particles now taken by the moving kinks of the lubricant-substrate interface. We suggest a possible realization in 2D optical lattice experiments.

Figure 1

Hysteresis in the $v_{\mathrm{cm}}-F_{\mathrm{ext}}$ characteristics for $(r_{+},r_{-})=(\varphi ,{\varphi }^2)$. Throughout, our units are $m=1$, $a_{+}=1$, $F_{+}=2\pi U_{+}/a_{+}=1$, $K$ is measured in $F_{+}/a_{+}$, $\gamma$ in $\sqrt{m{F}_{+}/a_{+}}$, velocities in $\sqrt{a_{+}{F}_{+}/m}$, and we choose $F_{-}=2\pi U_{-}/a_{-}=F_{+}$ [8]. The behavior is shown for fast ($v_{\mathrm{ext}}=0.1$, upper panel) and slow ($v_{\mathrm{ext}}=0.01$, lower panel) drive. Adiabatic increase and decrease of $F_{\mathrm{ext}}$ is denoted by triangles and circles, respectively. A characteristic multistep feature appears when decreasing adiabatically $F_{\mathrm{ext}}$. Here $\gamma =0.1$ and $K=4$.

Figure 2

(a) The critical depinning ($F_c^{+\uparrow }$, $F_c^{-\downarrow }$) and repinning ($F_c^{+\downarrow }$, $F_c^{-\uparrow }$) forces, as functions of the chain stiffness $K$, for $(r_{+},r_{-})=(\varphi ,{\varphi }^2)$ (i.e., commensurate $\theta =1$, solid symbols, pinning extending to infinitely large $K$) and for $(r_{+},r_{-})=(\varphi ,{\varphi }^3)$ (i.e., incommensurate $\theta =\varphi$, open symbols). In all calculations $v_{\mathrm{ext}}=0.1$, $\gamma =0.1$. For $\theta =1$: (b) the width $F_c^{+\uparrow }-F_c^{+\downarrow }$ of the bistability region as a function of ($K_{*}-K$) (crosses), compared to a fitted power law $F_c^{+\uparrow }-F_c^{+\downarrow }=B(K_{*}-K{)}^{\alpha }$, with $\alpha \simeq 2.7$, $K_{*}\simeq 135$; (c) the large-$K$ depinning force, compared to the expected power law $(F_c^{+\uparrow }+F_w)=A{K}^{-1}$, Eq. (3).

Figure 3

Time evolution of the positions $x_i-v_{\mathrm{plateau}}·t$ of 7 contiguous chain particles. $t$ is measured in units of $\sqrt{m{a}_{+}/F_{+}}$. The plots refer to three dynamical regimes observed along the adiabatic decrease of $F_{\mathrm{ext}}$, at the points indicated by (3a)–(3c) in Fig. 1: (a) free sliding at $F_{\mathrm{ext}}=0.07$, (b) dynamic stick slip at $F_{\mathrm{ext}}=0.045$, and (c) quantized sliding state at $F_{\mathrm{ext}}=0.02$. Same parameters as in Fig. 1, with $v_{\mathrm{ext}}=0.1$.

Figure 4

Hysteresis, as $v_{\mathrm{ext}}$ is cycled, in the underdamped regime.