The breakdown of superlubricity by driving-induced commensurate dislocations

Abstract

In the framework of a Frenkel-Kontorova-like model, we address the robustness of the superlubricity phenomenon in an edge-driven system at large scales, highlighting the dynamical mechanisms leading to its failure due to the slider elasticity. The results of the numerical simulations perfectly match the length critical size derived from a parameter-free analytical model. By considering different driving and commensurability interface configurations, we explore the distinctive nature of the transition from superlubric to high-friction sliding states which occurs above the critical size, discovering the occurrence of previously undetected multiple dissipative jumps in the friction force as a function of the slider length. These driving-induced commensurate dislocations in the slider are then characterized in relation to their spatial localization and width, depending on the system parameters. Setting the ground to scale superlubricity up, this investigation provides a novel perspective on friction and nanomanipulation experiments and can serve as a theoretical basis for designing high-tech devices with specific superlow frictional features.

Figure 1

(a) Sketch of the Frenkel-Kontorova-like model discussed in the work. (b) Instantaneous friction force normalized by the average single-particle viscous force for different chain lengths. (c) Average friction force per particle normalized by the viscous force as a function of the chain length, the colored squares refer to panel (b). The inset shows the average friction force as a function of the chain length. The parameter values used in these simulations are u₀ = 0.02, η = 3.2 × 10⁻², v₀ = 1.6 × 10⁻² and k_dr = 0.1.

Figure 2. Breakdown of superlubricity for a chain of N = 1000 particles subjected to a pulling or pushing driving in overdense or underdense condition.

(a–d) average power dissipated P(i) by the i-th particle as a function of the particle position in the chain, the inset schematize the arrangement of the particles in the wall region. (e–h) color maps showing the behavior of the commensuration index d(i) in time. (i–l) two dimensional plots of d(i) in the steady sliding state. The model parameters are the same of Fig. 1, for the underdense case we have chosen while for the overdense one .

Figure 3

(a,c) Comparison between the average friction force per particle 〈F〉/N calculated numerically and given by eq. (8) for the pulling driving in the overdense and underdense case respectively. The fitting parameter is α₁ = 2.64 ± 0.11 for the overdense case and α₁ = 2.69 ± 0.15 for the underdense one. (b,d) Comparison between the critical number of particles N_cr calculated numerically and given by eq. (11) for the pulling driving in the overdense and underdense case respectively. Black squares, red circles, blue and green triangle are obtained by varying one parameter at a time in the range 0.02 < U₀ < 0.5, 5 < K < 100, 0.005 < V₀ < 0.1 and 0.01 < η < 0.2 keeping the others set to U₀ = 0.2, K = 10, V₀ = 0.05 and η = 0.1. The dashed black lines are the bisectors of the plots, i.e. the set of points where 〈F〉_num = 〈F〉_th and .

Figure 4

Comparison between the simulated F(t) and the theoretical prediction of from eq. (10) for different values of (a) K, (b) U₀. The curves show numerical results for F(t) and the dashed lines are the theoretical values from eq. (10). (a) The red, green and blue curves are calculated for K = 20 N = 2000, K = 10 N = 2000 and K = 5 N = 360 respectively. (b) The red, green and blue curves correspond to U₀ = 0.5 N = 450, U₀ = 0.2 N = 1000 and U₀ = 0.1 N = 1000 respectively.

Figure 5

Comparison between the simulated F(t) and the theoretical prediction of from eq. (10) for different values of (a) η, (b) V₀ and (c) N. The curves show numerical results for F(t) and the dashed lines are the theoretical values from eq. (10). (a) the black, green, red and blue curves are calculated for η = 0.2 N = 540, η = 0.1 N = 1000, η = 0.05 N = 2000 and η = 0.01 N = 9000 respectively. (b) The black, green, red and blue curves correspond to V₀ = 0.1 N = 560, V₀ = 0.05 N = 1000, V₀ = 0.03 N = 1600 and V₀ = 0.01 N = 4600 respectively. (c) The black, blue and green curves correspond to N = 1000, 2000 and 100000 respectively.

Figure 6

(a) Single particle potential energy U(i) as a function of the particle index for a pulled overdense chain with N = 1000. The three different plots correspond to δ = 7.0 × 10⁻⁴ (green), δ = 8.35 × 10⁻⁴ (red) and δ = 1.0 × 10⁻³ (blue). (b) Single particle force F^sub(i) obtained differentiating the curves of panel (a), to cancel the incoherent contributions and highlight only the regions where the force is significantly different from zero, the force value at every point i as been obtained averaging over the five neighboring points (average filtering). (c) Single particle potential energy U(i) for larger δ values showing the onset of multiple domain walls with a periodicity inversely proportional to . The three different plots correspond to δ = 3.0 × 10⁻³ (green), δ = 5.0 × 10⁻³ (red) and δ = 7.0 × 10⁻³ (blue). (d) Single particle force F^sub(i) for δ = 7.0 × 10⁻⁴ and different chain lengths. The three different plots correspond to N = 1000 (green), N = 4000 (red) and N = 8000 (blue).

Figure 7. Comparison of the numerically calculated slope δ with the theoretical prediction of eq. (14) with N_cr corresponding to the theoretical value of eq. (11).

Black squares, red circles, blue and green triangles obtained by varying 0.01 < V₀ < 0.03, 20 < K < 100, 0.02 < U₀ < 0.1 and 0.01 < η < 0.05 respectively while keeping the other parameters constant at the values U₀ = 0.2, K = 10, V₀ = 0.05 and η = 0.1.

Figure 8

(a) Width of the dissipation peak of Fig. 2(a) as a function of the chain and substrate stiffness. (b) Integral dissipated power as a function of the chain and substrate stiffness. Red circles and blue triangles are obtained by varying 5 < K < 100 and 0.02 < U₀ < 0.5 respectively while keeping constant the other parameters at the values U₀ = 0.2, K = 10, V₀ = 0.05 and η = 0.1.

Figure 9. Onset of a second domain wall during the pulling of a long overdense chain.

(a) Jumps in the total friction force as the commensurate domain wall regions nucleates at the driving edge. (b) Color map showing the time evolution of the commensuration index d(i). (c) Commensuration index d(i) in the steady sliding state. (d) Average power dissipated P(i) by the i-th particle as a function of the particle position in the chain. The parameters used in this simulation are the same as in Figs 1 and 2, only the total length has been increased up to N = 4000.