^{1}, Rosario Capozza

^{2}, Andrea Vanossi

^{2,3,a)}, Giuseppe E. Santoro

^{3,4}, Nicola Manini

^{1,3}and Erio Tosatti

^{3,4}

### Abstract

In the framework of Langevin dynamics, we demonstrate clear evidence of the peculiar quantized sliding state, previously found in a simple one-dimensional boundary lubricated model [A. Vanossi *et al.*, Phys. Rev. Lett.97, 056101 (2006)], for a substantially less idealized two-dimensional description of a confined multilayer solid lubricant under shear. This dynamical state, marked by a nontrivial “quantized” ratio of the averaged lubricant center-of-mass velocity to the externally imposed sliding speed, is recovered, and shown to be robust against the effects of thermal fluctuations, quenched disorder in the confining substrates, and over a wide range of loading forces. The lubricant softness, setting the width of the propagating solitonic structures, is found to play a major role in promoting in-registry commensurate regions beneficial to this quantized sliding. By evaluating the force instantaneously exerted on the top plate, we find that this quantized sliding represents a dynamical “pinned” state, characterized by significantly low values of the kinetic friction. While the quantized sliding occurs due to solitons being driven gently, the transition to ordinary unpinned sliding regimes can involve lubricant melting due to large shear-induced Joule heating, for example at large speed.

This work was supported by CNR, as a part of the European Science Foundation EUROCORES Programme FANAS. R.C. and A.V. acknowledge gratefully the financial support by the Regional Laboratory InterMech—NetLab “Surfaces & Coatings for Advanced Mechanics and Nanomechanics” (SUP&RMAN), and of the European Commissions NEST Pathfinder program TRIGS under Contract No. NEST-2005–PATHCOM-043386.

I. INTRODUCTION

II. THE 2D CONFINED LUBRICATED MODEL

III. THE ROBUSTNESS OF THE PLATEAU DYNAMICS

IV. SOLITON COVERAGE AND LUBRICANT SOFTNESS

V. FRICTION

VI. INTERMITTENT DYNAMICS AT THE PLATEAU STATE

VII. DISCUSSION AND CONCLUSIONS

### Key Topics

- Friction
- 27.0
- Thermal models
- 7.0
- Brownian dynamics
- 5.0
- Langevin equation
- 5.0
- Boltzmann equations
- 4.0

## Figures

A sketch of the model with the rigid top (solid circles) and bottom (open) crystalline sliders (of lattice spacing and , respectively), the former moving at externally imposed -velocity . One or more solid lubricant layers (shadowed) of rest equilibrium spacing are confined in between.

A sketch of the model with the rigid top (solid circles) and bottom (open) crystalline sliders (of lattice spacing and , respectively), the former moving at externally imposed -velocity . One or more solid lubricant layers (shadowed) of rest equilibrium spacing are confined in between.

The time-averaged velocity ratio of a single lubricant layer as a function of the adiabatically increased top-slider velocity for different temperatures of the Langevin thermostat [panel (a)] and for distinct degrees of quenched disorder in the bottom substrate [panel (b)]; atomic random displacements are taken in a uniform distribution in the interval horizontally and vertically away from the ideal positions of Eq. (4). All simulations are carried out with a model composed by 4, 29, and 25 atoms in the top lubricant and bottom layers respectively, with an applied load . The plateau velocity ratio (dot-dashed line) is [Eq. (11)].

The time-averaged velocity ratio of a single lubricant layer as a function of the adiabatically increased top-slider velocity for different temperatures of the Langevin thermostat [panel (a)] and for distinct degrees of quenched disorder in the bottom substrate [panel (b)]; atomic random displacements are taken in a uniform distribution in the interval horizontally and vertically away from the ideal positions of Eq. (4). All simulations are carried out with a model composed by 4, 29, and 25 atoms in the top lubricant and bottom layers respectively, with an applied load . The plateau velocity ratio (dot-dashed line) is [Eq. (11)].

The critical depinning velocity for the breakdown of the quantized plateau as a function of the applied load per top-layer atom . For the one-layer curves, the same geometry as in Fig. 2, temperatures (circles) and 0.25 (diamonds) are considered. The two-layer curve (squares) is computed with a doubled number of lubricant atoms (58 rather than 29) confined between the same substrates in the same -range.

The critical depinning velocity for the breakdown of the quantized plateau as a function of the applied load per top-layer atom . For the one-layer curves, the same geometry as in Fig. 2, temperatures (circles) and 0.25 (diamonds) are considered. The two-layer curve (squares) is computed with a doubled number of lubricant atoms (58 rather than 29) confined between the same substrates in the same -range.

Variation of the plateau boundary velocity as a function of the kink coverage for a lubricant monolayer and bilayer. Calculations show local maxima of for commensurate values of for both and 2; except at kink coverage is generally larger for than . Simulations are carried out with , , and .

Variation of the plateau boundary velocity as a function of the kink coverage for a lubricant monolayer and bilayer. Calculations show local maxima of for commensurate values of for both and 2; except at kink coverage is generally larger for than . Simulations are carried out with , , and .

Variation of the plateau boundary velocity as a function of the kink coverage for a lubricant monolayer showing different interaction energies with the substrates. Simulations are carried out with the same conditions as in Fig. 4, but with varied , (but unchanged interparticle interaction ).

Variation of the plateau boundary velocity as a function of the kink coverage for a lubricant monolayer showing different interaction energies with the substrates. Simulations are carried out with the same conditions as in Fig. 4, but with varied , (but unchanged interparticle interaction ).

Snapshot of the dynamically pinned state of the 2D lubricant, showing the bond lengths as a function of the bond horizontal position along the slider. Shorter bonds identify kink regions, while longer bonds belong to in-register regions. Larger interaction with the substrates favors the commensurate in-register region.

Snapshot of the dynamically pinned state of the 2D lubricant, showing the bond lengths as a function of the bond horizontal position along the slider. Shorter bonds identify kink regions, while longer bonds belong to in-register regions. Larger interaction with the substrates favors the commensurate in-register region.

Critical depinning velocity as a function of the numbers of lubricant layers. All simulations are carried out in a condition that favors the quantized-velocity sliding state: the model is composed by 4, , and 25 atoms in the top lubricant and bottom layers respectively (thus ), with an applied load and . The data show an optimal dynamical pinning at and a tendency for to drop considerably as the lubricant thickness increases beyond that value. For no quantized plateau could be detected.

Critical depinning velocity as a function of the numbers of lubricant layers. All simulations are carried out in a condition that favors the quantized-velocity sliding state: the model is composed by 4, , and 25 atoms in the top lubricant and bottom layers respectively (thus ), with an applied load and . The data show an optimal dynamical pinning at and a tendency for to drop considerably as the lubricant thickness increases beyond that value. For no quantized plateau could be detected.

The tribological properties of the same model as in Fig. 2. As a function of the top-layer velocity adiabatically increased (circles) or decreased (squares) the three panels report: (a) the average velocity ratio , compared to the plateau value , Eq. (11); (b) the average friction force experienced by the top layer; and (c) the effective lubricant temperature, computed using the average kinetic energy in the frame of reference of the instantaneous lubricant CM [Eq. (23)].

The tribological properties of the same model as in Fig. 2. As a function of the top-layer velocity adiabatically increased (circles) or decreased (squares) the three panels report: (a) the average velocity ratio , compared to the plateau value , Eq. (11); (b) the average friction force experienced by the top layer; and (c) the effective lubricant temperature, computed using the average kinetic energy in the frame of reference of the instantaneous lubricant CM [Eq. (23)].

Results of a model composed by 4, 21, and 25 atoms in the top, lubricant and bottom chains, , which according to Eq. (11), produces perfectly quantized dynamics at a *negative* , a dot-dashed line in panel a: this backward lubricant motion is caused by forward-dragged antikinks. The other simulation parameters are , . As a function of the top-layer velocity adiabatically increased (circles) or decreased (squares) the three panels report: (a) the average velocity ratio ; (b) the average friction force experienced by the top layer; and (c) the effective lubricant temperature [Eq. (23)].

Results of a model composed by 4, 21, and 25 atoms in the top, lubricant and bottom chains, , which according to Eq. (11), produces perfectly quantized dynamics at a *negative* , a dot-dashed line in panel a: this backward lubricant motion is caused by forward-dragged antikinks. The other simulation parameters are , . As a function of the top-layer velocity adiabatically increased (circles) or decreased (squares) the three panels report: (a) the average velocity ratio ; (b) the average friction force experienced by the top layer; and (c) the effective lubricant temperature [Eq. (23)].

The average velocity ratio as a function of the adiabatically increased (circles) or decreased (squares) top-layer velocity . The model parameters are the same as in Fig. 8, but with no thermal effects .

The average velocity ratio as a function of the adiabatically increased (circles) or decreased (squares) top-layer velocity . The model parameters are the same as in Fig. 8, but with no thermal effects .

The time evolution of the average velocity ratio and the corresponding kinetic friction for the four dynamical states [(a)–(d)] marked in Fig. 10. The first three panels, referring to quantized sliding states, display a typical intermittent stick-slip dynamics with small amplitude fluctuations; the last panel exhibits large chaotic jumps in both and , as typical of the “hot” high-speed nonquantized sliding. The dot-dashed lines highlight the quantized plateau value .

The time evolution of the average velocity ratio and the corresponding kinetic friction for the four dynamical states [(a)–(d)] marked in Fig. 10. The first three panels, referring to quantized sliding states, display a typical intermittent stick-slip dynamics with small amplitude fluctuations; the last panel exhibits large chaotic jumps in both and , as typical of the “hot” high-speed nonquantized sliding. The dot-dashed lines highlight the quantized plateau value .

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