Onset of Sliding Friction in Incommensurate Systems

Abstract

We study the dynamics of an incommensurate chain sliding on a periodic lattice, modeled by the Frenkel Kontorova hamiltonian with initial kinetic energy, without damping and driving terms. We show that the onset of friction is due to a novel kind of dissipative parametric resonances, involving several resonant phonons which are driven by the (dissipationless) coupling of the center of mass motion to the phonons with wavevector related to the modulating potential. We establish quantitative estimates for their existence in finite systems and point out the analogy with the induction phenomenon in Fermi-Ulam-Pasta lattices.PACS numbers: 45.05.+x, 46.55.+d, 46.40.Ff The possibility of measuring friction at the atomic level provided by the Lateral Force Microscopes [1] and Quartz Crystal Microbalance [2] has stimulated intense research on this topic [3]. Phonon excitations are the dominant cause of friction in many cases [4]. Most studies are carried out for one-dimensional non-linear lattices [5][6][7][8][9][10][11][12] and in particular for the Frenkel-Kontorova (FK) model [13], where the surface layer is modeled by a harmonic chain and the substrate is replaced by a rigid periodic modulation potential. The majority [6][7][8][9][10][11][12] examines the steady state of the dynamical FK model in presence of dissipation representing the coupling of phonons to other, undescribed degrees of freedom.We study the dynamics of an undriven incommensurate FK chain. Our aim is to ascertain whether the experimentally observed superlubricity [14] can be due to the blocking of the phonon channels caused by an incommensurate contact of the two sliding surfaces. Therefore we do not include any explicit damping of the phonon modes, since we wish to find out if they can be excited at all by the motion of the center of mass (CM). In an earlier study, Shinjo and Hirano [5] found a superlubric regime for this model, where the chain would slide indefinitely without dynamic friction but with a recurrent exchange of kinetic energy between CM and a single internal mode. We will show that their finding is oversimplified by either too short simulation times or too small system sizes. The inherent non-linear coupling of the CM to the phonons leads to an irreversible decay of the CM velocity, albeit with very long time scales in some windows. The dissipative mechanism is driven by the coupling of the CM to the modes with modulation wavevector q or its harmonics, ω nq , and consists in a novel kind of parametric resonances with much wider windows of instabilities than those deriving from the standard Mathieu equation [15]. The importance of harmonic resonances at ω nq has been pointed out before [6,8,10], with the suggestion [10] that they could be absent in finite systems due to the discreteness of the phonon spectrum. However, it has not been realized that they act as a driving term for the onset of dissipation via subsequent complex parametric excitations which we shall describe, establishing quantitative estimates for their existence in ...

Predicting the lifetime of superlubricity

“…However, no proof has yet been available to establish the superlubric state in a practical length and time scale for real-world applications. Theoretical studies [7][8][9] show that even for atomistically smooth surfaces in contact, catastrophic breakdown of the superlubric state could occur by leaking a significant amount of kinetic energy in the mechanical motion of operation to the rest or, in other words, heat. Based on their studies on a Frenkel-Kontorova (F-K) chain model, Consoli et al [7] concluded that dissipative parametric resonance between vibrational modes in the chain results in the onset of friction in an incommensurate system.…”

“…Theoretical studies [7][8][9] show that even for atomistically smooth surfaces in contact, catastrophic breakdown of the superlubric state could occur by leaking a significant amount of kinetic energy in the mechanical motion of operation to the rest or, in other words, heat. Based on their studies on a Frenkel-Kontorova (F-K) chain model, Consoli et al [7] concluded that dissipative parametric resonance between vibrational modes in the chain results in the onset of friction in an incommensurate system. This phenomenon was later observed in the relative sliding of concentric graphitic walls in double-walled carbon nanotubes (DWCNTs) [8], and the origin of the identified catastrophic kinetic energy leaking is attributed to the resonant coupling between the translational motion and radial breathing modes.…”

Dynamical transitions in incommensurate systems

“…-The Frenkel-Kontorova (FK) model [1] describes the interaction of a harmonic chain of atoms with a rigid substrate with period incommensurate to the lattice parameter of the chain. Its generality makes it a powerful model to investigate many different physical systems [2,3], and in particular microscopic friction between contacting surfaces [4][5][6]. The static version of the model is characterized by the Aubry transition [7], from a floating to a pinned state, for a critical value λ c of the substrate modulation potential.…”

“…The static version of the model is characterized by the Aubry transition [7], from a floating to a pinned state, for a critical value λ c of the substrate modulation potential. Using the undamped dynamical version of this model, we addressed in previous papers the topic of "dissipation" (in the sense of transfer of energy from the center of mass to phonon modes) in incommensurate structures: we have studied the mechanism (parametric resonances) that governs the onset of sliding friction [6], and the conditions under which a new conserved quantity can be defined, which can be seen as a Generalized Angular Momentum (GAM) in the complex plane [8].In this work, we present new results, showing that, in the dynamics, a floating-to-pinned transition, analogous to the static Aubry transition, is found for all values of the potential λ < λ c . The transition is characterized by a region of critical times, with a remarkably complex behavior.…”

“…The ratio between l and the modulation potential period m is chosen to be an irrational number (in our case, m = 1 and l = τ = ( √ 5 + 1)/2, the golden mean), in order to simulate an incommensurate (IC) modulated structure. Numerically, we use a finite chain of length N l and use ratio of consecutive Fibonacci numbers to approximate incommensurability [6]. The Aubry transition can be monitored by the modulation function (see e.g.…”