Emergent friction in the conservative incommensurate Frenkel–Kontorova model: the low-velocity limit and the phonon-resonance peak structure

In the Hamiltonian Frenkel–Kontorova (FK) model with an incommensurate golden-mean contact and coupling below the Aubry threshold, static friction vanishes, yet sliding is not frictionless: parametric excitation of the chain's phonons converts center-of-mass kinetic energy into internal vibrations, and at finite temperature the resulting dynamics is effectively viscous, with a damping coefficient that is non-monotonic in velocity, peaks near phonon resonances, grows with temperature, and — in the simulations of van den Ende, de Wijn and Fasolino (2012) — approaches a finite nonzero constant as the sliding velocity tends to zero, implying preservation of superlubricity. Both statements rest on finite-time, finite-size simulations. Two questions remain open: (i) the asymptotic low-velocity limit — whether the emergent damping genuinely survives at arbitrarily long times and large system sizes, or whether very-low-velocity sliding is either eventually arrested by parametric phonon growth on time scales beyond simulation reach or exactly superlubric; and (ii) a microscopic quantitative theory of the damping coefficient, in particular of the measured rightward shift of the resonance peaks relative to the free-chain resonance condition. Later work has confirmed the phenomenology in two dimensions and with defects, produced linear-response theories for related slider–chain geometries and for thermostatted superlubric 2D interfaces, and extended experiments to 0.1 mm/s, but neither the conservative-chain asymptotic question nor the peak-shift theory has been settled.

Background

Friction between two atomically flat crystalline surfaces depends sensitively on the geometric relation of their lattice periodicities. When the ratio of the periods is irrational (an incommensurate contact), atoms of one surface explore all relative positions against the other, and the lateral force can average to zero. Shinjo and Hirano, who coined the term superlubricity, studied this in the Frenkel–Kontorova (FK) model — a chain of particles of mass $m$ coupled by harmonic springs of stiffness $K$, resting on a rigid periodic substrate potential of period $a_s$ — and proposed a phase diagram in the plane of coupling versus sliding velocity in which a weakly coupled incommensurate chain, once set into motion, first becomes frictional and then superlubric again at higher velocities.

The FK model has a dimensionless coupling $\lambda = U_0/(K a_s^2)$, where $U_0$ measures the substrate potential depth. A structural transition (the Aubry transition) separates a pinned phase with static friction from a floating phase without static friction; the critical coupling $\lambda_c$ is largest when the period ratio equals the golden mean, $\lambda_c \approx 0.1546$, and numerical work realizes this ratio by finite chains whose particle and substrate periods are successive Fibonacci numbers. Static friction is the threshold force needed to start sliding; kinetic friction is the force opposing motion once sliding. Whether vanishing static friction implies vanishing kinetic friction is a separate dynamical question. Consoli, Knops and Fasolino showed that the superlubric sliding found in early studies is transient: the oscillating center of mass parametrically excites long-wavelength acoustic phonons (vibrational normal modes) of the chain, which grow exponentially and eventually convert the center-of-mass kinetic energy into internal vibrations, arresting the chain. In a chain sliding at velocity $p_0$ the substrate modulation acts at the washboard frequency $\Omega = 2\pi p_0$, and strong parametric resonances occur when $\Omega$ matches combinations of phonon frequencies — approximately $n p_0 \sim \omega_{nq}/(2\pi)$ with $q = 2\pi a_0/a_s$ the wavevector of the substrate modulation, $\omega_k$ the phonon dispersion of the free ($\lambda = 0$) chain, and $n$ a positive integer. In a driven, damped FK model at zero temperature, Strunz and Elmer found peaks of the inherent friction precisely at these resonance velocities.

van den Ende, de Wijn and Fasolino studied the conservative (undriven, undamped, purely Hamiltonian) incommensurate FK model at finite temperature. The chain is first brought to thermal equilibrium by the sliding itself (verified through Maxwell–Boltzmann statistics of the center-of-mass velocity and equipartition of energy over the phonon modes); a small uniform velocity increment $p^+$ is then added, and the center-of-mass velocity, averaged over many equilibrium configurations, decays exponentially, $\langle P(\tau)\rangle \propto e^{-\eta\tau}$, defining an effective viscous damping coefficient $\eta$ that emerges from the microscopic dynamics without any phenomenological friction term — behavior expected on general statistical-mechanical grounds for many-particle conservative systems. They found that $\eta$ is non-monotonic in velocity, with peaks near the phonon resonances that broaden with temperature; that $\eta$ grows with temperature, because thermal fluctuations distort the incommensurate contact; and that at low velocities $\eta$ approaches a finite nonzero constant, so the friction force $F_{\rm fric} = -\eta M P$ vanishes linearly as $v \to 0$ and superlubricity is preserved in the zero-velocity limit. Parameter estimates for Xe monolayers on Ag(111) place the resonance features at sliding velocities of order 30–70 m/s, far above atomic-force-microscope speeds and at the limit of quartz-crystal-microbalance experiments.

Both conclusions rest on finite-time, finite-size simulations ($N = 144$ particles). The onset time of parametric growth grows rapidly at lower velocities, and the simulations cannot determine the fate of sliding below the lowest accessible resonance: the authors state verbatim that they cannot rule out superlubric motion at very low velocities where the parametric growth of internal vibrations could occur either on time scales too long for the simulations or not occur at all. The positions of the friction peaks are shifted to velocities higher than the resonance condition above, which uses the free-chain phonon dispersion; the authors conjecture that the shift reflects finite-coupling corrections to the dispersion and support this with a single observation at smaller $\lambda$, but give no quantitative theory. Real-system complications — surface roughness, electronic friction, three-dimensional geometry, and overdamped colloidal realizations of the FK model — are excluded by construction; for colloidal FK realizations with overdamped phonon modes see Bohlein, Mikhael and Bechinger, and for the stability of superlubric sliding of graphite flakes against rotations see de Wijn, Fusco and Fasolino.

Problem Statement

Determine, from the microscopic Hamiltonian dynamics of the conservative (undriven, undamped) incommensurate Frenkel–Kontorova chain in its floating regime below the Aubry transition, the asymptotic form of the emergent friction at low sliding velocity and its resonant fine structure. Two linked deliverables define completion.

(1) Low-velocity limit and asymptotic stability. Establish whether the effective damping coefficient $\eta(v,T)$ — defined by the exponential decay of the ensemble-averaged center-of-mass velocity after a small uniform velocity increment $p^+$ is applied to a chain in thermal equilibrium at temperature $T$ — remains finite and nonzero as $p^+ \to 0$ at each $T > 0$, so that the friction force $F_{\rm fric} = -\eta M P$ vanishes linearly with velocity and superlubricity is preserved. This requires settling the fate of sliding at velocities below the lowest parametric resonance: does parametric phonon growth eventually arrest the chain on time scales that diverge as $v \to 0$ (making the numerically observed low-velocity plateau of $\eta$ a transient), or does a genuinely non-decaying superlubric regime exist there? The answer must specify whether the viscous regime is stable under extension of the observation time and under growth of the system size (the thermodynamic limit along Fibonacci approximants of the golden mean), and how the $T \to 0$ and $v \to 0$ limits commute or fail to commute.

(2) Quantitative theory of the resonance structure of $\eta(v,T)$. Derive the velocity and temperature dependence of $\eta$ near the phonon resonances — in particular the positions of the friction peaks — from the microscopic dynamics without approximating the phonon dispersion by its free-chain ($\lambda = 0$) form. The theory must reproduce the observed rightward shift of the peaks relative to the resonance condition $n p_0 \sim \omega_{nq}/(2\pi)$ with $q = 2\pi a_0/a_s$, its dependence on the coupling $\lambda$, and the thermal broadening of the peaks.

Deliverables (1) and (2) are two facets of one overarching objective, a microscopic quantitative theory of the emergent friction coefficient $\eta(v,T)$ of the conservative incommensurate Frenkel–Kontorova chain: (1) fixes its low-velocity asymptotics and stability, (2) its resonant structure. The model deliberately excludes roughness, electronic friction, and other external dissipation channels; extending to those channels, to overdamped colloidal platforms, or to experimental observation of the resonance regime are recorded as separate follow-on questions, not parts of this problem's completion boundary.

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