Frenkel-Kontorova Model Research Articles

Shapiro steps in the commensurate structures with integer value of the winding number

Abstract Dynamical mode locking phenomena and the appearance of Shapiro steps are studied in commensurate structures with integer values of winding number in the dc- and ac-driven overdamped Frenkel-Kontorova model. While in the standard case with sinusoidal substrate potential, the system reduces to the single particles model in which only harmonic steps exist and analytical form for the step size can be revealed, in the case of deformable potential, the presence of many degrees of freedom strongly influences the Shapiro steps. Whole series of subharmonic steps appear, and the two types of response functions, the one for the commensurate structures with odd and the one for the commensurate structures with even winding number have been observed.

Predicting the lifetime of superlubricity

The concept of superlubricity has recently called upon notable interest after the demonstration of ultralow friction between atomistically smooth surfaces in layered materials. However, the energy dissipation process conditioning the sustainability of a superlubric state has not yet been well understood. In this work, we address this issue by performing dynamic simulations based on both full-atom and reduced Frenkel-Kontorova models. We find that the center-of-mass momentum autocorrelation of a sliding object can be used as a statistical indicator of the state of superlubricity. Beyond a critical value of it, the sliding motion experiences a catastrophic breakdown with a dramatically high rate of energy dissipation, caused by the inter-vibrational-mode coupling. By tracking this warning signal, one can extract heat from modes other than the translation to avoid the catastrophe and extend the lifetime of superlubricity. This concept is demonstrated in double-walled carbon-nanotubes–based nanomechanical devices with the indicator-based feedback design implemented.

Comment on “Lattice vibrations in the Frenkel-Kontorova model. I. Phonon dispersion, number density, and energy”

Comment on “Lattice vibrations in the Frenkel-Kontorova model. I. Phonon dispersion, number density, and energy”

Phonon drag force acting on a mobile crystal defect: Full treatment of discreteness and nonlinearity

Phonon scattering calculations predict the drag force acting on defects and dislocations rises linearly with temperature, in direct contradiction with molecular dynamics simulations that often finds the drag force to be independent of temperature. Using the Mori-Zwanzig projection technique, with no recourse to elasticity or scattering theories, we derive a general Langevin equation for a crystal defect, with full treatment of discreteness and non-linearity in the defect core. We obtain an analytical expression for the drag force that is evaluated in molecular statics and molecular dynamics, extracting the force on a defect directly from the inter-atomic forces. Our results show that a temperature independent drag force arises because vibrations in a discrete crystal are never independent of the defect motion, an implicit assumption in any phonon-based approach. This effect remains even when the Peierls barrier is effectively zero, invalidating qualitative explanations involving the radiation of phonons. We apply our methods to an interstitial defect in tungsten and solitons in the Frenkel-Kontorova model, finding very good agreement with trajectory-based estimations of the thermal drag force.

Secondary invariants of Birkhoff minimizers and heteroclinic orbits

For the generalized Frenkel–Kontorova model on high-dimensional lattices, we study the classification of Birkhoff minimizers with the same rotation vector which is rationally dependent. We introduce secondary invariants to distinguish Birkhoff minimizers with the same rotation vector and show that the collection of Birkhoff minimizers with the same secondary invariants is totally ordered. We also construct heteroclinic connections between neighboring Birkhoff minimizers which are more periodic.

Velocity tuning of friction with two trapped atoms

Friction is the basic, ubiquitous mechanical interaction between two surfaces that results in resistance to motion and energy dissipation. In spite of its technological and economic significance, our ability to control friction remains modest, and our understanding of the microscopic processes incomplete. At the atomic scale, mismatch between the two contacting crystal lattices can lead to a reduction of stick-slip friction (structural lubricity), while thermally activated atomic motion can give rise to a complex velocity dependence, and nearly vanishing friction at sufficiently low velocities (thermal lubricity). Atomic force microscopy has provided a wealth of experimental results, but limitations in the dynamic range, time resolution, and control at the single-atom level have hampered a full quantitative description from first principles. Here, using an ion-crystal friction emulator with single-atom, single substrate-site spatial resolution and single-slip temporal resolution, we measure the friction force over nearly five orders of magnitude in velocity, and contiguously observe four distinct regimes, while controlling temperature and dissipation. We elucidate the interplay between thermal and structural lubricity in a system of two coupled atoms, and provide a simple explanation in terms of the Peierls-Nabarro potential. This extensive control at the atomic scale paves the way for fundamental studies of the interaction of many-atom surfaces, as for example in the Frenkel-Kontorova model, and possibly into the quantum regime.

Experimental Study of Spectral Properties of a Frenkel-Kontorova System.

We report on microwave emission from linear parallel arrays of underdamped Josephson junctions, which are described by the Frenkel-Kontorova (FK) model. Electromagnetic radiation is detected from the arrays when biased on current singularities (steps) appearing at voltages V(n)=Φ(0)(nc̅/L), where Φ(0)=2.07×10(-15) Wb is the magnetic flux quantum, and c̅, L, and n are, respectively, the speed of light in the transmission line embedding the array, L its physical length, and n an integer. The radiation, detected at fundamental frequency c̅/2L when biased on different singularities, indicates shuttling of bunched 2π kinks (magnetic flux quanta). Resonance of flux-quanta motion with the small-amplitude oscillations induced in the arrays gives rise to fine structures in the radiation spectrum, which are interpreted on the basis of the FK model describing the resonance. The impact of our results on design and performances of new digital circuit families is discussed.

Theory of the deformation of aligned polyethylene.

Solitons are proposed as the agents of plastic and viscoelastic deformation in aligned polyethylene. Interactions between straight, parallel molecules are mapped rigorously onto the Frenkel–Kontorova model. It is shown that these molecular interactions distribute an applied load between molecules, with a characteristic transfer length equal to the soliton width. Load transfer leads to the introduction of tensile and compressive solitons at the chain ends to mark the onset of plasticity at a well-defined yield stress, which is much less than the theoretical pull-out stress. Interaction energies between solitons and an equation of motion for solitons are derived. The equation of motion is based on Langevin dynamics and the fluctuation–dissipation theorem and it leads to the rigorous definition of an effective mass for solitons. It forms the basis of a soliton dynamics in direct analogy to dislocation dynamics. Close parallels are drawn between solitons in aligned polymers and dislocations in crystals, including the configurational force on a soliton. The origins of the strain rate and temperature dependencies of the viscoelastic behaviour are discussed in terms of the formation energy of solitons. A failure mechanism is proposed involving soliton condensation under a tensile load.

Stability of synchronization in dissipatively driven Frenkel-Kontorova models.

We rigorously show that dissipatively driven Frenkel-Kontorova models with either uniform or time-periodic driving asymptotically synchronize for a wide range of initial conditions. The main tool is a new Lyapunov function, as well as a 2D representation of the attractor. We then characterize dynamical phase transitions and outline new algorithms for determining them.

Minimizers with Bounded Action for the High-Dimensional Frenkel–Kontorova Model

In Aubry–Mather theory for monotone twist maps or for one-dimensional Frenkel–Kontorova (FK) model with nearest neighbor interactions, each global minimizer (minimal energy configuration) is naturally Birkhoff. However, this is not true for the one-dimensional FK model with non-nearest neighbor interactions or for the high-dimensional FK model. In this paper, we study the Birkhoff property of minimizers with bounded action for the high-dimensional FK model.

Rectification and phase locking of graphite

Rectification phenomena and the phase locking in a two-dimensional overdamped Frenkel–Kontorova model with a graphite periodic substrate were studied. The presence of dc and ac forces in the longitudinal direction causes the appearance of dynamicalmode locking and the steps in the response function of the system. On the other hand, the presence of an ac force in the transverse direction causes the appearance of rectification,even though there is no net dc force in the transverse direction. It is found that whereas the longitudinal velocity increases in a series of steps, rectification in the transverse direction can occur only between two neighbor steps. The amplitude and phase of the external ac driving force affect the depinning force,rectification of the system and particles trajectories.

Lattice vibrations in the Frenkel-Kontorova model. I. Phonon dispersion, number density, and energy

We studied the lattice vibrations of two inter-penetrating atomic sublattices via the Frenkel-Kontorova (FK) model of a linear chain of harmonically interacting atoms subjected to an on-site potential, using the technique of thermodynamic Green's functions based on quantum field-theoretical methods. General expressions were deduced for the phonon frequency-wave-vector dispersion relations, number density, and energy of the FK model system. In addition, as the application of the theory, we investigated in detail cases of linear chains with various periods of the on-site potential of the FK model. Some unusual but interesting features for different amplitudes of the on-site potential of the FK model are discussed. In the commensurate structure, the phonon spectrum always starts at a finite frequency, and the gaps of the spectrum are true ones with a zero density of modes. In the incommensurate structure, the phonon spectrum starts from zero frequency, but at a non-zero wave vector; there are some modes inside these gap regions, but their density is very low. In our approximation, the energy of a higher-order commensurate state of the one-dimensional system at a finite temperature may become indefinitely close to the energy of an incommensurate state. This finding implies that the higher-order incommensurate-commensurate transitionsmore » are continuous ones and that the phase transition may exhibit a “devil's staircase” behavior at a finite temperature.« less

Lattice vibrations in the Frenkel-Kontorova Model. II. Thermal conductivity

We applied the formulas for the phonon spectral-density function that we presented in the previous paper of this series to analyze the thermal conductivity of the lattice in the framework of the Frenkel-Kontorova (FK) model. We found that two extra mechanisms of phonon scattering (different from the point impurities, three-phonon processes, and boundary scattering typical of all crystals), viz., resonance, and anharmonic scattering, that mainly influences the thermal conductivity of the lattice. The frequencies of resonance scattering are discrete, and their number increases from a finite number to infinity with their transition from the commensurate to the incommensurate state. Changing the amplitude and period of the FK model changes the frequencies and the frequency number of resonance scattering and the intensity of anharmonic scattering. We analyze these changes in detail. Our theory can explain all existing numerical results on this problem and suggest strategies to reduce the thermal conductivity of the lattice of layered materials.

Farey sequence in the appearance of subharmonic Shapiro steps.

The largest Lyapunov exponent has been examined in the dynamical-mode locking phenomena of the ac+dc driven dissipative Frenkel-Kontorova model with deformable substrate potential. Due to deformation, large fractional and higher order subharmonic steps appear in the response function of the system. Computation of the largest Lyapunov exponent as a way to verify their presence led to the observation of the Farey sequence. In the standard regime, the appearance of half-integer and other subharmonic steps between the large harmonic steps, and their relative sizes follow the Farey construction. In the nonstandard regime, however, the half-integer steps are larger than harmonic ones, and Farey construction is only present in the appearance of higher order subharmonic steps. The examination of Lyapunov exponents has also shown that regardless of the substrate potential or deformation, there was no chaos in the system.

How Does Dissipation Affect the Transition from Static to Dynamic Macroscopic Friction?

Description of the transitional process from a static to a dynamic frictional regime is a fundamental problem of modern physics. Previously, we developed a model based on the well-known Frenkel-Kontorova model to describe dry macroscopic friction. Here, this model has been modified to include the effect of dissipation in derived relations between the kinematic and dynamic parameters of a transition process. The main (somewhat counterintuitive) result is a demonstration that the rupture (i.e., detachment front) velocity of the slip pulse which arises during the transition does not depend on friction. The only parameter (besides the elastic and plastic properties of the medium) controlling the rupture velocity is the shear to normal stress ratio. In contrast to the rupture velocity, the slip velocity does depend on friction. The model we have developed describes these processes over a wide range of rupture and slip velocities (up to 7 orders of magnitude) allowing, in particular, the consideration of seismic events ranging from regular earthquakes, with rupture velocities on the order of a few km/s, to slow slip events, with rupture velocities of a few km/day.

Discrete model of dislocations in fractional nonlocal elasticity

Discrete models of dislocations in fractional nonlocal materials are suggested. The proposed models are based on fractional-order differences instead of finite differences of integer orders that are usually used. The fractional differences allow us to describe long-range interactions in materials. In continuous limit the suggested discrete models give continuum models of dislocations in nonlocal continua. Fractional generalization of the Frenkel–Kontorova model by using long-range interactions is suggested. We also propose a fractional generalization of interacting atomic chains (IAC) model of dislocations by considering long-range interacting chains.

Propagation and Interaction of Edge Dislocation (Kink) in the Square Lattice**Supported by the National Magnetic Confinement Fusion Science Program of China under Grant No 2014GB104002, the Strategic Priority Research Program of Chinese Academy of Sciences under Grant No XDA03030100, the National Natural Science Foundation of China under Grant Nos 11275156 and 11304324, the Open Project Program of State Key Laboratory

The propagation of kink or edge dislocations in the underdamped generalized two-dimensional Frenkel–Kontorova model with harmonic interaction is studied with numerical simulations. The obtained results show that exactly one line of atoms can be inserted into the lattice, which remains at standstill. However, if more than one line of atoms are inserted into the lattice, then they will split into several lines with α = 1, where α presents the atoms inserted. In other words, only the kink with α = 1 is stable, while the other kinks are unstable, and will split into α = 1 kinks, which remain at standstill.

Huge metastable axial strain in ultrathin heteroepitaxial vertically aligned nanowires

Strain engineering is a powerful tool to tailor the physical properties of materials coherently stacked in an epitaxial heterostructure. Such an approach, applied to the mature field of planar heteroepitaxy, has yielded a variety of new phenomena and devices. Recently, heteroepitaxial vertically aligned nanocomposites have emerged as alternatives to planar structures. Owing to the peculiar geometry of such nanoarchitectures, efficient strain control can be achieved, opening the way to novel functionalities. In this paper, we report a very large tensile axial strain in epitaxial transition metal nanowires embedded in an oxide matrix. We show that axial strains in excess of 1.5% can be sustained over a large thickness (a few hundred nanometers) in epitaxial nanowires having ultrasmall diameters (∼3–6 nm). The axial strain depends on the diameter of the nanowires, reflecting its epitaxial nature and the balance of interface and elastic energies. Furthermore, it is experimentally shown that such strain is metastable, in agreement with the calculations performed in the framework of the Frenkel-Kontorova model. The diameter dependence and metastability provide effective ways to control the strain, an appealing feature for the design of functional nanoarchitectures.

Friction phenomena in the overdamped three-layer model.

An overdamped three-layer model consisting of two harmonic chains of interacting particles, representing the upper and the middle layers, which move over the substrate potential, is studied in the present paper. A dc+ac force is applied only on the upper harmonic chain, and dynamics of both layers are investigated. The results show that the dynamical mode locking and Shapiro steps appear not only in the upper layer but also in the middle one. It is noted that the motion of particles in the upper layer corresponds to the standard Frenkel-Kontorova model. The dependence of the Shapiro steps of the middle layer on the system parameters are determined. It is shown that the height of the first Shapiro step of the upper layer is unrelated to the interaction parameters of the particles of both the upper and the middle layers, while the height of the first Shapiro step of the middle layer depend only on the interaction parameters of the particles of the middle layers. Two critical forces which transfer from locked state to the sliding one of both the upper and the middle layers are also studied. They depend on the amplitude and the frequency of the external ac driving force.

Critical length limiting superlow friction.

Since the demonstration of superlow friction (superlubricity) in graphite at nanoscale, one of the main challenges in the field of nano- and micromechanics was to scale this phenomenon up. A key question to be addressed is to what extent superlubricity could persist, and what mechanisms could lead to its failure. Here, using an edge-driven Frenkel-Kontorova model, we establish a connection between the critical length above which superlubricity disappears and both intrinsic material properties and experimental parameters. A striking boost in dissipated energy with chain length emerges abruptly due to a high-friction stick-slip mechanism caused by deformation of the slider leading to a local commensuration with the substrate lattice. We derived a parameter-free analytical model for the critical length that is in excellent agreement with our numerical simulations. Our results provide a new perspective on friction and nanomanipulation and can serve as a theoretical basis for designing nanodevices with superlow friction, such as carbon nanotubes.