Frenkel-Kontorova Model Research Articles

Thermal rectification in segmented Frenkel–Kontorova lattices with asymmetric next-nearest-neighbor interactions

In this work, we conduct an extensive study of the asymmetric heat flow, i.e. thermal rectification, present in the two-segment Frenkel Kontorova model with both nearest-neighbor (NN) and next-nearest-neighbor (NNN) interactions. We have considered systems with both high and low asymmetry and determined that, in the weak-coupling limit, thermal rectification is larger when NNN interactions are relevant. The behavior of the heat fluxes as a function of the coupling strength between the two segments is largely consistent with a well-defined rectification for larger system sizes. The local heat fluxes present a very different behavior for systems with high and low asymmetry. The results of this work may help in the design of molecular bridges, which have recently been shown to be able to function as thermal rectification devices.

Rotation numbers and bounded deviations for quasi-periodic monotone recurrence relations

It is known that monotone recurrence relations are defined by finding equilibria of generalized Frenkel-Kontorova models. In this paper, we intend to study quasi-periodic monotone recurrence relations. By introducing a countable set consisting of integers with bounded distances and improving a method of Angenent used for studying periodic monotone recurrence relations, we derive a criterion for the existence of solutions with bounded deviations from the rigid rotation. With the help of the criterion, a range of parameters is provided to guarantee the quasi-periodic standard map has orbits with bounded deviations from the rigid rotation corresponding to each given rotation number.

Bicrystallography-informed Frenkel–Kontorova model for interlayer dislocations in strained 2D heterostructures

In recent years, van der Waals (vdW) heterostructures and homostructures, which consist of stacks of two-dimensional (2D) materials, have risen to prominence due to their association with exotic quantum phenomena originating from correlated electronic states harbored by them. Atomistic scale relaxation effects play an extremely important role in the electronic scale quantum physics of these systems, providing means of manipulation of these materials and allowing them to be tailored for emergent technologies. We investigate such structural relaxation effects in this work using atomistic and mesoscale models, within the context of twisted bilayer graphene — a well-known heterostructure system that features moiré patterns arising from the lattices of the two graphene layers. For small twist angles, atomic relaxation effects in this system are associated with the natural emergence of interface dislocations or strain solitons, which result from the cyclic nature of the generalized stacking fault energy (GSFE), that measures the interface energy based on the relative movement of the two layers. In this work, we first demonstrate using atomistic simulations that atomic reconstruction in bilayer graphene under a large twist also results from interface dislocations, although the Burgers vectors of such dislocations are considerably smaller than those observed in small-twist systems. To reveal the translational invariance of the heterointerface responsible for the formation of such dislocations, we derive the translational symmetry of the GSFE of a 2D heterostructure using the notions of coincident site lattices (CSLs) and displacement shift complete lattices (DSCLs). The workhorse for this exercise is a recently developed Smith normal form bicrystallography framework. Next, we construct a bicrystallography-informed and frame-invariant Frenkel–Kontorova model, which can predict the formation of strain solitons in arbitrary 2D heterostructures, and apply it to study a heterostrained, large-twist bilayer graphene system. Our mesoscale model is found to produce results consistent with atomistic simulations. We anticipate that the model will be invaluable in predicting structural relaxation and for providing insights into various heterostructure systems, especially in cases where the fundamental unit cell is large and therefore, atomistic simulations are computationally expensive.

Controlled synchronization of coupled pendulums by Koopman Model Predictive Control

We propose a method to solve a class of control problems arising from a system of coupled pendulums. The system considered in this work is a one-dimensional array of pendulums pivoting around a single axis with adjacent pendulums coupled through torsion springs. Only a single torque motor attached to one of the two boundary pendulums actuates the system. This setup of coupled pendulums is a mechanical realization of the Frenkel–Kontorova (FK) model – a spatially discrete version of the sine-Gordon equation describing (nonlinear) waves. The main challenges of controlling this system are high order (the number of pendulums can be high), nonlinear and oscillatory dynamics, and only one actuator. The proposed class of problems can be characterized as controlled synchronization – designing a closed-loop controller that synchronizes the motion of the pendulums. Controlled synchronization is a special case of reference tracking, where all pendulums reach a common point or a trajectory. One can formulate many practically motivated problems within this class; here, we identify three problems: a vibration control of a flexible structure, swing-up control of all pendulums in the array, and low-friction sliding of an atomic-scale structure. We show that the presented problems can be dealt with by the Koopman Model Predictive Control (KMPC). The KMPC allows for controlling nonlinear systems by combining the classical linear model predictive control (MPC) with the Koopman operator approach for nonlinear dynamical systems. The main idea of the method is to construct a linear predictor of a nonlinear system in a higher-dimensional, lifted space and use the predictor within the linear MPC. The optimization problem formulation within the KMPC can then be tailored to a specific problem in the class. Simulations and experiments on a hardware platform realizing the FK model show the effectiveness of the method.

Role of DNA-DNA sliding friction and nonequilibrium dynamics in viral genome ejection and packaging.

Many viruses eject their DNA via a nanochannel in the viral shell, driven by internal forces arising from the high-density genome packing. The speed of DNA exit is controlled by friction forces that limit the molecular mobility, but the nature of this friction is unknown. We introduce a method to probe the mobility of the tightly confined DNA by measuring DNA exit from phage phi29 capsids with optical tweezers. We measure extremely low initial exit velocity, a regime of exponentially increasing velocity, stochastic pausing that dominates the kinetics and large dynamic heterogeneity. Measurements with variable applied force provide evidence that the initial velocity is controlled by DNA-DNA sliding friction, consistent with a Frenkel-Kontorova model for nanoscale friction. We confirm several aspects of the ejection dynamics predicted by theoretical models. Features of the pausing suggest that it is connected to the phenomenon of 'clogging' in soft matter systems. Our results provide evidence that DNA-DNA friction and clogging control the DNA exit dynamics, but that this friction does not significantly affect DNA packaging.

Modulation instability gain and localized waves in the modified Frenkel–Kontorova model with high-order nonlinearities

In this paper, modulation instability and nonlinear supratransmission are investigated in a one-dimensional chain of atoms using cubic–quartic nonlinearity coefficients. As a result, we establish the discrete nonlinear evolution equation by using the multi-scale scheme. To compute the modulation instability gain, the linearizing technique is employed. The effect of the higher nonlinear component on modulation instability is particularly examined. Following that, full numerical integration was performed to identify modulated wave patterns as well as the appearance of a rogue wave. Through the nonlinear supratransmission phenomenon, one end of the discrete model is driven into the forbidden bandgap. As a consequence, for driving amplitudes above the supratransmission threshold, the bright soliton and modulated wave patterns are satisfied. An important behavior is observed in the transient range of time of propagation when the bright soliton wave turns into a chaotic soliton wave. These results corroborate our analytical investigations on the modulation instability and show that the one-dimensional chain of atoms is a fruitful medium to generate long-lived modulated waves.

Injection and nucleation of topological defects in the quench dynamics of the Frenkel-Kontorova model

Topological defects have strong impact on both elastic and inelastic properties of materials. In this article, we investigate the possibility to controllably inject topological defects in quantum simulators of solid state lattice structures. We investigate the quench dynamics of a Frenkel-Kontorova chain, which is used to model discommensurations of particles in cold atoms and trapped ionic crystals. The interplay between an external periodic potential and the inter-particle interaction makes lattice discommensurations, the topological defects of the model, energetically favorable and can tune a commensurate-incommensurate structural transition. Our key finding is that a quench from the commensurate to incommensurate phase causes a controllable injection of topological defects at periodic time intervals. We employ this mechanism to generate quantum states which are a superposition of lattice structures with and without topological defects. We conclude by presenting concrete perspectives for the observation and control of topological defects in trapped ion experiments.

Largest Lyapunov exponent as a tool for detecting relative changes in the particle positions.

Dynamics of the driven Frenkel-Kontorova model with asymmetric deformable substrate potential is examined by analyzing response function, the largest Lyapunov exponent, and Poincaré sectionsfor two neighboring particles. The obtained results show that the largest Lyapunov exponent, besides being used for investigating integral quantities, can be used for detecting microchanges in chain configuration of both damped Frenkel-Kontorova model with inertial term and its strictly overdamped limit. Slight changes in relative positions of the particles are registered through jumps of the largest Lyapunov exponent in the pinning regime. The occurrence of such jumps is highly dependent on type of commensurate structure and deformation of substrate potential. The obtained results also show that the minimal force required to initiate collective motion of the chain is not dependent on the number of Lyapunov exponent jumps in the pinning regime. These jumps are also registered in the sliding regime, where they are a consequence of a more complex structure of largest Lyapunov exponent on the step.

Noise and generation effects in parallel Josephson junction chains

We consider mutual effects of soliton dynamics and noise in a chain of Josephson junctions in the framework of the Frenkel-Kontorova model. The investigations are performed by the analysis of experimentally relevant quantities such as oscillation power and spectral linewidth, which both can be derived from the power spectral density of the system as, respectively, its integral and its width. Taking into account both ohmic and surface losses together with bias inhomogeneity, and considering dense soliton chain under the effect of external magnetic field, it is demonstrated that increase of the system discreteness (decrease of coupling between neighboring elements) leads to the oscillation power reduction and spectral linewidth increase, which signals about increase of the total noise in the system. Drastic increase of the linewidth is observed when the distance between elements becomes comparable with the soliton size, which is explained by an increase of stochasticity due to reflections of solitons from the system edges and noise.

Solid-state thermal rectification of bilayers by asymmetric elastic modulus.

A highly efficient thermal rectification applicable to large panels still needs to be developed. Here, we experimentally achieve a high thermal rectification efficiency of 33% by carefully engineering elastic modulus asymmetry in a centimeter-scale bilayered silver-graphene oxide sponge. The thermal conduction primarily occurs in the out-of-plane direction, and the forward heat flow direction is from the hard silver to the soft graphene oxide. Surprisingly, the forward heat flow direction is reversed when a silver layer is formed on a harder polystyrene foam. The forward direction is always from the harder side to the softer side, and the asymmetry in elastic modulus is suggested as a possible mechanism based on the one-dimensional Frenkel-Kontorova (FK) model. The finite element analysis indicates that other mechanisms such as temperature-dependent thermal conductivity and radiation asymmetry cannot explain the high rectification efficiency. This scalable work over a wide temperature range may find immediate industrial applications.

Directed Motion and Pinning of Atomic Lattices at Disordered Interfaces

Novel directed motion of an atomic lattice on a substrate with impurity-like disorder is studied using numerical simulations based on an extended Frenkel-Kontorova model. The directed motion of the lattice is caused by the spatiotemporal modulation of natural length between atoms. Even in the presence of surface disorder due to impurities the directed motion occurs when the strength of impurity potential is smaller than a certain value. For strong impurity potential cases, however, the effect of impurities is dominant for the behaviors of the atomic lattice, and then no directed motion occurs. Using a scaling analysis based on a impurity pinning theory, the physical nature of the directed motion of atomic lattices in the disordered interfacial system is clarified.

An Analysis of Some Properties and the Use of the Twist Map for the Finite Frenkel–Kontorova Model

We discuss the twist map, with a special interest in its use for the finite Frenkel–Kontorova model. We explain the meaning of the tensile force in some proposed models. We demonstrate that the application of the twist map for the finite FK model is not correct, because the procedure ignores the necessary boundary conditions.

Shapiro steps and chaos in the Frenkel-Kontorova model with substrate lateral vibration.

Numerical simulations are used to examine the dynamics of the dc-driven Frenkel-Kontorova model with an oscillation substrate subjected to lateral periodic excitations in overdamped and underdamped cases, respectively. The results reveal that the system exhibits frequency locking and chaotic behaviors due to the fact that the lateral vibration of the substrate potential introduces an additional frequency and degree of freedom into the system. In the overdamped case, we show that the appearance of subharmonic Shapiro steps can be attributed to the deformation of the substrate potential or inertia. The characteristics of the steps are significantly affected by the amplitude and frequency of the lateral vibration. When the vibration frequency is relatively high, the change of the width of the first harmonic Shapiro step with increasing amplitude exhibits a Bessel-type oscillation, but the oscillation deviates from the Bessel curve at lower frequencies. In the amplitude dependence, although the oscillatory behavior of the critical depinning force at the high frequency is anomalous, local maxima (minima) of the first step width correspond to local minima (maxima) of the critical depinning force, and the largest Lyapunov exponent obtained in the pinned state represents a mirror relationship of the critical depinning force. In contrast to the overdamped system, the underdamped one exhibits both subharmonic Shapiro steps and chaotic behaviors. We show the increased inertia of the latter system plays an important role in suppressing the emergence of subharmonic steps, which is opposite to the result of the former. When the dc force changes, chaos appears not only between adjacent subharmonic Shapiro steps but also on some specific steps where chaos should be avoided. The variation regularity of the first step width and the critical depinning force is thus annihilated in vibration amplitude and frequency dependence. However, superlubricity can be achieved by careful adjustment of vibration amplitude and frequency. The findings can serve as a theoretical guideline for technological applications such as device building and voltage standards.

Statistical mechanics of rate-independent stick-slip on a corrugated surface composed of parabolic wells

The stick-slip phenomenon, at the basis of friction, is crucial for several applications ranging from nanotechnology and biophysics to mechanics and geology. Deep understanding of friction mechanisms and, in particular, the methodologies for its reduction must be sought in its nanoscopic nature, where atomic interactions and stick-slip processes play a crucial role. At this scale, thermal fluctuations clearly have a major effect on the physics of the problem. Hence, we develop here a theory for rate-independent stick-slip, based on equilibrium statistical mechanics. In particular, we introduce suitably modified Prandtl–Tomlinson and Frenkel–Kontorova models in order to study the system with one particle and the chain with N particles, respectively. The adopted corrugated substrate is composed of a sequence of quadratic wells. Interestingly, the calculation of corresponding partition functions shows a conceptual link with the theory of Jacobi and Riemann theta functions, allowing an efficient determination of the average static frictional force and other relevant quantities. We show some applications including the study of structural lubricity and thermolubricity.

A generalized Frenkel–Kontorova model for a propagating austenite–martensite phase boundary: revisited numerically

We explain the propagating austenite–martensite phase boundary by a Frenkel–Kontorova model for a chain of meshes along a ledge of the phase transitions. We demonstrate such steps for example chains of 16 and 47 meshes. We can represent a Langevin solution which describes possible cases of a consecutive excitation of a higher phase under a low external force.Graphic abstract

Relativity 4-ever?

This essay is about superluminal motion. It is generally believed that special relativity prohibits movements faster than the speed of light. It is explained which motion is actually forbidden by special relativity and why. Tachyons are breafly discussed and it is explained that, due to internal instability, tachyon fields cannot be used to transmit information faster than the speed of light. However, as John Stuart Bell remarked, “what is proved by the impossibility proofs is lack of imagination”. Inspired by the Frenkel–Kontorova model of crystal dislocations, we demonstrate the way to overcome the light speed barrier by introducing elvisebrions. Elvisebrions are defined as objects that can exist in the case when some hidden sectors, very weakly interacting with the visible sector, are either not Lorentz invariant, or Lorentz invariant but with different limiting velocities. In this case, elvisebrions can move in a superluminal manner without violating our existing physical models.

Nearly free phonons in a weak soliton potential and the case of twisted bilayer graphene

Structural relaxation in slightly misaligned bilayer graphene leads to the formation of domains separated by topological solitons. The influence of such a soliton lattice on phonons is investigated in the framework of the Frenkel-Kontorova model, where a solution is derived in the continuum limit. A nearly free phonon model is developed to explain the characteristic features in the phonon band structure derived from the Frenkel-Kontorova model. Its 2D extension is discussed and applied on the specific case of twisted bilayer graphene. This model explains a number of computational results reported until now and can be used directly to investigate other misaligned two-dimensional networks as found in van der Waals heterostructures.

Mountain pass solutions to a generalized Frenkel-Kontorova model

Mountain pass solutions to a generalized Frenkel-Kontorova model

Nano-friction phenomenon of Frenkel–Kontorova model under Gaussian colored noise

The nano-friction phenomenon in a one-dimensional Frenkel–Kontorova (FK) model under Gaussian colored noise is investigated by using the molecular dynamic simulation method. The role of colored noise is analyzed through the inclusion of a stochastic force via a Langevin molecular dynamics method. Via the stochastic Runge–Kutta algorithm, the relationship between different parameter values of the Gaussian colored noise (the noise intensity and the correlation time) and the nano-friction phenomena such as hysteresis, the maximum static friction force is separately studied here. Similar results are obtained from the two geometrically opposed ideal cases: incommensurate and commensurate interfaces. It was found that the noise strongly influences the hysteresis and maximum static friction force and with an appropriate external driving force, the introduction of noise can accelerate the motion of the system, making the atoms escape from the substrate potential well more easily. Interestingly, suitable correlation time and noise intensity give rise to super-lubricity. It is noteworthy that the difference between the two circumstances lies in the fact that the effect of the noise is much stronger on triggering the motion of the FK model for the commensurate interface than that for the incommensurate interface.

Structure and energetics of the elbows in the Au(111) herringbone reconstruction

We study the structure of the threading edge dislocations, or ``elbows,'' which are an essential component of the well-known herringbone reconstruction of the (111) surface of Au. Previous work had shown that these dislocations can be stabilized by long-range elastic relaxations into the bulk. However, the validity of the harmonic spring model that had been used to estimate the energies of the dislocations is uncertain. To enable a more refined model of the dislocation energetics, we have imaged the atomic structure of these dislocations using scanning tunneling microscopy. We find that the harmonic spring model does not adequately reproduce the observed structure. We are able to reproduce the structure, however, with a two-dimensional Frenkel-Kontorova (FK) model that uses a pairwise Morse potential to describe the interactions between the top layer Au atoms on a rigid substrate. The parameters of the potential were obtained by fitting the energy of uniaxially compressed phases, or ``stripes'', computed with density functional theory, as a function of surface Au density. Within this model, the formation of the threading dislocations remains unfavorable. However, the large forces on the substrate atoms near the threading-dislocation cores, render the assumption of a completely rigid substrate questionable. Indeed, if the FK parameters are modified to account for the relaxation of just one more atomic layer, threading dislocations can, in principle, become favorable, even without bulk elastic relaxations. Additional evidence for a small elbow energy is that our computed change in the Au(111) surface stress tensor caused by the $(\sqrt{3}\ifmmode\times\else\texttimes\fi{}22)$ reconstruction is considerably smaller than previous estimates.