Frenkel-Kontorova Model Research Articles

Influence of impurities on solitons in the nonlinearLCtransmission line

This paper studies the propagation of solitons in the nonlinear LC transmission line (NLCTL) with capacitor impurity. Based on Kirchhoff's laws, the numerical simulation shows that the amplitude of the soliton will be increased or decreased when it is close to the positive or negative impurity. Then, it will be split into reflected and transmitted waves by both the positive and negative impurities. Furthermore, their final amplitude and propagating speeds are almost independent of the impurity polarity. The observations near the impurity can be understood in the physical picture of the linear uncoupled energy absorption. By these results, we find that the impurity-soliton interactions (ISIs) in NLCTLs for both inductance and capacitance impurities, which have been seen to be different before, actually can be unified. They also indicate that the ISIs in NLCTLs essentially can be integrated with those in many other soliton systems, such as the Frenkel-Kontorova model and hydrodynamics. Moreover, the impurity-induced influence on the NLCTL solitons can also be well interpreted in the framework of the nonlinear Schrödinger model with an impurity term derived from the discrete voltage propagation equations by means of the perturbation method.

Anisotropic character of atoms in a two-dimensional Frenkel—Kontorova model

The dynamics of a certain density of interacting atoms arranged on a two-dimensional square lattice, which is made to slide over a two-dimensional periodic substrate potential with also the quare lattice symmetry, in the presence of dissipation, by an externally applied driving force, is studied. By rotating the misfit angle θ, the dynamical behaviour displays two different tribological regimes: one is smooth, the other becomes intermittent. We comment both on the nature of the atomic dynamics in the locked-to-sliding transition, and on the dynamical states displayed during the atom motion at different values of the driving force. In tribological applications, we also investigate how the main model parameters such as the stiffness strength and the magnitude of the adhesive force affect the static friction of the system. In particular, our simulation indicates that the superlubricity will appear.

Phase transition of atomic chain in the one-dimensional Frenkel-Kontorova model

The effect of the initial velocity of boundary atom on the motion of an atomic chain is studied by the one-dimensional Frenkel-Kontorova model. The obatined results show that the three phases could be observed in our simulation, these being harmonic phase when vv1, choatic phase when v1vv2 and uniformity phase when vv2. It is aslo shown that the two critical velocites(v1,v2) are strongly influenced by the number of atoms and the depth of the substrate potential.

Difference of Directions Between External Driving Force and Movement Direction of Center of Mass

Driven dynamics of a two-dimensional Frenkel-Kontorova model is studied in the paper. In our numerical simulations, it is found that the movement direction of the center of mass is not consistent with that of the external driving force except for some special symmetric directions at the lower driving force. Our results also indicate that the movement direction of the center of mass strongly depends on both the magnitude and the direction of the external driving force as well as the misfit angle between two layers.

Friction phenomena and phase transition in the underdamped two-dimensional Frenkel-Kontorova model

Locked-to-sliding phase transition has been studied in the driven two-dimensional Frenkel-Kontorova model with the square symmetric substrate potential. It is found that as the driving force increases, the system transfers from the locked state to the sliding state where the motion of particles is in the direction different from that of driving force. With the further increase in driving force, at some critical value, the particles start to move in the direction of driving force. These two critical forces, the static friction or depinning force, and the kinetic friction force for which particles move in the direction of driving force have been analyzed for different system parameters. Different scenarios of phase transitions have been examined and dynamical phases are classified. In the case of zero misfit angle, the analytical expressions for static and kinetic friction force have been obtained.

Uniform sliding states in the undamped Frenkel–Kontorova model

The uniform sliding states of the undamped Frenkel–Kontorova model are reduced to the periodic solutions of an advance-delay differential equation, the existence of which is investigated in this paper by employing variational methods.

Strain-Driven Formation of Nanostripes on In/Si(113) Surface

Periodic nanostripe arrays are observed on In/Si(113) surface using scanning tunneling microscopy. The stripe superstructures are identified as N × 1 reconstructions elongating in [2̄1̄1] or [1̄2̄1] direction and consisting of one vacancy line, one Si adatom row, and N − 2 In rows, in which N = 5 is predominant. The vacancy line formation relieves the strain induced by the Si adatom row and In rows, and plays an important role in stabilizing the stripe structures. The stability of nanostripe structures is demonstrated by analyzing the strain-mediated interaction of vacancy lines in the framework of the Frenkel–Kontorova model, which indicates that the predominant vacancy line period of N = 5 corresponds to the minimum Frenkel–Kontorova energy.

The effect of ac-driven force on superlubricity in a two-dimensional Frenkel–Kontorova model

By using the molecular dynamic simulation method with a fourth-order Runge–Kutta algorithm, a two-dimensional dc- and ac-driven Frenkel–Kontorova model with a square symmetry substrate potential for a square lattice layer has been investigated in this paper. For this system, the effects of many different parameters on the static friction force have been studied in detail. It was found that not only the amplitude and frequency of the ac-driven force, but also the direction of dc- and ac-driven forces and the misfit angle between two layers have a strong influence on the static friction force. This indicated that the phenomenon of superlubricity appears easily with larger ac amplitude and smaller ac frequency for some special direction of the external driving force and misfit angle.

Application of Two-Dimensional Frenkel–Kontorova Model to Nanotribology

The dry friction force between two contacting surface layers is studied. The upper layer is arranged on a two-dimensional square lattice and driven by an external driving force. The lower layer is approximated by a two-dimensional periodic substrate potential. This model, usually called two-dimensional Frenkel–Kontorova model, is applied to study the friction forces in this paper. The behaviors of different substrate potential strongly affect the static friction force. It is found that the system has strong anisotropic character. The possibility to obtain superlubricity is suggested.

On the controlled propagation of wave signals in a sinusoidally forced two-dimensional continuous Frenkel–Kontorova model

The nonlinear processes of supratransmission and infratransmission are employed here in order to show numerically that binary information may be transmitted into (2 + 1)-dimensional, continuous Frenkel–Kontorova media spatially defined on a square, by perturbing harmonically two adjacent boundaries by means of Neumann conditions. The presence of these nonlinear phenomena is established numerically through a computational method that preserves the positivity of the energy operators, and that consistently approximates the solution of the model, the local energy density, the total energy and its dissipation; the existence of a region of bistability where two regimes coexist (conducting and insulating) is established as a corollary. The transmission of binary information is accomplished by fixing a frequency in the forbidden band-gap of the system, and modulating the amplitude of the signals as the sum of a seed (whose amplitude oscillates sinusoidally between the supratransmission and infratransmission thresholds) and small, positive, constant perturbations associated to nonzero bits. Our simulations show that a reliable transmission of information is indeed feasible.

Dynamics of the Frenkel–Kontorova model with irrational mean spacing

In this paper we focus on the dynamics of the overdamped limit case of the Frenkel–Kontorova (F–K) model with irrational mean spacing. We extend the results of Baesens and MacKay on rational mean spacing case to irrational case: there exists a depinning force Fd ⩾ 0 such that the F–K model possesses an equilibrium with a given mean spacing if the constant driving force F ∊ [0, Fd] or it admits a uniform sliding solution provided F > Fd. Moreover, the average velocity of the particles exists and is unique, and it is non-decreasing and continuous with respect to F ∊ [0, +∞). In particular, we prove that the average velocity depends continuously upon the mean spacing.

Reflection Wave Suppression in Frenkel-Kontrova Model Simulation: Analysis Based on the GKYP Lemma

The Frenkel-Kontorova model is a nonlinear lattice model to represent extremely complicated behaviors such as a dislocation in crystal and DNA structures. This model is given as the dynamical syste...

Heat conduction in the nonlinear response regime: Scaling, boundary jumps, and negative differential thermal resistance

We report a numerical study on heat conduction in one-dimensional homogeneous lattices in both the linear and the nonlinear response regime, with a comparison among three prototypical nonlinear lattice models. In the nonlinear response regime, negative differential thermal resistance (NDTR) can occur in both the Frenkel-Kontorova model and the phi4 model. In the Fermi-Pasta-Ulam- beta model, however, only positive differential thermal resistance can be observed, as shown by a monotonous power-law dependence of the heat flux on the applied temperature difference. In general, it was found that NDTR can occur if there is nonlinearity in the onsite potential of the lattice model. It was also found that the regime of NDTR becomes smaller as the system size increases, and eventually vanishes in the thermodynamic limit. For the phi4 model, a phenomenological description of the size-induced crossover from the existence to the nonexistence of a NDTR regime is provided.

Properties of the Shapiro steps in the ac driven Frenkel-Kontorova model with deformable substrate potential

Properties of the dynamical-mode-locking phenomena are studied in the ac driven overdamped Frenkel-Kontorova model with deformable substrate potential. Appearance of very large subharmonic steps due to deformation of the substrate potential significantly influences the stability and existence of harmonic steps. Strong correlation among harmonic and subharmonic steps has been observed in which the larger the width of half-integer steps, the smaller that of harmonic steps. Amplitude dependence of harmonic steps significantly changes with the deformation of the potential where deviation from the well-known Bessel-like oscillations appears. Strong influence of the frequency of the ac driving force on the appearance and size of subharmonic steps has been observed.

Ab initio multi-string Frenkel–Kontorova model for a b = a/2[111] screw dislocation in bcc iron

We formulate a multi-string Frenkel–Kontorova (MSFK) model for a a/2[111] screw dislocation in bcc iron, and investigate the occurrence of degenerate and non-degenerate dislocation core structures as functionals of the law of interaction between the [111] strings of atoms forming the crystal. By comparing the effective inter-string interaction laws derived from ab initio density functional calculations and from semi-empirical interatomic potentials for α-iron, we show that it is the form of the function determining how the atomic strings interact with each other as a function of their relative one-dimensional displacement in the [111] direction that determines whether a degenerate or a non-degenerate screw dislocation core configuration has lower energy. We show that by constructing a one-dimensional inter-string interaction law, and by solving the MSFK equations, it is possible to easily predict the nature of the screw dislocation core, hence providing a simple yet effective check to aid the development of short-range semi-empirical interatomic potentials for bcc transition metals. Finally, we analyse the relation between the inter-string interaction law, and the shape and the height of the Peierls energy barriers separating the adjacent equilibrium configurations for a migrating screw dislocation.

Friction phenomena in a two-dimensional Frenkel–Kontorova model

By using the molecular dynamic simulation method with a fourth-order Runge–Kutta algorithm, a two-dimensional dc- and ac-driven Frenkel–Kontorova (FK) model with a square symmetry substrate potential for a square lattice layer has been investigated in this paper. For this system, the effects of many different parameters on the average velocity and the static friction force have been studied. It is found that not only the amplitude and frequency of ac-driven force, but also the direction of the external driving force and the misfit angle between two layers have some strong influences on the static friction force. It can be concluded that the superlubricity phenomenon appears easily with a larger ac amplitude and lower ac frequency for some special direction of the external force and misfit angle.

Superglide at an Internal Incommensurate Boundary

The intriguing possibility of frictionless gliding of one solid surface on another has been predicted for certain incommensurate interfaces in crystals, based on Aubry's solution to the Frenkel-Kontorova model of a harmonic chain in a periodic potential field. Here we test this prediction for grain boundaries by comparing atomistic simulations with direct experimental observations on the structure and load-deformation behavior of gold nanopillars containing a root-two incommensurate grain boundary. The simulations show supergliding at this boundary limited by finite-size effects which cause edges to act as defects of the incommensurate structure. Structural relaxation at the edges generates stacking faults, dislocations, and asymmetric surface steps. These features as well as the related load-displacement behavior are replicated by experimental observations on the compression of nanopillars using a quantitative nanoindentation device inside a transmission electron microscope. The good agreement between the observed and predicted behavior suggests that incommensurate interfaces could play an important role in the deformation of polycrystalline materials.

Effect of the substrate potential on the static friction force in the Frenkel-Kontorova model

By using the one-dimensional Frenkel-Kontorova (FK) model, we study the maximum static friction force between the two surface layers which are in contact. The relation between the substrate potential and the static friction force is studied in this paper. The effect of parameters of the upper layer such as the elastic coefficient K on the static friction force are also investigated.Results are obtained for three cases for which the ratio of the atomic distance to the period of potential field is commensurate, or equal to the golden mean, or the spiral mean respectively. The results show that both the form of the substrate potential and the commensurability have sigificant effect on the static friction force.

Rotating modes in the Frenkel-Kontorova model with periodic interaction potential

Employing a homotopy argument and the Leray-Schauder degree theory, we show the existence of rotating modes for the Frenkel-Kontorova model with periodic interaction potential. The solutions describing rotating modes are periodic and called rotating oscillating solutions, in which the phase of a fixed rotator increases by $2\pi$ per period, while its neighbors oscillate with small amplitudes around their equilibrium positions. We also discuss a fundamental difference between the Frenkel-Kontorova model with periodic interaction potential and that with convex interaction potential by demonstrating the nonexistence of the rotating modes for the latter case.

From the Frenkel-Kontorova Model to Josephson Junction Arrays

The Frenkel Kontorova (FK) model is known to exhibit the so called Aubry's transition which is a jamming or frictional transition at zero temperature. Recently we found similar transition at zero and finite temperatures in a super-conducting Josephson junction array (JJA) on a square lattice under external magnetic field. In the present paper we discuss how these problems are related.