One-dimensional Frenkel-Kontorova Model Research Articles

Nano-friction phenomena in driven Frenkel-Kontorova model with stochastic excitation

In this paper, the effects of a Gaussian white noise excitation on the one-dimensional Frenkel-Kontorova (FK) model are studied by the stochastic Runge-Kutta method under two different types of substrate cases, i.e. incommensurate case and commensurate case. The noise excitation is considered through the inclusion of a stochastic force via a Langevin molecular dynamics approach, and we uncover the mechanism of nano-friction phenomenon in the FK model driven by the stochastic force. The relationship between the noise intensity and the nano-friction phenomenon, such as hysteresis, maximum static friction force, and the super-lubricity, is investigated by using the stochastic Runge-Kutta algorithm. It is shown that with the increase of noise intensity, the area of the hysteresis becomes smaller and the maximum static friction force tends to decrease, which can promote the generation of super-lubricity. Similar results are obtained from the two cases, in which the ratios of the atomic distance to the period of the substrate potential field are incommensurate and commensurate, respectively. In particular, a suitable noise density gives rise to super-lubricity where the maximum static friction force vanishes. Hence, the noise excitation in this sense is beneficial to the decrease of the hysteresis and the maximum static friction force. Meanwhile, with the appropriate external driving force, the introduction of a noise excitation can accelerate the motion of the system, making the atoms escape from the substrate potential well more easily. But when the chain mobility reaches a saturation state (<i>B</i> = 1), it is no longer affected by the stochastic excitation. Furthermore, the difference between the two circumstances lies in the fact that for the commensurate interface, the influence of the noise is much stronger and more beneficial to triggering the motion of the FK model than for the incommensurate interface since the atoms in the former case are coupled and entrapped more strongly by the substrate potential.

Thermoelectric properties of Wigner crystal in two-dimensional periodic potential

We study numerically transport and thermoelectric properties of electrons placed in a two-dimensional (2D) periodic potential. Our results show that the transition from sliding to pinned phase takes place at a certain critical amplitude of lattice potential being similar to the Aubry transition for the one-dimensional Frenkel-Kontorova model. We show that the 2D Aubry pinned phase is characterized by high values of Seebeck coefficient S = 12. At the same time we find that the value of Seebeck coefficient is significantly influenced by the geometry of periodic potential. We discuss possibilities to test the properties of 2D Aubry phase with electrons on a surface of liquid helium.

一次元Frenkel-Kontorovaモデルにおける超潤滑・摩擦の発生機構

一次元Frenkel-Kontorovaモデルにおける超潤滑・摩擦の発生機構

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.

Effect of the oscillation of substrate potential in driven Frenkel-Kontorova chains

In this paper, the effect of the oscillation of the substrate potential in a one-dimensional Frenkel-Kontorova model is considered. The relationship between the oscillating amplitude, frequency of the substrate and the nanofriction phenomena such as hysteresis, maximum static friction force, super-lubricity are investigated. Similar results are obtained for the two cases in which the ratios of the atomic distance to the period of potential field of the substrate potential field are incommensurate and commensurate respectively. The results show that on one hand, with the appropriate frequency, the area of the hysteresis will decrease while the amplitude increases, and the tendency of the decrease depends on the frequency. In particular, suitable frequency and amplitude give rise to super-lubricity. However, when the frequency is too high, the result is the same as those in the case without oscillation. On the other hand, fixing the amplitude, the area of the hysteresis will increase with the increase of frequency in spite of tendencies being different. At the same time, on a whole, the maximum static friction force has an increasing tendency. Interestingly and importantly, for a certain amplitude, as the frequency increases, the maximum static friction force first decreases to zero (corresponding to super-lubricity), and then increases. That is, there is an optimum oscillating frequency which makes the system have the minimum static friction force. Furthermore, the difference between the above two circumstances lies in that for commensurate interfaces, there are the same start-up velocities for a certain frequency and various small amplitudes, which is different from the incommensurate mating contacts. Hence, it shows that the latter has a more complex dynamic behavior under the same hypothesis.

712 温度制御下Frenkel-Kontorova原子論的摩擦モデルの摩擦相図(OS8-3 表面の機能化と薄膜・トライポロジー(3),OS8 表面の機能化と薄膜・トライポロジー)

712 温度制御下Frenkel-Kontorova原子論的摩擦モデルの摩擦相図(OS8-3 表面の機能化と薄膜・トライポロジー(3),OS8 表面の機能化と薄膜・トライポロジー)

Prediction of reconstruction in heteroepitaxial systems using the Frenkel-Kontorova model

Many heteroepitaxial metal-on-metal systems reconstruct into patterns of alternating domains of stacking faults separated by partial misfit dislocations. Here, we use two approaches to investigate the question of whether these can be predicted and controlled: (i) We map the system onto a one-dimensional Frenkel-Kontorova model, and then obtain a simple criterion to determine whether or not the surface will reconstruct; this had earlier been done for homoepitaxial systems, but is here generalized to the heteroepitaxial case. (ii) The two-dimensional Frenkel-Kontorova model is solved numerically by performing quenched molecular dynamics simulations. The necessary microscopic parameters are obtained by performing ab initio density functional theory calculations on the unreconstructed systems. The systems considered are overlayers of Fe, Co, Pt, Ag, Au, and Pb on a Ru(0001) substrate, as well as clean Ru(0001). The predictions of the two approaches agree with one another as well as with experiment. Both the presence and periodicity of the reconstruction are very sensitive to the value of ``chemical potential'' $\ensuremath{\Gamma}$. Accordingly, we suggest that $\ensuremath{\Gamma}$ can be used to tune the periodicity so as to obtain a desired nanotemplate for subsequent growth of self-organized nanostructures.

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.

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.

Strain waves, earthquakes, slow earthquakes, and afterslip in the framework of the Frenkel-Kontorova model

The one-dimensional Frenkel-Kontorova (FK) model, well known from the theory of dislocations in crystal materials, is applied to the simulation of the process of nonelastic stress propagation along transform faults. Dynamic parameters of plate boundary earthquakes as well as slow earthquakes and afterslip are quantitatively described, including propagation velocity along the strike, plate boundary velocity during and after the strike, stress drop, displacement, extent of the rupture zone, and spatiotemporal distribution of stress and strain. The three fundamental speeds of plate movement, earthquake migration, and seismic waves are shown to be connected in framework of the continuum FK model. The magnitude of the strain wave velocity is a strong (almost exponential) function of accumulated stress or strain. It changes from a few km/s during earthquakes to a few dozen km per day, month, or year during afterslip and interearthquake periods. Results of the earthquake parameter calculation based on real data are in reasonable agreement with measured values. The distributions of aftershocks in this model are consistent with the Omori law for temporal distribution and a 1/r for the spatial distributions.

The dynamic characters of excitations in a one-dimensional Frenkel–Kontorova model

A one-dimensional (1D) Frenkel–Kontorova (FK) model is studied numerically in this paper, and two new analytical solutions (a supersonic kink and a nonlinear periodic wave) of the Sine–Gordon (SG) equation (continuum limit approximation of the FK model) are obtained by using the Jacobi elliptic function expansion method. Taking these new solutions as initial conditions for the FK model, we numerically find there exist the supersonic kink and the nonlinear periodic wave in these systems and obtain a lot of interesting and significant results. Moreover, we also investigate the subsonic kink and the breather in these systems and obtain some new feature.

Subharmonic Shapiro Steps in the Frenkel-Kontorova Model

The dynamical mode-locking phenomena in the overdamped one-dimensional Frenkel-Kontorova model subjected to a parametrized deformable periodic substrate potential and driven by an ac force are examined. When the shape of the substrate potential starts to deviate from the standard one, new subharmonic steps are found to appear in the response function, even in the structures with an integer value of the average interparticle distance, while the critical depinning force can even decrease for some values of the system parameters.

Dynamical mode locking in commensurate structures with an asymmetric deformable substrate potential

The overdamped dynamics in the commensurate structures of the one-dimensional Frenkel-Kontorova model subjected to a parametrized deformable periodic substrate potential and driven by a periodic force is examined. It was found that when the shape of the substrate potential starts to deviate from the standard one, new subharmonic steps appear in the response function even in the structures with an integer value of average interparticle distance while the critical depinning force can even decrease for some values of system parameters. These novel phenomena could be particularly relevant for the charge-density wave systems, vortex lattices, and systems of Josephson-junction arrays.

Localized transition waves in bistable-bond lattices

Discrete two-dimensional square- and triangular-cell lattices consisting of point particles connected by bistable bonds are considered. The bonds follow a trimeric piecewise linear force-elongation diagram. Initially, Hooke's law is valid as the first branch of the diagram; then, when the elongation reaches the critical value, the tensile force drops to the other. The latter branch can be parallel with the former (mathematically this case is simpler) or have a different inclination. For a prestressed lattice the dynamic transition is found analytically as a wave localized between two neighboring lines of the lattice particles. The transition wave itself and dissipation waves carrying energy away from the transition front are described. The conditions are determined which allow the transition wave to exist. The transition wave speed as a function of the prestress is found. It is also found that, for the case of the transition leading to an increased tangent modulus of the bond, there exists nondivergent tail waves exponentially localized in a vicinity of the transition line behind the transition front. The previously obtained solutions for crack dynamics in lattices appear now as a partial case corresponding to the second branch having zero resistance. At the same time, the lattice-with-a-moving-crack fundamental solutions are essentially used here in obtaining those for the localized transition waves in the bistable-bond lattices. Steady-state dynamic regimes in infinite elastic and viscoelastic lattices are studied analytically, while numerical simulations are used for the related transient regimes in the square-cell lattice. The numerical simulations confirm the existence of the single-line transition waves and reveal multiple-line waves. The analytical results are compared to the ones obtained for a continuous elastic model and for a related version of one-dimensional Frenkel–Kontorova model.

Mode entanglement of electrons in the one-dimensional Frenkel-Kontorova model

We study the mode entanglement in the one-dimensional Frenkel-Kontorova model, and found that quantum entanglement is distinct before and after the transition by breaking of analyticity. We show that the more extended the electron is, the more entangled the corresponding state. Finally, a quantitative relation is given between the average square of the concurrence quantifying the degree of entanglement and the participation ratio characterizing the degree of localization.

Localized transition waves in bistable-bond lattices

Discrete two-dimensional square- and triangular-cell lattices consisting of point particles connected by bistable bonds are considered. The bonds follow a trimeric piecewise linear force-elongation diagram. Initially, Hooke's law is valid as the first branch of the diagram; then, when the elongation reaches the critical value, the tensile force drops to the other. The latter branch can be parallel with the former (mathematically this case is simpler) or have a different inclination. For a prestressed lattice the dynamic transition is found analytically as a wave localized between two neighboring lines of the lattice particles. The transition wave itself and dissipation waves carrying energy away from the transition front are described. The conditions are determined which allow the transition wave to exist. The transition wave speed as a function of the prestress is found. It is also found that, for the case of the transition leading to an increased tangent modulus of the bond, there exists nondivergent tail waves exponentially localized in a vicinity of the transition line behind the transition front. The previously obtained solutions for crack dynamics in lattices appear now as a partial case corresponding to the second branch having zero resistance. At the same time, the lattice-with-a-moving-crack fundamental solutions are essentially used here in obtaining those for the localized transition waves in the bistable-bond lattices. Steady-state dynamic regimes in infinite elastic and viscoelastic lattices are studied analytically, while numerical simulations are used for the related transient regimes in the square-cell lattice. The numerical simulations confirm the existence of the single-line transition waves and reveal multiple-line waves. The analytical results are compared to the ones obtained for a continuous elastic model and for a related version of one-dimensional Frenkel–Kontorova model.

Relaxation and reconstruction on (111) surfaces of Au, Pt, and Cu

We have theoretically studied the stability and reconstruction of (111) surfaces of Au, Pt, and Cu. We have calculated the surface energy, surface stress, interatomic force constants, and other relevant quantities by ab initio electronic structure calculations using the density functional theory in a slab geometry with periodic boundary conditions. We have estimated the stability towards a quasi-one-dimensional reconstruction by using the calculated quantities as parameters in a one-dimensional Frenkel-Kontorova model. On all surfaces we have found an intrinsic tensile stress. This stress is large enough on Au and Pt surfaces to lead to a reconstruction in which a denser surface layer is formed, in agreement with experiment. The experimentally observed differences between the dense reconstruction pattern on Au(111) and a sparse structure of stripes on Pt(111) are attributed to the details of the interaction potential between the first layer of atoms and the substrate.

The Frenkel–Kontorova Model for Studying Suprastructures with Incommensurate Elements

Incommensurability of interacting sublattices in inorganic suprastructures can lead to formation of defects, including fragmentation of one of the sublattices. A theoretical approach is proposed, which is based on the analytical and numerical studies of the static one-dimensional Frenkel–Kontorova model for chains of finite length. This approach provides the possibility of relating geometric and energy parameters of the interacting sublattices to the values of bond deformations that arise. These deformations, in turn, may result in the fragmentation of one of the sublattices and formation of imperfect structures.

Can aperiodicity cause superlubricity?

The ground state of incommensurate structures is determined by the modulation due to the substrate potential. This causes an interplay between the motion of the center of mass and frequencies related to the modulation wavevector q. We build for the one-dimensional Frenkel-Kontorova model an analytical framework which enables us to interpret the initial behavior of the system for a broad range of initial velocities in terms of these relevant phonon modes, and to investigate the occurrence of superlubricity.

Simple variational approach to the quantum Frenkel-Kontorova model.

We present a simple and complete variational approach to the one-dimensional quantum Frenkel-Kontorova model. Dirac's time-dependent variational principle is adopted together with a Hartree-type many-body trial wave function for the atoms. The single-particle state is assumed to have the Jackiw-Kerman form. We obtain an effective classical Hamiltonian for the system, which is simple enough for a complete numerical solution for the static ground state of the model. Numerical results show that our simple approach captures the essence of the quantum effects first observed in quantum Monte Carlo studies.