One-dimensional Frenkel-Kontorova Research Articles

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.

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.

Low-energy channel for mass transfer in Pt crystal initiated by molecule impact

Crystal surface bombardment by atoms or molecules, neutral or ionized, occurs both in ambient conditions and in many technological operations, such as surface plasma treatment, ion implantation, etc. Recently, it was established that the impact of a molecule initiates the mass transfer in the one-dimensional Frenkel-Kontorova atomic chain more efficiently than that of a single atom. This is explained by the fact that the atom can initiate only a very sharp, fast-moving crowdion (anti-kink), which requires relatively high energy, while the molecule is able to initiate a less localized crowdion with considerably lower velocity and energy. In the current study, by means of molecular dynamics simulation, for the first time, this phenomenon is studied for a realistic 3D model of platinum crystal. We compare the efficiency of single Pt atom impact and Pt2 molecule impact on the (101) surface of fcc Pt crystal for the initiation of mass transfer in the material by crowdions. It is revealed that in order to generate a crowdion moving inside the crystal, the properly oriented molecule needs an order of magnitude smaller energy than single atom. This considerable reduction of required energy happens when the molecule is oriented perpendicularly to the crystal surface and hits the crystal along a close-packed atomic row. Furthermore, it is revealed for the first time that the molecule with sufficiently large velocity can initiate the so-called supersonic 2-crowdion, which travels longer distances in the crystal than the classical supersonic crowdion having same or even higher energy. Our results can be useful for understanding and prediction the mass transfer during technological applications where bombardment by atomic clusters is employed to modify and improve mechanical or functional properties of surfaces.

Thermal resonance and energy transport in a biharmonically driven Frenkel–Kontorova lattice

In this work, we study the heat conduction properties of a one-dimensional Frenkel–Kontorova lattice driven by an external, time-periodic biharmonic force applied locally at one boundary and in contact with two heat reservoirs operating at different temperature by means of molecular dynamics simulations. In the single-frequency externally driven case already studied it was observed that there is a value of the driving frequency at which the heat flux takes its maximum value, a phenomenon termed as thermal resonance. It was also determined that it is possible to direct the heat flow against the imposed temperature bias by adjusting the frequency of the single harmonic driving force. With the implementation of the biharmonic forcing we have explored the temperature range at which thermal resonance effect is present. Furthermore, we have determined that by changing the relative amplitude of both harmonic components as well as the frequency of the second, taken always as a multiple, not necessarily integer, of the first one, we can adjust the frequency at which the studied effect is present in the proposed model.

Analysis of the Motion of Frenkel-Kontorova Dislocations in Single Crystals of Aluminum with Allowance for the Peierls Barrier

The regularities of the motion of a one-dimensional Frenkel-Kontorova dislocation in pure aluminum at helium temperatures are studied. Computer simulation was carried out using the sine Gordon equation, written in dimensionless variables. It is proven that when the transition to dimensionless variables the discreteness of the model is preserved. The dependence of the true values of stresses on deformation in the Euler variables, as well as the velocity distribution of the dislocation fragments along the coordinate for successive instants of time, are obtained. It is shown that under these conditions dislocation motion is realized by quantum tunneling of the dislocation bends. The quantum-mechanical estimate confirms the possibility of quantum tunneling of the kink of dislocations in aluminum at low temperatures.

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 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.

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.

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.

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.

The Peierls stress for a wide dislocation

The Peierls stress σ P of a wide dislocation with a core w atoms wide is small. However, estimates of the asymptotic dependence of σ P on w in a crystal of shear modulus μ range from σ P ≈ μ/ w to σ P ≈ μ/ exp(2 πw). Existing calculations are summarized, and some new calculations are presented, but a clear picture does not emerge. It seems that the two-dimensional Peierls model and the one-dimensional Frenkel-Kontorova (F-K) model behave rather similarly, even though the relations between w and the ratio of μ to the critical shear stress τ crit of the perfect crystal are difference in the two cases. An exponential decrease in σ P with increasing w is found in calculations in which the energy associated with a smooth potential is sampled at a set of points which are carried through the lattice by a rigid displacement function, and also in Hobart's computation for the F-K model. A power law dependence is found in models using a piecewise linear force law for which the second derivative of the potential is discontinuous. A treatment in the spirit of second-order perturbation theory, in which the displacement function is modulated as the dislocation moves through the lattice, supports Lifshits' proposal that σ P ≈ μ/ w 5 for any smooth potential. Attention is drawn to old arguments showing that the presence of kinks can lead to anelastic effects and to microstrain, but not to extensive plastic strain. The work of Dietze and of Kuhlmann-Wilsdorf shows that thermal vibrations will reduce σ P by a factor exp(− πwT/ mt m), where T m is the melting temperature and m is respectively 1.5 and less than 10 in the two calculations.