Two-dimensional Frenkel-Kontorova Model Research Articles

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.

A quantitative rigidity result for a two-dimensional Frenkel–Kontorova model

We consider a Frenkel–Kontorova system of harmonic oscillators in a two-dimensional Euclidean lattice and we obtain a quantitative estimate on the angular function of the equilibria. The proof relies on a PDE method related to a classical conjecture by E. De Giorgi, also in view of an elegant technique based on complex variables that was introduced by A. Farina.In the discrete setting, a careful analysis of the reminders is needed to exploit these types of methodologies inspired by continuum models.

The superlubricity of the special material with hexagonal symmetry in a two-dimensional Frenkel–Kontorova model

Based on a two-dimensional Frenkel–Kontorova model, the interacting atoms arranged on a two-dimensional hexagonal lattice are studied. It is made to slide by the application of an external driving force in the presence of dissipation over a two-dimensional periodic substrate potential with the hexagonal symmetry. A critical depinning force is defined. It depends on the direction and the magnitude of external driving force, the adhesive force from the substrate, the interaction strength between atoms in the upper layer and especially the misfit angle $$\theta $$ between two layers. The dependence of the state transition from the lock to sliding on the driving force is studied. For the special case of zero misfit angle, the analytical expressions for the critical depinning force are obtained which are in a good agreement with numerical ones. It seems that the superlubricity can appear.

Production of screw dislocations by the external dc and ac forces

The effects of both the ac and the dc external driving forces on the production of screw dislocations are studied by employing the two-dimensional Frenkel–Kontorova model. The ac force is introduced in the model as a way to describe external periodic disturbances or artificial periodic forces. We find that the appearance of the screw dislocation is influenced by the frequency and amplitude of the ac force. It is shown that the screw dislocation can be easily produced by either the lower frequency or the larger amplitude of the ac force.

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.

Ratchet effect and amplitude dependence of phase locking in a two-dimensional Frenkel-Kontorova model

We demonstrate the ratchet and phase locking effects in a two-dimensional overdamped Frenkel-Kontorova model with a square symmetric periodic substrate when both a longitudinal dc drive and a circular ac drive are applied. Besides the harmonic steps, the large half integer steps can also clearly be seen in the longitudinal (x) direction. These half integer steps are directly correlated to the appearance of positive and negative ratchet effects in the transverse (y) direction due to the symmetry breaking in the combination of the dc and ac drives. The angle between the net displacement and the longitudinal direction is analytically obtained in a single period of the ac drive. In the examination of the amplitude dependence of the ac drive, the maxima decrease monotonically with the amplitude, while the anomalies occur for the critical depinning force and the harmonic steps due to the spatial symmetry breaking of orbits in the presence of the ac drive.

Friction phenomena in two-dimensional Frenkel-Kontorova model with hexagonal symmetry lattice

Locked-to-sliding phase transition is studied based on the two-dimensional Frenkel-Kontorova model in this paper. The method of molecular dynamics simulation is used. The effect of the static friction force on system parameter is investigated numerically when the upper layer atoms are of the hexagon symmetric structure.

The influence of the disorderded substrate potential on static friction force

In order to better simulate a real model which can be realized in experiment, we study a two-dimensional(2D) Frenkel-Kontorova(FK) model driven by the disordered substrate potential which is charactered by the sum of identical Gaussian functions. Commensurate and incommensurate interfaces are discussed. In the paper, we mainly analyse the mechanism of friction, and the effects of static friction force on the disordered substrate potential.

Effect of molecular structure on liquid slip

Slip behavior of three liquids with distinct molecular shapes--linear (hexadecane), branched (pentaerythritol tetra), and a chain of rings (polyphenylether)--is studied using molecular dynamics simulation and reduced-order modeling. Slip at a liquid-solid interface is shown to be affected by the molecular structure of the liquid. A two-dimensional Frenkel-Kontorova model captures the fundamental structural features of the liquid molecules and gives insight into how molecules flex and slip along the surface. We formulate an approximation to the Peierls-Nabarro energy which incorporates both the position of liquid atoms relative to substrate atoms and molecular flexibility. We find that increased molecular flexibility can lead to reduced slip by allowing the liquid to conform epitaxially to the substrate with only a small energetic penalty. Liquid molecules which are less flexible can conform to the substrate only with greater expense of conformational energy, and so exhibit larger slip.

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.

Ground-State Transition in a Two-Dimensional Frenkel—Kontorova Model

The ground state of a generalized Frenkel—Kontorova model with a transversal degree of freedom is studied. When the coupling strength, K, and the frequency of a single-atom vibration in the transversal direction, ω0y, are increased, the ground state of the model undergoes a transition from a two-dimensional configuration to a one-dimensional one. This transition can manifest in different ways. Furthermore, we find that the prerequisite of a two-dimensional ground state is θ ≠ 1/q.

Surface reconstruction transition of metals induced by molecular adsorption

Two surface reconstruction patterns, strikingly different from the Au(111) intrinsic herringbone reconstruction, have been observed at the interfaces between Au(111) and the self-assembled monolayer of perylene or iron phthalocyanine by scanning tunneling microscopy (STM). These interfacial structures, elucidated by using the two-dimensional Frenkel-Kontorova model and density-functional theory (DFT) calculations, are attributed to the anisotropic surface stress induced by interfacial charge transfer. The length scales of the induced reconstruction periodicities are theoretically estimated, in quantitative agreement with experiments. Our findings afford the possibility of tailoring the contact structure of molecular electronic devices by molecular and surface engineering.

Resonance and Rectification in a Two-Dimensional Frenkel—Kontorova Model with Triangular Symmetry

The mode-locking phenomena in the dc- and ac-driven overdamped two-dimensional Frenkel-Kontorova model with triangular symmetric structures are studied. The obtained results show that the transverse velocity 〈vx〉 increases in a series of quantized steps nμx (n=0, 1, 2, 3 ···). Moreover, the positive or negative rectification of longitudinal velocity 〈vy〉 can occur when n is an odd number. It is also found in our simulations that the critical depinning force oscillates with the amplitude of ac-driven force, i.e., the system is dominated by the ac-driven force. The oscillatory behavior is strongly determined by the initial phase of ac-force.

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.

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.

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.

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.

Investigation of superlubricity in a two-dimensional Frenkel–Kontorova model with square lattice symmetry

A two-dimensional Frenkel–Kontorova model with a square symmetry substrate potential for a square lattice layer driven by an external driving force with an arbitrary direction α and an arbitrary misfit angle θ between upper and lower layers is presented in this paper. The effects of the system parameters have been investigated. The application of our results to the tribology is discussed and the dependence of the static friction force on the system parameters is studied. How to make the material with superlubricity is suggested.