Numerical Study of Dynamical Frictional Phenomena Based on the Two-Dimensional Frenkel-Kontorova Model with Impurities

Abstract

We study numerically dynamical frictional phenomena using the Frenkel-Kontorova model with impurities. Even in the presence of randomness sliding states have a quasi long range order, which changes with the sliding velocity. Such an order affects the velocity dependence of the kinetic frictional force and produces peculiar transverse pinning effects. We discuss them in the relation with the moving Bragg glass theory developed for vortex lattices in type II superconductors. Frictional phenomena have common importance in various physical systems such as vortex lattices in type-II superconductors, charge and spin density waves, interfacial friction of solids and so on. 1),2) We investigate here dynamical frictional phenomena in two-dimensional elastic lattice systems based on the Frenkel-Kontorova (FK) model, 3) and focus especially on the kinetic friction and the transverse pinning effect. The FK model employed here consist of a discrete square lattice with harmonic interatomic force and a substrate lattice with impurities.The periodic potential due to the regular part of the substrate lattice has a two-dimensional sinusoidal form, of which principal axes coincide with those of the lattice on it.Impurities are distributed randomly in the substrate and the potential of each impurity has a Gaussian form.The amplitude of the periodic (impurity) potential is denoted by K(W ).We study overdamped dynamics of the FK model driven by an external force along one of the principal axes.Incommensurate systems are investigated here. The periodic potential pins the lattice even in an incommensurate case when the amplitude K is greater than a critical value Kc ≈ 0.23.The number of sites of a lattice is set to be N =8 9 2 (or 144 2 ) and the impurity density is fixed at about 0.21. A periodic boundary condition is imposed. The kinetic frictional force Fkin is plotted against the sliding velocity v in Fig.1. In the low velocity regime, v � 10, the kinetic frictional force is almost constant and not affected by the periodic potential.In the high velocity regime, however, it decreases with increasing velocity and depends on the periodic potential.The effectiveness of the periodic potential depends on the sliding velocity.This behavior will relate to the change of the order of the sliding lattice as mentioned later. We investigate next the response transverse to the sliding direction.According to the theory of the moving Bragg glass (MBG) developed by Giamarchi and Le Doussal, 4) a quasi long range order of the lattice exists in the transverse direction.