THE INFLUENCE OF SURFACE ENERGY ON THE INCOMMENSURATE SUPERSTRUCTURE FORMATION PROCESSES

Abstract

The influence of surface energy on the phase portraits and the Lyapunov’s exponents for an incommensurate superstructure, which are described by second order differential equations was studies. The phase portraits of nonlinear dynamic systems in Python software using JiTCDDE library were constructed. Second-order differential equations were solved by the BDF method.

The influence of surface energy on the incommensurate superstructure which causes the transition of the system from a heterogeneous state to a homogeneous one was described. It is established that under the condition when T = 1, the increase of the surface energy effect is accompanied by the decrease in the number of bifurcations, which indicates that the degeneration of the incommensurate superstructure is eliminated.

Phase portraits for various parameter values, which describe the stability of the incommensurate superstructure (T) and anisotropic interaction (K) are described. This system of differential equations describes the process of the appearing of the incommensurate superstructure from the spatial regions of correlated motion of tetrahedral groups for the small values of the T and K parameters. The spatial regions of the correlated motion of tetrahedral groups are characterized by slightly different modulation parameters due to the existence of uncontrollable defects and impurities in the crystal. Appeared stresses on the boundaries of such spatial regions, in the process of formation of the incommensurate superstructure lead to the appearance of the harmonics of a wave of the incommensurate superstructure at small values of the T and K (T, K <0.001) parameters. The changing of the phase and the amplitude of the order parameter under the influence of external factors and surface energies does not cause the system to switch to the chaotic mode with the appearing of a chaotic phase.

The first discovered that portraits of attractors of the perturbed system with small values of the T and K parameters are characterized by one attractor. The reason is the small scale of the attraction pools. It is noted that this phenomenon refers to crisis states in which the attractor undergoes changes in parameters caused by collision with the boundary of the basin when changing the T and K parameters. As a result, the chaotic attractor does not arise from the cascade of doubling of the period.

It is shown that the change of the phase and the amplitude of the order parameter under the influence of external factors and surface energies at small values of the T and K parameters, does not cause the system to switch to a chaotic mode with the appearing of the chaotic phase.

Key words: incommensurate superstructures, phase portraits, surface energy.