Physico-Chemical nanomaterials science

The influence of surface energy on incommensurate superstructure
Sveleba S.A., Karpa I.V., Katerynchuk I.M., Kunyo I.M.
Ivan Franko National University of Lviv, 107 Tarnavsky St., Lviv, Ukraine.
E-mail: incomlviv@gmail.com 

In this work, the construction of phase portraits of nonlinear dynamic systems, in the Python software environment using the scipy library [1]. The obtained phase portraits of the investigated system of differential equations, is depending on the parameters T and K. The stability of the initial phase and anisotropic interaction are described by them, for system determined by the Dzyaloshinskii’s invariant. The n = 4, T <0.04 and K <0.01 given one trajectory of equilibrium of the system. If T> 0.04 and K> 0.01 the limit cycle loses its stability, which is manifested in the change of its form. That is, the appearance of the attractor in the form of a loop, due to the soft bifurcation of the birth of the torus. Further growth of the value of the parameter of anisotropic interaction is accompanied by the appearance of a bifurcation cascade of doubling cycles. Such an infinite cascade of doubling the period ends with the appearance of a chaotic attractor. Consequently, the anisotropic interaction described by the Dzyaloshinskii’s invariant leads to a violation of the spatial periodicity of the superstructure with the appearance of the amplitude and frequency modulation of the superstructure.
The surface energy removes the degeneration of this system, thus transforming it into a homogeneous state. The question arises whether the influence of surface energy on the incommensurate superstructure can be replaced by the influence of an electric field on the magnitude of spatially modulated spontaneous polarization.
Therefore, the influence of the electric field applied to the crystal in the direction of the existence of spatially modulated magnitude of spontaneous polarization was investigated in the work. With the increase in the magnitude of the intensity of the electric field of the system, it becomes a metastable state.
Consequently, it can be assumed that in the surface layer due to the fact that the superficial energy is greater than the energy of the incommensurate superstructure, the latter is absent. The bulk part of the crystal is characterized by a higher energy of the incommensurate superstructure relative to the surface energy. When reducing the bulk part of the crystal, the energy of the incommensurate superstructure decreases. When the energy of the incommensurate superstructure becomes equal to the surface energy in a thin-layer (or micro-sized) crystal, there is a transition from the inhomogeneous state to the homogeneous state.