Investigation of thermodynamic properties of single-layer two-component coatings

The results of studying the dynamics of disordered alloys by the method of collective variables are presented. The densities of phonon states of disordered alloys with BCC structure of the kxrb1-x system are calculated. The contribution of phonons to the free energy and entropy of the alloy at different temperatures is determined. It is shown that the contribution of phonons to the total free energy of the alloy decreases with decreasing temperature. The contribution of phonons to the entropy of alloys at 300 K is crucial.


Introduction
Recently, there has been a sharp increase in interest in studying the dynamics of disordered systems, the simplest example of which is binary substitution alloys. Two directions can be distinguished in the theory of dynamic properties of refractory alloys. One of them is associated with the use of various modifications of the theory of a self-consistent field, and the second -with numerical methods. Each approach has its own advantages and limitations. For example, the coherent potential method does not take into account the effects of fluctuations in the local environment in the vicinity of the substitution atoms of the matrix crystal. The introduction of local corrections (clusters, etc.) leads to excessive complications of the method. Implementation of machine experiment methods is associated with great computational difficulties.
In this paper, the lattice dynamics of disordered alloys is studied by the method of collective variables. In this case, within the framework of a unified approach, it is possible to consistently take into account the effects of correlations and ordering.
The Hamiltonian of a binary alloy in the approximation of paired interionic interactions can be written as follows [1]: The first term in (1) is the kinetic energy, and the second is the potential energy of the alloy. The following notation is introduced: Мχ -is the mass of an ion of the χ type (χ = 1, 2); δRi -the displacement vector of the ion located at the site Ri from the equilibrium position R 0 i; Vχχ'(q) -is the Fourier transform of the effective pair interaction of ions of grades χ, χ'. Explicit expressions for Vχχ`(q) are given in [1]. The occupation number operator σjtakes values 1 or -1, depending on whether the site Ri is occupied by an ion of type 1 or 2; N -is the number of lattice sites. To simplify the entries, (1) presents the Hamiltonian of a substitution alloy whose unit cell contains one atom (alloys of BCC and FCC structures).
We restrict ourselves to the harmonic approximation and turn to general variables [2].

Numerical results
In this paper the results of the theory are illustrated by calculating the density of disordered alloys with a bcc structure of the system KxRb1-x. The concentration x varied in the entire range of values 0≤x≤1. It was of interest to clarify the relative contribution of atomic vibrations to the thermodynamic properties of alloys. Alloys of the K-Rb system in the rigid lattice approximation were studied by us in [1,3]. This work is a natural continuation of the works [3,4]. As a first step, the calculation of the density of phonon states was performed in Matlab. The calculations were carried out for three temperatures: 300, 200, 100 K. At these temperatures, the system forms a continuous series of solid solutions. A model pseudopotential [10] was used in calculating the Fourier transforms of the effective interionic potentials Vχχ'(q) (χ,χ' = К, Rb).
When the alloy concentration and temperature change, the shape of the dispersion curves of the middle crystal does not change. Only the values of the frequencies � change, passing at х=0 to the spectrum of pure rubidium, and at х=1to pure potassium. Based on the curves obtained for � the density of phonon statesg ( )was calculated. The values of g( )are normalized to unity: where ωmax -maximum frequency. In PVCs, the frequencies ωkλ, and hence g(ω), depend on temperature indirectly, via the equilibrium atomic volume. If the density of phonon states is known, then it is easy to calculate the contribution of atomic vibrations to various thermodynamic functions.
The contribution of phonons to the free energy F and the entropy of the alloy S at 100, 200, and 300 K was studied. The calculation of Fph, Sph was carried out by the formulas (per atom).
where kB -is the Boltzmann constant. The expression for the free energy F0 (the alloy atoms are in the equilibrium position) is given in [3,4]. The table shows the results of calculations. It is seen that the contribution of phonons to the total free energy of the alloy F=F0+Fph decreases with decreasing temperature, as it should be. The relative share of Fph in F is about 2-3%. The contribution of phonons to the entropy of alloys at 300 K is decisive. In the region of low concentrations ( ≤ 0,1) and at other temperatures, the configurational component of the entropy Sconf is significantly less than Sph. The role of configuration entropy increases with decreasing temperature. For example, at 100 K Sph and Sconf are comparable in magnitude (see table 1).
The study of the role of fluctuation effects in the dynamics of disordered systems will be the subject of the next article.