Calculation of Thermodynamic Properties of Metals and Their Binary Alloys by the Perturbation Theory

Abstract

This paper presents the results of calculating the thermodynamic properties of aluminum, copper, and their binary alloys under isothermal and shock compression. The calculations were performed by a theoretical equation of state based on perturbation theory. The pair Morse potential was used to describe the intermolecular interaction in metals. The calculation results are in good agreement with the experimental data and the results of molecular dynamics modeling performed in this work using the LAMMPS software package. Furthermore, it is shown that the equation of state based on the perturbation theory with the corresponding potential of intermolecular interaction can be used to calculate the thermodynamic properties of gaseous (fluid) systems and pure metals and their binary alloys.

1. Introduction

Thermodynamic modeling of complex chemical reacting systems has been used to solve practical problems in various fields of science and technology for many years [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]. At the same time, realistic modeling of the thermodynamic properties and the composition of multicomponent and multiphase products of complex chemical reacting systems in a wide range of pressures and temperatures is significant.

Basically, semi-empirical equations of state (EOS) models are often used for practical thermodynamic calculations to describe the behavior of metals. A common disadvantage of such EOS is their insufficient physical validity, which is why the results of thermodynamic calculations often turn out to be unrealistic, especially in those regions of pressures and temperatures where empirical constants were not chosen.

At present, for thermodynamic modeling of the properties of dense fluid systems, theoretically based equations of state are widely used, built based on modern methods of statistical mechanics and realistic potentials of the interaction of molecules. Such EOS provides good agreement with the results of Monte Carlo (MC) and molecular dynamics (MD) simulations over a wide range of pressures and temperatures.

Most of the currently existing theoretically substantiated EOS models that allow predicting the thermodynamic parameters of fluids and fluid mixtures can be based on MCRSR (Mansoori–Canfield–Rasaiah–Stell–Ross) [3,4], thermodynamic perturbation theories [5,6,7], and integral equations for molecular distribution functions in the HMSA (hypernetted-chain/soft-core Mean Spherical Approximation) [8,9]. The main idea of each of these theories is to express the excess Helmholtz energy of the mixture in the form of a Taylor series for the basic hard-sphere fluid. The difference between the KLRR perturbation theory [5,6,7], the variational theory MCRSR [3,4], and the HMSA theory [8,9] is the absence of a correcting function. This function corrects the excess Helmholtz energy so that the excess pressure and internal energy are in good agreement with the results of the Monte Carlo computer experiments. A detailed description of thermodynamic theories and a comparison of their accuracy are presented in [10].

Theoretical EOS are also used to calculate the thermodynamic properties of pure metals [17], the structural characteristics of binary alloys [18,19], and the thermodynamic properties of two-component alloys [20,21].

At present, one of the best theories for obtaining the EOS of fluids both at high pressures and at lower temperatures and densities is the KLRR perturbation theory [5]. The modified version of the KLRR-T [14,15] has higher accuracy and speed than the original version. In reference [16], based on the KLRR-T perturbation theory, an EOS model was developed and implemented in the form of a computational program, making it possible to calculate the thermodynamic properties of both one-component systems and binary mixtures. The results of calculating the thermodynamic parameters of one- and two-component systems [14,16,22,23] due to isothermal and shock compression by the EOS based on the KLRR-T perturbation theory agree with the data of computer simulation by MC and MD methods and experimental data.

Perturbation theory applies to any gaseous or condensed systems in which the interatomic potential describes interactions. Therefore, this paper shows the application of the EOS, developed based on the KLRR-T perturbation theory, with the corresponding intermolecular interaction potential for calculating the thermodynamic properties of aluminum, copper, and their binary alloys under isothermal compression.

2. Intermolecular Interaction Potential

For the reliability of thermodynamic modeling, it is necessary to use potentials that realistically describe the nature of intermolecular forces in the pressure and temperature range of interest.

The pair Morse potential is widely used to describe the intermolecular interaction of metal molecules:

φ(r) = ε{exp[−2α(r − r_m)] − 2exp[−α(r − r_m)]},

(1)

where ε > 0 is the well depth; r_m is the distance between the centers of molecules, at which the potential energy is minimal; and α—parameter characterizes the width of the well depth.

The potential parameters for the substances studied in work are borrowed from reference [24] and are presented in

Table 1. Potential parameters.

Metal	ε/kB, K	rm, Ǻ	α, 1/Ǻ
Al	3083	3.20	1.21
Cu	2417	2.71	1.42

.

In binary mixtures, interactions also occur between pairs of unlike molecules (i ≠ j). The unlike potential parameters for such pairs of molecules can either be specified explicitly, or formally expressed in terms of the potential parameters for the corresponding pairs of molecules of the same name:

$$r_{m,ij}=k_{ij}\frac{r_{m,ii}+r_{m,jj}}{2}; {\epsilon }_{ij}=l_{ij}\sqrt{{\epsilon }_{ii}{\epsilon }_{jj}} ; {\alpha }_{ij}=m_{ij}\sqrt{{\alpha }_{ii}{\alpha }_{jj}}$$

(2)

In Equation (2), k_ij, l_ij, and m_ij are correction factors, the values of which are usually close to one. These factors can be determined from the available experimental data or taken equal to k_ij = l_ij = m_ij = 1, which corresponds to the classical Lorentz–Berthelot mixing rules. In this work, additive parameters are used to describe the cross interactions of binary alloys of aluminum and copper, i.e., k_ij = l_ij = m_ij = 1 in Equation (2).

3. Modeling the Thermodynamic Parameters of Isothermal Compression of Metals Al and Cu and Their Binary Alloys

3.1. Modeling Isothermal Compression of Aluminum and Copper

Molecular dynamics modeling was additionally carried out using the LAMMPS software package [25,26] to study the possibility of using the theoretical EOS for modeling the properties of metals.

The calculated supercells of aluminum and copper were obtained by multiplying along three spatial coordinates, 7 × 7 × 7, of the corresponding unit four-atomic cell and contained 1372 atoms.

Modeling of isothermal compression of aluminum was carried out at temperatures of 298 K and 673 K, with copper at 298 K. For modeling, an NPT ensemble with periodic boundary conditions was used. This means that particles interact across the boundary, and they can exit one end of the box and re-enter the other end. To suppress fluctuations of thermodynamic parameters in the system, a thermostat and drag 1.0 parameter were used, which does not significantly affect the simulation results. The integration step was selected on the basis of the required calculation accuracy, and amounted to 0.0001 ps. The calculation duration was 400,000 steps, which corresponds to a time interval of 40 ps.

By the EOS of two-component systems developed based on perturbation theory [16], the parameters of isothermal compression of aluminum at temperatures of 298 K and 673 K with copper at 298 K were calculated. The results are presented graphically in PV coordinates in Figure 1 and Figure 2, where the results also show calculations of the analytical EOS [27], empirical EOS [28], experimental data [29,30,31], and the results of MD simulation.

As can be seen from Figure 1 and Figure 2, the results of calculations of the isothermal compression of aluminum and copper by the EOS based on the perturbation theory agree with the experimental data, as well as the results of calculations based on the analytical EOS and the data of molecular dynamics simulation.

3.2. Modelling Isothermal Compression of Binary Al-Cu Alloys

The possibility of using the theoretical EOS [16] for calculating the thermodynamic properties of binary alloys was investigated in this work on binary alloys of aluminum and copper AlxCu100−x of various molar compositions: 90–10, 80–20, 70–30, 60–40, and 50–50.

The size of the computational domain was selected based on the crystal lattice parameters of the individual substances used in the work, which is 33 Å × 33 Å × 33 Å. When carrying out MD modeling in three spatial directions, periodic boundary conditions were used. The time step was 0.1 fs, and the number of steps in the calculation was 200000, which corresponds to the total calculation time of 20 ps. The full calculation time is optimal for mixing substances in the melt and bringing the supercell fluctuations to a value that does not affect the accuracy of further calculations. Molecular dynamics modeling of isothermal compression of the alloy for the ratios Al₅₀Cu₅₀, Al₆₀Cu₄₀, Al70Cu30, Al₈₀Cu₂₀, and Al₉₀Cu₁₀ was carried out in a series of numerical experiments on isotherms: 1073 K, 1273 K, and 1523 K in the pressure range 200–2000 MPa. In the specified range of thermodynamic parameters of the system, the investigated alloy is in the liquid phase.

Thermodynamic modeling of isothermal compression of binary alloys of aluminum and copper was carried out based on EOS [16]. The results of the calculations agree with the data of MD simulation and are presented graphically for three compositions—Al₅₀Cu₅₀, Al₇₀Cu₃₀, and Al₉₀Cu₁₀—in the form of the dependence of density on pressure isotherms 1073 K (Figure 3a) and 1523 K (Figure 3b).

The properties of the Al60Cu40 binary alloy were considered in more detail as a function of the temperature at the zero isobars for comparison with the results of molecular dynamics modeling using the modified EAM potential (MEAM) [34], calculations based on the plane-wave formulation of the density functional theory (DFT-MD) [35], and the Gupta potential [34].

Molecular dynamics modeling in this work was carried out at a pressure of 2000 MPa, since zero pressure is unattainable for the EAM potential due to the disintegration of the supercell. However, the simulation results indicate an insignificant difference (<1%) in the density of the alloy at pressures of 0 bar and 2000 bar.

In turn, thermodynamic modeling based on the EOS [16] was carried out at a pressure of 100 Pa. The calculation results are presented graphically in the form of the temperature dependence of the alloy density in Figure 4, which also shows the results of the MD simulation of our work and other authors [34,35].

Figure 4 shows the agreement of the density values calculated using the EOS with the simulation results performed in this work and works [34,35]. In the considered temperature range 923–1523 K, the deviations of the density values obtained based on the EOS calculation [16] in this work are no more than 3%. This result confirms the applicability of the theoretical EOS based on perturbation theory for modeling the properties of binary alloys.

4. Modeling Al and Cu Shock Hugoniot

The investigation of the range of melting of aluminum and copper in a shock wave, and the calculation of the thermal properties of metals at high pressures and temperatures, is relevant in solving scientific and practical problems [36,37,38].

For molecular dynamics modeling of shock compression of metals, the Hugoniostat method was used, implemented in the LAMMPS software package [25,26]. The calculated supercell of aluminum and copper was formed at an initial temperature of 298.15 K and a pressure of 1 kbar.

Thermodynamic modeling of shock-wave compression of aluminum and copper was carried out based on the theoretical EOS [16]. The calculation results are presented in a PV diagram in Figure 5 and Figure 6 for aluminum and copper, respectively. The figures also show the results of MD simulation [36,38], calculations based on the analytical EOS [27], and experimental data [39,40,41,42,43,44].

Figure 5 and Figure 6 show the agreement of the results of calculations of the pressure during shock-wave compression of aluminum and copper with experimental data, and the results of calculations based on the analytical EOS and the data of molecular dynamics modeling.

5. Discussion

Based on the perturbation theory, an equation of state was developed for calculating the thermodynamic properties of fluids. In this work, it is shown that this equation of state can be used to calculate the properties of condensed substances, including metals, if a suitable interatomic interaction potential is used. Calculations of the thermodynamic parameters of the isothermal compression of aluminum and copper—as well as their binary alloys, based on the equation of state developed using perturbation theory [16] using the Morse intermolecular interaction potential—are consistent with experimental data, calculations based on analytical EOS, and the results of molecular dynamics modeling.

Some difference between the results of calculations based on the theoretical equation of state from the data of molecular dynamics modeling in our work and in the work of other authors is explained by the fact that the calculation method using perturbation theory does not take into account the real structure of metals and their alloys or the specific arrangement of atoms in the crystal lattice, but uses interatomic pair potential Morse. However, the calculated results show that the Morse potential describes the thermodynamic properties of metals and alloys under isothermal and shock-wave compression with good accuracy.

6. Conclusions

The developed equation of state based on the KLRR perturbation theory using the Morse pair interaction potential is applicable for calculating the thermodynamic parameters of isothermal compression of aluminum and copper metals.

The equation of state can be used to calculate the thermodynamic parameters of gaseous (fluid) systems and condensed media, including binary metal alloys.