Next Article in Journal
Fast Image Classification for Grain Size Determination
Next Article in Special Issue
Machine-Learning-Based Model of Elastic—Plastic Deformation of Copper for Application to Shock Wave Problem
Previous Article in Journal
Effect of Carbon Content and Intercritical Annealing on Microstructure and Mechanical Tensile Properties in FeCMnSiCr TRIP-Assisted Steels
Previous Article in Special Issue
Molecular Insight into the Deformation of Single Crystal Copper Loaded by High-Speed Shock Wave
 
 
Search for Articles:
Title / Keyword
Author / Affiliation / Email
Journal
Article Type
 
 
Advanced Search
 
Section
Special Issue
Volume
Issue
Number
Page
 
Journals
Metals
Volume 11
Issue 10
10.3390/met11101548
metals-logo
Submit to this Journal Review for this Journal Propose a Special Issue
► Article Menu

Article Menu

share Share announcement Help format_quote Cite question_answer Discuss in SciProfiles thumb_up
...
Endorse textsms
...
Comment
first_page
settings
Order Article Reprints
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

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

by
Youlia Andreevna Bogdanova
*,
Sergey Aleksandrovich Gubin
and
Irina Vladimirovna Maklashova
National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Moscow 115409, Russia
*
Author to whom correspondence should be addressed.
Metals 2021, 11(10), 1548; https://doi.org/10.3390/met11101548
Submission received: 30 August 2021 / Revised: 18 September 2021 / Accepted: 24 September 2021 / Published: 28 September 2021
(This article belongs to the Special Issue Shock-Wave Loading of Metallic Materials)
Browse Figures
Versions Notes

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 − rm)] − 2exp[−α(r − rm)]},
where ε > 0 is the well depth; rm 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.
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 , i j = k i j r m , i i + r m , j j 2 ;   ε i j = l i j ε i i ε j j     ;   α i j = m i j α i i α j j
In Equation (2), kij, lij, and mij 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 kij = lij = mij = 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., kij = lij = mij = 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.

Author Contributions

Conceptualization, Y.A.B. and I.V.M.; methodology, Y.A.B. and S.A.G.; software, Y.A.B. and I.V.M.; validation, Y.A.B.; formal analysis, Y.A.B.; investigation, Y.A.B. and I.V.M.; resources, I.V.M.; data curation, S.A.G.; writing—original draft preparation, Y.A.B.; writing—review and editing, I.V.M. and S.A.G.; visualization, Y.A.B.; supervision, S.A.G.; project administration, Y.A.B.; funding acquisition, S.A.G. All authors have read and agreed to the published version of the manuscript.

Funding

This research was supported by The Ministry of Science and Higher Education of the Russian Federation (Agreement with Joint Institute for High Temperatures RAS, No. 075-15-2020-785, dated 23 September 2020).

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Borisov, A.A.; Gubin, S.A.; Shargatov, V.A. Applicability of a chemical-equilibrium model to explosion products. Dyn. Detonations Explos. Explos. Phenom. 1991, 134, 138–153. [Google Scholar] [CrossRef]
  2. Gelfand, B.E.; Gubin, S.A.; Mihalkin, V.N.; Shargatov, V.A. On the calculations of flows with detonation-waves. Khimicheskaya Fizika 1984, 3, 683–690. [Google Scholar]
  3. Ross, M. The repulsive forces in dense argon. J. Chem. Phys. 1980, 73, 4445–4450. [Google Scholar] [CrossRef]
  4. Ross, M. A high-density fluid-perturbation theory based on an inverse 12th-power hard-sphere reference system. J. Chem. Phys. 1979, 71, 1567–1571. [Google Scholar] [CrossRef]
  5. Kang, H.S.; Lee, C.S.; Ree, T.; Ree, F.H. A perturbation theory of classical equilibrium fluids. J. Chem. Phys. 1985, 82, 414–423. [Google Scholar] [CrossRef]
  6. Henderson, D.; Barker, J.A. Perturbation theory and equation of state for fluids. II. A successful theory of liquids. J. Chem. Phys. 1967, 47, 4714–4721. [Google Scholar] [CrossRef]
  7. Weeks, J.D.; Chandler, D.; Andersen, H.C. Role of repulsive forces in determining the equilibrium structure of simple liquids. J. Chem. Phys. 1971, 54, 5237–5247. [Google Scholar] [CrossRef]
  8. Zerah, G.; Hansen, J.-P. Self-consistent integral equations for fluid pair distribution functions: Another attempt. J. Chem. Phys. 1986, 84, 2336–2344. [Google Scholar] [CrossRef]
  9. Fried, L.E.; Howard, W.M. An accurate equation of state for the exponential-6 fluid applied to dense supercritical nitrogen. J. Chem. Phys. 1998, 109, 7338–7349. [Google Scholar] [CrossRef]
  10. Gubin, S.A.; Victorov, S.B. The accuracy of the theories based on statistical physics for the thermodynamic modeling of state parameters of dense pure gases (fluids). J. Phys. Conf. Ser. 2019, 1205, 012020. [Google Scholar] [CrossRef]
  11. Oliveira, J.P.; Shen, J.; Zeng, Z.; Park, J.M.; Choi, Y.T.; Schell, N.; Maawad, E.; Zhou, N.; Kim, H.S. Dissimilar laser welding of a CoCrFeMnNi high entropy alloy to 316 stainless steel. Scr. Mater. 2022, 206, 114219. [Google Scholar] [CrossRef]
  12. Conde, F.F.; Escobar, J.D.; Oliveira, J.P.; Béreš, M.; Jardini, A.L.; Bose, W.W.; Avila, J.A. Effect of thermal cycling and aging stages on the microstructure and bending strength of a selective laser melted 300-grade maraging steel. Mater. Sci. Eng. A 2019, 758, 192–201. [Google Scholar] [CrossRef]
  13. Escobar, J.D.; Poplawsky, J.D.; Faria, G.A.; Rodriguez, J.; Oliveira, J.P.; Salvador, C.A.F.; Mei, P.R.; Babu, S.S.; Ramirez, A.J. Compositional analysis on the reverted austenite and tempered martensite in a Ti-stabilized supermartensitic stainless steel: Segregation, partitioning and carbide precipitation. Mater. Des. 2018, 140, 95–105. [Google Scholar] [CrossRef]
  14. Victorov, S.B.; El-Rabii, H.; Gubin, S.A.; Maklashova, I.V.; Bogdanova, Y.A. An accurate equation-of-state model for thermodynamic calculations of chemically reactive carbon-containing systems. J. Energ. Mater. 2010, 28, 35–49. [Google Scholar] [CrossRef]
  15. Victorov, S.B.; Gubin, S.A. Thermodynamic Modeling of Complicated Chemical Systems at High Pressures and Temperatures; NRNU MEPhI: Moscow, Russia, 2016. (In Russian) [Google Scholar]
  16. Bogdanova, Y.A.; Gubin, S.A.; Victorov, S.B.; Gubina, T.V. Theoretical model of the equation of state of a two-component fluid with the Exp-6 potential based on perturbation theory. High Temp. 2015, 53, 481–490. [Google Scholar] [CrossRef]
  17. Zhang, S.; Morales, M.A. First-principles equations of state and structures of liquid metals in multi-megabar conditions. AIP Conf. Proc. 2020, 2272, 090004. [Google Scholar] [CrossRef]
  18. Dai, J.; He, D.; Song, Y. Correlations of Equilibrium Properties and Electronic Structure of Pure Metals. Materials 2019, 12, 2932. [Google Scholar] [CrossRef] [PubMed] [Green Version]
  19. Lalnuntluanga, C.; Mishra, R.K. Temperature effect on structural and transport coefficient of liquid copper under square-well interaction. AIP Conf. Proc. 2021, 2327, 020043. [Google Scholar] [CrossRef]
  20. Dubinin, N.E.; Vatolin, N.A.; Filippov, V.V. Thermodynamic perturbation theory in studies of metal melts. Russian Chem. Rev. 2014, 83, 987–1002. [Google Scholar] [CrossRef]
  21. Dubinin, N.E.; Yuryev, A.A.; Vatolin, N.A. Straightforward calculation of the WCA entropy and internal energy for liquid metals. Thermochim. Acta 2011, 518, 9–12. [Google Scholar] [CrossRef]
  22. Bogdanova, Y.A.; Maklashova, I.V.; Gubin, S.A.; Amir, Z.A. The influence of type of the intermolecular interaction potential on transport properties of helium. J. Phys. Conf. Ser. 2020, 1686, 012075. [Google Scholar] [CrossRef]
  23. Bogdanova, Y.A.; Gubin, S.A.; Amir, Z.A. Modeling of Thermophysical Properties and Transport Properties of Basic Combustion Products of Organic Substances. Phys. Atom. Nucl. 2020, 83, 1563–1568. [Google Scholar] [CrossRef]
  24. Selezenev, A.A.; Aleynikov, A.Y.; Gantchuk, N.S.; Yermakov, P.V.; Labanowski, J.K.; Korkin, A.A. SageMD: Molecular-dynamic software package to study properties of materials with different models for interatomicinteractions. Comput. Mater. Sci. 2003, 28, 107–124. [Google Scholar] [CrossRef]
  25. MCCCS Towhee. Available online: http://towhee.sourceforge.net (accessed on 5 September 2005).
  26. Martin, M.G. MCCCS Towhee: A tool for Monte Carlo molecular simulation. Mol. Simulat. 2013, 39, 1212–1222. [Google Scholar] [CrossRef]
  27. Gubin, S.A.; Maklashova, I.V.; Melnikova, K.S. The thermophysical and mechanical properties of a composite of aluminum and aluminum oxide-based additive mixing model. Combust. Explos. 2012, 5, 297–301. (In Russian) [Google Scholar]
  28. Murnagan, F.D. The Compressibility of Media under Extreme Pressures. Proc. Natl. Acad. Sci. USA 1944, 30, 244–248. [Google Scholar] [CrossRef] [PubMed] [Green Version]
  29. Vaboya, S.N.; Kennedy, G.C. Compressibility of 18 metals to 45 kbar. J. Phys. Chem. Solids 1970, 31, 2329–2345. [Google Scholar] [CrossRef]
  30. Syassen, K.; Holzapfel, W.B. Isotermal compression of Al and Ag to 120 kbar. J. Appl. Phys. 1978, 49, 4427–4430. [Google Scholar] [CrossRef]
  31. Mao, H.K.; Bell, P.M. Specific volume measurement of Cu, Mo, Pd and Ag and calibration of the ruby R1 fluorescence pressure gauge from 0.06 to 1 Mbar. J. Appl. Phys. 1978, 49, 3276–3283. [Google Scholar] [CrossRef]
  32. Dewaele, A.; Loubeyre, P.; Mezouar, M. Equations of state of six metals above 94 GPa. Phys. Rev. B. 2004, 70, 094112. [Google Scholar] [CrossRef] [Green Version]
  33. Litygina, L.M.; Malyushtskaya, Z.V.; Pashkina, T.A.; Kabalkina, S.S. Isotermal compression of Al to 10 GPa at 673 K. Phys. Stat. Sol. Ser. A 1982, 69, 147–150. [Google Scholar] [CrossRef]
  34. Dziedzic, J.; Winczewski, S.; Rybicki, J. Structure and properties of liquid Al–Cu alloys: Empirical potentials compared. Comput. Mater. Sci. 2016, 114, 219–232. [Google Scholar] [CrossRef] [Green Version]
  35. Wang, S.Y.; Kramer, M.J.; Xu, M.; Wu, S.; Hao, S.G.; Sordelet, D.J.; Ho, K.M.; Wang, C.Z. Experimental and ab initio molecular dynamics simulation studies of liquid Al60Cu40 alloy. Phys. Rev. B. 2009, 79, 144205. [Google Scholar] [CrossRef]
  36. Gubin, S.A.; Maklashova, I.V.; Selezenev, A.A.; Kozlova, S.A.; Demidenko, T.S. Molecular dynamics simulation and visualization of melting aluminum crystal in shock wave. Sci. Vis. 2014, 6, 14–23. [Google Scholar]
  37. Gubin, S.A.; Maklashova, I.V.; Selezenev, A.A.; Kozlova, S.A. Molecular-Dynamics Study Melting Aluminum at High Pressures. Phys. Procedia 2015, 72, 338–341. [Google Scholar] [CrossRef] [Green Version]
  38. Gubin, S.A.; Maklashova, I.V. The Hugoniot adiabat of crystalline copper based on molecular dynamics simulation and semiempirical equation of state. J. Phys. Conf. Ser. 2018, 946, 012098. [Google Scholar] [CrossRef]
  39. Marsh, S.P. (Ed.) LASL Shock Hugoniot Data; University California Press: Berkeley, CA, USA, 1980. [Google Scholar]
  40. Mitchell, A.C.; Nellis, W.J. Shock compression of aluminum, copper and tantalum. J. Appl. Phys. 1981, 52, 3363–3374. [Google Scholar] [CrossRef]
  41. Trunin, R.F.; Gudarenko, L.F.; Zhernokletov, M.V.; Simakov, G.V. Experimental Data on Shock Compressibility and Adiabatic Expansion of Condensed Substances; RFNC: Sarov, Russia, 2001. (In Russian) [Google Scholar]
  42. Al'tshule, L.V.; Kormer, S.B.; Brazhnik, M.I.; Vladimirov, L.A.; Speranskaya, M.P.; Funtikov, A.I. The isentropic compressibility of aluminum, copper, lead at high pressures. Zh. Eksp. Teor. Fiz. 1960, 38, 1061–1073. [Google Scholar]
  43. Isbell, W.H.; Shipman, F.H.; Jones, A.H. Hugoniot Equation of State Measurements for Eleven Materials to Five Megabars; Report MSL-68-13; General Motors Corp., Mat. Sci. Lab.: Warren, MI, USA, 1968. [Google Scholar]
  44. Van Thiel, M. (Ed.) Compendium of Shock Wave Data; Report UCRL-50108; Lawrence Livermore Laboratory: Livermore, CA, USA, 1977. [Google Scholar]
Metals 11 01548 g001 550
Figure 1. Pressure dependence on the degree of compression of the aluminum crystal lattice at (a) T = 298 K and (b) 673 K. Initial state: V0 = 0.372 cm3/g. Lines in the figures: 1—calculation based on the theoretical EOS [16] in this work; 2—analytical EOS [27]; 3—empirical EOS of Murnaghan [28]. Symbols: 4, 5, 7—experimental data [30,32,33]; 6—MD simulation results in this work.
Figure 1. Pressure dependence on the degree of compression of the aluminum crystal lattice at (a) T = 298 K and (b) 673 K. Initial state: V0 = 0.372 cm3/g. Lines in the figures: 1—calculation based on the theoretical EOS [16] in this work; 2—analytical EOS [27]; 3—empirical EOS of Murnaghan [28]. Symbols: 4, 5, 7—experimental data [30,32,33]; 6—MD simulation results in this work.
Metals 11 01548 g001
Metals 11 01548 g002 550
Figure 2. Pressure dependence on the degree of compression of the crystal lattice of copper at T = 298 K. Initial state: V0 = 0.112 cm3/g. Lines in the figures: 1—calculation based on the theoretical EOS [16] in this work; 2—analytical EOS [27]; 3—empirical EOS of Murnaghan [28]. Symbols: 4—experimental data [31]; 5—MD simulation results in this work.
Figure 2. Pressure dependence on the degree of compression of the crystal lattice of copper at T = 298 K. Initial state: V0 = 0.112 cm3/g. Lines in the figures: 1—calculation based on the theoretical EOS [16] in this work; 2—analytical EOS [27]; 3—empirical EOS of Murnaghan [28]. Symbols: 4—experimental data [31]; 5—MD simulation results in this work.
Metals 11 01548 g002
Metals 11 01548 g003 550
Figure 3. Dependence of the density of AlxCu100-x alloys on pressure on isotherms: (a) T = 1073 K and (b) 1523 K. The lines on the figures are the results of calculations based on the theoretical EOS [16] in this work; the symbols are the results of MD simulation in this work.
Figure 3. Dependence of the density of AlxCu100-x alloys on pressure on isotherms: (a) T = 1073 K and (b) 1523 K. The lines on the figures are the results of calculations based on the theoretical EOS [16] in this work; the symbols are the results of MD simulation in this work.
Metals 11 01548 g003
Metals 11 01548 g004 550
Figure 4. Dependence of density on the temperature at zero isobars for binary alloy Al60Cu40. Line in the figure: 1—calculation based on the theoretical EOS [16] in this work. Symbols: 2—MD simulation results in this work; 3–5—simulation results [34,35].
Figure 4. Dependence of density on the temperature at zero isobars for binary alloy Al60Cu40. Line in the figure: 1—calculation based on the theoretical EOS [16] in this work. Symbols: 2—MD simulation results in this work; 3–5—simulation results [34,35].
Metals 11 01548 g004
Metals 11 01548 g005 550
Figure 5. Dependence of pressure on the degree of compression of the aluminum crystal lattice under compression in a shock wave. Lines in the figures: 1—calculation based on the theoretical EOS [16] in this work; 2—analytical EOS [27]. Symbols: 3—MD simulation results [36]; 4–6—experimental data [39,40,41].
Figure 5. Dependence of pressure on the degree of compression of the aluminum crystal lattice under compression in a shock wave. Lines in the figures: 1—calculation based on the theoretical EOS [16] in this work; 2—analytical EOS [27]. Symbols: 3—MD simulation results [36]; 4–6—experimental data [39,40,41].
Metals 11 01548 g005
Metals 11 01548 g006 550
Figure 6. Dependence of pressure on the degree of compression of the crystal lattice of copper during compression in a shock wave. Lines in the figures: 1—calculation based on the theoretical EOS [16] in this work. Symbols: 2—MD simulation results [38]; 3–8—experimental data [39,40,41,42,43,44].
Figure 6. Dependence of pressure on the degree of compression of the crystal lattice of copper during compression in a shock wave. Lines in the figures: 1—calculation based on the theoretical EOS [16] in this work. Symbols: 2—MD simulation results [38]; 3–8—experimental data [39,40,41,42,43,44].
Metals 11 01548 g006
Table
Table 1. Potential parameters.
Table 1. Potential parameters.
Metalε/kB, Krm, Ǻα, 1/Ǻ
Al30833.201.21
Cu24172.711.42
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Share and Cite

MDPI and ACS Style

Bogdanova, Y.A.; Gubin, S.A.; Maklashova, I.V. Calculation of Thermodynamic Properties of Metals and Their Binary Alloys by the Perturbation Theory. Metals 2021, 11, 1548. https://doi.org/10.3390/met11101548

AMA Style

Bogdanova YA, Gubin SA, Maklashova IV. Calculation of Thermodynamic Properties of Metals and Their Binary Alloys by the Perturbation Theory. Metals. 2021; 11(10):1548. https://doi.org/10.3390/met11101548

Chicago/Turabian Style

Bogdanova, Youlia Andreevna, Sergey Aleksandrovich Gubin, and Irina Vladimirovna Maklashova. 2021. "Calculation of Thermodynamic Properties of Metals and Their Binary Alloys by the Perturbation Theory" Metals 11, no. 10: 1548. https://doi.org/10.3390/met11101548

APA Style

Bogdanova, Y. A., Gubin, S. A., & Maklashova, I. V. (2021). Calculation of Thermodynamic Properties of Metals and Their Binary Alloys by the Perturbation Theory. Metals, 11(10), 1548. https://doi.org/10.3390/met11101548

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Metrics

Back to TopTop