Thermal Equation Of State Research Articles

Thermodynamic properties of real gases and BWR equation of state

The fundamental base for the calculation of the thermodynamic properties of materials is thermal equation of state and dependence of some of the basic specific heat capacities on temperature. The dependence of the specific thermal capacity on the second independent variable (for example on the volume) it is already possible to deduce from the thermal equation of state. The aim of this paper is to assess the compliance values of specific heat capacity which was calculated using the BWR thermal equation of the state and experimentally obtained known values of specific heat capacity for the substance, whose characteristics are available in a wide range of state space.

Computation for Equation of States of (Mg<sub>0.92</sub>, Fe<sub>0.08</sub>)SiO<sub>3 </sub>Perovskite Based on Mie-Grüneisen-Debye Model

In this paper, the thermal equation of state (EOS) of (Mg0.92, Fe0.08)SiO3is computed by Birch-Murnaghan and Mie-Grüneisen-Debye equations and the related parameters are also analyzed. The value of and has little effect on EOS of (Mg0.92, Fe0.08)SiO3perovskite. The effect of EOS of (Mg0.92, Fe0.08)SiO3perovskite is mainly from the temperature under high pressure. The temperature is higher; the deviation of EOS relative to the PREM model is bigger. The thermal EOS complies with PREM model at T=2000K. The thermal pressure of (Mg0.92, Fe0.08)SiO3perovskite a constant only related to temperature at the lower mantle conditions. At the same time, the EOS of (Mg0.92, Fe0.08)SiO3perovskite is insensitive to the data of and at T=2000K, but when and the thermal EOS is more agreement with PREM model. That is to say, when the value of the and is in the range of 253~273 GPa and 3.69~4.23, (Mg0.92, Fe0.08)SiO3is the perovskite phase, and (Mg0.92, Fe0.08)SiO3perovskite structure remains stable at the mantle conditions.

Phase diagram and thermodynamic properties of AIPO4 based on first-principles calculations and the quasiharmonic approximation

We calculated the phase diagram of \(\hbox {AlPO}_{4}\) up to 15 GPa and 2,000 K and investigated the thermodynamic properties of the high-pressure phases. The investigated phases include the berlinite, moganite-like, \(\hbox {AlVO}_{4},\, P2_1/c\), and \(\hbox {CrVO}_{4}\) phases . The computational methods used include density functional theory, density functional perturbation theory, and the quasiharmonic approximation. The investigated thermodynamic properties include the thermal equation of state, isothermal bulk modulus, thermal expansivity, and heat capacity. With increasing pressure, the ambient phase berlinite transforms to the moganite-like phase, and then to the \(\hbox {AlVO}_{4}\) and \(P2_1/c\) phases, and further to the \(\hbox {CrVO}_{4}\) phase. The stability fields of the \(\hbox {AlVO}_{4}\) and \(P2_1/c\) phases are similar in pressure but different in temperature, as the \(\hbox {AlVO}_{4}\) phase is stable at low temperatures, whereas the \(P2_1/c\) phase is stable at high temperatures. All of the phase relationships agree well with those obtained by quench experiments, and they support the stabilities of the moganite-like, \(\hbox {AlVO}_{4}\), and \(P2_1/c\) phases, which were not observed in room-temperature compression experiments.

Equation of state, composition, and conductivity of dense metal-vapor plasma

The caloric and thermal equations of state, composition, and conductivity of dense vapor plasma of different metals (Cu, Ag, Au, Ti, Ni, and B) have been calculated for the temperature range of 10000–50000 K and for pressures up to 20000 atm. An ion-molecular chemical model of a nonideal gas-plasma mixture, which was previously proposed for an aluminum vapor plasma, has been used in the calculations. Results obtained for the equation of state and conductivity (resistance) of metal vapor in the range of applicability of the model are in fairly good agreement with the experimental data. A comparison of the calculation and experimental results for different metals suggests that the model proposed is universal for metal-vapor plasmas.

High Pressure Structural Instability and Thermal Properties of Rutile TiO2 from First-principles

We report a first-principles calculation to investigate the structural instability of rutile TiO2. The high pressure structural parameters are well reproduced. The calculated phonon dispersion curves agree with experiments at zero pressure. Under compression, we capture a large softening around Г point, which indicates the structural instability. From the high pressure elastic constants, we find that the rutile TiO2 is unstable when the applied pressure is larger than 17.7 GPa. Within the quasi-harmonic approximation, the thermal equation of state, thermal expansion coefficient, bulk modulus, and entropy are well reproduced. The thermal properties confirm the available experimental data and are extended to a wider pressure and temperature range.

Molecular dynamics study of melting curve, entropy of fusion and solid–liquid interfacial energy of cobalt under pressure

We performed molecular dynamics (MD) simulations with the two embedded atom method (EAM) potentials to calculate the melting curves of cobalt over a wide range of pressure. Zhou׳s EAM potential can produce satisfying results, in better agreement with the experiment compared with Pun׳s. Based on Zhou׳s potential, we simulated the melting of Co with two approaches, i.e., the one-phase (hysteresis) method and two-phase (solid–liquid coexistence) method. Both approaches can effectively reduce the superheating, and their results are in the close proximity at the applied pressures. With the one-phase method, during the investigation of the entropy of fusion of Co, we found that with the pressure increasing, the entropy of fusion decreases rapidly first and then oscillates with pressure; when the pressure is beyond 100GPa, the entropy of fusion shows less pressure effect. When taking account of the solid–liquid interfacial energy at different pressures, we found that it increases monotonically with pressure, and can be well described as a fifth-order polynomial relation. Moreover, the thermal equation of state (EOS) and the temperature dependence of atomic structures of Co have been obtained successfully.

Phase transition and thermodynamic properties of beryllium from first-principles calculations

We have investigated the phase transition, electronic properties, phonon dispersion curves and thermodynamic properties of beryllium (Be) at high pressures and high temperatures using the first-principles calculations based on the density functional theory (DFT). The transition pressure of Be from hexagonal-closed-packed (hcp) structure to body-centered-cubic (bcc) structure occurs at 388GPa. The calculated phonon dispersion curves agree with experimental results. Under compression, the phonon dispersion curves of hcp Be do not show any anomaly or instability. At low pressure the phonon dispersion curves of bcc Be display imaginary frequencies along Γ–N symmetry line. Within the quasi-harmonic Debye model, the thermal equation of state and other properties including the thermal expansion coefficient, isothermal bulk modulus and its first pressure derivative, heat capacity and entropy of hcp and bcc Be are predicted.

Structural and dynamical properties of MgSiO3 melt over the pressure range 200–500 GPa: Ab initio molecular dynamics

Ab initio molecular dynamics simulations on MgSiO3 melt have been performed over the pressure range 200–500GPa. The results calculated by local density approximation (LDA) and generalized gradient approximation (GGA) were compared. Our results showed that compared with experimental data, LDA can yield more accurate pressure and thus the thermal equation of state and Hugoniot curve than GGA. The reason is that in MgSiO3 melt GGA overestimates the average length of MgO bonds than LDA, while they yield nearly the same results for SiO and OO bonds. Over the pressure range 230.1–464.7GPa at 13,000K, the calculated pair correlation function changes gradually, the mean SiO coordination number increases nearly linearly from 6.4 to 7.3 while the MgO and OO coordination numbers are almost unchanged, indicating no signature of a first-order liquid–liquid phase transition. The calculated distribution of OSi coordination number presents evidence for the presence of oxygen triclusters (oxygen coordinated with three Si frameworks). The calculated diffusion coefficient decreases from about 4 to 2×10−4cm2/s but the shear viscosity estimated using Zwanzig's formula increases from about 4 to 10cP with increasing pressure from 230.1 to 464.7GPa at 13,000K.

Concentration-dependent crystal structure, elastic constants and electronic structure of Zr x Ti1−x alloys under high pressure

The physical properties of Zr x Ti1−x (x = 0.0, 0.33, 0.5, 0.67, 0.75 and 1.00) alloys were simulated by virtual crystal approximation (VCA) methods which is generally used for disordered solid solutions modeling. The elastic constant, electronic structure and thermal Equation of state (EOS) of disordered Zr x Ti1−x alloys under pressure are investigated by plane-wave pseudo-potential method. Our simulations reveal increasement of variations of the calculated equilibrium volumes and decreasement of Bulk modulus as a function of the alloy compositions. Lattice parameters a and c of alloys with different Zr concentrations decrease linearly with pressure increasing, but the c/avalues are increasing as pressure increases, indicating no phase transitions under pressure from 0 GPa to 100 GPa. The elastic constants and the Bulk modulus to the Shear modulus ratios (B/G) indicate good ductility of Zr, Zr0.33Ti0.67, Zr0.5Ti0.5, Zr0.75Ti0.25 and Ti, but the Zr0.67Ti0.33 alloy is brittle under 0 K and 0 GPa. The metallic behavior of these alloys was also proved by analyzing partial and total DOS.

Thermal equation of state and thermodynamic properties of iron carbide Fe3C to 31 GPa and 1473 K

AbstractResent experimental and theoretical studies suggested preferential stability of Fe3C over Fe7C3at the condition of the Earth's inner core. Previous studies showed that Fe3C remains in an orthorhombic structure with the space groupPnmato 250 GPa, but it undergoes ferromagnetic (FM) to paramagnetic (PM) and PM to nonmagnetic (NM) phase transitions at 6–8 and 55–60 GPa, respectively. These transitions cause uncertainties in the calculation of the thermoelastic and thermodynamic parameters of Fe3C at core conditions. In this work we determinedP‐V‐Tequation of state of Fe3C using the multianvil technique and synchrotron radiation at pressures up to 31 GPa and temperatures up to 1473 K. A fit of ourP‐V‐Tdata to a Mie‐Gruneisen‐Debye equation of state produce the following thermoelastic parameters for the PM‐phase of Fe3C:V0 = 154.6 (1) Å3,KT0= 192 (3) GPa,KT′ = 4.5 (1),γ0= 2.09 (4),θ0 = 490 (120) К, andq = −0.1 (3). Optimization of theP‐V‐Tdata for the PM phase along with existing reference data for thermal expansion and heat capacity using a Kunc‐Einstein equation of state yielded the following parameters:V0 = 2.327 cm3/mol (154.56 Å3),KT0 = 190.8 GPa,KT′ = 4.68, ΘE10 = 305 K (which corresponds toθ0 = 407 K), γ0 = 2.10,e0 = 9.2 × 10−5K−1,m = 4.3, andg = 0.66 with fixed parametersmE1 = 3n = 12, γ∞ = 0,β = 0.3, anda0 = 0. This formulation allows for calculations of any thermodynamic functions of Fe3C versusTandVor versusTandP. Assuming carbon as the sole light element in the inner core, extrapolation of our equation of state of the NM phase of Fe3C suggests that 3.3 ± 0.9 wt % С at 5000 К and 2.3 ± 0.8 wt % С at 7000 К matches the density at the inner core boundary.

Thermal equation of state and thermodynamic properties of molybdenum at high pressures

A comprehensive P-V-T dataset for bcc-Mo was obtained at pressures up to 31 GPa and temperatures from 300 to 1673 K using MgO and Au pressure calibrants. The thermodynamic analysis of these data was performed using high-temperature Birch-Murnaghan (HTBM) equations of state (EOS), Mie-Grüneisen-Debye (MGD) relation combined with the room-temperature Vinet EOS, and newly proposed Kunc-Einstein (KE) approach. The analysis of room-temperature compression data with the Vinet EOS yields V0 = 31.14 ± 0.02 Å3, KT = 260 ± 1 GPa, and KT′ = 4.21 ± 0.05. The derived thermoelastic parameters for the HTBM include (∂KT/∂T)P = −0.019 ± 0.001 GPa/K and thermal expansion α = a0 + a1T with a0 = 1.55 ( ± 0.05) × 10−5 K−1 and a1 = 0.68 ( ± 0.07) × 10−8 K−2. Fitting to the MGD relation yields γ0 = 2.03 ± 0.02 and q = 0.24 ± 0.02 with the Debye temperature (θ0) fixed at 455-470 K. Two models are proposed for the KE EOS. The model 1 (Mo-1) is the best fit to our P-V-T data, whereas the second model (Mo-2) is derived by including the shock compression and other experimental measurements. Nevertheless, both models provide similar thermoelastic parameters. Parameters used on Mo-1 include two Einstein temperatures ΘE10 = 366 K and ΘE20 = 208 K; Grüneisen parameter at ambient condition γ0 = 1.64 and infinite compression γ∞ = 0.358 with β = 0.323; and additional fitting parameters m = 0.195, e0 = 0.9 × 10−6 K−1, and g = 5.6. Fixed parameters include k = 2 in Kunc EOS, mE1 = mE2 = 1.5 in expression for Einstein temperature, and a0 = 0 (an intrinsic anharmonicity parameter). These parameters are the best representation of the experimental data for Mo and can be used for variety of thermodynamic calculations for Mo and Mo-containing systems including phase diagrams, chemical reactions, and electronic structure.

Structure and thermodynamic properties of BeO: Empirical corrections in the quasiharmonic approximation

The equilibrium lattice parameters, bulk modulus, and phase transition of BeO are investigated by using the density functional theory with the Perdew-Burke-Ernzerhof (PBE) and Perdew and Zunger (PZ) functionals. With two different exchange-correlation functionals, we predict the similar results that BeO is stable in hexagonal wurtzite (B4) phase up to pressure of 100 GPa and then transforms directly into the rocksalt (B1) phase. The calculated phonon dispersion curves of the B4 phase BeO are in excellent agreement with the experimental data. Under compression, the phonon dispersion curves of BeO in the B4 phase do not show any anomaly or instability. Within the quasiharmonic approximation (QHA) plus empirical energy corrections (EEC) calculations, the thermal equation of state and thermodynamic properties of BeO are obtained. The EECs improve the systematic deviations of PBE and PZ functionals and reproduce the experimental results in the range of the validity of the QHA. Moreover, the effect of EECs on the phase stability has been investigated.

High-temperature compression experiments of CaSiO3 perovskite to lowermost mantle conditions and its thermal equation of state

In order to examine pressure–volume–temperature (P–V–T) relations for CaSiO3 perovskite (Ca-perovskite), high-temperature compression experiments with in situ X-ray diffraction were performed in a laser-heated diamond anvil cell (DAC) to 127 GPa and 2,300 K. We also employed an external heating system in the DAC in order to obtain P–V data at a moderate temperature of 700 K up to 113 GPa, which is the reference temperature for constructing an equation of state. The P–V data at 700 K were fitted to the second-order Birch–Murnaghan equation of state, yielding K 700,1bar = 207 ± 4 GPa and V 700,1bar = 46.5 ± 0.1 A3. Thermal pressure terms were evaluated in the framework of the Mie–Gruneisen–Debye model, yielding γ 700,1bar = 2.7 ± 0.3, q 700,1bar = 1.2 ± 0.8, and θ 700,1bar = 1,300 ± 500 K. A thermodynamic thermal pressure model was also employed, yielding α700,1bar = 5.7 ± 0.5 × 10−5/K and (∂K/∂T) V = −0.010 ± 0.004 GPa/K. Computed densities along a lower mantle geotherm demonstrate that Ca-perovskite is denser than the surrounding lower mantle, suggesting that Ca-perovskite-rich rocks do not rise up through the lower mantle. One of such rocks might be a residue of partial melting of subducted mid-oceanic ridge basalt (MORB) at the base of the mantle. Since the partial melt is FeO-rich and therefore denser than the mantle, all the components of subducted MORB may not return to shallow levels.

Shock-Induced Phase Transitions in Systems of Hard Spheres with Attractive Interactions

Shock-induced phase transitions are studied by adopting the recently-developed theoretical framework, which is applicable for shock waves in three phases (gas, liquid, and solid), based on the system of hard spheres with mutual attractive interactions. The Rankine-Hugoniot conditions derived from the system of Euler equations with caloric and thermal equations of state are studied, and the admissibility (stability) of a shock wave is analyzed. Two typical scenarios of the shock-induced phase transitions from gas phase to solid phase are found. A scenario of shock-induced phase transitions involving three phases simultaneously near the triple point is also found.

An ion-molecular chemical model of dense aluminum vapor plasma

A chemical model of nonideal gas-plasma mixture is proposed to describe a dense plasma of aluminum vapors. The molecules and the molecular ions are taken into consideration in addition to the conventionaly utilized set of electrons, ions and atoms to advance to low temperatures; doubly and triply ionized ions are considered for the high-temperature region. The effects of density are taken into account in a virial approximation. The virial corrections for the charge-charge and the charge-atom interactions are critically analyzed and chosen. The caloric and thermal equation of state, as well as the plasma composition have been obtained. It has been shown that an important role play molecules and molecular ions of aluminum especially in the early stage of heating of metal vapors. A satisfactory agreement with experimental results is obtained in the range of applicability of the model.

Extended thermodynamics of dense gases

We study extended thermodynamics of dense gases by adopting the system of field equations with a different hierarchy structure to that adopted in the previous works. It is the theory of 14 fields of mass density, velocity, temperature, viscous stress, dynamic pressure, and heat flux. As a result, most of the constitutive equations can be determined explicitly by the caloric and thermal equations of state. It is shown that the rarefied-gas limit of the theory is consistent with the kinetic theory of gases. We also analyze three physically important systems, that is, a gas with the virial equations of state, a hard-sphere system, and a van der Waals fluid, by using the general theory developed in the former part of the present work.

Thermal equations of state and phase relation of PbTiO3: A high P-T synchrotron x-ray diffraction study

The phase relation of tetragonal and cubic PbTiO3 and their unit-cell parameters have been determined by synchrotron x-ray diffraction at pressures up to 7.8 GPa and temperatures up to 1074 K with a cubic anvil apparatus. From these measurements, a pressure-temperature phase boundary between the tetragonal and cubic phases has been established. With increasing temperature or pressure, the c/a ratio of the ferroelectric, tetragonal PbTiO3 becomes closer to unity, suggesting that both heating and compression favor the paraelectric, cubic structure. Using a modified high-T Birch-Murnaghan equation of state and a thermal-pressure approach, we have derived the thermoelastic parameters of tetragonal and cubic PbTiO3, including the ambient bulk modulus K0, temperature derivative of bulk modulus at constant pressure, volumetric thermal expansivity, pressure derivative of thermal expansion, and temperature derivative of bulk modulus at constant volume. Our obtained K0 value for tetragonal PbTiO3 is consistent with previously reported results, while that for cubic PbTiO3 is smaller than earlier results probably due to differences in the experimental techniques used (cubic anvil apparatus versus diamond anvil cell) and related stress conditions of the samples. All other thermoelastic parameters for both tetragonal and cubic PbTiO3 have been determined for the first time. Compared with previous high-temperature data at atmospheric pressure, our P-V-T dataset for tetragonal PbTiO3 infers a pressure-induced crossover in volumetric thermal expansion from negative to positive between 0 and 1 GPa, an phenomenon that is of fundamentally interest and practically important.

Shock-Induced Phase Transitions from Gas Phase to Solid Phase

A theoretical framework for analyzing shock waves in three phases (gas, liquid, and solid) in a unified way is developed by adopting the system of hard spheres with mutual attractive interactions. The Rankine–Hugoniot conditions are derived from the system of Euler equations with caloric and thermal equations of state, and the admissibility (stability) of shock waves is investigated. It is found that there can exist two typical scenarios of the shock-induced phase transitions from gas phase to solid phase. For these scenarios the internal degrees of freedom of a particle f should be selected suitably. The necessary value of f is larger than O (10 1 ) or O (10 3 ) depending on the scenario.

FIRST-PRINCIPLES HIGH-PRESSURE ELASTIC AND THERMODYNAMIC PROPERTIES OF TANTALUM

The all-electron full-potential linearized muffin-tin orbital method, by means of quasi-harmonic Debye model, is applied to investigate the elastic constant and thermodynamic properties of body-centered-cubic tantalum (bcc Ta). The calculated elastic constants of bcc Ta at 0 K is consistent with the previous experimental and theoretical results. Our calculations give the correct trends for the pressure dependence of elastic constants. By using the convenient quasi-harmonic Debye model, we refined the thermal equations of state. The thermal expansivity and some other thermal properties agree well with the previous experimental and theoretical results.

Thermal equation of state of CaIrO3 post-perovskite

The pressure-volume-temperature (P-V-T) relation of CaIrO3 post-perovskite (ppv) was measured at pressures and temperatures up to 8.6 GPa and 1,273 K, respectively, with energy-dispersive synchrotron X-ray diffraction using a DIA-type, cubic-anvil apparatus (SAM85). Unit-cell dimensions were derived from the Le Bail full profile refinement technique, and the results were fitted using the third-order Birth-Murnaghan equation of state. The derived bulk modulus KT0 at ambient pressure and temperature is 168.3 ± 7.1 GPa with a pressure derivative K 0 T0 = 5.4 ± 0.7. All of the high temperature data, combined with previous experimental data, are fitted using the high-temperature Birch-Murnaghan equation of state, the thermal pressure approach, and the Mie-Grunei- sen-Debye formalism. The refined thermoelastic parameters for CaIrO3 ppv are: temperature derivative of bulk modu- lus ðoKT =oTÞ P =- 0.038 ± 0.011 GPa K -1 , aKT = 0.0039 ± 0.0001 GPa K -1 , oKT =oT ðÞ V =- 0.012 ± 0.002 GPa K -1 , and o 2 P=oT 2 �� V = 1.9 ± 0.3 9 10 -6