High-temperature Equation Of State Research Articles

Thermodynamic Characterization and Equation of State for Solid and Liquid Lead

A high-temperature equation of state (EoS) for the fcc phase of solid lead and liquid lead was developed herein using experimental data on thermodynamic properties, volumetric thermal expansion, compressibility, temperature-dependent bulk modulus, and sound velocity from ultrasonic measurements and melting curve. The whole totality of experimental data was optimized using the temperature-dependent Murnaghan EoS over a pressure range of 0–130 kbar. The temperature dependences of thermodynamic and thermophysical parameters were described herein using an expanded Einstein model. The resultant EoS describes well the whole set of available experimental data within measurement uncertainties of individual parameters.

Hybrid ab-initio/experimental high temperature equations of state: Application to the NaCl pressure scale

Equations of state (EoS) are a fundamental subject in high pressure/temperature (PT) research. Ab initio calculations based on density functional theory (DFT) can provide valuable information about a material's EoS at PT conditions that cannot be easily accessed experimentally. However, these calculations have systematic errors due to (1) a lack of a precise description of the exchange correlation energy, (2) methodological limitations in the way temperature is addressed, for instance, anharmonicity at high temperatures in quasiharmonic calculations. To address the first issue, we have improved, developed, and tested correction schemes aiming to remove DFT errors and to produce predictive low temperature EoS with accuracy comparable to experiments. We have investigated four schemes and applied them to three different functionals. The second issue has been addressed with a simple anharmonic correction that effectively removed high temperature anharmonic errors.

Dynamic stability of Zr under high pressure and high temperature

The phase transitions and structure stabilities of materials have always attracted much attention of the experimental and theoretical investigators. When calculating the phonon dispersion of the cubic structure of the transition metal Zr (β -Zr), the traditional methods always give the negative phonon frequencies. So the quasi-harmonic approximation cannot solve this kind of problem. We obtain the phonon dispersion of β -Zr at high pressure and high temperature by using the newly developed self-consistent ab initio lattice dynamics method, which can well consider the phonon-phonon interactions. And then the stable region of β -Zr in the high pressure and high temperature phase diagram is predicted. The full phase diagram of Zr is also predicted. We also obtain the high temperature equation of state (EOS) and thermal expansion of β -Zr, which can help to construct the EOS data base of Zr.

Numerical research on the ion-beam-driven hydrodynamic motion of fissile targets for nuclear safety studies

AbstractAs a method to evaluate high-temperature equation of state (EOS) data of fissile materials precisely and safely, we numerically examined an experimental setup based on a sub-range fissile target and a high-intensity short-pulsed heavy-ion beam. As an example, we calculated one-dimensional hydrodynamic motion of a uranium target with ρ = 0.03ρsolid(ρsolid ≡ solid density = 19.05 g/cm3) induced by a pulsed23Na+beam with a duration of 2 ns and a peak power of 5 GW/mm2. The projectile stopping power was calculated using a density- and temperature-dependent dielectric response function. To heat the target uniformly, we optimized the experimental condition so that the energy deposition could occur almost at the top of the Bragg peak. The energy deposition inhomogeneity could be reduced to ±5% by adjusting the incident energy and the target thickness to be 2.02 MeV/u and 180 μm, respectively. The target could be heated homogeneously up tokT=7 eV well before the arrival of the rarefaction waves at the center of the target. In principle, the EOS data can be evaluated by iteratively adjusting the data embedded in the hydro code until the measured hydrodynamic motion is reproduced by the calculation. This method is consistent with the conditions of nuclear nonproliferation, because a very small amount of fissile material is enough to perform the experiment, and no shock compression occurs in the target.

Hugoniot-Measurement Experiments of the Heated Samples

Pressure calibration in static compression experiments have been undertaken on the basis of the equation of state (EOS) derived from the Hugoniot data of the pressure scale materials such as Au, Pt and MgO. However, room-temperature isothermal compression curve and further high temperature compression curve have been derived by using the assumed Grüneisen parameters, which cause the larger error at the higher temperature. If the Hugoniot data of the heated sample are measured, the accurate high-temperature EOS can be obtained, and the Grüneisen parameter (γ) can be directly discussed. We have measured Hugoniot data of Cu, W, Au, etc. by using the high-time resolution streak camera system equipped with a powder gun and two-stage light gas gun. In this study, the Hugoniot-measurement technique of the elevated temperature sample using high-frequency heating apparatus was established equipped with a powder gun. We succeeded in the measurement of the Hugoniot data (shock-velocity and particle-velocity) of the heated sample at 800°C on W.

High-pressure and high-temperature equation of state of cobalt oxide: Implications for redox relations in Earth's mantle

The high-pressure and high-temperature equation of state of rock salt-structured cobalt oxide was measured up to 65 GPa and 2600 K using synchrotron X-ray diffraction in conjunction with the laser heated diamond-anvil cell. Fitting a Mie-Gruneisen-Debye model to the data we find best-fit parameters V0 = 77.4 (fixed) A3, K0 = 190 (1) GPa, K' = 3.49 (4), γ0 = 1.54 (4), q = 2.87 (15), and θ0 = 517.8 K (fixed). We use this newly determined equation of state in conjunction with existing measurements of the thermoelastic parameters of cobalt metal to calculate the Gibbs free-energy difference between the cobalt oxide and cobalt metal phases as a function of pressure and temperature. A comparison of the energetics of the Co/CoO system with the Ni/NiO system predicts that below 58 GPa CoO+Ni is stable relative to NiO+Co, while above 58 GPa the reverse is true. This tipping point in energy can be mapped as a crossing point in the electrochemical potential of the two metal ions, suggesting that cobalt becomes more siderophile than nickel with increasing pressure. This result is in qualitative agreement with existing measurements of nickel and cobalt partition coefficients between mantle and core materials.

Experimental Measurement of Speeds of Sound in Dense Supercritical Carbon Monoxide and Development of a High-Pressure, High-Temperature Equation of State

We report the adiabatic sound speeds for supercritical fluid carbon monoxide along two isotherms, from 0.17 to 2.13 GPa at 297 K and from 0.31 to 3.2 GPa at 600 K. The carbon monoxide was confined in a resistively heated diamond-anvil cell, and the sound speed measurements were conducted in situ using a recently reported variant of the photoacoustic light scattering effect. The measured sound speeds were then used to parametrize a single site dipolar exponential-6 intermolecular potential for carbon monoxide. PρT thermodynamic states, sound speeds, and shock Hugoniots were calculated using the newly parametrized intermolecular potential and compared to previously reported experimental results. Additionally, we generated an analytical equation of state for carbon monoxide by fitting to a grid of calculated PρT states over a range of 0.1-10 GPa and 150-2000 K. A 2% mean variation was found between computed high-pressure solid-phase densities and measured data-a surprising result for a spherical interaction potential. We further computed a rotationally dependent fluid to β-solid phase boundary; results signal the relative magnitude of short-range rotational disorder under conditions that span existing phase boundary measurements.

TEMPERATURE DEPENDENCE OF ELASTIC CONSTANTS OF ALKALINE EARTH OXIDE SOLIDS

A simple and straightforward theoretical model is developed to investigate the elastic constants of alkaline earth oxide solids under the effect of temperature. The calculation are performed with the help of high temperature equation of state (EOS) derived from Tallon's method based on thermodynamic analysis. The results obtained for these solids are discussed and compared with experimental data under the effect of high temperature. The results are found to be in good agreement with available experimental results

Lattice dynamics and thermodynamic properties of (Mg,Fe2+)SiO3postperovskite

[1] Phonon dispersion relations and vibrational density of states of (Mg0.9375Fe0.0625)SiO3 postperovskite have been determined by direct first-principles calculations of the dynamical matrix up to 150 GPa. Incorporation of iron in the postperovskite phase, irrespective of the two investigated configurations and the spin state, was found to decrease the acoustic phonon frequencies but to have a minor effect on the optic modes at high frequencies. The phonon dispersion curves exhibit negative phonon frequencies below 10 GPa when iron is incorporated in the high or low spin state and indicate unstable dynamic structures. Then, the calculated phonon frequencies of dynamically stable structures are used to determine vibrational contributions to the Helmholtz free energy within the quasi-harmonic approximation. The high temperature equation of state and several thermodynamic properties are then derived for (Mg0.9375Fe0.0625)SiO3 postperovskite and compared with those of pure MgSiO3 postperovskite. The results show that a low concentration of iron, irrespective of high spin or low spin, in MgSiO3 postperovskite has a minor effect on the thermodynamic properties at pressure-temperature conditions of the lowermost part of the Earth's mantle.

High pressure, high temperature equation of state for Fe2SiO4ringwoodite and implications for the Earth's transition zone

[1] We measured the density of iron-ringwoodite and its pressure and temperature dependence at conditions of the mantle transition zone using the laser-heated diamond anvil cell in conjunction with X-ray diffraction. Our new data combined with previous measurements constrain the thermoelastic properties of ringwoodite as a function of pressure and temperature throughout the transition zone. Our best fit Mie-Gruneisen-Debye equation of state parameters for Fe end-member ringwoodite are V0 = 42.03 cm3/mol, K0 = 202 (4) GPa, K′ = 4, γ0 = 1.08 (6), q = 2, and θD = 685 K. This new equation of state revises calculated densities of the Fe end-member at transition zone conditions upwards by ∼0.6% compared with previous formulations. We combine our data with equation of state parameters across the Mg-Fe compositional range to quantify the effect of iron and temperature on the density and bulk sound velocity of ringwoodite at pressure and temperature conditions of the Earth's transition zone. The results show that variations in iron content and temperature have opposing effects on density and bulk sound velocity, suggesting that compositional (iron content) and temperature variations in the transition zone may be distinguished using seismic observables.

High temperature equation of state of metallic hydrogen

The equation of state of liquid metallic hydrogen is solved numerically. Investigations are carried out at temperatures from 3000 to 20 000 K and densities from 0.2 to 3 mol/cm3, which correspond both to the experimental conditions under which metallic hydrogen is produced on earth and the conditions in the cores of giant planets of the solar system such as Jupiter and Saturn. It is assumed that hydrogen is in an atomic state and all its electrons are collectivized. Perturbation theory in the electron-proton interaction is applied to determine the thermodynamic potentials of metallic hydrogen. The electron subsystem is considered in the randomphase approximation with regard to the exchange interaction and the correlation of electrons in the local-field approximation. The proton-proton interaction is taken into account in the hard-spheres approximation. The thermodynamic characteristics of metallic hydrogen are calculated with regard to the zero-, second-, and third-order perturbation theory terms. The third-order term proves to be rather essential at moderately high temperatures and densities, although it is much smaller than the second-order term. The thermodynamic potentials of metallic hydrogen are monotonically increasing functions of density and temperature. The values of pressure for the temperatures and pressures that are characteristic of the conditions under which metallic hydrogen is produced on earth coincide with the corresponding values reported by the discoverers of metallic hydrogen to a high degree of accuracy. The temperature and density ranges are found in which there exists a liquid phase of metallic hydrogen.

A high temperature equation of state for the H 2O-CaCl 2 and H 2O-MgCl 2 systems

An equation of state (EOS) is developed for salt-water systems in the high temperature range. As an example of the applications, this EOS is parameterized for the calculation of density, immiscibility, and the compositions of coexisting phases in the CaCl 2-H 2O and MgCl 2-H 2O systems from 523 to 973 K and from saturation pressure to 1500 bar. All available volumetric and phase equilibrium measurements of these binaries are well represented by this equation. This EOS is based on a Helmholtz free energy representation constructed from a reference system containing hard-sphere and polar contributions plus an empirical correction. For the temperature and pressure range in this study, the electrolyte solutes are assumed to be associated. The water molecules are modeled as hard spheres with point dipoles and the solute molecules, MgCl 2 and CaCl 2, as hard spheres with point quadrupoles. The free energy of the reference system is calculated from an analytical representation of the Helmholtz free energy of the hard-sphere contributions and perturbative estimates of the electrostatic contributions. The empirical correction used to account for deviations of the reference system predictions from measured data is based on a virial expansion. The formalism allows generalization to aqueous systems containing insoluble gases (CO 2, CH 4), alkali chlorides (NaCl, KCl), and alkaline earth chlorides (CaCl 2, MgCl 2). The program of this model is available as an electronic annex (see EA1 and EA2) and can also be downloaded at: http://www.geochem-model.org/programs.htm.

High-pressure and high-temperature equation of state and phase diagram of solid benzene

The high-pressure structural properties of benzene have been investigated up to the occurrence of the chemical reaction by means of x-ray diffraction and infrared absorption techniques. The infrared spectra show that sample annealing above 500 K produces pure phase-II crystals. X-ray diffraction patterns collected at 540 K on these crystals allowed the equation of state of benzene to be obtained. These results indicate the stability of phase II up to the pressure where benzene reacts and no evidence of the III and ${\mathrm{III}}^{\ensuremath{'}}$ crystal structures is gained. On these bases the existing phase diagram of benzene is discussed and a simplified one is proposed.

Laser heated diamond cell system at the Advanced Photon Source for in situ x-ray measurements at high pressure and temperature

We describe a laser heated diamond anvil cell system at the GeoSoilEnviroCARS sector at the Advanced Photon Source. The system can be used for in situ x-ray measurements at simultaneously ultrahigh pressures (to >150 GPa) and ultrahigh temperatures (to >4000 K). Design goals of the laser heating system include generation of a large heating volume compared to the x-ray beam size, minimization of the sample temperature gradients both radially and axially in the diamond anvil cell, and maximization of heating stability. The system is based on double-sided laser heating technique and consists of two Nd:YLF lasers with one operating in TEM00 mode and the other in TEM01* mode, optics to heat the sample from both sides, and two spectroradiometric systems for temperature measurements on both sides. When combined with an x-ray microbeam (3–10 μm) technique, a temperature variation of less than 50 K can be achieved within an x-ray sampled region for longer than 10 min. The system has been used to obtain in situ structural data and high temperature equations of state on metals, oxides, and silicates to 3500 K and 160 GPa.

Anharmonic properties of Mg2SiO4-forsterite measured from the volume dependence of the Raman spectrum

The effect of temperature on the Raman spectrum of Mg2SiO4-forsterite is investigated between 100 and 2000 K. The data, combined with those obtained at high pressure, are used to discuss the intrinsic anharmonicity of the Raman-active modes. Anharmonicity of a mode of frequency v(i) is measured through a parameter (a(i)) = (partial derivative ln v(i)/partial derivative T)(v) calculated from the measured temperature and pressure shifts of the vibrational frequency. In forsterite, these parameters are negative, of the order of -10(-5) K-1. Those related to the internal modes of the SiO4 tetrahedra are significantly different (-0.5 10(-5) K-1) from those of the lattice modes (-3 10(-5) K-1). Because of the large temperature range investigated in this study, the temperature and volume dependencies of the a(i) parameters could be estimated for the first time in a mineral, and are shown to be negligible, at least up to 2000 K. The significant non-zero values of the al parameters combined with their small volume dependencies imply that intrinsic anharmonicity in forsterite contributes significantly to entropies and specific heats, even at high pressures, but has negligible effects on thermal pressure and high-temperature equations of state.

Estimation of properties of rare-gas solids from compression characteristics of closed-shell ions

The authors present a unified approach to repulsion in ionic and van der Waals solids based on a compressible-ion/atom model. Earlier studies have shown that repulsion in ionic crystals can be viewed as arising from the compression energy of ions, described by two parameters per ion. Here they obtain the compression parameters of the rare-gas atoms Ne, Ar, Kr and Xe by interpolation using the known parameters of related equi-electronic ions (e.g. Ar from S2-, Cl-, K+ and Ca2-). These parameters fit the experimental zero-temperature interatomic distances and compressibilities of the rare-gas crystals satisfactorily. At high-temperature equation of state based on an Einstein model of thermal motions is used to calculate the thermal expansivities, compressibilities and their temperature derivatives for Ar, Kr and Xe. It is argued that an instability at higher temperatures represents the limit to which the solid can be superheated, beyond which sublimation must occur.

Quantum-mechanical high temperature Equation of state: Perturbation theory with finite range Kac potential

Quantum-mechanical high temperature Equation of state: Perturbation theory with finite range Kac potential

Equation of state of cubic solids; some generalizations

The equations of state (pressure-volume-temperature relationships) for a number of cubic solids can be approximated at high temperature ( T > θ D ) by a temperature-independent isothermal bulk modulus B T which varies linearly with the applied pressure. This is demonstrated by an analysis of both piston-displacement and ultrasonic data, as well as by an elementary calculation based on the Mie-Grüneisen equation of state. These generalizations also should be valid for B T at low temperatures to within one or two percent except for the inert gas solids. The high temperature equation of state which follows from this analysis contains the following features: (a) the Grüneisen constant γ is a temperature-independent linear function of volume, (b) B T is a linear function of temperature at constant pressure, (c) (∂( B S − B T )/∂ T) V = β B T γ and (∂( B S − B T )/∂ P) T = −βγ T, where β is the volume thermal expansion coefficient. These relationships should hold as a first approximation for most cubic solids.

High-Temperature Equation of State

An equation of state is obtained in closed form for a gas composed of molecules with a Lennard-Jones (n, 6) potential. It is derived by expanding the partition function in powers of n−1 and expanding the attractive part of the potential in powers of T−1. The resulting equation is of the form P/ρkT=A+B/T+ClogT where A, B, and C can be simply expressed in terms of known hard-sphere quantities such as the compressibility factor and the radial distribution function as functions of density. The equation contains no adjustable parameters and reliably reproduces PVT data up to densities of 40 mole/liter and reduced temperatures as low as 3.

High-Temperature Equation of State by a Perturbation Method. II. Polar Gases

The perturbation theory described in a previous article is applied to the calculation of the equation of state at high temperature of a polar gas. The electrostatic dipolar potential is considered to be a perturbation on the nonpolar Lennard-Jones potential. The dipole contribution to the equation of state is expressed in terms of properties of the unperturbed system. These properties are obtained from experimental data on argon and xenon. Numerical tables of the reduced equation of state of the polar gas, over a wide temperature and density range, are presented. The theory is tested on water vapor and on methyl fluoride. Theoretical predictions of the pressure agree with experimental data over a wide density range (up to 0.25 g/cc for water) to within several percent.