Equation Of State Data Research Articles

Ab initio calculation of thermodynamic potentials and entropies for superionic water.

We construct thermodynamic potentials for two superionic phases of water [with body-centered cubic (bcc) and face-centered cubic (fcc) oxygen lattice] using a combination of density functional theory (DFT) and molecular dynamics simulations (MD). For this purpose, a generic expression for the free energy of warm dense matter is developed and parametrized with equation of state data from the DFT-MD simulations. A second central aspect is the accurate determination of the entropy, which is done using an approximate two-phase method based on the frequency spectra of the nuclear motion. The boundary between the bcc superionic phase and the ices VII and X calculated with thermodynamic potentials from DFT-MD is consistent with that directly derived from the simulations. Differences in the physical properties of the bcc and fcc superionic phases and their impact on interior modeling of water-rich giant planets are discussed.

Buoyancy and g-modes in young superfluid neutron stars

We consider the local dynamics of a realistic neutron star core, including composition gradients, superfluidity and thermal effects. The main focus is on the gravity g-modes, which are supported by composition stratification and thermal gradients. We derive the equations that govern this problem in full detail, paying particular attention to the input that needs to be provided through the equation of state and distinguishing between normal and superfluid regions. The analysis highlights a number of key issues that should be kept in mind whenever equation of state data is compiled from nuclear physics for use in neutron star calculations. We provide explicit results for a particular stellar model and a specific nucleonic equation of state, making use of cooling simulations to show how the local wave spectrum evolves as the star ages. Our results show that the composition gradient is effectively dominated by the muons whenever they are present. When the star cools below the superfluid transition, the support for g-modes at lower densities (where there are no muons) is entirely thermal. We confirm the recent suggestion that the g-modes in this region may be unstable, but our results indicate that this instability will be weak and would only be present for a brief period of the star's life. Our analysis accounts for the presence of thermal excitations encoded in entrainment between the entropy and the superfluid component. Finally, we discuss the complete spectrum, including the normal sound waves and, in superfluid regions, the second sound.

Measuring high pressure equation of state of polystyrene using laser driven shock wave

High precision polystyrene equation of state data were measured using laser-driven shock waves with pressures from 180 GPa to 700 GPa. α quartz was used as standard material, the shock wave trajectory in quartz and polystyrene was measured using the Velocity Interferometer for Any Reflector (VISAR). Instantaneous shock velocity in quartz and polystyrene was obtained when the shock wave pass the interface. This provided ~1% precision in shock velocity measurements.

Decomposition of supercritical ammonia and modeling of supercritical ammonia–nitrogen–hydrogen solutions with applicability toward ammonothermal conditions

Ammonothermal growth of nitrides occurs at temperatures in excess of 800K and pressures greater than 150MPa. For this region, no experimentally verified equation of state (EOS) exists for ammonia, nor is there any accurate description available for the equilibrium constant of the ammonia decomposition reaction for pressures in excess of 100MPa. To fill this void, experimental P-v-T data was collected for pure ammonia at T>700K and compared to extrapolated data from the reference EOS for ammonia. For the extrapolated region, the reference EOS provided excellent agreement to within error of the collected experimental data. A simplified EOS based on the Beattie–Bridgeman (BB) EOS was derived and fit to calculated and, for ammonia, extrapolated reference EOS data for ammonia, hydrogen, and nitrogen (T<1000K, P<300MPa). By applying mixing rules with separated contributions for polar and non-polar interactions, an EOS was derived for NH3–N2–H2 mixtures. With these expressions, an accurate description for the equilibrium constant for the ammonia decomposition reaction as a function of pressure and temperature was derived and verified against experimental data determined for total system pressures of 92, 151 and 210MPa at T ∼810K. Coupling of the EOSs with the equilibrium constant permitted accurate modeling of a sealed autoclave filled with pure ammonia, after incorporating corrections for the expansion of the internal free volume due to thermal expansion and elastic strain response of the autoclave walls due to internal pressure buildup. Calculated total system pressure and equilibrium ammonia density at various temperatures and initial ammonia fill densities are in very good to excellent agreement with experimental data. This paper thus provides a simple EOS for ammonia, hydrogen, nitrogen and NH3–N2–H2 mixtures accurate to within approximately 1–2% in pressure for temperatures greater than 700K. The equilibrium constant for the ammonia decomposition reaction includes non-ideal mixing contributions from the second virial coefficient with a resulting accuracy in the equilibrium mole fraction of ammonia of approximately 2% for T<850K.

Thermodynamics of iron at extreme pressures and temperatures

A thermodynamic database on all iron phases (BCC, FCC, HCP and melt) has been created using thermochemical and equations of state data from experiments and theory. The database permits the calculation of the phase diagram of iron to physical conditions of the Earth's core (pressure of 365GPa and temperature of 6453K). If the inner core were all iron, its upper temperature would be 6453 (500)K. The average heat capacity of a pure iron HCP inner core is calculated as 29.4J/mol/K with an entropy of 92J/mol/K and a gruneisen parameter of 1.81.

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.

Inter-atomic potential energy and Grüneisen parameter: A new method for equation of state of solids

This paper presents a rational foundation for the computation of equation of state (EOS) data for solids at high pressure. We demonstrate a new method which makes use of an accurate relation expressing the Grüneisen parameter γ (as a function of the specific volume V of the material) in terms of the specific inter-atomic potential energy ϕ(V). Existing expressions for γ in the literature are usually approximations in terms of total pressure P. There is a variety of such “(γ,P)” formulas and one needs experience in deciding which to use in any particular application. The alternative “(γ,ϕ)” relationship presented here is both unique and exact within the Debye and harmonic approximations and allows the individual terms of the EOS to be determined separately. It is rigorously derived in this paper and solved numerically, using experimental input data for aluminium, copper, lead and gold, to predict ϕ, dϕ/dV, γ, dγ/dV and EOS data for the four metals. Comparison is made with existing computations in the literature showing good agreement. The method can be applied to any metal or non-metal using experimental input data from any suitable volume-dependent locus.

AB INITIO EQUATIONS OF STATE FOR HYDROGEN (H-REOS.3) AND HELIUM (He-REOS.3) AND THEIR IMPLICATIONS FOR THE INTERIOR OF BROWN DWARFS

We present new equations of state (EOS) for hydrogen and helium covering a wide range of temperatures from 60 K to 10$^7$ K and densities from $10^{-10}$ g/cm$^3$ to $10^3$ g/cm$^3$. They include an extended set of ab initio EOS data for the strongly correlated quantum regime with an accurate connection to data derived from other approaches for the neighboring regions. We compare linear-mixing isotherms based on our EOS tables with available real-mixture data. A first important astrophysical application of this new EOS data is the calculation of interior models for Jupiter and the comparison with recent results. Secondly, mass-radius relations are calculated for Brown Dwarfs which we compare with predictions derived from the widely used EOS of Saumon, Chabrier and van Horn. Furthermore, we calculate interior models for typical Brown Dwarfs with different masses, namely Corot-3b, Gliese-229b and Corot-15b, and the Giant Planet KOI-889b. The predictions for the central pressures and densities differ by up to 10$\%$ dependent on the EOS used. Our EOS tables are made available in the supplemental material of this paper.

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.

Molecular mechanics and equation of state modeling of compressible polyolefin solutions: Impact of pressure and cut-off radius of intermolecular potentials

This paper reports on the density of solutions of polyethylene (PE) in hexane using molecular dynamics (MD) simulations based upon the accurate OPLS-AA force field at high pressures and 425K. The NPT–MD simulations are carried out at polymer concentration of 20wt% in hexane in the pressure range from 100 to 3000bar. For PE solutions of 10 and 20wt% the pressure is varied from 100 to 1000bar. Additionally, the PE solution densities are calculated based on the modified Sanchez–Lacombe (MSL) equation of state (EOS) model to examine the accuracy of the MD computations. The simulated densities increase monotonically with increasing external pressure and compare quite favorably with the experimental and EOS data. It is also revealed that the MSL EOS model produces identical mixture densities regardless of the type of the b parameter. The effect of cut-off radius to density is investigated and it is shown that the solution density increases as cut-off radius increases. A minimum cut-off radius of 1.1nm is suggested for the intermolecular forces for accurate densities at pressures below 100bar. For higher pressures density and non-bonded interactions display less sensitivity to cut-off distance. Analysis of the pair distribution function versus pressure is carried out where the height of the first peak increases and the radial distribution function shifts to shorter separations reflecting structural change of the condensed phase. The molecular modeling approach employed in this research provides a good insight into the polymer–polymer, polymer–solvent, and solvent–solvent interactions. The implemented methodology using the OPLS-AA force field and constant pressure/temperature algorithms compare well with the literature data, suggesting the validity of the proposed method.

Thermodynamic properties of HFO-1243zf and their application in study on a refrigeration cycle

Thermodynamic properties of HydroFluoro-Olefin, 3,3,3-trifluoropropene (HFO-1243zf) are firstly modeled with an accurate molecular based PC-SAFT equation of state (EOS). The average absolute deviations (AAD) between vapor pressures, saturated liquid densities from this developed PC-SAFT EOS and experimental data are 0.15% and 0.80%, respectively. The average absolute deviations between predicted gaseous pressures, sub-cooled liquid densities from the PC-SAFT EOS and experimental data are 0.91% and 0.92%, respectively. In this study, the developed PC-SAFT EOS for HFO-1243zf and other EOSs for R-134a, R22, and R32 are used in the investigation of a refrigeration cycle for air conditioning application. The results show that cycle using R1243zf as refrigerant has the highest coefficient of performance (COP) and it is a potential candidate for the replacement of R134a in a refrigeration cycle.

Ab initiosimulations of MgO under extreme conditions

We determined the phase diagram of magnesium oxide with finite-temperature density functional theory molecular dynamics simulations up to temperatures and pressures as relevant for the deep interior of super-Earths and in rocky cores of giant planets such as Jupiter. The equation of state data, the Hugoniot, and a ramp compression curve are computed and compared to earlier results from diamond anvil cell and (decaying) shock wave experiments. In addition, the dynamical electrical conductivity and the reflectivity along the experimental Hugoniot curve are calculated in order to characterize electronic structure changes under compression. The structural properties of MgO are identified using pair correlation functions and self-diffusion coefficients. The solid-solid coexistence line is calculated by comparing the free enthalpies of the B1 and the B2 phase. The free energy of the solid phases is determined via thermodynamic relations using the ab initio simulation results and phonon calculations in the harmonic approximation. Our results indicate that the solid B2 phase of MgO does not occur in the interior of the Earth but may play an important role in super-Earths and in rocky planetary cores.

Equations of state for HFO-1234ze(E) and their application in the study on refrigeration cycle

This paper aims to describe the thermodynamic properties of the HFO-1234ze(E) with the molecular based BACKONE and PC-SAFT equations of state (EOS) that can be easily extended to mixtures. This paper also evaluates the accuracies of the BACKONE, PC-SAFT, and multi-parameter EOSs. For the data used in the construction of the EOS, the average absolute deviations (AAD) between experimental vapour pressures, experimental saturated liquid densities and values from the BACKONE EOS are 0.34% and 0.42%, respectively. The AAD between experimental vapour pressures, experimental saturated liquid densities and those from the PC-SAFT EOS are 0.16% and 1.51%, respectively. The AAD between predicted compressed liquid densities, predicted isobaric heat capacities, predicted pressures in gaseous phase from the BACKONE EOS and experimental data are 0.60%, 2.29%, and 0.99%, respectively. The AAD between predicted compressed liquid densities, predicted isobaric heat capacities, predicted pressures in gaseous phase from the PC-SAFT EOS and experimental data are 1.35%, 2.91%, and 1.84%, respectively. An independent investigation for data used in the construction of the multi-parameter EOS shows that the AAD between experimental vapour pressures, experimental saturated liquid densities, experimental densities in compressed liquid, experimental isobaric heat capacities, experimental pressures in gaseous phase and those from the multi-parameter EOS are 0.55%, 0.17%, 0.07%, 2.21%, and 0.11%, respectively. The BACKONE, PC-SAFT, and multi-parameter EOSs are used in the investigation of a refrigeration cycle using HFO-1234ze(E) as refrigerant for air-conditioning application. The study shows that the relative differences of all investigated characteristics calculated from thermodynamic properties from these EOSs are mostly within 1%. The thermodynamic properties of HFO-1234ze(E), HFO-1234yf, R-134a, R22, and R32 from the most accurate EOSs are used in the investigation of a refrigeration cycle. The results show that the cycle using R1234ze(E) as refrigerant has the highest coefficient of performance.

Simulation of warm dense matter in intense bubble collapse

Previous work by the authors includes computational study of shock-bubble interaction. This work demonstrated strong compression and heating within the bubble, with gas reaching densities of order of magnitude 1 g/cc and temperatures of 10 eV. These conditions correspond to the warm dense matter regime. This paper addresses limitations of previous work through utilization of various equations of state (EOS) appropriate for the modeling of dense plasma. This is achieved through the design and implementation of a generic interface based on tabulated EOS data. Any EOS may be utilized through the framework, requiring only knowledge of pressure and energy as functions of density and temperature. The solutions to various issues such as table interpolation, tabulated change of variables, arbitrary calculation of entropy, and calculation of thermodynamic derivatives are presented. In addition, the trade-offs between CPU time, memory requirement and computational accuracy are discussed. Validation work is presented and a comparison of different EOS is also explored. The EOS used include but are not limited to the EOS for air utilized by Moss et al. (1994) to study SBSL, SESAME tabulated EOS and QEOS-type formulations. Finally, conditions attained during shock-bubble interaction are re-examined.

Compressibility of mimetite and pyromorphite at high pressure

High pressure X-ray diffraction experiments on mimetite [Pb5(AsO4)3Cl] and pyromorphite [Pb5(PO4)3Cl] were performed up to 14.1 and 14.9 GPa, respectively, at 300 K, using in situ angle-dispersive X-ray diffraction and a diamond anvil cell. No phase transition of mimetite and pyromorphite was observed within the experimental pressure range. Fitting the P–V data under hydrostatic stress conditions with a third-order Birch–Murnaghan Equations of State (BM-EoS) we obtained: K 0=46(7) GPa, V 0=680(2) Å3, and K 0′=15(4) for mimetite; K 0=44(5) GPa, V 0=636(1) Å3, and K 0′=15(3) for pyromorphite. The axial compressibility was also calculated with a third-order ‘linearized’ BM-EoS. We obtained K 0a :K 0c =1:1.00 for mimetite, and K 0a :K 0c =1:1.28 for pyromorphite, indicating that mimetite and pyromorphite are elastically isotropic and slightly anisotropic, respectively. Comparing the previous equation of state data of vanadinite [Pb5(VO4)3Cl] with the current results of mimetite [Pb5(AsO4)3Cl] and pyromorphite [Pb5(PO4)3Cl], we found that the substitution of V5+ by As5+ and P5+ has an insignificant effect on the bulk modulus, but has a greater effect on the axial parameters, compression ratio, and elastic anisotropy.

A multiscale framework for high-velocity impact process with combined material point method and molecular dynamics

The equation of state (EOS) plays an important role in high-velocity impact process since phase transformation, melting, and even vaporization may happen under such extreme loading conditions. It is desired to adopt an accurate EOS covering a large range of points in the phase space. This paper proposes a combined molecular dynamics and material point method approach to simulate the high-velocity impact process. The EOS data are first obtained from a series of molecular dynamics computations, and the parameters are fitted. Then the EOS parameters are adopted in the material point method simulation to model the impact process. Simulation results show that the fitted EOS can be very accurate compared to experimental results. The shape of the debris cloud obtained by our multiscale method agrees well with that of the experiments. An empirical equation is also proposed to predict the fraction of melting material in the high-velocity impact process.

An Equation of State for thermodynamic equilibrium of gas mixtures and brines to allow simulation of the effects of impurities in subsurface CO2 storage

Comprehensive understanding and prediction of chemical and reactive processes during and following injection of CO2 in depleted gas reservoirs and saline aquifers is important for the assessment of the performance and impacts of planned and existing Carbon Capture and Storage (CCS) projects. Over the last decade significant improvements have been made in numerical modelling of the complex, coupled processes involved. Among the many remaining issues where progress is still called for, is the consistent simulation of impacts of gas mixtures. In particular the presence of ‘impurities’ or ‘co-contaminants’ in the injected CO2 stream that are retained from the original flue-gases, such as H2S and SO2, have the potential, upon dissolution in the pore water, to alter aqueous and water–mineral reactions. Moreover, presence of these and other injected or in situ gases (e.g. CH4) affect CO2 solubility and thermodynamic properties of the fluid and gas phases, which, in turn, impact transport processes.As an important step towards evaluation of the impact of gas mixtures on these processes, a new Equation of State (EOS) has been developed which allows accurate and efficient modelling of thermodynamic equilibrium of gas mixtures and brines over a large range of pressure, temperature and salinity conditions. Presently the new EOS includes CO2, SO2, H2S, CH4 and N2. This model is based on equating the chemical potentials in the system, using the Peng–Robinson EOS to calculate the fugacity of the gas phase. The model performs favourably with respect to existing EOS's and experimental data for single gas systems and accurately reproduces available data sets for gas mixtures. Preliminary analysis shows, amongst others, that CO2 solubility is most sensitive to CH4 admixture and least sensitive to the presence of SO2 in the injected gas.

BADGER v1.0: A Fortran equation of state library

The BADGER equation of state library was developed to enable inertial confinement fusion plasma codes to more accurately model plasmas in the high-density, low-temperature regime. The code had the capability to calculate 1- and 2-T plasmas using the Thomas–Fermi model and an individual electron accounting model. Ion equation of state data can be calculated using an ideal gas model or via a quotidian equation of state with scaled binding energies. Electron equation of state data can be calculated via the ideal gas model or with an adaptation of the screened hydrogenic model with ℓ-splitting. The ionization and equation of state calculations can be done in local thermodynamic equilibrium or in a non-LTE mode using a variant of the Busquet equivalent temperature method. The code was written as a stand-alone Fortran library for ease of implementation by external codes. EOS results for aluminum are presented that show good agreement with the SESAME library and ionization calculations show good agreement with the FLYCHK code. Program summaryProgram title: BADGERLIB v1.0Catalogue identifier: AEND_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEND_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 41480No. of bytes in distributed program, including test data, etc.: 2904451Distribution format: tar.gzProgramming language: Fortran 90.Computer: 32- or 64-bit PC, or Mac.Operating system: Windows, Linux, MacOS X.RAM: 249.496 kB plus 195.630 kB per isotope record in memoryClassification: 19.1, 19.7.Nature of problem:Equation of State (EOS) calculations are necessary for the accurate simulation of high energy density plasmas. Historically, most EOS codes used in these simulations have relied on an ideal gas model. This model is inadequate for low-temperature, high-density plasma conditions; the gaseous and liquid phases; and the solid phase. The BADGER code was developed to give more realistic EOS data in these regimes.Solution method:BADGER has multiple, user-selectable models to treat the ions, average-atom ionization state and electrons. Ion models are ideal gas and quotidian equation of state (QEOS), ionization models are Thomas–Fermi and individual accounting method (IEM) formulation of the screened hydrogenic model (SHM) with l-splitting, electron ionization models are ideal gas and a Helmholtz free energy minimization method derived from the SHM. The default equation of state and ionization models are appropriate for plasmas in local thermodynamic equilibrium (LTE). The code can calculate non-LTE equation of state (EOS) and ionization data using a simplified form of the Busquet equivalent-temperature method.Restrictions:Physical data are only provided for elements Z=1 to Z=86. Multiple solid phases are not currently supported. Liquid, gas and plasma phases are combined into a generalized “fluid” phase.Unusual features:BADGER divorces the calculation of average-atom ionization from the electron equation of state model, allowing the user to select ionization and electron EOS models that are most appropriate to the simulation. The included ion ideal gas model uses ground-state nuclear spin data to differentiate between isotopes of a given element.Running time:Example provided only takes a few seconds to run.

Determination of CO2 Solubility in Saturated Liquid CH4 + N2 and CH4 + C2H6 Mixtures above Atmospheric Pressure

Accurate CO2 solubility data are crucial for putting a proposal of pressurized liquefied natural gas (PLNG) project into practice. A solid–liquid equilibrium (SLE) apparatus, which is based on the static–analytic method, has been set up to measure the solubility of carbon dioxide in saturated liquid CH4/CH4 + N2/CH4 + C2H6 in the pressure region from atmospheric up to 3 MPa, corresponding to the temperature region from (112 to 170) K. In addition, Peng–Robinson (PR) and Suave–Redlich–Kwong (SRK) equations-of-state (EOS) are selected to calculate the solubility of carbon dioxide in CH4 + N2 and CH4 + C2H6 mixtures, and the results are consistent with the experimental data in the whole region. Two temperature-dependent correlations for the interaction coefficients kij on CH4 + CO2 system are derived by the experimental data for PR and SRK EOS, which can improve the calculation accuracy of the binary and ternary solid–liquid phase equilibrium.

Development of EOS data for granular material like sand by using micromodels

Detonations in soil can occur due to several reasons: e.g. land mines or bombs from the Second World War. Soil is also often used as a protective barrier. In all cases the behaviour of soil loaded by shock waves is important. The simulation of shock wave loaded soil using hydro-codes like AUTODYN needs a failure model as well as an equation of state (EOS). The parameters for these models are often not known. The popular material law for sand from Laine and Sandvik [1], e.g., is a first approximation, but it can only be used for dry sand with a certain grain grading. The parameters porosity, grain grading, and humidity have a big influence on the material behaviour of cohesive soils. Micro-mechanic models can be used to develop the material behaviour of granular materials. EOS data can be obtained by numerically loading micro-mechanically modelled grains and measuring the density under a certain pressure in the finite element model. The influence of porosity, grain grading, and humidity can be easily investigated. EOS data are determined in this work for cohesive soils depending on these parameters.