Equation Of State Data Research Articles

Extension of the SpK atomic physics code to generate global equation of state data

Global microphysics models are required for the modelling of high-energy-density physics (HEDP) experiments, the improvement of which are critical to the path to inertial fusion energy. This work presents further developments to the atomic and microphysics code, SpK, part of the numerical modelling suite of Imperial College London and First Light Fusion. We extend the capabilities of SpK to allow the calculation of the equation of state (EoS). The detailed configuration accounting calculations are interpolated into finite-temperature Thomas–Fermi calculations at high coupling to form the electronic component of the model. The Cowan model provides the ionic contribution, modified to approximate the physics of diatomic molecular dissociation. By utilising bonding corrections and performing a Maxwell construction, SpK captures the EoS from states ranging from the zero-pressure solid, through the liquid–vapour coexistence region and into plasma states. This global approach offers the benefit of capturing electronic shell structure over large regions of parameter space, building highly-resolved tables in minutes on a simple desktop. We present shock Hugoniot and off-Hugoniot calculations for a number of materials, comparing SpK to other models and experimental data. We also apply EoS and opacity data generated by SpK in integrated simulations of indirectly-driven capsule implosions, highlighting physical sensitivities to the choice of EoS models.

Analysis of Neutron Star f-mode Oscillations in General Relativity with Spectral Representation of Nuclear Equations of State

We study quasi-normal f-mode oscillations in neutron star (NS) interiors within a linearized general relativistic formalism. We utilize approximately 9000 nuclear equations of state (EOSs) using spectral representation techniques, incorporating constraints on nuclear saturation properties, chiral effective field theory for pure neutron matter, and perturbative quantum chromodynamics for densities pertinent to NS cores. The median values of the f-mode frequency, ν f (damping time, τ f ) for NSs with masses ranging from 1.4 to 2.0 M ⊙ lie between 1.80 and 2.20 kHz (0.13–0.22 s) for our entire EOS set. Our study reveals a weak correlation between f-mode frequencies and individual nuclear saturation properties, prompting the necessity for more intricate methodologies to unveil multiparameter relationships. We observe a robust linear relationship between the radii and f-mode frequencies for different NS masses. Leveraging this correlation alongside NICER observations of PSR J0740+6620 and PSR J0030+0451, we establish constraints that exhibit partial and minimal overlap for observational data from Riley et al. and Miller et al., respectively, with our nucleonic EOS data set. Moreover, NICER data align closely with the radius and frequency values for a few hadron–quark hybrid EOS models. This indicates the need to consider additional exotic particles such as deconfined quarks at suprasaturation densities. We conclude that future observations of the radius or f-mode frequency for more than one NS mass, particularly at the extremes of the viable NS mass scale, would either rule out nucleon-only EOSs or provide definitive evidence in its favor.

Deep energy-pressure regression for a thermodynamically consistent EOS model

In this paper, we aim to explore novel machine learning (ML) techniques to facilitate and accelerate the construction of universal equation-Of-State (EOS) models with a high accuracy while ensuring important thermodynamic consistency. When applying ML to fit a universal EOS model, there are two key requirements: (1) a high prediction accuracy to ensure precise estimation of relevant physics properties and (2) physical interpretability to support important physics-related downstream applications. We first identify a set of fundamental challenges from the accuracy perspective, including an extremely wide range of input/output space and highly sparse training data. We demonstrate that while a neural network (NN) model may fit the EOS data well, the black-box nature makes it difficult to provide physically interpretable results, leading to weak accountability of prediction results outside the training range and lack of guarantee to meet important thermodynamic consistency constraints. To this end, we propose a principled deep regression model that can be trained following a meta-learning style to predict the desired quantities with a high accuracy using scarce training data. We further introduce a uniquely designed kernel-based regularizer for accurate uncertainty quantification. An ensemble technique is leveraged to battle model overfitting with improved prediction stability. Auto-differentiation is conducted to verify that necessary thermodynamic consistency conditions are maintained. Our evaluation results show an excellent fit of the EOS table and the predicted values are ready to use for important physics-related tasks.

Construction of Numerical PVT-Models for the Bulla-Daniz Gas-Condensate Field Based on Laboratory Experiments on Reservoir Fluid Samples

PVT analysis is important for field-wide optimization and development. This is because we must understand the fluid's overall behavior from the reservoir to the production and processing facilities, and ultimately to the refinery. Modern computer software that uses equation of state (EOS) models to simulate experiments and illustrate fluid phase characteristics has contributed to its growth as a distinct field of study. To find the operating parameters that will maximize the surface liquid content and prolong the production plateau duration at the lowest feasible cost, PVT simulations are run. These simulations employ laboratory-derived data to fine-tune the EOS models, with the outcomes being integrated into reservoir simulation and research. The quality of the data is crucial to getting a good match between EOS and laboratory data, and for retrograde gas condensates, this can be particularly difficult because of their complex phase behavior. When utilized in reservoir simulations, an inadequate match leads to computational mistakes and unrepresentative findings, endangering the reservoir management decisions that depend on it.

Optical full-field strain measurement within a diamond anvil cell.

Digital image correlation computations are run on optical images of iron samples within a diamond anvil cell to obtain in-plane strain components at the surface of the sample up to 17GPa. The α-Fe → ε-Fe transition onset pressure and phase coexistence pressure domain can be identified from the evolution of the surface average of strain components. Strain fields exhibit localizations for both direct and reverse transition; they coincide with the approximate boundary locations of reversion variants inside the microstructure of a single crystal sample. The so-called DICDAC (Digital Image Correlation within a Diamond Anvil Cell) setup is then a suitable tool for the investigation of phase transformations strains under pressure. In addition, specific volumes that are deduced from strain out of the transition pressure domains agree within ΔV/V = 0.4% with the equation of state data from the literature.

Physics-enhanced neural networks for equation-of-state calculations

Rapid access to accurate equation-of-state (EOS) data is crucial in the warm-dense matter (WDM) regime, as it is employed in various applications, such as providing input for hydrodynamic codes to model inertial confinement fusion processes. In this study, we develop neural network models for predicting the EOS based on first-principles data. The first model utilises basic physical properties, while the second model incorporates more sophisticated physical information, using output from average-atom (AA) calculations as features. AA models are often noted for providing a reasonable balance of accuracy and speed; however, our comparison of AA models and higher-fidelity calculations shows that more accurate models are required in the WDM regime. Both the neural network models we propose, particularly the physics-enhanced one, demonstrate significant potential as accurate and efficient methods for computing EOS data in WDM.

PlanetProfile: Self‐Consistent Interior Structure Modeling for Ocean Worlds and Rocky Dwarf Planets in Python

AbstractThe open‐source PlanetProfile framework was developed to investigate the interior structure of icy moons based on collectively matching their observed properties and comparative planetology. The software relates observed and measured properties, assumptions such as the type of materials present, and laboratory equation‐of‐state (EOS) data through geophysical and thermodynamic models to evaluate radial profiles of mechanical, thermodynamic, and electrical properties, as self‐consistently as possible. We have created a Python version of PlanetProfile. In the process, we have made optimization improvements and added parallelization and parameter‐space search features to utilize fast operation for investigating unresolved questions in planetary geophysics, in which many model inputs are poorly constrained. The Python version links to other scientific software packages, including for evaluating EOS data, magnetic induction calculations, and seismic calculations. Physical models in PlanetProfile have been reconfigured to improve self‐consistency and generate the most realistic relationships between properties. In this work, we focus on the description of the software design and algorithms in detail, summarize model results for major moons across the outer solar system, and discuss new inferences about the interior structure of several bodies. The high values and narrow uncertainty ranges reported for the axial moments of inertia for Callisto, Titan, and Io are difficult to reconcile with the realistic physical and chemical approximations applied, requiring highly porous rock layers equivalent to incomplete differentiation for Callisto and Titan, and a high rock melt fraction for Io. Radial profiles for each model and comparison to prior work are provided as Zenodo archives.

Laser-driven ramp-compression experiments on the national ignition facility

This report details the analyses and related uncertainties in measuring longitudinal-stress–density paths in indirect laser-driven ramp equation-of-state (EOS) experiments [Smith et al., Nat. Astron. 2(6), 452–458 (2018); Smith et al., Nature 511(7509), 330–333 (2014); Fratanduono et al., Science 372(6546), 1063–1068 (2021); and Fratanduono et al., Phys. Rev. Lett. 124(1), 015701 (2020)]. Experiments were conducted at the National Ignition Facility (NIF) located at the Lawrence Livermore National Laboratory. The NIF can deliver up to 2 MJ of laser energy over 30 ns and provide the necessary laser power and control to ramp compress materials to TPa pressures (1 TPa = 10 × 106 atmospheres). These data provide low-temperature solid-state EOS data relevant to the extreme conditions found in the deep interiors of giant planets. In these experiments, multi-stepped samples with thicknesses in the range of 40–120 µm experience an initial shock compression followed by a time-dependent ramp compression to peak pressure. Interface velocity measurements from each thickness combine to place a constraint on the Lagrangian sound speed as a function of particle velocity, which in turn allows for the determination of a continuous stress–density path to high levels of compressibility. In this report, we present a detailed description of the experimental techniques and measurement uncertainties and describe how these uncertainties combine to place a final uncertainty in both stress and density. We address the effects of time-dependent deformation and the sensitivity of ramp EOS techniques to the onset of phase transformations.

Phase equilibria and interface properties of hydrocarbon propellant–oxygen mixtures in the transcritical regime

Vapor–liquid equilibria and fluid interface properties of binary mixtures containing either methane or cyclohexane representing fuel and nitrogen or oxygen are reported. The mixtures are studied at different temperatures and pressures, which are chosen such that the temperature of the fuel component is subcritical, while that of the gaseous component is mainly supercritical. Data are obtained from molecular dynamics (MD) simulation, as well as density functional theory and density gradient theory in combination with the PC-SAFT equation of state (EOS). The studied interface properties include surface tension, interface thickness, enrichment, and relative adsorption. Furthermore, bulk vapor–liquid equilibrium properties are computed with two distinct MD methods as well as the PC-SAFT EOS. All approaches are compared to data from highly accurate empirical EOS. Despite the fundamental differences between these methods, very good agreement between the results of MD, density functional theory, and density gradient theory and EOS data is observed for the phase equilibria and interface properties, reinforcing the present predictions. It is found that the equivalence of nitrogen to oxygen is rather limited, in particular for the methane propellant. The disparities are particularly pronounced for low temperatures, where the compositions of the bulk phases differ significantly. As a result, enthalpy of vaporization as well as surface tension attains much higher values for mixtures containing oxygen.

Yukawa-Friedel-tail pair potentials for warm dense matter applications.

Accurate equationsof state (EOS) and plasma transport properties are essential for numerical simulations of warm dense matter encountered in many high-energy-density situations. Molecular dynamics (MD) is a simulation method that generates EOS and transport data using an externally provided potential to dynamically evolve the particles without further reference to the electrons. To minimize computational cost, pair potentials needed in MD may be obtained from the neutral-pseudoatom (NPA) approach, a form of single-ion density functional theory (DFT), where many-ion effects are included via ion-ion correlation functionals. Standard N-ion DFT-MD provides pair potentials via the force matching technique but at much greater computational cost. Here we propose a simple analytic model for pair potentials with physically meaningful parameters based on a Yukawa form with a thermally damped Friedel tail (YFT) applicable to systems containing free electrons. The YFT model accurately fits NPA pair potentials or the nonparametric force-matched potentials from N-ion DFT-MD, showing excellent agreement for a wide range of conditions. The YFT form provides accurate extrapolations of the NPA or force-matched potentials for small and large particle separations within a physical model. Our method can be adopted to treat plasma mixtures, allowing for large-scale simulations of multispecies warm dense matter.

First-principles equationof state of CHON resin for inertial confinement fusion applications.

A wide-range (0 to 1044.0 g/cm^{3} and 0 to 10^{9}K) equation-of-state (EOS) table for a CH_{1.72}O_{0.37}N_{0.086} quaternary compound has been constructed based on density-functional theory (DFT) molecular-dynamics (MD) calculations using a combination of Kohn-Sham DFT MD, orbital-free DFT MD, and numerical extrapolation. The first-principles EOS data are compared with predictions of simple models, including the fully ionized ideal gas and the Fermi-degenerate electron gas models, to chart their temperature-density conditions of applicability. The shock Hugoniot, thermodynamic properties, and bulk sound velocities are predicted based on the EOS table and compared to those of C-H compounds. The Hugoniot results show the maximum compression ratio of the C-H-O-N resin is larger than that of CH polystyrene due to the existence of oxygen and nitrogen; while the other properties are similar between CHON and CH. Radiation hydrodynamic simulations have been performed using the table for inertial confinement fusion targets with a CHON ablator and compared with a similar design with CH. The simulations show CHON outperforms CH as the ablator for laser-direct-drive target designs.

Linking Viscosity to Equations of State Using Residual Entropy Scaling Theory

In our previous work (J Chem Eng Data 2021, 66(3):1385–1398), a residual entropy scaling (RES) approach was developed to link viscosity to residual entropy [a thermodynamic property calculated with an equation of state (EoS)] using a simple polynomial equation for refrigerants. Here, we present an extension of this approach to a much wider range of fluids: all pure fluids and their mixtures whose reference EoS and experimental viscosity data are available. A total of 84 877 experimental points for 124 pure fluids and 351 mixtures are collected from 1846 references. The investigated pure fluids contain a wide variety of fluids from light gases with quantum effects at low temperatures to dense fluids and fluids with strong intermolecular association. More than 68.2 % (corresponding to the standard deviation) of the evaluated experimental data agree with the RES model within 3.2 % and 8.0 % for pure fluids and mixtures, respectively. Compared to the recommended models implemented in the REFPROP 10.0 software (the state-of-the-art for thermophysical property calculation), if the dilute gas viscosity is calculated in the same way, our RES approach yields similar statistical agreement with the experimental data while having a much simpler formulation and fewer parameters. To use our RES model, a software package written in Python is provided in the supporting information.Graphical

Quick Guides for Use of the CompOSE Data Base

We present a combination of two quick guides aimed at summarizing relevant information about the CompOSE nuclear equation of state repository. The first is aimed at nuclear physicists and describes how to provide standard equation of state tables. The second quick guide is meant for users and describes the basic procedures to obtain customized tables with equation of state data. Several examples are included to help providers and users to understand and benefit from the CompOSE database.

Two-dimensional simulation of microsecond-timescale underwater electrical explosion of a copper wire

Underwater electrical wire explosion (UEWE) is an efficient source of underwater shock waves (SWs). In order to efficiently simulate the interaction between the UEWE SW and structures, a coupled model that includes the electric circuit, the exploding wire and the surrounding water is established based on user-subroutines provided by the commercial explicit dynamics software ANSYS AUTODYN. The modeling starts from room temperature by using the tabular wide-range metal equation of state (EOS) and conductivity data. Experimental validation is performed with copper wires exploded by a μs-timescale pulsed discharge. The numerical results show satisfactory consistency with experiments in terms of the current and voltage waveforms, the wire expansion trajectory, the evolution of SW front, the interaction between SW and electrodes and the SW pressure profiles. The main discrepancy lies in the SW amplitude that is ∼20% higher in the calculation and the possible reasons are discussed in detail. Based on this approach and with proper modifications to the metal EOS and conductivity data, the interaction between UEWE SWs and structures can be efficiently modeled in 2D and 3D for practical applications.

Shock initiation of low density polymer bonded explosive LX-14: A study of two morphologies

A series of six shock initiation experiments have been carried out on low density LX-14 powder in order to simulate a shock insult on the heavily damage polymer bonded explosive, LX-14. Two distinct morphologies were studied (tap density molding powder and machined swarf), both at the same density of 0.942 g/cm3, or 50.1% theoretical maximum density. The purpose of these experiments was to provide data to help make an assessment of the effects that damage has on the material sensitivity to a planar shock. This was achieved primarily by providing shock sensitivity data in the form of a Pop plot, and also reactants equation of state data, which aids in the determination of input conditions for both the experiments performed in this work and also future experiments of this material type. The experiments were of a cut-back format, consisting of four sample heights on each shot and diagnosed with optical velocimetry. The experiments were carried out at the Technical Area 40 Chamber 9 gas gun facility at the Los Alamos National Laboratory, where the LX-14 targets were subjected to Al 6061 and Oxygen Free High Conductivity copper impactors launched to velocities up to 2.14 km/s. Time corrected reactive growth wave profiles are presented in this paper along with the derivation of the following Hugoniot parameters for this explosive, where the molding powder and machine swarf are represented by the following linear equations, respectively, US=1.64(±0.64)up+2.25(±0.49) and US=0.87(±0.60)up+4.08(±0.49). The results show that the steady increase in shock sensitivity with increasing void fraction reaches an inflection point beyond which the shock sensitivity begins to decease. This inflection point lies between 65% and 50% of the theoretical maximum density of the LX-14. In both cases, the damaged LX-14 was not as sensitive as expected, with a relative shock sensitivity of the molding powder being less than the pressed LX-14, and the machined swarf having a shock sensitivity that is comparable to pressed LX-14.

Determination of the symmetry groups of radiation hydrodynamics equations and the compatible equations of state and opacities

The purpose of this study was to find all the symmetry groups of the radiation hydrodynamics equations with no a priori assumptions on the equations of state (EOS) and opacities. As shown in earlier works, the application of the Lie group technique to such a system of equations leads to invariance conditions in the form of linear differential equations, which, up until now, were only partially solved. In this paper, using the same technique and under the same assumptions, but with a simpler formulation, we show that these equations can be entirely solved analytically. This result enables us to list all the one-parameter groups that may be symmetry groups of the system. To be actually so, they must be associated with suitable EOS and opacities whose general expressions are also given. The interesting point is that some of them can be chosen so as to fit realistic data for EOS and opacities. Using this property, we propose a method to design low-scale experiments to simulate radiative processes, which would involve too much energy to have experimented with at their full scale. In addition, we derive the reduced systems associated with the one-parameter symmetry groups found. We show that some classical self-similar problems can be extended to more general EOS and opacities, and we treat in detail the self-similar expansion of a semi-infinite medium submitted to an internal source of energy.

Equilibrium state thermodynamic properties of rare gas dimers and trimers obtained from equations of state and statistical thermodynamics: Application to neon, argon, krypton and xenon

The equilibrium thermodynamic properties of gas phase dimerization and trimerization of Ne, Ar, Kr, and Xe are calculated from equations of state (EOS). The equilibrium constants, changes in enthalpy, Gibbs energy, entropy and heat capacity are determined over 10-kelvin ranges are reported. These quantities are calculated from accurate and precise equations of state available from the NIST Thermophysical Tables. In addition, the molar entropies and heat capacities are calculated for the dimers, and molar entropies for the trimers. Statistical thermodynamic (ST) methods are used to complement these calculations and employ dimer partition functions obtained from a sum-over-state approach that includes both bound and metastable rovibrational states. For the trimers, such calculations are carried out only for neon, for which the rovibrational energies of the bound states are available. The results of the EOS and ST calculations for the Ne – Xe dimers are in good agreement with each other, while those for the neon trimer are less so. The effect of excluding the metastable states from the dimer partition functions is demonstrated. The results indicate that the thermodynamic properties of dimerization/trimerization for polyatomic molecules may be obtained if accurate EOS data is available, suggesting that the method may be applied to polyatomic molecules. Relationships between the enthalpies of dissociation of the dimers and trimers and the enthalpies of vaporization of the respective liquids are demonstrated.

Ab Initio Phase Diagram of Copper

Copper has been considered as a common pressure calibrant and equation of state (EOS) and shock wave (SW) standard, because of the abundance of its highly accurate EOS and SW data, and the assumption that Cu is a simple one-phase material that does not exhibit high pressure (P) or high temperature (T) polymorphism. However, in 2014, Bolesta and Fomin detected another solid phase in molecular dynamics simulations of the shock compression of Cu, and in 2017 published the phase diagram of Cu having two solid phases, the ambient face-centered cubic (fcc) and the high-PT body-centered cubic (bcc) ones. Very recently, bcc-Cu has been detected in SW experiments, and a more sophisticated phase diagram of Cu with the two solid phases was published by Smirnov. In this work, using a suite of ab initio quantum molecular dynamics (QMD) simulations based on the Z methodology, which combines both direct Z method for the simulation of melting curves and inverse Z method for the calculation of solid–solid phase boundaries, we refine the phase diagram of Smirnov. We calculate the melting curves of both fcc-Cu and bcc-Cu and obtain an equation for the fcc-bcc solid–solid phase transition boundary. We also obtain the thermal EOS of Cu, which is in agreement with experimental data and QMD simulations. We argue that, despite being a polymorphic rather than a simple one-phase material, copper remains a reliable pressure calibrant and EOS and SW standard.

Development of semiempirical equation of state of binary functionally graded materials and its influence on generation of ramp compression: Comparison with bilayer graded density impactors

Ramp wave study involving graded impactors with discrete or continuous variation of material density is of immense interest for investigation of shock induced structural changes in solids. Hydrodynamic simulation of ramp wave generated by continuously changing functionally graded materials (FGM) requires accurate knowledge of its equation of state (EOS) parameters. However, theoretical as well as experimental EOS data for the same are not readily available. Current work is an attempt towards development of semiempirical EOS model for binary FGM (bFGM) wherein the density is a continuous function of position, either linearly or quadratically. This has been achieved by deriving analytical functions for density dependence of hugoniot parameters using the recently proposed kinetic energy average (KEA) model as well as commonly employed interpolation based methods. Six bFGMs with Al, Mg, and paraffin as low-impedance components and Cu and W as high-impedance ones are considered for the present study. It is shown that shock impedance of linear and quadratic bFGM can be expressed as superlinear/superquadratic functions of position. Subsequently, hydrodynamic simulation results of impact loading of thus constructed bFGMs on ultrathin Ta target are reported. The time profile of target pressure displayed signatures of quasi-isentropic compression (shock ramp). Ramp profiles obtained in our simulation are compared with those generated by recently reported [Phys. Rev. B 99, 214105 (2019)] alternate material bilayer graded density impactor (blGDI) with programmable layer thicknesses. Time profile of ramp pulse is shown to have direct correlation with spatial profile of shock impedance. Influence of associated physical parameters, such as impedance of front layer material, impedance ratio of two components, impact velocity, and mixture model of EOS on shock-ramp adiabats are explored in great detail. The merits of using low and high impedance material as one component of bFGM or blGDI in lowering target heating and enhancing peak ramp pressure are investigated theoretically. Study reveals that paraffin based bFGM with quadratic density function produces the lowest temperature and entropy.

Metallization of Shock-Compressed Liquid Ammonia.

Ammonia is predicted to be one of the major components in the depths of the ice giant planets Uranus and Neptune. Their dynamics, evolution, and interior structure are insufficiently understood and models rely imperatively on data for equation of state and transport properties. Despite its great significance, the experimentally accessed region of the ammonia phase diagram today is still very limited in pressure and temperature. Here we push the probed regime to unprecedented conditions, up to ∼350 GPa and ∼40 000 K. Along the Hugoniot, the temperature measured as a function of pressure shows a subtle change in slope at ∼7000 K and ∼90 GPa, in agreement with abinitio simulations we have performed. This feature coincides with the gradual transition from a molecular liquid to a plasma state. Additionally, we performed reflectivity measurements, providing the first experimental evidence of electronic conduction in high-pressure ammonia. Shock reflectance continuously rises with pressure above 50GPa and reaches saturation values above 120GPa. Corresponding electrical conductivity values are up to 1 order of magnitude higher than in water in the 100GPa regime, with possible significant contributions of the predicted ammonia-rich layers to the generation of magnetic dynamos in ice giant interiors.