Atom-in-jellium equations of state for cryogenic liquids

TL;DR

This paper applies an efficient atom-in-jellium model to predict equations of state for cryogenic liquids nitrogen, oxygen, and fluorine, showing good agreement with experimental shock data and advanced simulations, and improving high-pressure predictions.

Contribution

The study extends the atom-in-jellium EOS approach to cryogenic liquids, providing more accurate high-pressure models for nitrogen, oxygen, and fluorine compared to previous methods.

Findings

EOS predictions align with shock data and PIMC simulations

Systematic deviations from Thomas-Fermi EOS due to shell ionization

Enhanced high-pressure accuracy for nitrogen and oxygen EOS

Abstract

Equations of state (EOS) calculated from a computationally efficient atom-in-jellium treatment of the electronic structure have recently been shown to be consistent with more rigorous path integral Monte Carlo (PIMC) and quantum molecular dynamics (QMD) simulations of metals in the warm dense matter regime. Here we apply the atom-in-jellium model to predict wide-ranging EOS for the cryogenic liquid elements nitrogen, oxygen, and fluorine. The principal Hugoniots for these substances were surprisingly consistent with available shock data and Thomas-Fermi (TF) EOS for very high pressures, and exhibited systematic variations from TF associated with shell ionization effects, in good agreement with PIMC, though deviating from QMD and experiment in the molecular regime. The new EOS are accurate much higher in pressure than previous widely-used models for nitrogen and oxygen in particular, and should allow much more accurate predictions for oxides and nitrides in the liquid, vapor, and plasma regime, where these have previously been constructed as mixtures containing the older EOS.

Equations

$${\rho }_0{u}_s$$

$$p_0+{\rho }_0{u}_s^2$$

$$\left[p_0+{\rho }_0{e}_0+\frac{1}{2}{\rho }_0{u}_s^2\right]u_s$$

$$\Delta e=e_{\text{mbox}ref}({\rho }_0,T_0^{\prime })-e_{\text{mbox}ref}({\rho }_0,T_0).$$

Full text

equation of state, electronic structure

I Introduction

Many of the most common planet-forming substances contain oxygen, including silica and silicates, oxides such as alumina, MgO, and FeO, carbonates, water, and CO2. Some contain nitrogen, in particular ammonia. Equations of state (EOS) used to understand planetary impacts and giant planets and exoplanets are typically constructed in the warm, dense matter regime by mixing elemental EOS. EOS for constituents mixed in this way should be reasonably accurate at atomic volumes and temperatures corresponding to high pressure compression of the corresponding cryogenic liquids. The third second-row diatomic cryogenic liquid, fluorine, is of practical interest as a component of SF6, used in dielectric breakdown switches in which it forms a plasma during operation, and of LiF, which is widely used in high pressure experimental studies as an optical window or tamper. Fluorine is also a constituent of various polymers and chemical explosives, and is a component of ‘flibe,’ a possible coolant and in-situ breeder of tritium in thermonuclear reactors, where it would be subjected to heating and compression.

Shock wave data, usually used to calibrate high pressure EOS, are limited in range for the cryogenic liquids oxygen and nitrogen, and non-existent for fluorine. Widely-used semi-empirical EOS from the sesame and leos libraries sesame ; leos are thus limited in data for calibration or validation. The sesame EOS for oxygen and nitrogen were constructed only for the molecular regime, using a model of molecular vibrations (MV) for the ion-thermal energy Kerley1980 . While tabulated to $10^{8}$ K for use in wide-ranged applications, they are not valid for temperatures over $10^{4}$ K, as they do not include the effects of molecular dissociation and ionization. The sesame EOS for fluorine Crockett2006 was constructed using a different procedure than other EOS in the sesame library, largely because of the lack of shock data. An accurate but narrow-range experimentally-derived, thermophysical EOS was used to constrain a Mie-Grüneisen model for the ions, blended into Thomas-Fermi-Dirac (TFD) calculations at high compressions. TFD was used also for the electron-thermal energy, as in most sesame EOS.

Recently, the rigorous but computationally expensive techniques of path integral Monte Carlo (PIMC) pimc and quantum molecular dynamics (QMD) qmd have shown that all-electron average-atom calculations, in which the ions surrounding the atom are simplified as a uniform ‘jellium’ Liberman1979 , can give accurate predictions of electronic excitation contributions to the EOS of dense plasmas Benedict2014 ; Driver2017 . We have found further that, for a set of well-studied reference elements, when atom-in-jellium electronic states are combined with a perturbative approach to estimate the ion-thermal contribution to the EOS Liberman1990 , they give reasonably accurate predictions of the complete EOS in the liquid phase as well as the dense plasma regime Swift2018 ; Swift2019 .

In the work reported here, we apply these same atom-in-jellium techniques to predict the EOS of nitrogen, oxygen, and fluorine, and compare the results against previous work, including combined QMD and PIMC studies Driver2016 , and shock wave experiments Zubarev1962 ; Dick1970 ; Nellis1980 ; Nellis1991 ; Trunin2008 ; Mochalov2010 ; Hamilton1988 , which have been reported for both nitrogen and oxygen but not for fluorine. We pay particular attention to the prediction of the shock Hugoniot for initial states in the cryogenic liquid. While the atom-in-jellium calculations are inaccurate where molecular bonding occurs, we find that a correction to the atom-in-jellium energy in the initial state allows us to make accurate predictions of the shock Hugoniot for final states in the warm dense matter regime of interest for astrophysical impacts and other applications of high energy density.

II Atom-in-jellium calculations

The cryogenic liquids are an interesting test of electronic structure calculations of EOS. The atom-in-jellium model was originally expected to be suitable only for close-packed metals, as it represents the electron distribution with spherical symmetry and does not capture the relative orientation of neighboring atoms, smearing their charge into the uniform background jellium outside the Wigner-Seitz sphere Liberman1979 . However, atom-in-jellium calculations were subsequently found to be similarly accurate for the EOS of non-close-packed metals, and even for carbon and silicon Swift2018 , despite the importance of directional bonding in these elements near ambient conditions.

The diatomic cryogenic liquids are notable in that saturated interatomic bonds and van der Waals forces are essential to their behavior at low pressures and temperatures; these behaviors are not captured by the present atom-in-jellium model. The atom-in-jellium EOS are thus not expected to be accurate at low pressures, and it is therefore of interest to know how closely they match existing data at higher pressures asnd temperatures, where directional chemical bonding becomes less important. Additionally, we are interested in EOS predictions that can be made at even higher pressures, where the EOS is currently unconstrained by experimental data, as well as comparisons with EOS predicted using other approaches such as the Mie-Grüneisen or Thomas-Fermi (TF) models tf .

The average-atom calculations reported here were performed using the same prescription as in our previous study Swift2018 . For each element, atom-in-jellium calculations were made over a range and density of tabulation suitable for a general-purpose EOS: mass density $\rho$ from $10^{-4}$ to $10^{3}\rho_{0}$ with 20 points per decade, and temperature $T$ from $10^{-3}$ to $10^{5}$ eV with 10 points per decade. The reference density $\rho_{0}$ was chosen to be that of the liquid cryogen; it should be noted that the choice of this density is purely a convenience in constructing a tabular EOS, where it is useful for the tabulation to include the starting state in typical applications of the EOS model, in order to reduce the sensitivity to interpolating functions. The EOS were not adjusted to reproduce any empirical data.

As was found in the previous study Swift2018 , the electronic wavefunctions were computed reliably down to 10 K or less for densities corresponding to condensed matter, and to 100 K or less for densities down to 0.1% of the ambient solid. At lower densities, calculations completed successfully only for temperatures of several eV or more. In contrast to the previous study, calculations of the restoring force for infinitesimal displacements of the nucleus gave an imaginary Einstein frequency, for densities slightly above that of the cryogenic liquid. This behavior indicates the localization of electrons to an atom, and reflects the role of bonding orbitals and the van der Waals interaction in stabilizing the liquid cryogen at low pressures.

The results of the atom-in-jellium calculations were, for each state of mass density $\rho$ and temperature $T$: electronic contributions to the Helmholtz free energy $f$; an ionic Einstein temperature $\theta_{E}$ and estimated Debye temperature $\theta_{D}$; the mean square displacement of the atom as a fraction of the Wigner-Seitz radius $f_{d}$; and the ionic contribution to $f$ using the generalized Debye model with asymptotic ionic freedom Swift2018 ; Swift2019 . The total electronic energy was used, including the ground state energy at zero temperature: this convention combines components sometimes presented as a cold compression curve and separate electron-thermal energy. For isolated states where the atom-in-jellium calculation failed to converge, polynomial interpolation from the surrounding states was used to complete the EOS table. For each state, the total Helmholtz free energy $f$ was calculated, and then differentiated using a quadratic fit to the three closest states in $\rho$ to determine the pressure $p(\rho,T)$ in tabular form. Similarly, quadratic fits in $T$ were differentiated to find the specific entropy $s$, and hence the specific internal energy $e(\rho,T)$ in tabular form. Together, these tables constitute an EOS suitable for use in multi-physics hydrodynamic simulations.

III Principal Hugoniots for cryogenic liquids

Although, without further processing, the atom-in-jellium EOS can be interrogated in compressed and heated states where the calculations completed, the ionic calculation was neither robust nor meaningful in cryogenic liquid states. This is a problem for their use in calculating the shock Hugoniot for an initially liquid sample, as a physical initial state is needed when solving the Rankine-Hugoniot equations rh . An optimal solution to this problem would be to improve the EOS calculation to give a physical ion-thermal energy, or to combine the calculation with a more reliable technique for this region of state space. However, this is a difficult problem in quantum chemistry, outside the scope of the present work. As long as our main interest is high-pressure states such as shocks into the warm dense matter regime, a simpler approach is possible.

The Rankine-Hugoniot equations describe the conservation of mass, momentum, and energy across a steady shock wave:

[TABLE]

where $u_{s}$ is the shock speed and $u_{p}$ the particle speed. They are typically used to deduce the Hugoniot, or locus of states accessible from a given initial state by the passage of a single, steady shock wave, for matter described by an EOS of the form $p(\rho,e)$. The initial state enters only through the quantities $\rho_{0}$, $e_{0}$, and $p_{0}$, so the EOS need not be valid near the unshocked state if these quantities can be obtained in a different way. This aspect is used in calculations of detonating high explosives, where the EOS of the reaction products can be used without any physical representation of the unshocked explosive he . For the present EOS, defined via the Helmholtz free energy, the temperature $T$ was eliminated when solving the Rankine-Hugoniot equations rhnew .

To deduce a Hugoniot from atom-in-jellium EOS with inaccurate or undefined states around that of the initial cryogenic liquid, $(\rho_{0},T_{0})$, we first explored higher initial temperatures $T_{0}^{\prime}$ at the same mass density, until a usable state was found. At this state $(\rho_{0},T_{0}^{\prime})$, the specific internal energy from the atom-in-jellium calculation took some value $e_{0}^{\prime}$. A Hugoniot could be calculated from this initial state. Next, we used a suitable previously-developed reference EOS to calculate a difference in specific internal energy between the states:

[TABLE]

We then recalculated the atom-in-jellium Hugoniot, defining the initial energy to be $e_{0}^{\prime}-\Delta e$. This approach essentially uses the specific heat capacity from the reference EOS to correct the atom-in-jellium state from a reliable value at $T_{0}^{\prime}$ to the desired $T_{0}$.

In the following comparisons for each element, experimental shock data were taken from the Marsh and van Thiel compendia Marsh1980 ; vanThiel1966 . As well as the principal Hugoniot from the cryogenic liquid state, we consider the cold compression curve, which explores higher densities, and the isochore passing through the cryogenic liquid state, which explores higher temperatures relevant to ablation and expansion following a shock.

III.1 Nitrogen

For nitrogen, apart from deviations around the initial state, the atom-in-jellium calculations of the cold curve and principal isochore lie close to those of the TF-based leos 70 leos , even down to the change in gradient around 300 eV on the isochore. Although the isochore from the MV-based sesame 5000 Kerley1980 falls well below that from the other EOS, its cold curve is remarkably consistent with the TF-based model over the full range considered. (Figs 1 and 2.)

The atom-in-jellium Hugoniot passes through the experimental shock measurements Zubarev1962 ; Dick1970 ; Nellis1980 ; Nellis1991 ; Trunin2008 ; Mochalov2010 around 50 GPa. At higher pressures, the Hugoniot is significantly stiffer than either the MV-based or the TF-based EOS. The MV EOS omits dissociation and ionization, and shows a Grüneisen-like behavior where the shock density approaches a limit asymptotically. The latter, TF-based, model exhibits a peak compression at a similar pressure to the atom-in-jellium EOS, except at a much higher density. The atom-in-jellium Hugoniot also shows a clear feature corresponding to ionization of the outer electrons around 3.5 g/cm3 The atom-in-jellium result follows QMD/PIMC predictions of the Hugoniot Driver2016 much more closely, except that it fails to reproduce the depression between 2 and 3 g/cm3 corresponding to dissociation of the N2 molecules. The QMD Hugoniot reproduces this plateau well, though it predicts the subsequent onset of stiffening at a lower density than experiments using a spherically-converging shock wave Trunin2008 ; Mochalov2010 . (Fig. 3.)

III.2 Oxygen

For oxygen, comparisons of the cold curves and isochores are much the same as for nitrogen. Apart from deviations around the initial state, the atom-in-jellium calculations lie close to those of the TF-based leos 80 leos , again replicating the change in gradient around 300 eV on the isochore. The isochore from the MV-based sesame 5010 Kerley1980 again falls well below that from the other EOS, but the cold curve is similar to that from the TF-based EOS. (Figs 4 and 5.)

For oxygen, the atom-in-jellium Hugoniot passes slightly above the experimental data, and then closely tracks the Hugoniot for the MV-based EOS to 1.5 TPa. At higher pressures, it exhibits a peak compression at 50 TPa, then approaches the TF-based EOS for pressures above 1000 TPa. As with nitrogen, the atom-in-jellium calculations exhibit a distinct feature around 5 g/cm3 from ionization of the outer electrons. The maximum compression, although at a similar pressure to the TF-based EOS, is at a significantly higher density. As for nitrogen, the atom-in-jellium results follow QMD/PIMC calculations of the Hugoniot Driver2016 , which in this case do not predict as strong of a dissociation feature. In oxygen, the contribution of spin to the covalent bond is important Militzer2003 ; the QMD calculations were spinless and thus underpredict the dissociation feature. Experimental data in the region of dissociation are relatively sparse, but measurements Nellis1980 ; Hamilton1988 made consistently with those for nitrogen suggest a similar plateau. (Fig. 6.)

III.3 Fluorine

For fluorine again, the atom-in-jellium cold curve and isochore are similar to the corresponding curves from the TFD-based EOS sesame 5040 Crockett2006 , apart from deviations around the initial state, and once more replicating the change in isochore gradient around 300 eV. (Figs 7 and 8.)

The atom-in-jellium Hugoniot again exhibits a distinct feature around 7 g/cm3 corresponding to ionization of the outer electrons, and a peak compression around 100 TPa. The peak compression is at a lower density than in the TFD-based EOS, and more localized in pressure. The atom-in-jellium EOS was constructed completely consistently with those for nitrogen and oxygen, with no empirical parameters. We therefore deem it highly likely that the Hugoniot from sesame 5040 is up to $\sim$20% too dense. (Fig. 9.)

IV Discussion

It is instructive to compare the systematic behavior of the atom-in-jellium EOS with previous, semi-empirical EOS for the Hugoniots of these cryogenic liquids. Where experimental Hugoniot data are available, the atom-in-jellium EOS is inaccurate at low pressures, as expected given its inability to capture the details of interatomic forces, but becomes close to or coincident with shock data at pressures of several tens to $O(100)$ GPa. As the shock pressure increases, the atom-in-jellium EOS was found to track surprisingly close to the molecular EOS in the sesame library, deviating when effects from the excitation of electron shells became significant and the molecular EOS would then be inaccurate. As has been found for other elements, the atom-in-jellium Hugoniot exhibited a relatively sharp peak in compression compared with the broader peak characteristic of a TF treatment of the electrons. For the elements considered here, this feature originates from ionization of the electrons in the $K$-shell. Interestingly, the peak compression in the TF-based leos models varied non-systematically from the atom-in-jellium calculations: higher for nitrogen but lower for oxygen, and with much greater difference than was found previously for Be, Al, Si, Fe, and Mo Swift2018 , which are solid at STP and more widely studied; Al and Mo being regarded as standards for high pressure work. Indeed, unlike these other elements, the TF calculation of post-peak compression only matched the atom-in-jellium calculation for oxygen. Shock Hugoniots derived from different EOS models were found to differ significantly, even when the cold curves and principal isochores were much more similar: the interplay between compression and heating in shock states can magnify the difference between EOS models.

In practice, because TF theory is inadequate around ambient conditions, TF-based EOS are typically a combination of an empirical Mie-Grüneisen fit to properties at STP together with shock and isothermal compression data, coupled to TF theory at higher temperatures. The markedly different behavior of shock Hugoniots derived from TF-based EOS in comparison with our atom-in-jellium EOS reflects the scarcity of shock data for these cryogenic liquids. This observation highlights the value of the atom-in-jellium calculations: although inadequate in the cryogenic liquid state, their theoretical underpinnings are inherently more self-consistent, so it is not necessary to make empirical adjustments to match shock data above a few hundred gigapascals. As such, they are likely to be more accurate in regimes where the EOS is not constrained directly by experimental data, particularly at elevated pressures and temperatures. However, it is important to stress that atom-in-jellium calculations do not describe molecular bonding, and thus the shock Hugoniots derived from these EOS lack features associated with molecular dissociation. The shock Hugoniot of fluorine should also exhibit this feature, though likely less pronounced than in oxygen, as the trend should correlate with the energy of the covalent bond: 9.79, 5.15, and 1.63 eV for N2, O2, and F2 respectively, at STP. Recent spinless QMD results reproduced the dissociation feature fairly well in nitrogen but appeared to underpredict it in oxygen Driver2016 .

In accord with recent results from PIMC and QMD simulations where available, our atom-in-jellium EOS predictions of the principal Hugoniots of nitrogen, oxygen, and fluorine indicate that EOS models for these elements currently in wide use have significant inaccuracies in the dense plasma regime, densities of several grams per cubic centimeter and temperatures above a few electron-volts.

V Conclusions

We have constructed wide-ranging EOS for nitrogen, oxygen, and fluorine using the atom-in-jellium model, including ion-thermal and cold curve energies as well as the electron-thermal energy for which atom-in-jellium calculations have been used most widely. Although the ion-thermal calculations are not valid around the cryogenic liquid states for these elements, which are stabilized by interatomic bonding and van der Waals forces not captured by the atom-in-jellium model, it is possible to calculate shock Hugoniots from these initial states by correcting preheated calculations to obtain a reasonable starting energy.

Apart from a regime affected by dissociation in N2, the atom-in-jellium Hugoniots match predictions based on QMD and PIMC where they are available.

As was found with elements which are solid at STP, despite disagreements at low pressures, shock Hugoniots derived from the atom-in-jellium EOS are close to or match shock data at pressures approaching around 100 GPa. At higher pressures, where no shock data currently exist, there are substantial differences between the Hugoniots from atom-in-jellium EOS and those from other EOS using a Thomas-Fermi treatment of the electrons or neglecting dissociation and ionization. As the atom-in-jellium EOS were calculated self-consistently, without any empirical adjustment to match data for any of these elements, this deviation is likely to reflect inaccuracies in the previous EOS models. Such inaccuracies imply that EOS for planetary materials such as oxides and silicates, constructed in the warm dense matter regime by mixing previous elemental EOS such as these, may well be inaccurate for applications such as the internal structure of giant planets and the dynamics of planetary collisions.

Acknowledgments

Kevin Driver kindly provided results from his PIMC and QMD simulations.

The work at LLNL was performed under the auspices of the U.S. Department of Energy under contract DE-AC52-07NA27344. B.M. was supported by the U.S. Department of Energy (Grant DE-SC0016248) and by the University of California through the multi-campus research Award 0013725.