Equation-of-state, sound speed, and reshock of shock-compressed fluid carbon dioxide

The covalent double-bonds that bind the atoms in a CO₂ molecule at ambient conditions are among the strongest of chemical bonds, but at pressures reaching tens of GPa, the compression energy (PΔV) becomes comparable to this bonding energy (hundreds of kJ/mol), and the previously stable molecule exhibits complex chemical behavior.^1,2 Laser-heated diamond-anvil cell experiments have characterized the solid phase diagram of CO₂ up to 120 GPa, which exhibits five molecular crystalline polymorphic phases before transforming into both crystalline and amorphous polymeric phases.^1–7 

The fluid phase diagram of CO₂ has been experimentally explored to 1 TPa (Refs. 8–14) and is proposed to exhibit similar chemical complexity to the solid phase diagram.¹⁵ When shock compressed, molecular liquid CO₂ (Fluid-I) is stable up to 40 GPa,^8,10 above which it transforms into an insulating 3- and 4-coordinated polymeric fluid (Fluid-II).^12,15 Above 100 GPa, CO₂ transitions into the Fluid-III phase and begins to ionize.¹⁴ The present work is a study of the Fluid-III phase of CO₂.

The pressure, density, temperature, and reflectivity of shocked CO₂ have been measured to 1 TPa and 93 000 K in Ref. 14. Experimental evidence indicates that CO₂ at the highest pressures and temperatures studied is in a complex bonded state as opposed to the previously predicted¹³ fully atomic C, O fluid. This work reports further details of the study presented in Ref. 14, and additionally reports the experimentally determined Grüneisen parameter and isentropic sound speed of shocked CO₂, and the mechanical behavior of CO₂ under reshock.

These experiments took place on the OMEGA Laser System at the Laboratory for Laser Energetics in Rochester, NY.¹⁶ The laser-shocked diamond-sapphire anvil cell^17–21 containing the precompressed CO₂ sample is depicted in Fig. 1. The CO₂ samples were cryogenically loaded into cells comprising 350- or 560-μm-thick diamond and 5000-μm-thick sapphire anvils. The CO₂ was contained between the anvils in a stainless-steel gasket. The gasket was initially 250-μm thick; after compression, the gasket thickness was reduced to approximately 150 μm. The anvils were mounted in tungsten carbide (WC) seats, the diamond side seat had a 900-μm lateral window, and the sapphire side seat had an 800-μm window. The diamond was coated with a 15-μm plastic ablator and 3 μm of gold; the gold served to absorb x-rays produced at the ablation front and prevent photoionization of the sample. The sample-side of the drive diamond was coated with a 0.1-μm Ti flash coating. Two 25-μm-thick α-quartz pieces, referred to as the pusher and the window, were located in the sample chamber with the CO₂. The quartz pusher served as an impedance matching,²² reflectivity, and temperature standard^23–26 for singly shocked CO₂. The quartz rear window allowed for the determination of a reshock state in the CO₂.

The loaded cell was mechanically compressed to pressures of 0.36–1.16 GPa at ambient temperature. These pressures correspond to densities in the CO₂ ranging from 1.35 g/cm³ (Ref. 27) to 1.74 g/cm³ (Refs. 5 and 7), and densities in the quartz ranging from 2.68 g/cm³ and 2.73 g/cm³ (Ref. 25). Above 0.5 GPa, CO₂ crystallizes into solid phase-I.⁶ The formation of crystals in the CO₂ was observed in cells with the highest precompression. The precompression pressures were measured using calibrated fluorescence spectroscopy of ruby beads within the cell,²⁸ and the density of the CO₂ was then determined from a 295 K isotherm.^5–7,27

The OMEGA laser irradiated the plastic ablator with intensities between 1.2 × 10¹⁴ W/cm² and 10.0 × 10¹⁴ W/cm²; these experiments used 12 beams with up to 480 J per beam (5760 J total) in an 865-μm focal spot and a 1-ns pulse duration. The laser ablation of the CH layer drove shock waves through the diamond anvil and quartz pusher into the liquid or solid CO₂ sample. The pressures in these experiments were sufficiently high to ionize the CO₂ and produce an optically reflective shock front. In these experiments, the transit times through the CH ablator, Au layer, and drive diamond ranged from 9 to 22 ns, depending on the laser intensity and drive diamond thickness. The 1 ns pulse shape leads to decaying shocks in the CO₂ samples, from which a range of shock states could be observed in a single experiment.

The velocity of the reflecting shock front was measured with a dual-channel line-imaging velocity interferomer system for any reflector (VISAR).²⁹ The Ti flash coating was opaque to VISAR, so the velocity of the shock front was not measured until the shock broke out into the quartz window. This allowed for finer time resolution, as the zero-velocity phase is a necessary reference.²⁹ The apparent velocity from VISAR is corrected for the precompressed refractive index of quartz²⁵ and CO₂.^6,7 The dual-interferometer system allows one to resolve 2π fringe ambiguities and determine a unique velocity solution. Integrating the velocity as a function of time must yield the thickness of the quartz and CO₂ sample chamber; this serves as another check on the velocity solution.

Impedance matching²² was performed at the quartz pusher/CO₂ interface using the Rankine–Hugoniot conditions for conservation of mass, momentum, and energy to calculate the particle velocity, pressure, density, and internal energy in the shocked CO₂. A Mie–Grüneisen linear release^23,25 was used to model the release of the higher-impedance quartz into the lower-impedance CO₂. Additionally, the intensity of the VISAR signal is used to determine the reflectivity of the shocked CO₂ at 532 nm by referencing to the known quartz reflectivity as a function of shock velocity.^25,26

Simultaneously with the VISAR, the self-emission (590–850 nm) from the shock front was measured using streaked optical pyrometry (SOP),^30,31 from which a brightness temperature was determined. Brightness temperature is inferred from the measured emission with the assumption that the shock front emits as a gray body with reflectivity as measured with VISAR. While particle velocity, density, and pressure are determined from impedance matching only at the instant the shock wave is transmitted from the quartz pusher into the CO₂, shock velocity, temperature, and reflectivity are tracked continuously through the shock transit of the entire experiment.

The pressure and density results from these experiments are plotted in Figs. 2(a) and 2(b) (triangles), along with previous CO₂ single-shock data.^8–13 We performed a linear fit to the shock velocity (⁠$US$⁠) vs particle velocity (⁠$UP$⁠) data between 189 and 995 GPa (this work and Ref. 13) with a linear term to account for variation in initial density ρ₀,

The data all lie within 2-σ of the fit. Other fits, including quadratic, cubic, and exponential, were performed, but statistical analysis showed that the data did not justify a fit more complex than linear. Parameters and covariance matrix elements for Eq. (1) are given by: $c0= −1.88 km/s, s=1.29, a=3.36 km/s cm3/g, σc02=7.06× 10−02, σs2=7.85× 10−05, σa2=3.30× 10−02, σc0σs=−1.30× 10−03, σc0 σa= −4.21× 10−02$⁠, and $σs σa=1.32×10−04$⁠.¹⁴ 

This fit was converted to pressure vs density using the Rankine–Hugoniot conservation relations and plotted in Fig. 2 with solid lines, and extrapolated to lower pressure with dotted lines. Quantum mechanical calculations [density function theory, (DFT),³² dashed–dotted lines], benchmarked in Ref. 12 (squares), predict significant curvature between 50 and 500 GPa due to changes in the molecular bonding of CO₂.^13,15,32 Our measurements from initially 1.4 g/cm³ (green) do not support such curvature and are in better agreement with LEOS models.³³ LEOS table 2274, constructed using the quotidian equation-of-state methodology, expresses the Helmholtz free energy as a function of volume and temperature, and includes a dissociation term and a non-dissociation term in the ion free energy.³³ These terms are coupled by the molar mass. It is significant that the LEOS table represents the mechanical behavior of the data in the present work, but does not represent the increase in compressibility seen in Ref. 12.

There exists a clear anomaly in the low-pressure gas-gun data from Ref. 12 (blue squares) and Ref. 8 (green pentagons) seen in Fig. 2. For a given shock pressure, it is expected that CO₂ of a higher initial density will have a higher final density due to reduced heating. The low-pressure data demonstrate this behavior until 42 GPa; above this pressure, the data from Ref. 12 (blue squares) show a higher final density for a given shocked pressure than the data from Ref. 8, despite having a lower initial density. The high-pressure data (this work and Ref. 13, 189–995 GPa) demonstrate the expected behavior. The anomaly in the gas-gun data has led to disagreement in the modeling of CO₂, and the present measurements support LEOS³³ over current DFT³² calculations in the high-pressure fluid regime.

Variation in initial density was leveraged to measure multiple Hugoniot curves, from which derivative quantities were probed using a difference method.²⁶ The Grüneisen parameter $γ=V∂P∂E |V$ was determined from the mechanical equation-of-state given in Eq. (1) and a difference method between initially 1.17 and 1.4 g/cm³; 1.17 and 1.7 g/cm³; and 1.4 and 1.7 g/cm³ Hugoniots. The Grüneisen parameter shows little dependence on initial density; all three difference methods yield the same result within 10%. The averaged result is plotted in Fig. 3(a) with a 1-σ confidence interval based on the uncertainty in the $US$–$UP$ fit. Additionally plotted are predictions of γ from tabular equations-of-state SESAME 5212 and LEOS 2274.³³ Theoretical γ is systematically higher than that of the experimental result, but all curves tend to the ideal gas limit of 2/3.

The Eulerian sound speed can be directly calculated from the measured Hugoniot and Grüneisen parameter γ with³⁴ 

$Cs$ was calculated from Eqs. (1) and (2) for three different initial densities: 1.17 g/cm³, 1.4 g/cm³, and 1.7 g/cm³. The sound speed on the Hugoniot from each initial density differed by less than 3%, showing even less dependence on initial density than the Grüneisen parameter. The averaged result is plotted in Fig. 3(b) with a 1-σ confidence interval propogated from the uncertainty in the parameters in Eq. (1) and the uncertainty in γ with a 100 000 trial Monte Carlo method. SESAME 5212, a single-phase equation-of-state, underpredicts the sound speed of shocked CO₂, but LEOS 2274³³ shows an excellent agreement with our experimental data. This is expected given the good agreement between LEOS 2274³³ and our Hugoniot data.

When the shock wave traverses the CO₂ sample and enters the higher-impedance quartz rear window, a second shock (reshock) is launched back into the CO₂ sample. Impedance matching²² is performed at the CO₂/window interface to determine the pressure, density, and internal energy of this reshock in CO₂. The shock velocity of the CO₂ (U_S,CO2) and the quartz window (U_S,Q) are measured on either of the interface with VISAR. From the known quartz Hugoniot^24,25 and Eq. (1) for an initial density of $ρ0,CO2$⁠, the pressure, density, and particle velocity on either side of the interface for both CO₂ (P $CO2, ρCO2, UP,CO2$⁠) and quartz (⁠$PQ, ρQ, UP,Q$⁠) are also known. By impedance matching, the pressure of the reshocked CO₂ (P_R) must be equal to P_Q. By the Rankine–Hugoniot conservation relations, the density of the reshocked CO₂ (⁠$ρR$⁠) is given by

The present work measured four reshock states in CO₂, which are summarized in Table I. The reshock results from this work and from Ref. 13 are summarized in Figs. 4 and 5. The experimental observables $US,CO2$ and U_S,Q are plotted in Fig. 4 along with LEOS curves³³ based on intersection of the modeled reshock with the experimental quartz Hugoniot.^24,25 This work shows strong agreement with LEOS, as does most of the data from Ref. 13.

Figure 5(a) represents the reshock data in the pressure-density plane. The pressure along a reshock curve is related to the pressure on the Hugoniot at the same specific volume V (inverse of density, ρ) by assuming a constant γ equation of state³⁵ 

where V₀ is the initial specific volume (1/ρ₀), P_R is the pressure on the reshock curve at volume V from initial state P₁,V₁ on the principle Hugoniot, P_H is the pressure on the principle Hugoniot at volume V, and γ is the Grüneisen parameter. Because the single-shock and reshocked state of CO₂ were determined with impedance matching, Eq. (4) serves as an independent way to determine γ, plotted in Fig. 3(a) (points). The uncertainty in γ represents how much γ can vary and still yield the measured reshocked state in the CO₂ within the error bar. The results are consistent with those obtained from a difference method applied to the CO₂ single-shock equation-of-state [Fig. 3(a) solid line], but P_R in Eq. (4) does not provide strong constraints on γ given present uncertainties in the measured density of reshocked CO₂. Reshock curves for the best value of γ are plotted in Fig. 5 (dotted).

There are three pairs of data points in Fig. 5(a) that reach the same pressure-density state following different thermodynamic paths. As shown in Fig. 5(b), when plotted as internal energy vs density, those points no longer overlap, implying that the states are at different temperatures. Internal energy was determined from the Rankine–Hugoniot condition for conservation of energy. To account for the different initial energy arising from different initial conditions, the initial energy for initially liquid points [blue and green in Fig. 5(b)] was taken from Ref. 36 [–421 J/g for initially 220 K and 1.167 g/cm³ (blue) and –393 J/g for initially 295 K and 1.39 g/cm³ (green)]. The initial energy in the initially solid (orange) points was then found by shifting from the initially liquid (green) points on a 295 K isotherm by integrating the pressure–volume curves from Refs. 5 and 6, and adding the latent heat of fusion from Ref. 37, to yield –552 J/g for initially 295 K and 1.67 g/cm³. In all cases, the initial internal energy of the CO₂ was approximately 50% of the error bar of the final shocked internal energy.

Because Eq. (4) relates pressure on the Hugoniot to pressure on the reshock curve at the same volume, the model fails beyond maximum compression of the principle Hugoniot. For this reason, Eq. (4) cannot be used to determine γ from the reshock data in Ref. 13. There are reshock data reported on CO₂ in Ref. 10 from aluminum and stainless-steel anvils, but the data have significant scatter and no reported uncertainty, so were not included here.

To summarize, this work provides additional details on recently published¹⁴ equation-of-state measurements of shock compressed CO₂ to 1 TPa and 93 000 K from varying initial densities, and presents new information on the Grüneisen parameter, sound speed, and reshock behavior of high-pressure shocked CO₂. We find that the compressibility, Grüneisen parameter, and sound speed of shocked CO₂ are well represented by LEOS;³³ this work does not support the extreme curvature in compressibility modeled by DFT.³² Notably, lower-pressure gas-gun data support DFT over LEOS. We discuss an anomaly in the lower-pressure CO₂ data, which has led to disagreement among models. This complexity in the compressibility behavior of shocked CO₂ warrants further study, since there is currently a gap between 71 and 189 GPa where no data exist to constrain theory. We report four reshock states of CO₂, and discuss the effect of the Grüneisen parameter on the reshock curve. This work provides significant new benchmarks for theoretical calculations of fluids in the warm-dense-mater regime.

This material is based upon the work supported by the Department of Energy National Nuclear Security Administration under Award No. DE-NA0003856, the University of Rochester, the New York State Energy Research and Development Authority, NNSA support to the University of California, Berkeley, and NSF 19–528 Physics Frontier Centers, Award No. 2020249, Center for Matter at Atomic Pressures. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. DE-AC52–07NA27344 and was supported by the LLNL-LDRD Program under Project No. 12-SI-007.

This report was prepared as an account of work sponsored by an agency of the U.S. Government. Neither the U.S. Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the U.S. Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the U.S. Government or any agency thereof.

The data that support the findings of this study are available from the corresponding author upon reasonable request.