Shock Hugoniot Data Research Articles

Shock compression of crystalline TeO2 to the high-pressure fluid regime: Insights from ab initio molecular dynamics simulations

The shock response of fully-dense and porous crystalline tellurium dioxide (TeO2) to the high-pressure and high-temperature fluid regime was investigated within the framework of density functional theory with Mermin’s generalization to finite temperatures. The principal and porous shock Hugoniot curves were predicted from canonical ab initio molecular dynamics (AIMD) simulations, with the phase space sampled along isotherms up to 80 000 K, for densities ranging from ρ=3 to 17 g/cm3. The polymorphs investigated are α-TeO2 paratellurite (P41212), TeO2 cotunnite (Pnma), and TeO2 post-cotunnite (P21/m). Based on the discontinuity found in the calculated Us−up slope of TeO2 post-cotunnite at a shock velocity of Us≃8.35 km/s and a particle velocity of up≃3.64 km/s, the shock melting temperature and pressure are predicted to be ≃6500 K and ≃170 GPa. Results from the AIMD simulations are in line with the static compression data of TeO2 paratellurite and cotunnite, and with the recent shock Hugoniot data for single-crystal α-TeO2 for pressures up to 85 GPa, obtained using the inclined-mirror method and the velocity interferometer system for any reflector combined with powder gun and two-stage light-gas gun.

Shock temperature of liquid nitrogen under pressure using a combination of multi-channel pyrometer and Doppler pin system

A comprehensive understanding of the phase transitions and physical properties of diatomic molecules under extreme pressures and temperatures is necessary. Utilizing a Doppler pin system and a multi-channel optical pyrometer, shock Hugoniot data and temperature states were obtained. First- and second-shock temperatures, ranging from 6000 K to 9000 K, were determined through spectral radiance analysis. The shock cooling phenomenon at second-shock against LiF was observed in each experiment, which established experimental reproducibility. The recorded radiance outputs were found in excellent order, resulting in precise shock temperatures and generating experimental uncertainties of less than 5%. Transformed velocity histories facilitated an accurate depiction of time-varying shock transmittance within compressed liquid and corroborated the book transparency of LiF under shock conditions. This work highlights the efficacy of combining interferometry and pyrometry for obtaining precise Hugoniot state temperatures and investigating shock cooling phenomena while accounting for LiF's melting behavior.

Learning thermodynamically constrained equations of state with uncertainty

Numerical simulations of high energy-density experiments require equation of state (EOS) models that relate a material’s thermodynamic state variables—specifically pressure, volume/density, energy, and temperature. EOS models are typically constructed using a semi-empirical parametric methodology, which assumes a physics-informed functional form with many tunable parameters calibrated using experimental/simulation data. Since there are inherent uncertainties in the calibration data (parametric uncertainty) and the assumed functional EOS form (model uncertainty), it is essential to perform uncertainty quantification (UQ) to improve confidence in EOS predictions. Model uncertainty is challenging for UQ studies since it requires exploring the space of all possible physically consistent functional forms. Thus, it is often neglected in favor of parametric uncertainty, which is easier to quantify without violating thermodynamic laws. This work presents a data-driven machine learning approach to constructing EOS models that naturally captures model uncertainty while satisfying the necessary thermodynamic consistency and stability constraints. We propose a novel framework based on physics-informed Gaussian process regression (GPR) that automatically captures total uncertainty in the EOS and can be jointly trained on both simulation and experimental data sources. A GPR model for the shock Hugoniot is derived, and its uncertainties are quantified using the proposed framework. We apply the proposed model to learn the EOS for the diamond solid state of carbon using both density functional theory data and experimental shock Hugoniot data to train the model and show that the prediction uncertainty is reduced by considering thermodynamic constraints.

Chemical evolution in nitrogen shocked beyond the molecular stability limit.

Evolution of nitrogen under shock compression up to 100GPa is revisited via molecular dynamics simulations using a machine-learned interatomic potential. The model is shown to be capable of recovering the structure, dynamics, speciation, and kinetics in hot compressed liquid nitrogen predicted by first-principles molecular dynamics, as well as the measured principal shock Hugoniot and double shock experimental data, albeit without shock cooling. Our results indicate that a purely molecular dissociation description of nitrogen chemistry under shock compression provides an incomplete picture and that short oligomers form in non-negligible quantities. This suggests that classical models representing the shock dissociation of nitrogen as a transition to an atomic fluid need to be revised to include reversible polymerization effects.

A wide-range semiclassical self-consistent average atom model

The discovery of material properties at extremes, which are essential for high energy density physics development, requires the most advanced experimental facilities, theories, and computations. Nowadays, it is possible to model properties of matter in such conditions using the state-of-the-art density functional theory (DFT) or path-integral Monte Carlo approaches with remarkable precision. However, fundamental and computational limitations of these methods impede their practical usage, while wide-range thermodynamic and transport models of plasma are required. As a consequence, an average atom (AA) framework is still relevant today and has been attracting more and more attention lately. The self-consistent field and electron density in an atomic cell is usually obtained using the Thomas–Fermi (TF), Hartree–Fock, Kohn–Sham approaches, or their extensions. In this study, we present the AA model, where semiclassical wave functions are used for bound states, while free electrons are approximated by the TF model with a thermodynamically consistent energy boundary. The model is compared in various regions of temperatures and pressures with the reference data: the Saha model for rarefied plasma, DFT for warm dense matter, and experimental shock Hugoniot data. It is demonstrated that a single AA model may provide a reasonable agreement with the established techniques at low computational cost and with stable convergence of the self-consistent field.

Hugoniot and released state of calcite above 200 GPa with implications for hypervelocity planetary impacts

Carbonate minerals, for example calcite and magnesite, exist on the planetary surfaces of the Earth, Mars, and Venus, and are subjected to hypervelocity collisions. The physical properties of planetary materials at extreme conditions are essential for understanding their dynamic behaviors at hypervelocity collisions and the mantle structure of rocky planets including Super-Earths. Here we report laboratory investigations of laser-shocked calcite at pressures of 200–960 GPa (impact velocities of 12–30 km/s and faster than escape velocity from the Earth) using decay shock techniques. Our measured temperatures above 200 GPa indicated a large difference from the previous theoretical models. The present shock Hugoniot data and temperature measurements, compared with the previous reports, indicate melting without decomposition at pressures of ~110 GPa to ~350 GPa and a bonded liquid up to 960 GPa from the calculated specific heat. Our temperature calculations of calcite at 1 atm adiabatically released from the Hugoniot points suggest that the released products vary depending on the shock pressures and affect the planetary atmosphere by the degassed species. The present results on calcite newly provide an important anchor for considering the theoretical EOS at the extreme conditions, where the model calculations show a significant diversity at present.

Measurement of shock and re-shock Hugoniot data of liquid nitrogen

ABSTRACT In this work, the equation-of-state data for liquid nitrogen shock compressed to 43 GPa along the principal Hugoniot and reflected-shock data up to 91 GPa were reported. A cryogenic target was used to liquefy the gas that was then further compressed by high-speed impactors. The first- and second-shock states were observed by a high-precision Doppler pin system (DPS). Optical wavefrom of DPS resulted in the first-shock velocity from which other derivative quantities i.e. particle-velocity, specific-volume, internal-energy, and Grüneisen-parameter, were determined. Our results and published equation-of-state data were used to extrapolate the principal Hugoniot. Their comparison showed softening above 27 GPa, attributed to the absorption of thermal energy that dissociates the molecular nitrogen. Our single-shock data provided a good match to available Hugoniot data. The reduced velocity profiles allowed us to detect optical reflectance of the dissociated liquid phase, and the apparent shock-velocity was used to determine the true shock-velocity in dissociation threshold.

Shock Hugoniot Data for Water up to 5 Mbar Obtained with Quartz Standard at High-Energy Laser Facilities

In this work, we present experimental results on the behavior of liquid water at megabar pressure. The experiment was performed using the HIPER (High-Intensity Plasma Experimental Research) laser facility, a uniaxial irradiation chamber of GEKKO XII (GXII) at the Institute of Laser Engineering (ILE), and the PHELIX at GSI (GSI Helmholtz Centre for Heavy Ion Research), a single-beam high-power laser facility, to launch a planar shock into solid multilayered water samples. Equation-of-state data of water (H2O are obtained in the pressure range 0.50–4.6 Mbar by tuning the laser-drive parameters. The Hugoniot parameters (pressure, density, etc.) and the shock temperature were simultaneously determined by using VISAR and SOP as diagnostic tools and quartz as the standard material for impedance mismatch experiments. Finally, our experimental results are compared with hydrodynamic simulations tested with different equations of state, showing good compatibility with tabulated SESAME tables for water.

Thermal electronic properties of aluminum under pressure: The role of sp to 3d electron transfer

As a simple metal, aluminum's valence band is usually described as a free-electron gas with three electrons per atom. The discrepancies between the experimental electronic Gr\uneisen parameter and heat capacity and their free-electron-gas counterparts are usually attributed to electron-phonon coupling. We recently calculated thermal electronic contributions to aluminum's internal energies with our average-atom code paradisio and obtained results that contradict this point of view. Our code also pointed out the overlap of the $sp$ valence band by the $3d$ one, resulting in an $sp$ to $d$ electron transfer. Applying Sommerfeld's temperature expansion method to the electron-electron Coulomb part of the internal energy, we relate the electronic Gr\uneisen parameter, $T=0$ K isotherm, and thermal contributions to the internal energy to a parameter $\ensuremath{\alpha}$ describing the fraction of $d$ electrons resulting from $p$ to $d$ transfer. Finally, we find a unique value of this parameter that provides a consistent explanation for the experimental Gr\uneisen parameter, for $T=0$ K energy deduced from experimental shock Hugoniot data, as well as for our average-atom thermal contributions to internal energy.

Shock-induced phase transition of g-C3N4 to a new C3N4 phase

In this study, phase transition of graphitic carbon nitride (g-C3N4) was investigated using the shockwave compression technique. Firstly, the shock Hugoniot data of g-C3N4 were obtained using a bore propellant gun and a light gas gun under impact velocities of 1.208–4.982 km/s, revealing one phase transition pressure of g-C3N4 at 22.4 GPa. Then, a series of shock recovery experiments was carried out with a pressure range of 17.0–62.1 GPa. The recovered samples were characterized by various techniques, including X-ray diffraction, transmission electron microscopy, energy dispersive X-ray spectroscopy (EDX), and X-ray photoelectron spectroscopy (XPS). The measured d-values of the recovered samples were compared with those from the previous reported results, revealing a new carbon nitride phase synthesized by the shockwave compression technique. The new phase is indexed as a triclinic cell with a = 0.481 nm, b = 0.353 nm, c = 0.285 nm, α = 67.52°, β = 100.75°, γ = 106.47°, and Vcell = 0.043 nm3. EDX and XPS spectra reveal the existence of C and N elements with an atomic ratio of 0.754, also confirming the presence of a new C3N4 phase obtained via a g-C3N4 phase transition induced by shockwave compression with a pressure range of 29.3–62.1 GPa. These sample results are in good agreement with the shock Hugoniot data.

Characterization of the super-short shock pulse generated by an exploding foil initiator

Exploding foil initiator (EFI) uses a thin flyer to initiate an explosive pellet. The shock pulse generated by EFI is very difficult to measure and calculate accurately, which is crucial to the operation of EFI. The velocity of the thin flyer driving by electrical explosion was obtained in order to quantitate the input of the impact process. Replacing the explosive pellet by a LiF window, the super-short shock pulse generated by a thin flyer (25 μm) was measured using the interface particle velocity method by a Photonic Doppler Velocimetry (PDV). The maximum particle velocity is 1.716 km/s in the experimental condition, meanwhile the shock duration is 4.5 ns. The impedance matching technique and the shock Hugoniot data were used to calculate the shock pressure and shock duration of the impact process, however the calculation results have distinct error comparing to the experimental ones. Utilizing the elastic-plastic model of flyer which considers the kinematic hardening plasticity effect, the simulations were carried out to predict the particle velocity and shock duration. The results show good consistency of the experimental and simulated particle velocities, which infers that the material model of flyer can describe its impact behaviours accurately.

Theoretical and experimental investigation of the equation of state of boron plasmas.

We report a theoretical equation of state (EOS) table for boron across a wide range of temperatures (5.1×10^{4}-5.2×10^{8} K) and densities (0.25-49 g/cm^{3}) and experimental shock Hugoniot data at unprecedented high pressures (5608±118 GPa). The calculations are performed with first-principles methods combining path-integral Monte Carlo (PIMC) at high temperatures and density-functional-theory molecular-dynamics (DFT-MD) methods at lower temperatures. PIMC and DFT-MD cross-validate each other by providing coherent EOS (difference <1.5 Hartree/boron in energy and <5% in pressure) at 5.1×10^{5} K. The Hugoniot measurement is conducted at the National Ignition Facility using a planar shock platform. The pressure-density relation found in our shock experiment is on top of the shock Hugoniot profile predicted with our first-principles EOS and a semiempirical EOS table (LEOS 50). We investigate the self-diffusivity and the effect of thermal and pressure-driven ionization on the EOS and shock compression behavior in high-pressure and -temperature conditions. We also study the sensitivity of a polar direct-drive exploding pusher platform to pressure variations based on applying pressure multipliers to LEOS 50 and by utilizing a new EOS model based on our ab initio simulations via one-dimensional radiation-hydrodynamic calculations. The results are valuable for future theoretical and experimental studies and engineering design in high-energy density research.

Extended Rice-Walsh equation of state for metals based on shock Hugoniot data for porous samples

The dimensionless material parameter R introduced by Wu and Jing into the Rice-Walsh equation of state (EOS) deduced from the shock Hugoniot data for porous Al and Cu was shown to be well approximated by a function of pressure alone. This data processing was extended to several metals including Mo, Ni, Co, Cr, Pb, Ta, W, and Fe using porous shock data basically published by Russian researchers. It was again found that the parameter R/p for all metals decays smoothly with shock pressure p and displays small experimental scatter in the high pressure region. It is then possible to construct the Rice-Walsh EOS using the empirically determined function R(p) and its known full-density shock Hugoniot and to reproduce porous shock Hugoniot for these metals. For most degrees of porosity, the agreement between the porous data and the calculated Hugoniots using the empirical function described was very good, while anomalous Hugoniots of Mo and Ni with extremely high porosity exhibit appreciable discrepancies. In this paper, the Wu-Jing parameter as a function of pressure together with thermal contribution was introduced to extend the Rice-Walsh EOS to explain Hugoniots with extremely high porosities. Extended EOS was also formulated with newly introduced thermal variables, which enabled the calculation of the cold compression curve for these metals. The present analysis suggested that the Rice-Walsh type EOS is, in most cases, a preferable choice for analysis with its simple form, pressure-dependent empirical Wu-Jing parameter and its compatibility with porous shock data. Extended Rice-Walsh EOS is only necessary for anomalous Hugoniots.

A complete equation of state for polyethylene based on Helmholtz free energy

Polyethylene (PE) is an important kind of plastic, which plays a significant role as the shell material of the fuel capsule, light weight structural element subjected to intense mechanical impact and explosion load. And it is well accepted that semi-empirical three-term equation of state (EOS) is one of the most widely used EOSs in practical work. Therefore, studies of semi-empirical three-term EOS of PE are significant for accurately predicting and analyzing the physical processes and experimental results under high pressure compression. A semi-empirical three-term complete EOS of PE based on the model of Helmholtz free energy is established in this work. According to the EOS model, the Helmholtz free energy is composed of cold energy, thermal contribution of atoms and thermal excitation of electrons. The cold energy is calculated by using the Mie potential. The optical frequency branch of atomic vibration and the thermal contribution of electrons are neglected in the calculation at temperatures below 104 K. The parameters of Helmholtz free energy are calculated by using the shock Hugoniot data and thermal parameters at ambient state. And then, the application pressure range and reliability of the semi-empirical three-term EOS of PE are evaluated. Shock Hugoniot, shock wave temperature and Grneisen coefficient of PE are deduced from the EOS. The results show that shock Hugoniot and shock wave temperature are consistent well with the experimental data and the first-principle calculation in a pressure range of 150 GPa. Because the specific volume of PE does not change obviously in the melting and chain dissociation process, the assumption of linear Hugoniot relation of PE is valid for calculating the cold energy parameters. The calculation results deviate from the experimental results at about 150 GPa while the compression lasts up to the chemical bond dissociation pressure of PE. In addition, the value of buck modulus and its derivative with respect to pressure at zero pressure and temperature depend strongly on Hugoniot parameters. Therefore, the parameter of Helmholtz free energy in this work is only valid for compression. In conclusion, the Helmholtz free energy model and parameters can well reproduce the experimental data and reasonably describe the thermodynamic state of PE at its dissociation pressure. Moreover, it should be pointed out that a more refined model of phase transition and thermal contribution of atoms and electrons should be considered when extrapolated to higher pressure.

Stability analysis of interfacial Richtmyer-Meshkov flow of explosion-driven copper interface

In this paper, a stability analysis is given to study the unstable mechanism of the Richtmyer-Meshkov flow of explosion-driven copper interface. The Richtmyer-Meshkov flow refers as an interfacial instability growth under shockwave incident loading. Numerical investigations are performed to check the applicability of the two-dimensional hydrocode, which is named AFE2D, and the physical models of detonation waves propagating in the high explosives, equations of state and the constitutive behaviors of solids in the analysis of Richtmyer-Meshkov flow problems. Here we theoretically analyze the two key issues of the unstable mechanism in Richtmyer-Meshkov flow in solids. The unstable mechanism includes temperature related melting mechanism and the plastic evolution related tensile fracture mechanism. In the analysis of the temperature related unstable mechanisms, the calculated temperature increase during the shockwave compression from the shock Hugoniot data in the shockwave physics is not enough to melt the material near the perturbed interface. On the other hand, the temperature increase from the translation of plastic work during perturbation growth which relats to the distribution of the cumulative effective plastic strain is also not enough to supply the thermal energy which is needed to melt the crystal lattice of solid, either. Therefore, the temperature related melting mechanism is not the main factor of the unstable growth of copper interface under explosion driven. In the analysis of the plastic tensile fracture related unstable mechanism, a scaling law between the maximum cumulative effective plastic strain and the scaled maximum amplitude of spikes is proposed to describe the relationship between the plastic deformation of material and the perturbation growth of interface. Combined with a critical plastic strain fracture criterion, the unstable condition of the scaled maximum amplitude of spikes is given. If the spikes grow sufficiently to meet the unstable condition, the interfacial growth will be unstable. Numerical simulations with varying initial configurations of perturbation and yield strength of materials show good agreement with the theoretical stability analysis. Finally, a criterion to judging whether the growth is stable is discussed in the form of competition between the temperature related unstable mechanism and the tensile fracture unstable mechanism.

Formulation of the Rice-Walsh equation of state based on Shock Hugoniot data for porous metals

The dimensionless material parameter R introduced by Wu and Jing into the Rice-Walsh equation of state (EOS) has been deduced from the LASL shock Hugoniot data for porous Al and Cu. It was found that the parameter R/p decays smoothly with shock pressure p and displays small experimental scatter in the high pressure region. This finding led to the conclusion that the parameter has only a weak temperature dependence and is well approximated by a function of pressure alone, and the Grüneisen parameter should be temperature dependent under compression. The thermodynamic formulation of the Rice-Walsh EOS for Al and Cu was realized using the empirically determined function R(p) for each material and their known shock Hugoniot. It was then possible to reproduce porous shock Hugoniot for these metals. For most degrees of porosity, agreement between the porous data and the calculated Hugoniots using the empirical function described was very good. However, slight discrepancies were seen for Hugoniots with very high porosity. Two new thermal variables were introduced after further analysis, which enabled the calculation of the cold compression curve for these metals. The Grüneisen parameters along full-density and porous Hugoniot curve were calculated using a thermodynamic identity connecting R and the Grüneisen parameter. It was shown that the Grüneisen parameter is strongly temperature dependent. The present analysis suggested that the Rice-Walsh type EOS is a preferable choice for the analysis with its simple form, pressure-dependent empirical Wu-Jing parameter, and its compatibility with porous shock data.

Behaviour of moist and saturated sand during shock and release

Relatively little is known about the changes that occur in the shock compaction and release of granular matter with varying levels of moisture. Here, we report a series of plate impact experiments giving shock Hugoniot and release data for a well characterized sand at dry, 10% moist, and saturated water contents. The results reveal that at low moisture content the shock impedance is slightly reduced, while the release remains predominantly inelastic. Close to saturation, much more substantial changes occur: the shock impedance stiffens substantially, the Hugoniot appears to split into two branches, and the release becomes almost completely elastic. We discuss mechanisms underpinning these changes in behavior.

Hugoniot curve calculation of nitromethane decomposition mixtures: A reactive force field molecular dynamics approach**Project supported by the National Natural Science Foundation of China (Grant No. 11374217) and the Shandong Provincial Natural Science Foundation, China (Grant No. ZR2014BQ008).

We investigate the Hugoniot curve, shock–particle velocity relations, and Chapman–Jouguet conditions of the hot dense system through molecular dynamics (MD) simulations. The detailed pathways from crystal nitromethane to reacted state by shock compression are simulated. The phase transition of N2 and CO mixture is found at about 10 GPa, and the main reason is that the dissociation of the C–O bond and the formation of C–C bond start at 10.0–11.0 GPa. The unreacted state simulations of nitromethane are consistent with shock Hugoniot data. The complete pathway from unreacted to reacted state is discussed. Through chemical species analysis, we find that the C–N bond breaking is the main event of the shock-induced nitromethane decomposition.

Shock Compression Modeling of Distended Mixtures

Development of material models to describe the thermodynamic shock states of distended mixtures is motivated by the need to understand how these materials respond over large compression ranges starting from low-pressure mechanical crush, extending into extreme thermodynamic states, and subsequent release to low-pressure. A material modeling approach is presented, which is comprised of a thermodynamically consistent equation-of-state (EOS) within a mixture-modeling framework. This modeling approach enables the investigator to describe the dynamic response of distended mixtures over a large compression range. The EOS model is applied to shock Hugoniot data on tungsten carbide, tantalum pentoxide, and calcite-water mixtures.

Verification of conventional equations of state for tantalum under quasi-isentropic compression

Shock Hugoniot data have been widely used to calibrate analytic equations of state (EOSs) of condensed matter at high pressures. However, the suitability of particular analytic EOSs under off-Hugoniot states has not been sufficiently verified using experimental data. We have conducted quasi-isentropic compression experiments (ICEs) of tantalum using the compact pulsed power generator CQ-4, and explored the relation of longitudinal stress versus volume of tantalum under quasi-isentropic compression using backward integration and characteristic inverse methods. By subtracting the deviatoric stress and additional pressure caused by irreversible plastic dissipation, the isentropic pressure can be extracted from the longitudinal stress. Several theoretical isentropes are deduced from analytic EOSs and compared with ICE results to validate the suitability of these analytic EOSs in isentropic compression states. The comparisons show that the Gruneisen EOS with Gruneisen Gamma proportional to volume is accurate, regardless whether the Hugoniot or isentrope is used as the reference line. The Vinet EOS yields better accuracy in isentropic compression states. Theoretical isentropes derived from Tillotson, PUFF, and Birch-Murnaghan EOSs well agree with the experimental isentrope in the range of 0–100 GPa, but deviate gradually with pressure increasing further.