Hugoniot Calculations Research Articles

An abinitio approach to the Hugoniot.

The Hugoniot is the equation of state of a shock-compressed material and is a key part of high-pressure physics. One way of calculating it is via the Hugoniostat that has significant computational advantages over direct calculation via non-equilibrium molecular dynamics. We introduce a number of improvements to the Hugoniostat, which significantly reduce the run time and the number of atoms required for converged results. Consequently, abinitio Hugoniot calculations are tractable. We illustrate the benefits through simple model potentials and with density functional theory calculations of argon.

Tunable noninteracting free-energy density functionals for high-energy-density physics applications

In this work, we introduce the concept of a tunable noninteracting free-energy density functional and present two examples realized: (i) via a simple one-parameter convex combination of two existing functionals and (ii) via the construction of a generalized gradient approximation (GGA) enhancement factor that contains one free parameter and is designed to satisfy a set of incorporated constraints. Functional (i), constructed as a combination of the local Thomas–Fermi and a pseudopotential-adapted GGA for the noninteracting free-energy, has already demonstrated its practical usability for establishing the high temperature end of the equation of state of deuterium [Phys. Rev. B 104, 144104 (2021)] and CHON resin [Phys. Rev. E 106, 045207 (2022)] for inertial confinement fusion applications. Hugoniot calculations for liquid deuterium are given as another example of how the application of computationally efficient orbital-free density functional theory (OF-DFT) can be utilized with the employment of the developed functionals. Once the functionals have been tuned such that the OF-DFT Hugoniot calculation matches the Kohn–Sham solution at some low-temperature point, agreement with the reference Kohn–Sham results for the rest of the high temperature Hugoniot path is very good with relative errors for compression and pressure on the order of 2% or less.

Shock Hugoniot Calculations of Newly Designed Thermoplastic Elastomers and Comparison with Classical Binder Estane.

Our purpose is to design excellent binder candidates used in polymer-bonded explosives (PBX) according to the calculated shock of Hugoniot. Here, we mainly examined the thermoplastic elastomer (TPE) binders commonly used in PBX formulations. Equilibrium molecular dynamics (MD) simulation and mixing rule methods were used to calculate the shock Hugoniot values of 180 newly designed TPEs. We focused on the influence of the polymerization degree, contents, and types of soft and hard segments composed of TPEs on the shock Hugoniot and compared them with the classic PBX binder, Estane. The results show that the hard segment has an effect on the Hugoniot curve, which gradually diminishes as the degree of polymerization increases. The underlying physical mechanism can be attributed to the presence of a large number of hydrogen bonds in hard segment domains. The shock Hugoniot of TPEs also depends on the type of soft segments. The volume compression rate of TPEs decreases with increasing content of hard segments under a given shock. By comparing with Estane, a TPE binder commonly used in PBX, we ultimately chose several new TPEs with the potential to serve as PBX binders in terms of shock performance.

Shock compression behavior of stainless steel 316L octet-truss lattice structures

Lattice structures offer desirable mechanical properties for applications of energy absorption and impact mitigation but limited research has been carried out on their shock compression behavior. In this work, the shock compression behavior of stainless steel 316L (SS316L) octet-truss lattice structures was investigated through experimental techniques and numerical simulations. Plate impact experiments with high-speed imaging were conducted at impact velocities of 270–390 m/s on lattice specimens with 5 × 5 × 10 unit cell geometries additively manufactured (AM) using direct metal laser sintering. High-speed imaging together with digital image correlation was used to extract full-field measurements and define a two-wave structure consisting of an elastic wave and planar compaction (shock) wave which propagated along the impact direction. A linear shock velocity versus particle velocity relation was found to approximate the measurements with a unit slope and a linear fit constant equal to the crushing speed. The shock velocity versus particle velocity relation, full-field measurements, and elastic limit together with the Eulerian form of the Rankine–Hugoniot jump conditions were used to find relations for the stress and internal energy behind the shock. Stress behind the shock increased with relative density and particle velocity, and specific internal energy converged to a single curve similar to that of bulk AM SS316L. Explicit finite element analysis using the Johnson–Cook constitutive model demonstrated similar shock behavior observed in experiments and a linear shock velocity versus particle velocity relation and corresponding Hugoniot calculations were found to be in agreement with experimental results. Numerical simulations confirmed negligible effects of exterior versus interior measurements and further validated the application of one-dimensional shock theory.

First-principles equation of state database for warm dense matter computation.

We put together a first-principles equation of state (FPEOS) database for matter at extreme conditions by combining results from path integral Monte Carlo and density functional molecular dynamics simulations of the elements H, He, B, C, N, O, Ne, Na, Mg, Al, and Si as well as the compounds LiF,B_{4}C,BN,CH_{4},CH_{2},C_{2}H_{3},CH,C_{2}H,MgO,andMgSiO_{3}. For all these materials, we provide the pressure and internal energy over a density-temperature range from ∼0.5 to 50 gcm^{-3} and from ∼10^{4} to 10^{9}K, which are based on ∼5000 different first-principles simulations. We compute isobars, adiabats, and shock Hugoniot curves in the regime of L- and K-shell ionization. Invoking the linear mixing approximation, we study the properties of mixtures at high density and temperature. We derive the Hugoniot curves for water and alumina as well as for carbon-oxygen, helium-neon, and CH-silicon mixtures. We predict the maximal shock compression ratios of H_{2}O, H_{2}O_{2}, Al_{2}O_{3},CO,andCO_{2} to be 4.61, 4.64, 4.64, 4.89, and 4.83, respectively. Finally we use the FPEOS database to determine the points of maximum shock compression for all available binary mixtures. We identify mixtures that reach higher shock compression ratios than their end members. We discuss trends common to all mixtures in pressure-temperature and particle-shock velocity spaces. In the Supplemental Material, we provide all FPEOS tables as well as computer codes for interpolation, Hugoniot calculations, and plots of various thermodynamic functions.

Effect of radiation on compressibility of hot dense sodium and iron plasma using improved screened hydrogenic model with l splitting

High pressure investigations of matter involve the study of strong shock wave dynamics within the materials which gives rise to many thermal effects leading to dissociation of molecules, ionization of atoms, and radiation emission, etc. The response of materials experiencing a strong shock can be determined by its shock Hugoniot calculations which are frequently applied in numerical and experimental studies in inertial confinement fusion, laboratory astrophysical plasma, etc. These studies involve high energy density plasmas in which the radiation plays an important role in determining the energy deposition and maximum compressibility achieved by the shock within material. In this study, we present an investigation for the effect of radiation pressure on the maximum compressibility of the material using shock Hugoniot calculations. In shock Hugoniot calculations, an equation of state (EOS) is developed in which electronic contributions for EOS calculations are taken from an improved screened hydrogenic model with –l splitting (I-SHML) [High Energy Density Physics (2018) 26 48] under local thermodynamic equilibrium (LTE) conditions. The thermal ionic part calculations are adopted from the state of the art Cowan model while the cold ionic contributions are adopted from the scaled binding energy model. The Shock Hugoniot calculations are carried out for sodium and iron plasmas and our calculated results show excellent agreement with published results obtained by using either sophisticated self-consistent models or the first principle study.

Interpreting Experimental Results from Shock Impacts on Single Crystal PETN in the Context of Continuum Models

AbstractIn this paper we interpret previous experimental measurements of impacts on single crystal PETN in the context of a simple continuum model. Hugoniot calculations using the simple model framework under various limiting assumptions envelope measured Hugoniot states reasonably well and suggest that non‐negligible deviatoric elastic strains may exist in the shock compressed state for weak impacts. An accompanying simple kinematic model for lattice deformation was used to estimate the expected shifts in X‐ray diffraction peaks for the (211) plane under impacts normal to (110) planes. The results suggest that negligible relaxation of the deviatoric lattice deformation occurs on the short (10–20 ns) time and wave propagation (40–80 m) scales of laser‐driven shock experiments, and that the corresponding elastic strains may be large. Because of the potentially large deviatoric elastic strains, the specific finite deformation strain measure used within thermomechanical single crystal models is of significant consequence, which is explored numerically by the comparison of isothermal purely elastic responses under uniaxial and simple shear deformations.

Hugoniot equation of state of Si-doped glow discharge polymer and scaling to other plastic ablators

Pressure, density, and temperature were measured along the principal Hugoniot of the Si-doped Glow Discharge Polymer used in Inertial Confinement Fusion (ICF) capsules up to 5 Mbar, covering conditions beyond the first shock in a full-scale Inertial Confinement Fusion (ICF) capsule. The experiments were performed using the GEKKOXII laser at the Institute of Laser Engineering at Osaka University in Japan. Results are in good agreement with predictions obtained from ab initio Hugoniot calculations, but softer than the quotidian equation of state average atom model. Ab initio calculations show that dissociation of carbon bonds need to be taken into account in order to explain Hugoniot compressibility.

Effect of a core-softened O–O interatomic interaction on the shock compression of fused silica

Isotropic soft-core potentials have attracted considerable attention due to their ability to reproduce thermodynamic, dynamic, and structural anomalies observed in tetrahedral network-forming compounds such as water and silica. The aim of the present work is to assess the relevance of effective core-softening pertinent to the oxygen-oxygen interaction in silica to the thermodynamics and phase change mechanisms that occur in shock compressed fused silica. We utilize the MD simulation method with a recently published numerical interatomic potential derived from an ab initio MD simulation of liquid silica via force-matching. The resulting potential indicates an effective shoulder-like core-softening of the oxygen-oxygen repulsion. To better understand the role of the core-softening we analyze two derivative force-matching potentials in which the soft-core is replaced with a repulsive core either in the three-body potential term or in all the potential terms. Our analysis is further augmented by a comparison with several popular empirical models for silica that lack an explicit core-softening. The first outstanding feature of shock compressed glass reproduced with the soft-core models but not with the other models is that the shock compression values at pressures above 20 GPa are larger than those observed under hydrostatic compression (an anomalous shock Hugoniot densification). Our calculations indicate the occurrence of a phase transformation along the shock Hugoniot that we link to the O–O repulsion core-softening. The phase transformation is associated with a Hugoniot temperature reversal similar to that observed experimentally. With the soft-core models, the phase change is an isostructural transformation between amorphous polymorphs with no associated melting event. We further examine the nature of the structural transformation by comparing it to the Hugoniot calculations for stishovite. For stishovite, the Hugoniot exhibits temperature reversal and associated phase transformation, which is a transition to a disordered phase (liquid or dense amorphous), regardless of whether or not the model accounts for core-softening. The onset pressures of the transformation predicted by different models show a wide scatter within 60-110 GPa; for potentials without core-softening, the onset pressure is much higher than 110 GPa. Our results show that the core-softening of the interaction in the oxygen subsystem of silica is the key mechanism for the structural transformation and thermodynamics in shock compressed silica. These results may provide an important contribution to a unified picture of anomalous response to shock compression observed in other network-forming oxides and single-component systems with core-softening of effective interactions.

Reaction Behavior of Polytetrafluoroethylene/Al Granular Composites Subjected to Planar Shock Wave

The shock-compression responses of PTFE (polytetrafluoroethylene)/Al granular composites subjected to planar shock waves of various pressures are investigated. A 57-mm diameter single-stage gas-gun and 50-mm diameter plane-wave lenses are employed to perform planar shock wave experiments. High frequency manganin piezoresistance stress gauges are used to monitor the stress (regarded as pressure in consideration of the high pressure state) at four Lagrangian positions of the PTFE/Al granular composites specimens. Planar shock wave experiments show characteristics of densification at measured input pressure of 0.5 GPa to 1.27 GPa using single-stage gas-gun and shock-induced reaction (SIR) indicated by growth of shock pressure and specific volume expansion at measured input pressure of 7.29 GPa to 12.25 GPa using plane-wave lenses. The pressure and relative volume states behind the shock wave front are calculated from the experimental recorded pressure profiles using Lagrangian analysis method, which are used to determine the reaction ratios under different shock pressures by comparing with partial reacted Hugoniot calculations. It was shown that the reaction ratios obtained in this research have good agreement with the thermochemical modeling calculations. The corresponding results indicate that the shock-induced reactions of PTFE/Al granular composites occur in the shock wave rising period and the reaction ratios are intimately related to the shock wave pressure.

First-principles simulations and shock Hugoniot calculations of warm dense neon

All-electron path integral Monte Carlo (PIMC) and density functional theory molecular dynamics (DFT-MD) simulations provide a consistent, first-principles investigation of warm dense neon plasmas in the density-temperature range of 1--15 g ${\mathrm{cm}}^{\ensuremath{-}3}$ and ${10}^{4}--{10}^{9}$ K. At high temperatures, DFT-MD becomes intractable because of too many partially occupied bands, while at lower temperatures, PIMC is intractable because of the free-particle approximation of fermion nodes. In combination, PIMC and DFT-MD pressures and internal energies provide a coherent equation of state with a region of overlap in which the two methods cross-validate each other. Pair-correlation functions at various temperatures and densities provide details of the plasma structure and the temperature-driven ionization process. The electronic density of states of neon shows that a gap persists for the highest density-temperature conditions studied here with DFT-MD. Finally, the computed shock Hugoniot curves show an increase in compression as the first and second shells are ionized.

Combining Kohn-Sham and orbital-free density-functional theory for Hugoniot calculations to extreme pressures.

The shock Hugoniot for lithium 6 deuteride ((6)LiD) was calculated via first principles using Kohn-Sham density-functional theory molecular dynamics (KSMD) for temperatures of 0.5-25 eV. The upper limit of 25 eV represents a practical limit where KSMD is no longer computationally feasible due to the number of electronic bands which are required to be populated. To push the Hugoniot calculations to higher temperatures we make use of orbital-free density-functional theory molecular dynamics (OFMD). Thomas-Fermi-Dirac-based OFMD gives a poor description of the electronic structure at low temperatures so the initial state is not well defined. We propose a method of bootstrapping the Hugoniot from OFMD to the Hugoniot from KSMD between 10 and 20 eV, where the two methods are in agreement. The combination of KSMD and OFMD allows construction of a first-principles Hugoniot from the initial state to 1000 eV. Theoretical shock-compression results are in good agreement with available experimental data and exhibit the appropriate high-temperature limits. We show that a unified KSMD-OFMD Hugoniot can be used to assess the quality of the existing equation-of-state (EOS) models and inform better EOS models based on justifiable physics.

Phase transformations of nanometer size carbon particles in shocked hydrocarbons and explosives

Estimates for the displacement of the phase equilibrium lines for small carbon particles containing from several hundred to several tens of thousands of atoms are made, and an error analysis of the uncertainties in these estimates is derived and evaluated using available experimental data. Hugoniot calculations for methane, benzene, polyethylene, and polybutene, based on a carbon particle surface energy adjusted equation of state, are in better agreement with shock pressure-volume and temperature data than those obtained with a bulk carbon equation of state. The results suggest that carbon particles, of order 103–104 atoms, can exist in the liquid state at lower temperatures than bulk carbon.

An equation of state code for hot dense matter, based on the QEOS description

An equation of state code for hot dense matter, based on the QEOS description [More et al. Phys. Fluids 31 (10), 3059 (1988)] is presented. Results of Hugoniot calculations and hydrodynamic simulations, based on this EOS, are shown. The purpose of this work is to make QEOS available in the form of a computer program.

Shock-induced inorganic reactions and condensed phase detonations

Existing hugoniot calculations of inorganic chemical reactions in powder mixtures are found to contain errors and ambiguities. Most of the problems were caused by improper identification of the thermodynamic reference state. The derivations have been revised using two different integration paths: i) constant volume, and ii) constant pressure. Sample calculations with a Ni-Al system show that the two schemes yield close results. The constant pressure scheme presents the possibility of self-sustained chemical reactions (detonations) occurring in condensed phase. An example of such a detonation in the Ni-Al system is numerically demonstrated with hydrocode calculations.

The shock wave equation of state of brucite Mg(OH)2

New equation of state (EOS) data for brucite Mg(OH)2 shocked between 12 and 60 GPa are reported. When combined with earlier data of Simakov et al. (1974), it is found that brucite EOS data between 12 and 97 GPa can be fit with a single linear Us‐ up relationship: Us = 4.76(0.11) + 1.35(0.05) up. The third order Birch‐Murnaghan equation parameters are: Kos = 51 ± 4 GPa and K′os = 5.0 ± 0.4. The lack of a Us‐up discontinuity indicates that no phase transformation with a significant volume change occurs to at least 97 GPa. However, thermodynamic and theoretical Hugoniot calculations suggest brucite may dehydrate with only a small volume change. A lower bound for this dehydration pressure under shock conditions is inferred to be 26 GPa. We report the first partial release states measured for this material. The data are in quantitative agreement with earlier shock recovery experiments (Lange and Ahrens, 1984). Volatilization upon release begins at pressures as low as 12 GPa, much less than predicted by the shock entropy method. Calculated phase boundaries using the present EOS data are consistent with experimental data and indicate that brucite is unlikely to be stable under lower mantle conditions. However, brucite data, in conjunction with data for silicates and oxides, can be used to infer the effect of H2O on lower mantle properties. At high pressure, bulk sound velocities calculated for MgO and Mg(OH)2 are very similar, indicating that the presence of hydrous assemblages in the lower mantle may not produce anomalous bulk seismic velocities. Comparison of densities in brucite and other high‐pressure phases under mantle conditions indicates that the water content of the lower mantle is between 0 and 3 wt %.

Shock melting and Hugoniot calculations for gold

Shock Hugoniot calculations are carried out for gold using both solid and liquid state theories. The result suggests that shock melting begins to take place at a pressure of 1.2 Mbar. At about 2.1 Mbar, the melting transition is complete. The shock temperature of gold in the liquid phase is substantially lower than that in the corresponding solid phase. The validity of the calculations is tested using equilibrium volume and bulk modulus data, deduced from diamond anvil cell experiments, Hugoniot data and the most recently reported shock temperature data at ∼6 Mbar.

Shock Hugoniot calculations and melting in Pb

Shock Hugoniot calculations are carried out for lead (Pb) using both solid and liquid state theories. The Hugoniots in solid and liquid cases are in mutual agreement within the experimental uncertainties. However, the shock temperatures are quite different as computed from solid and liquid state theories. This fact can perhaps be used to detect melting along the shock Hugoniot provided the shock temperatures are accurately measured.

The calculation of Hugoniots in ionic solids

The authors demonstrate the quantitative prediction of Hugoniots for an ionic crystal (NaCl) using the shell model with the best available potentials. The calculations show that these shock wave data can be predicted quite accurately by relatively simple use of established computer codes. The Hugoniot results, together with static phase-change data, also provide a very severe test of interatomic potentials. The two sets of empirical potentials ((i) Catlow, Diller and Norgett (1977), (ii) Sangster and Atwood (1978)) both perform much better than electron gas potentials. The same approach can be used to test potentials and to make predictions for the behaviour of solids for extreme conditions of temperature and pressure.

Thes−dtransition in compressed lanthanum

Calculations of the pressure-volume isotherms for fcc La have been carried out up to pressures of 2 Mbars and temperatures up to 3.4 eV, using the self-consistent linear-muffin-tin-orbital method. The isothermal bulk modulus shows an anomalous stiffening over the pressure range 320-560 kbars, due to termination of the $6s$ to $5d$ electronic transition. The effect is calculated to be most pronounced at zero temperature and is largely washed out by a temperature of 3.4 eV. These results lead to a temperature-dependent peak in the lattice Gr\"uneisen parameter, based on a simple Slater-model analysis. Hugoniot calculations using this Gr\"uneisen parameter are in good agreement with experiment and reproduce the anomalous stiffening observed in the shock compression data for La. While we also find both significant overlap of the rare-gas cores and melting to occur in the vicinity of the shock anomaly, the present calculations identify the principal cause of the anomaly observed in the La shock data to be the termination of the $6s\ensuremath{-}5d$ electronic transition in this material.