Shock Hugoniot Research Articles

Recreating Giants Impacts in the Laboratory: Shock Compression of MgSiO3 Bridgmanite to 14 Mbar

AbstractUnderstanding giant impacts requires accurate description of how extreme pressures and temperatures affect the physical properties of the constituent materials. Here, we report shock experiments on two polymorphs of MgSiO : enstatite and bridgmanite (perovskite) crystals. We obtain pressure‐density shock equation of state to 14 Mbar and more than 9 g/cm , a 40% increase in density from previous data on MgSiO . Density‐functional‐theory molecular dynamics (DFT‐MD) simulations provide predictions for the shock Hugoniot curves for bridgmanite and enstatite and suggest that the Grüneisen parameter decreases with increasing density. The good agreement between the simulations and the experimental data, including for the shock temperature along the enstatite Hugoniot reveals that DFT‐MD simulations reproduce well the behavior of dense fluid . We also reveal a high optical reflectance indicative of a metal‐like electrical conductivity which supports the hypothesis that magma oceans may contribute to planetary magnetic field generation.

The Riemann problem for the Eulerian droplet model with buoyancy and gravity forces

Under the premise of considering buoyancy and gravity forces, this paper makes a more detailed study on the one-dimensional Eulerian droplet model from a theoretical aspect focusing on exploring its Riemann problem. This model can be considered the pressureless Euler system with two external forces. The Riemann solutions are constructively obtained. They involve delta-shock solution and vacuum solution. Using generalized Rankine–Hugoniot relation, we derive in greater detail the position, propagation speed and strength of delta-shock wave under a suitable entropy condition. Furthermore, we investigate the limit behaviors of the Riemann solutions when external forces vanish partially or completely.

Rankine–Hugoniot Relations Including Pickup Ions

Abstract The Rankine–Hugoniot relations are consequences of the global, magnetohydrodynamic conservation laws of mass, momentum, energy, and magnetic flux at shocks in plasma. They provide us with the boundary conditions, although these are insufficient to quantify the energy partition in substantially non-Maxwellian plasmas in the presence of nonthermal, pickup ions (PUIs). Ion dynamics inside the shock front is essentially nonadiabatic and gyrophase-dependent. The shock jump conditions are typically evaluated using the upstream and downstream distributions in the regions where they already become gyrotropic. The relation between the gyrophase-dependent motion inside the shock front and the gyrophase-averaged moments of the ion distributions is established using numerical ion tracing across a model shock profile, which includes the major basic features of a collisionless shock. The upstream distribution of PUIs is taken to be a filled shell. The obtained moments of the PUI distribution function are used to construct the Rankine–Hugoniot relations for a gyrotropic plasma regime. The quality of numerical ion tracing is checked by comparing it with full particle simulations. The moments of the distribution in the isotropic region are estimated.

Numerical Modelling of Shock Initiation of a HNIW‐Based Explosive

AbstractTo investigate shock‐to‐detonation transition (SDT) characteristics of a HNIW‐based explosive, a rate law based on the Forest Fire model was calibrated by conducting five wedge tests. In the wedge tests, initial shock pressure at wedge‐shaped test explosive charges was varied from 5 to 11 GPa. In each test, the shock trajectory along the wedge‐shaped explosive charge was recorded with a streak camera. The streak records showed that sharp SDT occurred in the test explosive. The shock Hugoniot of the test explosive was determined from the free‐surface velocity of PMMA attenuators measured with a VISAR instrument and the early shock velocity in the wedges, determined from the shock trajectories. The Pop plot for the test explosive was obtained from run distances to detonation estimated based on the shock trajectories. The Forest Fire rate of the test explosive was calibrated from the experimental Hugoniot and the Pop plot. The wedge tests were numerically simulated with the calibrated rate by using a one‐dimensional Lagrangian hydrodynamic code. The calculated shock trajectories closely reproduced experimental observations. The rate was further applied to two‐dimensional numerical simulations of large‐scale gap tests (LSGTs). The estimated critical gap thickness by the LSGT simulations was 43.7 mm, which was in good agreement with experimental data of 46.67 mm. The results of these two types of numerical simulations suggested that the calibrated rate could be used to model most SDT‐related applications.

Wave speeds in single-crystal and polycrystalline copper

There is a need in the scientific community to improve predictive hydrodynamic codes. Empirical data are necessary to improve and constrain these models. This work considers the potential differences between the shock Hugoniots of single-crystal and polycrystalline copper. Polycrystalline materials are composed of grains of randomly oriented single crystals, so knowledge of these is necessary to build mesoscale representations of bulk materials. In this work, we measure the shock speeds of three different orientations of single-crystal copper, [100], [110], and [111], as well as polycrystalline copper. Symmetric planar impacts with a “top hat” target design provide timing information for the elastic and plastic waves. We use a comparison of ultrasonically measured ambient longitudinal sound speeds with the elastic wave speeds at pressure to constrain error bars for our analysis. Our results show that differences between the various orientations of single-crystal copper and polycrystalline copper are within the error of the experiment.

Riemann Problem for the Aw–Rascle Model of Traffic Flow with General Pressure

Riemann problem for the Aw–Rascle model of traffic flow with general pressure law is solved analytically. The Riemann solutions with three or four kinds of different structures involving the delta shock wave in three cases are obtained. For the delta shock wave, the generalized Rankine–Hugoniot relation and entropy condition are clarified. The theoretical analysis is confirmed by the numerical results finally.

Path integral Monte Carlo and density functional molecular dynamics simulations of warm dense MgSiO3

In order to provide a comprehensive theoretical description of MgSiO$_3$ at extreme conditions, we combine results from path integral Monte Carlo (PIMC) and density functional molecular dynamics simulations (DFT-MD) and generate a consistent equation of state for this material. We consider a wide range of temperature and density conditions from 10$^4$ to 10$^8$ K and from 0.321 to 64.2 g$\,$cm$^{-3}$ (0.1- to 20-fold the ambient density). We study how the L and K shell electrons are ionized with increasing temperature and pressure. We derive the shock Hugoniot curve and compare with experimental results. Our Hugoniot curve is in good agreement with the experiments, and we predict a broad compression maximum that is dominated by the K shell ionization of all three nuclei while the peak compression ratio of 4.70 is obtained when the Si and Mg nuclei are ionized. Finally we analyze the heat capacity and structural properties of the liquid.

Universal relationship of compression shocks in two-dimensional Yukawa systems.

Using molecular dynamical simulations, compressional shocks in two-dimensional (2D) dusty plasmas are quantitatively investigated under various conditions. A universal relationship between the thermal and the drift velocities after shocks is discovered in 2D Yukawa systems. Using the equation of state of 2D Yukawa liquids, and the obtained pressure from the Rankine-Hugoniot relation, an analytical relation between the thermal and the drift velocities is derived, which well agrees with the discovered universal relationship for various conditions.

Delta-Shock Solution to the Eulerian Droplet Model by Variable Substitution Method

Abstract By introducing a special kind of variable substitution, we skillfully solve the delta-shock and vacuum solutions to the one-dimensional Eulerian droplet model. The position, propagation speed, and strength of the delta shock wave are derived under the generalised Rankine–Hugoniot relation and entropy condition. Moreover, we show that the Riemann solution of the Eulerian droplet model converges to the corresponding the pressureless Euler system solution as the drag coefficient goes to zero.

Shock Hugoniot and Mie-Grüneisen EOS of TiAl alloy: A molecular dynamics approach

Molecular dynamics (MD) simulations coupling multi-scale shock technique (MSST) are performed to study the shock Hugoniot and Mie-Grüneisen equation of state (EOS) for TiAl alloy. The accuracy of the EAM potential for TiAl is examined by both melting temperature at a free pressure and Hugoniot curve. The material composition seriously affects the linear relation of shock wave velocity and particle velocity, Hugoniot curve, and internal energy. The Born-Mayer potential and Morse potential both can describe the cold curve at lower compressions, while the cold curve described via the Born-Mayer potential is larger than that of Morse potential at higher compressions. The shock melting temperature (Tm) is totally determined by Grüneisen coefficient γ, which shows an opposite trend as compared with the increasing Tm at the whole compressions. Mie-Grüneisen EOS presents a concave surface in pressure – specific volume – internal energy (P-V-E) space, which is independent of the reference curves utilized the cold curve and Hugoniot curve.

Erratum: “Equation-of-state, critical constants, and thermodynamic properties of lithium at high energy density” [Phys. Plasmas 26, 122702 (2019)

High-density lithium plasmas are expected to be generated in the Inertial Confinement Fusion (ICF) reactor chamber due to energy deposition of the prompt X-rays and ion debris in the first wall. These dense plasmas are encountered in many other applications as well. The design and optimization of such lithium-based applications requires information about the thermodynamic properties of Li fluid over a wide range of parameters including the high-energy-density regime. A model that takes into account all essential features of strongly-coupled plasmas of metal vapors is developed and used to predict the equation-of-state (EOS), critical constants, thermodynamic functions, and principal shock Hugoniot of hot dense lithium fluid. Predictions and computational results from the present model are analyzed and presented and are available in electronic form suitable for use to interested researchers and scientists.

Shock and Spall in the Low-alloy Steel AF9628

The shock and spall behavior of tempered AF9628, an ultra-high strength (UHSS) martensitic low-alloy carbon steel, is investigated using gun-driven, symmetric, planar impacts. The study utilized laser velocimetry to obtain transmitted wave velocity histories that were analyzed to obtain the Hugoniot Elastic Limit (HEL), the shock Hugoniot, and the spall strength. The work was limited to the stress range below 13 GPa (below the e phase transition), and an HEL of 2.35 GPa and a spall strength of 6.8 GPa are reported. The HEL and spall strength of AF9628 are both above the typical values for inexpensive low-alloy steels, and only slightly below those for a much more expensive Aermet steel. The dynamic compressive yield strength (as calculated from the HEL) is only slightly higher than the quasi-static compressive yield strength when compared using a similar yield assignment convention, indicating a small, but positive, value of strain-rate sensitivity. The Hugoniot for AF9628, US = 0.67uP + 5.01 km/s, is in good agreement with other low-alloy martensitic steels for which data is available, and distinctly different than the Hugoniot for ferritic steels. The Hugoniot extrapolates to a bulk sound speed significantly higher than what is measured at ambient conditions, an incongruity that is unusual among metals and that the authors attribute to the unsteady nature of the shock waves generated in this work. Finally, it is suggested that there may be a relationship between dynamic compressive yield strength (calculated from the HEL) and dynamic tensile failure strength (the spall strength).

Magnesium oxide at extreme temperatures and pressures studied with first-principles simulations.

We combine two first-principles computer simulation techniques, path integral Monte Carlo and density functional theory molecular dynamics, to determine the equation of state of magnesium oxide in the regime of warm dense matter, with densities ranging from 0.35 to 71 g cm-3 and temperatures ranging from 10 000 K to 5 × 108 K. These conditions are relevant for the interiors of giant planets and stars as well as for shock wave compression measurements and inertial confinement fusion experiments. We study the electronic structure of MgO and the ionization mechanisms as a function of density and temperature. We show that the L-shell orbitals of magnesium and oxygen hybridize at high density. This results in a gradual ionization of the L-shell with increasing density and temperature. In this regard, MgO behaves differently from pure oxygen, which is reflected in the shape of the MgO principal shock Hugoniot curve. The curve of oxygen shows two compression maxima, while that of MgO shows only one. We predict a maximum compression ratio of 4.66 to occur for a temperature of 6.73 × 107 K. Finally, we study how multiple shocks and ramp waves can be used to cover a large range of densities and temperatures.

Equation-of-state, critical constants, and thermodynamic properties of lithium at high energy density

High-density lithium plasmas are expected to be generated in the Inertial Confinement Fusion reactor chamber due to energy deposition of the prompt X-rays and ion debris in the first wall. These dense plasmas are encountered in many other applications as well. The design and optimization of such lithium-based applications require information about the thermodynamic properties of Li fluid over a wide range of parameters including the high-energy-density regime. A model that takes into account all essential features of strongly coupled plasmas of metal vapors is developed and used to predict the equation-of-state, critical constants, thermodynamic functions, and principal shock Hugoniot of hot dense lithium fluid. Predictions and computational results from the present model are analyzed and presented and are available in electronic form suitable for use to interested researchers and scientists.

Riemann Problem and Wave Interactions for a Class of Strictly Hyperbolic Systems of Conservation Laws

A class of strictly hyperbolic systems of conservation laws are proposed and studied. Firstly, the Riemann problem with initial data of two piecewise constant states is constructively solved. The solutions involving contact discontinuities and delta shock waves are obtained. The generalized Rankine–Hugoniot relation and entropy condition for the delta shock wave are clarified and the existence and uniqueness of the delta-shock solution is proved. Furthermore, the global structure of solutions with five different configurations is constructed via investigating the interactions of delta shock waves and contact discontinuities. Finally, we present a typical example to illustrate the application of the system introduced.

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.

Generalized plane delta shock waves for the n-dimensional zero-pressure gas dynamics with energy conservation law

By virtue of the generalized plane wave solution, we study a type of generalized plane delta shock wave for the n-dimensional zero-pressure gas dynamics governed by the conservation of mass, momentum and energy. It is found that a special kind of generalized plane delta shock wave on which both state variables simultaneously contain the Dirac delta functions appears in Riemann solutions, which is significantly different from the customary ones on which only one state variable contains the Dirac delta function. The generalized Rankine-Hugoniot relation of the generalized plane delta shock wave is derived. Under a suitable entropy condition, we further solve a kind of n-dimensional Riemann problem with Randon measure as initial data, and four different explicit configurations of solutions are constructively established. Finally, the overtaking of two plane delta shock waves is analyzed.

Rankine–Hugoniot Shock Conditions for Space and Astrophysical Plasmas Described by Kappa Distributions

Abstract This paper provides the set of Rankine–Hugoniot (R–H) jump conditions for shocks in space and astrophysical plasmas described by kappa, distributions. The characteristic result is the development of a new R–H condition that transforms the values of kappa upstream and downstream the shock. The kappa index parameterizes and labels kappa distributions, and it is necessary for characterizing the thermodynamics of space plasmas. This first approach is restricted to non-magnetized plasmas, and the whole achievement is derived by following first principles of statistical mechanics and thermodynamics. The results show that, depending on the shock strength, the kappa indices across the shock may decrease or increase, indicating cases of shock acceleration or deceleration, respectively.

Construction of the equations of state for polycrystalline solids for the purpose of the numerical solution of problems of continuous medium mechanics

The equations of state for polycrystalline solids which are in region of compression are developed on experimentally defined shock Hugoniot adiabat and dependence of specific heat on density and temperature. These equations are convenient for the numerical solution of problems of continuous medium mechanics. The numerical method for definition of specific cold energy and the Gruneizen function at T = 0 K depending on compression ratio is proposed. Results of comparison for temperatures behind shock wave in sapphire by means of the developed equations of state and other methods are given.

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.