Shock-wave Equation Of State Research Articles

Shock‐induced melting of MgSiO3 perovskite and implications for melts in Earth's lowermost mantle

New shock wave equation of state (EOS) data for enstatite and MgSiO3 glass constrain the density change upon melting of Mg‐silicate perovskite up to 200 GPa. The melt becomes denser than perovskite near the base of Earth's lower mantle. This inference is confirmed by shock temperature data suggesting a negative pressure‐temperature slope along the melting curve at high pressure. Although melting of Earth's mantle involves multiple phases and chemical components, this implies that the partial melts invoked to explain anomalous seismic velocities in the lowermost mantle may be dynamically stable.

Thermoelastic model of minerals: application to Al 2 O 3

A thermoelastic model for calculating the high-pressure and high-temperature properties of isotropic solids is presented by extending the formalism by Thomsen and combining the resulting one with the Vinet model for static lattice and the Debye model for lattice vibration. Applying it to polycrystalline corundum, we have shown that the calculated values of entropy and heat capacity at constant pressure are in agreement with literature values to 2325 K at zero pressure and that the calculated values of thermal expansivity agree reasonably with experimental data to 1100 K at zero pressure. The model reproduces experimental data of sound velocities vp and vs of compressional and shear waves to 1825 K at zero pressure and those to 62 GPa at room temperature, and it reproduces also experimental shock-wave equation of state to 150 GPa. The velocity correlation (∂ln vs/∂ln vp)S was found to have weak pressure and temperature dependences and the results under lower mantle conditions are compared with those of magnesian and calcium silicate perovskites and magnesiowustite, and the PREM values of the Earth's lower mantle.

General shock wave equation of state for solids

A theoretical shock wave equation of state has been derived based on the Morse potential function, known from classical spectroscopy. Using this potential function and an interstitial-electron model, a linear relation between the shock wave velocity and the mass displacement velocity can be found, analogous to the common empirical relation, but with parameters derived from easily accessible nonshock-wave data. The equation is applicable to a wide range of inorganic materials, as will be demonstrated in two forthcoming articles on metals and ceramics. Examples are shown for silver and copper. Calculations are within 5% average error between experimental data and the model.

Shock wave profile study of tuff from the Nevada Test Site

Shock wave equation of state and release behavior of tuff from the Nevada Test Site were measured in the range 8 to 107 GPa by recording shock wave profiles produced by impact of projectiles accelerated by a two‐stage light‐gas gun. Shock wave profiles were measured with a fast optical interferometer. Hugoniot, sound speed, and effective Gruneisen parameter data were determined from the profiles. The specimen geometry allows direct observation of the shape of the leading edge of the shock, in which complex compressive dynamics caused by pore crushing or phase transitions can manifest themselves. The tuff specimens were tested in both a water‐saturated and semidry state. Hydration of the rock increases both the shock and sound speeds for a given compression. The Us‐Up relation for our water‐saturated tuff is slightly nonlinear and can be approximated as Us = 2.00 + 1.42 Up for 0 < Up < 6 km/s, and Us = 2.29 + 1.29Up for 0 < Up < 3 km/s. For semidry tuff at pressures less than 18 GPa, Us = 1.70 + 1.28Up.

Shock wave equation of state of powder material

A model is proposed to predict the following quantities for powder materials compacted by shock waves: the pressure, the specific volume, the internal energy behind the shock wave, and the shock-wave velocity Us. They are calculated as a function of flyerplate velocity up and initial powder specific volume V00. The model is tested on Cu, Al2024, and Fe. Calculated Us vs up curves agree well with experiments provided V00 is smaller than about two times the solid specific volume. The model can be used to predict shock-wave state points of powder or solid material with a lower or higher initial temperature than room temperature.

Shock wave equation of state of muscovite

Shock wave data to provide an equation of state of muscovite (initial density: 2.835 g/cm3) were determined up to a pressure of 141 GPa. The shock velocity (Us) versus particle velocity (Up) data are fit with a single linear relationship: Us=4.62(±0.12) +1.27(±0.04)Up (km/s). Third‐order Birch‐Murnaghan equation of state parameters (isentropic bulk modulus and isentropic pressure derivative of bulk modulus) are Kos=52±4GPa and K'os=3.2±0.3. The pressure‐temperature relation along the Hugoniot suggests that muscovite may dehydrate to KAlSi3O8 (hollandite), corundum, and water, with a small volume change, above 80 GPa. Thermodynamic calculations of the equilibrium pressure for the dehydration yields a significantly lower value. Observed unloading paths from shock pressures up to about 80 GPa are steeper in a density‐pressure plane than the Hugoniot and become shallower with increasing shock pressure above that pressure. The changing slope may indicate that devolatilization occurs during unloading above 80 GPa. The present equation of state data for muscovite are compared with results of previously reported recovery experiments.

Shock wave equation of state of serpentine to 150 GPa: Implications for the occurrence of water in the Earth's lower mantle

The shock equation of state of serpentine has been determined to 150 GPa. Four distinct regions occur along the Hugoniot: a low‐pressure phase, a mixed phase region, a high‐pressure phase, and a very high pressure phase. The low‐pressure phase (LPP) exists under shock pressures to about 40 GPa. This material exhibits shock properties that are partially consistent with those of low pressure serpentine, but steep release paths and a low value of K′ = 2.77 suggest transformation to another, possibly amorphous, assemblage. Thermodynamic calculations indicate that under equilibrium conditions, serpentine would decompose to oxides plus water at conditions below 10 GPa along the Hugoniot. A mixed phase region begins at 40 GPa with complete transition to a high‐pressure phase occurring by about 55 GPa. The high‐pressure phase (HPP) occurs at shock pressures between 55 GPa and 125 GPa. Model Hugoniots based on perovskite plus periclase plus water and brucite plus periclase plus stishovite reproduce the serpentine HPP Hugoniot within experimental error, so definitive identification of the HPP as a distinct hydrous mineral phase or as a free water containing mixture is not possible. Above 125 GPa a transition to a very compressible phase, possibly a hydrous partial melt, occurs. The serpentine HPP Hugoniot is about 15–20 % less dense than the Earth's lower mantle. Models of the lower mantle based on shock equations of state for olivine, pyroxene, and serpentine indicate that for an atomic Mg/(Mg+Fe) ratio of 0.80, the presence of 2 wt % H2O is consistent with seismically determined lower mantle density estimates. Greater amounts of H2O can be accommodated if accompanied by an increase in Fe content. Calculated Hugoniot sound speeds of the serpentine HPP, although poorly constrained, are broadly consistent with lower mantle sound speeds. Thus the high‐pressure density and sound speed of an H2O‐rich magnesium silicate determined from shock equation of state experiments indicate that the observed seismic properties of the lower mantle allow the existence of several weight percent of water in the lower mantle.

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 wave equation of state and finite strain theory : Jeanloz, R J Geophys ResV94, NB5, May 1989, P5873–5886

Shock wave equation of state and finite strain theory : Jeanloz, R J Geophys ResV94, NB5, May 1989, P5873–5886

A thermochemical model for shock-induced reactions (heat detonations) in solids

Recent advances in studies of shock-induced chemistry in reactive solids have led to the recognition of a new class of energetic materials which are unique in their response to shock waves. Experimental work has shown that chemical energy can be released on a time scale shorter than shock-transit times in laboratory samples. However, for many compositions, the reaction products remain in the condensed state upon release from high pressure, and no sudden expansion takes place. Nevertheless, if such a reaction is sufficiently rapid, it can be modeled as a type of detonation, termed ‘‘heat detonation’’ in the present paper. It is shown that unlike an explosive detonation, an unsupported heat detonation will decay to zero unless certain conditions are met. An example of such a reaction is Fe2 O3 +2Al+shock→Al2 O3 +2Fe (the standard thermite reaction). A shock-wave equation of state is determined from a mixture theory for reacted and unreacted porous thermite. The calculated shock temperatures are compared to experimentally measured shock temperatures, demonstrating that a shock-induced reaction takes place. Interpretation of the measured temperature history in the context of the thermochemical model implies that the principal rate-controlling kinetic mechanism is dynamic mixing at the shock front. Despite the similarity in thermochemical modeling of heat detonations to explosive detonations, the two processes are qualitatively very different in reaction mechanism as well as in the form the energy takes upon release, with explosives producing mostly work and heat detonations producing mostly heat.

Shock wave equation of state and finite strain theory

The linear shock‐velocity (Us), particle‐velocity (up) relation, which is known to be exceptionally successful in describing a wide variety of Hugoniot equation of state measurements, is shown to be virtually indistinguishable from the Birch‐Murnaghan (third‐order Eulerian finite strain) equation of state. For typical values of zero‐pressure parameters, specifically for the pressure derivative of the adiabatic bulk modulus K′0S ≃ 3 to 6, the Grüneisen parameter γ0 ≃ 1.5 (±1.0), and the logarithmic volume derivative of the Grüneisen parameter q ≃ 1.0 (±1.0), the Eulerian finite strain formulation yields the appropriate compressional moduli at infinitesimal strains, and it closely reproduces the Hugoniot at large compressions when compared with the linear Us‐up relation. Thus shock wave measurements provide strong empirical support for the success of the Eulerian finite strain equation of state in describing the compression of condensed matter to high pressures.

First-principles calculation of the shock-wave equation of state of isotopic lithium hydrides

The equations of state (EOS) of solid LiH in the B1 (NaCl-type) and B2 (CsCl-type) structures have been calculated within the local-density approximation (LDA), using the augmented-plane-wave method with Ceperley-Alder's form for the exchange-correlation energy. Using the EOS obtained by the LDA, the volume dependencies of the Debye temperatures for both phases have been determined. The calculated Debye temperatures in the B1 phase at zero pressure and zero temperature are 1246 K for $^{7}\mathrm{LiH}$ and 972 K for $^{7}\mathrm{LiD}$ in agreement with the experimental values in the T=0 K limit at normal pressure which are 1190\ifmmode\pm\else\textpm\fi{}80 K for $^{7}\mathrm{LiH}$ and 1030\ifmmode\pm\else\textpm\fi{}50 K for $^{7}\mathrm{LiD}$, respectively. The calculated lattice constants for isotopic LiH at zero pressure are in agreement with the experimental values at T=83 K by 1.9%. With these results the shock-wave equations of state (the Hugoniot) for both phases have been calculated. The Hugoniots for the B1 phase are in agreement with the experiment by Marsh.

Experimental and theoretical equation of state of cubic boron nitride

The phase diagram of boron nitride (BN) is similar to that of carbon, incorporating phases at high temperatures and pressures whose structures and physical properties resemble diamond1–9. Cubic zinc-blende-structured BN is especially important because it is extremely hard—second only to diamond. Here we report the first measurement of the 300-K equation of state of this material to ultrahigh pressures (115 GPa), and obtain a zero-pressure bulk modulus of 369 ± 14 GPa. A theoretical equation of state derived from first-principles pseudopotential calculations yields a 300-K isotherm that agrees with our experimental results to better than 2.5% in volume and 2.0% in bulk modulus. The high-pressure Hugoniot (shock-wave equation of state) calculated from our equation of state for BN is in good agreement with existing shock-wave data. Our study illustrates the reliability of current experimental techniques (such as the ruby-fluorescence calibra-tion) and theoretical methods (pseudopotentials) for characterizing the behaviour of superhard, incompressible materials under high pressure.

Shock wave equation of state of enstatite

Shock compression data are reported for hot‐pressed Bamble bronzite (En86) loaded to pressures between 104 and 161 GPa. When compared to earlier shock wave data on En90 at lower pressures and to static compression data, our data require the presence of a phase change. In P‐ρ space the data yield two distinct trajectories, which cannot be explained by experimental error. The higher‐density data, corrected for porosity and a small amount of metallic iron impurity, agree with a theoretical En86 high‐pressure phase Hugoniot calculated from static compression equation of state data for perovskite (pv) structure silicates when experimental errors and uncertainties in the equation of state parameters are considered. All the En86 data can be described by a calculated Hugoniot if the first pressure derivative of the MgSiO3 (pv) bulk modulus is taken as 4.5 ± 1.0. Combining the present preferred data with recent shock wave data for single‐crystal forsterite, we find that En86 is slightly more dense than Fo86 at pressures above 110 GPa. Comparison of the forsterite and enstatite data with the Preliminary Reference Earth Model (PREM) lower mantle densities, with corrections applied for the higher shock temperatures relative to lower mantle temperatures, shows that PREM densities are satisfied by olivine or pyroxene stoichiometries with Mg mole fractions from 0.82 to 0.90. These values are lower than estimates of 0.90 to 0.95 developed from extrapolating static compression data to lower mantle conditions.

Shock wave equations of state using mixed‐phase regime data

A method is given that uses Hugoniot data in the mixed‐phase regime to constrain further equation of state (EOS) parameters of low‐ and high‐pressure phases of materials under‐going phase transformations on shock loading. We compute the relative proportion of low‐ and high‐pressure phases present in the mixed‐phase region and apply additional tests to the EOS parameters of the separate low‐ and high‐pressure phases by invoking two simple requirements: the fraction of high‐pressure phase (1) must increase with increasing shock pressure, and (2) must approach one at the high‐pressure end of the mixed‐phase regime. We apply our analysis to previously published data for potassium thioferrite, KFeS2, and pyrrhotite, Fe0.9S. We find that including the mixed‐phase regime data in the KFeS2 analysis requires no change in the published high‐pressure EOS parameters. For Fe0.9S we must modify the high‐pressure phase EOS parameters to account for both the mixed‐phase and high‐pressure phase Hugoniot data. Our values of zero‐pressure density, bulk modulus and first pressure derivative of the bulk modulus of the high‐pressure phase of Fe0.9S are 5.3 Mg/m3, 106 GPa, and 4.9, respectively.

Reduction of shock-wave equations of state to isothermal equations of state

A method is presented whereby shock-wave data can be reduced from the experimentally determined Hugoniot equations of state to isothermal equations of state. The method is similar to that used by Takeuchi and Kanamori (1966) but is mathematically simpler and more readily adapted to numerical computation. Unlike other treatments of shock-wave data, no extrapolations to zero temperature are required.