Vinet Equation Research Articles

Multidimensional Rietveld refinement of high-pressure neutron diffraction data of PbNCN.

High-pressure neutron powder diffraction data from PbNCN were collected on the high-pressure diffraction beamline SNAP located at the Spallation Neutron Source (SNS) of Oak Ridge National Laboratory (Tennessee, USA). The diffraction data were analyzed using the novel method of multidimensional (two dimensions for now, potentially more in the future) Rietveld refinement and, for comparison, employing the conventional Rietveld method. To achieve two-dimensional analysis, a detailed description of the SNAP instrument characteristics was created, serving as an instrument parameter file, and then yielding both cell and spatial parameters as refined under pressure for the first time for solid-state cyanamides/carbodi-imides. The bulk modulus B 0 = 25.1 (15) GPa and its derivative B'0 = 11.1 (8) were extracted for PbNCN following the Vinet equation of state. Surprisingly, an internal transition was observed beyond 2.0 (2) GPa, resulting from switching the bond multiplicities (and bending direction) of the NCN2- complex anion. The results were corroborated using electronic structure calculation from first principles, highlighting both local structural and chemical bonding details.

Effect of mixed carbon phase state on detonation parameters and reaction zone of explosives under extreme condition

A precise description of the thermodynamic states of gaseous and solid detonation products is essential when using thermodynamic calculations to determine the detonation performance and destructive power of explosives. For high oxygen-lean explosives (the oxygen contained in explosives is insufficient to completely oxidize combustible elements and excess solid carbon will be generated in the detonation products), the phase state of solid carbon product affects the Chapman–Jouguet (CJ) detonation performance parameters, reaction zone, and energy release process. However, the recovery of detonation products demonstrates that the actual detonation carbon product is primarily a mixed state of diamond/graphite stack, as opposed to the existing thermodynamic codes, which essentially treat detonation carbon as single-phase carbon. To understand the thermodynamic effect of the mixed carbon phase state on the non-ideal detonation behavior, in this work, the matching relationship among the VINET equation of state parameters, thermodynamic potential parameters of the solid products of the equivalent system and the phase mixed system was constructed by using the nonlinear fitting method. The relationship between the carbon phase composition at the CJ point and the explosive composition structure was researched. Investigations were conducted into how the mixed carbon phase affected the volume and content of gas products as well as the composition at CJ points. Diamond formation in products is good for enhancing explosive's working capacity. Based on mixed-state potential parameters, the correlation mechanism between the mixed carbon phase and the chemical reaction zone was investigated, and it was found that intramolecular carbon/intermolecular carbon and more detonation graphite/diamond products all would lead to the extension of the reaction zone.

Phase diagram and thermo-elastic properties of Fe-S compounds up to 15 GPa: Thermodynamic constraints on the core of medium-sized telluric planets

The Fe-FeS binary is largely seen as the archetypal system to model the properties of the core of small to medium-sized telluric planetary bodies. Noteworthy, while both at the low pressures characteristic of the Moon, and at the very high pressure pertinent to the Earth or Venus, the Fe-FeS is a simple binary eutectic, in the intermediate range relevant for planets such as such as Mercury or Mars, or satellite such as Ganymede, Io, and Europa, the Fe-S phase diagram is quite complex, with intermediate compounds of narrow stability field that incongruently melt.Properties of Fe-FeS compounds have been here studied using in situ X-ray diffraction in the pressure range of 11–15 GPa, between room temperature to solidus (around ∼1100 K). Results show that Fe + FeS mixture transforms to Fe3S2 at 850 K and 12.1 GPa, adopting an orthorhombic crystal structure. Fitting the unit-cell volumes at 940 K to 2nd order Birch-Murnaghan and Vinet equation of state yields to V0 = 372.2 ± 0.8 or 367.3 ± 0.6 Å3 and K0 = 130 ± 4 or 185 ± 4 GPa, respectively, with K′ fixed to 4. The thermal expansion around 14 GPa has been estimated to be 26.6 × 10−5K−1 based on the temperature evolution of the unit-cell volume from 850 K to 1100 K. Moreover, the lattice parameters of FeS were obtained in the range 11–15 GPa between 470 K and 1100 K, and the compressional behavior was studied.To model the properties of the solid portion of the cores of telluric planets or exoplanets in the 12–21 GPa range, Fe3S2 should be used as the end-member together with Fe or FeS depending on whether the S content is below or above 27 wt%. While Fe3S2 is a potential crystallizing product of S-rich telluric bodies, it is not expected to form in the core of Europa, Io, or Ganymede as too small, nor in that of Mercury as most likely too poor in S. When found in a meteorite, the Fe3S2 phase could be used to infer the size and temperature of the parent body.

Equation of state of KI up to 150 GPa

ABSTRACT The equation of state of potassium iodide, KI, has been determined up to 156 GPa at 298 K by angular-dispersive X-ray diffraction in diamond-anvil-cell. Helium pressure transmitting medium was used to minimize non-hydrostatic stress on the sample. The B2 phase is stable at least between 2 and 156 GPa. Pressure–volume data were fitted using a Rydberg–Vinet equation of state. The volume, isothermal bulk modulus and its pressure derivative of B2-KI at ambient pressure are , GPa and . It allows using KI as a pressure calibrant in high pressure experiments.

Sound Velocity and Equation of State of Ballistic Gelatin by Brillouin Scattering.

Brillouin scattering spectroscopy with diamond anvil cells was used by measuring the pressure dependence of the sound-relevant polymer material, glass-forming liquid, and H2O (water and ice VII) velocities of the material from ambient pressure to 12 GPa at room temperature. Measurements of 20%, 10%, and 4% gelatin solutions were performed. For comparison purposes, we also measured the pressure dependence of the sound velocity of animal tissue up to 10 GPa. We analyzed the Brillouin data using the Tait and Vinet equations of state. We discussed the possible influence of frequency dispersion on bulk modulus at low pressure. We compared the elastic moduli obtained for gelatin to those of several other polymers.

Single crystal elasticity and equation of state of tantalum up to 54 GPa

We measured single crystal elasticity of Ta under high pressures up to 54 GPa at room temperature using inelastic x-ray scattering at room temperature. Simultaneously, we measured the density of Ta using x-ray diffraction. Combining the bulk modulus and density, we obtain an equation of state of Ta as a primary scale. The Vinet equation was fitted to the pressure–volume data and we found consistency with previous work including experimental static and shock compressions and theoretical calculation. We proposed a parameter set for the Vinet equation [K0 = 191.1(3) GPa, K′0 = 4.006(2)] which is consistent with the pressure based on extrapolated velocities within 2% up to ∼80 GPa. Furthermore, we found the present scale to be consistent with a recent ruby scale (Ruby2020) up to ∼50 GPa.

Thermodynamic behavior of Na-majorite and knorringite-majorite garnet systems

The behavior of thermal parameters pressure (P) and thermoelastic parameters i.e. Isothermal bulk modulus (K_T), its pressure derivative (K'T) and Thermal expansivity (α) has been analyzed for Na-majorite and knorringite-majorite systems at different compressions down to a maximum value of 0.3 for temperature ranges from 300 K to 1500 K. The pressure dependent compressibility of these garnets have been studied. The comprehensive study of thermodynamic and thermoelastic behavior at different isotherms and isobars respectively has been achieved by employing both Vinet equation of state and Anderson-Isaak equation. The study of thermal expansivity (α) using Anderson-Isaak and Stacy-Davis approach shows a similar trend of MgO system. Based on the validation of Kumar's approach, the lattice parameter of garnet system is computed using Kumar and modified Kumar's approach at high temperature and is found to be very less deviated from the value of lattice parameter at room temperature.

Equations of state of iron and nickel to the pressure at the center of the Earth

Synchrotron radiation x-ray diffraction investigations of iron (Fe) and nickel (Ni) are conducted at pressures up to 354 and 368 GPa, respectively, and the equations of state (EOSs) at 298 K for the two elements are obtained for data extending to pressures as high as those at the center of the Earth, using the latest Pt-EOS pressure scale. From a least-squares fit to the Vinet equation using the observed pressure–volume data, the isothermal bulk modulus K0 and its pressure derivative K0′ are estimated to be 159.27(99) GPa and 5.86(4) for hcp-Fe, and 173.5(1.4) GPa and 5.55(5) for Ni. By comparing the present EOSs and extrapolated EOSs reported in the literature for Fe and Ni, the volumes of Fe and Ni at 365 GPa are found to be 2.3% and 1.5% larger than those estimated from extrapolated EOSs in previous studies, respectively. It is concluded that these discrepancies are due to the pressure scale. The present results suggest that the densities of Fe and Ni at a pressure of 365 GPa corresponding to the center of the Earth are 2.3% and 1.5%, respectively, lower than previously thought.

Equation of State Determination for Rhenium Using First-Principles Molecular Dynamics Calculations and High-Pressure Experiments

The room-temperature bulk modulus of rhenium (Re) was measured in the pressure range 0 to 115 GPa using a laser-annealing diamond anvil cell and the synchrotron X-ray diffraction method. Thermal properties of Re were investigated up to 4000 K based on first-principles molecular dynamics calculations, and the equation of state for Re was determined using experimental and calculated data. A Vinet equation of state fitted to the 300 K data yielded a bulk modulus of KT0 = 384 GPa and a pressure derivative of K T 0 ′ = 3.26. The contribution of thermal pressure was determined to have the form ΔPth = [αKT(Va) + (∂KT/∂T)Vln(Va/V)]ΔT. When αKT(Va) was assumed to be constant, the fit to the data yielded αKT(Va) = 0.0056 GPa/K. In contrast, the volume dependence of the thermal pressure was very small, and fitting yielded a value of (∂KT/∂T)V = −0.00042.

Density determination of liquid iron-nickel-sulfur at high pressure

Abstract The density of liquid iron-nickel-sulfur (Fe46.5Ni28.5S25) alloy was determined at pressures up to 74 GPa and an average temperature of 3400 K via pair distribution function (PDF) analysis of synchrotron X-ray diffraction (XRD) data obtained using laser-heated diamond-anvil cells. The determined density of liquid Fe46.5Ni28.5S25 at 74 GPa and 3400 K is 8.03(35) g/cm3, 15% lower than that of pure liquid Fe. The obtained density data were fitted to a third-order Vinet equation of state (EoS), and the determined isothermal bulk modulus and its pressure derivative at 24.6 GPa are KTPr = 110.5(250) GPa and KTPr′ = 7.2(25), respectively, with a fixed density of rPr = 6.43 g/cm3 at 24.6 GPa. The change in the atomic volume of Fe46.5Ni28.5S25 upon melting was found to be ~10% at the melting temperature, a significantly larger value than that of pure Fe (~3%). Combined with the above EoS parameters and the thermal dependence reported in the literature, our data were extrapolated to the outer core conditions of the Earth. Assuming that S is the only light element and considering the range of suggested Ni content, we estimated a 5.3–6.6 wt% S content in the Earth’s outer core.

Calculating detonation performance of explosives by VLWR thermodynamics code introduced with universal VINET equation of state

Thermodynamic calculation is the theoretical basis for the study of initiation and detonation, as well as the prerequisite for forecasting the detonation performance of unknown explosives. Based on the VLWR(Virial-Wu) thermodynamic code, this paper introduced the universal solid equation of state (EOS) VINET. In order to truly reflect the compressibility of nanocarbon under the extremely high-temperature and high-pressure environment in detonation, an SVM (support vector machine) was utilized to optimize the input parameters of carbon. The detonation performance of several explosives with different densities was calculated by the optimized universal EOS, and the results show that the thermodynamic code coupled with the universal solid EOS VINET can predict the detonation performance parameters of explosives well. To investigate the application of the thermodynamic code with the improved VINET EOS in the working capacity of explosives, the interrelationship between pressure P-particle velocity u and pressure P-volume V were computed for the detonation products of TNT and HMX-based PBX (HMX: binder: insensitive agent = 95:4.3:0.7) in the CJ isentropic state. A universal curve proposed by Cooper was used to compared the computed isentropic state, where the ratio of pressure to CJ state were plotted against the ratio of velocity to CJ state. The parameters of the JWL(Jones-Wilkins-Lee) EOS for detonation products were obtained by fitting the P–V curve. The cylinder tests of TNT and HMX-based PBX were numerically simulated using the LS-DYNA, it is verified that, within a certain range, the improved algorithm has superiority in describing the working capacity of explosives.

The ultrahigh pressure stability of silver: An experimental and theoretical study

We measured the atomic volume of Ag in a toroidal diamond anvil cell to a maximum pressure of 416 GPa and calculated the atomic volume and elastic constants of Ag up to 750 and 460 GPa, respectively. Our density functional theory calculations at 0 K utilize an all-electron fully relativistic method and agree well with our volume measurements, particularly at pressures above ∼75 GPa. We corrected our experimental results for non-hydrostaticity using a line shift analysis, and the resulting Vinet equation of state (EOS) parameters are reported. We find that the uniaxial stress sustained by Ag increases linearly up to 4.5 GPa at a pressure of 416 GPa. Our experimental results indicate that the fcc structure of Ag remains stable to at least 416 GPa at room temperature. Our theoretical results show that C44 increases as pressure increases and reaches a maximum at ∼100 GPa above which it begins to decrease, a sign that the fcc structure of Ag is becoming unstable, and at V/V0 = 0.30, the bcc structure is lower in energy than fcc.

P-V-Tmeasurements of Fe3C to 117 GPa and 2100 K: Implications for stability of Fe3C phase at core conditions

AbstractWe report the thermal Equation of State (EoS) of the non-magnetic Fe3C phase based on in situ X-ray diffraction (XRD) experiments to 117 GPa and 2100 K. High-pressure and temperature unit-cell volume measurements of Fe3C were conducted in a laser-heated diamond-anvil cell. Our pressure-volume-temperature (P-V-T) data together with existing data were fit to the Vinet equation of state with the Mie-Grüneisen-Debye thermal pressure model, yielding V0 = 151.6(12) Å3, K0 = 232(24) GPa, K0′= 5.09(46), γ0 = 2.3(3), and q = 3.4 (9) with θ0 = 407 K (fixed). The high-T data were also fit to the thermal pressure model with a constant αKT term, PTh = αKT(ΔT), and there is no observable pressure or temperature dependence, which implies minor contributions from the anharmonic and electronic terms. Using the established EoS for Fe3C, we made thermodynamic calculations on the P-T locations of the breakdown reaction of Fe3C into Fe7C3 and Fe. The reaction is located at 87 GPa and 300 K and 251 GPa and 3000 K. An invariant point occurs where Fe, Fe3C, Fe7C3, and liquid are stable, which places constraints on the liquidus temperature of the outer core, namely inner core crystallization temperature, as the inner core would be comprised by the liquidus phase. Two possible P-T locations for the invariant point were predicted from existing experimental data and the reaction calculated in this study. The two models result in different liquidus “phases” at the outer core-inner core boundary pressure: Fe3C at 5300 K and Fe7C3 at 3700 K. The Fe7C3 inner core can account for the density, as observed by seismology, while the Fe3C inner core cannot. The relevance of the system Fe-C to Earth’s core can be resolved by constructing a thermodynamic model for melting relations under core conditions as the two models predict very different liquidus temperatures.

Influence of Thermal and Mechanical Stimuli on the Behavior of Al-CAU-13 Metal-Organic Framework.

The response of the metal–organic framework aluminum-1,4-cyclohexanedicarboxylate or Al-CAU-13 (CAU: Christian Albrecht University) to the application of thermal and mechanical stimuli was investigated using synchrotron powder X-ray diffraction (SPXRD). Variable temperature in situ SPXRD data, over the range 80–500 K, revealed a complex evolution of the structure of the water guest containing Al-CAU-13•H2O, the dehydration process from ca. 310 to 370 K, and also the evolution of the guest free Al-CAU-13 structure between ca. 370 and 500 K. Rietveld refinement allowed this complexity to be rationalized in the different regions of heating. The Berman thermal Equation of State was determined for the two structures (Al-CAU-13•H2O and Al-CAU-13). Diamond anvil cell studies at elevated pressure (from ambient to up to ca. 11 GPa) revealed similarities in the structural responses on application of pressure and temperature. The ability of the pressure medium to penetrate the framework was also found to be important: non-penetrating silicone oil caused pressure induced amorphization, whereas penetrating helium showed no plastic deformation of the structure. Third-order Vinet equations of state were calculated and show Al-CAU-13•H2O is a hard compound for a metal–organic framework material. The mechanical response of Al-CAU-13, with tetramethylpyrazine guests replacing water, was also investigated. Although the connectivity of the structure is the same, all the linkers have a linear e,e-conformation and the structure adopts a more open, wine-rack-like arrangement, which demonstrates negative linear compressibility (NLC) similar to Al-MIL-53 and a significantly softer mechanical response. The origin of this variation in behavior is attributed to the different linker conformation, demonstrating the influence of the S-shaped a,a-conformation on the response of the framework to external stimuli.

Elasticity of single-crystal NaCl under high-pressure: simultaneous measurement of x-ray inelastic scattering and diffraction

ABSTRACT Elastic stiffness constants, C ijs, of single crystal of NaCl B1 phase have been determined at pressure up to 20 GPa using inelastic x-ray scattering to measure elastic wave velocities and x-ray diffraction to measure densities in the same experimental setup. The Voigt-Reuss-Hill average velocities calculated from the present Cijs are in good agreement with previous ultrasonic measurements of polycrystalline samples. Absolute pressures based on the present elastic constants and densities are proposed. In order to fit our results an equation of state with at least three parameters (an addition to a reference volume), such as the fourth-order form of Birch–Murnaghan equation or the higher order form of Vinet equation, is required, as fewer parameters do not reproduce the observed pressure dependence of the volume and bulk modulus.

Observation of high-pressure bcc phase of titanium at 243 GPa

In this study, the pressure-induced phase transition of the 3d transition metal titanium (Ti) has been investigated at pressures up to 290 GPa at room temperature using an x-ray diffraction technique with a micro-beam. From the analysis of powder x-ray diffraction patterns, the high-pressure δ phase (orthorhombic, Cmcm) was found to transform to the β phase (body-centered cubic: bcc, Im3m) at 243 GPa. Although the volume reduction at this phase transition was less than 1%, the coexistence of the δ phase suggests that the transition is of the first order. The isothermal bulk modulus at zero pressure, B0, and the pressure derivative of B0, B0′, were estimated to be 125(2) GPa and 3.46(6) GPa, respectively, by least squares fitting to the Vinet equation of state based on the P–V data of the δ and β phases. This transition to the bcc phase verified predictions from theoretical studies and demonstrated the systematics of the pressure-induced structural phase transition in transition metals.

In Situ High Pressure Structural Investigation of Sm-Doped Ceria

As a result of the lattice mismatch between the oxide itself and the substrate, the high-pressure structural properties of trivalent rare earth (RE)-doped ceria systems help to mimic the compressive/tensile strain in oxide thin films. The high-pressure structural features of Sm-doped ceria were studied by X-ray diffraction experiments performed on Ce1−xSmxO2−x/2 (x = 0.2, 0.3, 0.4, 0.5, 0.6) up to 7 GPa, and the cell volumes were fitted by the third order Vinet equation of state (EoS) at the different pressures obtained from Rietveld refinements. A linear decrease of the ln B 0 vs. ln ( 2 V a t ) trend occurred as expected, but the regression line was much steeper than predicted for oxides, most probably due to the effect of oxygen vacancies arising from charge compensation, which limits the increase of the mean atomic volume ( V a t ) vs. the Sm content. The presence of RE2O3-based cubic microdomains within the sample stiffens the whole structure, making it less compressible with increases in applied pressure. Results are discussed in comparison with ones previously obtained from Lu-doped ceria.

Equation of state of hexagonal-close-packed rhenium in the terapascal regime

All-electron density functional theory calculations were performed aiming to determine the equation of state and the dependence with pressure of the $c/a$ ratio and the anisotropic compressibility for hexagonal-close-packed rhenium. Calculations were carried out up to maximum compression of 0.46, and the resulting total energy versus volume is well described by a Vinet equation of state with ${B}_{0}$ = 367(5) GPa and ${B}_{0}^{\ensuremath{'}}$ = 4.64(3), valid up to 1.5 TPa. The agreement with two recent experimental studies [S. Anzellini et al., Equation of state of rhenium and application for ultra high pressure calibration, J. Appl. Phys. 115, 043511 (2014); T. Sakai et al., High pressure generation using double-stage diamond anvil technique: Problems and equations of state of rhenium, High Pressure Res. 38, 107 (2018)] supports their conclusion that the pressure in previous experiments with a double-stage diamond-anvil cell [L. Dubrovinsky et al., Implementation of micro-ball nanodiamond anvils for high-pressure studies above 6 Mbar, Nat. Commun. 3, 1163 (2012)] was significantly overestimated.

Effect of high pressure on the typical 2D hydrogen-bonded crystal azodicarbonamide

We have performed in situ Raman scattering and synchrotron angle-dispersive X-ray diffraction (ADXRD) investigations to explore high-pressure behaviors of azodicarbonamide (C2N4O2H4, ADC) to 21.6 and 23.8 GPa, respectively. ADC exhibits the representative two-dimensional (2D) hydrogen-bonded networks, and is the most hugely used foaming agent in industry both under ambient and high pressures. Careful identification of external modes and indexation of Bragg diffraction peaks under different pressures demonstrate ADC crystal remains the P21/c symmetry in this study. The bulk modulus (B0) and its pressure derivative (B0’) are determined to be 13.2(7) GPa and 8.2(2) by fitting the isothermal third Vinet equation of state (EOS). The conformation change of ADC molecule has been observed at 12.7 GPa. And this is evidenced by the splitting of skeleton atoms vibrations and the discontinuous evolutions with respect to hydrogen bond donor (NH) vibrations. First principle calculation reveals the observed confirmation change arises from the discontinuous variation of the torsion angel between H2NC=O and CN=NC skeleton groups. Hirshfeld surface analysis indicates that the extensive hydrogen bonds within 2D networks dominate the intermolecular close contacts even under high pressures. The intermolecular interaction energy calculation also implies that the energy between neighboring molecules in the 2D layer dominates the networks stability, whereas the energy between neighboring molecules in the second nearest adjacent layer is the primary factor for crystal instability. The cooperativity of the molecular flexibility and directional hydrogen bonds is responsible for structural stability under high pressures.

Static compression of B2 KCl to 230 GPa and its P-V-T equation of state

The pressure-volume-temperature (P-V-T) measurements of the B2 (CsCl-type) phase of KCl were performed at 9–61 GPa/1500–2600 K and up to 229 GPa at room temperature, based on synchrotron X‑ray diffraction measurements in a laser-heated diamond-anvil cell (DAC). The nonhydrostatic stress conditions inside the sample chamber were critically evaluated based on the platinum pressure marker. With thermal annealing by laser after each pressure increment, the deviatoric stress was reduced to less than 1% of the sample pressure even at the multi-megabar pressure range. The obtained P-V-T data were fitted to the Vinet equation of state with the Mie-Grüneisen-Debye model for thermal pressure. The thermal pressure of KCl was found to be as small as ~10 GPa even at 3000 K at any given volume, which is only half of that of common pressure markers (i.e. Pt, Au, or MgO). Such a low-thermal pressure validates the use of a KCl pressure medium as a pressure marker at high temperatures.