Two universal equations of state for solids satisfying the limiting condition at high pressure

In this paper it is shown that the relationship of bulk modulus with pressure, B=f(P), should be linear both at low and high-pressure limiting conditions. Because most of present equations of state (EOS) for solids cannot satisfy such linear relationship at high pressure, a new function f(P) is proposed to satisfy the linearity. By integrating the bulk modulus, an EOS with three parameters and satisfying the quantum-statistics limitation is derived. It is shown that the EOS can be reduced to two-parameter EOS approximately satisfying the limiting condition. By applying the two EOSs and other three typical EOSs to 50 materials, it is concluded that for materials at low and middle-pressure regimes, the limiting condition does not operate, the Baonza EOS gives the best results, but it cannot provide analytic expression for cohesive energy. The Vinet and our second EOSs are slightly inferior, both EOSs can provide analytic expression for cohesive energy, and for materials at high-pressure regimes our second EOS gives the best results. The Holzapfel and our first EOSs give the worst results, although they strictly satisfy the limiting condition. For practical applications, the limiting condition is not important because it only operates as V→0.     

Structure and high pressure behaviour of organic crystals based on aromatically substituted 1,3,4-oxadiazoles

The crystalline structure of a new compound containing the 1,3,4-oxadiazole moiety, 4-(5-methyl-1,3,4-oxadiazole-2yl-)-N,N′-dimethyl-phenylamine (MODPA) was determined. It shows a monoclinic structure with space group P2₁/c and lattice parameters: a=1.02997(6), b=0.64840(4), c=1.58117(10) nm and β=99.4820(10)°. To study the intermolecular interactions in oxadiazole containing organic crystals, X-ray studies on MODPA and 2,5-diphenyl-1,3,4-oxadiazole (DPO) were performed up to 5 GPa at room temperature. The Murnaghan equation of state is used to describe the compression behaviour of both substances. From these results, the bulk modulus and its pressure derivative were determined. The values obtained are: K₀=6.3 GPa and K′₀=6.8 for MODPA and K₀=7.3 GPa and K′₀=6.7 for DPO. Additionally, measurements under increasing temperature at ambient pressure were carried out to evaluate the thermal expansion coefficient: α=1.8×10⁻⁴ K⁻¹ for MODPA and α=1.9×10⁻⁴ K⁻¹ for DPO.     

The temperature dependence of the equation of state at high pressures revisited: a universal model for solids

A general method to include temperature effects into the equation of state (EOS) of solids is discussed. A universal model based on a pseudo-spinodal approach is used to predict the pressure and temperature dependencies of the thermodynamic properties for a variety of solids: n-H₂, Ar, Kr, Xe, NaCl, LiF, NaF, KCl, CsCl, Li, Na, K, Rb, Cs, Al, Fe, Cu, Zn, Ag, Cd, Pt, Au, and Pb. The predictive capabilities of the complete EOS are discussed and compared with available models.     

Impurity effects on the phase transformations and equations of state of zirconium metals

The in situ high P-T X-ray diffraction experiments were conducted at pressures up to 17 GPa and temperatures up to 1273 K to study the phase transformations and equations of state for two grades of zirconium metals. At ambient temperature, our results reveal significant differences in both the transition pressure and kinetics of the α-ω phase transformation between the ultra-pure Zr (35 ppm Hf and <50 ppm O) and impure Zr (1.03 at% Hf and 4.5 at% O). These observations indicate that impurities, particularly oxygen ions, play important roles in the transformation mechanisms as well as crystal stability. On the other hand, impurities have no measurable effects on either the elastic bulk moduli of both α and ω phases or the volume change across the α-ω phase transformation. At elevated temperature, both impure and ultra-pure Zr show similar transition temperatures for the ω-β phase boundary over a pressure range of 6-16 GPa, suggesting that impure oxygen and hafnium ions can only be an α-Zr stabilizer; they do not seem to significantly increase the stability of the ω-Zr relative to the β-Zr.     

Analysis of Suzuki, Shanker, and Kumar formulations under strong compressions

A critical analysis of the Suzuki, Shanker, and Kumar formulations is presented by studying different classes of materials under high pressure. A similar trend for all the materials studied in the present work, demonstrates that Suzuki formulation is not capable to yield compression behaviour of solids. The Shanker formulation improves the results obtained by the Suzuki formulation in small compression range (0.9<V/V₀<1). For further compressions Shanker formulation also fails. On the other hand, the Kumar formulation is found to work well for the entire range of pressure. The reasons for the failure of Suzuki and Shanker formulations are discussed.     

Explicit Gibbs free energy equation of state for solids

Semi-empirical equations of state (EOS) are used for interpolation and extrapolation of experimental data and/or electronic structure calculations. For calculation of phase equilibria, it is preferable to use an explicit Gibbs free energy EOS, that is, to express the Gibbs free energy directly as a function of the pressure and temperature. Existing explicit Gibbs free energy EOS formulations often give unphysical predictions at high pressures. The origins of these problems are internal inconsistencies and uncontrolled extrapolations. A set of conditions is put forward, that should be fulfilled by semi-empirical EOS formulations in order to constrain them to known physical behaviour, e.g., to the Thomas-Fermi and quasi-harmonic models at high pressures. A new alternative integration path is devised that eliminates the need for the problematic extrapolation of the heat capacity to high temperatures at low pressures. Based on these developments, a new explicit Gibbs free energy EOS is formulated which is suitable for computational applications. The new EOS may be fitted to represent the thermophysical properties of solids with a reasonably small number of adjustable parameters. A sample application for MgO is presented.     

X-ray diffraction study of molybdenum diselenide to 35.9 GPa

In situ high-pressure angle dispersive synchrotron X-ray diffraction studies of molybdenum diselenide (MoSe₂) were carried out in a diamond-anvil cell to 35.9 GPa. No evidence of a phase transformation was observed in the pressure range. By fitting the pressure-volume data to the third-order Birch-Murnaghan equation of state, the bulk modulus, K_0T, was determined to be 45.7±0.3 GPa with its pressure derivative, K′_0T, being 11.6±0.1. It was found that the c-axis decreased linearly with pressure at a slope of −0.1593 when pressures were lower than 10 GPa. It showed different linear decrease with the slope of a −0.0236 at pressures higher than 10 GPa.     

High pressure equation of state studies using methanol-ethanol-water and argon as pressure media

We have measured the equation of state of the intermetallic compound AuIn₂ up to 20 GPa and Cd_0.8Hg_0.2 up to 50 GPa using methanol-ethanol-water solution or argon as pressure media. In the experiments performed with argon as pressure medium, we minimized non-hydrostatic conditions by thermally annealing the sample. We present data revealing compressibility anomalies in AuIn₂ at 2.7 GPa and in Cd_0.8Hg_0.2 near 8, 18 and 34 GPa with methanol-ethanol-water and argon. At pressures above 5 GPa the P-V data for AuIn₂ and Cd_0.8Hg_0.2 from experiments preformed with argon as a pressure medium start deviating from those using methanol-ethanol-water, and the equation of state based on experiments in argon is stiffer compared with that in methanol-ethanol-water. This behavior is consistent with the relative merits of the two pressure transmitting media as documented in the literature. We also provide a brief summary of the results of electronic structure calculations that associate these anomalies with electronic topological transitions.     

Single-crystal elastic constants of magnesium difluoride (MgF2) to 7.4 GPa

The six independent elastic constants (C₁₁, C₁₂, C₁₃, C₃₃, C₄₄, and C₆₆) of single-crystal MgF₂ in the rutile structure have been measured by Brillouin spectroscopy at room temperature from ambient conditions to 7.4 GPa. Measurements were performed on two monocrystals with perpendicular faces, (001) and (100). A quasi-linear fit from finite strain theory was applied to the experimental data revealing the pressure dependence of the six elastic constants of MgF₂. The shear modulus C_S=1/2(C₁₁−C₁₂), and the aggregate shear (Voigt–Reuss–Hill) modulus G show a softening with increasing pressure, indicating the approach of the rutile-to-CaCl₂-type structural phase transition at P~9 GPa. The adiabatic bulk modulus (Reuss average) and its pressure derivative have been determined: K_0S=105.1±0.3 GPa, (∂K_0S/∂P)_T=4.14±0.05. The pressure–volume equation of state of MgF₂ was computed self-consistently from the Brillouin data. Our results are in good agreement with X-ray diffraction data. As the phase transition is approached, MgF₂ becomes strongly anisotropic and develops partially auxetic behavior (a negative Poisson's ratio in certain directions).     

Three equations of state for solids considering thermal effect

 The Einstein model to consider thermal effect in universal equations of state (UEOS) is modified. It is proposed that the zero-point vibration term should be deleted in a thermal UEOS, and the parameters cannot be directly taken as experimental data at a reference temperature, V_R, B_R, and , but their values at absolute zero temperature, V₀, B₀, and . An approach is proposed to solve V₀, B₀, and from V_R, B_R, and . The approaches are applied to three typical universal EOSs, including the Baonza, mGLJ and Morse EOSs. The numerical results show that the solved values of parameters are almost identical for different EOSs. And the thermo-physical properties predicted through different EOSs are almost identical at zero- and low-pressure conditions, once the same approach and input experimental data are used to solve the parameters. It is concluded that the prediction of thermo-physical properties at zero- and low-pressure conditions cannot be taken as the criteria to judge the applicability of a universal EOS.     

Revisiting the thermostatistics of the Grüneisen parameters and applications to quasiharmonic solids

We present an integrated thermostatistical study of the Grüneisen parameter and its application to a quasiharmonic solid. This analysis comprises two parts. The first part considers the connections between different thermodynamic formulations of the Grüneisen parameter, including those arising from the Mie-Grüneisen formula and related equations of state. We also establish the most general consequences of the so-called Grüneisen׳s rule. In the second part, Grüneisen׳s rule is used to establish the thermostatistics of a solid that obeys the quasiharmonic approximation. Both cases are relevant to the application of fundamental microscopic models of the heat capacity of solids.     

Analysis of thermoelastic behaviour of MgO at high temperatures and high pressures

The thermoelastic behaviour of MgO has been studied for the temperature range (300-3000 K) under different compressions down to V/V₀=0.3. It has been shown that a comprehensive study of the thermoelastic properties of MgO can be made with the help of the Anderson-Isaak equation for thermal expansivity and the Vinet equation of state taken together. We have estimated the values of thermal expansivity α, isothermal bulk modulus K_T, their variations with pressure and temperature, the Anderson-Gruneisen parameter and the change in entropy with compression for MgO along isotherms at different temperatures. The results have been discussed and compared with the corresponding values reported in the recent literature.     

Pressure and volume dependence of Grüneisen parameter of solids

Using the assumption δ_T=δ_T0(V/V₀)^k and the Grüneisen parameter γ macroscopic definition expression, we obtained a relationship for the volume dependence of the Grüneisen parameter γ. We have calculated the Grüneisen parameter γ with this relationship for NaCl and ε-Fe at high pressure under study. The calculated values of γ are found to show fairly in good agreement with available experimental data.     

Structural stability of tin clathrates under high pressure

High pressure Raman scattering experiments have been performed for Rb₈Sn₄₄□₂ in order to investigate the pressure induced phase transition. At pressures of 6.0 and 7.5 GPa, Raman spectrum was drastically changed, indicating the phase transitions. The irreversibility of the spectral change and the disappearance of Raman peak observed at 7.5 GPa strongly suggest the occurrence of irreversible amorphization.     

Structure and high-pressure behavior of 2,5-di-(4-aminophenyl)-1,3,4-oxadiazole

The crystalline structures of two modifications of a compound containing the oxadiazole ring, 2,5-di-(4-aminophenyl)-1,3,4-oxadiazole (DAPO) were determined. One of these modifications contains water molecules in the crystal structure, which is observed for the first time for an oxadiazole crystal. Both crystals show an orthorhombic structure. The water free modification, DAPO I, belongs to the space group Pbca (61) and has the lattice parameters: a=13.461(5), b=7.937(3) and c=22.816(8) Å (CCDC 246608). The water containing pseudo-polymorph, DAPO II, has the space group Cmcm (63) and the lattice parameters: a=16.330(5), b=12.307(2) and c=6.9978(14) Å (CCDC 246609). To gain information on the inter molecular interactions within the crystals, X-ray experiments under compression at ambient temperature and under heating at vacuum conditions were performed. Neither DAPO I nor DAPO II undergo phase transitions in the ressure range up to 5 GPa, as could be concluded from X-ray and Raman experiments. X-ray and calorimetric studies indicate that DAPO II dehydrates into DAPO I under increasing temperature. Structural considerations suggest a two-stage process. The compression behavior of both substances is well described by the Murnaghan equation of state (MEOS) and the values of the bulk modulus and its pressure derivative are determined for these crystals. Additionally, in the case of DAPO I, also the thermal expansion coefficient α₀ was measured.     

High pressure study of structural and electronic properties of magnesium telluride

The full-potential linear muffin-tin orbital method (FP-LMTO) within the local density approximation (LDA) is used to calculate the electronic band structures and the total energies of MgTe in its stable (NiAs-B8) and high pressure phases. The latter provide us with the ground state properties such us lattice parameter, bulk modulus and its pressure derivatives. The transition pressure at which this compound undergoes the structural phase transition from the NiAs to CsCl phase is calculated. The energy band gaps and their volume and pressure dependence in the stable NiAs-B8 phase are investigated. The ground state properties, the transition pressure are found to agree with the experimental and other theoretical results. The elastic constants at equilibrium in both NiAs and CsCl structure are also determined.     

High pressure behaviour of GaCMn3

Results of X-ray powder diffraction measurements up to 35 GPa carried out on GaCMn₃ are presented. GaCMn₃ does not undergo any structural transition in this pressure region. However, the pressure–volume data can be fitted to two straight line segments in the pressure regions; one from 0 to 5 GPa and another from 5 to 30 GPa respectively.     

Sulfur-catalyzed phase transition in MoS2 under high pressure and temperature

We report phase transition and stability of MoS₂ with and without the presence of sulfur melt under high-pressure and high-temperature conditions. Rhombohedral (3R) phase is found to be a high-temperature phase of MoS₂ at high pressures. Excess sulfur melt catalyzes the hexagonal (2H) to rhombohedral (3R) phase transformation and lowers the conversion temperature by more than 280 K. Boundary between 2H and 3R phases has been delineated with a negative slope. Based on experimental observations, sulfur-catalyzed 2H→3R transformation mechanisms are proposed involving atomic exchange between MoS₂ and sulfur, which is different from the case of without excess sulfur that proceeds through rotation and translation of the S–Mo–S sandwich layers.     

Raman spectra for hydrogen hydrate under high pressure: Intermolecular interactions in filled ice Ic structure

Raman studies of a high-pressure structure of hydrogen hydrate, a filled ice Ic structure, were performed using a diamond anvil cell in the pressure range 3.2-44.1 GPa. The Raman spectra of a vibron revealed that extraction of hydrogen molecules from the filled ice Ic structure occurred above 20 GPa. In addition, the Raman spectra of a roton revealed that a rotation of hydrogen molecules in the filled ice Ic structure was suppressed at around 20 GPa and then the rotation recovered, and the rotation of hydrogen molecules was suppressed again above 35.5 GPa. These results indicate that intermolecular interactions increased between guest hydrogen molecules and host water molecules at around 20 and 35.5 GPa. These intermolecular interactions were considered to be induced to stabilize the filled ice Ic structure. Above 40 GPa, symmetrization of hydrogen bond was considered to contribute to the stability of hydrogen hydrate.     

High pressure phase transition and elastic properties of thorium chalcogenides

The structural and elastic properties of thorium chalcogenides at high pressure, have been investigated using a suitable inter-ionic potential. The calculated equation of state, phase transition pressures for B1-B2 transition and bulk moduli for ThX (X=S,Se,Te) compounds agree well with the experimental results. ThTe, which crystallizes in the CsCl structure, does not show any structural transition up to 48 GPa. The present analysis does not show any anomalous features in elastic properties arising from ‘f’ electrons.