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.     

Infinite pressure behaviour of certain inverted type equations of state for solids

It is shown that a generalized version of inverted type equation of state (EOS) yields the recently proposed Saxena EOS [J. Phys. Chem. Solids 65 (2004) 1561] for a particular value of the parameter related to , the pressure derivative of bulk modulus at extreme compression. It is found that some other inverted type EOS due to earlier workers can also be derived from a generalized version. A common feature of the inverted equations considered in the present paper is that they yield negative values for . This is in strong contradiction and violation of the theory of infinite pressure extrapolation due to Stacey.     

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.     

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.     

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.     

Thermodynamics of ice at high pressures and low temperatures

A new equation of state of ice Ih recently proposed by Feistel and Wagner [J. Phys. Chem. Ref. Data 35 (2006) 1021-1047] is used to study the phenomena related to the equilibrium isentropic compression of an ice-water mixture and dynamic loading of solid ice. New results are presented concerning the properties of the new equation of state, equilibrium solid-liquid phase transitions and Hugoniots of low-temperature (100 K) and temperate (263 K) shock-compressed ice.     

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 thermoelastic constants of solids and minerals at high temperatures

In the present communication, the thermoelastic constants of ionic solids NaCl and KCl and the minerals MgO, CaO and Mg₂SiO₄ are analyzed using the tabulated data compiled by Anderson and Isaak. It is found that there exists a precise linear relationship between the thermoelastic constant and the thermal pressure. The proposed empirical relationship provides a method to estimate the thermoelastic properties out side the experimental data range.     

Structural, high pressure and elastic properties of transition metal monocarbides: A FP-LAPW study

The structural, elastic and thermal properties of four transition metal monocarbides ScC, YC (group III), VC and NbC (group V) have been investigated using full potential linearized augmented plane wave (FP-LAPW) method within generalized gradient approximation (GGA) both at ambient and high pressure. We predict a B₁ to B₂ structural phase transition at 127.8 and 80.4 GPa for ScC and YC along with the volume collapse percentage of 7.6 and 8.4%, respectively. No phase transition is observed in case of VC and NbC up to pressure 400 and 360 GPa, respectively. The ground state properties such as equilibrium lattice constant (a₀), bulk modulus (B) and its pressure derivative (B′) are determined and compared with available data. We have computed the elastic moduli and Debye temperature and report their variation as a function of pressure.     

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 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.     

Measurement of Seebeck effect (thermoelectric power) at high pressure up to 40 GPa

The paper reports details of a high-pressure thermoelectric power (Seebeck effect) technique up to 40 GPa. Several different types of high-pressure cells with anvil insets are presented. The technique was applied for measurements of pressure dependence of the thermopower of several substances including elemental metals (lead, Pb; indium, In), cerium-nickel alloy, Ce-Ni and sulphur, S. Two peculiarities in the pressure dependences of the thermopower of CeNi were found and attributed to structural transformations, near ∼5 and ∼10 GPa. These transitions were confirmed in direct X-ray diffraction studies. Sulphur compressed to 40 GPa exhibited a hole type conductivity and the thermopower value was about ∼+1 mV/K. Additionally, as an example of pressure calibration, the data on the electrical resistivity of zinc selenide, ZnSe, are given in a range of 0-23 GPa. These data suggest three possible scenarios of phase transitions from a rock salt (RS) high-pressure phase of ZnSe under decompression: RS→zinc blende (ZB), RS→cinnabar→ZB, and RS→wurtzite.     

Electrical property and phase transition of CdSe under high pressure

In situ electrical resistivity measurement of CdSe was performed under high pressure and moderate temperature using a diamond anvil cell equipped with a microcircuit. With the pressure increasing, a sharp drop in resistivity of over two orders of magnitude was observed at about 2.6 GPa, it was caused by the transition to the rock-salt CdSe. After that, the resistivity decreased linearly with pressure. However, in different pressure range, the decreasing degree was obviously different. This attributed to the different electron structures. By fitting to the curve of pressure dependence of resistivity in different pressure range, the relationship of the band gap to pressure was given and the metallization pressure was speculated to be in the range of 70-100 GPa. The temperature dependence of resistivity showed that in the experimental temperature and pressure range the resistivity had a positive temperature coefficient.     

The elastic behavior of dense C3N4 under high pressure: First-principles calculations

We have carried a detailed theoretical study on the geometry, density of states, elastic properties, sound velocities and Debye temperature of α-, β-, c- and p-C₃N₄ compounds under a maximum of pressure up to 100 GPa by using first principles calculations. The optimized lattice constants under zero pressure and zero temperature agreed well with the previous experimental and theoretical results. The band gaps of the four types of dense C₃N₄ were widened gradually with the increase of pressure. The calculated Poisson’s ratio γ and B/G values suggest α-, c- and p-C₃N₄ are brittle materials under 0–100 GPa, whereas β-C₃N₄ will become a ductile material as external pressure reaches 57 GPa. We found that the Debye temperature of the four dense C₃N₄ gradually reduces in the order of c-C₃N₄>p-C₃N₄>α-C₃N₄>β-C₃N₄ at 0 GPa and 0 K. However, the Debye temperature of c-C₃N₄ was lower than p-C₃N₄ when external pressure exceeds 6.3 GPa. It may hint that the results could be served as a valuable prediction for further experiments.     

First-principles calculations of structural stability and mechanical properties of tungsten carbide under high pressure

The structural stability and mechanical properties of WC in WC-, MoC- and NaCl-type structures under high pressure are investigated systematically by first-principles calculations. The calculated equilibrium lattice constants at zero pressure agree well with available experimental and theoretical results. The formation enthalpy indicates that the most stable WC is in WC-type, then MoC-type finally NaCl-type. By the elastic stability criteria, it is predicted that the three structures are all mechanically stable. The elastic constants C_ij, bulk modulus B, shear modulus G, Young?s modulus E and Poisson?s ratio ν of the three structures are studied in the pressure range from 0 to 100 GPa. Furthermore, by analyzing the B/G ratio, the brittle/ductile behavior under high pressure is assessed. Moreover, the elastic anisotropy of the three structures up to 100 GPa is also discussed in detail.     

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).     

Analysis of structural properties of refractory compounds at high temperature and pressure using a potential model

In this paper we focused on the structural and elastic properties of four transition metal mononitrides (TMNs) (M=Ti, Nb, Hf and Zr) by using realistic three body interaction potential (RTBIP) model, including the role of temperature. These TMN compounds have been found to undergo NaCl (B1) to CsCl (B2) phase transition, at a pressure quite high as compared to other binary systems. We successfully obtained the phase transition pressures and volume changes at different temperatures. In addition, elastic constants of TMNs at different temperatures are discussed. The present theoretical results have been compared with the available experimental data and predictions of LDA theory.     

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.     

Behavior of decomposed ammonia borane at high pressure

High pressure behavior of ammonia borane after thermal decomposition was studied by Raman spectroscopy at high pressure up to 10 GPa using diamond anvil cell (DAC). The ammonia borane was decomposed at around 140 °C under the pressure at ∼0.7 GPa. Raman spectra show the hydrogen was desorbed within 1 h. The hydrogen was sealed in DAC well and cooled down to room temperature. Applying higher pressure up to ∼10 GPa indicates interactions between the products and loss of dihydrogen bonding. No rehydrogenation was detected in the pressure range investigated.     

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.