Isothermal Equation Of State Research Articles

Analysis of isothermal equations of state for ionic solids and geophysical minerals at high temperatures

We present an analysis of various isothermal equations of state (EOSs) such as the Birch-Murnaghan EOS, the Vinet EOS, the Shanker EOS, and the logarithmic EOS by studying the pressure-volume relationships, the isothermalbulk modulus K T , and its first pressure derivative KT for ionic solids such as NaCl, KCl, MgO, CaO and geophysical minerals such as MgSiO 3 perovskite at temperatures ranging from room temperature up to their melting temperatures under high compressions down to V/V 0 =0.6 where V 0 is the volume V at P=0. The results are compared with the corresponding values derived from the Hama-Suito EOS based on the augmented-plane-wave (APW) method and the quantum statistical model (QSM). The comparison reveals that the Vinet EOS and the Shanker EOS yield the closest agreement with the ab-initio results. At higher temperatures and higher compressions the values of P, K T , and K' T obtained from the Birch-Murnaghan EOS are somewhat higher and those obtained from the logarithmic EOS are lower than the values obtained from the Hama-Suito EOS. The results obtained in the present study compare well with the available experimental P-V-T data for MgO and MgSiO 3 .

Die Packungsdichte der Atome in Modifikationen von Kohlenstoff und Silicium

Abstract The crystal chemistry of carbon and silicon is discussed with respect to the packing density of atoms in their crystalline modifications. The reduced atomic volumes of metallic carbon as calculated from diamond, hexagonal diamond and buckminsterfullerene C60 agree within less than 0.2, indicating the correctness of the reducing procedure. A much lower value calculated from graphite, rhombohedral graphite and a low-temperature modification of buckminsterfullerene remains unexplained. Two proposed crystal structures of a cubic body-centered form of carbon appear unacceptable. The reduced atomic volumes of diamond-type silicon and metastable γ-Si are in good agreement. Their values, calculated from six allotropic modifications (Si I, II, XI, V, VI, VII, and Si X) at pressures up to ≈200 GPa, are represented by an empirical exponential function of pressure. From this function, an isothermal equation of state is derived including the packing density of the atoms.

High-pressure single-crystal X-ray and powder neutron study of F,OH/OD-chondrodite: Compressibility, structure, and hydrogen bonding

The crystal structure of a natural F-bearing chondrodite [Mg4.64Fe0.28Mn0.014Ti0.023(Si1.01O4)2F1.16OH0.84; a = 4.7328(1), b = 10.2749(2), c = 7.8756(1) Å, a = 109.065(1)∞; space group P21/b (a unique)] from the Tilly Foster mine (Brewster, New York) and the structure of a deuterated synthetic analogue [Mg5(SiO4)2F1.10OD0.90] were refined at high pressure using single-crystal X-ray and time-of-flight neutron powder diffraction data, respectively. The isothermal equation of state was determined from single-crystal high-precision X-ray diffraction measurements of unit-cell parameters as a function of pressure up to 9.63(1) GPa. A third-order Birch-Murnaghan equation-of-state fit to pressurevolume data results in V0 = 362.00(2) Å3, K0,T = 117.0(4) GPa and KT' = 5.6(1). The larger value of K0,T relative to that determined for the OH end-member confirms that F makes the chondrodite structure less compressible. The axial compressibilities behave anisotropically with fitted parameters a0 = 4.73282(9) Å, K(a)0 = 458(3) GPa, and K(a)' = 28.9(9); b0 = 10.2759(3) Å, K(b)0 = 298(1) GPa, and K(b)' = 13.4(4); c0 = 7.8759(2) Å, K(c)0 = 316(2) GPa, and K(c)' = 16.7(5). X-ray structural investigations at 0.0001, 4.952(7), and 9.550(7) GPa in a diamond-anvil cell reveal a complex compression mechanism. The greatest changes with pressure occur in the M2 octahedra, which are located at the flexion points of the octahedral chain. These octahedra have the longest bonds at ambient conditions due to cation-cation repulsion, which displaces the cation from the centroid of the octahedron. The application of pressure overcomes the effect of cation-cation repulsion inducing a strong bond-length compression. The neutron powder data were collected at 0.0001, 1.26(1), 2.89(2), 3.87(1), 5.27(2), and 7.04(2) GPa in the Paris-Edinburgh cell using a new gasket design and a fluid pressure medium. Changes in the O-D···O/F hydrogen bond geometry with pressure can be interpreted in terms of a rotation of the OD vector and a reduction in the degree of static disorder associated with the hydrogen bond acceptor/donor site. These new crystallographic data are discussed in terms of corresponding spectroscopic measurements reported recently in the literature

Pressure Dependence of Elastic Constants and Debye Temperature for Zr50.5Ti4.8Cu19.0Ni11.4Al14.3Bulk Metallic Glass

The microstructure of Zr50.5Ti4.8Cu19.0Ni11.4Al14.3bulk metallic glass (BMG) was investigated by high-resolution electron microscopy. The microstructure of the BMG is of homogeneous composition and short-range ordered microstructure. Using an ultrasonic technique with the pulse echo overlap method, the pressure-dependent acoustic velocities of the BMG have been measured up to 0.5 GPa. The elastic constants and their pressure dependence have been obtained. The isothermal equation of state (EOS) of the BMG is determined in terms of the Murnaghan form. With the results of the elastic constants and the EOS for those of its metallic components, the atomic configuration of the BMG is discussed.

Equation of state of dense hydrous magnesium silicate phase A, Mg7Si2O8(OH)6

The isothermal equation of state (EoS) of phase A, Mg 7 Si 2 O 8 (OH) 6 , has been determined using high-pressure single-crystal X-ray diffraction. A third-order Birch-Murnaghan EoS fit to pressure-volume data collected from room pressure and temperature to 7.6 GPa results in V 0 = 512.56(3) A 3 , K 0 = 97.5(4) GPa, and K ′ = 5.97(14). Compression of the hexagonal ( P 6 3 ) structure is anisotropic with the c axis, which is perpendicular to the distorted close-packed planes of anions, approximately 23% less compressible than the a axis: K a = 90.1(5) GPa with K a ′ = 5.4(2) and K c = 116.8(9) GPa with K ′ c = 7.5(3). The bulk modulus of phase A is intermediate between those of brucite (Br) and forsterite (Fo) and less than those of hydroxylclinohumite and hydroxylchondrodite, in a manner that is entirely consistent with its water content and density in relation to the Fo-Br series of minerals.

Analysis of some two-parameter equations of state for the NaCl crystal

An isothermal equation of state (EOS) for ionic crystal solids is derived by starting from the theory of interionic potential and the simple Born - Mayer potential model. A critical analysis of the two-parameter equationsof state: Birch-Murnaghan EOS, Kumar EOS, Vinet EOS, Shanker EOS, and the one derived here, D-Y EOS, is presented by investigating the pressure-volume and isothermal bulk modulus data for the NaCl crystal. The results show that the D-Y EOS is in good agreement with the other EOSs and with experimental data up to very high pressures.

An Isothermal Mganetohydrodynamic Code and Its Application to the Parker Instability

As a companion to an adiabatic version developed by Ryu and his coworkers, we have built an isothermal magnetohydrodynamic code for astrophysical flows. It is suited for the dynamical simulations of flows where cooling timescale is much shorter than dynamical timescale, as well as for turbulence and dynamo simulations in which detailed energetics are unimportant. Since a simple isothermal equation of state substitutes the energy conservation equation, the numerical schemes for isothermal flows are simpler (no contact discontinuity) than those for adiabatic flows and the resulting code is faster. Tests for shock tubes and Alfven wave decay have shown that our isothermal code has not only a good shock capturing ability, but also numerical dissipation smaller than its adiabatic analogue. As a real astrophysical application of the code, we have simulated the nonlinear three-dimensional evolution of the Parker instability. A factor of two enhancement in vertical column density has been achieved at most, and the main structures formed are sheet-like and aligned with the mean field direction. We conclude that the Parker instability alone is not a viable formation mechanism of the giant molecular clouds.

Cosmic gas density profiles on fractal space

A model is proposed to account for the spatial and temporal mass and density profiles of the interstellar gas. In this model, instead of solving the hydrodynamics of the flow field with an isothermal equation of state, the evolution of the collapsing gas is followed kinematically using fractal geometry. The scaling of the density and mass profiles of the collapsing gas with the reduced radius λ= r/ r 0 were reproduced. The agreement between these predictions and the gas self-similar solutions is excellent. In addition, the fractal dimension of an attractor characterizing the turbulent and chaotic structure of the interstellar medium was deduced. This model supports the idea that density profiles lie on fractal space which is characterized by a fractal dimension.

Compression of synthetic hydroxylclinohumite [Mg9Si4O16(OH)2] and hydroxylchondrodite [Mg5Si2O8(OH)2

The isothermal equations of state (EoS) of synthetic hydroxylclinohumite, Mg9Si4O16(OH)2, and synthetic hydroxylchondrodite, Mg5Si2O8(OH)2, have been determined using high-pressure singlecrystal X-ray diffraction, carried out in a diamond anvil cell under hydrostatic conditions. Both humites are monoclinic (space group P21/b with a unique): a = 4.7490(3) Å, b = 10.2861(4) Å, c = 13.6991(11) Å and α = 100.649(6)° for hydroxylclinohumite, and a = 4.7449(2) Å, b = 10.3464(2) Å, c = 7.8990(6) Å, and α = 108.681(3)° for hydroxylchondrodite. A third-order Birch-Murnaghan EoS was determined from unit-cell volume data to 8.1 GPa for hydroxylclinohumite: V0 = 657.69(5) Å3, KT = 119.4(7) GPa and K' = 4.8(2). A similar analysis of hydroxylchondrodite for data collected to 7.8 GPa resulted in V0 = 367.36(3) Å3, KT = 115.7(8) GPa and K' = 4.9(2). Axial compression is anisotropic with the direction perpendicular to the close-packed anion layer, i.e., the a axis, being the least compressible. Axial moduli and their pressure derivatives are: Ka = 162(1) GPa, Ka' = 6.7(3), Kb = 97.9(5) GPa, Kb' = 4.0(1), Kc = 111.1(7) GPa and Kc' = 4.2(2) for hydroxylclinohumite. For hydroxylchondrodite: Ka = 149(1) GPa, Ka' = 6.8(3), Kb = 101.3(4) GPa, Kb' = 4.3(1), Kc = 102.4(6) GPa, and Kc' = 4.1(2). Comparison of the bulk moduli of these phases with other phases along the Mg2SiO4-Mg(OH)2 join shows that the bulk modulus increases systematically with density (ρ) and can be approximated by, KT (GPa) = 97(6) × ρ - 186(17). The bulk modulus also decreases systematically with water content: KT (GPa) = 127.9 (16) - 2.75 (11) × wt% H2O.

An Isothermal Equation of State of Solid

In this paper, an isothermal three-parameter equation of state (EOS) of solid is proposed in the form V/V 0 = f(P), with pressure P as the independent and relative volume V/V 0 as the dependent variable. The proposed EOS uses three parameters expressible in terms of B 0 , By and B 0 , denoting bulk modulus and its first and second pressure derivatives at zero pressure. The new model is applied to the isotherms of ionic, metallic, quantum and rare-gas solid, with pressures ranging from zero to variable maximum pressures of up to 1 TPa. The fits are uniformly excellent. Root-mean-square deviations between data and fits are computed and compared with the three-parameter empirical EOS proposed by Kumari and Dass [J. Phys.: Condens. Matter 2, 3219 (1990)]. It is shown that our new form yields a decisively superior fit. Furthermore, it is shown that our proposed equation of state has an advantage for some close-packed materials because it allows B'∞ = (δB s /δP) s (P → ∞) to be fitted, and this is where the usual standard equations fail badly.

An isothermal equation of state of solid

An isothermal equation of state of solid

On the enhancement of diffusion kinetics in nanocrystalline materials

It is well known that volume diffusivity in nanocrystalline materials is orders of magnitude greater than that in coarse grained materials of identical chemistry. The present study offers a theoretical explanation of this enhancement by attempting to correlate the diffusivity with the excess free volume of the grain boundary atoms in the nanocrystalline materials. Assuming a simplified geometry of the nanocrystalline aggregate, the excess free volume of the grain boundary atoms has been expressed as a function of grain size. Applying the concept of isothermal equation of state, it is shown that the effect of negative hydrostatic pressure on the diffusion kinetics in nanocrystalline materials at a given temperature is equivalent to the same obtained for coarse grained materials at an effective elevated temperature. The predicted results have been compared with the earlier published relevant experimental data from several Cu-based systems.

Elastic constants of Pd39Ni10Cu30P21 bulk metallic glass under high pressure

The pressure-dependent acoustic velocities of a Pd39Ni10Cu30P21 bulk metallic glass (BMG) have been measured up to 0.5 GPa by using an ultrasonic technique with the pulse echo overlap method. The elastic constants, the Debye temperature, and their pressure dependence are obtained. The isothermal equation of state (EOS) of the BMG is established in terms of the Murnaghan form. The atomic configurations of the BMG are discussed by comparing the elastic constants and the EOS with those of its metallic component and of other amorphous materials.

Equation of state of phase E

AbstractThe isothermal equation of state (EoS) of phase E, Mg1.96(7)Fe0.072(5)Si1.04(5)H3.7(8)O6, has been determined using high-pressure single-crystal X-ray diffraction. A third-order Birch Murnaghan EoS fit to pressue-volume data collected from room pressure and temperature to 6.7 GPa reveals that phase E has the lowest bulk modulus, KT = 92.9(7) GPa, and highest pressure derivative of the bulk modulus, K' = 7.3(2), for any dense hydrous magnesium silicate (DHMS) yet measured. A parameterized third-order Birch-Murnaghan EoS was also fit to the unit-cell parameters which display significant curvature with increasing pressure. This analysis shows that the c-axis (Kc = 89.1(10) GPa) is 6% more compressible than the a-axis (Ka = 94.8(6) GPa), with little of the anisotropy commonly observed in other layered structures. The high K' is indicative of the similarity to layers of the brucite structure. The introduction of interlayer cation polyhedra to the structure serves to reduce both the anisotropy, by reducing the compressibility perpendicular to the sheets, and the ability to shear, by increasing the coherence between layers.

The volume dependence of thermal pressure in perovskite and other minerals

This is a review paper concerning the thermal pressure, P TH, of solids and the conditions under which it is independent of volume. When P TH is independent of V, the general equation of state (EoS) reduces from P(V,T)=P 1(V,0)+P TH (V,T) to P(V,T)=P 1(V,0)+P TH V 0,T , and thus is separated into two independent mathematical functions. P 1( V,0) is the isothermal EoS. Four tests of thermoelastic data are shown to determine the T and V range in which P TH is independent of volume. Eighteen solids are examined. Most of these are minerals, but two metals, three alkali metals and three noble gases are also included. The focus is on three lower mantle minerals, MgSiO 3, MgO, and CaSiO 3. For these three minerals (∂ P TH/∂ V) T vanishes at conditions of the lower mantle, but P TH is a function of V at ambient conditions. However, for most solids, (∂ P TH/∂ V) T becomes zero at high temperature. The behavior of (∂ P TH/∂ V) T is apparently not correlated with such properties as crystal class, chemical composition, bonding type, and anharmonicity. The vanishing of (∂ P TH/∂ V) T is strictly a high temperature property of solids.

Tidally Induced Gap Formation in Protostellar Disks: Gap Clearing and Suppression of Protoplanetary Growth

We present the results of numerical simulations of protostellar accretion disks that are perturbed by a protoplanetary companion that has a much smaller mass than the central object. We consider the limiting cases where the companion is in a coplanar, circular orbit and is initially embedded in the disk. Three independent numerical schemes are employed, and generic features of the flow are found in each case. In the first series of idealized models, the secondary companion is modeled as a massless, orbiting sink hole able to absorb all matter incident upon it without exerting any gravitational torque on the disk. In these simulations, accretion onto the companion induces surface density depression and gap formation centered on its orbital radius. After an initial transitory adjustment, the accretion rate onto the sink hole becomes regulated by the rate at which viscous evolution of the disk can cause matter to diffuse into the vicinity of the sink hole orbit, and thus the sink hole grows on a disk viscous timescale. In the second series of comprehensive models, the companion's gravity is included. When the tidal torque exerted by the companion on the disk becomes important, the angular momentum exchange rate between the companion and the disk induces the protoplanetary accretion to reduce markedly below that in the idealized sink hole models. Whether this process is effective in inhibiting protoplanetary accretion depends on the equation of state and disk model parameters. For polytropic or isothermal equations of state, we find, in basic agreement with earlier work, that when the mean Roche lobe radius of the companion exceeds the disk thickness and when the mass ratio, q, between the companion and the central object exceeds ~40/, where is the effective Reynolds number, a clean deep gap forms. Although precise estimation is rendered difficult due to the limitation of numerical schemes in dealing with large density contrasts, the generic results of three series of simulations indicate that accretion onto sufficiently massive protoplanets can become ineffective over the expected disk lifetimes in their neighborhood. We suggest that such a process operates during planetary formation and is important in determining the final mass of giant planets.

Thermal effect and equation of state in solids

In the present paper, an attempt has been made to include the thermal effect into an isothermal equation of state using the Debye approximation representing volume as a function of pressure. The calculations are done in case of NaCl, CsCl, Mo, and W. The present calculations are in good agreement with the reported results.

Analysis of isothermal equations of state for magnesium oxide at high temperatures

Analysis of isothermal equations of state for magnesium oxide at high temperatures

Simulations of Collisions between Two Gas‐rich Galaxy Disks with Heating and Cooling

Particle hydrodynamics (SPH) simulations are presented of direct collisions between two model galaxies, most consisting of a rigid halo and a gas disk. Local self-gravity is also computed in the gas. The companion galaxy in these simulations is about one-third of the mass of the primary, and its disk is half the size. An adiabatic equation of state is combined with simple approximations for the effects of radiative cooling and local heating due to young star activity, which allows a continuous range of thermal phases to develop. These terms and multiple phases have not generally been included in galaxy collision simulations to date. Their effects are assessed in part by repeating runs with an isothermal equation of state and comparing the results. One model with a star plus gas disk is also included for comparison. These models are most relevant to interactions involving low surface brightness, or other late-type galaxies with extensive gas disks, including the precursors to well-known ring galaxies like the Cartwheel and VII Zw 466. In the simulations, the companion impact is slightly off center in the target disk, as is probably the case in these systems. In all cases, clear ring waves develop in the primary despite the disruption of parts of the disk by impact shocks. The gas density in the disk of the primary is initialized to values slightly below the gravitational instability threshold throughout, and the ring waves induce star formation in all the heating and cooling models. The structure of the waves and other interaction morphologies are found to be quite similar on large scales in both isothermal and heating/cooling cases, despite the fact that at certain stages large quantities of gas are heated above the initial temperature in the latter. On a finer scale, there are clear differences, including the fact that star formation heating in ring waves increases the vertical scale height of the primary gas disk and delays spoke development. The companion disk is largely disrupted in most of these simulations, and a substantial mass of gas is splashed out into a bridge connecting the two potential centers. The companion disk reforms by accreting gas out of the bridge, though generally in a different plane than its initial one. There is also a good deal of infall back onto the primary disk. Although heated by impact, the gas in the bridge cools rapidly. However, kinematic expansion prevents it from reaching threshold density, and there is no star formation heating there. A comparison run with a diskless companion produced no significant bridge, so in this type of collision the bridge is primarily a hydrodynamic phenomenon. The amount of material pushed out into the splash bridge and how much of it comes from each galaxy depends on the relative orientation of the disks at impact. This orientation also affects how much bridge material accretes onto each galaxy. The onset of accretion is initially delayed but then accelerates to a peak and declines thereafter in both galaxies. The infall is spatially asymmetric and is primarily located in well-defined streams. Most of the accreted gas ends up in the central regions of the model galaxies, but only after spiraling around the center and passing through one or more shocks. Accretion heating is substantial, and is shown to inhibit or delay global star formation enhancements. The thermal effects of the impact between galaxies are short-lived, but the models predict that accretion and young star heating effect the global thermal phase balance for a much longer period. The magnitude and duration of these effects also depend on the relative orientation of the disks at impact. Thus, the postcollision Hubble type of the companion is a sensitive function of initial orientation.

THE VOLUME DEPENDENCE OF THERMAL PRESSURE IN SOLIDS

I investigate the conditions under which the thermal pressure, PTH, is independent of volume in solids. When PTH is independent of V, the equation of state (EoS) reduces fromP(V,T)=P1(V,0)+PTH(V,T) toP(V,T)=P1(V,0)+PTH(V0,T) and thus is separated into two independent functions. P1(V, T) is the isothermal EoS. It is shown that PTH is independent of volume at ambient conditions for NaCl and Mg2SiO4, but not for MgO, gold, or Al2O3.The approach is to establish the differential equation relating αKT and V,V∂(αKT)∂V=αKT(δT−K′). and then find the V, T conditions under which δT = K′. Here α is the thermal expansivity; KT is the isothermal bulk modulus; δT is a thermoelastic parameter known as the Anderson-Grüneisen parameter, δT = −(1/αKT)(∂KT/∂T)P = (∂ ln α/∂ ln V)T; and K′ is (∂KT/∂T)T. The main effort is used in finding the V, T dependence of the dimensionless quantities, δT and K′ and especially the difference, δT − K′ in terms of V and T. © 1997 Elsevier Science Ltd. All rights reserved