Approximate Equation Of State Research Articles

Converging shock flows for a Mie-Grüneisen equation of state

Previous work has shown that the one-dimensional (1D) inviscid compressible flow (Euler) equations admit a wide variety of scale-invariant solutions (including the famous Noh, Sedov, and Guderley shock solutions) when the included equation of state (EOS) closure model assumes a certain scale-invariant form. However, this scale-invariant EOS class does not include even simple models used for shock compression of crystalline solids, including many broadly applicable representations of Mie-Grüneisen EOS. Intuitively, this incompatibility naturally arises from the presence of multiple dimensional scales in the Mie-Grüneisen EOS, which are otherwise absent from scale-invariant models that feature only dimensionless parameters (such as the adiabatic index in the ideal gas EOS). The current work extends previous efforts intended to rectify this inconsistency, by using a scale-invariant EOS model to approximate a Mie-Grüneisen EOS form. To this end, the adiabatic bulk modulus for the Mie-Grüneisen EOS is constructed, and its key features are used to motivate the selection of a scale-invariant approximation form. The remaining surrogate model parameters are selected through enforcement of the Rankine-Hugoniot jump conditions for an infinitely strong shock in a Mie-Grüneisen material. Finally, the approximate EOS is used in conjunction with the 1D inviscid Euler equations to calculate a semi-analytical Guderley-like imploding shock solution in a metal sphere and to determine if and when the solution may be valid for the underlying Mie-Grüneisen EOS.

Evaluation of the enthalpy of formation of carbon nanotubes and their phase diagram

The enthalpy of formation of single-wall carbon nanotubes (SWCNTs) and multiwall carbon nanotubes (MWCNTs) of different geometric sizes is evaluated. The region of phase stability of carbon nanotubes (CNTs) of different geometric shapes (with various lengths, inner radii, and numbers of carbon layers) is determined based on the approximate equation of state of nanocarbon. The estimates do not contradict the experimentally measured parameters of phase transformation of MWCNTs into nanodiamonds. The calculated melting point of nanotubes turned out to be higher than that of graphite. The region of phase stability of CNTs may turn out to be wider than the region of thermodynamic stability of graphite.

Enthalpy of formation of single- and multilayer graphene

An approximate equation of state of nanocarbon is used to estimate changes in the enthalpy of formation of single- and multilayer graphene and the graphene-nanodiamond phase equilibrium pressure as a function of geometrical shape, number of layers, and size of a graphene nanoparticle. It is found that upon an increase in the number of layers in graphene with particle sizes of 100–300 nm, the phase equilibrium pressure falls and tends to that of graphite-diamond equilibrium. An agreement between the estimations, the data from MD simulations of the graphene-nanodiamond phase equilibrium, and experimentally measured parameters of the graphene-nanodiamond phase transition was obtained.

Publisher's Note: “Approximate equations of state in two-temperature plasma mixtures” [Phys. Plasmas 21, 022706 (2014)

Publisher's Note: “Approximate equations of state in two-temperature plasma mixtures” [Phys. Plasmas 21, 022706 (2014)

Approximate equations of state in two-temperature plasma mixtures

Approximate thermodynamic state relations for multicomponent atomic and molecular gas mixtures are often constructed by artificially partitioning the mixture into its constituent materials and requiring the separated materials to be in temperature and pressure equilibrium. Iterative numerical algorithms have been employed to enforce this equilibration and compute the resulting approximate state relations in single-temperature mixtures. In partially ionized gas mixtures, there is both theoretical and empirical evidence that equilibrating the chemical potentials, number densities, or partial pressures of the free electrons is likely to produce more accurate results than equilibrating the total pressures. Moreover, in many situations of practical interest the free electrons and heavy particles have different temperatures. In this paper, we present a generalized algorithm for equilibrating the heavy-particle and electron temperatures and a third user-specified independent thermodynamic variable in a two-temperature plasma mixture. Test calculations based on the equilibration of total pressure vs. electron pressure are presented for three different mixtures.

Ab initio prediction of the first and second pressure derivatives of isothermal bulk modulus for the high-pressure rocksalt phase of ZnO

Because of the significance of the first and second pressure derivatives of the isothermal bulk modulus, a detailed theoretical study of the high-pressure rocksalt phase of ZnO under high pressure and temperature has been carried out by means of first-principles density functional theory calculations combined with the quasi-harmonic Debye model in which the phononic effects are considered. Particular attention is paid to the predictions of the isothermal bulk modulus and its pressure derivatives which play central role in the formulation of approximate equations of state for the first time.

Thermal behaviour of chlorite: an in situ single-crystal and powder diffraction study

The high-temperature behaviour of two natural chlorites has been studied by in situ X-ray diffraction: on a single-crystal of clinochlore, triclinic polytype II b -4, space group C 1, with pseudomonoclinic metric and composition (Mg 9.14 Fe 2+ 1.02 Fe 3+ 0.01 Mn 0.01 Ti 0.01 Al 1.76 ) ∑=11.95 (Si 6.32 Al 1.68 ) ∑=8 O 20 (OH) 16 ; by powder diffraction for a chamosite with composition (Mg 3.52 Fe 2+ 4.33 Fe 3+ 1.15 Al 2.85 ) ∑=11.85 (Si 5.45 Al 2.55 ) ∑=8 O 20 (OH) 16 and same symmetry. Unit-cell parameters were measured for both samples up to 550 °C. Diffraction data for structural refinement of clinochlore were collected at 25, 301, 399 and 502 °C. Room temperature 57 Fe Mossbauer spectroscopy measurements were performed on the chamosite sample before and after heating, showing complete Fe oxidation above 500 °C. Mean thermal expansion coefficients of the two samples were: α a = 1.05(3) × 10 −5 , α b = 1.02(3) × 10 −5 , α c = 0.99(5) × 10 −5 and α V = 3.07(7) × 10 −5 /°C for clinochlore and α a = 2.47(25) × 10 −5 , α b = 0.93(5) × 10 −5 , α c = 0.93(12) × 10 −5 and α V = 4.38(25) × 10 −5 /°C for chamosite. These results confirm that the expansion of chlorite depends on the composition, but the dependence is not simply related to the Fe content, as suggested by previous studies. The Si, Al tetrahedral volumes of clinochlore do not significantly change in the T range 25–550 °C. The expansion of M1, M2, M3 and M4 octahedra is 1.5, 1.9, 3.0 and 3.4 % respectively. The interlayer OH–O distances increase of about 1 %, the mean thermal expansion coefficient being 4.5 × 10 −6 /°C, indicating that the hydrogen bonds maintain their moderate strengths also at high temperature. The tetrahedral rotation angle α decreases from 6.38 to 3.34°. A comparison with the modifications induced by pressure on the same sample of clinochlore confirms that the response of the structure to T and P is only approximately opposite. By combining the effects on cell volume, it is possible to formulate the following approximate equation of state: V = V 0 (1 – 1.14 × 10 −2 Δ P + 3.07 × 10 −5 Δ T ), where P is in GPa and T in °Celsius. The equation is valid for a triclinic polytype of chlorite having clinochlore composition.

Thermodynamics of strongly coupled repulsive Yukawa particles in ambient neutralizing plasma: Thermodynamic instability and the possibility of observation in fine particle plasmas

The thermodynamics is analyzed for a system composed of particles with hard cores, interacting via the repulsive Yukawa potential (Yukawa particulates), and neutralizing ambient (background) plasma. An approximate equation of state is given with proper account of the contribution of ambient plasma and it is shown that there exists a possibility for the total isothermal compressibility of Yukawa particulates and ambient plasma to diverge when the coupling between Yukawa particulates is sufficiently strong. In this case, the system undergoes a transition into separated phases with different densities and we have a critical point for this phase separation. Examples of approximate phase diagrams related to this transition are given. It is emphasized that the critical point can be in the solid phase and we have the possibility to observe a solid-solid phase separation. The applicability of these results to fine particle plasmas is investigated. It is shown that, though the values of the characteristic parameters are semiquantitative due to the effects not described by this model, these phenomena are expected to be observed in fine particle plasmas, when approximately isotropic bulk systems are realized with a very strong coupling between fine particles.

Equation of state in the quasi-classical approximation

An approximate equation of state is considered that contains a correction for electron-gas inhomogeneity in the exchange-correlation and kinetic energies in the quasi-classical approximation. The problem of finding the model parameters satisfying the condition P=0 has been solved for metals at a normal density. The results of numerical solution of the model equations are presented for degrees of compression. The values of the quasi-classical parameter determining the approximation accuracy are found. The results are illustrated by the dependences on the degree of compression plotted in the range ρ/ρ0=1–106 and by the spatial dependences. Our approach to construction of the equation of state provides for a quite simple but sufficiently accurate approach to the experimental values of the model parameters. Moreover, the proposed model gives much more self-consistent information as compared to conventional approximations. In particular, the ionicity is calculated for an increase in the density. The model equation can also be used instead of the well-known effective bound-electron potential approximation.

Structural thermal behavior of paragonite and its dehydroxylate: a high-temperature single-crystal study

The lattice constants of paragonite-2M1, NaAl2(AlSi3)O10(OH)2, were determined to 800 °C by the single-crystal diffraction method. Mean thermal expansion coefficients, in the range 25–600 °C, were: αa = 1.51(8) × 10−5, αb = 1.94(6) × 10−5, αc = 2.15(7) × 10−5 °C−1, and αV = 5.9(2) × 10−5 °C−1. At T higher than 600 °C, cell parameters showed a change in expansion rate due to a dehydroxylation process. The structural refinements of natural paragonite, carried out at 25, 210, 450 and 600 °C, before dehydroxylation, showed that the larger thermal expansion along the c parameter was mainly due to interlayer thickness dilatation. In the 25–600 °C range, Si,Al tetrahedra remained quite unchanged, whereas the other polyhedra expanded linearly with expansion rate proportional to their volume. The polyhedron around the interlayer cation Na became more regular with temperature. Tetrahedral rotation angle α changed from 16.2 to 12.9°. The structure of the new phase, nominally NaAl2 (AlSi3)O11, obtained as a consequence of dehydroxylation, had a cell volume 4.2% larger than that of paragonite. It was refined at room temperature and its expansion coefficients determined in the range 25–800 °C. The most significant structural difference from paragonite was the presence of Al in fivefold coordination, according to a distorted trigonal bipyramid. Results confirm the structural effects of the dehydration mechanism of micas and dioctahedral 2:1 layer silicates. By combining thermal expansion and compressibility data, the following approximate equation of state in the PTV space was obtained for paragonite: V/V0 = 1 + 5.9(2) × 10−5T(°C) − 0.00153(4) P(kbar).

Conservative High-Order Finite-Difference Schemes for Low-Mach Number Flows

Three finite-difference algorithms are proposed to solve a low-Mach number approximation for the Navier–Stokes equations. These algorithms exhibit fourth-order spatial and second-order temporal accuracy. They are dissipation-free, and thus well suited for DNS and LES of turbulent flows. The key ingredient common to each of the methods presented is a Poisson equation with variable coefficient that is solved for the hydrodynamic pressure. This feature ensures that the velocity field is constrained correctly. It is shown that this approach is needed to avoid violation of the conservation of kinetic energy in the inviscid limit which would otherwise arise through the pressure term in the momentum equation. An existing set of finite-difference formulae for incompressible flow is generalized to handle arbitrary large density fluctuations with no violation of conservation through the non-linear convective terms. An algorithm which conserves mass, momentum, and kinetic energy fully is obtained when an approximate equation of state is used instead of the exact one. Results from a model problem are used to show both spatial and temporal convergence rates and several test cases are presented to illustrate the performance of the algorithms.

Radiation hydrodynamics of x-ray driven targets

A multigroup radiation transport algorithm is implemented in planar and spherical geometry and coupled to a one-dimensional Lagrangian hydrodynamic code. The radiative transfer equations are derived under the assumptions of local thermodynamic equilibrium (LTE), no scattering, the neglect of Doppler shifts and the neglect of the time derivative. A second-order form of the transfer equation which reproduces the correct result at high and small optical depths is used. The curvature in spherical geometry is dealt with using a ray trace scheme that is particularly suited to very anisotropic incident x-ray sources and the study of preheat at the centre of spherical targets. Preliminary simulations of an indirectly driven National Ignition Facility (NIF) design target using an approximate equation of state, show the performance is degraded because of radiative heating of the shell from the central hot spot just before the time of peak compression. This reduces the maximum density achieved. The maximum temperature is limited by the radiative cooling of the central hot fuel.

Numerical simulation of the instability of an inviscid liquid jet in a coflowing compressible airstream

The nonlinear interfacial instability of a liquid jet in a coflowing compressible airstream is studied numerically. A high-resolution scheme which has second-order accuracy in space and time is coupled with a Lagrangian marker particle algorithm to visualize the large-scale motion of the interfaces in compressible flow. A numerical algorithm based on an approximate equation of state of a compressible liquid is developed to allow this two-fluid system to be governed by the nonlinear unsteady Euler equations in conservative form. The initial growth of small disturbances given by the simulations agrees well with linear theory. The process of jet disruption in compressible flow is demonstrated to consist of the formation of liquid spikes, interweaving of the gas and liquid and stretching and detachment of the liquid main center core.

Oscillation of the radial distribution function

We prove that the radial distribution function oscillates at low density in a system with a short-range nonnegative potential and investigate the branching of the solutions of an approximate equation of state.

Theory and simulation of hard-chain mixtures: Equations of state, mixing properties, and density profiles near hard walls

A combination of theoretical modeling and computer simulation is used to study the equation of state of binary mixtures of hard chains, where each chain is modeled by a series of freely jointed, tangent, hard spheres. Three approximate equations of state are derived, based on our previous work on one-component fluids. These equations contain no adjustable parameters and relate properties of the chain mixture to properties of pure monomer and dimer fluids at the same total volume fraction. Their predictions are tested against Monte Carlo results for the pressure of mixtures of 8-mers and monomers and mixtures of 8-mers and 4-mers, obtained using a hard-wall technique. Very good agreement is obtained using an equation of state developed here, in which the compressibilty factor of the mixture is set equal to the molar average of the compressibility factors of the pure components at the same overall volume fraction, as well as from Wertheim’s second-order thermodynamic theory of polymerization (TPT2). Using the equations developed here, we also examine the mixing properties of hard-chain fluids. For mixing at constant pressure, the free energy and entropy of mixing range from ideal-solution behavior at low pressures to Flory–Huggins behavior at high pressures. For mixing at constant volume fraction, the free energy and entropy of mixing reduce directly to the Flory–Huggins result without recourse to the usual lattice approximations. Site-density profiles obtained from the simulations indicate that chains are depleted near the walls at low densities and are enhanced near the wall at high densities; monomers, by contrast, are enhanced near the walls at all densities.

Oscillation of the radial distribution function and singularity of the approximate equation of state for the hard sphere model

Abstract We prove that the radial distribution function oscillates at low density in a system with a short-range nonnegative potential.

Computer simulations to study the explosive consolidation of powders into rods

In the direct method of explosive consolidation, metal or ceramic powder to be consolidated is placed in a metal tube, which in turn is surrounded by a concentric cylinder of explosive. The explosive is detonated at one end so that the detonation front moves axially along the powder-filled tube. The desired result is a crack-free, uniformly consolidated rod with the density of the starting material. At present, the wall thickness of the hollow cylinder of explosive that best produces the desired result is determined by experiment. We present the results of computer simulations revealing details of this dynamic process that are difficult to obtain experimentally. The simulations show that the best thickness depends on the equations of state of both explosive and powder. Such simulations can guide changes to the experimental geometry, even when approximate equations of state are used, although the best explosive and its thickness must ultimately be determined by experiment.

Roughening transition temperature in the presence of an adsorbing fluid.

The effect on the roughening temperature ${T}_{R}$ of an adsorbing fluid above the substrate is calculated. The partition function is evaluated for three models, the Ising model, the absolute solid-on-solid model, and the discrete Gaussian models, with the addition of a second species in vapor form above the surface. Then with the use of the duality transformation of Jos\'e et al. and the renormalization-group method of Kosterlitz, a relationship between ${T}_{R}$ and the chemical potential of the fluid is obtained. ${T}_{R}$ is then expressed in terms of the ambient pressure by using approximate equations of state. It is found that a significant decrease in ${T}_{R}$ may occur as the pressure is increased, and the conditions under which this effect might be observed are discussed. .Ire ranges the heat capacity, thermal expansion, and heat conduction are calculated using the same parameters for the soliton representation as were used to calculate the temperature dependence of the proton spin-lattice relaxation time. Discontinuities in the thermal properties at the glass transition temperature ${T}_{g}$ are attributed to an effective-mass change of the low-energy solitons. This work was conducted to test, with the use of nuclear magnetic resonance, some recent ideas in the dynamics of amorphous materials.

Hard and Soft-Core Equations of State for Simple Fluids. X. Characteristic Curves and Loci of Cp Extrema for the Lennard–Jones 6–12 Equation of State

Abstract Characteristic curves and loci of Cp extrema are constructed for the Lennard-Jones 6–12 equation of state, using fitted and approximate equations of state, and the exact virial expansion at low densities. The shape of the Cp locus near its termination temperature is analyzed via the third and fourth ‘pressure’ virial coefficients for hard-core and Lennard-Jones systems.

Liquid fluorine properties by the hardsphere model with attractive interactions

We test the validity of an approximate equation of state of real fluids and the expression for surface tension of hardsphere liquids with attractive interactions derived by Sharma and others by obtaining expressions for isothermal compressibility,C 1-parameter and acoustical parameter and relate them to microscopic Gruneisen parameter. The calculated values for liquid fluorine show fair agreement with experimental values.