Equations of state for strong compression

Abstract

Various higher order equations of state (EOS) are compared with theoretical predictions for strong compression as well as with different linearization schemes. One specially convenient EOS together with a conjugated linearization scheme is presented as a series expansions with the appropriate asymptotic behaviour at very strong compression. Tests for the uncertainties in EOS extrapolations are discussed. A comparison with literature data for highly compressible solids illustrates the advantages of the present EOS with respect to many other commonly used forms.     

Equation of state for inert gas solids

The equation of state is a fundamental relation to analyse the thermophysical properties of different class of solids and it plays a key role in basic and applied condensed matter physics research. A lot of work has been done in the field of ionic solids, minerals and metals but a very little work is done in the field of inert gas solids. Most of the equations of state failed to explain the properties of inert gas solid because of their abnormal behavior in the low temperature range. In the present paper, Singh-Gupta equation of state has been used to study the properties of these solids. The results obtained using these equations have shown a good agreement with available experimental results. Thus it is shown that these equations of states successfully explain the behavior of inert gas solids.      

An isothermal equation of state for solids

An isothermal equation of state (EOS) for solids, recently suggested by the authors in the realistic form, V/V₀=f(P), with relative volume as the dependent and the pressure as the independent variable, was shown to have an advantage for some close-packed materials in that it allows B′_∞=(∂B_s/∂P)_s(P→∞) to be fitted, and this is where the usual standard equations fail. In the present study, our EOS is applied to a number of inorganic as well as organic solids, including alloys, glasses, rubbers and plastics; varying widely in their bonding and structural characteristics, as well as in their bulk modulus values. A very good agreement is observed between the data and fits. The results obtained are compared with those from two well-known equations, expressible in the realistic form, proposed by Murnaghan and Luban. Further, the results are also compared with those from the widely used two- and three-parameter EOSs, expressible in the unrealistic form only, P=f(V/V₀), proposed by Birch—and also with those from the EOS model of Keane in which B′_∞ is explicitly expressed as an equation of state parameter. The results obtained from our model compare well to these EOSs. Our EOS, in general, yields the smallest mean-squared deviations between data and fits. The values of B′_∞calculated from our EOS are compared with those from Keane's model. Further, we have studied the variation of B′_∞with temperature using the experimental isotherms of Mo and W at 10 different temperatures ranging from 100 to 1000 K, and observed that the values of B′_∞ yielded by our model and that of Keane vary, as expected, within a narrow range. Furthermore, our EOS is applied to study the stability of the fit parameters with variation in the pressure ranges with reference to the isothermal compression data on Mo and W—and also to study the variation of isothermal bulk modulus with pressure, with reference to the ultrasonic data on NaCl and noted a very good agreement with experiment. In addition, our model is applied, with B₀ and B′₀ constrained to the theoretical values, to the five theoretical isotherms of MgO at 300, 500, 1000, 1500 and 2000 K obtained on the basis of a first principles approach—a good agreement is observed with the predictions, and the values of B′_∞ inferred at different temperatures tend to converge to a constant value.     

Applicability of equation of state in extreme compression region and study of diatomic solids under pressure

In the present study, the expression of first pressure derivative of bulk modulus B′?earlier derived by Goyal and Gupta EoS [7] is corrected and the validity of the EoS is then verified in extreme compression region. The expressions of second and third order pressure derivative $(B^{{}^{″}},B^{″\prime })$of bulk modulus are obtained. The values of B′, B²B? to BB″,?Grunesien parameters?λ_∞,?γ_∞,?q_∞ at infinite pressure (P?→?∞)?are calculated using the identities [11–13] to check the validity of the equation in extreme compression region. The Goyal and Gupta EoS is then used to study the volume compression in diatomic solids, LiH and MgO. The values of pressure, bulk modulus and its first order pressure derivative at different compressions are calculated and compared with the results obtained from Hama–Suito EoS. The results justify the validity of the present EoS in high pressure region and the results for diatomic solids are also found in good consistency with the compared results.     

Similarity criteria for Debye temperatures of simple solids at compression

A similarity criterion for the volume dependence of Debye temperatures of simple solids at compression is presented. It is shown that the volume dependences of the characteristic Debye temperatures of various solids fall on a single common curve in dimensionless coordinates. The validity of the model is tested on examples of solids with various types of interatomic forces – molecular crystal, ionic crystal, metals and covalent crystal.     

An equation of state for anisotropic solids under shock loading

An anisotropic equation of state is proposed for accurate extrapolation of high-pressure shock Hugoniot states to other thermodynamics states for shocked single crystals and polycrystalline alloys. The proposed equation of state represents mathematical and physical generalization of the Mie-Grüneisen equation of state for isotropic material and reduces to this equation in the limit of isotropy. Using an anisotropic nonlinear continuum framework and generalized decomposition of a stress tensor [Int. J. Plasticity 24, 140 (2008)], the shock waves propagation along arbitrary directions in anisotropic solids of any symmetry can be examined. The non-associated strength model includes the distortion effect of the yield surface which can be used to describe the anisotropic strength differential effect. A numerical calculation showed that the general pulse shape, Hugoniot Elastic Limits (HELs), and Hugoniot stress levels for aluminum alloy 7010-T6 agree with the experimental data. The results are presented and discussed, and future studies are outlined.     

New isothermal equation of state of solids applied to high pressures

In the present paper, a new two-parameter inverted equation of state (EOS) is developed which is found to be working very well in the high-pressure region. To check its success and validity, this EOS has been applied in a number of solids. The computed volume compression is found to be in very good agreement with the experimental data in the whole range of pressure in all the solids. The minimum and the maximum pressure range used in the present study is 0–320 kbar and 0–3000 kbar, respectively.     

岩盐结构氧化锌物态方程的分子动力学模拟

利用分子动力学方法和有效经验对势模型对ZnO岩盐结构高温高压下的物态方程进行了研究， 发现分子动力学方法得到的ZnO岩盐结构的摩尔体积(300?1273 K，3.2?10.4 GPa)和实验结果吻合；另外，基于经验势模型的可靠性预测了1373?2273 K和0? 50 GPa的ZnO岩盐结构的P-V -T关系，并利用相应的热力学公式拟合得到了ZnO岩盐结构常态下的线性热膨胀系数、等温体模量及其对压力的一阶导数等重要的热力学参量．     

Empirical equations of state for solids

A comparison of three one-parameter empirical equations of state due to Lennard-Jones, Bardeen, and Birch using Bridgman's experimental compression data shows that they do not generally fit the data within experimental error. Heuristic arguments are used to derive a well known three-parameter equation of state which includes these equations as special cases. Several simple two-parameter equations of state, including the Murnaghan equation, can be obtained as special cases of the general form. All of these two-parameter equations are shown to fit Bridgman's data within experimental error with about equal success. It is also shown that the general three-parameter equation, and hence its special cases, is consistent with the observation of Grover, Getting, and Kennedy that the log of the bulk modulus of solids is a linear function of compression. Several of the two-parameter special cases are shown to give satisfactory extrapolations.     

Higher-order implicit strong numerical schemes for stochastic differential equations

Higher-order implicit numerical methods which are suitable for stiff stochastic differential equations are proposed. These are based on a stochastic Taylor expansion and converge strongly to the corresponding solution of the stochastic differential equation as the time step size converges to zero. The regions of absolute stability of these implicit and related explicit methods are also examined.     

Modeling pressure-ionization of hydrogen in the context of astrophysics

Abstract

The recent development of techniques for laser-driven shock compression of hydrogen has opened the door to the experimental determination of its behavior under conditions characteristic of stellar and planetary interiors. The new data probe the equation of state (EOS) of dense hydrogen in the complex regime of pressure ionization. The structure and evolution of dense astrophysical bodies depend on whether the pressure ionization of hydrogen occurs continuously or through a “plasma phase transition” (PPT) between a molecular state and a plasma state. For the first time, the new experiments constrain predictions for the PPT. We show here that the EOS model developed by Saumon and Chabrier can successfully account for the data, and we propose an experiment that should provide a definitive test of the predicted PPT of hydrogen. The usefulness of the chemical picture for computing astrophysical EOS and in modeling pressure ionization is discussed.     

First-principles equations of state for Al,Cu, Mo,and Pb to ultrahigh pressures

Abstract

Using accurate electronic-structure, interatomic-potential, and statistical methods, first-principles theoretical equations of state (EOSs) have been developed for the prototype metals Al, Cu, Mo, and Pb over wide ranges of volume and temperature. Emphasis is given to both the cold compressed solid, relevant to diamond-anvil-cell (DAC) experiments, and to the hot compressed liquid, relevant to multimegabar shock-wave experiments. The A1 EOS has recently been used to analyze nuclear-impedance-match (NIM) shock data on AI-Cu, AI-Mo, and A1-Pb sample pairs in the 5–30 Mbar regime [J. Appl. Phys. 69, 2981 (1991)]. Below 10 Mbar, calculated theoretical Hugoniots are in very good agreement with both absolute and NIM shock data for all four metals. Above 10 Mbar, some quantitative uncertainties remain, especially for Mo. Useful analytic representations of the calculated 300-K isotherms below 10 Mbar have also been developed for fcc Al, Cu, and Pb and for bcc Mo. The theoretical isotherms are in excellent agreement with a variety of ultrasonic, DAC, and reduced-shock data obtained in the assumed phases below 3 Mbar. At higher pressures complicating solid-solid phase transitions are expected and/or observed in Al, Mo, and Pb, but not in Cu, which is consequently a good candidate as a static ultrahigh-pressure EOS standard.     

Note on the quantum displacement of the critical point

The quantum corrections to the law of corresponding states are studied by calculating the critical pressure, temperature, and density to first order in Planck's constanth on an exactly soluble model. The ratio of the critical parameters to the corresponding classical values are found to be (p _c/p _c ⁰)^1/2=_c/_c ⁰ = T_c/T_c ⁰ = 1–0.67, with=h _c ^1/3(mkT _c)^–1/2. The critical ratio is independent ofh to first order. The results are compared with critical data for noble gases and hydrogen isotopes.     

Vibrational relaxation of a molecule in strong interaction with a reservoir: Nonmonotonic temperature dependence

This theoretical study of the vibrational relaxation of a molecule in interaction with a reservoir uncovers a noteworthy temperature (T) dependence of the time evolution of the relaxation. Its rate increases with T in one interval but decreases in another. The feature arises not for a weak molecule-reservoir interaction but only for coupling strong enough to require polaronic dressing transformations. Our treatment, based on a recent generalization of the well-known Montroll–Shuler equation for relaxation and an explicit calculation of bath correlations from the microscopically specified Hamiltonian, could provide an alternative explanation of an “inverted” T-dependence of relaxation in an experimental report by Fayer and collaborators on W(CO)₆ dissolved in CHCl₃.     

Assessing the equation of state and comparing it with other relationships used for determining the surface tension of solids

Some facts regarding the equation of state (EQS) in calculating the surface tension of solids by means of contact angle measurements were manifested. In the present investigation, it was mathematically proved that the surface tension of a solid as estimated by the EQS is in fact equivalent to the Zisman critical surface tension for that same solid. Additionally, the applicability of the EQS's approach in attaining the surface tension of powdered solids by the aid of the capillary rise procedure is also discussed and its limitations are clarified. Furthermore, a methodology was devised so that the surface tension of solids as determined by the EQS could be compared with those calculated by approaches using components of surface tension. This methodology revealed that the applications of approaches based on the geometric mean (i.e. Owens/Wendt and van Oss et al. relationships) are restricted to achieving only high surface tensions of solids.     

Equation of state for the Universe from similarity symmetries

In this paper we proposed to use the group of analysis of symmetries of the dynamical system to describe the evolution of the Universe. This method is used in searching for the unknown equation of state. It is shown that group of symmetries enforce the form of the equation of state for noninteracting scaling multifluids. We showed that symmetries give rise to the equation of state in the form p =-Λ + w ₁ρ(a) + w ₂ a ^β + 0 and energy density ρ = Λ+ρ₀₁ a ^-3(1+w) +ρ₀₂ a ^α +ρ₀₃ a ^-3, which is commonly used in cosmology. The FRW model filled with scaling fluid (called homological) is confronted with the observations of distant type Ia supernovae. We found the class of model parameters admissible by the statistical analysis of SNIa data.We showed that the model with scaling fluid fits well to supernovae data. We found that Ω_m,0 ≃ 0.4 and n ≃ -1 (β = -3n), which can correspond to (hyper) phantom fluid, and to a high density universe. However if we assume prior that Ω_m,0 = 0.3 then the favoured model is close to concordance ΛCDM model. Our results predict that in the considered model with scaling fluids distant type Ia supernovae should be brighter than in the ΛCDM model, while intermediate distant SNIa should be fainter than in the ΛCDM model. We also investigate whether the model with scaling fluid is actually preferred by data over ΛCDM model. As a result we find from the Akaike model selection criterion: it prefers the model with noninteracting scaling fluid.     

Analysis of equation of state for nanomaterials

An equation of state (EOS) recently proposed for nanomaterials is discussed critically. Different possible forms of the EOS are discussed with their correlations. We have considered 20 nanomaterials for this purpose, viz. CdSe, Rb₃C₆₀, carbon nanotube, γ-Fe₂O₃, ε-Fe (hexagonal iron), MgO, γ-Al₂O₃ (67 nm), α-Fe₂O₃, α-Fe (filled nanotube), TiO₂ (anatase ), 3C-SiC (30 nm), TiO₂ (rutile phase), Zr_0.1Ti_0.9O₂, γ-Si₃N₄, Ni-filled MWCNT, Fe-filled MWCNT, CeO₂ (cubic fluorite phase and orthorhombic phase), germanium (49 nm), GaN (wurtzite phase) and SnO₂ (rutile phase) (14 nm). It is found that the change in the form of EOS does not improve the results. This demonstrates the validity of the EOS proposed for nanomaterials. The EOS is also used to study the effect of temperature on compression of Ni (20 nm). It is found that there is small shift in isotherm due to increase in the temperature. The results have been found to present a good agreement with the available experimental data.     

用多次冲击压缩方法研究稠密氢氦等摩尔混合气体的物态方程

采用液氮冷却以及气体增压技术，制备了～30MPa及～90K初始状态的氢、氦等摩尔混合气体样品.以二级轻气炮作为加载工具，用不同灵敏度设置的两套多通道瞬态高温计系统获得完整、清晰的稠密氢、氦混合气体多次冲击压缩过程的光谱辐射强度信号.并建立起相应的实验数据处理和分析技术，获得了5—140GPa范围内氢、氦混合气体一至五次冲击雨贡纽物态方程，以及一次、二次和四次冲击温度实验数据.流体变分理论和离解模型用来分析和解释所获得的测量结果. 关键词： 氢、氦混合气体 多次冲击压缩 光谱辐射强度历史 物态方程     

A law of large numbers and a central limit theorem for the Schrödinger operator with zero-range potentials

We consider the Schrödinger operator with zero-range potentials onN points of three-dimensional space, independently chosen according to a common distributionV(x). Under some assumptions we prove that, whenN goes to infinity, the sequence converges to a Schrödinger operator with an effective potential. The fluctuations around the limit operator are explicitly characterized.