Analytical Equation Of State Research Articles

TAMie Force Field for Alkanethiols: Multifidelity Gaussian Processes for Dealing with Scarce Experimental Data.

This study extends the transferable anisotropic Mie potential (TAMie) to alkanethiols. The force field parameters are optimized by using an analytic equation of state as a surrogate model. Given the lack of experimental density data at elevated temperatures where Monte Carlo simulations have high statistical precision, the equation of state is supplemented by a linear multifidelity Gaussian process approach to bridge the temperature gap. Force field parameters are adjusted by minimizing squared deviations of calculated vapor pressures and liquid densities from experimental data of 1-propanethiol, 1-butanethiol and 1-pentanethiol leading to small mean absolute relative deviations in liquid densities and vapor pressures. The force field is transferable to higher 1-thiols, as shown for 1-hexanethiol and 1-octanethiol. Individual parameter sets are provided for methanethiol and ethanethiol. The shear viscosity of pure substances is predicted in fair agreement with experimental data, considering that it is not included in the parametrization. Further, the phase behavior of binary mixtures of alkanethiols with alkanes is studied, and predictions of the TAMie model are found in excellent agreement with experimental data.

NNEoS : Neural network-based thermodynamically consistent equation of state for fast and accurate flash calculations

Equations of state (EOS) correlate thermodynamic properties and are essential for flash calculations. However, solving for an EOS can be time-consuming, and EOS do not precisely represent physical reality, causing the deviation of flash results from phase equilibrium data. In this work, we propose a neural network-based EOS (NNEoS) inherently satisfying thermodynamic consistency. NNEoS first predicts the residual Gibbs energy and then derives other thermodynamic properties through differentiation. NNEoS can be trained using an analytical EOS and then serve as a reliable, computationally efficient substitute. NNEoS can also be fine-tuned with experimental data to better match flash results to experimental data. We evaluate the performance of NNEoS against analytical EOS on three case studies, including binary and multicomponent mixtures with and without cross-association. The results show that NNEoS achieves significantly faster flash calculations via GPU-based parallel computing and offers superior predictive accuracy after fine-tuning compared to analytical EOS.

Molecular Dynamics simulations and discrete perturbation theory for systems interacting via the parabolic-well pair potential

In this work, we performed Molecular Dynamics (MD) simulations to investigate the thermodynamic properties, including vapor pressures, critical points, pressure, excess energy, and vapor-liquid equilibrium, for a continuous version of the parabolic-well (PW) pair potential within the attractive range of 1.5 ≤λ≤3.0. In addition to conducting extensive MD simulations, we have developed an analytical equation of state for the Helmholtz free energy of particles interacting via the PW pair potential. This development was accomplished within the framework of discrete perturbation theory, involving discrete steps to accurately represent the general shape of the PW potential. Remarkably, our theoretical approach based on discrete perturbation theory demonstrates excellent agreement with MD simulations across a wide range of densities, temperatures, and pressures for the PW pair potential. Our findings demonstrate that PW fluids, characterized by long-range attractive interactions, deviate from the principle of corresponding states. However, for short-range attractive interactions, the fluids moderately follow the Ising universality class. Finally, the integration of discrete perturbation theory with MD simulations proves to be an efficient method for exploring the behavior of molecular fluids and colloidal systems.

An analytic and complete equation of state for condensed phase materials

Analytic equations of state (EOS) are intended to reproduce theoretical and experimental data in a single phase portion of the thermodynamic space. We devise a complete and thermodynamically consistent model with four distinct features: (1) a reference isotherm that remains thermodynamically stable, (2) a flexible specific heat model based on a fourth-order rational polynomial, (3) a Grüneisen parameter that depends on specific volume and temperature, and (4) pressure and internal energy functions that can be inverted analytically in temperature. The model aims to improve the accuracy of existing equations of state while remaining computationally efficient. To demonstrate its features, we include calibrations for single-crystal pentaerythritol tetranitrate (PETN), liquid nitromethane (NM), and hexagonal close-packed beryllium (Be) metal. The parameter optimization uses the specific heat capacity, Grüneisen parameter, and static compression curves obtained from density functional theory for the crystalline solids and molecular dynamics simulations for liquid NM. We also present a velocity autocorrelation function that yields accurate phonon densities of states for the EOS calibration from the molecular dynamics trajectories. Each of the three calibrations is constrained to enforce the ambient state from experimental measurements and validated against experimental Hugoniot data from multiple sources. We also include one-dimensional hydrodynamic simulations of the isentropic compression experiments for beryllium conducted at the Z facility.

The role of detonation condensates on the performance of 1,3,5-triamino-2,4,6-trinitrobenzene (TATB) detonation

Thermochemical models of detonation are widely used to estimate energy delivery, but they are based on the assumption that the carbon-rich condensates (soot) formed during detonation are very similar to bulk carbon. We present an analytic equation of state (EOS) based on experimental detonation data for soot formed during the detonation of triaminotrinitrobenzene (TATB)-based high explosives. X-ray photoelectron spectra of several detonation soots are used to determine the elemental nitrogen abundance, with surprisingly high values for TATB. The proposed TATB soot EOS is highly compressible at low pressures and shares some features of glassy carbon, exhibiting graphite- and diamond-like behavior as a function of pressure. We demonstrate the influence of formed soot on detonation performance, including a lowering of the detonation velocity at typical charge densities, and a more compressive product Hugoniot at overdriven conditions. The soot model improves the accuracy of thermochemical calculations for TATB-based explosives across a wide range of states. Detonation velocity predictions for HMX (cyclotetramethylene-tetranitramine)-TATB blends with 80% or more TATB content, as well predictions for 1,3-diamino-2,4,6-trinitrobenzene (DATB) and 3-nitro-1,2,4-triazol-5-one (NTO), which share some features with TATB, are also improved.

Development of a fully analytical equation of state using ab initio interaction potentials. Application to pure simple fluids: Noble gases Ne, Ar, Kr, and Xe

In this work, we have extended the first-order mean spherical approximation (FMSA) equation of state (EOS) of Tang and Lu [J. Chem. Phys., 99, 9828 (1993)] in order to predict accurately fluid phase diagrams using ab initio potentials. At this level of development, we restricted to simple spherical compounds.The original equation based on hard-core two-Yukawa (HC2Y) was improved by the following:•Adding a new approximate simple correction to include higher orders in the development of Tang and Lu to get an analytical model closer to the full MSA.•Including a term to a better account for the long-range behavior of two-body potentials.•Adding a three-body interaction term (1st order perturbation) based on Axilrod-Teller potential.A general recommendation for setting the hard-sphere diameter value was also given.The predictive capability of the EOS was tested in a systematic manner. First, we checked that various simple theoretical potential (Mie n-6 and Exp n-6) phase diagrams are well reproduced. Second, the EOS was used for predicting fluid phase diagrams of the pure noble gases fluid series Ne, Ar, Kr, and Xe using realistic (i.e. derived from ab initio) two-body and three-body interaction potentials. The model is in good agreement with the experimental phase diagrams.As a conclusion, the simple statistical-mechanics-based fully analytical EOS developed in this work predicts accurately fluid phase diagrams of simple spherical compounds using ab initio potentials. No adjustable parameter was used. This EOS constitutes a strong basis for future development of a predictive model based on ab initio potentials and applicable to more complex compounds.

A unified model for the evolution of cataclysmic variables

ABSTRACT We give an updated version of the analytical equation of state used in the Cambridge stellar evolution code (stars) as a free to use open-source package that we have used to model cool white dwarfs (WDs) down to temperatures $\log _{10}(T_\mathrm{eff}/\mathrm{K})\, =\, 3$. With this update in the stars code, we model the secular evolution of cataclysmic variable (CV) stars using a double dynamo model wherein there is an interplay between two α − Ω dynamos, one in the convective envelope and the other at the boundary of a slowly rotating shrinking radiative core and the growing convective envelope. We confirm that this model provides a physical formalism for the interrupted magnetic braking paradigm. In addition, our model also provides a mechanism for extra angular momentum loss below the period gap. We construct the relative probability distribution of orbital periods Porb using the mass distribution of WDs in CVs and find that our model excellently reproduces the period gap and the observed period minimum spike in CV distribution. We also compare the evolutionary trajectories from our model with those of other empirical models and find agreement between the two. We also report good agreement between our modelled systems and observational data.

Analytic equation of state for the Buckingham Exp(α, m) fluids at high temperatures and densities based on the Ross variation perturbation theory

High temperatures and densities analytical thermodynamic properties are derived for the general Buckingham Exp(α, m) fluids by: a) using the Ross analytical (RA) variation perturbation theory developed by Sun [Mol. Phys. 105 (2007) 3139] for multi-Yukawa potentials; b) using a two-Yukawa potential matching the Exp(α, m), and c) introducing correcting terms which account for the discrepancies between the original Exp(α, m) potential and its two-Yukawa representation. We call this method the Ross analytical corrected theory (RAC), which leads to analytic expressions for the excess Helmholtz free energy, excess internal energy and the compressibility factor Z (i.e. the equation of state). They are valid at high temperatures and densities, since they are obtained from the Ross theory, which extends the range of validity of perturbation theories to higher densities. The matching two-Yukawa potential can also be considered as a reference potential for a first-order perturbation theory, which after further simplification leads to an analytic augmented van der Waals (avdW) theory for the above thermodynamic properties of the Exp(α, m) fluids. For the Exp(11.5, 6) and Exp(14.5, 6) fluids, the excess internal energy, compressibility factor Z and reduced pressure obtained from the RAC and avdW theories are compared with Monte Carlo (MC) simulation results corresponding to both the Exp and TY potentials. The overall accuracy of the RAC and avdW theories for the Exp(11.5, 6) fluid is also assessed in comparison to the Ross numerical (RN) theory (called also MCRSR for Mansoori, Canfield, Rasaiah, Stell and Ross) and to more complex methods such as the KLRR-T (Kang, Lee, Ree, Ree-Theostar) and the HMSA/C (hypernetted mean spherical approximation with Chebyshev approximants) theories, which require more extensive numerical calculations.

NUMERICAL SIMULATION OF SUPERCRITICAL AND SUBCRITICAL INJECTION OF CRYOGENIC NITROGEN BASED ON THE HOMOGENEOUS EQUILIBRIUM MODEL OF TWO-PHASE FLOW

Multidimensional numerical simulations of the outflow of the dense turbulent jet of cryogenic nitrogen into a chamber filled with nitrogen at normal temperature are performed using the homogeneous equilibrium model of the two-phase flow and the new accurate analytical equation of state (EOS) of real gas for nitrogen. The applicability of the new EOS is verified in a wide range of density (from 0 to the liquid density in the triple point) and temperature (from 100 K to 5000 K). The results of calculations are compared with available experimental data on nitrogen density variation in the jets. The satisfactory agreement of the results is obtained. The developed EOS allows the identification of two-phase flow regions and the mass fractions of liquid and gas based on the local instantaneous values of density and temperature in the jet without using the two-fluid flow model.

Thermodynamics of equilibrium alkali plasma. Simple and accurate analytical model for non-trivial case

Analytical equation of state for pure alkali metals (lithium, sodium, potassium, rubidium and cesium) in equilibrium gas phase is proposed. This equation has a simple form, generalizes the equation of state for a perfect gas and is universal for all alkali elements. It correctly reproduces the experimental data for the equilibrium gas phase over a wide range of pressures (up to ∼105 Pa) and temperatures (up to ∼3⋅103 K). On the basis of this equation of state, expressions for thermal and caloric coefficients as well as for other physical characteristics are obtained. The results of this study confirm feasibility of the principle of corresponding states in relation to the group of alkali elements.

Multiphase Coexistences in Rod-Polymer Mixtures.

Using recently derived analytical equations of state for hard rod dispersions, we predict the phase behavior of athermal rod–polymer mixtures with free volume theory. The rods are modeled as hard spherocylinders, while the nonadsorbing polymer chains are described as penetrable hard spheres. It is demonstrated that all of the different types of phase states that are stable for pure colloidal rod dispersions can coexist with any combination of these phases if polymers are added, depending on the concentrations, rod aspect ratio, and polymer–rod size ratio. This includes novel two-, three-, and four-phase coexistences and isostructural coexistences between dilute and concentrated phases of the same kind, even for the more ordered (liquid) crystal phases. This work provides insight into the conditions at which particular multiphase coexistences are expected for well-defined model colloidal rod–polymer mixtures. We provide a quantitative map detailing the various types of isostructural coexistences, which confirms an early qualitative hypothesis by Bolhuis et al. (J. Chem. Phys.107, 19971551).

The EOS/resolution conspiracy: convergence in proto-planetary collision simulations

ABSTRACT We investigate how the choice of equation of state (EOS) and resolution conspire to affect the outcomes of giant impact (GI) simulations. We focus on the simple case of equal-mass collisions of two Earth-like 0.5-M⊕ proto-planets showing that the choice of EOS has a profound impact on the outcome of such collisions as well as on the numerical convergence with resolution. In simulations where the Tillotson EOS is used, impacts generate an excess amount of vapour due to the lack of a thermodynamically consistent treatment of phase transitions and mixtures. In oblique collisions this enhances the artificial angular momentum (AM) transport from the planet to the circum-planetary disc reducing the planet’s rotation period over time. Even at a resolution of 1.3 × 106 particles, the result is not converged. In head-on collisions, the lack of a proper treatment of the solid/liquid-vapour phase transition allows the bound material to expand to very low densities, which, in turn, results in very slow numerical convergence of the critical specific impact energy for catastrophic disruption $Q_{\rm {RD}}^{*}$ with increasing resolution as reported in prior work. The simulations where ANalytic Equation Of State (ANEOS) is used for oblique impacts are already converged at a modest resolution of 105 particles, while head-on collisions converge when they evidence the post-shock formation of a dense iron-rich ring, which promotes gravitational re-accumulation of material. Once sufficient resolution is reached to resolve the liquid-vapour phase transition of iron in the ANEOS case, and this ring is resolved, the value of $Q_{\rm {RD}}^{*}$ has then converged.

Global equation of state and the phase transitions of the hard disc system

The hard disc system plays a fundamental role in the study of two-dimensional matters. High-precision compressibility data and a complete picture of the phase transitions have been reported for all the phases and phase transition regions for the hard disc system with numerical simulation results. However, thorough descriptions of the system and phase transitions depend on analytical equations of state (EoS) over the entire density range, such that various thermodynamic properties can be calculated with the EoS. While majority of EoS’s published are for the stable fluid region only, few attempted the liquid-hexact transition region. All the EoS’s currently available are incapable of quantitative descriptions of the phase transitions. Here we construct a simple EoS to reproduce high-precision simulation data for all the stable liquid, liquid-hexatic transition region and hexatic phase. A global EoS is then obtained when the new EoS is smoothly united with a revisited EoS for the solid phase. Using this global equation, we are able to accurately identify all the phases and the phase transitions from the stable liquid to hexatic, then to solid phases.

Analytic equation of state for Mie(α, β) fluids based on an improved Ross variation perturbation theory

The analytic expressions of thermodynamic properties obtained by Sun for multi-Yukawa fluids from the Ross variation perturbation theory (Mol. Phys. 105 (2007) 3139) can be used to study other practical fluids over wide ranges of temperature and density by mimicking their intermolecular potentials with two-Yukawa (TY) functions. Using the Lennard-Jones LJ (12, 6) as an example, Sun noticed that the accuracy obtained is sensitive to the choice of the fitted TY potential. To remedy this drawback we propose in this work to introduce into Sun Ross analytical (RA) theory correction terms that take into account the discrepancies between the more general Mie(α, β) potential (the LJ is the Mie(12, 6)) and its two-Yukawa representations leading to an improved Ross analytical corrected (RAC) theory. By using three different TY potentials fitted on the LJ from the literature the corresponding excess Helmholtz free energy, excess internal energy and compressibility factor Z obtained from the Ross analytical corrected (RAC) theory are seen to be all in good agreement with the values obtained from the original Ross numerical (RN) theory applied directly to the LJ potential for a wide range of temperatures and densities. These results show that the introduction of these analytical correcting terms increases the accuracy of Sun Ross analytical (RA) theory, which also becomes less dependent on the choice of the TY parameters.

Rigidly rotating perfect fluid stars in 2+1 dimensions

Cataldo has found all rigidly rotating self-gravitating perfect fluid solutions in 2+1 dimensions with a negative cosmological constant $\Lambda$, for a density that is specified a priori as a function of a certain radial coordinate. We rewrite these solutions in standard polar-radial coordinates, for an arbitrary barotropic equation of state $p(\rho)$. For any given equation of state, we find the two-parameter family of solutions with a regular centre and finite total mass $M$ and angular momentum $J$ (rigidly rotating stars). For analytic equations of state, the solution is analytic except at the surface, but including at the centre. Defining the dimensionless spin $\tilde J:=\sqrt{-\Lambda}\,J$, there is precisely one solution for each $(\tilde J,M)$ in the region $|\tilde J|-1<M<|\tilde J|$, which consists of parts of the point particle region $M<-|\tilde J|$ and overspinning regions $|\tilde J|>|M|$. In an adjacent compact part of the black hole region $|\tilde J|<M$ (whose extent depends on the equation of state), there are precisely two solutions for each $(\tilde J,M)$. Hence exterior solutions exist in all three classes of BTZ solution (black hole, point particle and overspinning), but not all possible values of $(\tilde J,M)$ can be realised as stars. Regardless of the values of $\tilde J$ and $M$, the causal structure of all stars for all equations of state is that of anti-de Sitter space, without horizons or closed timelike curves.

Analytical evaluation of third and fourth virial coefficients for hard disk fluids in narrow channels and equation of state

The third and fourth virial coefficients of hard disks in narrow channels have been evaluated as a series in powers of the channel width deviations from the one dimensional limit. We have obtained analytically the coefficients of the power series to very high order. These virial coefficients are used in truncated virial series to provide approximate analytical equation of state (EOS). The EOS is shown to accurately describe Monte Carlo simulations over a wide range of pressures and densities near the one dimensional limit. Explicit expressions for the free energy, chemical potential, compressibility and EOS as a function of the channel widths are given. These results represent progress toward understanding the long standing problem of the thermodynamic crossover from a one dimensional to quasi one dimensional fluid. They are also useful in a number of applications. We mention the examples of thermodynamic perturbation theory and stochastic dynamic mobility in anomalous single-file diffusion.

An Eulerian multimaterial framework for simulating high-explosive aquarium tests

Aquarium tests of cylindrical high-explosive charges provide optical data of the detonation front velocity and shape, propagation of the shock wave in the surrounding water, and expansion rates of the detonation products behind the front. Data from aquarium experiments is often used for calibration of reactive burn models based on phenomenological equations of state (EOS) and reaction rate laws. This paper presents a multimaterial numerical modeling framework to solve the 2D axisymmetric reactive Euler equations for high-explosive aquarium tests, in particular for ammonium nitrate - fuel oil (ANFO) explosives. An extension of the Ghost Fluid Method (GFM) is used to handle the dynamic material interfaces for the ANFO explosion products, the charge-confining material (polymethyl methacrylate PMMA), and the surrounding water. This study analyzes the sensitivity of calculations (both computational efficiency and numerical accuracy) to different algorithms for the material interface models including the original GFM versus Riemann solver-based strategies. A novel method for defining the left and right states in the interfacial Riemann problem eliminates the need for sorting or nodal interpolation during the projection along the material interface. Numerical tests indicate that populating the interface node values using the Riemann solution mitigate the overheating error observed in steady-state calculations. Solution convergence and computational efficiency are explored as a function of the spatial and temporal order of the schemes. Results from the computational model with analytical equations of state and fitted reaction rate parameters show very good quantitative agreement with experimentally observed detonation front velocity, reaction products expansion, and shock wave propagation in the surrounding water for a cylindrical ANFO charge encased in PMMA. The proposed modeling framework, in conjunction with experimental tests, provides a reliable tool to assess equations of state and reaction rate expressions for reactive burn models of confined high explosives.

A new equation of state applied to planetary impacts

We present a new analytical equation of state (EOS), which correctly models high pressure theory and fits well to the experimental data of ɛ-Fe, SiO2, Mg2SiO4, and the Earth. The cold part of the EOS is modeled after the Varpoly EOS. The thermal part is based on a new formalism of the Gruneisen parameter, which improves behavior from earlier models and bridges the gap between elasticity and thermoelasticity. The EOS includes an expanded state model, which allows for the accurate modeling of material vapor curves. The EOS is compared to both the Tillotson EOS and ANEOS model, which are both widely used in planetary impact simulations. The complexity and cost of the EOS is similar to that of the Tillotson EOS, while showing improved behavior in every aspect. The Hugoniot state of shocked silicate material is captured relatively well and our model reproduces vapor curves similar to that of the ANEOS model. To test its viability in hydrodynamical simulations, the EOS was applied to the lunar-forming impact scenario and the results are presented in Paper II and show good agreement with previous work.

Discrete potential fluids in the supercritical region

Supercritical fluids are an interesting area of research because their use in industrial processes has environmental benefits. Since their processes are low energy consumption they are considered as green solvents of the future. However, the study of the supercritical region remains a controversial issue, particularly, the possibility to find three different regime: gas-like, liquid-like and a mixture of them. The existence of a line (Widom line) or a region (delta Widom) that delimit this behavior is still under discussion. In this work we assume that this border is a line, like a continuation of a vapor-liquid saturation curve in the pressure-temperature plane. The Widom line is defined by the locus of points of maximum correlation length. Taking advantage of the relation that exists between the correlation length with the response functions and scalar curvature in the vicinity of the critical point, we build the Widom line with the locus of extrema of response functions and scalar curvature, and use the coincidence of these curves to determine the end of the Widom line. The Widom lines for Square-Well of variable range and Hard-Core Lennard-Jones systems are obtained using analytical equations of state. We propose a quantitative criterion to determine the end of this line. Besides, we give a new and alternative way of determining this line using the locus of the pressure inflection points (obtained in the density-pressure plane) and look for the coincidence of these points with the locus of the maxima of the isothermal compressibility in the temperature-pressure plane.

Main and Interactive Effects of Four Factors on CO2 Storage in Fractured Nanopores

In this paper, main and interactive effects of four important factors, temperatures, adsorption thicknesses, fracture apertures, and feed CO2 concentrations, on the thermodynamic phase behavior for CO2 storage processes in the fractured nanopores of shale/tight reservoirs with adsorptions are investigated. First, a modified analytical equation of state is developed by considering the effects of confinements and intermolecular interactions, which is applied to predict the confined pure/mixing fluid phase behavior in fractured nanopores coupled with a novel empirical correlation for the adsorption thickness and the fracture geometry equation. Second, the aforementioned four important factors are studied to evaluate their main and interactive effects on the phase behavior of pure CO2, N2, O2, Ar, alkanes of C1–10, and their mixtures. It is found that all the pure and mixing fluid pressures monotonically and linearly increase to different extents with an increasing temperature. Moreover, the pressures/critical shifts and critical properties perform downward and upward parabola curves with respect to the adsorption thicknesses, respectively. On the other hand, the pressures/critical shifts are monotonically decreased, while the critical properties increase with the fracture apertures increasing from 0.01 up to 10 nm and remain constant afterward. By increasing the feed CO2 concentrations, the critical shifts and pressures for all of pure and mixing fluids are increased, while both the critical pressures and temperatures decrease. In addition, three interactive effects on the phase behavior are analyzed that the effect of a single factor behaves differently with the variations of any other factors. Finally, the amounts of the N2, O2, Ar, and light alkanes of C1–4 are suggested to be controlled for CO2 storage in the fractured nanopores because additions of these components strongly affect the mixture phase behavior.