Hard-sphere Equation Of State Research Articles

A Perturbed Hard-Sphere Equation of State for Alkali Metal Alloys

A perturbed hard-sphere equation of state for liquid alkali metals has been employed to calculate the liquid density of alkali metal alloys over a wide range of temperature. Two temperature-dependent parameters appear in the equation of state, which are universal functions of the reduced temperature, i.e., two scale parameters are sufficient to calculate the temperature-dependent parameters, and hence, to predict the equation of state. In this article, calculated results of the liquid density of binary molten alloys of Na–K and K–Cs over the whole range of concentration at temperatures from the freezing point up to several hundred kelvin above the boiling point reproduce accurately the experimental PVT data points.

SIMPLIFIED HARD-SPHERE AND HARD-SPHERE CHAIN EQUATIONS OF STATE FOR ENGINEERING APPLICATIONS

A simple hard-sphere equation of state is proposed. This hard-sphere equation is a ratio of second-order polynomials that meets the ideal gas and close-packed density limits. It predicts the compressibility of hard-sphere fluids at low and medium densities to within a degree of quality similar to the widely used Carnahan-Starling equation. In addition, the proposed equation performs better at high densities, particularly near the close-packed density. An expression is also derived to relate the site-site correlation function at contact for hard dimers with the site-site correlation function at contact for hard spheres. With this relationship, the thermodynamic perturbation-dimer theory (TPT-D) of hard-sphere chains is simplified. The new theory performs comparably with the TPT-D when the compressibility factors of hard-sphere chain fluids containing up to 201-mer are predicted, however, it has the advantages of both simplicity and accuracy. From a practical perspective, this theory can be used to construct equations of state for polymer solutions or fluid systems containing short- and long-chain molecules.

Thermodynamic properties of inverse power fluids

The local scaling behavior of the radial distribution function of the soft sphere or inverse power, r-n potential, fluid leads to a formula for the equation of state. From this formula different analytic forms for the compressibility factor, Z, have been derived. In the first, Z is expressed as a product of three functions, the hard sphere equation of state and two other functions incorporating the effects of the potential softness. In the second formula, the compressibility factor is cast in terms of the position and height of the first peak in the radial distribution function. In the final form, Z can be expressed as an exponential function which depends entirely on a combination of the virial coefficients. In each case Z is an explicit expression which has the correct low density limiting behavior and is accurate up to the freezing density for all packing fractions and circa n>or=12. Expressions are derived for the various component functions required for the different forms of Z, and relations between them are established. The compressibility factor manifests a maximum value or "ridge" when plotted as contours on the density-softness plane. It starts for the softer fluids at lower densities, increases with particle stiffness, and crosses the freezing line at n congruent with 33. From the compressibility factor other thermodynamic quantities can be obtained and the density-softness dependence of the infinite frequency limit elastic properties been determined. A self-consistent expression is derived for the effective hard sphere packing fraction (or equivalently, diameter), valid for all packing fractions and circa n>12. The effective hard-sphere diameter is compared with the formulas of Barker and Henderson, and Wheatley.

Are the energy and virial routes to thermodynamics equivalent for hard spheres?

The internal energy of hard spheres (HS) is the same as that of an ideal gas, so that the energy route to thermodynamics becomes useless. This problem can be avoided by taking an interaction potential that reduces to that of the HS in certain limits. In this paper, the square-shoulder (SS) potential characterized by hard-core diameter σ′, soft-core diameter σ > σ′ and shoulder height ε is considered. The SS potential becomes that of the HS if (i) ε → 0, or (ii) ε → ∞, or (iii) σ′ → σ or (iv) σ′→0 and ε → ∞. The energy-route equation of state for the HS fluid is obtained in terms of the radial distribution function for the SS fluid by taking limits (i) and (ii). This equation of state is shown to exhibit, in general, an artificial dependence on the diameter ratio σ′/σ. If, furthermore, the limit σ′/σ → 1 is taken, the resulting equation of state for HS coincides with that obtained through the virial route. The necessary and sufficient condition to obtain thermodynamic consistency between both routes for arbitrary σ′/σ is derived.

A comparison between semi-empirical and molecular-based equations of state for describing the thermodynamic of supercritical micronization processes

Recently, supercritical fluids have been widely studied as media for fine particle formation and the thermodynamic description of multi-phase and multi-component systems are of fundamental importance for exploiting the use of supercritical fluid processes at industrial level. This paper compares the capability of three equations of state, based on different physical models, for describing phase equilibrium of systems containing compressed gases and supercritical fluids. In particular, a semi-empirical Peng–Robinson, a lattice based Sanchez–Lacombe and an off-lattice theory, such as perturbed-hard-sphere-chain theory, were used. Calculations were performed for vapor–liquid equilibria for organic solvent–supercritical anti-solvent systems, solid–fluid and solid–liquid–fluid equilibria for solute–supercritical fluid systems, and solid–liquid–fluid equilibria for solute–solvent–anti-solvent ternary systems. For binary organic solvent–supercritical fluid systems, all equations of state fairly reproduce both equilibrium and volumetric properties. For the solid-supercritical fluid systems, the binary interaction parameters were obtained by fitting solubility experimental data; these values fairly reproduce melting point depression at high-pressure only when cubic equation of state and off-lattice model were used. Phase behaviors of ternary systems were calculated using binary interaction parameters fitted on binary systems. Solubility of high molecular weight compounds in solvent–anti-solvent systems are fairly reproduced only by the perturbed hard sphere equation of state. The comparison between the three models was performed in terms of statistical errors and methods for parameter determination. The results here reported may give useful information in order to select the more appropriate and suitable model for reasonably representing the phase behaviors involved in micronization processes.

Equation of state and structural properties of the Weeks-Chandler-Andersen fluid

Molecular dynamics simulations have been carried out for the equation of state and percolation properties of the Weeks-Chandler-Andersen (WCA) system in its fluid phase as functions of density and temperature. The compressibility factor Z collapses well for the various isotherms, using an effective particle diameter for the WCA particle which is (in the usual WCA reduced units) sigma(e)=2(16)(1+T)(16), where T is the temperature. A corresponding "effective" packing fraction is zeta(e)=pisigma(e) (3)N6V, for N particles in volume V, which therefore scales out the effects of temperature. Using zeta(e) the simulation derived Z can be fitted to a simple analytic form which is similar to the Carnahan-Starling hard sphere equation of state and which is valid at all temperatures and densities where the WCA fluid is thermodynamically stable. The data, however, are not scalable onto the hard sphere equation of state for the complete packing fraction range. We explored the continuum percolation behavior of the WCA fluids. The percolation distance sigma(p) for the various states collapses well onto a single curve when plotted as sigma(p)sigma(e) against zeta(e). The ratio sigma(p)sigma(e) exhibits a monotonic decrease with increasing zeta(e) between the percolation line for permeable spheres and the glass transition limit, where sigma(p)sigma(e) approximately 1. The percolation packing fraction was calculated as a function of effective packing fraction and fitted to an empirical expression. The local coordination number at the percolation threshold showed a transition between the soft core and hard core limits from ca. 2:74 to 1:5, as previously demonstrated in the literature for true hard spheres. A number of simple analytic expressions that represent quite well the percolation characteristics of the WCA system are proposed.

Equation of state for hard-spheres and excess function calculations of binary liquid mixtures

Using our previously proposed matrix method, an equation of state for hard spheres is presented, which can reproduce the exact values of the first-eight virial coefficients. This equation meets both the low density and the close-packed limits and can predicts the first order fluid–solid phase transition of hard spheres. The results obtained show that the new equation of state can correlate the simulation data of compressibility factor up to high densities better than other equations of state. The new equation of state is extended to mixtures of hard spheres and excess functions of various binary liquid mixtures are calculated using the perturbation theory of Leonard–Henderson–Barker. The results are compared with existing theoretical and experimental data and with those calculated by other hard-sphere equations of state. It is seen that the results obtained by the new equation of state is quite satisfactory compared to other equations of state for the hard spheres and mixture of hard spheres.

The Nonequilibrium Phase and Glass Transition Behavior of β-Lactoglobulin

Concentrated solutions of bovine β-lactoglobulin were studied using osmotic stress and rheological techniques. At pH 6.0 and 8.0, the osmotic pressure was largely independent of NaCl concentration and could be described by a hard sphere equation of state. At pH 5.1, close to the isoelectric point, the osmotic pressure was lower at the lower NaCl concentrations (0 mM, 100 mM) and was fitted by an adhesive hard sphere model. Liquid-liquid phase separation was observed at pH 5.1 at ionic strengths of 13 mM and below. Comparison of the liquid-liquid and literature solid-liquid coexistence curves showed these solutions to be supersaturated and the phase separation to be nonequilibrium in nature. In steady shear, the zero shear viscosity of concentrated solutions at pH 5.1 was observed at shear rates above 50 s −1. With increasing concentration, the solution viscosity showed a progressive increase, a behavior interpreted as the approach to a colloidlike glass transition at ∼60% w/w. In oscillatory shear experiments, the storage modulus crossed the loss modulus at concentrations of 54% w/w, an indication of the approaching glass transition. Comparison of the viscous behavior with predictions from the Krieger-Dougherty equation indicates the hydrodynamic size of the protein decreases with increasing concentration, resulting in a slower approach to the glass transition than a hard sphere system.

Evaluation of the acoustic non-linearity parameter of a liquid using hard sphere equation of state

A general equation for the non-linearity parameter has been derived in terms of the compressibility factor, such that it can be evaluated using any hard sphere equation of state. Then the Carnahan–Starling equation of state was used to obtain B/A values. The method was found to be useful for determining the values of the ratio B/A for liquids in which a molecule can be regarded as a hard sphere. The results of the calculations obtained are compared with values in the literature for pure liquids, and satisfactory agreement is found.

A perturbed hard-sphere equation of state for refractory metals

A perturbed hard-sphere equation of state developed previously for liquid alkali metals has been employed to calculate the liquid density of refractory metals over a wide range of temperatures and pressures. Two temperature-dependent parameters appear in the equation of state, which are universal functions of the reduced temperature, i.e., two scale parameters are sufficient to calculate the temperature-dependent parameters, and hence, to predict the equation of state. It is shown that the scale constants correlate with the normal boiling point parameters of metals. Our calculations on the liquid density of tantalum, rhenium, molybdenum, titanium, zirconium, hafnium, and niobium from undercooled temperatures up to several hundred degrees above the boiling point and pressures ranging from 0 up to 200 MPa reproduces very accurately the experimental PVT data.

Static properties and time correlation functions of fluids with steeply repulsive potentials

There has been much interest in recent years in the properties of fluids composed of particles that interact through potentials with a ‘tunable’ softness, particularly for highly repulsive particles close to the hard-sphere limit. Much of this interest is driven by the granular media and colloid communities, but there are fundamental issues concerning the physics of fluids in general that can be addressed with such model systems. In this report we continue our series of investigations into the properties of an example of such a fluid, the so-called inverse power or soft-sphere fluid which is composed of particles interacting through a pair potential, ϕ (r)=ϵ (σ/r) n , where n measures the steepness or stiffness of the potential. We review the current state of our knowledge of the properties of such fluids and point out some still unresolved areas. We present the results of new computations for a range of n values and densities. Interest is focused on the time correlation functions as exemplified by the force autocorrelation function, CF (t), and the shear stress correlation function, C s(t). Powles and Heyes [POWLES, J. G. and HEYES, D. M., 2000, Molec. Phys., 98, 917.] showed that at short times the shear stress autocorrelation function fits quite closely to the analytic form, where x is a reduced time that incorporates the effects of n and temperaure, x = (T)*1/2 nt*. T* and T* are the reduced temperature and time in particle units consisting of particle mass and the pair potential parameters, ϵ and σ. We give further supporting evidence for this functional form, both for C s and CF . However, we still lack a theoretical explanation for this behaviour. A formal time expansion of CF (t) with similar approximations to those used in our previous publications for the O(t 2) coefficient predicts a Gaussian analytic form for C(t), therefore posing an unresolved paradox at present. We present a general procedure by which we can derive formally exact expressions for static properties (such as the interaction energy, pressure and infinite frequency elastic moduli) that can be conveniently reduced to relatively simple approximate expressions involving n and an equivalent hard-sphere density and equation of state, which become more accurate in the hard-sphere limit. We also derive a simple (approximate) analytic formula for the mean square force on a particle in terms of n and the hard-sphere equation of state, which is accurate in the hard-sphere limit. We examine the radial distribution functions of these fluids for various n and density, and compare their forms close to contact with the hard-sphere radial distribution functions with appropriate diameters.

Equation of state of inverse power fluids

A new equation of state for the inverse power, r − n potential, fluid is proposed. It is derived on the basis of the local scaling behaviour of its structural properties, without referring to any perturbative scheme, and therefore recourse to an effective hard sphere diameter. It is shown that the general formula for the compressibility factor can be expressed as the product of three functions. The first represents the hard-sphere equation of state at the same packing fraction, and the other two incorporate the effects of the potential softness, again as functions of density. Using computer simulation results, explicit forms for these soft parts have been established, to give an approximate analytic expression for the r − n fluid equation of state. Two different regions, characterized by positive and negative softness ‘compressibility’ have been found.

P.1.077 Escitalopram vs venlafaxine XR treatment ofmajor depressive disorder

In this study, twenty-five hard sphere equations of state (EOSs) were applied to evaluate the compressibility factor of fifty different single electrolyte solutions (Z1). Then by using two well-known mixing rules (Barrio Solana (BS) and Santos et al. (S)) the compressibility factor (Zm) was calculated for 1086 binary mixtures of electrolytes. The obtained results were compared with the simulation data and it was concluded that the Dehghani–Modarress (DM) EOS when applied along with the Santos et al. mixing rule has the highest accuracy (ARD%Z1DM=0.124 and ARD%ZmDM−S=0.466). Also, the mean ionic activity coefficients of a number of 1:1 electrolytes were calculated by applying the hard sphere EOSs with the mixing rules based on the mean spherical approximation (MSA) model where the concentration of cation was considered a function of electrolyte concentration. By comparing these results with reported experimental data it was found that the Ree–Hoover (RH) EOS with Santos et al. mixing rule predicts the mean ionic activity coefficients with the highest accuracy (ARD% γ(RH − S) = 0.206). In continuation of this study the compressibility factor and mean ionic activity coefficient were recalculated by means of the artificial neural network (ANN) method. The optimum architectures of ANN models for predicting the compressibility factor of single and binary mixtures of electrolytes as well as the mean ionic activity coefficient of 1:1 electrolytes were respectively obtained as 2:4:1, 3:3:1 and 3:5:1. The calculated error in the results showed that the mathematical models of the ANN method predicted the abovementioned parameters with better accuracy in comparison with the EOSs and MSA models (ARD%Z1ANN=0.047, ARD%ZmANN=0.022 and ARD % γ(ANN) = 0.014).

Application of the non-primitive MSA-based models in predicting the activity and the osmotic coefficients of aqueous electrolyte solutions

The non-primitive mean spherical approximation (NP-MSA) based models were used to correlate the individual and the mean ionic activity coefficients of the symmetric and asymmetric aqueous electrolyte solutions. The results of the models for the mean ionic activity coefficients were directly used to calculate the osmotic coefficient for the electrolyte solutions studied in this work. In the NP-MSA models, the Ghotbi–Vera and the Boublik–Mansoori–Carnahan–Starling–Leland hard sphere equation of state as the reference system coupled with the non-primitive mean spherical approximation model (NP-GV-MSA and NP-BMCSL-MSA models, respectively). In correlating the mean ionic activity coefficients of electrolyte solutions, while the hard sphere diameters for anion were treated as a concentration-independent adjustable parameter, the diameters for cation were assumed to be concentration dependent. In the use of the non-primitive MSA-based models in correlating the individual ionic activity coefficients of electrolyte solutions, both anion and cation diameters were considered to be concentration dependent. The results for the mean ionic activity coefficients obtained from the NP-GV-MSA model compared favorably with those of the NP-BMCSL-MSA model produced in this work and also with those of NP−BMCSL-MSA available in the literature. It was shown that the NP-GV-MSA model more accurately calculate the mean ionic activity and the osmotic coefficients than those obtained from the NP−BMCSL-MSA models.

Equation of state for hard-spheres

A new equation of state is presented for hard-spheres, which can reproduce the exact values of the first seven virial coefficients. These values are in agreement with the molecular dynamic calculations of Alder and Wainwright [J. Chem. Phys. 33, 1439 (1960)]. Comparisons with the existing equations of state show that the new equation possesses superior ability to describe the hard-sphere phase behavior.

Scaled hard-sphere equations of state for linear fused hard-sphere pure fluids and fluid mixtures

The performance of a theoretically based equation of state, previously developed for hard molecular fluids, is analysed on the basis of computer-simulation data which have recently become available for fluids of linear fused hard-sphere molecules with up to eight monomers. The equation allows one to obtain the excess compressibility factor of hard-body fluids by scaling the excess compressibility factor of an equivalent hard-sphere fluid by means of a parameter which can be obtained from the geometry of the molecules. The agreement between theory and simulation was found to be excellent. The extension of the equation of state to fluid mixtures of hard molecules expresses the excess compressibility factor of the mixture as a scaling of the excess compressibility factor of an equivalent hard-sphere fluid mixture. The scaling parameter is again obtained from the geometry of the molecules involved in the mixture. The equation of state obtained in this way is compared with NPT Monte Carlo simulations for several fluid mixtures of linear fused hard-sphere molecules which are also reported.

A General Expression for the Ordered-Packed Volume Fraction of Hard Spheres of Different Diameters

An expression is proposed to evaluate the ordered-packed volume fraction of hard spheres of unequal diameters in multicomponent mixtures. This value can be used with different mixing rules and with any one-component hard-sphere equation of state (EOS) meeting the correct close-packed limit. It is shown that the new method improves the results produced by conventional mixing rules for the individual radial distribution functions (RDFs) at contact value of hard-sphere mixtures. Particular attention is given to the case of the large diameter ratio especially when the larger spheres are at low concentration. Based on the observation that, in a binary mixture, the value of the RDF at contact value of the larger spheres approaches the value of the RDF at contact value of a one-component hard-sphere fluid, the proposed expression is used to correct the RDF at contact value of a pure hard-sphere fluid in the Santos et al. mixing rule. The EOS obtained by combining the Ghotbi−Vera one-component EOS with the correc...

Equations of state for hard spheres. A review of accuracy and applications

Twenty two different analytical expressions for the compressibility factor of the hard sphere system have been collected and reviewed in order to perform two applications: Firstly, the accuracy of these expressions in reproducing six available computer simulation data sets for the hard sphere system, some of which are very recent, is tested. It is shown that there is not a direct relation between the simplicity or complexity of their analytical form and their accuracy. Secondly, the expressions, together with the Verlet–Weis formula for the temperature dependence of the molecular diameter, have been used to reproduce our molecular dynamics results for the pressure of the Lennard-Jones reference system proposed in the Weeks–Chandler–Andersen theory. We conclude that the approximation of using hard sphere equations of state together with the Verlet–Weis formula is very accurate at temperatures and densities near the Lennard-Jones critical point. At densities close to the triple point or to the liquid curve in the liquid–vapour equilibrium, the coincidence is adequate (deviations around 3%) only if the appropriate hard sphere equation of state is used.

Performance of three mixing rules using different equations of state for hard-spheres

The predictions of the pair radial distribution function (RDF) at contact value, compressibility factor, and chemical potentials for binary and ternary hard-sphere mixtures obtained with the Barrio–Solana, the Santos et al., and the generalized Lebowitz mixing rules are compared using the Carnahan–Starling, Kolafa, and Khoshkbarchi–Vera one-component hard-sphere EOS. An expression for the pair RDF at contact value for Barrio–Solana mixing rule and the expressions for the chemical potentials in multicomponent hard-sphere fluids are derived. As a general rule, the equations of state obtained based on the generalized Lebowitz mixing rule predict better the pair RDF at contact value simulated data. The compressibility factor and the chemical potentials simulation results are generally better predicted by either the Santos et al. or the Barrio–Solana mixing rule, depending on the packing fraction, the composition, and the diameter ratio values. For a binary hard-sphere fluid with large diameter ratio, σ 2/ σ 1=20, in the high-density region and at low mole fractions of the larger component, the Barrio–Solana mixing rule predicts the compressibility factor simulation results considerably more accurately than the other mixing rules studied here.

Application of hard sphere equations of state to the Weeks–Chandler–Andersen reference system

Six analytical expressions (four of which have been published recently) for the equation of state of a hard sphere system, together with two different expressions for the molecular diameter, have been tested in order to reproduce molecular dynamics results for pressure and potential energy of the Weeks–Chandler–Andersen reference system for the Lennard-Jones potential. The best choices for the combination of equations of state and molecular diameters in the calculation of those properties are given. It is shown how that choice may differ for different ranges of temperatures and densities, and how there is not a direct relation between the simplicity or complexity of the analytical expressions and their accuracy.