A simplified hard-sphere equation of state meeting the high and low density limits

A simplified hard-sphere equation of state has been developed, which meets the correct limit for close-packed conditions. It is shown that the proposed equation of state for hard spheres can represent accurately the computer simulation compressibility factor data and virial coefficients over a wide density range. The comparison of the results of the calculations using this equation, the Carnahan-Starling equation, and the two equations proposed by Iglesias-Silva and Hall, shows that the equation proposed here represents the compressibility factor data and the virial coefficients with better accuracy.     

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.     

Virial series for fluids of hard hyperspheres in odd dimensions

A recently derived method [R. D. Rohrmann and A. Santos, Phys. Rev. E 76, 051202 (2007)] to obtain the exact solution of the Percus-Yevick equation for a fluid of hard spheres in (odd) d dimensions is used to investigate the convergence properties of the resulting virial series. This is done both for the virial and compressibility routes, in which the virial coefficients B(j) are expressed in terms of the solution of a set of (d-1)/2 coupled algebraic equations which become nonlinear for d>/=5. Results have been derived up to d=13. A confirmation of the alternating character of the series for d>/=5, due to the existence of a branch point on the negative real axis, is found and the radius of convergence is explicitly determined for each dimension. The resulting scaled density per dimension 2eta(1/d), where eta is the packing fraction, is wholly consistent with the limiting value of 1 for d-->infinity. Finally, the values for B(j) predicted by the virial and compressibility routes in the Percus-Yevick approximation are compared with the known exact values [N. Clisby and B. M. McCoy, J. Stat. Phys. 122, 15 (2006)].     

Rescaled density expansions and demixing in hard-sphere binary mixtures

The demixing transition of a binary fluid mixture of additive hard spheres is analyzed for different size asymmetries by starting from the exact low-density expansion of the pressure. Already within the second virial approximation the fluid separates into two phases of different composition with a lower consolute critical point. By successively incorporating the third, fourth, and fifth virial coefficients, the critical consolute point moves to higher values of the pressure and to lower values of the partial number fraction of the large spheres. When the exact low-density expansion of the pressure is rescaled to higher densities as in the Percus-Yevick theory, by adding more exact virial coefficients a different qualitative movement of the critical consolute point in the phase diagram is found. It is argued that the Percus-Yevick factor appearing in many empirical equations of state for the mixture has a deep influence on the location of the critical consolute point, so that the resulting phase diagram for a prescribed equation has to be taken with caution.     

On the relation between virial coefficients and the close-packing of hard disks and hard spheres

The question of whether the known virial coefficients are enough to determine the packing fraction η(∞) at which the fluid equation of state of a hard-sphere fluid diverges is addressed. It is found that the information derived from the direct Pade? approximants to the compressibility factor constructed with the virial coefficients is inconclusive. An alternative approach is proposed which makes use of the same virial coefficients and of the equation of state in a form where the packing fraction is explicitly given as a function of the pressure. The results of this approach both for hard-disk and hard-sphere fluids, which can straightforwardly accommodate higher virial coefficients when available, lends support to the conjecture that η(∞) is equal to the maximum packing fraction corresponding to an ordered crystalline structure.     

Equation of state of a seven-dimensional hard-sphere fluid. Percus-Yevick theory and molecular-dynamics simulations

Following the work of Leutheusser [Physica A 127, 667 (1984)], the solution to the Percus-Yevick equation for a seven-dimensional hard-sphere fluid is explicitly found. This allows the derivation of the equation of state for the fluid taking both the virial and the compressibility routes. An analysis of the virial coefficients and the determination of the radius of convergence of the virial series are carried out. Molecular-dynamics simulations of the same system are also performed and a comparison between the simulation results for the compressibility factor and theoretical expressions for the same quantity is presented.     

Extended pressure—consistent equation for simple fluids

Abstract

A previous generalization of the Percus-Yevick (PY) and hypernetted chain (HNC) equations for simple fluids, involving a density- and temperature-dependent coefficient m, is extended by including a spatial dependence in m. The new approximation yields an exact fourth virial coefficient and, by further requirement, a consistent equation of state from both the virial and compressibility forms. Comparison of calculated results for the hard sphere potential shows an improvement over the PY, HNC, and previous pressure-consistent equations     

‘Exact’ integral equation theory and local formulation for excess thermodynamic properties of hard spheres

A hybrid hard sphere bridge function is proposed, which, in combination with the standard Ornstein–Zernike integral equation, can predict extremely accurately hard sphere compressibility, virial pressure, and correlation function. Second, a local formulation for determination of excess chemical potential is derived out, which, in combination with the present hybrid hard sphere bridge function and OZ integral equation, can predict the excess chemical potential also extremely accurately. The resultant excess entropy is in excellent agreement with that from the Carnahan–Starling equation of state. The present formalism performs excellently over the whole density range, i.e. from zero to freezing density, and is largely superior to a formalism available in the literature.     

Inverse power potentials: virial coefficients and a general equation of state

Virial coefficients up to the seventh are calculated for pair potentials depending on inverse powers of separation, for inverse powers from 5 to 80. Unlike the limiting (infinite inverse power) hard-sphere potential, some virial coefficients for finite inverse power potentials are found to be negative. This makes resummation of the virial series for general inverse power potentials more difficult than that for hard spheres, and some alternative resummation methods are presented and compared. A general equation of state is proposed for fluids of particles interacting through inverse power pair potentials, for inverse powers greater than about 10. This includes the "molecular" inverse power of 12, for which the current results support and extend the results of previous studies.     

Thermodynamic properties of polar fluids from a perturbed-dipolar-hard-sphere equation of state

Based on theoretical results for a system of hard spheres with dipoles, a new equation of state is applied to the correlation of thermodynamic properties for four fluids: argon, ammonia, water and acetonitrile. The reference system has the same dependence on density as that given by the Carnahan-Starling equation, but the coefficients are now functions of temperature through the reduced dipole moment. These coefficients are chosen to match the Padé approximant developed by Rushbrooke, Stell and Hoye for the Helmholtz energy of dipolar hard spheres. The reference system proposed here shows a phase transition for reduced dipole moments greater than 1.9. A simple, empirical perturbation term is added to the reference system to account for induction and dispersion forces. For polar fluids, the equation gives results significantly better than those obtained from conventional cubic equations of state, when using the same limited experimental data for determining equation-of-state parameters.     

Semiclassical statistical mechanics of hard-body fluid mixtures

The thermodynamic properties of semiclassical hard-body fluid mixtures are studied. Explicit expressions are given for the free-energy, equation of state and virial coefficients of the classical hard convex-body fluid mixtures. The numerical results are discussed under different conditions. The agreement with the exact data is good in all cases. The first-order quantum corrections are also studied. The quantum effects depend on the condition, shape parameters L11* and L22*, and concentrations x1 and x2 in general and increase with an increase of packing fraction eta, in particular.     

Higher virial coefficients of four and five dimensional hard hyperspheres

The seventh and eighth virial coefficients for hard hyperspheres are calculated by Monte Carlo techniques. It is found that B(7)/B(2) (6)=0.001 43+/-0.000 13 and 0.000 44+/-0.000 12 in four and five dimensions, respectively, and that B(8)/B(2) (7)=0.000 414+/-0.000 20 in four dimensions. These values are used to investigate various proposed equations of state. Comparisons against the molecular dynamics calculations of Luban and Michels show that their proposed semiempirical form is excellent at higher densities. Moreover, we confirm Santos observation in five dimensions that a suitable linear combination of the Percus-Yevick compressibility and virial equations of state fits the molecular dynamics data nearly as well as any other proposed form.     

The equation of state of isotropic fluids of hard convex bodies from a high-level virial expansion

We have calculated virial coefficients up to seventh order for the isotropic phases of a variety of fluids composed of hard aspherical particles. The models studied were hard spheroids, hard spherocylinders, and truncated hard spheres, and results are obtained for a variety of length-to-width ratios. We compare the predicted virial equations of state with those determined by simulation. We also use our data to calculate the coefficients of the y expansion [B. Barboy and W. M. Gelbart, J. Chem. Phys. 71, 3053 (1979)] and to study its convergence properties. Finally, we use our data to estimate the radius of convergence of the virial series for these aspherical particles. For fairly spherical particles, we estimate the radius of convergence to be similar to that of the density of closest packing. For more anisotropic particles, however, the radius of convergence decreases with increased anisotropy and is considerably less than the close-packed density.     

Thermodynamic properties of van der Waals fluids from Monte Carlo simulations and perturbative Monte Carlo theory

Computer simulations have been performed for fluids with van der Waals potential, that is, hard spheres with attractive inverse power tails, to determine the equation of state and the excess energy. On the other hand, the first- and second-order perturbative contributions to the energy and the zero- and first-order perturbative contributions to the compressibility factor have been determined too from Monte Carlo simulations performed on the reference hard-sphere system. The aim was to test the reliability of this "exact" perturbation theory. It has been found that the results obtained from the Monte Carlo perturbation theory for these two thermodynamic properties agree well with the direct Monte Carlo simulations. Moreover, it has been found that results from the Barker-Henderson [J. Chem. Phys. 47, 2856 (1967)] perturbation theory are in good agreement with those from the exact perturbation theory.     

Shape factors in equations of state. Part II. Repulsion phenomena in multicomponent chain fluids

An equation of state for the multicomponent fluid phase of nonattracting rigid particles of arbitrary shape is presented. The equation is a generalization of a previously presented equation of state for pure fluids of rigid particles; the approach describes the volumetric properties of a pure fluid in terms of a shape factor, zeta, which can be back calculated by scaling the volumetric properties of pure fluids to that of a hard sphere. The performance of the proposed equation is tested against mixtures of chain fluids immersed in a "monomeric" solvent of hard spheres of equal and different sizes. Extensive new Monte Carlo simulation data are presented for 19 binary mixtures of hard homonuclear tangent freely-jointed hard sphere chains (pearl-necklace) of various lengths (three to five segments), with spheres of several size ratios and at various compositions. The performance of the proposed equation is compared to the hard-sphere SAFT approach and found to be of comparable accuracy. The equation proposed is further tested for mixtures of spheres with spherocylinders. In all cases, the equation proved to be accurate and simple to use.     

Calculation of the second virial coefficients of alkali metals by modified Peng–Robinson equation

In this work, the Peng–Robinson (P–R) equation of state has been modified by proposing a new α function for calculating the second virial coefficients of alkali metals. The relationship between α^0.5 and (1???T _r ^0.5 ) is a nonlinear function. The correlation between the second virial coefficient and P–R equation was presented by expanding the P–R equation into its Taylor series form. For P–R equation, the linear correlation between parameters C₁ and C₂ of α function and acentric factors \( \omega \) of alkali metals was proposed. The new α function and its first, second and third derivatives are continuous. The average standard deviations of compressibility factor which calculated by modified P–R equation are less than 4.3%. The second virial coefficients of alkali metals were calculated over the temperature range 600–3000 K by using the modified P–R equation. Comparison with literature data, the new equation provides more reliable and accurate second virial coefficient predictions for alkali metals than the original P–R equation. It is useful to guide and improve calculation of the second virial coefficients of other metal vapors for design and operation of separation processes in vacuum metallurgy.     

Liquid-vapor interface of square-well fluids of variable interaction range

Properties of the liquid-vapor interface of square-well fluids with ranges of interaction lambda=1.5, 2.0, and 3.0 are obtained by Monte Carlo simulations and from square-gradient theories that combine the Carnahan-Starling equation of state for hard spheres with the second and third virial coefficients. The predicted surface tensions show good agreement with the simulation results for lambda=2 and for lambda=3 in a temperature range reasonably close to the critical point, 0.8相似文献   

The equation of state of hard hyperspheres in nine dimensions for low to moderate densities

The equation of state of hard hyperspheres in nine dimensions is calculated both from the values of the first ten virial coefficients and from a Monte Carlo simulation of the pair correlation function at contact. The results are in excellent agreement. In addition, we find that the virial series appears to be dominated by an unphysical singularity or singularities on or near the negative density axis, in qualitative agreement with the recently solved Percus-Yevick equation of state in nine dimensions.     

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.     

Generalized boublick equation: an accurate expression for the equation of state of hard fused spheres

Abstract

A simple expression to calculate the shape factor of hard bodies is proposed. Introducing this factor in the Boublik equation of state, very good results are obtained for hard dumbells and more complicated systems of linear homonuclear hard fused spheres. Agreement with available Monte Carlo results are also satisfactory enough for heteronuclear molecules. Furthermore, the new expression is reduced to the classical shape factor for hard convex bodies and provides a common basis to manage to concave and convex hard bodies.