State Of Hard-sphere Fluid Research Articles

Analytical accurate expressions for radial distribution function and equation of state of hard sphere fluid based on the Percus-Yevick exact solution and a new unifying closure

ABSTRACT New accurate analytic expressions for the radial distribution function and equation of state of hard sphere fluid are obtained by use of its analytic solution from Percus-Yevick closure and by a new unifying closure of the Ornstein-Zernike integral equation. This will be possible by linearisation of the closure to get a set of one-parameter structural functions incorporating both Percus-Yevick and Hyper-Netted Chain closures. This will enable us to deliver very general expressions for the structural functions and equation of state for a potential and fluid. For the most important case of hard sphere fluid the method can be complemented with its known virial coefficients to get accurate analytical expressions for its radial distribution function and equation of state. The expressions are then successfully compared with those of computer simulations for hard sphere fluid at all densities with great accuracy.

Correction: Equations of the state of hard sphere fluids based on recent accurate virial coefficients B5-B12.

Correction for 'Equations of the state of hard sphere fluids based on recent accurate virial coefficients B5-B12' by Jianxiang Tian et al., Phys. Chem. Chem. Phys., 2019, 21, 13070-13077. DOI.

Equations of the state of hard sphere fluids based on recent accurate virial coefficients B5-B12.

By using the recently published simulation data for the compressibility factor (M. N. Bannerman, L. Lue and L. V. Woodcock, J. Chem. Phys., 2010, 132, 084507; S. Pieprzyk, M. N. Bannerman, A. C. Branka, M. Chudak and D. Heyes, Phys. Chem. Chem. Phys., 2019, 21, 6886) and the recently published numerical virial coefficients B5-B12 (R. J. Wheatley, Phys. Rev. Lett., 2013, 110, 200601; C. Zhang and B. M. Pettitt, Mol. Phys., 2014, 112, 1427; A. J. Schultz and D. A. Kofke, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2014, 90, 023301) of hard sphere fluids, we herein report the updated versions of the equations of the state of hard sphere fluids based on the asymptotic expansion method (J. Tian, H. Jiang, Y. Gui and A. Mulero, Phys. Chem. Chem. Phys., 2009, 11, 11213), the Padé approximants (N. Clisby and B. McCoy, J. Stat. Phys., 2006, 122, 15) and the exponential approximants (N. S. Barlow, A. J. Schultz, S. J. Weinstein and D. A. Kofke, J. Chem. Phys., 2012, 137, 204102) and compare them with other recently published equations. It is found that the so-called mKLM equation describes the relation of the compressibility factor versus the packing fraction with the highest accuracy. The asymptotic expansion method based equation Z(-5, 2) is recommended to well describe both the virial coefficients and the compressibility factor.

Virial series expansion and Monte Carlo studies of equation of state for hard spheres in narrow cylindrical pores.

In this paper, the virial series expansion and constant pressure Monte Carlo method are used to study the longitudinal pressure equation of state for hard spheres in narrow cylindrical pores. We invoke dimensional reduction and map the model into an effective one-dimensional fluid model with interacting internal degrees of freedom. The one-dimensional model is extensive. The Euler relation holds, and longitudinal pressure can be probed with the standard virial series expansion method. Virial coefficients B_{2} and B_{3} were obtained analytically, and numerical quadrature was used for B_{4}. A range of narrow pore widths (2R_{p}), R_{p}<(sqrt[3]+2)/4=0.9330... (in units of the hard sphere diameter) was used, corresponding to fluids in the important single-file formations. We have also computed the virial pressure series coefficients B_{2}^{'}, B_{3}^{'}, and B_{4}^{'} to compare a truncated virial pressure series equation of state with accurate constant pressure Monte Carlo data. We find very good agreement for a wide range of pressures for narrow pores. These results contribute toward increasing the rather limited understanding of virial coefficients and the equation of state of hard sphere fluids in narrow cylindrical pores.

A semi empirical compact equation of state for hard sphere fluids at any density

We propose a new semi empirical expression of the equation of state for a hard sphere fluid which is valid in the disordered phases over the whole density range below and above the freezing point. Starting from the existing numerical results for the virial coefficients, we elaborate a compact expression for the equation of state which is compatible both with the low and medium density behaviour (disordered stable phase) and with the asymptotic high density behaviour (disordered metastable phase). The resulting equation of state has a compact form and exhibits a simple pole at the close random packing density and a double pole at a density equal to 1. That equation of state only depends on the following quantities: the virial coefficients B2, B3 ,B4, which can be exactly computed, the random packing density ξ0, which is imposed by statistical geometry, and the residue of the pole in ξ0. The results are in a fairly good agreement with the numerical data.

A comprehensive approach to an equation of state for hard spheres and Lennard Jones fluids

We present a simple method of obtaining various equations of state for hard sphere fluid in a simple unifying way. We will guess equations of state by using suitable axiomatic functional forms (n = 1, 2, 3, 4, 5) for surface tension Smn(r), r ≥ d/2 with intermolecular separation r as a variable, where m is an arbitrary real number (pole). Among the equations of state obtained in this way are Percus—Yevick, scaled particle theory and Carnahan—Starling equations of state. In addition, we have found a simple equation of state for the hard sphere fluid in the region that represents the simulation data accurately. It is found that for both hard sphere fluids as well as Lennard—Jones fluids, with m = 3/4 the derived equation of state (EOS) gives results which are in good agreement with computer simulation results. Furthermore, this equation of state gives the Percus—Yevick (pressure) EOS for the m = 0, the Carnahan—Starling EOS for m = 4/5, while for the value of m = 1 it corresponds to a scaled particle theory EOS.

Equation of state for hard-sphere fluids offering accurate virial coefficients

The asymptotic expansion method is extended by using currently available accurate values for the first ten virial coefficients for hard sphere fluids. It is then used to yield an equation of state for hard sphere fluids, which accurately represents the currently accepted values for the first sixteen virial coefficients and compressibility factor data in both the stable and the metastable regions of the phase diagram.

New equations of state for hard disk fluid by asymptotic expansion method

Using the newly introduced asymptotic expansion method to obtain equations of state for hard sphere fluid, new simple equations of state for hard disks based on known virial coefficients are derived. Comparison of the obtained equations of state with computer simulation data shows that they are accurate in the whole fluid region.

A simple method of generating equations of state for hard sphere fluid

We present in this paper a simple method of obtaining various equations of state for hard sphere fluid in a simple unifying way. Using the first several virial coefficients of hard sphere fluid, we will guess equations of state by using the asymptotic expansion method. Among the equations of state obtained in this way are Percus–Yevick, Scaled Particle Theory, and Carnahan–Starling equations of state. Also by combining the Monte Carlo results on hard sphere fluid with the asymptotic expansion method many other equations of state for hard sphere fluid can be found where all of them give essentially similar results in the region of isotropic hard sphere liquid, i.e., up to η < 0.5, in which η is the packing fraction. In addition we have found a simple equation of state for the hard sphere fluid in the metastable region which represents the simulation data accurately.

Thermodynamically consistent equation of state of hard sphere fluids

The Wiener–Hopf technique has been been applied to solve the Ornstein–Zernike equation for hard sphere fluids and to calculate thereby a thermodynamically consistent equation of state. An analytic form of a thermodynamically consistent equation of state for hard sphere fluids is obtained in which the correlation range is treated as an adjustable parameter. With a suitable choice of the range parameter the equation of state presented is found to be numerically comparable to the Carnahan–Starling equation of state in accuracy.

Equations of state for hard-sphere fluids

Equations of state and contact values of hard-sphere radial distribution functions (rdf's) which are given by a linear combination of the Percus— Yevick and scaled-particle virial expressions are considered. In the one-component case the mixing coefficientθ(η) is, in general, a function of the volume fractionη. In mixtures the coefficientθ(η i ,d i ), in general, depends upon the volume fractionη i , and diameterd i , of each species,i andj. For the contact valuesY ij of the rdf's, the mixing coefficientsΘ ij (η k ) also depend on speciesi andj. Density expansions for the exactθ for the one-component hard-sphere fluid are obtained and compared with several approximations made in earlier works and in our own work, as well as with simulations. For a mixture, it turns out that one cannot obtain the exact fourth virial coefficient by using a linear combination of the Percus-Yevick and scaled-particle virial expressions forY ij unless one allowsΘ ij to depend on mole fractionsx i even at the zeroth order of its density expansion. We also find thatΘ ij must depend on particle speciesi andj in order to satisfy the exact limits obtained earlier by Sung and Stell. A new equation of state for the binary hard-sphere mixture which satisfies all the exact limits we have considered is suggested.

Theory of the Two- and One-Dimensional Rigid Sphere Fluids

The approximate theory of the three-dimensional hard sphere fluid developed by Reiss, Frisch, and Lebowitz has given astonishingly good predictions with little labor. In an attempt to investigate the reason for this result we adduce, in this paper, further evidence for the internal consistency of the approximations of this theory. Thus it is noted that the same equation of state of hard sphere fluid is obtained when one used the ``integral condition'' as when the ``infinity condition'' is used. We have then applied the theory to study the thermodynamic properties, in particular the equation of state, of the rigid sphere fluid in two and one dimensions. The approximate equation of state of the two-dimensional rigid sphere fluid is in good agreement over the range of fluid densities with the results of the machine Monte Carlo calculations by Jacobson and Wood and dynamical machine calculations of Wainwright and Alder. The exact Tonks' equation of state of the one-dimensional rigid sphere fluid is derived in a particularly simple manner.