Hard-sphere expansion

The hard-sphere expansion (HSE) theory is incorporated in the Kirkwood-Buff fluctuation integral to predict solubilities of solids in supercritical carbon dioxide and ethylene (Kwon and Mansoori, 1993). 

A general method of predicting the effective molecular diameters and the thermodynamic properties for fluid mix-tures based on the hard-sphere expansion conformal solution theory is developed. The method of Verlet and Weis produces effective hard-sphere diameters for use with this method for those fluids whose intermolecular potentials are known. For fluids with unknown potentials, a new method has been developed for obtaining the effective diameters from isochoric behavior of pure fluids. These methods have been extended to polar fluids by adding a new polar excess function, to account for polar contributions in a mixture. A new set of pseudo parameters has been developed for this purpose. The calculation of thermodynamic properties for several fluid mixtures including CH —C02 has been carried out successfully. 

It is important to realize that the diameters needed for thermodynamic calculations do not necessarily represent a true minimum attainable separation distance between molecules. The objective is rather to determine optimal or effective diameters which give best results when used with a particular method of dealing with the contributions of molecular attraction. In this chapter the effective diameters sought are to be used specifically with the hard-sphere expansion (HSE) conformal solution theory of Mansoori and Leland (3). This theory generates the proper pseudo parameters for a pure reference fluid to be used in predicting the excess of any dimensionless property of a mixture over the calculated value of this property for a hard-sphere mixture. The value of this excess is obtained from a known value of this type of excess for a pure reference fluid evaluated at temperature and density conditions made dimensionless with the pseudo parameters. For example, if Xm represents any dimensionless property for a mixture of n nonpolar constituents at mole fractions xu x2,. . . x -i at temperature T and density p, then ... 

Chang, J. I. C. Improvement of the Hard Sphere Expansion Conformal... 

This leads to the third virial coefficient for hard spheres. In general, the nth virial coefficient of pairwise additive potentials is related to the coefficient7) in the expansion of g(r), except for Coulombic systems for which the virial coefficients diverge and special teclmiques are necessary to resiim the series. 

The high-temperatiire expansion, truncated at first order, reduces to van der Waals equation, when the reference system is a fluid of hard spheres. 

In the numerical solution the matrix structure is evaluated from Eqs. (44)-(46). Then Eqs. (47)-(49) with corresponding closure approximations are solved. Details of the solution have been presented in Refs. 32 and 33. Briefly, the numerical algorithm uses an expansion of the two-particle functions into a Fourier-Bessel series. The three-fold integrations are then reduced to sums of one-dimensional integrations. In the case of hard-sphere potentials, the BGY equation contains the delta function due to the derivative of the pair interactions. Therefore, the integrals in Eqs. (48) and (49) are onefold and contain the contact values of the functions... 

In this section we restrict ourselves to solvent effects that are due to the first term in the expansion of AG in Eq. (9.4.2). This is equivalent to the assumption that all the particles involved are hard particles, hence only their sizes affect the solvation Gibbs energies. We shall also assume for simplicity that the solvent molecules are hard spheres with diameter a. All other molecules may have any other geometrical shape. 

A quite different approach to the detonation product state has been to treat it as solidlike. Jones and Miller6 performed equilibrium calculations on TNT with this idea in mind. They used an equation in which the volume was a virial expansion in the pressure. Other solidlike equations are cited in Ref. 2, but these have mostly been used for computing the state parameters with an assumed product state. The modified Kistia-kowsky-Wilson equation of interest to us liere appears to be one of several possible compromises between the hard-sphere molecule approach and the solid state approach. 

The concept of free volume varies on how it is defined and used, but is generally acknowledged to be related to the degree of thermal expansion of the molecules. When liquids with different free volumes are mixed, that difference contributes to the excess functions (Prausnitz et al., 1986). The definition of free volume used by Bondi (1968) is the difference between the hard sphere or hard core volume of the molecule (Vw= van der Waals volume) and the molar volume, V ... 

The volume exclusion effect has received recently renewed attention.5 9 Paunov et al.5 argued that the excluded volume should be taken as eight times the real volume of the ions (this corresponds to the first-order correction in the virial expansion for a hard-sphere fluid)... 

The perturbation theories [2, 3] go a step beyond corresponding states the properties (e.g., Ac) of some substance with potential U are related to those for a simpler reference substance with potential Uq by a perturbation expansion (Ac = Aq + A + Aj + ). The properties of the simple reference fluid can be obtained from experimental data (or from simulation data for model fluids such as hard spheres) or corresponding states correlations, while the perturbation corrections are calculated from the statistical mechanical expressions, which involve only reference fluid properties and the perturbing potential. Cluster expansions involve a series in molecular clusters and are closely related to the perturbation theories they have proved particularly useful for moderately dense gases, dilute solutions, hydrogen-bonded liquids, and ionic solutions. 

The extension of the ROZ formalism to confined molecular fluids has recently been carried out for adsorbed diatomic molecules [6] and dipolar fluids confined in hard sphere matrices [18, 19], In the case of ionic matrix, new features of the system have to be taken into account. On one hand, we have now a two component matrix (with positive and negative ions). This case was already considered in [14, 15] for the primitive model electrolyte adsorbed in an electroneutral charged matrix. On the other hand, we have to deal with two different temperatures the matrix temperature, (h (at which the ionic fluid is equilibrated before quenched) and the fluid temperature fi, at which the fluid is adsorbed in the solid matrix. As usual when dealing with molecular fluids one starts with an expansion of the correlation functions in terms of spherical harmonics as follows,... 

The experimental observations show that stratification is always observed when spherical colloidal particles are present in the film at a sufficiently high volume fraction therefore, a realistic explanation can be that the stepwise transitions are manifestations of the oscillatory structural forces. The role of the hard spheres this time is played by the colloidal particles rather than by the solvent molecules. The mechanism of stratification was studied theoretically in Reference 346, where the appearance and expansion of black spots in the stratifying films were described as being a process of condensation of vacancies in a colloid crystal of ordered micelles within the film. 

Note that Equations 5.480a, 5.480b, and 5.480e are exact, while in Equations 5.480c and 5.480d the terms up to xr in a series expansion are taken into account. For hard spheres, we can calculate ... 

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