Hard sphere term

There are several ways of obtaining functionals for nonideal systems. In most cases the free energy functional is expressed as the sum of an ideal gas term, a hard-sphere term, and a term due to attractive forces. Below, I present a scheme by which approximate expression for the free energy functional may be obtained. This approach relies on the relationship between the free energy functional and the direct correlation function. Because the direct correlation functions are defined through functional derivatives of the excess free energy functional, that is,... 

Table IV summarizes the critical parameters obtained from RPM-based analytical theories. We have referred all values for DH pairing theories to expressions including a Carnahan (CS) hard-sphere term [68]. To obtain a well-balanced view, it is certainly not sufficient to refer to critical point predictions only. Nevertheless, some conclusions from such an analysis are pertinent ...

B. A. Cosgrove and J. Walkley, Can.]. Chem., 60,1896 (1982). Scaled Particle Theory of Gas Solubility and Inclusion of the Temperature Dependent Hard Sphere Term. 

The complete DLVO pair potential, dlvo(/), is the sum of the repulsive electrostatic (uy(r)) and the attractive ( w(r)) pair potentials, plus the excluded volume or hard-sphere term mhs(/)> i-e-... 

The value of DFT is evidently dependent on the accessibility and accuracy of the grand potential functional, Si [p(r)]. The usual practice is to treat the molecules as hard spheres and divide the fluid-fluid potential into attractive and repulsive parts. A mean field approximation is used to simplify the former by the elimination of correlation effects. The hard sphere term is further divided into an ideal gas component and an excess component (Lastoskie etal., 1993). The ideal component is considered to be exactly local, since this part of the Helmholtz free energy per molecule depends only on the density at a particular value of r. 

It is instructive to consider some other possible versions [7,31,32,52]. The combination of the DH limiting law for the ionic tree energy with the ideal gas free energy does not yield a phase transition. This is also true, if the electrostatic energy is used instead of the tree energy. In both cases, however, phase transitions are obtained with the vdW hard-core tree energy or the Camahan-Starling hard sphere term. However, then the critical density and the critical temperature come out nearer to that of vdW fluids [7],... 

The osmotic pressure can be written as the sum of two terms a hard-sphere term and an attractive perturbation. Such a perturbative treatment leads to introduce a hard-sphere radius RHg (2), (6). 

This paper is organized as follows. In Sec. 2 we review the MVDW model proposed by Skibinsky et al. ° and take into account more exact hard sphere term from Liu s paper It is demonstrated that improvement of repulsive part doesn t change a topological portrait of phase behavior of polymorphic fluid in the wide range of thermodynamic variables. Section 3 displays the picture of the phase behavior for different parameters of MVDW model and third critical point which didn t observed earlier for this model is clearly established. It... 

We now consider modifications to the repulsive term in the van der Waals equation. Although the van der Waals hard-sphere term is correct at low densities. Figure 4.13 shows that it quickly becomes erroneous as the density is increased the excluded volume is not constant, but depends on density in some complicated way. Therefore we can improve the equation of state by using the Carnahan-Starling form (4.5.4) for Z/jg. Our modified Redlich-Kwong (mRK) equation of state is then [29]... 

In the case of polar fluids, a fluid of dipolar spheres provides the simplest model for discussion, in which the reference potential < o(l 2) is a hard-sphere term and the perturbing w(l 2) is an ideal dipole-dipole term ... 

In eq 6.54 equation Z is the compressibility factor pVjRT, gy is the radial distribution function which gives the probability of finding a molecule of type i at a distance r from a central molecule of type j, Uy is the intermolecular potential whose parameters are and Cy, the prime denotes differentiation with respect to distance r, k is Boltzmann s constant and p the number density. In the development of the VDW-1 theory the intermolecular potential is assumed to be composed of a hard-sphere term plus a long-range attraction, that is given by. 

In the molecular picture behind SAFT a chain consists of ttH hard-sphere segments. These hard-sphere segments are bonded by covalent bonds. The hard-sphere term of both SAFT versions is the sum of two contributions a hard-sphere contribution and a term due the connectivity of these hard-sphere segments, as... 

Gilman [124] and Westwood and Hitch [135] have applied the cleavage technique to a variety of crystals. The salts studied (with cleavage plane and best surface tension value in parentheses) were LiF (100, 340), MgO (100, 1200), CaFa (111, 450), BaFj (111, 280), CaCOa (001, 230), Si (111, 1240), Zn (0001, 105), Fe (3% Si) (100, about 1360), and NaCl (100, 110). Both authors note that their values are in much better agreement with a very simple estimate of surface energy by Bom and Stem in 1919, which used only Coulomb terms and a hard-sphere repulsion. In more recent work, however, Becher and Freiman [126] have reported distinctly higher values of y, the critical fracture energy. ... 

Rasaiah J C and Stell G 1970 Upper bounds on free energies in terms of hard sphere results Mol. Phys. 18 249... 

The above argument shows that complete overlap of coil domains is improbable for large n and hence gives plausibility to the excluded volume concept as applied to random coils. More importantly, however, it introduces the notion that coil interpenetration must be discussed in terms of probability. For hard spheres the probability of interpenetration is zero, but for random coils the boundaries of the domain are softer and the probability for interpenetration must be analyzed in more detail. One method for doing this will be discussed in the next section. Before turning to this, however, we note that the Flory-Huggins theory can also be used to yield a value for the second virial coefficient. 

However, before proceeding with the description of simulation data, we would like to comment the theoretical background. Similarly to the previous example, in order to obtain the pair correlation function of matrix spheres we solve the common Ornstein-Zernike equation complemented by the PY closure. Next, we would like to consider the adsorption of a hard sphere fluid in a microporous environment provided by a disordered matrix of permeable species. The fluid to be adsorbed is considered at density pj = pj-Of. The equilibrium between an adsorbed fluid and its bulk counterpart (i.e., in the absence of the matrix) occurs at constant chemical potential. However, in the theoretical procedure we need to choose the value for the fluid density first, and calculate the chemical potential afterwards. The ROZ equations, (22) and (23), are applied to decribe the fluid-matrix and fluid-fluid correlations. These correlations are considered by using the PY closure, such that the ROZ equations take the Madden-Glandt form as in the previous example. The structural properties in terms of the pair correlation functions (the fluid-matrix function is of special interest for models with permeabihty) cannot represent the only issue to investigate. Moreover, to perform comparisons of the structure under different conditions we need to calculate the adsorption isotherms pf jSpf). The chemical potential of a... 

Now consider the loss term C[f] in equation 9.29. The probability of a loss is proportional to four terms (1) the number of type-1 hard-spheres already in the volume (= /i) (2) the number of type-2 hard-spheres entering the volume (= /2 d V2) (3) the probability, P[ ff ] = P vi,V2, V, V2), that two hard-spheres with velocities V, V2 will actually collide and make transitions to states with velocities hi, V2 t (4) the total volume of the allowed outgoing velocity-space (= d v dPv2)-Integrating over all possible momenta, we find that the loss term is given by... 

Our final form for the Boltzman equation is then obtained by substituting these last two expressions for the gain and loss terms into equation 9.29 and adding the term Vijf x, v, t) to the LHS for the case where there is an external force F Vjj is the gradient operator with respect to v and m is the hard-sphere mass) ... 

To compare molecular theoretical and molecular dynamics results, we have chosen the same wall-particle potential but have used the 6 - oo fluid particle potential. Equation 14, Instead of the truncated 6-12 LJ potential. This Is done because the molecular theory Is developed In terms of attractive particles with hard sphere cores. The parameter fi n Equation 8 Is chosen so that the density of the bulk fluid In equilibrium with the pore fluid Is the same, n a = 0.5925, as that In the MD simulations. 

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