Hard sphere fluids approximation

The assumption of Gaussian fluctuations gives the PY approximation for hard sphere fluids and tire MS approximation on addition of an attractive potential. The RISM theory for molecular fluids can also be derived from the same model. 

In applying this approach to the equation of state of the hard-sphere fluid [57], it was found that the molecular-field approximation... 

In the present article, we focus on the scaled particle theory as the theoretical basis for interpreting the static solution properties of liquid-crystalline polymers. It is a statistical mechanical theory originally proposed to formulate the equation of state of hard sphere fluids [11], and has been applied to obtain approximate analytical expressions for the thermodynamic quantities of solutions of hard (sphero)cylinders [12-16] or wormlike hard spherocylinders [17, 18]. Its superiority to the Onsager theory lies in that it takes higher virial terms into account, and it is distinctive from the Flory theory in that it uses no artificial lattice model. We survey this theory for wormlike hard spherocylinders in Sect. 2, and compare its predictions with typical data of various static solution properties of liquid-crystalline polymers in Sects. 3-5. As is well known, the wormlike chain (or wormlike cylinder) is a simple yet adequate model for describing dilute solution properties of stiff or semiflexible polymers. 

There exists another prescription to extend the hydrodynamical modes to intermediate wavenumbers which provides similar results for dense fluids. This was done by Kirkpatrick [10], who replaced the transport coefficients appearing in the generalized hydrodynamics by their wavenumber and frequency-dependent analogs. He used the standard projection operator technique to derive generalized hydrodynamic equations for the equilibrium time correlation functions in a hard-sphere fluid. In the short-time approximation the frequency dependence of the memory kernel vanishes. The final result is a... 

Here ji(qa) is the spherical Bessel function of order l,g(a) is the radial distribution function at contact, and f = /fSmn/Anpo2g a) is the Enskog mean free time between collisions. The transport coefficients in the above expressions are given only by their Enskog values that is, only collisional contributions are retained. Since it is only in dense fluids that the Enskog values represents the important contributions to transport coefficient, the above expressions are reasonable only for dense hard-sphere fluids. Earlier Alley, Alder, and Yip [32] have done molecular dynamics simulations to determine the wavenumber-dependent transport coefficients that should be used in hard-sphere generalized hydrodynamic equations. They have shown that for intermediate values of q, the wavenumber-dependent transport coefficients are well-approximated by their collisional contributions. This implies that Eqs. (20)-(23) are even more realistic as q and z are increased. 

The excess chemical potential for a dilute hard-sphere fluid in the Paunov et al. approximation (eqs 1 and 6f) is given by5... 

The quantities K R) describe occupancy transformations fully involving the solution neighborhood of the observation volume. These coefficients are known only approximately. Building on the preceding discussion, however, we can go further to develop a self-consistent molecular field theory for packing problems in classical liquids. We discuss here specifically the one component hard-sphere fluid this discussion follows Pratt and Ashbaugh (2003). 

Percus-Yevick theory giving the approximate PY of a hard sphere fluid... 

This approximation is known as the mean spherical approximation (MSA). For the case of a hard-sphere fluid for which u r) = 0, the MSA is equivalent to the PY approximation. For the case that the hard spheres have embedded point charges, the function u(r) is simply Coulomb s law. Although the MSA provides the least detailed expression for c(r), it is popular because the OZ equation can often be solved using this approximation to yield an analytical expression for g(r). The equation for g(r) within a hard sphere is... 

Wertheim s formulation of his SSC approximation, which we have already discussed in the context of nonpolarizable fluids in Sections II and III, applies to the more general case of polar-polarizable fluids. In describing this case we use his notation. For polarizable dipolar hard spheres, the approximation is defined by the integral equations ... 

The use of (4.67) in (4.64) appears to yield accurate S2 up to densities of around pa s j at typical liquid temperatures. As p increases beyond this, S2 rapidly begins to be overestimated by the use of (4.67), as shown in Fig. 24 for a hard-sphere fluid. (As seen in the figure, however, for realistic values of a/a, S2 is extremely small in the first place.) An approximation that is far better at typical liquid densities (and nearly as good at lower densities) is one introduced by Stell and Hoye" in their study of the critical behavior of 82-... 

Basic questions of the equilibrium theory of fluids are concerned with (1) an adequately detailed description of the emergence of a fluid phase from a solid or the transition between a hquid and its vapor, the phase transition problem, and (2) the prediction from first principles of the bulk thermodynamic properties of a fluid over the whole existence region of the fluid. We will consider primarily the second of these questions. All bulk thermodynamic properties of monatomic fluids follow from a knowledge of the equation of state. This chapter will review certain recent developments in the approximate elucidation of the equation of state of a particularly simple fluid, the classical hard sphere fluid. This fluid is composed of identical particles or molecules, obeying classical mechanical laws, which are rigid spheres of diameter a. Two such molecules interact with one another only when they collide elastically. 

The procedure for finding the approximate equation of state of the two-dimensional hard sphere fluid follows almost step by step that outlined above for the three-dimensional fluid except that the curvature term in Eq. (67) can be neglected. Without entering into further details of the calculations we merely quote the result... 

The approximate theory outlined in this section while giving excellent numerical agreement and considerable physical insight into the hard sphere fluid does not suggest an obvious answer to why the approximation made is so successful. For example, the limiting surface tension, a p), given by Eq. (69) agrees exactly with the known first term of the density expansion of ... 

We have already in this and the previous sections made a number of comparisons between the various theories of fluids and the machine computations for the hard sphere system. Unfortunately, many recent developments in theory have been evaluated numerically only to the extent that the fourth and fifth virial coefficients can be compared. The table below lists the values of the fourth and fifth virial coefficients for the three-dimensional hard sphere fluid in units of the second virial coefficient b [cf. Eq. (33)]. The bases of calculation have been identified already in Section III except for the older "netted-chain approximation of Rushbrooke and Scoins. ... 

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