The Hard-Sphere Fluid

The simplest fluid to be studied through molecular simulation is the hard-sphere one, whose intermolecular potential is described in Example 17.2. The radial distribution function and pressure for this fluid have been established as functions of density, the latter from molecular dynamics calculations. The most successful analytical expression for the pressure of a hard-sphere fluid (P ) is that proposed by Carnahan and Starling (1969 1972)  

V is the molar volume and Vq is the hard-sphere molar volume at closest packing, equal to  

The usefulness of the hard-sphere fluid is based on the evidence that for dense fluids, such as liquids, the attractive forces between molecules serve primarily to determine the volume, i.e. the density. Otherwise they play a minor role, as compared to the strong rq)ulsive forces, in determining its structure, i.e. the radial distributionfimction (Reed and Gubbins, p.249). The hard-sphere fluid represents, therefore, an attractive choice as a base on which to build semiempirical equations of state, or as a reference fluid for perturbation calculations. 

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Crooks G E and Chandler D 1997 Gaussian statistics of the hard sphere fluid Phys. Rev. E 56 4217... 

Figure B3.3.8. Insertion probability for hard spheres of various diameters (indieated on the right) in the hard sphere fluid, as a fiinetion of paeking fraetion p, predieted using sealed partiele theory. The dashed line is a guide to the lowest aeeeptable value for ehemieal potential estimation by the simple Widom method.

FIG. 7 Values of the density profile at eontaet for hard spheres in a sht of width H as a funetion of H. The density of the hard sphere fluid that is in equilibrium with the fluid in the slit is pd = 0.6. The solid eurve gives the lOZ equation results obtained using the PY elosure. The broken and dotted eurves give the results of the HAB equation obtained using the HNC and PY elosures, respeetively. The results obtained from the HAB equation with the MV elosure are very similar to the solid eurve. The eireles give the simulation results. 

Fig. 1.20. Velocity autocorrelation function for the hard sphere fluid at a reduced density p/po = 0.65 as a function of time measured in t units [75].

Pratt, L. R. LaViolette, R. A. Gomez, M. A. Gentile, M. E., Quasi-chemical theory for the statistical thermodynamics of the hard-sphere fluid, J. Phys. Chem. B 2001,105, 11662-11668... 

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

Figure 8 Compressibility factor P/fiksT versus density p = pa3 of the hard-sphere system as calculated from both free-volume information (Eq. [8]) and the collision rate measured in molecular dynamics simulations. The empirically successful Camahan-Starling84 equation of state for the hard-sphere fluid is also shown for comparison. (Adapted from Ref. 71).

Figure 9 Properties of the attractive colloidal fluid investigated in Ref. 75 (a) self-diffusivity and (b) average free volume versus strength of the interparticle attraction (c) self-diffusivity versus average free volume for the hard-sphere fluid (open circles) and the attractive colloidal fluid (closed circles). Data compiled from Ref. 75.

The program of calculating the BO-level potentials from Schroedinger level cannot often be carried through with the accuracy required for the intermolecular forces in solution theory. (9.) Fortunately a great deal can be learned through the study of BO-level models in which the N-body potential is pairwise additive (as in Eq. (3)) and in which the pair potentials have very simple forms. (2, 3, 6) Thus for the hard sphere fluid we have, with a=sphere diameter,... 

The 6-12 potential is only qualitatively like the realistic potentials that can be derived by calculations at Schroedinger level for, say, Ar-Ar interactions. But it requires careful and detailed study to see how real simple fluids (i.e. one component fluids with monatomic particles) deviate from the behavior calculated from the 6-12 model. Moreover the principal structural features of simple fluids are already quite realistically given by the hard sphere fluid. 

We separate in a natural fashion the system in two parts, namely the hard-sphere fluid and the tube itself. Let F, Ft, and Fs denote the free energies of the system, the tube, and the spheres, respectively (all of which depend on n), such that we can write F = Ft+ Fs- The most probable value for n is obtained by... 

One conclusion from this study is that although the hard-sphere fluid has been very successful as a reference fluid, for example, in developing analytical equations of state, it is unrealistic in representing the dynamical relaxation processes in real systems, even with very steeply repulsive potentials. Owing to the discontinuity in the hard-sphere potential, this fluid, in fact, is not a good reference fluid for the short time (fast or j9 ) viscoelastic relaxation aspects of rheology. 

In order to use the above expressions for calculating the thermodynamic properties, appropriate expressions for the radial distribution function and for the equation of state for the hard-sphere reference system are required which are given in Appendix A. Fortunately, accurate information for the hard-sphere fluid as well as for the hard-sphere solid is available and this enables the determination of the properties of the coexisting dilute and concentrated phases of colloidal dispersions. 

Consider the hard-sphere fluid and establish the zero separation theorem... 

Table 7.1 gives quasi Monte Carlo estimates of the Kn for the case of hard spheres. For the hard-sphere fluid, the predicted distributions for two densities are shown in Figs. 7.8 and 7.9. The predicted occupancy distributions are physically faithfiil to the data. 

Figure 7.12 Excess chemical potential of the hard-sphere fluid as a function of density. The open and filled circles correspond to the predictions of the primitive quasi-chemical theory and the self-consistent molecular field theory, respectively. The solid and dashed lines are the scaled-particle (Percus-Yevick compressibility) theory and the Carnahan-Starling equation of state, respectively (Pratt and Ashbaugh, 2003).

J. J. Erpenbeck, Phys. Rev. Lett., 52,1333 (1984). Shear Viscosity of the Hard-Sphere Fluid Via Nonequilibrium Molecular Dynamics. 

Here p(r) is the smoothed density and A is the thermal de Broglie wavelength. The repulsive part of the Helmholtz free energy is usually calculated by the Carnahan-Starling equation derived for the hard sphere fluid [80] ... 

At this concentration the hard-sphere fluid has properties which are significantly different from those of an ideal fluid PV/ Nk T) = 1). 

Just as the ideal gas forms a convenient point of reference in discussing the properties of real gases, so does the hard-sphere fluid in discussing the properties of liquids. This is especially true at low densities, where the role of intermolecular forces in real systems is not so important. In this limit, the hard-sphere model is useful in developing the theory of solutions, as will be seen in chapter 3. 

Figure 1. The pressure dependence of the glass temperature in model and experimental fluids. The dashed-dotted curve (red) is for the Lennard-Jones with a purely repulsive interaction (UR) fluid, thin solid curve (red) is for the Lennard-Jones fluid (U) and the black dotted curve is for the hard spheres fluid with a. square-well (SW) potential. These curve recall results of...

A commonly used reference fluid is the hard-sphere fluid expressed by the Camahan-Starling (CS) equation of state ... 

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