Hard-sphere liquids

The hypothesis proposed in [76] implies that collective effects change the collisional event statistics in a hard sphere liquid. The collisional distribution tp(t) dt is assumed to deviate from the famous Poissonian law i/>(t) = exp (—t/To)/r0 corresponding to the flow of events being uniform in time. The distribution... 

One of the further refinements which seems desirable is to modify Eq. (9) so that it has wiggles (damped oscillatory behavior). Wiggles are expected in any realistic MM-level pair potential as a consequence of the molecular structure of the solvent (2,3,10,11,21,22) they would be found even for two hard sphere solute particles in a hard-sphere liquid or for two H2I80 solute molecules in ordinary liquid HpO, and are found in simulation studies of solutions based on BO-level models. In ionic solutions in a polar solvent another source of wiggles, evidenced in Fig. 2, may be associated with an oscillatory nonlocal dielectric function e(r). ( 36) These various studies may be used to guide the introduction of wiggles into Eq. (9) in a realistic way. 

Alexander, M., Rojas-Ochoa, L.F., Leser, M.E., and Schurtenberger, P. (2002). Stmcture, dynamics and optical properties of concentrated milk suspensions an analogy to hard-sphere liquids. J. ColloidInterf. Sci. 253, 35- 6. 

Except for extremely simple potential models (essentially hard spheres), liquid properties cannot be calculated by theoretical methods and one has to resort on computer simulation methods as Monte Carlo (MC) and molecular dynamics (MD) [55] or to integral equation methods [56]. In principle, simulation techniques are able to provide essentially exact results for the model, i.e. for a given potential function, so they are an ideal tool to test the ability of the potential to reproduce experimental data. As far as the nature of... 

Estimate the packing fraction for a hard-sphere liquid with a density of 21.25 atoms nm and a hard-sphere diameter of 350 pm. Use this result to calculate the Percus-Yevick product for the system at 85 K using the Carnahan-Starling equation of state (equation (2.9.11)). 

The free-volume models reviewed here and in a later section are based on Cohen and Turnbull s theory (18) for diffusion in a hard-sphere liquid. These investigators argue that the total free volume is a sum of two contributions. One arises from molecular vibrations and cannot be redistributed without a large energy change, and the second is in the form of discontinuous voids. Diffusion in such a liquid is not due to a thermal activation process, as it is taken to be in the molecular models, but is assumed to result from a redistribution of free-volume voids caused by random fluctuations in local density. 

A well-known and simple theory for describing molecular transport in a liquid is the free-volume theory of Cohen and Turnbull [1959, 1970]. Employing statistical mechanics, these authors showed that the most probable size distribution of the free volume per molecule in a hard sphere liquid may be described by an exponential decreasing function. It was assumed that diffusion of the hard-spheres can only take place when, due to thermal fluctuations, holes are formed whose size is greater than a critical volume. When applying this theory to a structural relaxation process in a liquid, its (circular) frequency o) = r = 2jtv is expressed by... 

The scaled particle theory SPT) was developed mainly for the study of hard-sphere liquids. It is not an adequate theory for the study of aqueous solutions. Nevertheless, it has been extensively applied for aqueous solutions of simple solutes. The scaled particle theory (SPT) provides a prescription for calculating the work of creating a cavity in liquids. We will not describe the SPT in detail only the essential result relevant to our problem will be quoted. Let aw and as be the effective diameters of the solvent and the solute molecules, respectively. A suitable cavity for accommodating such a solute must have a radius of c ws = ((Tw + cTs) (Fig. 3.20b). The work required to create a cavity of radius a s at a fixed position in the liquid is the same as the pseudo-chemical potential of a hard sphere of radius as. The SPT provides the following approximation for the pseudochemical potential ... 

Fig. 7.12. Pair potential for a hard sphere liquid (one component).

One liquid system of considerable heuristic value is a hypothetical one composed of a dense assembly of hard spheres. We shall find that such a system represents a useful reference liquid with which to compare real liquids. A hard sphere liquid is clearly a simple liquid within the framework outlined in Section 7.2.1. the form of 0 is given in Figure 7.12 and the corresponding potentials for a binary mixture are illustrated in Figure 7.13. 

The PY equation for a binary hard sphere liquid has been considered by Lebowitz (1964). Lebowitz obtains the Laplace transforms of rg (r) exactly, and these have been inverted and evaluated numerically by Throop and Bearman (1965). However, contact between theory and experiment is most readily established in q space and so, following Enderby and North (1968), we return to an intermediate step in Lebowitz s argument in which the direct correlation functions are rigorously derived. These are as follows ... 

If the analytical interference function were obtained, a more realistic X-ray scattering pattern could be generated using Equation (4). It is, however, very difficult, if not impossible, to derive an analytical solution for the interference function of such a complicated system. A successful example of deriving an analytical solution of the interference function is based on a much simpler system. Percus and Yevick have derived an analytical solution of the interference function for a hard-sphere liquid system through so-called TMrect Correlation Function. ... 

The surface concentration dependence of the lateral mobility of Fig. 7 was analyzed in terms of the free-volume theory of hard sphere liquids of Cohen and Turnbull [55, 56], as well as in view of the Enskog theory of dense gases [57] extended by Alder s molecular dynamics calculations to liquid densities [58]. The latter approach was particularly successful. It revealed that the lateral diffusion constant of the Fc amphiphiles does follow the expected linear dependence on the relative free area, Af/Ao, where Af = A — Ao, A = MMA, and Aq is the molecular area of a surfactant molecule. It also revealed that the slope of this dependence which is expected to inversely depend on the molecular mass of a diffusing particle, was more than 3 orders of magnitude smaller [54]. Clearly, this discrepancy is due to the effect of the viscous drag of the polar head groups in water, a factor not included in the Enskog theory. 

A liquid state theory has been developed on the basis of an ideal liquid, which is a hard-sphere liquid. Usually, thus, a random disordered structure of liquid has been assumed. This is the basis for the description of liquid by the two-body density correlator, or the radial distribution function g r). Recent studies indicate this picture is not sufficient even for a hard-sphere liquid [46,47], The assumption of a disorder structure of a liquid is always correct as the zeroth order approximation. However, we believe that a physical description beyond this is prerequisite for understanding unsolved fundamental problems in a liquid state, which include thermodynamic and kinetic anomalies of water type liquids, liquid-liquid transition, liquid glass transition, and crystal nucleation. 

From statistical mechanical theory, a simple model for a hypothetical hard-sphere liquid (spherical molecules of finite size without attractive intermolecular forces) gives the following expression for the Helmholtz energy with its natural variables T, V, and n as the independent... 

The functional that reproduces the scaled particle theory and the Percus-Yevick copressibility equation of state for uniform hard sphere liquids is... 

It is a question of some interest whether the solvent can be presumed to stick to the solvated ion as it diffuses through the solution. If instead it is assumed that the solvent slips by the solute the factor 6jt in (3.20) is reduced to 4n. This has no great qualitative effect on the picture to be developed. For a derivation of Stokes law see R. M. Fuoss and F. Accascina, Electrolyte Conductance (New York Interscience, 1959), pp. 53-59. Theoretical studies on hard-sphere liquids indicate that Stokes law applies if it is assumed that solvent slips by the molecules [B. J. Alder, D. M. Gass, and T. E. Wain-wright, /. Chem. Phys. 53, 3813 (1970)]. 

As in the Wertheim equation of state, Chiew also made use of the Carnahan-Starling expression for the pressure of the corresponding hard-sphere liquid. Similar agreement between TPTl and the Chiew theory was found with Monte Carlo simulations of Dickman and Hall for the pressure. 

Easteal, A.J. Woolf, L.A. (1990). Tracer diffusion in hard sphere liquids from molecular dynamics simulations. Chem. Phys. Lett., 167,329-333. 

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