Two-dimensional hard-sphere fluid

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... 

Fig. 12. Comparison of the theoretical equation of state with machine computations for a two-dimensional hard sphere fluid.

More generally, a spherical cavity of radius r = a + b)j2 acts like a fixed solute hard sphere of diameter b in the remaining (solvent) hard sphere molecules each of diameter a. The equations of state of the three-, two- and one-dimensional hard sphere fluids can be written in terms of the appropriate G as [cf. Eqs. (26)-(28)]... 

Equation (45) already sufl ces to specify exactly the equation of state of a one-dimensional hard sphere fluid. To see this we note that in one dimension (see Fig. 10) no two molecules lying... 

Adsorption of hard sphere fluid mixtures in disordered hard sphere matrices has not been studied profoundly and the accuracy of the ROZ-type theory in the description of the structure and thermodynamics of simple mixtures is difficult to discuss. Adsorption of mixtures consisting of argon with ethane and methane in a matrix mimicking silica xerogel has been simulated by Kaminsky and Monson [42,43] in the framework of the Lennard-Jones model. A comparison with experimentally measured properties has also been performed. However, we are not aware of similar studies for simpler hard sphere mixtures, but the work from our laboratory has focused on a two-dimensional partly quenched model of hard discs [44]. That makes it impossible to judge the accuracy of theoretical approaches even for simple binary mixtures in disordered microporous media. 

Our presentation here has been limited to the 3D case of a hard-sphere fluid. However, the GPRG method is equally successful in predicting the compressibility factor for a two-dimensional (2D) fluid of hard disks [29]. 

T 0 this point, we have discussed only the adsorbed hard sphere fluid, but it is well known that hard spheres undergo freezing in both two and three dimensions. One expects to find analogous behavior in adsorption systems, especially when the fluid is confined to a monolayer or several layers (i.e., stratified). This can occur in a very narrow hard-wall slit pore, or it can be produced by the presence of a strong holding potential at the surface (which we discuss below). In fact, there has been great interest in the melting behavior of simple two-dimensional systems. 

This is an indication of the collective nature of the effect. Although collisions between hard spheres are instantaneous the model itself is not binary. Very careful analysis of the free-path distribution has been undertaken in an excellent old work [74], It showed quite definite although small deviations from Poissonian statistics not only in solids, but also in a liquid hard-sphere system. The mean free-path X is used as a scaling length to make a dimensionless free-path distribution, Xp, as a function of a free-path length r/X. In the zero-density limit this is an ideal exponential function (Ap)o- In a one-dimensional system this is an exact result, i.e., Xp/(Xp)0 = 1 at any density. In two dimensions the dense-fluid scaled free-path distributions agree quite well with each other, but not so well with the zero-density scaled distribution, which is represented by a horizontal line (Fig. 1.21(a)). The maximum deviation is about... 

FIG. 14 Phase diagram of a system of hard-spheres between two parallel walls. In the three-dimensional limit (/c -> oo) the system is fluid-like for 3D < 0.5. When the walls separation is comparable to the particles size (/c 1) the system can undergo disorder-order phase transition. Adapted from Chavez-Paez et al. [39]. 

Computer simulations also constitute an important basis for the development of the molecular theory of fluids. They could be regarded as quasiexpeiimental procedures to obtain datasets that connect the fluid s microscopic parameters (related mainly to the structure of the system and the molecular interactions) to its macroscopic properties (such as equation of state, dynamic coefficients, etc.). In particular, some of the first historical simulations were performed using two-dimensional fluids to test adaptations of commonly used computer simulation methods [14,22] Monte Carlo (MC) and molecular dynamics (MD). In fact, the first reliable simulation results were obtained by Metropolis et al. [315], who applied the MC method to the study of hard-sphere and hard-disk fluids. 

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