Hard sphere fluid

The entropically driven disorder-order transition in hard-sphere fluids was originally discovered in computer simulations [58, 59]. The development of colloidal suspensions behaving as hard spheres (i.e., having negligible Hamaker constants, see Section VI-3) provided the means to experimentally verify the transition. Experimental data on the nucleation of hard-sphere colloidal crystals [60] allows one to extract the hard-sphere solid-liquid interfacial tension, 7 = 0.55 0.02k T/o, where a is the hard-sphere diameter [61]. This value agrees well with that found from density functional theory, 7 = 0.6 0.02k r/a 2 [21] (Section IX-2A). 

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. 

Reiss H 1977 Scaled particle theory of hard sphere fluids Statistical Mechanics and Statistical Methods in Theory and Application ed U Landman (New York Plenum) pp 99-140... 

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.

Attard P 1993 Simulation of the chemical potential and the cavity free energy of dense hard-sphere fluids J. Chem. Phys. 98 2225-31... 

In the theory of the liquid state, the hard-sphere model plays an important role. For hard spheres, the pair interaction potential V r) = qo for r < J, where d is the particle diameter, whereas V(r) = 0 for r s d. The stmcture of a simple fluid, such as argon, is very similar to that of a hard-sphere fluid. Hard-sphere atoms do, of course, not exist. Certain model colloids, however, come very close to hard-sphere behaviour. These systems have been studied in much detail and some results will be quoted below. 

Biben T and Flansen J P 1991 Phase separation of asymmetrio binary hard-sphere fluids Phys. Rev. Lett. 66 2215-18... 

FIGURE 2.2 Radial distribution functions for (a) a hard sphere fluid, (A) a real gas, (c) a liquid, (li) a crystal. 

Stratification, as illustrated by the plots in Fig. 6, is due to eonstraints on the paeking of moleeules next to the wall and is therefore largely determined by the repulsive part of the intermoleeular potential [55]. It is observed even in the absenee of intermoleeular attraetions, sueh as in the ease of a hard-sphere fluid eonfined between planar hard walls [42,90-92]. For this system Evans et al. [93] demonstrated that, as a eonsequenee of the damped oseil-latory eharaeter of the loeal density in the vieinity of the walls, F-, is a damped oseillatory funetion of s., if is of the order of a few moleeular diameters, whieh is eonfirmed by Fig. 5. 

To illustrate the effects of nonplanarity of the substrate on fluid structure, a hard-sphere fluid exposed to a periodic array of wedges (see Fig. 13) is... 

T. Biben, J.-P. Hansen. Phase separation of asymmetric binary hard-sphere fluids. Phys Rev Lett (5(5 2215-2218, 1991. 

T. Biben, P. Bladon, D. Frenkel. Depletion effects in binary hard-sphere fluids. J Phys Condens Matter 2.T0799-10821, 1996. 

I. K. Snook, D. Henderson. Monte Carlo study of a hard-sphere fluid near a hard wall. J Chem Phys (55 2134-2139, 1978. 

M. Schoen, S. Dietrich. Structure of a hard-sphere fluid in hard wedges. Phys 7 cv 5(5 499-510, 1997. 

D. Henderson, S. Sokolowski, D. Wasan. Structure of a hard-sphere fluid near a rough surface a density-functional approach. Phys Rev E 57 5539-5543, 1998. 

FIG. 3 The functions g r) and y r) for a hard sphere fluid. The broken curve gives PY results and the sohd curve gives the results of a fit of the simulation data. The circle gives the simulation results. The point at r = 0 gives the result obtained from Eq. (36), using the CS equation of state. 

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. 

Now, let us consider a model in which the association site is located at a distance slightly larger than the hard-core diameter a. The excess free energy for a hard sphere fluid is given by the Carnahan-Starling equation [113]... 

The density profiles are shown in Fig. 7(a). Fig. 7(b), however, illustrates the dependenee of the degree of dimerization, x( ) = P i )lon the distance from the wall. It ean be seen that, at a suffieiently low degree of dimerization (s /ksT = 6), the profile exhibits oseillations quite similar to those for a Lennard-Jones fluid and for a hard sphere fluid near a hard wall. For a high degree of dimerization, i.e., for e /ksT = 10 and 11.5, we observe a substantial deerease of the eontaet value of the profile in a wide layer adjacent to a hard wall. In the ease of the highest assoeiation energy,... 

The results for the chemical potential determination are collected in Table 1 [172]. The nonreactive parts of the system contain a single-component hard-sphere fluid and the excess chemical potential is evaluated by using the test particle method. Evidently, the quantity should agree well with the value from the Carnahan-Starling equation of state [113]... 

In Sec. 3 our presentation is focused on the most important results obtained by different authors in the framework of the rephca Ornstein-Zernike (ROZ) integral equations and by simulations of simple fluids in microporous matrices. For illustrative purposes, we discuss some original results obtained recently in our laboratory. Those allow us to show the application of the ROZ equations to the structure and thermodynamics of fluids adsorbed in disordered porous media. In particular, we present a solution of the ROZ equations for a hard sphere mixture that is highly asymmetric by size, adsorbed in a matrix of hard spheres. This example is relevant in describing the structure of colloidal dispersions in a disordered microporous medium. On the other hand, we present some of the results for the adsorption of a hard sphere fluid in a disordered medium of spherical permeable membranes. The theory developed for the description of this model agrees well with computer simulation data. Finally, in this section we demonstrate the applications of the ROZ theory and present simulation data for adsorption of a hard sphere fluid in a matrix of short chain molecules. This example serves to show the relevance of the theory of Wertheim to chemical association for a set of problems focused on adsorption of fluids and mixtures in disordered microporous matrices prepared by polymerization of species. 

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. 

Imagine a system of matrix hard spheres of diameter cr = 5oy (the diameter of fluid species is taken as a length unit, oy = 1). The fluid to be adsorbed is a hard sphere fluid. The essence of our modeling is in the fluid-matrix potential. It is chosen in the following form [53]... 

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... 

Finally, let us discuss the adsorption isotherms. The chemical potential is more difficult to evaluate adequately from integral equations than the structural properties. It appears, however, that the ROZ-PY theory reflects trends observed in simulation perfectly well. The results for the adsorption isotherms for a hard sphere fluid in permeable multiple membranes, following from the ROZ-PY theory and simulations for a matrix at p = 0.6, are shown in Fig. 4. The agreement between the theoretical results and compu-... 

In the GCMC simulations we are considering a fluid of hard spheres with diameter such that it equals the diameter of monomers belonging to chains, i.e., = cfq. The density of a hard sphere fluid in the presence of... 

The adsorption isotherms have been obtained according to the procedure of Ford and Glandt for a hard sphere fluid in a hard sphere matrix [23]. Let us denote the packing fraction of matrix species by rjf rj = -nNc Ma /6). 

We would like to discuss consistently the results obtained in the theory and simulations for a hard sphere fluid adsorbed in a matrix of chains with four, eight, and sixteen monomer beads (m = M = 4 8 16). 

FIG. 5 Adsorption isotherms for a hard sphere fluid from the ROZ-PY and ROZ-HNC theory (solid and dashed lines, respectively) and GCMC simulations (symbols). Three pairs of curves from top to bottom correspond to matrix packing fraction = 0.052, 0.126, and 0.25, respectively. The matrix in simulations has been made of four beads (m = M = 4). 

We present some of the results obtained for equal sized hard-sphere fluid and hard-sphere matrix in Figs. 12 and 13. In these figures we show the... 

Our main focus in this chapter has been on the applications of the replica Ornstein-Zernike equations designed by Given and Stell [17-19] for quenched-annealed systems. This theory has been shown to yield interesting results for adsorption of a hard sphere fluid mimicking colloidal suspension, for a system of multiple permeable membranes and for a hard sphere fluid in a matrix of chain molecules. Much room remains to explore even simple quenched-annealed models either in the framework of theoretical approaches or by computer simulation. 

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