Mixtures hard sphere

The CS pressures are close to the machine calculations in the fluid phase, and are bracketed by the pressures from the virial and compressibility equations using the PY approximation. Computer simulations show a fluid-solid phase transition tiiat is not reproduced by any of these equations of state. The theory has been extended to mixtures of hard spheres with additive diameters by Lebowitz [35], Lebowitz and Rowlinson [35], and Baxter [36]. 

The method has been extended to mixtures of hard spheres, to hard convex molecules and to hard spherocylinders that model a nematic liquid crystal. For mixtures m. subscript) of hard convex molecules of the same shape but different sizes. Gibbons [38] has shown that the pressure is given by... 

Lebowitz J L 1964 Exact solution of the generalized Percus-Yevick equation for a mixture of hard spheres Phys. Rev. 133 A895... 

Lebowitz J L and Rowlinson J S 1964 Thermodynamic properties of hard sphere mixtures J. Chem. Phys. 41 133... 

Another triek is applieable to, say, a two-eomponent mixture, in whieh one of the speeies. A, is smaller than the other, B. From figure B3.3.8 for hard spheres, we ean see that A need not be particularly small in order for the test partiele insertion probability to elimb to aeeeptable levels, even when insertion of B would almost always fail. In these eireumstanees, the ehemieal potential of A may be detemiined direetly, while that of B is evaluated indireetly, relative to that of A. The related semi-grand ensemble has been diseussed in some detail by Kofke and Glandt [110]. 

As shown in section C2.6.6.2, hard-sphere suspensions already show a rich phase behaviour. This is even more the case when binary mixtures of hard spheres are considered. First, we will mention tire case of moderate size ratios, around 0.6. At low concentrations tliese fonn a mixed fluid phase. On increasing tire overall concentration of mixtures, however, binary crystals of type AB2 and AB were observed (where A represents tire larger spheres), in addition to pure A or B crystals [105, 106]. An example of an AB2 stmcture is shown in figure C2.6.11. Computer simulations confinned tire tliennodynamic stability of tire stmctures tliat were observed [107, 1081. 

Bartlett P, Ottewill R FI and Pusey P N 1990 Freezing of binary mixtures of colloidal hard spheres J. Chem. Phys. 93 1299-312... 

Eldridge M D, Madden P A and Frenkel D 1993 Entropy-driven formation of a superlattioe in a hard-sphere binary mixture Mol. Phys. 79 105-20... 

Imhof A and Dhont J K G 1995 Experimental phase diagram of a binary oolloidal hard-sphere mixture with a large size ratio Phys. Rev. Lett. 75 1662-5... 

Oykstra M, van Roi] R and Evans R 1999 Direot simulation of the phase behaviour of binary hard-sphere mixtures test of the depletion potential desoription Phys. Rev. Lett. 82 117-20... 

P. Attard, G. N. Patey. Hypemetted-chain closure with bridge diagrams. Asymmetric hard sphere mixtures. J Chem Phys 92 4970-4982, 1990. 

R. Dickman, P. Attard, V. Simonian. Entropic forces in binary hard sphere mixtures. J Chem Phys 707 205-213, 1997. 

The g r) that results from the modified Verlet (MV) closure is very close to the simulation results in Figs. 2 and 3. The MV results for g d), or equivalently, y d), are plotted in Fig. 4(a). The resulting equation of state is similar to the CS expression. An even more demanding test is an examination of the MV results for y r) for r < d. As is seen in Fig. 4(b), the MV results for y(0) are quite good [25], and are better than the PY and HNC results. Some results have also indicated that the MV closure gives quite accurate results for a mixture of hard spheres [26]. 

One should perhaps mention some other closures that are discussed in the literature. One possibility is to combine the PY approximation for the hard core part of the potential and then use the HNC approximation to compute the corrections due to the attractive forces. Such an approach is called the reference hypernetted chain or RHNC approximation [48,49]. Recently, some new closures for a mixture of hard spheres have been proposed. These include one by Rogers and Young [50] (RY) and the Martynov-Sarkisov [51] (MS) closure as modified by Ballone, Pastore, Galli and Gazzillo [52] (BPGG). The RY and MS/BPGG closure relations take the forms... 

The MS closure results from s = 2. The HNC closure results from s = 1. In the latter two expressions, additional adjustable parameters occur, namely ( for the RY closure and for the BPGG version of the MS approximation. However, even when adjustable, these parameters cannot be chosen at will, as they should be chosen such that they eliminate the so-called thermodynamic inconsistency that plagues many approximate integral equations. We recall that a manifestation of this inconsistency is that there is a difference between the pressure as computed from the virial equation (10) and as computed from the compressibility equation (20). Note that these equations have been applied to a very asymmetric mixture of hard spheres [53,54]. Some results of the MS closure are plotted in Fig. 4. The MS result for y d) = g d) is about the same as the MV result. However, the MS result for y(0) is rather poor. Using a value between 1 and 2 improves y(0) but makes y d) worse. Overall, we believe the MS/BPGG is less satisfactory than the MV closure. 

Attard has obtained OZ2 results for hard spheres. Henderson and Sokolowski have obtained results for hard spheres [111], an LJ fluid [112], and a mixture of LJ fluids [113]. The results are very good. The values of y r) obtained from the OZ2 equation with the PY closure are very similar to the... 

Our interest is in the model of hard spheres with spherieally symmetrie assoeiative interaetions. This has been proposed and well studied by Cummings and Stell [25-27]. The model represents a two-eomponent mixture of hard sphere speeies, a and /5, with equal diameters, = [Pg.178]

Let us thus consider a model in which the association energy depth changes when two reacting particles are approaching the surface see Refs. 86,90. If in the vicinity of the surface the binding energy is lower than it is far from the surface, the probability of the chemical reaction to occur in the surface zone decreases. Similarly to the previous case, we consider an equimolar mixture of associating hard spheres of equal diameters. The interaction between the species a and (3 is assumed in the form... 

As we have noted in Sec. II, one of the methods leading to the so-called singlet equations for the density profiles, originally initiated for simple fluids in Ref. 24, starts from considering a mixture of fluid particles and another species of hard spheres at density pq and diameter Dq, taking next the limit Pg - 0,Dg- oo. 

As in Sec. II, we consider a mixture composed of a dimerizing one-component fluid and a giant hard sphere [21,119]. We begin with the multidensity Ornstein-Zernike equation for the mixture... 

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. 

First, we would like to eonsider a simple hard sphere model in a hard sphere matrix, similar to the one studied in Refs. 20, 21, 39. However, our foeus is on a very asymmetric hard sphere mixture adsorbed in a disordered matrix. Moreover, having assumed a large asymmetry of diameters of the eomponents and a very large differenee in the eoneentration of eomponents, here we restriet ourselves to the deseription of the struetural properties of the model. Our interest in this model is due, in part, to experimental findings eoneerning the potential of the mean foree aeting between eolloids in a eolloidal dispersion in the presenee of a matrix of obstaeles [12-14]. 

FIG. 4 The calculated internal energy of a 1-1 salt (line) is compared with the corresponding simulation results (open circles) obtained by Van Megen and Snook (Ref. 20). The Debye-Hiickel (DH, dashed line) and corrected Debye-Hiickel (CDH, full line) theory were used together with a GvdW(I) treatment of the uncharged hard-sphere mixture. The ion diameter was 4.25 A, the temperature was 298 K and the dielectric constant e was 78.36. 

Interactions of such glassy polymeric particles should resemble the collisions of hard spheres. Phase diagrams of the type shown in Fig. 36 have been obtained for various polymer-organic solvent mixtures [85,94,345-353]. 

States Hard-Sphere Model for the Diffusion-Coefficients of Binary Dense-Plasma Mixtures. 

We also adopt the above combination rule (Eq. [6]) for the general case of exp-6 mixtures that include polar species. Moreover, in this case, we calculate the polar free energy contribution Afj using the effective hard sphere diameter creff of the variational theory. 

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