Mixtures of Hard Spheres

This exercise was reported in reference [72] for the simplest example of a colloidal mixture, namely, a binary mixture of hard-spheres with diameters Oj and 02, and number concentrations and within the PY [79] approximation for 5 p( ). The asymmetry parameter 6 (= 0 /02 1), the total volume fraction ( ) = ), -f )2 (with ( ) = nria yil6), and the molar fraction Xj = + Wj) of the 

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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... 

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 1992 Superlattice formation in binary mixtures of hard-sphere colloids Phys. Rev. Lett. 68 3801-4... 

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. 

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]

The ideal gas free energy functional is defined exactly from statistical mechanics, dropping the temperature-dependent terms that do not affect the fluid structure. Free energy functional contribution due to the excluded volume of the segments is calculated from Rosenfeld s (1989) DFT for a mixture of hard spheres. The functional derivatives of these free energy functional contributions, which are actually required to solve the set of Euler-Lagrange equations, are straightforward. 

In obtaining the expression for the activity coefficient part of the chemical potential, we have considered droplets of a single size represented by the most populous size (corresponding to the maximum in the size distribution). A more formal equation allowing for droplets of various sizes can be written according to the Mansoori—Carnahan—Starling equation of state for mixtures of hard spheres.26 The results based on such an expression are not expected to be essentially different from those obtained on the basis of a single droplet size. 

Wertheim, M.S. Fluids of dimerizing hard spheres, fluid mixtures of hard spheres and dispheres, 7. Chem. Phys. 1986, 85, 2929-2936. 

L. Blum, Mean Spherical Model for a Mixture of Hard Spheres and Hard Dipoles, Chem. Phys. Lett. 26 200 (1976). 

Figure 4. Experimental nonlinearities of Dj - CH pure V-V transfer rate in argon solution as a function of global density p at 152 K ( ) and 200 K (o). Full lines are calculated using a model fluid composed of a mixture of hard spheres.

The exact solution of the Percus-Yevick (PY) equation is known for a one-component system of hard spheres (Wertheim 1963 Thiele 1963) and for mixtures of hard spheres (Lebowitz 1964). Numerical solutions of the PY equation (for Lennard-Jones particles) have been carried out by many authors, e.g., Broyles (1960, 1961), Broyles et al. (1962), Throop and Bearman (1966), Baxter (1967), Watts (1968), Mandel et al. (1970), Grundke and Henderson (1972a, b)... 

The scaled particle theory was extended to mixture of hard spheres by Lebowitz et al. (1965). In a one-component system of hard spheres of diameter a, placing a hard particle of radius RHs produces a cavity of radius r= RHs + a/2. When there is a mixture of hard spheres of diameters a the radius of a cavity produced by a hard sphere of radius RHs depends on the species i, i.e., Y = Rhs + ai/2. 

Thus, in mixture of hard spheres, it is meaningful to place a hard sphere of radius i HS at some fixed position. However, the size of the cavity is different for the different species. 

The extension of these ideas to fluids with multiple attraction sites has been given by Wertheim and used by him to develop a theory of equilibrium polymerization. The difficulty with extension of the single-attraction site theory to multiple attraction sites is in incorporating the increasingly complex steric incompatibility (SI). In particular, the difficulty commonly encountered is SI3, which physically corresponds to the absence of self-hindrance, where the rigidity of an s-mer prevents two of its component molecules from encountering each other. In another paper, Wertheim has considered fluids of dimerizing hard spheres and fluid mixtures of hard spheres and diatomics. 

As in the case for pure fluids, the perturbation theory is useful both as a calculational tool and as a guide to the development of empirical equations. If we use a mixture of hard spheres as our reference fluid... 

A much better procedure is that of Snider and Herrington (27) who used the PY result for P0 for a mixture of hard spheres. For a binary mixture the PY result is ... 

The problem of finding effectively the equation of state of a mixture of hard spheres of different diameters, incidentally, is of considerable interest in a number of applications, e.g., for finding the high temperature equation of state of mixtures of real gases and the surface tension of mixtures, amoi other things. While a number of the theories of fluids mentioned in Section IV of this chapter can also be reformulated " formally for mixtures,... 

Thus g is negative at low densities as is the molecular excess volume = d jdp. A high density estimate of g is obtained by (1) assuming that the triplet correlation function of any particles 1, 2,3 is such that particles 2 and 3 are uncorrelated except for the fact that they may not overlap, and (2) the use of Eqs. (68) and (45) for the pair correlation function on contact of the hard spheres of diameter 1. This approximate computation indicates that the sign of g (and v ) undergoes an inversion as the density increases. This implies that a mixture of hard spheres will separate into two phases at sufficiently high p and low T, e.g., the predicted miscibility gap for p = 0.1 occurs for n 1.1. ... 

A critical review of these results as well as several other novel approaches to theories of mixtures evaluated for mixtures of hard spheres has been presented. by Salsbuig and Fickett. These authors also compare the various theories numerically particularly for the diameter ratio 5/3 used by Smith for his preliminary (low density) Monte Carlo calculations of binary hard sphere mixtures. Salsburg and Fickett conclude for these systems that... 

The behavior of one-dimensional mixtures of hard spheres of different diameters a- and follows directly from the exact validity of the quasi-chemical approximation for the one-dimensional combinatorial factor for particles interacting only with their nearest neighbois. Unfortunately, this result is valid only for one-dimensional systems. It would be of considerable interest to extend the scaled particle theory to deal with mixtures of hard spheres. 

The Camahan-Starling equation of state (4.5.4) has been extended by Mansoori et al. [16] to binary mixtures of hard spheres having different diameters. Binary mixtures of hard spheres exhibit fluid-solid phase transitions at packing fractions somewhat larger than that for the pure substance that is, at T) > 0.5. The exact state for the transition depends on composition and on the relative sizes of the spheres. We expect the density of the transition to increase as the size disparity increases the limited computer simulation data available support this expectation [17]. Certain kinds of hard-sphere mixtures are the simplest substances to exhibit a fluid-fluid phase transition [17], but those phase transitions are more like liquid-liquid than vapor-liquid. Analytic representations of the Z(r ) for hard-sphere and other hard-body fluids have been critically reviewed by Boublik and Nezbeda [18]. 

Other models for ternary amphiphilic systems are based on mixtures of hard spheres and ellipsoids with Lennard-Jones interactions [58] or on mixtures of hard spheres and diatomic hard-sphere molecules [59]. Such models have been studied by molecular dynamics simulations. 

To describe the measured cloud-points, the SAFT equation of state (eos) has been used. The SAFT eos [6] is based on the perturbation theory (see Chapter 3), and, in spite of the rather complex derivation of the model equations, the basic idea and the application of the model is less complex. The SAFT eos can be written as a sum of Helmholtz energies. The first contribution is the Helmholtz energy of an ideal gas, followed by a correction for a mixture of hard spheres, a correction for chain formation, and a correction for the dispersion and association forces ... 

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