Hard spheres systems binary mixtures

Many of the papers on DFT have focused primarily on the hard-sphere system, and it is for this system that most success has been achieved. However, DFT has also been applied to the Lennard-Jones 12-6 system, binary mixtures, nonspherical molecules, and coulombic systems. We will discuss some of these applications later in the chapter as we review what is known about the phase diagrams of various models systems. 

In view of the success of the methods based on hard-sphere theories for the accurate correlation and prediction of transport properties of single-component dense fluids, it is worthwhile to consider the application of the hard-sphere model to dense fluid mixtures. The methods of Enskog were extended to mixtures by Thome (see Chapman Cowling 1952). The binary diffusion coefficient >12 for a smooth hard-sphere system is given by... 

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... 

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. 

In order to close these expressions for particulate pressures, we also need equations for the variance of total particle volume concentration in an assemblage of particles belonging to the two different types. For an arbitrary polydisperse particulate pseudo-gas, variances of partial volume concentrations for different particles can be evaluated on the basis of the thermodynamical theory of fluctuations. According to this theory, these variances are expressible in terms of the minors of a matrix that consists of the cross derivatives of the chemical potentials for particles of different species over the partial number concentrations of such particles [39]. For a binary pseudo-gas, these chemical potentials can be expressed as functions of number concentrations using the statistical theory of binary hard sphere mixtures developed in reference [77]. However, such a procedure leads to a very cumbersome and inconvenient final equation for the desired variance. To simplify the matter, it has been suggested in reference [76] to ignore a slight difference between this variance and the similar quantity for a monodisperse system of spherical particles of the same volume concentration. This means that the variance under question may be approximately described by Equation 7.4 even in the case of binary mixtures. 

In his original works, Rosenfeld considered hard spheres, soft spheres, Lennard-Jones system, and one-component plasma [52,53]. Thereafter, the excess entropy scaling was applied to many different systems, including core-softened liquids [17,18,51,54,55], liquid metals [56,57], binary mixtures [58,59], ionic liquids [60,61], network-forming liquids [54,60], water [62], chain fluids [63], and bounded potentials [51,64,65]. 

The possibility of entropy-driven phase separation in purely hard-core fluids has been of considerable recent interest experimentally, theoretically, and via computer simulations. Systems studied include binary mixtures of spheres (or colloids) of different diameters, mixtures of large colloidal spheres and flexible polymers, mixtures of colloidal spheres and rods," and a polymer/small molecule solvent mixture under infinite dilution conditions (here an athermal conformational coil-to-globule transition can occur)." For the latter three problems, PRISM theory could be applied, but to the best of our knowledge has not. The first problem is an old one solved analytically using PY integral equation theory by Lebowitz and Rowlinson." No liquid-liquid phase separation... 

The calculation of TCF in multicomponent systems has been done only for spherical cavities with the formalism developed by Lebowitz et al. Methods IV and V can in principle be extended for TCF calculation for nonspherical cavities in multicomponent systems. An artificial binary system (benzene-water) was selected here to illustrate the computational methodology. In practice, these two solvents mix very little, and their mixture can be of little interest, but they are quite different in their chemical nature and this makes such a system interesting. The method is by no means limited to certain mixtures and is universally applicable to any mixture if the molecular and physical parameters of the pure components are known (hard sphere diameter, number density, thermal expansion coefficient, dielectric constant). Figure 9 displays TCF calculated as a function of solvent mole fractions for a spherical cavity of cyclohexane size created in a hypothetical water-benzene mixture. Gc (Figure 9) increases with the increase of water mole fraction, but there is little difference between pure benzene and a mixture containing around 50% water as far as solvation of cyclohexane is concerned. 

Once the hard sphere fluid has been appropriately defined, the free energy of the repulsive system may be described using the equation of state of hard sphere binary mixtures proposed by Boublik [302] ... 

The effect of a structured surface on the crystallization of hard-sphere colloids has been extensively studied in experiments [87, 88, 89, 90], These experiments indicate that crystallization on a template is induced at densities below freezing. This finding is supported by computer simulations of hard spheres in contact with a patterned substrate, by Heni and Lowen [91], These simulations indicate that surface freezing already sets in 29% below the coexistence pressure. Furthermore the effect of a surface on crystallization has also been studied in mixtures of binary hard-spheres [92, 93] and colloid-polymer mixtures [94, 95, 96], In both systems surface crystallization was found to take place before bulk fluid-solid coexistence. In the systems studied in Refs. [92, 93, 94, 95, 96], depletion forces favor the accumulation of the larger component on the wall, and this should facilitate surface crystallization [97]. 

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