Binary hard sphere mixtures

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... 

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

Solid-fluid phase diagrams of binary hard sphere mixtures have been studied quite extensively using MC simulations. Kranendonk and Frenkel [202-205] and Kofke [206] have studied the solid-fluid equilibrium for binary hard sphere mixtures for the case of substitutionally disordered solid solutions. Several interesting features emerge from these studies. Azeotropy and solid-solid immiscibility appear very quickly in the phase diagram as the size ratio is changed from unity. This is primarily a consequence of the nonideality in the solid phase. Another aspect of these results concerns the empirical Hume-Rothery rule, developed in the context of metal alloy phase equilibrium, that mixtures of spherical molecules with diameter ratios below about 0.85 should exhibit only limited solubility in the solid phase [207]. The simulation results for hard sphere tend to be consistent with this rule. However, it should be noted that the Hume-Rothery rule was formulated in terms of the ratio of nearest neighbor distances in the pure metals rather than hard sphere diameters. Thus, this observation should be interpreted as an indication that molecular size effects are important in metal alloy equilibria rather than as a quantitative confirmation of the Hume-Rothery rule. 

DFT studies of binary hard-sphere mixtures predate the simulation studies by several years. The earliest work was that of Haymet and his coworkers [221,222] using the DFT based on the second-order functional Taylor expansion of the Agx[p]- Although this work has to some extent been superceded, it was a significant stimulus to much of the work that followed both with theory and computer simulations. For example, it was Smithline and Haymet [221] who first analyzed the Hume-Rothery rule in the context of hard sphere mixture behavior and who first investigated the stability of substitutionally ordered solid solutions. The most accurate DFT results for hard-sphere mixtures have come from the WDA-based theories. In particular the results of Denton and Ashcroft [223] and those of Zeng and Oxtoby [224] give qualitatively correct behavior for hard spheres forming substitutionally disordered solid solutions. 

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... 

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 1964 Lebowitz and Rowlinson [6] showed that, within the Percus-Yevick treatment of hard sphere fluids [7], binary hard sphere mixtures are completely miscible for aU concentrations and size ratios. This proof was later extended by Vrij [8] to hard sphere mixtures with an arbitrary number of components. Up till 1990, it was indeed generally beUeved that hard sphere mixtures do not phase... 

B. B. Laird and A. D. J. Haymet, Calculation of the entropy of binary hard sphere mixtures from pair correlation functions. J. Chem. Phys., 97, 2153,1992. 

Kozina A, Sagawe D, Diaz-Leyva P, Bartsch E, Palberg T (2012) Polymer—enforced crystal-hzation of a eutectic binary hard sphere mixture. Soft Matter 8 627-630... 

Foffi, G., Gotze, W., Sciortino, R, Tartaglia, R, and Voigtmann, Th. 2004. a-relaxation processes in binary hard-sphere mixtures. Phys. Rev. E 69 011505. 

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... 

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. 

Although there has not been much theoretical work other than a quantitative study by Hynes et al [58], there are some computer simulation studies of the mass dependence of diffusion which provide valuable insight to this problem (see Refs. 96-105). Alder et al. [96, 97] have studied the mass dependence of a solute diffusion at an infinite solute dilution in binary isotopic hard-sphere mixtures. The mass effect and its influence on the concentration dependence of the self-diffusion coefficient in a binary isotopic Lennard-Jones mixture up to solute-solvent mass ratio 5 was studied by Ebbsjo et al. [98]. Later on, Bearman and Jolly [99, 100] studied the mass dependence of diffusion in binary mixtures by varying the solute-solvent mass ratio from 1 to 16, and recently Kerl and Willeke [101] have reported a study for binary and ternary isotopic mixtures. Also, by varying the size of the tagged molecule the mass dependence of diffusion for a binary Lennard-Jones mixture has been studied by Ould-Kaddour and Barrat by performing MD simulations [102]. There have also been some experimental studies of mass diffusion [106-109]. 

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]. 

P p2) using a binary hard-sphere model and obtained P p2)l uW which agrees well with the measurement. The data in Fig. 4 thus demonstrate that the binary hard-sphere model" can indeed describe the depletion effect in our colloid/PEP mixture. 

Wichert, J. M. Gulati, H. S. Hall, C. K. J. (1996). Binary hard chain mixtures. I. Generalized Flory equations of state. /. Chem. Phys., Vol. 105, 7669-7682 Woodward, C. E. Forsman, J. (2008). Density functional theory for polymer fluids with molecular weight polydispersity. Phys. Rev. Lett., Vol. 100,098301 Woodward, C. E. (1991). A density functional theory for polymers Application to hard chain-hard sphere mixtures in slitlike pores. /. Chem. Phys., Vol. 94,3183-3191 Woodward, C. E. Forsman, J. (2009). Interactions between surfaces in polydisperse semiflexible polymer solutions. MacromoL, Vol. 42, 7563-7570 Woodward, C. E. Yethiraj A. (1994). Density functional theory for inhomogeneous polymer solutions. /, Chem. Phys., Vol. 100,3181-3186... 

For equimolar binary mixtures where the effects of molecular interactions are not strong, it was found (Easteal Woolf 1984) that experimental interdiffusion coefficients (Dij) were generally in close agreement with values computed for the hard-sphere mixture for the appropriate molecular mass and size ratio. The Enskog intradiffusion coefficients [(AO Je were calculated from... 

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]. 

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... 

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

N. S. Snider and T. M. Herrington,/. Chem. Phys., 47,2248 (1967). Hard Sphere Model of Binary Liquid Mixtures. 

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