Phase diagram binary hard spheres

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. 

FIGURE 8.1 Phase diagram of binary hard spheres with a small-to-large size ratio of 0.82. The phase diagram is shown in the composition x, reduced pressure p, representation, where x = N /iN + Nj) is the number fraction of small spheres. The labels Tccl and fccs denote the fee crystals of large and small particles, respectively. (From Hynninen AP et al. 2007. Nature Materials 6 202-205. With permission.)... 

Dinsmore A D, Yodh A G and Pine D J 1995 Phase diagrams of nearly hard-sphere binary oolloids Phys. Rev. E 52 4045-57... 

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... 

Colloidal crystals . At the end of Section 2.1.4, there is a brief account of regular, crystal-like structures formed spontaneously by two differently sized populations of hard (polymeric) spheres, typically near 0.5 nm in diameter, depositing out of a colloidal solution. Binary superlattices of composition AB2 and ABn are found. Experiment has allowed phase diagrams to be constructed, showing the crystal structures formed for a fixed radius ratio of the two populations but for variable volume fractions in solution of the two populations, and a computer simulation (Eldridge et al. 1995) has been used to examine how nearly theory and experiment match up. The agreement is not bad, but there are some unexpected differences from which lessons were learned. 

FIGURE 7.10 Pseudo-binary phase diagram of the water, HNO3/DMDBTDMA, dodecane system to identify the third-phase limit. Experimental points (circles) and theory (lines) obtained from Baxter sticky hard-sphere approach. The theoretical line is obtained with the experimental determination of the linear variation of the stickiness parameter tt1 versus [HN03]/[DMDBTDMA]. The different lines illustrate the impact of the error in this T-1 experimental linear law. (From C. Erlinger, L. Belloni, T. Zemb, and C. Madic, Langmuir, 15 2290-2300, 1999. With permission). 

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. 

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