Hard spheres, equilibrium phase diagrams

Figure 13.4 Equilibrium phase diagram for uniformly sized hard spheres. The liquid-crystal coexistence region is 0.494 < c ) < 0.545. Face-centered cubic structure for volume fraction 4) > 0.545 and regular close packing 0.740. (From Mau, S.C. and Huse, D.A., Phys. Rev.E, 59, 4396,1999.)...

Figure 3.10. Phase diagrams of attractive monodisperse dispersions. Uc is the contact pair potential and (j) is the particle volume fraction. For udk T = 0, the only accessible one-phase transition is the hard sphere transition. If Uc/hgT 0, two distinct scenarios are possible according to the value of the ratio (range of the pair potential over particle radius). For < 0.3 (a), only fluid-solid equilibrium is predicted. For % > 0.3 (b), in addition to fluid-solid equilibrium, a fluid-fluid (liquid-gas) coexistence is predicted with a critical point (C) and a triple point (T).

Figure 2.1 The hard-sphere phase diagram. Below volume fraction < (f>] = 0.494, the suspension is a disordered fluid. Between <) >i = 0.494 and 02 = 0.545, there is coexistence of this disordered phase with a colloidal crystalline phase with FCC (or HCP) order the colloidal crystalline phase is the equilibrium one up to the maximum close-packing limit of 0cp = 0.74. Nonequilibrium colloidal glassy behavior can also occur between

Fig. 9. Two-parameter ordering phase diagram for a system of 500 identical hard spheres (Truskett et ai, 2000 Torquato et ai, 2000). Shown are the coordinates in structural order parameter space (r, ) for the equilibrium fluid (dot-dashed), the equilibrium FCC crystal (dashed), and a set of glasses (circles) produced with varying compression rates. Here, r is the translational order parameter from (26) and is the bond-orientational order parameter Q( from (25) normalized by its value in the perfect FCC crystal ( = Each circle...

The application of this approach to the hard-sphere system was presented by Ree and Hoover in a footnote to their paper on the hard-sphere phase diagram. They made a calculation where they used Eq. (2.27) for the solid phase and an accurate equation of state for the fluid phase to obtain results that are in very close agreement with their results from MC simulations. The LJD theory in combination with perturbation theory for the liquid state free energy has been applied to the calculation of solid-fluid equilibrium for the Lennard-Jones 12-6 potential by Henderson and Barker [138] and by Mansoori and Canfield [139]. Ross has applied a similar approch to the exp-6 potential. A similar approach was used for square well potentials by Young [140]. More recent applications have been made to nonspherical molecules [100,141] and mixtures [101,108,109,142]. 

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 4.8 Phase diagram for a pure substance composed of hard spheres. The fluid-phase Z was computed from the Carnahan-Starling equation (4.5.4) the solid-phase Z was taken from the computer simulation data of Alder et al. [14]. The broken horizontal line at Zt = 6.124 connects fluid (T = 0.494) and solid (t = 0.545) phases that can coexist in equilibrium, as computed by Hoover and Ree [12].

In Fig. 4.3 we also plot the (equilibrium) binodals using FVT outlined in Chap. 3 for hard spheres plus penetrable hard spheres with diameters of 2Rg. Qualitatively, the phase diagram topology is quite well predicted. For q = 0.08, only equilibrium fluid, crystal and fluid + crystal regions are found and predicted. Both for q = 0.57 and 1 the phase diagram contains fluid, gas, liquid and crystalline (equilibrium) phases. In the different unmixing regions one now finds gas-liquid coexistence with a critical point, three-phase gas-liquid-crystal and... 

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