Hard spheres, equilibrium phase diagrams liquid phases

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).

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]. 

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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