Perturbation theory Phase points

Up to this point we have focused on the use of a single theory to describe both phases in a calculation of SFE. A somewhat less ambitious approach is to use a theory appropriate to each phase in the calculations. In particular, the cell theory can be used for the solid phase and a liquid state theory (e.g., a hard-sphere equation of state or thermodynamic perturbation theory) used for the fluid phase. This approach turns out to be at least as accurate as either the two-phase cell theory or DFT approaches described above and is often more accurate. Moreover, it has been more successful in the treatment of systems more complex than hard spheres. 

Although most of the studies of this model have focused on the fluid phase in connection with the theory of electrolyte solutions, its solid-fluid phase behavior has been the subject of two recent computer simulation studies in addition to theoretical studies. Smit et al. [272] and Vega et al. [142] have made MC simulation studies to determine the solid-fluid and solid-solid equilibria in this model. Two solid phases are encountered. At low temperature the substitutionally ordered CsCl structure is stable due to the influence of the coulombic interactions under these conditions. At high temperatures where packing of equal-sized hard spheres determines the stability a substitutionally disordered fee structure is stable. There is a triple point where the fluid and two solid phases coexist in addition to a vapor-liquid-solid triple point. This behavior can be qualitatively described by using the cell theory for the solid phase and perturbation theory for the fluid phase [142]. Predictions from density functional theory [273] are less accurate for this system. 

Fig. 6.21 Phase diagrams of hard spherocylinders LjD = 5) mixed with penetrable hard spheres in the reservoir-phs representation for three q-values as indicated. The curves were obtained from perturbation theory with simulation results of the reference system. Redrawn and converted from [46]. The thick straight lines represent triple coexistences. Data points in the upper graph are direct Monte Carlo computer simulation results from [47]. Filled circle critical point, open circles Ii + I2 coexistence...

Experimental data including the acidic species in the vapor phase within the above concentration range are scarce. Only very few publications of VLE data in that range are available [168, 173]. In contrast, numerous vapor pressure curves are accessible in literature. Chemical equilibrium data for the polycondensation and dissociation reaction in that range (>100 wt%) are so far not published [148]. However, a starting point to describe the vapor-Uquid equilibrium at those high concentratirMis is given by an EOS which is based on the fundamentals of the perturbation theory of Barker [212, 213]. Built on this theory, Sadowski et al. [214] have developed the PC-SAFT (Perturbed Chain Statistical Associated Fluid Theory) equation of state. The PC-SAFT EOS and its derivatives offer the ability to be fuUy predictive in combination with quantum mechanically based estimated parameters [215] and can therefore be used for systems without or with very little experimental data. Nevertheless, a model validation should be undertaken. Cameretti et al. [216] adopted the PC-SAFT EOS for electrolyte systems (ePC-SAFT), but the quality for weak electrolytes as phosphoric... 

Fig. 10. Densities of coexisting phases for a fluid obeying the 6-12 potential according to a statistical mechanical perturbation theory developed by Barker and Henderson. The points are a mixture of machine calculations and actual experimental data (from Barker and Henderson, 1967).

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