Thermodynamic perturbation theory, solid-fluid

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. 

In order to determine the thermodynamic properties by means of the perturbation theory, the thermodynamic properties of the reference system are needed. Here, the expressions for the equation of state and the radial distribution function of a system of hard spheres are included for both the fluid and solid reference states. A face-centred-eubic arrangement of the particles at closest packing is assumed for the solid phase. 

Figure 12. The hard-sphere equation-of-state (d = 3.9 A) as a function of reduced fluid density computed for polyethylene at T = 430 K and N = 6429 by various thermodynamic routes free energy (upper solid), compressibility (lower solid), wall (dashed), and GFD (short/long dash). The inset includes attractions by perturbation theory using the GFD curve as the reference system the points represent experimental results. ...

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