The Properties of a Hard-Sphere Fluid

It is useful to examine the properties of a fluid made up of non-interacting hard spheres. Such a system may be regarded as an important reference liquid, albeit fictitious, with respect to which the properties of real systems can be compared. Its properties are most easily obtained on the basis of the Percus-Yevick (PY) approximation. Since the spheres do not interact, the interaction energy u(r) is zero outside any sphere  

Since the spheres are hard, that is, they do not penetrate one another, 

It is apparent that the interaction energy is discontinuous at the boundary of a sphere. From the discussion in section 2.6, discontinuities are also possessed by the pair correlation function g r) and the direct correlation function c(r) at the boundary of a sphere. In order to remove problems associated with these discontinuities, one introduces a new function, y(r) defined as follows  

Outside of a sphere where u(r) is zero, one finds that 

Referring to the PY approximation (equation (2.6.5)), inside a sphere where g(r) is zero, 

---

Before concluding this discussion of surface tension let us make a more careful analysis of these intercorrelations of properties for which the scaled particle theory has been used. The approach assumes that the thermodynamic derivatives can be interrelated by the properties of a hard-sphere fluid. In particular, the compressibility, thermal expansion, and surface tension can be related to one another. The ability to relate the surface tension to the other properties is unique to the scaled particle theory and it is not possible to test the consistency of the approach to that relation. However, for the hard-sphere scaled particle solution to relate the thermal expansion coefficient and the compressibility we must have d UjdV )x T d SldV )rp, This requirement is obvious when the thermal expansion... 

---