Closure approximation hypemetted chain

This approximation amounts to truncating the functional expansion of the excess free energy at second order in the density profile. This approach is accurate for Lennard-Jones fluids under some conditions, but has fallen out of favor because it is not capable of describing wetting transitions and coexisting liquid-vapor phases [105-107]. Incidentally, this approximation is identical to the hypemetted chain closure to the wall-OZ equation [103]. 

From the various possible closures, the mean spherical approximation (MSA) [189] has found particularly wide attention in phase equilibrium calculations of ionic fluids. The Percus-Yevick (PY) closure is unsatisfactory for long-range potentials [173, 187, 190]. The hypemetted chain approximation (HNC), widely used in electrolyte thermodynamics [168, 173], leads to an increasing instability of the numerical algorithm as the phase boundary is approached [191]. There seems to be no decisive relation between the location of this numerical instability and phase transition lines [192-194]. Attempts were made to extrapolate phase transition lines from results far away, where the HNC is soluble [81, 194]. 

Two of the classic integral equation approximations for atomic liquids are the PY (Percus-Yevick) and the HNC (hypemetted chain) approximations that use the following closures... 

Closure approximations to the PRISM equation are generally developed via an analogy with atomic liquids. Three of the common closures for hard spheres are the Percus-Yevick (PY), hypemetted chain (HNC), and Martynov-Sarkisov (MS) closures. It has been shown that the PY closure is the most accurate of the three, and in fact the HNC and MS closures have either no solution or unphysical solutions at low densities. The PY closure is given by. 

---