Inability of the hypernetted chain integral equation to exhibit a spinodal line

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Abstract
The hypernetted chain (HNC) integral equation is solved for a variety of systems including the simple fluid of hard spheres with attractive Yukawa tail, the symmetrical and highly asymmetrical mixtures of charged particles, and the mixtures of hard spheres with positively nonadditive diameters. All these systems exhibit a liquid–gas or fluid–fluid phase separation. We are concerned with the behavior of the HNC solution near the two‐phase region in the phase diagram. In all cases, the HNC equation does not have solutions inside a certain region whose boundary line is not a spinodal line. As the boundary is approached, the isothermal compressibility deduced from the compressibility route does not diverge, but tends to finite values. The isotherms and isochores terminate at the boundary line in square root branch points. This behavior is directly correlated to the existence of multiple HNC solutions. Physical and nonphysical solutions coincide on the boundary line. This universal HNC behavior is compared with that of the mean spherical approximation and Percus–Yevick integral equations.