Liquid-liquid phase transition for an attractive isotropic potential with wide repulsive range

We investigate how the phase diagram of a repulsive soft-core attractive potential, with a liquid-liquid phase transition in addition to the standard gas-liquid phase transition, changes by varying the parameters of the potential. We extend our previous work on short soft-core ranges to the case of large soft-core ranges, by using an integral equation approach in the hypernetted-chain approximation. We show, using a modified van der Waals equation we recently introduced, that if there is a balance between the attractive and repulsive part of the potential this potential has two fluid-fluid critical points well separated in temperature and in density. This implies that for the repulsive (attractive) energy U-R(U-A) and the repulsive (attractive) range w(R)(w(A)) the relation U-R/U-A proportional to w(R)/w(A) holds for short soft-core ranges, while U-R/U-A proportional to 3w(R)/w(A) holds for large soft-core ranges.