The critical endpoint in phase diagrams of attractive hard spheres

We calculate the critical endpoint (cep) in phase diagrams of attractive hard spheres (HS), for both a Yukawa attraction and a polymer depletion potential, using approximate analytical theories. The cep corresponds to the minimum attraction range and the maximum contact potential for which liquid can exist. For both sytems we find essentially the same result: in the cep the attraction range is about one-third of the particle radius, and the attraction strength is around 2 kT. This same cep is found from simulations on particles interacting through a Lennard-Jones-type potential. It thus seems that the cep is essentially independent of the details of the pair potential for any system with a smooth attractive pair potential.