Criticality of colloids with three distinct interaction patches: As simple as A,B,C?

We systematically study the phase behavior of a simple model of associating fluids which consists of hard spherical particles with three short-ranged attractive sites on their surfaces (sticky spots or patches), of types A,B, and C, that can form bonds with energy ε_{ij} (i,j=A,B,C). We consider realizations of the model with one, two, or three nonzero ε_{ij}. Using Wertheim's first order perturbation theory of association, we establish the minimum requirements on the bond energies for the model to exhibit a liquid-vapor critical point, and investigate the nature of criticality in each case. As a preliminary, we rigorously show that, within this theory, particles with M identical sites do not condense if M<3, a result that was previously conjectured, but never proved.