  We study the interplay between phase separation and self-assembly in chains,
rings and branched structures in a model of particles with dissimilar patches.
We extend Wertheim's first order perturbation theory to include the effects of
ring formation and theoretically investigate the thermodynamics of the model.
We find a peculiar shape for the vapor-liquid coexistence, featuring re-entrant
behavior in both phases and two critical points, despite the single-component
nature of the system. The emergence of the lower critical point is caused by
the self-assembly of rings taking place in the vapor, generating a phase with
lower energy and lower entropy than the liquid. Monte Carlo simulations of the
same model fully support these unconventional theoretical predictions.
