  We investigate the structural and thermodynamic properties of a model of
particles with $2$ patches of type $A$ and $10$ patches of type $B$. Particles
are placed on the sites of a face centered cubic lattice with the patches
oriented along the nearest neighbor directions. The competition between the
self-assembly of chains, rings and networks on the phase diagram is
investigated by carrying out a systematic investigation of this class of
models, using an extension of Wertheim's theory for associating fluids and
Monte Carlo numerical simulations. We varied the ratio
$r\equiv\epsilon_{AB}/\epsilon_{AA}$ of the interaction between patches $A$ and
$B$, $\epsilon_{AB}$, and between $A$ patches, $\epsilon_{AA}$ ($\epsilon_{BB}$
is set to $0$) as well as the relative position of the $A$ patches, i.e., the
angle $\theta$ between the (lattice) directions of the $A$ patches. We found
that both $r$ and $\theta$ ($60^\circ,90^\circ,$ or $120^\circ$) have a
profound effect on the phase diagram. In the empty fluid regime ($r < 1/2$) the
phase diagram is re-entrant with a closed miscibility loop. The region around
the lower critical point exhibits unusual structural and thermodynamic behavior
determined by the presence of relatively short rings. The agreement between the
results of theory and simulation is excellent for $\theta=120^\circ$ but
deteriorates as $\theta$ decreases, revealing the need for new theoretical
approaches to describe the structure and thermodynamics of systems dominated by
small rings.
