Three-dimensional patchy lattice model: ring formation and phase   separation
  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. 
J.M. Tavares, N.G. Almarza and M.M. Telo da Gama
cond-mat.soft