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.
PACS numbers:Understanding the competition between self-assembly and phase-separation is central in today's research in different fields encompassing biology, soft-matter, material science, statistical mechanics. Self-assembly of finitesize aggregates requires strong interaction energies compared to the thermal energy to guarantee that the generated structure is persistent. As a result, self-assembly competes with the ubiquitous macroscopic phase separation, the low-temperature tendency common to atoms, molecules and larger particles to maximize the number of bonded neighbours and minimize the potential energy, giving rise to a condensed (liquid) state [1]. When particles can arrange themselves into low energy and weakly interacting finite size aggregates, self-assembly can completely suppress phase separation [2][3][4]. Soft matter and biology offers several examples of three-dimensional stable aggregates (e.g. micelles, vesicles, capsids [5,6]) which constitute the stable phase in wide temperature and density regions.The possibility of hampering the formation of large amorphous aggregates in favor of ordered structures is often facilitated by the presence of strong, directional and saturable interactions (limited valence) [7][8][9]. The renewed focus on the role of directional interactions, stimulated by the synthesis of new-generation patchy colloids [10][11][12][13], has deepened our understanding not only of the role of the valence on the gas-liquid phase separation [14] but also on the competition between selfassembly and phase separation. Two recent investigations have provided insights particularly relevant for this work: (i) a numerical study of Janus colloids in which a gas-liquid critical point and a self-assembly process are simultaneously observed [15,16]. In this model, the formation of energetically stable vesicles stabilizes at low temperature T the gas-phase; (ii) a study of particles with dissimilar patches [17][18][19], promoting respectively chaining and branching, specifically designed to reproduce a mean-field model introduced by Safran and Tlustly [20] to describe the phase behavior of dipolar fluids. Here branching produces a gas-liquid critical point, but on cooling the formation of energetically favored chains stabilizes, thi...