We study a family of lattice-gas systems defined on semiregular grids, obtained by projecting the vertices of three different geodesic icosahedra onto a spherical surface. By using couplings up to third neighbors, we explore various interaction patterns, ranging from core-corona repulsion to square-well attraction and short-range attractive, long-range repulsive potentials. The relatively small number of sites in each grid $(\sim 100)$ enables us to compute the exact statistical properties of the systems as a function of temperature and chemical potential by Wang-Landau sampling. For each case considered, we highlight the existence of distinct low-temperature “phases”, featuring, among others, regular-polyhedral, cluster-crystal, and wormlike structures. We highlight similarities and differences between these motifs and those observed on the triangular lattice under the same conditions. Finally, we discuss the relevance of our results for the bottom-up realization of spherical templates with desired functionalities.

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• Classical statistical mechanics
• Phase diagrams
• Phase transitions