Cluster crystals with combined soft- and hard-core repulsive interactions

Particle systems interacting with a soft repulsion, at thermal equilibrium and under some circumstances, are known to form cluster crystals, i.e., periodic arrangements of particle aggregates. We study here how these states are modified by the presence of an additional hard-core repulsion, accounting for particle size. To this end we consider a two-dimensional system of Brownian particles interacting through a potential which includes hard-core and soft-core (of the GEM-α type) repulsive terms. The system shows different phases, and we focus in the regime where cluster crystals form. We consider two situations: the low-temperature one in which particles inside the clusters also show an ordered structure (crystal cluster-crystal phase), and the one occurring at higher temperature in which particles within the clusters are spatially disordered (fluid cluster crystal). An explicit expression for the energy in terms of the typical distance between clusters and the typical distance of the particles within the clusters is obtained for vanishing temperature, from which mean mean inter- and intracluster distances are obtained. Finite-temperature corrections are also discussed, considering explicitly the role of entropy.