Self-Assembly of Particles on a Curved Mesh

Discrete statistical systems offer a significant advantage over systems defined in the continuum, since they allow for an easier enumeration of microstates. We introduce a lattice-gas model on the vertices of a polyhedron called a pentakis icosidodecahedron and draw its exact phase diagram by the Wang-Landau method. Using different values for the couplings between first-, second-, and third-neighbor particles, we explore various interaction patterns for the model, ranging from softly repulsive to Lennard-Jones-like and SALR. We highlight the existence of sharp transitions between distinct low-temperature "phases", featuring, among others, regular polyhedral, cluster-crystal-like, and worm-like structures. When attempting to reproduce the equation of state of the model by Monte Carlo simulation, we find hysteretic behavior near zero temperature, implying a bottleneck issue for Metropolis dynamics near phase-crossover points.