Chain formation in a two-dimensional system of hard spheres with a short-range anisotropic interaction

We analyze a generalization of the hard-sphere dipole system in two dimensions in which the range of the interaction can be varied. We focus on the system in the limit in which the interaction becomes increasingly short ranged, while the temperature becomes low. By using a cluster expansion and taking advantage of low temperatures to perform saddle-point approximations, we argue that a well-defined double limit exists in which the only structures which contribute to the free energy are chains. We then argue that the dominance of chain structures is equivalent to the dominance of chain diagrams in a cluster expansion, but only if the expansion is performed around a hard-sphere system (rather than the standard ideal gas). We show that this leads to nonstandard factorization rules for diagrams and use this to construct a closed-form expression for the free energy at low densities. We then compare this construction to several models previously developed for the hard-sphere dipole system in the regime where chain structures dominate and argue that the comparison provides evidence in favor of one model over the others. We also use this construction to incorporate some finite-density effects though the hard-sphere radial distribution function and analyze how these effects impact chain length and the equation of state.