Lattice hard spheres

Lattice hard spheres refers to athermal lattice gas models, in which pairs of sites separated by less than some short distance $\sigma$ cannot be simultaneously occupied.

Brief description of the models

Basically the differences with the standard lattice gas model (Ising model) are:

• An occupied site excludes the occupation of some of the neighboring sites.

• No energy interactions between pairs of occupied sites -apart of the hard core interactions- are considered.

These systems exhibit phase (order-disorder) transitions

Three-dimensional lattices

• See Ref. 1 for some results of three-dimensional lattice hard sphere systems (on a simple cubic lattice)

The model defined on a simple cubic lattice with exclusion of ony the nearest neighbour positions of an occupied site, presents a continuous transition

If also next-nearest neighbours are excluded, then the transition becomes first order (See Ref 1).

Two-dimensional lattices

Square lattice

• See Ref 2. for results of two-dimensional systems (lattice hard disks) on a square lattice.

Triangular lattice

The hard hexagon lattice model belongs to this kind of models. In this model an occupied site excluded the occupation of nearest neighbour positions. This model exhibits a continuous transition. (See references in the entry: hard hexagon lattice model).

Other models defined on the triangular lattice (with more excluded positions) have been studied theoretically and by Monte Carlo simulation (Refs 3-4). It seems (see Ref. 3) that the model that includes first and second neighbour exclusion presents also a continuous transition, whereas if third neigbours are also excluded the transition becomes first order.

References

1. A. Z. Panagiotopoulos, "Thermodynamic properties of lattice hard-sphere models", J. Chem. Phys. 123, 104504 (2005)
2. Heitor C. Marques Fernandes, Jeferson J. Arenzon, and Yan Levin "Monte Carlo simulations of two-dimensional hard core lattice gases" J. Chem. Phys. 126, 114508 (2007).
3. N. C. Bartelt and T. L. Einstein, "Triangular lattice gas with first- and second-neighbor exclusions: Continuous transition in the four-state Potts universality class" Phys. Rev. B 30, 5339 - 5341 (1984).
4. Chin-Kun Hu and Kit-Sing Mak, "Percolation and phase transitions of hard-core particles on lattices: Monte Carlo approach", Phys. Rev. B 39, 2948 - 2951 (1989).