Leaky cell model of hard spheres

Abstract

We study packings of hard spheres on lattices. The partition function, and therefore the pressure, may be written solely in terms of the accessible free volume, i.e. the volume of space that a sphere can explore without touching another sphere. We compute these free volumes using a leaky cell model, in which the accessible space accounts for the possibility that spheres may escape from the local cage of lattice neighbors. We describe how elementary geometry may be used to calculate the free volume exactly for this leaky cell model in two-and three-dimensional lattice packings and compare the results to the well-known Carnahan-Starling and Percus-Yevick liquid models. We provide formulas for the free volumes of various lattices and use the common tangent construction to identify several phase transitions between them in the leaky cell regime, indicating the possibility of coexistence in crystalline materials.

Cavity Volume and Free Energy in Many-Body Systems

“…The theory presented within this work concerns cavity volume, which is related, although distinct from, the well-studied theory of free volume (Buehler et al 1951;Eyring and Hirschfelder 1937;Fai et al 2021;Devonshire 1937, 1938). Given a configuration of hard particles, the free volume of a particular particle is the volume that it may access by continuous motion while holding all other particles in place, without particle interpenetration.…”

“…We constrast our results with the scaled particle theory equation of state (Helfand et al 1961), which gives the compressibility factor in terms of the packing fraction η as Z = (1 − η) −2 , which is expected to be accurate before the freezing transition at around η ≈ 0.706. Furthermore, we make comparison with the leaky cell theory for an hexagonal lattice (Fai et al 2021), which is an extension of the classical cell theory (Buehler et al 1951), to dilute regimes. The cell theory approaches are expected to be better approximations in dense regimes.…”

“…Calculating averages of free volumes is a many-body problem, making explicit closedform solutions generally unobtainable. One way of vastly simplifying these calculations is the cell theory, where the system is presumed to be well-approximated by a lattice, whose lattice parameters can be derived as a function of the number density, with the consequence that complex multi-particle interactions may be reduced to investigating the local environment of a single particle within the lattice (Eyring and Hirschfelder 1937;Fai et al 2021;Hirschfelder et al 1937;Devonshire 1937, 1938). The cell theory is sufficiently simple to provide closed-form results and generally performs well in the dense regime, as dense hard particle systems are lattice-like in two and three dimensions.…”

Cavity volume and free energy in many-body systems

“…The theory presented within this work concerns cavity volume, which is related, although distinct from, the well studied theory of free volume [7,13,14,27,28]. Given a configuration of hard particles, the free volume of a particular particle is the volume that it may access by continuous motion while holding all other particles in place, without particle interpenetration.…”

“…Calculating averages of free volumes is a many-body problem, making explicit closedform solutions generally unobtainable. One way of vastly simplifying these calculations is the cell theory, where the system is presumed to be well approximated by a lattice, whose lattice parameters can be derived as a function of the number density, with the consequence that complex multiparticle interactions may be reduced to investigating the local environment of a single particle within the lattice [13,14,20,27,28]. The cell theory is sufficiently simple to provide closed-form results and generally performs well in the dense regime, as dense hard particle systems are lattice-like in two and three dimensions.…”