Systematic Approach to Explanation of the Rigid Disk Phase Transition

Abstract

By classifying particle center positions with a hexagonal grid, evaluation of the two-dimensional rigid sphere partition function is reduced to a special lattice statistics problem, with precisely defined nearest-neighbor effective pair interactions. The hexagonal cell size is chosen to be the maximum consistent with no more than double occupancy. Since the resulting lattice partition function (with three states per site) contains a collectively determined many-cell effective interaction Δ*, as well as nearest-neighbor contributions, it becomes necessary to examine in detail the statistical geometry of available phase space for the original spheres, under varying restraints of nearest-neighbor cell distribution. Accordingly, we obtain for the first time an unambiguous definition of ``random close-packed'' or ``glassy'' arrangements of spheres (which however are not themselves equilibrium states), and to relations between properties of these arrangements, and of Δ*. The key features which subsequently allow description of rigid sphere order—disorder behavior are: (1) the observation that certain nearest-neighbor cell pairs which occur in the glassy state (both unoccupied and both doubly occupied) are geometrically excluded completely in the ordered, close-packed arrangement; (2) Δ* sensitively depends upon these pair distributions. In spite of the fact that Δ* is thereby assigned a specially generalized free-volume form, the theory leads to a proper virial series development at low density. In addition, we report some preliminary results for the effective cell interactions, for glassy state parameters, and some calculations designed to reveal the structure of Δ*. Although this analysis does not yet represent a full quantitative theory of the two-dimensional rigid disk system, it does lead to a novel qualitative explanation of how a fluid—solid transition can occur, and suggestions are given for completing the quantification.