Frank H. Stillinger,^1,2 Dorothea K. Stillinger,¹ Salvatore Torquato,^2,3,4 Thomas M. Truskett,⁵
and Pablo G. Debenedetti⁵

¹ Bell Laboratories, Lucent Technologies, Murray Hill, NJ 07974.
² Princeton Materials Institute, Princeton University, Princeton, NJ 08544.
³ Department of Civil Engineering and Operations Research, Princeton University, Princeton, NJ, 08544.
⁴ Institute for Advanced Study, Princeton, NJ 08540.
⁵ Department of Chemical Engineering, Princeton University, Princeton, NJ 08544.

The distribution function f ⁽³⁾ for triplets of mutual nearest neighbors offers a description of local order for many-particle systems confined to a plane. This paper proposes a self-consistent theory for f ⁽³⁾ in the case of the classical rigid disk model, using three basic identities for closure. Numerical analysis of the resulting coupled nonlinear integral equations yields predictions for the pressure, the boundary tension, and the Kirkwood superposition defect for three disks in mutual contact. The approximation employed implicitly constrains the disk system to remain in the fluid phase at all densities up to close packing. The pressure and boundary tension agree reasonably well with the corresponding predictions of the two-dimensional scaled particle theory, but the former agrees even better with a rational approximant due to Sanchez that reproduces eight virial coefficients.

I. Introduction

The two-dimensional rigid disk system possesses value as a model for planar-surface adsorption phenomena, while offering conceptual simplicity. But in spite of that simplicity it continues to provide substantial challenges to theory, particularly regarding its freezing behavior under lateral compression. It has yet to be conclusively determined whether that freezing transition is a simple first-order phase change in the conventional large-system limit, whether a single higher-order transition is involved, or whether an intervening hexatic phase appears between the isotropic fluid and the triangular crystal. Furthermore, the question of the possible existence of a well-defined metastable extension for the fluid into the compressed density regime beyond freezing constitutes another significant theoretical problem.

The purpose of this presentation is to utilize the distribution function for nearest-neighbor triangles to investigate the fluid-phase equation of state and local order in the classical rigid disk system. As will be seen below, this automatically includes a metastable extension into the high-compression regime. The strategy follows a prior feasibility study¹ that developed earlier ideas advanced by Collins,^2,3 and which indicated that given a suitable closure approximation, a self-consistent theory could be constructed for the nearest-neighbor triangle distribution function.

The next Section II defines the nearest-neighbor triangle distribution function, and reviews the identities it must satisfy, one of which connects to the pressure equation of state.¹ Section III introduces a generic form for the distribution that provides the basis for a self-consistent closed theory. Numerical results for this self-consistent theory appear in Section IV; these include the pressure, the straight boundary line tension, and the Kirkwood superposition defect for disk triplets in mutual contact. A concluding Section V discusses the results obtained as well as opportunities for extending the present approach.