Statistical geometry of particle packings. I. Algorithm for exact determination of connectivity, volume, and surface areas of void space in monodisperse and polydisperse sphere packings

Geometry plays an important role in determining the physical properties of many systems composed of large numbers of particles. In continuum systems, the study of geometry in a large number of cases involves the analysis of configurations of spheres, which may or may not intersect each other. In many cases, the representation of structure in terms of assemblies of spheres is dictated by the inherent properties of the component particles, as in the case of monatomic liquids @1‐3#, colloidal suspensions @4#, etc. In other cases, such as the modeling of porous media @5#, powders @6#, etc. arrangements of spheres serve as useful and reasonable models that facilitate theoretical analysis. In many problems, the void region ~the region that lies outside an appropriately defined occupied space associated with the spheres; see discussion below! and the interface between the void and occupied regions are of specific interest. An important example is the hard sphere system. It has been shown @7,8# that the thermodynamics of a system composed of hard spheres is determined entirely by the volume and surface area of the available space, which is defined as the set of all allowed positions for the center of an additional hard sphere; the occupied space in this case is the union of spheres around each hard sphere center, whose radius equals the diameter of the hard spheres. Note that unlike the hard spheres themselves, the relevant exclusion spheres can overlap with each other. Another interesting example is the estimation of the solvation free energy of proteins, which depends on the accessible surface area @9#. A detailed analysis of various particle and void correlation functions has been pursued in recent years, as a useful approach to quantifying the morphology and properties of inhomogeneous media @10‐12#. The nearest neighbor void distribution functions @11# for a given configuration of sphere centers may be related to the volume fraction and surface areas of the void region, by defining exclusion spheres of variable radius around sphere centers. The above remarks serve to illustrate the usefulness of knowing the volumes and surface areas of the void region in packings of spheres. They also indicate that the general problem of interest is the calculation of these quantities in configurations of overlapping spheres. In such configurations, the void space is in general composed of disconnected regions, or cavities. Some of the void quantities mentioned above can be calculated straightforwardly using standard Monte Carlo sampling methods. However, such estimation becomes highly unsatisfactory on account of statistical uncertainty when the volume fraction of the void region is small @13#, e.g., for high densities in the hard sphere system. Hence it is desirable to have a method that permits an exact decomposition of void space into cavities and the calculation of the volume and surface areas associated with such cavities. While such methods have previously been developed for two dimensions @13‐15#, no exact method has been available for threedimensional systems. Further, no method has been available for such calculation in polydisperse packings in either twoor three-dimensional systems. In this paper, we present a method for calculating the volumes and surface areas of cavities in arbitrary monodisperse and polydisperse sphere packings. In the companion part II @16#, we use the algorithm to study the statistical geometry of inherent structures @17# in a system composed of atoms interacting with the Lennard-Jones potential. The paper is organized as follows. In Sec. II we state the problem considered and outline the algorithm, for the monodisperse case. In Sec. III we discuss the network mapping that allows the identification of cavities. In Sec. IV we describe the identification of polyhedra that enclose the cavity volumes. In Sec. V we discuss the calculation of cavity volumes and surface areas within such enclosing polyhedra. Some caveats that pertain to periodic arrangements of spheres are discussed in Sec. VI. The generalization to the polydisperse case is detailed in Sec. VII. Section VIII contains a summary and concluding remarks.