The puzzle-interlayer model: an approach to the analysis of tightly packed arrangements of hard particles

The small-angle scattering (SAS) structure functions are analyzed for an idealized, random two-phase system: a stationary arrangement of hard puzzle particles, separated by a relatively thin interspace, which can be approximated by an interlayer. The detailed shape of the interlayer is defined by the shape of the particles themselves: The starting point for producing these initial particles is a three-dimensional initial puzzle P0 in the state of tessellation. Its pieces, homogeneous particles of random shape, fit together filling the space. In a second step, an expanded puzzle P is constructed by translating the initial particles by a certain length (relative to one another). The whole tightly packed particle arrangement depends on P0. The interlayer region between the particles is a connected, homogeneous region. The SAS intensity of P depends on the parameter and on the typical shape and size of the pieces of P0. Chord length distributions (CLDs) are used in the description. The random shape of the pieces of P0 possesses a CLD 1. The CLD 2 of the intermediate spaces is approximated by that of an idealized layer of constant thickness . The scattering of P results in terms of the CLDs of both phases. The approach can be applied to many types of P0. Two initial tessellations of P0 are studied, a ‘dead leaves’ tessellation produced by spherical primary grains and the Poisson plane mosaic.