Surface correlation functions of dead-leaves models

TL;DR

This paper derives exact analytical expressions for surface correlation functions in dead-leaves models, applicable to any grain shape and dimension, enhancing understanding of material structures in scattering theory.

Contribution

It provides the first general analytical formulas for surface correlation functions in dead-leaves models, including arbitrary grain shapes and dimensions.

Findings

Analytical expressions match well with numerical results for spherical grains.

Dead-leaves structures have similar surface-surface but different pore-surface correlations compared to Debye media.

General formulas for Boolean models' correlation functions are also derived.

Abstract

The pore-surface and surface-surface correlation functions are structural characteristics that play an important role in theoretical materials science and in small-angle scattering theory. Exact analytical expressions for the surface correlation functions are available only for very few models, and we here derive such expressions for the general class of dead-leaves models. Within these models, a two-phase pore/solid structure is created by sequentially and randomly filling space with pore-like or solid-like grains that overlap any pre-existing structure, in the same way as dead leaves fall on the ground. The obtained mathematical expressions are valid for any grain shape, in arbitrary dimension. The results are illustrated with monodispersed spherical grains,as well as with a dead-leaves realization of a Debye random medium. In the latter case, the size distribution of the grains is designed to produce a structure having exponential two-point correlation function. Compared to Debye random media obtained by numerical reconstruction, the dead-leaves structure has almost identical surface-surface correlation function, but distinctly different pore-surface correlation function. As a byproduct of our analysis, we also submit a general expression for the pore-surface and surface-surface correlation functions of the Boolean model, valid for arbitrary grains.