The importance of the local structure of fractal aggregates

Abstract

The pair correlation function, g(r), is a fundamental descriptor of the inner structure of fractal aggregates of monomers. It provides a natural tool for studying physical properties involving two-point interaction (e.g. optics of aggregates). Several domains of distances between pairs of monomers have been identified. The fractal domain (in which g(r) is a power-law) is generally dominant for large aggregates. We show here that the local behavior of g(r)—involving monomers tangent to a given monomer—is necessary for most of the quantitative applications, even if that local domain is not directly related to fractal morphology. We derive a simple generic pair correlation function for fractal aggregates, depending on three structural parameters only: the fractal dimension, df ; the prefactor, kf ; the local mean coordination number, Zˉ . Unlike the fractal dimension, the prefactor and the mean coordination number are not universal since they depend on many parameters of the generation process. We discuss the impact of the definite shape of g(r) on the aggregate structure factor profile (a measurable quantity through small-angle x-ray scattering). Improvement from the new g(r) shape is also illustrated deriving an analytical expression of the geometric cross section, G, of aggregates for fractal dimension <2 . Accuracy is checked by comparing with numerical data from aggregates of fractal dimension df=1.9 . On the up-to-date analytical expression of G, the contributions of the short- and long-range behaviours of g(r) are well separated, and it is clear that the local behavior of the pair correlation function is required to obtain accurate values of the geometric cross section. Thus, in addition to the fractal structure of aggregates, the local structure of aggregates (to which the prefactor kf and the g(r) divergence at short distances are related) also appears important to accurately describe their physical features.