Structural Properties of Vicsek-like Deterministic Multifractals

Abstract

Deterministic nano-fractal structures have recently emerged, displaying huge potential for the fabrication of complex materials with predefined physical properties and functionalities. Exploiting the structural properties of fractals, such as symmetry and self-similarity, could greatly extend the applicability of such materials. Analyses of small-angle scattering (SAS) curves from deterministic fractal models with a single scaling factor have allowed the obtaining of valuable fractal properties but they are insufficient to describe non-uniform structures with rich scaling properties such as fractals with multiple scaling factors. To extract additional information about this class of fractal structures we performed an analysis of multifractal spectra and SAS intensity of a representative fractal model with two scaling factors—termed Vicsek-like fractal. We observed that the box-counting fractal dimension in multifractal spectra coincide with the scattering exponent of SAS curves in mass-fractal regions. Our analyses further revealed transitions from heterogeneous to homogeneous structures accompanied by changes from short to long-range mass-fractal regions. These transitions are explained in terms of the relative values of the scaling factors.

Keywords

• fractals
• small-angle scattering
• form factor
• structural properties
• dimension spectra
• pair distance distribution function