A model of small-angle scattering from three-phase fractal systems

Abstract

In small-angle scattering (SAS) intensities on a double logarithmic scale, one generically observes "knees" arising from changes of the power-law exponents. While the appearance of power-law behavior is associated usually with a fractal structure, the explanation of the knees is a more intricate problem in general. It is shown that the crossover between the exponents can be accounted for by various levels of the structures with different scattering length densities (SLD). Thus, the whole system is a multiphase system, composed of the homogeneous structures embedded into each other. In this way, one can explain the crossover between arbitrary exponents whose values vary from −4 to 0. In order to confirm this conclusion, we calculate the SAS intensity from a set of non-interacting, randomly oriented and uniformly distributed three-phase fractals with controllable Hausdorff dimension. We observe the appearance of crossover (knee), which depends on the values of the SLD of each phase and estimate its position. Random fractals are described as well within the developed model by taking the fractal sizes at random, which leads to polydispersity and, as a consequence, to smearing of the SAS intensity curves.