The structure of deterministic mass and surface fractals: theory and methods of analyzing small-angle scattering data

Abstract

Small-angle scattering (SAS) of X-rays, neutrons or light from ensembles of randomly oriented and placed deterministic fractal structures are studied theoretically. In the standard analysis, a very few parameters can be determined from SAS data: the fractal dimension, and the lower and upper limits of the fractal range. The self-similarity of deterministic structures allows one to obtain additional characteristics of their spatial structures. The paper considers models which can describe accurately SAS from such structures. The developed models of deterministic fractals offer many advantages in describing fractal systems, including the possibility to extract additional structural information, an analytic description of SAS intensity, and effective computational algorithms. Generalized Cantor fractals and few of its variants are used as basic examples to illustrate the above concepts and to model physical samples with mass, surface, and multi-fractal structures. The differences between the deterministic and random fractal structures in analyzing SAS data are emphasized. Several limitations are identified in order to motivate future investigations of deterministic fractal structures.

Studies of Electroconductive Magnetorheological Elastomers

“…For small values of the radius of gyration, the curves clearly show the appearance of a plateau (at about 4 Â 10 À2 ≲ q ≲ 10 À1 ), which may indicate that the sizes of the scattering units are much smaller than the distances between them. Similar behavior of the scattering curve has been observed also in [40][41][42][43][44][45][46][47][48]. Figure 5 shows the SANS curves on a double logarithmic scale, corresponding to a polymer matrix consisting of Stomaflex creme and silicone oil (Left part), as well as the polymer matrix in which were embedded Fe particles (Right part) (see Ref.…”

Small-Angle Scattering from Fractals: Differentiating between Various Types of Structures

“…Therefore, the multifractal form factor can be expressed through a recurrence relation between subsequent iterations. For an arbitrarily two-scale multifractal, one can write [62,82]:…”

Structural Properties of Vicsek-like Deterministic Multifractals

“…At m = 2, while repeating the same procedure for each disk, one obtains k 2 disks of radius β 2 s2 r 0 , k 1 disks of radii β s1 β s2 r 0 , k 2 1 disks of radii β 1 s2 r 0 , k 1 and so on. Thus, at an arbitrarily iteration m, we can write the corresponding form factor in terms of a recurrence relation of the form [32]:…”

“…For ESS structures, the main advantage of SAS relies on its ability to distinguish between mass and surface fractals through the value of the scattering exponent τ in the fractal region [28][29][30][31]. More recently, it has been shown that SAS can also differentiate between ESS and statistically self-similar (SSS) structures [32] as well as between regular and fat fractals, that is, those with positive Lebesgue measure [33].…”

“…This is mainly due to the fact that the vast majority of physical materials are heterogeneous at nano-and micro-scales, thus requiring models with at least two scaling factors. Although a first step in this direction was done by obtaining the SAS spectrum from a multifractal structure generated using the chaos game algorithm [34], an expression for the scattering intensity was derived only recently [32]. Moreover, in Reference [34] it has been shown that, for the investigated structure, the oscillations in the fractal region are not very pronounced leading to difficulties in recovering the scaling factors from experimental data.…”