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

Abstract

Small-angle scattering (SAS; X-rays, neutrons, light) is being increasingly used to better understand the structure of fractal-based materials and to describe their interaction at nano- and micro-scales. To this aim, several minimalist yet specific theoretical models which exploit the fractal symmetry have been developed to extract additional information from SAS data. Although this problem can be solved exactly for many particular fractal structures, due to the intrinsic limitations of the SAS method, the inverse scattering problem, i.e., determination of the fractal structure from the intensity curve, is ill-posed. However, fractals can be divided into various classes, not necessarily disjointed, with the most common being random, deterministic, mass, surface, pore, fat and multifractals. Each class has its own imprint on the scattering intensity, and although one cannot uniquely identify the structure of a fractal based solely on SAS data, one can differentiate between various classes to which they belong. This has important practical applications in correlating their structural properties with physical ones. The article reviews SAS from several fractal models with an emphasis on describing which information can be extracted from each class, and how this can be performed experimentally. To illustrate this procedure and to validate the theoretical models, numerical simulations based on Monte Carlo methods are performed.

Structure, self-assembly, and properties of a truncated reflectin variant

“…3E. The unified fit's low-q (<0.07 Å −1 ) Guinier region revealed an R g value of 16.6 ± 0.3 Å, which correlated with an experimental R g /R H ratio of 1.18 that was indicative of deviation from a globular shape for RfA1TV (36,47), and the fit's high-q (>0.07 Å −1 ) Porod region revealed a Porod exponent of ∼3, which was likewise indicative of a complex shape for the protein (48)(49)(50). The Kratky plot of I(q) × q 2 as a function of q for our scattering intensity profile is shown in Fig.…”

Montmorillonite nanoclay aggregated in a fractal structure in an acrylic‐styrene matrix, slowed the chain dynamics and increased an order of magnitude the tensile modulus

“…The long‐range structure in the nanostructured films was investigated by SAXS, and the log–log plot of intensity as a function of the magnitude of the scattering angle | q | is shown in Figure 4; the magnitude of the scattering vector is defined as q = | q | = 4π·sinθ/λ 21 . These results show that the scattered intensity of the nanocomposite films obeys a power‐law scaling which is typical of fractal structures 26,27 The scaling power D varies from 1.9 (5 wt% MMT) to 2.3 (25 wt% MMT), and the average for all the nanocomposites is 2.10 ± 0.17. This scaling corresponds to that of randomly oriented lamellae or platelets 26 and indicates that MMT is aggregated approximately into the same self‐similar structure regardless of MMT content (within the range of this investigation).…”

Analysis of the sol and gel structures of potato starch over a wide spatial scale

“…The k ‐value was assumed to depend on the scattering of molecular networks, which might be the diffusion‐limited aggregation cluster. For example, the formation of a 2D diffusion‐limited aggregation cluster presented by Anitas ( 2020 ) was attributed to the particles undergoing a random walk (attributable to Brownian motion), eventually clustering into aggregates. The k ‐value obtained by us agrees well with the analytical value obtained by Muthukumar ( 1983 ) using the mean‐field theory for diffusion‐limited cluster formation.…”