Small-angle scattering from multiphase fractals

Abstract

Small-angle scattering (SAS) intensities observed experimentally are often characterized by the presence of successive power-law regimes with various scattering exponents whose values vary from -4 to -1. This usually indicates multiple fractal structures of the sample characterized by different size scales. The existing models explaining the crossover positions (that is, the points where the power-law scattering exponent changes) involve only one contrast parameter, which depends solely on the ratio of the fractal sizes. Here, a model that describes SAS from a multi-phase system with a few contrast parameters is described, and it is shown that the crossover position depends on the scattering length density of each phase. The Stuhrmann contrast variation method is generalized and applied to experimental curves in the vicinity of the crossover point beyond the Guinier region. The contrast variation is applied not to the intensity itself but to the model parameters, which can be found by fitting the experimental data with the suggested interpolation formula. The model supplements the existing two-phase models and gives the simple condition of their inapplicability: if the crossover point depends on the contrast then a two-phase model is not relevant. The developed analysis allows one to answer the qualitative question of whether one fractal 'absorbs' another one or they are both immersed in a surrounding homogeneous medium like a solvent or solid matrix. The models can be applied to experimental SAS data where the absolute value of the scattering exponent of the first power-law regime is higher than that of the subsequent second power-law regime, that is, the scattering curve is 'convex' near the crossover point. As is shown, the crossover position can be very sensitive to contrast variation, which influences significantly the length of the fractal range.

Small-Angle Scattering from Mass and Surface Fractals

“…In particular, if the triangles composing the system are arranged in such a way that they form a Sierpinski gasket, the intermediate region will correspond to a mass fractal region [29][30][31]. When the triangles have a power-law distribution in their sizes, the intermediate region will correspond to a surface fractal one [37,38].…”

“…The last years have brought a breakthrough in the theoretical analysis of SAXS/SANS experimental data, allowing for the extraction of additional structural information and detailed modeling of fractals using a deterministic approach [29][30][31][32][33][34][35][36][37][38][39][40]. This progress was stimulated by recent advances in nanotechnology, which allows preparation of both mass and surface deterministic fractals at sub-micrometer scale [15-18, 41, 42], as well as by instrumentation which allows novel structural features to be recorded in experimental data [43].…”

High-throughput SANS experiment on two-detector system of YuMO spectrometer

“…The optimal Q-range has especially to be taken into account when measuring samples with «fractal» properties. The fractal structures with determined dimension have a linear behavior of scattering curve in log-log scale [40][41][42][43][44][45]. Boundary conditions determine the size-range where the fractal structure could be observed.…”

Structural Analysis of Deterministic Mass Fractals Using Small- Angle Scattering and Lacunarity Techniques

“…However, rapid progress in the field of materials science [5] allows creating exact deterministic fractal structures [6][7][8][9]. Since the influence of the fractal structure on the physical properties of the system is of significant research interest [11], investigations concerning structural properties of deterministic fractals have been recently suggested [14,30,31].…”

“…Then, in order to obtain the fractal structure, we establish the rule of evolution (generator) [30,31], shown in Figure 3. The rule is the following: scale the initial disk by the factor of β s = 1/3 and make four copies of it, so the length size of the squares in which the disks at m = 1 are inscribed as a 1 = a 0 β s .…”