A. Yu. Cherny, E. M. Anitas, V. A. Osipov, A. I. Kuklin
Small-angle scattering (SAS) intensities observed experimentally, are very often characterized by the presence of successive power-law regimes with various scattering exponents whose values varies 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 and, hence, it depends solely on the ratio of the fractal sizes. Here, we develop a model which describe SAS from multi-phase system with a few contrast parameters and show that the crossover position depends on the scattering length density of each phase. We generalize the Stuhrmann contrast variation method and show how it can be 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 developed analysis allows us to answer the qualitative question 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.
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http://arxiv.org/abs/1307.2351
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