The structural characterization of deterministic mass fractals at nano- and microscales is presented in this chapter using two complementary techniques in both reciprocal and real spaces. In the former case, fractal and geometrical features are obtained from the small-angle scattering (SAS) (neutrons, X-rays, light) spectrum in the reciprocal space. The lacunarity technique is considered to extract structural properties and differentiate textures of fractals in real space. We present and discuss various types of mass fractals, such as thin and fat fractals, as well as fractals generated with the Chaos game representation (CGR). We show how the main structural properties of the fractals, such as the fractal dimension, the iteration number, the scaling factor, the overall size of the fractal, and the size of the basic units of the fractal, can be extracted by using SAS and lacunarity techniques.
Part of the book: Chaos Theory
The concepts of mass and surface fractals are introduced, and the corresponding small-angle scattering (SAS; X-rays, neutrons) intensities are computed. It is shown how to resolve the fractal structure of various complex systems from experimental scattering measurements, and how obtained data are related to specific features of the fractal models. We present and discuss various mass and surface fractal structures, including fractals generated from iterated function systems and cellular automata. In addition to the fractal dimension and the overall fractal size, the suggested analysis allows us to obtain the iteration number, the number of basic units which form the fractal and the scaling factor.
Part of the book: Complexity in Biological and Physical Systems
Electroconductive magnetorheological elastomers (MREs) have attracted a wide scientific attention in recent years due to their potential applications as electric current elements, in seismic protection, in production of rehabilitation devices, and sensors or transducers of magnetic fields/mechanical tensions. A particular interest concerns their behavior under the influence of external magnetic and electric fields, since various physical properties (e.g., rheological, elastic, electrical) can be continuously and/or reversibly modified. In this chapter, we describe fabrication methods and structural properties from small-angle neutron scattering (SANS) of various isotropic and anisotropic MRE and hybrid MRE. We present and discuss the physical mechanisms leading to the main features of interest for various medical and technical applications, such as electrical (complex dielectric permittivity, electrical conductivity) and rheological (viscosity) properties.
Part of the book: Electric Field
Structural analysis of fractals generated using one-dimensional additive cellular automata (ACA) is presented in this chapter. ACA is a dynamical system that evolves in discrete steps and generates two-dimensional self-similar structures. We investigate the structure of M-state ACA Rule 90 and Rule 150 using small-angle scattering (SAS; X-rays, neutrons, light) technique and multi-fractal analysis. We show how the scattering data from ACA can provide information about the overall size of the system, the number of total units, the number of rows, the size of the basic fractal units, the scaling factor, and the fractal dimension. In this case, when a particular row number reproduces a complete structure of the fractals, we can also obtain the fractal iteration number. We show that subsets of different states of M-state ACA can manifest both mono- and multi-fractal properties. We provide some useful relations between structural parameters of ACA that can be obtained experimentally from SAS.
Part of the book: Small Angle Scattering and Diffraction