Mass Fractal Structure Functions

To describe the structure of systems without apparent correlation peaks, many studies employ simplifying approximations that hold only in a limited -range, such as the Guinier approximation for 0  

The gyration radius Rg is determined from a Zimm or Guinier (ln/( ) versus plot. Similarly, in the fractalic region of the scattering curve, roughly in the range 2tc/ q 27i ro, the scattering curve can be fitted to the power-law relationship  

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Much of the current interest in fractal geometry stems from the fact that fractal dimensions are experimentally accessible quantities. For polymers and colloids, the measurement techniques of choice are scattering experiments using X-rays, neutrons, or light. These measurements may be made on liquids or solids and can be performed readily as a function of time and temperature. Both mass and surface fractal structures yield scattering curves that are power laws, the exponents of which depend on the fractal dimension (6). For mass fractal structures the relation is... 

The fractal structure of the interconnected network has been confirmed through numerical analysis of the TEM micrographs [279]. For a homogeneous medium, the mass, M(r), increases as a function of distance (r) from the origin in two dimensions M(r) == r. For a fractal structure, however. 

In all the cases examined so far, it is the matter distribution of the object that has exhibited the property of self-similarity. These objects are called mass fractals. Other situations are encountered, where it is not the matter distribution which has self-similarity, but rather the pore distribution in these cases, we speak of pore volume fractals. Some structures are found in which only the contour or surface manifests scale invariance these are called boundary or surface fractals, and the exponent we need to know is the boundary fractal dimension. To obtain the corresponding exponents, we calculate the autocorrelation function, the mass distribution or the number of boxes, restricting ourselves to the relevant subsets (the points occupied by matter, the points in the pore volume, or the points lying in the interface). 

In relation to aggregate structures, it is classified as a mass fractal when D < 3 and a surface fractal when D > 3 [21]. In particular, when D = 4, it is termed the Porod region and is regarded as a smooth interface. The D value during the gelation has been predicted [22]. For the -dependency of f, a combination of the the OZ and DB equations, the equation that incorporates elongation function, and OZ and DB equations that contain fiactal dimension have been proposed [16]. 

Morphology of Dealloyed Structures on 2-D Lattices The roughness of the dissolution front is characterized by its fractal dimension, d (estimated from the mass theorem [190]). Variation of as a function of p is shown in Figure 31, displaying the classical transition toward the dimension of the infinite cluster at the percolation threshold d = 1.60). 

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