Fractal scattering

In the case of mass fractals, scattering will occur from the bulk of an object instead of surface. As a consequence, when q >... 

The close similarity of and qj t is characteristic for the fractal scattering regime, i.e. for Xagg scattering intensity scales with N while the aggregate volume scales with N-y (cf. Eqs. (4.45) and (4.61)). Differences between q and qj t, thus, mainly reflect the polydispersity of the primary particles Xp (Babick et al. 2012a). 

Mass fractal scattering, on the other hand, is associated with the structure of the inhomogeneities and not their surface. Physically, mass fractal scattering is observed when small particles agglomerate together to form a non-aystalline superstructure similar to that shown in Fig. 6.7. These structures are known to exist in aerogels, whereby particles are very close to each other but do not possess any order. The structure of these particles can be described as a fractal-like object [50]. The degree to which... 

Power-law scattering features will be discussed in relation to mass-fractal scaling laws. Fractal scaling concepts used to interpret the power-law decay are well published in the literature. 

The power, [P], in the fractal power-law regime gives as the fractal dimension, d(. P = —df for each level of the fit, the parameters obtained using the unified model are G, Rg, B, and P. P is the exponent of the power-law decay. When more than one level is fitted, numbered subscripts are used to indicate the level—i.e., G —level 1 Guinier pre-factor. The scattering analysis in the studies summarized here uses two-level fits, as they apply to scattering from the primary particles (level 1) and the aggregates (level 2). 

Figure 8.8. Important components of diffuse scattering in the SAXS of polymer materials with two or more phases. Only for fractals Porod s law is fundamentally changed...

Experimentally accessible is D by means of scattering methods [144], The corresponding fractal analysis of scattering data is gaining special attractivity from its intriguing simplicity. In a double-logarithmic plot of I (s) v.v. s the fractal dimension is directly obtained from the slope of the linearized scattering curve. It follows from the theory of fractals that... 

Figure 8.14. Sketch of diffuse scattering in the double-logarithmic plot that is used for the determination of the fractal dimension. Superior and lower cut limit the fractal region that should be followed by an interval in which POROD s law is valid... 

The foregoing treatment can be extended to cases where the electron-ion recombination is only partially diffusion-controlled and where the electron scattering mean free path is greater than the intermolecular separation. Both modifications are necessary when the electron mobility is - 100 cm2v is-1 or greater (Mozumder, 1990). It has been shown that the complicated random trajectory of a diffusing particle with a finite mean free path can have a simple representation in fractal diffusivity (Takayasu, 1982). In practice, this means the diffusion coefficient becomes distance-dependent of the form... 

In t-DVB/S copolymerization, Antonietti and Rosenauer isolated microgels slightly below the macrogelation point [221]. Using small angle neutron scattering measurements they demonstrated that these microgels exhibit fractal behavior, i.e. they are self-similar like the critically branched structures formed close to the sol-gel transition. 

Other possible methods of obtaining fractal dimensions are light- and small-an-gle-X-ray-scattering (SAXS). The first method gives information about mass fractality of aggregates and aggregation kinetics (Fleischmann et al., 1990), whereas the latter allows mass and surface fractals to be differentiated (Schmidt, 1990). 

Fractal dimensions of synthetic goethite obtained by small angle X-ray scattering (SAXS) were in agreement with the values calculated using the equation of Avnir and Jaroniec (Weidler et al., 1995 a). Values of D, mentioned in later section have been obtained using this equation. 

Schlichting, E. (1965) Die Raseneisenbildung in der nordwestdeutschen Podsol-Gley-Land-schaft. Chemie der Erde 24 11-26 Schmalz, R.F. (1959) Formation of red beds in modern and ancient deserts. Discussion. Geol. Soc. Am. Bull. 79 277-280 Schmidt, P.W. (1990) Small-angle scattering studies of disordered, porous and fractal systems. J. Appl. Cryst. 24 414-435 Schmucki, P. Virtanen, S. Davenport, A. Vitus, C.M. (1996) J. Electrochem. Soc. 144 ... 

The dependence of the electron ion recombination rate constant on the mean free path for electron scattering has also been analyzed on the basis of the Fokker Planck equation [40] and in terms of the fractal theory [24,25,41]. In the fractal approach, it was postulated that even when the fractal dimension of particle trajectories is not equal to 2, the motion of particles is still described by difihsion but with a distance-dependent effective diffusion coefficient. However, when the fractal dimension of trajectories is not equal to 2, the motion of particles is not described by orthodox diffusion. For the... 

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