Fractal mass fractal

Fractal objects are quantified by their fractal dimension, dj. For linear-like stmctures, 1 < <2. FractaUy rough stmctures have a mass fractal... 

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. 

FIGURE 17.4 USAXS and SALS results for samples Al, A2, and A3 described in the text. Each sample shows four structural levels with the i g for some of the levels indicated in the graph. The power-law value for the second level, corresponding to the mass-fractal dimension, d(, is also indicated. 

Table 17.1 also displays the mass-fractal dimension, df, given by the power-law value, P, for the second level (aggregate). The df= 1.63 for A1 is less than that, and is less than 2.37 for A2 and 2.1 for A3. The lower value of df could indicate a more open structure of the A1 sample, " but it could also arise from less interpenetration of the silica aggregates occurring during drying. ... 

Figure 8.13. Different kinds of fractals, (a) Surface fractal, (b) Mass fractal, (c) Pore fractal... 

Characteristic for a fractal structure is self-similarity. Similar to the mentioned pores that cover all magnitudes , the general fractal is characterized by the property that typical structuring elements are re-discovered on each scale of magnification. Thus neither the surface of a surface fractal nor volume or surface of a mass fractal can be specified absolutely. We thus leave the application-oriented fundament of materials science. A so-called fractal dimension D becomes the only absolute global parameter of the material. 

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). 

In Table 29.5 sorption capacity of a fibrous carbon (activated carbon felt, MAST Carbon International), experimental deliganding carbon felt and mass-fractal (a = - 2.39) HSGD carbon towards methylene blue and unconjugated bilirubin, measured in batch experiments are compared. 

In the context of particle aggregation, the particular fractal structures called mass fractals are usually considered. Mass fractals correlate the location of mass with radius, as opposed to other types of fractal constructs such as boundary fractals, surface fractals, and pore fractals (see Ref. 17). The power law which correlates the size of an aggregate with the number of primary particles in the aggregate is given by... 

It is shown below that p fulfills a scaling relation which involves the size and mass fractal dimension of the primary aggregates. Due to significant deviations of the solid fraction p from 1, the filler volume fraction of carbon black in rubber composites has to be treated as an effective one in most applications, i.e., 0eff=0/0p (compare [22]). 

The exponent df is denoted mass fractal dimension or simply fractal dimension. It characterizes the mass distribution in three dimensional space and can vary between lfractal analysis of furnace blacks was performed, e.g., by Herd et al. [108] or Gerspacher et al. [109, 110]. The solid volume Vp of primary aggregates is normally determined (ASTM 3849) from the cross-section area A and the perimeter P of the single carbon black aggregates by referring to a simple Euclidean relation [108] ... 

Here, Ap is the cross section area of a single primary particle. Dependent on the application of Eqs. (12) or (13), respectively, significantly different values for the mass fractal dimension are obtained. 

Therefore, it appears likely that the approach considering the solid volume of primary aggregates, as evaluated from the two-dimensional cross-section area by Eq. (12), leads to an overestimation of the mass fractal dimension. A more realistic estimate is obtained with Eq. (13). By referring to Eq. (12), the data obtained by Herd et al. [108] show a successively increasing value of the mass fractal dimension from to df 2.8 with increas-... 

The proportionality constant Nf in Eq. (21) is a generalized Flory-Number of order one (Np=l) that considers a possible interpenetrating of neighboring clusters [22]. For an estimation of cluster size in dependence of filler concentration we take into account that the solid fraction of fractal CCA-clusters fulfils a scaling law similar to Eq. (14). It follow directly from the definition of the mass fractal dimension df given by NA=( /d)df, which implies... 

---