Mass fractal dimension

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. ... 

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... 

A very helpful approach to understand the formation of particle clusters was given by Meakin [2], who uses the concept of mass fractal dimension ) , which relates the size cf of a particle cluster to its mass M (Eq.l). 

Any discussion of fumed silica particle structure has to take into account this enormous difference. The approach of mass fractal dimension may provide a rough but helpful estimation. A real mass fractal is limited by the size of the cluster as an upper limit and the size of the particles as a lower limit. Then, the density of the cluster pduster be calculated from the true density of the particle Pparticio fhe ratio of the cluster size c/ciuster to the particle size /particle and the mass fractal dimension /) of the cluster (Eq. 2) ... 

In the flame process firm aggregates are formed from primary particles by DLA followed by agglomeration of these aggregates by RLCA, according to mass fractal dimensions of 2 5... 

The scaling properties of fractal structures, both surface and mass structures, are quantified by their fractal dimensions. A mass fractal dimension (Df) may be defined in the following way. The mass (M) of an object may be expressed as a power law of its radius (R). 

Mass fractal dimensions are always less than the dimension of the space in which the fractal object exists. Therefore, as a fractal structure grows, its mass increases less rapidly than the volume it occupies. Therefore, the density of a fractal object is not constant but decreases with increasing size. Surface fractal dimensions, on the other hand, must lie in the range of one less than the dimension of space up to the dimension of space. The surface area of these objects increases with increasing mass at a faster rate than for Euclidian objects. As a result, the surfaces are very convoluted. Both types of structures are observed in silica polymers. 

Koylu et al. used extensive TEM studies to derive the mass fractal dimension, Df, of flame-made soot particles from Equation A1 (see the appendix) by extracting the number of primary particles per agglomerate, N, the primary particle size, (ip, and the radius of gyration of the agglomerate, R. The constant fractal prefactor, A, as well as the fractal dimension, Dj, were then determined by regression. 

Ultra small-angle x-ray scattering (USAXS) is capable of determining all characteristics of agglomerate and primary particles from a single scattering experiment mass fractal dimension, Dj, radii of gyration (primary particles and... 

The mass-fractal dimension of the ramified agglomerates is determined from the slope of the weak power law decay in between the power law regimes that follow Porod s law (Equation 10.9) ... 

Beaucage, G., Small-angle scattering from polymeric mass fractals of arbitrary mass-fractal dimension, J. Appl. Crystallogr, 29, 134, 1996. 

There are a number of different fractal dimensions commonly used to describe a specific property of a system. These fractal dimensions and methods to obtain them are explained in detail in Boundary and Surface Fractal Dimensions and Mass Fractal Dimension. Fractal dimensions used in specific applications will be shown also in the related section. 

A three-dimensional mass fractal dimension. Dm, describes the packing of particles forming an aggregate. Its value varies from 1 to 3. Unlike the Ds, which ascribes a low value to a smooth surface, the higher the value of Dm, the more densely packed is the aggregate. Mass fractal dimension of 3 corresponds to a solid structure. A lower Dm shows a loose and commonly branchier structure of the fractal aggregate (Fig. 3). 

Box counting method is commonly used to obtain mass fractal dimension from an aggregate s projected area. 

---