Mass fractal dimension aggregates

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). 

Bower et al measured the boundary fractal dimension of lactose aggregates in 1,1,2-trichlorotrifluoroethane using serial perimeter dilation and Adler and Hancock s distance transform function [56]. They found that it decreased as shear rate and aggregate mass fractal dimension increased. 

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] ... 

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-... 

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... 

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

Electrical sensing zone technique commonly used to determine equivalent volume diameter, required in Eq. (19), might be problematic. The error associated with this technique is contributed by the breakup of aggregates and inclusion of pores in volume measurement. With this technique, an aggregate will have to be suspended in a liquid. The challenge is to preserve the structure of aggregates. Hence the first method is preferred to obtain the mass fractal dimension of aggregates in situ. 

Laser diffraction technique is able to generate the envelope diameter distribution of the aggregates in the aerosol cloud, as well as the surface and mass fractal dimensions as explained in Light Scattering under both Boundary and Surface Fractal Dimensions and Mass Fractal Dimension. However, validation of Eq. (25) by comparison studies between the impaction and laser diffraction techniques is required. 

Techniques have been developed to determine the fractal dimension of experimental aggregate particles in solution using small-angle scattering techniques, from x-ray, light, or even neutron sources. In these techniques the scattering intensity, I q), is proportional to the scattering vector, q, raised to the mass fractal dimension by ... 

Figure 3.1 A log-log plot of number of particles iVo.a in an aggregate against radins of gyration of the aggregate. The data are obtained from computer simulations and show three different mass fractal dimensions, of 1.74, 2.11 and 2.41.

Figure 3.2 Example aggregates drawn from classes having mass fractal dimension (from top to bottom) of 1.74, 2.11 and 2.41. Each aggregate contains 2000 primary particles. Note the increasing compactness as the mass fractal dimension increases.

Figure 34 A typical structure factor S q) for a fractal aggregate of monodisperse spheres (solid line) and the form factor P q) for a collection of unaggregated monodisperse spheres. In this example the spheres have unit radius, the mass fractal dimension of the aggregate is 1.8 and the radius of gyration of the aggregate is 1000/ q.

In some ways, a more robust method of using small-angle light scattering to measure the mass fractal dimension of particle aggregates is to rely on the changing projected area of an assembly of particles as they aggregate, an idea first used by Volker Oles in 1992 [20]. 

Figure 3.6 A log-log plot of relative obscuration against aggregate radius of gyration for the three different simulated aggregate systems shown in Figure 3.1. From top to bottom, the mass fractal dimensions of the systems are 1.74 (triangles), 2.11 (squares) and 2.41 (stars).

Kusters et al. [22] analysed polystyrene latex particles of about 1 p,m diameter aggregated in salt solution in a stirred tank, with samples withdrawn from the tank by pipette for light-scattering measurements. They found mass fractal dimensions of around 2.5 using the volume obscuration method. 

Figure 3.9 Box-counting analysis of a projection of a computer-generated diffusion-limited cluster aggregate of 10 000 particles with mass fractal dimension of 1.88. The images have box sizes L of (top left to bottom right) 1, 2,4,8,16, 32 and 64 pixels and require 205 245,59519, 17 062, 4895,1436, 462 and 135 squares respectively to cover the image.

Dinsmore and Weitz (2002) [60] examined a model system of polymethylmethacrylate particles dyed with rhodamine in a density- and refractive-index-matched solution of decalin and cyclohexylbromide. They used CLSM to follow the (very slow) aggregation in real time and determined the particle positions in full three-dimensional detail. From this they performed a comprehensive structural analysis of the gels, including measurement of coordination numbers and backbone fractal dimension of the structures, as well as the much more commonly measured mass fractal dimension. 

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