Particle aggregation fractal dimension

In principle, however, the effective surface area for heat transfer from the single monomer is decreased by the contact area between the particles. The fractal dimension of the aggregates varies in different systems and depends on the evolution process. For dense aggregates, heat transfer rates are decreased due to the reduced heat exchange area compared to primary particles only connected by point contact in a chain like structure (Liu et al., 2006a-c). This leads to an overestimation of... 

FIGURE 6.21 Computer generated aggregate assuming particle-partide aggregation. Fractal dimension of 1.5 in two dimensions made to simulate a fractal dimension of 2.5 in three dimensions. Printed, with permission from Sutherland [72]. 

Particle size and aggregate fractal dimension were measured with a Malvern Instruments Mastersizer/E. A 100 mm lens was used to measure particle sizes between 200 nm and 110 im (the 45 mm lens allowed particle analysis from 50 nm to 80 pm). Measurements were taken immediately after filling the cell to avoid settling effects, and no stirring was applied. 

In Fig. 1.2, the images of the nanocomposites studied, obtained in the force modulation regime, and corresponding to them nanoparticles aggregate, fractal dimension (d distributions are shown. As it follows Ifom the deduced values of dj = 2.40-2.48), nanofiller particles aggregates in the... 

Figure 5.12 Rg versus m I m for aggregates of N primary particles. The circle diameters are directly proportional to the aggregate fractal dimensions A Reproduced by permission of Elsevier.

Using this method. Park et al. [81] analysed TEM images of diesel particles and showed that the projected area equivalent diameter nearly equals the mobility diameter in the mobility size range from 50 to 220 nm. Doubly charged particles and possible fragments were observed for the DMA-classified particles. The fractal dimension calculated from the TEM images of mobility-classified aggregates was 1.75. The... 

The aggregate fractal dimension dp is a function of size distribution of constituting particles, solids concentration, temperature, pH, ionic strength and polymer concentration [14, 33-35]. Computer simulation has been applied extensively [36] to study both DLA and RLA and predict aggregate structure by manipulating simulation parameters such as the random walk size [37, 38]. 

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). 

Unlike the simulations which only consider particle-cluster interactions discussed earlier, hierarchical cluster-cluster aggregation (HCCA) allows for the formation of clusters from two clusters of the same size. Clusters formed by this method are not as dense as clusters formed by particle-cluster simulations, because a cluster cannot penetrate into another cluster as far as a single particle can (Fig. 37). The fractal dimension of HCCA clusters varies from 2.0 to 2.3 depending on the model used to generate the structure DLA, RLA, or LTA. For additional details, the reader may consult Meakin (1988). 

Fig. 41. Typical 2D fractal structure obtained by aggregation of particles in the journal bearing flow. Fractal dimension of the cluster is 1.54 (Hansen and Ottino, 1996b).

Computer simulations combined with experiments have also shown that one can deduce from the fractal dimension the nature of nucleation and growth of particles and what chemical and physical mechanisms control the formation of particle aggregates. We consider this briefly before proceeding to other topics. 

Debye X/[4tt sin (0/2)] = s Particle size or aggregate size see Equation (57) The information obtained depends on the magnitude of the ratio (Lch/Lyd). Depending on the value of this ratio, details on particle shape or fractal dimension can be obtained see Section 5.6... 

The development leading to Equation (66) in this section related the form factor P(6) to the radius of gyration Rg, which is one measure of the structure of a particle. We can actually get much more information from the form factor. In the following section, we discuss this and illustrate the use of P(d) for measuring the fractal dimension (defined in Chapter 1, Section 1.5b. 1) of an aggregate. 

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