Fractal Dimension of Particles

The electron-ion recombination in high-mobility systems has also been analyzed in terms of the fractal theory [24,25]. 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 diffusion but with a distance-dependent effective diffusion coefficient D r) = D(H-// where the parameter / is proportional to the mean free path X [24]. However, when the fractal dimension of trajectories is not equal to 2, the motion of particles is not described by orthodox diffusion. 

Hydrolysis of tetraalkoxysilanes in pure water is usually incomplete, but it is more effectively carried out in an alcohol-water mixture and can be catalyzed by H+ or weak bases such as ammonia. Polymerization occurs in the range pH 2-7 under neutral conditions, the rate of polymerization is limited by the slowness of hydrolysis, whereas in acidic media hydrolysis is complete before polymerization begins. The nanostructure of the gel (density of particles, fractal dimensions of particle clusters, and degree of cross-linking of particles to form a network) is controlled by these conditions. 

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]. 

Ersahin, S., H. Gunal, T. Kutlu, B. Yetgin, and S. Coban. 2006. Estimating specific surface area and cation exchange capacity in soils using fractal dimension of particle-size distribn-tion. Geoderma 136, no. 3 588-597. doi 10.1016/j.geoderma.2006.04.014. 

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).

In practical application, it was reported that the platinum particles dispersed in highly porous carbonized polyacrylonitrile (PAN) microcellular foam used as fuel-cell electrocatalyst160 have the partially active property. The fractal dimension of the platinum particles was determined to be smaller than 2.0 by using the potentiostatic current transient technique in oxygen-saturated solutions, and it was considered to be a reaction dimension, indicating that not all of the platinum particle surface sites are accessible to the incoming oxygen molecules. 

P(r) can be transformed into a distribution of the particle size as defined by the hydrodynamic radius Rh. But only for TDFRS, and not for PCS, a particle size distribution in terms of weight fractions can be obtained without any prior knowledge of the fractal dimension of the polymer molecule or colloid, which is expressed by the scaling relation of Eq. (39). This can be seen from the following simple arguments ... 

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... 

In the following table are data on the dependence of the average floccule radius, achieved after 500 s of flocculation, on the initial primary particle number density for a transport-controlled flocculation process. Estimate the fractal dimension of the floccules formed. (Answer D 1.8, based on a log-log plot.)... 

Measurements of the average floccule radius in a suspension of colloids, using light-scattering techniques, indicated the following time dependence R(t) = Ro(l + yty, where Ro = 5 nm, y = 9.3 s 1, and j8 = 0.56. Derive this equation, and estimate the initial number density of primary particles as well as the fractal dimension of the floccules formed. (Answer p0 = 7.5 x 1017 m 3 and D = 1.78.)... 

Kaye BH (1986) Image analysis procedures for characterising the fractal dimension of fine particles. Proc Part Technol Conf Nurnberg... 

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