Fractal aggregates

Porosity within a particle is a manifestation of the shape of a particle. Fractal particles will have internal porosity as a result of their shapes. Fractal particles with low fractal dimensions (i.e., <2.0) will have a broad pore size distribution, where the largest pore approaches the size of the aggregate. Fractal particles with large fractal dimensions (i.e., >2.0) will have narrower pore size distributions with most of the porosity occurring at a size much smaller than that of the aggregate. Calcination of metal salt particles or metal hydroxides to produce oxides is another common method to produce internal porosity. In the gas evolution that takes place in transformation to the oxide, pores are opened up in the particle structure. The opening of pores in a hydrous alumina powder can increase its surface area from 0.5 m /gm (its external area) to 450 mVgm (its internal pore area). 

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... 

Nevertheless, occluded rabber must not be confused with the polymer part whose molecular mobility is changed by adsorption. Occluded rubber, which is mainly trapped in aggregate fractal sites, only represents a part of the volume of elastomer whose molecular motion is slowed down. 

If we have an aggregating fractal system, the generalized radius of aggregates is related to the number of particles in an aggregate Ao.a by... 

This relationship was used to calculate aggregate fractal dimensions from previous settling rate studies. However, this model was not found to describe the settling behaviour of fractal aggregates adequately due to the perhaps unwarranted assumption that and / are the only variables affecting U, which implies that the drag coefficient will be the same for fractals as for impermeable spheres in the laminar flow region 1). 

Logan and Wilkinson [116] improved the approach of Li and Ganczarczyk [117] by incorporating into the model a relationship between the aggregate projected area A, /, and the 2d aggregate fractal dimension d2f to account for surface area differences between fractals and equivalent-sized spheres. Since the settling velocity U... 

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.

Raper J.A. and Amal, R. (1993). Measurement of aggregate fractal dimension using static light scattering. Part. Part. Syst. Charact., 10, 239-245. 

Monodisperse particles correspond to PD < 0.02, and PD between 0.02 and 0.08 represents a narrow distribution. Notice that features of aggregation (aggregation rates) of the particles and aggregate fractality depend strongly on initial PD of particles, and preparation of monodisperse uniform inorganic colloid dispersions has a great practical importance. 

D. Coelho, J.-F. Thoveit, R. Thou, P.M. Adler, Hydrodynamic drag and electrophoresis of suspensions of fractal aggregates. Fractals 5(3), 507-522 (1997). doi 10.1142/ S0218348X97000401... 

Fractal aggregate, fractal agglomerate aggregates or agglomerates with a non-uniform distribution of the constiment particles, which typically coincides with a very porous, branch-like morphology fractal aggregates are characterised by a power-law decrease of the pair-correlation density function g(v) (Eq. (4.8)) and a power-law relationship between mass and size (Eq. (4.9)), in which the exponent is less than the Euclidean dimension fractal aggregates are not ideal fractal objects, but rather obey the fractal relationships only in a statistical sense (cf. Sect. 4.2.1). 

Figure la. Fractal aggregate, constructed by computer simulated diffusion limited aggregation. Fractal dimension D = 1.44... 

FIGURE 6.2.11 Self-similar behavior of aggregating fractal clusters obtained by scaling. The scaling size is obtained by taking the ratio of the second moment to the first (i) mass-independent diffusion (ii) mass-dependent diffusion. Notice different properties of the self-similar distribution at the origin for the two cases. (From Wright et al, 1992.)... 

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]. 

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