Aggregation of fractals

Illustration Aggregation of fractal structures in chaotic flows. In a... 

Illustration Aggregation of fractal structures in chaotic flows. In a further study of aggregation in two-dimensional chaotic flows, the passively convected clusters retain their geometry after aggregation, i.e., fractal structures are formed. A typical fractal cluster resulting from these simulations is shown in Fig. 41. 

Other approaches to reducing the membrane pore size are being investigated. Many of them are based on the sol-gel process or chemical vapor deposition as discussed in Chapter 3. An example is the preparation of small-pore silica membranes. Amorphous silica membranes have been prepared from solutions of silicate-based polymers. More specifically, some strategies are employed aggregation of fractal polymeric clusters, variation of sol composition, the use of organic molecular templates and modification of pore surface chemistry [Wallace and Brinker, 1993]. 

We now provide an example of such an inversion from the work of Wright et al (1992) in which spatial computer simulations were used to generate data on the aggregation of fractal clusters formed by Brownian motion of colloidal particles. We consider three-dimensional diffusion under two circumstances (i) that in which the diffusion coefficient of the cluster is independent of its mass and (ii) that in which the diffusion coefficient, decreases with increasing mass. The simulated process automatically produces noisy data and the number density in cluster mass is presented in Figure 6.2.10 at three different times for both cases (i) and (ii). 

FIGURE 6.2.12 Aggregation frequency obtained by soving the inverse problem for the aggregation of fractal clusters (i) mass-independent diffusion (ii) mass-dependent diffusion. (From Wright et al, 1992.)... 

Aggregation of Fractals. Under many conditions of silicate sol synthesis, the primary species are characterized by a mass fract ctimension D (2) that relates the polymer mass M to its radius rc as ... 

Acetylene-substituted Si-N-P compounds, synthesis, 236-237 Activated aluminas, description, 165 Aerogels, definition, 127 Aggregation of fractals, 104,106 Alcohol-substituted polymers from aldehydes and ketones, deprotonation-substitution reactions, 249-250 Alkenylborazine copolymers, quantitative reactivity studies of copolymerization reactions, 394... 

Figure C2.6.7. Fractal aggregate of gold particles with a = 12 0.7 nm, obtained under DLCA conditions, with = 1.74 (reproduced with pennission from [65]. Copyright 1984 Elsevier Science Publishers B.V).

Meakin P 1983 Formation of fractal clusters and networks by irreversible diffusion-limited aggregation Phys. Rev. Lett. 51 1119-22... 

T. Irisawa, M. Uwaha, Y. Saito. Scaling laws in thermal relaxation of fractal aggregates. Europhys Lett 50 139, 1985. 

In the case of amphiphilic molecules, characterized by the coexistence of spatially separated apolar (alkyl chains) and polar moieties, both parts cooperate to drive the intermolecular aggregation. This simple but pivotal peculiarity makes amphiphilic molecules soluble in both polar and apolar solvents and able to realize, in suitable conditions, an impressive variety of molecular aggregates characterized by spatially separated apolar and polar domains, local order at short times and fluidity at long times, and differences in size, shape (linear or branched chains, cyclic or globular aggregates, extended fractal-like molecular networks), and lifetime. 

Fig. 31. Fractal structures obtained experimentally at different stages of aggregation of a colloidal monolayer of 1 /im sulfonated polystyrene particles on the surface of an aqueous calcium chloride solution, initially uniformly distributed (Robinson and Earnshaw, 1992).

Rupture of fractal (flocculated) aggregates of polystyrene latices in simple shear flow and converging flow was studied by Sonntag and Russel (1986, 1987b). For simple shear flow and low electrolyte concentrations, the critical fragmentation number decreases sharply with agglomerate radius (R) as... 

Most recently, Horwatt, Manas-Zloczower, and Feke (1992a, b) carried out a numerical study of the rupture of fractal clusters generated by different aggregation protocols (more details of these protocols are given in Section... 

At the simplest level, the rate of flow-induced aggregation of compact spherical particles is described by Smoluchowski s theory [Eq. (32)]. Such expressions may then be incorporated into population balance equations to determine the evolution of the agglomerate size distribution with time. However with increase in agglomerate size, complex (fractal) structures may be generated that preclude analysis by simple methods as above. 

In this section, we consider flow-induced aggregation without diffusion, i.e., when the Peclet number, Pe = VLID, where V and L are the characteristic velocity and length and D is the Brownian diffusion coefficient, is much greater than unity. For simplicity, we neglect the hydrodynamic interactions of the clusters and highlight the effects of advection on the evolution of the cluster size distribution and the formation of fractal structures. 

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).

The aggregation of spherical particles such as small particles of silica have been modelled by a fractal approach.16,17 Fractal mathematics can... 

Ying Q, Zhang J, Liang D, Nakanishi W, Isobe H, Nakamura E, Chu B (2005) Fractal behavior of functionalized fullerene aggregates. I. Aggregation of two-handed tetraaminofullerene with DNA. Langmuir 21 9824-9831. 

M. Carpineti and M. Giglio Spinodal-Type Dynamics in Fractal Aggregation of Colloidal Clusters. Phys. Rev. Lett 68, 3327 (1992). 

Several forms of the modified Stake s law have been used in a series of studies concerning aggregates in the liquid phase (Li and Yuan, 2002). In these studies, the particle-liquid density difference has been further modified and adopted for the cases of impermeable bio-logical/microbial aggregates, permeable aggregates, and fractal aggregates. 

Konstandopoulos, A. G. Convective-diffusive deposition of fractal-like aggregates and the microstructure and properties of the resulting nanostructured deposits (to be submitted) (2007). 

Solution Assume that the aggregates are fractals with the mass ma(d) of an aggregate of diameter d being given by... 

Variation of specific surface area with geometry Surface area of fractal aggregates Polydispersity of a synthetic polymer... 

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