Fractals cluster structures

So far we have described the growth of a compact cluster. But, as we have seen in Fig. 5, some clusters show more fllamental structures under speciflc circumstances, and this type of form is called fractal. The simplest growth process leading to fractal structures is the so-called diffusion-limited... 

Matsushita, M Hayakawa, Y. and Sawada, Y. (1985) Fractal structure and cluster statistics of zinc-metal trees deposited on a line electrode. Phys. Rev. A, 32, 3814-3816. 

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. 

We focus on aggregation in model, regular and chaotic, flows. Two aggregation scenarios are considered In (i) the clusters retain a compact geometry—forming disks and spheres—whereas in (ii) fractal structures are formed. The primary focus of (i) is kinetics and self-similarity of size distributions, while the main focus of (ii) is the fractal structure of the clusters and its dependence with the flow. 

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 situ SAXS investigations of a variety of sol-gel-derived silicates are consistent with the above predictions. For example, silicate species formed by hydrolysis of TEOS at pH 11.5 and H20/Si = 12, conditions in which we expect monomers to be continually produced by dissolution, are dense, uniform particles with well defined interfaces as determined in SAXS experiments by the Porod slope of -4 (non-fractal) (Brinker, C. J., Hurd, A. J. and Ward, K. D., in press). By comparison, silicate polymers formed by hydrolysis at pH 2 and H20/Si = 5, conditions in which we expect reaction-limited cluster-cluster aggregation with an absence of monomer due to the hydrolytic stability of siloxane bonds, are fractal structures characterized by D - 1.9 (Porod slope — -1.9) (29-30). 

Once interpenetration occurs the resistance to deformation increases markedly, so for example we would expect compaction of a sediment to become limited, as would further concentration in a filter press. It is worth emphasising the point that this is a simplistic approach, as prior to interpenetration the clusters undergo structural rearrangements changing their fractal index at a critical volume fraction. A typical data set for yield stress is shown in Figure 6.16.19... 

The porous structure of a cluster fractal can be quantified by estimating its number density at any stage of growth. For example, the number density of any cluster in Fig. 6.10 can be calculated with the equation... 

A characteristic feature of the carbon modifications obtained by the method developed by us is their fractal structure (Fig. 1), which manifests itself by various geometric forms. In the electrochemical cell used by us, the initiation of the benzene dehydrogenation and polycondensation process is associated with the occurrence of short local discharges at the metal electrode surface. Further development of the chain process may take place spontaneously or accompanied with individual discharges of different duration and intensity, or in arc breakdown mode. The conduction channels that appear in the dielectric medium may be due to the formation of various percolation carbon clusters. 

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. 

TEM and ultracentrifuge results showed (see Fig. 16) that this process results in effective encapsulation of the carbon with practically complete yield only rather small hybrid particles, but no free carbon or empty polymer particles, were found. It has to be stated that the hybrid particles with high carbon contents do not possess spherical shape, but adopt the typical fractal structure of carbon clusters, coated with a thin but homogeneous polymer film. The thickness of the monomer film depends on the amount of monomer, and the exchange of monomer between different surface layers is - as in miniemulsion polymerization - suppressed by the presence of an ultrahydrophobe. 

Rajca, A. (2002). Organic spin clusters, fractals, and networks with very high-spin. In Hyper-structured Molecules III, Sasabe, H. (ed.), Chapter 3, pp. 46-60, Taylor Francis, London... 

A typical situation is realized in such cases - fractal structure properties appear on a range scale which is limited by the dimensions of the particles forming the aggregate at the beginning, and at the end, by the dimensions of the initial fractal clusters. Typical particle dimensions are 1-10 nm aggregate dimensions are 10-1000 nm. The fractal dimension, df, depends on the conditions of aggregate formation and, as a rule, lies within a range of df = 2 — 2.9. 

Fractal structures have been examined, in particular, in diffusion-controlled aggregation process (polymerization) [7-9], in colloids (aggregates of particles) [10-13], and in percolation clusters [1-3],... 

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