Fractal clusters

The power-law relation in Eq. 6.1 can be interpreted physically as indicative of a cluster fractal.12 The exponent D is then termed the cluster fractal dimension. Some basic concepts about cluster fractals are introduced in Special Topic 3 at the end of this chapter. Suffice it to say here that Eq. 6.1 can be pictured as a generalization of the geometric relation between the number of primary particles in a cluster that is d-dimensional (d = 1, 2, or 3) and the d-dimensional size of the cluster. For example, if a cluster is one-dimensional (d = 1), it can be portrayed as a straight chain of, say, circular primary particles of diameter L0. The number of particles in a chain of length L is... 

Cluster fractals that are created by diffusion-limited flocculation processes are described mathematically by power-law relationships like those in Eqs. 6.1 and 6.5. These relationships are said to have a scaling property because they satisfy what in mathematics is termed a homogeneity condition 22... 

If the von Smoluchowski rate law (Eq. 6.10) is to be consistent with the formation of cluster fractals, then it must in some way also exhibit scaling properties. These properties, in turn, have to be exhibited by its second-order rate coefficient kmn since this parameter represents the flocculation mechanism, aside from the binary-encounter feature implicit in the sequential reaction in Eq. 6.8. The model expression for kmn in Eq. 6.16b, for example, should have a scaling property. Indeed, if the assumption is made that DJRm (m = 1, 2,. . . ) is constant, Eq. 6.16c applies, and if cluster fractals are formed, Eq. 6.1 can be used (with R replacing L) to put Eq. 6.16c into the form... 

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... 

Family F., and D. P. Landau, Kinetics of Aggregation and Gelation, North-Holland, Amsterdam, 1984. An excellent compendium of articles on cluster fractals—both theory and experiment. 

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... 

Therefore, the stated above results have confirmed again that D, values distribution is the main reason of microgels stracture variation, characterized by its fractal dimension Dp Dp change at reaction duration growth is well described quantitatively within the frameworks of aggregation mechanism cluster-cluster. Fractal space, in which curing reaction proceeds, is formed by the stracture of the largest cluster in system [55],... 

Clerc J P, Giraud G, Laugier J M and Luck J M (1990) The electrical conductivity of binary disordered systems, percolation clusters, fractals and related models, Adv Phys 39 191-209. Balberg I (1987) Tunneling and nonuniversal conductivity in composite materials, Phys Rev Lett 59 1305-1308. 

Clearly there is need for spin-glass theories beyond mean-field. One approach in this direction is presented by Malozemoff et al. (1983) and Malozemoff and Barbara (1985). They propose a critical fractal cluster model of spin glasses which is able to describe the essential features of the phenomena occurring near the spin-glass transition and to account for the static critical exponents. The basic assumption of this fractal model is the existence of a temperature- and magnetic-field-dependent characteristic cluster size on which all relevant physical quantities depend and which diverges at the transition temperature Tj. It is related to the correlation length and the cluster fractal dimension D by More... 

Fig. 1. Illustration of the selfsimilarity property in the case of (a) a deterministic and (b) a disordered cluster-cluster fractal aggregate. Both aggregates have almost the same fractal dimension D. = 1.5, d = 2). The top frame shows the original aggregate. The frames below show successive enlargements of the central region (from R. Jullien, et al. (1987)).

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