Fractal aggregates objects

This theory was first developed for colloidal aggregate networks and was later adapted to fat crystal networks (52-54). In colloidal systems (with a disordered distribution of mass and statistical self-similar patterns), the mass of a fractal aggregate (or the distribution of mass within a network), M, is related to the size of the object or region of interest (R) in a power-law fashion ... 

The success of fractal models applied to the physics of disordered media may be explained first of all by the fact that fractal forms are characteristic of a huge number of processes and structures because many diverse models of the formation and growth of disordered objects of disparate nature may ultimately be reduced to a transition model—namely a connected set and an unconnected set—and to a limited diffusive aggregation [1-6]. In the first case a fractal percolation cluster is formed in the second case a fractal aggregate is formed. 

Because it is likely that aggregates have significant internal flow through their structure, aggregate permeability must be considered. Fractal aggregates are expected to behave like objects that are smaller than equivalent spheres with reduced drag effects. Indeed, simulations of hydrodynamic friction using the Stokes model overestimate the friction of fractal objects. 

The parameter used to describe the structure of fractal aggregates is the fractal dimension Df [4-8]. This parameter is directly associated with the dimension of the object, and is different from the space dimension in which the object is contained [18]. The relationship between Df and the aggregate geometry as described by the volume V and radius R is given by... 

In contrast to a compact aggregate in which the number N of primary particles varies with the third power of the radius R of the aggregate, in a fractal aggregate N scales with I. The exponent/is the so-called fractal dimension. Figure 16.13 illustrates the nature of a fractal aggregate. Close inspection of such objects reveals... 

The first of these factors is filler particles aggregation. The detailed description of this effect influence on structure and properties of composites, having glassy matrix, is given in review [3]. For the considered composites this effect is important first of all for determination of filler volume fraction cp/, which is calculated by division of filler mass fraction into its density. However, filler particles in real composites exist in the form of fractal aggregates, which density can be more smaller than monolithic object of such filler. Below the example of value cp/calculation for concrete composite will be given. 

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 14.4. Regular fractal aggregate constructed in two dimensions (d = 2) and in three dimensions (3d). The aggregate d = 3 (b) is shown along its principal diagonal The fractal dimension of such an object is D = Log (2d + l)/Log 3 [JUL 87]...

For compact, homogeneous objects in tliree dimensions, p= 3. Colloidal aggregates, however, tend to be ratlier open, fractal stmctures, witli 3. For a general introduction to fractals, see section C3.6 and [61]. 

Avnir et al. llbl have examined the classical definitions and terminology of chirality and subsequently determined that they are too restrictive to describe complex objects such as large random supermolecular structures and spiral diffusion-limited aggregates (DLAs). Architecturally, these structures resemble chiral (and fractal) dendrimers therefore, new insights into chiral concepts and nomenclature are introduced that have a direct bearing on the nature of dendritic macromolecular assemblies, for example, continuous chirality measure44 and virtual enantiomers. ... 

This structure is generated via the modified diffusion-limited aggregation (DLA) algorithm of [205] using the law p = a (m/N). Here, N = 2, 000 (the number of particles of the DLA clusters), a = 10 and ft = 0.5 are constants that determine the shape of the cluster, p is the radius of the circle in which the cluster is embedded, pc = 0.1 is the lower limit of p (always pc < p), and to is the number of particles sticking to the downstream portion of the cluster. This example corresponds to a radial Hele-Shaw cell where water has been injected radially from the central hole. Due to heterogeneity a sample cannot be used to calculate the dissolved amount at any time, i.e., an average value for the percent dissolved amount at any time does not exist. This property is characteristic of fractal objects and processes. 

For a quantitative analysis of the structure of carbon blacks as shown in Fig. 17 it is useful to consider the solid volume Vp or the number of primary particles Np per aggregate in dependence of aggregate size d. In the case of fractal objects one expects the scaling behavior [1, 2]... 

Fractals are self-similar objects, e.g., Koch curve, Menger sponge, or Devil s staircase. The self-similarity of fractal objects is exact at every spatial scale of their construction (e.g., Avnir, 1989). Mathematically constructed fractal porous media, e.g., the Devil s staircase, can approximate the structures of metallic catalysts, which are considered to be disordered compact aggregates composed of imperfect crystallites with broken faces, steps, and kinks (Mougin et al., 1996). 

A useful (also extreme) counterpart to the also idealized linear geometry is fractal geometry which plays a key role in many non-linear processes.280 281 If one measures the length of a fractal interface with different scales, it can be seen that it increases with decreasing scale since more and more details are included. The number which counts how often the scale e is to be applied to measure the fractal object, is not inversely proportional to ebut to a power law function of e with the exponent d being characteristic for the self-similarity of the structure d is called the Hausdorff-dimension. Diffusion limited aggregation is a process that typically leads to fractal structures.283 That this is a nonlinear process follows from the complete neglect of the back-reaction. The impedance of the tree-like metal in Fig. 76 synthesized by electrolysis does not only look like a fractal, it also shows the impedance behavior expected for a fractal electrode.284... 

Diffusion-limited aggregation of particles results in a fractal object. Growth processes that are apparendy disordered also form fractal objects (30). Sol—gel particle growth has also been modeled using fractal concepts (3,20). The nature of fractals requires that they be invariant with scale, ie, the fractal must look similar regardless of the level of detail chosen. The second requirement for mass fractals is that their density decreases with size. Thus, the fractal model overcomes the problem of increasing density of the classical models of gelation, yet retains many of its desirable features. The mass of a fractal, Af, is related to the fractal dimension and its size or radius, R, by equationS ... 

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