Fractal Structure of Aggregates

One way of measuring the fractal dimension of aggregates is discussed in Chapter 5 (See Section 5.6a and Example 5.4). In the example below, we illustrate the relation between the fractal structure of aggregates and the surface area of the aggregates. 

Q. Jiang, Fractal Structure of Aggregates Induced by Shear Motion, Ph.D. Dissertation, Department of Civil Engineering and Engineering Mechanics, University of Arizona, 1993. 

Figure 11.7 Fractal structure of aggregates of gold particles (TEM recordings) (a)-(d) aggregates at various resolutions. Reprinted from Hiemenz and Rajagopalan (1997), with permission from Elsevier...

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. 

Parameter that provides a mathematical description of the fractal structure of a polymer network, an aggregated particulate sol, or of the particles that comprise them. 

Amal, R. Raper, J.A. Waite, T.D. (1990) Fractal structure of hematite aggregates. J. Colloid Interface Sd. 140 158-168 Amal, R. Raper, J.A. Waite, T.D. (1992) Effect of fijlvic add adsorption on the aggregation... 

This is a good place to draw attention to x-ray and neutron scattering since we have already introduced a combination of x-ray and light scattering to examine the fractal structure of silica aggregates in Example 5.4. 

Simulations may be grouped according to the general uses for which they were developed. As such, the first to be developed, and the simplest simulation techniques, are those investigating the structure of aggregates. These models first appeared in 1963 [18], and after Mandelbrot s seminal work on fractal geometry [45], development of these types of simulations increased dramatically, spurred by the well-known Witten-Sander model [46]. 

Electrical sensing zone technique commonly used to determine equivalent volume diameter, required in Eq. (19), might be problematic. The error associated with this technique is contributed by the breakup of aggregates and inclusion of pores in volume measurement. With this technique, an aggregate will have to be suspended in a liquid. The challenge is to preserve the structure of aggregates. Hence the first method is preferred to obtain the mass fractal dimension of aggregates in situ. 

The modeling also showed the importance of some of the underdeveloped aspects of coagulation theory, particularly with regard to the interaction of the hydrodynamics and the fractal structure of the aggregates on the coagulation kernels. Traditional coagulation theory emphasized understanding the interactions of two solid spheres. Some work has been done on the flow in and... 

Chapters 4 and 5 [34, 35], A number of computer simulations and statistical models have been proposed to explain growth, polymerization and aggregation processes that lead to the formation of fractal structures that closely resemble those found in nature. Attempts have been made to relate the fractal dimensions of aggregates to their formation mechanisms and aggregation kinetics. 

---