Fractal Dimension of Aggregates

Another reason for the difficulties in defining fractal dimension is that there is no experimental technique that is able to reveal the exact configuration of colloidal aggregates. Instead, the structural properties have to be derived from images (e.g. obtained with electron microscopes) or from physical properties that reflect the aggregate structure to a certain degree. Indeed, there are different experimental methods which are employed for the determination of the fractal dimension. Hence, they should always be indicated when stating values of df. 

The scattering and hydrodynamic behaviour of aggregates are considered in detail in Sects. 4.3 and 4.4, respectively. 

One of the most popular features of fractal aggregates is the independence of the aggregate stmcture on size. However, Gmachowski (2002) pointed out that this does not hold tme for small fractal aggregates. In particular, he examined the correlation between the aggregation number N and the geometric aggregate radius 

Nevertheless, the implications of the densificalion have to be considered for both types of aggregates because the assumption of a size-independent fiactal dimension can considerably mislead the computation of physical aggregate properties or the interpretation of experimental data. 

In summary, the statistic variation of the fractal dimension within a population of aggregates is a function of the aggregation number While for large N a uniform structure can be assumed, the variation of the fractal dimension is not negligible for small aggregation numbers (N 100) and has to be appropriately considered when studying the physical properties of aggregates. 

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Jiang, Q., and Logan, B. E., Fractal dimensions of aggregates from shear devices. J. AWWA, February, pp. 100-113 (1996). 

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. 

Bushell, G., Amal, R., and Raper, J., The effect of polydispersity in primary particle size on measurement of the fractal dimension of aggregates. Part. Syst. Charact., 15, 3, 1998. 

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. 

This technique has been successfully applied in numerous studies to determine the fractal dimension of aggregate particles. 

Jiang Q., Logan B.E. (1991), Fractal dimensions of aggregates determined from steady-state size distributions. Environ. Sci. TechnoL, 25,12, 2031-2038. 

The fractal dimensions of aggregates characteristically indicate highly variable colloidal dynamics in sea water [13]. Colloid numbers increase nearly logarithmically with decreasing size, creating a continuum of particle sizes that links nanometer-size matter to large particles, which ultimately remove matter from the ocean by sinking. 

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. 

Tence, M., Chevalier, J. P. and Jullien, R. (1986). On the measurement of the fractal dimension of aggregated particles hy electron-microscopy - experimental-method, corrections and comparison with numerical-models. J. Phys., 41, 1989-1998. 

DLCA and RLCA processes represent universality classes for homoaggregation (aggregation involving similar particles), i.e. the aggregates display some characteristic features for which the kinetics of particle coagulation and the fractal structure are independent of the details of the system. One line of evidence that supports this hypothesis is the observation that the fractal dimensions of aggregates formed by RLCA are remarkably constant, as are those formed by DLCA, for many different particle types [39,40]. 

The most common techniques that are used to measure fractal dimensions of aggregates in solution are scattering techniques (visible light, neutrons and X-rays) see Chapter 2. These techniques allow the calculation of the so-caUed distribution function P r), which scales as a power law of the distance r within a fractal object ... 

Thus the fractal dimension of aggregates can be easily measured at small scattering angles using a laser diffraction instrument [86]. 

Bushell, G. C., Amal, R., Raper, J. A., The Effect of Polydispersity in Primary Particle Size on Measurement of the Fractal Dimension of Aggregates, Part Part Syst Charact, 1998, 15,3-8. 

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