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Figure 4.1 Key mechanisms controlling the fate and transport of colloidal matter and associated trace compounds in natural waters.

induced by perikinetic aggregation (bridging flocculation by polymers, salt-induced coagulation, heteroaggregation, etc.), in addition to addressing fundamental issues such as fractal growth. Microscopic observations of natural colloids and model systems have concluded that the formation of aggregates in aquatic systems is mainly controlled by three types of colloid: compact inorganic colloids, large rigid biopolymers, and aquagenic refractory organic matter in variable proportions, sizes, and chemical properties [1].

This chapter describes some of the simulation and theoretical models used to investigate aggregation processes. The models depend upon whether the problem under consideration is defined at a microscopic or mesoscopic level and on the appropriate degree of complexity and rigour that is required to solve real practical problems of interest. Several techniques and examples will be described that are applicable to coagulation/flocculation processes in colloidal dispersions. Owing to the fact that such processes are now recognized to lead to the formation of fractal objects, fractal concepts will be discussed.

4.2 NATURE AND MORPHOLOGY OF AQUATIC PARTICLES/COLLOIDS: USE OF FRACTAL CONCEPTS

Aggregates formed in natural waters (rivers, lakes and oceans) and in wastewater treatment systems, as well as in controlled laboratory experiments, yield geometrical properties that can be conveniently described using fractal geometry [1-12]. The fractal concept was introduced in the 1970s by Mandelbrot [13]. His approach, essentially based on geometry, quickly became popular and is now successfully used in many areas of science for the description of complex geometrical structures that cannot be described by the classical Euclidean geometry. Full lines, surfaces and volumes have integer linear dimensions of 1, 2 and 3, respectively, whereas open, porous, complicated structures have noninteger dimensions, i.e. fractal dimensions. The fractal dimension is thus a quantitative measure of the more or less compact nature of the structure under consideration.

Fractal structures can be divided into two groups: regular and irregular. They all exhibit a remarkable property: a self-similarity or invariance with changes in scale or size. Self-similarity means that if we enlarge a given portion of a fractal object then the new enlarged object is identical to the initial one. Regular fractal objects exhibit a regular shape with well-defined positions between branches (Figure 4.2a). They are built with deterministic and iterative rules. Irregular fractals, for which some randomness occurs during their growth, are more representative of natural aggregation processes. In that case, the self-similarity property is valid on average, i.e. the new picture obtained after enlarging a part of an irregular fractal is only statistically equivalent to the original form (Figure 4.2b).

Figure 4.2 Illustration of the self-similarity property of fractal aggregates. (a) Aregular fractal aggregate: when the central (grey) part of the aggregate is enlarged (as displayed on the right), the same figure as the original one is obtained and scale invariance is verified. (b) An irregular (disordered) fractal with some randomness included in its growth where the self-similarity property now is only valid on average.

Figure 4.2 Illustration of the self-similarity property of fractal aggregates. (a) Aregular fractal aggregate: when the central (grey) part of the aggregate is enlarged (as displayed on the right), the same figure as the original one is obtained and scale invariance is verified. (b) An irregular (disordered) fractal with some randomness included in its growth where the self-similarity property now is only valid on average.

Quantitatively, the fractal character of a single large aggregate is often demonstrated by apower-law behaviour of its mass distribution in space [14,15]. If one measures the mass m contained in a sphere of radius r centred at a given point (usually the centre of mass) of an irregular fractal aggregate, the following scaling relationship is obtained:

where Dm represents the mass aggregate fractal dimension (Figure 4.3).

Figure 4.3 (a) Three-dimensional computer-generated aggregate (Witten and Sanders Diffusion Limited Aggregate) consisting of 10 000 particles. In the presence of random motion of the particles, single particles are irreversibly stuck to a growing germ. (b) Log-log variation of the aggregate mass versus the aggregate size. From the slope, one can already see the fractal character of the structure. From the slope, the aggregate fractal dimension was calculated as 2.5.

Figure 4.3 (a) Three-dimensional computer-generated aggregate (Witten and Sanders Diffusion Limited Aggregate) consisting of 10 000 particles. In the presence of random motion of the particles, single particles are irreversibly stuck to a growing germ. (b) Log-log variation of the aggregate mass versus the aggregate size. From the slope, one can already see the fractal character of the structure. From the slope, the aggregate fractal dimension was calculated as 2.5.

The mass fractal dimension gives a quantitative description of the spatial mass repartition within an aggregate, a higher fractal dimension being representative of more compact structures. Table 4.1 presents some fractal dimensions for some computer-generated and 'natural' aggregates. The mass fractal dimension Dm usually

Table 4.1 Examples of various mass fractal dimension Dm for computer-simulated, biological and inorganic aggregates in two (2D) and three dimensions (3D).

2D 3D Ref.

Computer-generated aggregates Witten-Sanders aggregates (DLA)a CCA aggregates (DLCA)a CCA aggregates (RLCA)a Regular fractal aggregate (Figure 4.2)

Natural aggregates

Marine and oceanic snow Hematite particles

Hematite particles with schizophyllan

Inorganic aggregates Gold colloids (DLCA) Ludox silica spheres Polystyrene spheres Latex spheres with sodium chloride

a See Section 4.3.

ranges between 1.4 and 2.5 for aggregates produced in natural waters, sediments, soils and in vitro.

If a system containing a number of aggregates is considered, the aggregate mass distribution is also expected to follow a power-law relationship:

where l represents a characteristic length. It is important to note that the characteristic length l for the determination of Dm is not necessarily the maximum length of the aggregate. For example, the aggregate radius of gyration Rg is defined by

where RG corresponds to the position of the centre of mass of the aggregate (Figure 4.4), N is the total number of particles within the aggregate and Rn is the position of the particle n. The minimum or maximum aggregate diameter and the aggregate equivalent sphere area diameter can also be used as characteristic lengths to derive the aggregate fractal dimensions [16-18].

Because of the inhomogeneous distribution of particles within a fractal, an important feature of fractal aggregates is the apparent decrease in the mean aggregate density p with the increase of the size R of aggregates:

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