Applying Fractal Geometry To Quantify Growth Polymerization And Aggregation

An example of how fractal geometry can be used to better understand complex environmental processes is briefly described here and more thoroughly detailed in

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.

Two distinct models, particle-cluster aggregation and cluster-cluster aggregation [36], have been proposed to describe aggregation processes. The particle-cluster aggregation model is based on the sequential addition of particles to a growing cluster and is not believed to be relevant for the majority of colloidal aggregation processes that occur in nature. Aggregation in natural systems is best simulated using a cluster-cluster aggregation (CLA) model [36] in which a large number of initially dispersed and randomly moving particles collide and stick to form small clusters that grow following collision with other particles and clusters.

CLA can be classified on the basis of its rate-limiting process [36]: (a) ballistic CLA, for which particles and clusters follow linear or ballistic trajectories and join at their point of first contact; (b) diffusion-limited CLA (DLCA), where particles and clusters follow random walks (Brownian motion) and the fractal dimensions of the generated clusters range between 1.75 and 1.80; (c) a reaction-limited CLA (RLCA), where a short-range repulsive barrier must be overcome before direct contact and sticking can occur between particles and/or clusters. In this final CLA class, on average, a large number of collisions are required before aggregation can take place. In RLCA, aggregates have fractal dimensions that range between ~1.9 and 2.1 for d = 3 and ~1.5 and 1.6 for d = 2 [36, 37].

Aggregation processes are time dependent; thus, in addition to the geometric aspects, kinetic aspects must also be considered. In real systems, two distinct regimes of colloidal aggregation are generally distinguished, i.e. fast and slow aggregation. Each regime has a different rate-limiting step, and the resulting aggregates have different fractal dimensions and size distributions. In order to make the models time dependent, a sticking coefficient a, which represents the collision efficiency between two particles and/or clusters, is defined. The sticking coefficient specifies the aggregation rate and depends on the value of the energy barrier that must be overcome to obtain aggregation. Values of a range between a = 0 and a = 1 for 0 % or 100 % collision efficiency respectively [38]. When a is large and close to 1, aggregation is fast, limited only by the diffusion of the particles and/or clusters. Under these conditions, loose aggregates, characterized by fractal dimensions between 1.75 and 1.80, are formed (Figure 1.4) [38], in agreement with the DLCA model. In contrast, when a is small (low collision efficiency), the aggregation rate is slow and the process is mainly controlled by the energy barrier that must be overcome for particle/cluster sticking to occur. In this case, dense aggregates are formed with high fractal dimensions (Figure 1.4) [38] and aggregation is best described by the RLCA model. The sticking probability and aggregation rate can be increased by a decrease of the repulsive component of the energy barrier. In practice, the repulsive barrier is decreased: (i) by decreasing the pH of the medium, with a resulting decrease in the negative o o

Figure 1.4 Schematic representation of a diffusion-limited and a reaction-limited cluster-cluster aggregation process leading to aggregates of either low fractal dimension (loose aggregate) or higher fractal dimension (denser aggregate). Reproduced with permission from [19], copyright American Chemical Society.

Figure 1.4 Schematic representation of a diffusion-limited and a reaction-limited cluster-cluster aggregation process leading to aggregates of either low fractal dimension (loose aggregate) or higher fractal dimension (denser aggregate). Reproduced with permission from [19], copyright American Chemical Society.

charge on the interacting particles and clusters; (ii) by increasing the ionic strength of the medium, resulting in an increased charge screening of the particles/clusters; or (iii) by increasing temperature and/or stirring to provide greater kinetic energy to the aggregating system.

Most aggregation processes that occur in natural aquatic systems can be described adequately by the simple, limiting-case models outlined above. However, several natural aggregation processes are much more complex, in which case a variety of factors may affect the fractal dimension. For example, the fractal dimension can be significantly reduced by the effects of long-range attractive or repulsive interactions. Furthermore, in the often fragile aggregates that form in aqueous media, a number of processes, including restructuring by bending, folding and twisting and even bond breaking, can occur, resulting in a fractal dimension that is somewhat larger than that expected from simple models [36]. For fragile aggregates, defloc-culation and dispersion may also result from changing environmental conditions, such as pH, concentration, salinity, etc., further complicating the interpretation of the fractal dimension that is measured for these systems. Owing to all of the above factors, it is clear that much care is required when interpreting fractal measurements of aggregation in natural systems.

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