## Clustercluster Aggregation Processes

The cluster-cluster aggregation (CCA) model, was introduced simultaneously, but independently, in 1983 by Jullien and co-workers [19] and Meakin [20] as a realistic model to describe aggregation, such as for gold particles [21]. The CCA model begins with a collection of elementary spherical particles which are randomly distributed in a box. Particles are moved randomly in all directions in order to mimic Brownian motion (random walk with periodic boundary conditions). When two particles come into contact, they are assumed to stick together irreversibly and form a new rigid aggregate (dimer) that diffuses according to a diffusion coefficient that is related to its size and geometry (small aggregates move faster than large aggregates). With time, larger, rigid aggregates are formed by the irreversible reactivity of the small clusters (Figure 4.5). Model iterations are stopped when only a single aggregate remains in the simulation box. Different extensions of the model have been developed to include intrinsic anisotropy, readjusting effects and polarizability [22].

Figure 4.5 Four stages of an off-lattice diffusion-limited CCA process using 10 000 elementary spherical particles. In this simulation, the aggregate diffusion coefficient is assumed to be controlled by the aggregate mass using D ~ D0m-1 where m represents the aggregate mass and D0 the diffusion coefficient of a single particle. The aggregates generated by this model are self-similar with a fractal dimension of 1.8.

Figure 4.5 Four stages of an off-lattice diffusion-limited CCA process using 10 000 elementary spherical particles. In this simulation, the aggregate diffusion coefficient is assumed to be controlled by the aggregate mass using D ~ D0m-1 where m represents the aggregate mass and D0 the diffusion coefficient of a single particle. The aggregates generated by this model are self-similar with a fractal dimension of 1.8.

### 4.3.1 Particle-Particle Interactions

If only the attractive van der Waals forces were operating on suspended particles in water, then one might expect the particles to stick together immediately and coagulate. Nonetheless, particles suspended in water or in solutions of high dielectric constant are usually charged due to the ionization or dissociation of surface sites or the adsorption of charged entities (multivalent ions, polyelectrolytes, humic or fulvic acids). Hence, particles generally undergo repulsive electrostatic forces originating from chemical reactions occurring at the particle-solution interface. Particle aggregation is expected to occur when the attractive van der Waals forces (which are always present) exceed the repulsive electrostatic interactions between the particles, as shown in Figure 4.6.

The Derjarguin, Landau, Verwey, Overbeck (DLVO) theory [23-25] has established the potential energy-distance relationship between two particles as a function of the characteristics of both the particles and the suspending solution. In natural systems, this approach requires compilation [26-28] of the major key physicochem-ical parameters that characterize the colloid material, including: (a) colloid shapes

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