Coagulation Dynamics

During coagulation, the destabilized particles collide with each other to form clusters and continue to grow as flocs when they encounter other particles or clusters.

For efficient coagulation, it is important to understand the process of particles movement during aggregation (i.e., dynamics of particles aggregation). Since particles aggregation is a kinetic and non-equilibrium growth process, the formation of clusters or flocs is through random collisions, and the structure of flocs is inherently related to the dynamics of aggregation, which in turn, is reflected by the time evolution of the cluster mass distribution. On the other hand, the breakage of flocs can be induced by a given shear force and the breakage behavior is affected by ambient physical and chemical conditions (Clark and Flora, 1991; Oles, 1992;

Gregory and Dupont, 2001). As flocs grow larger, they increasingly become porous due to random collisions and collision efficiency comes lower (Brakalov, 1987).

Once aggregate approach the length scales of turbulent eddies, hydrodynamic stress induced by mixing can fragment fragile flocs. However, the flocs strength is directly related to flocs structure and the inter-particle bonds between the components of flocs (Bache et al., 1997). Fractal dimension, which represents the scale-invariant branched structure formed as symmetric dilation (Mandelbrot, 1982), can be utilized to quantify the structure of flocs in coagulation process (Aubert and Cannell, 1986;

Meakin, 1988; Lin et al., 1989). Intrinsically, the flocs with a fractal structure will form when the equilibrium between aggregation and breakage of flocs approach, which determines their strength during shearing. In this section, the studies about the dynamics of aggregation, the breakage of flocs and restructuring will be extensively reviewed.

2.3.1 Floc Formation

Witten and Sander (1981) have introduced the growth of flocs is dictated entirely by diffusion controlled aggregation model (i.e., diffusion-limited aggregation, DLA) in particle aggregation. In DLA, there is no repulsion between colloidal particles, so that particles attach permanently to other particles at the first contact and then grow to clusters or flocs via rapid random-walk trajectories (i.e., ‘rapid’ perikinetic aggregation) originating from outside of region occupied by the cluster. In this model, the sticking probability is equal to one and every collision between particles is effective. The fractal structures of flocs formed in this way are random ramified and insensitive to the sticking probability. Another model is the reaction-limited aggregation (RLA), for which a considerable number of collision (i.e., ‘slow’

orthokinetic aggregation) are needed between particles before sticking (Meakin and Witten, 1983). Obviously, in this model the sticking probability is small than one and only a small fraction of particles collisions lead to a permanent contact. This process produces more compact clusters or flocs whose density is irrelevant to distance. During coagulation, DLA and RLA will dominate over the fractal growth of particles, which results in different flocs structure (Elimelech et al., 1995). RLA produces a much narrower aggregate size distribution than DLA (Kostansek, 2004).

For difference in size and structure of flocs, particle-cluster and cluster-cluster aggregation provide a good explanation, illustrated in Fig. 2.7. In particle-cluster aggregation, a particle is able to penetrate into a cluster before encountering another particle and sticking, which leads to a more compact structure, as seen in Fig. 2.7 (a).

As shown in Fig. 2.7 (b), tow cluster could collide and then stick at the first contact before the clusters have interpenetrated to a significant extent, resulting in much more open structure. Torres et al. (1991) have found that flocs formed by cluster-cluster

aggregation are very similar to those formed by DLA. At RLA, particles need to collide many times before sticking occurs due to low collision efficiency, which increases opportunities for particle interpenetration. For this reason, flocs formed by RLA are more compact than those formed by DLA (Lin et al., 1989).

Moreover, flocs structure may change during coagulation, which gives more compact forms. Aubert and Cnnell (1986) have found that silica particle aggregation induced by DLA initially gives loose flocs with low fractal dimension, but denser flocs with higher fractal dimension are observed after a period of time. It is likely that shear induced by mixing can cause some deformation and rearrangement of flocs, leading to some compact flocs. On the other hand, it has been generally acknowledged that the turbulent energy dissipation with increasing floc size as a result of increasing porosity (Tambo and Watanabe, 1979; Kusters, 1991), and the porosity within the flocs could be depicted by use of fractal concepts in shearing (Sonntag and Russell, 1986).

(a) (b )

Fig. 2.7 Fractal flocs formed by (a) particle-cluster aggregation (b) cluster-cluster aggregation.

2.3.2 Floc Breakage and Restructuring

Because of the inherent complexity, fragility of flocs and variation in floc size, shape and composition, the breakage behavior of flocs is not easily identified. It has been generally accepted that two model of flocs rupture induced by shear stress, including splitting and erosion (Francois, 1987; Mikkelsen and Keiding, 2002).

These stresses have been manifested as splitting (i.e., large-scale fragmentation) and surface erosion, as demonstrated in Fig. 2.8.

Splitting is due to the instantaneous velocity differences across the body of the flocs, which produce several flocs fragments of a size similar to the parent flocs (Thomas, 1965). In addition, since fluid drag forces can strip primary particles or small clusters from the surface of flocs, called surface erosion, leading to an increase in the small particle size ranges. Glasgow and Luecke (1980) have observed experimentally that splitting is the dominant mechanism for flocs fragmentation.

However, Williams et al. (1992) have suggested that more compact flocs structures were more likely to suffer surface erosion, while more open flocs would fracture by splitting. On the other hand, Potanin (1993) has modeled the shear-induced fragmentation of fractal flocs and advocated a combination of soft and rigid characteristics of actual flocs. From it, two or more fragments with mean daughter floc size are observed. These fragments are denser than parent flocs and there is no direct relationship between parent and daughter structure.

During coagulation, irregular flocs with a fractal structure suffered any shearing is probably to produce packed structure when particle-particle links shift to location with higher effective coordination numbers. This can not only occur in fragmentation, but also a change in fractal structure can often happen in the period of coagulation. Jiang and Logan (1996) have proposed that the fractal dimension of

flocs can increase with increasing floc size in coagulation. In other case, however, the fractal dimension of flocs can decrease over time in the initial stages of flocs formation for alum coagulation of latex mircrospheres (Chakraborti et al., 2003).

The discrepancy can be attributed to restructuring of flocs that is the most prevalent compaction mechanism as a steady state is approached between aggregation and fragmentation during coagulation (Spicer and Pratsinis, 1996).

Tensile stress

S h ea r stress (b) Surface Erosion

(a) Sp l it t ing

Fig. 2.8 Two proposed mechanisms for the breakage of aggregates at different shear conditions: (a) Splitting (b) Surface Erosion. Redrawn from Jarvis et al.

(2005)