Kinetics of Particle Agglomeration

The rate of particle agglomeration depends on the frequency of collisions and on the efficiency of particle contacts (as measured experimentally, for example, by the fraction of collisions leading to permanent agglomeration). We address ourselves first to a discussion of the frequency of particle collision. 

Frequency of Collisions between Particles. Particles in suspension collide with each other as a consequence of at least three mechanisms of particle transport  

1) Particles move because of their thermal energy (Brownian motion). Coagulation resulting from this mode of transport is referred to as perikinetic. 

2) If colloids are sufficiently large or the fluid shear rate high, the relative motion from velocity gradients exceeds that caused by Brownian (thermal) effects (orthokinetic agglomeration). 

3) In settling, particles of different gravitational settling velocities may collide (agglomeration by differential settling). 

---

Many investigators have studied models of the von Foerster type (see Trucco (4) for several references and properties), which belong to the general population balance category used in kinetic theory, particle agglomerization, crystallization, etc. (see Himmelblau and Bischoff (5) for a textbook treatment). We will use the model of Rubinow (6) for reasons discussed below. 

Methods. Determination of Agglomeration Rate. The kinetics of PEO agglomeration may be described in a wide initial range (up to 50-70% of total agglomeration) by all three forms of Smoluchowski s equation, in which the reciprocal particle number (1 /N), the mean particle volume (u), and the reciprocal cube of the particle surface (1/23) vary linearly with time (t) (14,... 

It is, thus, evident, that the agglomeration rate depends on the fraction of outer particle surface covered with binder. This fraction does not immediately result from how much liquid is sprayed into the bed, because sprayed liquid may be located both on the surface or in the interior of porous agglomerates. Liquid on the surface can lead to further aggregation, whereas liquid trapped in agglomerate voids is not accessible and, therefore, inactive for agglomeration (steric hindrance). Fractional surface coverage and accessible binder fraction depend on the properties of primary particles, so that these properties are expected to also influence the kinetics of the agglomeration process. 

Heterodisperse Suspensions. The rate laws given above apply to monodisperse colloids. In polydisperse systems the particle size and the distribution of particle sizes have pronounced effects on the kinetics of agglomeration (O Melia, 1978). For the various transport mechanisms (Brownian diffusion, fluid shear, and differential settling), the rates at which particles come into contact are given in Table 7.2. 

As discussed in Chapter 15, the size distribution of particles in an agglomeration process is essentially determined by a population balance that depends on the kinetics of the various processes taking place simultaneously, some of which result in particle growth and some in particle degradation. In a batch process, an equilibrium condition will eventually be established with the net rates of formation and destruction of particles of each size reaching an equilibrium condition. In a continuous process, there is the additional complication that the residence time distribution of particles of each size has an important influence. 

When two similarly charged colloid particles, under the influence of the EDL, come close to each other, they will begin to interact. The potentials will detect one another, and this will lead to various consequences. The charged molecules or particles will be under both van der Waals and electrostatic interaction forces. The van der Waals forces, which operate at a short distance between particles, will give rise to strong attraction forces. The potential of the mean force between colloid particle in an electrolyte solution plays a central role in the phase behavior and the kinetics of agglomeration in colloidal dispersions. This kind of investigation is important in these various industries ... 

The equation derived by Troelstra and Kruyt is only valid for coagulating dispersions of colloids smaller than a certain maximum diameter given by the Rayleigh condition, d 0.10 A0. Equation 4 applies in cases where particles are transported solely by Brownian motion. Furthermore, the kinetic model (Equations 2 and 3) has been derived under the assumption that the collision efficiency factor does not change with time. In the case of some partially destabilized dispersions one observes a decrease in the collision efficiency factor with time which presumably results from the increase of a certain energy barrier as the size of the agglomerates becomes larger. 

The reaction model assumed is one in which free-radical polymerisation is compartmentalised within a fixed number of reaction loci, all of which have similar volumes. As has been pointed out above, new radicals are generated in the external phase only. No nucleation of new reaction loci occurs as polymerisation proceeds, and the number of loci is not reduced by processes such as particle agglomeration. Radicals enter reaction loci from the external phase at a constant rate (which in certain cases may be zero), and thus the rate of acquisition of radicals by a single locus is kinetic-ally of zero order with respect to the concentration of radicals within the locus. Once a radical enters a reaction locus, it initiates a chain polymerisation reaction which continues until the activity of the radical within the locus is lost. Polymerisation is assumed to occur almost exclusively within the reaction loci, because the solubility of the monomer in the external phase is assumed to be low. The volumes of the reaction loci are presumed not to increase greatly as a consequence of polymerisation. Two classes of mechanism are in general available whereby the activity of radicals can be lost from reaction loci ... 

The kinetics of a granulation process can be dramatically improved by the addition of recycled seed agglomerates. These nuclei readily pick up the fine feed material in layer growth since the pendular liquid bond between two particles increases in strength as the size difference between the particles becomes larger [15]. It has been suggested [16] that the coalescence proba-... 

The particle size distribution produced during precipitation is a result of the relative rates of reaction, nudeation, growth, and agglomeration, as well as the degree of backmixing in the precipitator. The kinetics of each of these steps will be discussed next. 

---