Smoluchowski’s theory

As the particles in a coUoidal dispersion diffuse, they coUide with one another. In the simplest case, every coUision between two particles results in the formation of one agglomerated particle,ie, there is no energy barrier to agglomeration. Applying Smoluchowski s theory to this system, the half-life, ie, the time for the number of particles to become halved, is expressed as foUows, where Tj is the viscosity of the medium, k Boltzmann s constant T temperature and A/q is the initial number of particles. 

According to Smoluchowski s theory (equation 6.53), the agglomeration rate increases proportional with the fluid shear rate 7... 

The collision frequency between drops may be estimated by means of Smoluchowski s theory (see, for example, Levich, 1962). The collision frequency (w = number of collisions per unit time per unit volume) for randomly distributed rigid equal-size spheres, occupying a volume fraction , is given by... 

At the simplest level, the rate of flow-induced aggregation of compact spherical particles is described by Smoluchowski s theory [Eq. (32)]. Such expressions may then be incorporated into population balance equations to determine the evolution of the agglomerate size distribution with time. However with increase in agglomerate size, complex (fractal) structures may be generated that preclude analysis by simple methods as above. 

Rate of aggregation of compact clusters in well-mixed systems follows Smoluchowski s theory. 

In liquids, collisional energy transfer takes place by multistep diffusion (the rate determining step) followed by an exchange interaction when the pair is very close. The bimolecular-diffusion-controlled rate constant is obtained from Smoluchowski s theory the result, including the time-dependent part, may be written as... 

In reality, the diffusional rate constant is time-dependent, as explained at the end of Section 4.2.1, and should be written as ki(t). Several models have been developed to express the time-dependent rate constant (see Box 4.1). For instance, in Smoluchowski s theory, ki (t) is given by... 

In contrast to Smoluchowski s theory, the rate constant for the intrinsic reaction has a finite value kR when A and B are at the distance of closest approach... 

The rate of flocculation of selenium sols by solutions of potassium or barium chloride of various concentrations at temperatures between 15° and 20° C. has been determined.8 Results show that a very high concentration of these electrolytes is necessary for rapid flocculation. Smoluchowski s theory holds when the velocity of flocculation is not far removed from that obtaining when the colloidal particles are totally discharged. The results, however, deviate largely from this theory when the concentrations of the electrolytes are lower.3... 

Particle Size Distribution. The effect of particle size distribution was roughly in accordance with the theory of Mueller who extended Smoluchowski s theory to polydisperse systems (33). According to this theory, a particle diameter ratio of 1 10 in the polydisperse system reduces the time of halfcoagulation by the ratio 1.0 to 0.4. 

Fortelny et al. [1988, 1990] assumed that the Brownian motion is the principal driving force for coalescence in polymer blends. Applying Smoluchowski s theory, the authors obtained ... 

This expression becomes that of Smoiuchowski when Ka = 0. Smoluchowski s theory predicts that for other than the most dilute dispersion, flocculation would be expected to be fast on any practical time scale, e.g. for aqueous dispersions at 25°C with initially N° particles percm, x 10"/ ° sec, and therefore instant sedimentation of the disperse phase of a practical system (containing 1-10% solids) would occur provided no other factors oppose the collision process. 

Smoluchowski s theory assumed that coagulation has been going on for a long time and the system is near steady state. Flocculation is considered irreversible. Van den Tern-pel (216) assumed fliat the rate of coalescence is proportional to the number of points of contact between the particles in an aggregate. He considers flocculation and coalescence to occur simultaneously. The number of primary particles not yet combined into aggregates is ... 

Under these assumptions, the particle s velocity is linear in the applied electric field, V = hE, where the electrophoretic mobility b = e frj is given by the permittivity e and viscosity rj of the fluid and the zeta potential of the surface. In Smoluchowski s theory, the latter is equal to the voltage across the double layer, which is proportional to the surface charge at low voltage. 

In this equation, the first term relates to agglomeration effects independent of power input and the second term in brackets to agglomeration due to shear as described in Smoluchowski s theory. The third term describes the reduced agglomeration efficiency found experimentally at higher shear rates and is consistent with the findings of Mumtaz etal. (1997). 

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