Spherical flocculation

An interesting example of a large specific surface which is wholly external in nature is provided by a dispersed aerosol composed of fine particles free of cracks and fissures. As soon as the aerosol settles out, of course, its particles come into contact with one another and form aggregates but if the particles are spherical, more particularly if the material is hard, the particle-to-particle contacts will be very small in area the interparticulate junctions will then be so weak that many of them will become broken apart during mechanical handling, or be prized open by the film of adsorbate during an adsorption experiment. In favourable cases the flocculated specimen may have so open a structure that it behaves, as far as its adsorptive properties are concerned, as a completely non-porous material. Solids of this kind are of importance because of their relevance to standard adsorption isotherms (cf. Section 2.12) which play a fundamental role in procedures for the evaluation of specific surface area and pore size distribution by adsorption methods. 

Because most shear-thinning fluids, particularly polymer solutions and flocculated suspensions, have high apparent viscosities, even relatively coarse particles may have velocities in the creeping-flow of Stokes law regime. Chhabra(35,36) has proposed that both theoretical and experimental results for the drag force F on an isolated spherical particle of diameter d moving at a velocity u may be expressed as a modified form of Stokes law ... 

Flocculation kinetics can be described in different ways. Here we introduce a treatment first suggested by Smoluchowski [547], and described in Ref. [538], p. 417. The formalism can also be used to treat the aggregation of sols. A prerequisite for coalescence is that droplets encounter each other and collide. Smoluchowski calculated the rate of diffusional encounters between spherical droplets of radius R. The rate of diffusion-limited encounters is SttDRc2, where c is the concentration of droplets (number of droplets per unit volume). For the diffusion coefficient D we use the Stokes-Einstein relation D = kBT/finr/R. The rate of diffusion-limited encounters is, at the same time, the upper limit for the decrease in droplet concentration. Both rates are equal when each encounter leads to coalescence. Then the rate of encounters is given by... 

Surfactants are used for stabilization of emulsions and suspensions against flocculation, Ostwald ripening, and coalescence. Flocculation of emulsions and suspensions may occur as a result of van der Waals attraction, unless a repulsive energy is created to prevent the close approach of droplets or particles. The van der Waals attraction Ga between two spherical droplets or particles with radius R and surface-to-surface separation h is given by the Hamaker equation,... 

Different concentration limits of the filler arise from the CCA concept [22]. With increasing filler concentration first an aggregation limit O is reached. For >+, the distance of neighboring filler particles becomes sufficiently small for the onset of flocculation and clusters with solid fraction A are formed. Dependent on the concentration of filler particles, this flocculation process leads to spatially separated clusters or, for 0>0, a through going filler network that can be considered as a space-filling configuration of fractal CCA-clusters. The different cases for spherical filler particles are shown schematically in Fig. 1. 

Application of the electric double layer theory to soil minerals at a quantitative level is difficult because soil mineral surfaces at the microscopic scale are not well defined, that is, they are neither perfectly spherical nor flat, as the double layer requires. However, application of the double layer theory at a qualitative level is appropriate because it explains much of the behavior of soil minerals in solution, for example, dispersion, flocculation, soil permeability, and cation and/or anion adsorption. When equilibrium between the counterions at the surface (near the charged surface) and the equilibrium solution is met, the average concentration of the counterions at any... 

A final assumption made in the derivation of Stokes law was that the particles of interest were spheres. In many cases this is not true. Particles may have irregular shapes, depending on how they were formed and the amount of agglomeration which may have taken place. Liquid aerosols are always spherical, so that for liquid aerosols the assumption of sphericity holds. For isometric particles this assumption can also be used with little error. For long chains of particles or flocculated particles, large deviations from Stokes law are possible. 

Flow of any concentrated suspension will become impossible when the solid particles can form a continuous three-dimensional network of contacts throughout the sample. This so-called maximum packing fraction 4> depends mainly on the particle size distribution and the particle shape. Broader particle size distributions result in lower values of 4>m, because the smaller particles can fill the gaps between the bigger ones, and a deviation from spherical shape results in lower values of 4>m due to steric hindrance of packing. Also flocculation will result in a decrease in the value of 4>in, because the individual floes are only loosely packed. 

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