Fractal structures particles

Fig. 31. Fractal structures obtained experimentally at different stages of aggregation of a colloidal monolayer of 1 /im sulfonated polystyrene particles on the surface of an aqueous calcium chloride solution, initially uniformly distributed (Robinson and Earnshaw, 1992).

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. 

Fig. 41. Typical 2D fractal structure obtained by aggregation of particles in the journal bearing flow. Fractal dimension of the cluster is 1.54 (Hansen and Ottino, 1996b).

In situ SAXS investigations of a variety of sol-gel-derived silicates are consistent with the above predictions. For example, silicate species formed by hydrolysis of TEOS at pH 11.5 and H20/Si = 12, conditions in which we expect monomers to be continually produced by dissolution, are dense, uniform particles with well defined interfaces as determined in SAXS experiments by the Porod slope of -4 (non-fractal) (Brinker, C. J., Hurd, A. J. and Ward, K. D., in press). By comparison, silicate polymers formed by hydrolysis at pH 2 and H20/Si = 5, conditions in which we expect reaction-limited cluster-cluster aggregation with an absence of monomer due to the hydrolytic stability of siloxane bonds, are fractal structures characterized by D - 1.9 (Porod slope — -1.9) (29-30). 

Parameter that provides a mathematical description of the fractal structure of a polymer network, an aggregated particulate sol, or of the particles that comprise them. 

If the dispersion particles are attracted to each other, they tend to flocculate and form a structure. At low concentrations the particles form open aggregates, which give a fractal structure (93,94). At higher concentrations a network structure results, which can be so pronounced that the mixture has a yield point and behaves like a solid when at rest. Shearing breaks up this structure, and viscosity decreases. 

In our opinion, this book demonstrates clearly that the formalism of many-point particle densities based on the Kirkwood superposition approximation for decoupling the three-particle correlation functions is able to treat adequately all possible cases and reaction regimes studied in the book (including immobile/mobile reactants, correlated/random initial particle distributions, concentration decay/accumulation under permanent source, etc.). Results of most of analytical theories are checked by extensive computer simulations. (It should be reminded that many-particle effects under study were observed for the first time namely in computer simulations [22, 23].) Only few experimental evidences exist now for many-particle effects in bimolecular reactions, the two reliable examples are accumulation kinetics of immobile radiation defects at low temperatures in ionic solids (see [24] for experiments and [25] for their theoretical interpretation) and pseudo-first order reversible diffusion-controlled recombination of protons with excited dye molecules [26]. This is one of main reasons why we did not consider in detail some of very refined theories for the kinetics asymptotics as well as peculiarities of reactions on fractal structures ([27-29] and references therein). 

In the context of particle aggregation, the particular fractal structures called mass fractals are usually considered. Mass fractals correlate the location of mass with radius, as opposed to other types of fractal constructs such as boundary fractals, surface fractals, and pore fractals (see Ref. 17). The power law which correlates the size of an aggregate with the number of primary particles in the aggregate is given by... 

TEM and ultracentrifuge results showed (see Fig. 16) that this process results in effective encapsulation of the carbon with practically complete yield only rather small hybrid particles, but no free carbon or empty polymer particles, were found. It has to be stated that the hybrid particles with high carbon contents do not possess spherical shape, but adopt the typical fractal structure of carbon clusters, coated with a thin but homogeneous polymer film. The thickness of the monomer film depends on the amount of monomer, and the exchange of monomer between different surface layers is - as in miniemulsion polymerization - suppressed by the presence of an ultrahydrophobe. 

Now various structures—for example, aggregates of particles in colloids, certain binary solutions, polymers, composites, and so on—can be conceived as fractal. Materials with a fractal structure belong to a wide class of inhomogeneous media and may exhibit properties differing from those of uniform matter, like crystals, ordinary composites, or homogeneous... 

Although DLS is most often used to size solid colloidal particles, the technique has also been applied to characterize aerosols [78,86,87], emulsion droplets [88,89], amphiphilic systems [90-92], and macromolecular solutions [12,16,93]. Another common application is the study of the fractal structure and kinetics of colloidal aggregation [94-102], More information about dynamic light scattering and its applications can be found in Refs. 23. 103 (104), and 105, in reviews, Refs. 11, 13, 36, 37, 49, 50, and 106, and in collections of papers Refs. 12. 14. 16. 93 (107), 105, and 108-114. 

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