Size distribution of aggregates

Considerable attention has been devoted to aggregation processes, the physics of which is well described by the Smoluchowski equation with so-called homogeneous kernels (i.e., with Kaitaj=a.xKitj). Here, parameter A is the degree of homogeneity. To describe the size distribution of aggregates in a finite system, functions have been proposed [67] in the form... 

Figure 5. Size distribution of aggregated latex sample. The 2 1 size ratio suggests the formation of doublets.

Micellar aggregates are considered in chapter 3 and a critical concentration is defined on the basis of a change in the shape of the size distribution of aggregates. This is followed by the examination, via a second order perturbation theory, of the phase behavior of a sterically stabilized non-aqueous colloidal dispersion containing free polymer molecules. This chapter is also concerned with the thermodynamic stability of microemulsions, which is treated via a new thermodynamic formalism. In addition, a molecular thermodynamics approach is suggested, which can predict the structural and compositional characteristics of microemulsions. Thermodynamic approaches similar to that used for microemulsions are applied to the phase transition in monolayers of insoluble surfactants and to lamellar liquid crystals. 

This approach can be used to give critical micellizatlpn concentrations as well as size distributions of aggregates — but is, of course, unable to distinguish between different forms of the association structures. In order to achieve this packing, restraints must be introduced. 

Electron microscopy allows one to analyze the average particle size, the number of particles per agglomerate, and the projected area from which a calculation of the void volume of each aggregate can be done. Centrifugal sedimentation allows direct measurement of the size distribution of aggregates... 

On the basis of experimental observations of aggregation, including electron microscopy and QELS, Matsushita (15) concluded that at least some colloidal aggregates can be well described in terms of fractal geometry and that the size distribution of aggregates and the kinetics of their aggregation can also be related to their fractal structure. 

The relation (5.149) holds for the current concentrations cj in the course of the slow process in the regions of premicellar aggregates and full micelles because we assumed a quasi-equilibrium size distributions of aggregates in these regions. Then we can rewrite this relation for the concentration perturbations in the following form... 

These processes are effective if q and r are outside the minimum region of the size distribution of aggregates. This means that q and r belong to the range of premicellar aggregates and j to the range of full micelles (Fig. 12). The linearised kinetic equation has the following form... 

Concluding the discussion of micellisation kinetics it is necessary to note that a new theory based on the ideas of nucleation kinetics have been proposed recently by Kuni et al. [174,175]. The nucleation theory allows to study in detail the size distribution of aggregates on the basis of a thermodynamic analysis and to obtain more general kinetic equations. 

When we consider again the time interval essentially exceeding ti, the condition of the quasi-stationary flow of monomers in the minimum region of the size distribution of aggregates is justified and the relations (5.216) to (5.220) hold. Then, substituting expressions (5.217) for Sc, 8Cp, and (5.218) for 8cj into equation (5.225), summing up from j = s to j = jmax and using relations (5.221), (5.222) we get... 

The grain size distribution of aggregate should be carefully designed and checked if concrete of good quality is to be prepared. The grading is established after sieve analysis using sieves in accordance with local standards usually separate sieve systems are used for coarse and fine aggregates. 

The size distribution of aggregates can be calculated in terms of the total mole fraction of monomers, which is by definition ... 

FIG. 8—Grain size distribution of aggregates found in historic Roman cement mortars from casts, in situ run elements and renders mean values of a total of 39 samples measured by sieve-fractioning. 

Figure 13.2 Size distribution of aggregates after perturbation of a micellar solution governed by fast micellar kinetics.

Equation (2.96) shows that the asymptotic size distribution of aggregates at =0 can be written in the form ... 

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