The Kinetics of Aggregation

Imagine two primary particles which collide, sticking together and forming a doublet. The reaction is represented schematically by  

By analogy with classical chemical kinetics, denoting the number of primary particles per unit volume (equivalent to a concentration) by Ni, the rate of disappearance of primary particles will be 

This simple result shows that the stability (interpreted, for example, as an acceptable storage time) is inversely proportional to the particle concentration. However, all the quantitative information is contained in kn, which reflects the mutual reactivity of the particles. We have assumed here that the doubletforming reaction is irreversible, i.e., two particles coming close enough together are irremediably stuck to one another. Naturally, ku is larger if the particles attract than if they repel each other. 

A particularly simple case occurs when the particles are indifferent to each other, so that their interaction energy is zero at all distances but becomes inflnite upon contact. In Fig. 3.4, the energy would be zero for all s down to the contact distance. In such a model, the kinetics are governed wholly by diffusion reactions and it can be shown that 

The kinetics are much more difficult to handle when the particles repel one another (in Fig. 3.3, this is the curve with an energy threshold denoted T max)- It turns out that the aggregation rate for zero interaction energy is proportional to that observed when there is an energy threshold E  

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The remainder of this contribution is organized as follows. In section C2.6.2, some well studied colloidal model systems are introduced. Methods for characterizing colloidal suspensions are presented in section C2.6.3. An essential starting point for understanding the behaviour of colloids is a description of the interactions between particles. Various factors contributing to these are discussed in section C2.6.4. Following on from this, theories of colloid stability and of the kinetics of aggregation are presented in section C2.6.5. Finally, section C2.6.6 is devoted to the phase behaviour of concentrated suspensions. 

For a more complete understanding of colloid stability, we need to address the kinetics of aggregation. The theory discussed here was developed to describe coagulation of charged colloids, but it does apply to other cases as well. First, we consider the case of so-called rapid coagulation, which means that two particles will aggregate as soon as they meet (at high salt concentration, for instance). This was considered by von Smoluchowski 1561 here we follow [39, 57]. 

Weitz D A and Huang J S 1984 Self-similar structures and the kinetics of aggregation of gold colloids Kinetics of Aggregation and Geiationed F Family and D P Landau (Amsterdam North-Holland) pp 19-28... 

More detaiied predictions of the stabiiity and aggregation behaviour of particuiate suspensions require consideration of the kinetics of aggregation and their dependence on fiuid-particie hydrodynamics. 

Ve have chosen this binary systera to study the kinetics of aggregation of a low molecular weight amphiphile. During this aggregation process, gelation occurs. 

The kinetics of stabilization must match the kinetics of aggregation. While ion diffusion and adsorption is fast (electrostatic stabilization), the diffusion and adsorption of polymers (steric stabilization) is slow and may be a limiting factor which can only be overcome by applying high polymer concentrations. The stability of the electrostatic stabilized suspension in moderately fast processes can for example be monitored and controlled in-line by modem electro-acoustic techniques [13]. 

Payens (1976, 1977) realized that in order to derive an expression for the clotting time of milk, a kinetic description of the enzyme reaction must be combined with the kinetics of aggregation of destabilized colloidal particles. In early versions of his theory, the enzymatic reaction was described by Michaelis—Menten kinetics with the... 

Discussion of these results is directed at the following question. How do these field results compare with present theories for the kinetics of aggregation and sedimentation in aquatic systems In answering this question, some laboratory determinations of attachment probabilities will be used in modelling simulations of the kinetics and effects of colloid chemical processes in lakes. 

A model for the kinetics of aggregation and sedimentation in lakes has been presented elsewhere (O Melia, 1980) a short summary is given here. The approach begins with a particle balance for the epilimnion of a lake ... 

Here nh np and nk are the number concentrations of particles of sizes i, j, and k in the epilimnion and nfc in is the number concentration of fc-size particles in river inflows. The term X(i,j)s incorporates most of the effects of physical processes on the rate at which particles of size i and j come into close proximity. The subscript S is used to indicate that Smoluchowski s approach (1917) to the kinetics of particle transport has been adopted. Smoluchowski did not consider hydro-dynamic retardation in his early analysis, and it has not been included here in Mi,j)s. A more rigorous approach is possible (Valiolis and List, 1984a, b). The term a(i J)s incorporates chemical factors that retard the kinetics of aggregation between particles of size i and j and also those aspects of the kinetics of particle transport that are not included in Smoluchowski s analysis. The Stokes settling velocity of a particle of size k is denoted as vk the mean depth of the epilimnion is zc qin and qoul refer to river flows into and out of the lake expressed as volume of water per unit of lake surface area and time (the sum of such inflows or outflows is also termed the areal hydraulic loading of the lake). The symbol W refers to all processes of production or destruction of particles in the epilimnion it can include a variety of chemical and biological processes. 

There is extensive evidence from freshwaters, estuaries, and the oceans that the surface properties, colloidal stability, and the kinetics of aggregation reactions in natural waters are affected by naturally occurring organic substances dissolved in these waters. These effects of natural organic substances in establishing colloidal stability in aquatic systems are anticipated to occur in subsurface environments and to affect the kinetics of particle and pollutant passage and retention in subsurface systems. The kinetics, extent, and significance of... 

While compaction effects may act to minimise some of the differences, it appears Likely that the differences in size and fractal properties are induced by the kinetics of aggregation of hematite particles in suspension. Slowly formed aggregates possess a cohort of sub-micron sized aggregates of relatively high fractal dimension which form a reasonably impermeable cake while rapidly formed a regates are generally of larger size and lower fractal dimension and thus create a substantially more permeable cake. 

Dynamic versions of the simulations reviewed in this chapter can also be used to follow the kinetics of aggregate formation and breakup and the like, but the types of dynamic algorithms needed for such an effort need to examined as well. 

If the depth of the primary minimum (that on the left from the maximum in Fig. 6a) is not so great, i.e., the attractive force which keeps the drops together is weaker, then the floes formed are labile and can disassemble into smaller aggregates. This is the case of reversible flocculation (3). For example, a floe composed of i+j drops can be split into two floes containing i and j drops. We denote the rate eonstant of this reverse process by (see Fig. 20a). In the present case bofli the straight process of flocculation (Fig. 19) and the reverse process (Fig. 20a) take simultaneously plaee. The kinetics of aggregation in this more general and eomplex case is described by the Smoluchowski set of equations, Eq. (96), where one is to substitute ... 

The vCH2 temperature dependence has been used in determining the melting point of a wide range of LB films [503, 515-519] and SAMs [370, 520]. The vCH2 molecular area dependences have been used to study the mesophase transformations in L monolayers [508, 521-524], while the VCH2 time dependences have been helpful in studies of the kinetics of aggregation (self-assembly)... 

Thus, the relationship between aggregation and precipitation rate may be used in calculating the kinetics of aggregate formation and growth in an electric... 

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