Kinetics aggregation

The half-time for aggregation ti/2 defined as the time it takes to reduce Af by a factor of two, is then given by 

The rate constant k is determined by the interaction between the particles. Let us first consider the simple case of particles that do not interact except at zero separation where they form a permanent bond. The aggregation rate of such indifferent sticky particles is governed by the rate of transport toward each other. The transport may be dominated by an imposed external force field or it may be the result of only diffusion. In the former case, the aggregation process is called orthokinetic and in the latter perikinetic. Here, we focus on perikinetic aggregation. 

During aggregation, the number of kinetic particles adjacent to the aggregate is decreased and the resulting concentration difference is responsible for the diffusion of kinetic particles toward the growing aggregate. For two identical spherical particles, it has been derived that 

Another remarkable consequence of Equation 16.30 is that the rate constant does not depend on the particle size. This independency results from the cancellation of the increased probability of larger particles to collide by the decreased collision probability due to a smaller value of the diffusion coefficient. 

The diffusion-controlled aggregation rate is referred to as rapid aggregation and the rate constant is usually denoted by k. For a 1% (v/v) dispersion of particles having radii of lOOnm in an aqueous medium of 20 C (q = 0.001 N s), it can thus be calculated that the half-time of the aggregation is less than a tenth of a second. 

In the absence of flow, the aggregation kinetics of interacting dipolar particles has been studied by See and Doi (1991) (see Fig. 8-5). The time-dependence of the aggregation process seems to be reasonably well described by a hierarchical model, in which singlets form doublets after a time ti, doublets form quadruplets after a time t2, and so on. By 

This prediction agrees with both simulations in two dimensions and experiments on three-dimensional systems (Vorob eva and Vlodavets 1974). The dependence on concen-tration and time is predicted to be Save oc three dimensions 

For the sake of simplicity and clarity, the kinetics of aggregation processes are usually examined for monodisperse particles in the absence of nucleation, dissolution, and aggregate breakage. In that case, clusters of dimensionless mass m (i.e. the number of particles in the aggregate) result from the aggregation of two sub-clusters with masses i and j = m - i. At the same time, clusters of m join with other clusters. Hence, the mass balance of aggregates m takes the form  

This is the population balance equation (PBE). Its principal form was first given by Smoluchowski (1916). In Eq. (4.1), denotes the aggregate number concentration, while Kjj is a kinetic constant related with the collision between clusters of mass i and j, and Etj is the sticking probability of the sub-clusters. Commonly, it is assumed that any successful collision (i.e. surface contact) leads to a stable particle bond and that Ey can be, thus, set to 1. 

The collision kernels Kjj depend on the particle trajectories and the particle interaction. For purely diffusing particles and clusters without any particle interaction (i.e. diffusion-limited cluster aggregation—DLCA), Smoluchowski (1916) derived  

In the presence of long-ranging repulsive interactions. Brownian collision is considerably hampered and the kernel Kij is reduced by a factor Wy, which can be calculated from the energy potential of the acting conservative forces (Fuchs 1934) and the hydrodynamic hindrance functions B(r), which accounts for viscous lubrication (Honig et al. 1971 cf. Sect. 5.2.3)  

Repulsive interactions cannot be ignored for electrically charged aerosols or for suspended colloids with high surface charges. For colloidal suspensions, it is frequently assumed that the double layer is relatively thin (xa 1). Thus, the interaction between two sub-clusters is determined by the closest pairs of primary particles and the kernel Ky is approximated by  

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In tills section we focus on tlie tlieory of stability of charged colloids. In section C2.6.5.1 it is shown how particles can be made to aggregate by adding sufficient electrolyte. The associated aggregation kinetics are discussed in section C2.6.5.2, and tlie stmcture of tlie aggregates in section C2.6.5.3. For more details, see tlie recent reviews [53, 54 and 55], or tlie colloid science textbooks [33, 39]. 

A combination of equation (C2.6.13), equation (C2.6.14), equation (C2.6.15), equation (C2.6.16), equation (C2.6.17), equation (C2.6.18) and equation (C2.6.19) tlien allows us to estimate how low the electrolyte concentration needs to be to provide kinetic stability for a desired lengtli of time. This tlieory successfully accounts for a number of observations on slowly aggregating systems, but two discrepancies are found (see, for instance, [33]). First, tire observed dependence of stability ratio on salt concentration tends to be much weaker tlian predicted. Second, tire variation of tire stability ratio witli particle size is not reproduced experimentally. Recently, however, it was reported that for model particles witli a low surface charge, where tire DL VO tlieory is expected to hold, tire aggregation kinetics do agree witli tire tlieoretical predictions (see [60], and references tlierein). 

Altliough tire tlieories of colloid stability and aggregation kinetics were developed several decades ago, tire actual stmcture of aggregates has only been studied more recently. To describe tire stmcture, we start witli tire relationship between tire size of an aggregate (linear dimension), expressed as its radius of gyration and its mass m ... 

Hidalgo-Alvarez R, Martin A, Fernandez A, Bastes D, Martinez F and de las Nieves F J 1996 Electro kinetic properties, colloidal stability and aggregation kinetics of polymer colloids Adv. Colloid Interface Sc/. 67 1-118... 

Wright, H. and Ramkrishna, D., 1992. Solutions of inverse problems in population balance aggregation kinetics. Computers and Chemical Engineering, 16(2), 1019-1030. 

Mass transfer for this technique was examined by studying the protonation and aggregation kinetics of 5,10,15,20-tetraphenylporphyrin (H2TTP) at the dodecane-aqu-eous interface [61]. The rate law for the diffusion-controlled protonation of H2TTP at the interface was derived. 

Meakin, P., Models for colloidal aggregation. Ann. Rev. Phys. Chem. 39, 237-269 (1988). Meakin, P Aggregation kinetics. Physica Scripta 46, 295-331 (1992). 

Ferrone F. Analysis of protein aggregation kinetics. Methods Enzymol 1999 309 256-274. 

In vitro, fibril formation by several proteins displays an initial lag phase, followed by a rapid increase in aggregation (reviewed in Rochet and Lansbury, 2000). Introduction of fibrillar seeds eliminates the lag phase. These cooperative aggregation kinetics suggest that fibril formation begins with the formation of a nucleus and proceeds by fibril extension. The structure of the nucleus must therefore act as a template for the protein s conformation in the fibril. As the structural requirements for templating are unclear, it is difficult to assess the consistency of the model classes with this feature of fibril formation. We have described one possible templating mechanism for the cross-/ spine of GNNQQNY (Nelson et al., 2005). 

Andrade SM, Costa SMB (2002) Aggregation kinetics of meso-tetrakis(4-sulfonatophenyl) porphine in the presence of proteins temperature and ionic strength effects. J Fluoresc 12 77-82... 

The development of various techniques has led to important advances. The possibility to measure intermolecular and intercolloidal forces directly represents a qualitative change from the indirect way such forces had been inferred in the past from aggregation kinetics or from bulk properties such as the compressibility (deduced from small angle scattering) or phase behavior. Both static (i.e., equilibrium) and dynamic (e.g., viscous) forces can now be directly measured, providing information not only on the fundamental interactions in liquids but also on the structure... 

Besides, comparisons with other non-macromolecular gelling systems are in progress. Specially, we can co.mpare with a square planar copper complex, which aggregates in linear chains to gelify the cyclohexane (l ). It is immediatly noticed that characteristic times of the aggregation kinetics are correlated to the complexity of the molecular aggregation mechanism involved. 

Other possible methods of obtaining fractal dimensions are light- and small-an-gle-X-ray-scattering (SAXS). The first method gives information about mass fractality of aggregates and aggregation kinetics (Fleischmann et al., 1990), whereas the latter allows mass and surface fractals to be differentiated (Schmidt, 1990). 

Aggregation kinetics depends on transport mechanisms bringing the particles in contact and on particle-particle... 

To additionally account for the effect of the aggregate structure on the aggregation kinetics, a colhsion diameter xc is introduced which can be calculated based on the fractal dimension Z)F of the aggregate (Eq. (5)). 

Concerning aggregation kinetics and stabilization it was found in case of barium sulfate that barium ions adsorb more readily than sulfate ions [11], Thus, increasing the ratio R of barium to sulfate ions in the educt composition and thereby increasing the excess of barium ions after precipitation leads to high surface potentials and stabilization. Fig. 5 shows measured particle size distributions of continuously precipitated barium sulfate, measured based on quasi-elastic fight scattering (UPA 150 by Microtrac) within 3 min after pie-... 

H.C. Schwarzer, W. Peukert, Prediction of aggregation kinetics based on surface properties of nanoparticles, Chem. Eng. Sci., in press. 

For a given, low applied shear rate, perikinetic collision rates tend to be higher than orthokinetic rates when the dispersed species are quite small, of the order of 100 nm or less. The orthokinetic collision rates tend to dominate for larger-sized dispersed species, of the order of several pm or more. For more information on aggregation kinetics see Refs. [27,292,318,319]. 

Observe the aggregation kinetics as point defects coalesce to form clusters or other phases. 

Honeyman, B.D., and Santschi, P.H. (1991) Coupling of trace metal adsorption and particle aggregation kinetics and equilibrium studies using 59Fe-labelled hematite. Environ. Sci. Technol. 25, 1739-1747. 

R. M. Ziff, E. D. McGrady, and P. Meakin, On the validity of Smoluchowski s equation for cluster-cluster aggregation kinetics, J. Chem. Phys. 82 5269 (1985). 

Investigation of aggregation kinetics benefits from visualizing the changes of these discrete size distributions over time. This can be accomplished simply by adding the time axis, the results of which are depicted in Fig. 6. Here it is relatively simple to follow the evolution of the particle size distribution over time, as predicted by the von Smoluchowski analytical solution subject to the aforementioned constraints. Any section at a discrete point in time will yield the particle size distributions shown in Fig. 5. 

FIG. 7 Aggregation kinetics of hematite (diameter = 70 nm) under favorable chemical conditions and in quiescent fluid, (a) First 30 min of aggregation showing the smallest size fractions, (b) full 150-min experiment showing the smallest size fractions, and (c) Complete data set. 

FIG. 10 (a) Slow hematite aggregation kinetics measured in the laboratory... 

Domination of aggregation kinetics by differential settling implies more significant influences of hydrodynamics. Hydrodynamic corrections on the order of 10-4 were demonstrated by Han and Lawler [3], indicating a significant portion of the collision efficiency may be attributable to hydrodynamic forces. 

In contrast to the modeling methods described above, simulation methods approach the mathematical description of colloid aggregation kinetics from a fundamentally different viewpoint these methods track particle and aggregate movement over one-, two-, or three-dimensional space. This chapter will only provide a brief introduction and overview of the types of simulation methods that have been developed, as this is a broad and growing field of research worthy of numerous volumes alone. The following discussion will proceed by defining four categories of simulations as follows, and as outlined in Table 3. 

K. H. Gardner, Aggregation Kinetics of Colloidal Particles in Aqueous Solution, Ph.D. Dissertation, Clarkson University, Potsdam, New York, 1996. 

Shear aggregation brings an added complexity to the modeling of aggregation kinetics. This complication due to shear is not discussed further. 

Moskovits M, Vlckova B (2005) Adsorbate-induced silver nanoparticle aggregation kinetics. J Phys diem B 109 14755... 

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