Aggregation Smoluchowski model

Illustrative Cases. Three cases are illustrated in Figure 9, marked by the circles labeled A, B, and C. Case A refers to classical experiments by Swift and Friedlander (27) on the coagulation of monodisperse latex particles (diameter = 0.871 Jim) in shear flow and in the absence of repulsive chemical interactions. Considering a velocity gradient of 20 s-1, HA is 0.0535, log HA is — 1.27, and djd- is 1.0 for these experimental conditions. The circle labeled A in Figure 9 marks these conditions and indicates that the hydrodynamic corrections to Smoluchowskis model predict a reduction of about 40% in the aggregation rate by fluid shear. The experimental measurements by Swift and Friedlander showed a reduction of 64%. This observed reduction from Smoluchowskis rectilinear model was therefore primarily physical or hydro-dynamic and consistent with the curvilinear model. 

When the colloid destabilization occurs, the kinetics of collapse is often fast. Let us consider that we have a colloid system containing originally n particles per volume. Once destabilized, the particles aggregate by colliding with each other and thus the concentration of particles drops with time. Typieally this phenomenon is represented by the seeond-order equation, often called Smoluchowski model ... 

A probabilistic kinetic model describing the rapid coagulation or aggregation of small spheres that make contact with each other as a consequence of Brownian motion. Smoluchowski recognized that the likelihood of a particle (radius = ri) hitting another particle (radius = T2 concentration = C2) within a time interval (dt) equals the diffusional flux (dC2ldp)p=R into a sphere of radius i i2, equal to (ri + r2). The effective diffusion coefficient Di2 was taken to be the sum of the diffusion coefficients... 

Solutions of the Smoluchowski equation are of interest in all branches of physics where aggregation processes take place. There is a vast literature on its application including several reviews [5-7]. Here, we concentrate on those applications which can be used to model the polymerization processes. 

In the past, the equivalence between the size distribution generated by the Smoluchowski equation and simple statistical methods [9, 12, 40-42] was a source of some confusion. The Spouge proof and the numerical results obtained for the kinetics models with more complex aggregation physics, e.g., with a presence of substitution effects [43,44], revealed the non-equivalence of kinetics and statistical models of polymerization processes. More elaborated statistical models, however, with the complete analysis made repeatedly at small time intervals have been shown to produce polymer size distributions equivalent to those generated kinetically [45]. Recently, Faliagas [46] has demonstrated that the kinetics and statistical models which are both the mean-field models can be considered as special cases of a general stochastic Markov process. 

The Smoluchowski coagulation equation describes the rate of formation of acyclic aggregates. Only then it describes the evolution of a Markovian distribution [34]. Strictly speaking the Smoluchowski equation simply disregards any cycle formation. For polymers this is true for models with high functionality... 

There is essentially a single modeling approach that has been developed, referred to here as the von Smoluchowski approach, and this method will be presented first. The von Smoluchowski approach requires analytical expressions to represent particle collision rates, to calculate collision efficiencies, and to dictate aggregate structure formation. These individual components are discussed in the subsequent sections, followed by analytical and numerical techniques of solving the von Smoluchowski equation. 

From the discussion of various simulation methods, it is clear that they will continue to play an important role in further development of aggregation theories as they have advanced the state of knowledge over the last 20 years. The major limitation of the precise methods of Molecular and Brownian Dynamics continues to be difficulty associated with treatment of aggregates with complex geometry the same topic that limits the ability to model these systems using von Smoluchowski s approach. Research needs to be conducted on the hydrodynamics of interactions between fractal aggregates of increasing complexity in order to advance the current ability to describe these types of systems. 

This kinetic model, proposed by Danov, Ivanov, Gurkov, and Borwankar, is called the DIGB model for the sake of brevity. Danov et al. (39) generalized the Smoluchowski scheme (Fig. 2a) to account for droplet coalescence within floes. Any aggregate (floe) composed oik particles can partially coalesce to become an aggregate of i particles (1 < i < k), with the rate constant being (Fig. 2b). This aggre-... 

The study of the rates of aggregation of colloidal particles owing to Brownian motion has a long history and was initiated by von Smoluchowski in 1916. Several solutions to the Smoluchowski equation have been attempted since then and two will be sketched here. The problem is to establish the time evolution of the number and size of the agglomerates in terms of the number of monomer particles. The expressions are typical for simplified models as is usual in physical chemistry. 

Diffusion-limited aggregation (DLA) is often referred to as a fast process because the above model predicts the maximum aggregation rate. Reaction-limited aggregation (RLA) rates are slower than predicted by the DLA model. The effect of the slow attachment reaction rate on the aggregation rate is expressed as a stability ratio (W) and is defined as the ratio of the Smoluchowski rate constant (ks) to the observed rate constant (ko). 

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