Smoluchowski equation model

The von Smoluchowski equation must be corrected when the partners are ions to account for attractive or repulsive forces. They can be approximated by an electrostatic model. The quantity by which Eq. (9-10) or (9-13) is to be multiplied is... 

All models described up to here belong to the class of equilibrium theories. They have the advantage of providing structural information on the material during the liquid-solid transition. Kinetic theories based on Smoluchowski s coagulation equation [45] have recently been applied more and more to describe the kinetics of gelation. The Smoluchowski equation describes the time evolution of the cluster size distribution N(k) ... 

The two major theories of flocculation, the bridging model (1) and the electrostatic patch model (2, 3 ), provide the conceptual framework for the understanding of polymer-aided flocculation, but they do not directly address the kinetics of the process. Smellie and La Mer (4) incorporated the bridging concept into a kinetic model of flocculation. They proposed that the collision efficiency in the flocculation process should be a function of the fractional surface coverage, 0. Using a modified Smoluchowski equation, they wrote for the initial flocculation rate... 

The coagulation, flocculation, and adsorption processes were modeled mathematically using classical coagulation theory as a starting point. The Smoluchowski equation for orthokinetic coagulation in laminar flow is written (18)... 

New difficulties arise when we try to take into account the dynamical interaction of particles caused by pair potentials U(r) mutual attraction (repulsion) leads to the preferential drift of particles towards (outwards) sinks. This kind of motion is described by the generalization of the Smoluchowski equation shown in Fig. 1.10. In terms of our illustrative model of the chemical reaction A + B —> B the drift in the potential could be associated with a search of a toper by his smell (Fig. 1.12). An analogy between Schrodinger and Smoluchowski equations is more than appropriate indeed, it was used as a basis for a new branch of the chemical kinetics operating with the mathematical formalism of quantum field theory (see Chapter 2). 

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. 

The same recurrence equation is obtained by solving successive terms of the Smoluchowski equation with 2 S C = Hx equal to //(1 + ft) and 1 for models 1 and 2, respectively. 1... 

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. 

Physically, it is the fraction of functional groups available to reaction (and hence equivalent to Hx in Eqs. (65) and (72)). Equation (99) reveals that for this version of the FS model the time units are nine times longer" than those from direct application of the original Smoluchowski equation. 

The Smoluchowski equation was thought of as failing beyond the gel point [22]. Dusek [53] and Ziff [54] have demonstrated this not to be true for the Flo-ry-Stockmayer model. At the same time Leyvraz and Tschudi [32] presented an elegant and more general derivation of the solutions of the Smoluchowski equation both prior to and beyond the gel point. 

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. 

This equation, known as the Helmholtz-Smoluchowski equation, relates the potential at a planar bound surface region to an induced electro-osmosis fluid velocity 6. Recall that in the previous section surface charge was related to a potential in solution. In the following section surface charge will be related to the chemistry of the surface. A model for the development of surface charge in terms of acid-base dissociation of ionizable surface groups is introduced. 

A further gain is undoubtedly the inclusion of the concept of reptation diffusion into the gel effect model, largely due to Tulig and Tirrell and their co-workers [24, 44, 45], Solving the kinetic scheme (similar to that in Sect. 1.1) with the use of the Smoluchowski equation [46], they derived a relationship analogous to eqn. (30) (based on rheologic concepts)... 

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