Size methods Smoluchowski

At the simplest level, the rate of flow-induced aggregation of compact spherical particles is described by Smoluchowski s theory [Eq. (32)]. Such expressions may then be incorporated into population balance equations to determine the evolution of the agglomerate size distribution with time. However with increase in agglomerate size, complex (fractal) structures may be generated that preclude analysis by simple methods as above. 

In this result, the condition of small particles means that the actual size of the particles (which is often difficult to obtain) is not required. For reasons to be discussed later, we will call the potential obtained by this method the zeta potential (Q rather than the surface potential. In the following section we consider the alternative case of large colloidal particles, which leads to the Smoluchowski equation. 

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. 

There is a close parallel between this development and the microscopic theory of condensed-phase chemical reactions. First, the questions one asks are very nearly the same. In Section III we summarized several configuration space approaches to this problem. These methods assume the validity of a diffusion or Smoluchowski equation, which is based on a continuum description of the solvent. Such theories will surely fail at the close encounter distance required for reaction to take place. In most situations of chemical interest, the solute and solvent molecules are comparable in size and the continuum description no longer applies. Yet we know that these simple approaches are often quite successful, even when applied to the small molecule case. Thus we again have a microscopic relaxation process exhibiting a strong hydrodynamic component. This hydrodynamic component again gives rise to a power law decay in the rate kernel (cf. 

The frequency of encounters between spherical particles of different size has been treated by Debye (4), following a method first used by Smoluchowski (10). The number of collisions is obtained from the diffusion equation for particles diffusing steadily into a hole surrounding the particle in question. For spherical particles the equation can be solved and the well known Expression 1 is found. 

In order to apply the Smoluchowski equation (Equations (1.3), (2.1), (3.29)), we need values for the least distance of approach (rAn) and the diffusion coefficient (Dab)- The value of tab can be estimated from molecular volumes (Section 2.5.1.2). The diffusion coefficient can be determined by various methods, but experimental values are available only for a minority of the myriad possible situations. A common practice is to use the Stokes-Einstein relation (Section 1.2.3), which rests on the assumption that solute molecules in motion behave like macroscopic particles to which classical hydrodynamic theory can be applied. We shall first outline (a) the relation between the diffusion coefficient D and the mechanics of motion of particles in fluids, leading to the Stokes-Einstein equation relating D to solute size and solvent viscosity and (b) the direct experimental determination of D. We shall then (c) compare the results and note the reservations that are required in relying on the Stokes-Einstein estimates of D in various cases. 

Notice that, unlUce the case of the Henry equation, in the simple cases of Htickel and Smoluchowski equations the size of particles (radius) is not required. There are alternative graphical ways for correcting for the size of particles (Pashley and Karaman, 2004). Although the Htickel and Smoluchowski equations are very useful, it can be shown that they only cover a very small part of the colloidal domain. In most cases, corrections are needed, e.g. via the use of Henry equation or other graphical methods where the correction factor, f, can be estimated. Negative values of y/o can be obtained if p is negative. 

There are some limits where the Smoluchowski s equation can be solved by simple methods. First one has to discuss the kernels. Consider a diffusion-controlled reaction where the clusters grow in time. Then the cluster size increases and the typical radius of a cluster can be written as R(i) where df is the fractal dimension of the cluster containing i monomers. With increasing size the diffusion constant decreases according to Stoke s law and one can assume a power law for the diffusion constant of the cluster, i.e. The reaction constants are then of the... 

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