The Smoluchowski theory

The Smoluchowski theory for diffusion-controlled reactions, when combined with the Stokes-Einstein equation for the diffusion coefficient, predicts that the rate constant for a diffusion-controlled reaction will be inversely proportional to the solution viscosity.16 Therefore, the literature values for the bimolecular electron transfer reactions (measured for a solution viscosity of r ) were adjusted by multiplying by the factor r 1/r 2 to obtain the adjusted value of the kinetic constant... 

What is meant by rapid coagulation What is the basic principle behind the Smoluchowski theory of rapid coagulation What is the rate coefficient for rapid coagulation How is it defined, and what properties of the dispersion determine its magnitude What are the limitations of this theory as presented in the text ... 

Over molecular length scales, the diffusion distances become very short (< 1 nm) so that only very rapid events can be influenced by these short diffusion times. Necessarily, this limits the number of systems to only relatively few, where the rate at which the reactants can approach one another is slow or comparable with the rate at which the reactants react chemically with each other. Some typical systems which have been studied are discussed in Sect. 2. The Smoluchowski [3] theory of reactions in solution, which occur at a rate limited solely by how fast the reactants can approach each other (sufficiently closely to react chemically almost instantaneously) is discussed in Sect. 3. If the chemical reaction is not so rapid, the observed rate of reaction may be influenced by both the rate of approach and the rate of subsequent chemical reaction. Collins and Kimball [4], and later Noyes [5], have extended the Smoluchowski theory (1917) to consider this situation (Sect. 4). In light of these quantitative theoretical models of diffusion-limited rate processes, some of the more recent and careful experiments on diffusion-controlled reactions in solution are considered briefly in Sect. 5. As the Smoluchowski theory... 

Ruthenium(II) bipyridyl and Cr(III) aquo complexes luminesce strongly when photostimulated. The emission of light can be quenched effectively by such species as oxygen, paraquat, Fe(II) aquo complexes, Ru(II) complexes and Cr(NCS)i (Sutin [15]). Pfeil [16] found that the quenching rate coefficients are typically a third to a half of the value which might be predicted from the Smoluchowski theory [3]. 

The Smoluchowski theory of diffusion-limited (or controlled) reactions relies heavily on the appropriateness of the inital condition [eqn. (3)]. Though the initial condition does not determine the steady-state rate coefficient [eqn. (20)] because diffusion of B in towards the reactant A is from large separation distances (>10nm) in the steady-state, it does directly determine the magnitude of the transient component of the rate coefficient because this is due to an excess concentration of B present initially compared with that present in the steady-state. As a first approximation to the initial distribution, the random distribution is intuitively pleasing and there is little experimental evidence available to cast doubt upon its appropriateness. Section 6.6 and Chap. 8 Sect. 2.2 present further comments on this point. 

A comparison of experimental results with the Smoluchowski theory... 

The remainder of this section considers several experimental studies of reactions to which the Smoluchowski theory of diffusion-controlled chemical reaction rates may be applied. These are fluorescence quenching of aromatic molecules by the heavy atom effect or electron transfer, reactions of the solvated electron with oxidants (where no longe-range transfer is implicated), the recombination of photolytically generated radicals and the reaction of carbon monoxide with microperoxidase. 

As in the previous chapter on the Smoluchowski theory and its extensions, similar boundary and initial conditions may be used. The reaction of a species A with a vast excess of B (yet still sufficiently dilute to ensure that Debye- Hiickel screening is unimportant) can be considered as one where the A species are statistically independent of each other and are surrounded by a sea of B species. An ionic reactant A has a rate of reaction with all the B reactants equal to the sum of the rates of reaction of individual A—B pairs. This rate for large initial separations of A and B is... 

The rate of reaction of methyl radicals is in excellent agreement with the predictions of the Smoluchowski theory (see Chap. 2, Sect. 2.6). Consequently, it appears that geminate radicals move towards and away from each other at a diffusion-limited rate. Once an encounter pair is formed, reaction is very rapid (primary recombination). Furthermore, the encounter pair is held together for a considerable time (< 0.1ns in mobile solvents) because the surrounding solvent molecules hinder their separation (solvent caging). There is much evidence which lends some support for this view the most important influences on the recombination probability are listed below. 

As an example of the application this work, Kapral [285] and Pagistas and Kapral [37] have considered the reaction rate between iodine atoms (or some other similar species) effectively distributed uniformly in solution. They compared their calculations with those of the diffusion equation analysis and with the molecular pair approach rather than compare rate coefficients, Kapral [285] compared the rate kernels (which are approximately the time derivatives of rate coefficients). Over long times, these kinetic theory and molecular pair rate kernels both reduce to the typical form of the Smoluckowski rate kernel. However, with parameters such as R — 0.43 nm and D = 6 x 10 9m2s 1, the time beyond which the rate kernels of kinetic theory and the Smoluchowski theory are in reasonably close agreement is 20 ps, a time much longer than the velocity... 

However, the important consequences of this analysis are that the complications of the reduction in the density of B between A reactants only develops during the decay of the non-steady-state density of B towards the steady state. The average concentration of B after reaction has begun is less than the initial (or bulk) value [B]0 usually used in the Smoluchowski theory. The diffusion of B towards A is driven by the larger concentration of B at considerable distances from A than the concentration of B nearer to A. The concentration or density gradient of B at A is decreased slightly by this competition between different A reactants for B. Hence, the current B towards one A remains almost as it was in the Smoluchowski theory [eqn. (18)], viz. 

This expression is equivalent to that of eqns. (123) and (143) for the survival probability in the Smoluchowski theory. 

The rate coefficient, k(p)> increases with sink density because, on average, the microscopic density gradient around each sink is reduced less then the microscopic density due to the neighbouring sinks. From Fig. 47, the extent of the increase in k(p) is quite small and would be difficult to observe experimentally. A reaction between a stationary species, A, and a diffusing species, B, occurs at a rate k [ A) [B], where k is the second-order rate coefficient, ft(p), above. If the Smoluchowski theory had been used instead, it would have a rate coefficient fe(0) and the concentration of the diffusing species B remains at its initial volume [B]0. The rate of reaction is (Q) [A] [B]0. These reaction rates are approximately the same, because [B]0/[B] fe(p)/fe(0). Under the circumstances where [B] > A], the importance of these competitive effects is small. 

Many years ago Polya [20] formulated the key problem of random walks on lattices does a particle always return to the starting point after long enough time If not, how its probability to leave for infinity depends on a particular kind of lattice His answer was a particle returns for sure, if it walks in one or two dimensions non-zero survival probability arises only for the f/iree-dimensional case. Similar result is coming from the Smoluchowski theory particle A will be definitely trapped by B, irrespectively on their mutual distance, if A walks on lattices with d = 1 or d = 2 but it survives for d = 3 (that is, in three dimensions there exist some regions which are never visited by Brownian particles). This illustrates importance in chemical kinetics of a new parameter d which role will be discussed below in detail. 

The consistent derivation and analysis of the Waite-Leibfried equations is presented in Chapter 4. We show that this theory is the linear approximation of the exact many-particle formalism. Its relation to the Smoluchowski theory is also established. 

The simplest class of bimolecular reactions involves only one type of mobile particles A and could result either in particle coagulation (coalescence, fusion) A + A —> A, or annihilation, A + A — 0 (inert product). Their simplicity in conjunction with the simple topology of d = 1 allows us to solve the problem exactly, which makes it very attractive for testing different approximations and computer simulations. In the standard chemical kinetics (i.e., mean-field theory, Section 2.1.1) we expect in d = 2 and 3 for both reactions mentioned trivial behaviour quite similar to the A+B — 0 reaction, i.e., tia( ) oc t-1, as t — oo. For d = 1 in terms of the Smoluchowski theory the joint density obeys respectively the equation (4.1.56) with V2 = and D = 2Da. 

Despite the fact that (5.3.3) reveals the same asymptotic behaviour (nA(t) oc f-1/2, as t —> oo), the relative concentration is always smaller than predicted by the Smoluchowski theory for nA — l/2nA(0), the discrepancy is 9%. In other words, the Smoluchowski approach slightly underestimates a real reaction rate due to its neglect of reactant density fluctuations stimulating. (Note that in the case of second reaction, A + A — A, the reaction rate K(t) in the Smoluchowski approach has to be corrected by a factor 1/2... 

According to the Smoluchowski theory of diffusion-controlled bimolecular reactions in solutions, this constant decreases with time from its kinetic value, k0 to a stationary (Markovian) value, which is kD under diffusional control. In the contact approximation it is given by Eq. (3.21), but for remote annihilation with the rate Wrr(r) its behavior is qualitatively the same as far as k(t) is defined by Eq. (3.34)... 

Smoluchowski approximation for noninteracting B s and independent AB pairs turned out to be non-trivial problem (see also [76]). Even when the Smoluchowski theory for irreversible pseudo-first-order reactions is exact, no rigorous theory that is valid for an arbitrary set of kinetic ptirameters was developed in [70] the additional assumption was made that every time a bound AB pair dissociates forming an unbound pair at contact, this pair behaves as if it was surrounded by an equilibrium distribution of B s independent of the history of previous associations and dissociations (see also [77]). 

Most numerical techniques employed for aggregation simulation are based on the equilibrium growth assumption and on the Smoluchowski theory. As shown in Meakin (1988, 1998), analytical solutions for the Smoluchowski equation have been obtained for a variety of different reaction kernels these kernels represent the rate of aggregation of clusters of sizes x and y. In most cases, these reaction kernels are based on heuristics or semi-empirical rules. 

In the approach adopted in my first edition, the derivation and use of the general dynamic equation for the particle size distribution played a central role. This special form of a population balance equation incorporated the Smoluchowski theory of coagulation and gas-to-panicle conversion through a Liouville term with a set of special growth laws coagulation and gas-to-particle conversion are processes that take place within an elemental gas volume. Brownian diffusion and external force fields transport particles across the boundaries of the elemental volume. A major limitation on the formulation was the assumption that the panicles were liquid droplets that coalesced instantaneously after collision. 

Chemical reaction kinetics proceeds on the (often implicit) assumption that the reaction mixture is ideally mixed, and does not consider the time needed for reacting species to encounter each other by diffusion. The encounter rate follows from the theory of Smoluchowski. It turns out that most reactions in fairly dilute solutions follow chemical kinetics, but that reactions in low-moisture foods may be diffusion controlled. In the Bodenstein approximation, the Smoluchowski theory is combined with a limitation caused by an activation free energy. Unfortunately, the theory contains several uncertainties and unwarranted presumptions. 

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