The Smoluchowski Effect

Here, as in the discussion of the Smoluchowski effect, we have made expHdt reference to the discrete, periodic stracture of the lattice. The symmetry ofa periodic lattice is substantially lower than the continuous translational and rotational symmetry of the uniform jeUium model. Only translations involving a Bravais lattice vector and a discrete number of point symmetry operations remain and the constant potential has to be replaced by a crystal potential, which reflects the lowered symmetry. This has profound consequences for the electronic structure. 

Xd takes into account the Smoluchowski effect. X is the heat conductivity of the continuum gas, a the mean free path of the molecules, y he accomodation coefficient and xp is the average gap width between the particles with respect to heat conduction in a ratified gas and is to be calculated from i=n f j... 

Smoluchowski theory [29, 30] and its modifications fonu the basis of most approaches used to interpret bimolecular rate constants obtained from chemical kinetics experiments in tenus of difhision effects [31]. The Smoluchowski model is based on Brownian motion theory underlying the phenomenological difhision equation in the absence of external forces. In the standard picture, one considers a dilute fluid solution of reactants A and B with [A] [B] and asks for the time evolution of [B] in the vicinity of A, i.e. of the density distribution p(r,t) = [B](rl)/[B] 2i ] r(t))l ] Q ([B] is assumed not to change appreciably during the reaction). The initial distribution and the outer and inner boundary conditions are chosen, respectively, as... 

The Smoluchowski-Levich approach discounts the effect of the hydrodynamic interactions and the London-van der Waals forces. This was done under the pretense that the increase in hydrodynamic drag when a particle approaches a surface, is exactly balanced by the attractive dispersion forces. Smoluchowski also assumed that particles are irreversibly captured when they approach the collector sufficiently close (the primary minimum distance 5m). This assumption leads to the perfect sink boundary condition at the collector surface i.e. cp 0 at h Sm. In the perfect sink model, the surface immobilizing reaction is assumed infinitely fast, and the primary minimum potential well is infinitely deep. 

The experimental and simulation results presented here indicate that the system viscosity has an important effect on the overall rate of the photosensitization of diary liodonium salts by anthracene. These studies reveal that as the viscosity of the solvent is increased from 1 to 1000 cP, the overall rate of the photosensitization reaction decreases by an order of magnitude. This decrease in reaction rate is qualitatively explained using the Smoluchowski-Stokes-Einstein model for the rate constants of the bimolecular, diffusion-controlled elementary reactions in the numerical solution of the kinetic photophysical equations. A more quantitative fit between the experimental data and the simulation results was obtained by scaling the bimolecular rate constants by rj"07 rather than the rf1 as suggested by the Smoluchowski-Stokes-Einstein analysis. These simulation results provide a semi-empirical correlation which may be used to estimate the effective photosensitization rate constant for viscosities ranging from 1 to 1000 cP. 

The Peclet number compares the effect of imposed shear (known as the convective effect) with the effect of diffusion of the particles. The imposed shear has the effect of altering the local distribution of the particles, whereas the diffusion (or Brownian motion) of the particles tries to restore the equilibrium structure. In a quiescent colloidal dispersion the particles move continuously in a random manner due to Brownian motion. The thermal motion establishes an equilibrium statistical distribution that depends on the volume fraction and interparticle potentials. Using the Einstein-Smoluchowski relation for the time scale of the motion, with the Stokes-Einstein equation for the diffusion coefficient, one can write the time taken for a particle to diffuse a distance equal to its radius R, as... 

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 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. 

In Chaps. 3 and 4, estimates of encounter distances and mutual diffusion coefficients from similar experiments to those of Buxton et al. [18] are discussed. The complications to the analysis of diffusion-controlled rate processes in solution when the reactants interact strongly with one another or the reaction can occur over distances much larger than typical encounter distances do not lead to markedly different time-dependent rate coefficient expressions to the Smoluchowski form. Indeed, replacing R in eqn. (29) by an effective encounter distance, Reff, allows the compactness of the Smoluchowski rate coefficient to be extended to other situations. Means of estimating Reff are discussed in Chaps. 3, 4, 5 (Sect. 4.3), 8 (Sect. 2.6) and 9 (Sects. 4 and 6). 

In order to solve for the survival and recombination probabilities, p and q in eqn. (126), it is necessary to solve eqn. (122) for p(r, f]r0, f0) and use eqn. (123) to find p or eqn. (125) for q. Again, the boundary and initial conditions are required. Before the pair is formed (f < and ttf is slightly less than f0), the density p is zero, of necessity. The boundary conditions are closely related to the Smoluchowski conditions [eqns. (5), (22), (46) and (47)]. As the radicals approach each other they have a probability of reacting, which can be related to an effective second-order rate coefficient, fcact> f°r the activation-limited process of recombination by... 

By contrast, when both the reactive solute molecules are of a size similar to or smaller than the solvent molecules, reaction cannot be described satisfactorily by Langevin, Fokker—Planck or diffusion equation analysis. Recently, theories of chemical reaction in solution have been developed by several groups. Those of Kapral and co-workers [37, 285, 286] use the kinetic theory of liquids to treat solute and solvent molecules as hard spheres, but on an equal basis (see Chap. 12). While this approach in its simplest approximation leads to an identical result to that of Smoluchowski, it is relatively straightforward to include more details of molecular motion. Furthermore, re-encounter events can be discussed very much more satisfactorily because the motion of both reactants and also the surrounding solvent is followed. An unreactive collision between reactant molecules necessarily leads to a correlation in the motion of both reactants. Even after collision with solvent molecules, some correlation of motion between reactants remains. Subsequent encounters between reactants are more or less probable than predicted by a random walk model (loss of correlation on each jump) and so reaction rates may be expected to depart from those predicted by the Smoluchowski analysis. Furthermore, such analysis based on the kinetic theory of liquids leads to both an easy incorporation of competitive effects (see Sect. 2.3 and Chap. 9, Sect. 5) and back reaction (see Sect. 3.3). Cukier et al. have found that to include hydrodynamic repulsion in a kinetic theory analysis is a much more difficult task [454]. 

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... 

It is interesting to note that eqn. (190) is reminiscent of the steady-state Collins and Kimball rate coefficient [4] [eqn. (27)] with kact replaced by kacig R) and 4ttRD by eqn. (189). Equation (190) for the rate coefficient is significantly less than the Smoluchowski rate coefficient on two counts hydrodynamics repulsion and rate of encounter pair reaction. Had experimental studies shown that a measured rate coefficient was within a factor of two of the Smoluchowski rate coefficient, it would be tempting to invoke partial diffusion control of the reaction rate. The reduction of rate due to hydrodynamic repulsion should be included first and then the effect of moderately slow reaction rates between encounter pairs. 

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