Einstein-Smoluchowski diffusion theory

More recently an alternative explanation of the complex excimer behaviour observed in polymer systems has been proposed whereby the close proximity of some fluorophores leads to a time dependent rate of quenching analogous to that predicted by the Einstein-Smoluchowski diffusion theory for low molecular weight systems. This predicts a fluorescence response function of the form... 

Perrin model and the Johansson and Elvingston model fall above the experimental data. Also shown in this figure is the prediction from the Stokes-Einstein-Smoluchowski expression, whereby the Stokes-Einstein expression is modified with the inclusion of the Ein-stein-Smoluchowski expression for the effect of solute on viscosity. Penke et al. [290] found that the Mackie-Meares equation fit the water diffusion data however, upon consideration of water interactions with the polymer gel, through measurements of longitudinal relaxation, adsorption interactions incorporated within the volume averaging theory also well described the experimental results. The volume averaging theory had the advantage that it could describe the effect of Bis on the relaxation within the same framework as the description of the diffusion coefficient. 

The relaxation process may be accompanied by diffusion. Consequently, the mean relaxation time for such kinds of disordered systems is the time during which the relaxing microscopic structural unit would move a distance R. The Einstein-Smoluchowski theory [226,235] gives the relationship between x and R as... 

Brownian motion theory was verified by many scientists (T. Svedberg, A. Westgren, J.Perrin, L.de Broglie and others), who both observed individual particles and followed the diffusion in disperse systems [5]. The influence of various factors, such as the temperature, dispersion medium viscosity, and particle size on the value of the Brownian displacement, was evaluated. It was shown that the Einstein-Smoluchowski theory describes the experimental data adequately and with high precision. 

The fundamental theory of electron escape, owing to Onsager (1938), follows Smoluchowski s (1906) equation of Brownian motion in the presence of a field F. Using the Nemst-Einstein relation p = eD/kRT between the mobility and the diffusion coefficient, Onsager writes the diffusion equation as... 

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... 

Hasinoff noted that the rate coefficient of formation of encounters pairs, fcD, was smaller than predicted from the Smoluchowski—Stokes—Einstein rate coefficient [eqn. (29)]. In aqueous glycerol, this reduction was by 0.14 times, in aqueous polyethylene glycol by 0.30 times, and in aqueous ethylene glycol by 0.11 times. Hasinoff compared these reductions in rate of diffusive rate of formation of encounter pairs with three theories of anisotropic reactivity due to Weller [262], Schmitz and Schurr [257] and... 

Kinetics is concerned with many-particle systems which require movements in space and time of individual particles. The first observations on the kinetic effect of individual molecular movements were reported by R. Brown in 1828. He observed the outward manifestation of molecular motion, now referred to as Brownian motion. The corresponding theory was first proposed in a satisfactory form in 1905 by A. Einstein. At the same time, the Polish physicist and physical chemist M. v. Smolu-chowski worked on problems of diffusion, Brownian motion (and coagulation of colloid particles) [M. v. Smoluchowski (1916)]. He is praised by later leaders in this field [S. Chandrasekhar (1943)] as a scientist whose theory of density fluctuations represents one of the most outstanding achievements in molecular physical chemistry. Further important contributions are due to Fokker, Planck, Burger, Furth, Ornstein, Uhlenbeck, Chandrasekhar, Kramers, among others. An extensive list of references can be found in [G.E. Uhlenbeck, L.S. Ornstein (1930) M.C. Wang, G.E. Uhlenbeck (1945)]. A survey of the field is found in [N. Wax, ed. (1954)]. 

This form is useful in the generalization of Eq. (485) to fractional diffusion. The investigation of the diffusion equation (485) began when Louis Bachellier (Jules Poincare s student) wrote his thesis in 1900. It was called The Theory of Speculations and was devoted to the evolution of the stock market. Many of the most famous scientists have contributed to our knowledge of diffusion processes, amongst them Einstein, Langevin, Smoluchowski, Fokker, Planck, Levy, and others. 

The simplest form of the Fokker-Planck equation is that for diffusion in configuration space, namely the Smoluchowski equation, which was originally derived by Einstein [13] in 1905 in the context of the theory of the Brownian movement of a particle in one dimension under no external forces. We have seen earlier that this equation is (Eq. 2.50)... 

The fundamental rate expression to be considered is the Smoluchowski relation k = 4n iVDAB AB (Equation (2.1)). The derived expression ART/r] (Equation (2.3a)), is a useful approximation, but deviations from it are observed, because the Stokes-Einstein equation which is involved is derived by hydrodynamic theory for spherical particles moving in a continuous fluid, and does not accurately represent the measured values of translational diffusion coefficients in real systems. Although the proportionality Da 1 /rj is indeed a reasonable approximation for many solutes in common solvents, the numeral coefficient 1 /4 is subject to uncertainty. In the first place, this theoretical value derives from the assumption that in translational motion there is no friction between a solute molecule and the first layer of solvent molecules surrounding it, i.e., that slip conditions hold. If, however, one assumes instead that there is no slipping ( stick conditions), so that momentum is... 

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. 

The rate of diffusion controlled reaction is typically given by the Smoluchowski/Stokes-Einstein (S/SE) expression (see Brownian Dynamics), in which the effect of the solvent on the rate constant k appears as an inverse dependence on the bulk viscosity r), i.e., k oc (1// ). A number of experimental studies of radical recombination reactions in SCFs have found that these reactions exhibit no unusual behavior in SCFs. That is, if the variation in the bulk viscosity of the SCF solvent with temperature and pressure is taken into accounL the observed reaction rates are well described by S/SE theory. However, these studies were conducted at densities greater than the critical density, and, in fact, the data is inconclusive very near to the critical density. Additionally, Randolph and Carlier have examined a case in which the observed diffusion controlled, free radical spin exchange rates are up to three times faster than predicted by S/SE theory, with the deviations becoming most pronounced near the critical point. This deviation was attributed to some sort of solvent-solute clustering effect. It is presently unclear why this system is observed to behave differently from those which were observed to follow S/SE behavior. Possible candidates are differences in thermodynamic conditions or molecular interactions, or even misinterpretation of the data arising from other possible processes not considered. 

Stokes-Einstein relation is inversely proportional to the solvent viscosity. However, in Smoluchowski theory this independence is extended to the frame of reference on the A as well each B particle diffuses relative to the A particle with a mutual diffusion coefficient D. The flux of B particles per unit area (Jb) is known to be dependent on the gradient operator (Vc) with respect to the coordinates relative to the position of A and mutual diffusion through the relation... 

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