Smoluchowski-Einstein theory

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. 

Eventually, the answer was found by Albert Einstein and the Polish physicist Marian Smoluchowski (1872-1917), then a professor at the University of Lviv. The title of one of Einstein s papers on the theory of Brownian motion is rather telling On the motion of particles suspended in resting water which is required by the molecular-kinetic theory of heat . Einstein and Smoluchowski considered chaotic thermal motion of molecules and showed that it explains it all a Brownian particle is fidgeting because it is pushed by a crowd of molecules in random directions. In other words, you can say that Brownian particles are themselves engaged in chaotic thermal motion. Nowadays, science does not make much distinction between the phrases Brownian motion and thermal motion — the only difference lies back in history. The Einstein-Smoluchowski theory was confirmed by beautiful and subtle experiments by Jean Perrin (1870-1942). This was a long awaited, clear and straightforward proof that all substances are made of atoms and molecules. ... 

Certainly, no microscope would let you see the twists and turns of an individual molecule s path. However, the Einstein-Smoluchowski theory tells us how to spot the difference between a fuzzy line which consists of a great number of tiny random kinks, and an ordinary smooth curve, even though we cannot discern the individual kinks. (We do not always need to see everything, e.g. we can happily tell water from alcohol even though the individual molecules are invisible ) In the same way, a poljrmer chain looks nothing like a shape stretched in a certain direction. And the path of a man in a forest would depend quite noticeably on whether he is equipped with a compass or not ... 

The Einstein-Smoluchowski theory leads to the value = (rnksT) / / i Kr r) for spherical Brownian particles of radius r and mass m moving in a liquid of viscosity 77 at a temperature T. 

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 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... 

We have in this way obtained a generalization of Einstein s theory of the interaction between matter and radiation including multiple photon processes and involving transition probabilities. But there is a basic difference. The operator definite positive. We no longer have a simple addition of transition probabilities. This corresponds exactly to the interference of probabilities discussed in Section IV. The process is not of the simple Chapman-Smoluchowski-Kolmogoroff type (Eq. (11)) the operator transition probability. As the result, the second of the two sequences discussed above may decrease the effect of the first one. It is very interesting that even in the limit of classical mechanics (which may be performed easily in the case of anharmonic oscillators) this interference of probabilities persists. This is in agreement with our conclusion in Section IV. 

Formula (483) was first obtained by Albert Einstein (1879-1955) in 1905 and bears his name. Independently of Einstein, the theory of the Brownian motion was developed by Marian von Smoluchowski (1872-1917) in 1905-1906. The expression obtained by him agrees with formula (483) with a constant multiplier equal to one. 

Equation (1) may be derived using a variety of microscopic models of the relaxation process. In the derivation of Eq. (1), Debye [1] used the theory of the Brownian motion developed by Einstein and Smoluchowski. Einstein s theory of Brownian motion [2] is based on the notion of a discrete time walk. The walk may be described in simple schematic terms as follows. Consider a two-dimensional lattice then, in discrete time steps of length At, the random walker is assumed to jump to one of its nearest-neighbor sites, displayed, for example [7], on a square lattice with lattice constant Ax, the direction being random. Such a process, which is local both in space and time, can be modeled [7] in the one-dimensional analogue by the master equation... 

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... 

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... 

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... 

Peter Debye in 1944 further extended the work of Rayleigh and the fluctuation theory of Smoluchowski and Einstein to include the measurement of the scattering of light by macromolecular solutions for determining molecular size. 

To obtain a correct form of Eq. (22) allowing for thermodynamic non-ideality of the solution, fluctuation theory originally developed by Einstein, Zernicke, Smoluchowski and Debye has been adapted to polymer solutions. 

According to the theory developed by Smoluchowski and by Einstein, if a spherical particle of radius r rotates in a liquid of viscosity i), in a short time A/, by an angle Aa, then the mean value of angular rotation A is given by the Brownian equation for rotational motion ... 

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... 

The depolarization of light by dense systems of spherical atoms or molecules has been known as an experimental fact for a long time. It is, however, discordant with Smoluchowski s and Einstein s celebrated theories of light scattering which were formulated in the early years of this century. These theories consider the effects of fluctuation of density and other thermodynamic variables [371, 144]. 

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)]. 

Let us consider now behaviour of the gas-liquid system near the critical point. It reveals rather interesting effect called the critical opalescence, that is strong increase of the light scattering. Its analogs are known also in other physical systems in the vicinity of phase transitions. In the beginning of our century Einstein and Smoluchowski expressed an idea, that the opalescence phenomenon is related to the density (order parameter) fluctuations in the system. More consistent theory was presented later by Omstein and Zemike [23], who for the first time introduced a concept of the intermediate order as the spatial correlation in the density fluctuations. Later Zemike [24] has applied this idea to the lattice systems. 

---