Fluctuations Einstein-Smoluchowski theory

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. 

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. 

Fluctuation theory by Smoluchowski and Einstein see the first footnote of Part I. 

Smoluchowski (1908), Einstein (1910), Ornstein Zernike (1914, 1918). In a textbook on scattering HIGGINS BENOIT ([136], Sect. 7.6) consider the fluctuation theory from a different point of view. 

Investigation of the Brownian motion of dispersed particles made it possible to experimentally verify the theory of fluctuations, also formulated by Einstein and Smoluchowski. Svedberg s observations of Brownian motion indicated that the number of particles confined within a small... 

The theory of thermodynamic fluctuations in the context of light scattering was introduced first by Smoluchowski and Einstein [12, 13]. Einstein considered that the scattering of light in a solution arises from local density and concentration fluctuations. However, it was only till the late 40s and early 50s that Brinkman and Hermans [14], Stockmayer [15], and Kirkwood and Goldberg [16] adopted this approach in the context of light scattering by poiymer solutions. Here, we summarize only the basic results for a binary mixture. 

The theory that describes these fluctuations was provided by Einstein and Smoluchowski. The nucleation rate depends on the interfacial energy and the supersaturation, which influence the differences in the chemical potential of both phases. This theory employs the fact that the nucleation rate is the same in the whole volume under consideration, i.e., so-called homogenous nucleation (Kondepudi 2008). The kinetic factor of the process can be expressed by (20.1) the nucleation rate (ki) that affects (20.2) cluster growth rate (kj) (Figure 20.4) ... 

Marian von Smoluchowski (1872-1917). .. was a Polish physicist whose research on discrete state matter is still highly valued in modem science. He is particularly acknowledged for his theory on Brownian motion, which he developed independently of Einstein and which laid the foundation for the theory of stochastic processes. A similar rank is deserved by his discovery of density fluctuations in liquids and gases and their relevance for macroscopic scattering— most prominently explained by the phenomenon of critical opalescence. Both works proved veiy influential for the understanding of colloidal suspensions. Furthermore, he did pioneering work on the quantification of particle aggregation as well as in the field of electrokinetic phenomena. 

The fluctuation theories of Einstein and Smoluchowski lead to the following general expression... 

According to the Smoluchowski-Einstein theory, the free energy (G) of a system increases spontaneously by a value (SG) in response to a fluctuation of any of its variables or characteristics. For the variable z of a system undergoing a fluctuation Sz, 8G is expressed as... 

Smoluchowski-Einstein theory of fluctuations gives an expression for the Rayleigh ratio identical to that previously established [equation (6.12)] (see Appendix). This means that the intensity scattered by a collection of Nt particles of gas is merely the sum of the intensity scattered by individual molecules, with the scattering phenomenon resulting from the random presence/movement of a gas molecule. 

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