Brownian motion theory, Smoluchowski

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... 

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... 

Studies on orthokinetic flocculation (shear flow dominating over Brownian motion) show a more ambiguous picture. Both rate increases (9,10) and decreases (11,12) compared with orthokinetic coagulation have been observed. Gregory (12) treated polymer adsorption as a collision process and used Smoluchowski theory to predict that the adsorption step may become rate limiting in orthokinetic flocculation. Qualitative evidence to this effect was found for flocculation of polystyrene latex, particle diameter 1.68 pm, in laminar tube flow. Furthermore, pretreatment of half of the latex with polymer resulted in collision efficiencies that were more than twice as high as for coagulation. 

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)]. 

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... 

In the present section, it is demonstrated how the linear response of an assembly of noninteracting polar Brownian particles to a small external field F applied parallel and perpendicular to the bias field Fo may be calculated in the context of the fractional noninertial rotational diffusion in the same manner as normal rotational diffusion [8]. In order to carry out the calculation, it is assumed that the rotational Brownian motion of a particle may be described by a fractional noninertial Fokker-Planck (Smoluchowski) equation, in which the inertial effects are neglected. Both exact and approximate solutions of this equation are presented. We shall demonstrate that the characteristic times of the normal diffusion process, namely, the integral and effective relaxation times obtained in Refs. 8, 65, and 67, allow one to evaluate the dielectric response for anomalous diffusion. Moreover, these characteristic times yield a simple analytical equation for the complex dielectric susceptibility tensor describing the anomalous relaxation of the system. The exact solution of the problem reduces to the solution of the infinite hierarchies of differential-recurrence equations for the corresponding relaxation functions. The longitudinal and transverse components of the susceptibility tensor may be calculated exactly from the Laplace transform of these relaxation functions using linear response theory [72]. 

There exists an extensive literature on the theory of coagulation (Fuchs, 1964 Zebel, 1966 Hidy and Brock, 1970 Twomey, 1977), and we can treat here only the most salient features. In the absence of external forces, the aerosol particles undergo collisions with each other due to their thermal (Brownian) motion. The mathematical description of thermal coagulation goes back to the classical work of Smoluchowski (1918) on hydrosols. Application to aerosols seems to have been made first by Whitlaw-Gray and Patterson (1932). Let dN, = f(r,) dr, and dN2=f(r2) dr2 describe the number densities of particles in the size intervals r, + dr, and r2+dr2,... 

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... 

Fortelny et al. [1988, 1990] assumed that the Brownian motion is the principal driving force for coalescence in polymer blends. Applying Smoluchowski s theory, the authors obtained ... 

We begin by examining the rate of collision of suspended spherical particles in a static fluid due to Brownian motion. This theory was first put forward by the great Polish physicist M. Von Smoluchowski, to whom we have often referred. In consequence of the equivalence between diffusion and Brownian motion, we consider the relative motion between the particles as a diffusion process. The particles are assumed to be in sufficiently dilute concentration that only binary encounters need be considered. To further simplify the calculation, we consider the suspension to be made up of only two different-sized spherical particles, one of radius a, and the other of radius... 

About the turn of the century cuid shortly thereafter, certain developments in mathematical physics and in physical chemistry were realized which were to prove important in the theory of mass and charge transport in solids, later. Einsteinand Smoluchowski( ) initiated the modern theory of Brownian motion by idealizing it as a problem in random flights. Then some seventeen years or so later, Joffee A proposed that interstitial defects could form inside the lattice of ionic crystals and play a role in electrical conductivity. The first tenable model for ionic conductivity was proposed by Frenkel, who recognized that vaccin-cies and interstitials could form internally to account for ion movement. 

We must now relate the energetics to the kinetics. The theory of rapid flocculation was first proposed by Smoluchowski , who treated the problem as one of diffusion (Brownian motion) of the spherical particles of an initially monodisperse dispersion, with every collision, in the absence of a repulsive force, leading to a permanent contact. In general, we may express the rate of flocculation, i.e. the rate of decrease of the total number of particles, as... 

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. ... 

Equation (8.1) is correct only for 1. To discuss the general case, we have to study the Smoluchowski equation for the rotational Brownian motion. This equation can be derived straightforwardly according to the Kirkwood theory described in Section 3.8. Such a derivation is given in Appendix 8.1. Here we derive it by an elementary method to clarify the underlying physics. 

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. 

Once formed, particles in solution interact with each other because of Brownian motion. The theory of rapid flocculation was first proposed by Smoluchowski,... 

Marian Smoluchowski (1872-1919) a Polish scientist and pioneer of statistical physics. He described Brownian motion, worked on the kinetic theory at the same time as Albert Einstein, and presented an equation that became the basis of the theory of stochastic processes. 

In this section the foundations of the theory underlying chemical kinetics are presented. Based on the diffusion equation to describe Brownian motion together with Smoluchowski s theory [ 1, 2], a thorough derivation of the bulk reaction rate constant for neutral species for both diffusion and partially diffusion controlled reactions is presented. This theory is then extended for charged species in subsequent sections. 

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 ... 

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