Smoluchowski

The simple treatment of this and of other electrokinetic effects was greatly clarified by Smoluchowski [69] for electroosmosis it is as follows. The volume flow V (in cm /sec) for a tube of radius r is given by applying the linear velocity V to the body of liquid in the tube... 

The situation in electroosmosis may be reversed when the solution is caused to flow down the tube, and an induced potential, the streaming potential, is measured. The derivation, again due to Smoluchowski [69], begins with the assumption of Poiseuille flow such that the velocity at a radius x from the center of the tube is... 

There are two approaches to the kinetics of emulsion flocculation. The first stems from a relationship due to Smoluchowski [52] for the rate of diffusional encounters, or flux ... 

Using W2 = 17jP2, (A3.2.81 and (A3.2.9) may be used to satisfy the Smoluchowski equation, (A3.2.2). another necessary property for a stationary process. Thus u(t) is an example of a stationary Gaussian-Markov... 

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... 

Diflfiision-controlled reactions between ions in solution are strongly influenced by the Coulomb interaction accelerating or retarding ion diffiision. In this case, die dififiision equation for p concerning motion of one reactant about the other stationary reactant, the Debye-Smoluchowski equation. 

This ensures the correct connection between the one-dimensional Kramers model in the regime of large friction and multidimensional imimolecular rate theory in that of low friction, where Kramers model is known to be incorrect as it is restricted to the energy diflfiision limit. For low damping, equation (A3.6.29) reduces to the Lindemann-Flinshelwood expression, while in the case of very large damping, it attains the Smoluchowski limit... 

In the Smoluchowski limit, one usually assumes that the Stokes-Einstein relation (Dq//r7)a = C holds, which fonns the basis of taking the solvent viscosity as a measure for the zero-frequency friction coefficient appearing in Kramers expressions. Here C is a constant whose exact value depends on the type of boundary conditions used in deriving Stokes law. It follows that the diffiision coefficient ratio is given by ... 

Non-parabolic barrier tops cause the prefactor to become temperahire dependent [48]. In the Smoluchowski... 

In the Smoluchowski limit the reaction is by definition the slow coordinate, such that y(kr) y(0) = dL yW Lrand Agii fO). Though the time-dependent friction... 

Smoluchowski Mv 1918 Versuch einer mathematischen Theorie der Koagulationskinetik kolloider Ldsungen Z. Rhys. Ohem. 92 129-39... 

Smoluchowski Mv 1915 Uber Brownsche Molekularbewegung unter Einwirkung auperer Krafte und deren Zusammenhang mit der verallgemeinerten Diffusionsgleichung Ann. Rhys. 48 1103-12... 

For a more complete understanding of colloid stability, we need to address the kinetics of aggregation. The theory discussed here was developed to describe coagulation of charged colloids, but it does apply to other cases as well. First, we consider the case of so-called rapid coagulation, which means that two particles will aggregate as soon as they meet (at high salt concentration, for instance). This was considered by von Smoluchowski 1561 here we follow [39, 57]. 

In a homogeneous system, tire rate of mixing is governed by Smoluchowski s equations [77], according to which tire diffusion-limited association rate of S and L (equation (C2.14.13)), supposed uncharged, equals tliat of tire flux and is... 

In HMC the momenta are constantly being refreshed with the consequence that the accompanying dynamics will generate a spatial diffusion process superposed on the ini rtial dynamics, as in BGK or Smoluchowski dynamics. It is well known from the theory of barrier crossing that this added spatial... 

The flocculation rate is deterrnined from the Smoluchowski rate law which states that the rate is proportional to the square of the particle concentration by number inversely proportional to the fluid viscosity, and independent of particle size. 

There are two general theories of the stabUity of lyophobic coUoids, or, more precisely, two general mechanisms controlling the dispersion and flocculation of these coUoids. Both theories regard adsorption of dissolved species as a key process in stabilization. However, one theory is based on a consideration of ionic forces near the interface, whereas the other is based on steric forces. The two theories complement each other and are in no sense contradictory. In some systems, one mechanism may be predominant, and in others both mechanisms may operate simultaneously. The fundamental kinetic considerations common to both theories are based on Smoluchowski s classical theory of the coagulation of coUoids. 

As the particles in a coUoidal dispersion diffuse, they coUide with one another. In the simplest case, every coUision between two particles results in the formation of one agglomerated particle,ie, there is no energy barrier to agglomeration. Applying Smoluchowski s theory to this system, the half-life, ie, the time for the number of particles to become halved, is expressed as foUows, where Tj is the viscosity of the medium, k Boltzmann s constant T temperature and A/q is the initial number of particles. 

The size-dependent agglomeration kernels suggested by both Smoluchowski and Thompson fit the experimental data very well. For the case of a size-independent agglomeration kernel and the estimation without disruption (only nucleation, growth and agglomeration), the least square fits substantially deviate from the experimental data (not shown). For this reason, further investigations are carried out with the theoretically based size-dependent kernel suggested by Smoluchowski, which fitted the data best ... 

For stirrer speeds of 4.2, 8.4, 16.7, 25 and 33.4Fiz, agglomeration kernels obtained in this study vary from 0.01 to 183 s . Unfortunately, no other measured data for agglomeration of calcium oxalate analysed using Smoluchowski s kernel were found in the literature. The corresponding values reported by Wojcik and Jones (1997) for calcium carbonate, however, cover a range from 0.4 to 16.8s-. ... 

According to Smoluchowski s theory (equation 6.53), the agglomeration rate increases proportional with the fluid shear rate 7... 

Smoluchowski, M.V., 1916. Drei Vortrage iiber Diffusion, Brownsche Molekularbewe-gung mid Koagulation von Kolloidteilchen. Physik Zeitung, XVII, 557-599. 

Smoluchowski, M.V., 1917. Mathematical theory of the kinetics of coagulation of colloidal systems. Zeitschrift fur Physikalische Chemie, 92, 129-168. 

Using a Smoluchowski rate-equation approach [71], we can write a system of nonlinear differential equations... 

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