Brownian coagulation diffusion

Consider particles of radius a Ao and assume that in the course of their motion in the liquid, they are completely entrained by turbulent pulsations that play the basic role in the mechanism of mutual approach of suspended particles. Then it can be assumed that particle transport is performed via isotropic turbulence. Since particles move chaotically in the liquid volume, their motion is similar to Brownian one and can be considered as diffusion with some effective factor of turbulent diffusion Dr. In the same manner as in the case of Brownian coagulation, it is possible to consider the diffusion flux of particles of radius U2 toward the test particle of radius Uj. The distribution of particles U2 is characterized by the stationary diffusion equation... 

In slow coagulation, particles have to diffuse over an energy barrier (see the previous section) in order to aggregate. As a result, not all Brownian particle encounters result in aggregation. This is expressed using the stability ratio IV, defined as... 

Smoluchowski, M.V., 1916. Three lectures on diffusion. Brownian movement and coagulation of colloidal systems. Physik Zeitung, 17, 557. 

The polymer radius has to be larger than 80% of the particle radius to avoid adsorption limitation under orthokinetic conditions. As a rule of thumb a particle diameter of about 1 pm marks the transition between perikinetic and orthokinetic coagulation (and flocculation). The effective size of a polymeric flocculant must clearly be very large to avoid adsorption limitation. However, if the polymer is sufficiently small, the Brownian diffusion rate may be fast enough to prevent adsorption limitation. For example, if the particle radius is 0.535 pm and the shear rate is 1800 s-, then tAp due to Brownian motion will be shorter than t 0 for r < 0.001, i.e., for a polymer with a... 

A probabilistic kinetic model describing the rapid coagulation or aggregation of small spheres that make contact with each other as a consequence of Brownian motion. Smoluchowski recognized that the likelihood of a particle (radius = ri) hitting another particle (radius = T2 concentration = C2) within a time interval (dt) equals the diffusional flux (dC2ldp)p=R into a sphere of radius i i2, equal to (ri + r2). The effective diffusion coefficient Di2 was taken to be the sum of the diffusion coefficients... 

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)]. 

The second method for aerosol coagulation in turbulent flows arises because of inertial differences between particles of different sizes. The particles accelerate to different velocities by the turbulence depending on their size, and they may then collide with each other. This mechanism is unimportant for a monodisperse aerosol. For a polydisperse aerosol of unspecified size distribution, Levich (1962) has shown that the agglomeration rate is proportional to the basic velocity of the turbulent flow raised to the 9/4 power, indicating that the agglomeration rate increases very rapidly with the turbulent velocity. Since very small particles are rapidly accelerated, this mechanism also decreases in importance as the particle size becomes very small, being most important for particles whose sizes exceed 10-6 to 10"4 cm in diameter. In all cases brownian diffusion predominates when particles are less than 10-6 cm in diameter. 

Frictional electrification, 183-184 Frictional resistance, 50 Friedlander, S. K, 312, 316 Fuchs, N. A., 61 and brownian rotation, 138 and coagulation, 313-314 and diffusion charging, 188 and diffusion and collisions, 304 and equilibrium charge distributions, 204-205... 

If the energy barrier to aggregation is removed (e.g., by adding excess electrolyte) then aggregation is diffusion controlled only Brownian motion of independent droplets or particles is present. For a monodisperse suspension of spheres, Smoluchowski developed an equation for this rapid coagulation ... 

The difference between elastic and "quasielastic" measurements is that in the latter, small changes in the frequency due to the translational ("Brownian") movement of the scattering particles are also measured. The broadness of the intensity distribution of the emitted light for frequencies around the primary monocluomatic beam frequency is directly related to the diffusion coefficient of the particles, which can then be related to the hydrodynamic radius if a model for the particle shape is available Dynamic light scattering can thus be used to follow the kinetics of particle coagulation by following the decrease in diffusion coefficient as the particle size increases. ... 

Particles smaller than about I /im collide as a result of their Brownian motion most of the theoretical and experimental studies of coagulation have been concerned with (his mechanism. For particles much larger than the mean free path of the ga.s. there is experimental evidence that the collision process is dilTusion-limited. Consider a sphere of radius at, fixed at (he origin of the coordinate system in an infinite medium containing suspended spheres of radius Particles of radius aj are in Brownian motion and diffuse to... 

In the approach adopted in my first edition, the derivation and use of the general dynamic equation for the particle size distribution played a central role. This special form of a population balance equation incorporated the Smoluchowski theory of coagulation and gas-to-panicle conversion through a Liouville term with a set of special growth laws coagulation and gas-to-particle conversion are processes that take place within an elemental gas volume. Brownian diffusion and external force fields transport particles across the boundaries of the elemental volume. A major limitation on the formulation was the assumption that the panicles were liquid droplets that coalesced instantaneously after collision. 

The interaction in a two-body collision in a dilute suspension has been expanded to provide a useful and quantitative understanding of the aggregation and sedimentation of particulate matter in a lake. In this view, Brownian diffusion, fluid shear, and differential sedimentation provide contact opportunities that can change sedimentation processes in a lake, particularly when solution conditions are such that the particles attach readily as they do in Lake Zurich [high cc(i,j)exp]. Coagulation provides a conceptual framework that connects model predictions with field observations of particle concentrations and size distributions in lake waters and sediment traps, laboratory determinations of attachment probabilities, and measurements of the composition and fluxes of sedimenting materials (Weilenmann et al., 1989). 

M. V. Smoluchowski, Phys. Z., 17, 557 (1916). Three Lectures on Diffusion, Brownian Movement, and Coagulation of Colloidal Particles. 

The electrostatic interaction between diffuse layers of ions surrounding particles is one of the most thoroughly theoretically developed factors of colloid stability. The theory of electrostatic factor is, essentially, the basis for the quantitative description of coagulation by electrolytes. This theory was developed in the Soviet Union by B.V. Derjaguin and L.D. Landau in 1935 -1941, and independently by the Dutch scientists E.Verwey and T. Overbeek, and is presently known by the initial letters of their names as the DL VO theory [44,45]. The DLVO theory is based on comparison of molecular interaction between the dispersed particles in dispersion medium and the electrostatic interaction between diffuse layers of ions, with Brownian motion of particles taken into account (in the simplest version of theory this is done on a qualitative level). 

Let us now examine the theory of coagulation in a greater detail. In agreement with the theory of random processes, one of the two particles involved in Brownian motion may be viewed as stationary. It is thus possible for the one to bind the coordinate system origin to this stationary n-dimensional particle and say that the diffusion coefficient of the second particles is given by the mutual diffusion coefficient, Dmn = Dm + Dn (see... 

The physical meaning of this can be explained as follows. As we have seen the diffusion equations can be applied to Brownian motion only for time intervals that are large compared to the relaxation time, r, of the particles or for distances that are large compared to the aerosol mean free path kp. Diffusion equations cannot describe the motion of particles inside a layer of thickness kp adjacent to an absorbing wall. If the size of the absorbing sphere is comparable to kp, this layer has a substantial effect on the kinetics of coagulation. 

The adopted diffusion model of Brownian motion allows to us consider the collision frequency of particles of radius U2 with the test particle of radius ai as a diffusion flux of particles U2 toward the particle a. Assume the surface of the particle ai to be ideally absorbing. It means that as soon as the particle U2 will come into contact with the particle ai, it will be absorbed by this particle. In other words, absorption occurs as soon as the center of the particle U2 reaches the surface of a sphere of radius Rc = a U2. The quantity Rc is called the coagulation radius. Hence, the concentration of particles a2 should be equal to zero at... 

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