Brownian coagulation model

Sitarski and Seinfeld (6) were the first to provide a theoretical basis for Fuch s semi-empirical formula, by solving the Fokker-Plank equation by means of Grad s (7) 13-moment method. Their solution was further improved by Mork et al. (8). The Brownian coagulation coefficient predicted by these models agrees fairly well with the Fuchs interpolation formula. However, the model does not predict the proper free molecular limit. The validity of the Fuchs semiempirical formula was further reinforced, by the Monte Carlo simulations of Brownian coagulation, by Nowakowski and Sitarski (9). 

A model for Brownian coagulation of equal-sized electrically neutral aerosol particles is proposed. The model accounts for the van der Waals attraction and Born repulsion in the calculation of the rate of collisions and subsequent coagulation. In this model, the relative motion between two particles is considered to be free molecular in the neighborhood of the sphere of influence. The thickness of this region is taken to be equal to the correlation length of the relative Brownian motion. The relative motion of the particles outside this region is described... 

Fig. 10.4 The model of Brownian coagulation of particles according to Smoluchowski.

In a simple model, when the considered volume is filled with identical particles engaged in the process of Brownian coagulation, and the suspension is assumed to remain monodisperse, the change of number concentration of particles with time is described by the balance equation for the particles ... 

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... 

The equation derived by Troelstra and Kruyt is only valid for coagulating dispersions of colloids smaller than a certain maximum diameter given by the Rayleigh condition, d 0.10 A0. Equation 4 applies in cases where particles are transported solely by Brownian motion. Furthermore, the kinetic model (Equations 2 and 3) has been derived under the assumption that the collision efficiency factor does not change with time. In the case of some partially destabilized dispersions one observes a decrease in the collision efficiency factor with time which presumably results from the increase of a certain energy barrier as the size of the agglomerates becomes larger. 

As pointed out earlier, the present treatment attempts to clarify the connection between the sticking probability and the mutual forces of interaction between particles. The van der Waals attraction and Bom repulsion forces are included in the analysis of the relative motion between two electrically neutral aerosol particles. The overall interaction potential between two particles is calculated through the integration of the intermolecular potential, modelled as the Lennard-Jones 6-12 potential, under the assumption of pairwise additivity. The expression for the overall interaction potential in terms of the Hamaker constant and the molecular diameter can be found in Appendix I of (1). The Brownian motions of the two particles are no longer independent because of the interaction force between the two. It is, therefore, necessary to describe the relative motion between the two particles in order to predict the rate of collision and of subsequent 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. ... 

Aerosols are unstable with respect to coagulation. The reduction in surface area that accompanie.s coalescence corresponds to a reduction in the Gibbs free energy under conditions of constant temperature and pressure. The prediction of aerosol coagulation rates is a two-step process. The first is the derivation of a mathematical expression that keeps count of particle collisions as a function of particle size it incorporates a general expression for tlie collision frequency function. An expression for the collision frequency based on a physical model is then introduced into the equation Chat keep.s count of collisions. The collision mechanisms include Brownian motion, laminar shear, and turbulence. There may be interacting force fields between the particles. The processes are basically nonlinear, and this lead.s to formidable difficulties in the mathematical theory. 

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). 

At lowest shear stresses the behavior of bentonite clays may be the same as that of a solid-like system with high viscosity, which is consistent with the Kelvin model and corresponds to region I. The investigation of relaxation properties of coagulation structures forming in these moderately concentrated dispersions of bentonite clays revealed the existence of an elastic aftereffect at low shear stresses. This aftereffect is related to mutual coorientation of anisometric particles that are capable of taking part in rotational Brownian motion without any rupture of contacts. Consequently, the nature of elastic aftereffect is entropic. In such systems high viscosities are related... 

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... 

Pnueli D., Gutfinger C., Fidrman M., A turbulent-brownian Model for Aerosol Coagulation, Aerosol Sd. Technol, 1991, Vol. 14, p. 201-209. 

Another aspect of ionic liquids is that the thermal movement of the colloidal nanop>artides is suppressed due to the high viscosity of the surrounding medium minimizing the probabihty of close contacts. Let us analyse this aspect in further detail. Let us assume that nanopartides with size a = 3 nm have just formed and are dispersed at room temperature (20°C) in a medium with viscosity 77 at a volume fraction q) of 0.01. To assess qualitatively their half-life time, we assume here a very simple model of rapid random coagulation, where every collision of two nanopartides immediately leads to coagulation and, in consequence, agglomeration. The number of collisions v that one nanoparticle experiences per time unit with other nanopartides can be expressed by eq. 4,( 1 which can be obtained based on the Einstein-Smoluchowski 1 7, es] formalism of Brownian motion of colloidal particles. 

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