Polydisperse Coagulation

In the previous sections, we focused our attention on calculation of the instantaneous coagulation rate between two monodisperse aerosol populations. In this section, we will develop the overall expression describing the evolution of a polydisperse coagulating aerosol population. A good place to start is with a discrete aerosol distribution. 

These compressed air nebulizers produce polydisperse aerosols. After the aerosol is produced, the size distribution may change due to evaporation of liquid from the droplets. In addition, the particles may be electrically charged due to an ion imbalance in the droplets as they form if such charges become further concentrated due to evaporation, the particle may break up into smaller particles. Thus electrical neutralization of the aerosol, for example, by exposure to a radioactive source, is usually necessary to prevent electrostatic effects from dominating the particle motion, coagulation, and other behavior. 

All these methods generally give (< 1 /xm) polydisperse aerosols of the solid particles and, unless rapid air dilution is provided, coagulation leads to large agglomerates of the small primary particles. 

The DLS measurements can also be used in more complicated situations, for example, (a) when interparticle interactions are important, (b) for dispersions with particles of other shapes, (c) for monitoring coagulation, and (d) when the dispersion is polydisperse. In all these cases, a significant amount of modeling is often necessary to interpret the measured autocorrelation function, and we do not consider them here. Instead, we restrict our attention to a brief discussion of item (d) above, namely, polydisperse systems, since it is concerned with a problem of more routine interest. 

What are the factors that are relevant for extending the theories of coagulation presented here to (a) polydisperse colloids, (b) nonspherical colloids, and (c) conditions for which fluid flow is important ... 

According to this kinetic model the collision efficiency factor p can be evaluated from experimentally determined coagulation rate constants (Equation 2) when the transport parameters, KBT, rj are known (Equation 3). It has been shown recently that more complex rate laws, similarly corresponding to second order reactions, can be derived for the coagulation rate of polydisperse suspensions. When used to describe only the effects in the total number of particles of a heterodisperse suspension, Equations 2 and 3 are valid approximations (4). 

Experimental data regarding the coagulation coefficients of Pt aerosols, measured by Nolan and Keenan (15) are compared with different models in Table IV. The corrections for polydispersity in the above data, as reported by Mercer (16), are accounted for in the experimental values of the coagulation... 

In this system using a polystyrene containing block copolymer, the polystyrene segment should readily partition into the lipophilic polystyrene particle core while the poly(FOA) or PDMS block is solubilized in the CO2 continuous phase to provide steric stabilization and prevent coagulation. In comparison of the polystyrene-b-poly(FOA) diblock copolymers to the polystyrene-b-PDMS diblock copolymers, it was found that the use of a polystyrene-b-PDMS stabilizer gives much more monodisperse particles. This most likely arises from the synthetic technique employed in the surfactant synthesis. The blocks in the polystyrene-b-PDMS block copolymers have a much narrower polydispersity than the blocks in the polystyrene-b-poly(FOA) block copolymers. It was noted that the particles obtained in... 

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. 

Point charge, field strength of, 210 Poiseuille flow, 156-157 Poisson s equation, 212, 215-216 Polarization force, 218 Polarization of light, 290 incident, 282 scattered, 284 Pollen spores, 319-320 Pollution, air, 225 Polonium-210 particles, 140 Polydisperse aerosols, 3,13 and coagulation, 308 concentration of, 88 and scattering, 289-294 size of, 4... 

Equation 18.12 shows that the inverse of the concentration at any time is a linear function of time, the slope of the line being determined by the coagulation constant. Experimental data from both mono-disperse and polydisperse aerosols follow this general form, at least initially. As will be discussed later, the coagulation constant may be appreciably larger than the theoretical value. If th is defined as the half-value time, i.e., the time in which the concentration decreases by a factor of 2, then... 

The most important physical characteristic of polydisperse sy.stems is their particle size distribution. This distribution can take two forms The first is the discrere dtstrihuiion in which only certain "allowed particle sizes are considered. Consider a suspension that consists of aggregates of unitary particles formed by coagulation. All particles will then be composed of integral numbers of these unitary particles, and the size distribution can be... 

To simplify the calculations, it is assumed that the concentrations, mobilities, and other properties of the positive and negative ions are equal and that the concentrations of the tons and charged particles have reached a steady slate. We consider a group of particles of uniform size no coagulation occurs,. so a polydisperse aerosol can be treated as a set of uncoupled monodisperse particles. The rates at which ions of both signs attach to particles are assumed to be independent ol each other. In the steady state, ions are generated and destroyed at the same rate by attachment to particles. Calculations indicate that ion recombination is not an important mechanism for ion loss in the atmosphere (Bricard and Pradel, Ii966). 

Several other chapters have been substantially rewritten to reflect the sharpened focus on aerosol dynamics. For example, the chapter on optical properties has been expanded to include more applications to polydisperse aerosols. It help.s support the chapter that follows on experimental methods in which coverage of instrumentation for rapid size distribution measurements has been augmented. Methods for the rapid on-line measurement of aerosol chemical characteristics are discussed in the chapters on optical properties and experimental methods. This chapter has been strongly influenced by the work of the Minnesota group (B. Y. H. Liu, D. Y. H, Pui, P, McMuny, and their colleagues and students) who continue to invent and perfect advanced aerosol instrumentation. Discussions of the effects of turbulence have been substantially expanded in chapters on coagulation and gas-to-particle conversion. 

The investigations showed that the given block- copolymers are characterised by a sufficiently low polydispersity. It is shown that the threshold of the coagulation BSN-7D increases with the increase of the content of the chloride-bearing bisphenol, which is apparently connected with the polarity >C=CCl2 - group, and thus, a better solubility of these polymers. 

It is concluded on the basis of equation [4.S] that the intensity of the particle loss due to the thermal coagulation is directly proportional to square of the particle concentration, while the coagulation efficiency increases with decreasing particle radius. This means that the coagulation of small particles at a high concentration is a very rapid process. Equation [4.S] is valid only for monodisperse aerosols, i.e. aerosols composed of particles of uniform size. However, the same qualitative conclusion can also be drawn in the case of polydisperse systems. 

If the polydispersity of bubbles generated in air-dissolved flotation or electroflotation is high, there is no need for additional introduction of centimicron bubbles. Optimal flow of two-stage flotation corresponds to the maximum attainable degree of monodispersity of bubbles. In this case the ratio between volume fractions of micro- and macrobubbles and collision efficiencies of the processes of particle capture by small bubbles and bubble coagulation must be such that the particle capture process outweighs the process of coalescence. 

As earlier, consider ti to be the characteristic coagulation time of a polydisperse ensemble of drops, caused by the mechanism of turbulent diffusion due to the forces of hydrodynamic and molecular interactions. This time should be estimated. For typical values of the flow, Pq = 40 kg m , 2o = 5 x 10 m, Pq = 1.2 X 10 Pa-s, W = 5 X 10 m /m and distribution parameters of = 10 m, k = 3, one obtains 1/ti = 0.257 s. Thus, a twofold increase in drop radius occurs in a time t of 7 s. This time is almost two orders of magnitude higher than for a monodisperse distribution without regard to hydrodynamic and molecular forces. Such a big difference in characteristic times is undoubtedly caused not by taking into account the polydispersivity of the distribution, but as a result of considering the interaction forces. 

The solution obtained corresponds to a monodisperse distribution of drops without regard for coagulation. We consider a possible solution taking into account a polydisperse distribution and the coagulation of drops. 

We consider now in tandem the taking into account of the polydispersiveness of drop distribution and the probability of drop coagulation. 

Simultaneously taking into account polydispersiveness and the coagulation of the drops, the following system of equations is obtained ... 

Coagulation rate are proportional to the square of number of concentration (weak dependence forces on the particle size). Nanoparticles in the gas phase always agglomerate or may be redispersed with polydispersity. Hard particles (partially sintered) can agglomerate by dispersing the particles due to size affections or by controlling the reaction s nucleation at room temperature. 

Tuorila, P, The rapid and slow coagulation of polydispersed systems gold and alumina dispersions, Kolloidchem. Beihefte, 22, 191, 1926. 

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