The Kinetics of Coagulation

In the general case, trajectories of particles entrained by the flow are determined from the equation of motion, which can be written as 

Here it is supposed that particles motion obeys the Stokes law, and U is the flow velocity of the carrying liquid. 

The dimensionless parameter S is referred to as the Stokes number. It characterizes the measure of particle s inertia. At S 1, the particle s inertia is small, and it follows from (10.119) that the trajectory of the particle is dose to the streamline. At S 1, one finds from (10.119) that d rfdt x 0, i.e. trajectories are the straight lines. 

Having found the law describing the flow around the body, we can obtain from (10.119) the family of particle trajectories, and then determine the limiting trajectory and cross section of particles collisions with the body. 

A notable feature of inertial deposition of partides on the body is the existence of the Stokes critical number such that at S So- the capture of particles by the body is impossible in other words, a particle s inertia is not sufficient to overcome particle s entrainment by the flow of carrying liquid [68]. The values of Scr for the cylinder are as follows 0.0625 for potential flow, approximate solution [58] 0.1 for potential flow, numerical solution. If we take into account formation of the boundary layer at the surface, the resulting value will be So- 0.25 [68]. 

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

Colloidal solutions are characterized by the degree of stability or instability. This is related to the fact that both kinds of properties in everyday phenomena need to be understood. The kinetics of coagulation is studied using different methods. The number of particles, Np, at a given time is dependent on the diffusion-controlled process. The rate is given by... 

W is the stability ratio, i. e. the factor by which the coagulation velocity is reduced due to interparticle repulsion. It is related to the height of the energy barrier. When coagulation is fast, W = 1. Various aspects of slow coagulation are still not fully understood (O Melia, 1987). Several theories of the kinetics of coagulation are discussed by Grand et al. (2001). 

Liang, L. (1988) Effects of surface chemistry on the kinetics of coagulation of submicron iron oxide particles (a-Fe203) in water. Ph.D. Thesis Caltech. USA... 

The first group, consisting of Sections 2.2-2.4, covers sedimentation. After some preliminaries, we discuss Stokes s law, a hydrodynamic equation that will appear again when we discuss electrokinetic phenomena in Chapter 12 and the kinetics of coagulation in Chapter 13. Stokes s law is a key relationship in understanding the rate of sedimentation and is used in the derivation of the sedimentation equation for spherical particles. Following this, the equation for the sedimentation coefficient, a... 

EXAMPLE 13.4 Change of Stability Ratio with Ionic Concentration. Colloidal gold stabilized by citrate ions and having a mean particle radius of 103 A was coagulated by the addition of NaCI04. The kinetics of coagulation were studied colorimetrically and the stability ratio W for different NaCI04 concentrations was determined (Enustun and Turkevich 1963) ... 

The Role of Adsorption of Hydrolyzed Aluminum in the Kinetics of Coagulation... 

Technically, this result follows only if kmn < A(m + n), where A is a positive constant. If this condition is not met, cluster growth to a unit comprising all the primary particles can occur after a finite time interval (cf. Eq. 6.20b), corresponding to gel formation and a nonconstant M,. For a discussion of this point, see F. Leyvraz, Critical exponents in the Smoluchowski equations of coagulation, pp. 201-204 in F. Family and D. P. Landau, op. cit.7 and F. Leyvraz and H. R. Tschudi, Singularities in the kinetics of coagulation processes, J. Phys. A 14 3389 (1981). 

Smoluchowski (16) has treated the kinetics of coagulation assuming that the particles which collide during their Brownian motion coagulate. The rate of collision was computed assuming quasi-steady dif-... 

Keeping the notation similar to that used in the previous sections, although the quantities involved have somewhat different meanings, it is easy to extend some of the previous results to the kinetics of coagulation. The conservation equation can be written as... 

Similarly, a weak repulsive steric interaction in combination with a strong van der Waals interaction will result in flocculation. The kinetics of coagulation and flocculation in the presence of a total interaction energy profile either electrostatic or steric is discussed next. 

Below, in Sections 5.2 and 5.3, we consider effects related to the surface tension of surfactant solution and capillarity. In Section 5.4 we present a review of the surface forces due to intermo-lecular interactions. In Section 5.5 we describe the hydrodynamic interparticle forces originating from the effects of bulk and surface viscosity and related to surfactant diffusion. Section 5.6 is devoted to the kinetics of coagulation in dispersions. Section 5.7 regards foams containing oil drops and solid particulates in relation to the antifoaming mechanisms and the exhaustion of antifoams. Finally, Sections 5.8 and 5.9 address the electrokinetic and optical properties of dispersions. 

The discussion of the preceding sections leads immediately to a consideration of the kinetics of coagulation. Two cases are important, depending on whether or not coagulation is inhibited by an energy barrier. 

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. 

Kaeding U. (1998), The kinetics of coagulation of natural organic matter with alum, CRC for Water Quality and Treatment, Postgraduate Student Conference 1998,109-114. 

Various experimental methods for monitoring the kinetics of coagulation in emulsions have been developed, such as the electroacoustic method (167), direet video-enhaneed microscopic investigation (168), and ultrasonie attenuation spectroscopy (169). 

An electroacoustic method based on the application of a sonic field and detection of the electric signal proved to be a reliable technique for measuring the kinetics of coagulation/coalesence process of colloidal systems in non-polar and non-transparent media like water-in-crude oil emulsion. 

The kinetics of coagulation were first analyzed by Smoluchowski [32], who assumed that no repulsive barrier was present, and that aggregation occurred by the attachment of single particles to clusters (ignoring cluster-cluster aggregation). The theory was further developed by Fuchs [33], who showed that a repulsive potential F(r) would reduce the coagulation rate by the faetor JV, called the stability ratio ... 

Some general rules apply to the kinetics of coagulation of colloids, which can also be used in the specific case of emulsions discussed in Chapter 12. 

The stability of colloids is not a question of quality to be answered simply by yes or but a question of quantity A measure of the stability is found in the veiedQ of coagulation, which may vary from very quick (flocculation within a few seconds) to immeasurably slow Consequently for a complete discussion of the stability the kinetics of coagulation have to be taken into account ... 

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