Kinetics of coagulation

Colloidal particles exhibit random Brownian motion as a result of which a net diffusive flux can be generated. If the thermodynamic force on each particle which causes such a flux is the gradient of chemical potential it must, under steady 

Equation 3.72 is known as the Stokes-Einstein equation, which is reconsidracd in detail in Chapter 8. 

If external forces act on the particles, as electrical fields will for charged colloidal particles. Equation 3.71 is modified to 

For equilibrium situations, the flux is zero, and it follows from Equation 3.73 that the concentration distribution is of the Boltzmann type  

A simple case where flocculation takes place among single colloidal spheres of initial concentration is considered below. The concentration decreases with time because the collisions between two spheres lead to flocculation. The analysis is restricted to collisions between two individual particles and thdr consequences. Hence the solution is vahd only at short times. 

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. 

In the absence of a barrier to coagulation, and if the primary minimum is deep, every collision between a particle and a floe will lead to the growth of the floe. The rate of coagulation is then controlled entirely by the kinetics of the diffusion process leading to particle-particle collision. The theory of fast coagulation was developed originally by Smoluchowski (1918) and elaborated by Muller (1926). The rate equation has the same form as that for a bimolecular reaction  

At a concentration of 10l(l particles em f (or 10lf m i) the half-life is about 20 s. 

This theory takes account only of collisions between single particles and neglects the collisions of single particles with aggregates and of aggregates with one another. It applies, therefore, strictly only to the early stages of coagulation. When such effects are taken into account, it is found that the process still follows overall bimoleeular kinetics, but v in equation (9.16) is now the total number of particles irrespective of their size and fy now refers to the time at which the total number of particles has been halved. 

If the adherence between two particles which collide is inhibited by an energy barrier, the process resembles a bimoleeular chemical reaction involving tin activation energy. In this case wc have to deal with slow coagulation. On the basis of this analogy one might expect therefore that 

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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) ... 

There are a number of issues related to kinetics of coagulation that are not discussed in the previous sections. For example,... 

Kruyt, H. R. (Ed.), Colloid Science. Vol. 1. Irreversible Systems, Elsevier, Amsterdam, Netherlands, 1952. (Graduate and undergraduate levels. A classic reference on colloids. Chapters 6-8, by Professor J. Th. G. Overbeek, present the classical DLVO theory of colloidal forces and their application to kinetics of coagulation.)... 

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

Kinetics of Coagulation Particle Transport as Rate Determining Step... 

Fig. 6.8. Log-log plot of Wrap versus electrolyte concentration for hematite ( Fe203) colloids suspended in either CaCl2 or NaCI solution at pH 10.5. Arrows indicate critical coagulation concentrations [Eq. 6.76 data from L Liang, Effects of surface chemistry on kinetics of coagulation of submicron iron oxide particles (a-Fe2Oi) in water, Ph.D. dissertation, California Institute of Technology, Pasadena, CA, 1988. Environmental Quality Laboratory Report No. AC-5-88].

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... 

L. Liang, Effects of Surface Chemistry on Kinetics of Coagulation of Submicron Iron Oxide Particles in Water, Ph.D. Thesis, California Institute of Technology, Pasadena, California, 1988. 

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. 

The present paper deals with kinetics of coagulation of Phthallylsulfathiazole stabilized xylene in water emulsion in the presence of some cationic detergents. Rate of flocculation, rate of coalescence and rate of creaming have been determined. To estimate the stability of the present systems their zeta potentials have been measured and stability factors calculated. Temperature effect on the system was also studied. 

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