Coagulation concentration, critical particles

Figure 3.34 Optical absorbance against log[BaCl2/mol dm ] two hours after adding electrolyte (O) electrostatically-stabilized PS particles with marking the critical coagulation concentration ( ) PS particles with grafted MeOPEG chains...

Critical coagulation concentrations for spherical particles of a given material should be proportional to e3 and independent of particle size. 

Some other practical situations where particle aggregation is important include the precipitation of colloidal mud at the mouth of a river due to the salinity of the sea-water exceeding the critical coagulation concentration, land (e.g. mountainside) stability, building and road foundations, the retention of a porous structure in filtration, mineral processing117 and paper making. Control of particle aggregation is also of primary importance in adhesives, inks, pharmaceuticals, cosmetics, foodstuffs and lubricants. 

The Onset of Instability - The Critical Coagulation Concentration. Provided that the magnitude of the primary maximum is substantial, then the probability of the transition of the approaching particle into the primary minimum is small. However,... 

The transitions from stable dispersion to aggregation just described in terms of the critical coagulation concentrations and the Schulze-Hardy rule, apply best to suspensions in which the particles have only one kind of charge. However, clay particles can carry positive and negative charges at the same time, on different parts of the particle. See Section 5.6.2. 

The sensitivity of the stability ratio to chemical or particle interaction factors can be illustrated by an examination of the model expression for Wn in Eq. 6.75. For example, if temperature and the particle interaction parameters are fixed, then Wn will vary with the concentration, c (also included in /c), of Z-Z electrolyte. At low values of c, k is also small, and the first equality in Eq. 6.75 indicates that Wu will take on its largest values. (Decreasing c also provokes an increase in dm because of Eq. 6.73, but this effect is dominated by that of k.40) Conversely, as c increases, the value of Wu will drop until it achieves its minimum, Wn = 1.0, when Z dm = 2 (Eq. 6.75). At this concentration, termed the critical coagulation concentration (ccc), or flocculation value, the flocculation process has become transport-controlled and therefore is rapid. Thus in general... 

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].

On the other hand, if the rate of formation of a doublet is much smaller than its rate of dissociation, the doublet is unstable. It will be shown later that the unstable doublets reach a dynamic equilibrium with the singlets in an extremely short time (of the order of the time scale of dissociation). The equilibrium concentration of these unstable doublets is small and depends upon the relative magnitudes of the rates of formation and dissociation. Since the dissociation rate of a doublet decreases rather dramatically with increasing particle size (because of the rapid increase in the depth of the interaction potential well with increasing particle size), there exists a critical particle size above which the coagulated particle pair is stable, i.e., its rate of formation is much greater than its rate of dissociation. This critical particle size is analogous to the critical cluster size... 

A comparison between the rates of formation and dissociation of doublets as well as the calculation of the critical particle size are presented in the next section. The subsequent section discusses the implications of the above two mechanisms of aerosol growth on the calculation of the Brownian coagulation coefficient from the evolution of particle number concentration. 

In this chapter, mathematical procedures for the estimation of the electrical interactions between particles covered by an ion-penetrable membrane immersed in a general electrolyte solution is introduced. The treatment is similar to that for rigid particles, except that fixed charges are distributed over a finite volume in space, rather than over a rigid surface. This introduces some complexities. Several approximate methods for the resolution of the Poisson-Boltzmann equation are discussed. The basic thermodynamic properties of an electrical double layer, including Helmholtz free energy, amount of ion adsorption, and entropy are then estimated on the basis of the results obtained, followed by the evaluation of the critical coagulation concentration of counterions and the stability ratio of the system under consideration. 

The result for rigid particles [38] can be recovered as a special case of the present problem. Suppose that the surface potential remains constant (the rate of surface dissociation reactions is fast). The critical coagulation concentration ratio of cations can be obtained by replacing i//d with i//0 in Eq. (95). We have... 

R scaled center-to-center distance between two particles Rc value of R at which critical coagulation concentration occurs... 

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