Coagulation, stability ratios

C. Kinetic Aspects of Suspension Coagulation Stability Ratio 179... 

In slow coagulation, particles have to diffuse over an energy barrier (see the previous section) in order to aggregate. As a result, not all Brownian particle encounters result in aggregation. This is expressed using the stability ratio IV, defined as... 

The principle of this method is that the initial slope (time = zero) of the optical density-time curve is proportional to the rate of flocculation. This initial slope increases with increasing electrolyte concentration until it reaches a limiting value. The stability ratio W is defined as reciprocal ratio of the limiting initial slope to the initial slope measured at lower electrolyte concentration. A log W-log electrolyte concentration plot shows a sharp inflection at the critical coagulation concentration (W = 1), which is a measure of the stability to added electrolyte. Reerink and Overbeek (12) have shown that the value of W is determined mainly by the height of the primary repulsion maximum in the potential energy-distance curve. 

Summary plot of experimentally derived stability ratios, Wexp, of hematite suspensions, as a function of added electrolyte or adsorbate concentration at pH around 6.5 (pH = 10.5 for Ca2+ and Na+). Hematite concentration is about 10-20 mg/ . The stability ratio, Wexp, was determined from measurements on the coagulation rate it is the reciprocal of the experimentally determined collision efficiency factor, a. 

Experimental measurements in each lake included particle concentration and size measurements in the water column, sedimentation fluxes in sediment traps, and chemical and size characteristics of materials recovered from sediment traps. The colloidal stability of the particles in the lake waters was determined with laboratory coagulation tests. Colloidal stability was described by the stability ratio (a). For a perfectly stable suspension, a = 0 for a complete unstable one, a = 1.)... 

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). 

If the electrostatic barrier is removed either by specific ion adsorption or by addition of electrolyte, the rate of coagulation (often followed by measuring changes in turbidity) can be described fairly well from simple diffusion-controlled kinetics and the assumption that all collisions lead to adhesion and particle growth. Overbeek (1952) has derived a simple equation to relate the rate of coagulation to the magnitude of the repulsive barrier. The equation is written in terms of the stability ratio ... 

The stability ratio W is defined as the ratio of the rate of fast coagulation to that of slow coagulation and is given by... 

The stability of a dispersion against coagulation is expressed quantitatively by what is known as the stability ratio, usually denoted by W. The stability ratio is defined as... 

Equation (51) shows that Wis a sensitive function of max, the maximum in the interaction potential, which in turn is a very sensitive function of properties such as p0, electrolyte concentration, and so on. As a consequence, the stability ratio decreases rapidly with, for example, added electrolyte, and the dispersion coagulates beyond a threshold value of electrolyte concentration known as the critical coagulation concentration, as we saw in Section 13.3b.1. 

Using the approach developed in Example 13.3 and interaction energy expressions for spherical particles, it has been possible to predict how the stability ratio W varies with electrolyte concentration according to the DLVO theory. Since W can be measured by experimental studies of the rate of coagulation, this approach allows an even more stringent test of the DLVO theory than CCC values permit. We shall not bother with algebraic details, but instead go directly to the final result ... 

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 new Chapter 13 collects the material on colloid stability previously distributed between old Chapters 11 and 12 and integrates it with new material on stability ratio and slow and fast coagulation, polymer-induced forces, and polymerinduced stabilization and destabilization. 

Figure 8,7 Theoretical dependence of stability ratio on electrolyte concentration calculated from equation (8.2) for a = 1CT8 m, A = 2 x 10 19 J and fa = 76.8 mV = 3kT/e> At high electrolyte concentrations W < 1 owing to coagulation being accelerated by van der Waals attractive forces (reduced flow rate in the narrow inter-particle gap has not been allowed for) (By courtesy of Elsevier Publishing Company)...

Darling and van Hooydonk (1981) also considered how to reduce the diffusional collision rate to obtain slow coagulation and used the classical approach of Fuchs (Reerink and Overbeek, 1954), whereby an activation energy is computed from the pair interaction free energy of the aggregating particles. The reaction kernel is given by Eq. (6) divided by the stability ratio W,... 

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... 

The experimental stability ratio is defined generally as the rate of fast flocculation divided by the rate of slow coagulation (see, e.g., Section 7.8 in R. J. Hunter, op. cit.1). Equation 6.70 is a mathematical interpretation of this definition in terms of experimentally accessible quantities related to floccule size. 

Figure 5, in which the stability ratio is plotted against electrolyte concentration, for cations of valencies 1,2, and 3 and monovalent anions, shows that the critical coagulation concentration increases with decreasing cation valency (Figure 6). This rule is not valid for the critical stabilization 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. 

Application of Smoluchowski s equations typically results in an overestimation of the growth rate of aggregates due to the assumption that all collisions result in permanent attachment. Recognition of the importance of surface properties in the aggregation of small particles prompted Fuchs [6] to develop expressions to modify Smoluchowski s equations. Fuchs described the effect of the repulsive electrostatic interaction between two particles, which is a function of the particle separation distance, as a reduction in particle coagulation rate, Wy, termed the stability ratio. 

When there are energy barriers between the particles, for example, attractive and repulsive interaction energy beirriers like those discussed in the previous section, Fuchs [57] showed that the rate of coagulation, J, should be divided by a factor W, the colloid stability ratio, where W is given by [58,59]... 

In this equation, is the total interaction energy between the two colliding particles defined in the previous section. The stability ratio, W, for the system gives the ratio of rapid coagulation, Jp, to slow coagulation, J[= J W], DQi) is the position-dependent diflusion equation. This diffusion coefficient ratio is a factor that decreases the collision rate because of the difficulty in draining the liquid between the two solid surfaces. This diffiision coefficient ratio is given by [60,61]... 

The W values [65] for a dispersion of AI2O3 as a function of pH and KNO3 salt concentration are shown in Figure 10.27. The AI2O3 particles are colloidally stable far away from their isoelectric point (i.e., pH 8.9). As the salt concentration is increased the zeta potential decreases and the colloid stability ratio, W, decreases. Near the isoelectric point there is no electrostatic repulsion, giving a rapid coagulation. 

LOG MOLAR CONCENTRATION FIGURE 10J29 Colloid stability ratio for different salt concentrations showing the critical coagulation concentration, CCC. Data from Barringer [25]. 

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