The Stability Ratio

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... 

Then, introducing the stability ratio yj and instability ratio yJJ, the stable component is written as... 

V. Mishra, S. M. Kresta, J. H. Masliyah 1998, (Self-preservation of the drop size distribution function and variation in the stability ratio for rapid coalescence of a polydisperse emulsion in a simple shear field), J. Colloid Interface Sci. 197, 57. 

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. 

However in Table IV we see no increase in W at 1%, and only a small increase at 2% of dispersant. The value of W increases rapidly at about the same concentration that the conductivity increases, the counterion concentration increases and the zeta-potential increases. At OLOA-1200 levels of 3.5% and higher the stability ratio exceeds 5x10, with half-times in excess of seven months these stability ratios developed when zeta-potentials were -120 mV or more. 

Figure 12. Comparison of the dispersant concentration and dependence of the stability ratio W the conductivity of carbon black dispersions in dodecane. Reproduced with permission from Ref. (16). Copyright 1983, Elsevier Science Publishers.

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 ... 

A major outcome of the abovementioned studies is the importance of pH and notably of the salinity of the groundwater controlling colloid concentrations and, consequently, the relevance of colloids for radionuclide transport. The pH-dependent colloid stability varies considerably for different colloid types. Experimental data for the relationship of the stability ratio W... 

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

A more quantitative measure of stability, known as the stability ratio, can be obtained by setting up and solving the equation for diffusive collisions between the particles. Quantitative formulations of stability, known as the Smoluchowski and Fuchs theories of colloid stability, are the centerpieces of classical colloid science. These and related issues are covered in Section 13.4. 

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. 

This expression can be used for arriving at the stability ratio for charged particles in nonaqueous media in which the repulsion can be modeled using a simple Coulombic expression (see Problem 3 at the end of the chapter). 

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) ... 

What is the stability ratio W1 What values of W correspond to a stable dispersion Why ... 

How is the stability ratio related to the interparticle interaction energy Under what conditions can it be written in Arrhenius form ... 

What is the relation between the stability ratio and the DLVO theory How would you use the DLVO theory to predict the stability ratio of a dispersion ... 

Discuss how the stability ratio varies with dispersion properties such as electrolyte concentration, pH, surface potential, Hamaker constant, particle size, and so on. 

Discuss the agreements and disagreements between the theoretical predictions for the stability ratio with what is observed experimentally. 

Draw a sketch of the integral for the stability ratio W in Example 13.3 and verify that the assumptions made there to evaluate the integral are correct when the maximum in the interaction energy is large. 

Arachidic acid sols were studied with different concentrations of La3+ added. The stability ratio W and the direction of particle migration in an electric field (i.e., particle charge) were observed and the following results obtained ... 

Kitahara and Ushiyamaf flocculated a polystyrene latex of radius 665 A with KC1. The stability ratio Wwas found to vary with the KC1 concentration as follows ... 

The kH/k0Mc reactivity ratios do not parallel the stability ratios. The low rates of formation of 5 and 8 can neither be attributed to factor (i), which seems to stabilize the final product, nor to factor (///), which would also facilitate the reaction. Because factor (v) appears to be weak in the product, it should be even weaker in the transition state. 

The results were expressed in terms of the stabilization ratio s, given by... 

For the three systems considered, the stabilization ratio s was determined as a function of the total photocurrent density jph = Ipf/A (A = surface area) and the concentration of reducing agent y. In all cases, s was found to increase with increasing... 

Table I. Expressions of the stabilization ratio for different reaction mechanisms. (AB)s + h+ Xx ...

In order to take particle-particle interactions into account, a stability ratio W is included which relates the collision kernel /So to the aggregation kernel /3agg. The stability ratio W depends on the interaction potential aggregation rate without to the rate with interactions additional to the omnipresent van der Waals forces. For Brownian motion as dominant reason for collisions, the stability ratio W can be calculated according to Eq. (6) taken from Fuchs [ 10]. In case of shear as aggregation mechanism, the force dip/dr relative to the friction force should rather be considered instead of the ratio of interaction energy relative to thermal energy. 

When there is a repulsive energy barrier, only a fraction 1/W of the encounters between particles lead to permanent contact. W is known as the stability ratio - i.e. 

A theoretical expression relating the stability ratio to the potential energy of interaction has been derived by Fuchs110 ... 

Theoretical relationships between the stability ratio and electrolyte concentration can be obtained by numerical solution of this integral for given values of A and if/a- Figure 8.7 shows the results of... 

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