Stability ratio

An important measure of how fast a coagulation will be is the so-called stability ratio. It is defined as 

It can be shown that the stability ratio is, approximately, directly related to the maximum (barrier) of the potential energy function. This is because in the slow-coagulation regime the electrolyte concentration is such that the diffuse layer is very compressed. In the more general case, when we know the potential-distance (V-H) function, the Fuchs equation can be used but this often requires numerical solutions for a given Hamaker/surface potential values  

If only the y iax value is known then the Reerink-Overbeek equation can be used  

A dilute dispersion that rapidly coagulated in about 1 s can become stable with coagulation limes of several months if the is about 20 ksT. As a mle of thumb, we can mention that Vmax values of about (15-25) gr are often sufficient for stable dispersions provided that the Debye length is also rather large. 

It is always important to remember that the Vmax poses a kinetic barrier, because the potential is lowest in the primary minimum. Eventoally, even if it will take long time, the dispersion will be destroyed (complete coagulation) because some particles will have sufficient kinetic energy to overcome the barrier ksT is an average of the natural kinetic energy of the particles). If there is no secondary minimum and the potential barrier is sufficiently large compared to the thermal energy (ksT), then very few molecules will be able to fall into the primary 

The examination of the free energy of two interacting particles reveals the main factors that govern the suspension stabiUly and provide criteria for classifying a colloidal system as stable or not. However, in order to understand how the particle-particle interactions quantitatively affect the suspension stability, it is necessary to consider their influence on the particle aggregation rate. 

Smoluchowski neglected any kind of interaction between the particles, which means that the diffusion of a particle is not affected by the presence of other particles and that each event of collision leads to a stable aggregate bond. Yet, as Fuchs (1934) argued for the stability of charged aerosols, any attractive or repulsive interaction affects the diffusive approaching of two particles and Smoluchowski s aggregation rate is changed by a factor W  

The factor W describes the retardation of aggregation due particle interactions it has values below 1 when attractive interactions prevail (Vtot 0), whereas for repulsive interactions (Vtot 0), W amounts to values above 1. For this reason, it is called stability ratio. A colloidal suspension can be considered as to be long-term stabilised when the stability ratio exceeds values of 10 -10 (Hunter 1988, p. 239 Ramakrishnan et al. 1998).  

For the derivation of Eq. (5.30) Fuchs assumed that the diffusion coefficients of two approaching particles remain constant. However, due to viscous interaction the particles have a mutual impact on their mobilities The closer the two particles approach, the larger the additional resistance is which is related to the squeezing of the fluid. 

Honig et al. (1971) and Spielman (1970) independently worked on this aspect and included the viscous interactions in the calculation of the stability ratio W. 

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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... 

A combination of equation (C2.6.13), equation (C2.6.14), equation (C2.6.15), equation (C2.6.16), equation (C2.6.17), equation (C2.6.18) and equation (C2.6.19) tlien allows us to estimate how low the electrolyte concentration needs to be to provide kinetic stability for a desired lengtli of time. This tlieory successfully accounts for a number of observations on slowly aggregating systems, but two discrepancies are found (see, for instance, [33]). First, tire observed dependence of stability ratio on salt concentration tends to be much weaker tlian predicted. Second, tire variation of tire stability ratio witli particle size is not reproduced experimentally. Recently, however, it was reported that for model particles witli a low surface charge, where tire DL VO tlieory is expected to hold, tire aggregation kinetics do agree witli tire tlieoretical predictions (see [60], and references tlierein). 

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.

The electrostatic barrier developed only after enough dispersant adsorbed that a concentration of dissolved dispersant of about 0.1% or more remained in the oil phase, where counterions developed as evidenced by increased conductivity, the development of large negative zeta potentials, steeply rising stability ratios, and complete deflocculation. 

Comparison of hematite surface charge, [coul g 1], electrophoretic mobility, and stability ratio, Wexp, as a function of pH. Note that at pHpzc the net surface charge and mobility are both zero, and the stability is a minimum. 

The experimental stability ratio (W), the potentiometrically-determined surface charge, and the electro-kinetic mobility of 70 nm particles over the pH range from 3 to 11 are shown. The drawn-out line in Fig. c summarizes experiments obtained with I = 0.05 - 0.1. (Modified from Liang and Morgan, 1990.)... 

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. 

Experimentally derived stability ratio, Wexp, of hematite suspensions, plotted as a function of fatty acid concentration at pH 5.2. The ionic strength is 50 milimolar NaCI and hematite concentration is 34.0 mg/ . Laurie acid is denoted by C, capric acid by C10, caprylic add by Cs and propionic acid by C3. (From Liang and Morgan, 1990)... 

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.)... 

Comparing ligands 22 and 38 which contain the same number of binding sites, we note that the special complexation features of 38 result from the cryptate nature of its complexes. Indeed, whereas in complexes of 22 polar solvent molecules may approach the cation from top and bottom, it is much more shielded in complexes of 38. This difference in behaviour is reflected in the corresponding change in ligand thickness (Table 12). The results in Table 11 also display the expected decrease in M2+/M+ stability ratio as the dielectric constant decreases from water to methanol. 

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... 

Fig. 4. Stability ratio (W) determined for Z1O2 (Bitea et al. 2003a) and smectite colloids (Missana Adell 2000) as a function of pH and ionic strength.

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. 

EXAMPLE 13.3 Expression for Stability Ratio in Terms of max. By replacing r by rm, the center-to-center distance of separation at the maximum in the potential energy curve and using a truncated Taylor series about the maximum to estimate (r), show that W potential energy barrier at the maximum. Comment briefly on these and any other assumptions or approximations involved. 

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) ... 

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