Particle 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... 

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). 

Ni-YSZ cermet anodes satisfy most of the basic requirements for SOFC anodes. The effective conductivity of a Ni-YSZ cermet anode increases with the Ni to YSZ volume ratio, relative density, and decreasing the particle size ratio of NiO to YSZ. While coarse YSZ powders may result in poor mechanical strength and low stability, coarse NiO powders may lead to poor effective conductivity. The effective conductivity increases with the temperature at which the NiO is reduced to Ni metal in a reducing atmosphere. Further, very low reduction temperatures (e.g., below 400°C) may result in not only low electrical conductivity, but also poor mechanical strength. 

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.)... 

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.)... 

Particle stability is governed by the three parameters, p, S, and VJV. Note that since the latter ratio appears as an addition to the ratio in Eq. (17),... 

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 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. 

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) ... 

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

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 ... 

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. 

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 most common scenario for measuring the experimental stability ratio involves its initial value (t =0). If the initial state of a suspension is arranged to comprise only primary particles, then Eq. 6.58 applies and Eq. 6.70 reduces to the expression ... 

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... 

Strong specific anion effects were reported particularly at low electrolyte concentrations (10 4—10 2 M),1 a range in which the DLVO theory is considered accurate. However, as shown later, the present experimental data cannot be reproduced by the traditional theory in this range of electrolyte concentrations. In the past, no agreement could be obtained, on the basis of the traditional theory, because small changes in the values of the parameters, caused by the nonuniformity of the particles, affected strongly the stability ratio.18 The polarization model provides similar results in the above range of electrolyte concentrations, when the dipole densities are sufficiently low and cannot explain the data. 

Figure 2. Experimental values of the stability ratio of protein-covered latex particle as a function of electrolyte concentration, at pH = 10.0, reported by Lopez-Leon et al.,1 compared to those calculated from the polarization-based hydration model, for the following parameter values NA = 1.2 x 1018 sites/m2, NB = 1.62 x 1018 sites/m2, A, = 0.9 x lO 20 J, KH = lCL6 M, Aon = 8.95 x 10 8 M, KNh = 0.021 M, (p/e )Na = 1.8 D (1) Ka = 0.76 M, (p/e)ci = 2.3D (2)Kno = 0.62M,(p/e )no3 = -1.8D stars, NaN03 squares, NaCL...

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 stability ratio decreases with the increase in the radius of the rigid core of a particle. This is consistent with the DLVO model together with the Fuches theory [8] for the case of rigid particles, and with the result obtained by Taguchi et al. [12] for ion-penetrable particles at low electrical potentials. 

If the radius of the rigid core of a particle is fixed, the stability ratio increases with the density of fixed charge. This is expected, since the higher the latter, the greater the electrical repulsion force between two particles, and, therefore, the more stable a colloidal suspension. 

For particles of different sizes, the greater the difference between the radii of two particles, the smaller the stability ratio. This implies that polydispersed particles are more unstable than monodispersed particles. This is because that the greater the difference in the radii of two interacting particles, the grater the absolute van der Waals attraction energy. 

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. 

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