Fuchs’ stability ratio

Both these equations neglect the effects of interparticle forces. In the case of a static fluid, the effects of interparticle forces can be included by dividing the Smoluchowski relation [Eq. (69)] by the Fuchs stability ratio W as given by Spielman (80). Thus W depends on properties of the solution including its composition as well as properties of particle surface. When the fluid is in motion, the stability ratio IE becomes dependent on the flow as well... 

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. 

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. 

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 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. 

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]... 

The colhsion efficiency, Ey, in Equation 5.321 accounts for the interactions (of both hydrodynamic and intermolecular origin) between two colhding particles. The inverse of Ey is often caUed the stability ratio or the Fuchs factor and can be expressed in the following general form ... 

In Figure 2 three curves are shown, for particle radii of 500, 1000, and 1500A, and the stability ratio W calculated for each by the Fuchs equation (14)... 

The relative stability of colloidal dispersions near to incipient coagulation can be characterized by their stability ratio IV where (Fuchs, 1934)... 

We now consider the kinetics of the flocculation of a system for which there is an energy barrier. The rate is a function of the probability of particles having sufficient energy to overcome this barrier. The problem is again one of diffusion and was first solved by Fuchs . His theory leads to a factor W by which the rapid rate (Smoiuchowski) is reduced by the presence of a repulsive force W is called the stability ratio and is related to the height of the potential energy barrier by... 

Last but not least, it should be noted that Fuchs equation ignores concentration effects, i.e. manybody interaction. This concerns the viscous interaction as well as long ranging double layer interaction. Additionally, the mean distance to the next interacting particle cannot be considered infinite for dense suspensions. Hutter (1999) showed by means of simulation that particle concentration adversely affects the stability ratio W in dense suspensions. 

The total energy of interaction between the double layers on adjacent solid particles is the sum of the attractive (and repulsive (contributions. It is this potential K which retards the process of rapid coagulation by a factor IF, the stability ratio of Fuchs [8], given by,... 

The kinetics of coagulation were first analyzed by Smoluchowski [32], who assumed that no repulsive barrier was present, and that aggregation occurred by the attachment of single particles to clusters (ignoring cluster-cluster aggregation). The theory was further developed by Fuchs [33], who showed that a repulsive potential F(r) would reduce the coagulation rate by the faetor JV, called the stability ratio ... 

Equation (2.33) tends to overestimate the particle flocculation rate for the electrostatically or sterically stabilized colloidal dispersions due to the potential energy barrier established between the approaching particles against particle flocculation. Fuchs [45] was the first to introduce the stability ratio (W) to take into account the fact that not aU the collision events lead to successful coagulation of colloidal particles. 

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 ... 

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