Slow coagulation

Consider now coalescence taking into account the molecular and electrostatic forces. Of the greatest interest is dependence of stability factor F on parameters k, //, Sa, Sr, t, y, a, and definition of the criterion for transition from slow to fast coagulation. In Section 11.5 the condition of transition from slow to fast coagulation was considered within the framework of DLVO theory of identical colloid particles without taking into account hydrodynamic interaction 

The condition (13.112) corresponds to the maximum on the potential energy curve of particle s interaction, where the curve is tangent to the abscissa axis. If we introduce the dimensionless potentials Va and Vr, 

In the experiment with an anticoagulant a mixture of 1 mole/1 HCl and 5.6 mole/1 CaCl2, has been used the average size of particles was 0.6 pm, volume concentration of anticoagulant varied from 1% up to 10%. The data correspond 

For the latex particles used in experiment, F = 10 J and Sa = 5 10 . To determine parameter Sr, it is necessary to know surface potential of particles. From experimental dependence F(Wd), where Wd is volume concentration of anticoagulant, the unique value = 13 mV Sr = IJ) has been determined, for which the theoretical dependence gives the best approach to experimental data (Fig. 13.34). 

It should be noted, that the considered range of r at Sa = 5 10 and Sr = 1.7 falls into the area of decrease of function F(r). The maximum value of F on the theoretical curve is attained at r 400, corresponding to concentration of electrolyte Wd 0.005. The experiments were carried out at concentrations Wd 0.01, 

If some energy barrier to particle contact and adherence exists—that is, if some of the collisions are not sticky —the collision process can be seen as 

FIGURE 10.14. Experimentally, the rate of particle flocculation can be determined by measuring the change in turbidity of the system as a function of time and extrapolating back to zero time. A typical turbidity curve would have the form shown. 

Equation 10.26 would be valid if colloidal diffusion processes were exactly analogous to those for individual molecules. However, the interactions between particles in colloidal systems tend to extend over distances much greater than those involved in the formation of atomic or molecular activated complexes (say, 10-100 run vs. O.l-l.O nm). As a result, the effects of those interactions will begin to be felt by the particles well before they approach to the critical distance r. Their mutual diffusion rate will therefore be reduced and the collision frequency will drop accordingly. The collision frequency will also be reduced by the hydrostatic effect mentioned above for rapid coagulation. 

A more accurate expression for the stability ratio, W, taking into consideration the above retarding effect of the interaction potential on collision frequency, is 

For charged particles experiencing electrostatic repulsion, an approximate equation for the stability ratio is 

---

The second case concerns situations where not all particle encounters result in aggregation. This is known as slow coagulation. This was addressed first by Fuchs [ ] again we follow [39, 57]. 

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... 

As two particles approach in a liquid their charge fields may interact and form two minima as depicted in Figure 6.8. If the particles approach to a distance Li, known as the primary minimum they aggregate to form a configuration with minimum energy - and rapid coagulation is said to take place. On the other hand, if the particles remain separated at a distance L2, the secondary minimum, loose clusters form which do not touch. This is known as slow coagulation and is the more easily reversed. 

Aerosols are solid or liquid particles, suspended in the liquid state, that have stability to gravitational separation over a period of observation. Slow coagulation by Brownian motion is implied. 

Our objective is thus to characterize the structure of the polymer network formed during coagulation in the spinning process, as well as the structure obtained by slow coagulation, and to consider the formation of such structures in terms of the phase transformations in rigid polymer solutions. 

The nature of the phase formed in the slow coagulation experiment can be deduced on the basis of its thermal transitions, and from its X-ray diffraction pattern. In the DSC trace of the slowly coagulated PBT-MSA-water system shown in Figure 6, three endothermic transitions are observed between 90 °C and 240 °C. Corresponding transitions were observed by optical microscopy using a hot stage. Solid PBT fibers or films do not exhibit thermal transitions below about 650°C (1). 

Figure 4 Optical micrograph of spherulites formed by slow coagulation of an isotropic solution of PBT in MSA, viewed between crossed polarizers.

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). 

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

Examples are readily found in which the observed rate constant for coagulation is several orders of magnitude smaller than the rate constant we have been discussing. These are cases of slow coagulation and imply a component of net repulsion between the dispersed particles. We continue with the analysis of the kinetics of slow coagulation at this point. This is the topic of the next section. 

This is the Arrhenius form to which the example refers. In it, the height of the maximum in a net potential energy curve plays the role of the activation energy. In the next chapter we see how this method has been used to evaluate W for systems in which the overlapping ion atmospheres of approaching colloidal particles provides the repulsion needed to give slow coagulation. 

What is meant by slow coagulation What is the basic principle behind the Fuchs theory of slow coagulation What is the rate coefficient for slow coagulation How is it defined, and what properties of the dispersion determine its magnitude What are the limitations of this theory as presented in the text ... 

The rate constant kp is given in terms of physical parameters (Boltzmann Constant KB, the absolute temperature T, and the absolute viscosity rj) that characterize these transport conditions. In the case of not completely destabilized colloids, when according to v. Smoluchowski so-called slow coagulation is observed, the rate constant contains in addition the collision efficiency factor, p, the fraction of collisions leading to permanent attachment under perikinetic conditions ... 

---