Flocculation rates

For example, van den Tempel [35] reports the results shown in Fig. XIV-9 on the effect of electrolyte concentration on flocculation rates of an O/W emulsion. Note that d ln)ldt (equal to k in the simple theory) increases rapidly with ionic strength, presumably due to the decrease in double-layer half-thickness and perhaps also due to some Stem layer adsorption of positive ions. The preexponential factor in Eq. XIV-7, ko = (8kr/3 ), should have the value of about 10 " cm, but at low electrolyte concentration, the values in the figure are smaller by tenfold or a hundredfold. This reduction may be qualitatively ascribed to charged repulsion. 

The preceding treatment relates primarily to flocculation rates, while the irreversible aging of emulsions involves the coalescence of droplets, the prelude to which is the thinning of the liquid film separating the droplets. Similar theories were developed by Spielman [54] and by Honig and co-workers [55], which added hydrodynamic considerations to basic DLVO theory. A successful experimental test of these equations was made by Bernstein and co-workers [56] (see also Ref. 57). Coalescence leads eventually to separation of bulk oil phase, and a practical measure of emulsion stability is the rate of increase of the volume of this phase, V, as a function of time. A useful equation is... 

The flocculation rate is deterrnined from the Smoluchowski rate law which states that the rate is proportional to the square of the particle concentration by number inversely proportional to the fluid viscosity, and independent of particle size. 

Viscosity Increase. The flocculation rate of an emulsion is iaversely proportional to the viscosity of the continuous phase and an iacrease of the viscosity from 1 mPa-s (=cP) (water at room temperature) to a value of 10 Pa-s (100 P) (waxy Hquid) reduces the flocculation rate by a factor of 10,000. Such a change would give a half-life of an unprotected emulsion of a few hours, which is of Httle practical use. 

The foundations of the theory of flocculation kinetics were laid down early in this century by von Smoluchowski (33). He considered the rate of (irreversible) flocculation of a system of hard-sphere particles, i.e. in the absence of other interactions. With dispersions containing polymers, as we have seen, one is frequently dealing with reversible flocculation this is a much more difficult situation to analyse theoretically. Cowell and Vincent (34) have recently proposed the following semi-empirical equation for the effective flocculation rate constant, kg, ... 

The two major theories of flocculation, the bridging model (1) and the electrostatic patch model (2, 3 ), provide the conceptual framework for the understanding of polymer-aided flocculation, but they do not directly address the kinetics of the process. Smellie and La Mer (4) incorporated the bridging concept into a kinetic model of flocculation. They proposed that the collision efficiency in the flocculation process should be a function of the fractional surface coverage, 0. Using a modified Smoluchowski equation, they wrote for the initial flocculation rate... 

This approach is based on the assumption that polymer adsorption is fast ("instantaneous") compared with flocculation. In other words the surface coverage is taken to be constant during the flocculation process. Equation (1) states that the flocculation rate tends to zero when 0 tends to 0 or 1. The maximum rate occurs at 0 = 0.5, i.e., at 50% surface coverage. 

The flocculation rate dependency on the fractional surface coverage 0 in Equation (1) has been qualitatively confirmed (13, 14), although the maximum rate appears to occur for a surface coverage of less than 50%. The adsorption rate is also a function of 0, and it has been shown (15) for adsorption onto a smooth solid surface that the rate is proportional to the fraction of polymer-free surface area, 1-0. This approach has not... 

In summary, polymeric flocculants generally increase peri-kinetic flocculation rates compared with perikinetic coagulation rates. This is not necessarily true for orthokinetic flocculation, and experimental results in the literature are seemingly in conflict. Collision rate theory predicts that the polymer adsorption step may become rate limiting in orthokinetic flocculation. The present study was designed to elucidate the relationship between polymer adsorption rates and particle flocculation rates under orthokinetic conditions. 

It was also assumed that a successful collision in flocculation can only occur if a polymer-free area on one floe hits a polymer-covered area on another floe or vice versa. The complete dimensionless flocculation rate equation is given by Equation (5) below... 

Particle collision frequency due to Brownian motion was estimated to be less than 1% of the collision frequency due to shear. The effects of Brownian motion could therefore be neglected in the flocculation rate calculations. However, for the smallest molecular size, radius of gyration 14 nm (see Table I), the effect of Brownian motion on the particle-polymer collision efficiency was of the same order of magnitude as the effect of shear. These two contributions were assumed to be additive in the adsorption rate calculations. Additivity is not fundamentally justified (23) but can be used as an interpolating... 

It is seen that the flocculation rates are generally considerably lower than the coagulation rate. A "pseudo" optimum flocculation concentration of 6 OFC units is found for short flocculation times, but for longer times it is clear that the suspension is stabilized and no further flocculation occurs. At higher initial doses restabilization becomes even more pronounced. ... 

Adsorption rates were not significantly affected by molecular weight, but flocculation was about 25% faster for the high molecular weight polymer. Two shear rate levels were tested 1800 s-1 and 8000 s-. The absolute adsorption and flocculation rates increased with shear rate as expected. The "pseudo" OFC appeared to be shifted to a higher value for the higher shear rate. Collision efficiencies were affected by both molecular weight and shear rate, as discussed below. 

Flocculation rate limitation. The adsorption step was rate limiting for the overall flocculation process in this system. Polymer adsorption rate measurements for dispersed systems reported in the literature (2,26) do not lend themselves to direct comparisons with the present work due to lack of information on shear rates, flocculation rates, and particle and polymer sizes. Gregory (12) proposed that the adsorption and coagulation halftimes, tA and t, respectively, should be good indications of whether or not the adsorption step is expected to be rate limiting. The halftimes, tA and t, are defined as the times required to halve the initial concentrations of polymer and particles, respectively. Adsorption should not limit the flocculation rate if... 

It is reasonable to assume, at least for oppositely charged polymers and particles, that aA > ag, which means that the adsorption time is always expected to be shorter than the coagulation time under perikinetic conditions. Consequently, perikinetic flocculation rates are very likely not to be adsorption rate limited. The ratio of orthokinetic adsorption time to orthokinetic coagulation time is... 

A more thorough analysis of flocculation is given by Fuchs (F3) and by Zebel (Z1,Z2), both of whom arrived at essentially the same result. The flocculation rate can be written in the form... 

Thus, within this precision, it is apparent from Eq. (16) that the flocculation rate is an additive function of ordinary diffusion or Smoluchowski flocculation and electrostatic attraction. The ordinary diffusion rate is given by Eq. (16) for kem = 1, whereas the flocculation rate due to electrostatic... 

Fig. 3. Effect of particle number concentration on flocculation rate.

Emulsion stability of SEDDS is usually good because the droplets are small and have narrow size distributions. However, stability can be measured by determining the flocculation rates, degree of separation, or changes in the diameter of droplets formed on dilution over time during storage under various conditions. 

---