Rate of aggregation

Hounslow, M.J., Mumtaz, H.S., Collier, A.P., Barrick, J.P. and Bramley, A.S., 2001. A micro-mechanical model for the rate of aggregation during precipitation from solution. Chemical Engineering Science, 56, 2543-2552. 

Analytical approaches to obtain the agglomerate size distribution are possible for well-mixed systems and when the rate of aggregation of clusters is defined by simple functions. In general, irreversible aggregation in well-mixed systems is described by Smoluchowski s coagulation equation, which... 

Hence, the polydispersity [Eq. (60)] is independent of time, provided Jo 4>(ri)drj is bounded. Moreover, if = e as is expected for a rate of aggregation independent of mass (Schumann, 1940), the polydispersity is equal to 2. Also note that... 

Rate of aggregation of compact clusters in well-mixed systems follows Smoluchowski s theory. 

Thus, the source terms for each environment S(c) and Sk ((/)) will be closed. Of particular interest are the local nucleation rates /(c ). As discussed in Wang and Fox (2004), due to poor micromixing the local nucleation rates can be much larger than those predicted by the average concentrations /((c)). This results in a rapid increase in the local particle number density mo due to the creation of a very large number of nuclei. As discussed below, this will have significant consequences on the local rate of aggregation. 

The effect of aggregation of the subsurface solid phase on kerosene volatilization was studied by Fine and Yaron (1993), who compared the rate of aggregation in two size fractions of a vertisol soil the <1 mm fraction and 2 mm aggregates. The total porosity of these two fractions was similar (53% and 55% of the total volume, respectively). Differences in aggregation are reflected in the air permeability that is, their respective values were 0.0812 0.009 cm and 0.145 0.011 cm Figure 8.10 presents the volatilization of kerosene as affected by the soil aggregation, when the initial amount applied was equivalent to the retention capacity. The more permeable fraction releases kerosene faster and thus enhances volatilization. 

Effect of Cation Concentrations and Temperature on the Rates of Aggregation of Acidic Phospholipid Vesicles Application to Fusion... 

We have another purpose in studying the temperature dependence of the rate of aggregation. It may be expected that the close approach of vesicles is the first stage in their fusion. The process of membrane fusion is known to be sensitive to temperature (14,15,16). In order to elucidate the temperature dependence of the rate of fusion per se, we have first to determine the temperature dependence of the process of close approach. We provide here light-scattering measurements which were designed to test our predictions for the aggregation of sonicated... 

The applicability of Equation 11 improves when the rate of aggregation is not too slow, e.g., when the surface potentials (calculated from the modified Gouy Chapman equation (see Refs. 28 and 29)) are not too high. These are the only cases of interest for quantitative comparison with experiment, since detection becomes impossible when the rate of aggregation is too slow. We point out below how we estimate rates for other cases. 

As shown in the section dealing with the expressions for the rate of aggregation of vesicles, the sensitivity of rates of aggregation to... 

Temperature Dependence of Rates of Aggregation General Analysis. In this section we present a general analysis, although the numbers appearing pertain mostly to PS vesicles. To simplify the analysis, we use W as given by Equation 6. From Equations 4 and 6 we have... 

The above analysis indicates that it is possible to choose systems of sonicated vesicles for which an appreciable increase of the rate of aggregation with temperature is predicted, provided that VT does not increase significantly with temperature. 

Temperature Dependence of Vt. The magnitude of the attractive VDW interactions at a distance of separation of 10 A between sonicated phospholipid vesicles is approximately 1 kT0. Hence, for cases of slow aggregation where VT 15AT0, the VDW interaction would have little effect on the rate of aggregation or its temperature dependence. Our calculations indicate (see expressions in Ref. 33) that the change of Vw with temperature may be neglected given the available accuracy of the VDW theory. 

As we noted before, the rates of aggregation are expected to increase with temperature provided that Vr remains constant or does not increase significantly with temperature. This behavior is illustrated in Table III. 

Table II. Temperature Dependence of Quantities Affecting Rates of Aggregation of Vesicles0 b...

Ve with temperature. Regarding the temperature dependence of the binding of Ca2+ or Mg2+ to PS, preliminary results (54) indicate that the measured ratios Ca2+/PS and Mg2+/PS do not vary siginificantly or even slightly increase with temperature over the range of 0°-37°C. Therefore, the order of magnitude increase in the rates of aggregation which is seen in Cases 2 and 3 in Table III is predicted by the present theory to be observed for PS vesicles. 

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