Coalescence rate constants

P(u, v) Coalescence rate constant for collisions 1/sec 1/sec Ap Relative fluid density with respect to gm/cm... 

Coalescence rate constant was estimated with the help of... 

It is evident from the date recorded in Table I that at higher temperature coalescence rate constant is higher, that is in order of kc30° < kc40° < c50° The temperature effect for all detergents was found to be in the order LPC < CTAB < CPC < CPB. This implies that in the presence of CPB the coalescence rate constant of the emulsions increases enormously (e.g. four to six times for every 10°C rise of temperature) with the rise of temperature whereas in the presence of LPC the increase in the rate is not that pronounced. 

Figure 5. Plot of coalescence rate constant (Kc) against detergent concentration.

Table I. Effect of Temperature on Coalescence Rate Constant for the System...

Provided that the emulsion does not undergo any flocculation, the coalescence rate can be measured simply by following the number of droplets or average diameter as a function of time. For this, a given volume of the emulsion is carefully diluted into the Isotone solution of the Coulter counter, and the number of droplets is measured. The average diameter can be obtained using laser diffraction methods (e.g., with the Master Sizer). By following this procedure at various time periods, the coalescence rate constant K can be obtained. 

Ivanov et al.4 >6 5oo.5oi,562 jj yg proposed a semiquantitative theoretical approach that provides a straightforward explanation of the Bancroft rule for emulsions. This approach is based on the idea of Davies and RideaP that both types of emulsions are formed during the homogenization process, but only the one with lower coalescence rate survives. If the initial drop concentration for both emulsions is the same, the coalescence rates for the two emulsions — (Rate)i for emulsion 1 and (Rate)2 for emulsion 2 (Figure 5.44) — will be proportional to the respective coalescence rate constants, and, 2 (ss Section 5.6, below), and inversely proportional to the film lifetimes, Xj and X2 ... 

FIGURE 5.56 Relative change in the total number of drops, vs. time, t initial number of primary drops o = 10 cm- coalescence rate constant k = lO Vs. Curve 1 numerical solution of Equation 5.336. Curve 2 output of the model of Borwankar et al. Curve 3 output of the model of van den Tempel. The values of the flocculation rate constant are (a) = 10 " cmVs (b) a = 10 cmVs (c) (sy= 10 cmVs. 

On the other hand, if xq is unchanged and sq different, as in Figure 6 compared with Figure 4, the critical time is the same (-40 ms) because it corresponds to the time required to reach the critical nuclearity fixed by the donor potential, and the growth kinetics depend only on the initial monomer concentration xq and on the coalescence rate constant fed ... 

Foaming agent Mecm expansion K Coalescence rate constants 10" s" Stability periods 10 s ... 

For clusters of higher nuclearity too, the kinetic method for determining the redox potential (M /Mn) is based on electron transfer, for example, from mild reductants of known potential which are used as reference systems, towards charged clusters MjJ. [31] Note that the redox potential differs from the microelectrode potential °(M, M /M ) by the adsorption energy of Mon M (except for = 1). The principle [31 ] is to observe at which step n of the cascade of coalescence reactions, a reaction of electron transfer occurring between a donor S and the cluster could compete with the coalescence. Indeed, n is known from the coalescence rate constant value, measured in the absence of S, and from the time elapsed from the atom appearance to the start of coalescence. The donor S is produced by the same pulse as the atoms M°, the radiolytic radicals being shared between M and S. One form at least in the couple S/S should possess intense optical absorption properties to permit a detailed kinetics study. 

Figure 3 Relative change in the total number of droplets vs. time initial number of primary particles Njq = 1 x 10 cm flocculation rate constant Kp= 1 x 10 cmVs curve 1, the numerical solution ofthe set Eq. (37) curve 2, the model of Borwankaref al. (38) for diluted emulsions curve 3, the model of van den Tempel (71) (a) coalescence rate constant = 1 x iQ- s" (b) = 1 x iQ - s" (c) Kg =...

Here, K = coalescence rate constant, and T= absolute temperature. When the rates are equal for both eoalescence and flocculation, a=10 to 30><10 cm s. When eoalescence is slow the collision frequency and the duration of collisions are more important. In this case, mixing enhances... 

They showed that the coalescence rate constant, K, increases while the flocculation rate constant decreases with increased demulsifier concentration. Flocculation is high at low demulsifier concentration. At increased concentration it breaks the interfacial film and promotes coalescence. A plot of initial coalescence rate constant versus dosage indicates that the demulsification of this system was in a flocculation-rate controlling state, within its environment. Aggregation is reversible and the drop identity is not lost. 

Figure 22 The total number of constituent drops in a flocculating emulsion, decreases with time, t, because of a parallel process of coalescence. The curves are calcualted for the following parameter values initial number of constituent drops iiq = lO cm coalescence rate constant F = 10 s h Curve 1 is a numbeiical solution to Eq. (121) Curves 2 and 3 are the results predicted by the models of Bor-wankar et al. (194) and van den Tempel (193), respectively. The values of the flocculation rate constant are (a) ar= 10 " cmVs (b) ar=...

Despite the commercial importance of PVC particle morphology to its end-use applications, there has been little work done on the development of quantitative models relating the size evolution of primary particles in terms of process conditions. Kiparissides [57] developed a population balance model to describe the time evolution of the primary particle size distribution as a function of the process variables, such as temperature and ionic strength of the medium. However, for the solution of the population balance model, the coalescence rate constant between the primary particles needs to be known. This, in turn, requires the calculation of electrostatic and steric stabilization forces acting on these particles. 

The term /3(m, v) represents the coalescence rate constant of two colloidal particles of volume u and v. Note that the initial particle growth occurs mainly by particle aggregation and, to a smaller extent, by polymerization of the adsorbed monomer in the polymer-rich phase [58]. Thus, knowledge of analytical expressions for the coalescence rate constant is of profound importance to the solution of the population balance model (Equation 4.46), describing the time evolution of the primary particle size distribution. Such expressions have been derived by Kiparissides et al. [57, 59]. 

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