Smith-Ewart time-dependent

In that publication a dependence of the shape of the rate-time function on such parameters as initial monomer concentration, emulsifier concentration, and dose rate was shown for the methyl acrylate system. The behavior of this system tentatively was explained by assuming a strong gel effect even at low conversions, of prolonged particle formation, and some kind of interparticle radical termination—all factors which are included neither in the Harkins view nor in the classical Smith-Ewart theory. 

The conversion-time curves appear to be very similar to the shape typical of emulsion polymerization, i.e., an S-shaped curve is attributed to the autoacceleration caused by the gel effect (Smith-Ewart 3 kinetics, n>>l). The rate of polymerization-conversion dependence is described by a curve with two rate maxima. The decrease in the rate after passing through the first maximum is ascribed to the decrease of the monomer concentration in particles. Particle nucleation ends between 40 and 60% conversion, beyond the second rate maximum. This is explained by the presence of coemulsifier which stabilizes the monomer droplets against diffusive degradation. 

The time-dependent Smith-Ewart differential difference equations methods available for their solution... 

We have also been able to obtain an explicit analytic solution to eqn (4), and hence to the general time-dependent Smith-Ewart differential difference equations, for the case where the rate of formation of new radicals in the external phase is zero, i.e., cr = 0. Of course, if no radicals ever have been generated within the external phase of the reaction system, then the problem becomes trivial and admits of an obvious and simple solution, namely, that all loci are at all times devoid of propagating radicals, and the rate of polymerisation is always zero. This solution is clearly of no interest. The case which is of interest is that of a reaction system in which radicals have been generated within the external phase, so that a certain rate of polymerisa-... 

Equation (58) indicates that an increase in initiatior concentration will not enhance the rate of polymerization. It can be used for estimating the molecular mass of the polymer assuming, of course, the absence of transfer. The ratio N/q corresponds to the mean time of polymer growth and molecular mass is equal to the product of the number of additions per unit time and the length of the active life time of the radical, kpN/e. An increase in [I] also means a higher value of q, and thus a shortening of the chains. As in Phase II, the polymerized monomer in the particles is supplemented by monomer diffusion from the droplets across the aqueous phase a stationary state is rapidly established with constant monomer concentration in the particle. The rate of polymerization is then independent of conversion (see, for example the conversion curves in Fig. 7). We assume that the Smith-Ewart theory does not hold for those polymerizations where the mentioned dependence is not linear [132], The valdity of the Smith-Ewart theory is limited by many other factors. 

The Time-Dependent Smith-Ewart Differential Difference Equations Methods Available for Their Solution. 

The time-dependent Smith-Ewart differential difference equations can also be derived by an alternative method first used by OToole (196S) for the steady state. In this methnd, one considers the rates of transition of locus populations across a notional harrier situated between two neighboring states of radical occupancy. The harrier is illustrated in Fig. 3, where the... 

The feasibility of the matrix approach to solving the time-dependent Smith-Ewart differential difference equations depends on the fact that each of these equations is linear in certain <. Thus, the typical equation of the set [Eq. (2)j can be rearranged to read... 

Considerable progress has been made in recent years in obtaining solutions to the time-dependent Smith-Ewart differential difference equa> tions for various special types of reaction system in the nonsteady state. Although it has so far not proved possible to give an entirely general solution to these equations, it has proved possible to obtain a general solution to a modified set of equations which, under certain circumstances, approximate to the exact set of equations. 

Fig. 11. Predictions for aveiasc number of radicals per locus, 7(t), ns a function of time t obtained by numerical solution of ttme-dependent Smith-Ewart differential diKience equations, showing effect of decreasing k and increasing x, keeping cr constant. Value taken for o is lx I0 sec . The values taken for k and y are ns follows A k = lx I0 sec, ...

Acres and Dalton (1963a) also studied the emulsion polymerization of methyl methacrylate initiated by Co y radiation using a recording dilatometer. Only the conversion-time curves were measured with constant dose rate, varying monomer concentration, and with constant monomer concentration at different dose rates. Except at the lowest monomer concentration a clear gel effect was observed, with linear rates up to that point. The linenr rates increased with increasing monomer concentration up to about 0.4 mol/liter and then leveled oif. The dependence of the rate, before the gel effect, on the dose rate was 0.4 and, unlike their findings with styrene, not dependent on the monomer concentration. Their results were consistent with those of Hummel ei al. that methyl methacrylate follows, with y radiation, the generally accepted Smith-Ewart Case 2 kinetics except for the marked gd effect. 

III The Time-Dependent Smith-Ewart DiffBieniial Difference Equations ... 

The Smith-Ewart kinetic theory of emulsion polymerization is simple and provides a rational and accurate description of the polymerization process for monomers such as styrene, butadiene, and isoprene, which have very limited solubility in water (less than 0.1%). However, there are a number of exceptions. For example, as we indicated earlier, large particles (> 0.1 to 0.5 cm diameter) may and can contain more than one growing chain simultaneously for appreciable lengths of time. Some initiation in, followed by polymer precipitation from the aqueous phase may occur for monomers with appreciable water solubility (1 to 10%), such as vinyl chloride. The characteristic dependence of polymerization rate on emulsifier concentration and hence N may be altered quantitatively by the absorption of emulsifier by these particles. Polymerization may actually be taking place near the outer surface of a growing particle due to chain transfer to the emulsifier. 

The time-dependent behavior of emulsion polymerization arises due to variation in monomer concentration, changes in the number of polymer particles Nt, or both. We have aheady observed that N, changes due to nucleation in stage I of emulsion polymerization and this normally ends at about 10-15% conversion. However, when the monomer-to-water ratio MIW) is high or the monomer is more than sparingly soluble, the constancy of N, cannot be assiuned up to conversions as large as 50%. If the monomer droplets are sufficiently small, they also become the loci of particle formation and, in such circumstances, the Smith-Ewart theory is inadequate to explain the experimental phenomena. We now present the outline of a mathematical model of emulsion polymerization that is... 

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