Smith-Ewart differential difference equations

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-... 

In reviewing the cases for which explicit analytic solutions have so far been obtained, it is helpful to recall that the Smith-Ewart differential difference equations are derived on the assumption that the state of radical occupancy of a reaction locus can change as a result of three distinct types of process ... 

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... 

Recent interest in this aspect of the subject commenced with the appearance in 1974 of a paper by Gilbert and Napper, in which solutions were reported for a special case of the Smith-Ewart differential difference equations, namely, the case where the loss of radicals from reaction loci occurs exclusively by processes which are kinetically of first order in radical concentration within the loci. Loss through bimolecular mutual termination of radicals is assumed either not to occur at all or, at most, to account for the loss of a negligible proportion of the propagating radicals. Radical loss is assumed to occur almost exclusively by processes such as diffusion from the reaction loci back into the external... 

Other recent contributions to this aspect of the subject include those of Brooks and Qureski and Brooks. The former of these papers is concerned with the distribution and desorption of free radicals during the emulsion polymerization of styrene. The latter gives a simplified treatment of the type of problem which has been dealt with by Birtwistle and Blackley, and the Napper group in essence, the Brooks treatment involves truncation of the infinite series of Smith-Ewart differential difference equations. 

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. 

Compartmentalized Free-radical Polymerization.—Considerable interest has been shown in recent years in the solution of the differential difference equations which are obtained when the theory of Smith and Ewart is applied to reaction systems which contain a fixed number of reaction loci, but in which a steady state for the various locus populations has yet to be established. An example of such a reaction system would be a seeded emulsion polymerization system within whose external phase new radicals suddenly begin to be generated, and which does not contain sufficient surfactant to permit the nucleation of new particles. The theory which has been developed is concerned with the question of the nature of the approach to the steady-state distribution of locus populations, and with what might be learned from accurate measurements made during the approach to the steady state. 

The resultant single differential equation can then readily be transformed into Eq. (18). Thus, it becomes clear that the O Toole (1965) formulation of the steady-state problem is exactly equivalent to that of Smith and Ewart (1948), notwithstanding that the approach to the problem is somewhat different. 

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