Smith-Ewart recursion equation

This leads to the well known Smith-Ewart recursion equation which has been solved by Stockmayer (14) extended by O Toole (15) and further by Ugelstad et alia (IST to the case where radicaT reabsorption occurs. This expression has been solved in detail by computer for this paper in the manner presented by Ugelstad et alia (16). In particular the average number of radicals per par-ticle, TT, has been related to the quantities m and a, defined below, for the case where no aqueous termination or growth takes place... 

Figure 11. Solutions of the Smith-Ewart recursion equation for the case of no aqueom propagation or termination. Dotted line m = 0 (Smith-Ewart Case II). Curve 1 (m = 10 ) depicts typical styrene-like polymerization. Curve 2(m = 0.01) depicts radiation initiated emulsion polymerization of vinyl chloride. Curve 3 (m > 1.0) depicts chemically initiated emulsion polymerization of vinyl chloride.

Stockmayer (1957) was the first to present a general solution to the Smith-Ewart recursion equation for . This solution is reproduced below for the case m which free radical escape from particles is not possible. 

The second part of the R problem, the determination of is the subject of numerous papers (5.10,12,13). Most of these papers are concerned with obtaining solutions to the Smith-Ewart recursion relationship given by Equation 9 ... 

Stockmayer [88] obtained a general solution for the Smith-Ewart recursion equation (Equation (4.6) in Appendix 4.1) by searching for a generating function... 

These unusual variations were explained by a comparison of the kinetics of persulfate-initiated solution and emulsion polymerization. According to the Stockmayer solution of the Smith-Ewart recursion equation (56) adapted for single radical entry into the particles, and no loss of radicals by diffusion out of the particles... 

The second part of the Smith—Ewart theory concentrates on calculating the average number of radicals per particle. As long as the monomer concentration in the particles is constant, as may often be the case in Interval 11, this number then yields the rate of polymerisation. Smith and Ewart did this by means of a recursion equation that is valid for the situation prevailing after particle formation is finished. 

The two other cases occur when the left side (Case 1) or the right side (Case 3) of equation (1.4) is not fulfilled, giving negative or positive deviations from the 0.5 value. Smith and Ewart did not treat these cases completely, re-absorption of radicals was only included for the case when termination in the particles was dominating (their Case IB) and particles with more than one radical (Case 3) were only considered when desorption was negligible. Also they did not give the full solution of the recursion equation (1.3). This was not solved until 1957 by Stockmayer (1957). If desorption is neglected, the solution is... 

Stockmayer also presented solutions for the case that takes into account desorption of radicals. This solution, however, is wrong for the most important range in desorption rates. But Sto ckmayer s solution(s) lead the way for the possibility of exact mathematical solution of emulsion polymerisation kinetics at a time when digital computers were not yet very important in chemical computations. The general solution when desorption is taken into account was presented by O Toole (1965). He applied a modified form of the Smith-Ewart recursion equation that gave the solution... 

This equation is valid if the number of particles is large. The factor of 2 does not appear with 2 because only one particle having n radicals is formed from one having n + 2 radicals. Equation (7.3.5) is the basic recursion relation for emulsion polymerization derived by Smith and Ewart. Many refinements to this equation have been suggested by several authors, but their net result is similar... 

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