Reactivity ratio confidence limits

One more method which is not subject to reindexing errors and which allows, like the two above procedures, to determine the confidence limits of the reactivity ratios was proposed by Joshi and Joshi (JJ) [218]. This method being a further modification of the intersection procedure (IP) is free of its intrinsic disadvantages. 

Numerous reports are available [19,229-248] on the development and analysis of the different procedures of estimating the reactivity ratio from the experimental data obtained over a wide range of conversions. These procedures employ different modifications of the integrated form of the copolymerization equation. For example, intersection [19,229,231,235], (KT) [236,240], (YBR) [235], and other [242] linear least-squares procedures have been developed for the treatment of initial polymer composition data. Naturally, the application of the non-linear procedures allows one to obtain more accurate estimates of the reactivity ratios. However, majority of the calculation procedures suffers from the fact that the measurement errors of the independent variable (the monomer feed composition) are not considered. This simplification can lead in certain cases to significant errors in the estimated kinetic parameters [239]. Special methods [238, 239, 241, 247] were developed to avoid these difficulties. One of them called error-in-variables method (EVM) [239, 241, 247] seems to be the best. EVM implies a statistical approach to the general problem of estimating parameters in mathematical models when the errors in all measured variables are taken into account. Though this method requires more information than do ordinary non-linear least-squares procedures, it provides more reliable estimates of rt and r2 as well as their confidence limits. 

Another important recent contribution is the provision of a good measurement of the precision of estimated reactivity ratios. The calculation of independent standard deviations for each reactivity ratio obtained by linear least squares fitting to linear forms of the differential copolymer equations is invalid, because the two reactivity ratios are not statistically independent. Information about the precision of reactivity ratios that are determined jointly is properly conveyed by specification of joint confidence limits within which the true values can be assumed to coexist. This is represented as a closed curve in a plot of r and r2- Standard statistical techniques for such computations are impossible or too cumbersome for application to binary copolymerization data in the usual absence of estimates of reliability of the values of monomer feed and copolymer composition data. Both the nonlinear least squares and the EVM calculations provide computer-assisted estimates of such joint confidence loops [15]. 

Figure 13 shows the compositional drift for a polymerization of acrylamide with DMAEM at 50 C. The quaternary ammonium monomer is consumed much faster than the acrylamide. Included in the figure is the predicted copolymer composition based on the reactivity ratios described in the second part of this chapter. With one exception, all the data are contained within the 95% confidence limits. We can, therefore, conclude... 

Figure 13. Experimental monomer composition (o) for an AAM-DMAEM inverse microsuspension copolymerization at 50 C. The reaction conditions are the same as in Figure 12. The dashed line is the predicted compositional drift based on the reactivity ratios measured in solution polymerization. The solid lines are the 95% confidence limits.

Copolymer-reactivity ratios obtained from the feed and copolymer composition data with linearized equations, as in the Fineman-Ross procedure, do not allow proper weighting of the experimental data, and cannot provide a proper estimate of the precision of the parameters, which, being interdependent, have joint confidence limits. Computer-based methods for determining reactivity ratios have been summarized and non-linear least squares methods described.. Errors in the dependent variables were included by Yamada... 

The reactivity ratio and Q,e values for were obtained in styrene (M2) copolymerizations using mol percents of in the feed between 3.2 and 90.9. The values of rj = 0.16 (0.13-0.18) and T2 1.55 (1.41-1,71) exhibited a rather large 95% joint confidence limit, but it is again quite clear that the vinyl group is very electron-rich (e = -1.98). We therefore see a remarkable similarity in the value of e" to the values obtained with every other n -cyclo-pentadienyl monomer studied to date. Also, in agreement with other such monomers, a large value of Q (1.66) was found. 

Recalculations yield negative reactivity ratios in several cases. We are aware that this is a physically unrealistic artifact. When the 95% confidence limits are applied to these negative numbers, a value of zero usually falls within these limits. It may be noted that a single reference sometimes contains a variety of reactivity ratios for one monomer pair. The reader m assume that these result from a change of polymerization conditions eg., different polymerization temperatures or polymerization media of varying polarity. 

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