Macromolecules 1989, 22, 1145-1147
1145
How Powerful Are Composition Data in Discriminating between the Terminal and Penultimate Models for Binary Copolymerization? Graeme Moad,* David H. Solomon, and Thomas H. Spurling CSIRO, Division of Chemicals and Polymers, GPO Box 4331 Melbourne, Victoria 3001, Australia Richard A. Stone* CSIRO, Division of Mathematics and Statistics, Clayton, Victoria 3168, Australia. Received February 24, 1988 ABSTRACT: The terminal model for binary copolymerization is often accepted simply on the grounds that it provides a satisfactory fit to a set of experimental data. The penultimate (and other) models are often not considered or are rejected as making no significant improvement. It is argued that such inferences are invalid unless reference is made to the power of the experiments in estimating penultimate unit effects. A method for assessing the possible magnitude of penultimate unit effects in binary copolymerization is offered. This involves using a nonlinear least-squares method to fit the penultimate model and constructing an approximate 95% joint confidence interval for rl/rl' and r,jri. The method is illustrated with reference to literature data for methyl methacrylatejstyrene copolymerization. An analysis of the composition data of Fukuda et al. for this system shows that while the terminal model is a possible solution, the size of the 95% joint confidence interval shows that penultimate unit effects could be substantial and the model may still give an adequate fit to the experimental composition data (0.4 Irl/r{ I2.7 and 0.3 5 rz/r2/ I2.2). Design criteria for evaluation of reactivity ratios which take into account the need for model discrimination are discussed. An understanding of the factors which influence the composition of copolymers is of significance both for industrial application and for the advancement of polymer science. Of particular importance is a need to have reliable reactivity ratios with which to predict the course of copolymerization and the composition and microstructure of the copolymers produced. At present this need is not satisfied. A variety of factors contribute to this situation and discussions on this can be found in the polymer literature.' In this paper2 we are concerned specifically with discriminating between the terminal and penultimate models used to describe binary copolymerization and with assessing the power of copolymer composition data in this context. In much of the literature on copolymerization, it is presumed that the terminal model [so called because the propagation rate constants are dependent solely upon the nature of the monomer and the last added (ultimate) unit in the polymer chain] is applicable unless proven otherwise. While this might be consistent with the application of Ockham's razor,3 it is our view that the influence of remote unit effects on the propagation (and other steps) should never be i g n ~ r e d . ~ It is well recognized that copolymer composition data are often not sufficiently sensitive to enable unambiguous discrimination between polymerization mechanism^.^ In some cases, the answer may be to turn to other methods of measuring reactivity ratios (e.g., from sequence distribution information) .l However, often such methods are not applicable and composition measurements are all that is available. For a n y data set, the four-parameter (penultimate) model will.give an improved fit over the two-parameter (terminal) model. Cases have been reported where the improvement in fit is substantial and it is clear that the penultimate model is more appropriate. An example is the copolymerization of styrene and a~rylonitrile.~*' However, for most cases, the improvement in fit is not statistically significant and estimates of penultimate model reactivity ratios are of low precision. In this circumstance it is common practice to use terminal model reactivity ratios. However, we believe that before using terminal model 0024-9291/89/2222-ll45$01.50/0
reactivity ratios for predicting monomer sequence distributions or certain aspects of polymerization kinetics, it is essential that we first assess the power of the experimental data used to derive the reactivity ratios. It is not sufficient simply to establish that the terminal model f i t s the data; it is necessary to establish t h a t remote unit effects are neg 1igible. One way to demonstrate whether a given set of experimental data are able to discriminate between the terminal and penultimate models and gain some knowledge on the error that may be introduced by choosing the simpler model is to establish what magnitude the penultimate unit effects might assume while remaining consistent with the experimental data. We suggest that a reasonable approach to this end is to obtain estimates of penultimate unit effects, rl/r< and r2/r2/ (where rl, r
