(432c) A Systematic Approach to Estimate Reactivity Ratios in Multicomponent Polymerization Systems

Terpolymerization systems (as representative of multicomponent polymerization systems that are largely unstudied) are undoubtedly of great importance in both academia and industry. There is always a need for understanding the underlying kinetics of such complex reaction systems and obtaining highly accurate values of rate parameters that govern these reactions and in turn determine the polymer composition and other physico-chemical properties. Reactivity ratios in multicomponent polymerization systems are critical parameters for describing chain microstructure and hence determining polymer chain composition, sequence length distribution, polymerization rate, molecular weight distribution, etc.

At the same time, there is no doubt that reactivity ratios for multicomponent polymerizations (the simplest case being the copolymerization/binary case) have been estimated  incorrectly for several decades and are still being handled inappropriately (from a statistical estimation perspective) in the scientific literature. This has resulted in a database of biased and unreliable reactivity ratios. Based on the analogy between terpolymerization and copolymerization mechanisms, reactivity ratios are extracted from existing corresponding binary reactivity ratios in the literature and very few attempts have been made for direct estimation in the literature (essentially related to the lack of trustworthy data and systematic modeling efforts for such large systems). Given that there is a lot of uncertainty regarding published binary reactivity ratios, using these values for terpolymerizations would only expand the sources of uncertainty in the terpolymerization composition equations. Moreover, the fact that one can find widely differing sets of binary reactivity ratio values for the same copolymerization system in the literature,  begs the question as to which set of values should be used. Finally, the addition of the third monomer to a binary copolymerization affects the values of monomer reactivity ratios, and hence it justifies using terpolymer data directly in order to include all the available process information in the parameter estimation scheme.

Therefore, using binary reactivity ratios seems to be an oversimplification, not only with respect to the values themselves, but also with respect to not including measures (and hence the effect) of their uncertainty. Thus, evaluating ternary reactivity ratios based directly on terpolymerization composition data can minimize or eliminate error propagation from inconsistent binary reactivity ratio analysis.

What is the correct approach for estimating reactivity ratios?The problem of reactivity ratio estimation, among several other nonlinear parameter estimation problems, where all variables (both dependent and independent) contain error, is encountered frequently in science and engineering, including process engineering studies, medical applications, polymerization reactors, thermodynamic models, and so on. For such cases, results from basic nonlinear regression, where only dependent variables contain considerable amounts of error, would yield imprecise and biased parameter estimates. A relatively recent approach is the error-in-variables-model (EVM) that is probably the most complete approach for situations where the dependent and independent variables do not need to be distinguished. This feature makes EVM the perfect method for estimating reactivity ratios in multicomponent polymerizations.

Examining the literature shows that for terpolymerization studies very little work has been done for estimation of the reactivity ratios directly from terpolymerization experimental data sets. And of course, designing terpolymerization experiments for such a purpose (i.e., optimal selection of the location of the data points along the experimental operating region) has not been studied at all. Duever et al. (1983) used the Alfrey-Goldfinger model (instantaneous terpolymerization composition equation, analogous to the Mayo-Lewis model in copolymerization) and implemented the EVM parameter estimation technique for this problem. The reactivity ratio estimates from this and a few more earlier, yet incomplete, approaches show lack of precision and reliability. Within our investigations, since the EVM approach is the most appropriate technique for estimating these parameters, we have attributed such problems to the lack of informative experimental data, adequate information about the error structure of the collected data, lack of optimal design of experiments, and even the structure of the Alfrey-Goldfinger equation itself. Each one of these problems can severely affect the reliability of the results and our analysis showed that all of these factors are partially responsible for lack of reliability in the results.  To overcome these problems, we have investigated:

(1)     Firstly, the effect of different error structures. It is highly important that the correlations among different measurements are considered for parameter estimation. As well, the question about whether all three components are measured or their values are normalized has to be clarified. These details are all incorporated into the variance-covariance matrix of measurements that is given to the EVM procedure.

(2)     Secondly, the amount of information contained in the experimental data. Since collecting experimental data that result in precise parameter estimates is a resource-intensive task, there is always a need for designing experiments in an optimal fashion, thus minimizing overall effort and maximizing the information from the process in question. Such an approach has never been pursued in the literature to our knowledge and any parameter estimation based on non-designed (arbitrary) experiments will show a great deal of uncertainty in the results. So, we looked at the amount of information provided by different sets of experimental data as well as simulating optimal ternary experiments to find out about their influence on the estimated reactivity ratios.

(3)     Finally, different forms of the instantaneous terpolymer composition equation were considered. The results of reactivity ratio estimation based on the classical Alfrey-Goldfinger equation always show considerable level of correlation among parameter estimates. Also, since this model consists of two ratios of terpolymer compositions (i.e., F_i/F_j and F_i/F_k), depending on the combination of the mole fraction ratios that are taken into account in this equation, the level of uncertainty of our parameter estimation results varies. These results clearly state that working with the Alfrey-Goldfinger equation will always lead to questionable reactivity ratios. So, we derived a different form of this composition equation in which F_i, F_j, and F_kstand alone and implemented the EVM technique on this new ternary copolymer composition expression.  

Our intention is to evaluate how significantly the quality of reactivity ratio estimates can be improved if terpolymerization data are used directly.  To do so, we wanted first to successfully estimate reactivity ratios from existing ternary experimental data in the literature; and subsequently, to compare these ternary-based reactivity ratios with binary ones that were reported in the literature. This approach has been applied for several ternary systems such as acrylonitrile/styrene/methyl methacrylate, ethylene/vinyl acetate/ methyl acrylate, indene/methyl methacrylate/acrylonitrile, acrylonitrile/styrene/maleic anhydride, etc. The observations from these analyses point to very important remarks and insights about estimating ternary reactivity ratios from terpolymerization data, which are outlined in the following:

(1)     Utilizing the new terpolymerization expression resolved the problem of uncertainties in reactivity ratio estimates for the same system when using the Alfrey-Goldfinger equation with different combinations of mole fractions ratios.

(2)     The correct error structure for this problem involves the correlation between measurements for the terpolymer composition (e.g., mole fractions F_i and F_j). Ternary reactivity ratios can be estimated when this negative correlation ranges from -0.2 to -0.8. The magnitude of this correlation can be calculated either from replicated experiments.

(3)     The reactivity ratio estimates are in most cases based on very limited experimental data and therefore their precision is not acceptable; these parameters are often highly correlated, which can be related to non-designed experimentation.

(4)     As process responses, either two out of three mole fractions in the terpolymer composition are measured or all three mole fractions are acquired independently. This choice affects both the EVM model as well as the variance-covariance matrix of measurements and of course the estimated reactivity ratios.

(5)     Using simulated ternary data without any ambiguity regarding the error structure and the measurement responses, resolved the problems around the selection of measurements and the correlations between those measurements.

(6)     Choosing rich information content experimental data sets and utilizing ‘anti-correlation’ design of experiments provides results that are of higher precision.

All these results, therefore, point out that ternary reactivity ratios can be estimated from ternary experimental data, and that binary reactivity ratios are not representative for a ternary systems’ kinetic characteristics. However, since the success of the overall analysis strongly depends on the information contained in the data, data accuracy and experimental design are extremely important for parameter estimation and thus they can affect the conclusions drawn. Therefore, terpolymerization experiments with replicates and optimal experimental designs could greatly enhance parameter estimation. More details on the optimal design of experiments will be presented at the time of the conference.