Penultimate model reactivity ratios

If the terminal model adequately explains the copolymer composition, as is often the case, the terminal model is usually assumed to apply. Even where statistical tests show that the penultimate model does not provide a significantly better fit to experimental data than the tenninal model, this should not be construed as evidence that penultimate unit effects are unimportant. It is necessary to test for model discrimination, rather than merely for fit to a given model. In this context, it is important to remember that composition data are of very low power when it comes to model discrimination. For MMA-S copolymerization, even though experimental precision is high, the penultimate model confidence intervals are quite large 0.4 rAAB BAB 2.7, The terminal model 

Triad information is more powerful, but typically is subject to more experimental error and signal assignments are often ambiguous (Section 7,3.3.2). Triad data for the MMA-S system are consistent with the terminal model and support the view that any penultimate unit effects on specificity are small. 

Further evidence that penultimate unit effects are small in the MMA-S system comes from comparing the reactivities of small model radicals with the reactivity ratios (Section 7.3.1.2.2 and Table 7.4). 

The kinetics of many eopolymerizations have now been examined with absolute (overall) propagation rate constants being determined by the rotating sector, PLP or ESR methods. A similar situation as pertains for the MMA-S 

The values of arid are not well defined by kinetic data. The wide variation in a a and i o for MMA-S copolymerization shown in Table 7.5 reflects the large uncertainties associated with these values, rather than differences in the rate data for the various experiments. Partly in response to this, various simplifications to the implicit penultimate model have been used (e.g. /a3 b.a= a a.vh and. s A=- ii). These problems also prevent trends in the values wdth monomer structure from being established. 

The Chemistry of Radical Polymerization Table 7.5. Implicit Penultimate Model Reactivity Ratios 

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Similarly, penultimate model reactivity ratios can be estimated from initial monomer feed composition and triad concentrations using eqs. 50-53. 

We have previously reviewed ( 1, 2) the methods used to calculate structural features of copolymers and terpolymers from monomer reactivity ratios, monomer feed compositions and conversions, and have written a program for calculating structural features of copolymers from either terminal model or penultimate model reactivity ratios (3). This program has been distributed widely and is in general use. A listing of an instructive program for calculating structural features of instantaneous terpolymers from monomer feed compositions and terminal model reactivity ratios was appended to one of our earlier reviews (.1). 

Scheme 7.t. b Poly(AN-erj-S) chain, c Value for I-plienylethyl radical from Table. t.6 3.4.1.1. Value from penultimate model reactivity ratios for AN-S copolymerizafion at 60... 

Table 8.5 Solvent Dependence of Penultimate Model Reactivity Ratios for S-AN ...

Relationships between second-order Markov addition probabilities and penultimate model reactivity ratios have also been derived and can be found in [5]. Similarly, for the complex participation model, Cais et al have derived expressions which enable calculation of any given sequence from a knowledge of reactivity ratios and the equilibrium constant for complex formation determined by other methods [6]. 

The penultimate model reactivity ratios obtained from the mathematical analyses of the compositions and the triad fractions are given in Table 1, together with the standard error, Sy, in the fitted data parameter. The two sets of reactivity ratios are in excellent agreement. The fit of the model to the experimental data is demonstrated in Figures 3 and 4. 

Penultimate model reactivity ratios at 60"C for AN and S polymerized in bulk. The standard error in the experimental data (Sy) is calculated from the curve fit. 

Mathematical analysis of the copolymer composition data and the triad fraction data indicates that the copolymerizations are best described by a penultimate model. The penultimate model reactivity ratios obtained from the analysis of the copolymer compositions are given in Table 3 for each solvent. 

Penultimate model reactivity ratios calculated from analysis of copolymer compositions for copolymerizations at 60°C. 

It is also possible to process copolymer composition data to obtain reactivity ratios for higher order models (e.g. penultimate model or complex participation, etc.). However, composition data have low power in model discrimination (Sections 7.3.1.2 and 7.3.1.3). There has been much published on the subject of the design of experiments for reactivity ratio determination and model discrimination.49 "8 136 137 Attention must be paid to the information that is required the optimal design for obtaining terminal model reactivity ratios may not be ideal for model discrimination.49... 

The apparent terminal model reactivity ratios are then r => aK and c =rR, K It follows that rABVBf = rABrBA - const. The bootstrap effect does not require the terminal model and other models (penultimate, complex participation) in combination with the bootstrap effect have been explored.103,1 4215 Variants on the theory have also appeared where the local monomer concentration is a function of the monomer feed composition.11[Pg.431]

This program functions in essentially the same manner as Program A, except it can accept as input 18 reactivity ratios, corresponding to a situation where all monomers exhibit penultimate effects. As is also the case in Program A, terminal model reactivity ratios can be used, however. The programming involves the calculation of 27 conditional probabilities. These probabilities are calculated in the same way that P(AA/AA), P(AA/BA) and P(AA/CA) are calculated in the case of Program A. 

The early kinetic models for copolymerization, Mayo s terminal mechanism (41) and Alfrey s penultimate model (42), did not adequately predict the behavior of SAN systems. Copolymerizations in DMF and toluene indicated that both penultimate and antepenultimate effects had to be considered (43,44). The resulting reactivity model is somewhat compHcated, since there are eight reactivity ratios to consider. 

Cases have been reported where the application of the penultimate model provides a significantly better fit to experimental composition or monomer sequence distribution data. In these copolymerizations raab "bab and/or C BA rBBA- These include many copolymerizations of AN, 4 26 B,"7 MAH28" 5 and VC.30 In these cases, there is no doubt that the penultimate model (or some scheme other than the terminal model) is required. These systems arc said to show an explicit penultimate effect. In binary copolynierizations where the explicit penultimate model applies there may be between zero and three azeotropic compositions depending on the values of the reactivity ratios.31... 

The arrangement of monomer units in copolymer chains is determined by the monomer reactivity ratios which can be influenced by the reaction medium and various additives. The average sequence distribution to the triad level can often be measured by NMR (Section 7.3.3.2) and in special cases by other techniques.100 101 Longer sequences are usually difficult to determine experimentally, however, by assuming a model (terminal, penultimate, etc.) they can be predicted.7 102 Where sequence distributions can be accurately determined Lhey provide, in principle, a powerful method for determining monomer reactivity ratios. 

A general method has been developed for the estimation of model parameters from experimental observations when the model relating the parameters and input variables to the output responses is a Monte Carlo simulation. The method provides point estimates as well as joint probability regions of the parameters. In comparison to methods based on analytical models, this approach can prove to be more flexible and gives the investigator a more quantitative insight into the effects of parameter values on the model. The parameter estimation technique has been applied to three examples in polymer science, all of which concern sequence distributions in polymer chains. The first is the estimation of binary reactivity ratios for the terminal or Mayo-Lewis copolymerization model from both composition and sequence distribution data. Next a procedure for discriminating between the penultimate and the terminal copolymerization models on the basis of sequence distribution data is described. Finally, the estimation of a parameter required to model the epimerization of isotactic polystyrene is discussed. 

The implicit penultimate model was proposed for copolymerizations where the terminal model described the copolymer composition and monomer sequence distribution, but not the propagation rate and rate constant. There is no penultimate effect on the monomer reactivity ratios, which corresponds to... 

The reader is cautioned that literature references prior to 1985-1990 did not distinguish between the explicit and implicit penultimate models. The prior penultimate model did not correspond to either the explicit or implicit penultimate models. The pre-1985-1990 penultimate model contained only the four monomer reactivity ratios (Eq. 6-74) with no radical... 

Application of the penultimate model to the experimental results of [64] resulted in the following reactivity ratios ... 

The calculations of the statistical characteristics of such polymers within the framework of the kinetic models different from the terminal one do not present any difficulties at all. So in the case of the penultimate model, Harwood [193-194] worked out a special computer program for calculating the dependencies of the sequences probabilities on conversion. Within the framework of this model, Eq. (5.2) can be integrated in terms of the elementary functions as it was done earlier [177] in order to calculate copolymer composition distribution in the case of the simplified (r 2 = Fj) penultimate model. In the framework of the latter the possibility of the existence of systems with two azeotropes was proved for the first time and the regions of the reactivity ratios of such systems [6] were determined. In a general version of the penultimate model (2.3-24) the azeotropic compositions x = 1/(1 + 0 ) are determined [6] by the positive roots 0 =0 of the following... 

The kinetic copolymerization models, which are more complex than the terminal one, involve as a rule no less than four kinetic parameters. So one has no hope to estimate their values reliably enough from a single experimental plot of the copolymer composition vs monomer feed composition. However, when in certain systems some of the elementary propagation reactions are forbidden due to the specificity of the corresponding monomers and radicals, the less number of the kinetic parameters is required. For example, when the copolymerization of two monomers, one of which cannot homopolymerize, is known to follow the penultimate model, the copolymer composition is found to be dependent only on two such parameters. It was proposed [26, 271] to use this feature to estimate the reactivity ratios in analogous systems by means of the procedures similar to ones outlined in this section. 

Table 6.8 Parameters (2.4) of the penultimate model (2.3) describing copolymerization of styrene M, with acrylonitrile M2 in toluene solution at T = 60 °C. The values of reactivity ratios were obtained [283] from the data on copolymer composition (I) and triad distribution (II)...

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