Terminal and Penultimate Models

Such dependencies are quite usual for the terminal and penultimate models since in these cases the sequence distributions are described by Markov statistics [51-53, 6]. In the former case this description is carried out by means of the Markov chain, the states S of which correspond to the individual monomer units... 

The terminal and penultimate models then correspond to first- and second-order Markovian statistics, respectively. But you don t actually have to know this, in the sense that we can just proceed using common sense. For example, the probability of finding the sequence ABABA in a system obeying first-order Markovian statistics (i.e., copolymerization where the terminal model applies) is given in Equations 6-31. 

Burke, A. L., T. A. Duever, and A. Penlidis. Model discrimination via designed experiments Discriminating between the terminal and penultimate models on the basis of composition data. Macromolecules, 27. 386-399 (1994). 

In the early 1940s when the polymerization theory was developed, tiie ideal, terminal, and penultimate models for fhe copolymerization were established also the possible distribution laws for the monomer sequence along the copolymer chains were defined Bemoullian, firsf- and second-order Markoffian. ... 

Rg. 1.3. Comparison of the molar ratio in the polymer as a function of the molar ratio in the feed for the terminal and penultimate models with two sets of reactivity ratios. The reactivity ratios for the terminal model were r =0.1 and re = 10. For the penultimate model, the reactivity ratios were ka = 0.94, ta = 0.01, ru = 0.9, and rp = 5.0. (Reproduced with permission from Ref. [9]. 1964 John Wiley and Sons, Inc.)... 

This figure shows the molar ratio in the polymer versus the molar ratio in the feed, jt, calculated for reactivity ratios in the intermediate range for the terminal and penultimate models. This figure clearly shows the small differences (2%) in composition between the two mechanisms of polymerization. This result might seem unimportant. Let us examine the sequence distribution using the same reactivity ratios for the two models. The number distribution of sequences is shown in Fig. 1.4, and the weight-average distribution is shown in Fig. 1.5 [9]. 

This final question asks you to try and determine if a set of experimental data fits the terminal or penultimate model. For a copolymerization of A and B monomers (at low degrees of conversion) the following data were obtained. 

A flrst important question concerns whether the goal is to discriminate between competing models (i.e., terminal vs penultimate model kinetics) or to seek the best parameter estimates. We flrst assume that terminal model kinetics are being considered and later discuss implications regarding the assumption of penultimate model kinetics. As seen in the previous section, for terminal model kinetics, reactivity ratios are typically estimated using the instantaneous copolymer composition equation or the Mayo-Lewis equation, expressed in two common forms. Equations 6.7 and 6.11. 

Assuming terminal propagation kinetics, the best fit for was found to be much greater than unity such that kt was greater than either homo-termination value, a non-sensical result as is diffusion-controlled. When the deviation of propagation kinetics from the terminal model is taken into account, however, the estimates for kt become well-behaved and bounded by the homo-termination values [26]. Various penultimate models that account for the influence of polymer composition on segmental diffusion have been proposed to fit low-conversion kt data. Equation 3.49 emphasizes the role of the whole chain composition [26], while other formulations use both the terminal and penultimate units to represent the conformational characteristics of the last portion of the polymer chain [33, 34]. 

In the explicit penultimate model, it is assumed that both the terminal and penultimate units of a polymer radical may affect the rate of the propagation reaction. As in the terminal model, side reactions are considered to be insignificant. The explicit penultimate model was first suggested in 1946 by Merz and co-workers (10), who derived equations for predicting the composition and sequence distribution under this model. A full description of the model—including an expression for ( p)— has since been provided by Fukuda and co-workers (11), and it is their notation that is used below. In addition, penultimate model equations for the case of terpolymerization have been published by Coote and Davis (12). 

Brar and Sunita [58] described a method for the analysis of acrylonitrile-butyl acrylate (A/B) copolymers of different monomer compositions. Copolymer compositions were determined by elemental analyses and comonomers reactivity ratios were determined using a non-linear least squares errors-in-variables model. Terminal and penultimate reactivity ratios were calculated using the observed distribution determined from C( H)-NMR spectra. The triad sequence distribution was used to calculate diad concentrations, conditional probability parameters, number-average sequence lengths and block character of the copolymers. The observed triad sequence concentrations determined from C( H)-NMR spectra agreed well with those calculated from reactivity ratios. 

In eqn [24], fi is the mole fraction of monomer i in the monomer mixture. In the penultimate models, two monomer-derived segments at each radical chain end are taken into account, for example, by the rate coefficient kuy which refers to addition of a monomer molecule j to a radical chain end where both terminal and penultimate units consist of spedes i. Equation [24] is used to derive q and also q =fejj/feji from measured compositions of monomer mixture and copolymer, fi and Fi, respectively. 

These observations suggest how the terminal mechanism can be proved to apply to a copolymerization reaction if experiments exist which permit the number of sequences of a particular length to be determined. If this is possible, we should count the number of Mi s (this is given by the copolymer composition) and the number of Mi Mi and Mi Mi Mi sequences. Specified sequences, of any definite composition, of two units are called dyads those of three units, triads those of four units, tetrads those of five units, pentads and so on. Next we examine the ratio NmjMi/Nmi nd NmjMiMi/NmiMi If these are the same, then the mechanism is shown to have terminal control if not, it may be penultimate control. To prove the penultimate model it would also be necessary to count the number of Mi tetrads. If the tetrad/triad ratio were the same as the triad/dyad ratio, the penultimate model is proved. 

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. 

The first quantitative model, which appeared in 1971, also accounted for possible charge-transfer complex formation (45). Deviation from the terminal model for bulk polymerization was shown to be due to antepenultimate effects (46). Mote recent work with numerical computation and C-nmr spectroscopy data on SAN sequence distributions indicates that the penultimate model is the most appropriate for bulk SAN copolymerization (47,48). A kinetic model for azeotropic SAN copolymerization in toluene has been developed that successfully predicts conversion, rate, and average molecular weight for conversions up to 50% (49). 

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... 

For many systems, the copolymer composition appears to be adequately described by the terminal model yet the polymerization kinetics demand application of the penultimate model. These systems where rAAB=rliAR and aha bba hut sAfsB are said to show an implicit penultimate effect. The most famous system of this class is MMA-S copolymerization (Section 7.3.1.2.3). 

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... 

More recent work has shown that the observed variation in propagation rate constants with composition is not sufficient to define the polymerization rates.5" 161,1152 There remains some dependence of the termination rate constant on the composition of the propagating chain. Thus, the chemical control (Section 7.4.1) and the various diffusion control models (Section 7.4.2) have seen new life and have been adapted by substituting the terminal model propagation rate constants (ApXv) with implicit penultimate model propagation rate constants (kpKY -Section 7.3.1.2.2). 

More complex models for diffusion-controlled termination in copolymerization have appeared.1 tM7j Russo and Munari171 still assumed a terminal model for propagation but introduced a penultimate model to describe termination. There are ten termination reactions to consider (Scheme 7.1 1). The model was based on the hypothesis that the type of penultimate unit defined the segmental motion of the chain ends and their rate of diffusion. 

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 is the simplest of the models where violation of the Flory principle is permitted. The assumption behind this model stipulates that the reactivity of a polymer radical is predetermined by the type of bothjts ultimate and penultimate units [23]. Here, the pairs of terminal units MaM act, along with monomers M, as kinetically independent elements, so that there are m3 constants of the rate of elementary reactions of chain propagation ka ]r The stochastic process of conventional movement along macromolecules formed at fixed x will be Markovian, provided that monomeric units are differentiated by the type of preceding unit. In this case the number of transient states Sa of the extended Markov chain is m2 in accordance with the number of pairs of monomeric units. No special problems presents writing down the elements of the matrix of the transitions Q of such a chain [ 1,10,34,39] and deriving by means of the mathematical apparatus of the Markov chains the expressions for the instantaneous statistical characteristics of copolymers. By way of illustration this matrix will be presented for the case of binary copolymerization ... 

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 precision and accuracy of the experimental data must be sufficient to allow one to discriminate between the terminal, explicit penultimate, and implicit penultimate models, [Burke et al., 1994a,b, 1995 Landry et al., 2000]. This has not always been the case, especially in the older literature, and the result has sometimes been contradictory reports. Penultimate effects are most easily detected in experiments carried out by including data at very low or very high f values. 

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