Penultimate effect model

Termination scheme 11 applies to the geometric mean and phi factor models and scheme 12 Is required for the penultimate effect model. All the above reaction models were used In attempts to simulate kinetic data. 

A similar logic can be applied to copolymers. The story is a bit more complicated to tell, so we only outline the method. If penultimate effects operate, then the probabilities Ph, Pi2> and so on, defined by Eqs. (7.32)-(7.35) should be replaced by conditional probabilities. As a matter of fact, the kind of conditional probabilities needed must be based on the two preceding events. Thus reactions (7.E) and (7.F) are two of the appropriate reactions, and the corresponding probabilities are Pj n and V i2 - Rather than work out all of the possibilities in detail, we summarize the penultimate model as follows ... 

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). 

Penpenultiraale and higher order remote unit effect models may also affect the outcome of copolymerizations. However, in most eases, experimental data, that are not sufficiently powerful to test the penultimate model, offer little hope of testing higher order models. The importance of remote unit effects on copolymerization will only be fully resolved when more powerful analytical techniques become available. 

Analysis of the data collected by Hill et al. illustrates how a four parameter problem can be reduced to one involving only two parameters by appropriately formulating the problem. In addition an approach to fitting models directly to triad fraction data is shown. Our analysis confirms the conclusion made by Hill et al. that a penultimate effect does exist for both styrene and acrlyonitrile. 

More complex schemes have been proposed, such as second-order Markov chains with four independent parameters (corresponding to a copolymerization with penultimate effect, that is, an effect of the penultimate member of the growing chain), the nonsymmetric Bernoulli or Markov chains, or even non-Maikov models a few of these will be examined in a later section. Verification of these models calls for the knowledge of the distribution of sequences that become longer, the more complex the proposed mechanism. Considering only Bernoulli and Markov processes it may be said that at the dyad level all models fit the experimental data and hence none can be verified at the triad level the Bernoulli process can be verified or rejected, while all Markov processes fit at the tetrad level the validity of a first-order Markov chain can be confirmed, at the pentad level that of a second-order Maikov chain, and so on (10). 

The original mathematical treatment of the penultimate effect was presented by Merz and coworkers [Barb, 1953 Ham, 1964 Merz et al., 1946], Fukuda and coworkers developed a more extensive treatment, which distinguished between two penultimate models of copolymerization behavior—the explicit penultimate and implicit penultimate models [Coote and Davis, 1999, 2000 Davis, 2001 Fukuda et al., 1985, 1987, 1992, 2002 Ma et al., 2001], The explicit penultimate model for copolymerization involves the use of eight propagation reactions... 

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. 

Figures 6-12 and 6-13 shows plots of copolymer composition and propagation rate constant, respectively, versus comonomer feed composition for styrene-diethyl fumarate copolymerization at 40°C with AIBN [Ma et al., 2001]. The system follows well the implicit penultimate model. The copolymer composition data follow the terminal model within experimental error, which is less than 2% in this system. The propagation rate constant shows a penultimate effect, and the results conform well to the implicit penultimate model with si = 0.055, S2 — 0.32.

The ability to determine which copolymerization model best describes the behavior of a particular comonomer pair depends on the quality of the experimental data. There are many reports in the literature where different workers conclude that a different model describes the same comonomer pair. This occurs when the accuracy and precision of the composition data are insufficient to easily discriminate between the different models or composition data are not obtained over a wide range of experimental conditions (feed composition, monomer concentration, temperature). There are comonomer pairs where the behavior is not sufficiently extreme in terms of depropagation or complex participation or penultimate effect such that even with the best composition data it may not be possible to conclude that only one model fits the composition data [Hill et al., 1985 Moad et al., 1989]. 

BD/St-copolymers were also prepared by the use of the catalyst system Nd(OCOCCl3)3/TIBA/DEAC [503]. The copolymer exhibited 79% cis-1,4-structure in the BD units at a content of incorporated styrene of 23 mol%. In this study diades were also determined. According to the authors BD moieties which are adjacent to St moieties predominantly exhibit a transit-configuration whereas BD moieties in BD-BD-diades exhibit a cis-1,4-structure [503]. It therefore can be concluded that the microstructure of an entering BD monomer is controlled by a penultimate effect. This effect can be best described by a model in which backbiting coordination of a penultimate BD unit to Nd is involved [177,367]. 

Variants of the above have been used to explain unusual data where it was suspected that monomer units penultimate to the chain end (or even further back) were affecting the reaction rate constants (6). Such treatments suffer from a degree of arbitrariness in that the experimental data may not provide an adequate test of the kinetic model (3). In some cases, penultimate unit effect models have been used to interpret unusual data where one might expect depropagation to be important. 

Another series of papers [296-298] should be mentioned, where low-molecular model compounds are used to prove the correctness of the penultimate model of copolymerization. Japanese scientists by means of the ESP-method [297-298] managed to observe a noticeable penultimate effect for the acrylate radical reactivity. 

This is the case, for example, in the copolymerization of carbon monoxide and ethylene where the CO will not add to itself but does copolymerize with the olefin monomer. General theoretical treatments have been developed for such cases, taking into account temperature and penultimate effects. Again, the superiority of these more complicated theories over the simpler copolymer model is not proved for all systems to which they have been applied. 

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 experimental copolymer composition data for styrene(Mi)-fumaroni-trile(M2) give a good fit to Eq. (7.86) with rj = 0.072 and r[ = 1.0 [33], but deviate markedly from the behavior predicted by the st-order Markov model with ri = 0.23. Penultimate effects have been observed in a number of other systems. Among these are the radical copolymerizations of ethyl methacrylate-styrene, methyl methaciylate-4-vinyl pyridine, methyl acrylate-l,3-butadiene, and other monomer pairs. 

IPUE implicit penultimate unit effect (model for copolymerisation)... 

Penultimate Model Some copolymerization systems in which the values for reactivity ratios measured at different compositions are inconsistent can be adequately represented by the penultimate model [86]. In this case, the reactivity of the propagating chain depends on the chemical nature of the last two monomeric units the one at the active end and the previous one (penultimate) [87, 88], This is common in systems in which the monomers contain bulky substituents such as the fumaronitrile-styrene copolymerization [89], In other systems, the penultimate effect has been reported to be limited [90], The penultimate model can be formulated as follows. Consider the reaction of a growing chain having a penultimate unit j and terminal unit i with a monomer n, M ... 

While copolymer composition is well-described by the terminal model, the copolymer-averaged propagation rate coefficient (kp. Equation 3.45) for many common systems [10, 26, 27] is not. The measured kp values can be higher or lower than the terminal model predictions, with the deviation substantial in some cases. The implicit penultimate unit effect model, which accounts for the influence of the penultimate monomer-unit of the growing polymer radical on the propagation kinetics [26, 27], provides a good representation of this behavior ... 

---