Penultimate effects

The mathematical treatment of the penultimate effect [33,34] in a binary copolymerization system involves the use of eight propagating reactions  

Each monomer is thus characterized by two monomer reactivity ratios. One of these represents the propagating species in which the penultimate and terminal monomer units are the same, while the other represents the propagating species in which the penultimate and terminal units differ. The latter monomer reactivity ratios are denoted by prime notations (r[, r ). 

Following a procedure similar to that used in deriving Eq. (7.11), the copolymer composition equation with a kinetic penultimate effect present is obtained as 

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. 

The copolymerization data for the styrene(Mi)-fumaronitrile(M2) system indicate that there are also effects due to remote monomer units preceding the penultimate unit. The effect of remote units has been treated by further expansion of the copolymer composition equation by the use of greater number of monomer reactivity ratios for each monomer [34]. However, the utility of the resulting expression is limited due to the large number of variables involved. 

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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 ... 

Methyl-2-furaldehyde gave a similar overall behaviour, but a penultimate effect was observed in its copolymerization with isopropenylbenzene whereby two molecules of the aldehyde could add together if the penultimate unit in the growing chain was from the olefin. This was borne out by the copolymers composition and spectra. The values of the reactivity ratios showed this interesting behaviour rx = 1.0 0.1, r2 = 0.0 0.1. An apparent paradox occurred the aldehyde, which could not homo-polymerize, had equal probability of homo- and copolymerization and the olefin, which homopolymerized readily, could only alternate. The structure arising from this situation was close to a regular sequence of the type ... 

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). 

Some theoretical justifications for the prevalence of systems which show an implicit penultimate effect have appeared. These are summarized in the recent reviews by Cootc and Davis.2,3... 

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. 

Derive the equivalent of Equation (13.41) when penultimate effects are considered. 

Figure 5 shows the joint 95% posterior probability region for the two parameter functions. Shimmer bands are also indicated at the 95% probability level. This analysis confirms the results of Hill et al. that both styrene and acrlyonitrile exhibit a penultimate effect. 

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. 

Monomer sequence length distribution and penultimate effect in ethylenc-cycloolefln copolymers synthesized over homogeneous metallocene catalysts... 

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... 

Penultimate effects have been observed for many comonomer pairs. Among these are the radical copolymerizations of styrene-fumaronitrile, styrene-diethyl fumarate, ethyl methacrylate-styrene, methyl methacrylate l-vinylpyridine, methyl acrylate-1,3-butadiene, methyl methacrylate-methyl acrylate, styrene-dimethyl itaconate, hexafluoroisobutylene-vinyl acetate, 2,4-dicyano-l-butene-isoprene, and other comonomer pairs [Barb, 1953 Brown and Fujimori, 1987 Buback et al., 2001 Burke et al., 1994a,b, 1995 Cowie et al., 1990 Davis et al., 1990 Fordyce and Ham, 1951 Fukuda et al., 2002 Guyot and Guillot, 1967 Hecht and Ojha, 1969 Hill et al., 1982, 1985 Ma et al., 2001 Motoc et al., 1978 Natansohn et al., 1978 Prementine and Tirrell, 1987 Rounsefell and Pittman, 1979 Van Der Meer et al., 1979 Wu et al., 1990 Yee et al., 2001 Zetterlund et al., 2002]. Although ionic copolymerizations have not been as extensively studied, penultimate effects have been found in some cases. Thus in the anionic polymerization of styrene t-vinylpyri-dine, 4-vinylpyridine adds faster to chains ending in 4-vinylpyridine if the penultimate unit is styrene [Lee et al., 1963]. 

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]. 

What are the differences between the two treatments (kinetic penultimate effect and depropagation) used to account for the deviations observed in the copolymer composition equation ... 

In some cases, the reactivity of the growing chain end depends on the nature of the last but one monomer unit. So, eight propagation constants have to be considered. This so-called penultimate effect can be the reason why the binary copolymerization cannot be described precisely enough by Eq. 3.18. 

The composition of copolymer and distribution of units in copolymer molecule can be predicted as follows. Let us designate two types of comonomer molecules as A and B and the respective radicals as A and b1 The symbols with an asterisk deal with the process proceeding on the template. In addition, let us assume that we can neglect the penultimate effect. In this case, the process of propagation is expressed by the following set of reactions and respective rates and rate constants ... 

In the copolymerization of ethene with 4-methyl-l-pentene, a penultimate effect has been observed (8). The 4-methyl-l-pentene unit in the penultimate position causes a remarkable decrease of the ethene reactivity. For this reason, isolated units of ethene are decreased in the copolymer. Actually, just the reverse would be expected from the concept of steric hindrance. However, the particular microstructure is produced by highly isospecific catalytic sites. 

Actually polymerizations in the second group above give the most highly syndiotactic polymers and the sequence distribution is not the simple Bemoullian one expected from such a mechanism. No attempt seems to have been made to formulate a transition state for this type of polymerization. It would obviously lead to difficulties if expressed in terms of penultimate effects as the solvents used are strongly solvating and secondary interaction with penultimate units would not be favoured. 

COLEMAN and Fox (18) have pointed out that the non Bernoullian sequence distribution observed in some of these systems can be formed without the hypothesis of penultimate effects. All that is required is that two or more types of active species be present which do not rapidly interconvert. Each can add monomer at its own rate and with its own characteristic regulating effect. No penultimate effect is necessary but the sequence distribution will be non-Bernoullian. This type of mechanism is particularly attractive in the explanation of stereoblock polymer formation in the lithium alkyl systems in toluene with small amounts of ether present. The presence of at least two species of active centres has been inferred from an examination of polymer fractions obtained from butyllithium initiated polymerizations (19) in toluene. The change in molecular weight distribution with time suggests the presence of two... 

They correspond to a first-order Markov process for the stereocontrol— i.e., a penultimate effect of the last diad on stereocontrol. 

The copolymerization equation is valid if all propagation steps are irreversible. If reversibility occurs, a more complex equation can be derived. If the equilibrium constants depend on the length of the monomer sequence (penultimate effect), further changes must be introduced into the equations. Where the polymerization is subjected to an equilibrium, a-methylstyrene was chosen as monomer. The polymerization was carried out by radical initiation. With methyl methacrylate as comonomer the equilibrium constants are found to be independent of the sequence length. Between 100° and 150°C the reversibilities of the homopolymerization step of methyl methacrylate and of the alternating steps are taken into account. With acrylonitrile as comonomer the dependence of equilibrium constants on the length of sequence must be considered. 

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]. 

Some quantitative data (r1 r2, Q, e values) for the copolymerization of captodative olefins are available, mainly for the copolymerization with styrene. They are compiled in Tables 11-14. In Tables 11 and 12, the data for the copolymerization of captodative olefins with styrene are given for comparison together with the values for other 1,1-disubstituted olefins. However, these data have to be considered with caution because of the possibility that some propagation steps could be reversible and because of possible penultimate effects [95]. 

For the above reactions, it is assumed that the reactivity of the propagating radical is dependent only on its terminal radical unit. However, the rate of addition of a monomer to the growing radical depends on the type of monomer in the penultimate position. The importance of the penultimate effects has not been widely investigated. As a result, it is assumed that the simple copolymerization equations given above are valid. 

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. 

When the reactivity of the centre is determined not only by the last added unit but also by the last but one unit, we speak of the penultimate effect. Merz et al. treated this problem using eight independent reactions [200, 201 ]. 

Living polymerizations continue to attract the attention of theorists studying transfer. Chinese authors have analyzed the penultimate unit effect on transfer to a monomer mathematically [53]. According to their conclusions, the penultimate effect is important when the activities of the growth centres on the polymer chain and on the monomer (after transfer) are widely different, otherwise it can be neglected. 

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