Chain copolymerization penultimate model

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 general formulae (5.1), (5.3), and (5.7) are still valid under the transition to the more complicated models described in Sect. 2. In the case of the penultimate model it concerns also the dynamic Eqs. (5.2) into which now one should substitute the dependence j (i) obtained after the solution of the problem of the calculation of the stationary vector tE(x) of the Markov chain corresponding to this model. Substituting the function X1(x1) obtained via the above procedure (see Sect. 3.1) into Eq. (5.2) for the binary copolymerization we can find its explicit solution expressed through the elementary functions. However, this solution is rather cumbersome and has no practical importance. It is not needed even for the classification of the dynamic behavior of the systems, which can be carried out only via analysis of Eq. (5.5) by determining the number n = 0,1,2, 3 of the inner azeotropes in the 2-simplex [14], The complete set of phase portraits of the binary... 

You have no doubt been asking yourself Why are these dirty, rotten swine doing this to me Step back and think about what we ve said about copolymerization. There is something called the terminal models where the rate constants depend only on the nature of the terminal group, and the penultimate model, where there is a dependence on the character of the final two units in the chain (Figure 6-17). The use of conditional probabilities is the starting point for distinguishing between these experimentally, as we will see. 

The simple copolymer model, with two reactivity ratios for a binary comonomer reaction, explains copolymer composition data for many systems. It appears to be inadequate, however, for prediction of copolymerization rates. (The details of various models that have been advanced for this purpose are omitted here, in view of their limited success.) Copolymerization rates have been rationalized as a function of feed composition by invoking more complicated models in which the reactivity of a macroradical is assumed to depend not Just on the terminal monmomer unit but on the two last monomers in the radical-ended chain. This is the penultimate model, which is mentioned in the next Section. 

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 ... 

In these models, the complex formed by the monomer pair competes with the individual monomer molecules for the propagation reaction with the radicals. There are two variations of this approach in the complex participation model, the pair of monomers form a complex and are added to the chain radical [106-109]. On the other hand, in the complex dissociation model, the complex participates in the propagation process, but dissociates upon reaction and only one of the monomers is added to the chain [101, 103]. Although there is ample experimental evidence for the existence of such complexes in these copolymerizations (such as the bright colors associated with them) [76], it is questionable whether the complexes actually participate in the propagation step [76]. Additionally, for several years, Hall and Padias have accumulated experimental and theoretical evidence that refutes the validity of the models based on complex participation [76, 77]. Both the complex participation and the penultimate models were combined in the so-called comppen model [110]. 

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. ... 

Figure 1 shows the NMR spectra of poly( -co- ) whose monomer ratios ( and ) were 1 9, 1 1, and 9 1. Since monomer 3 is very bulky, the propagation process of this copolymerization is considered to be influenced by not only a terminal monomer unit but also the penultimate one of the growing chain, and is interpreted as a penultimate model. 

The polymerization model most commonly adopted for olefin copolymerization is the terminal model, particularly for studies of polymerization kinetics. In the terminal model, only the last monomer molecule added to the chain end influences polymerization and transfer rates. Besides the fact that it is logically expected, there is also significant experimental evidence supporting the terminal model for olefin polymerization. Since monomer propagation and chain-transfer reactions take place by insertion between the chemical bond formed by the metal in the active site and the polymer chain end, it is certainly reasonable to assume that both the nature of the active site and the type of monomer last added to the chain will affect these reactions. On the other hand, higher-order models such as the penultimate and pen-penultimate models have not found widespread use in coordination polymerization. 

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. 

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 ... 

This assumption is implicitly present not only in the traditional theory of the free-radical copolymerization [41,43,44], but in its subsequent extensions based on more complicated models than the ideal one. The best known are two types of such models. To the first of them the models belong wherein the reactivity of the active center of a macroradical is controlled not only by the type of its ultimate unit but also by the types of penultimate [45] and even penpenultimate [46] monomeric units. The kinetic models of the second type describe systems in which the formation of complexes occurs between the components of a reaction system that results in the alteration of their reactivity [47-50]. Essentially, all the refinements of the theory of radical copolymerization connected with the models mentioned above are used to reduce exclusively to a more sophisticated account of the kinetics and mechanism of a macroradical propagation, leaving out of consideration accompanying physical factors. The most important among them is the phenomenon of preferential sorption of monomers to the active center of a growing polymer chain. A quantitative theory taking into consideration this physical factor was advanced in paper [51]. 

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 simple copolymer model is a first-order Markov chain in which the probability of reaction of a given monomer and a macroradical depends only on the terminal unit in the radical. This involves consideration of four propagation rate constants in binary copolymerizations, Eqs. (7-2)-(7-4). The mechanism can be extended by including a penultimate unit effect in the macroradical. This involves eight rate constants. A third-order case includes antepenultimate units and 16 rate coefficients. A true test of this model is not provided by fitting experimental and predicted copolymer compositions, since a match must be obtained sooner or later if the number of data points is not saturated by the adjustable reactivity ratios. 

Even though the discussion has been mainly on homopolymerization, the same polymerization mechanism steps are valid for copolymerization with coordination catalysts. In this case, for a given catalyst/cocatalyst system, propagation and transfer rates depend not only on the type of coordinating monomer, but also on the type of the last monomer attached to the living polymer chain. It is easy to understand why the last monomer in the chain will affect the behavior of the incoming monomer as the reacting monomer coordinates with the active site, it has to be inserted into the carbon-metal bond and will interact with the last (and, less likely, next-to-last or penultimate) monomer unit inserted into the chain. This is called the terminal model for copolymerization and is also commonly used to describe free-radical copolymerization. In the next section it will be seen that, with a proper transformation, not only the same mechanism, but also the same polymerization kinetic equations for homopolymerization can be used directly to describe copolymerization. 

This model for the copolymerization described by Eq. (3.17) is called the terminal model. The model needs four propagation steps to describe the rate of copolymerization, implying that only the last unit in the growing chain determines the reactivity. However, it is known in literature (9) that for a system like, e.g., styrene-methylmethacrylate, the expression for the propagation rate [Eq. (3.17)] is not sufficient to give a good description of the polymerization. Not only the last unit, but also the one before the last unit may influence the reactivity. This eflect is known as the penultimate ejfect. 

The ATRP is based on a sequence of ATRA reactions and ATRA can be used to prepare various dimeric species that model the expected chain ends in a copolymerization reaction. An ESR study of radicals generated from these mixed dimers can be used to clarify the penultimate unit effects relevant to copolymerization. The ESR spectra of monomeric radicals of (meth)acrylates were previously investigated by Fischer et Gilbert et al., and Matsumoto et al. However, no ESR... 

The basic kinetic equations for chain addition copolymerization are given in Table I for three termination models geometric mean (GM), phi factor (PF) and penultimate effect (PE). It Is important to note the symmetry in form created by confining the effect of choice of termination model to a single factorable function H. 

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