Copolymerization sequence distribution

For a growing radical chain that has monomer 1 at its radical end, its rate constant for combination with monomer 1 is designated and with monomer 2, Similady, for a chain with monomer 2 at its growing end, the rate constant for combination with monomer 2 is / 22 with monomer 1, The reactivity ratios may be calculated from Price-Alfrey and e values, which are given in Table 8 for the more important acryUc esters (87). The sequence distributions of numerous acryUc copolymers have been determined experimentally utilizing nmr techniques (88,89). Several review articles discuss copolymerization (84,85). 

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). 

Copolymerizations are processes that lead to the formation of polymer chains containing two or more discrete types of monomer unit. Several classes of copolymer that differ in sequence distribution and/or architecture will be... 

Any understanding of the kinetics of copolymerization and the structure of copolymers requires a knowledge of the dependence of the initiation, propagation and termination reactions on the chain composition, the nature of the monomers and radicals, and the polymerization medium. This section is principally concerned with propagation and the effects of monomer reactivity on composition and monomer sequence distribution. The influence of solvent and complcxing agents on copolymerization is dealt with in more detail in Section 8.3.1. 

Tire simplest model for describing binary copolyinerization of two monomers, Ma and Mr, is the terminal model. The model has been applied to a vast number of systems and, in most cases, appears to give an adequate description of the overall copolymer composition at least for low conversions. The limitations of the terminal model generally only become obvious when attempting to describe the monomer sequence distribution or the polymerization kinetics. Even though the terminal model does not always provide an accurate description of the copolymerization process, it remains useful for making qualitative predictions, as a starting point for parameter estimation and it is simple to apply. 

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... 

Methods for evaluation of reactivity ratios comprise a significant proportion of the literature on copolymerization. There are two basic types of information that can be analyzed to yield reactivity ratios. These are (a) copolymer composition/convcrsion data (Section 7.3.3.1) and (b) the monomer sequence distribution (Section 7.3.3.2). The methods used to analyze these data are summarized in the following sections. 

The solvent in a bulk copolymerization comprises the monomers. The nature of the solvent will necessarily change with conversion from monomers to a mixture of monomers and polymers, and, in most cases, the ratio of monomers in the feed will also vary with conversion. For S-AN copolymerization, since the reactivity ratios are different in toluene and in acetonitrile, we should anticipate that the reactivity ratios are different in bulk copolymerizations when the monomer mix is either mostly AN or mostly S. This calls into question the usual method of measuring reactivity ratios by examining the copolymer composition for various monomer feed compositions at very low monomer conversion. We can note that reactivity ratios can be estimated for a single monomer feed composition by analyzing the monomer sequence distribution. Analysis of the dependence of reactivity ratios determined in this manner of monomer feed ratio should therefore provide evidence for solvent effects. These considerations should not be ignored in solution polymerization either. 

Harwood112 proposed that the solvent need not directly affect monomer reactivity, rather it may influence the way the polymer chain is solvated. Evidence for the proposal was the finding for certain copolymerizations, while the terminal model reactivity ratios appear solvent dependent, copolymers of the same overall composition had the same monomer sequence distribution. This was explained in... 

The effect of a catalyst is important in cationic copolymerizations. Epoxides and /3-lactones form random copolymers only with trialkyl aluminum catalysts. Unusual sequence distributions were observed in the cationic copolymerization of epoxides or lactones using Lewis acids175-177) have been attributed to the di-... 

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. 

In their paper Hill and coworkers discriminate between alternative copolymerization models by fitting the models to composition data and then predicting sequence distributions based on the fitted models. Measured and fitted sequence distributions are then compared. A better approach taken here is to fit the models to the sequence distribution data directly. 

The instantaneous composition of a copolymer X formed at a monomer mixture composition x coincides, provided the ideal model is applicable, with stationary vector ji of matrix Q with the elements (8). The mathematical apparatus of the theory of Markov chains permits immediately one to wright out of the expression for the probability of any sequence P Uk in macromolecules formed at given x. This provides an exhaustive solution to the problem of sequence distribution for copolymers synthesized at initial conversions p l when the monomer mixture composition x has had no time to deviate noticeably from its initial value x°. As for the high-conversion copolymerization products they evidently represent a mixture of Markovian copolymers prepared at different times, i.e. under different concentrations of monomers in the reaction system. Consequently, in order to calculate the probability of a certain sequence Uk, it is necessary to average its instantaneous value P Uk over all conversions p preceding the conversion p up to which the synthesis was conducted. 

Monomer reactivity ratios and thus comonomer sequence distributions in copolymers can vary with copolymerization reaction conditions. The comonomer distribution could affect the geometry of the adsorbed polymer - mineral complex and the fines stabilization properties. 

It was in article [52] where the main reason responsible for the above-mentioned peculiarities was explicitly formulated and substantiated. Its authors related these peculiarities with partitioning of monomer molecules between the bulk of a reaction mixture and the domain of a growing polymer radical. This phenomenon induced by preferential sorption of one of the monomers in such a domain is known as the bootstrap effect. This term was introduced by Harwood [53], because when growing a polymer radical can control under certain conditions its own microenvironment. This original concept enabled him to interpret many interesting features peculiar to this phenomenon. Particularly, he managed to qualitatively explain the similarity of the sequence distribution in copolymerization products of the same composition prepared in different solvents under noticeable discrepancies in composition of monomer mixtures. 

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 sequence distributions expected for the different models have been described [Hill et al., 1982, 1983 Howell et al., 1970 Tirrell, 1986] (Sec. 6-5a). Sequence distributions obtained by 13C NMR are sometimes more useful than composition data for discriminating between different copolymerization models. For example, while composition data for the radical copolymerization of styrene-acrylonitrile are consistent with either the penultimate or complex participation model, sequence distributions show the penultimate model to give the best fit. 

The synthesis of biopolymers in vivo leads to macromolecules with a defined sequence of units. This effect is very important for living organisms and is different in comparison with random copolymerization in which sequences of units are distributed according to stochastic rules. On the other hand, the predicted sequence of units can be achieved by a set of successive reactions of respective monomer molecule addition. In template copolymerization, the interaction between comonomers and the template could pre-arrange monomer units defining sequence distribution in the macromolecular product. 

The conclusion can be drawn that in template copolymerization reactivity ratios depend on the nature and concentration of the template used. Template controls composition and sequence distribution of monomer units in copolymers obtained. 

A very interesting modification of the system was examined by Ferguson and McLeod. The authors replaced poly(vinyl pyrrolidone) with copolymers vinyl pyrrolidone-styrene or vinyl pyrrolidone-acrylamide. It was found that the mechanism of polymerization is the same as in the presence of homopolymer (PVP). However, the rate of polymerization decreases rapidly when vinyl pyrrolidone concentration in copolymer decreases. The concentration of vinyl pyrrolidone residues was kept equimolar to the concentration of acrylic acid. It was stressed that structure of template and, in the case of copolymeric template, sequence distribution of units play an important role in template effect. 

For a detailed analysis of monomer reactivity and of the sequence-distribution of mers in the copolymer, it is necessary to make some mechanistic assumptions. The usual assumptions are those of binary, copolymerization theory their limitations were discussed in Section III,2. There are a number of mathematical transformations of the equation used to calculate the reactivity ratios and r2 from the experimental results. One of the earliest and most widely used transformations, due to Fineman and Ross,114 converts equation (I) into a linear relationship between rx and r2. Kelen and Tudos115 have since developed a method in which the Fineman-Ross equation is used with redefined variables. By means of this new equation, data from a number of cationic, vinyl polymerizations have been evaluated, and the questionable nature of the data has been demonstrated in a number of them.116 (A critique of the significance of this analysis has appeared.117) Both of these methods depend on the use of the derivative form of,the copolymer-composition equation and are, therefore, appropriate only for low-conversion copolymerizations. The integrated... 

It should be emphasized that copolymerizations that conform to the premises of binary-copolymerization theory produce copolymers of well defined structure. The kinetics of the competitive propagation-reactions determine not only the copolymer composition but also the sequence distribution. The mathematical procedures needed for calculating number-average sequence-lengths of mers, and sequence length-distributions of mers, are well known and have been... 

RIS theory is used to calculate mean-square unperturbed dimensions 0 and dipole moments

of ethylene-vinyl chloride copolymers as a function of chemical composition, chemical sequence distribution, and stereochemical composition of the vinyl chloride sequences. As was previously found for several other copolymeric chains, is much more sensitive to chemical composition and chemical sequence distribution than is 0. The present calculations also indicate that both and are most strongly dependent on chemicel sequence distribution for ethylene-vinyl chloride chains having vinyl chloride sequences which are significantly syndiotactlc in structure. 

The copolymer generator calculates a series of "mini-batch" copolymerizations and has been described in more detail by Molau (19) and Meyer and Lowry (20). The sequence distribution generator uses Harwood s (14) run number approach to calculate triad functions. 

A typical example showing that we are able to build macromolecules at will is given by C. P. Pinazzi and co-workers in the first chapter of the second section, Chapter 27. They report how model polyenes can be built and how they react. In Chapter 28 K. F. O Driscoll illustrates the limitations in polymerization. For every vinyl monomer, a ceiling temperature exists, above which depropagation exceeds polymerization. If two vinyl monomers are copolymerized at a temperature at which one depropa-gates, the polymer formed will have an unusual composition and sequence distribution. 

For every vinyl monomer there exists a ceiling temperature above which it is thermodynamically impossible to convert monomer into high polymer because of the depropagation reaction. If two vinyl monomers are copolymerized under conditions such that one or both may depropagate, the resultant polymer will have an unusual composition and sequence distribution. Existing theoretical and experimental works are reviewed which treat of copolymer composition, rate of copolymerization, and degree of copolymerization. 

In contrast to the above mentioned models, the similar statistical description of the products of the complex-radical copolymerization occurring through the scheme (2.5) has been carried out quite recently [37, 49, 55-60]. Within the framework of this Seiner-Litt model, both copolymer composition [37,49, 55-58] and fractions of the different triads and blocks of the monomer units in the macromolecules were calculated [57]. The probability approaches which were applied in these works, are regarded as being of limited applicability in contrast to the general statistical method [49, 59, 60], By means of the latter method, the sequence distribution and composition inhomogeneity of the copolymer were completely described [49, 60] and also thorough calculations of its microstructure with the account for the tactidty were carried out [59, 60]. 

The Markovian character of the sequence distribution statistics in the macromolecules results [6, 94] from assumption about the steady-state of the radical concentrations, which usually holds with a high degree of accuracy in the copolymerization processes [6, 95], It is worth mentioning that along with such kinetic stationarity one should usually speak about the statistical stationarity. It means that when the number of the units in copolymer molecules exceeds 10-15, their composition practically becomes independent on degree of polymerization and is indistinguishable from the value predicted by the stationary Markov chain theory. This conclusion is supported by the theoretical [96,97,6] and experimental [98] evidence. 

For the quantitative description of the sequence distribution in the multicomponent copolymers the statistical characteristics similar to the ones applied for the description of the binary copolymerization products were used [45, 104-110]. In their well-known paper [45] Alfrey and Goldfinger stated an exponential character of the run distribution f (n) for length n with copolymerization of any number m of monomer types. Besides this distribution and its statistical moments [104-107] other parameters as alternation degree were introduced [108,107], which equals the overall fraction of all heterotriads P(M Mj) (i 4= j), and also the parameter with the similar meaning called alternating order [109]. Tosi [110] suggested to use informational entropy as a quantitative measure of the randomness of the multicomponent copolymers ... 

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