Reaction, chain, copolymer statistics

Copolymerization is a variant of polymerization where macromolecules composed of two or more kinds of monomer are formed. According to the frequency of entry of various monomers into the chains, copolymers (and even the reactions by which they are formed) are sometimes more closely specified by special attributes. Thus copolymers may be unspecified, statistical, random, periodic, alternating, block or graft. 

The classi cation of copolymers according to structural types and the nomenclature for copolymers have been described previously in Chapter 1. The present chapter is primarily concerned with the simultaneous polymerization of two monomers by free-radical mechanism to produce random, statistical, and alternating eopolymers. Copolymers having completely random distribution of the different monomer units along the copolymer chain are referred to as random copolymers. Statistical copolymers are those in which the distribution of the two monomers in the chain is essentially random but in uenced by the individual monomer reactivities. The other types of copolymers, namely, graft and block copolymers, are not synthesized by the simultaneous polymerization of two monomers. These are generally obtained by other types of reactions (see Section 7.6). 

Copolymerization of substituted and unsubstituted monomers was also investigated [74]. The density of side-chains along the PPP backbone was slowly decreased by varying the ratio of substituted to non-substituted monomers in the reaction mixture. Copolymers exhibit statistically distributed repeating units no evidence for alternation or for block-sequences was found. Their was a marked effect on the UV spectra, indicative of a modification in the extent of conjugation between adjacent benzene rings induced by side-chains. This effect is linked to steric hindrance and it shows that a disubstituted benzene unit acts as an electronic insulator between unsubstituted blocks of PPP. DPs of 50 are reported for these copolymers. 

It should be emphasized that for Markovian copolymers a knowledge of the values of structural parameters of such a kind will suffice to find the probability of any sequence Uk, i.e. for an exhaustive description of the microstructure of the chains of these copolymers with a given average composition. As for the composition distribution of Markovian copolymers, this obeys for any fraction of Z-mers the Gaussian formula whose covariance matrix elements are Dap/l where Dap depend solely on the values of structural parameters [2]. The calculation of their dependence on time, and the stoichiometric and kinetic parameters of the reaction system permits a complete statistical description of the chemical structure of Markovian copolymers to be accomplished. The above reasoning reveals to which extent the mathematical modeling of the processes of the copolymer synthesis is easier to perform provided the alternation of units in macromolecules is known to obey Markovian statistics. 

Monomer concentrations Ma a=, ...,m) in a reaction system have no time to alter during the period of formation of every macromolecule so that the propagation of any copolymer chain occurs under fixed external conditions. This permits one to calculate the statistical characteristics of the products of copolymerization under specified values Ma and then to average all these instantaneous characteristics with allowance for the drift of monomer concentrations during the synthesis. Such a two-stage procedure of calculation, where first statistical problems are solved before dealing with dynamic ones, is exclusively predetermined by the very specificity of free-radical copolymerization and does not depend on the kinetic model chosen. The latter gives the explicit dependencies of the instantaneous statistical characteristics on monomers concentrations and the rate constants of the elementary reactions. 

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 ... 

Thus, as can be inferred from the foregoing, the calculation of any statistical characteristics of the chemical structure of Markovian copolymers is rather easy to perform. The methods of statistical chemistry [1,3] can reveal the conditions for obtaining a copolymer under which the sequence distribution in macromolecules will be describable by a Markov chain as well as to establish the dependence of elements vap of transition matrix Q of this chain on the kinetic and stoichiometric parameters of a reaction system. It has been rigorously proved [ 1,3] that Markovian copolymers are formed in such reaction systems where the Flory principle can be applied for the description of macromolecular reactions. According to this fundamental principle, the reactivity of a reactive center in a polymer molecule is believed to be independent of its configuration as well as of the location of this center inside a macromolecule. 

Secondary metathesis reactions are sometimes encountered during metathesis copolymerization, leading to a reshuffling of the units in the chain and eventually to a random distribution for example in the copolymerization of 248 and 258 using RUCI3 as catalyst, statistical copolymers are produced no matter whether the monomers are mixed initially or added sequentially576. See also the copolymers of 128 Section Vm.B.6. 

It should be emphasized that for the Markovian copolymers, the knowledge of these structure parameters will suffice for finding the probabilities of any sequences LZ, i.e., for a comprehensive description of the structure of the chains of such copolymers at their given average composition. As for the CD of the Markovian copolymers, for any fraction of Z-mers it is described at Z 1 by the normal Gaussian distribution with covariance matrix, which is controlled along with Z only by the values of structure parameters (Lowry, 1970). The calculation of their dependence on time and on the kinetic parameters of a reaction system enables a complete statistical description of the chemical structure of a Markovian copolymer. It is obvious therewith to which extent a mathematical modeling of the processes of the synthesis of linear copolymers becomes simpler when the sequence of units in their macromolecules is known to obey Markov statistics. 

The consistent kinetic analysis of the copolymerization with the simultaneous occurrence of the reactions (2.1) and (2.5) leads to the conclusion that the probabilities of the sequences of the monomer units M, and M2 in the macromolecules can not be described by a Markov chain of any finite order. Consequently, in this very case we deal with non-Markovian copolymers, the general theory for which is not yet available [6]. However, a comprehensive statistical description of the products of the complex-radical copolymerization within the framework of the Seiner-Litt model via the consideration of the certain auxiliary Markov chain was carried out [49, 59, 60]. 

Let us consider the generation of long chains of a statistical binary copolymer at low conversions. The reaction of the active centre with the monomer exhibits the characteristic molar heat of reaction H and molar entropy change S... 

When 1,3-dioxolane was added to the solution of living (nontermi-nated) poly(l,3-dioxepane) or vice versa, further polymerization ensued and the increase of molecular weight indicated that polymerization of added monomer proceeded exclusively on living active species of the former monomer. The isolated copolymer was analyzed by l3C NMR spectroscopy and it was found that, instead of a block copolymer, the copolymer with nearly statistical distribution of DXL and DXP units was formed practically from the beginning of the process. This is a clear indication that chain transfer to polymer leads to branched oxonium ions, which participate in further reactions with a rate comparable to the rate of propagation. 

This special case is called an ideal copoly-merization. As in special case 3, the distribution of monomers in the copolymer is. random, but the copolymer composition is not usually the same as that of the monomers in the reaction mass. It has been our experience that many students have a bit of trouble seeing why this condition should give a random copolymer. A lot of students tend to associate the word random with a 50 50 composition where the monomers are distributed according to coin toss statistics. However, a 90 10 distribution of, say, monomer 1 and monomer 2 can be random. Certainly, there will be blocks of Mt units, just because there are so many of them relative to M s, but the copolymer is random if the probability of finding a 1 at any point in the chain is 0.9, while the probability of finding a 2 is 0.1. 

---