Statistical binary copolymers

To a first approximation, it is possible to take for the binary statistical copolymer with units A and B... 

Statistical copolymers containing repeating units each with a different functional group can be obtained using appropriate mixture of monomers. For example, a polyestermide can be synthesized from a ternary mixture of a diol, diamine, and diacid or a binary mixture of a diacd and amine-alcohol [East et al., 1989]. 

In the case of statistic copolymers of two monomers (binary copolymers) the glass transition temperature steadily changes with the molar amounts of the two monomers. In many cases, a similar behavior is observed with some mechanical properties (tensile strength, impact strength, stiffness, and hardness) (see Chap. 1). Deviations can occur in copolymers, which contain only a few percent of one comonomer. 

Chains containing two types of structural unit are described as copolymers or binary copolymers. In the illustration 15 structural units of each type are used. When all copolymer molecules have exactly the same number of structural units they all have the same degree of polymerization, the same molar mass, and the same composition. The repeating units can be distributed along linear polymer chains in a manner which obeys some statistical law (statistical copolymers), randomly distributed (random copolymers), arrayed alternately along the chain (alternating copolymers) or combined in two distinct blocks (block copolymers). When two different monomers are used to produce branched or block copolymers a considerable variety of structures and composition distributions is possible. 

Intramolecular Repulsive Interactions. Miscible blends can also be achieved in absence of specific interactions, by exploiting the so-called intramolecular repulsive effect. This is observed in mixtures where at least one of the components is a statistical copolymer miscibility is restricted to a miscibility window, that is, it takes place within a well-defined range of copolymer composition. For example, poly(styrene-co-acrylonitrile) (SAN) and poly(methyl methacrylate) form miscible blends for copolymer compositions in the range 9-39% acrylonitrile (26,27). Miscibility in these systems is not a result of specific interactions but it is due to the intramolecular repulsive effect (28) between the two monomer units in the copol5uner such that, by mixing with a third component, these imfavorable contacts are minimized. The same situation is encoimtered in binary mixtures of two copol5uners (29). 

In general, for a binary mixture of two statistical copolymers poly(A-co-B) and poly(C-co-D), the interaction parameter is (30) ... 

Based on the thermodynamic theory of the glass transition, Couchman derived relations to predict the Tg composition dependence of binary mixtures of miscible high polymers (113) and other systems (114-116). The treatment that follows is easily generalized to the case for statistical copolymers (113). 

When applied to systems containing statistical copolymers, the SLCT loses its enormous analytical and physical simplicity due to the greater complexity of these systems compared to binary homopolymer blends. The lack of mathematical simplicity in describing copolymer blends arises, in part, from the dependence of the free energy on the monomer sequence. Therefore, a further approximation is introduced into the SLCT to generate a theoretical approach that is simple and easy to use but is devoid of a serious deficiency of the extensions of FH theory to random copolymer systems, namely, the neg-... 

Statistical characteristics of the second type define the microstructure of copolymer chains. The best known characteristics in this category are the fractions P [/k) (probabilities) of sequences Uk involving k monomeric units. The simplest among them are the dyads U2, the complete set of which, for example, for a binary copolymer is composed of four pairs of monomeric units M2M, M2M2. The number of the types of k-ad in chains of m-component copolymers grows exponentially as mk so that with practical purposes in mind it is generally enough to restrict the consideration to sequences Uk] with moderate values of k. Their calculation turns out to be rather useful... 

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 ... 

Eigenvalues of the operator Qr are real while the largest of them, Af, equals unity by definition. As a result, in the limit n-> oo all items in the sum (Eq. 38), excluding the first one, Q Q f = Xr/Xfh will vanish. In this case, chemical correlators will decay exponentially along the chain on the scale n 1/ In AAt values n < n the law of the decay of these correlators differs, however, from the exponential one even for binary copolymers. This obviously testifies to non-Markovian statistics of the sequence distribution in molecules (see expression Eq. 11). The closer is to unity, the greater are the values of n. The situation when n 1 corresponds to proteinlike copolymers. 

The second type of nonideal models takes into account the possible formation of donor-acceptor complexes between monomers. Essentially, along with individual entry of these latter into a polymer chain, the possibility arises for their addition to this chain as a binary complex. A theoretical analysis of copolymerization in the framework of this model revealed (Korolev and Kuchanov, 1982) that the statistics of the succession of units in macromolecules is not Markovian even at fixed monomer mixture composition in a reactor. Nevertheless, an approach based on the "labeling-erasing" procedure has been developed (Kuchanov et al., 1984), enabling the calculation of any statistical characteristics of such non-Markovian copolymers. 

Different Methods for die Statistical Description of Binary Copolymers... 

The parameters a = l/rij5 the number of which equals m(m — IX are reciprocal reactivity ratios (2.8) of binary copolymers. Markov chain theory allows one, without any trouble, to calculate at any m, all the necessary statistical characteristics of the copolymers, which are formed at given composition x of the monomer feed mixture. For instance, the instantaneous composition of the multicomponent copolymer is still determined by means of formulae (3.7) and (3.8), the sums which now contain m items. In the general case the problems of the calculation of the instantaneous values of sequence distribution and composition distribution of the Markov multicomponent copolymers were also solved [53, 6]. The availability of the simple algebraic expressions puts in question the expediency of the application of the Monte-Carlo method, which was used in the case of terpolymerization [85,99-103], for the calculations of the above statistical characteristics. Actually, the probability of any sequence MjMjWk. .. Mrl 4s of consecutive monomer units, selected randomly from a polymer chain is calculated by means of the elementary formula ... 

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 ... 

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... 

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