Polycondensation statistics

Gordon, M., and Scantlebury, G. R., Non-random polycondensation, statistical theory of the substitution effect. Trans. Faraday Soc. 60, 604-621 (1964). 

As early as 1952, Flory [5, 6] pointed out that the polycondensation of AB -type monomers will result in soluble highly branched polymers and he calculated the molecular weight distribution (MWD) and its averages using a statistical derivation. Ill-defined branched polycondensates were reported even earlier [7,8]. In 1972, Baker et al. reported the polycondensation of polyhydrox-ymonocarboxylic acids, (OH)nR-COOH, where n is an integer from two to six [ 9]. In 1982, Kricheldorf et al. [ 10] pubhshed the cocondensation of AB and AB2 monomers to form branched polyesters. However, only after Kim and Webster published the synthesis of pure hyperbranched polyarylenes from an AB2 monomer in 1988 [11-13], this class of polymers became a topic of intensive research by many groups. A multitude of hyperbranched polymers synthesized via polycondensation of AB2 monomers have been reported, and many reviews have been published [1,2,14-16]. 

As the result of theoretical consideration of polycondensation of an arbitrary mixture of such monomers it was proved [55,56] that the alternation of monomeric units along polymer molecules obey the Markovian statistics. If all initial monomers are symmetric, i.e. they resemble AaScrAa, units Sa(a=l,...,m) will correspond to the transient states of the Markov chain. The probability vap of transition from state Sa to is the ratio Q /v of two quantities Qa/9 and va which represent, respectively, the number of dyads (SaSp) and monads (Sa) per one monomeric unit. Clearly, Qa(S is merely a ratio of the concentration of chemical bonds of the u/i-ih type, formed as a result of the reaction between group Aa and Ap, to the overall concentration of monomeric units. The probability va0 of a transition from the transient state Sa to an absorbing state S0 equals l-pa where pa represents the conversion of groups Aa. 

Noteworthy that all the above formulated results can be applied to calculate the statistical characteristics of the products of polycondensation of an arbitrary mixture of monomers with kinetically independent groups under any regime of this process. To determine the values of the elements of the probability transition matrix of corresponding Markov chains it will suffice to calculate only the concentrations Q()- of chemical bonds (ij) at different conversions of functional groups. In the case of equilibrium polycondensation the concentrations Qy are controlled by the thermodynamic parameters, whereas under the nonequilibrium regime of this process they depend on kinetic parameters. 

The rate constants in the reactions (29) may be conveniently envisaged as elements of symmetric matrix k. In order to calculate the statistical characteristics of a particular polycondensation process along with matrix k parameters should be specified which characterize the functionality of monomers and their stoichiometry. To this end it is necessary to indicate the matrix f whose element fia equals the number of groups A in an a-th type monomer as well as the vector v with components Vj,... va,..., v which are equal to molar fractions of monomers M1,...,Ma,...,M in the initial mixture. The general theory of polycondensation described by the ideal model was developed more than twenty years ago [2]. Below the key results of this theory are presented. 

Random hyperbranched polymers are generally produced by the one-pot polymerization of ABX monomers or macromonomers involving polycondensation, ring opening or polyaddition reactions hence the products usually consist of broad statistical molecular weight distributions. 

The microstructural approach can be extended to the polymers of 1,2-disub-stituted ethylenes, to diene polymers, and to those obtained by ring-opening polymerization and by polycondensation. Differences with regard to vinyl and vinylidene polymers are observed in the type and number of the sequences, their nomenclature and the statistic model for the quantitative interpretation of their distribution. 

A polymer derived from the polycondensation of a single actual monomer, the molecules of which terminate in two different complementary functional groups (e.g. 6-aminohexanoic acid) is, by definition, a (regular) homopolymer. When two different monomers of this type react together, the product is a copolymer that can be named in appropriate fashion. For example, if 6-aminohexanoic acid is copolycondensed with 7-aminoheptanoic acid, leading to a statistical distribution of monomeric units, the product is named poly[(6-aminohexanoic acid)-stoi-(7-aminoheptanoic acid)]. 

The statistical treatment of random stepwise crosslinking reactions (e.g. polycondensation) neglecting ring formation originates from Stock-mayer and Flory and is explained in Flory s book (55) on a number of examples. Using simple probability statistics, it is possible to calculate the molecular size distribution in the sol and in the gel, fractions of sol and gel, the crosslinking density and the fraction of free functionalities in... 

Fig. 5. Interrelation between critical concentrations of crosslink forming (a) and ring forming (a) functionalities at the gel point calculated on the basis of random flight statistics for trifunctional polycondensation (/ = 3) and different values of parameters A and 91 [Gordon and Scantleburry (72)]...

In other words, each generation is weighted after differentiation with a function that depends on the path length. One easily verifies Eq. (C.84) for the random polycondensates when a - Gaussian statistics) is assumed. 

The above-described "labeling-erasing" procedure is in common use in statistical chemistry of polymers (Kuchanov, 2000). It gives a chance to obtain a number of important theoretical results under kinetic modeling of polymerization and polycondensation processes, where the deviation from their description in terms of the ideal kinetic model is due to the short-range effects. 

The above reasoning allows a conclusion that once a researcher has decided upon the particular ideal kinetic model of polycondensation, he or she will be able to readily calculate any statistical characteristics of its products. The only thing he or she is supposed to do is to find the solution of a set of several ordinary differential equations for the concentrations of functional groups, using then the expressions known from literature. 

---