Predicting and Measuring the Sequence Distribution of Condensation Polymers - Mathematical Approaches to Polymer Sequence Analysis and Related Problems - page 230

Carlo computations performed by Chaumont, Gnagou et al. and the theory proposed
by Lopez-Serrano, Macosko et al. (with the derivation of explicit formulas for
<n A >

the number-average sequence length).
In poly(ethylene-terephthalate), poly(butylene-adipate), and other polymers, the
monomers are paired and thus the repeat unit is made of two monomers, namely
ethylene glycol and terephthalic acid, butanediol and adipic acid, more in general,
a diol and a diacid. The development of a mathematical model which uses two
paired monomers as a single entity is certainly possible (it is identical to the models
developed for addition copolymers). However, it cannot account for a very well-
known phenomenon, namely stoichiometric unbalance (in fact, paired monomers
are necessarily stoichiometrically balanced). To get high-molecular copolyconden-
sates, it is necessary to conduct the reaction up to nearly complete conversion. In
that case, the copolymer average composition, cA will be virtually the same as that
of the initial monomer mixture, and thus the problem of its determination ceases
to be actual. Thus, the Bernuolli model holds. It predicts that the abundance of an
oligomer is equal to the product of the molar fractions of the monomers. For in-
stance, the abundance of oligomer AABBAACDDCAAB is given by the following
product:

, <n B >

I AABBAAAAAB D c A c A c B c B c A c A c A c A c A c A c B (11.5)

The quantities of interest are the molar fraction, I.A m B n / , of the oligomer A m B n ,

the molar fraction, I.A m B n C p / , of oligomer A m B n C p , and the molar fraction,

I.A m B n C p D q / A m B n C p D q . They are obtained by combinatorial cal-

culus, summing all contributions:

, of oligomer

.m C n/Š

mŠ nŠ .c A / m .c B / n

I.A m B n / D (11.6)

.m C n C p/Š

mŠ nŠ pŠ .c A / m .c B / n .c C / p

I.A m B n C p / D (11.7)

.m C n C p C q/Š

mŠ nŠ pŠ qŠ .c A / m .c B / n .c C / p .c D / q

I.A m B n C p D q / D (11.8)

The above equations are often referred to as the Newton and the Liebnitz formu-
las, respectively. Figure 11.2 reports the theoretical mass spectra for a copolymer,
a terpolymer, and a tetrapolymer, respectively. The masses correspond to units of
butyleneadipate butylenesuccinate, butylenesebacate, and butyleneterephthalate [ 6 ].
In this way, the comparison with experiment is simple. It can be seen that the number
of peaks grows as the size of the oligomer grows. In the case of binary copolymers,
the number of peaks [ 7 ]growsas

.s C 1/

. In the case of tetrapolymers, the spectrum
becomes crowded at the trimer level (see figure).