Predicting and Measuring the Sequence Distribution of Addition Polymers - Mathematical Approaches to Polymer Sequence Analysis and Related Problems - page 205

This is a very general result, valid for all Markovian chains P AA ;P AB ;P BA ;P BB

which are not infinite. The result can be stated saying that in markovian chains,
finite-size effects produce chains of the type

A m B n and the instantaneous bivariate

distribution is a Gaussian curve centered at c A . For dimers, trimers and tetramers,

the bivariate distribution is quite broad [ 20 ]. However, the variance changes with
chain size in a very strong manner (it scales as s 1 ,see( 10.21 )). In the original

derivation (by Stockmayer), this result was hidden. In fact, an equivalent expression
for

2 wasusedinwhich

P AA P AB P BA P BB do not appear.

10.6 The Penultimate Model

In the penultimate model, the rate constants of monomer addition to the grow-
ing copolymer radical depends on its terminal unit, its penultimate unit and the
monomer to be added [ 11 ]. Thus, there are eight different propagation reactions
which define eight different rate constants of propagation

.
The sequence is much less intuitive than the previous one. In fact, it is described
by a second-order Markov chain in which the states S1, S2, S3, S4, are pairs of
monomer units. The chain has an associated

k pqr :::; p; q; r D 1; 2

Q -matrix. Eight Q -matrix ele-
ments are zero, whereas other the eight Q -matrix elements are nonzero. In this case,
there are four reactivity ratios

4 4

r 11 ;r 12 ;r 21 and r 22 ,definedas:

k 111
k 112 ;r 21 D

k 211
k 212 ;r 22 D

k 222
k 221 ;r 12 D

k 122
k 121

r 11 D (10.22)

the relation between c A and f A for this model is

c A D ˚ penu .f A ;r 11 ;r 12 ;r 21 ;r 22 /; (10.23)

where ˚ penu denotes a relatively simple dependence. Penlidis and coworkers [ 21 ]

investigated the role of impurities in free-radical copolymerization and they noted
that ˚ penu can be cast in a compact form, structurally similar to ( 10.15 ) introducing

two new variables,

r 6 and r 7 ,definedas:

r 21 .f A r 11 C f B /
f A r 21 C f B

r 12 .f B r 22 C f A /
f B r 12 C f A

r 6 D ;r 7 D (10.24)

The penultimate model is slightly less popular than the terminal model, due to its
higher complexity. Losio et al. [ 22 ] used it for NMR sequencing of copolymers with
units of ethylene and 4-methyl-1-pentene, whereas Tritto et al. [ 23 ] and Yamada
et al. [ 24 ] used it for NMR sequencing of ethylene-norbornene and of vinylacetate-
vinylpivalate copolymers, respectively.