Biomedical Engineering Reference
In-Depth Information
where
. In this case, it is very useful to define eight reactivity ratios
r
A
;r
0
A
;r
0
A
;r
00
A
;r
B
;r
0
B
;r
0
B
and
p; q; r; s D 1; 2
r
00
B
. The instantaneous composition equation for this
model is:
c
A
D ˚
antep
.f
A
;r
A
;r
0
A
;r
0
A
;r
00
A
;r
B
;r
0
B
;r
0
B
;r
000
/
(10.30)
B
An exact expression for
˚
antep
has been derived [
32
]. The sequence is described
by a third-order Markov chain with an associated
T
-matrix. The relations between
the 16 rate constants
k
pqrs
in (
10.29
) and the Markov
T
-matrix elements are highly
nonlinear relations. Nevertheless, using the eight reactivity ratios, they become al-
most linear [
32
]. The model suffers of a most evident limitation, namely too many
degrees of freedom. Park et al. developed a simplified pen-penultimate model and
used it for NMR sequencing of ethylene-norbornene copolymers [
33
].
10.9
The Complex Participation Model
In some processes, such as the free-radical copolymerization of maleic anhydride
and styrene [
34
] or acrylonitrile and styrene [
35
,
36
], a peculiar effect arises, namely
the formation of donor-acceptor complexes. These complexes are hold together by
bonds which are very different form standard covalent bonds, and therefore they will
be denoted as
M
1
:::M
2
. In most cases,
M
1
is electron-rich and
M
2
is electron-poor.
k
11
;k
12
;k
21
;k
22
Four additional rate constants of propagation,
, are required to de-
scribe how the donor-acceptor complex is inglobated into the growing chain. As a
result, the complex participation model kinetic scheme is made of eight equations.
The molar fraction of A units in the copolymer is given by:
D ˚
partic
.f
A
;k
11
;k
12
;k
21
;k
22
;k
11
;k
12
;k
21
;k
22
/;
c
A
(10.31)
where
˚
partic
denotes a quite involved functional dependence which is discussed
in some detail in Kuchanov's review. Seiner and Litt used the complex partici-
pation model for NMR sequencing of methyl acrylate-diphenylethylene [
37
]and
styrene-tetrachlorocyclopropene [
38
]. The consistent kinetic analysis [
39
]ofthe
copolymerization (with the simultaneous occurrence of all the above reactions)
leads to the conclusion that the probabilities of the sequences of the monomer units
M
1
in the macromolecules cannot be described by a Markov chain of any
finite order. Consequently, in this very case, we deal with non-Markovian copoly-
mers the general theory for which is not yet available. Since this conclusion has
important mathematical consequences, some details of the derivation are in place.
The authors [
39
] introduce a conditional coloring procedure and they distinguish
monomer units as colored black or white. When this simple procedure is applied to
sequences of monomer units, it allows to distinguish, besides their type, the man-
ner of their adding to a polymer chain. The unit
and
M
2
M
i
is black if the corresponding
monomer
M
i
is added to the radical as the first monomer of the complex. The unit
M
i
is white in all the other cases, namely when the corresponding monomer
M
i
is
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