Biomedical Engineering Reference
In-Depth Information
is a first-order Markov chain with an associated
2 2
P
-matrix and that the four
P
-matrix elements are
P
AA
D k
11
f
A
=g
2a
P
AB
D k
12
f
B
=g
2a
(10.12)
P
BA
D k
21
f
A
=g
2b
P
BB
D k
22
f
B
=g
2b
;
(10.13)
where
g
2b
D k
21
f
A
C k
22
f
B
. In order to proof this
intuitive result in a formal manner, it is necessary to introduce definitions for the
statistical process and to demonstrate that it converges towards a stationary vector.
The stationary sequence possesses an associated composition:
g
2a
D k
11
f
A
C k
12
f
B
and
c
A
=c
B
D P
AB
=P
BA
(10.14)
The above equation does not lend itself to immediate comparison with the equation
c
A
ยค f
A
(see (
10.7
)). Fortunately, it can be recast in an equivalent (more explicit)
form:
r
1
f
A
.t /
2
C f
A
.t /f
B
.t /
r
1
f
A
.t /
2
C r
2
f
B
.t /
2
C 2f
A
.t /f
B
.t /
;
c
A
D
(10.15)
where
f
A
is now time-dependent (the sum
f
A
.t / C f
B
.t /
always equals one) and
where
r
2
are the so-called
reactivity ratios
[
1
-
3
]. Equation (
10.15
)isthe
well-known
instantaneous composition equation for the terminal model
. The termi-
nal model is extremely popular. Equations have been used in thousands of reports.
Brar and coworkers used the terminal model in a systematic manner for copolymer
sequencing by NMR and published more than 50 reports. Among others, reports
on three copolymers containing
r
1
and
-vinylcarbazole units, on three copolymers con-
taining trans-4-acryloyloxyazobenzene units, on acrylonitrile-hexyl methacrylate
[
12
], on vinyl acetate-glycidyl methacrylate [
13
], on acrylonitrile-methyl acrylate
[
14
], on acrylic acid - vinylacetate [
15
] on hydroxyethyl methacrylate - methyl
acrylate [
16
].
N
10.5
The Bivariate Distribution
Let
W.m;n/
denote the weight of the copolymer chains of the type
A
m
B
n
(which
are made of
m
repeat units of type
A
and
n
repeat units of type
B
). The overall
mass of the chain is
B
are the molar masses of
the two repeat units. One usually assumes that all copolymer molecules produced
instantaneously have the same composition. However, this is not true. Since the size
s D m C n
g
all
D
m
A
C
n
B
,where
A
and
of a copolymer is finite, the compositions cannot be all identical. In
order to quantify this effect, one can introduce
y
, the deviation from the average
m
s
composition, defined as
. In order not to loose adherence with experi-
mental measurements, it is mandatory to transform
y D c
A
, the weight
(in grams or milligrams) of the polymeric chains which possess a size in the range
W.m;n/
into
W
s
.y/
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