Biomedical Engineering Reference
In-Depth Information
closed form. For instance, the number-average block lengths
<n
A
,
<n
B
>
,
<n
C
>
,
<n
D
>
are defined by formulas similar to (
10.25
)and(
10.26
). The latter are given
by the ratio of two geometrical series. Summing them and simplifying, the number-
average block lengths,
<n
C
>
and
<n
D
>
become:
1
1 P
CC
<n
D
>D
1
1 P
DD
<n
C
>D
(10.35)
The availability of simple algebraic expressions put in question the expediency of
the application of the Monte Carlo method to compute the above values [
44
-
50
].
When
z
, it becomes unpractical not to use the vector notation. The feed and
the composition are simply
f
ave
and
c
ave
.When
z
>4
D 7
, the components of
c
ave
are
.c
A
;c
B
;c
C
;c
D
;c
E
;c
F
;c
G
/
. The bivariate distribution is a hyper-gaussian in
z
dimen-
sions centered at the average composition vector.
10.11
Copolymerization at High Conversion
Free-radical copolymerization is often conducted in batch, up to high
values
(
is the monomer-to-polymer conversion). The theory predicts that the resulting
copolymer will be a mixture of the copolymer produced at time
t
1
,
t
2
,
t
3
,etc.More
t D t
nomore
, the abundance of MPREF (the monomer which is
preferentially included in the copolymer chain) becomes zero. The reaction medium
becomes dense, the reaction accelerates (Trommdsdorff effect) and thus the sum
k
ptot
D k
11
C k
12
C k
21
C k
22
of the rate constants of propagation in (
10.11
)
increases by an order of magnitude. The theory predicts that the compositional dis-
tribution histogram in (
10.16
) will loose its symmetry, it will become skewed and it
will be at least a factor two or three wider than the Stockmayer's value in (
10.21
).
With such values of the CODIHI's variance, it is absolutely apparent that copoly-
merization at high conversion cannot be described by a Markovian chain of any
order. In fact, the feed
specifically, at time
f
A
will change with time, and thus copolymer chains pro-
duced at different time will have different compositions. The system is governed by
a set of ordinary differential equations. Kuchanov studied them in some detail and
reported the results in his review. The composition can be represented as a point in a
.
dimensional phase space. It is possible to follow its changes by drawing their
trajectories in a multidimensional phase space. In the case of ABC copolymers, the
space is a triangle with unit-length sides.
The operation of averaging a quantity over conversion will be denoted with
ahat:
z
1/
Z
1
c
A
D
c
A
d
0
(10.36)
0
The integration of the instantaneous composition over reaction times can be
performed symbolically and the result can be cast in a closed form. However, the
Search WWH ::
Custom Search