Chemistry Reference
In-Depth Information
The purpose of the following treatment is to define the conditions under which
indefinitely large chemical structures, or infinite networks, will occur. To this end
we seek the answer to the question: Under what conditions is there a finite
probability that an element of the structure selected at random occurs as part of an
infinite network? …. First of all, it is necessary to determine the branching
coefficient a which is defined as the probability that a given functional group of a
branching unit leads via chain of bifunctional units to another branch unit.
Flory outlined the above given reaction scheme (here ''I'' may have any value
from 0 to infinity) as basis for his further considerations: ''…the probability that the
first ''a'' group of the chain shown on the right, has reacted is given by p
a
, the fraction
of all ''a'' groups which have reacted; similarly, the probability that the ''b'' group on
the right of the first b-b unit has reacted, is given by p
b
. Let q present the ratio of ''a''
s (reacted and unreacted) belonging to branch units to the total number of ''a'' s in the
mixture. Then the probability that a ''b'' group has reacted with a branch unit is p
b
q;
the probability that it is connected to a bifunctional a-a unit pb(1-q). Hence, the
probability that the ''a'' group of a branch unit is connected to the sequence of units
shown in the preceding formula is given by'':
p
a
½p
b
ð
1
q
Þ
p
a
i
p
b
q
''The probability a that the chain in a branch unit regardless of the number ''i''
of pairs of bifunctional units is given by the sum of each expression having i = 0,
1, 2, 3….etc., respectively. That is'':
a ¼ R½p
a
p
b
ð
1
q
Þ
i
p
a
p
b
q
ð
4
:
33
Þ
or
a ¼ p
a
p
b
q
=
1
p
a
p
b
1
q
½
ð
Þ
ð
4
:
34
Þ
''If we let the ratio of ''a'' t o '' b'' groups initially present be presented by ''r,''
as in the case of ''ii'' type linear polymers (i.e. polymers based on a
2
? b
2
monomers), then; p
b
= rp
a
Substitution in Eq. (
4.33
) to eliminate either p
b
or p
a
'' :
a
¼
rpa
2
q
1
rp
a
1
q
ð
Þ
ð
4
:
35
Þ
or
a ¼ p
b
q
r
p
b
ð
1
q
Þ
ð
4
:
36
Þ
Flory then mentions that r and q can experimentally be determined via the feed
ratio and the titration of end groups. Furthermore, he considered three special
cases:
(1) When there are no a
2
units, r = 1 and
a ¼ rp
a
¼ p
b
r
ð
4
:
37
Þ
