Chemistry Reference
In-Depth Information
syntheses performed in Carother's group. Polycondensations conducted in the
presence or absence of protic catalysts, such as H
2
SO
4
or 4-toluene sulfonic acid,
were compared. Several decades later Costa and Villermaux [
22
-
24
] presented a
detailed kinetic study of polyester syntheses based on tri(ethylene glycol) and
adipic acid.
4.3 Three-dimensional Polycondensations of ab
f
(ab
n
)
Monomers
Whereas the discussion of three-dimensional polycondensations in Flory's first
review article [
6
] is limited to a
3
? a
2
? b
2
monomer combinations, his topic
''Principles of….'' [
4
] also includes a detailed theoretical description of poly-
condensations based on ab
f
monomers (Ref.
4
pp. 361-383). Flory postulated and
(theoretically) demonstrated for the first time that polycondensations of ab
f
monomers free of side reactions necessarily results in randomly branched (hy-
perbranched = hb) polymers, and that regardless of the conversion no gelation
occurs. He defined ''f'' as the number of ''b'' groups in the monomer, p
a
as the
conversion of ''a'' groups and ''x'' as DP of individual chains (as before).
Regardless of x, all hb polymers can only have one ''a'' end group, and the number
of unreacted ''b'' is correlated with x according to (f-2) x-1. He then introduced
the branching probability ''a'' which equals p
b
, the fraction of ''b'' groups reacted:
p
b
¼ p
a
=
f
1
ð
Þ
ð
4
:
24
Þ
a ¼ pa
=
f
1
ð
Þ
ð
4
:
25
Þ
With N
0
as the total number of units, the total number of molecules is N
0
(1-p)
and the DP
n
obeys Eq. (
4.26
):
DP
n
¼ 1
=
1
p
ð
Þ
¼1
=
1
a f
1
½
ð
Þ
ð
4
:
26
Þ
For calculation of the MWD Flory defined x
x
as the total number of configu-
rations (meaning isomers of branching; Ref. [
4
] p. 365). He then formulated Eqs.
(
4.27
) and (
4.28
) as the number (frequency) distribution, where n
x
is the mole
fraction of an x-mer in analogy to linear polymers:
n
x
¼
x
x
a
x
1
ð
1
a
Þ
fx
2x
þ
1
ð
4
:
27
Þ
n
x
¼½
ð
1
a
Þ=
a
x
x
b
x
with b ¼ a
ð
1
a
Þ
ð
4
:
28
Þ
On the basis of numerous assumptions and mathematical operations (not pre-
sented here) the mass distribution was formulated according to:
x
x
b
x
w
x
¼
ð
1
a
Þ=
a
½
1
a f
1
½
ð
Þ
ð
4
:
29
Þ
