Biomedical Engineering Reference
In-Depth Information
DP
w
¼
1
þ
p
1
p
:
ð
3
:
1
Þ
Here p is the proportion of chemically reacted
groups.
The derivation of this equation explained how and why it was extremely dif
-
in this case diol
-
cult to
obtain very high molecular mass product for such a reaction, since even when p = 0.999
-
an arguably unfeasible amount
the degree of polymerization is only c.2000, so the
weight average molecular mass M
w
is typically of order 100 000 g mol
−
1
.
The expansion of this approach to non-linear (branched and cross-linked) systems has
much more relevance to the present volume. Work by Flory (
1941
,
1942
) and extensions
by Stockmayer (
1943
) identi
-
cance for these systems.
Unlike the linear systems, for non-linear reactions, when there are more than two
potential reactive groups, very high molecular masses are quite easily achieved, and
for a simple symmetric triol system, when p becomes greater than 0.5, the Flory
ed that p has still more signi
-
Stockmayer (FS) theory predicts that DP
w
will diverge to in
nity.
The non-linear analogue of (
3.1
)is
1
þ
p
DP
w
¼
Þ
:
ð
3
:
2
Þ
1
p
ð
f
1
In this case, f is the functionality
in this case simply the number, 3, of reactant groups on
the triol. The denominator of this equation becomes zero when the product p(f
-
−
1) = 1,
and this marks the so-called Flory
-
Stockmayer gel point, so that p
c
, the critical degree of
conversion, is given by
1
p
c
¼
Þ
:
ð
3
:
3
Þ
ð
f
1
Of course substituting f = 2 in (
3.2
) simply returns (
3.1
).
The applicability of this approach to vulcanization systems is also of interest. Flory
deduced that if a preformed chain had n
0
possible cross-linking sites distributed along the
polymer chain backbone, then trivially f
n
0
. A consequence of this approach is that a
fully cross-linked elastomer can be formed if, on average, each chain is linked to at least
one other. Since n is typically quite large
≈
in the absence of so-called
wastage reactions, only 1 in 1000 units in the chain needs to have been cross-linked to
pass the gel point.
Early efforts to test this theory used f-functional (
-
1000 or more
-
) end-linking of
esters and alcohols, for example the reaction of a multifunctional f
m
-ol with, say, a multi-
functional f
n
-acid, e.g. pentaerythritol (f
m
= 4) and the dibasic adipic acid (f
n
= 2), as used
in a series of experiments by Stockmayer and Weil (Flory,
1953
).
The prime purpose of this work was to establish whether or not the degree of conversion
p
c
at the gel point was as given by (
3.3
). These and many other systems were used, and in
most cases it was found that measured values of p
c
were greater than the value predicted,
but it was soon realized that the main reason for this was that, as well as intermolecular
bonds being formed, a small proportion of intramolecular reaction took place, forming
cycles. Consequently later work concentrated on other systems, including those from
'
polycondensation
'
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