Page 563 - Carrahers_Polymer_Chemistry,_Eighth_Edition

526 Carraher’s Polymer Chemistry

Flory and others derived expressions describing the extent of reaction at the gel point using sta-
at an extent reaction p can be described by the expression
tistical methods. In general, the M w
Mw,o (1 + p)
M w,p = -------------------------- (16.5)
(1 – p) (f – 1)

where M is the weight-average molecular weight at no reaction, and M is the weight-average
w,p
w,o
molecular weight at an extent of reaction p and f is the functionality. At the gel point, molecular

weight becomes quite large approaching infinity, giving eventually
p = 1
gel f −
1 (16.6)
For a system containing two types of functional groups

1
2
(p ) = -------------------------------- (16.7)
gel
r (f′ – 1) (f′′ – 1)
where r is the ratio of the two types of functionality f′ and f′.
This equation is valid when the ratio of the two types of reactant groups is about equal and where

the reactivity of members within each of the two types is about the same, and finally, when the reac-
tivity of the two types of sites does not change during the process.
For the case where cross-linking occurs with the cross-linking sites contained in already formed
polymer chains, average functionalities can be calculated based in terms of the weigh-average or
number-average molecular weight. For the weight-average molecular weight, the average degree of
is the weight-average molecular weight between
polymerization, DPw , can be described, where Mc
cross-linkable sites and p is the fraction of repeat units reacted, that is, the degree of conversion.
Mw f(1 + p)
DP = -------- = -------------------------- (16.8)
w
Mc (1 – p) (1 – f)

If DP w,o is the weight-average degree of polymerization of the initial mixture of long chains
then

DP (1 + p)
w,o
DP = ------------------------------------- (16.9)
w
(1 – p) ( DP – 1)
w,o
and at the gel point the degree of polymer approaches infi nity giving
p = 1
gel
DP w,o − 1 (16.10)

Equating the two equations for p gives
gel
DP – 1 = f – 1 (16.11)
w,o
Similar expressions can be derived based on the number-average molecular weight so that at the
gel point

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