General case Flory-Stockmayer Theory
a general image of multifunctional branch unit,
a
f
{\displaystyle a_{f}}
, reacting bifunctional monomers , b functional groups form step-growth polymer.
the flory-stockmayer theory predicts gel point system consisting of 3 types of monomer units
linear units 2 a-groups (concentration
c
1
{\displaystyle c_{1}}
),
linear units 2 b groups (concentration
c
2
{\displaystyle c_{2}}
),
branched units (concentration
c
3
{\displaystyle c_{3}}
).
the following definitions used formally define system
f
{\displaystyle f}
number of reactive functional groups on branch unit (i.e. functionality of branch unit)
p
a
{\displaystyle p_{a}}
probability has reacted (conversion of groups)
p
b
{\displaystyle p_{b}}
probability b has reacted (conversion of b groups)
ρ
=
f
c
3
2
c
1
+
f
c
3
{\displaystyle \rho ={\frac {fc_{3}}{2c_{1}+fc_{3}}}}
ratio of number of groups in branch unit total number of groups
r
=
2
c
1
+
f
c
3
2
c
2
=
p
b
p
a
{\displaystyle r={\frac {2c_{1}+fc_{3}}{2c_{2}}}={\frac {p_{b}}{p_{a}}}}
ratio between total number of , b groups.
p
b
=
r
p
a
.
{\displaystyle p_{b}=rp_{a}.}
the theory states gelation occurs if
α
>
α
c
{\displaystyle \alpha >\alpha _{c}}
, where
α
c
=
1
f
−
1
{\displaystyle \alpha _{c}={\frac {1}{f-1}}}
is critical value cross-linking ,
α
{\displaystyle \alpha }
presented function of
p
a
{\displaystyle p_{a}}
,
α
(
p
a
)
=
r
p
a
2
ρ
1
−
r
p
a
2
(
1
−
ρ
)
{\displaystyle \alpha (p_{a})={\frac {rp_{a}^{2}\rho }{1-rp_{a}^{2}(1-\rho )}}}
or, alternatively, function of
p
b
{\displaystyle p_{b}}
,
α
(
p
b
)
=
p
b
2
ρ
r
−
p
b
2
(
1
−
ρ
)
{\displaystyle \alpha (p_{b})={\frac {p_{b}^{2}\rho }{r-p_{b}^{2}(1-\rho )}}}
.
one may substitute expressions
r
,
ρ
{\displaystyle r,\rho }
definition of
α
{\displaystyle \alpha }
, obtain critical values of
p
a
,
(
p
b
)
{\displaystyle p_{a},(p_{b})}
admit gelation. gelation occurs if
p
a
>
α
c
r
(
α
c
+
ρ
−
α
c
ρ
)
.
{\displaystyle p_{a}>{\sqrt {\frac {\alpha _{c}}{r(\alpha _{c}+\rho -\alpha _{c}\rho )}}}.}
alternatively, same condition
p
b
{\displaystyle p_{b}}
reads,
p
b
>
r
α
c
α
c
+
ρ
−
α
c
ρ
{\displaystyle p_{b}>{\sqrt {\frac {r\alpha _{c}}{\alpha _{c}+\rho -\alpha _{c}\rho }}}}
the both inequalities equivalent , 1 may use 1 more convenient. instance, depending on conversion
p
a
{\displaystyle p_{a}}
or
p
b
{\displaystyle p_{b}}
resolved analytically.
trifunctional monomer difunctional b monomer
a trifunctional branch unit functional group reacting bifunctional branch unit b functional group, forming continuous step-growth polymer molecule.
α
c
=
1
f
−
1
=
1
3
−
1
=
1
2
{\displaystyle \alpha _{c}={\frac {1}{f-1}}={\frac {1}{3-1}}={\frac {1}{2}}}
since functional groups trifunctional monomer, ρ = 1 and
α
=
p
b
2
ρ
r
1
−
p
b
2
r
(
1
−
ρ
)
=
p
b
2
r
{\displaystyle \alpha ={\frac {\frac {p_{b}^{2}\rho }{r}}{1-{\frac {p_{b}^{2}}{r(1-\rho )}}}}={\frac {p_{b}^{2}}{r}}}
therefore, gelation occurs when
p
b
2
r
>
α
c
{\displaystyle {\frac {p_{b}^{2}}{r}}>\alpha _{c}}
or when,
p
b
>
r
2
{\displaystyle p_{b}>{\sqrt {\frac {r}{2}}}}
similarly, gelation occurs when
p
a
>
1
2
r
{\displaystyle p_{a}>{\sqrt {\frac {1}{2r}}}}
^ cite error: named reference flory, p.j. invoked never defined (see page).
^ cite error: named reference stauffer, dietrich, et al. invoked never defined (see page).
^ flory, p.j.(1941). molecular size distribution in 3 dimensional polymers ii. trifunctional branching units . j. am. chem. soc. 63, 3091
^ flory, p.j. (1941). molecular size distribution in 3 dimensional polymers iii. tetrafunctional branching units . j. am. chem. soc. 63, 3096
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