10.569 Synthesis of Polymers
Prof. Paula Hammond
Lecture 8: Network Formation, Statistical Approach, Pw Based, A Word on MWD for Nonlinear Polymerization Network formation
Consider first case:
a—b + a—a
a
[a3]
monomers so far have been difunctional
O
H2N R
O
O
HOC
HO
COH
R'
COH
R"
difunctional = 2 groups that can react
(participate in rxn)
OH
trifunctional:
HO
R"
OH
OH
a—b a—a b—a b—a b—a
a
b
a
b
a
b
a
Always end in ‘a’ group
Forms branches, not crosslinks
What is the molecular weight or MW distribution?
(Peebles, Schaefgen, Flory) for a—b + af systems
f = func of af
π = conversion of b-groups (b=minority)
(N B )o
r=
≤ 1.0
(N A )o
frπ + 1 − rπ
1 − rπ
( f − 1)2 (rπ )2 + (3 f − 2 )rπ + 1
pw =
( frπ + 1 − rπ )(1 − rπ )
pn =
Citation: Professor Paula Hammond, 10.569 Synthesis of Polymers Fall 2006 materials, MIT
OpenCourseWare (http://ocw.mit.edu/index.html), Massachusetts Institute of Technology, Date.
pw
pn
=z=
1 + frπ
( frπ + 1 − rπ )2
polydispersity
Consider limit as r → 1.0 lim z =
r→1.0
pw
pn
= 1+
1
f
⎛ r → 1.0 ⎞
⎜⎜
⎟⎟
⎝ π = 1.0 ⎠
Specific cases:
Let f = 1 ⇒ a—b + a—x
end capper
same as a—b + end capper
⇒ z = 2.0 ⇒ same result discussed for general difunctional systems
let f = 2 → a—b + a—a
linear polymer
z = 1 + ½ = 1.5 → narrower MWD
does this make sense?
Recall: a—b + a—a : b—a b—a b—a b—a b a—a b—a b—a b—a Join 2 chains together with a—a
Get rid of extremeties of MWD. Longer chains are joined together with short chain. Long chains can join w/long chains but much less likely. Consider an example with af:
1 mol a—b + 10-2 mol a3:
(0.01) f=3
1 b group to 1.03 a groups
Note: for r = 0.97 and
typical a—b polymerization:
p n = 66 =
10.569, Synthesis of Polymers, Fall 2006
Prof. Paula Hammond
1+ r
1− r
Lecture 8
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Citation: Professor Paula Hammond, 10.569 Synthesis of Polymers Fall 2006 materials, MIT
OpenCourseWare (http://ocw.mit.edu/index.html), Massachusetts Institute of Technology, Date.
r=
1
≅ 0.97
1.03
p n =
3(0.97 ) − 0.97 + 1
≅ 99
1 − 0.97
Systems forming networks
Soluble fraction → “sol”
Polymer chains, oligomers, monomers not connected to network Gel fraction → insoluble, intractable ⇒ constitutes network
Gel point: π at which an infinite network is formed ⇒ πc
Above gel point πc: gel fraction ↑
sol fraction ↓
Determine πc as:
Carothers
pn → ∞
pw → ∞
pn → ∞
Consider simplest case:
Equal # of a,b functional groups Define f AVG =
∑N f
∑N
i i
i
fi = functionality of given monomer
Ni = # of molecules with fi
O
O
3 HOC
COH
f AVG =
fi ≥ 1.0
3(2 ) + 1(3) + 1.5(2 ) 12.0
=
= 2.18
3 + 1 + 1.5
5.5
OH
1 HO
1.5
CH2CH CH2OH
HOCH2CH2OH
Can define conversion
No = initial # of monomer molecules
N = # of remaining molecules
Initial # of functional groups = fAVGNo
10.569, Synthesis of Polymers, Fall 2006
Prof. Paula Hammond
Lecture 8
Page 3 of 6
Citation: Professor Paula Hammond, 10.569 Synthesis of Polymers Fall 2006 materials, MIT
OpenCourseWare (http://ocw.mit.edu/index.html), Massachusetts Institute of Technology, Date.
O
6 HCOH
6 OH r=1.0 ⇒
# of functional groups reacted = 2(No-N)
For each a—b reaction, lose 1a, 1b ⇒ 2 functional groups For each a—b reaction, decrease # of molecules by 1 # of functional groups reacted 2(N o − N )
=
a π =
total # of functional gruops
N o f AVG
b pn =
N o
(see original definition)
N
# avg degree of polymerization
Rearrange
N o f AVG π = 2 N o − 2 N
a :
N o (πf AVG − 2 ) = −2 N
No
−2
=
N
πf AVG − 2
Universal expression
Carothers Equation
Works for all the cases, but fAVG
must be adjusted when r ≠ 1.0
N
2
p n =
=
N o 2 − πf AVG
Rearrange Carothers:
π =
2
f AVG
−
2
p n f AVG
Consider gel point:
pn → ∞
⇒ π c =
e.g.
2
f AVG
π at gel
point
general expression
for Carothers
more functionality → gel at lower conversion
f AVG = 2.18
2
πc =
= 0.92
2.18
Consider less perfect case:
Ni fi
(Na )o
f AVG =
only time for r =
= 1.0
(Nb)o
Ni
∑
∑
If r ≠ 1.0 ⇒ only gain in increased MW + crosslinking when using –a + –b
(i.e. deficient group quantity determines how many of these
reactions take place)
10.569, Synthesis of Polymers, Fall 2006
Prof. Paula Hammond
Lecture 8
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Citation: Professor Paula Hammond, 10.569 Synthesis of Polymers Fall 2006 materials, MIT
OpenCourseWare (http://ocw.mit.edu/index.html), Massachusetts Institute of Technology, Date.
Excess of b only decreases p n by end capping
Count only a groups + 2
f AVG =
2
(∑ N f )
∑N
ai ai
i
f A,i = function in - a of monomer i
N A,i = # of molecules of monomer i
Example:
HOCH2CH2OH
A
O
B
fi
2
fA = fa,i = 2
4
fB = fb,i = 2
1
fC = fa,i = 3
O
HOC
COH
OH
C
N
HOCH2CH2CHCH2OH
Enter body copy in Verdana 10.
8b, 7a deficient
where
a = OH
O
b = COH
f AVG =
πc =
2(N a f a + N c f c ) 2((2 )(2 ) + (1)(3))
= 2.0
=
2 + 4 +1
Na + Nb + Nc
2
= 1.0
2.0
Only at full conversion do you form network.
π c > 1.0 ⇒ physically impossible to create network
Case of exact stoichiometry: doesn’t matter which is deficient.
Let NB = 3.5
πc =
2
= 0.93
2.15
Other case:
Flory-Stockmayer
pw → ∞
Generalized cases:
10.569, Synthesis of Polymers, Fall 2006
Prof. Paula Hammond
Lecture 8
Page 5 of 6
Citation: Professor Paula Hammond, 10.569 Synthesis of Polymers Fall 2006 materials, MIT
OpenCourseWare (http://ocw.mit.edu/index.html), Massachusetts Institute of Technology, Date.
πc =
1
{r ( f w,A − 1)( f w,B − 1)}1/ 2
f w,A
∑f
=
∑f
f w,B
∑f
=
∑f
A,i
N A,i
i
A,i N A,i
B,i
r=
2
∑f
∑f
2
where fw,i can be ≥ 1.0
N B,i
i
B,i N B,i
A,i N A,i
B,i N B,i
For our earlier example,
Carothers:
≤ 1.0
π c = 0.90
(NB = 3.5)
π c = 0.93
Lower πc
→ longest chains form more of
infinite network
10.569, Synthesis of Polymers, Fall 2006
Prof. Paula Hammond
Lecture 8
Page 6 of 6
Citation: Professor Paula Hammond, 10.569 Synthesis of Polymers Fall 2006 materials, MIT
OpenCourseWare (http://ocw.mit.edu/index.html), Massachusetts Institute of Technology, Date.