10.569 Synthesis of Polymers
Prof. Paula Hammond
Lecture 13: Thermodynamics of Free Radical Polymerizatioins, Ceiling T’s,
Tromsdorff Effect, Instantaneous P n
Δ H o − T Δ S o = Δ G o = − RT ln K eq
Δ G = Δ G o + RT ln K eq
in general but at equilibrium:
Δ G = 0 = Δ G o + RT ln K eq so Δ H o − T c
Δ S o = − RT c ln
M
1
[ ] c
at equilibrium where [M] c
= ceiling monomer con’c
Solve for T c
:
T c
=
Δ H o
Δ S o + R ln[ M ] c
[ ] c
=
[ ] eq
[ ] o s
Equilibrium
S.S. in mol/L (standard state in mol/L)
[ ] o s
⇒ bulk concentration of monomer or 1 mol/L (depending on
We will use 1 mol/L
Carbonyl: CH
2
=O
CH
3
—CH=O
CCl
3
—CH=O
Cyclics:
O
THF
Other carbonyls
O O
O
O O trioxane unzips via
H
C
2
O n as well
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.

Monomer [M] c
@ 25 o C (mol/L)
Vinyl acetate
H
1x10 -9
O
O
H
3
C
Methyl metacrylate (MMA)
CH
3
T c
(pure monomer) (
∼ 400 o C or higher
1x10 -3 220 o C o C)
O
H
3
CO
α -methyl styrene
2.2 61 o C
ν =
R p
R t
= k k p t
[ ]
[ ]
= (for thermally diss. initiator) = k p
[ ]
(
2 k d fk t
) 1
2 p n
= 2 a ν
Statistical Approach p = probability of growth of polymer chain
= “propagation probability” p = k M k p
[ ]
[ ]
+ 2 k t
[
M ⋅
]
=
R p
R
+ R t p
(
+ R tr
)
if needed (for chain transfer)
1-p = probability of chain termination
1 − p = k p
2 k t
[ ]
[ ]
+ 2 k t
X i
= number fraction of radicals having reached degree of polymerization = i
X i
= p (i-1) (1-p)
for chain end
10.569, Synthesis of Polymers, Fall 2006
Prof. Paula Hammond
Lecture 13
Page 2 of 5
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.

p n
=
∞
∑ iXi i = 1
∑
Xi
=
∞
∑ iXi =
(
1 − p
) ∑ ip i = 1
∞ i = 1
( i − 1
)
1.0
⇒ p n
=
1
1 − p here p ≠ π as in step growth p n
= now
1 −
1 p
=
R p
+ R t
R t
→ 1.0 (i.e. R p
>> R t
) p =
R
R p p
+ R t
⇒ p n
→ ν
⇒ p n
→
R p
R t p w
=
∑ iWi =
∑
∑ i
2
Xi iXi
=
∑ p i
2 n
Xi
apply series defn weight fraction
⇒ p w
=
1 + p
1 − p
PDI: z = 1+p } disproportionation as p → 1.0 z → 2.0
This all assumes termination by disproportionation
If we assume coupling, account for ∼ X ⋅ + ⋅ Y ∼
⇒ p n
=
2
1 − p p w
=
2 + p
1 − p as 1.0 z → 3/2 z = p w p n
=
2 + p
2
10.569, Synthesis of Polymers, Fall 2006
Prof. Paula Hammond
Lecture 13
Page 3 of 5
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 p
= k p
⎜ fk d
⎝ k t
⎠
1
2 [ ] 1
2
[ ]
= − d
[ ] dt
instantaneous expression
Express things in terms of π (monomer conversion)
π =
[ ] o
−
[ ] o
[ ]
= 1 −
[ ] o d π dt
= −
M
1
[ ] o dt d π dt
= k p ⎜
⎛ fk d k t
⎟
1
⎞
2
⎟
[ ] 1
2
[
1 − π
] integrate assuming a constant [I] = [I] o
π ( )
= 1 − exp
⎡
⎢ k p
⎛
⎝ fk k t d
⎞
⎠
1
2 [ ] o t
⎤
⎥
⎥
⎥ so applies to batch polymer [M] ↓ but [I] constant ⇒ e.g. early stages of polymer
If [I] is not held constant: d dt
= − k d
[ ] [ ] o exp
(
− k t
) d π dt
= k
⎛ p
⎝ fk k t d
⎞
1
2 [ ] 1
2 exp
⎛
⎝
− k t
2
⎟
⎠
(
1 − π
)
Integrate from t=0 to t:
π t
= 1 − exp
⎧
⎨
⎛
⎜⎜
2 k k d p
⎞
⎠⎝
⎛ fk
⎜⎜ k t d
⎠
1
⎞
2
⎟⎟
[ ] 1
2
⎡
⎢ exp ⎜
⎣ ⎝ k d
2 t
⎟
⎞ −
⎠
1 ⎥
⎤
⎫
⎦⎪
global → conv at that time
10.569, Synthesis of Polymers, Fall 2006
Prof. Paula Hammond
Lecture 13
Page 4 of 5
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.

[M], [I] ↓ ⇒ R p
↓
Max conversion is batch situation: t
So that all of I is consumed
π max
= 1 − exp
⎧
⎨−
⎩
⎛
⎜⎜
2 k k d p
⎞
⎟⎟
⎛
⎠⎝ fk d k t
⎞
⎠
1
2
1
2 o
⎫
⎪
⎬
⎭ e.g. Polystyrene, 1 wt% AIBN
⇒ π max
= 96.7% p n
= f ( t ) p n becomes fxn of time p n
( )
=
R p
R i t
( )
( )
[M]
[I]
Time avg value: p n
( )
= tot mol of monomer polym tot mol of "dead" chains (formed)
= o
π o
(
1 − exp
(
− k d t
))
10.569, Synthesis of Polymers, Fall 2006
Prof. Paula Hammond
Lecture 13
Page 5 of 5
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.