1
FREE-RADICAL POLYMERIZATION
I. Basics
A. Free-radical polymerization is a type of chain polymerization (like ionic
polymerization).
B. Free radical monomer attacks double bond of another monomer to form bond
and propagate the free radical.
C. Most vinyl monomers are amenable to free-radical polymerization.
II. Kinetics
A. Initiation
1.
Decomposition is first-order in initiator.
ν d = k d [ I]
ν d = k d [ I]
Initiators are characterized by Td and t½
a. Td – decomposition temperature
i. Temperature where decomposition of initiator undergoes autoacceleration.
I → 2 R•
2.
ii. Thermal energy released from decomposition an initiator
molecule aids decomposition another initiator molecule.
b. t½ - half-life
i. Time needed for initial amount of initiator to decrease
concentration in half-life.
ii.
Related to decomposition rate constant
iii. Very temperature sensitive
3. Initiation is first-order in monomer and initiator radical.
M + R• → M•
ν i = k i [ R i ][ M ]
2
4.
Initiator efficiency – f
a. Not all initiator radicals initiate polymerization
b.
Possible fates of initiator radical after decomposition
i. Recombination in solvent cage
ii. Recombination outside of solvent cage
iii. Combination of initiator radical with polymer chain radical
iv. Reaction with initiator (R• + I → I + R•)
v.
Hydrogen abstraction from chain
vi. Reaction (hydrogen abstraction, possibly) with solvent
vii. Chain initiation ☺
c. Efficiency is ratio of number of chains started to total number of
initiator radicals
d. Initiator efficiencies range from 0.1 to 0.8
5.
Initiators
a.
Peroxides
i. Benzoyl peroxide (Td = 30 °C)
[Td is the self-accelerated decomposition temperature is the
temperature where the rate of enthalpy generated from the
decomposition of the peroxide serves to provide sufficient
energy for further decomposition.]
O
C
O
O
O
O
C
C
O
+
ii. Diacetyl peroxide (Td = 35 °C)
O
H 3C
C
O
O
O
C
CH3
+ CO2
3
iii. Di t-butylperoxide (Td = 80 °C)
CH3
H3C
C
CH3
O
O
C
CH3
CH3
CH3
iv. Cumyl peroxide (Td = 115 °C)
CH3
C
O
O
H
CH3
b.
Azo compounds
i. 2,2′-azobisisobutylnitrile (AIBN) (Td = 25 °C)
CH3
N
C
C
CH3
N
CH3
N
C
CH3
C
N
2
N
CH3
C
C
CH3
ii. Phenylazotriphenylmethane (Td = 40 °C)
The triphenylmethylradical is
very stable.
N
N
C
C
+
+ N2
+ N2
4
c. Redox initiators
i. Persulfate
S2O82- + S2O32- → SO4-• + SO42- + S2O3-•
ii. Fe2+ w/peroxide
a.) Fe2+ acts as a catalyst to lower activation energy and the
decomposition temperature.
H2O2 + Fe2+ → HO• + HO- + Fe3+
d.
Photoinitiators (for room temperature polymerizations)
i. Sulfide
R
O
OH
C
C
S
S
R
2 R
S
ii. Benzoin λ = 3600 Å
O
OH
C
HC
+
iii. Benzil
O
O
O
O
C
C
C
C
+
iv. Benzophenone
O
O
C
C
+
5
B. Propagation
1.
Assumptions about propagation
a. Propagation is independent of chain length (chain length is large)
b.
No chain transfer (to be considered later)
c. Only one free radical per chain
d.
Steady state approximation
i. Concentration of radical constant throughout reaction
ii. Initiation rate equals termination rate
2. First-order in radical and monomer.
ν p = k p [ M ][ M i ]
M + M• → M•
3. Head-to-tail or head-to-head propagation possible, though exclusive
head-to-head and tail-to-tail propagation is rare.
4. Inhibitors
a. Necessary for storage of most vinyl monomers
b.
Unwanted radical reacts with inhibitor
c. Inhibitor radical does not react with monomer
d.
Retarders are radicals that have slower reaction rate with monomer.
e. Examples
i. nitrobenzene, trinitrobenzene
ii. 2,5-di t-butyl-p-cresol
CH3
OH
H3C
H3C
CH3
H3C
CH3
CH3
6
C. Termination
1. Combination (coupling)
ν tc = k tc [ M i]
2
M• + •M → M – M
2. Disproportionation
R
H
H
C
C
H
H
+
H
H
C
C
H
H
R
R
H
H
C
C
ν td = k td [ M i]
2
H
+
H
H
H
C
C
H
H
3. Some polymerizations prefer combination (polystyrene) while others
prefer disproportionation (PMMA).
Termination preference of different polymers at 60 °C.
Disproportionation
≈ 0%
23%
79%
≈ 100%
Polyvinyl cyanide
Polystyrene
Polymethylmethacrylate
Polyvinylacetate
Combination
≈ 100%
77%
21%
≈ 0%
4. The total rate of termination is the sum of the combination and the
disproportionation rate. kt = ktc + ktd
D. Degree of Polymerization
1. Rate of propagation is first-order in monomer and radical.
ν p = k p [ M ][ M i ]
2. Apply the steady state approximation to find the concentration of chain
radical.
d [ Mi]
dt
= k i [ R i ][ M ] − 2k t [ M i] = 0 ⇒
2
d [ R i]
dt
⇒
[ Mi] =
= 2k d f [ I ] − k i [ R i ][ M ] = 0 ⇒
[ Mi] =
k i [ R i ][ M ]
2k t
=
[ R i] =
k i 2k d f [ I ][ M ]
2k t k i [ M ]
f – efficiency of initiator decomposition
=
k i [ R i][ M ]
2k t
2k d f [ I]
ki [M]
k d f [ I]
kt
R
7
3. Calculate the propagation rate.
ν p = k p [ M ][ M i] = k p [ M ]
k d f [ I]
kt
1
= k′ [ M ][ I ] 2
4. Calculate the termination rate.
⎛ k f [ I] ⎞
2
ν t = k t [ M i] = k t ⎜ d
⎟ = k d f [ I]
⎝ kt ⎠
5. Calculate the degree of polymerization (ratio of propagation rate to
termination rate).
1
k f
k p d [ M ][ I ] 2
ν
k p [M]
kt
=
DP = p =
1
νt
k d f [ I]
kdktf I 2
[]
E. Measuring the rate of polymerization (dilatometry)
1. Polymerization rate is the rate of monomer consumption.
−d [ M ]
dt
2. The average initial rate can be calculated as:
ν poly =
ν 0poly =
[ M ]0 − [ M ]
t
3. Direct measurement of the reaction rate (stop reaction at different times,
measure amount of monomer) is possible, but time-consuming.
4. Measuring a change in a physical property would be much easier.
5. Measuring change in volume as reaction proceeds is dilatometry.
a. Volume of reaction mixture is measured via the height of a capillary
tube.
b. Total volume is the volume of monomer, polymer and solvent.
V = w m v m + w p v p + w s vs
c. Assume the weight of the polymer is consumed monomer.
V − w 0m vp − w s vs
0
wp = wm − wm ⇒ wm =
vm − vp
d. At time zero, volume is from monomer and solvent.
V0 = w 0m vm + w s vs
e. Assume at infinite time, volume is from polymer and solvent.
Note: w ∞p = w 0m
V∞ = w 0m vp + w s vs
8
f. Subtract V0 - V∞.
V0 − V∞ = w 0m vm + w s vs − w 0m vp − w s vs = w 0m vm − w 0m vp = w 0m ( vm − vp )
wm =
V − w 0m vp − w s vs
vm − vp
=
V − ( w 0m vp + w s vs )
V0 − V∞ w
0
m
= w 0m
V − V∞
V0 − V∞
g. Yield of reaction (Y) can be written in terms of monomer weight and
consequent volume (or height) in dilatometer.
Δh ( t )
w 0m − w m V0 − V
Y=
=
=
w 0m
V0 − V∞ Δh ( ∞ )
h.
Thus rate is
ν poly = Y
[ M ]0
=
t
Δh ( t ) [ M ]0
Δh ( ∞ )
t
F. Average degree of polymerization and number-average molecular weight
1. The rate of polymer formation depends on counting the polymer from
disproportionation and combination correctly.
d [P]
2
= ( k tc + 2k td ) [ M i]
dt
DP =
νp
νt
=
k p [ M ][ M i]
k p [M]
−d [ M ] dt
=
=
2
d [ P ] dt
( k tc + 2k td ) [ Mi] ( k tc + 2k td ) [ Mi]
η=
DP =
νp
νi
=
k p [ M ][ M i]
2k t [ M i ]
2
=
kp [M]
2k t [ M i]
kp
2η ( k tc + k td )
[M] =
η 2k t
2η k t
=
=
( k tc + 2k td ) [ Mi] ( k tc + 2k td ) k p ( k tc + 2k td ) ( k tc + 2k td )
kp
η is the kinetic chain length
2. The rate of polymer formation depends on counting the polymer from
disproportionation and combination correctly.
a. If combination dominates, i.e., ktc >> ktd
DP =
2η ( k tc + k td ) 2η ( k tc )
≈
= 2η
( k tc + 2k td )
( k tc )
b. If disproportionation dominates, i.e., ktd >> ktc
DP =
2η ( k tc + k td ) 2η ( k td )
≈
=η
( k tc + 2k td ) ( 2k td )
3. Number-average molecular weight depends on degree of polymerization.
M n = M DP
9
G. Distribution of DP
1. To consider the distribution of polymer chain lengths, we must begin
making a distinction between different polymer chain lengths. Thus the rate
of formation of polymer chain with “n” units is
d [ M n i]
= k p [ M ][ M n −1 i ] − k p [ M ][ M n i] − 2k t [ Mi ][ M n i] = 0
dt
2. The ratio of polymer with “n” units to “n-1” units is
−1
k p [M]
k p [M] ⎛
[ M n i] =
1
=
⎜⎜
[ M n −1 i] k p [ M ] + 2k t [ Mi] k p [ M ] ⎝ 1 + 2k t [ Mi]
⎞ ⎛ 2k t [ M i ] ⎞
⎟ = ⎜1 +
⎟
k p [ M ] ⎟⎠ ⎜⎝
k p [ M ] ⎟⎠
3. Recalling the definition of the kinetic chain length, η = ν p ν i = k p [ M ] 2k t [ M i ]
[ M n i] = ⎛1 + 1 ⎞
[ M n −1 i] ⎜⎝ η ⎟⎠
−1
4. Multiplying successive “n-1” ratios together yields
[ M n i] = [ M n i] [ M n −1 i] [ M n −2 i] [ M n −3 i] [ M 2 i] = ⎛1 + 1 ⎞ (
[ M1 i] [ M n −1 i] [ M n −2 i] [ M n −3 i] [ M n −4 i] [ M1 i] ⎜⎝ η ⎟⎠
− n −1)
5. The fraction of propagating radicals that have a degree of polymerization “n” is
(
M n i ] [ M1 i] ⎛ 1 ⎞
[
=
fn =
1+
[ Mi] [ Mi] ⎜⎝ η ⎟⎠
1− n )
6. An alternative approach involves consideration of the steady-state
approximation.
d [ Mi]
2
= νi − 2k t [ M i] = 0
dt
d [ M1 i ]
= νi − k p [ M ][ M1 i ] − 2k t [ M i ][ M1 i ] = 0
dt
7. Substitution and rearrangement yields
[ M1 i] = 1 ⎛1 + 1 ⎞
[ Mi] η ⎝⎜ η ⎠⎟
−1
8. Thus the fraction of propagating radicals with DP of “n” is
[ M n i] = 1 ⎛1 + 1 ⎞
fn =
[ Mi] η ⎜⎝ η ⎟⎠
−n
10
9. Using the above expression we will find mole fractions of polymer having
DP of “n”.
d [ Pn ] dt
d [ Pn ] dt
Xn =
=
∑ d [ Pn ] dt d [ P] dt
n
10. To do the summations, we must be careful about the distinction between
termination via combination versus termination via disproportionation.
a. In combination, a simple summation would count each termination twice
except for two radicals that had n/2 units.
M1 + Mn-1 → Pn
M2 + Mn-2 → Pn
M3 + Mn-3 → Pn
etc …
Mn/2 + Mn/2 → Pn
d [ Pn ]
dt
= k tc,nn ⎡⎣ M i n 2 ⎤⎦ ⎡⎣ M i n 2 ⎤⎦ +
1 n −1
∑ k tc,mn [ Mi n −m ][ Mi m ]
2 m =1
b. In disproportionation, counting arguments (ignoring rate constants) tell
us that a polymer of chain length “n” is made twice as fast as in
combination.
d [ Pn ]
= 2k td [ M i n ][ M i ]
dt
c. Thus together the rate of formation of polymer with chain length “n” is
d [ Pn ]
1 n −1
′k tc,mn [ Mi n −m ][ Mi m ] + 2k td [ Mi n ][ Mi]
dt
2Σ
m =1
d. The rate constant for combination of unlike radicals is twice the rate of
combination of identical radicals. (Again: a simple counting argument.)
ktc, mn = ktc,nn
n −1
d [ Pn ]
m
= k tc ∑ [ M i n − m ][ M i m ] (1) + 2k td [ Mi n ][ Mi ]
dt
m =1
= k tc,nn ⎡⎣ M i n 2 ⎤⎦ ⎡⎣ Mi n 2 ⎤⎦ +
11. To do the summations, we must be careful about the distinction between
termination via combination versus termination via disproportionation.
d [ Pn ]
dt
[ Mi ] ⎫ 2
⎪⎧ n −1 [ M i n − m ] [ M i m ] m
= ⎨ k tc ∑
(1) + 2k td n ⎪⎬ [ Mi]
[ Mi] ⎪⎭
⎪⎩ m =1 [ M i] [ M i]
[ M n i] = 1 ⎛1 + 1 ⎞
12. Substituting for the fraction of radicals f n =
[ Mi] η ⎜⎝ η ⎟⎠
d [ Pn ]
dt
⎡ k ⎛ 1 ⎞ − n n −1 m 2k
= ⎢ tc2 ⎜1 + ⎟ ∑ (1) + td
η
⎢⎣ η ⎝ η ⎠ m =1
⎛ 1 ⎞⎤
2
⎜1 + ⎟ ⎥ [ Mi]
⎝ η ⎠ ⎥⎦
−n
11
n −1
13. The summation yields
∑ (1)
m
= n −1 .
m =1
d [ Pn ] 1 ⎛ 1 ⎞ ⎛ n − 1
⎞
2
= ⎜1 + ⎟ ⎜ k tc
+ 2k td ⎟ [ M i ]
dt
η⎝ η⎠ ⎝
η
⎠
−n
14. The mole fraction for each polymer chain length is
−n
⎛ n − 1 ⎛ k td ⎞ ⎞
⎞
1 ⎛ 1 ⎞ ⎛ n −1
2
+ 2⎜
1 + ⎟ ⎜ k tc
+ 2k td ⎟ [ M i ]
⎟⎟
−n ⎜
⎜
d [ Pn ] dt η ⎝ η ⎠ ⎝
η
1⎛ 1⎞ ⎝ η
⎝ k tc ⎠ ⎠
⎠
Xn =
=
= ⎜1 + ⎟
2
d [ P ] dt
η⎝ η⎠
⎛
⎛ k td ⎞ ⎞
( k tc + 2k td ) [ Mi]
⎜1 + 2 ⎜
⎟⎟
⎝ k tc ⎠ ⎠
⎝
15. Termination is exclusively combination, i.e., ktd = 0
n −1 ⎛ 1 ⎞
X n = 2 ⎜1 + ⎟
η ⎝ η⎠
−n
16. Termination is exclusively disproportionation, i.e., ktc = 0
1⎛ 1⎞
X n = ⎜1 + ⎟
η⎝ η⎠
−n
17. Additional manipulation yields the weight fractions
⎛ n − 1 ⎛ k td ⎞ ⎞
+ 2⎜
⎜
⎟⎟
k tc ⎠ ⎟
Xn n
n ⎛ 1⎞ ⎜ η
⎝
Wn =
=
⎜1 + ⎟
⎛k ⎞ ⎟
DP 2η2 ⎝ η ⎠ ⎜
⎜⎜ 1 + ⎜ td ⎟ ⎟⎟
⎝ k tc ⎠ ⎠
⎝
H. Effect of chain transfer
1. The production of polymer with chain length “n” can also occur via chain
transfer.
−n
Mn• + X → Pn + X•
2. Thus the expression for the rate of formation of polymer and degree of
polymerization given above must be modified.
d [ P]
d [ Pn ]
=∑
= ( k x [ X ] + k tc [ M i] + 2k td [ M i ]) [ M i]
dt
dt
n
DP =
k p [M]
k x [ X ] + ( k tc + 2k td ) [ M i ]
12
3. For mathematical convenience, we’ll consider the inverse of DP.
( DP )
−1
=
k x [ X ] ( k tc + 2k td ) ν p
( k tc + 2k td )
+
=
2
2
k p [M]
2η ( k tc + k td )
k p [M]
4. Chain transfer can occur with monomer(M), polymer(P), solvent(S),
initiator(I) or a specifically added chain transfer agent(Y).
k x [ X ] = k M [ M ] + k S [S] + k I [ I ] + k polymer [ P ] + k Y [ Y ]
5. Define chain transfer constants as
k
Ci = i
kp
6. Thus the inverse of the degree of polymerization can be written as
−1
−1
[ I] + C [S] + C [ Y ] + C
[P]
DP = DP + CI
S
Y
polymer
0
[M]
[M]
[M]
[M]
( ) ( )
a.
( DP )
−1
0
is the inverse of DP without chain transfer.
( DP )
−1
0
= CM +
( k tc + 2k td ) rp
2
k 2p [ M ]
7. The fraction of propagating radicals with DP of “n” is now written as
[ R ] 1 + γη ⎛1 + 1 + γ ⎞
fn = n =
⎜
⎟
η ⎝ η
[R ]
⎠
−n
a. where γ is the chain transfer term defined as
[ I] + C [S] + C [ Y ]
γ = CM
[M] S [M] Y [M]
8. The mole fraction for each polymer chain length is
⎡
⎛ ⎛ k ⎞ ⎛ k ⎞ 1 ⎞⎤
1 + γη
+ 2 γη ⎜1 + ⎜ td ⎟ + ⎜ td ⎟ ⎟ ⎥
⎢ ( n − 1)
−n
η
d [ Pn ] dt ⎛ 1 + γη ⎞ ⎢
1⎞
⎝ k tc ⎠ ⎝ k tc ⎠ γη ⎠ ⎥ ⎛
⎝
Xn =
=⎜
⎟
⎥ ⎜1 + γ + η ⎟
d [ P ] dt ⎝ η ⎠ ⎢
⎛ ⎛ k td ⎞ ⎛ k td ⎞ 1 ⎞
⎠
⎢
⎥⎝
1 + 2γη ⎜1 + ⎜
⎟+⎜
⎟ ⎟
⎝ ⎝ k tc ⎠ ⎝ k tc ⎠ γη ⎠
⎣⎢
⎦⎥
13
I. Kinetics Summary
1
1. Propagation rate: ν p = k ′ [ M ][ I ] 2
2. Termination rate: ν t = k d f [ I ]
3. Number-average molecular weight
kp
[ M ] ⎛1 + 1 − f comb. ⎞
Mn = M
⎟
1
1 ⎜
1 ⎠
⎝
2
2
2k
k
f
I
( d t ) []
fcomb – fraction of combination (versus disproportionation)
4. The first chain initiated rapidly forms a high molecular weight chain.
5. Monomer concentration steadily decreases (in contrast to step
polymerization).
6. ki, kp, kt follow Arrhenius behavior
7. Increasing temperature increases initiation thus decreases molecular
weight.
8. Above a ceiling temperature, chains will depolymerize.
Enthalpies and entropies of polymerization at 25 °C
1,3 -butadiene
Ethylene
Isoprene
Methyl Methacrylate
Stryene
ΔH (kJ./mol)
- 78
- 109
- 75
- 65
- 70
ΔS (J./mol)
109
155
101
117
104
Tc (K)
585
610
466
198
395
9. Molecular weights can be limited by viscosity of reaction mix.