2
Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on September 10, 2015 | http://pubs.acs.org Publication Date: September 24, 1982 | doi: 10.1021/bk-1982-0197.ch002
Simulation of the Kinetics of Styrene Polymerization L. A. CUTTER and T. D. DREXLER United States Steel Corporation, Research Center, Monroeville, PA 15146
A mathematical model for styrene polymerization, based on free-radical kinetics, accounts for changes in termination coefficient with increasing conversion by an empirical function of viscosity at the polymerization temperature. Solution of the differential equations results in an expression that calculates the weight fraction of polymer of selected chain lengths. Conversions, and number, weight, and Z molecular -weight averages are also predicted as a function of time. The model was tested on peroxide-initiated suspension polymerizations and also on batch and continuous thermally initiated bulk polymerizations.
0097-6156/82/0197-0013$06.00/0 © 1982 American Chemical Society
In Computer Applications in Applied Polymer Science; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1982.
14
COMPUTER APPLICATIONS IN APPLIED POLYMER SCIENCE
Bamford e t al.(J_) have presented the b a s i c k i n e t i c s f o r f r e e r a d i c a l p o l y m e r i z a t i o n of styrene/ and Hamielec e t a l . ( 2 J employing Bamford's nomenclature/ developed d i f f e r e n t i a l equations that are the s t a r t i n g p o i n t f o r the mathematical model. #
Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on September 10, 2015 | http://pubs.acs.org Publication Date: September 24, 1982 | doi: 10.1021/bk-1982-0197.ch002
The r a t e s ( i n m o l e s / l i t r e sec) of the component r e a c t i o n s i n the o v e r a l l f r e e - r a d i c a l chain scheme are as f o l l o w s : The
r a t e of i n i t i a t i o n / I = 2»f»k »In + k »M a t i
X
where f = the e f f i c i e n c y of the i n i t i a t o r i n s t a r t i n g r e a c t i o n chains. k^ = r a t e c o e f f i c i e n t f o r decomposition of the i n i t i a t o r / s e c " In = i n i t i a t o r c o n c e n t r a t i o n i n m o l e s / l i t r e M = monomer c o n c e n t r a t i o n i n m o l e s / l i t r e x = exponent f o r thermal i n i t i a t i o n / assumed = 3. The
r a t e of chain propagation
of a r a d i c a l of chain
1
length
0
r = Kp*M«R r k
= r a t e c o e f f i c i e n t f o r propagation
u
= c o n c e n t r a t i o n of r a d i c a l s of chain length r m o l e s / l i t r e .
r»p
R
r
l i t r e / m o l e sec.
The r a t e of t e r m i n a t i o n o f r a d i c a l s of chain length r = k »R °*R° (Termination i s assumed t o be by combination o n l y ) , tc r k. bP
c
The
= r a t e c o e f f i c i e n t f o r t e r m i n a t i o n l i t r e / m o l e sec. = t o t a l c o n c e n t r a t i o n of f r e e r a d i c a l s m o l e / l i t r e .
r a t e s of chain t r a n s f e r a r e : 0 #
To s o l v e n t = k »R S fs r To monomer = k »R *M f r k
= r a t e c o e f f i c i e n t f o r chain t r a n s f e r t o solvent/ l i t r e / m o l e sec. S = solvent concentration/ m o l e s / l i t r e , kf = rate c o e f f i c i e n t f o r chain t r a n s f e r t o monomer/ l i t r e / m o l e f g
The d i f f e r e n t i a l equations used i n t h i s model are the same as those presented by Hamielec.(2) The r a t e o f change of concent r a t i o n of r a d i c a l s of chain length one (R °) i s given by
In Computer Applications in Applied Polymer Science; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1982.
2.
dR. — — dt
The
Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on September 10, 2015 | http://pubs.acs.org Publication Date: September 24, 1982 | doi: 10.1021/bk-1982-0197.ch002
Simulation of Styrene Polymerization
CUTTER AND DREXLER
0
Kinetics
0 0 0 = I - k • M • R„ + ( k • S + k • M) • (R - R. ) p 1 fs f 1 0 0 -k R. • R tc 1
15
£
(
1
)
rate of change f o r R ° and longer chains i s 2
0 dR
r
- T —
dt
0 0 0 = k • M • R « -k • M • R - ( k . • S + k • M) • R p r-1 p r fs f r (2) - k • R •R tc r 0
The
0
o v e r a l l r a t e of change of r a d i c a l i s 0 X—^dR ^
.
d R
dt
dt
1
w
0o (R )
• tc
(3)
~r=l' The
r a t e of monomer consumption (polymerization rate) i s given by dM 0 - — = I + k • M • R + k dt p f
The
r a t e of formation
0 • M • R
(4)
of dead polymer of chain length r i s r-1
0 dP^ 0 — = (k • S + k • M) R + 1/2«k • dt fs f r tc
0 0
r
> /
R j n
0 0 • R r-n
(5)
n=1 I f the r a t e of formation constant
of r a d i c a l / 1/ i s taken as a
f o r a short p e r i o d of time/ Equation
3 can be i n t e g r a t e d
d i r e c t l y f o r t h i s short p e r i o d t o give
tanh ( / l • k tc J
• t)
(6)
tc
For the p o l y m e r i z a t i o n of styerene/ v l • k^_ i s about 0.1 so t h a t • t ) *1 and f o r w i t h i n a few seconds of r e a c t i o n time tanh f./ l H k t
a l l p r a c t i c a l purposes 0
/ I
In Computer Applications in Applied Polymer Science; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1982.
16
COMPUTER APPLICATIONS IN APPLIED POLYMER SCIENCE Equation 4 can be i n t e g r a t e d by the c o n s i d e r a t i o n
long chains I i s n e g l i g i b l e compared with k^ M R ° .
that f o r
The r e s u l t i s
t
f oJ
M = M «e Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on September 10, 2015 | http://pubs.acs.org Publication Date: September 24, 1982 | doi: 10.1021/bk-1982-0197.ch002
(k + k j • R o f p
0
• dt
(8)
f
0 dR A f u r t h e r s i m p l i f i c a t i o n can be made because ^ i s very nearly equal t o zero, g i v i n g r i s e t o a pseudo steady s t a t e . Under t h i s c o n d i t i o n the d e r i v a t i v e s can be eliminated from Equations 1 and 2, making them a l g e b r a i c . These equations give a r e c u r r i n g r e l a t i o n s h i p between R and R--| • r
k
0 R.
r
•
= R
•S + k fs
I
f >M+k
k = R
^
|
•M
f
^ k
(9)
• S + k • M + / ik fs f tc
0 R
•M + / i •k tc
| (10)
•M + k • S + k «M + / i •k fs f tc
p Now l e t
k
•M 2
k
» M + k
—
» S + k
p
fs
—
(11)
•M + / i •k f tc
where £ i s the p r o b a b i l i t y f a c t o r f o r the p r o b a b i l i t y that a f r e e r a d i c a l w i l l propagate rather than enter a termination r e a c t i o n . With t h i s d e f i n i t i o n , Equation 10 can be w r i t t e n 0 R and
r
0 — R r-1
• C
(12)
Equation 9 becomes 0 R
1
0 = R • (1-C)
(13)
Equations 12 and 13 can be manipulated t o give 0 R
r
0 = R • (1-C)
• C
In Computer Applications in Applied Polymer Science; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1982.
(14)
2.
Simulation
CUTTER AND DREXLER
of Styrene Polymerization
Kinetics
17
With the use of Equation 14, Equation 5 can be w r i t t e n d
P
/ ( fs* S + k
r
dt
=
• R°
+ 1/2 • k tc Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on September 10, 2015 | http://pubs.acs.org Publication Date: September 24, 1982 | doi: 10.1021/bk-1982-0197.ch002
0 _! • Mj • R • (1-C) • C x
k
2
f
r
• (r-1) • ( 1 - ? )
2
d5)
r?~
•
2
I f Equation 15 i s i n t e g r a t e d and summed over a l l the s p e c i e s , we get op
t r~ P
r -
k
/
|( fs * S + k
/ [C +
L
J
k
f s
2
»R
• (4 ^4 + 7«C 7»£ + + C ) 2
dt
(22)
2d-er
The weight f r a c t i o n of polymers of given chain length can be c a l c u l a t e d from r*P w
r " Zr~Z Mo-M
< >
This r a t i o can be obtained f o r s e l e c t e d chain lengths by i n t e g r a t i n g Equation 15.
In Computer Applications in Applied Polymer Science; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1982.
23
2.
CUTTER AND DREXLER
Simulation of Styrene Polymerization
Kinetics
19
Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on September 10, 2015 | http://pubs.acs.org Publication Date: September 24, 1982 | doi: 10.1021/bk-1982-0197.ch002
Computer Model A digital-computer program was w r i t t e n t o solve the equations and c a l c u l a t e the conversion at any time i n a batch polymeriz a t i o n . Polymerization r a t e s , instantaneous and cumulative molecular-weight d i s t r i b u t i o n s , and molecular-weight averages are a l s o c a l c u l a t e d a t t h i s time. Numerical i n t e g r a t i o n i s performed as i n d i c a t e d i n the previous s e c t i o n on Equations 8, 15, 20, 21, and 22 at constant temperature over time i n t e r v a l s s e l e c t e d by the user. Up t o 10 step changes i n temperature can be made i n the course of a batch p o l y m e r i z a t i o n . P r o v i s i o n i s made f o r up to f i v e i n i t i a t o r s ( p r i m a r i l y f o r suspension p o l y m e r i z a t i o n ) ; i n a d d i t i o n thermally i n i t i a t e d p o l y m e r i z a t i o n can be c a l c u l a t e d with or without initiators. For i n i t i a l t e s t i n g of the model, values of the r a t e c o e f f i c i e n t s were taken from the l i t e r a t u r e ; m o d i f i c a t i o n s were made t o b e t t e r f i t the data as shown i n Table I . As noted by e a r l i e r i n v e s t i g a t o r s , a c o r r e c t i o n must be a p p l i e d t o the termination c o e f f i c i e n t k t o allow f o r the decrease i n termination rate caused by the r e d u c t i o n i n r a t e s of d i f f u s i o n of the polymer r a d i c a l s as conversion and v i s c o s i t y i n c r e a s e . An e m p i r i c a l c o r r e c t i o n f u n c t i o n was developed t o correct k f o r the increased v i s c o s i t y a t higher conversions. t c
fcc
K
1 + c •exp(log n-c \ (corrected) = K - — j-± r-^tc t c 1 + exp ( l o g ^ n - c ^ J
where n = zero-shear v i s c o s i t y i n poises c = constant - 4 -5 c = constant = e i
2
At low conversions and v i s c o s i t i e s K ( c o r r e c t e d ) = K the expression 1+exp( l o g ^ n - 4 ) i n the denominator i s the main correction to K t o make K decrease t o conform t o the experimental data. Making the expression 1+exp(log^-C^) minimizes the e f f e c t of v i s c o s i t y at low conversions, where v i s c o s i t y i s changing r a p i d l y , but has a r e l a t i v e l y small e f f e c t on the r a t e s and molecular weights. The c o r r e c t i o n term i n the numerator was included t o moderate the e f f e c t of the denominator at very high v i s c o s i t i e s and conversions. The zero-shear v i s c o s i t y was c a l c u l a t e d by Hoffman (_3) adapting the g e n e r a l i z e d equation of Berry and Fox (4) fcc
fcc
t c
In Computer Applications in Applied Polymer Science; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1982.
fcc
20
COMPUTER APPLICATIONS IN APPLIED POLYMER SCIENCE u c -i D x x a d) a; x CD rH rH C rH C •H rH rH X X X X X rH in co in in rH r- in C CM CO o CM • • • • • rH rH CM rH CM 4J
I in in CM CM a in in x o • . CD . o o CO vo c
in rrCM
in \ o o CO o in rH r> 1 H —' 1
EH
CD
\
» •
a x a a; x in rH CH rH rH C X X
o oriH ) o X X rH X —1 in in in c CM rH •
•H 4->
E
Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on September 10, 2015 | http://pubs.acs.org Publication Date: September 24, 1982 | doi: 10.1021/bk-1982-0197.ch002
\
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0> O.
4J
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rr.
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(0 0)(0 e o o 0 0 0 O 4 J a. - H c x N 4J o o c
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i u (0'H C • H C 4 J 4 J 4J CD a £ - H •H CC cDc c 4J C C 1 -H
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