Chapter 21
Network Formation in Free-Radical Copolymerization Pseudo Kinetic Rate Constant Method for
Copolymers
with Long Branches
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H. Tobita and A. E . Hamielec McMaster Institute for Polymer Production Technology, Department of Chemical Engineering, McMaster University, Hamilton, Ontario L8S 4L7, Canada Kinetic models for network formation based on the pseudo-kinetic rate constant method are proposed both for pre- and post-gelation periods. These models which consider a l l of the important reactions in free radical copolymerization are quite general and r e a l i s t i c . The crosslinking density distribution, which i s a new concept in the modeling of network formation, is also proposed. The present models are easy to apply and provide greater insight into the phenomena which occur during network formation in free radical polymerization. R e c e n t l y t h e p o l y m e r i c network ( g e l ) h a s become a v e r y a t t r a c t i v e r e s e a r c h a r e a c o m b i n i n g a t t h e same t i m e fundamental and a p p l i e d t o p i c s of great i n t e r e s t . Since the p h y s i c a l p r o p e r t i e s of p o l y m e r i c networks s t r o n g l y depend on t h e p o l y m e r i z a t i o n k i n e t i c s , an u n d e r s t a n d i n g o f t h e k i n e t i c s o f network f o r m a t i o n i s i n d i s p e n s a b l e f o r d e s i g n i n g network s t r u c t u r e . V a r i o u s models have been proposed f o r t h e k i n e t i c s o f network f o r m a t i o n s i n c e t h e p i o n e e r i n g work o f F l o r y ( 1_) and Stockmayer (2) / b u t t h e i r predictions are, quite often unsatisfactory, e s p e c i a l l y f o r a free r a d i c a l p o l y m e r i z a t i o n system. These systems a r e o f s i g n i f i c a n t commercial i n t e r e s t . I n o r d e r t o account f o r t h e s p e c i f i c r e a c t i o n scheme o f 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 , i t w i l l be n e c e s s a r y t o c o n s i d e r a l l o f t h e important elementary r e a c t i o n s . I n t h i s p a p e r , t h e p s e u d o - k i n e t i c r a t e c o n s t a n t method i n w h i c h t h e k i n e t i c t r e a t m e n t o f a multicomponent p o l y m e r i z a t i o n reduces t o t h a t o f a homopolymerization i s e x t e n s i v e l y a p p l i e d f o r t h e s t a t i s t i c a l c o p o l y m e r i z a t i o n o f v i n y l / d i v i n y l monomers and a p p l i c a t i o n s t o t h e p r e - and p o s t - g e l a t i o n p e r i o d s a r e i l l u s t r a t e d . Pseudo-Kinetic
R a t e C o n s t a n t Method f o r L i n e a r Copolymer
The p s e u d o - k i n e t i c r a t e c o n s t a n t method f o r multicomponent p o l y m e r i z a t i o n has been a p p l i e d i n some c o p o l y m e r i z a t i o n s t u d i e s ( 3 - 5 ) , and i t s d e r i v a t i o n and s p e c i f i c a p p r o x i m a t i o n s have been made c l e a r ( 6 , 7 ) . The p s e u d o - k i n e t i c r a t e c o n s t a n t s b a s i c a l l y 0097-6156/89/0404-0242$06.00/0 © 1989 American Chemical Society
Provder; Computer Applications in Applied Polymer Science II ACS Symposium Series; American Chemical Society: Washington, DC, 1989.
21.
243
Network Formation
ΤΟΒΓΓΑ & HAMIELEC
c o n s i s t of the product of the i n t r i n s i c k i n e t i c r a t e constant, the mole f r a c t i o n o f monomer o f t y p e i ( f ^ ) and/or t h e mole f r a c t i o n o f polymer r a d i c a l o f t y p e j ( φ·· ) w h i c h a r e i n c l u d e d i n t h e p a r t i c u l a r r e a c t i o n . F o r example, t h e p s e u d o - k i n e t i c r a t e c o n s t a n t f o r p r o p a g a t i o n (kp) f o r a b i n a r y system i s g i v e n by; φ
kp = ( f c f + f c f ) 1
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1 1
1
1 2
2
+
( k
f
21 l
+ k
f
22 2
)
*2
( 1 )
Symbols used a r e d e f i n e d a t t h e end o f t h i s paper. The d e f i n i t i o n s o f o t h e r p s e u d o - k i n e t i c r a t e c o n s t a n t s c a n be found i n e a r l i e r papers (6,7). Necessary c o n d i t i o n s f o r t h e v a l i d i t y o f t h e p s e u d o - k i n e t i c r a t e constants are; (1) The t e r m i n a l model f o r c o p o l y m e r i z a t i o n i s a p p l i c a b l e . (2) The v a r i a n c e o f c o m p o s i t i o n a l d i s t r i b u t i o n f o r i n s t a n t a n e o u s l y formed polymers i s n e g l i g i b l y s m a l l . F o r l o n g l i n e a r c h a i n s t h e second c o n d i t i o n i s s u p p o r t e d by t h e Stockmayer b i v a r i a t e d i s t r i b u t i o n (8,9) w h i c h shows t h e b i v a r i a t e d i s t r i b u t i o n o f c h a i n l e n g t h and c o m p o s i t i o n i s t h e p r o d u c t o f b o t h d i s t r i b u t i o n s , and t h e c o m p o s i t i o n a l d i s t r i b u t i o n i s g i v e n by t h e normal d i s t r i b u t i o n whose v a r i a n c e i s i n v e r s e l y p r o p o r t i o n a l t o chain length. Applying the pseudo-kinetic rate constants, the e x p l i c i t f o r m u l a t i o n o f t h e k i n e t i c s o f a multicomponent p o l y m e r i z a t i o n reduces t o t h a t o f a homopolymerization. Branched and C r o s s l i n k e d Polymers C r o s s l i n k i n g D e n s i t y D i s t r i b u t i o n . L e t us c o n s i d e r t h e s t a t i s t i c a l c o p o l y m e r i z a t i o n o f v i n y l / d i v i n y l monomers w i t h o u t c h a i n t r a n s f e r t o polymer f o r s i m p l i c i t y . I n t h i s c a s e t h e c r o s s l i n k i n g d e n s i t y Ρ i s d e f i n e d as f o l l o w s . Ρ _ 2 (number o f c r o s s l i n k a g e s ) ~~ (number o f u n i t s bound i n t h e c h a i n ) = 2Ρ /(Ν χ) χ
0
(P) S
where U= [ 1 - ^(θ,ψ) W (e,if) ]/[ΐ(θ)+ β(θ)+ Ρ (θ,ψ) ί^(θ,ψ)] . θ
g
1 r
3 0 1
ΓΨ / Γ
(ψ)= t
W
3 ο 1
(Θ,Ψ)
Κ_(Θ,Ψ) d 9
0
Provder; Computer Applications in Applied Polymer Science II ACS Symposium Series; American Chemical Society: Washington, DC, 1989.
(29)
251
Network Formation
21. ΤΟΒΓΓΑ & HAMIELEC
* Number- and Weight-Average C h a i n L e n g t h o f t h e S o l F r a c t i o n .
•?
δο1 η
(ψ)= [1/τ
- sol
( + ) a
(ψ)]- [ Ρ| (Ψ)/2]
(30)
ο 1
VP
, 1-
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3 0 1 η ρ
(31)
p f !(ψ) Γ ^ ί ψ )
* Number-Average C h a i n L e n g t h o f t h e P r i m a r y M o l e c u l e s w h i c h B e l o n g to the Gel Fraction. T h i s p r o p e r t y may be u s e f u l when one uses t h e c l a s s i c a l e q u a t i o n f o r s w e l l i n g d e r i v e d by F l o r y .
θ 1
Γ9
(Θ,Ψ)=
2 S [τ(θ) + (Ρ(θ)/2) S]
P
(32)
where S= Ρ (Θ,Ψ) Κ ( θ , Ψ ) / [ τ ( θ ) + β(θ) + Ρ (Θ,Ψ) W (e,
