Kinetic Model of Cure Reactions - American Chemical Society

16 Kinetic Model of Cure Reactions Aids to Property and Processing Predictions

Downloaded by EAST CAROLINA UNIV on May 30, 2014 | http://pubs.acs.org Publication Date: August 29, 1982 | doi: 10.1021/bk-1982-0227.ch016

H. S.-Y. HSICH, R. M . ZURN, and R. J. AMBROSE Lord Corporation Mechanical Group, Material and Process Research and Development, Erie, PA 16514

A generalized kinetic model of cure is developed from the aspect of relaxation phenomena. The model not only can predict modulus and viscosity during the cure cycle under isothermal and non-isothermal cure conditions, but also takes into account filler effects on cure behavior. The increase of carbon black filler loading tends to accelerate the cure reaction and also broadens the relaxation spectrum. The presence of filler reduces the activation energy of viscous flow, but has little effect on the activation energy of the cure reaction. Understanding and p r e d i c t i n g cure k i n e t i c s o f e l a s t o m e r i c o r t h e r m o s e t t i n g m a t e r i a l s i s of p r a c t i c a l i n t e r e s t both i n the manufacturing process and i n end-product performance/ r e l i a b i l i t y . U n f o r t u n a t e l y , i n most p o l y m e r i c systems, the c u r e r e a c t i o n s are q u i t e complex; t h e r e f o r e , i t i s d i f f i c u l t f o r one t o develop a k i n e t i c model which can e x p l a i n and p r e d i c t changes of p h y s i c a l and mechanical p r o p e r t i e s o f the polymer d u r i n g the cure r e a c t i o n . Although t h e r e have been many s t u d i e s on cure k i n e t i c s , most o f them are l i m i t e d t o the method o f c a l o r i m e t r y [ 1 - 5 ] , such as d i f f e r e n t i a l scanning c a l o r i m e t r y (DSC) o r d i f f e r e n t i a l thermal a n a l y s i s (DTA). In those s t u d i e s , the d e f i n i t i o n o f the s t a t e of cure i s not d i r e c t l y c o r r e l a t e d t o the p h y s i c a l , mechanical, o r r h e o i o g i c a l p r o p e r t i e s o f polymers. T h e r e f o r e , k i n e t i c models which are developed from the c a l o r i m e t r i c method are unable t o p r e d i c t p r o p e r t i e s , such as v i s c o s i t y and dynamic modulus, which are used t o determine the manufacturing o p e r a t i o n and end-product performance of p o l y m e r i c systems. C o n v e n t i o n a l l y , r u l e s o f thumb have been w i d e l y used i n the i n d u s t r y f o r c u r i n g r u b b e r s . I t has been assumed t h a t the r a t e of cure i s doubled f o r every 10*C i n c r e a s e i n cure temperature. 0097-6156/83/0227-0263$07.00/0 © 1983 American Chemical Society In Chemorheology of Thermosetting Polymers; May, C.; ACS Symposium Series; American Chemical Society: Washington, DC, 1982.

264

CHEMORHEOLOGY OF THERMOSETTING

POLYMERS

In a d d i t i o n , cure time i s i n c r e a s e d f i v e minutes f o r every 0.25 inches o f t h i c k n e s s o f a molding [ 6 , 7 ] . In g e n e r a l , these r u l e s do not apply t o most p o l y m e r i c systems because t h e phenomena o f heat t r a n s f e r and cure k i n e t i c s have been o v e r - s i m p l i f i e d . The cure r a t e depends on t h e b a s i c polymers, c u r a t i v e s , cure temperature, and f i l l e r l o a d i n g . The p r e d i c t i o n o f cure r a t e w i l l be d i s c u s s e d from a new model o f cure k i n e t i c s which i s developed from t h e concept o f a n o n - e q u i l i b r i u m thermodynamic f l u c t u a t i o n t h e o r y o f chemical r e l a x a t i o n .


KINETIC THEORY OF THE CURE REACTION The k i n e t i c model f o r p r e d i c t i n g v i s c o s i t y and modulus d u r i n g cure o f elastomers o r thermosets has been given by R o l l e r [ 8 ] , C r a i g [ 9 ] , and M u s a t t i and Macosko [ 1 0 ] . U n f o r t u n a t e l y , these models can o n l y p r e d i c t a narrow range o f data d u r i n g c u r e ; a l s o these models do not i n c l u d e p o l y m e r - f i l l e r i n t e r a c t i o n s . Because of t h e shortcomings o f these models, a g e n e r a l i z e d k i n e t i c model of cure was r e c e n t l y proposed [ 1 1 ] . As has been d i s c u s s e d by Hsich [ 1 2 , 1 3 ] , any chemical r e l a x a t i o n ( r e a c t i o n ) o r s t r u c t u r a l r e l a x a t i o n can be e x p l a i n e d from i r r e v e r s i b l e thermodynamic f l u c t u a t i o n t h e o r y i n which changes o f p h y s i c a l and mechanical p r o p e r t i e s d u r i n g t h e r e l a x a t i o n process can be i n t e r p r e t e d and p r e d i c t e d from t h e mean square f l u c t u a t i o n s o f thermodynamic o r d e r i n g parameters. Then the p h y s i c a l o r mechanical p r o p e r t i e s a t any g i v e n cure time can be w r i t t e n as a r e l a x a t i o n f u n c t i o n : p

-

+

p

o ( co-

p 0

)

{

1 - e x p

[

"

}

( 1 )

where P ( t ) i s t h e p h y s i c a l o r mechanical p r o p e r t y a t time, t , and P and Poo are t h e i n i t i a l and f i n a l v a l u e s o f t h e p r o p e r t y . 3 i s a constant d e s c r i b i n g t h e width o f t h e r e l a x a t i o n spectrum, and τ i s t h e r e l a x a t i o n time which i s a f u n c t i o n o f temperature, T, and a c t i v a t i o n energy, H, o f c u r e , τ can be d e f i n e d a s : 0

τ

= τ

0

exp [ - ^ - ]

(2)

Where τ i s a c o n s t a n t which r e p r e s e n t s t h e r e l a x a t i o n time a t Τ-χ» and R i s t h e gas c o n s t a n t . The model d e s c r i b e d i n E q . ( l ) not o n l y can p r e d i c t t h e cure behavior measured by standard curometers, but a l s o can e x p l a i n f i l l e r e f f e c t s on t h e cure r e a c t i o n . The model enables one t o p r e d i c t scorch time and cure time o f elastomers a t v a r i o u s f i l l e r l o a d i n g s and cure temperatures. In t h e f o l l o w i n g d i s c u s s i o n , t h i s k i n e t i c model o f cure w i l l be extended t o e x p l a i n and p r e d i c t t h e modulus o r v i s c o s i t y of elastomers/thermosets d u r i n g 0

In Chemorheology of Thermosetting Polymers; May, C.; ACS Symposium Series; American Chemical Society: Washington, DC, 1982.

16.

HSICH E T AL.

265

Kinetic Model of Cure Reactions

c u r e . The model f o r p r e d i c t i n g the shear modulus can be r e w r i t t e n as: p

t -t

Where G r e p r e s e n t s the shear modulus, and G and G» are the minimum and maximum v a l u e s of the shear modulus d u r i n g c u r e , r e s p e c t i v e l y . The i n d u c t i o n time, t , has been mentioned p r e v i o u s l y [ 1 1 ] . T h i s i n d u c t i o n time i s d e f i n e d as the time needed f o r polymer systems t o reach the temperature necessary f o r cure onset. ( T h i s i s why the cure curve decreases t o a minimum value before i t i n c r e a s e s . ) For p r e d i c t i n g v i s c o s i t y d u r i n g c u r e , one o n l y needs t o s u b s t i t u t e v i s c o s i t y f o r shear modulus i n Eq. ( 3 ) . S i n c e non-isothermal cure i s of p r a c t i c a l importance i n cure c o n t r o l , one would l i k e t o extend the cure model t o i n c l u d e non-isothermal cure k i n e t i c s . T h i s m o d i f i c a t i o n i s shown below: 0


0

ρ 6· ( t ) - 8 e x p [ ^ ] + [ G - G e

:

ι e

β

ρ // χρ(^)]α^χρ[-^

d(t - t

p

h J ]}(4)

where G' i s a c o n s t a n t which r e p r e s e n t s the v a l u e of the shear s t o r a g e modulus at temperature, T-*«>. Ε i s the a c t i v a t i o n energy which i s a s c r i b e d t o i n t e r m o l e c u l a r f o r c e s [14]· T i s the e q u i l i b r i u m cure temperature, t i s the i n d u c t i o n time at T and t f i s the f i n a l cure time. Under i s o t h e r m a l c o n d i t i o n s , τ i s a f u n c t i o n of temperature o n l y . However, under non-isothermal c o n d i t i o n s as i n Eq. ( 4 ) , τ i s a f u n c t i o n of temperature which i n t u r n i s a f u n c t i o n of time. T h i s can be expressed as: e

c

0

c

In Eq. ( 4 ) , G' exp [ E / R T ( t ) ] d e s c r i b e s the e f f e c t of temperature on the p r o p e r t y of concern i n the absence of c u r e . The remaining p a r t of Eq. (4) d e s c r i b e s the e f f e c t of the c u r i n g r e a c t i o n on the p r o p e r t y being c o n s i d e r e d . In the f o l l o w i n g d i s c u s s i o n , Eqs. (3) and (4) w i l l be used f o r p r e d i c t i n g isothermal and non-isothermal cure c u r v e s . e


266


POLYMERS

S i n c e v i s c o s i t y i s o f v i t a l importance i n m a t e r i a l p r o c e s s i n g , one would l i k e t o p r e d i c t v i s c o s i t y d u r i n g c u r e . I n doing so, Eq. (3) i s m o d i f i e d as f o l l o w s :

n(t)

t -t = n + ( n - n ) { 1- exp [ - i - ^ ) Q

w

Q

8

] ) (6)

Here, v i s c o s i t y , η, can be e i t h e r complex v i s c o s i t y o r r e a l - p a r t v i s c o s i t y . F o r o u r s t u d i e s , we have used the complex v i s c o s i t y . The parameters n and r\ are the minimum and maximum v a l u e o f η on t h e cure curve. For non-isothermal cure c o n d i t i o n s , Eq. (4) can be m o d i f i e d as f o l l o w s f o r p r e d i c t i n g v i s c o s i t y d a t a : Downloaded by EAST CAROLINA UNIV on May 30, 2014 | http://pubs.acs.org Publication Date: August 29, 1982 | doi: 10.1021/bk-1982-0227.ch016

0

c

w

ρ

n(t)-n e x p C ^ H n . e

//

exp(^)]{l-exp[-[4

- t j f —r^) ]>(7)

d(t o

where n i s a c o n s t a n t which r e p r e s e n t s t h e v a l u e o f v i s c o s i t y e

a t temperature l+ ο

106-

TIME (MIN)

ι ι ι ι ι ι ι ι ι I

I I I I

Jι ι ι ι Iι ι ι ι I ι ι ι ι I »ι 20

Figure 13. Shear storage modulus v s . isothermal cure c o n d i t i o n s .

I

1

1

1

' I'

40

1

'

1

I

1

1

1

1

cure time under non-

heating rate 12° C/min (22°Cto150°C)

30 phr. β = 2.51. Ε = 4.00 Kcal/mole

10 phr. β = 3.26. Ε = 5.6 Kcal/mole

T _

T 20

r

Figure 14. Complex v i s c o s i t y v s . cure c o n d i t i o n s .

cure time under non-isothermal


278


2)


3)

POLYMERS

The a c t i v a t i o n energy of v i s c o u s f l o w w i l l be decreased by an i n c r e a s e i n f i l l e r l o a d i n g and, t h e r e f o r e , t h e v i s c o s i t y and modulus o f f i l l e d polymer systems a r e l e s s temperature dependent. On t h e o t h e r hand, f i l l e r l o a d i n g has l i t t l e e f f e c t on t h e a c t i v a t i o n energy o f cure. I n c r e a s i n g f i l l e r l o a d i n g broadens t h e r e l a x a t i o n spectrum o f t h e c u r e r e a c t i o n . Broadening t h e r e l a x a t i o n spectrum by f i l l e r l o a d i n g a l s o has been found i n t h e mechanical spectrum o f cured rubber from the g l a s s t r a n s i t i o n r e g i o n t o rubbery p l a t e a u r e g i o n [15].

Literature Cited

1. Fara, R. Α., Polymer, 9, 137 (1968). 2. Acitelli, Μ. Α., Prime, R. B. and Sacher, E., Polymer, 12, 335 (1971). 3. Prime, R. B., Polym. Eng. and Sci., 13, 365 (1973). 4. Sacher, E., Polymer, 14, 91 (1973). 5. Kamal, M., Sourour, R. S. and Ryan, M., Soc. Plast. Eng., 19, 187 (1973). 6. Wise, R. W., "Rubber Technology," Chapter 4, edited by M. Morton, Van Nostrand Reinhold Co., New York (1973). 7. Pyne, J. R., "Determination of Cure Cycles for Rubber Products," Proceedings of a RAPRA Seminar held at Shawbury (May, 1979). 8. Roller, M. B., Polym. Eng. and Sci., 15 (6), 406 (1975). 9. Craig, D., Soc. Plast. Eng., 18, 533 (1972). 10. Mussatti, F. G., and Macosko, C. S., Polym. Eng. and Sci., 13, 236 (1973). 11. Hsich, H. S.-Y., J. Mater. Sci., 13, 2560 (1978). 12. Hsich, H. S.-Y., J. Mater. Sci., 15, 1194 (1980). 13. Hsich, H. S.-Y., J. App. Polym. Sci., in press. 14. Hsich, H. S.-Y., J. Mater. Sci., 17, 438 (1982).


16.

HSICH E T A L .

Kinetic Model of Cure Reactions

15. To be published. 16. Tung, C.-Y. M, and Dynes, P. J., J. Appl. Polym. Sci., 27, 569 (1982). 17. Bueche, F., "Physical Properties of Polymer", (John Wiley Interscience, New York, 1962), pp.168-221. 31, 1983


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Kinetic Model of Cure Reactions - American Chemical Society

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