Characterization of Highly Cross-linked Polymers - American

4 Elastomeric Poly(dimethylsiloxane) Networks with Numerous Short-Chain Segments Downloaded by UNIV OF TENNESSEE KNOXVILLE on November 13, 2016 | http://pubs.acs.org Publication Date: February 16, 1984 | doi: 10.1021/bk-1984-0243.ch004

J. E. MARK Department of Chemistry and Polymer Research Center, The University of Cincinnati, Cincinnati, OH 45221 J. G. CURRO Physical Properties of Polymers Division, Sandia National Laboratories, Albuquerque, NM 87185

End-linking techniques may be used to prepare (unfilled) elastomeric networks of polydimethylsiloxane) (PDMS) [Si(CH ) O-] which contain a large mol fraction of unusually short chains along with chains of the lengths usually associated with rubberlike materials. Such bimodal networks frequently have very attractive mechanical properties, in particular high extensibility at relatively high average degrees of cross-linking, and thus are unusually tough elastomers. Measurements of stress-strain isotherms over a wide range in temperature and degree of swelling, stresstemperature coefficients, and birefringencetemperature coefficients indicate the improve ments in properties to be intramolecular, specifically non-Gaussian effects related to limited chain extensibility. These effects are explored using a theory of rubberlike elasticity based on network distribution functions generated from the rotational isomeric state model for PDMS chains. 3

2

Elastomeric networks of known s t r u c t u r e may be prepared by the e n d - l i n k i n g of f u n c t i o n a l l y terminated chain molecules (1-6) rather than by the usual random procedure (7-8) of l i n k i n g chains through repeat u n i t s with a r b i t r a r y l o c a t i o n s w i t h i n the chain s t r u c t u r e s . These h i g h l y s p e c i f i c chemical r e a c t i o n s can thus be used to provide networks of any d e s i r e d molecular weight Mc between c r o s s - l i n k s , by simply end-linking such chains having number-average molecular weights M equal to Mc» n

0097-6156/ 84/0243-0047S06.00/0 © 1984 American Chemical Society

American Chemical Society Library Labana and Dickie; Characterization 1155 16th St. of N.Highly W. Cross-linked Polymers ACS Symposium Series; American Chemical Society: Washington, DC, 1984. Washington. 0. G. 20036

HIGHLY CROSS-LINKED POLYMERS

Downloaded by UNIV OF TENNESSEE KNOXVILLE on November 13, 2016 | http://pubs.acs.org Publication Date: February 16, 1984 | doi: 10.1021/bk-1984-0243.ch004

48

A network obtained i n t h i s manner, o f course, has a network chain length d i s t r i b u t i o n which i s a l s o the same as that o f the polymer from which i t was prepared. The technique thus permits the p r e p a r a t i o n o f elastomeric m a t e r i a l s having d i s t r i b u t i o n s which are multimodal (e.g. bimodal) as w e l l as the usual unimodal type (7,8). Recently, particular interest has focused on bimodal p o l y ( d i m e t h y l s i l o x a n e ) (PDMS) networks (9-17) c o n t a i n i n g large mol f r a c t i o n s o f very short chains as w e l l as chains o f the usual lengths required f o r rubberlike e l a s t i c i t y (7,8,18). T y p i c a l values o f ^ are a few hundred g mol"*l and 18,000 g mol~l, respectively. The i n t e r e s t i s due to the f a c t that these elastomers, i n the u n f i l l e d s t a t e , f r e q u e n t l y have e x c e l l e n t mechanical p r o p e r t i e s . T h e i r r e l a t i v e l y high values of both the maximum e x t e n s i b i l i t y and ultimate strength combine to y i e l d large values o f the energy required f o r rupture; i . e . , such bimodal networks are unusually "tough" elastomers. The most i n t e r e s t i n g aspect o f t h i s toughening e f f e c t from the fundamental point o f view i s the very large increase i n modulus which can occur a t high extensions (9-17). A variety of studies (16) demonstrated that t h i s non-Gaussian e f f e c t shown by the bimodal PDMS networks could properly be a t t r i b u t e d to l i m i t e d chain e x t e n s i b i l i t y . The r e s u l t s included s t r e s s s t r a i n isotherms over a wide range i n temperature, s t r e s s temperature coefficients, and birefringence-temperature coefficients. The molecular o r i g i n o f the non-Gaussian e f f e c t having been e s t a b l i s h e d , i t becomes important to develop a molecular theory f o r i t s e l u c i d a t i o n . Non-Gaussian t h e o r i e s o f r u b b e r l i k e e l a s t i c i t y c u r r e n t l y available (8,19,20) g e n e r a l l y have the disadvantage of containing parameters which can be determined only by comparisons between theory and experiment. The approach taken in the present i n v e s t i g a t i o n avoids this shortcoming by u t i l i z i n g the wealth o f information which r o t a t i o n a l isomeric s t a t e theory provides on the s p a t i a l c o n f i g u r a t i o n s o f chain molecules (21), i n c l u d i n g most o f those used i n elastomeric networks. S p e c i f i c a l l y , Monte Carlo c a l c u l a t i o n s (22-24) based on the r o t a t i o n a l isomeric s t a t e approximation (21) are used to simulate spatial configurations, and thus distribution functions f o r the end-to-end separation r o f the network chains. These d i s t r i b u t i o n functions may be used i n place o f the Gaussian f u n c t i o n to give a molecular theory o f r u b b e r l i k e elasticity which i s unique to the p a r t i c u l a r polymer o f interest, and a p p l i c a b l e to the regions o f very large deformation, where the bimodal networks e x h i b i t t h e i r h i g h l y unusual p r o p e r t i e s . Short-Chain

unimodal PDMS Networks

Information

on the conformational

preferences o f the PDMS chain

Labana and Dickie; Characterization of Highly Cross-linked Polymers ACS Symposium Series; American Chemical Society: Washington, DC, 1984.


4.

M A R K AND CURRO

Poly(dimethylsiloxane) Networks

49

was obtained from previous experimental and t h e o r e t i c a l i n v e s t i g a t i o n s (21,25). The Monte Carlo method was used i n conjunction with t h i s information to generate large numbers o f typical s p a t i a l c o n f i g u r a t i o n s , at 110°C (23), f o r chains having a s p e c i f i e d number η o f s k e l e t a l bonds (18,22-24). The c o n f i g u r a t i o n s were grouped according to t h e i r values o f the end-to-end separation r , and the r e s u l t s c u r v e - f i t t e d using a c u b i c - s p l i n e least-squares technique (26). The d i s t r i b u t i o n f u n c t i o n thus obtained was then used i n the standard " t h r e e c h a i n " model approach (8) to r u b b e r l i k e e l a s t i c i t y , i n order to estimate the entropy o f deformation i n the a f f i n e l i m i t , and from that the nominal s t r e s s f * Ξ f/A* (where A* i s the undeformed c r o s s - s e c t i o n a l area). The r e s u l t i n g values o f f were normalized by VkT, where V i s the number density o f network chains, k i s the Boltzmann constant, and Τ i s the absolute temperature. Figure 1 presents the r e s u l t s f o r the i l l u s t r a t i v e cases η = 20 and 40 s k e l e t a l bonds, as a f u n c t i o n o f the elongation Ot « L / L i , where L and L£ are the stretched and unstretched sample lengths, r e s p e c t i v e l y . An a l t e r n a t i v e r e p r e s e n t a t i o n o f the same r e s u l t s i s i n terms o f the reduced s t r e s s or modulus defined by (27,28)

(1) These results are t y p i c a l l y plotted against reciprocal elongation, as suggested by the semiempirical equation o f Mooney and R i v l i n (8,28-30) (2) i n which 20\ and 2C2 are constants independent o f a. Such a p l o t o f these t h e o r e t i c a l r e s u l t s i s shown i n Figure 2. The curves i n this f i g u r e are q u i t e s i m i l a r to experimentally obtained r e s u l t s on PDMS networks, as i s i l l u s t r a t e d by some o f the curves i n Figure 3(12). The curves i n these f i g u r e s show upturns i n the s t r e s s and modulus as the e l o n g a t i o n i n c r e a s e s . The upturns are due to the r a p i d l y d i m i n i s h i n g number o f c o n f i g u r a t i o n s c o n s i s t e n t with the required large values o f r , and thus, correspondingly large decreases i n the entropy o f the network chains, and increases i n f * and [f*]» Bimodal PDMS Networks The long and short chains i n the bimodal PDMS networks were assumed to have values o f η o f 20 and 250, r e s p e c t i v e l y (9-16). The long chains were modeled as Gaussian chains, whereas the d i s t r i b u t i o n f u n c t i o n f o r the short chains was determined from Monte Carlo c a l c u l a t i o n s as already described. The entropy o f the bimodal network was then taken to be the sum o f




PDMS

Figure 1. T h e o r e t i c a l curves of s t r e s s against s t r a i n for unimodal PDMS networks c o n s i s t i n g of chains with 20 and 40 s k e l e t a l bonds, r e s p e c t i v e l y .

PDMS

Figure 2. The r e s u l t s of Figure 1 represented as suggested by the semi-empirical Mooney-Rivlin r e l a t i o n s h i p (8,28-30) which u s u a l l y gives a modulus [ f * ] decreasing l i n e a r l y with decreasing o f ( 2 8 ) . 1



M A R K AND CURRO


1

a Figure 3. Some typical stress-strain isotherms experimentally obtained on bimodal PDMS networks consisting of very short and relatively long chains having molecular weights of 220 and 18,500 g mol"*, respectively ( 12). The mol % of short chains i s used to label each isotherm, and the f i l l e d circles locate results obtained out of sequence to test for reversibility.


52


contributions volume,

from V

T

long chains and V

AS - A S

L

c

short chains per u n i t

+ ASg

(3)


In order to couple the l o c a l deformation to the macroscopic deformation we make the p r e l i m i n a r y but crude assumption that the average deformation i s a f f i n e . This implies that +

x

a

ss

(4)

where and X are the mol f r a c t i o n s o f long and short chains, respectively. The deformation i s then p a r t i t i o n e d n o n - a f f i n e l y between the long and short chains i n order to maximize the entropy o f the network. Some t y p i c a l r e s u l t s are presented i n Figure 4. I t can be seen that the t h e o r e t i c a l curves show increased steepness at high elongation as the f r a c t i o n o f short chains i s increased, i n agreement with experimental observations (12,14). The p o s i t i o n o f the upturn, however, does not appreciably change with composition as i t does i n the experimental curves. This d i f f e r e n c e i s almost c e r t a i n l y due to the a f f i n e (average) deformation assumption made i n the present theory. I t may be p o s s i b l e to r e f i n e the theory to take approximate account o f the nonaffineness o f the e l a s t i c deformation, which becomes p a r t i c u l a r l y important i n the region o f very high elongations g

(31-33).

PDMS,

*

N=20

10-

Figure 4. T h e o r e t i c a l curves o f stress against s t r a i n for bimodal PDMS networks c o n s i s t i n g o f long PDMS chains (n » 250 s k e l e t a l bonds), and very short chains (n = 20) present to the extent o f the values o f the volume f r a c t i o n X s p e c i f i e d f o r each curve. g


4.

M A R K AND CURRO


53


Acknowledgment s It is a pleasure to acknowledge several very helpful discussions with Professor Paul J . Flory o f Stanford U n i v e r s i t y , and the f i n a n c i a l support provided by the N a t i o n a l Science Foundation through Grant DMR 79-18903-03 (Polymers Program, D i v i s i o n o f M a t e r i a l s Research) and the Department o f Energy through Contract DE-AC04-76-DP00789. JEM a l s o wishes to thank the Sandia N a t i o n a l L a b o r a t o r i e s f o r t h e i r h o s p i t a l i t y during a v i s i t when much o f t h i s work was c a r r i e d out.

Literature Cited 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

J. E. Mark and J. L. Sullivan, J. Chem. Phys., 66, 1006 (1977). J. E. Mark, Makromol.Chem.,Suppl. 2, 87 (1979). J. E. Mark and M. A. Llorente, J. Am. Chem. Soc., 102, 632 (1980). A. L. Andrady, M. A. Llorente, M. A. Sharaf, R. R. Rahalkar, J. E. Mark, J. L. Sullivan, C. U. Yu, and J. R. Falender, J. Appl. Polym. Sci., 26, 1829 (1981). J. E. Mark, Pure Appl. Chem., 53, 1495 (1981). J. E. Mark, Rubber Chem. Technol., 54, 809 (1981). P. J. Flory, "Principles of Polymer Chemistry", Cornell University Press, Ithaca, N.Y., 1953, ch. XI. L. R. G. Treloar, "The Physics of Rubber Elasticity", 3rd Ed., Clarendon Press, Oxford, 1975. A. L. Andrady, M. A. Llorente, and J. E. Mark, J. Chem. Phys., 72, 2282 (1980). A. L. Andrady, M. A. Llorente, and J. E. Mark, J. Chem. Phys., 73, 1439 (1980). J. E. Mark and A. L. Andrady, Rubber Chem. Technol., 54, 366 (1981). M. A. Llorente, A. L. Andrady, and J. E. Mark, J. Polym. Sci., Polym. Phys. Ed., 19, 621 (1981). M. A. Llorente, A. L. Andrady, and J. E. Mark, Colloid and Polym. Sci., 259, 1056 (1981). J. E. Mark, Adv. Polym. Sci., 44, 1 (1982). J. E. Mark, in "Elastomers and Rubber Elasticity", ed. by J. E. Mark and J. Lal, American Chemical Society, Washington, D.C., 1982. Z.-M. Zhang and J. E. Mark, J. Polym. Sci., Polym. Phys. Ed., 20, 473 (1982). S.-J. Pan and J. E. Mark, Polym. Bulletin, 7, 553 (1982). J. E. Mark, J. Chem. Educ., 58, 898 (1981). K. J. Smith, Jr., J. Polym. Sci., A-2, 9, 2119 (1971). J. Kovac and C. C. Crabb, Macromolecules, 15, 537 (1982). P. J. Flory, "Statistical Mechanics of Chain Molecules", Interscience, New York, 1969.


54 22. 23. 24.


25. 26. 27. 28. 29. 30. 31. 32. 33.

HIGHLY C R O S S - L I N K E D P O L Y M E R S

D. Y. Yoon and P. J . Flory, J . Chem. Phys., 61, 5366 (1974). P. J . Flory and V. W. C. Chang, Macromolecules, 9, 33 (1976). J . C. Conrad and P. J. Flory, Macromolecules, 9, 41 (1976). P. J . Flory, V. Crescenzi, and J . E. Mark, J . Am. Chem. Soc., 86, 146 (1964). C. H. Reinsch, Numerishe Mathematik, 10, 177 (1967). J . E. Mark and P. J . Flory, J . Appl. Phys., 37, 4635 (1966). J . E. Mark, Rubber Chem. Technol., 48, 495 (1975). M. Mooney, J. Appl. Phys., 19, 434 (1948). R. S. Rivlin, Phil. Trans. R. Soc. London, Ser. A, 241, 379 (1948). G. Ronca and G. Allegra, J . Chem. Phys., 63, 4990 (1975). P. J . Flory, Proc. R. Soc. London, Ser. A, 351, 351 (1976). P. J . Flory and B. Erman, Macromolecules, 15, 800 (1982).

RECEIVED October 13, 1983


Characterization of Highly Cross-linked Polymers - American

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