Elastomers and Rubber Elasticity - American Chemical Society

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Bimodal Elastomeric Networks J. E. MARK The University of Cincinnati, Department of Chemistry and Polymer Research Center, Cincinnati, OH 45221

Elastomeric networks may be prepared by end -linking mixtures of very short and relatively long polymer chains, for example, hydroxyl-terminated polydimethylsiloxane chains. Such networks (in the unfilled state) frequently show marked increases i n modulus at very high elongations. Experimental results are cited to show how this unusual behavior depends on composition, chain length, spatial het erogeneity, junction functionality, temperature, and swelling. It i s concluded that the observed increases in modulus are due to the limited exten s i b i l i t y of the network chains. These elastomers may therefore be used to study limited chain extensibility and to evaluate non-Gaussian theories of rubberlike e l a s t i c i t y . Another, more practical application i s the achievement of ultimate prop erties significantly better than those of networks having the usual unimodal distribution of chain lengths. There are now a number of techniques which may be used to prepare elastomeric networks of known structure (1-8). Two par t i c u l a r l y useful and convenient ones involve the multi-functional end-linking of hydroxyl-terminated (4-16) or vinyl-terminated polydimethylsiloxane (PDMS) chains (3,17-21), and the crosslinking of PDMS chains through vinyl side groups present i n known amounts and i n known locations along the chains (4,18,22-25). A typical reaction of this type i s

0097-6156/82/0193-0349$06.00/0 © 1982 American Chemical Society Mark and Lal; Elastomers and Rubber Elasticity ACS Symposium Series; American Chemical Society: Washington, DC, 1982.


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in which H O y v O H represents a hydroxyl-terminated PDMS chain. Characterizing the uncross-linked chains with respect to molec ular weight and molecular weight d i s t r i b u t i o n and then running the specified reaction to completion gives networks having known values of the cross-link functionality φ and average chain length, and detailed information on the chain length d i s t r i b u t i o n as well. Similar reactions have been carried out using hydroxylterminated chains of poly(ethylene oxide)(26), poly(tetramethylene oxide)(26), poly(caprolactone)(27), and poly(propylene oxide)(28), and a triisocyanate end-linking agent. A related reaction involving the end-linking of vinyl-terminated PDMS chains with a multifunctional silane i s somewhat more v e r s a t i l e in that a wider range of values of cross-link functionality may be achieved. Because of their known structures, such "model networks have been extremely useful for the elucidation of mol ecular aspects of rubberlike e l a s t i c i t y . In several of these studies (12-16,20), bimodal PDMS net works were prepared by end-linking mixtures of very short and r e l a t i v e l y long chains, as i s i l l u s t r a t e d i n Figure 1. These networks, i n the u n f i l l e d state, were found to be unusually tough elastomers. Of particular interest i s the fact that they have values of the modulus [f*] which increase very substantially at high elongations, thus giving unusually large values of the u l timate strength. This i s rather surprising since usually an elastomer w i l l have good ultimate properties only when r e i n forced with some mineral f i l l e r (or hard, glassy domains i n the case of a multi-phase polymer), or when i t can generate i t s own reinforcement through strain-induced c r y s t a l l i z a t i o n (29-32). Part of these increases i n modulus and ultimate strength are due to the low incidence of dangling-chain i r r e g u l a r i t i e s i n such model networks i n general (19). Another contribution could be non-Gaussian effects arising from limited chain e x t e n s i b i l i t y (12-16,20). I t i s not known, however, whether these are the only effects occurring at very low temperatures. Such networks could also, for example, undergo strain-induced c r y s t a l l i z a t i o n , with consequent improvements i n ultimate properties from cry s t a l l i t e reinforcement (29-32). PDMS has a very low melting point (-40°C) (33) and, although elongation s i g n i f i c a n t l y raises i t (34,35), u n f i l l e d networks of this polymer are thought to re main amorphous well below room temperature (35,36,37). Unusually low temperatures must therefore be employed i n any search for strain-induced c r y s t a l l i z a t i o n . There are very few relevant data in the l i t e r a t u r e and, of course, no relevant results whatever are available for the new types of networks with the very pecul iar distributions of interest here. This question of possible strain-induced c r y s t a l l i z a t i o n and the associated network r e i n forcement assumes particular importance because of the very unusual and attractive properties of these materials. A d e f i n i t i v e answer to these questions i s obtained i n this review by analysis of how the relevant elastomeric properties 11

Mark and Lal; Elastomers and Rubber Elasticity ACS Symposium Series; American Chemical Society: Washington, DC, 1982.


Figure 1. Sketch of a portion of a bimodal network. The very short polymer chains are repre sented by heavy lines and the relatively long chains by thinner lines.


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depend on composition, chain length, spatial heterogeneity, junction functionality, temperature, and swelling. The r e s u l t ing molecular interpretation of these unusual properties of b i modal networks permits the u t i l i z a t i o n of these elastomers i n a variety of applications, a number of which are discussed i n some d e t a i l .


Representations of the Network Moduli i n Elongation The most convenient measure of the stress exhibited by an elongated elastomeric network i s the nominal or engineering stress f* = f/A*, where f i s the equilibrium value of the force and A* i s the cross-sectional area of the undeformed sample. The s t r a i n i s given by the r e l a t i v e length or elongation α = L/L where L and L- are the lengths of the sample i n the deformed and undeformed states, respectively. Dividing the stress f/A* by the strain function (a - a ) indicated i n the simplest molecular theories of rubberlike e l a s t i c i t y (38,39) then gives the e l a s t i c modulus or reduced stress (38-42) 2

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(1)

For rubberlike materials, the modulus thus defined generally shows an additional dependence on elongation (39,40,42), appar ently because of increase i n the non-affineness of the deforma tion as the elongation increases (43,44). This dependence i s frequently represented (approximately) by the semiempirical equation of Mooney and R i v l i n [f*] = 2

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i n which 2C and 2C are constants independent of α (39,40,42, 45,46). Thus, the value of the modulus i s 2C i n the l i m i t at large deformation (α*" -*. 0), and 2C + 2C i n the l i m i t at small deformation (of- •> 1). 2

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Dependence of Properties on Network Structure and Experimental Variables Composition. Some representative isotherms for (unfilled) bimodal networks prepared from very short and r e l a t i v e l y long hydroxyl-terminated PDMS chains mixed i n various proportions are shown i n Figure 2(15). In this and i n some of the subsequent Figures, the modulus i s plotted against reciprocal elongation, as suggested by Equation (2). The bimodal networks are seen to have very good (maximum) e x t e n s i b i l i t y for these r e l a t i v e l y high values of the modulus [ f * ] . Furthermore, the networks contain ing a large mol % of the short chains show a marked increase i n [f*] at high elongations, which corresponds to a significant


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Bimodal Elastomeric Networks

increase i n ultimate strength. I t i s the nature of this increase and i t s molecular o r i g i n which are the main subjects of the investigations described i n this review. The results presented in Figure 2 portray the dependence of this effect on network composition. S p e c i f i c a l l y , increase i n the number of short chains i n the networks gives a more pronounced increase i n [f ], thus underscoring the importance of the short-chain component i n this regard. Downloaded by UNIV OF MISSOURI COLUMBIA on April 14, 2017 | http://pubs.acs.org Publication Date: July 19, 1982 | doi: 10.1021/bk-1982-0193.ch018

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Chain Length. I f i t i s the short chains which give the improvements i n ultimate properties, decrease i n their average length should give even more pronounced upturns i n [f ] at high elongations. This i s indeed found to be the case, as i s i l l u s trated i n Figure 3(15). These stress-strain isotherms are f o r PDMS networks similar to those described i n Figure 2 but with short chains having = 220 rather than 660 g mol-'-. -

Spatial Heterogeneity. Some insight into the mechanism through which the short chains operate may be obtained using bimodal networks which are made s p a t i a l l y as well as compositiona l l y heterogeneous i n a two-state reaction i n which some of the very short chains are prereacted so as to form clusters or do mains of high cross-link density. I f the increases i n [f*] were due to some intermolecular organization such as a " f i l l e r " effect (47-50) or strain-induced c r y s t a l l i z a t i o n (29-32,37), then seg regating the short chains should enhance the modulus. Such networks have been prepared using short chains having values of of either 1,100 (14) or 220 g mol" (51), and have been studied with regard to their values of the high elongation mod ulus 2Ci and the modulus [f*] at rupture. The values of 2C and [ f * J for the 1,100-18,500 PDMS networks are shown as a func tion of the extent of prereaction Xp i n Figure 4(14). In both cases, increase i n appears to give a small i n i t i a l increase in these two quantities; unfortunately, the increases are very small and i n fact are within the usual error l i m i t s i n these types of measurements. Thus, there does not appear to be any s i g n i f i c a n t reinforcing effect brought about by the clustering of the very short chains i n the s p a t i a l l y inhomogeneous crossl i n k i n g process. This suggests that the increases i n [f*] are primarily intramolecular rather than intermolecular. 1

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Junction Functionality. Some results bearing on the possi ble effect of cross-link functionality on the upturn i n modulus are shown i n Figure 5(20). The magnitude of the increase i n [f ] does not show any obvious correlation with φ, which again suggests the predominant importance of the intramolecular char a c t e r i s t i c s of the short chains. At least from the evidence at hand (20), the functionality of the junction points seems r e l a t i v e l y unimportant i n this regard. Temperature. The effect of temperature on the stress-strain isotherms i s of particular importance with regard to the



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,-ι Figure 2. Typical stress-strain isotherms for PDMS networks prepared by tetrafunctionally end-linking very short and relatively long chains. Number-average molecular weights are M = 660 and 18,500 gl mol, respectively (15). Measurements were carried out on the unswollen networks, in elongation at 25°C. Data plotted as suggested by Mooney-Rivlin representation of reduced stress or modulus (Eq. 2). Short extensions of the linear portions of the isotherms locate the values of a at which upturn in [/*] first becomes discernible. Linear portions of the isotherms were located by least-squares analysis. Each curve is labelled with mol percent of short chains in network structure. Vertical dotted lines indicate rupture points. Key: O, results obtained using a series of increasing values of elongation; . , results obtained out of sequence to test for reversibility. n



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Figure 3. Results for PDMS networks similar to those described in Figure 2, but with the short chains having M = 220 g/mol (15). Lower part of the ordinate refers only to lowest isotherm in the series. See key to Fig. 2. n



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Figure 4. The elastic properties of some bimodal PDMS networks. Short chains were segregated by pre-reacting them with a limited amount of (tetrafunctional) end-linking agent (14). Short and long chains had M = 1,100 and 18,500 g I mol, respectively. Composition was 82 mol % short chains. High deformation modulus 2C and ultimate strength as measured by the modulus U*] at rupture, are shown as a function of the extent of pre-reaction. n

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Figure 5. Stress-strain isotherms obtained for bimodal (600-11,300), PDMS net works containing 75.2 mol % short chains (20). Chains were vinyl-terminated, and end-linked using a silane having values of the func tionality φ.


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p o s s i b i l i t y of strain-induced c r y s t a l l i z a t i o n i n the networks. Typical results of this type are shown i n Figure 6(16). Of particular interest here are the temperature dependence of the elongation O y at which the upturn i n [f*] becomes discernible, of the elongation a at which rupture occurs, and of the t o t a l increase A [ f * ] i n modulus up to the rupture point. I f the i n crease i n [f ] had been due to strain-induced c r y s t a l l i z a t i o n , α would have decreased with decrease i n temperature, and Oj. and A [ f * ] would have increased. These qualities are seen to be r e l a t i v e l y insensitive to temperature, which confirms the con clusion that the anomalous behaviour i s not due to s t r a i n induced c r y s t a l l i z a t i o n . Also relevant here are some force-temperature ("thermoelast i c ) results obtained at elongations s u f f i c i e n t l y large to give large increases i n [f*] i n the s t r e s s - s t r a i n isotherm (16). Such curves, i l l u s t r a t e d i n Figure 7, show no deviations from l i n e a r i t y which could be attributed to strain-induced c r y s t a l l i z a t i o n . Similarly, birefringence-temperature measurements also carried out at α > a show no deviations from l i n e a r i t y that could be attributed to c r y s t a l l i z a t i o n , or to other i n t e r molecular orderings of the network chains. Typical results of this type are shown i n Figure 8 (16). r

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Swelling. Work i n progress (52) indicates that the upturns in [f ] for PDMS bimodal networks are not decreased by swelling, which would have diminished any strain-induced c r y s t a l l i z a t i o n (36,37). A l l of the above evidence indicates that the increase i n modulus and improvements i n ultimate properties are due to intramolecular effects, s p e c i f i c a l l y to non-Gaussian effects arising from l i m i t e d chain e x t e n s i b i l i t y . A network chain near i t s maximum e x t e n s i b i l i t y can no longer increase i t s end-to-end separation by configurational changes, i . e . , by simple rotations about i t s s k e l e t a l bonds. Deformations of bond angles (and possibly even bond lengths) would be required, and the energies for these processes are much greater than those for configurat i o n a l changes. This i s presumably the o r i g i n of the very marked increases i n the modulus at high elongations and the much improved ultimate properties. Some Applications Characterization of Limited Chain E x t e n s i b i l i t y . The mol ecular o r i g i n of the unusual properties of bimodal PDMS networks having been elucidated at least to some extent, i t i s now possi ble to u t i l i z e these materials i n a variety of applications. The f i r s t involves the interpretation of limited chain e x t e n s i b i l i t y i n terms of the configurational characteristics of the PDMS chains making up the network structure (5,12,13).



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The f i r s t important characteristic of limited chain extensi b i l i t y i s the elongation a at which the increase i n [f ] f i r s t becomes discernible. Values of this minimum elongation are readily obtainable from stress-strain isotherms such i s those shown i n Figures 2, 3, 5, and 6. Although the deformation i s non-affine i n the v i c i n i t y of the upturn, i t i s possible to pro vide at least a semi-quantitative interpretation of such results in terms of the network chain dimensions (5,12,13). At the beginning of the upturn, the average extension r of a network chain having i t s end-to-end vector along the direction of stretching i s simply the product of the unperturbed dimension < r 2 1 / and a (12). Similarly, the maximum e x t e n s i b i l i t y r i s the product of the number η of skeletal bonds and the factor 1.34 A which gives the a x i a l component of a skeletal bond i n the most extended h e l i c a l form of PDMS, as obtained from the geomet r i c analysis summarized i n Figure 9(12). The r a t i o r / r at a thus represents the fraction of the maximum extensibili?y occur ring at this point i n the deformation. The values obtained indicate that the upturn i n modulus generally begins at approxi mately 60-70% of maximum chain e x t e n s i b i l i t y (5,12,13). This i s approximately twice the value which had been estimated prev iously (39), i n a misinterpretation of stress-strain isotherms of elastomers undergoing strain-induced c r y s t a l l i z a t i o n . It i s also of interest to compare the values of r / r at the beginning of the upturn with some theoretical results by Flory and Chang (53) on distribution functions for PDMS chains of f i n i t e length. Of relevance here are the calculated values of r / r at which the Gaussian d i s t r i b u t i o n function starts to over estimate the probability of extended configurations, as judged by comparisons with the results of Monte Carlo simulations. The theoretical results most relevant to the experimental results on the bimodal PDMS networks are shown i n Figure 10(53). They sug gest, for example, that a network of PDMS chains having η = 53 skeletal bonds should show an upturn at a value of r / r a l i t t l e less than O.80. The observed value was O.77 (12), which i s thus in excellent agreement with theory. A second important characteristic i s the value o t of the elongation at which rupture occurs. The corresponding Values of r / r show that rupture generally occurred at approximately 80-90% of maximum chain e x t e n s i b i l i t y (12). These quantitative results on chain dimensions are very important but may not apply d i r e c t l y to other networks, i n which the chains could have very different configurâtional characteristics and i n which the chain length d i s t r i b u t i o n would presumably be quite different from the very unusual bimodal distribution intentionally produced i n the present networks.


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Evaluation of Non-Gaussian E l a s t i c i t y Theories. There are now numerous theories of rubberlike e l a s t i c i t y which use nonGaussian d i s t r i b u t i o n functions to take into account limited



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Figure 8. Representative birefringence-temperature relations for the 220-18,500 PDMS networks. Filled circles locate results obtained to check for reversibility (16).

Figure 9. The end-to-end distance per skeletal bond η for regular conformations of polydimethylsiloxane and polyethylene network chains (12). Maximum extensi bility r of this chain molecule occurs at r /n = 1.34 A. m

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ELASTOMERS AND RUBBER ELASTICITY

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Radial distributions (in A' ) for PDMS chains having η = 40 skeletal bonds (each of length I = 1.64 A) (53).


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chain e x t e n s i b i l i t y (39). Unfortunately, the various theories have generally been u n c r i t i c a l l y applied to e l a s t i c i t y results in which the increases i n modulus were due largely if not entirel y to strain induced c r y s t a l l i z a t i o n , as was pointed out previously (11,16,37). results on the bimodal PDMS networks do not suffer from this complication, and are apparently the only r e l i a b l e experimental results available at the present time for the evaluation of the non-Gaussian theories. Such evaluations are i n progress (54). Downloaded by UNIV OF MISSOURI COLUMBIA on April 14, 2017 | http://pubs.acs.org Publication Date: July 19, 1982 | doi: 10.1021/bk-1982-0193.ch018

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Improvements i n Ultimate Properties. In this application, i t i s most illuminating simply to plot the nominal stress f/A against elongation. Typical results are shown i n Figure 11 (15). This type of representation has the advantage of having the area under each curve correspond to the network rupture energy (per unit undeformed cross-sectional area and per unit i n i t i a l length). This energy E (in joules per mm-* of network sample) required for rupture was taken as a measure of "toughness" of the PDMS elastomers. In the case of the unimodal networks, E i s r e l a t i v e l y small. As can be seen from the Figure, this i s due to the small maximum e x t e n s i b i l i t y i n the case of small M , and to the small maximum values of the nominal stress i n the case of large W. Thus, u n f i l l e d elastomers are generally very weak materials (31,32). The bimodal networks have improved ultimate properties i n that they can be prepared so as to have r e l a t i v e l y large values of the nominal stress without the usual corresponding decrease i n maximum e x t e n s i b i l i t y . This shortchain reinforcing effect i s very s t r i k i n g i n that E can be increased by a factor of nearly 5 (15) i n going from 0 to 90 mol% of the 220 chains, and this corresponds to an increase of only 9.7 wt%! Strain-induced c r y s t a l l i z a t i o n would presumably further improve the ultimate properties of a bimodal network. I t would therefore obviously be of considerable importance to study the effect of chain length d i s t r i b u t i o n on the ultimate properties of bimodal networks prepared from chains having melting points well above the very low value characteristic of PDMS. Studies of this type are being carried out on bimodal networks of polyethylene oxide) (55), poly(caprolactone) (55), and polyisobutylene (56). r

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Acknowledgements It i s a pleasure to acknowledge that much of the author's work on elastomeric materials has been supported by grants from the National Science Foundation (Polymers Program, Division of Materials Research). Fellowhship support for students has also been generously provided by the Dow Corning Corporation and the IBM Corporation.


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α Figure 11. Typical plots of nominal stress against elongation, for tetrafunctional PDMS networks at 25° C (15). All but three networks are bimodal. Number-average molecular weight of relatively long chains is M — 18,500 gl mol. Key for the networks where short chains had M (g/mol): Δ , 1,100; O, 660; Φ, 220. Curves n

n

are labelled with the mol percent of short chains in the network. Area below curves represents the rupture energy per unit initial cross sectional area and per unit initial length.


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Literature Cited 1. 2. 3.


4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

20. 21. 22. 23. 24. 25. 26. 27.

Kraus, G.; Moczvgemba, G. A. J . Polym. Sci. Part A 1964, 2, 277. Rempp, P.; Herz, J. E. Angew. Makromol. Chemie 1979, 76/77, 373 and pertinent references cited therein. Valles, E. M.; Macosko, C. W. Macromolecules 1979, 12, 673 and pertinent references cited therein. Mark, J. E. Makromol. Chem., 1979, Suppl. 2, 87 and pertinent references cited therein. Mark, J. E. Pure Appl. Chem. 1981, 53, 1495. Mark, J. E. Rubber Chem. Technol. 1981, 54, 809. Mark, J. E. J. Chem. Educ. 1981, 58, 898. Mark, J . E. Adv.Polym. Sci. 1982, 00, 000. Llorente, Μ. Α.; Mark, J. E. J . Chem. Phys. 1979, 71, 682. Mark, J. E.; Llorente, M. A. J . Am. Chem. Soc. 1980, 102, 632. Llorente, Μ. Α.; Mark, J. E. J. Polym. Sci., Polym. Phys. Ed. 1980, 18, 181. Andrady, A. L.; Llorente, Μ. Α.; Mark, J . E. J. Chem. Phys. 1980 , 72, 2282. Andrady, A. L.; Llorente, Μ. Α.; Mark, J. E. J . Chem. Phys. 1980, 73, 1439. Mark, J . E.; Andrady, A. L. Rubber Chem. Technol. 1981, 54, 366. Llorente, Μ. Α.; Andrady, A. L.; Mark, J. E. J. Polym. S c i . , Polym. Phys. Ed. 1981, 19, 621. Zhang, Z.-M.; Mark, J. E. J . Polym. Sci., Polym. Phys. Ed. 1981, 19, 000. Llorente, Μ. Α.; Mark, J . E. Macromolecules 1980, 13, 681. Llorente, Μ. Α.; Andrady, A. L.; Mark, J. E. J . Polym. S c i . , Polym. Phys. Ed. 1980, 18, 2263. Andrady, A. L.; Llorente, Μ. Α.; Sharaf, Μ. Α.; Rahalkar, R. R.; Mark, J . E.; Sullivan, J . L.; Yu, C. U.; Falender, J. R. J . Appl. Polym. Sci. 1981,26, 1829. Llorente, Μ. Α.; Andrady, A. L.; Mark, J. E. Colloid Polym. Sci. 1981, 259, 000. Meyers, K. O.; Bye, M. L.; M e r r i l l , E. W. Macromolecules 1980, 13, 1045. Falender, J . R.; Yeh, G. S. Y.; Mark, J . E. J . Chem. Phys. 1979, 70, 5324. Falender, J . R.; Yeh, G. S. Y.; Mark, J . E. J. Am. Chem. Soc. 1979, 101, 7353. Falender, J . R.; Yeh, G. S. Y.; Mark, J. E. Macromolecules 1979, 12, 1207. Llorente, Μ. Α.; Mark, J. E. Rubber Chem. Technol. 1980, 53, 988. Mark, J. E.; Sung, P.-H. Eur. Polym. J . 1980, 16, 1223. Sung, P.-H.; Mark, J. E. Polym. J . 1980, 12, 835.


366 28. 29. 30.


31. 32. 33. 34. 35.

36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56.

E L A S T O M E R S A N D R U B B E R ELASTICITY

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Elastomers and Rubber Elasticity - American Chemical Society

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