Elastomers and Rubber Elasticity - American Chemical Society


Kuhn, W. Kolloid-Z. u. Z. Polymere 1934, 68, 2 ; 1936, 87,258. 23. Wall, F. T. J. Chem. Phys. 1942, 10, 132; 1943, 11, 527. 24. Treloar, L. R. G. Tran...
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16 Elasticity and Structure of Cross-linked Polymers : Networks with Comblike Cross-links 1

Downloaded by COLUMBIA UNIV on January 23, 2018 | http://pubs.acs.org Publication Date: July 19, 1982 | doi: 10.1021/bk-1982-0193.ch016

WILHELM OPPERMANN and GUNTHER

REHAGE

Institute of Physical Chemistry, T.U. Clausthal, Adolf-Römer-Strasse 2a, 3392 Clausthal-Zellerfeld, Federal Republic of Germany

The equilibrium modulus of poly(dimethylsiloxane) (PDMS)networks having comb-like crosslinks changes systematically with the structure of the cross­ links, even if the number of chains per volume re­ mains constant. At high branching densities, the experimentally observed moduli are three times greater than those calculated from the theory of phantom networks. As the branching density decrea­ ses, this ratio also decreases and tends to approach unity. These observations, though incompatible with either the phantom theory or the affine theory alone, are explained by a transition between the two. The Mooney-Rivlin constant 2C , as determined by stress­ -strain measurements, considerably exceeds the modu­ lus calculated from phantom network theory. More­ over, 2C does not vanish at high crosslink function­ a l i t i e s . This indicates that Flory's theory i s not applicable to these complicated networks. 1

2

During the l a s t f i v e years, new aspects have been introduced i n t o the molecular theory o f rubber e l a s t i c i t y Q--4). The o l d e r theory s u c e s s f u l l y p r e d i c t s such p r o p e r t i e s as t h e r m o e l a s t i c i t y , b i r e ­ f r i n g e n c e , and s w e l l i n g , but was unable t o account f o r the s t r a i n dependence o f the modulus. The r e v i s e d theory claims t o overcome with t h i s i n s u f f i c i e n c y . I t o r i g i n a t e s from the understanding t h a t the two c l a s s i c a l approaches, namely the model u s i n g a f f i n e l y transposed c r o s s l i n k s and the model u s i n g f l u c t u a t i n g c r o s s l i n k s , represent extremes. Real networks should e x h i b i t intermediate be­ haviour, a t r a n s i t i o n being induced by s t r a i n . P a r a l l e l t o the development o f the new t h e o r e t i c a l approaches considerable experimental work was done on model networks e s p e c i a l ­ l y synthesized, t o show the e f f e c t s o f pendent chains, loops, d i s ­ t r i b u t i o n o f chain length, f u n c t i o n a l i t y o f c r o s s l i n k s , e t c . on p r o p e r t i e s (5-21). In some i n s t a n c e s , the p r o p e r t i e s turned out 1

This is Part II of a series. For Part I, cf. Ref. 21.

0097-615 6/82/0193-0309$06.00/0 © 1982 American Chemical Society

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

310

ELASTOMERS AND RUBBER ELASTICITY

Downloaded by COLUMBIA UNIV on January 23, 2018 | http://pubs.acs.org Publication Date: July 19, 1982 | doi: 10.1021/bk-1982-0193.ch016

to depend on the chemical composition o f the network, and a l s o on the f r e q u e n t l y n e g l e c t e d m i c r o s t r u c t u r e i n other i n s t a n c e s . We have i n v e s t i g a t e d the s t a t i c and dynamic mechanical prop­ e r t i e s o f networks of d i f f e r e n t chemical and t o p o l o g i c a l s t r u c ­ tures (19,20). In a previous paper, we r e p o r t e d r e s u l t s obtained on networks with c r o s s l i n k f u n c t i o n a l i t y four . In the p r e s ­ ent study, we i n v e s t i g a t e d the e f f e c t o f the s t r u c t u r e o f junc­ t i o n s on the mechanical behaviour o f PDMS. Rather uncommon net­ works with comb-like c r o s s l i n k s were employed, i n t e n d i n g t h a t these would be most c h a l l e n g i n g t o t h e o r e t i c a l p r e d i c t i o n s . Theoretical Considerations In our f i r s t paper, the molecular theory o f rubber e l a s t i c i t y was b r i e f l y reviewed, e s p e c i a l l y the b a s i c assumptions and t o p i c s s t i l l subject t o d i s c u s s i o n (21) . We w i l l now focus on the e f f e c t s of the s t r u c t u r e and the f u n c t i o n a l i t y f o f the c r o s s l i n k s and the r e l e v a n t theory. In a network where the c r o s s l i n k s move a f f i n e l y t o the macro­ scopic s t r a i n , the f u n c t i o n a l i t y o f the c r o s s l i n k s i s i n s i g n i f i ­ cant. The modulus o f such, a network o n l y depends on the number o f c h a i n s , no matter how they are connected (22-27) 2

G

2

= vkT /

(1)

c

Τ i s the absolute temperature, k the Bo^tzmann constant, ν the number o f network chains per volume, the mean-square end-toend d i s t a n c e o f chains i n the undeformed network and the same q u a n t i t y f o r the f r e e , disconnected c h a i n s . I f the f l u c t u a t i o n s o f c r o s s l i n k s are considered, a f u n c t i o n ­ ality-dependent f a c t o r , 1 - 2 / f , has t o be a p p l i e d (28-31) 0

2

2

G . = d-|) vkT / pn f

0

(2)

In order t o enable these f l u c t u a t i o n s t o occur, the network chains are assumed t o be "phantom" i n nature; i . e . t h e i r m a t e r i a l p r o p ­ e r t i e s are dismissed and they a c t only t o e x e r t f o r c e s on the j u n c t i o n s t o which they are attached. With common networks having t e t r a f u n c t i o n a l j u n c t i o n s , the r e s u l t s o f the two approaches d i f ­ f e r by a f a c t o r o f two. I d e n t i c a l r e s u l t s are only obtained from both t h e o r i e s , when the f u n c t i o n a l i t y i s i n f i n i t e . From a p r a c t i ­ c a l viewpoint, however, a value o f about 20 f o r f can already be equated t o i n f i n i t y because c r o s s l i n k d e n s i t i e s can h a r d l y be ob­ t a i n e d with an accuracy b e t t e r than ± 10%. A p p l i c a t i o n o f both approaches t o d e s c r i b e simple e l o n g a t i o n experiments y i e l d s t h a t e i t h e r theory p r e d i c t s the s o - c a l l e d r e ­ duced s t r e s s σ t o be equal t o the shear modulus G and t o be i n ­ dependent o f s t r a i n . σ

= red

=

λ-λ

G

(3)

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

16.

O P P E R M A N N A N D REHAGE

Structure of Cross-linked Polymers

311

The reduced s t r e s s i s d e f i n e d as the f o r c e p e r cross-seçtional area o f the undeformed sample, d i v i d e d by the term λ-λ~ with λ being the r e l a t i v e e l o n g a t i o n L / L . With u n d i l u t e d rubber, t h i s i s not found e x p e r i m e n t a l l y . In most cases, however, the e l a s t i c behaviour i n a moderate e l o n g a t i o n range i s s a t i s f a c t o r i l y de­ s c r i b e d by the e m p i r i c a l Mooney-Rivlin equation, which p r e d i c t s a l i n e a r dependence o f ^ ^ on r e c i p r o c a l e l o n g a t i o n (32-34) 0

r e (

0

a

+

a

2 C

2

/ A

( 4 )

n

The i d e n t i t y o f ^ G a c c o r d i n g t o Eq. (3) i s then only v a l ­ id i n the l i m i t i n g case λ a 1. The recent development o f molecu­ l a r theory by Ronca and A l l e g r a and by F l o r y i s i n approximate accordance with the observed s t r a i n dependence (J^-4). I t i s a s ­ sumed t h a t the two c l a s s i c a l approaches mentioned above r e p r e s e n t extremes. A r e a l network should e x h i b i t intermediate behaviour, dependent upon the extent o f deformation, s t a t e o f d i l u t i o n , and presumably other parameters which can i n f l u e n c e the m o b i l i t y o f the j u n c t i o n s . The model u s i n g a f f i n e l y d i s p l a c e d c r o s s l i n k s g i v e s the upper bound o f the reduced s t r e s s σ ^ f o r u n d i l u t e d rubber a t small s t r a i n s ; the model u s i n g f l u c i u a t i n g c r o s s l i n k s g i v e s the reduced s t r e s s a t i n f i n i t e t e n s i l e s t r a i n . The r a t i o of the Mooney-Rivlin parameters, C^/C^, i s p r e d i c t e d by t h i s theory t o depend on c r o s s l i n k f u n c t i o n a l i t y a c c o r d i n g t o r

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2 C

red = 1

e

d

Hence the measurement o f t h i s r a t i o f o r networks with d i f f e r e n t f u n c t i o n a l i t i e s can serve t o t e s t the theory. Note t h a t C^/C^ should be independent o f temperature ( t h i s i s confirmed by measurements) and a l s o independent o f c r o s s l i n k d e n s i t y (however, a decrease o f CJC. with i n c r e a s i n g ν i s found e x p e r i m e n t a l l y

(10,34-36) Experimental On poly(dimethy1siloxane) (PDMS) networks having comb-like c r o s s l i n k s , t o r s i o n a l v i b r a t i o n experiments and s t a t i c s t r e s s s t r a i n measurements a t small deformations were performed as a f u n c t i o n o f temperature, t o r s i o n a l v i b r a t i o n s a l s o as a f u n c t i o n of frequency. The networks s t u d i e d were s y n t h e s i z e d by a c r o s s l i n k i n g me­ chanism v i a e n d - l i n k i n g o f r e l a t i v e l y short c h a i n s . The chemical process used, i s the h i g h l y s e l e c t i v e r e a c t i o n o f v i n y l groups, which are the t e r m i n a l groups o f a polymer, with a c t i v e hydrogen atoms on a s i l a n e - t y p e c r o s s l i n k i n g agent. T h i s r e a c t i o n goes e s s e n t i a l l y t o completion with no s i d e r e a c t i o n s o r i s o m e r i z a t i o n . In the present study, c r o s s l i n k i n g agents are used which are polymers themselves, namely p a r t l y hydrogenated PDMS: In a s t r i c t nomenclature, t h i s has t o be considered a d i m e t h y l s i l o x a n e

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

312

ELASTOMERS A N D R U B B E R ELASTICITY

(DMS,-Si(CH ) 0-)-hydrogenmetnylsiloxane (HMS,-SiH(CH )0-)-copolymer. The structure of the crosslinking molecules was varied extensively, with respect to chain length and with respect to the content of HMS units. The characterisation of the crosslinking agents used to synthesize the samples Bl - B12 is given in Table I: column 2 gives the fraction of HMS units, column 3 the molecular weight and column 4 the degree of polymerisation. The content of active hydrogen varied from one on every monomer unit (HMS fraction 1) to one on every 40th monomer unit on the aver­ age ( HMS fraction 0.025); the degree of polymerisation covers the range from 35 to 600, corresponding to molecular weights from 2100 to 44200 g-mol" . 3 2

3

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1

Table I Characterisation data for the synthesis of the networks studied Characterisation of Crosslinking Molecules

OdIIip±t3

Portions in Reaction Mixture, Weight-% of

Extract. Portion, Weight-%

Fraction Molecular Degree Crossl. Divinyl of HMS Weight of Pol. Component Component PDMS-Bl -B2 -B3

1 1 1 1 0.50 0.33 0.25 0.20 0.33 0.33 0.14 0.025

-B4 -B5 -B6

-B7 -B8 -B9 -B10 -Bll -B12

2100 4500 7200 9000 6700 6900 7000 7100 10400 17300 11500 44200

35 75 120 150 100 100 100 100 150 250 160 600

0.91 0.91 0.91 0.91 2.02 3.09 4.13 5.18 3.09 3.09 7.18 31.20

0.3 0.3 0.3 0.3

99.09 99.09 99.09 99.09 97.98 96.91 95.87 94.82 96.91 96.91 92.82 68.80

0.3 0.8 0.8 1.0 0.9 0.7 2.3 n. d.

We calculated the molecular weight of the crosslinking molecules from the ratio of monofunctional (Si(CH ) Cl) and bifunctional (Si(CH ) Cl or SiH(CH )Cl ) units present during the hydrolysis of chlorosiloxanes, which forms the polymerisation process. It was checked by viscosity, measured on the undiluted material at 25°C, according to the equation 3

3

2

2

3

3

2

log η = 1 + 0.0123 M

(6)

given by Barry (37). Satisfactory agreement between the two meth­ ods was obtained, except for samples with the largest HMS frac­ tion. This was expected because Eq. (6) holds only for PDMS. The copolymer-crossiinking agents were made either by acid equilibration of PHMS and PDMS in suitable proportions (Fig. la) or by cohydrolysis of the corresponding chlorosilanes (Fig. lb). Equilibration gave a small amount (