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luliiiiiiiiii o sC o •-> o M Ui -o IN) UI vjn M BIBLIOTHEEK TU Delft P 1316 6255ELASTICITY AND STRUCTURE
OF POLYURETHANE NETWORKS
ELASTICITY AND STRUCTURE
OF POLYURETHANE NETWORKS
PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE
TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE
HO-GESCHOOL DELFT OP GEZAG VAN DE RECTOR MAGNIFICUS
DR. IR. C. J. D. M. VERHAGEN,
HOOGLERAAR IN DE AFDELING DER TECHNISCHE
NA-TUURKUNDE, VOOR EEN COMMISSIE UIT DE SENAAT TE
VERDEDIGEN OP WOENSDAG 17 JANUARI 1968 TE 16 UUR
DOOR
REIJER BLOKLAND
SCHEIKUNDIG INGENIEUR
1968
DIT PROEFSCHRIFT IS GOEDGEKEURD
DOOR DE PROMOTOR
PROF. DR. W. PRINS
/^
X
A c k n o w l e d g e m e n t
The author e x p r e s s e s his gratitude to the Algemene Kunstzijde Unie N. V . , Arnhem, for offering the opportunity to c a r r y out the work described in this t h e s i s .
The a s s i s t a n c e of M e s s r s . C . J . F r a n c k e , J . H . Gouda and Y. T i m m e r m a n , and B . J . Spit in the optical investigations, a s well a s the a s s i s t a n c e of J . G . Kennedy, H . C . Nieuwpoort and C D . van Beelen in the construction of the m e a s u r i n g a p p a -r a t u s , is acknowledged. Thanks a -r e due to M -r . H . J . Dekker for p r e p a r i n g the drawings and p i c t u r e s , and last but not l e a s t to my wife for p r e p a r i n g the m a n u s c r i p t .
T A B L E O F C O N T E N T S
1. INTRODUCTION, AIM OF THIS STUDY 1 2. LONG-RANGE ELASTICITY, THEORY AND EXPERIMENTS 3
2 . 1 Molecular t h e o r i e s of ideal e l a s t o m e r s 3
2 . 1 . 1 Introduction 3 2 . 1 . 2 The t h e o r i e s of Hermans, Flory and Wall 6
2 . 1 . 3 The theory of J a m e s and Guth 8
2 . 1 . 4 Concluding r e m a r k s 10 2.2 The phenomenological theory of l a r g e elastic d e f o r - 11
mations
2 . 3 L i t e r a t u r e data 14 2 . 3 . 1 Discrepancies between theory and experiment 14
2 . 3 . 2 Unidirectional extension 16 2 . 3 . 3 Other types of deformation 21 3. EXPERIMENTS ON POLYURETHANE ELASTOMERS 23
3.1 Introduction 23 3.2 P r e p a r a t i o n of p o l y m e r s 24 3 . 2 . 1 Chemistry 24 3 . 2 . 2 Materials 26 3 . 2 . 3 Methods of preparation 28 3.3 Unidirectional extension 30 3 . 3 . 1 Apparatus 30 3. 3,2 Experimental technique 30
3.4 Simple shear and unidirectional compression 31
3 . 4 . 1 Apparatus 31 3 . 4 . 2 Experimental technique 34
4 . RESULTS 36 4 . 1 Characterization of networks 36
4. 2 Unidirectional extension 36 4 . 3 Unidirectional extension and compression 43
4 . 4 Simple s h e a r 49 4. 5 Thermoelasticity 50 5. ON IDEAL AND REAL ELASTOMER NETWORKS 53
5.1 P r o p e r t i e s of ideal networks 53 VIII
5 . 1 . 1 Gaussian polymer chains 53
5 . 1 . 2 Energy effects 55 5 . 1 . 3 Affine deformation 56 5 . 1 . 4 Relaxation effects 56 5 . 1 . 5 Elastically active chains 57 5.2 Previous explanations of d i s c r e p a n c i e s between e x p e r i - 57
ments and ideal elastic theory
5 . 2 . 1 Non-Gaussian networks 58
5 . 2 . 2 Energy effects 58 5 . 2 . 3 Non-affine deformation 61
5 . 2 . 4 Relaxation effects 62 5 . 2 . 5 The theory of Volkenstein, et a l . 63
5 . 2 . 6 The theory of Dobson and Gordon 65
5 . 2 . 7 The theory of DiMarzio 67 5 . 2 . 8 The theory of Jackson, Shen and McQuarrie 69
5.3 Conclusions 71 6. DIRECT EVIDENCE FOR STRUCTURING E F F E C T S IN 73
NETWORKS 6.1 Introduction 73 6.2 Photoelasticity 74 6.3 Light scattering 76 6.4 Electron microscopy 78 6.5 Conclusions 80 7. THE STRESS - STRAIN BEHAVIOUR OF STRUCTURED 81
NETWORKS
7.1 Introduction 81
7.2 The free energy of a network with bundle s t r u c t u r e s 81
7.3 Comparison with experiment 87 7 . 3 . 1 Unidirectional extension 89 7 . 3 . 2 Unidirectional c o m p r e s s i o n 92 7 . 3 . 3 Simple shear 94 7 . 3 . 4 Degree of swelling 94 7 . 3 . 5 Concluding r e m a r k s 94 SUMMARY 99 SAMENVATTING 101 LIST OF SYMBOLS 103 REFERENCES 105
Chapter 1
I N T R O D U C T I O N . A I M O F T H I S S T U D Y
In comparison with other m a t e r i a l s rubberlike substances p o s s e s s some peculiar p r o p e r t i e s . In the first place, a rubber s t r i p can easily be reversibly elongated to s e v e r a l t i m e s its original length. Moreover, the r e t r a c t i v e force i n c r e a s e s when the elongated e l a s t o m e r is heated at constant length. In three-dimensional c o m p r e s s i o n , on the other hand, a rubber behaves like a normal liquid. The compression modulus is about 10* t i m e s the Young's modulus for a typical e l a s t o m e r .
If one uses the word ' r u b b e r ' one is inclined to think of natural r u b b e r , i.e. the product of the t r e e Hevea B r a z i l i e n s i s . Since any polymer netwerk, consisting of long chains with sufficiently weak i n t r a - and intermolecular interaction, within a c e r t a i n t e m p e r a t u r e range can behave r u b b e r l i k e , henceforth the word
' e l a s t o m e r ' will be used.
The phenomenon of longrange elasticity found a qualitative m o -lecular b a s i s only after the ideas of Staudinger (l) about the existence of long chain molecules came to be accepted. In the e a r l y t h i r t i e s it was recognized by s e v e r a l w o r k e r s (2-4) that the constituents of an e l a s t i c network, i.e. chain molecules build up from a large number of s m a l l molecules, a r e able to a s s u m e an enormous number of conformations. The different conformations a r e obtained by rotation around the chain bonds under the influence of t h e r m a l motion. In an undistorted polymer network, the net-work chains can be c h a r a c t e r i z e d by the mean square end-to-end distance, which can in principle be calculated from molecular p a r a m e t e r s (5).
If one adopts the model of 'freely jointed statistical l i n k s ' for a network chain, the most probable end-to-end distance is the one where the number of model chain configurations is at a maximum. Stretching of the network will on the average i n c r e a s e the end-to-end distance of a network chain. A number of chain configurations then becomes inaccessible. As a r e s u l t the polymer molecules, and consequently the network, will tend to r e s u m e its original dimensions under influence of an entropy force (6-9). An e l a s t i c polymer network can thus in principle be considered as an a s s e m -bly of entropy springs. The concepts outlined above provide the b a s i s for all c u r r e n t molecular t h e o r i e s of long-range elasticity. A - phenomenological - approach to d e s c r i b e large r e v e r s i b l e 1
deformations has been given by Mooney (10) and later in g r e a t detail by Rivlin and his associates (11). Rivlin was able to extend the c l a s s i c a l theory of elasticity in such a way that also large elastic deformations could be dealt with.
The molecular theory of highly elastic substances will be p r e -sented in more detail in Chapter 2. The molecular theory leads to relatively simple s t r e s s - s t r a i n relations for ideal e l a s t o m e r net-works. F o r r e a l e l a s t o m e r s , however, always two kinds of deviations from theoretical behaviour a r e observed. The first deviation, occurring at high deformation r a t i o ' s X, ( X>2), i s in principle u n derstood. The second discrepancy between theory and e x p e r i -ment, occurring most clearly in unidirectional elongation at moderate extensions, ( 1 . 2 < X < 2 ) , has not yet been given a s a t i s -factory molecular explanation. The investigation of this significant gap in understanding of rubberlike elasticity is the aim of this study.
Chapter 2 also provides a synopsis of the phenomenological theory of large elastic deformations because the r e s u l t s of this theory a r e needed further on. Finally the r e s u l t s of published e x p e r i -ments will be reviewed.
F o r the experimental part of our study, polyurethane networks w e r e selected. Chapter 3 gives the c h e m i s t r y and the preparation of these e l a s t o m e r s . Also the m e a s u r i n g apparatus and e x p e r i -mental techniques a r e described. In Chapter 4 s t r e s s - s t r a i n data, obtained by subjecting the e l a s t o m e r s to unidimensional extension, compression and simple shear a r e presented. F o r some e l a s t o m e r s the energy component of the stretching force is also determined.
In Chapter 5 a review is given of the v a r i o u s molecular explanations of deviaexplanations from the ideal network theory. Several s h o r t -comings of these explanations a r e discussed.
Chapter 6 gives experimental evidence from optical r a t h e r than mechanical behaviour demonstrating in a direct way the existence of some structuring in polyurethane networks. This is in line with previous suggestions, e.g. by Gee, as to the probable o c c u r -r e n c e of c o -r -r e l a t e d -regions o-r bundle s t -r u c t u -r e s in e l a s t o m e -r s .
In Chapter 7 the effect of bundle s t r u c t u r e s on the mechanical behaviour is investigated m o r e closely. An e l a s t o m e r network m o -del is used in which intermolecular structuring is taken into account through the obstruction this imposes upon the random walk of the chain elements. The theory can describe all observed phenomena collected during this study as well as those reported in the l i t e r a t u r e .
Chapter 2
L O N G - R A N G E E L A S T I C I T Y , T H E O R Y AND E X P E R I M E N T S
2.1 MOLECULAR THEORIES OF IDEAL ELASTOMERS 2.1.1 I n t r o d u c t i o n
A calculation of the exact s t r e s s - s t r a i n behaviour of e l a s t o m e r s has so far been impossible because of the lack of detailed infor-mation about the molecular s t r u c t u r e of the network.
Several simplifications concerning the network s t r u c t u r e have therefore been introduced. In spite of these unavoidable simplifi-cations, such i d e a l elastic network t h e o r i e s applied to r e a l e l a s t o m e r s yield quite reasonable r e s u l t s . Consequently, the ex-p e r i m e n t s justify to a g r e a t extent the aex-pex-proximations which had to be introduced. In this chapter we will not linger on the various assumptions which have to be made in transposing a r e a l e l a s t o m e r network into an ideal one. A detailed t r e a t m e n t of these assumptions is postponed to Chapter 5.
F i r s t of all let us focus our attention on an individual polymer chain. This chain is r e g a r d e d as an ideal chain, i.e. all interaction between chain e l e m e n t s , as well a s interaction with neighbouring chains, is neglected. An element of the chain is defined in Kuhn's s e n s e as a freely jointed statistical link. The first chain element is located at the origin of a Cartesian coordinate s y s t e m , the last (N-th) element has coordinates x, y, z. The probability of this situation is equal to the probability of reaching point x, y, z s t a r t i n g from the origin in a random walk in which N steps a r e taken. This probability P(x, y, z), i s given by the Gaussian e r r o r function.
P(x, y, z)dx dy dz =
= : j ^ ^ e x p [ - B ^ ( x ^ + y ^ + z ^ ) ] d x d y d z 2 , 1
in which e^=3/2NA^, and A is the length of a chain element. Equation 2.1 i s obtained if the volume exclusion of the chain elements i s neglected. Hence P(x, y, z) is at a maximum in the origin. The probability P(r) to observe a c e r t a i n end-to-end distance r is given by:
P ( r ) d r = - E — exp ( - p ^ r ^ ) 4TT r ^ d r
TT 7 2
2.2
w h e r e r ^ = x^+y^ + z^. The function Eq. 2.1 is spherically s y m -m e t r i c a l , its -maxi-mu-m is at r = o. The function Eq. 2.2 has a maximum at r = A(2N/3) ^/^. The r o o t - m e a n - s q u a r e value of r for the random chain i s :
< r ^ > J r ^ P ( r ) d r o
3 = NA'
2 3 ' 2.3
Both probability functions a r e shown in Fig. 2 . 1 .
-2.0
-^ r
Fig. 2.1. Distribution functions given by Eq. 2.1 (Fig. a) and Eq. 2.2 (Fig. b) ( g = 1 in both cases).
In an elastic network the long polymer chains between c r o s s l i n k s may be replaced by the imaginary ideal, Gaussian chains con-s i d e r e d above. In thicon-s cacon-se the number of configurationcon-s a network chain with endtoend distance r can a s s u m e is p r o p o r -tional to P(x, y, z).
Now, consider a unit cube cut from an ideal elastomer network strained by external f o r c e s . Its dimensions will change into
A.^ , \y and ^z • A polymer chain in the network is thought to follow this external deformation. The corresponding d e c r e a s e in the number of chain configurations can now easily be calculated from Eq. 2.1. The d e c r e a s e in entropy is given by:
AS = k i n TTTT ^ ^
\-2 , 4
= -ken(>?. -l)x^ +(\% -l)y^ + {\\- l)z^]
where the index o indicates the undeformed chain. For a random network in the unstrained state all directions a r e equally probable. Thus, on the average:
< x > = < y ^ > = < z § > = l<r^> 2.5 Next it is assumed that the set of free chains before crosslinking
r e m a i n s unchanged during the crosslinking p r o c e s s . O r , com-bining Eq. 2.3 with Eq. 2.5, we find:
. 2 • 1
< ^ o > = ^ 2 . 6
In addition energy changes during deformation a r e neglected. Since the network consists of v chains, and the chain entropy is additive, now the deformational free energy of the strained net-work can be expressed a s :
AF = - T A S = i v k T ( X ^ ^ + 4 + X^j - 3) 2.7
Unidirectional s t r e t c h or compression is characterized by:
- i / g
X, = X Xj, = \ , = \ ' 2 . 8
if the very minor volume change is neglected. Eq. 2.7, combined with Eq. 2.8 yield for the stretching force on the unit cube:
f = ~~- = v k T ( X - X - ^ ) 2.9 P u r e s h e a r , as well as simple s h e a r can be described by:
\^ =\ Xy = X"^ X, = 1 2.10 Introducing Eq. 2.10 into Eq. 2.7, the equation of state for p u r e
s h e a r becomes:
f^ = ^ ^ = v k T ( X - X"^) 2 . 1 1
o A.
In simple s h e a r , the amount of s h e a r , s, i.e. the tangent of the angle through which a v e r t i c a l edge is tilted, is equal to X - X~ •"• for moderate extensions(12). F r o m Eq. 2.7 the relation between s h e a r i n g force, fsj , and the amount of s h e a r is calculated to be,
f,3 = ^ ^ = v k T ( X - X - ^ ) 2.12 a s
All these types of deformation a r e shown in Fig. 2.2, a-f.
The simplified 'single chain' reasoning given above contains all e s s e n t i a l elements of the molecular theory of high elasticity. More c o r r e c t t h e o r i e s a r e due to H e r m a n s , Flory and Wall (13,14,15). The most complete version, however, is given by J a m e s and Guth (16,17,18), at least for constant volume deformations. Between the various t h e o r i e s some discrepancies still exist. These discrepancies only concern the value of the constant v k T , which is of little p r a c t i c a l importance in our considerations. 2.1.2 T h e t h e o r i e s of H e r m a n s , F l o r y a n d W a l l
In the undeformed network a c e r t a i n number of chains, v° , have an end-to-end distance r j . After deformation, this number has changed into Vj. The number of m i c r o s t a t e s of the undeformed network is given by:
V ' _ v°
Q o = 7 r - ^ i l P i 2.13
i
Pj i s the probability of a chain having an end-to-end vector equal to r j .
A /
1 a T- bX
X
1/2't'
Fig. 2. 2. Various types of strain
a. unstrained state b. three-dimensional s t r a i n c. unidirectional extension d. unidirectional compression e. pure shear f. simple shear 7
In the deformed state the number of m i c r o s t a t e s becomes:
n=^^iip,^' 2.14
n Vj 1 1
1
The d e c r e a s e in network entropy due to the deformation i s given by:
A S = k l n j ^ = k E v , l n ^ 2.15
" O ' ^ l
The network chains a r e again considered as ideal, Gaussian chains, but the assumption of affine deformation is introduced in a m o r e general way; Vj is calculated by maximizing Q under the con-ditions S vi = 0 and<x^>=X^^<XQ > (with s i m i l a r expressions for the y and z components). This yields:
AF = - T A S
1 2.16 = - v k T [ ( X = ; + X^y + \\ - 3) - 2 1 n X , X ^ X j
Flory and Wall later modified expression 2.16 slightly by i n t r o -ducing entropy effects due to the crosslinking p r o c e s s . In their theory the factor 2 before the logarithmic t e r m is replaced by4/f, in which f is the functionality of a crosslink.
2.1.3 T h e t h e o r y of J a m e s a n d G u t h
The statistical theory of highly elastic substances presented by J a m e s and Guth is the most general one. In a way this theory is a justification of the approaches mentioned above. In sketching this theory we first have to d i s c r i m i n a t e between fixed points and junctions in the e l a s t o m e r network. The fixed points a r e all p a r t s of the network which a r e immobilized by external constraints. The junctions a r e the crosslinks between the network chains. A network chain joining the points x.^, y.^., z^ and Xv, yvi ^v is again considered to be an ideal, Gaussian chain. Consequently the num-b e r of configurations of a chain is given num-by:
n (I r - r I) =
r-r is a vector indicating the position of point x x, y T> ^T . with r e s p e c t to a fixed origin in the network. By neglecting interaction between different chains the number of configurations, consistent with a given position of fixed points and junctions, is given by the products of the number of configurations possible for the individual network chains.
(r^, rp ' H' El )
n n Q( r - r ) 2.18
The v e c t o r s VQ,, r a indicate fixed points,the vectors r i . r g refer to junctions. The total number of configurations' can be obtained by integrating over all possible s e t s of positions of the junctions. By a r a t h e r lengthy mathematical reasoning J a m e s (19) was able to perform this integration. It was found that all v a r i a b l e s describing the junctions a r e eliminated. Only a Gaussian function of the fixed points is retained. Hence for the deformed network he finds:
n ( X , , X , , ^ ^ ) = C e x p ( - ^ Z ^ E ^ l r ^ - r p | = ) 2.19
The free energy of deformation follows from: 0 (X„ , X„ , X,)
AF = - k T l n . ' ; J \ ' ' 2.20 Q (1, 1, 1)
which leads to an equation s i m i l a r to Eq. 2.7.
J a m e s and Guth point out that the distribution of chain lengths most probably will change during the crosslinking p r o c e d u r e . Rejecting the assumption implied by Eq. 2.6 they present a n -admittedly crude - e s t i m a t e of the change of the modulus during cure. The Eq. 2.21 for the free energy of deformation is then obtained:
A F = i v k T ( X^^ + X^ + X^, - 3) 2 . 2 1
The modulus in Eq. 2.21 differs a factor 5 from the modulus in Eq. 2.7. It is of i n t e r e s t to note that Duiser and Staverman (20) as well a s Chömpff and Duiser (21) have derived the same factor 2 from a completely different type of reasoning, based upon an
tension of the Rouse theory to networks.
Some important r e s u l t s of the calculations of J a m e s and Guth which will be used later (Chapter 7) a r e listed h e r e :
1. The force between two adjacent c r o s s l i n k s , which exhibit Brownian motion compatible with the connecting network chains, is the same as if both crosslinks were fixed in the most probable positions.
2. If the network is subjected to s t r a i n , the most probable po-sitions of the c r o s s l i n k s will be displaced as if the c r o s s l i n k s were embedded in an elastic continuum. Hence the forces exerted by the network r e m a i n unchanc;ed whether the network is t r e a t e d as containing free c r o s s l i n k s , or as containing crosslinks fixed in its most probable positions.
3. F o r calculating convenience the Gaussian network can be treated as a cubical network, i.e. a network composed of t h r e e s e t s of independent chains, with t h e i r end-to-end vectors p a r a l l e l to the coordinate axes.
2.1.4 C o n c l u d i n g r e m a r k s
The various formulae presented above can be summarized by writing
A F = V k T [ ^ (X"^^ + X^ + X^, - 3) - B1 n X ^ Xy X J 2. 22 in which A and B a r e constants.
F o r all practical purposes the value of AvkT is r a t h e r uncertain. Apart from the questionable assumption implied by Eq. 2.6, or the crude estimation of modulus changes during c u r e , it is n e c e s s a r y to introduce a front factor < r f > / < ro > (22,23). The average
chain dimensions in the undeformed m a t e r i a l may not be actually the same as in the unstrained s t a t e , e.g. because the m e a s u r e m e n t s a r e performed at another t e m p e r a t u r e than the curing took place; < r ^ stands for the initial chain dimensions in the test piece at the t e m p e r a t u r e of the experiment, and is in general unknown. The recognition that < r^ > is in general a function of the t e m p e r a t u r e implies that the complete equation of state c a r r i e s a (small) energy t e r m in addition to the main entropy t e r m . In the previous discussion we have focussed our attention on the entropy t e r m only because of our interest in the X - dependent part of the free energy. Mukherji and P r i n s (24,25) have suggested that the value of A might be connected with deviations from Gaussian s t a t i s t i c s due to topological complexities.
All this makes that the absolute number of elastically active c r o s s l i n k s in the network in general cannot be determined
unam-biguously.
The logarithmic t e r m is of no importance if deformations at a l -most constant volume a r e considered, i.e. for all types of deforma-tions that will be investigated h e r e .
Bearing the above points in mind, we conclude that in s t r e s s -s t r a i n inve-stigation-s of i n t e r e -s t to u-s the different molecular t h e o r i e s lead to the same expression for the free energy of a deformed network, namely:
AF = i c ( X 2 ^ + X=y + X^, - 3) 2.23 in which C is a t e m p e r a t u r e dependent constant. Eqs. 2.9 and 2.12
in this notation then become:
f = C(X - X-=) and fss = C(X - X"'-) 2.23a
2.2 THE PHENOMENOLOGICAL THEORY OF LARGE ELASTIC DEFORMATIONS
In contrast to the development of molecular theories of high elasticity, the elastic theory of finite deformations has a quite different origin. The aim of this theory is to codify the p r o p e r t i e s of highly elastic polymer m a t e r i a l s , r e g a r d l e s s of the molecular mechanisms which cause the highly elastic behaviour. Hence this theory is purely phenomenological, and does not claim to offer an explanation for the origin of the observed phenomena. This approach to the problem of high elasticity does, however, provide a useful tool in the study of e l a s t o m e r deformations.
Rivlin (11) has c o n s t r u c t e d t h e t h e o r y of finite elastic deformations starting from two assumptions. F i r s t of all, the elastic m a t e r i a l is assumed to be isotropic in the undeformed s t a t e . Secondly volume changes at deformation a r e neglected. This also is quite a good approximation. If an e l a s t o m e r is deformed, a c e r t a i n amount of work is s t o r e d in the strained m a t e r i a l . Consequently, s t r e s s - s t r a i n behaviour can be described by a 'stored e n e r g y ' function W, depending upon the deformation. In the case of i s o -t h e r m a l deforma-tion -the s-tored energy W is really a free energy change AF. We have seen that in AF the entropy t e r m is p r e d o m i -nant. In this section we will retain the 'stored energy' notation, in o r d e r to distinguish phenomenological from molecular theory.
Let us consider a unit cube of e l a s t o m e r . External s t r e s s e s a r e assumed to deform the cube into a cuboid of dimensions Xjj , Xy and Xj . Obviously, W (X^.X^.X^) must i n c r e a s e from z e r o to a c e r t a i n positive value during the deformation p r o c e s s . In the
deformed state W (1,1,1) = 0. We note that W has to r e m a i n un-changed if any degree of deformation Xj is replaced by its negative value, X_i . Secondly W must be invariant to any permutation of the three principal extension r a t i o s . These statements follow from the fact that rotation does not, of c o u r s e , change the free energy of deformation. Thus W must be e x p r e s s i b l e as a function of any t h r e e independent s y m m e t r i c functions of X^^ , Xy and X^^ . According to Rivlin these three functions, the s t r a i n invariants, a r e as follows:
I I = ^ x + K + ^?
2.24
I 3 = ^ x ''^y ^ z
For an incompressible m a t e r i a l I 3 = 1, and W = W (11, Ig). In this case the second invariant Ig mostly is written as
4 ^ = x ; ^ + x : ^ + x ; ^ 2.25 1 3 '
Since the t r u e nature of W r e m a i n s unknown, it is usually a p -proximated by a power s e r i e s in I^ - 3 and Ig - 3.
W = S E d l - 3 ) ' ( I s - 3)^ 2.26
i = 0 ] = o
Eq. 2.24 satisfies the condition W (Ii= 3, Ig =3) = 0.
The cuboid under consideration is maintained in the strained state by the external forces fx . fy a n d f j (Fig. 2.b). All forces a r e thought to act perpendicular to the coordinate axes, consequently all tangential components a r e zero. The situation is not essentially different when the m o r e general c a s e , with forces acting under a a r b i t r a r y angle with the l a t e r a l faces of the cuboid, is considered. If the cuboid is subjected to a virtual deformation, the amount of work done by the external forces must equal the change dW in elastically stored energy. Thus for the compressible case:
I^XyX^dl^ + fy X^ X^dXy + f^X^Xyd Xj = ÖW , , ^ ÖW , , ^ ÖW , ,
Hence g X ; " * " ^y ^^ ^"^^ with s i m i l a r relations for the derivatives with r e s p e c t to Xy and
X J. Using the relation
ÖW ^ 5JA^ SJ^ a w öls , ö_W S13
s x , 3 i i öx^ s i ^ a x , öi3 sx ^"^'^^^
F o r the incompressible c a s e d(Xjj X, Xj) = 0, and W is a function of Ii and I2 only. Now the relations 2.29 become:
f x = 2 ( X ^ . — - X t f ^ ) - P 2.30 and corresponding e x p r e s s i o n s for f ^ and f z . The quantity p i s a
constant, which can be considered as a hydrostatic p r e s s u r e , F o r unidirectional extension as well as co.mpression (Fig. 2.2. c,d) f y a n d f j a r e zero. Now the s t r e s s - s t r a i n relation is found by eliminating p.
f = 2(X - X - = ) ( | ^ +\-^~) 2 . 3 1
o i l o I s
In the c a s e of pure s h e a r . Fig. 2.2 e,the relation between shearing force and degree of deformation i s :
f3 = 2 ( X - X - = ) ( | ^ + ^ ) • 2.32 In our c a s e only simple s h e a r will be studied. This type of d e
-formation is homogeneous, but not p u r e , since the shearing force does not act perpendicular to the coordinate directions. Conse-quently the principal axes rotate when the e l a s t o m e r is subjected to simple s h e a r . The amount of s h e a r , s , i s the tangent of the angle through which a v e r t i c a l edge is tilted. Rivlin has shown that the s t r e s s - s t r a i n relation i s given by:
aw aw
2.3 LITERATURE DATA
2.3.1 D i s c r e p a n c i e s b e t w e e n t h e o r y a n d e x p e r i m e n t As we have seen, all molecular t h e o r i e s yield Eq. 2.23 for the free energy of a deformed e l a s t o m e r network. F r o m the e a r l y forties on, numerous experiments have been conducted to test this relation. In practically all c a s e s the e l a s t o m e r s were investigated under unidirectional extension only. F o r this type of s t r a i n the theoretical relation between stretching force and extension is given by Eq. 2.9. With hardly any exception the experiments lead to the observations r e p r e s e n t e d in Fig. 2.3 (compare also F i g s . 4 . 1 , 4.2, 4.3).
Fig. 2.3. Unidirectional extension curve for natural rubber (taken from ref. 12, page 91)
At low extensions, i.e.X< 1.2, the theoretical curve coincides with the experimental one. At higher extensions, i.e. approximately 1 . 2 < X < 5 , the theoretical force exceeds the measured force. Moreover, the difference between experimental and measured force i n c r e a s e s with the extension ratio. At much higher extensions
the opposite is observed. In that region the experimental stretching force i n c r e a s e s sharply, while the theoretical force lags behind.
The difference between theory and experiment at high extensions is c l e a r l y understood nowadays. At extensions approaching the finite length of the polymer chains the chain length distribution function can no longer be approximated by the Gaussian e r r o r function. In this case the c o r r e c t Langevin distribution function should be used. With this distribution function the theoretical forceextension curve a g r e e s , at least qualitatively, with e x p e r i -ment (12, page/104 ff. 26). In addition most e l a s t o m e r s c r y s t a l l i z e when highly stretched. The increasing number of c r y s t a l l i t e s in the network plays the role of additional c r o s s l i n k s , and therefore r e s u l t s in an extra i n c r e a s e of the stretching force.
The departures from ideal Gaussian network t h e o r y at moderate extensions, hoVever, a r e not understood at all. As first pointed out by Mooney (10) and later verified extensively, unidirectionally stretched e l a s t o m e r s usually obey the e m p i r i c a l relation:
f = ( C i + X"^ C2)(X - X"^) 2.34 Cg almost always is a positive quantity, in magnitude comparable
to C i . Eq, 2.34, generally known as the Mooney-Rivlin equation, should be compared with the theoretical expression 2.9. C i and Cg a r e called the first and second Mooney-Rivlin constant respectively. It is seen that the one constant formula 2.9 is obtained from the phenomenological Eq. 2.26, if only the first t e r m of the s e r i e s expansion is retained, i = 1, j = 0. The Mooney-Rivlin equation Eq. 2.34 is obtained from Eq. 2.26 and Eq. 2.31, if the first as well as the second t e r m of the s e r i e s expansion a r e used, i = 1, j = 0 and i = o, j = 1, In this case the free energy of deformation b e c o m e s :
A F = C 1 ( X^^ + X^ + X^, - 3) + C 2 ( X-= + X-y= + X-2 - 3) 2. 35 Using Eq. 2.35, we obtain for the shearing force in simple s h e a r , (Eq. 2.33),
fB6= 2 ( C i + C 2 ) ( X - X-^) 2.36 The Cg constant is found to be most important in unidirectional
extension, although other types of deformation at moderate s t r a i n s can also be described m o r e accurately with the two constant formula Eq. 2.35 than with the one constant formula Eq. 2.23 (27).
We now have the opportunity to state accurately the aim of this study. As shown, the one constant formula for the free energy of a deformed network can be obtained from a satisfactory molecular
theory. The d e p a r t u r e s from the one constant free energy equation, a s e x p r e s s e d by the positive constant Cg so far have not been given a satisfactory molecular explanation. Our study then will attempt to provide such a molecular b a s i s for the experimental two constant formula for the free energy of a strained e l a s t o m e r network.
2.3.2 U n i d i r e c t i o n a l e x t e n s i o n
To the best of our knowledge every stretched e l a s t o m e r obeys the Mooney-Rivlin equation 2.34 at moderate extensions, i.e. 1.2 < X<2,0, Obviously, C i is always a positive quantity, its magnitude varying
from about 0. 2-10 kg/cm^. In one or two c a s e s Cg is found to be z e r o (28,29), but practically all investigations yield positive values for this quantity. A - perhaps spurious - negative Cg value has been observed on a lightly crosslinked polyethylene, measured above the melting point (29). F o r this special e l a s t o m e r , m o r e o v e r , energy changes during s t r e t c h s e e m to be quite substantial, and different from other - m o r e 'common' - e l a s t o m e r s . The excep-tional behaviour of linear polyethylene networks is also indicated by recent work of Gent and Vickzoyji (42), who m e a s u r e d the Mooney-Rivlin constants on samples of varying crosslink density. C2/C1 values ranging from 0.06 (at high crosslink density) to 7.0 (at low crosslink density) a r e r e p o r t e d , which values a r e r a t h e r unusual for e l a s t o m e r s .
In the Tables 2 . 1 - 2 . 7 force-extension data of s e v e r a l e l a s t o m e r s a r e summarized. It is not pretended with this survey to cover all published data. The data included, however, a r e sufficient to
outline the p r e s e n t knowledge about the two Mooney-Rivlin con-s t a n t con-s . We note the following factcon-s:
- the constant Ci i n c r e a s e s with increasing crosslink density, as well as with increasing t e m p e r a t u r e . On the other hand Ci de-c r e a s e s when the e l a s t o m e r under study is swollen.
- The constant C2 s e e m s to obey only one rule without exception. That is Cg as well as the r a t i o C g/C 1 d e c r e a s e when the e l a s t o m e r involved is swollen (see e.g. 43). The type of solvent used is i r r e l e v a n t , as long as no microphase separation is involved. All other r u l e s a r e subject to some doubt, since they a r e based on somewhat contradictory r e s u l t s . Possibly no general r u l e s can be given because C 2 may be related to the very s t r u c t u r e of the e l a s t o m e r involved. Moreover, this s t r u c t u r e may be a function of the network preparation conditions, as e.g. milling. Most p r o b -ably Cs i n c r e a s e s with increasing degree of crosslinking. C2 is not o r only slightly dependent upon t e m p e r a t u r e .
con-Table 2.1 Mooney-Rivlin constants for natural r u b b e r s , measured in extension. Blackwell (30). sulphur vulcanizates Gumbrell c.s. (31). sulphur vulcanizates Flory, Ciferri (32), Y -radiation vulcanizates sulphur vulcanizate Loan (33). peroxyde vulcanizates Ciferri (34). Y -radiation vulcanizate Roe, Krigbaum (35). electron radiation vulcanizate Smith, Puett (36). electron radiation vulcanizate B a r r i e , Standen (37). peroxyde vulcanizates t e m p e -rature,°C ambient ambient 34 59 34 34 59 35 34 ambient 27 47 67 30 40 50 60 30 50 70 25 70 25 70 25 70 C i kg/cm^ 3.2 2.9 2.6 2.9 3.8 1.00 1.17 1.43 1.73 1.75 2.54 3.08 0.90 0.88 1.52 1.92 2.06 2.00 2.00 1.06 1.48 1.58 1.78 1.98 2.499 2.578 2.654 2.731 2.08 2.24 2.39 1.4 2..1 1.9 2.1 1.7 1.9 C2 kg/cm^ 1.5 1.5 1.6 1.8 1.7 1.00 1.00 1.03 1.05 1.05 1.04 1.02 1.42 1.28 1.10 1.60 1.54 1.50 2.00 0.835 0.712 1.92 1.78 1.68 0.862 0.869 0.885 0.900 1.45 1.45 1.45 1.7 1.9 1.1 1.2 1.3 1.3 C 2 / C 1 0.47 0.52 0.61 0.62 0.46 1.00 0.86 0.72 0.61 0.60 0.41 0.33 1.58 1.45 0.72 0.75 0.75 0.75 1.00 0.79 0.48 1.22 1.00 0.85 0.345 0.337 0.332 0.329 0.70 0.65 0.61 1.21 0.90 0.58 0.57 0.76 0.68 17
Table 2.2 Mooney-Rivlin constants for silicone e l a s t o m e r s , measured in extension. C i f e r r i , Flory (32). peroxyde vulcanizate Y -radiation vulcanizate Thomas, (38). peroxyde vulcanizate Smith, (39). peroxyde vulcanizate t e m p e -r a t u -r e ,°C 30 34 94 149 30 28 -40 0 40 80 120 160 200 C i ^ k g / c m 0.70 0.72 0.88 0.98 1.00 0.447 0.543 0.582 0.567 0.75 0.84 0.90 1.0 1.05 1.14 1.2 Cs ^ kg/cm 0.62 0.58 0.62 0.60 0.80 0.194 0.357 0.388 0.408 0.28 0.28 0.28 0.28 0.28 0.28 0.28 C s / C i 0.89 0.80 0.71 0.61 0.80 0.43 0.65 0.66 0.72 0.37 0.33 0.31 0.28 0.27 0,25 0.23 Mark, Flory (40). different electron radiation vulcanizates 1.22 1.1 0.90 B a r r i e , Standen, (37) different peroxyde vulcanizates 30 25 70 25 70 25 70 25 70 1.00 1.02 0.75 0.81 0.79 0.30 0.32 0.34 0.35 0.48 0.53 0.55 0.61 1.0 0.8 0.85 0.50 0.40 0.25 0.26 0.24 0.25 0.50 0.54 0.36 0.37 1.00 0.79 1.13 0.62 0.51 0.83 0.81 0.73 0.71 1.04 1.02 0.65 0.61
Table 2.3 Mooney-Rivlin constants for polyurethane e l a s t o m e r s , measured in extension. t e m p e - C^ Cg r a t u r e , C k g / c m k g / c m ' E l a s t o m e r ingredients: toluene diisocyanate, toluene diamine and POPG 2000.
Havlik, Smith, (41) (calculated from their Fig. 5) C 2 / C 1 25 49 71 3.30 3.39 3.40 2.00 2.12 2.28 0.60 0.62 0.67 Adiprene C. Du Pont peroxyde vulcanizate. Loan (33). ambient 2.40 1.82 0.76
Table 2.4 Mooney-Rivlin constants for butyl e l a s t o m e r s , measured in extension.
C i f e r r i , Flory, (32) sulphur vulcanizate
Smith, (39)
sulphur vulcanizate (calculated from his Fig. 5) t e m p e -r a t u -r e , C 49 0 40 80 120 140 C i k g / c m ^ 1.24 2.11 2.39 2.70 3.00 3.18 Cs k g / c m ^ 1.50 1 . 5 + 0 . 1 1 . 5 1 0 . 1 1 . 5 + 0 . 1 1 . 5 1 0 . 1 1.5+ 0.1 C 2 / C 1 1.21 0.70 0.62 0.55 0.49 0.46 19
Table 2.5 Mooney-Rivlin constants for styrene-butadiene elastomers, measured in extension.
Gumbrell c.s.,(31) % styrene r e s p . 5, 10 15, 25, 30 and 35. sulphur vulcanizates Loan (33) 28% styrene, two different peroxyde vulcanizates. tempe-r a t u tempe-r e , C ambient ambient C i kg/cm^ 2.80 1.68 1.36 1.88 1.83 0.79 1.02 1.66 Cg ^ k g / c m 1.04 1.07 1.00 1.00 1.04 1.06 1.25 1.12 C g / C i 0.37 0.64 0.74 0.53 0.57 1.34 1.23 0.67
Table 2.6 Mooney-Rivlin constants for polyacrylate networks, measured in extension. C i f e r r i , Flory (32) Poly(methyl metha-crylate) Poly(ethyl acrylate) Table 2.7 Mooneyt e m p e -r a t u -r e , 145 175 0 49 65 °C Ci ^ k g / c m 1.00 0.60 1.12 1.54 1.56
Rivlin constants for some copolymers, B a r r i e , Standen,(37) t h r e e different s a m p l e s m e a s u r e d tempe-r a t u tempe-r e 55 100 55 100 55 100 in °C extensior Ci kg/cm^ 2.1 2.8 2.4 2.2 2.6 3.1 ^ = g k g / c m 4.80 2.60 1.26 0.70 1.40 ethylene-1. Cs ^ kg/cm^ 2.4 2.4 2.9 2.2 2.2 2.4 C2/C1 4.80 4.33 1.12 0.45 0.90 propylene Cg/Ci 1.14 0.86 1.21 1.00 0.85 0.77
stant formula for the free energy of deformation, greatly depends upon the c h a r a c t e r of the network. Especially for some p o l y -a c r y l -a t e s , dry -a s well -a s swollen, unusu-ally high C g / C i r -a t i o s -a r e reported (Table 2.6, 32,44). Even for the same tj^je of e l a s t o m e r very different C g / C i r a t i o s a r e reported. F o r natural rubber, for example.Cg/C i v a r i e s f r o m 0.3-1.3approximately (Table2.1). Hence we a r e forced to conclude that this quantity is influenced by the previous history of the e l a s t o m e r , its preparation, as well as by the conditions of the m e a s u r e m e n t . Probably C g / C i i n c r e a s e s with increasing molecular weight of the s t a r t i n g material (28, 30). R e -laxation during experiments, on the other hand, will diminish C g / C i ( 3 2 ) .
2.3.3 O t h e r t y p e s of d e f o r m a t i o n
Unfortunately only a few s t r e s s - s t r a i n investigations other than unidimensional extension have been published. T r e l o a r (12) gives some data on natural rubber, investigated in unidirectional ex-tension as well as in uniform two-dimensional exex-tension (i.e. uni-directional compression). Cg appears to be approximately z e r o for compression (12, Fig. 5.6), whereas in extension it is s u b -stantial. In the c a s e of simple s h e a r the theoretical expression Eq. 2.35 is obeyed up to large amounts of shear (12, Fig. 5.9). It should be pointed out, however, that the Ci +Cg is not the s a m e a s the C i + Cg obtained in extension. P u r e homogeneous s t r a i n could also be described b e t t e r with the two-constant formula for the free energy of deformation, C g / C i being approximately 0.05 (12,pag. 162).
Noteworthy a r e the experiments of Rivlin and Saunders (45). F r o m a batch of natural rubber they p r e p a r e d various s p e c i m e n s , a c c o r -ding to the same r e c i p e . The rubber specimens w e r e subjected to p u r e homogeneous s t r a i n , undirectional extension and compression, p u r e shear and torsion. All s t r e s s s t r a i n data could be r e p r e -sented by the general relation:
f = C i ( I i - 3) + f(l2 - 3) 2.37 f (Ig-3) denotes some function of I g - 3 .
Each deformation initself, however, could be described s a t i s f a c t o r i -ly by the two-constant free energy expression Eq. 2.34, i.e. by approximating the t e r m f (Ig-S) in this equation by the function Cg (l2-3). The r a t i o C g / C i was found to be strongly dependent upon the type of deformation. Table 2.8.
Table 2.8. C s / C i as a function of deformation, (45)
unidirectional unidirectional biaxial torsion extension compression homogeneous
s t r a i n C3_ C i 0.75 0 0.1 0.3^ 0 . 1 ^ . 0.15' 1. m e a s u r e d at increasing s t r a i n . 2. m e a s u r e d at decreasing s t r a i n .
3. R.S. Rivlin: J. Appl. Phys. 18, (1947), 444.
-In this connection we also draw the attention to a review of Mullins and Thomas (55), in which the following version of the Mooney-Rivlin equation is mentioned:
f = C i ( X x-=) Cs ( X - X"-2 X^ + X" - )
2 . 3 8
A different stored energy function, and consequently different s t r e s s -s t r a i n relation-ship-s, i-s al-so given in the -s a m e review. Thi-s func-tion i s :
W = W i ( I i 3) + W s l n ( ^ ) 2.39 where W^ and Wg a r e constants.
F r o m the above data one is forced to conclude that the two-constant Eq. 2.35 cannot be a universal function describing all types of deformation with the s a m e constants. This means also that the s e r i e s expansion Eq. 2.26 or the free energy relation Eq. 2.35 is an insufficient description of the functions W and AF.
Chapter 3
E X P E R I M E N T S ON P O L Y U R E T H A N E E L A S T O M E R S
3.1 INTRODUCTION
In studying the deformation of e l a s t o m e r s a m o r e complete picture is obtained when the m a t e r i a l is subjected to various types of s t r a i n (12, page 169 ff). In our c a s e e l a s t o m e r s a r e subjected to t h r e e types of s t r a i n , viz. unidirectional extension, unidirectional compression and simple s h e a r . Clearly, it is d e s i r a b l e to a s c e r -tain the different s t r e s s - s t r a i n relations on one and the s a m e e l a s t o m e r . In addition it is worthwile to obtain data concerning the relation between network s t r u c t u r e and deformation c h a r a c -t e r i s -t i c s . L i -t e r a -t u r e da-ta poin-t -to -the conclusion -tha-t d e p a r -t u r e s from the ideal network theory greatly depend upon the s t r u c t u r e of the elastomer under study. This a r e a of investigation i s , how-e v how-e r , not v how-e r y systhow-ematically how-explorhow-ed.
A type of m a t e r i a l which s e e m s to meet all r e q u i r e m e n t s ex-p r e s s e d above, is found among the ex-polyurethane comex-pounds.
Polyurethane e l a s t o m e r s can be p r e p a r e d by mixing suitable in-gredients, i.e. polyols of relatively high molecular weight and polyfunctional isocyanates. The reaction can take place at ambient t e m p e r a t u r e s . F r o m one and the s a m e batch of ingredients different specimens of identical s t r u c t u r e a r e easily p r e p a r e d by a casting technique. A great variety of chain s t r u c t u r e s in the network can be obtained by changing the nature and the r a t i o s of r e a c t a n t s . The amount of crosslinking can be controlled by using polyols of a functionality higher than two, as well as by the excess of isocyanate added. The extent of intermolecular interaction may be adjusted by introducing a different number of urethane groups in the network.
A minor disadvantage is found in the fact that m o r e than one r e -action can proceed simultaneously in the curing mixture. The number of crosslinks in the network is therefore not easily con-trolled. Our p r e p a r a t i o n s of the various e l a s t o m e r s , however, w e r e conducted as identically as possible. Moreover, e l a s t o m e r t r e a t m e n t s between preparation and experiment, for example s t o r a g e time and extraction, w e r e in our study equal for e l a s t o m e r s which were to be compared with each Other. In addition, t r e a t -ments like milling and incorporation of fillers w e r e avoided. Therefore our data can be compared with each other without introducing additional uncertainties. The importance of this s t a t e -23
ment is emphasized by the widely different C g / C i r a t i o s which have been reported for one and the s a m e type of e l a s t o m e r (sec-tion 2.3.2).
3.2 PREPARATION OF POLYMERS 3.2.1 C h e m i s t r y
Isocyanates will r e a c t with any m a t e r i a l containing active hydrogen a t o m s . At ambient t e m p e r a t u r e the complete reaction with com-pounds like a m i n e s , alcohols or water generally r e q u i r e s s e v e r a l days. By introducing a suitable catalyst, however, the reaction is possibly terminated in seconds. If an alcohol is allowed to r e a c t with an isocyanate, the active groups combine to produce a urethane group.
• ^NCO + HO ^ -— HNCOO^
isocyanate hydroxyl urethane The symbol—-—-indicates that the active groups a r e a p a r t of bigger molecules. F u r t h e r reaction between the urethane and remaining free isocyanate is possible, leading to allophanate s t r u c t u r e s . This p r o c e s s leads to a crosslinked polymer.
• —NCO + HNCOO • -^HNCONCOO^ allophanate The p r e s e n c e of water in the reacting mixture causes the following sequence of reactions :
— .NCO + HgO - • -NHCOOH - —NHg + COg t carbamic acid amine The unstable carbamic acid desintegrates into an amine and carbon dioxyde. The gas evolution gives r i s e to the production of foam. A cellular product can be obtained this way. The amine can r e a c t further with isocyanate under formation of a u r e a group. This group is generally m o r e reactive towards isocyanate than polyurethanes a r e . Consequently, if water is p r e s e n t in the curing m i x t u r e , biuret branch points will be formed m o r e easily than the allophanate branchings.
^NHg + NCO -> .HNCONH • u r e a .CONH HNCONH — + O C N - ^ - " N ^ ^ C O N H biuret
Poly(oxypropylene)glycols with molecular weights between 400 and 5000 a r e used for the preparation of e l a s t o m e r s . These polyols yield v e r y e l a s t i c m a t e r i a l s after reaction with a slight •excess of diisocyanate. This property can be attributed to the long molecules which a r e formed, but also to the high flexibility of the - C - O - C - group in the backbone of the network chains (46, pag. 271). Generally the e l a s t o m e r s produced can be stretched to s e v e r a l hundred percent. Even elongations up to tens of t i m e s the original length have been reported (41).
Polyurethane e l a s t o m e r s can be p r e p a r e d by two basic p r o c e s s e s . The usual method involves the reaction of a linear diol with an excess of diisocyanate to form an isocyanate terminated product, the prepolymer.
2 R (NCO)g + H O ^ — OH
-OCNRNHOCO-— —^OCONHRNCO prepolymer, viscous liquid or low melting solid R is an alkyl or aryl group. The, mostly volatile, R (NCO)2 com-binations a r e very toxic. P r e p o l y m e r s of substantially higher molecular weight, however, p r e s e n t practically no air contamina-tion problem. The second step in this e l a s t o m e r preparacontamina-tion, is chain extension and network formation with selected polyols or amines. This step is usually accompanied by allophanate, or biuret branch point formation. The curing is completed by heating.
The alternative p r o c e s s , the one shot or single stage p r o c e s s , eliminates the reaction step in which the prepolymer is formed. In this case all components a r e thoroughly dried and mixed in one stage at room t e m p e r a t u r e . After degassing the blend is poured into a mold and subsequently heated to complete the c u r e .
In this study all polymers w e r e p r e p a r e d according to the one shot p r o c e s s . Apart from being the simplest p r o c e d u r e , the one stage p r o c e s s also avoids the handling of very viscous or even solid p r e p o l y m e r s . In filling up the molds, and especially the apparatus designed for measuring simple shear only mixtures of low viscosity a r e suited.
All e l a s t o m e r s w e r e p r e p a r e d with c o m m e r c i a l toluene diisocy-anate, which is a mixture of i s o m e r s . The toluene diisocyanate-polyol mixture needs a long curing t i m e , which is not p r a c t i c a l . Introducing a too active catalyst, however, reduces the pot life of the mixture too drastically. To e n s u r e thorough mixing, as well as sufficient degassing after the diisocyanate is added, a pot life of
about half an hour is required. This could be effectuated by using triethylenediamine (1,4 - diaza - [2.2.2.] bicyclooctane) in a con-centration of about 200 ppm.
3.2.2 M a t e r i a l s 1. Toluene diisocyanate
This product was obtained from E.I. du Pont de Nemours and Co. The m a t e r i a l , t r a d e name Hylene TM, is reported to be an ortho-p a r a 80/20 i s o m e r mixture. It was ortho-purified by destination in vacuo. The destined m a t e r i a l was stored in a dry nitrogen atmosphere to prevent contamination by m o i s t u r e .
2. Poly(oxyethylene)glycols, (POEG)
Products of molecular weight 400, 1000 and 4000 w e r e purchased from B r i t i s h Drug House (Poole, England). The m a t e r i a l of the lowest molecular weight is a colourless liquid, the other two POEG's a r e low-melting wax like solids.
3. Poly(oxypropylene)glycols, (POPG)
The POPG's used w e r e 4 diols of molecular weight between 400 and 4000, and a t r i o l of molecular weight 2700, placed at our d i s -posal by the Dow Chemical Co. Rotterdam, Holland. F r o m Shell Chemie Nederland N.V. a diol of molecular weight 750, as well as a t r i o l of molecular weight 5000, w e r e obtained. In addition a t e t r a -and a hexafunctional POPG w e r e obtained from Marles-Kuhlmann (Chocques, France).
POPG's a r e p r e p a r e d by base-catalyzed polyaddition of propylene oxide to a polyfunctional low molecular weight alcohol. The initial m a t e r i a l s for POPG's of functionality 2,3,4 and 6 respectively a r e : propylene glycol, glycerol, pentaerythritol and sorbitol. All m a t e -r i a l s we-re analyzed fo-r hyd-roxyl content by the acetylation method (47). F r o m these analyses the number average molecular weights w e r e calculated. F o r this calculation all molecules were assumed to be of the s a m e functionality, and only hydroxyl terminated. Also molecular weights of s e v e r a l POPG's w e r e determined by vapour phase osmometry (48). The o s m o m e t r i c method yields lower values than the ones calculated from the endgroup analysis. Consequently, some low molecular weight m a t e r i a l must be p r e s e n t in the oligomers. Finally, the molecular weight distribution was analyzed by gel permeation chromatography. The resulting values of M^ / ^ indicate that the m a t e r i a l s consist of r a t h e r sharp fractions. Data concerning the POPG's which w e r e used a r e summarized in Table 3.1.
Table 3.1. Some data c o n c e r n i i ^ the poly(oxypropylene)gIycols used in this study. t r a d e name V o r a n o l : P 400 P 1200 P 2000 CP 2700 P 4000 C a r a d o l : 750 5000 P l u r a c o l ; P e P 4 5 0 SP 760 supplier Dow Chemicals Rotterdam
molecular weights obtained from hydroxyl water vapour phase hydroxyl number content OH-analysis osmometry number
Shell Nederland Chemie N.V. Maries -Kuhlmann Chocques F r a n c e 276.9 95.0 55.4 63.5 25.7 0.01 % 0.05 % 0.01 % 0.02 % 0.025 % 405 1204 2075 2615 4160 833 4920 417 713 405 1960 2160 710 3500 M ,
M„
1.11 1.07 1.14 function-ality 1181 2020 2650 4360 _ _ -1.13 1.10 1.18 1.06 1.4 2 2 3 2 2 3 4 6 to - 34. 1.1.1 Trimethylol propane
This triol was obtained from Fluka A.G. (Buchs, Switzerland). The p u r e m a t e r i a l was used as received.
5. Triethylenediamine
This catalyst was purchased from Theodor Schuchardt (Miinchen, Germany). The m a t e r i a l was used as received.
6. F e r r i c acetylacetonate
This very active catalyst, suggested to us by another study on polyurethanes (42), was p r e p a r e d from acetylacetone and f e r r i c chloride (49).
3.2.3 M e t h o d s of p r e p a r a t i o n P r e l i m i n a r y experiments
As already mentioned, the one shot p r o c e s s was adopted for e l a s -tomer p r e p a r a t i o n s . Initially POEG was chosen as polyol compo-nent, instead of the commonly used POPG. POEG will yield net-works in which the individual chains have a s i m p l e r s t r u c t u r e , since s t e r i c effects due the methyl side groups a r e absent. Howe v Howe r , during timHowe consuming p r Howe l i m i n a r y HowexpHowerimHowents no s a t i s f a c -tory elastomer specimens could be prepared from POEG based m i x t u r e s . This failure was due to the very short pot life of all m i x t u r e s . Since POEG's of higher molecular weight a r e solids, preparations had to be conducted at elevated t e m p e r a t u r e s . After addition of toluene diisocyanate, also a discolouration of the o r i g i nally clear mixture was observed. Numerous unsuccesful e x p e r i -ments with different ingredients, using both catalysts mentioned, as well as uncatalyzed m i x t u r e s , w e r e conducted. Finally, POEG was abandoned in favour of POPG. These polyols were found to be much better suited for elastomer p r e p a r a t i o n s in a single stage p r o c e s s .
IsooyanateAydroxyl r a t i o
Before adopting a standard procedure for all elastomer p r e p a r a -tions, the n e c e s s a r y excess amount of toluene diisocyanate had to be established. Viscoelastic tacky e l a s t o m e r s a r e obtained when the stoichiometric amount of diisocyanate is added, even when polyols with a functionality higher than two, i.e. crosslinking agents, a r e used. This phenomenon undoubtedly is due to the p r e
-sence of numerous unreacted hydroxyl groups, resulting from an isocyanate shortage. An excess of diisocyanate is n e c e s s a r y to e n s u r e complete reaction. This excess is related to the inaccuracy of the hydroxyl analysis (a few percent) and to the occurrence of isocyanate-consuming side reactions (see section 3.2.1). It was found that good e l a s t o m e r s can be p r e p a r e d by introducing at least a 10% excess of diisocyanate. Moreover, it has been reported that e l a s t o m e r p r o p e r t i e s as modulus and ultimate elongation become much l e s s sensitive to the isocyanate A y d r o x y l r a t i o at an isocyan-ate excess of approximisocyan-ately 1.05 (50). Bearing all this in mind, an excess amount of 15% toluene diisocyanate was chosen for our e l a s t o m e r p r e p a r a t i o n s .
P r e p a r a t i o n of specimens
The required amount of POPG, or generally a blend of polyols of different functionality, was weighed into a 100 cc round-bottom flask, fitted with a vacuum connection and a magnetic s t i r r e r . The catalyst, triethylenediamine, was added to a concentration of a p -proximately 200 ppm. Next the mixture was heated to 75 °C and s t i r r e d under vacuum for s e v e r a l hours. This procedure s e r v e d to degass and if n e c e s s a r y to dry the m a t e r i a l s before adding the toluene diisocyanate. After cooling to room t e m p e r a t u r e 1.15 t i m e s the stoichiometric amount of toluene diisocyanate was added quick-ly. The reaction mixture subsequently was s t i r r e d in vacuo until its viscosity i n c r e a s e d appreciably, generally after half an hour. After this mixing and final degassing p r o c e d u r e the blend was cast into molds so as to obtain s a m p l e s suitable for extension and com-p r e s s i o n excom-periments. The molds w e r e com-put into a vacuum oven and heated at 80°C for 3 days. Immediately after inserting the molds, the oven was evacuated and refilled with dry nitrogen. This p r o c e -dure was repeated a few t i m e s to e n s u r e removal of m o i s t u r e . In some cases m e a s u r e m e n t s of simple shear had to be performed with the same e l a s t o m e r , as was used for elongation and c o m p r e s -sion. Except filling up the molds, now also a measured quantity of the liquid mixture was delivered in the s h e a r - i n s t r u m e n t (Fig. 3.1) with the help of a hypodermic syringe. Crosslinking subsequently took place in situ. Curing conditions, in the apparatus as well as in the oven were kept as identical as possible. In the apparatus the curing e l a s t o m e r was s e p a r a t e d from the a t m o s p h e r e by a blanket of dry nitrogen.
Two types of molds w e r e used, open flat-bottom t r a y s of 100 mm d i a m e t e r and s m a l l tubes of 10 mm d i a m e t e r . All molds had to be constructed of Teflon, since the e l a s t o m e r s become strongly a t -tached to glass or metal s u r f a c e s . In the t r a y s t r a n s p a r a n t ,
usually bubble-free e l a s t o m e r sheets of about 1 mm thickness were obtained. The mixture in the tubes cured to cylindrical r o d s . Rods could be removed easily from the molds after cooling e l a s -t o m e r and mold in liquid ni-trogen. Removal of -the elas-tomer shee-ts from the trays did not r e q u i r e any special effort.
3.3 UNIDIRECTIONAL EXTENSION 3.3.1 A p p a r a t u s
Stress-extension c h a r a c t e r i s t i c s of the e l a s t o m e r s t r i p s w e r e determined with an apparatus described previously (29, page 4 8 -50). The s t r i p is mounted vertically between two invar steel clamps. The upper clamp is attached to the pan of an analytical balance, the lower clamp is connected to a m i c r o m e t e r . Connec-ting rods also a r e made of invar s t e e l , which has a v e r y low t h e r m a l expansion coefficient. The m i c r o m e t e r can be moved to adjust the desired degree of s t r e t c h .
Clamps and e l a s t o m e r s t r i p were both surrounded by a double-wall glass chamber. The t e m p e r a t u r e was kept constant to 0.1°C by circulating water from a thermostat around the measuring chamber. During experiments nitrogen of high purity was passed through the chamber, to avoid oxidative degradation,
3.3.2 E x p e r i m e n t a l t e c h n i q u e
E l a s t o m e r s t r i p s used in extension experiments were punched out of thin sheets, using a metal punch. Strip dimensions were approx-imately 50 X2 x i mm. Both clamped ends of the s t r i p a r e s e v e r a l times wider than the s m a l l middle section. This design prevented breakage and slip in the clamps completely.
Before experiments the m a t e r i a l was extracted for at least 3 days. The extraction liquid was tetrahydrofurane, which is a good solvent for polyurethanes. The extraction procedure served to r e -move any m a t e r i a l unattached to the network. This gel fraction was found to be 2% at most. In addition, deliberately added m o i s -t u r e in -the ex-trac-tion liquid comple-ted -the cure of -the e l a s -t o m e r s . The necessity of this 'post-curing' has been suggested by recent Japanese work (51), concerning changes in p r o p e r t i e s of freshly p r e p a r e d polyurethane e l a s t o m e r s , stored in moist air. This change is attributed to a reaction of remaining free isocyanate with m o i s -t u r e , diffusing in-to -the ne-twork.
In determining the force-extension relation the stretching force is measured to an accuracy better than 0.1%. Extension of the s t r i p
is obtained by m e a s u r i n g the distance between ink m a r k s with a cathetometer. The e r r o r in this m e a s u r e m e n t is 0.2%. The length of the unloaded s t r i p is determined by extrapolating the initial p a r t of the force-extension curve to z e r o force. At least 5 points a r e taken for this extrapolation. The best straight line connecting these points is found by the method of the least s q u a r e s .
All force-extension m e a s u r e m e n t s have been conducted at one or all of the t e m p e r a t u r e s 30, 40, 50 and 70°C. After mounting the s t r i p the g r e a t e s t extension r a t i o was imposed, together with the highest t e m p e r a t u r e used in a measuring cycle. Next the e l a s t o m e r was allowed to r e l a x for 20 h o u r s . Most of the e l a s t o m e r s reached a 'steady' r e t r a c t i v e force after this period, of about 1% lower than the original force. 'Steady' in this case denotes that after a m e a s u r i n g cycle of a few h o u r s , always conducted goipg from high to low extension and back, the initial high extension was observed again within experimental e r r o r . A few slightly crosslinked m a t e -r i a l s exhibited a s t -r e s s -relaxation of about 5% du-ring conditioning. F o r these e l a s t o m e r s the force m e a s u r e m e n t at the highest exten-sion after completing a cycle deviated 0.5 - 1.0% from the first m e a s u r e m e n t .
3.4 SIMPLE SHEAR AND UNIDIRECTIONAL COMPRESSION 3.4.1 A p p a r a t u s
Both simple shear and compression could be measured with the s a m e apparatus (Fig. 3.1). This apparatus consists of a cylindrical m e a s u r i n g chamber (Fig. 3.1 A) mounted on a steel support (Fig. 3.IB). In this chamber a piston can be moved freely. During simple s h e a r m e a s u r e m e n t s the piston (Fig. 3.1C) is a hollow cylinder which is screwed on a thin rod. The bottom consists of a perforated Teflon d i s c , of such a volume that the weight of the piston can be reduced to z e r o in m e r c u r y . The annular gap between piston and the wall of the m e a s u r i n g chamber is filled with a ring of e l a s t o -m e r , for-med in situ and therefore fir-mly attached to both walls. Subsequently a shearing force is imposed by p r e s s i n g down the piston.
F o r compression m e a s u r e m e n t s another, much lighter piston is applied (Fig. 3.1b). In this c a s e a cylindrical e l a s t o m e r specimen formed in a mold is placed between the piston and the bottom of the m e a s u r i n g chamber. Moving down the piston now r e s u l t s in uni-directional compression of the e l a s t o m e r .
elastomer mercury Level
"Teflon perforated disc •— water
ff'trnfJ.' ZZJj teflon cover
Fig. 3.1.
a. Apparatus for measuring simple shear and compression b. Piston used for compression m e a s u r e m e n t s .
A. measuring chamber B. frame
C. piston
D. clamp and guiding shaft
E. s t r a i n gauge F. clamp G. m i c r o m e t e r H. motor and gearbox
The stainless steel m e a s u r i n g chamber is 190 mm high and 57.0 mm in diameter. It is surrounded by a water jacket. During e x p e r i -ments the t e m p e r a t u r e is kept constant to within 0.2°C by circu-lating water from a thermostat. The chamber can partly be filled with m e r c u r y by means of a flexible tube connection to a s t o r a g e v e s s e l . The stainless steel piston used in simple s h e a r e x p e r i -ments floats in the m e r c u r y .
The lower p a r t of this piston is 110 mm high and 52.0 mm in dia-m e t e r , its total height being 280 dia-mdia-m. If the piston is dia-moved during a shearing experiment the buoyant force will change only very little. The related force change is negligible in comparison with the i m -posed shearing force.
The piston can be p r e s s e d downward by a motor of accurately con-stant speed (Fig. 3.1, H). The motor is provided with a gearbox (Fig. 3.1, H), permitting a number of d i s c r e t e displacement speeds between 0.01 and 10 c m A o u r . Motor action is t r a n s f e r r e d to the piston via a s t r a i n gauge (Hettinger, type Q l / 5 , D a r m s t a d t , Germany, Fig. 3.1, E), of 10 kgf. maximum load. Tilting of the piston is avoided by a guiding shaft on the upper p a r t (Fig. 3.1, D), which also can be used as a clamp. During an experiment the s t r a i n i m -posed on an e l a s t o m e r specimen i n c r e a s e s linearly with t i m e . The s t r e s s is measured by the s t r a i n gauge and after amplification of the signal r e c o r d e d against t i m e . Thus s t r e s s - s t r a i n curves a r e directly written on the r e c o r d e r . These curves still have to be c o r r e c t e d for the (linear) bending of the spring plate in the s t r a i n gauge, and also for the piston weight if compression m e a s u r e m e n t s a r e conducted.
P a s s i n g on from simple s h e a r to compression m e a s u r e m e n t s only two minor changes of the apparatus a r e n e c e s s a r y . F i r s t l y all m e r c u r y is removed from the measuring chamber, after which the bottom is covered with a fitting Teflon disk. Secondly the s t a i n l e s s steel piston is replaced by an aluminium one (Fig. 3.1, b). This second piston c o n s i s t s of an aluminium disk, 50 m m in d i a m e t e r , covered with a Teflon plate. After a cylindrical elastomer specimen i s inserted between piston and bottom of the measuring chamber, compression m e a s u r e m e n t s can be conducted.
Before the experiments the s t r a i n gauge was calibrated by loading the spring plate with known weights. It was found that the deforming force could be m e a s u r e d to an accuracy of 1%. The amount of s h e a r or the compression r a t i o can in principle be determined m o r e accurately, because the displacement can be m e a s u r e d to an accuracy of 0.002 mm. Due to the required corrections for bend-ing of the s t r a i n gauge's sprbend-ing plate, as well as for the piston weight the total e r r o r in the deformation ratio is only slightly s m a l l e r than the e r r o r in the force.
3.4.2 E x p e r i m e n t a l t e c h n i q u e Simple shear
At the beginning of an experiment the piston floats freely in m e r -cury, while the holes in the lower p a r t a r e partly i m m e r s e d . In this position the piston is fixed by means of the clamp. Next the m e r c u r y level is r a i s e d until the holes a r e below the surface. After this a known quantity of the liquid e l a s t o m e r mixture is delivered into the annular space between piston and measuring chamber. The mixture is than cured under a blanket of dry nitrogen. Great c a r e has been taken to duplicate the crosslinking conditions of the other samples from the batch involved. Curing r e s u l t s in a ring of e l a s -t o m e r of abou-t 5 mm heigh-t, which adheres s-trongly -to -the walls of both piston and m e a s u r i n g chamber.
After crosslinking the m e r c u r y level is reduced to its original position, i.e. the position in which the piston weight is exactly c o m -pensated. As a r e s u l t , during experiments the ring of e l a s t o m e r is approximately 20 mm above the m e r c u r y level. A number of holes, which have been drilled in the piston wall prevent the creation of a vacuum in the space between e l a s t o m e r and m e r c u r y . Next the clamp is loosened, and the network can be subjected to simple s h e a r by p r e s s i n g down the piston. After this measurement the piston is allowed to r e c o v e r its original position, the upward movement being conducted with the s a m e velocity as the movement downward. In all c a s e s downward and upward curves w e r e measured. Relaxation effects w e r e not detectable. Moreover, it was observed that the curves r e m a i n unchanged in a wide range of shearing velocities. All experiments w e r e c a r r i e d out at v e r y low shearing velocities, usually 1 m m A o u r , to e n s u r e m e a s u r e m e n t s of the equilibrium modulus throughout the cycle.
Unidirectional compression
Compression m e a s u r e m e n t s w e r e c a r r i e d out with cylindrical e l a s t o m e r specimens. Initially these specimens w e r e obtained by curing of the polymer mixture in closed molds of the desired d i -mensions. It was found, however, that bubble-free specimens could r a r e l y be p r e p a r e d this way. This difficulty was circumvented by cutting specimens from bubble-free sections of relatively long e l a s t o m e r r o d s , p r e p a r e d in open Teflon tubes. After some e x p e r i ments the following cutting technique appeared to be very s u c c e s s -ful. Elastomer rods w e r e fastened on a lathe, and slowly rotated. A r a z o r blade was used to cut the rod into parallel sections of the d e s i r e d thickness. To facilitate the cutting a liberal amount of
